” in case there is no e-mail address specified in to, cc or bcc fields of the received e-mail, except for the address of the server, then just the sms message will be sent to the specified phone numbers instead of the notification. to summarize, for this specific case, unimail allows to send sms via e-mail. the initiator of notifying about the e-mail is the sender. however, the receiver should have an opportunity to control the list of people who are permitted to send him sms notifications. to be able to control the lists, the user should tie his phone number with a personal email address. for this configuration, unimail requires a simple registration:  to: unimail@unimail.asnet.am fig. 3. output log file of unimail. a. mkhitaryan, a. nanassian and e. matveev 77  subject: *reg: * after sending the e-mail, an sms containing a randomly generated 4 or 5-digit number will be sent to the specified phone number. the user should send that number back to the server from the same e-mail address:  to: unimail@unimail.asnet.am  subject: ** once the registration is complete, the user’s e-mail address and phone number will be paired. the next operations are allowed for the registered users only. unimail supports two well-known systems of controlling permitted lists, i.e., black and white lists. any registered user can choose the appropriate type of control. by default, the mode being chosen is “a black list” mode. to change the mode the user should send an e-mail to the address of the unimail server with subject:  for switching to “white list” mode: o *use white list *  for switching to “black list” mode: o *use black list * to include or exclude someone in/out of the black or white lists, the user should send an e-mail to the same e-mail address with the following indicators:  subject: o *add white/black list * o *del white/black list *  body: o * / , / , …* the registered user can also block/unblock all the services provided by unimail by sending an email with subject:  * block sms *  * unblock sms * 4. conclusion unimail is an info-communication autonomous software resource, which allows you to send sms notifications in parallel to the e-mail. it uses a unique approach when the initiator of notification is the sender, who considers his e-mail to be important enough. at the same time, the receiver can easily control the list of people who are permitted to send him notifications. receivers can even turn off all the services provided by unimail. to sum up, unimail provides a complete package to control notification system of both senders and receivers. it operates with any e-mail provider and can notify anyone from the regional cellular network. references [1] д. геворкян, а. нанасян, к. хачатрян, «новые web ресурсы asnet.am», proceedings of international conference of computer science and information technօlogies, csit-2011, ереван, pp. 311-312, 2011. unimail info-communicational software 78 [2] d. gevorkyan, k. khachatryan, a. nanassian, a. petrosyan, g.petrosyan, v. sahakyan and e. vardanyan, “mail informerselective incoming instant phone notification system”, proceedings of international conference computer science and information technologies, csit, yerevan, armenia, pp. 466-467, 2009. [3] а. нанасян и к. хачатрян «mail2sms.asnet.am – система оповещения о входящих письмах», proceedings of international conference of computer science and information technologies csit-2013. ереван, 2013. [4] а. мхитарян, э. матвеев, а. нанасян, в. саакян и a. petrosyan, “гибридная инфокоммуникационная email/sms система unimail”, proceedings of international conference of computer science and information technologies, csit, yerevan, armenia, pp. 389-391, 2017. submitted 22.11.2017, accepted 12.02.2018. unimail-ի տեղեկատվական հաղորդակցման համակարգի ծրագրային ապահովում ա. մխիթարյան, ա. նանասյան և է. մատվեև ամփոփում աշխատանքում ներկայացված են sms և email տեխնոլոգիաների վրա հիմնված տեղեկատվական հաղորդակցման անկախ ծրագրային ռեսուրսը՝ unimail անվամբ, unimail սերվերի առանձին մասերը և դրանց պարտականությունները։ դիտարկվել են unimail-ի կառուցվածքային առանձնահատկությունները և աշխատանքի սկզբունքները։ նկարագրված է հրամանների ցանկը և կարգավորումների մանրամասները օգտատերերի համար։ программная поддержка инфокоммуникационной системы unimail а. мхитарян, а. нанасян и э. матвеев аннотация инфокоммуникационная система unimail независимый аппаратнопрограммный ресурс на основе технологий электронной почты и sms. приведены функциональные компоненты сервера unimail. обсуждаются особенности архитектуры и принципы работы, базовые команды системы. mathematical problems of computer science 46, 117–125, 2016. the parallel simulation method for d-dimensional abelian sandpile automata hayk e. nahapetyan, suren s. poghosyan, vahagn s. poghosyan and yuri h. shoukourian institute for informatics and automation problems of nas ra e-mail: povahagn@gmail.com, shouk@sci.am abstract in this paper, the star-packing problem introduced in [1] for a square lattice is generalized for d-dimensional lattice ld, d ∈ n. the problem is to pack the lattice ld with star graphs s2d. using the solution of this problem, a parallel algorithm for the simulation of d-dimensional cellular automata is developed. as an example of cellular automata, the relaxation process of unstable states of abelian sandpile model is considered. appropriate software packages have been developed using openmp and cuda technologies. the parallel simulation results, carried out for 3-dimensional lattices of different sizes, are presented. keywords: abelian sandpile model, dense packing problem, parallel algorithm, cellular automata. 1. introduction cellular automata (ca) are discrete models studied in computability theory, mathematics, physics, complexity science, theoretical biology and microstructure modeling. modern multicore computers with shared memory and multiprocessor clusters with distributed memory make it possible to efficiently parallelize simulation algorithms for ca. in the work [2], a cluster-based parallel algorithm and a package to simulate two-dimensional ca is introduced. the concept of self-organized criticality was first introduced by bak, tang and wiesenfeld in 1987 [4], and gave rise to growing interest in the study of self-organizing systems. bak et al. argued that in many natural phenomena, the dissipative dynamics of the system is such that it drives the system to a critical state, thereby leading to ubiquitous power law behaviors. this mechanism has been invoked to understand the power law distributions observed in turbulent fluids, earthquakes, distribution of visible matter in the universe, solar flares and surface roughening of growing interfaces. the sandpile models, being a class of cellular automata, are among the simplest theoretical models which show self-organized criticality. a special subclass of interest consists of so called abelian sandpile models (asm). the abelian property means that the final stable state of the ca is independent of the order in which the updates of cells are carried out. this property plays a key role during the numerical, as well as analytical studies of the asm [5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. in numerous works, in the absence of analytical relations for 117 118 the parallel simulation method for d-dimensional abelian sandpile automata various physical characterizers, different methods of statistical analysis have been applied for calculation or verification of hypothetical and analytical expressions for sandpile models. the luck of such systems, also the demand for accurate calculations of physical observables would dramatically increase the acceptable sizes of lattice as well as the time constraints for calculations. an approach of efficient parallel simulation of sandpile model on multicore computers with shared memory, which is based on the idea that there are few active cells (unstable vertices) at each moment of time, is given in [3]. in general, when the initial unstable configuration contains many unstable vertices, this approach does not give good results for big lattices. in what follows we present a more efficient parallel algorithm for the simulation of d-dimensional ca implemented on systems with shared memory, which is based on the star-packing of the d-dimensional lattices. a solution to the star-packing problem is given in [1] for the 2-dimensional square lattice. in this paper we generalize the solution to the case of general d dimensions. below a cellular automaton for implementing the abelian sandpile model on a d-dimensional lattice is considered aiming at the development of parallel simulation. 2. the star-packing problem consider a d-dimensional lattice ld of linear size n with periodic boundary conditions, which is a graph with the vertex set v = v (ld) = {v = (x1, x2, . . . , xd) : xi = 0, 1, 2, . . . , n − 1; i = 1, 2, . . . , d} (1) and the set of undirected edges e = e(ld) defined by the following rule. each vertex v = (x1, x2, . . . , xd) ∈ v is connected with 2d vertices given by adjacency list adj(v) = {(x1, x2, . . . , (xk ± 1) mod n, . . . , xd) : k = 1, 2, . . . , d}. (2) the number of vertices is |v | = nd, and the number of edges is |e| = d nd. two different vertices v1 and v2 are called independent, if they are not adjacent and they do not have any common adjacent vertex: v1 /∈ adj(v2), v2 /∈ adj(v1) and adj(v1) ∩ adj(v2) = ∅. (3) the star-packing problem is to pack the lattice ld with non-overlapping star graphs s2d. consider a star packing with a set of central vertices v ′ ⊂ v of stars. the following two rules define the set v ′: 1. full coverage condition: ( ∪ v∈v ′ adj(v) )∪ v ′ = v. 2. non-overlapping condition (independence of stars): any two different vertices v1, v2 ∈ v ′ are independent. it is obvious, that a star packing exists if |v | is dividable by 2d + 1. given a vertex v = (x1, x2, . . . , xd) ∈ v and a vector r⃗ = (p1, p2, . . . , pd) ∈ z. define the sum v + r⃗ to be a vertex in v with coordinates v + r⃗ ≡ ((x1 + p1) mod n, (x2 + p2) mod n, . . . , (xd + pd) mod n) ∈ v. (4) h. nahapetyan, su. poghosyan, v. poghosyan and yu. shoukourian 119 equivalently, the sum of a vector r⃗ and an arbitrary subset v ′ ⊆ v to be the following subset of v v ′ + r⃗ ≡ {v + r⃗ : v ∈ v ′} ⊆ v. (5) note that, given a star packing defined by a set v ′, the set v ′ + r⃗ also defines a star packing, where r⃗ ∈ z. theorem: consider a d-dimensional lattice ld of linear size n with periodic boundary conditions. assume that n is dividable by 2d + 1. then a star packing of ld exists, and the set of central vertices of stars is v ′ = {v = (x1, x2, . . . , xd) : (x1 + 2x2 + 3x3 + . . . + d xd) mod (2d + 1) = 0, v ∈ v }. (6) proof: given a vector r⃗ = (p1, p2, . . . , pd) ∈ z, define a function q(r⃗) = p1 + 2p2 + 3p3 + . . . + d pd. (7) let us introduce the d-dimensional basis vectors e⃗k = (0, 0, . . . , 0︸︷︷︸ k-1 , 1, 0, 0, . . . , 0︸︷︷︸ d-k ), k = 1, 2, 3, . . . , d, (8) and the null vector e⃗0 = (0, 0 . . . , 0). first, we prove the full coverage condition. given an arbitrary vertex v ∈ v \ v ′, it is necessary to prove that there exists a vertex u ∈ adj(v), which is a central vertex of some star, i.e., u ∈ v ′. in other words, ∀r⃗ = (p1, p2, . . . , pd) ∈ z, 0 ≤ pi ≤ n − 1, (9) one of the following 3 conditions holds 1. q(r⃗) mod (2d + 1) = 0. 2. ∃k = 1, 2, . . . , d; q(r⃗ − e⃗k) mod (2d + 1) = 0. 3. ∃k = 1, 2, . . . , d; q(r⃗ + e⃗k) mod (2d + 1) = 0. assume that q(r⃗) has the following form: q(r⃗) = (2d + 1)s + t, 0 ≤ t < 2d + 1, s ≥ 0. (10) let the value of k be k =   0, if t = 0; ⇒ q(r⃗) mod (2d + 1) = 0 t, if t < d; ⇒ q(r⃗ − e⃗k) mod (2d + 1) = 0 2d + 1 − t, if t > d; ⇒ q(r⃗ + e⃗k) mod (2d + 1) = 0 , (11) which proves the coverage condition. now we prove the non-overlapping condition. it states, ∀ r⃗ = (p1, p2, . . . , pd) ∈ z, 0 ≤ pi ≤ n − 1, i = 1, 2, . . . , d, which satisfies the condition q(r⃗) mod (2d + 1) = 0, (12) and ∀∆⃗r = (∆p1, ∆p2, . . . , ∆pd) with 0 < |∆p1| + |∆p2| + . . . + |∆pd| ≤ 2, (13) 120 the parallel simulation method for d-dimensional abelian sandpile automata we have q(r⃗ + ∆⃗r) mod (2d + 1) ̸= 0. (14) since the q function is linear, we have to show that q(∆⃗r) mod (2d + 1) ̸= 0. then, it is sufficient to prove the inequality 0 < |q(∆⃗r)| < 2d + 1. note that there are 3 possible cases, which satisfy the condition (13): 1. ∃k0 = 1, 2, . . . , d, for which ∆pk0 = ±2 and ∆ps = 0, ∀s ̸= k0. then 0 < |q(∆⃗r)| = 2k0 < 2d + 1. (15) 2. ∃k0 = 1, 2, . . . , d, for which ∆pk0 = ±1 and ∆ps = 0, ∀s ̸= k0. then 0 < |q(∆⃗r)| = k0 < 2d + 1. (16) 3. ∃k1, k2 = 1, 2, . . . , d; k1 ̸= k2, for which ∆pk1 = ±1, ∆pk2 = ±1 and ∆ps = 0, ∀s ̸= k1, k2. then 0 < |q(∆⃗r)| = |k1∆pk1 + k2∆pk2| ≤ k1 + k2 < 2d + 1. (17) 2 fig.1 and fig.2 illustrate the star packing of the 2 and 3-dimensional lattices, respectively. fig. 1. a star-packing of a 2-dimensional lattice. h. nahapetyan, su. poghosyan, v. poghosyan and yu. shoukourian 121 fig. 2. a star-packing of a 3-dimensional lattice. the layer z=4 is depicted separately. 3. abelian sandpile model consider an undirected graph g = (v, e) with vertex set v = {v1, v2, . . . , vn} and edge set e. a random variable hi, which takes integer values, is attached to each vertex vi ∈ v , representing the height of the sand at that vertex. hmaxi denotes the maximum allowed height of vertex vi of the graph g. for d-dimensional lattice we take h max i = 2d−1. ct denotes the collection of heights hi, which defines a configuration of the system at a given discrete time t . a configuration is called stable, if all heights satisfy hi ≤ hmaxi . vertex vi is called closed, if hmaxi = deg (vi) − 1, where deg (vi) indicates the number of adjacent vertices of vi. vertex vi is called open, if h max i ≥ deg (vi). the dynamics of the system is defined by the following rules. consider a stable configuration ct at a given time t . we add a grain of sand at a random vertex vi ∈ v by setting hi to hi + 1 (we assume that the vertex is chosen randomly with a uniform distribution on the set v ). this new configuration, if stable, defines ct+1. if hi > h max i , vi becomes unstable and topples losing h max i +1 grains of sand, while all neighbors of vi receive one grain. note that if the vertex is open, the system loses grains. during the toppling of the closed vertices, the number of grains is conserved. note also that toppling of a vertex may cause some of its neighboring vertices to become unstable. in this case those vertices also topple according to the same toppling rule. once all unstable vertices have been toppled, a new stable configuration ct+1 is obtained. if the finite connected graph g has at least one open vertex, then all vertices become stable after finite number of topplings. moreover, the new stable configuration is independent of the toppling order. therefore, the dynamics is well defined. let âi be an operator, which acts on sandpile configurations and adds a grain at vertex i. it can easily be shown that âiâj = âjâi. this is the reason why the sandpile model is called abelian. 122 the parallel simulation method for d-dimensional abelian sandpile automata 4. simulation results in this section we present the simulation results of the implemented software with built-in parallelization algorithms. the abelian sandpile model on the finite d-dimensional lattice of linear size n is considered. the number of nodes is n = nd. the value of maximal height at all vertices vi ∈ v is hmaxi = 2d − 1, i = 1, 2, . . . , n. to perform parallel simulation, the concurrently toppled unstable vertices should meet the requirement of being independent. partitioning the set of vertices into the sets of independent vertices is carried out by starpacking discussed above. let twait be the time needed for a cell to finish its job, and h denotes the initial number of grains at each node. in order to obtain a parallel software environment, the openmp and cuda technologies have been used, which was implemented on the cpu i7 2670qm by intel and geforce gt. simulation results include 2 types of cellular automata simulating sandpile model. difference is that for the second system an additional job is introduced during each toppling, which takes twait constant time to finish. four types of implementations have been analyzed and compared: 1. standard − topplings are implemented sequentially on cpu. there is a loop around all nodes, and once an unstable node is found, it becomes stable. 2. no omp − topplings are implemented sequentially on cpu. by a parallelization algorithm, we partition the set of nodes into 7 sections for a 3-dimensional lattice. there are two types of loops, one type over all groups, and the second type over all vertices for each group. 3. omp − topplings are implemented parallel for each group on cpu, also there is one loop over all groups. 4. gpu − topplings are implemented in parallel for each group on gpu, also there is one loop over all groups. the simulations have been done on the same cpu and gpu architecture. in figs. 3-7 the simulation results are presented. 140 210 280 n 5 10 15 20 25 30 computation time (ms) standard no-omp omp gpu fig. 3. sandpile on a 3-dimensional lattice. grains are added randomly with uniform distribution. total amount of added grains is 10 × n3. h. nahapetyan, su. poghosyan, v. poghosyan and yu. shoukourian 123 n=7,h=6 n=7,h=12 n=14,h=6 n=14,h=12 100 200 300 400 computation time (ms) standard no-omp omp gpu fig. 4. sandpile on a 3-dimensional lattice with initial height h=6, h=12. twait = 5ms. n=7,h=6 n=7,h=12 n=14,h=6 n=14,h=12 200 400 600 800 computation time (ms) standard no-omp omp gpu fig. 5. sandpile on a 3-dimensional lattice with initial height h=6, h=12. twait = 10ms. n=21,h=6 n=21,h=12 n=42,h=6 n=42,h=12 10000 20000 30000 40000 50000 computation time (ms) standard no-omp omp gpu fig. 6. sandpile on a 3-dimensional lattice with start height h=6, h=12. twait = 3ms. 5. conclusion in this paper, a parallel algorithm for simulating d-dimensional ca has been described. the algorithm is based on the idea of star-packing of d-dimensional lattices. delmas and perennes in their paper [15] found a star packing for 3-dimensional lattice of linear size n = 7i with integer i. we found an explicit star packing of d-dimensional lattice, which meets weaker restriction on n, namely, n mod (2d + 1) = 0. as the results show, the algorithm developed gains about 8× speedup with openmp, and more than 50× on cuda (depending on gpu architecture and cellular automata, it can be 100× and more faster than non-paralyzed 124 the parallel simulation method for d-dimensional abelian sandpile automata performance). our next aim is to use algorithms for computing different observables of the sandpile model on d-dimensional lattices, meanwhile analytical calculations have been obtained for 2-dimension. references [1] v. s. poghosyan, s. s. poghosyan and h. e. nahapetyan, “the investigation of models of self-organized systems by parallel programming methods based on the example of an abelian sandpile model”, proc. csit conference 2013, yerevan armenia, sept. 23-27, pp. 260–262, 2013. [2] e. davtyan, h. karapetyan and k. shahbazyan, “software tool for cluster-based modeling of 2d cellular automata”, proc. csit conferance, yerevan armenia, 28 sept. 2 oct., pp. 404–407, 2013. [3] s. frehmel, “the sandpile model: parallelization of efficient algorithms for systems with shared memory”, acri, lecture notes in computer science 6350, pp. 35–45, 2010. [4] p. bak, c. tang and k. wiesenfeld, “self-organized criticality: an explanation of the 1/f noise”, phys. rev. lett., vol.59, no. 4, pp. 381–384, 1987. [5] d. dhar, “self-organized critical state of sandpile automaton models”, phys. rev. lett., vol. 64, no. 14, pp. 1613–1616, 1990. [6] d. dhar, “theoretical studies of self-organized criticality”, physica a, vol. 369, no. 1, pp. 29–70, 2006. [7] p. grassberger and s. s. manna, “some more sandpiles”, j. phys. france, vol. 51, pp. 1077–1098, 1990. [8] v. s. poghosyan, s. y. grigorev, v. b. priezzhev and p. ruelle, “pair correlations in the sandpile model: a check of logarithmic conformal field theory”, phys. lett. b, vol. 659, pp. 768–772, 2008. [9] su. s. poghosyan, v. s. poghosyan, v. b. priezzhev and p. ruelle, “numerical study of correspondence between the dissipative and fixed-energy abelian sandpile models”, phys.rev. e, 84, 066119, 2011. [10] a. fey, l. levine, and d.b. wilson, “driving sandpiles to criticality and beyond”, phys. rev. lett., 104, 145703, 2010. [11] a. fey, l. levine, and d. b. wilson, “approach to criticality in sandpiles”, phys. rev. e 82, 031121 (2010). [12] v. s. poghosyan, s. y. grigorev, v. b. priezzhev and p. ruelle, “logarithmic two-point correlators in the abelian sandpile model”, j. stat. mech., vol. 2010, no. 07, p07025, 2010. [13] v. s. poghosyan and v. b. priezzhev, “the problem of predecessors on spanning trees”, acta polytechnica, vol. 51, no. 1, pp. 59–62, 2011. [14] s. n. majumdar and d. dhar, “height correlations in the abelian sandpile model”, j. phys. a: math. gen., vol. 24, no. 7, l357–l362, 1991. [15] o.delmas, s. perennes, “circuit-switched gossiping in 3-dimensional torus networks”, proc. euro-par, parallel-processing, pp. 370–373, 1996. submitted 10.06.2016, accepted 25.10.2016. h. nahapetyan, s. poghosyan, v. poghosyan and yu. shoukourian 1 2 5 ²í³½³ïáõûïç d-ã³÷³ýç ²µ»éû³ý ³íïáù³ïç ½áõ·³ñ»é ùá¹»é³íáñù³ý ù»ãá¹ ð. ü³ñ³å»ïû³ý, ê. äáõáëû³ý, ì. äáõáëû³ý ¨ úáõ. þáõùáõñû³ý ²ù÷á÷áõù ²ûë ñá¹í³íáõù µ»ñí³í ¿ ù³é³ïáõëç ó³ýó»ñç ñ³ù³ñ [1]-áõù ý»ñùáõíí³í ³ëïõ³ûçý í³íïáõûãç ëý¹ñç áý¹ñ³ýñ³óáõùá l d, d 2 n d -ã³÷³ýç ó³ýóç ñ³ù³ñ: êý¹çñá ï³û³ýáõù ¿ s2d ·ñ³ýáí l d ó³ýóç í³íïù³ý ù»ç: îíû³é ëý¹ñç éáõíù³ý ñçù³ý íñ³ ùß³ïí»é ¿ d-ã³÷³ýç µçç³ûçý ³íïáù³ïç ùá¹»é³íáñù³ý ½áõ·³ñ»é³óí³í ³é·áñçãù: àñå»ë µçç³ûçý ³íïáù³ïç ûñçý³ï ¿ ¹çï³ñïí»é ²µ»éû³ý ³í³½³ïáõûïç ³ýï³ûáõý íç×³ïý»ñç é»é³ùë³óçáý åñáó»ëá: ð³ù³å³ï³ëë³ý íñ³·ñ³ûçý ÷³ã»ãý»ñá ý³ë³·íí»é »ý openmp ¨ cuda ï»ëýáéá·ç³ý»ñç ïçñ³éù³ùµ: ´»ñí³í »ý ï³ñµ»ñ ã³÷»ñç »é³ã³÷ ó³ýó»ñç ½áõ·³ñ»é³óí³í ùá¹»é³íáñù³ý ³ñ¹ûáõýùý»ñá: ìåòîä ïàðàëëåëüíîé ñèìóëÿöèè d-ìåðíûõ àâòîìàòîâ àáåëåâîé ïåñî÷íîé ãîðêè à. íàãàïåòÿí, ñ. ïîãîñÿí, â. ïîãîñÿí è þ. øóêóðÿí àííîòàöèÿ â ýòîé ñòàòüå ïðèâåäåíî îáîáùåíèå ïðîáëåìû çâåçäíîãî ïîêðûòèÿ, âûäâèíóòîé â [1] äëÿ êâàäðàòíûõ ðåøåòîê. îáîáùåíèå äîñòèãíóòî äëÿ d-ìåðíîé ðåøåòêè l d, d 2 n. ïðîáëåìà ñîñòîèò â ïîêðûòèè l d ðåøåòêè çâåçäíûì ãðàôîì s2d. íà îñíîâàíèè ðåøåíèÿ äàííîé ïðîáëåìû, ðàçðàáîòàíà ïàðàëëåëèçîâàííàÿ ïðîãðàììà ñèìóëÿöèè d-ìåðíîãî êëåòî÷íîãî àâòîìàòà. â êà÷åñòâå ïðèìåðà êëåòî÷íîãî àâòîìàòà ðàññìîòðåí ïðîöåññ ðåëàêñàöèè íåñòàáèëüíûõ ñîñòîÿíèé àáåëåâîé ìîäåëè ïåñî÷íîé ãîðêè. ñîîòâåòñòâóþùèå ïðîãðàììíûå ïàêåòû ðàçðàáîòàíû ñ èñïîëüçîâàíèåì òåõíîëîãèé openmp è cuda. ïðåäñòàâëåíû ðåçóëüòàòû ïàðàëëåëüíîé ñèìóëÿöèè, ïðîâåäåííîé äëÿ 3-ìåðíîé ðåøåòêè ðàçëè÷íûõ ðàçìåðîâ. mpcs.pdf (p.1-8) vahagn_abstract.pdf (p.9) microsoft word 18_tadevos_baghdasaryan.doc mathematical problems of computer science 39, 135--145, 2013. 135 adapting rgt solver interface to management strategy search problems tadevos s. baghdasaryan institute for informatics and automation problems of nas of ra e-mail: tbs_@mail.ru abstract we develop a unified solver for the class of problems where the space of possible solutions can be specified by reproducible game trees (rgt) [11]. in this paper we adapt the interface of rgt solver for marketing and supply chain management (scm) problems of the class presented by valuewar and trading agents competition models [1, 2, 4]. particularly, we describe: the ways for construction of operating entities, actions and strategy plans for the valuewar tool, the ways for construction of operating entities, actions, moves and strategy plans for the tac scm game. keywords: combating and competing games, expert systems, optimal strategies, valuewar, tac. 1. introduction 1.1. definition of rgt class in the variety of problems we identify the class where the space of possible solutions can be specified by reproducible combinatorial game trees (rgt) and develop a unified software, rgt solver, for elaborating optimal strategies for any input specified problem of the class [11]. as it was demonstrated in [11], rgt is a spacious class of problems with only a few following requirements to belong to: there are (a) interacting actors (players, competitors, etc. performing (b) identified types of actions in the (c) specified moments of time and (d) specified types of situations; there are identified benefits for each of the actors; the situations the actors act in and transformed after the actions, can be specified by certain rules, regularities. we do solve games of rgt class with meanings we do specify by states, situations, actors, actions of players, evaluators of situations and regularities of transformation of situations [11]. many security and competition problems belong to rgt class since those problems always interact, and rgt requirements include the most common of them. specifically, these are network intrusion protection (ip), management in oligopoly competitions and chess-like combinatorial problems, various security problems. the unified rgt specification of problems makes possible to design a unified solver for the problems of the class [9, 10]. the solver of rgt problems is a adapting rgt solver interface to the management strategy search problem 136 package aimed to acquire strategic expert knowledge to become comparable with a human in solving hard combinatorial competing and combating problems. 1.2. rgt interpretation of management strategy search problem according to [11] market competitions by valuewar [1-3] and scm by trading agents competitions [4] (tac scm) will be the following: states are determined by the set of parameters of current competition and scenarios of competition situations are determined by the states and actions of competition actors: a company competing against a few others actions are changes of the product price and quality in valuewar case or operational moves in tac game evaluators: algorithms calculating for input situations regularities, or transformation rules, in valuewar case are determined by general microand macroeconomics laws, which are applied to the situations. for tac case these regularities are implemented on the tac server side. the shell of rgt solvers is developed to provide the user with friendly java environment for unified solution of any rgt problem. the aims of the paper are the following: to adapt marketing (by vw) and scm (by tac) problems into the framework of rgt solver to develop ways for the construction of marketing and scm strategies. in the paper we present first the adaptation of valuewar followed by tac scm. valuewar (vw) integration includes: presentation of its composing nucleous entities presentation of actions and strategy plans definitions of vw situations embedding vw interface into rgt solver tac scm integration includes: presentation of scm nucleous and composite entities presentation of moves (actions) and strategy plans definition of situation in tac scm embedding tac scm interface into rgt solver. 2. adapting rgt solver to valuewar 2.1. valuewar so far the existing framework of rgt solver was implemented for the chess problem. but it is being modified to be more flexible in order to include other problems of the rgt class as well, in particular case the marketing and supply chain management problems presented by valuewar tool and tac scm model [9, 12, 13]. valuewar is a tool for the marketing strategy analysis, which presents a model of oligopoly competition of a few companies competing within a specified market. every company competes for one of the predefined goals. each of them is assigned to one of the various strategy plans that describe qualitative changes of basic competition parameters price and quality of items (services) they produce [2]. running the simulation a number of times (every time with a different strategy plan) and having the needed results, it is possible to separate the most optimal strategy plan for the given competitor playing within the selected type of market [5]. t. baghdasaryan 137 2.2. valuewar nucleous entities as an equivalent of figure in chess application of rgt solver, in valuewar we represent operating parameters as price (p) and quality (q). valuewar operates with strategy plans that represent the qualitive changes of the operating parameters mentioned above. in the frames of the solver we describe each parameter as a nucleous element with a particular name, type and value range. namely, for the price it would look in the following way [10]: price: quality: name “price” value range >=0 name “quality” value range in [0,100] 2.3. strategy plans for valuewar another equivalent is the definition of action, which is represented as move in chess and as a concept of strategy plan in valuewar. an sp describes the manner of changing p and q operational parameters. a typical example of sp would be, for instance, “decrease p, increase q”. at this point the actions are increase and decrease. each action will change the operating parameter in a predefined manner and by some delta. we describe action objects with the solver and further add their implementations within each reality. in our case we should apply actions over price and quality, so the same action is implemented twice both for the price object and quality object. depending on the application the agent selects and implements the corresponding version, passing the object as a parameter. in other words, in real world we can create some set of verbs. and having some real objects we define how to apply a particular verb to a particular object. it is possible that some action cannot be applied to the given object. such a case brings to nonsense unless we describe an appropriate implementation for that case. here we define some set of actions (verbs) and some set of objects (realities). so, for each object we have to create an implementation of the same action and associations between the object’s instance of that verb and the original verb. decrease price price reality price > 0 set of actions applicable to price increase implementation decrease implementation maximize implementation keep_unchanged implementation ..... implementations decrease quality quality reality quality in [1..100] set of actions applicable to quality increase implementation decrease implementation maximize implementation keep_unchanged implementation ..... implementations action applied to quality action applied to price fig. 1. application of the same action to different entities: price and quality. price price > 0 quality quality > 0 adapting rgt solver interface to the management strategy search problem 138 at the association level when met a term like “action over an object” (“increase price”) the agent starts to find a link between the “action” and its corresponding instance, related to the given “object”. the presence of such a link means that the action is applicable to that object and the agent will implement that term, otherwise the term is interpreted as nonsense. thus, for example, “increase” action will work differently for p and q (assuming that it is implemented for both of them). so, having such a set of actions, we create a set of various strategy plans and control the way of changing the operational parameters (p and q). 2.4. situation of valuewar a situation in valuewar is considered as a current market state. all parameters that form the market model with their values in couple with parameters of the participating parties (competitors) represent a set that we will call a situation (state). in its turn, every single move of a competitor (in accordance with its strategy plan) affects the market and changes its state, in other words, the competitor’s move changes the situation. in order to run the created strategy plans (performing strategy plan simulation) in the frames of the rgt solver we created a simple market model, based on all the parameters present in the basic valuewar market model. also, the market model may be switched to perform different predefined market types. in initial valuewar there are about 70 parameters forming the model, including such factors as demand, supply, population, per-capita-income, sales, total sales, market share and others. each of the forming factors is in interconnection with one or more others and is calculated by its corresponding formulas. the detailed description of the model is out of the scope of this paper and in fact, there may be different variations of it. actually, any market model is acceptable for valuewar framework problem, the only requirement is that the model environment should develop and change, affected by two single driving parameters price and quality. 2.5. embedding valuewar interface into rgt solver the valuewar simulator is embedded into rgt solver framework as a separate tab, which contains the main valuewar gameboard and subwindows for displaying various parameters. the gameboard is composed of two axes, forming a square of [p, q] positions field. the competitors are presented on the board as smaller squares of predefined color and their positions on the board reflect their values of p and q. as the simulation is in process, their positions may change in accordance to their strategy plans and achieved [p,q] values. sp competitor parameters sp competitor parameters competitor parameters sp competitor parameters market parameters sp fig. 3. affection of competitors’ strategy plans on a valuewar market model. figure 2. set of valuewar strategy plans. strategy plan action for the price action for the quality strategy plan action for the price action for the quality strategy plan action for the price action for the quality t. baghdasaryan 139 before the simulation starts, the user selects a type of a market and strategy plans for each of competing parties. then the user may choose the simulation to proceed automatically or step-by-step. the simulation is splitted into n periods. during a single period each party makes one pq-move and after that the market changes are calculated. every party has its separate window that reflects its related market parameters. in addition, market has its own window for reflecting its specific factors’ values. the user can change the strategy for any competitor at any period of the simulation, as well as the market type itself. in both cases the simulation continues with the new changes, starting from the next period. 3. adapting rgt solver to tac scm 3.1. tac scm comparing to the valuewar, tac problem is more complex not only because of more parameters present, but also that in tac there is higher level of uncertainty. there are two separate parts of a game – the first onedealing with customers and the second onedealing with suppliers [4]. in the tac scm game several agents compete in the market of personal computers assembling pcs of 16 configurations. configurations depend on types of each 4 main supply components used: cpu, ram, motherboard and hdd. every main component is being produced by 2 supplier brands and available in 2 qualitative options: high and low (speed or capacity). besides, there are limitations in compatibility of cpus with mboards: an mboard of one brand must be equipped only by a cpu of the corresponding brand. the actors in tac scm are: customers, which compose the demand of different types of pcs. on the first d day of every tac cycle they send numbers of rfqs (requests for quotes) to agents. an rfq includes: a configuration of requested pc, quantity, desired price, delivery date and penalty for any delay. agents (which have assembly factories (with the given daily production capacity), bank accounts and warehouse), which receive rfq bundles, analyze them and send back to customers as offer bundles (an offer includes: proposed configuration, quantity, price and delivery date). as a result, on the d+1 second day of the cycle customers may order some of the offers to the agent. suppliers, which produce the supply components. there are 2 suppliers for each main component (cpu, hdd, ram and mboard). each supplier has its daily production capacity and may produce 2 qualities of the same supply component. suppliers receive rfq bundles figure 4. the valuewar gameboard. -p +q -q -p+q +p+q -p-q +p-q p1= x11 p2= x12 p3= x13 p4= x14 p1= x21 p2= x22 p3= x23 p4= x24 p1= x31 p2= x32 p3= x33 p4= x34 p1= x41 p2= x42 p3= x43 p4= x44 c4 c1 c2 c3 adapting rgt solver interface to the management strategy search problem 140 from agents on day d+1, just after getting orders from customers. an agent rfq includes: requested supply component, quantity, desired price and delivery date. on the next d+2 day suppliers send back offer bundles with available quantities and delivery dates of the requested supply to the agent. then the agent may order some of those offers from the supplier on the same day. when an agent gets all the necessary supplies needed for any particular customer’s order, it starts the assembling itself on its factory. the assembling may last a number of days, taking into account: number of production cycles required for a single pc of the type, quantity of ordered pcs of the type and daily production capacity of the factory, in cycles. after the order is completed it is delivered to the customers and the agent’s bank account is updated accordingly, subtracting penalties for any delay (if any). the minimal full tac cycle lasts for 6 days. the game duration is t periods (days). every next day a new main tac cycle starts, meanwhile the deals started in previous cycles are still being proceeded. after the last period the game is over and the agent with the biggest bank account is considered as a winner. in valuewar we had two driving parameters, p and q – let’s say two dimensions. in tac they are much more: 4 types of components x 2 brands for each type x 2 qualities of each component, also various dates, etc – so, due to this “multidimentional factor” there is no explicit gameboard applicable. 3.2. nucleous and composite entities of tac scm the main base nucleous entities for tac are: date, price, quantity, speed, base component type, base component quality, base component brand, factory capacity, factory utilization, bank account, etc. more complex (composite) entities are: assembled product, particular supply component, bank, factory, agent, customer, supplier, etc. [10]. let's take an example for component. there are 4 types of components: cpu, hdd, mboard and ram. each of them is manufactured by 2 component-specific brands, and each brand produces 2 qualities of that component. for cpu component there are 2 manufacturers, pintel and imd, which manufacture 2 qualities of cpu 5ghz and 2ghz. for hdd component there are 2 manufacturers (watergate and mintor) producing 2 qualities of them (300gb and 500gb). for ram component mec and quinmax produce 2 qualities of them (1gb and 2gb). finally, for motherboard component basus and macrostar produce motherboards for pintel and imd (there is a limitation that pintel cpus will work only on pintel motherboards and imd cpus will work only on imd motherboards). first of all we should describe the basic nucleous elements with solver (examples below describe brands, capacity, component type, mboard type, price, date, utilization, production capacity entities): then based on them, we can describe some specific elements (examples for capacities for different components, production-specific brands, types of motherboards): fig. 5. example tac nucleous elements. component type name component type value in [cpu, ram, hdd, mboard] component quality name speed value in [2ghz, 5ghz] brands name brand value in [pintel, imd, basus, macrostar, mec, queenmax, w atergate, mintor] component quality name capacity value in [1gb, 2gb, 300gb, 500gb] price name price value >= 0 qualtity name quantity value >= 0 utilization % name utilization value in[1, 100] production capacity name production capacity value > 0 date name date value >= 0 t. baghdasaryan 141 further, more complex and final entities will be described (particular supply components, like "imd cpu 2ghz with base price of $1000", "mintor hdd 500gb with base price of $300" or final entities like "cpu supplier #1 branded 'pintel', with production capacity of 2000 cycles"): following this path, we can describe all the entities present in the tac problem, which will be equivalent to "figure" concept in a chess problem. below in figure 8 some examples are given: fig. 8. examples of tac entities. fig.7. a complex tac entity “queenmax 2gb ram” as a supply component fig. 6. example of specific tac nucleous elements. component quality name capacity value in [1gb, 2gb, 300gb, 500gb] component capacity name ram capacity value = 15 capacity in [1gb, 2gb] component capacity name hdd capacity value = 16 capacity in [300gb, 500gb] brands name brand value in [pintel, imd, basus, macrostar, mec, queenmax, w atergate, mintor] component brand name hdd brand value =11 brand in [w atergate, mintor] component brand name ram brand value =12 brand in [mec, queenmax] component brand name cpu brand value =13 brand in [pintel, imd] component brand name mboard brand value =14 brand in [basus, macrosta] component type name mboard type value in [pintel board, imd board] component type =mboard component type name component type value in [cpu, hdd, mboard, ram] price name price value >= 0 supply components name supply value queenmax 2gb component type = ram ram capacity = 2gb ram brand = queenmax price = 200 component capacity name ram capacity value =15 capacity in [1gb, 2gb] component brand name ram brand value = 12 brand in [mec, queenmax] component type name component type value in [cpu, ram, hdd, mboard] factory name factory value =1 production capacity = 2000 utilization = 50 factory name factory value =1 production capacity = 2000 utilization = 50 supplier name supplier value cpu supplier #1 component type = cpu production capacity = 2000 brand = pintel component name basus pintel board value =200 component type =mboard mboard type =pintel board mboard brand =basus price =250 supply component name supply value pintel 5ghz component type =cpu speed =5ghz cpu brand =pintel price =1500 supply component name supply value imd 2ghz component type =cpu speed =2ghz cpu brand =imd price =1000 component name mintor 500gb value =401 component type =hdd hdd capacity =500gb hdd brand =mintor price =300 supply component name supply value queenmax 2gb ram component type = ram ram capacity = 2gb ram brand = queenmax price = 200 adapting rgt solver interface to the management strategy search problem 142 3.3. defining moves and strategy plans for tac scm as in valuewar, strategy plans for tac specify the qualitative changes of some operational parameter(s). in the frames of tac problem, we can separate two concepts from each other: move and strategy plan (sp). we call a statement as “move” which contains entities with all their parameters specified with exact values (equivalent to “figure move” in chess). in contrast, we call a statement as “sp” in which at least one parameter is not specified with its value. again, an sp describes qualitative action(s) over tac entity (entities), whereas a tac move describes an exact (quantitative) action. in case of move the agent directly executes the statement. in case of sp (with missing values of one or more parameters) the agent may generate several moves from the same sp by substituting the missing values with some others from their acceptable value range. in order to select the most optimal move from the generated set it applies a game subtree for that appeared intermediate problem with possible usage of the existing knowledge base. a typical example of tac move would be the following statement: “buy n quantity of queenmax 2gb ram by d date” [7]. here we have 3 predefined entities: “quantity”, “queenmax 2gb ram” supply component and “date”. all three entities are populated with their exact values: quantity=n, date=d, supply=“queenmax 2gb ram”. replacing at least a single value from any of parameters will force to interpret the given statement as a strategy plan as shown in examples: note, that in complex entities not just any parameter may be missed. if in the example above we replace the value of “component type” (“ram”), then the statement loses its meaning in tac context: in “buy n quantity of queenmax 2gb by d date” the agent will not understand the meaning of “queenmax 2gb” – that is ram, hdd, mboard or cpu. otherwise, it will require more complex searching mechanisms in the agent implementation, which is currently out of scopeof this project. therefore, a typical example of sp would be the following statements: “buy n quantity of queenmax by d date” (as queenmax produces rams of different capacity); “buy quantity of ram by d date”(quantity may be any number within its predefined range); supply component supply queenmax 2gb ram component type = ram ram capacity ram brand = queenmax price = 200 qualtity quantity date date d fig.9. a single missing parameter in quantity (above) or “supply component” (below) causes the statement to be treated as an sp. supply component supply queenmax 2gb ram component type = ram ram capacity ram brand = queenmax price = 200 qualtity quantity n date date d fig.8. entities with fully populated values from the move’s statement. supply component supply queenmax 2gb ram component type = ram ram capacity = 2gb ram brand = queenmax price = 200 qualtity quantity n date date d t. baghdasaryan 143 “buy n quantity of 2gb ram in advance” (date is specified to be any coming number of days within the acceptable range); etc. also in the example above there is an action – “buy”. as in valuewar, here in tac we also should describe action objects with a further addition of their implementations into each entity it may be applied to. in the case above the “buy” action should be described in general and its appropriate implementation should exist for “queennmax 2gb ram” entity. that action may also take “quality” and “date” entities as mandatory (the first) or optional (the second) parameters. if no “buy” action is defined for “queennmax 2gb ram” entity, then that statement loses its meaning and that move or sp will take no effect. note, that “buy” is not defined either for entities “quality” or “date”, so the separate statements “buy n quantity” or “buy d date” are senseless and will not work. generally, in real world we can create some vocabulary of verbs. having some real objects we define how to apply a particular verb to a particular object. it is possible that some action cannot be applied to the given object. such a case brings to nonsense unless we describe an appropriate implementation for that case. here we define some set of actions (verbs) and some set of objects (realities). so, for each object we have to create an implementation of the same action and associations between the object’s instance of that verb and the original verb. at the association level when met a term like “action over an object” (“increase price”) the agent starts to search a link between the “action” and its corresponding instance within the “object”’s action set. the presence of such a link means that the given action is applicable to that object so the agent will implement that term; otherwise the term is treated as nonsense. thus, for example, “increase” action will work differently for p and q (assuming that it is implemented for both of them). 3.4. defining situation of tac scm we do not consider the tac market model because it is implemented on the server side, and according to the rules is inaccessible, so we have to deal only on the tac agent’s side. that is why we will define a tac situation by including only the states of agent-related parameters. all the agent’s entities with their current conditions and values compose a situation for the tac agent at a specific period of time. it includes: factory utilization, warehouse of supplies with quantities of any present components, bank account, orders from customers, orders to suppliers, current date, start date, etc. so there is not any entity associated to a situation, it is represented as parameters set from other entities. having defined the concept of situation, we can use it when forming a knowledge base. the knowledge base, in turn, consists of a set of knowledge elements. and a knowledge element, in general, has the following structure: an initial situation, the applied action (or sp) and the resulting situation. moves (or strategy plans) in action may consider situations directly whenever they include any conditional content. fig. 10. implementation of “buy queenmax 2gb ram” action. action name buy value supply component supply queenmax 2gb ram component type = ram ram capacity = 2gb ram brand = queenmax price = 200 buy implementation module mandatory parameters optional parameters adapting rgt solver interface to the management strategy search problem 144 3.5. embedding tac scm interface into rgt solver the tac agent template has its own interface; we just duplicate its data monitors into our interface to track the game progress. for that we integrate a data transfer module into the rgt solver and the tac agent’s template in order to transfer the ongoing data between each other. the tac solver module handles all the objects defined for the tac by the solver interface (namely: entities, actions, moves, strategies), so it is actually considered as the agent’s “brain”. the module may be integrated either in the agent or in the rgt solver. on the first case the agent will transfer only the information to be displayed on the solver’s tac gameboard tab. on the second case at every game event the agent will pass the incoming information from the server to the solver to be handled. after that the solver processes the data and passes the results back to the agent for further sending to the tac server. conclusion the rgt solver framework with two integrated management-focused problems (the valuewar strategy assessment tool and the tac scm game) is represented. the ways for constructing objects, actions and strategies for the given management problems are described. particularly: the ways for construction of operating entities, actions and strategy plans for valuewar tool; the ways for construction of operating entities, actions, moves and strategy plans for tac scm game. also, we embedded the corresponding interfaces of each valuewar and tac scm problems into separate tabs within the main solver framework. references [1] m. chussil and d. reibstein, “putting the lessons before the test. in wharton on to analyse & develop competitive strategies”, john wesley & sons, pp 343-368, 1997 [2] m. chussil, d. reibstein, strategy analysis with value war, the scipress, 1994. 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[8] t. baghdasaryan, a. grigoryan and z. naghashyan, "development of a scripting language interpreter for acquisition of expert knowledge in a regular way", csit2009 7th international conference in computer science and information technologies, 5p., 2009. [9] z naghashyan and e. pogossian, "developing java software for representation, acquisition and management of strategy knowledge", mathematical problems of computer sciences, pp.187-195, 2010. [10] k. khachatryan and v. vagradyan, “graphical language interpreter unified for ssrgt problems and relevant complex knowledge”, csit2011 8th international conference in computer science and information technologies, pp. 178-182, 2011. [11] e. pogossian, “effectiveness enhancing knowledge based strategies for ssrgt class of defense problems”, proceedings of nato advanced study institute (asi), 2011. t. baghdasaryan 145 [12] k. khachatyan and s. grigoryan “java programs for presentation and acquisition of meanings in ssrgt games”, proceedings of seua annual conference, 7p., 2012. [13] k. khachatryan and s. grigoryan, “java programs for matching situations to the meanings of ssrgt games”, proceedings of seua annual conference, 5p., 2012 submitted 30.11.2012, accepted 02.02.2013. rgt solver համակարգի ինտերֆեյսի համակերպումը կառավարման ռազմավարությունների փնտրման խնդիրներին թ. բաղդասարյան ամփոփում մշակվում է rgt solver ունիֆիկացված ծրագրային համակարգ այն դասի խնդիրների լուծման համար, որում հնարավոր լուծումների տիրույթը ներկայացվում է վերարտադրվող խաղային ծառերի միջոցով: նկարագրվում է rgt solver համակարգի ինտերֆեյսի համակերպումը տվյալ դասին պատկանող մարքեթինգի և մատակարարման շղթայի կառավարման խնդիրներին, որոնք ներկայացված են համապատասխանաբար մարքեթինգի ռազմավարությունների վերլուծման valuewar գործիքային փաթեթով և մրցակցող առևտրային գործակալներ (tac scm) խաղով: մասնավորապես, նկարագրվում է հետևյալը. valuewar թաղանթի համար բաղադրիչների, գործողությունների և ռազմավարական պլանների նկարագրման եղանակները, tac scm մոդելի համար բաղադրիչների, գործողությունների, քայլերի և ռազմավարական պլանների նկարագրման եղանակները: адаптация решателя rgt-игр к задачам поиска стратегий менеджмента т. багдасарян аннотация мы разрабатываем унифицированный механизм (rgt solver) для решения класса задач, в которых пространство возможных решений представляется воспроизводимыми игровыми деревьями (rgt). в рамках статьи мы описываем адаптацию интерфейса rgt solver к задачам маркетинга и управления цепью снабжения, относящихся к данному классу и представленными, соответственно, моделями valuewar и конкурирующего торгового агента (tac scm). в частности, мы описываем: способы описания компонент, действий и стратегических планов для оболочки valuewar; способы описания компонент, действий, ходов и стратегических планов для модели tac scm. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 45, 122--126, 2016. symbiosis of email and sms arman h. harutyunyan1, sona h. gharagyozyan2 1yerevan state university 2institute for informatics and automation problems of nas ra e-mail:a_harutyunyan@ipia.sci.am, sona@ipia.sci.am abstract the specific features of the means of written communication through computer and mobile networks, as well as the prehistory of the establishment of hybrid communication system using ip and gsm network technologies have been examined. keywords: symbiosis, sms, email, predictive. 1. introduction the new means of written communicationemail and sms, appearing at the end of the twentieth century, pressed the traditional postal service keeping it mainly as a means to send official documents. in place of the past epistolary genre with its inherent personality came concise information emails. nevertheless, the speed of delivery of messages to the addressee provides undeniable advantages to the electronic means of written communication. at the same time, integrated systems using "one package," technology of email and sms, which are complementary to each other are of particular interest. without going into a detailed assessment of the advantages and disadvantages of these systems, we have pointed out only the specific functional features of email and sms technologies. 2. problems and solutions originally designed for the exchange of written communications between computers, email is considered as a system “on demand” despite of its rapid delivery to the addressee. the fact of reading delivered message depends on the time the addressee opens his mailbox. in contrast to email, sms is not only a mailing system: the volume of transmitted message is limited, the 122 mailto:a_harutyunyan@ipia.sci.am mailto:sona@ipia.sci.am a. harutyunyan, s. gharagyozyan 123 delivery of the message is only possible in the zone of mobile network coverage or according to the current roaming agreement of mobile operator. the message delivery to the “pocket” of the mobile subscriber is an indisputable advantage of sms technology. accordingly, either the message is delivered “on demand” almost anywhere in the world (email), or directly into the hands of the addressee, but for the above mentioned limitations (sms). the integrated solution to the “distance” problem of particular sms service is solved by using ip-communications, as a main environment followed by the locking of current mobile network and web technologies providing the possibility sending sms to a specified, local mobile network from computers from virtually any region of the world. another above discussed problem: problem “on demand” is associated with the integration of sms technology in email system in the form of smsnotification systems on receipt of emails in the subscriber's mailbox. such postal sms-informants analyze the incoming correspondence of the subscriber separating the letters with return addresses listed by the subscriber to the "white list" to which the subscriber receives smsnotification, in case of receiving correspondence [1-3].. in other systems of email-informants, the initiative of sending notification is given to the sender. the problems to create a "hybrid" system of internet/sms is connected to the specifics of displaying web "pictures" on the display of a mobile phone and with the difficulties inputting alphabetic information. if a modern smartphone with an informant and virtual keyboard is being used while working with the system, no special problems will arise. however, the possibility of a user-friendly operation in a system with other common models of mobile phones leads to the need for a special service software. mobile subscribers use a wide variety of models of mobile phones: from budget phone with a small screen and a set of keys, providing only the basic functions of mobile communication: a telephone communication and exchange of sms to multifunctional smartphones with display of 5-6" touchscreen, virtual keyboard, multi-core processors and with the possibility to use internet. there often exists the possibility of connecting to the internet in common models of mobile phones with screens of about 2" and button dialing keyboard. however, the phenomenon of screen with small format and resolution, as well as of touch-tone, makes it extremely inconvenient use these phones working with network resources. however, not having a wide range of additional functions specific smartphones (often redundant), and they are today in demand (the share of push-button telephones in russian mobile networks, for example, is more than 30%. in the first quarter of 2015 there have been sold more than 2.7 million of these mobile phones [4,5]) corresponding to the basic purpose of mobile phones and having the indubitable advantage of the "pocket" device, due to the size and the weight (unlike the majority of smartphones, which are considered as "wearable" devices). these circumstances determine the feasibility of studies and the development of new solutions providing information on mobile devices of a similar class, the development of adaptive interactive "comfort" systems graphical interaction of mobile phone users with communication and information resources of mobile and ip networks (it is evident that this interaction is provided by gsm / ip servers). these dialogue systems should be extremely concise, limitative as far as possible, the participation of the user answers "yes-no" considering the complexity of a set of queries and text on a mobile phone keypad. problems of writing text messages are solved using predictive typing systems anticipating in the already typed letters of the options of the current and the next word (or phrases) while user is typing using the built-in phone dictionary. such systems for mobile phones (designed mainly for typing sms) have been known since 1999 with the development and encapsulation of the t9 system in mobile phone software and similar functions of itap and ezi systems [6,7,8]. being developed by tegic communications, t9 is used in mobile phones by most of the major manufacturers (nokia, symbiosis of email and sms 124 samsung, sony ericsson, etc.). t9 is considered as the most popular system of predictive typing. unlike т9, itap attempts to predict short phrases as well, analyzing not only typed letters of the current word, but also the previous text. basic dictionaries of these predictive typing systems include 35-60 thousands of words and expressions, and support most of the european and some asian languages. unlike the mentioned predictive systems when typing the text of sms there can be offered an algorithm of serial-word letter by letter for the possible continuation of the typing word, using funded words database individual for each user according to the most frequently used words by the user while typing sms: "verbal image" of the lexicon used by the user stored on the sms server. this database is formed by accumulated words containing in the previous sms sent by the user. depending on the frequency of the repetition of words corresponding “weight” is attributed to each word determined by the number of repetitions of the word accumulated in the sms. such predictive system is installed not in the phone but in a centralized server. the system can be used with any mobile telephone having access to the internet, respectively, without requiring additional resources (memory, cpu) for its functioning. the system generates prediction not by semantic analysis of words and sentences comparing them with the information from the database but by the coincidence of images: putting in comparison the literal consistent of the typing string to the "images" of the vocabulary fund used by the user while sending sms accumulated in the individual database. the analysis on the coincidence of characters ("images of words") allows to use prediction system without being tied to a particular language. the database with which the system operates is located on a server in the form of individual lists for each registered user containing a sequence of characters (letters) forming a word. databases are automatically replenished with words from sms sent by the user. the use of automatically generated personal database "images of the words" increases the probability of a correct prediction at the initial stages of typing allowing, at the same time, significantly to reduce the size of the lexicon of the dictionary. while typing, the first two letters of the typing word are taken as a reference (base) letters of the word; further the letters set is compared with the dictionary with the issuance tips with the options of word completion. when typing messages a user can use both latin alphabet and cyrillic alphabet. accordingly, while sending sms there is a need to convert in cyrillic in the similar-sounding words, written in latin letters (operation of transliteration) for reliable message playback at any gadget. in asnet computer network users can access a number of services of infocommunication applications sharing web and sms technologies (www. asnet.am/sms applications),developed in the years from 2007 to 2013 at the institute for informatics and automation problems of national academy of sciences of the republic of armenia(iiap nas ra) [9,10]. the majority of the above mentioned services used as a basis for the development of info-communication mss system of iiap for mobile phones, providing the possibility to comfortably work with smsoip and emails. such system provides: 1. formation and sending of sms via ip network using web technologies 2. formation and sending of email with sms notification (mobile network of the region where the server is running) containing user-selected fragment of the letter. 3. the use of graphical dialog interface, automatically adapting to the class of the gadget served at the moment. 4. when accessing the system by a phone with a small screen and keyboard, automatic switching to the dialogue version, minimizing data volume, user typing in the formation a. harutyunyan, s. gharagyozyan 125 of sms/email at the same time the formation of the address part of email, sms in single request in a natural language 5. the mechanism of "prediction" typed by the user during the formation of the message. 6. the use of the centralized individual savings with no database on the server system (user requisites, tel. book, address book, dictionary of sms: "verbal images”, etc.). references [1] [online]. available: https://help.mail.ru/mail-help/settings/notifications [2] [online]. available: http://iglous.ru/besplatnyj-sposob-poluchat-sms-uvedomleniya-opochte/ [3] d. gevorkyan, k. khachatryan, a. nanassian, a. petrosyan, g. petrosyan, v. sahakyan and e.vardanyan “mail informerselective incoming e-mail instant phone notification system”, proceedings of international conference computer science and information technologies, yerevan, pp. 466-468, 2009. [4] [online]. available: http://secretmag.ru/news/2015/06/24/knopochnie-telefoni/ [5] [online]. available: http://sia.ru/?action=show_news&id=305085§ion=484 [6] [online]. available: http://www.ixbt.com/mobile/review/prtxtsms.shtml [7] [online]. available: http://www.genon.ru/getanswer.aspx?qid=29112f32-777b-4d02b3ab-bbec2afb72ef [8] [online]. available: http://solo-project.com/articles/category/12/message/1241/ [9] d. gevorkyan, а. nanassian and k. khachatryan “new web resources asnet.am”, computer science and information technologies, proceedings of international conference, yerevan, pp. 311-313, 2011. [10] а. nanassian and k. khachatryan “mail2sms.asnet.am – the alert system of incoming”, proceedings of international conference computer science and information technologies, pp. 459-462, 2013. submitted 04.11.2015, accepted 22.02.2016 էլ-փոստի և sms հաղորդագրության համակցում (symbiosis) ա. հարությունյան և ս. ղարագյոզյան ամփոփում դիտարկվել են համակարգչում և բջջային ցանցում հաղորդագրությունների փոխանակման տարածված համակարգերը, ստեղծելով տվյալների փոխանակման նախադրյալներ, որոնք օգտագործում են ip և gsm ցանցերը։ symbiosis of email and sms 126 симбиоз email и sms а. арутюнян и с. карагезян аннотация рассмотрены особенности распространенных систем обмена письменными сообщениями в компьютерных и сотовых сетях, предпосылки создания гибридных систем передачи сообщений, использующих технологии ip и gsm сетей. рассмотрены проблемы доступа к ресурсам подобных систем с мобильных гаджетов, пути их решений. microsoft word informstochnor.doc ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 30, 87--91, 2008. 87 оптимальная передача информации через пороговую систему со стохастическим резонансом арсен а. ахумян1, вардан ж. товмасян2, оганес c. ароян2 1 институт радиофизики и электроники нан армении, аштарак 2 ереванский государственный университет, армения e-mail: arsen@irphe.am, tvardan@mail.ru, hharoyan@ysu.am аннотация рассмотрена зависимость скорости передачи информации через нелинейный канал со стохастическим резонатором от уровня шума. выведена связь межу спектром сигнала и спектральной полосой шума, при котором соотношение сигнал/шум на выходе канала, а также скорость передачи информации имеют максимальное значение. литература [1] r. benzi, a. sutera and a. vulpiani, the mechanism of stochastic resonance, j. phys. a 14, pp 453–457, 1981. [2] b. mcnamara and k. wiesenfeld, theory of stochastic resonance, phys. rev. a 39, pp 4854–4869, 1989. [3] z. gingl, l. b. kiss and f. moss, non-dynamical stochastic resonance: theory and experiments with white and arbitrarily coloured noise, europhys. lett. 29, pp 191–196, 1995. [4] f. chapeau-blondeau and x. godivier, theory of stochastic resonance in signal transmission by static nonlinear systems, phys. rev. e 55, pp 1478–1495, 1997. [5] l. gammaitoni, p. hänggi, p. jung and f. marchesoni, stochastic resonance, revs. of modern physics 70, pp 223–287, 1998. [6] c. e. shannon and w. weaver, the mathematical theory of communication, the university of illinois press, 1949. [7] в. с. анищенко, а. б. нейман, ф. мосс, л. шиманский-гайер, стохастический резонанс как индуцированный шумом эффект увеличения степени порядка, уфн, том 169, №1, pp 7–38, 1999. оптимальная передача информации через пороговую систему со стохастическим резонансом 88 æýýáñù³óç³ûç ûåïçù³é ñ³õáñ¹áõùá ëïáë³ëïçï é»½áý³ýëáí ß»ù³ûçý ñ³ù³ï³ñ·áõù ². ð³ëáõùû³ý, ì. âáíù³ëû³ý, ð. ð³ñáû³ý ²ù÷á÷áõù ¸çï³ñïí³í ¿ çýýáñù³óç³ûç ñ³õáñ¹ù³ý ³ñ³·áõãû³ý ï³ëí³íáõãûáõýá ³õùáõïý»ñç ù³ï³ñ¹³ïçó ëïáë³ëïçï é»½áý³ïáñáí ï³åáõõáõù£ ¸áõñë ¿ µ»ñí³í ³½¹³ýß³ýç ¨ ³õùáõïç ëå»ïïñ³é é³ûýáõãûáõýý»ñç ùçç¨ ï³åá, áñç ¹»åùáõù »éù³ûçý ³½¹³ýß³ý ³õùáõï ñ³ñ³µ»ñáõãûáõýá ¨ çýýáñù³óç³ûç ñ³õáñ¹ù³ý ³ñ³·áõãûáõýá ëï³óíáõù »ý ³é³í»é³·áõûý£ d:\sbornik\...\hodvac1.dvi mathematical problems of computer science 24, 2005, 5{10. data flow analysis b y linear p r ogr amming m odel a r m e n h . a la ve r d ya n , v ilik m. k a r a kh a n ya n a n d v a r d a n r . to n o ya n institue for informatics and automation problems of nas of ra e-mail armen am@yahoo.com abstract the paper presents the general discussion of the data °ow algorithms structuring problem on one hand, and, as an example of that particular problem class, the analysis of the linear programming problem when the objective function coe±cients vary depending on the data °ow. the problem is in reconstruction of current result with such an approach, which is the most plain from the full solution of the problem by the dynamic data set °ow. refer ences [1 ] l . a s la n ya n , j. ca s t e lla n o s , f. min g o , h . s a h a kya n , v . r ya z a n o v, a lg o r it h m s fo r d a t a flo ws , in t e r n a t io n a l jo u r n a l in fo r m a t io n th e o r ie s a n d a p p lic a t io n s , is s n 1 3 1 0 -0 5 1 3 3 , v o lu m e 1 0 , n u m b e r 3 ( 2 0 0 3 ) , p p . 2 7 9 -2 8 2 . [2 ] b a r b a r a g. r yd e r a n d ma r vin c. p a u ll, in c r e m e n t a l d a t a -° o w a n a lys is a lg o r it h m s , a cm tr a n s a c t io n s o n p r o g r a m m in g l a n g u a g e s a n d s ys t e m s ( top l a s ) , v o lu m e 1 0 is s u e 1 ( 1 9 8 8 ) , p p . 1 -5 0 . [3] â. ê. ëåîíòüåâ, óñòîé÷èâîñòü â ëèíåéíûõ äèñêðåòíûõ çàäà÷àõ, ”ïðîáëåìû êèáåðíåòèêè”, áûï. 35, 1979. [4 ] a . a la ve r d ya n , l in e a r p r o g r a m m in g w it h ch a n g in g co e ± c ie n t s of ob je c t ive fu n c t io n , in tr a n s a c t io n s o f t h e in s t it u t e fo r in fo r m a t ic s a n d a u t o m a t io n p r o b le m s o f n a s r a , ma t h e m a t ic a l p r o b le m s o f co m p u t e r s c ie n c e , 2 0 0 5 , y e r e va n . [5 ] ch . p a p a d im it r iu , k . s t e ig lit z , co m b in a t o r ia l o p t im iz a t io n : a lg o r it h m s a n d co m p le xit y, p r e n t ic e -h a ll in c , 1 9 8 2 . [6] þ. êóçíåöîâ, â. êóçóáîâ, à. âîëîùåíêî, ìàòåìàòè÷åñêîå ïðîãðàììèðîâàíèå, âûñøàÿ øêîëà, ìîñêâà, 1980. [7] á. ìóðòàô, ñîâðåìåííîå ëèíåéíîå ïðîãðàììèðîâàíèå, òåîðèÿ è ïðàêòèêà, ìîñêâà, ìèð, 1984. 5 6 data flow analysis by linear programming model îíû³éý»ñç ñáëù»ñç í»ñéáõíáõùá áëï ·í³ûçý íñ³·ñ³íáñù³ý ùá¹»éç ². ð. ²é³í»ñ¹û³ý, ì. ø. î³ñ³ë³ýû³ý, ì. è. îáýáû³ý ²ù÷á÷áõù ðá¹í³íá ýíçñí³í ¿ ïíû³éý»ñç ñáëù»ñç ³é·áñçãùý»ñç ï³éáõóù³ý áý¹ñ³ýáõñ ëý¹ñç ùýý³ñïù³ýá ùç ïáõùçó, ¨ áñå»ë ³ûë ¹³ëç ù³ëý³íáñ ëý¹ñç ûñçý³ï‘ ·í³ûçý íñ³·ñ³íáñù³ý ëý¹ñç áõëáõùý³ëçñù³ýá, »ñµ ñáëù³ûçý ïíû³éý»ñçó ï³ëí³í ÷á÷áëíáõù »ý ýå³ï³ï³ûçý ýáõýïóç³ûç ·áñí³ïçóý»ñá: êý¹çñá áýã³óçï ³ñ¹ûáõýùç í»ñ³ï³éáõóù³ý ù»ç ¿ ³ûýåçëç ùç ùáï»óù³ùµ, áñá ³é³í»é å³ñ½ ¿ ëý¹ñç éç³ï³ï³ñ éáõíáõùçó áëï ñáëùç ¹çý³ùçï ïíû³éý»ñç µ³½ùáõãû³ý: mathematical problems of computer science 59, 57–68, 2023. doi: 10.51408/1963-0102 udc 004.934 making speaker diarization system noise tolerant davit s. karamyan1,2, grigor a. kirakosyan2,3 and saten a. harutyunyan2 1russian-armenian university, yerevan, armenia 2krisp.ai, yerevan 3institute of mathematics of nas ra, yerevan, armenia e-mail: {dkaramyan, gkirakosyan, sharutyunyan }@krisp.ai abstract the goal of speaker diarization is to identify and separate different speakers in a multi-speaker audio recording. however, noise in the recording can interfere with the accuracy of these systems. in this paper, we explore methods such as multi-condition training, consistency regularization, and teacher-student techniques to improve the resilience of speaker embedding extractors to noise. we test the effectiveness of these methods on speaker verification and speaker diarization tasks and demonstrate that they lead to improved performance in the presence of noise and reverberation. to test the speaker verification and diarization system under noisy and reverberant conditions, we created augmented versions of the voxceleb1 cleaned test and voxconverse dev datasets by adding noise and echo with different snr values. our results show that, on average, we can achieve a 19.1% relative improvement in speaker recognition using the teacher-student method and a 17% relative improvement in speaker diarization using consistency regularization compared to a multi-condition trained baseline. keywords: speaker recognition, speaker diarization, noise robustness, teacherstudent, consistency regularization. article info: received 9 january 2023; send to review 30 january 2023, received in revised form 11 april 2023; accepted 17 april 2023. acknowledgement: this research was supported by krisp.ai. 1. introduction and related work speaker recognition (sr) is a broad field of study that addresses two major tasks: speaker identification and speaker verification. speaker identification is the task of identifying a person, whereas speaker verification is the task of determining whether the speaker is who they claim to be. in this study, we focus on far-field, text-independent speaker recognition, where the speaker’s identity is determined by the speaking style rather than the content of the speech. typically, such speaker recognition systems operate on unconstrained speech utterances that are converted into a fixed-length vector known as speaker embedding. many speech0-processing tasks use speaker embedding such as speaker diarization (sd) [1, 2], automatic speech recognition (asr) [3], and speech synthesis [4, 5]. 57 58 making speaker diarization system noise tolerant in recent years, deep neural networks have actively been employed for speaker embedding extractors since d-vector [6] was proposed. subsequently, the x-vector [7] was widely used because of the superior performance achieved by employing statistical pooling and time delay neural network (tdnn). other architectures such as resnet-based convolutional neural networks and cnns with cross-convolutional layers [8, 9] were employed for capturing the traits of speech. in addition, to deal with variable-length inputs, transformer [10], cnn-lstm [11] and a slew of variants of tdnn [12] were applied for dnn-based speaker embedding extractors. finally, to reduce the computational complexity and make the models smaller, [13, 14] employed 1d depth-wise separable convolutions for the speaker recognition task. metric learning techniques have been successful in speaker recognition tasks. these methods aim to create speaker embeddings with small distances between embeddings of the same speaker and large distances between embeddings of different speakers since unsupervised clustering will be applied to embeddings later in the speaker diarization pipeline. the triplet loss was proposed in [15] which required a careful selection of a triplet because the effectiveness of the performance depended on the contrast between negative and query samples. the prototypical loss was proposed in [16], where many negative samples were used and the euclidean distance between the centroid of all negative samples and the query embedding was maximized. in the generalized end-to-end loss [17], every utterance in the mini-batch functions as a query as opposed to just one in the prototypical loss. the angular prototypical (ap) loss [18] used only one utterance from each class as the query like the prototypical loss, but with a cosine similarity-based metric. the primary use case for speaker embeddings is speaker diarization. speaker diarization is the process of dividing an input audio stream into homogeneous segments according to the speaker’s identity. a typical speaker diarization system usually consists of several steps: (1) speech segmentation, where the input audio is segmented into short sections that are assumed to have a single speaker, and the non-speech sections are filtered out by voice activity detection (vad), (2) speaker embedding extractor, where speaker embeddings are extracted from segmented sections, (3) clustering, where the extracted audio embeddings are grouped [1] into clusters based on the number of speakers present in the audio recording, and optionally, (4) resegmentation step is performed to further refine clustering results. in real-world environment, noise causes significant degradations to the performance of speaker diarization systems, and is, hence, a major problem requiring special attention. the goal of noise-tolerant speaker diarization is to achieve improved performance in noisy environments. a recent work [19] tackles this problem using the auto-encoder architecture as a dimensionality reduction module. they extract two low-dimensional codes from speaker embeddings, representing the speaker identity and irrelevant noise information, then remove the noise factors. to our knowledge, there hasn’t been a lot of research done in this particular area. asr systems also suffer deterioration due to audio noise, and this has been the subject of extensive research [20, 21, 22], some of which inspired us. in this paper, we explore several approaches, borrowed from unsupervised domain adaptation, to make the speaker recognition models noise tolerant. in particular, we apply teacherstudent and consistency regularization techniques on speaker recognition and diarization tasks and compare them with multi-condition training when various noise augmentations are used. we were inspired by the significant results of this work for teacher-student [22], where clean and noisy audios are fed to the teacher and the student, respectively, to enforce similarity between the output distributions. consistency regularization is a commonly-used d. karamyan, g. kirakosyan and s. harutyunyan 59 technique amongst a variety of tasks in machine learning. this work [20] applies it in a manner similar to that mentioned previously, only here clean and noisy inputs are both fed to the student model. in the paragraphs that follow, we’ll discuss in detail how we apply these concepts to obtain noise-robust speaker recognition and diarization. 2. improving noise robustness of speaker diarization system there are several ways to improve the performance of speaker diarization systems in noisy and reverberant environments. for instance, work in [1] proposed the sequence of refinement operations to smooth and denoise data in the similarity space. in this work, we will focus only on the speaker embedding extraction part, and we are going to use unsupervised domain adaptation techniques to make the model noise tolerant. given a training dataset consisting of pairs (xi, yi) where xi represents an audio signal and yi represents the speaker id. our goal is to learn a parametrized function fθ, which should be able to compress any given audio into a d-dimensional vector, also known as a speaker embedding. moreover, if two audio signals are spoken by the same speaker, then the cosine similarity between their corresponding embeddings should be higher. conversely, if the two audios are spoken by different speakers, the cosine similarity between their embeddings should be lower. the additive angular margin (aam) loss, as proposed in [23], is a prevalent method for training speaker embedding extractors. the aim of the aam loss is to minimize the angle between speaker embeddings belonging to the same speaker while simultaneously maximizing the angle between speaker embeddings belonging to different speakers. 2.1 consistency regularization the core idea behind consistency regularization (cr) is to make sure that the network produces similar embeddings for the augmented versions of the same unlabeled utterance [20, 24, 25]. it is enforced by an additional regularization term in the loss function: lcr = 1 n n∑ i=1 |fθ(a(xi)) − fθ(a(xi))|22, where fθ is an embedding extractor with parameters θ, n represents the total number of training examples within the dataset. by a(x) we denote a stochastic operation that augments the audio in such a way that the speaker identity remains the same. so the difference is most likely non-zero. the final form of loss is a weighted combination of laam and lcr as shown below: l = (1 − α)laam + αlcr, where α is a hyperparameter taking values between 0 and 1. 2.2 teacher-student one critical problem with lcr loss is that it is not stable because of unstable target. to mitigate unstable target problem, the teacher-student model was proposed in [26], where two separate models were used: a student network with θ parameters and a teacher with 60 making speaker diarization system noise tolerant θ′ parameters. on unlabeled examples, the teacher network provides the learning target for the student network: lts = 1 n n∑ i=1 |fstudentθ (a(xi)) − f teacher θ′ (a(xi))| 2 2. student is trained as usual. teacher model is not trained via back-propagation. instead, its weights are updated at each iteration using the weights from the student network. again, the final loss is a weighted combination of laam and lts as shown below: l = (1 − α)laam + αlts. 2.3 knowledge distillation if the teacher model is already trained, it is desirable that its weights remain constant. this training setup is known as ”knowledge distillation”, where the student model is trained to mimic a pre-trained, larger model [27]. 3. experiments 3.1 model architecture in all experiments, we will use the speakernet [13] architecture as the backbone model. speakernet models are made up of 1d depth-wise separable convolutional layers. on top of the model, a statistical pooling layer is used to obtain a fixed-length vector. the proposed variation of speakernet (speakernet-m) has fewer parameters (5m) when compared to sota and shows very similar performance on voxceleb1 [28] trial files when compared to sota systems. the model provides embeddings of size 256 for a given audio sample. in teacher-student experiments, both the teacher and the student have the same architecture. 3.2 datasets the voxceleb1 [28] and voxceleb2 [29] datasets are widely recognized benchmarks in the field of speaker recognition. these datasets have pre-defined development and test sets, which allow for an objective and consistent evaluation of speaker recognition models. we trained our speaker recognition models using only the development part, which consisted of 7205 distinct speakers. for evaluation of speaker embeddings quality, we use voxceleb1 cleaned test trial file. the test trial file contains a list of audio pairs, and the model’s performance is evaluated based on its ability to correctly determine whether the two recordings belong to the same speaker or not. to evaluate speaker diarization, we use the voxconverse [30] development set. the dataset statistics are shown in table 1. 3.3 metrics the equal error rate (eer) metric is used to evaluate the speaker verification. this is the rate used to determine the threshold value for a system when its false acceptance rate and d. karamyan, g. kirakosyan and s. harutyunyan 61 table 1: statistics of datasets used for training speakernet. dataset # speakers duration (h) # utterances voxceleb1 1211 340.4 148642 voxceleb2 5994 2359.77 1,092,009 false rejection rate are equal. we calculate eer on voxceleb1 cleaned test trial file under original, noisy and echo conditions. for diarization evaluation purposes, we used diarization error rate (der). this is the sum of three error terms: false alarm (fa), missed detection (ms) and speaker confusion error rate (cer). similar to the previous works [12, 14], we use collar 0.25 sec and ignore overlap speech regions for confusion error rate calculation. we test the diarization system in original, noisy, and echo scenarios, just like we do for speaker verification. both eer and der are calculated using the cosine similarity back-end. 3.4 experiment setup 3.4.1 input features our audio pre-processing procedure is identical to the one described in the speakernet paper [13]. for each frame window of 20 ms, shifted by 10 ms, 64-dimensional acoustic features were calculated from the speech recordings. each utterance fed to the encoder has a size t × 64, where t is the number of frames in a given audio sample. we crop speech segments into random chunks from 3 to 8 seconds. with larger chunks, the model converges faster. 3.4.2 clean teacher our first baseline is a clean teacher trained on voxceleb1 and voxceleb2 datasets with additive angular margin loss. we set the aam loss hyperparameters to s = 30 and m = 0.2, as it was shown in [13, 14], these values give the best results. to avoid overfitting, we added specaugment [31] to the training pipeline, which randomly masks blocks of frequency and time channels. 3.4.3 noisy teacher our second baseline is a noisy teacher trained with the same objective as a clean baseline, and with the additional augmentation steps described below: • no augment: leave the utterance unchanged • rir augment: reverberate an input audio using an impulse response from rirs dataset [32] • noise augment: add noise from musan [33] dataset with signal-to-noise (snr) values randomly chosen from 0-50db • rir-noise augment: apply noise and echo perturbations to the same audio at the same time 62 making speaker diarization system noise tolerant • speed augment: speed perturbation with 0.95x and 1.05x speeds rir, noise, and rir-noise augmentations all have a probability of 0.25 and are mutually exclusive. speed augmentation is applied independently with a probability of 0.1. 3.4.4 consistency regularization we add an extra mean squared loss between embeddings for the augmented and nonaugmented versions of the same utterance to the aam loss during training. we set the α hyperparameter in the final loss to 0.1. 3.4.5 teacher-student in order to supervise the student model, we choose our clean-teacher baseline as the teacher. we did not update teacher weights during the training and no perturbations were applied to the input of the teacher model. the flow chart of teacher-student training is presented in fig. 1. during the training procedure, in addition to the aam loss, the mean squared loss between the student and teacher-produced embeddings is minimized. we set the α hyperparameter in the final loss to 0.1. fig.1. flow chart of teacher-student learning for improving noise robustness of sr. 3.4.6 optimization all models are trained for 200 epochs with an sgd optimizer, with an initial learning rate (lr) of 0.08 using a cosine annealing lr scheduler on 4 a100 gpus. 3.5 evaluations 3.5.1 speaker verification all the experiment findings are displayed in table 2. the results of the original speakernet and the pre-trained checkpoint1 publicly released by nvidia are also provided for comparison. 1https: //catalog.ngc.nvidia.com/orgs/nvidia/teams/nemo/models/speakerverification_speakernet d. karamyan, g. kirakosyan and s. harutyunyan 63 the pre-trained checkpoint was trained solely with noise augmentation using the abovementioned datasets. in order to examine the speaker verification system under noisy and reverberant conditions, we created augmented versions of voxceleb1 clean test trials by injecting noise and echo with different snr values. table 2: comparison of different speaker verification models under noise and reverb conditions. the results are reported in equal error rates. the more aggressively noise has been applied, the lower the snr values were. a noise level of 0 db indicates that the sound and the noise have the same energy. model orig 0db 5db 10db rir speakernet [13] 2.14 speakernet (nvidia) 1.92 9.75 5.43 3.61 16.5 clean teacher 1.87 12.9 6.94 4.21 16.5 noisy teacher 2.6 9.35 5.84 4.23 12.74 consistency reg. 1.76 8.05 4.40 3.13 12.26 teacher-student 1.73 9.16 4.79 3.26 9.18 table 2 showcases the effectiveness of the methods applied. we can see that training the speakernet model with data augmentation (noisy teacher) improves the results in the noisy/reverberant environment with a small deterioration of eer on the original (not perturbed) audios. the teacher-student method achieves the lowest eer scores in original and reverberant cases (rir), whereas the consistency regularization method shows the best results for noisy audios. using the teacher-student method, we were able to improve the eer by an average of 19.1% compared to the multi-condition trained model. with consistency regularization, we were able to improve the eer by an average of 14.8% compared to the multi-condition trained model. 3.5.2 speaker diarization we employ our trained speakernet models for speaker diarization task to see which model has the smallest performance degradation in noisy conditions. we found that the optimal sliding window size and shift for speech segmentation are 1.5 and 0.5 seconds, respectively. in addition, diarization experiments are based on oracle vad to evaluate the vad-independent performance. the affinity matrix a is constructed using the cosine similarity between segment embeddings. we further apply the following sequence of refinement operations to the affinity matrix a: • row-wise thresholding: for each row, keep top-12 largest elements and set the rest to 0 • symmetrization: y = 1 2 (a + at ) • diffusion: y = aat we use the spectral clustering method [34] to obtain speaker labels. to get a full picture, we present the diarization results for both known (oracle) and unknown numbers of speakers. in the latter case, we utilize the maximal eigen-gap approach to determine the number of speakers [1]. 64 making speaker diarization system noise tolerant table 3: comparison of speaker diarization systems with various speaker embedding extractors under noise and reverberant conditions. the results are reported in diarization error rate (der). model known #speakers unknown #speakers 0db 5db 10db rir orig avg 0db 5db 10db rir orig avg clean teacher 12.13 4.48 1.96 2.44 1.26 4.45 15.44 7.59 2.74 4.48 1.78 6.40 noisy teacher 9.20 4.49 3.13 3.12 1.57 4.30 13.09 7.94 4.18 4.14 1.95 6.26 consistency reg. 9.50 3.46 2.0 2.50 1.45 3.78 13.40 4.90 2.57 3.45 1.67 5.20 teacher-student 9.84 3.41 2.11 2.43 1.36 3.83 13.99 6.17 3.09 3.52 1.61 5.67 in order to assess the performance of the speaker diarization system under noisy and reverberant conditions, we modified the voxconverse dev dataset by adding noise and echo at various signal-to-noise ratios. the results, shown in table 3, indicate that the teacher-student and consistency regularization methods generally outperform the multi-condition baseline model for both scenarios involving known and unknown numbers of speakers. in particular, when the number of speakers is unknown, we observed approximately 17% and 9.5% relative performance improvements for the consistency regularization and teacher-student methods, respectively, compared to the multi-condition baseline. however, it is worth noting that in certain specific scenarios, the baseline models may outperform the models with the overall best average performance. 4. conclusions in this research, we explore ways to increase the accuracy of speaker recognition and speaker diarization in noisy and reverberant environments, such as multi-condition, teacher-student, and consistency regularization. the key 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[30] j. chung, j. huh, a. nagrani, t. afouras and a. zisserman, “spot the conversation: speaker diarization in the wild”, arxiv preprint arxiv:2007.01216, 2020. [31] d. park, w. chan, y. zhang, c. chiu, b. zoph, e. cubuk and q. le, “specaugment: a simple data augmentation method for automatic speech recognition”, arxiv preprint arxiv:1904.08779, 2019. [32] t. ko, v. peddinti, d. povey, m. seltzer and s. khudanpur, “a study on data augmentation of reverberant speech for robust speech recognition”, proceedings of the ieee international conference on acoustics, speech and signal processing (icassp), pp. 5220-5224, 2017. davit karamyan, grigor kirakosyan and saten harutyunyan 6 7 [3 3 ] d . s n yd e r , g. ch e n a n d d . p o ve y, \ mu s a n : a m u s ic , s p e e c h , a n d n o is e c o r p u s " , arxiv p reprint arxiv:1510.08484, 2 0 1 5 . [3 4 ] u .v o n l u xb u r g , \ a t u t o r ia l o n s p e c t r a l c lu s t e r in g " , statistics and computing, vo l. 1 7 , p p . 3 9 5 -4 1 6 , 2 0 0 7 . ²õùï³¹çù³óïáõýáõãû³ý ³å³ñáíáõùá ëáëý³ïý»ñç ¹ç³ñç½³óç³ûç ñ³ù³ï³ñ·áõù 1ð³û-èáõë³ï³ý ñ³ù³éë³ñ³ý, ºñ¨³ý, ð³û³ëï³ý 2krisp.ai, ºñ¨³ý, ð³û³ëï³ý 3ðð ¶²² ù³ã»ù³ïçï³ûç çýëïçïáõï,ºñ¨³ý, ð³û³ëï³ý e -m a il: f d ka r a m ya n , s h a r u t yu n ya n , g kir a ko s ya n g@kr is p .a i ²ù÷á÷áõù ¸³íçã ê. ø³ñ³ùû³ý1;2, ¶ñç·áñ ². îçñ³ïáëû³ý2;3, ê³ã»ý ². ð³ñáõãûáõýû³ý2 êáëý³ïý»ñç ¹ç³ñç½³óç³ûç ýå³ï³ïá ³áõ¹çá ó³ûý³·ñáõãû³ý ù»ç ï³ñµ»ñ ëáëý³ïý»ñç ñ³ûïý³µ»ñáõùý áõ ³é³ýóý³óáõùý ¿: ²ûýáõ³ù»ý³ûýçí, ýáý³ûçý ³õùáõïá ï³ñáõ ¿ ³½¹»é ³ûë ñ³ù³ï³ñ·»ñç ×ß·ñïáõãû³ý íñ³: ²ûë ñá¹í³íáõù áõëáõùý³ëçñí»é »ý ³ûýåçëç ù»ãá¹ý»ñ, çýãåçëçù »ý` ï³ñµ»ñ ³áõ·ù»ýï³óç³ý»ñáí áõëáõóáõùá, ï³ûáõýáõãû³ý ï³ñ·³íáñáõùá (consistency regularization) ¨ áõëáõóçã-³ß³ï»ñï ù»ãá¹á` ëáëý³ïý»ñç ó³ûý³ûçý ñ³ïï³ýçßý»ñ ¹áõñë µ»ñáõ ùá¹»éç ï³ûáõýáõãûáõýá ³õùáõïç ýï³ïù³ùµ µ³ñóñ³óý»éáõ ñ³ù³ñ: üßí³í ù»ãá¹ý»ñç ³ñ¹ûáõý³í»ïáõãûáõýá ëïáõ·í»é ¿ ëáëý³ïý»ñç ýáõûý³ï³ý³óù³ý ¨ ¹ç³ñç½³óç³ûç ëý¹çñý»ñáõù ¨ óáõûó ¿ ïñí»é, áñ ¹ñ³ýù ñ³ý·»óýáõù »ý ï³ûáõýáõãû³ý µ³ñ»é³íù³ýá` ³õùáõïç ¨ ³ñó³·³ýùç ³éï³ûáõãû³ý ¹»åùáõù: êáëý³ïý»ñç ýáõûý³ï³ý³óù³ý ¨ ¹ç³ñç½³óç³ûç ñ³ù³ï³ñ·»ñá ³õùáõïç ¨ ³ñó³·³ýùç å³ûù³ýý»ñáõù ÷áñó³ñï»éáõ ñ³ù³ñ ëï»õíí»é »ý voxceleb1 ¨ voxconverse dev ïíû³éý»ñç ñ³í³ù³íáõý»ñç áý¹é³ûýí³í ï³ñµ»ñ³ïý»ñá` ³í»é³óý»éáí ï³ñµ»ñ snr ³ñå»ùý»ñáí ýáý³ûçý ³õùáõï ¨ ³ñó³·³ýù: êï³óí³í ³ñ¹ûáõýùý»ñá óáõûó »ý ï³éçë, áñ ùçççý ñ³ßíáí ï³ñ»éç ¿ ñ³ëý»é ëáëý³ïý»ñç ýáõûý³ï³ý³óù³ý ×ß·ñïáõãû³ý ñ³ñ³µ»ñ³ï³ý µ³ñ»é³íù³ýá` 1 9 ; 1 % -áí` û·ï³·áñí»éáí áõëáõóçã-³ß³ï»ñï ù»ãá¹á ¨ ëáëý³ïý»ñç ¹ç³ñç½³óç³ûç ×ß·ñïáõãû³ý ñ³ñ³µ»ñ³ï³ý µ³ñ»é³íù³ýá` 1 7 % áí` û·ï³·áñí»éáí ï³ûáõýáõãû³ý ï³ñ·³íáñù³ý ù»ãá¹á` ñ³ù»ù³ï³í ï³ñµ»ñ ³áõ·ù»ýï³óç³ý»ñáí í³ñå»óí³í ùá¹»éç ñ»ï: ´³ý³éç µ³é»ñ` ëáëý³ïý»ñç ýáõûý³ï³ý³óáõù, ëáëý³ïý»ñç ¹ç³ñç½³óç³, ³õùï³¹çù³óïáõýáõãûáõý, áõëáõóçã-³ß³ï»ñï, ï³ûáõýáõãû³ý ï³ñ·³íáñáõù: 6 8 making speaker diarization system noise tolerant îáåñïå÷åíèå øóìîóñòîé÷èâîñòè ñèñòåìû äèàðèçàöèè äèêòîðîâ äàâèä ñ. êàðàìÿí1;2, ãðèãîð à. êèðàêîñÿí2;3, ñàòåí à. àðóòþíÿí2 1ðîññèéñêî-àðìÿíñêèé óíèâåðñèòåò, åðåâàí, àðìåíèÿ 2krisp.ai, åðåâàí, àðìåíèÿ 3èíñòèòóò ìàòåìàòèêè íàí ðà, åðåâàí, àðìåíèÿ e-mail: dkaramyan, sharutyunyan, gkirakosyan@krisp.aig àííîòàöèÿ öåëüþ ñèñòåìû äèàðèçàöèè äèêòîðîâ ÿâëÿåòñÿ èäåíòèôèöèðîâàíèå è ðàçäåëåíèåðàçíûõ äèêòîðîâ â àóäèîçàïèñè. îäíàêî øóì â çàïèñè ìîæåò ïîâëèÿòü íà òî÷íîñòü ýòèõ ñèñòåì. â ýòîé ñòàòüå ìû èññëåäóåì òàêèå ìåòîäû, êàê îáó÷åíèå ñ ðàçëè÷íûìè àóãìåíòàöèÿìè, ðåãóëÿðèçàöèÿ ñîãëàñîâàííîñòè (consistency regularization) è ìåòîä ”ó÷èòåëü-ó÷åíèê”, ÷òîáû ïîâûñèòü óñòîé÷èâîñòü ýêñòðàêòîðîâ ðå÷åâûõ õàðàêòåðèñòèê ê øóìó. ìû ïðîâåðÿåì ýôôåêòèâíîñòü ýòèõ ìåòîäîâ â çàäà÷àõ ðàñïîçíàâàíèÿ äèêòîðîâ ïî ãîëîñó è äèàðèçàöèè äèêòîðîâ è äåìîíñòðèðóåì, ÷òî îíè ïðèâîäÿò ê óëó÷øåíèþ óñòîé÷èâîñòè ïðè íàëè÷èè øóìà è ðåâåðáåðàöèè. ×òîáû ïðîâåðèòü ñèñòåìó ðàñïîçíàâàíèÿ è äèàðèçàöèè äèêòîðîâ â óñëîâèÿõ øóìà è ðåâåðáåðàöèè, ìû ñîçäàëè ðàñøèðåííûå âåðñèè voxceleb1 è íàáîðîâ äàííûõ voxconverse dev, äîáàâèâ øóì è ýõî ñ ðàçíûìè çíà÷åíèÿìè snr. íàøè ðåçóëüòàòû ïîêàçûâàþò, ÷òî â ñðåäíåì ìû ìîæåì äîáèòüñÿ îòíîñèòåëüíîãî óëó÷øåíèÿ ðàñïîçíàâàíèÿ äèêòîðîâ íà 1 9 ; 1 % ñ èñïîëüçîâàíèåì ìåòîäà ”ó÷èòåëü-ó÷åíèê” è îòíîñèòåëüíîãî óëó÷øåíèÿ äèàðèçàöèè äèêòîðîâ íà 1 7 % ñ èñïîëüçîâàíèåì ìåòîäà ðåãóëÿðèçàöèè ñîãëàñîâàííîñòè ïî ñðàâíåíèþ ñ áàçîâîé ìîäåëüþ, îáó÷åííîé ñ ïîìîùüþ ðàçëè÷íûõ àóãìåíòàöèé. êëþ÷åâûå ñëîâà:ðàñïîçíàâàíèå ïî ãîëîñó, äèàðèçàöèÿ äèêòîðîâ, óñòîé÷èâîñòü ê øóìó, ó÷èòåëü-ó÷åíèê, ðåãóëÿðèçàöèÿ ñîãëàñîâàííîñòè. 06_karamyan_davit_59 (1) 06 d:\sbornik\...\sav.dvi mathematical problems of computer science 30, 18{24, 2008. e ±cient m ar ch-like algor ithm for detection of all t wo-oper ation dynamic faults fr om subclass sav h . a ve t is ya n y, g. h a r u t u n ya n z, v .a . v a r d a n ia n z y russian-armenian state university e-mail: hamazasp avetisyan@yahoo.com zvirage logic e-mail: fgurgen.harutyunyan, valery.vardaniang@viragelogic.com abstract this paper introduces an e±cient march-like algorithm for detection of the well known class sav of dynamic faults. sav is the subclass of all two-operation dynamic functional fault models that are sensitized by means of applying two consecutive operations, one applied on the aggressor cell and the second operation applied on the victim cell. earlier, only subclasses saa and svv were considered by a few authors when both sensitizing operations were applied either on the aggressor or victim cell, and march algorithms were developed by them. subclasses sav and sva were not considered due to their complexity. a larger class of march-like algorithms has to be considered for detection of those subclasses since march algorithms cannot detect them. it is shown that 392n operations are su±cient for detection of faults from sav. refer ences [1 ] a . j. va n d e go o r , " te s t in g s e m ic o n d u c t o r m e m o r ie s : th e o r y a n d p r a c t ic e " , j ohn w iley and sons, 1 9 9 1 . [2 ] s . h a m d io u i, a .j. va n d e go o r , m. r o d g e r s , " ma r c h s s : a t e s t fo r a ll s t a t ic s im p le fa u lt s " , r ecords of ie e e int. w orkshop m td t, p p . 9 5 -1 0 0 , 2 0 0 2 . [3 ] s . h a m d io u i, a .j. va n d e go o r , m. r o d g e r s , " l in ke d fa u lt s in r a n d o m a c c e s s m e m o r ie s : c o n c e p t , fa u lt m o d e ls , t e s t a lg o r it h m s , a n d in d u s t r ia l r e s u lt s " , ie e e trans. cad , vo l. 2 3 , n o . 5 , ma y p p . 7 3 7 -7 5 6 , 2 0 0 4 . [4 ] s . h a m d io u i, z. a l-a r s a n d a .j. va n d e go o r , " te s t in g s t a t ic a n d d yn a m ic fa u lt s in r a n d o m a c c e s s m e m o r ie s " , in p roc. of ie e e vl si test symposium, p p . 3 9 5 -4 0 0 , 2 0 0 2 . [5 ] r . d . a d a m s , e . s . co o le y, " a n a lys is o f d e c e p t ive r e a d d e s t r u c t ive m e m o r y fa u lt m o d e l a n d r e c o m m e n d e d t e s t in g " , p roc. of ie e e north atlantic test w orkshop, p p . 2 7 -3 2 , 1 9 9 6 . [6 ] s . h a m d io u i, r . w a d s wo r t h , j. d . r e ye s , a .j. va n d e go o r , " im p o r t a n c e o f d yn a m ic fa u lt s fo r n e w s r a m t e c h n o lo g ie s " in p roc. of ie e e e uropean test w orkshop, p p . 2 9 -3 4 , 2 0 0 3 . 1 8 h. avetisyan, g. harutunyan, v. vardanian 1 9 [7 ] s . h a m d io u i, g.n . ga yd a d jie v, a n d a .j. va n d e go o r , " a fa u lt p r im it ive b a s e d a n a lys is o f d yn a m ic m e m o r y fa u lt s " , ie e e 14th annual w orkshop on circuits, systems and signal p rocessing, v e ld h o ve n , t h e n e t h e r la n d s , p p . 8 4 -8 9 , 2 0 0 3 . [8 ] a . b e n s o , a . b o s io , s . d i ca r lo , g. d i n a t a le , p . p r in e t t o , " ma r c h a b , ma r c h a b 1 : n e w ma r c h t e s t s fo r u n lin ke d d yn a m ic m e m o r y fa u lt s " , itc, 2 0 0 5 . [9 ] a . b e n s o , a . b o s io , s . d i ca r lo , g. d i n a t a le , p . p r in e t t o , " a u t o m a t ic m a r c h t e s t g e n e r a t io n fo r s t a t ic a n d d yn a m ic fa u lt s , in s r a ms " , p roc. e ts 2005, tallinn, p p . 1 2 2 -1 2 7 , 2 0 0 5 . [1 0 ] g. h a r u t u n ya n , v . a . v a r d a n ia n , y . zo r ia n , " min im a l m a r c h t e s t s fo r d yn a m ic fa u lt s in r a n d o m a c c e s s m e m o r ie s " , j ournal of e lectronic testing: theory and applications, vo l. 2 3 , n u m b e r 1 , p p . 5 5 -7 4 , 2 0 0 7 . [1 1 ] g. h a r u t u n ya n , v .a . v a r d a n ia n , y . zo r ia n , " min im a l m a r c h t e s t s fo r d yn a m ic fa u lt s in r a n d o m a c c e s s m e m o r ie s " , in p roc. of ie e e e uropean test symposium, p p . 4 3 -4 8 , 2 0 0 6 . [1 2 ] a .j. va n d e go o r , i. s c h a n s t r a , " a d d r e s s a n d d a t a s c r a m b lin g : ca u s e s a n d im p a c t o n m e m o r y t e s t s " , p roc. ie e e w orkshop d e l ta, p p . 1 2 8 -1 3 6 , 2 0 0 2 . [1 3 ] j.-f. l i, k .-l . ch e n g , c.-t. h u a n g , a n d c.-w . w u , " ma r c h b a s e d r a m d ia g n o s t ic a lg o r it h m s fo r s t u c k-a t a n d c o u p lin g fa u lt s " , p roc. ie e e itc, p p . 7 5 8 -7 6 7 , 2 0 0 1 . [1 4 ] v . a . v a r d a n ia n , y . zo r ia n , " a m a r c h -b a s e d fa u lt lo c a t io n a lg o r it h m fo r s t a t ic r a n d o m a c c e s s m e m o r ie s " , p roc. ie e e int. w orkshop m td t, p p . 6 2 -6 7 , 2 0 0 2 . ²ñ¹ûáõý³í»ï ù³ñß³ïçå ³é·áñçãù ¹çý³ùçï ³ýë³ñùáõãûáõýý»ñç »ýã³¹³ëç µáéáñ ³ýë³ñùáõãûáõýý»ñç ñ³ûïý³µ»ñù³ý ñ³ù³ñ ð. ²í»ïçëû³ý, ¶. ð³ñáõãûáõýû³ý, ì. ì³ñ¹³ýû³ý ²ù÷á÷áõù ²ûë ñá¹í³íáõù ý»ñï³û³óíáõù ¿ ³ñ¹ûáõý³í»ï ù³ñß³ïçå ³é·áñçãù, áñá ï³ñáõ³ýáõù ¿ ñ³ûïý³µ»ñ»é ¹³ëç µáéáñ ¹çý³ùçï ³ýë³ñùáõãûáõýý»ñá. ¹³ëá »ñïáõ ·áñíáõõáõãû³ùµ ½·³ûáõý³óíáõ ¹çý³ùçï ³ýë³ñùáõãûáõýý»ñç »ýã³¹³ë ¿, ³ûëçýùý ³ýë³ñùáõãûáõýý»ñ, áñáýù ½·³ûáõý³óíáõù »ý ñçßáõáõãû³ý µççç ýï³ïù³ùµ ñ³çáñ¹³ï³ý »ñïáõ ·áñíáõáõãûáõý ï³ï³ñ»éçë, ³é³ççý ·áñíáõáõãûáõýá ³·ñ»ëáñ ¿, çëï »ñïñáñ¹á ½áñ ¿ µççç ýï³ïù³ùµ: ü³ëïçýáõù ¹çï³ñïí³í »ý »õ»é ¨ »ýã³¹³ë»ñá, áñáýó ñ³ù³ñ ñ»õçý³ïý»ñá ý»ñï³û³óñ»é ¿çý ù³ñß ³é·áñçãù»ñ: ºí ¹³ë»ñá áõëáõùý³ëçñí³í ã»ý »õ»é, ù³ýç áñ ýñ³ýó ñ³ù³ñ ñý³ñ³íáñ ã¿ ï³éáõó»é ù³ñß ³ý·áñçãù: ê³ï³ûý ³ûë ¹³ë»ñç ñ³ù³ñ ñý³ñ³íáñ ¿ ï³éáõó»é ù³ñß³ïçå ³ý·áñçãù»ñ: ðá¹í³íáõù ý»ñï³û³óí³í ¿ ù³ñß³ïçå ³é·áñçãù, áñá ï³ï³ñ»éáí ·áñíáõáõãûáõý ñ³ûïý³µ»ñáõù ¿ ¹³ëç µáéáñ ³ýë³ñùáõãûáõýý»ñá: microsoft word 10_zaslavski.doc mathematical problems of computer science 39, 81--87, 2013. 81 on the comparative complexity of primitive recursive arithmetical and string functions1 igor d. zaslavsky and mikayel h. khachatryan institute for informatics and automation problems of nas ra e-mail: zaslav@ipia.sci.am, mikayel.khachatur@gmail.com abstract formal languages la and lw are introduced as in [1] for the representation of primitive recursive arithmetical and string functions. shannon functions shaw and shwa describing the relations between the complexities of functions representations in these languages are defined as in [1]. a new proof of the upper bounds for shaw is presented; it is based on a new method giving in some cases new possibilities for applications in comparison with the methods considered in [1]. keywords: string function, arithmetical function, term, alphabetic enumeration, shannon function, primitive recursive function. investigations described in this paper may be considered as the continuation of those presented in [1]. let us recall definitions of some notions given in [1]. we suppose that an alphabet 1 2{ , ,..., },pa a a a where 1,p  is fixed. the set of all strings in this alphabet (including the empty string  ) is denoted by a*; the set of all k-tuples 1 2( , ,..., ),kq q q where * iq a for 1 ,i k  will be denoted by *( ) .ka the set of all non-negative integers {0, 1, 2, ... } will be denoted by n; the set of all k-tuples 1 2( , ,..., ),kx x x where ix n for 1 ,i k  will be denoted by .kn k-dimensional string function in a is defined ( [1], [2] ) as a mapping of *( )ka into a*; kdimensional arithmetical function is defined as a mapping of ( )kn into n. primitive recursive string functions in a as well as primitive recursive arithmetical functions are defined in a usual way as in [1] and [2]. the alphabetic enumeration of the set a* is defined as in [1] and [2]; let us recall that this enumeration defines a one-to-one correspondence between the sets a* and n. the non-negative integer, corresponding to a string q in the alphabetic enumeration is denoted by ( ).q the string in a* corresponding to the number n in this enumeration is denoted by ( )p n or .n the length of a string q is denoted by .q all these notations are used in [1]. the alphabetic enumeration of strings gives also a one-to-one correspondence between ndimensional string functions in a, and n-dimensional arithmetical functions. 1 this work is supported by the grant 11-1b 189 of the government of the republic of armenia. on the comparative complexity of primitive recursive arithmetical and string functions 82 namely, we say ( [1], [2] ) that an n-dimensional arithmetical function f represents an ndimensional string function f, if 1 2 1 2( , , ... , ) ( , , ... , )n nf x x x f x x x    for all 1 2, ,..., nx x x in n. in this case we say also that f and f correspond to one another. the mentioned correspondence gives also a one-to-one correspondence between primitive recursive string functions in a and primitive recursive arithmetical functions ( [1], [2] ). in [1] the formal languages la and lw are introduced for the representation of primitive recursive arithmetical functions and primitive recursive string functions. the formal expressions in these languages are said to be terms; by t la and r lw we denote the statements “t is a term in la”, “r is a term in lw”. in the definition of la the symbols s and r are used for the operators of superposition and primitive recursion of arithmetical functions; in the definition of lw the symbols s and r are used for the operators of superposition and alphabetic primitive recursion of string functions ( [1], [2] ). special notations for some modifications of the mentioned operators ( sbl, sbr, sel, ser, sb, se in la; sbl, sbr, sel, ser, sb, se in lw) are also included in la and lw ([1]). we shall consider below special cases of the implementation of the modifications sb and se of the operator s (see [1]); these cases are described in the following points (1), (2), (3). let us note that all the terms considered in (1), (2), (3) are terms in the language lw. (1) if f and g are terms expressing correspondingly a v  dimensional function f (where 2v  ) and a one-dimensional function g, then the term ( , )f g se expresses the v  dimensional function h such that 1 2 1 2 1( , ,..., ) ( , ,..., , ( ))v v vh q q q f q q q g q for all values of the variables 1 2, , ... , .vq q q (2) if f and g are terms expressing correspondingly a 2-dimensional function f and a k-dimensional function g (where 1k  ), then the term ( , )f g sb expresses the (k+1)dimensional function h such that 1 2 1 1 2 1( , ,..., ) ( ( , ,..., ), )k k kh q q q f g q q q q  for all values of the variables 1 2 1, , ... , .kq q q  (3) if 1 2, ,f g g   are terms expressing correspondingly a v  dimensional function f (where 2v  ) and one-dimensional functions 1g and 2 ,g then the term 1 2( , , )f g g  sb expresses the ( 1)v  dimensional function h such that 1 2 1 1 1 2 1 2 1( , ,..., ) ( ( ), ( ), ,..., )v vh q q q f g q g q q q  for all values of the variables 1 2 1, , ... , .vq q q  as it will be seen below, it is convenient to represent the list of variables for the function h in the following form: 3 4, , ,... , .vr q q q using this list, we can write the expression for h as follows: 3 4 1 2 3 4( , , ,... , ) ( ( ), ( ), , ,..., ).v vh r q q q f g r g r q q q in [1] shannon functions ( )awsh n and ( )wash n are introduced; these functions describe the relations between the lengths of terms expressing arithmetical functions (in la) and string functions (in lw) when the considered functions correspond to one another. namely, if t lw , then by ( )la t we denote the set of all terms in la expressing the arithmetical function corresponding to the string function expressed by t. similarly, if r la , then by ( )lw r we denote the set of all terms in lw expressing the string function corresponding to the arithmetical function expressed by r. now we can give (see [1]) the definitions of ( )awsh n and ( )wash n as follows: i. zaslavski and m. khachatryan 83  ( )( )&( )( ) max minwa r la tt lw t nsh n r  ;  ( )( )&( )( ) max min .aw t lw rr la r nsh n t  in [1] the following statement is established (see the main theorem in [1]): there are upper and lower bounss for ( )awsh n and ( )wash n such that each of them has the form ,cn d where c and d are some constants. we shall consider the function ( ).awsh n there are some defects in the proof of the upper bouns for this function in [1]; their removal requires essential changes in the proof. below we give another proof of the mentioned bouns based on a method which is different from those used in [1]. namely, we shall give a new proof of the following theorem. theorem. there are constants c and d such that for any non-negative integer n   .awsh n cn d  we shall use three lemmas in the proof given in [1] (similar statements are proved also in [2]). by ( )n we denote the function such that (0) ,   1 1 1( ) ... n n a a a   times for any positive integer n. lemma 1. there are constants c and d such that for any term t la expressing a function 1 2( , ,..., ),mx x x a term lw expressing some function 1 2( , ,..., )mq q q can be constructed such that the following conditions are satisfied: 1. 1 2 1 2( ( ), ( ),..., ( )) ( ( , ,..., )),m mx x x x x x      for any 1 2, ,..., mx x x in n. 2. ' '.c t d   lemma 2. there is a primitive recursive string function g such that   g m m  for any .m n lemma 3. the one-dimensional string function  ( ) ( )q q   is primitive recursive. proof of theorem. let t be any term in la expressing some function 1 2( , , ... , ).mx x x as it is proved in [1], the following inequality holds: .m t the string function corresponding to  let us denote by 1 2( , , ... , ).mq q q we shall construct a term  in lw having the length mentioned in theorem and expressing the function . using lemma 1 we construct a term φ in la such that ,c t d    where c and d are constants (fixed in lemma 1), and φ expresses a function  satisfying the condition 1 2 1 2( ( ), ( ),..., ( )) ( ( , ,..., ))m mx x x x x x      for any 1 2, ,..., mx x x in n. using lemmas 1 and 2 we obtain the following equalities               1 2 1 2 1 2 1 2 1 2 ( , ... ) ( ( ), ( )... ( )) ( ( ), ( )... ( )) ( ) , ( ) ... ( ) ( ), ( )... ( ) , m m m m m q q q q q q g q q q g q q q g q q q                             by g and  we denote the terms in lw expressing the functions g and . let us consider the well-known primitive recursive arithmetical functions c, l, r, defining a one-to-one correspondence between 2n and n. such functions we define by the following equalities: on the comparative complexity of primitive recursive arithmetical and string functions 84 ( )( 1) ( , ) , 2 x y x y c x y x      ( ( ), ( )) , ( ( , )) , ( ( , )) . c l z r z z l c x y x r c x y y    we consider also the following functions (where 2, 2n k n   ): 1 2 1 2 3 ( 1) times ( 1) times 1 ( ) times ( , ,..., ) (... ( ( , ), ),..., ); ( ) ( (... ( )...)); ( ) ( ( (... ( )...))). n n n n n n n k nk c x x x c c c x x x x c z l l l z c z r l l l z          obviously, for any 1 2, ,..., ,nx x x z in n and for 1 ,k n  the following equalities hold: 1 2 1 2 ( ( ), ( ),..., ( )) ; ( ( , ,..., )) . n n n nn n nk n k c c z c z c z z c c x x x x   using lemma 1 we construct string functions , ,    , such that for any x, y, z in n ( ( ), ( )) ( ( , )); ( ( )) ( ( )); ( ( )) ( ( )). x y c x y z l z z r z              let us note a peculiarity of these functions. if some strings 1 2, , q q q in a do not contain other letters except 1.a then the following equalities hold: 1 2 1 1 2 2( ( ), ( )) , ( ( , )) , ( ( , )) .q q q q q q q q q         however, in general such equalities are not valid. let us consider also the following string functions (where 2, 2n k n   ) 1 2 1 2 3 ( 1) times 1 ( ) times ( , ,..., ) (... ( ( , ), ),..., ); ( ) ( (... ( )...)); ( ) ( ( (... ( )...))). n n n n n n k nk q q q q q q q q q q q                     the terms in lw expressing the functions 1, , , , , n n nk      (where 2, 2n k n   ) we denote, correspondingly, by 1, , , , , . n n nk         if some strings 1 2, , ... , nq q q q in a do not contain other letters except 1.a then the following equalities hold (where 2, 2n k n   ): 1 2 1 1 2 1 1 2 ( ( ), ( ),..., ( )) , ( ( , ,..., )) ; ( ( , ,..., )) . n n n nn n n n n nk n k q q q q q q q q q q q q            in general such equalities are not valid. now in the case, when 2m  , let us construct the term m as follows: ( 1) times ( 2) times ( ( ,..., ( ( , ) , ))..., )) , ). m m m                 se sb se sb i. zaslavski and m. khachatryan 85 here the group of symbols ( ( ,se sb is repeated (m-1) times; after this the group ) is repeated once; after this the group , )) is repeated (m-2) times; finally, the group , ) is repeated once. it is easily seen that the length of the term m does not exceed 10 10 ,c m d where 10c and 10d are some constants. let us consider some subterms of the term m as well as functions expressed by them. it is easily seen that the following statements are valid. the term ( , ) sb expresses the function  1 2( ), .q q  the term 2 ( ( , ), )     se sb expresses the function  1 2( ), ( ) ,q q   that is, the function  2 1 2( ), ( ) .q q   the term ( , ( ( , ), ))     sb se sb expresses the function   1 2 3( ), ( ) , .q q q    the term 3 ( ( , ( ( , ), )), )         se sb se sb expresses the function   1 2 3( ), ( ) , ( ) ,q q q     that is, the function  3 1 2 3( ), ( ), ( ) .q q q    using similar considerations, we conclude that the term m expresses the function 1 2 3(... ( ( ( ), ( )), ( )),..., ( )),mq q q q       that is, the function 1 2 3( ( ), ( ), ( ),..., ( )). m mq q q q     further, let us construct the term m (where 1m  ) as follows: ( 1) times( 1) times ( ( ( , , ), , )..., , ). mm m                sb ...sb sb it is easily seen that the length of the term m does not exceed 11 11,c m d   where 11c and 11d are some constants. using the inequalities c t d    and m t we conclude that the length m does not exceed 12 12 ,c t d where 12c and 12d are some constants. let us consider some subterms of the term m , as well as functions expressed by them. it is easily seen that the following statements are valid. as it is said above, the term  expresses the function  depending on m variables. the function  we denote also by 0 . the term 1 is defined as the term which is equal to . the term 2 ( , , )     sb expresses some function 1 depending on ( 1)m  variables; the list of variables for this function we denote by 1 3, ,..., .mr q q using such notations we can represent the equality describing the function 1 1 3( , ,..., )mr q q as follows: 1 1 3 1 1 3( , ,..., ) ( ( ), ( ), ,..., ),m mr q q r r q q    that is 1 1 3 21 1 22 1 3( , ,..., ) ( ( ), ( ), ,..., ).m mr q q r r q q    the term 3 ( ( , , ), , )        sb sb expresses the function 2 2 4 5( , , ,..., )mr q q q depending on ( 2)m  variables; the equality describing this function can be represented as follows: 2 2 4 5 2 2 2 4 5( , , ,..., ) ( ( ( )), ( ( )), ( ), , ,..., ),m mr q q q r r r q q q       that is 2 2 4 5 31 2 32 2 33 2 4 5( , , ,..., ) ( ( ), ( ), ( ), , ,..., ).m mr q q q r r r q q q     on the comparative complexity of primitive recursive arithmetical and string functions 86 the term 4 ( ( ( , , ), , ), , )            sb sb sb expresses the function 3 3 5 6( , , ,..., )mr q q q depending on ( 3)m  variables; the equality describing this function can be represented as follows: 3 3 5 6 3 3 3 3 5 6( , , ,..., ) ( ( ( ( ))), ( ( ( ))), ( ( )), ( ), , ,..., ),m mr q q q r r r r q q q           that is 3 3 5 6 41 2 42 2 43 2 44 2 5 6( , , ,..., ) ( ( ), ( ), ( ), ( ), , ,..., ).m mr q q q r r r r q q q      using similar considerations, we conclude that the term m expresses the function ( 1)m  depending on one variable (we shall denote this variable by ( 1)mr  ). the equality describing this function can be represented as follows:  ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 2)( 1) ( 2) ( ) ( ( (... ( )...)) , ( ( (... ( )...))),..., ( )),m m m m m m mm m r r r r                    that is ( 1) ( 1) 1 ( 1) 2 ( 1) ( 1)( ) ( ( ), ( ),..., ( )).m m m m m m mm mr r r r         now let us construct the term ( , ).m m s this term expresses the function 1 1 2 2 1 2 1 2 ( ( ( ( ), ( ),..., ( ))), ( ( ( ), ( ),..., ( ))),... ..., ( ( ( ), ( ),..., ( )))). m m m m m m m mm m q q q q q q q q q                 but the strings 1 2( ), ( ),..., ( )mq q q   do not contain other letters except 1.a so, we can conclude that the function expressed by ( , ),m m s is equal to 1 2( ( ), ( ),..., ( )).mq q q    hence the term ( , ( , ))m mg   s s expresses the function 1 2( ( ( ), ( ),..., ( ))),mg q q q    that is, the function 1 2( , ,..., ).mq q q clearly, 13 13 ,c t d   where 13c and 13d are some constants. so, the statement of theorem is proved for 2.m  the cases 1m  and 0m  are considered in a similar way. this completes the proof of theorem. note. applying usual methods of the recursive functions theory, we can obtain essentially more simple and more natural expressions for the term  than those considered above, for example 1 2( , ( , ( , ), ( , ),..., , ( , ))), m m m mg i i i    s s s s s      where any term mki for 1 k m  expresses the function 1 2( , ,..., ) . m k m ki q q q q however, such expressions do not give the required bounds of . for this aim special methods should be used. one of such methods is implemented above. i. zaslavski and m. khachatryan 87 references [1] m. h. khachatryan. “on the representation of arithmetical and string functions in formal languages,” transactions of iiap of nas of ra, mathematical problems of computer science, vol. 27, pp. 37-53, 2006. [2] a. i. maltsev. algorithms and recursive functions. 2nd edition, moskow, nauka , 1986 (in russian). submitted 28.11.2012, accepted 30.01.2013. պարզագույն անդրադրարձ (ռեկուրսիվ) թվաբանական և բառային ֆունկցիաների համեմատական բարդության մասին ի. զասլավսկի և մ. խաչատրյան ամփոփում դիտարկվում են [1]-ում սահմանված պարզագույն անդրադարձ (ռեկուրսիվ) թվաբանական և բառային ֆունկցիաների ներկայացման la և lw ձևային լեզուները։ շենոնի awsh և wash ֆունկցիաները, որոնք բնութագրում են թվաբանական և բառային ֆունկցիաների ներկայացումների բարդությունների միջև եղած կապերը նշված լեզուներում, սահմանվում են, ինչպես [1]-ում։ մի նոր մեթոդով տրվում է awsh ֆունկցիայի վերին գնահատականի ապացույցը։ այդ մեթոդը որոշ դեպքերում ապահովում է կիրառությունների ավելի լայն հնարավորություններ, քան` [1]-ում դիտարկվող մեթոդները։ о сравнительной сложности примитивно рекурсивных арифметических и словарных функций и. д заславский и м. хачатрян аннотация рассматриваются формальные языки la и lw, введенные в [1] для представления примитивно рекурсивных арифметических и словарных функций. функции шеннона awsh и wash , выражающие соотношения между сложностями представления арифметических и словарных функций в этих языках, определяются так же, как в [1]. дается новое доказательство верхней оценки для awsh , основанное на методе, дающем в ряде случаев новые возможности для приложений по сравнению с методами, рассматриваемыми в [1]. d:\sbornik\...\article_eng.dvi mathematical problems of computer science 31, 5{15, 2008. on a class of i r r educible p olynomials over fp me ls ik k . k yu r e g ya n , e d it a y u . h a r u t yu n ya n a n d mika ye l g. e vo ya n y institute for informatics and automation problems of nas of ra yfaculty of informatics and applied mathematics, yerevan state university email: edita@ipia.sci.am, email:michael.ipm@gmail.com abstract the paper presents some results regarding constructive theory of synthesis of irreducible polynomials of degree pt over fp from the given primitive elements in fp, where p is an odd prime and t is an integer whose prime factors all divide p ¡ 1. keywor ds p r im it ive p o lyn o m ia l, e xp lic it ly c o n s t r u c t e d s e qu e n c e s , p e r io d , lin e a r o p e r a t o r , p r im it ive e le m e n t refer ences [1 ] a lb e r t a .a ., f undamental concepts of higher algebra, u n ive r s it y o f ch ic a g o p r e s s , ch ic a g o , 1 9 5 6 . [2 ] p e t e r s o n w . w ., w e ld o n e . i. e rror-correcting codes, 2 n d e d ., m.i.t. p r e s s , ca m b r id g e , ma s s ., 1 9 7 2 . [3 ] v a r s h a m o v r .r . on a m e t h o d o f c o n s t r u c t in g ir r e d u c ib le p o lyn o m ia ls o ve r ¯ n it e ¯ e ld s . d okladi akademii nauk of armenia, vo l. 7 9 , n o .1 , 1 9 8 4 , p p . 2 6 { 2 8 ( in r u s s ia n ) . fp ¹³ßïç íñ³ ³ýí»ñí³ý»éç µ³½ù³ý¹³ùý»ñç ùç ¹³ëç ù³ëçý ø. îûáõñ»õû³ý, ø. ¾íáû³ý, ¾. ð³ñáõãûáõýû³ý ²ù÷á÷áõù ðá¹í³íáõù ý»ñï³û³óí³í »ý ùç ß³ñù ³ñ¹ûáõýùý»ñ, áñáýù ³éýãíáõù »ý ³ýí»ñ³í»éç µ³½ù³ý¹³ùý»ñç ëçýã»½ç ïáýëïñáõïïçí ï»ëáõãû³ýá, ù³ëý³íáñ³å»ë fp ¹³ßïç íñ³ pt ³ëïç×³ýç ³ýí»ñ³í»éç µ³½ù³ý¹³ùý»ñç ï³éáõóù³ýá û·ï³·áñí»éáí fp ¹³ßïç åñçùçïçí ¾é»ù»ýïý»ñá, áñï»õ p-ý å³ñ½ ï»ýï ãçí ¿, çëï t-ý ³ùµáõç ãçí ¿, áñç µáéáñ å³ñ½ µ³½ù³å³ïïçãý»ñá µ³å³ýáõù »ý p ¡ 1 ãçíá : 5 mathematical problems of computer science 47, 37–49, 2017. statistical tests for mixmax pseudorandom number generator narek h. martirosyan, gevorg a. karyan and norayr z. akopov yerevan physics institute, alikhanian brothers street 2, yerevan, armenia e-mail: narek.h.martirosyan@gmail.com, narek@yerphi.am abstract the pseudo-random number generators (prngs) are key tools in monte carlo simulations. more recently, the mixmax prng has been included in root and class library for high energy physics (clhep) software packages and claims to be a state of the art generator due to its long period, high performance and good statistical properties. in this paper the various statistical tests for mixmax are performed. the results compared with those obtained from other prngs, e.g., mersenne twister, ranlux, lcg reveal better qualities for mixmax in generating random numbers. the mersenne twister is by far the most widely used prng in many software packages including packages in high energy physics (hep), however, the results show that mixmax is not inferior to mersenne twister. keywords: mixmax, statistical tests, mcmc. 1. introduction in recent years, there is a growing interest on prngs in different branches of physics and not only. a good prng is important to have guaranteed results of monte carlo(mc) methods. there are many software packages for mc simulations where prngs are the central components. among these packages one can mention the geant4/clhep[1], a widely used simulation toolkit in hep for modeling the passage of elementary particles through matter, also used for medical and space science simulations. prngs are also crucial in markov chain monte carlo (mcmc) methods which are used for sampling from desired probability distribution by constructing markov chain on state space whose stationary distribution is of interest[2, 3, 4]. uniform prngs play a central role in constructing such markov chains. most of mcmc algorithms are developed within random walk models. a widely used example of random walk monte carlo method is metropolishastings algorithm[3, 4, 5, 6] which is also included in the list of the top 10 algorithms[7]. mcmc methods are mainly used for sampling from large dimensional spaces and computing multidimensional integrals. for example, in statistical mechanics, one needs to compute thermal averages of quantities, such as the total energy, magnetization, etc. by performing multidimensional integration or summation over configuration space. however,, the total number of configurations can be very large, e.g., in 3-dimensional ising model the number of spin configurations with particles at n3 lattice sites is 2n 3 . in thermodynamic 37 38 statistical tests for mixmax pseudorandom number generator equilibrium the probabilities of occurring each configuration is represented by boltzmann distribution. thereby having samples drawn from boltzmann distribution one can compute expectation values of thermodynamic quantities. the necessity to have large amounts of simulated data imposes a strict requirements on prngs, such as statistical properties of generated numbers, swiftness in number generation, replicability, lengthiness of generated random cycle and independence of produced random numbers. to address these challenges the renewed version of mixmax prng[8, 9] based on anosov c-systems and kolmogorov k-systems has been introduced in [10, 11, 12]. the mixmax is matrix-recursive prng and it has been shown that the properties of the mixmax generator is improved with increasing the size n of mixmax matrix[12]. the period of mixmax is also increased with increasing n and it can reach up to 1057824, note that the period of commonly used version of mersenne twister based on mersenne prime has the period of 219937 − 1. while having a long period, however, statistical properties and time characteristics of prngs are crucial to consider a generator ”good” or ”bad”. in this paper we will present the results of the statistical tests performed with the matrix size of n = 256 which is considered to be a default dimension of mixmax matrix with flexibility to be further increased. 2. visual demonstration we can reveal the defect of uniform prngs simply plotting random points in highdimensional euclidean space, if these points form a lattice structure then to a first approximation we can say that prng has defects in generating random points since the space is not filled uniformly. the fig.1 shows the comparison of mixmax with the linear congruential generator(lcg), which is known to be defective prng. in contrast to lcg, mixmax does not form a lattice structure. we obtain these figures by generating two u(0, 1) random number sequences and assigning a point in two-dimensional space. fig. 1. random points in two-dimensional space generated by lcg(left) and mixmax(right). 3. statistical testing with testu01 most of prng algorithms produce numbers uniformly distributed in the interval of [0, 1], hence, prngs should pass statistical tests of uniformity. many empirical statistical testing n. martirosyan, g. karyan and n. akopov 39 packages implement tests for these purposes, some of which are [13, 16, 14, 15]. most of the statistical tests implemented in these packages are discussed in knuth’s book [17], e.g., the package [14] implements mainly the tests of knuth. currently, one of the well-known tools for statistical testing is testu01 software library which provides implementations of the empirical statistical tests for uniform prngs. it contains more than 160 different empirical tests and offers several batteries of tests including the most powerful one, i.e., the ’big crush’. when a specific statistical test is applied to random numbers produced by prng the p-value of the test is printed as a measure of deviation from null-hypothesis, which in our case is a uniform distribution of random numbers. in comparison with other libraries testu01 is more flexible and efficient, and it can deal with larger sample sizes and has a wider range of statistical tests than the other libraries. in table 1, the outcome of testu01 bigcrush suite applied on mixmax, mersenne twister and lcg is stored by using 64-bit computer with intel core i3 −4150 processor of clock speed 3.50 × 4 ghz. table 1: testu01 bigcrush suite results. prng total cpu time bigcrush failed test’(s) p value mixmax 2h 43m 51s all tests were passed mersenne twister 3h 19m 27s 3 0.9990, 1 − 10−15 lcg 3h 30m 33s 22 < 10−300 as we can see from the table the mixmax passes the same test suite faster than mersenne twister and does not fail any test. testu01 test suite has been applied to ranlux prng with its modifications ranlux24, ranlux48. it is observed that ranlux, though having good statistical properties is very slow at generating random numbers. comparing with mixmax, ranlux24 is 10 times slower and ranlux48 is 17 times slower. this fact makes it not convenient for the use in generation of large amount of random numbers. 4. kolmogorov-smirnov tests kolmogorov-smirnov (k-s) test is one of the powerful tools that can be used to examine the statistical features of prngs. though one-dimensional (1d) k-s test is already implemented in testu01, we perform ks test independently for various parameters of sample size (n) and extract the distribution of k-s test statistic. the idea behind the test is to calculate the maximum distance between the expected cumulative distribution function(cdf) f(x), f(x) = pr(x ≤ x) and measured or empirical cumulative distribution function(ecdf) fn(x) of n data points fn(x) = 1 n (number of xi ≤ x). (1) the null hypothesis h0 is whether the sample of n random numbers comes from the expected distribution f(x) or not. if data comes from f(x), then the strong law of large numbers provides fn(x) → f(x), as n → ∞. the latter is strengthened by the glivenko-canteli (g-c) theorem[18], which states that under h0 hypothesis pr( lim n→∞ supx |fn(x) − f(x)| = 0) = 1. (2) 40 statistical tests for mixmax pseudorandom number generator hence the difference between cdfs can be used as a measure of agreement between a data and a given distribution. there are several statistical tests based on (g-c) theorem known as cramer-von mises tests [19, 20]. the key feature of tests based on g-c theorem is that their distributions are independent of the hypothesized models under h0 when data sample is large. the one dimenisonal(1d) k-s test is defined as follows: dn = supx |fn(x) − f(x)|. (3) under null hypothesis the distribution of √ n · dn converges to kolmogorov distribution for sufficiently large n when f(x) is continuous[21] lim n→∞ pr( √ n · dn ≤ x) ≡ k(x) = 1 − 2 ∞∑ i=1 (−1)i−1e−2i 2x2. (4) it is of interest to note that for small values of n kolmogorov distribution is not adequate, but there is a way to compute p − value for randomly produced dn[21]. in fig. 4. the normalized histogram of √ n·dn data points for mixmax and mersenne twister is presented and compared with probability density function (pdf) of kolmogorov distribution: f(x) = k′(x) f(x) = 8x ∞∑ i=1 (−1)i−1i2e−2i 2x2. (5) the histograms in fig.(2–7) are normalized to unity dividing each bin entry by the product of sample size and bin width (n · width). visual comparison shows that under h0 the distribution of √ n · dn follows the pdf of kolmogorov distribution. due to fast convergence of the series of partial sums in eq.(4,5) it suffices to take a limited number of terms in the sum, e.g., the first 100 terms are enough. 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 x fh x l mersenne mixmax fig. 2. distribution of √ n · dn for mixmax and mersenne twister. the size of a samples is n = 108 and the number of different replicas is 104. the black curve is the pdf of kolmogorov distribution. two-level tests can be used on k-s test to give evidence of visual coincidence on fig. 4.. for this purpose the chi-square test(next section) is applied. it has been checked that both distributions agree with theoretical expectation (black curve) in 95% confidence level (cl). n. martirosyan, g. karyan and n. akopov 41 in multidimensional k-s test one has d-dimensional data (d ≥ 2) and to test h0 it is needed to compare d-variate ecdf with the hypothetical d-variate cdf. the complication in multidimensional case is caused by the ambiguity in definition of the cdf since there are 2d − 1 independent ways of defining cdfs. there have been proposed different ways to calculate the multidimensional k-s statistic [22],[23]. in [22] four quadrants around all combinations (xi, xj) of data points are considered and d is taken as the maximum of 4 differences between cdfs over all quadrants. therefore, this idea makes the test statistic independent of ordering the data. the number of all pairs (xi, xj) for n points is equal to n2, therefore, to calculate dn one needs to compute the differences between cdfs in 3n 2 quadrants(the probability for fourth quadrant is found from normalization). this method suffers from the computing time when n is large. in [23], it is proposed to consider only the observed points rather than all combinations, thereby reducing the computational time by computing the differences for 3n quadrants only. it is possible to compute dn with computationally higher efficiency introducing a binning technique applied to a continuous multidimensional data, i.e., discretizing the data space. the idea of binning technique is discussed in [24]. in [25] the algorithm for 2d k-s test is presented when only one cdf from all the possible configurations is taken into account. as a result of it, the procedure used to compute d evaluating the difference of cdfs is reduced to a small number of data points. in our studies we have extended the standard definition of one dimensional cumulative distribution to its two dimensional “analogue” and computed k-s statistic using the algorithm presented in [25](see fig. 4.). 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 x fh x l mersenne mixmax fig. 3. distribution of √ n · dn for mixmax and mersenne twister for 2-dimensional case. the size of a samples is n = 103, note that n is the total number of random points in 2-dimensional space, i.e., 106 numbers are generated by prngs, the number of different replicas is 104. the black curve is the pdf of kolmogorov distribution. the shape of histogram shows a clear shift from the kolmogorov distribution. mixmax and mersenne twister that have been used in these studies give almost the same distributions for √ n · dn which seems to be independent from the dimension of the kolmogorov-smirnov test. 42 statistical tests for mixmax pseudorandom number generator three-dimensional extension of k-s test is presented in [26], where 8 cdfs are considered, and using mc techniques the table of critical values are also presented in this paper. fig. 4. distribution of √ n · dn for the 1st and 31st projections of mixmax rng in comparison with the theoretical expectation (black curve). the sample size is n = 108 and for each projection 104 different replicas are generated. to detect possible non-uniformities in the multidimensional random sequences of mixmax prng, an arbitrary selected projection has been checked via kolmogorov-smirnov test and the results are compared with those of the first projection. in fig. 4., the probability density distributions of kolmogorov-smirnov statistic are presented for the 1st and the 31st projections and the comparison is provided with the expected distribution. 5. chi-square tests the chi-square χ2 test is one of the famous statistical tests which is found in many applications when one deals with grouped or binned data. the chi-square test is applied to categorical sample distributions unlike the k-s test which using each random point compares the continuous sample distributions with the hypothesized ones. the χ2 statistic has the following form [17, 27]: χ2 = k∑ i=1 (oi − ei)2 ei , (6) where oi is the observed number of data points in ith bin and ei = npi is the expected number of data points falling into ith bin, here pi is the probability that observation falls into ith bin. to apply the test to prngs [0,1] interval is divided into k bins and the χ2 statistic is computed noting that pi = 1/k. if h0 is true then statistics defined in (6) computed for random samples follows chi-squared distribution with ν = (k − 1) degrees of freedom gν(y) = 2− ν 2 e− y 2 y ν 2 −1 γ(ν 2 ) , (7) n. martirosyan, g. karyan and n. akopov 43 where γ(ν) is a gamma function. it is useful to introduce a new random variable x = y ν and consider the pdf of x denoted as fν(x). this enables to get rid of small numbers in pdfs when ν is big. since 1 = ∫ fν(x)dx = ∫ gν(y)dy it follows that fν(x) = νgν(νx) = 2− ν 2 ν ν 2 e− x 2ν x ν 2 −1 γ(ν 2 ) . (8) the new random variable introduced in (8) is called a reduced chi-square. the distribution of the reduced chi-square for mixmax and mersenne twister is shown in fig. 5. in comparison with the theoretical expectation. 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.0 0.5 1.0 1.5 2.0 x f ν hx l ν= 49 mersenne mixmax fig. 5. comparing denisty histogram of reduced chi-square for mixmax and mersenne twister with the pdf in (8). the size of a samples is n = 106 and the number of different replicas is 104. 6. serial tests the serial test also known as a chi-square test of independence is a multidimensional analogue of the chi-square test which checks the independence between two or more random variables [17, 28]. when the serial test is applied to prngs one divides the random sequence into groups of non-overlapping d-tuples (xid, xid+, ..., xid+k−1), where i = 1, 2, ..., n d , hence the elements of d-tuple are considered as realizations of d random variables and the relationship between them is of interest . if xis are u(0, 1) random variables then k-tuples are uniformly distributed in [0, 1]d. to check this each dimension of unit hypercube is divided into k bins and the data of d-tuples is binned into [0, 1]d. now chi-square statistic (6) is applied to this data comparing the number of observations falling in each sub-hypercube with theoretical expectation: ei,j,...,d = npi,j,...,d, where the joint probability pi,j,...,d of d-dimensional data point to fall into (i, j, ..., d) sub-hypercube is the product of probabilities of each individual coordinate to fall into an appropriate bin, which is the condition of independence. pi,j,...,d = d∏ n=1 pn = ( 1 k )d . (9) 44 statistical tests for mixmax pseudorandom number generator unlike the non-overlapping tuples, the overlapping d-tuples of random sequence fall on neighboring parallel planes. the largest distance between the adjacent parallel hyperplanes is called a spectral test statistic[17, 29, 30, 31, 32, 33]. if the largest distance is small then it implies that overlapping d-tuples are more uniformly distributed in unit hypercube, therefore, prng is considered good. fig. 6. shows d = 3 and d = 5 dimensional cases of serial test, where each dimension is divided into k = 10 bins and the histogram of the reduced chi-square test statistics is compared with the distribution of (8) with appropriate degrees of freedom. the reduced chi-square distribution reveals no significant distinction between mixmax and mersenne twister. note that the serial test here is applied to a single random stream and measures the correlations between adjacent random tuples. this test can be also applied to different streams to check the independence between them. 7. parallel streams of mixmax all tests described in previous sections use one stream generated by prng. however,, in multiprocessor stochastic computations it is important to have uniformly distributed and statistically independent simultaneous random streams partitioned across the processors[34, 35, 36, 37, 38]. different parallelisation approaches of prngs have been studied in literature[39, 40, 41, 38]. one trivial technique for parallelisation is to take random seeds on each processor, but since every prng has a finite number of states, one should be careful in order to avoid possible overlapping between different streams. mixmax has very large state space, therefore, even taking random seeds on each processor does not affect the independence between multiple streams. another approach is to take a single sequence and partition it into different processors. mixmax provides a skipping-ahead algorithm which enables to skip forward by large amount of numbers in sequence, this technique guarantees the non-collision of partitioned streams[39]. the check for randomness of each individual stream can be done via the standard chi-square or k-s tests. the test of independence of multiple streams can be done via parallel version of serial test simply forming d-tuples of random numbers taken from each of d streams at a time. however,, it is not practical to test empirically all random streams when the period is very large, different techniques for testing parallel streams can be found in [38]. the serial test up to dimensions d = 7 has been performed and it is observed that multiple streams of mixmax are statistically independent which is also guaranteed by underlying theory of mixmax. one can analyze the independence and uniformity of parallel streams using the fact that the sum of n independent u(0, 1) random variables follow the irwinhall distribution of order n[42]. under h0 if each stream of prng is generated from a uniform distribution then the random sequence resulted from the element-wise addition of multiple streams has irwinhall distribution. the irwinhall distribution has the following form: fn(x) = 1 2(n − 1)! n∑ i=1 (−1)i ( n i ) (x − i)n−1sgn(x − i), (10) where sgn(x − i) is a sign function. when n = 2, then (10) reduces to the well known n. martirosyan, g. karyan and n. akopov 45 0.8 0.9 1.0 1.1 1.2 0 2 4 6 8 x f ν hx l ν= 999 mersenne mixmax 0.98 0.99 1.00 1.01 1.02 0 20 40 60 80 x f ν hx l ν= 99999 mersenne mixmax fig. 6. comparing denisty histogram of reduced chi-square with the test distribution (8) for 3-dimensional(up) and 5-dimensional(down) cases. triangular distribution f2(x) = { x, 0 ≤ x < 1, 2 − x, 1 ≤ x ≤ 2. (11) in fig. 7. visual comparison of the data with irwinhall distribution is shown, where up to 15 random streams are taken to form the sum. to check visual consistency of the histogram and model prediction in fig.(7.) a chi square test is applied for comparison with irwin-hall distribution. table 2 represents p-values of chi square statistic. 46 statistical tests for mixmax pseudorandom number generator n = 2 n = 5 n = 10 n = 15 0 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 x f n hx l fig. 7. the distribution of irwin-hall of order (2,5,10,15) compared with density histogram of data. table 2: p-values from chi-square test. test n = 2 n = 5 n = 10 n = 15 p-value 0.76 0.97 0.39 0.37 8. conclusion this paper presents the study of newly released mixmax prng. the various statistical tests including a very high quality testu01 have been used to check the quality of mixmax compared with other generators, mainly with mersenne twister. the results show that mixmax is not inferior to mersenne twister and even better in the sense of speed and period. acknowledgement the authors would like to thank george savvidy for very useful discussions and comments. this project has received funding from the european union’s horizon 2020 research and innovation programme under the marie sḱlodowska-curie grant agreement no 644121. references [1] s. agostinelli et al., “geant4: a simulation toolkit”, nucl. instrum. meth. a ,vol. 506, no. 250, 2003. 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[42] n.l. johnson, s. kotz and n. balakrishnan, continuous univariate distributions, new york: wiley series in probability and mathematical statistics, 1995. submitted 25.10.2016, accepted 24.01.2017. n. martirosyan, g. karyan and n. akopov 4 9 êï³ïçëïçï³ï³ý ã»ëï»ñ åë¨¹á-å³ï³ñ³ï³ý ãí»ñç ·»ý»ñ³ïáñ mixmax-ç ñ³ù³ñ ü. ø³ñïçñáëû³ý, ¶. î³ñû³ý ¨ ü. ²ïáåáí ²ù÷á÷áõù äë¨¹á-å³ï³ñ³ï³ý ãí»ñç ·»ý»ñ³ïáñý»ñá ³é³ýóù³ûçý ·áñíçù »ý ñ³ý¹çë³ýáõù øáýï» î³ééá ù»ãá¹áí ùá¹»é³íáñù³ý ñ³ù³ñ: ì»ñç»ñë ýáñ ”mixmax” ·»ý»ñ³ïáñá áý¹·ñïí»ó root ¨ class library for high energy physics (clhep) íñ³·ñ³ûçý ÷³ã»ãý»ñç ù»ç »ñï³ñ å³ñµ»ñáõãû³ý, ³ñ³·³·áñíáõãû³ý ¨ é³í ëï³ïçëïçï ñ³ïïáõãûáõýý»ñç ßýáññçí: ü»ñï³û³óíáõ ñá¹í³íáõù áõëáõùý³ëçñí»é »ý “mixmax”ç ëï³ïçëïçï ñ³ïïáõãûáõýý»ñá ï³ñµ»ñ ëï³ïçëïçï³ï³ý ã»ëï»ñáí: ²ñ¹ûáõýùý»ñá ñ³ù»ù³ïí»é »ý ï³ñµ»ñ ·»ý»ñ³ïáñý»ñçó ëï³óí³í ³ñ¹ûáõýùý»ñç ñ»ï, ûñçý³ï` “mersenne twister”-ç, “ranlux”-ç ¨ “lcg”-ç: òáõûó ¿ ïñí»é áñ “mixmax”-ý áõýç ³í»éç é³í ñ³ïïáõãûáõýý»ñ å³ï³ñ³ï³ý ãí»ñç ·»ý»ñ³óù³ý ñ³ù³ñ: “mersenne twister” ·»ý»ñ³ïáñá ñ³ý¹çë³ýáõù ¿ ³ù»ý³ï³ñ³íí³í ·»ý»ñ³ïáñá ß³ï íñ³·ñ³ûçý ÷³ã»ãý»ñáõù, ë³ï³ûý ý»ñï³û³óí³í ³ñ¹ûáõýùý»ñá óáõûó »ý ï³éçë, áñ “mixmax”-á áã ùç³ûý ãç ½ççáõù “mersenne twister”-çý, ³ûé ý³¨ ·»ñ³½³ýóáõù ¿ ýñ³ý: còàòèñòè÷åñêèå òåñòû äëÿ ãåíåðàòîðà ïñåâäîñëó÷àéíûõ ÷èñåë mixmax í. ìàðòèðîñÿí, ã. êàðÿí è í. àêîïîâ àííîòàöèÿ ãåíåðàòîðû ïñåâäî-ñëó÷àéíûõ ÷èñåë (ãïñ×) ÿâëÿþòñÿ êëþ÷åâûìè èíñòðóìåíòàìè ìîäåëèðîâàíèÿ ïî ìåòîäó ìîíòå êàðëî. íåäàâíî íîâûé ãïñ× ”mixmax” áûë âêëþ÷åí â ïàêåòû ïðîãðàììíîãî îáåñïå÷åíèÿ root è class library for high energy physics (clhep) ââèäó òîãî, ÷òî ýòîò ãåíåðàòîð ïðèçíàí îäíèì èç ëó÷øèõ ñóùåñòâóþùèõ ãïñ× ïî äëèííîìó ïåðèîäó, âûñîêîé ïðîèçâîäèòåëüíîñòè, à òàêæå îòëè÷íûì ñòàòèñòè÷åñêèì ñâîéñòâàì. â ïðåäëàãàåìîé ñòàòüå ïðèâîäÿòñÿ ðåçóëüòàòû ðàçëè÷íûõ ñòàòèñòè÷åñêèõ òåñòîâ äëÿ ïðîâåðêè ãåíåðàòîðà ”mixmax”. ïðîâåäåíî ñðàâíåíèå õàðàêòåðèñòèê ”mixmax” ñ äðóãèìè ãïñ×, íàïðèìåð ñ ”mersenne twister”, ”ranlux” è ”lcg”, ïîêàçàíî, ÷òî ”mixmax” îáëàäàåò ëó÷øèìè ñâîéñòâàìè äëÿ ãåíåðàöèè ñëó÷àéíûõ ÷èñåë. ãåíåðàòîð ”mersenne twister” ÿâëÿåòñÿ îäíèì èç ñàìûõ èñïîëüçóåìûõ âî ìíîãèõ ïðèëîæåíèÿõ, âêëþ÷àÿ ïàêåòû äëÿ ôèçèêè âûñîêèõ ýíåðãèé, îäíàêî ïðèâåäåííûå ðåçóëüòàòû ïîêàçûâàþò, ÷òî ”mixmax” íè â ÷åì íå óñòóïàåò è äàæå ïðåâîñõîäèò ”mersenne twister”. narek_47_mpcs abstract d:\sbornik\...\article_eng.dvi mathematical problems of computer science 31, 116{121, 2008. n ecessar y and su±cient condition for e xistence of locally-balanced 2-par tition of a t r ee under the e xtended de¯nition of a n eighbour hood of a ver tex s u r e n v . b a likya n y a n d r a fa ye l r . k a m a lia n z y yerevan state university e-mail: suren.balikyan@gmail.com z russian-armenian state university e-mail: rrkamalian@yahoo.com abstract a necessary and su±cient condition is obtained for the problem of partitioning of the set of vertices of a tree g into two disjoint sets v1 and v2 such that it satis¯es the condition jj¸(v) \ v1j ¡ j¸(v) \ v2jj · 1 for any vertex v of g, where ¸(v) is the set of all vertices of g the distance of which from v does not exceed 1. refer ences [1 ] s .v . b a likya n , r .r . k a m a lia n , " on n p -c o m p le t e n e s s o f t h e p r o b le m o f e xis t e n c e o f l o c a lly-b a la n c e d 2 -p a r t it io n fo r b ip a r t it e gr a p h s g wit h ¢ ( g ) = 3 " , r eports of nas r a, vo l. 1 0 5 , n u m . 1 , p p . 2 1 { 2 7 , 2 0 0 5 . [2 ] s .v . b a likya n , r .r . k a m a lia n , " on n p -c o m p le t e n e s s o f t h e p r o b le m o f e xis t e n c e o f l o c a lly-b a la n c e d 2 -p a r t it io n fo r b ip a r t it e gr a p h s g wit h ¢ ( g ) = 4 u n d e r t h e e xt e n d e d d e ¯ n it io n o f t h e n e ig h b o u r h o o d o f a v e r t e x" , r eports of nas r a, vo l. 1 0 6 , n u m . 3 , p p . 2 1 8 { 2 2 6 , 2 0 0 6 . [3 ] s .v . b a likya n , " on l o c a lly-b a la n c e d 2 -p a r t it io n s o f s o m e b ip a r t it e gr a p h s " , abstracts of papers of 15th international conference "m athematics. computing. e ducation.", vo l. 1 5 , p . 7 , d u b n a , r u s s ia , ja n u a r y 2 8 fe b r u a r y 0 2 2 0 0 8 . [4 ] f. h a r a r y, graph theory, a d d is o n -w e s le y, r e a d in g , ma , 1 9 6 9 . [5 ] c. b e r g e , graphs and hypergraphs, e ls e vie r s c ie n c e l t d , 1 9 8 5 . 1 1 6 s. balikyan and r. kamalian 1 1 7 ì³éáõù éáï³é-ñ³í³ë³ñ³ïßéí³í 2-ïñáñù³ý ·áûáõãû³ý ñ³ù³ñ ³ýññ³å»ßï ¨ µ³í³ñ³ñ å³ûù³ý ·³·³ãç ßñç³ï³ûùç áý¹é³ûýí³í ë³ñù³ýù³ý ¹»åùáõù ê. ´³éçïû³ý, è. ø³ù³éû³ý ²ù÷á÷áõù êï³óí³í ¿ ³ýññ³å»ßï ¨ µ³í³ñ³ñ å³ûù³ý í³éç ·³·³ãý»ñç µ³½ùáõãû³ý v1 ¨ v2 ãñ³ïíáõ »ýã³µ³½ùáõãûáõýý»ñç ³ûýåçëç ïñáñù³ý ·áûáõãûáõýá å³ñ½»éáõ ñ³ù³ñ, áñ í³éç ûáõñ³ù³ýãûáõñ v ·³·³ãç ñ³ù³ñ ï»õç áõý»ý³ jj (̧ v ) \ v1j ¡ j¸ ( v ) \ v2jj · 1 ³ýñ³í³ë³ñáõãûáõýá, áñï»õ (v)-áí ýß³ý³ïí³í ¿ ³ûý ·³·³ãý»ñç µ³½ùáõãûáõýá, áñáýó ñ»é³íáñáõãûáõýá v-çó ãç ·»ñ³½³ýóáõù 1-çý: microsoft word tpel.doc математические вопросы кибернетики и вычислительной техники 31, 158—162, 2008. 158 квадратурная формула золотого сечения с.б. аллахвердян институт проблем информатики и автоматизации нан ра и ергу e-mail: souren@ipia.sci.am резюме получены формулы апpроксимирующие корни многочлена лежандра с помощью золотого сечения (фибоначчи). сравнены результаты квадратурной формулы золотого сечения с известными формуйами чебышева и лежандра. литература [1] н. м. воробьев, числа фибоначчи, москва, наука, 1992. [2] д. каханер, к. моулер, с. нэш. численные методы и программное обеспечение (пер. с англ.). м.: мир, 2001. [3] r. fischer, fibonacci applications and, strategies for traders, new york: wiley, 1993, p. 13. ”àëï» ñ³ïù³ý” ù³é³ïáõë³ûçý µ³ý³ó¨ ê. ²éé³ëí»ñ¹û³ý ²ù÷á÷áõù êï³óí³í »ý µ³ý³ó¨»ñ, áñáýù ùáï³ñïáõù »ý è»å³ý¹ñç µ³½ù³ý¹³ùç ³ñù³ïý»ñá ”áëï» ñ³ïù³ý” (üçµáý³ããçç ãí»ñç) ùççáóáí: ´»ñí³í »ý ³ñ¹ûáõýùç` ”áëï» ñ³ïù³ý” ù³é³ïáõë³ûçý µ³ý³ó¨ç ñ³ù»ù³ïáõùá â»µçß¨ç ¨ è»å³ý¹ñç ñ³ûïýç ù³é³ïáõë³ûçý µý³ý³ó¨»ñç ñ»ï: article_with_style.dvi mathematical problems of computer science 31, 73{78, 2008. application of ldt to m any h ypotheses optimal t esting for m ar kov chain l e a d e r n a va e i payame noor university (pnu), iran abstract the problem of many (l > 2) hypotheses testing on distributions of a ¯nite state markov chain is studied. we apply large deviations techniques (ldt). it is proved that this method of investigation in solving the problem of logarithmically asymptotically optimal (lao) hypotheses testing is easier than the procedure that was introduced by haroutunian. the matrix of exponents e = feljmg; m; l = 1; l, of error probabilities of the lao test eljm(á) = lim n !1 ¡ 1 n log ®ljm(án ); where ® n ljm(án ) for l 6= m is the probability to accept the hypothesis l, when the hypothesis m is true, is determined. refer ences [1 ] b la h u t r . e . \ p r in c ip le a n d p r a c t ic e o f in fo r m a t io n th e o r y" , r e a d in g , ma , a d d is o n we s le y, 1 9 8 7 . [2 ] cs is z ¶a r i. a n d s h ie ld s p . \ in fo r m a t io n th e o r y a n d s t a t is t ic s " , fu n d e m e n t a ls a n d tr e n d s in co m m u n ic a t io n s a n d in fo r m a t io n th e o r y, vo l. 1 , n o . 4 , 2 0 0 4 . [3 ] cs is z ¶a r i. a n d k äo r n e r j. \ in fo r m a t io n th e o r y: co d in g th e o r e m fo r d is c r e t e me m o r yle s s s ys t e m s " , a c a d e m ic p r e s s , n e wy o r k, 1 9 8 1 . [4 ] cs is z ¶a r i. \ me t h o d o f t yp e s " , ie e e tr a n s . in fo r m . th e o r y, vo l. 4 4 . n o . 6 . p p . 2 5 0 5 -2 5 2 3 , 1 9 9 8 . [5 ] d e m b o a . a n d ze it o u n i o. \ l a r g e d e via t io n s te c h n iqu e s a n d a p p lic a t io n s " , jo n s a n d b a r t le t . p u b lis h e r s , l o n d o n , 1 9 9 3 . [6 ] gu t m a n m. \ a s ym p t o t ic a lly o p t im a l c la s s ī c a t io n fo r m u lt ip le t e s t wit h e m p ir ic a lly o b s e r ve d s t a t is t ic s " , ie e e tr a n s . in fo r m . th e o r y, vo l. 3 5 , n o . 2 . p p . 4 0 1 -4 0 8 , 1 9 8 9 . [7 ] h a r o u t u n ia n e . a . \ on a s ym p t o t ic a lly o p t im a l t e s t in g o f h yp o t h e s e s c o n c e r n in g ma r ko v c h a in " , ( in r u s s ia n ) . iz ve s t ia a c a d . n a u k a r m e n ia n s s r . s e r ia ma t h e m . vo l. 2 2 , n o . 1 . p p . 7 6 -8 0 , 1 9 8 8 . [8 ] h a r o u t u n ia n e . a , h a r o u t u n ia n m. e a n d h a r u t yu n ya n a . n .\ r e lia b ilit y cr it e r ia in in fo r m a t io n th e o r y a n d in s t a t is t ic a l h yp o t h e s is te s t in g " , fo u n d a t io n s a n d tr e n d s in co m m u n ic a t io n s a n d in fo r m a t io n th e o r y, vo l. 4 , n o . 2 -3 , 2 0 0 7 . [9 ] k u llb a c k s . \ in fo r m a t io n th e o r y a n d s t a t is t ic s " , w ile y, n e w y o r k, 1 9 5 9 . [1 0 ] n a t a r a ja n s .\ l a r g e d e via t io n s , h yp o t h e s e s t e s t in g , a n d s o u r c e c o d in g fo r ¯ n it e ma r ko v c h a in " , ie e e tr a n s . in fo r m . th e o r y, vo l. 3 1 , n o . 3 , p p . 3 6 0 -3 6 5 , 1 9 8 5 . [1 1 ] n a va e i l .\ on m a n y h yp o t h e s e s l a o t e s t in g via t h e t h e o r y o f la r g e d e via t io n s " , fa r e a s t jo u r n a l o f ma t h e m a t ic a l s c ie n c e s , vo l. 2 5 , n o . 2 , p p . 3 3 5 -3 4 4 , 2 0 0 7 . 7 3 7 4 application of ldt to many hypotheses optimal testing for markov chain øþî ïçñ³éáõãûáõýá ø³ñïáíç ßõã³ý»ñç ñ³ù³ñ µ³½ù³ïç í³ñï³íý»ñç ûåïçù³é ï»ëï³íáñù³ýá è. ü³í³ûç ²ù÷á÷áõù àõëáõùý³ëçñí³í ¿ í»ñç³íáñ íç×³ïý»ñáí ø³ñïáíç ßõã³ûç ýï³ïù³ùµ µ³½ù³ïç í³ñï³íý»ñç ëïáõ·ù³ý ëý¹çñá: îçñ³éíáõù ¿ ù»í ß»õáõùý»ñç ï»ëýçï³ý (øþî): ²å³óáõóí»é ¿, áñ í³ñï³íý»ñç éá·³ñçãùáñ»ý ³ëçùåïáïáñ»ý ûåïçù³é (è²ú) ï»ëï³íáñù³ý ñ»ï³½áïù³ý ³ûë »õ³ý³ïá ³í»éç ñ»ßï ¿, ù³ý ð³ñáõãûáõýû³ýç ïáõùçó ý»ñùáõíí³íá: начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 46, 55--58, 2016. about some queueing models for computational grid systems vladimir g. sahakyan, yuri h. shoukourian and hrachya v. astsatryan institute for informatics and automation problems of nas ra e-mail: vladimir.sahakyan@sci.am, shouk@sci.am, hrach@sci.am abstract in this paper parametric models of the queues are proposed in grid systems. the models take into account the restrictions on the waiting and interval of execution time of the tasks. keywords: grid computing, queueing systems, resource allocation systems. the optimal usage of cpu times in multi-core computational grid infrastructures depends on several factors, such as the job scheduling, the possibility of dynamic allocation of computational resources, the possibility of job migration to the different phases of implementation in optimal requested environment, performing a stop of the job with the possibility of continuing, etc. the use of grid infrastructures to handle the information stream requires a flexible approach to the allocation of resources as well as for the timely performance of jobs. the job receiving process in the queue for further executing in a grid infrastructure plays an important role in the organization of the whole process. at this stage the incoming job service script is generated based on a maintenance data bursts. this requires synchronization of distributed processes to manage the resources of a grid system. a set of gram (grid resource allocation and management) services allows to manage the resources and jobs, job queues handling, ensuring their execution on the required computational resources. in addition to gram services, the schedulers play special roles. the adoption of the job queue for a service applies to the scheduler responsible for ensuring its timely implementation. nowadays the usage of modern schedulers, such as maui, condor, pbs (portable batch system) has opportunities to perform the planning tasks on specified priorities [1,2,3]. however, in complex cases (for instance, to run the job at a specified time and in specified resources, or to manage multi-stream queues) we need to implement a flexible approach for scheduling. 55 mailto:vladimir.sahakyan@sci.am mailto:shouk@sci.am mailto:hrach@sci.am about some queueing models for computational grid systems 56 the optimal run of the programs in multi-core environments involve selection of service discipline, delivering at least a "loss function", which characterizes the quality of the functioning of the process. as such a function can be considered some functional depending on the average waiting time, utilization of time, the number of denial of service and others. many authors have shown that the discipline of the relative priority is optimal for linear functional without service interruption. if prevent the implementation of programs interrupted, sometimes absolute priority disciplines are better. however, in systems with many devices maintenance procedure can be disrupted, while retaining the properties of known subjects. for example, in case of a fifo discipline, if a job is received after the service does not affect the incoming job before him, it may be served. this can be done using batchfill mechanism (check available periods for low-priority tasks). often it is required to provide manage the job not later or earlier, or in a specified period of time after entering the system. for a description of these conditions we consider the classification of the job queues on systems diagrams defining the service priorities. schemes depending on the order of receipt  mfifo (modified fifo) – received to the job system, it is queued in order of receipt and can serve, if it does not affect the start time of service to all previously received assignments;  mlifo (modified lifo) received to the job system, it is queued in order of receipt and can serve, if it does not affect the start of the service later he received jobs from the queue.  mrs (modified random service) received to the job system, it is queued and can cater for random selection depending on specific task parameters. schemes based on the level of required resources -ju (job up): received to the job system, it is queued in order of required resources for the service ([service time] * [number of required cores]) and can serve, if it does not affect the start of the maintenance of all available jobs in the queue ahead of its assignments. previously, all will go on maintenance task requires the most resources. -jd (job down): received to the job system, it is queued in order of increasing required for maintenance resources ([service time] * [number of required cores]) and can serve, if it does not affect the start of the maintenance of all available jobs in line behind its assignments. previously, all will go on maintenance task requires the least resources. schemes based on the number of processors required -pu (processor up): received to the job system, it is queued in order of processors required for service and can serve if it does not affect the start of the maintenance of all available jobs in the queue ahead of its assignments. previously, all will go on maintenance task requires the most resources. -pd (p down): received to the job system, it is queued in order of increasing required maintenance for the processor and can serve, if it does not affect the start of service available queued jobs behind it. previously, all will go on maintenance task requires the least resources. schemes, with a restriction on the waiting time -wtr (waiting time restriction): received to the job system, it is queued in order acceptable waiting time, i.e., the higher priority admission service has a job with less latency. -itr (interval time restriction): received to the job system, it is queued and must start the handling after a specified time, but no later than the specified timeout. the first goes on a mission handling shall start the service before anyone else. all queuing schemes permitted job handling from the queue, if its handling does not affect the maintenance of higher-priority jobs. v. sahakyan, yu. shoukourian and h. astsatryan 57 schemes are mutually exclusive and only one of the circuits can be selected and when queuing. however, restrictions on the waiting time can be in all schemes. in systems allowing job migrations, these schemes can be described as a service processes without interruption (relative priority) and with interruption (absolute priority). the job interruption may take place in case of entering higher priority jobs. we assume that the job under the new entry to the service will be continued (not started first). setting with interrupted service it is returned to the queue with the adjusted design parameters (service time, waiting time, etc.). for a more detailed formulation let’s consider a computational consisting of m (m ≥ 1) processors. when using systems based on virtual clusters, limiting the number of processors is arbitrary and depends on the performance of the basic configuration. we assume that the number of jobs that can be in the queue is not limited. the grounds of refusal of service can only serve as the impossibility of his service with the user-defined constraints (time, number of processors, etc.). let job stream be supplied. each job can be characterized by parameters (ν, β, ω, γ), where ν the number of processors required for the job, β the time required to run, ω permissible total residence time job in the queue, γ the time from the admission to the system after it is allowed to start the service. fig. 1. the example of task an acceptable execution. if γ = 0, the value of this parameter can be omitted. in systems with no limit on the waiting time the value of the parameter ω is also lowered. for jobs with a restriction on the waiting time for admission to the system is checked on the ability to perform a task and then either taken in turn to perform, or is refused by the implementation. the time required for maintenance is in some sense arbitrary, which means maximum allowable. in reality, it is random and may be less than the predetermined one. therefore, the order may vary both in case of receiving the jobs and finishing the maintenance of jobs. service denial gets jobs, if at the time of admission to the system it finds that it cannot serve the specified parameters. for example, to start services at a specified time. the system is considered at discrete points in time, the job proceeds to the queue or completion of service. we call these moments 0-momentums. in each 0-momentum it carries out queue recalculation and a new order of the queue in the computational system. for various schemes a multithreading queuing model is proposed. between the streams are set priorities, and inside threads can be used as one of the described schemes of service. about some queueing models for computational grid systems 58 for the above mentioned schemes, new algorithms have been developed to schedule jobs with mixed priorities. note that the algorithms can be used both in homogeneous and in lines with mixed schemas with a restriction on waiting time. references [1] i. foster, computational grid, morgan-kufman, 1999. [2] i. foster, c. kesseman and s. tuecke, “the anatomy of the grid”, international journal of hpc applications, vol. 15, no. 3, pp. 25-37, 2001. [3] f. xhafa, “batch mode scheduling in grid systems”, int.j. web and grid services, vol.3, no. 1, pp.19-37, 2007. submitted 04.07.2016, accepted 20.10.2016. հերթերի որոշ մոդելների մասին հաշվողական գրիդ– համակարգերում վ.սահակյան, յու.շուքուրյան և հ. ասցատրյան ամփոփում հոդվածում առաջարկվում է հաշվողական գրիդ–համակարգի համար հերթի պարամետրիկ մոդելը։ մոդելը հաշվի է առնում առաջադրանքի հերթում սպասման և կատարման թույլատրելի ժամանակները։ о некоторых моделях очередей в вычислительных гридсистемах в. саакян, ю. шукурян и г. асцатрян аннотация в статье предложена параметрическая модель очереди для вычислительной гридсистемы. модель учитывает ограничения на допустимые времена ожиданий и промежутков выполнения заданий. microsoft word pa;yan_14.doc математические вопросы кибернетики и вычислительной техники 34, 2010. 39 о необходимости восстановления специальности для подготовки программистов в бакалавриате технических вузов армении (ра) а. х. палян с 1992 по 2007 годы в государственном инженерном университете армении (гиуа) на факультете компьютерных систем и информатики велось обучение бакалавров по следующим специальностям в области информационных технологий (ит):  вычислительные машины, комплексы, системы и сети (вмксс)  программное обеспечение вычислительной техники и асу (повтас)  автоматизированные системы обработки информации и управления (асоиу). по этим же специальностям велась обучение в технических вузах российской федерации (рф). в 2001 году авторитетная организация acm (сша) объявила рекомендуемый перечень специальностей бакалавриата в области ит, который был расширен в 2005 году. перечень опубликован в документе computing curricula 2005 (http://www.acm.org/education/curric_vols/cc2005-march06final.pdf). в настоящее время в части технических вузов он включает:  computer engineering (вмксс)  software engineering (повтас)  information systems (асоиу)  information technology (асоиу). в скобках указано примерное соответствие вышеупомянутым специальностям. данный перечень принят в большинстве вузов сша в качестве стандарта де-факто . после 2001 года в рф ряд ученых, например, профессор мгу в. сухомлин, выступали за введение перечня специальностей acm в бакалавриате вузов рф. в настоящее время на сайте минобразования и науки рф опубликован перечень специальностей бакалавриата ( http://mon.gov.ru/dok/fgos/7198/). в данном перечне направление “информатика и вычислительная техника” в части технических вузов включает специальности :  информатика и вычислительная техника (computer engineering)  программная инженерия (software engineering)  информационные системы и технологии (information systems, information technology). очевидно, что в рф приняли за основу перечень специальностей acm, примерное соответствие которым указано в скобках. в ра в связи с переходом на кредитную систему обучения в 2007 году минобразования и науки ра ввело новый перечень специальностей для бакалавриата технических вузов (гиуа и других вузов):  информатика и вычислительная техника пстроение ассоциативных правил путем цепного раздробления n-мерного единичного куба цепями 40  информационные системы  информационные технологии. в этом перечене отсутствует специальность для подготовки программистов, что по некоторым сведениям является следствием принятого минобразования и науки ра курса на укрупнение специальностей. таким образом, специальность для подготовки программистов, существовавшая долгие годы в технических вузах ра под названием повтас, а в настоящее время в рф и сша под названием “программная инженерия (software engineering)”, оказалась утраченной. вследствие этого в гиуа бывшие специальности повтас и вмксс были включены в новую специальность “информатика и вычислительная техника” в статусе специализаций. программистская специальность в статусе специализации выглядит очень странно. дело в том, что программирование включает широкий набор дисциплин и в соответствии с мировой практикой включает ряд специализаций, как например, программирование баз данних, сетевое программирование, программирование операционных систем и т. д. снижение статуса подготовки программистов до уровня специализации, несомненно, скажется на качестве их обучения. студент, специализирующийся по программированию, должен изучать также предметы, относящиеся к общей специальности, в частности, к вмксс. общеизвестно, что программисты составляют большинство специалистов, работающих в области ит ра. снижение уровня подготовки программистов в гиуа и других вузах приведет к снижению их общего потенциала, что отрицательно скажется на состоянии приоритетной для ра области информационных технологий . учитывая вышеизложенное, представляется целесообразным восстановление специальности для подготовки программистов в бакалавриате технических вузов ра под названием “программная инженерия (software engineering)”. для сохранения общего числа специальностей можно было бы объединить “информационные системы” и “информационные технологии” в одну специальность так, как сделано в рф, достигая тем самым полного соответствия перечней ра и рф. палян арменак христафорович докт. техн. наук, проф. (гиуа) 4_rafayel_23--32_new.dvi mathematical problems of computer science 48, 23{32, 2017. about complexity of fft algor ithms for length of q £ 2p r a fa ye l v . b a r s e g h ya n institute for informatics and automation problems of nas ra e-mail: rafayelbarseghyan@ipia.sci.am abstract the paper presents logarithmic formula which allows to compute the exact number of necessary operations for computing the discrete fourier transform (dft) of an arbitrary q £ 2p length, where q is an odd integer. keywords: fast fourier transform (fft), split-radix algorithm, computational complexity. 1 . in t r o d u c t io n th e d is c r e t e fo u r ie r t r a n s fo r m ( d ft) h a s a wid e r a n g e o f a p p lic a t io n s in m a n y ¯ e ld s o f s c ie n c e a n d e n g in e e r in g [1 ],[2 ]. th e m a in r e a s o n fo r it s p o p u la r it y is t h e e xis t e n c e o f va r io u s a lg o r it h m s wh ic h a llo ws t o s ig n i¯ c a n t ly r e d u c e t h e c o m p u t a t io n a l c o m p le xit y. th e s e a lg o r it h m s a r e g e n e r a lly kn o wn a s fa s t fo u r ie r t r a n s fo r m s ( fft) . fa s t a lg o r it h m fo r e ± c ie n t c o m p u t a t io n o f d ft wa s ¯ r s t in t r o d u c e d b y cu le y a n d tu ke y b y t h e ir h is t o r ic a l p a p e r in 1 9 6 5 [3 ]. fft a lg o r it h m s a llo w t o c o m p u t e d ft o f s iz e n wit h o ( n lg n ) o p e r a t io n s in c o n t r a s t t o d ir e c t fo r m c o m p u t a t io n wh ic h r e qu ir e s o ( n 2 ) o p e r a t io n s . th e r e a r e a n u m b e r o f fft a lg o r it h m s , b u t t h e m o s t p o p u la r m e t h o d s a r e b a s e d o n ¯ xe d -r a d ix a n d s p lit -r a d ix a p p r o a c h e s . s p lit -r a d ix a lg o r it h m s h a ve b e e n c o n s id e r e d t o b e t h e m o s t c o m p u t a t io n a lly e ± c ie n t a n d s t r u c t u r a lly r e g u la r . s p lit -r a d ix a lg o r it h m wa s ¯ r s t in t r o d u c e d b y y a vn e [4 ] in 1 9 6 9 a n d la t e r b y va r io u s a u t h o r s [8 ]. s p lit -r a d ix a lg o r it h m a llo ws t o c o m p u t e d ft o f n = 2 p wit h 4 n lg n ¡ 6 n + 8 a r it h m e t ic o p e r a t io n s . in r e c e n t ye a r s b y va r io u s a u t h o r s [5 ], a n e w m o d i¯ c a t io n o f s p lit r a d ix a lg o r it h m wa s d e ve lo p e d wh ic h a llo ws t o p e r fo r m d ft o f n = 2 p wit h 34 9 n lg n ¡ 124 27 n ¡ 2 lg n ¡ 2 9 ( ¡ 1 ) lg n lg n + 16 27 ( ¡1 ) lg n + 8 a r it h m e t ic o p e r a t io n s . fo r a p p lic a t io n s , wh ic h n e e d t o p e r fo r m d ft o f s iz e s n 6= 2 p, u s u a lly s p e c ia lis t s u s e t h e z e r o p a d d in g t e c h n iqu e . it m e a n s t h a t t h e in p u t s e qu e n c e is ¯ lle d wit h z e r o e s u n t il it b e c o m e s a p o we r o f t wo le n g t h fo r p e r fo r m in g a n y a va ila b le fft a lg o r it h m . s u c h m e t h o d s ig n i¯ c a n t ly d e c r e a s e s t h e r e qu ir e d n u m b e r o f a r it h m e t ic o p e r a t io n s . d ft fo r in p u t s e qu e n c e s wh ic h h a s le n g t h n o n -p o we r -o f-2 is r e qu ir e d in m a n y p r a c t ic a l a p p lic a t io n s , it is a n im p o r t a n t p r o b le m . a lg o r it h m fo r c o m p u t in g d ft fo r s iz e s q £ 2 p, wh e r e q is a n o d d in t e g e r , wa s in t r o d u c e d b y b i a n d ch e n in 1 9 9 8 [6 ]. a lg o r it h m h a s a 2 =4 s p lit -r a d ix s t r u c t u r e a n d in c a s e o f q = 1 h a s t h e s a m e c o m p le xit y a s t h e c o n ve n t io n a l s p lit -r a d ix fft a lg o r it h m . a ft e r t h a t , in 2 0 0 4 b y b o u g u e z e l a n d e t . a l. [1 0 ] a n e w im p r o ve d a lg o r it h m fo r q£ 2 p le n g t h d ft wa s p r e s e n t e d . 2 3 2 4 about complexity of fft algorithms for length of q £ 2p a lg o r it h m is b a s e d o n 2 = 8 s p lit -r a d ix fft a lg o r it h m s c h e m e a n d im p r o ve s s u c h im p o r t a n t fa c t o r s a s d a t a t r a n s fe r , a d d r e s s g e n e r a t io n , t wid d le fa c t o r c o m p u t a t io n a n d a c c e s s t o t h e lo o ku p t a b le , b u t it d id n o t r e d u c e n u m b e r o f a r it h m e t ic o p e r a t io n s . in 2 0 1 0 b i a n d ch e n [7 ] p u b lis h e d a n e w p a p e r wh e r e t h e y p r e s e n t e d a u n i¯ e d m e t h o d fo r g e n e r a t io n o f 2 =2 a ( wh e r e a is a n in t e g e r a n d a > 1 ) s p lit -r a d ix a lg o r it h m s fo r q £ 2 p le n g t h d fts . in t h is p a p e r a g e n e r a l lo g a r it h m ic fo r m u la is d e r ive d fo r c a lc u la t in g n u m b e r o f a r it h m e t ic o p e r a t io n s fo r 2 = 4 a n d 2 =8 s p lit -r a d ix a lg o r it h m s fo r q £ 2 p le n g t h d fts [1 4 ]. fo r a ll q < 2 0 s p e c ia l c a s e s , fo r m u la s a r e d e ve lo p e d fo r c o u n t in g e xa c t n u m b e r o f a r it h m e t ic o p e r a t io n s a n d s o m e c a s e s a r e s h o wn , wh e r e c o m p u t a t io n a l e ®e c t ive n e s s is in ve r s e ly p r o p o r t io n a l t o t h e le n g t h o f t h e d ft-s iz e . 2 . ge n e r a l a lg o r it h m l e t x = fx0; x1; : : : ; xn¡1gt b e a c o m p le x va lu e d c o lu m n -ve c t o r o f le n g t h n, wh e r e n = q £ 2 p a n d q is a n o d d in t e g e r . th e d ft o f t h is ve c t o r a r e d e ¯ n e d a s x[k] = n ¡1x k=0 x[n]w nkn ; ( 1 ) wh e r e 0 · k · n ¡ 1 ; w nn = e xp ( ¡j 2¼n n) = c o s ( 2¼ n n) ¡ j s in ( 2¼ n n) ; j = p ¡ 1 : b e lo w t h e a lg o r it h m fr o m [7 ] is p r e s e n t e d . e ve n in d e xe s o f t h e t r a n s fo r m a r e c o m p u t e d b y x[2 k] = n=2¡1x n=0 ( x[n] + x[n + n=2 ]]) w nkn=2; ( 2 ) wh e r e x[2 k] is a n n=2 d ft. th e o d d in d e xe s a r e d e ¯ n e d b y x[2 ak + l] = n ¡1x n=0 x[n]w n(2ak+l) n ; ( 3 ) wh e r e 0 · k · ( n=2 a ) ¡ 1 , a n d a is a n in t e g e r ( a > 1 ) a n d l h a s a s e le c t e d o d d va lu e s s o t h a t 2 ak + l g e n e r a t e s n=2 o d d in t e g e r s t h a t c a n b e u n iqu e ly m a t c h e d t o a ll t h e o d d in d e x va lu e s b e t we e n 0 a n d n. w it h s o m e m a n ip u la t io n s b a s e d o n t h e p e r io d ic a n d s ym m e t r ic p r o p e r t ie s o f w n(2ak+l) n , ( 3 ) c a n b e r e p r e s e n t e d a s x[2 ak + l] = (n=2a)¡1x n=0 x0[n]w nln w nk n=2a + (n=2a)¡1x n=0 x0[n + n=2 a]w (n+n=2a)l n w nk n=2a + : : : + (n=2a)¡1x n=0 x0[n + ( a ¡ 1 ) n 2 a ]w (n+ ( a¡1) n 2a )l n w nk n=2a; ( 4 ) wh e r e n = 0 ; 1 ; 2 : : : ; n= 2 ¡ 1 a n d x0[n] = x[n] ¡ x[n + n=2 ]: ( 5 ) it c a n b e o b s e r ve d t h a t ( 4 ) is a le n g t h -n=2 a d ft t h e in p u t s e qu e n c e o f wh ic h is t h e r e s u lt o f t h e c o m p u t a t io n in s id e t h e b r a c ke t s o f ( 4 ) fo r n = 0 ; 1 ; 2 : : : ; n= 2 ¡ 1 . in s u m m a r y, t h e e ve n in d e xe d o u t p u t s o f ( 1 ) a r e o b t a in e d fr o m o n e le n g t h -n= 2 d ft d e ¯ n e d in ( 2 ) b a s e d o n r. barseghyan 2 5 t h e r a d ix-2 d e c o m p o s it io n , a n d t h e o d d in d e xe d o u t p u t s a r e o b t a in e d fr o m a le n g t h -n= 2 a d fts , b a s e d o n t h e r a d ix-2 a d e c o m p o s it io n . th e c o m p le xit y o f a lg o r it h m c a n b e c o m p u t e d b y t h e fo llo win g e xp r e s s io n s : c£n = c £ n=2 + ac £ n=2a + n 2a c£ + 2 n ¡ c£t ; c+n = c + n=2 + ac + n=2a + n 2a c+ + 3 n ¡ c+t ; ( 6 ) wh e r e b y c£ a n d c+ d e n o t e t h e n u m b e r o f r e a l m u lt ip lic a t io n s a n d a d d it io n s t h a t a r e u s e d fo r e a c h o f t h e in n e r s u m s d e ¯ n e d in ( 4 ) , a n d c£t a n d c + t is t h e n u m b e r o f r e a l m u lt ip lic a t io n s a n d a d d it io n s s a ve d fr o m a ll t h e t r ivia l t wid d le fa c t o r s w nln in ( 4 ) . 3 . 2 / 4 s p lit -r a d ix a lg o r it h m fo r a = 2 t h e a lg o r it h m b e c o m e s a m o d i¯ e d ve r s io n o f c o n ve n t io n a l 2 = 4 s p lit -r a d ix a lg o r it h m . in s e r t in g a = 2 in t o ( 4 ) we g e t x[4 k + l] = n=4¡1x n=0 w nln ( 1x n=0 x0[n + i n 4 ]w iln=4 ) w nk n=4 = = n=4¡1x n=0 w nln ( x 0[n] + ( ¡j ) lx0[n + n 4 ]) w nkn=4; ( 7 ) fr o m ( 7 ) we c a n s e e t h a t it b e c o m e s a c o n ve n t io n a l s p lit -r a d ix a lg o r it h m wh ic h is r e p o r t e d in [4 ],[8 ],[9 ]. to c o ve r a ll o d d in d e xe s we s e t l = f¡ 1 ; 1 g. in t h is c a s e , we h a ve a n a r it h m e t ic c o m p u t a t io n a l g a in o n ly in c a s e s o f n = 0 a n d n = n=8 ( w 0n a n d w ln=8 n t wid d le fa c t o r s b e c o m e t r ivia l) . n o w it is e a s y t o s e e t h a t t h e n u m b e r o f a r it h m e t ic o p e r a t io n s a r e c+n = c + n=2 + 2 c + n=4 + 4 n ¡ 4 q c£n = c £ n=2 + 2 c£ n=4 + 2 n ¡ 1 2 q: ( 8 ) u s in g t h e t h e o r y o f d i®e r e n c e e qu a t io n s a n d ma xim a [1 2 ] c o m p u t e r a lg e b r a s ys t e m , we g e t t h e n u m b e r o f a r it h m e t ic o p e r a t io n s r e qu ir e d fo r c o m p u t a t io n o f ( 7 ) in lo g a r it h m ic fo r m c+n = ¡ 2p(28q¡3c+2q¡3c + q ) 9 + (¡1)p(10q¡3c+2q+6c + q ) 9 + p2 p+3q 3 + 2 q; c£n = ¡ 2p(44q¡3c£2q¡3c £ q ) 9 ¡ (¡1) p (10q+3c £ 2q ¡6c £ q ) 9 + p2 p+2q 3 + 6 q; ( 9 ) wh e r e cq a n d c2q d e n o t e t h e c o m p le xit ie s o f q a n d 2 q le n g t h d fts , r e s p e c t ive ly. u s in g m e t h o d s fr o m [6 ] fo r c o m p u t in g 2 q-le n g t h d ft we o b t a in c+n = 2 c + q + 4 q; c£n = 2 c £ q ; ( 1 0 ) ¯ n a lly s u b s t it u t in g ( 1 0 ) in t o ( 9 ) a n d 2 p = n q we g e t c+n = 8 3 pq 2 p ¡ 2p 9 ( 1 6 q ¡ 9 c+q ) ¡ 29 q ( ¡ 1 ) p + 2 q; c£n = 4 3 pq 2 p ¡ 2p 9 ( 4 4 q ¡ 9 c£q ) ¡ 109 q ( ¡1 ) p + 6 q; ( 1 1 ) 2 6 about complexity of fft algorithms for length of q £ 2p 4 q-le n g t h d ft c a n b e c o m p u t e d b y c+4q = 4 c + q + 1 6 q c£4q = 4 c £ q u s in g it a n d ( 8 ) , ¯ n a lly we g e t t h e fo r m u la s wh ic h s h o w t h e n u m b e r o f r e a l a r it h m e t ic o p e r a t io n s o f q £ 2 p c+n = 8 3 pq 2 p ¡ 2p 9 ( 1 6 q ¡ 9 c+q ) ¡ 29 q ( ¡1 ) p + 2 q =; = 8 3 n lo g 2 ³ n q ´ ¡ n ³ 16 9 ¡ 1 q c+q ´ ¡ 2 9 q ( ¡ 1 ) log2 ( n q ) + 2 q; c£n = 4 3 pq 2 p ¡ 2p 9 ( 3 8 q ¡ 9 c£q ) + 29 q ( ¡1 ) p + 6 q =; = 4 3 n lo g 2 ³ n q ´ ¡ n ³ 38 9 ¡ 1 q c+q ´ + 2 9 q ( ¡ 1 ) log2 ( n q ) + 6 q: ( 1 2 ) if q = 1 a n d , t h e r e fo r e c+1 = 0 a n d c + 1 = 0 fr o m ( 1 2 ) we c a n g e t c+n = 8 3 n lo g 2 n ¡ 169 n ¡ 2 9 ( ¡1 ) log2 n + 2 ; c£n = 8 3 n lo g 2 n ¡ 389 n + 2 9 ( ¡1 ) log2 n + 6 : ( 1 3 ) ( 1 3 ) is t h e s a m e a s t h e n u m b e r o f r e a l a r it h m e t ic o p e r a t io n s c o u n t r e qu ir e d b y c o n ve n t io n a l s p lit -r a d ix a lg o r it h m . d o in g s o m e o p t im iz a t io n fr o m [6 ], we g e t t h e fo llo win g r e c u r r e n t e xp r e s s io n s fo r c o m p u t in g 8 q le n g t h d fts . c+n = c + 4q + 4 c + q + 3 6 q = 8 c + q + 5 2 q; c£n = c £ 4q + 2 c £ q + 2 c £ sq = 6 c £ q + 2 c £ sq; ( 1 4 ) wh e r e c£sq d e n o t e s t h e n u m b e r o f a r it h m e t ic o p e r a t io n s r e qu ir e d b y s c a le d d ft [6 ]. u s in g ( 1 4 ) , we g e t a n im p r o ve m e n t in t h e n u m b e r o f a r it h m e t ic o p e r a t io n s c+n = 8 3 pq 2 p ¡ 2p 9 ( 1 6 q ¡ 9 c+q ) ¡ 29 q ( ¡ 1 ) p + 2 q =; = 8 3 n lo g 2 ³ n q ´ ¡ n ³ 16 9 ¡ 1 q c+q ´ ¡ 2 9 q ( ¡ 1 ) log2 ( n q ) + 2 q; c£n = 4 3 pq 2 p ¡ 2p 18 ( 8 2 q ¡ 3 ( 5 c£q + c£sq ) ) + 29 ( 7 q + 3 ( c £ q ¡ c£sq ) ) ( ¡1 ) p + 6 q =; = 4 3 n lo g 2 ³ n q ´ ¡ n 18 ³ 8 2 ¡ 3 q ( 5 c£q + c £ sq ) ´ + 2 9 ( 7 q + 3 ( c£q ¡ c£sq ) ) ( ¡ 1 ) log2 ( nq ) + 6 q: ( 1 5 ) 3 .1 co m p le xit y o f 2 =4 s p lit -r a d ix a lg o r it h m fo r q < 2 0 in [6 ] a m e t h o d fo r c o m p u t in g 3 -p o in t d ft wit h o p t im iz a t io n in c a s e o f 2 4 -p o in t is p r e s e n t e d . u s in g t h a t r e s u lt we c a n g e t a c o m p le t e e xp r e s s io n wh ic h d e s c r ib e s t h e n u m b e r o f a r it h m e t ic o p e r a t io n s r e qu ir e d b y 3 £ 2 p-le n g t h d ft. c+n ( 3 £ 2 p ) = 8 p2 p + 203 2 p ¡ 2 3 ( ¡ 1 ) p + 6 = = 8 3 nlo g 2 ( n 3 ) + 20 9 n ¡ 2 3 ( ¡ 1 ) log2( n3 ) + 6 ; c£n ( 3 £ 2 p ) = 4 p 2 p ¡ 1 1 2 p + 2 ( ¡1 ) p + 1 8 = = 4 3 nlo g 2 ( n 3 ) ¡ 11 3 n + 2 ( ¡ 1 ) log2( n3 ) + 1 8 : ( 1 6 ) r. barseghyan 2 7 fo r q = 9 c+n ( 9 £ 2 p ) = 2 4 p 2 p + 6 8 2 p ¡ 2 ( ¡ 1 ) p + 1 8 = = 8 3 nlo g 2 ³ n 9 ´ ¡ 68 81 n ¡ 2 3 ( ¡ 1 ) log2 ( n 9 ) + 1 8 ; c£n ( 9 £ 2 p ) = 1 2 p 2 p ¡ 2 4 2 p + 1 0 ( ¡ 1 ) p + 5 4 = = 4 3 nlo g 2 ³ n 9 ´ ¡ 8 3 n + 1 0 ( ¡ 1 ) log2 ( n 9 ) + 5 4 : ( 1 7 ) fo r q = 1 5 c+n ( 1 5 £ 2 p ) = 4 0 p 2 p + 4243 2 p ¡ 10 3 ( ¡ 1 ) p + 3 0 = = 8 3 nlo g 2 ³ n 15 ´ ¡ 424 45 n ¡ 10 3 ( ¡ 1 ) log2 ( n 15 ) + 3 0 ; c£n ( 1 5 £ 2 p ) = 2 0 p 2 p ¡ 1093 2 p + 46 3 ( ¡ 1 ) p + 9 0 = = 4 3 nlo g 2 ³ n 15 ´ ¡ 109 45 n ¡ 2 3 ( ¡1 ) log2 ( n 15 ) + 9 0 : ( 1 8 ) fo r c o m p u t in g q = 7 le n g t h d ft, we u s e t h e m e t h o d p r e s e n t e d in [1 3 ]. th a t m e t h o d a llo ws t o c o m p u t e d ft wit h c+7 = 7 2 ; c £ 7 = 1 6 . fo r g e t t in g a fu ll lo g a r it h m ic e xp r e s s io n fo r 7 £ 2 p, we u s e ( 1 2 ) . c+n ( 7 £ 2 p ) = 73 2 3+pp + 67 9 2 3+p ¡ 14 9 ( ¡ 1 ) p + 1 4 = = 8 3 n lo g 2 ³ n 7 ´ + 536 63 n ¡ 14 9 ( ¡ 1 ) log2 ( n 7 ) + 1 4 ; c£n ( 7 £ 2 p ) = 73 2 2+pp ¡ 61 18 2 p + 14 9 ( ¡ 1 ) p + 4 2 = = 4 3 n lo g 2 ³ n 7 ´ ¡ 61 126 n + 14 9 ( ¡ 1 ) log2 ( n 7 ) + 4 2 : ( 1 9 ) in c a s e o f q = 1 1 fr o m [1 3 ], we h a ve c+11 = 1 6 8 ; c £ 11 = 4 0 c+n ( 1 1 £ 2 p ) = 113 2 3+pp + 167 9 2 3+p ¡ 22 9 ( ¡ 1 ) p + 2 2 = = 8 3 nlo g 2 ³ n 11 ´ + 1336 99 n ¡ 22 9 ( ¡ 1 ) log2 ( n 11 ) + 2 2 ; c£n ( 1 1 £ 2 p ) = 113 2 2+pp ¡ 29 9 2 1+p + 22 9 ( ¡ 1 ) p + 6 6 = = 4 3 nlo g 2 ³ n 11 ´ ¡ 58 99 n + 22 9 ( ¡ 1 ) log2 ( n 7 ) + 6 6 : ( 2 0 ) in c a s e o f q = 1 3 fr o m [1 3 ], we h a ve c+13 = 1 8 8 ; c £ 13 = 4 0 c+n ( 1 3 £ 2 p ) = 133 2 3+pp + 371 9 2 2+p ¡ 26 9 ( ¡ 1 ) p + 2 6 = = 8 3 nlo g 2 ³ n 13 ´ + 1484 117 n ¡ 26 9 ( ¡ 1 ) log2 ( n 13 ) + 2 6 ; c£n ( 1 3 £ 2 p ) = 133 2 2+pp ¡ 67 9 2 1+p + 26 9 ( ¡ 1 ) p + 7 8 = 4 3 nlo g 2 ³ n 7 ´ ¡ 134 117 n + 26 9 ( ¡1 ) log2 ( n 7 ) + 7 8 : ( 2 1 ) 2 8 about complexity of fft algorithms for length of q £ 2p in c a s e o f q = 1 7 fr o m [1 3 ], we h a ve c+17 = 2 7 4 ; c £ 13 = 8 2 c+n ( 1 7 £ 2 p ) = 173 2 3+pp + 1097 9 2 1+p ¡ 34 9 ( ¡ 1 ) p + 3 4 = = 8 3 nlo g 2 ³ n 17 ´ + 2194 153 n ¡ 34 9 ( ¡ 1 ) log2 ( n 17 ) + 3 4 ; c£n ( 1 7 £ 2 p ) = 173 2 2+pp + 23 9 2 2+p + 34 9 ( ¡1 ) p + 1 0 2 = = 4 3 nlo g 2 ³ n 17 ´ ¡ 92 153 n + 34 9 ( ¡ 1 ) log2 ( n 17 ) + 1 0 2 : ( 2 2 ) in c a s e o f q = 1 9 fr o m [1 3 ], we h a ve c+19 = 4 0 4 ; c £ 19 = 7 6 c+n ( 1 9 £ 2 p ) = 193 2 3+pp + 833 9 2 2+p ¡ 38 9 ( ¡1 ) p + 3 8 = = 8 3 nlo g 2 ³ n 19 ´ + 3332 171 n ¡ 38 9 ( ¡ 1 ) log2 ( n 19 ) + 3 8 ; c£n ( 1 9 £ 2 p ) = 193 2 2+pp ¡ 19 9 2 1+p + 38 9 ( ¡ 1 ) p + 1 1 4 = = 4 3 nlo g 2 ³ n 19 ´ ¡ 38 171 n + 38 9 ( ¡1 ) log2 ( n 19 ) + 1 1 4 : ( 2 3 ) 4 . 2 / 8 s p lit -r a d ix a lg o r it h m in c a s e o f a = 4 , t h e a lg o r it h m b e c o m e s 2 =8 s p lit -r a d ix a lg o r it h m . x[8 k + l] = n=4¡1x n=0 w nln ( 3x n=0 x0[n + i n 8 ]w il8 ) w nk n=8: ( 2 4 ) th e t o t a l n u m b e r s o f r e a l m u lt ip lic a t io n s a n d r e a l a d d it io n s n e e d e d b y t h e a lg o r it h m a r e c+n = c + n=2 + 4 c+ n=8 + 11 2 n ¡ c+t c£n = c £ n=2 + 4 c£ n=8 + 5 2 n ¡ c£t : ( 2 5 ) b e lo w t h e n u m b e r o f a r it h m e t ic o p e r a t io n s in lo g a r it h m ic fo r m a r e p r e s e n t e d c+n = 11 4 pq 2 p ¡ 55 16 q 2 p ¡ 1 8 2 p ³ c+t ¡ ( 2 c+q + c+2qc+4q ) ´ + 1 4 c+t + +( ¡ 1 ) p 2 p=2 [7 ( 1 2 c+q ¡ 2 ( c+2q + c+4q + c+t ) + 5 5 q ) c o s ( p a r c t a n ( p 7 ) ) + p 7 ( 4 c+q ¡ 2 ( 1 1 c+2q ¡ 5 c+4q ¡ c+t ) ¡ 9 9 q ) s in ( p a r c t a n ( p 7 ) ) ] c£n = 5 4 pq 2 p ¡ 25 16 q 2 p ¡ 1 8 2 p ³ c£t ¡ ( 2 c£q + c£2qc£4q ) ´ + 1 4 c£t + +( ¡ 1 ) p 2 p=2 [7 ( 2 ( c£t ¡ 6 c£q + c£2q + c£4q ) ¡ 2 5 q ) c o s ( p a r c t a n ( p 7 ) ) + p 7 ( c£t + 4 c £ q ¡ 2 2 c£2q + 5 c£4q ¡ 4 5 q ) s in ( p a r c t a n ( p 7 ) ) ]: ( 2 6 ) wh e r e b y cq,c2q a n d c4q d e n o t e n u m b e r o f a r it h m e t ic o p e r a t io n s r e qu ir e d fo r c o m p u t a t io n s o f q, 2 q a n d 4 q le n g t h d fts , r e s p e c t ive ly. fo r c o m p u t in g 2 q a n d 4 q le n g t h d ft, we c a n u s e r. barseghyan 2 9 m e t h o d s d e s c r ib e d in t h e p r e vio u s s e c t io n , wh ic h a llo ws t o r e wr it e ( 2 6 ) c+n = 11 4 pq 2 p ¡ 15 16 q 2 p ¡ 1 8 2 p ³ c+t ¡ 8 c+q ´ + 1 4 c+t + +( ¡ 1 ) p 2 p=2 [7 ( 1 5 q ¡ 2 c+t ) c o s ( p a r c t a n ( p 7 ) ) + p 7 ( 2 c+t ¡ 2 7 q ) s in ( p a r c t a n ( p 7 ) ) ] c£n = 5 4 pq 2 p ¡ 25 16 q 2 p ¡ 1 8 2 p ³ c£t ¡ 8 c£q ´ + 1 4 c£t + +( ¡ 1 ) p 2 p=2 [7 ( 2 5 q ¡ 2 c£t ) c o s ( p a r c t a n ( p 7 ) ) + p 7 ( 2 c£t ¡ 4 5 q ) s in ( p a r c t a n ( p 7 ) ) ]: ( 2 7 ) 4 .1 co m p le xit y o f 2 =8 s p lit -r a d ix a lg o r it h m fo r q < 2 0 in c a s e o f q = 1 5 , we h a ve c+15 = 1 6 8 ; c £ 15 = 3 0 a n d fr o m [7 ] c + t = 1 8 0 ; c £ t = 6 0 a n d t h e r e fo r e c+n = 11 4 pq 2 p ¡ 15 16 q 2 p ¡ 1 8 2 p ³ c+t ¡ 8 c+q ´ + 1 4 c+t + +( ¡ 1 ) p 2 p=2 [7 ( 1 5 q ¡ 2 c+t ) c o s ( p a r c t a n ( p 7 ) ) + p 7 ( 2 c+t ¡ 2 7 q ) s in ( p a r c t a n ( p 7 ) ) ] c£n ( 1 5 £ 2 p ) = 54 p 1 5 £ 2 p ¡ 17 16 1 5 £ 2 p ¡ 15 16 ( ¡1 ) p 2 p=2 ( ) + 4 5 +( ¡ 1 ) p 2 p=2 [7 ( 2 5 q ¡ 2 c£t ) c o s ( p a r c t a n ( p 7 ) ) + p 7 ( 2 c£t ¡ 4 5 q ) s in ( p a r c t a n ( p 7 ) ) ]: ( 2 8 ) 5 . co m p a r is o n th e n u m b e r o f a d d it io n s a n d m u lt ip lic a t io n s r e qu ir e d fo r c o m p u t in g d ft fo r va r io u s le n g t h s a r e p r e s e n t e d in ta b le 1 ( 2 =4 s p lit -r a d ix a lg o r it h m ) . a s a n e xa m p le a r a n g e fr o m 2 5 6 t o 2 0 4 8 is c h o s e n . u s in g o n ly c o n ve n t io n a l a lg o r it h m fo r 2 p we h a ve o n ly c o m p u t e d ft o f 2 5 6 ; 5 1 2 ; 1 0 2 4 ; 2 0 4 8 s iz e s . if s iz e is n o t e qu a l t o t h e s e va lu e s we n e e d t o p a d t h e in p u t d a t a u p t o n e xt 2 p. u s in g q £ 2 p a lg o r it h m a llo ws t o c o ve r t h e r a n g e 2 5 6 ¡ 1 0 2 4 wit h 2 7 n e w p o in t s . th is a p p r o a c h a llo ws t o s ig n ī c a n t ly r e d u c e t h e n u m b e r o f a r it h m e t ic o p e r a t io n s . to ¯ n d o u t t h e q fo r wh ic h t h e a lg o r it h m b e c o m e s t h e m o s t e ± c ie n t in t e r m s o f t h e n u m b e r o f a r it h m e t ic o p e r a t io n s , ¯ r s t o f a ll we c u t t h e va lu e s o f q fo r wh ic h cn1 > cn2, b u t n1 < n2, wh e r e cn = c + n +c £ n . it is e a s y t o s e e t h a t o n ly fo r 1 ; 3 ; 5 ; 9 ; 1 3 ; 1 5 c o n d it io n p r e s e n t e d a b o ve is t r u e . fo r g e t t in g m o r e a c c u r a t e r e s u lt s we c o m p a r e t h e va lu e o f en = cn n . fin a lly we g e t en ( 9 £ 2 p ) < en ( 9 £ 3 p ) < en ( 9 £ 1 5 p ) < en ( 1 £ 2 p ) < < en ( 5 £ 2 p ) < en ( 1 5 £ 2 p ) : 3 0 about complexity of fft algorithms for length of q £ 2p table 1: number of arithmetic operations required by 2=4 split-radix algorithm for the dft length 256 ¡ 1024 n q p add. mul. count 256 1 8 5380 1284 6664 272 17 4 6832 1720 8552 288 9 5 6036 1196 7232 304 19 4 9200 1672 10872 320 5 6 6736 1880 8616 352 11 5 9468 2204 11672 360 45 3 7812 1140 8952 384 3 7 8028 2192 10220 416 13 5 10852 2372 13224 448 7 6 10992 2760 13752 480 15 5 10956 2112 13068 512 1 9 12292 3076 15368 544 17 5 15092 4052 19144 576 9 6 13584 3136 16720 608 19 5 19996 4028 24024 640 5 7 15172 4580 19752 704 11 6 20784 5288 26072 720 45 4 17424 3000 20424 768 3 8 18096 5396 23492 832 13 6 23888 5784 29672 896 7 7 24364 6668 31032 960 15 6 24432 5460 29892 1024 1 10 27652 7172 34824 1088 17 6 33040 9464 42504 1152 9 7 30228 7724 37952 1216 19 6 43184 9576 52760 1280 5 8 33744 10840 44584 1408 11 7 45308 12380 57688 1440 45 5 38628 7620 46248 1536 3 9 40284 12816 53100 1664 13 7 52196 13700 65896 1792 7 8 53488 15688 69176 1920 15 7 53964 13344 67308 2048 1 11 61444 16388 77832 in ta b le 2 a r e p r e s e n t e d t h e c o m p a r is o n r e s u lt s fo r t h e va r io u s le n g t h d fts ( 2 =8 s p lit -r a d ix a lg o r it h m ) . r. barseghyan 3 1 table 2: number of arithmetic operations required by 2=4 split-radix algorithm for the dft length 256 ¡ 2048 n q p add. mul. count 256 1 8 5380 1284 6664 360 45 3 8048 1360 9408 480 15 5 11256 2520 13776 512 1 9 12292 3076 15368 720 45 4 18076 3620 21696 960 15 6 25212 6180 31392 1024 1 10 27652 7172 34824 1440 45 5 39696 8640 48336 1920 15 7 55824 14880 70704 2048 1 11 61444 16388 77832 6 . co n c lu s io n in c a s e o f lo o kin g fo r c o m p u t a t io n a lly e ± c ie n t a lg o r it h m in t e r m s o f n u m b e r o f m u lt ip lic a t io n s in g e n e r a l c a s e we n e e d t o c h o o s e 2 =8 s p lit -r a d ix fft a lg o r it h m , b e c a u s e c o e ± c ie n t o f n lo g 2 is s m a lle r . e ± c ie n c y o f a lg o r it h m s in t e r m s o f t o t a l n u m b e r o f a r it h m e t ic o p e r a t io n s is d is c u s s e d b e lo w. th e t o t a l n u m b e r o f a r it h m e t ic o p e r a t io n s r e qu ir e d b y 2 = 4 s p lit -r a d ix a lg o r it h m c a n b e c o m p u t e d u s in g ( 1 2 ) a n d is p r e s e n t e d b e lo w cn ( 2 =4 ) = 4 pq 2 p ¡ 2 p ( 6 q ¡ c+q ) + 8 q: ( 2 9 ) th e t o t a l n u m b e r o f a r it h m e t ic o p e r a t io n s r e qu ir e d fo r c o m p u t a t io n 2 =8 s p lit -r a d ix a lg o r it h m c a n b e r e t r ie ve d fr o m ( 2 8 ) cn ( 2 =8 ) = 4 pq 2 p ¡ 2 p ( 5 2 q ¡ ( 1 8 ct ¡ cq ) ) + 8 q+ +2 ( ¡ 1 ) p 2 p=2 £ [7 ( 2 0 q ¡ ct ) c o s ( ® ) + + p 7 ( ct ¡ 3 6 q ) s in ( ® ) :] ( 3 0 ) fo r g e t t in g a c o m p u t a t io n a lly e ± c ie n t a lg o r it h m , we n e e d t o c a lc u la t e cn ( 2 =4 ) ¡ cn ( 2 = 8 ) u s in g ( 2 9 ) a n d ( 3 0 ) . fo r s im p lic it y, o n ly t h e c o e ± c ie n t s fo r 2 p a n d q £ 2 p a r e in c lu d e d cn ( 2 =4 ) ¡ cn ( 2 =8 ) = ( 72 q ¡ 1 8 ct ) 2 p < 0 ct > 2 8 q: in o t h e r wo r d s we c a n s a y t h a t if ct is g r e a t e r t h a n 2 8 q, 2 =4 s p lit -r a d ix a lg o r it h m is m o r e e ± c ie n t c o m p a r e d t o cn ( 2 =8 ) . refer ences [1 ] e . o. b r ig h a m , the f ast f ourier applications, e n g le wo o d cli®s , n j, p r e n t ic e -h a ll, 1 9 8 8 . [2 ] j. k . e r s o y, f ourier-r elated transforms. f ast algorithms and applications, e n g le wo o d cli®s , n j, p r e n t ic e -h a ll,1 9 9 7 . 3 2 about complexity of fft algorithms for length of q £ 2p [3 ] j. w . co o le y a n d j. w . tu ke y, \ a n a lg o r it h m fo r t h e m a c h in e c o m p u t a t io n o f t h e c o m p le x fo u r ie r s e r ie s ," m ath. computation, p p . 2 9 7 { 3 0 1 , 1 9 6 5 . [4 ] r . y a vn e , \ a n e c o n o m ic a l m e t h o d fo r c a lc u la t in g t h e d is c r e t e fo u r ie r t r a n s fo r m ," in p roc. af ip s, vo l. 3 3 , p p . 1 1 5 { 1 2 5 , 1 9 6 8 . [5 ] m. fr ig o a n d s . g. jo h n s o n , \ a m o d i¯ e d s p lit -r a d ix fft wit h fe we r a r it h m e t ic o p e r a t io n s " , ieee trans. signal p rocessing, vo l. 5 5 , p p . 1 1 1 -1 1 9 , 2 2 0 7 . [6 ] g. b i a n d y . q. ch e n , \ fa s t d ft a lg o r it h m fo r le n g t h n = q £ 2 p," ie e e trans. circuits and systems ii, vo l. 4 5 , n o . 6 , p p . 6 8 5 6 9 0 , 1 9 9 8 . [7 ] g. b i, g. l i a n d x . l i, \ a u n i¯ e d e xp r e s s io n fo r s p lit -r a d ix d ft a lg o r it h m s ," p p . 3 2 3 3 2 6 , 2 0 1 0 . [8 ] p . d u h a m e l, \ im p le m e n t a t io n o f s p lit -r a d ix fft a lg o r it h m s fo r c o m p le x, r e a l, a n d r e a ls ym m e t r ic d a t a " , ie e e trans. acoust., speech, signal p rocess., vo l. 3 4 , p p . 2 8 5 2 9 5 , a p r il, 1 9 8 6 . [9 ] h . v . s o r e n s e n , m. t. h e id e m a n a n d c. s . b u r r u s , " on c o m p u t in g t h e s p lit -r a d ix fft," ie e e tr a n s . a c o u s t ., s p e e c h , s ig n a l p r o c e s s ., vo l. 3 4 ,p p . 1 5 2 1 5 6 , fe b . 1 9 8 6 . [1 0 ] s . b o u g u e z e l, m. om a ir a n d m. n . s . s wa m y, \ a n e w r a d ix-2 / 8 fft a lg o r it h m fo r le n g t h -q £ 2 m d fts ," ie e e trans. circuits and systems i, vo l. 5 1 , n o . 1 , p p . 1 7 2 3 1 7 3 2 , 2 0 0 4 . [1 1 ] s . b o u g u e z e l, m. om a ir a n d m. n . s . s wa m y, \ a g e n e r a l c la s s o f s p lit -r a d ix fft a lg o r it h m s fo r t h e c o m p u t a t io n o f t h e d ft o f le n g t h -2 m," ie e e trans. signal p rocessing, vo l. 5 5 , n o . 8 , p p . 4 1 2 7 4 1 3 8 , 2 0 0 7 . [1 2 ] on lin e : [a va ila b le ] h t t p :/ / m a xim a .s o u r c e fo r g e .n e t [1 3 ] i. s e le s n ic k a n d s . b u r r u s , \ p r o g r a m s fo r p r im e l e n g t h ffts ," h t t p :/ / c n x.o r g / c o n t e n t / m 1 8 1 3 7 / 1 .5 / . [1 4 ] r . b a r s e g h ya n , \ co m p le xit y o f t h e c o m p o s it e le n g t h fft a lg o r it h m s " , p roceedings of international conference csit 2015, y e r e va n , a r m e n ia , 2 0 1 5 . submitted 22.08.2017, accepted 06.12.2017. q £ 2 p »ñï³ñáõãû³ý ü²¼-µ³ñ¹áõãû³ý ù³ëçý è. ´³ñë»õû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ëï³óí³í ¿ éá·³ñçãù³ï³ý µ³ý³ó¨á, áñá ï³ù³û³ï³ý q £ 2 p-»ñï³ñáõãû³ý í»ïïáñý»ñç ñ³ù³ñ ãáõûé ¿ ï³éçë ñ³ßí³ñï»é üáõñû»ç ¹çëïñ»ï ó¨³÷áëáõãû³ý (ü²ò) ñ³ù³ñ ³ýññ³å»ßï ·áñíáõáõãûáõýý»ñç ×ß·ñçï ù³ý³ïá, áñï»õ q-ý ï»ýï ãçí ¿: î ñëîæíîñòè àëãîðèòìîâ áïô äëÿ äëèíû q £ 2 p ð. áàðñåãÿí àííîòàöèÿ â ýòîé ñòàòüå âûâåäåíà ëîãàðèôìè÷åñêàÿ ôîðìóëà, êîòîðàÿ ïîçâîëÿåò âû÷èñëèòü òî÷íîå êîëè÷åñòâî íåîáõîäèìûõ îïåðàöèé äëÿ âû÷èñëåíèÿ äèñêðåòíîãî ïðåîáðàçîâàíèÿ ôóðüå (dft) äëÿ âåêòîðîâ äëèíû q £ 2 p, ãäå q íå÷åòíîå öåëîå ÷èñëî. microsoft word robert.doc математические вопросы кибернетики и вычислительной техники 30, 92--104, 2008. 92 построение ассоциативных правил путем цепного раздробления n-мерного единичного куба цепями левон асланян, роберт хачатрян институт проблем информатики и автоматизации нан ра lasl@sci.am, robert@simartek.am аннотация в работе решена задача поиска ассотиативных правил и приведен альтернативный алгоритму apriori метод решения этой задачи, путем цепного раздробления n-мерного куба, по технике анселя. описаны инстументы для работы над цепями, выделенные из результатов тонояна. приведено краткое описание программной реализации альтернативного подхода. литература [1] коробков б. к., “о монотонных функциях алгебры логики”, сб. ‘проблемы кибернетики’, вып. 13, м., ‘наука’, стр. 5-28, 1965. [2] ансель ж., “о числе монотонных булевых функций n переменных”, ‘кибернетический сборник’, новая серия, вып. 5, м., ‘мир’, стр. 53-5, 1968. [3] тоноян г. п., “разбиение вершин n-мерного единичного куба на цепи и расшифровка монотонных булевых функций”, журнал вычислительной математики и математической физики, том. 19, n% 6, стр. 1532-1542, 1976. [4] kotsiantis s. and kanellopoulos d., “association rules mining: a recent overview”, gests international transactions on computer science and engineering, vol. 32 (1), pp. 71-82, 2006. л. асланян, р. хачатрян 93 ²ëáóç³ïçí ï³ýáýý»ñç ï³éáõóáõù` n-ã³÷³ýç ùç³íáñ ëáñ³ý³ñ¹á ßõã³ý»ñç ïñáñ»éáõ »õ³ý³ïáí è. ²ëé³ýû³ý, è.ê³ã³ïñû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù éáõíí³í ¿ ³ëáóç³ïçí ï³ýáýý»ñç ÷ýïñù³ý ëý¹çñá ¨ µ»ñí³í ¿ ñ³ûïýç apriori ³é·áñçãùç ³ûéáýïñ³ýù³ûçý ï³ñµ»ñ³ïá ³û¹ ëý¹ñç éáõíù³ý ñ³ù³ñ: ²ûý ñçùýí³í ¿ n-ã³÷³ýç ùç³íáñ ëáñ³ý³ñ¹á ßõã³ý»ñç ïñáñ»éáõ íñ³, ²ýë»éç ïáõùçó ³é³ç³ñïí³í »õ³ý³ïáí: üï³ñ³·ñí³í »ý ·áñíçùý»ñ ßõã³ý»ñç ñ»ï ³ßë³ï»éáõ ñ³ù³ñ, áñáýù ¹áõñë »ý µ»ñí»é îáýáû³ýç ïáõùçó ëï³óí³í ³ñ¹ûáõýùý»ñçó: ´»ñí³í ¿ íñ³·ñ³ûçý ñ³ù³ï³ñ·ç ï³ñ× ýï³ñ³·çñ, áñï»õ çñ³ï³ý³óí³í ¿ ëý¹ñç ³ûéáýïñ³ýù³ûçý éáõíù³ý ï³ñµ»ñ³ïá: microsoft word crom.doc ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 30, 47--53, 2008. 47 создание контрольных точек и восстановление mpi программ мгер ю. мовсисян ереванский физический институт им. а.и.алиханяна mher.movsisyan@gmail.com àííîòàöèÿ выполнение программ на вычислительных кластерах обычно занимает довольно большое время. в процессе выполнения может возникнуть потребность изменения физического местоположения отдельных процессов параллельной программы или временная остановка всей программы. в этой статье описана разработанная система crom (checkpointing and recovery of mpi), которая предоставляет возможность создания контрольных точек для остановки и последующего возобновления выполнения mpi программы. функциональность создания контрольных точек и восстановления реализована в виде дополнительных компонент mpich2 и не требует изменений в коде mpi программы. ëèòåðàòóðà [1] m. movsisyan, v. sahakyan, “transparent checkpointing protocol for mpi programs with decentralized initiator”, csit 2007, pp. 227-229. [2] message passing interface forum, “mpi: a message-passing interface standard”, version 1.1, june 1995. http://www.mpi-forum.org/docs/docs.html [3] message passing interface forum, “mpi-2: extensions to the message-passing interface”, july 1997, http://www.mpi-forum.org/docs/docs.html [4] mpich2, http://www-unix.mcs.anl.gov/mpi/mpich2/ [5] open mpi, http://www.open-mpi.org/ [6] m. chandy and l. lamport, “distributed snapshots: determining global states of distributed systems”, in acm transactions on computing systems, 3(1): pp. 63-75, 1985. [7] myrinet, http://www.myri.com/myrinet/overview/ [8] the mpich team argonne national laboratory, “process management in mpich2” draft 2.1. march 30, 2007. [9] berkeley lab checkpoint/restart (blcr), http://ftg.lbl.gov/checkpointrestart/checkpointrestart.shtml [10] h. hargrove and c. duell, “berkeley lab checkpoint/restart (blcr) for linux clusters”, in proceedings of scidac 2006: june 2006. создание контрольных точек и восстановление mpi программ 48 [11] j. duell, p. hargrove, and e. roman, “the design and implementation of berkeley lab’s linux checkpoint/restart”, technical report lbnl-54941, lawrence berkeley national laboratory, 2003. [12] m. elnozahy, l. alvisi, y. m. wang, and d. b. johnson, “a survey of rollback-recovery protocols in message passing systems”, technical report cmu-cs-96-181, school of computer science, carnegie mellon university, pittsburgh, pa, usa, 1996. êïáõ·ù³ý ï»ï»ñç ëï»õíáõùá ¨ í»ñ³ï³ý·ýáõùá mpi íñ³·ñ»ñáõù ø. øáíëçëû³ý ²ù÷á÷áõù ð³ßíáõ³ï³ý ïé³ëï»ñý»ñáõù íñ³·ñ»ñç ï³ï³ñáõùá ëáíáñ³µ³ñ ï¨áõù ¿ µ³í³ï³ýçý »ñï³ñ£ ìñ³·ñ»ñç ï³ï³ñù³ý å³ù³ý³ï ï³ñáõ »ý í³·»é ½áõ·³ñ»é íñ³·ñç áýã³óùý»ñç ýç½çï³ï³ý ¹çñùç ÷á÷áëáõãû³ý ï³ù ³ùµáõç íñ³·ñç ï³ï³ñù³ý å³ù³ý³ï³íáñ ¹³¹³ñ»óù³ý ³ýññ³å»ßïáõãûáõýý»ñ£ ²ûë ñá¹í³íáõù ýï³ñ³·ñíáõù ¿ ñ»õçý³ïç ïáõùçó ùß³ïí³í crom (checkpointing and recovery of mpi) ñ³ù³ï³ñ·á, áñá ñý³ñ³íáñáõãûáõý ¿ áýó»éáõù å³ñå³ý»é ï³ï³ñíáõ mpi íñ³·çñá ¨ í»ñ³ãáõ³ñï»é£ ä³ñå³ýù³ý ¨ í»ñ³ãáõ³ïù³ý ýáõýïóçáý³éáõãûáõýá çñ³·áñíí³í ¿ mpich2 µ³õ³¹ñçãý»ñç ï»ëùáí ¨ ãç å³ñ³ýçáõù mpi íñ³·ñç ïá¹ç ÷á÷áëáõãûáõý£ mathematical problems of computer science 45, 67--76, 2016. on transitive closures of two-dimensional strongly positive arithmetical sets1 seda n. manukian institute for informatics and automation problems of nas ra e-mail: zaslav@ipia.sci.am abstract the notions of positive and strongly positive arithmetical set are considered in ([1]-[3]). it is noted in [3] that the transitive closure of any 2-dimensional strongly positive set is primitive recursive. in this article a more strong statement is proved: the transitive closure of any 2-dimensional strongly positive set is defined by an arithmetical formula in the signature (0, =, <, 𝑆𝑆), where 𝑆𝑆(𝑥𝑥) = 𝑥𝑥 + 1. besides, it is proved that the class of two-dimensional strongly positive sets and the class of transitive closures of such sets do not coincide with the class of two-dimensional arithmetical sets expressible by the formulas in the signature (0, = , <, 𝑆𝑆). keywords: positive, strongly positive, arithmetical set, dimension, signature. 1. introduction the notion of strongly positive arithmetical set is defined and investigated in [3]. it is proved in [3] that for any 𝑛𝑛 ≥ 3 there exists a 2𝑛𝑛-dimensional strongly positive set such that its transitive closure is not recursive. it is noted in [3] (without a proof) that the transitive closure of any 2dimensional strongly positive set is primitive recursive. below a stronger statement is proved: the transitive closure of any 2-dimensional strongly positive set can be defined by an arithmetical formula in the signature (0, =, <, 𝑆𝑆), where 𝑆𝑆(𝑥𝑥) = 𝑥𝑥 + 1(see below, theorem 1). it is proved also that the class of the mentioned transitive closures does not coincide with the class of sets expressible by arithmetical formulas in the signature (0, =, <, 𝑆𝑆). for example, it is proved that 1 this work was supported by state committee of science, mess ra in frame of the research project scs 15t1b238. 67 mailto:zaslav@ipia.sci.am 68 on transitive closures of two-dimensional strongly positive arithmetical sets the set {(𝑥𝑥, 𝑦𝑦)/𝑦𝑦 = 𝑥𝑥 + 2} is not strongly positive and cannot be represented as the transitive closure of some strongly positive set (see below, theorem 2). 2. main definitions and results by 𝑁𝑁 we denote the set of all non-negative integers, 𝑁𝑁 = {0,1,2, … }. by 𝑁𝑁𝑛𝑛, where 𝑛𝑛 ≥ 1, we denote the set of 𝑛𝑛-tuples (𝑥𝑥1, 𝑥𝑥2, … 𝑥𝑥𝑛𝑛), where 𝑥𝑥𝑖𝑖 ∈ 𝑁𝑁 for 1 ≤ 𝑖𝑖 ≤ 𝑛𝑛. an 𝑛𝑛-dimensional arithmetical set, where 𝑛𝑛 ≥ 1, is defined as any subset of 𝑁𝑁𝑛𝑛. an 𝑛𝑛-dimensional arithmetical predicate is defined as a predicate 𝑃𝑃, which is true on some set 𝐴𝐴 ⊆ 𝑁𝑁𝑛𝑛 and false on the set 𝑁𝑁𝑛𝑛\𝐴𝐴. if the mentioned relation between 𝐴𝐴 and 𝑃𝑃 takes place, then we say that 𝑃𝑃 is the representing predicate for 𝐴𝐴, and 𝐴𝐴 is the set of truth for 𝑃𝑃. the notions of primitive recursive set and recursive set are defined in a usual way (see [4][6]). the notion of arithmetical formula on a given signature (on the base of logical operations &,∨, ⊃, ¬, ∀, ∃) is defined in a usual way, ([2]-[6]). we will consider arithmetical formulas in the signatures (0, =, 𝑆𝑆) and (0, =, <, 𝑆𝑆), where 𝑆𝑆(𝑥𝑥) = 𝑥𝑥 + 1 for 𝑥𝑥 ∈ 𝑁𝑁. the deductive systems in the signatures (0, =, 𝑆𝑆) and (0, =, <, 𝑆𝑆) are defined as in [6]; we will denote these deductive systems correspondingly by deds and dedl. as it is proved in [6], these deductive systems are complete. we say that the formulas 𝐹𝐹 and 𝐺𝐺 in the corresponding signatures are equivalent if the formula (𝐹𝐹 ⊃ 𝐺𝐺)&(𝐺𝐺 ⊃ 𝐹𝐹) is deducible in the corresponding deductive system. we will consider the formulas in the mentioned signatures up to their equivalence. the relation “𝑛𝑛-dimesional arithmetical set 𝐴𝐴 is defined by an arithmetical formula 𝐹𝐹” is given in a usual way (see [2]-[6]) (in [2] this relation is called as follows: “𝑘𝑘-dimensional arithmetical set 𝐴𝐴 is represented (or representable) by a formula 𝐹𝐹"). the notion of transitive closure 𝐴𝐴∗ for an arithmetical set 𝐴𝐴 having an even dimension 2𝑘𝑘 (where 𝑘𝑘 ≥ 1) is defined in a usual way (see, for example, [3], [8]). let us recall that the following statement holds (see [3], lemma 3.4, and [8], p.72): if 𝐴𝐴 is a 2𝑘𝑘-dimensional set, 𝐴𝐴 ⊆ 𝑁𝑁2𝑘𝑘, where 𝑘𝑘 ≥ 1, then (𝑥𝑥1, 𝑥𝑥2, … , 𝑥𝑥𝑘𝑘, 𝑦𝑦1, 𝑦𝑦2, … , 𝑦𝑦𝑘𝑘) ∈ 𝐴𝐴∗ if and only if there exists a sequence (𝑄𝑄1, 𝑄𝑄2, … , 𝑄𝑄𝑚𝑚) of 𝑘𝑘-tuples such that 𝑚𝑚 ≥ 2, 𝑄𝑄1 = (𝑥𝑥1, 𝑥𝑥2, … , 𝑥𝑥𝑘𝑘), 𝑄𝑄𝑚𝑚 = (𝑦𝑦1, 𝑦𝑦2, … , 𝑦𝑦𝑘𝑘), (𝑄𝑄𝑖𝑖, 𝑄𝑄𝑖𝑖+1) ∈ 𝐴𝐴 for 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚 − 1. below we will say that the sequence (𝑄𝑄1, 𝑄𝑄2, … , 𝑄𝑄𝑚𝑚), having the mentioned properties is a sequence establishing the value (𝑄𝑄1, 𝑄𝑄𝑚𝑚) ∈ 𝐴𝐴∗ of the transitive closure 𝐴𝐴∗ (or, shortly, etc-sequence). in what follows we will consider etc-secuences only for the case 𝑘𝑘 = 1. the notion of strongly positive arithmetical set is defined as in [3]. let us recall that an 𝑚𝑚-dimensional arithmetical set, where 𝑚𝑚 ≥ 1, is said to be strongly positive if it is defined by an arithmetical formula 𝐹𝐹 which is constructed by logical operators & and ∨ from subformulas having the forms (𝑥𝑥 = 𝑎𝑎) (where 𝑎𝑎 is a constant, 𝑎𝑎 ∈ 𝑁𝑁), 𝑥𝑥 = 𝑦𝑦, 𝑦𝑦 = 𝑆𝑆(𝑥𝑥), ¬(𝑥𝑥 = 0), where 𝑥𝑥 and 𝑦𝑦 are variables. s. manukian 69 theorem 1: the transitive closure of any 2-dimensional strongly positive set can be defined by an arithmetical formula in the signature (0, =, <, 𝑆𝑆). theorem 2: the set {(𝑥𝑥, 𝑦𝑦)/ 𝑦𝑦 = 𝑆𝑆(𝑆𝑆(𝑥𝑥))} is not strongly positive and cannot be represented as a transitive closure of some strongly positive set. 3. proofs of theorems we consider the properties of 2-dimensional strongly positive sets. let 𝜋𝜋 be any set of such kind. by 𝜂𝜂 we denote the representing predicate for 𝜋𝜋. using the definition of strongly positive set we conclude that the predicate 𝜂𝜂 can be expressed by an arithmetical formula 𝐹𝐹 having the form 𝐹𝐹1 ∨ 𝐹𝐹2 ∨ … ∨ 𝐹𝐹𝑚𝑚, where any 𝐹𝐹𝑖𝑖 is the conjuction of subformulas having the following forms: 𝑥𝑥 = 𝑎𝑎, 𝑦𝑦 = 𝑏𝑏, (where 𝑎𝑎 and 𝑏𝑏 are constants, 𝑎𝑎 ∈ 𝑁𝑁, 𝑏𝑏 ∈ 𝑁𝑁), 𝑥𝑥 = 𝑦𝑦, 𝑦𝑦 = 𝑆𝑆(𝑥𝑥), 𝑥𝑥 = 𝑆𝑆(𝑦𝑦), ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0). the predicate expressed by the formula 𝐹𝐹𝑖𝑖 we denote by 𝜂𝜂𝑖𝑖; the set of truth for 𝜂𝜂𝑖𝑖 we denote by 𝜋𝜋𝑖𝑖. the following equalities hold: 𝜂𝜂(𝑥𝑥, 𝑦𝑦) ≡ 𝜂𝜂1(𝑥𝑥, 𝑦𝑦) ∨ 𝜂𝜂2(𝑥𝑥, 𝑦𝑦) ∨ … ∨ 𝜂𝜂𝑚𝑚(𝑥𝑥, 𝑦𝑦); 𝜋𝜋 = 𝜋𝜋1 ∪ 𝜋𝜋2 ∪ … ∪ 𝜋𝜋𝑚𝑚. clearly, all the predicates 𝜂𝜂, 𝜂𝜂1, 𝜂𝜂2, … , 𝜂𝜂𝑚𝑚, and all the sets 𝜋𝜋, 𝜋𝜋1, 𝜋𝜋2, … , 𝜋𝜋𝑚𝑚 are expressible by arithmetical formulas in the signature (0, =, 𝑆𝑆). let us note that if some 𝐹𝐹𝑖𝑖 includes simultaneously some two subformulas of the forms 𝑥𝑥 = 𝑦𝑦, 𝑦𝑦 = 𝑆𝑆(𝑥𝑥), 𝑥𝑥 = 𝑆𝑆(𝑦𝑦), then the corresponding predicate 𝜂𝜂𝑖𝑖 is identically false, hence, 𝐹𝐹𝑖𝑖 can be deleted from the structure of 𝐹𝐹, similarly, 𝐹𝐹𝑖𝑖 can be deleted from the structure of 𝐹𝐹 if it includes subformulas of the forms 𝑥𝑥 = 𝑎𝑎1 and 𝑥𝑥 = 𝑎𝑎2 where 𝑎𝑎1 ≠ 𝑎𝑎2 or subformulas of the forms 𝑦𝑦 = 𝑏𝑏1 and 𝑦𝑦 = 𝑏𝑏2 where 𝑏𝑏1 ≠ 𝑏𝑏2. let us consider possible forms of the formula 𝐹𝐹𝑖𝑖. we do not consider the cases mentioned above when 𝐹𝐹𝑖𝑖 can be deleted from the structure of 𝐹𝐹. (case 1). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑥𝑥 = 𝑎𝑎, 𝑦𝑦 = 𝑆𝑆(𝑥𝑥), and, possibly, ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0). in this case 𝜋𝜋𝑖𝑖 is either empty, or contains the single pair (𝑎𝑎, 𝑎𝑎 + 1) (for example, 𝜋𝜋𝑖𝑖 is empty if 𝐹𝐹𝑖𝑖 has the form (𝑥𝑥 = 0 & 𝑦𝑦 = 𝑆𝑆(𝑥𝑥)& ¬(𝑥𝑥 = 0))). (case 2). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑦𝑦 = 𝑏𝑏, 𝑦𝑦 = 𝑆𝑆(𝑥𝑥), and, possibly, ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0). in this case 𝜋𝜋𝑖𝑖 is either empty or contains the single pair (𝑏𝑏 − 1, 𝑏𝑏), where 𝑏𝑏 > 0. (case 3). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑦𝑦 = 𝑆𝑆(𝑥𝑥), and, possibly, ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0) (we suppose that 𝐹𝐹𝑖𝑖 contains no subformula having one of the forms 𝑥𝑥 = 𝑎𝑎, 𝑦𝑦 = 𝑏𝑏, 𝑥𝑥 = 𝑦𝑦, 𝑥𝑥 = 𝑆𝑆(𝑦𝑦)). in this case all pairs of numbers having the form (𝑥𝑥, 𝑥𝑥 + 1), where 𝑥𝑥 > 0, belong to 𝜋𝜋𝑖𝑖. the statement (0,1) ∈ 𝜋𝜋𝑖𝑖 is true if and only if the subformula ¬(𝑥𝑥 = 0) is not included in 𝐹𝐹𝑖𝑖. (case 4). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑦𝑦 = 𝑏𝑏, 𝑥𝑥 = 𝑆𝑆(𝑦𝑦), and, possibly, ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0). in this case 𝜋𝜋𝑖𝑖 is either empty, or contains the single pair (𝑏𝑏 + 1, 𝑏𝑏). (case 5). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑥𝑥 = 𝑎𝑎, 𝑥𝑥 = 𝑆𝑆(𝑦𝑦), and, possibly, ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0). in this case 𝜋𝜋𝑖𝑖 is either empty, or contains the single pair (𝑎𝑎, 𝑎𝑎 − 1), where 𝑎𝑎 > 0. (case 6). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑥𝑥 = 𝑆𝑆(𝑦𝑦), and, possibly, ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0) (we suppose that 𝐹𝐹𝑖𝑖 contains no subformulas having one of the forms 𝑥𝑥 = 𝑎𝑎, 𝑦𝑦 = 𝑏𝑏, 𝑥𝑥 = 𝑦𝑦, 𝑦𝑦 = 𝑆𝑆(𝑥𝑥)). 70 on transitive closures of two-dimensional strongly positive arithmetical sets in this case all pairs of numbers having the form (𝑥𝑥 + 1, 𝑥𝑥), where 𝑥𝑥 > 0, belong to 𝜋𝜋𝑖𝑖. the statement (1,0) ∈ 𝜋𝜋𝑖𝑖 is true if and only if the subformula ¬(𝑦𝑦 = 0) is not included in 𝐹𝐹𝑖𝑖. (case 7). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑥𝑥 = 𝑎𝑎, 𝑦𝑦 = 𝑏𝑏, and, possibly, 𝑥𝑥 = 𝑦𝑦 ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0). in this case 𝜋𝜋𝑖𝑖 is either empty, or contains the single pair (𝑎𝑎, 𝑏𝑏). (case 8). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑥𝑥 = 𝑎𝑎, 𝑥𝑥 = 𝑦𝑦, and, possibly, ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0). in this case 𝜋𝜋𝑖𝑖 is either empty, or contains the single pair (𝑎𝑎, 𝑎𝑎). (case 9). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑦𝑦 = 𝑏𝑏, 𝑥𝑥 = 𝑦𝑦, and, possibly, ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0). in this case 𝜋𝜋𝑖𝑖 is either empty, or contains the single pair (𝑏𝑏, 𝑏𝑏). (case 10). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑥𝑥 = 𝑎𝑎, and, possibly, ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0) (we suppose that 𝐹𝐹𝑖𝑖 contains no subformulas having one of the forms, 𝑦𝑦 = 𝑏𝑏, 𝑥𝑥 = 𝑦𝑦, 𝑦𝑦 = 𝑆𝑆(𝑥𝑥), 𝑥𝑥 = 𝑆𝑆(𝑦𝑦)). in this case 𝜋𝜋𝑖𝑖 is empty when 𝑎𝑎 = 0 and the subformula ¬(𝑥𝑥 = 0) is included in 𝐹𝐹𝑖𝑖. in the opposite case 𝜋𝜋𝑖𝑖 contains all pairs (𝑎𝑎, 𝑦𝑦), where y> 0. the statement (𝑎𝑎, 0) ∈ 𝜋𝜋𝑖𝑖 is true (for 𝑎𝑎 > 0) if and only if the subformula ¬(𝑦𝑦 = 0) is not included in 𝐹𝐹𝑖𝑖. (case 11). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑦𝑦 = 𝑏𝑏, and, possibly, ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0) (we suppose that 𝐹𝐹𝑖𝑖 contains no subformulas having one of the forms, x= 𝑎𝑎, 𝑥𝑥 = 𝑦𝑦, 𝑦𝑦 = 𝑆𝑆(𝑥𝑥), 𝑥𝑥 = 𝑆𝑆(𝑦𝑦)). in this case 𝜋𝜋𝑖𝑖 is empty when 𝑏𝑏 = 0, and the subformula ¬(𝑦𝑦 = 0) is included in 𝐹𝐹𝑖𝑖. in the opposite case 𝜋𝜋𝑖𝑖 contains all pairs (𝑥𝑥, 𝑏𝑏), where x> 0. the statement (0, 𝑏𝑏) ∈ 𝜋𝜋𝑖𝑖 is true (for 𝑏𝑏 ≠ 0) if and only if the subformula ¬(𝑥𝑥 = 0) is not included in 𝐹𝐹𝑖𝑖. (case 12). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑥𝑥 = 𝑦𝑦, and, possibly, ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0) (we suppose that 𝐹𝐹𝑖𝑖 contains no subformulas having one of the forms, x= 𝑎𝑎, y = 𝑏𝑏, 𝑦𝑦 = 𝑆𝑆(𝑥𝑥), 𝑥𝑥 = 𝑆𝑆(𝑦𝑦)). in this case 𝜋𝜋𝑖𝑖 contains all the pairs having the form (𝑥𝑥, 𝑥𝑥), where x > 0. the statement (0,0) ∈ 𝜋𝜋𝑖𝑖 is true if and only if the subformulas ¬(𝑥𝑥 = 0) and ¬(𝑦𝑦 = 0) are not included in 𝐹𝐹𝑖𝑖. it is easily seen that all the variants of the structure of 𝜋𝜋𝑖𝑖 are exhausted in the cases 1-12. now we will consider the variants of the structure of 𝜋𝜋∗. as it is proved in [3] (see [3], lemma 3.4) the statement (𝑥𝑥, 𝑦𝑦) ∈ 𝜋𝜋∗is true if and only if there exists an etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) such that 𝑞𝑞1 = 𝑥𝑥, 𝑞𝑞𝑟𝑟 = 𝑦𝑦, (𝑞𝑞𝑖𝑖, 𝑞𝑞𝑖𝑖+1) ∈ 𝜋𝜋 for 1 ≤ 𝑖𝑖 < 𝑟𝑟. without loss of generality we may suppose that any considered etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) where 𝑟𝑟 ≥ 3, satisfies the condition 𝑞𝑞𝑖𝑖 ≠ 𝑞𝑞𝑗𝑗 when 𝑖𝑖 ≠ 𝑗𝑗 (otherwise the given etc-sequence may be replaced by a shorter sequence having the same properties). let us consider the number 𝑑𝑑 such that 𝑑𝑑 = 𝑑𝑑1 + 1, where 𝑑𝑑1 is the maximum of the numbers 𝑎𝑎 and 𝑏𝑏 in the formulas 𝑥𝑥 = 𝑎𝑎 and 𝑦𝑦 = 𝑏𝑏 included in 𝐹𝐹. if no formula of such forms is included in 𝐹𝐹, then we admit 𝑑𝑑 = 3. we will use below some classification of pairs (𝑥𝑥, 𝑦𝑦) such that 𝑥𝑥 ∈ 𝑁𝑁, 𝑦𝑦 ∈ 𝑁𝑁. we say that (𝑥𝑥, 𝑦𝑦) belongs to the subset 𝑆𝑆1 if 𝑥𝑥 ≤ 𝑑𝑑, 𝑦𝑦 ≤ 𝑑𝑑. in a similar way we define the subsets 𝑆𝑆2, 𝑆𝑆3, 𝑆𝑆4 as sets of pairs (𝑥𝑥, 𝑦𝑦) such that (𝑥𝑥, 𝑦𝑦) ∈ 𝑆𝑆2 if 𝑥𝑥 > 𝑑𝑑, 𝑦𝑦 ≤ 𝑑𝑑; (𝑥𝑥, 𝑦𝑦) ∈ 𝑆𝑆3 if 𝑥𝑥 ≤ 𝑑𝑑, 𝑦𝑦 > 𝑑𝑑; (𝑥𝑥, 𝑦𝑦) ∈ 𝑆𝑆4 if 𝑥𝑥 > 𝑑𝑑, 𝑦𝑦 > 𝑑𝑑. the sets 𝑆𝑆1∗, 𝑆𝑆2∗, 𝑆𝑆3∗, 𝑆𝑆4∗ are defined coorrespondingly as 𝑆𝑆1 ∩ 𝜋𝜋∗, 𝑆𝑆2 ∩ 𝜋𝜋∗, 𝑆𝑆3 ∩ 𝜋𝜋∗, 𝑆𝑆4 ∩ 𝜋𝜋∗. a pair (𝑥𝑥, 𝑦𝑦) is said to be increasing if 𝑥𝑥 < 𝑦𝑦 and decreasing if 𝑥𝑥 > 𝑦𝑦. lemma 3.1: if the number ℎ satisfies the condition ℎ ≥ 𝑑𝑑, and the pair (𝑥𝑥, 𝑦𝑦) ∈ 𝜋𝜋∗ satisfies the conditions 𝑥𝑥 ≤ ℎ, 𝑦𝑦 ≤ ℎ, then there exists an etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) for the value (𝑥𝑥, 𝑦𝑦) ∈ 𝜋𝜋∗ such that 𝑞𝑞1 = 𝑥𝑥, 𝑞𝑞𝑟𝑟 = 𝑦𝑦, (𝑞𝑞𝑖𝑖, 𝑞𝑞𝑖𝑖+1) ∈ 𝜋𝜋 for 1 ≤ 𝑖𝑖 < 𝑟𝑟, 𝑞𝑞𝑖𝑖 ≤ ℎ for 1 ≤ 𝑖𝑖 ≤ 𝑟𝑟. s. manukian 71 proof: as it follows from the condition (𝑥𝑥, 𝑦𝑦) ∈ 𝜋𝜋∗, there exists an etcsequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) such that 𝑞𝑞1 = 𝑥𝑥, 𝑞𝑞𝑟𝑟 = 𝑦𝑦, (𝑞𝑞𝑖𝑖, 𝑞𝑞𝑖𝑖+1) ∈ 𝜋𝜋 for 1 ≤ 𝑖𝑖 < 𝑟𝑟. if 𝑞𝑞𝑖𝑖 ≤ ℎ for 1 ≤ 𝑖𝑖 ≤ 𝑟𝑟, then the statement of lemma is satisfied. in the opposite case let 𝑘𝑘 be the minimal index in the sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) such that 𝑘𝑘 > 1, 𝑞𝑞𝑘𝑘 > ℎ. let 𝑙𝑙 be the minimal index such that 𝑙𝑙 ≥ 𝑘𝑘, 𝑞𝑞𝑙𝑙+1 ≤ ℎ. clearly, any number 𝑞𝑞𝑗𝑗, where 𝑘𝑘 ≤ 𝑗𝑗 ≤ 𝑙𝑙 satisfies the condition 𝑞𝑞𝑗𝑗 > ℎ(note that the case 𝑘𝑘 = 𝑙𝑙 is not excluded). the following statements hold: 𝑞𝑞𝑘𝑘−1 ≤ ℎ, 𝑞𝑞𝑙𝑙+1 ≤ ℎ, the pair (𝑞𝑞𝑘𝑘−1, 𝑞𝑞𝑘𝑘) is increasing, the pair (𝑞𝑞𝑙𝑙, 𝑞𝑞𝑙𝑙+1) is decreasing. but any increasing pair (𝑥𝑥, 𝑦𝑦) ∈ 𝜋𝜋 such that 𝑥𝑥 ≥ ℎ, 𝑦𝑦 ≥ ℎ, 𝑥𝑥 < 𝑦𝑦 should satisfy the conditions of (case 3) or (case 10) mentioned above. therefore, either 𝑞𝑞𝑘𝑘−1 = ℎ, 𝑞𝑞𝑘𝑘 = ℎ + 1 (case 3) or 𝑞𝑞𝑘𝑘−1 = 𝑎𝑎, where 𝑎𝑎 is the number contained in a formula 𝑥𝑥 = 𝑎𝑎 included in 𝐹𝐹. similarly, any decreasing pair (𝑥𝑥, 𝑦𝑦) ∈ 𝜋𝜋 such that 𝑥𝑥 ≥ ℎ, 𝑦𝑦 ≥ ℎ, 𝑥𝑥 > 𝑦𝑦, satisfies the conditions of (case 6) or (case 11) mentioned above. therefore either 𝑞𝑞𝑙𝑙 = ℎ + 1, 𝑞𝑞𝑙𝑙+1 = ℎ (case 6) or 𝑞𝑞𝑙𝑙+1 = 𝑏𝑏, where 𝑏𝑏 is the number contained in a formula 𝑦𝑦 = 𝑏𝑏 included in 𝐹𝐹 (case 11). now if 𝑞𝑞𝑘𝑘−1 = 𝑞𝑞𝑙𝑙+1 = ℎ, 𝑞𝑞𝑘𝑘 = 𝑞𝑞𝑙𝑙 = ℎ + 1, (case 3, case 6) then the segment (𝑞𝑞𝑘𝑘−1, 𝑞𝑞𝑘𝑘, 𝑞𝑞𝑘𝑘+1, … , 𝑞𝑞𝑙𝑙, 𝑞𝑞𝑙𝑙+1) of the etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) can be replaced by the single number 𝑞𝑞𝑘𝑘−1 = 𝑞𝑞𝑙𝑙+1 = ℎ. if 𝑞𝑞𝑘𝑘−1 = 𝑎𝑎, then the mentioned segment can be replaced by the segment (𝑎𝑎, 𝑞𝑞𝑙𝑙+1) (case 10). if 𝑞𝑞𝑙𝑙+1 = 𝑏𝑏, then the mentioned segment can be replaced by the segment (𝑞𝑞𝑘𝑘−1, 𝑏𝑏) (case 11). clearly the sequence obtained by these replacements is an etcsequence for the value (𝑥𝑥, 𝑦𝑦) ∈ 𝜋𝜋∗. transforming in this way any segment of the sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) containing members greater than ℎ, we obtain the etc-sequence satisfying the conditions of lemma. this completes the proof. corollary 1: there is only finite number of etc-sequences obtained by the transformations described in lemma 3.1. indeed, the length of such sequence (without repetitions of members) is ≤ ℎ + 1, and any member of such sequence is ≤ ℎ. corollary 2: the set 𝑆𝑆1∗ can be defined by arithmetical formula in the signature (0, =, <, 𝑆𝑆) (even in (0, =, 𝑆𝑆)). indeed, applying corollary 1 to the case when ℎ = 𝑑𝑑, we conclude that the set of pairs (𝑥𝑥, 𝑦𝑦) such that 𝑥𝑥 ≤ 𝑑𝑑, 𝑦𝑦 ≤ 𝑑𝑑, (𝑥𝑥, 𝑦𝑦) ∈ 𝜋𝜋∗ is finite, hence, it can be defined by a formula having the form �(𝑥𝑥 = 𝑥𝑥1)&(𝑦𝑦 = 𝑦𝑦1)� ∨ �(𝑥𝑥 = 𝑥𝑥2)&(𝑦𝑦 = 𝑦𝑦2)� ∨ … ∨ �(𝑥𝑥 = 𝑥𝑥𝑚𝑚)&(𝑦𝑦 = 𝑦𝑦𝑚𝑚)�, where all 𝑥𝑥𝑖𝑖 and 𝑦𝑦𝑖𝑖 are constants. if this set is empty, then it can be defined by the formula (𝑥𝑥 = 𝑦𝑦)&(𝑥𝑥 = 𝑆𝑆(𝑦𝑦)). note. if 𝑥𝑥 ≤ 𝑑𝑑, 𝑦𝑦 ≤ 𝑑𝑑, then the statement (𝑥𝑥, 𝑦𝑦) ∈ 𝜋𝜋∗ may be tested constructively. the method of testing is actually given in corollary 1. lemma 3.2: if some pair (𝑥𝑥0, 𝑦𝑦0), where 𝑥𝑥0 > 𝑑𝑑, 𝑦𝑦0 ≤ 𝑑𝑑 belongs to 𝜋𝜋∗, then any pair (𝑥𝑥, 𝑦𝑦0), where 𝑥𝑥 > 𝑑𝑑, belongs to 𝜋𝜋∗. proof: if some pair (𝑥𝑥0, 𝑦𝑦0) satisfies the conditions of lemma, then there exists an etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) such that 𝑞𝑞1 = 𝑥𝑥0, 𝑞𝑞𝑟𝑟 = 𝑦𝑦0, (𝑞𝑞𝑖𝑖, 𝑞𝑞𝑖𝑖+1) ∈ 𝜋𝜋 for 1 ≤ 𝑖𝑖 < 𝑟𝑟. let 𝑘𝑘 be the minimal index in the sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) such that 𝑘𝑘 < 𝑟𝑟, 𝑞𝑞𝑘𝑘+1 ≤ 𝑑𝑑 (the case 𝑘𝑘 = 1 is not excluded). without loss of generality we may suprose that 𝑞𝑞𝑖𝑖 ≤ 𝑑𝑑 for 𝑘𝑘 + 1 ≤ 𝑖𝑖 ≤ 𝑟𝑟 (indeed, in the opposite case the segment (𝑞𝑞𝑘𝑘+1, 𝑞𝑞𝑘𝑘+2, … , 𝑞𝑞𝑟𝑟) can be transformed by the method described in the proof of lemma 3.1). 72 on transitive closures of two-dimensional strongly positive arithmetical sets the pair (𝑞𝑞𝑘𝑘, 𝑞𝑞𝑘𝑘+1) is decreasing, hence, either 𝑞𝑞𝑘𝑘+1 = 𝑏𝑏, where 𝑏𝑏 is the number in a formula 𝑦𝑦 = 𝑏𝑏 included in 𝐹𝐹, (case 11 considered above), or 𝑞𝑞𝑘𝑘+1 = 𝑞𝑞𝑘𝑘 − 1 (case 6 considered above). now the etc-sequence for the pair (𝑥𝑥, 𝑦𝑦0) where 𝑥𝑥 > 𝑑𝑑 is obtained from the sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) as follows: if 𝑞𝑞𝑘𝑘+1 = 𝑏𝑏, then the segment (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑘𝑘+1) is replaced by the segment (𝑥𝑥, 𝑞𝑞𝑘𝑘+1) (see case 11); if 𝑞𝑞𝑘𝑘 = 𝑞𝑞𝑘𝑘+1 + 1, then any pair (𝑥𝑥 + 1, 𝑥𝑥), where 𝑥𝑥 > 0, belongs to 𝜋𝜋 (see case 6) hence, we can obtain the required etc-sequence replacing in the sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) the segment (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑘𝑘+1) by the segment (x, x – 1, … , 𝑞𝑞𝑘𝑘+1 + 1, 𝑞𝑞𝑘𝑘+1). it is easily seen that the sequence obtained by the mentioned replacements is an etc-sequence for the value (𝑥𝑥, 𝑦𝑦0) ∈ 𝜋𝜋∗. this complets the proof. corollary: the set 𝑆𝑆2∗ can be defined by an arithmetical formula in the signature (0, =, <, 𝑆𝑆). indeed, applying lemma 3.1 and its corollaries to the case when ℎ = 𝑑𝑑 + 1, we obtain the complete list of pairs (𝑥𝑥, 𝑦𝑦) ∈ 𝜋𝜋∗ such that 𝑥𝑥 ≤ 𝑑𝑑 + 1, 𝑦𝑦 ≤ 𝑑𝑑 + 1. in particular we obtain the complete list of pairs having the property (𝑑𝑑 + 1, 𝑦𝑦0) ∈ 𝑆𝑆2∗, where 𝑦𝑦0 ≤ 𝑑𝑑. using lemma 3.2 we conclude that any pair (𝑑𝑑 + 1, 𝑦𝑦0) ∈ 𝑆𝑆2∗ where 𝑦𝑦0 ≤ 𝑑𝑑 generates the set {(𝑥𝑥, 𝑦𝑦)/(𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 = 𝑦𝑦0)}, contained in 𝑆𝑆2∗, so the set 𝑆𝑆2∗ is the union of sets having this form for all 𝑦𝑦0 ≤ 𝑑𝑑 such that (𝑑𝑑 + 1, 𝑦𝑦0) ∈ 𝑆𝑆2∗. but any set {(𝑥𝑥, 𝑦𝑦)/(𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 = 𝑦𝑦0)} is defined by the formula (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 = 𝑦𝑦0), so the set 𝑆𝑆2∗ is defined by the disjunction of these formulas. this completes the proof. lemma 3.3: if some pair (𝑥𝑥0, 𝑦𝑦0), where 𝑥𝑥0 ≤ 𝑑𝑑, 𝑦𝑦0 > 𝑑𝑑, belongs to 𝜋𝜋∗, then any pair (𝑥𝑥0, 𝑦𝑦), where 𝑦𝑦 > 𝑑𝑑 belongs to 𝜋𝜋∗. the proof is similar to that of lemma 3.2. we use the etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) such that 𝑞𝑞1 = 𝑥𝑥0, 𝑞𝑞𝑟𝑟 = 𝑦𝑦0, (𝑞𝑞𝑖𝑖, 𝑞𝑞𝑖𝑖+1) ∈ 𝜋𝜋 for 1 ≤ 𝑖𝑖 < 𝑟𝑟. let 𝑙𝑙 be the maximal index in the sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) such that 𝑞𝑞𝑙𝑙 ≤ 𝑑𝑑. without loss of generality we may suppose that 𝑞𝑞𝑖𝑖 ≤ 𝑑𝑑 when 1 ≤ 𝑖𝑖 ≤ 𝑙𝑙 (otherwise the segment (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑙𝑙) may be transformed by the method used in the proof of lemma 3.1). the pair (𝑞𝑞𝑙𝑙, 𝑞𝑞𝑙𝑙+1) is increasing, therefore either 𝑞𝑞𝑙𝑙 = 𝑎𝑎, where 𝑎𝑎 is the number in a formula 𝑥𝑥 = 𝑎𝑎 included in 𝐹𝐹 (case 10) or 𝑞𝑞𝑙𝑙+1 = 𝑞𝑞𝑙𝑙 + 1 (case 3). the etc-sequence for establishing the statement (𝑥𝑥0, 𝑦𝑦) ∈ 𝜋𝜋∗ (where 𝑦𝑦 > 𝑑𝑑) is obtained from the sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) by the following replasements: either the segment (𝑞𝑞𝑙𝑙, 𝑞𝑞𝑙𝑙+1, … , 𝑞𝑞𝑟𝑟) is replaced by the segment (𝑎𝑎, 𝑦𝑦) (case 10), or this segment is replased by the segment (𝑞𝑞𝑙𝑙, 𝑞𝑞𝑙𝑙 + 1, … , 𝑦𝑦 − 1, 𝑦𝑦) (case 3). this completes the proof. corollary: the set 𝑆𝑆3∗ can be defined by an arithmetical formula in the signature (0, =, <, 𝑆𝑆). the proof is similar to the proof of corollary of lemma 3.2. in what follows we will say that a formula having the form 𝑦𝑦 = 𝑆𝑆(𝑥𝑥), 𝑥𝑥 = 𝑆𝑆(𝑦𝑦), or 𝑥𝑥 = 𝑦𝑦 is contained in a special way in some 𝐹𝐹𝑖𝑖 if this formula is contained in 𝐹𝐹𝑖𝑖 and the conditions described correspondingly in (case 3), (case 6) or (case 12) mentioned above are satisfied. lemma 3.4: if the formula 𝑦𝑦 = 𝑆𝑆(𝑥𝑥) is contained in a special way in some 𝐹𝐹𝑖𝑖 then any pair (𝑥𝑥, 𝑦𝑦) such that 𝑥𝑥 < 𝑦𝑦, 𝑥𝑥 > 0, belongs to 𝜋𝜋∗. indeed, for establishing this statement it is sufficient to consider the etc-sequence (𝑥𝑥, 𝑥𝑥 + 1, … , 𝑦𝑦 − 1, 𝑦𝑦). lemma 3.5: if the formula 𝑥𝑥 = 𝑆𝑆(𝑦𝑦) is contained in a special way in some 𝐹𝐹𝑖𝑖, then any pair (𝑥𝑥, 𝑦𝑦) such that 𝑥𝑥 > 𝑦𝑦, 𝑦𝑦 > 0 belongs to 𝜋𝜋∗. s. manukian 73 for establishing this statement it is sufficient to consider the etc-sequence (𝑥𝑥, 𝑥𝑥 − 1, … , 𝑦𝑦 − 1, 𝑦𝑦). lemma 3.6: if the formula 𝑦𝑦 = 𝑆𝑆(𝑥𝑥) is contained in a special way in some 𝐹𝐹𝑖𝑖, and the formula 𝑥𝑥 = 𝑆𝑆(𝑦𝑦) is contained in a special way in some 𝐹𝐹𝑗𝑗, where 𝑖𝑖 ≠ 𝑗𝑗, then any pair (𝑥𝑥, 𝑦𝑦), where 𝑥𝑥 > 0, 𝑦𝑦 > 0 belongs to 𝜋𝜋∗. for establishing this statement it is sufficient to consider the etc-sequence (𝑥𝑥, 𝑥𝑥 + 1, … , 𝑧𝑧, … , 𝑦𝑦 − 1, 𝑦𝑦), where 𝑧𝑧 = max (𝑥𝑥, 𝑦𝑦) + 1. lemma 3.7: the set 𝑆𝑆4∗ can be defined by an arithmetical formula in the signature (0, =, <, 𝑆𝑆). proof: if the set 𝑆𝑆4∗is empty, then it is defined, for example, by the formula (𝑥𝑥 = 𝑦𝑦)&(𝑦𝑦 = 𝑆𝑆(𝑥𝑥)). otherwise, there exists a pair (𝑥𝑥0, 𝑦𝑦0) ∈ 𝑆𝑆4∗, that is 𝑥𝑥0 > 𝑑𝑑, 𝑦𝑦0 > 𝑑𝑑, (𝑥𝑥0, 𝑦𝑦0) ∈ 𝜋𝜋∗. hence, there exists an etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) such that 𝑞𝑞1 = 𝑥𝑥0, 𝑞𝑞𝑟𝑟 = 𝑦𝑦0, (𝑞𝑞𝑖𝑖, 𝑞𝑞𝑖𝑖+1) ∈ 𝜋𝜋 for 1 ≤ 𝑖𝑖 < 𝑟𝑟. we will distinguish two cases: (𝛼𝛼) there exists such 𝑖𝑖 that 1 ≤ 𝑖𝑖 ≤ 𝑟𝑟, 𝑞𝑞𝑖𝑖 ≤ 𝑑𝑑. (𝛽𝛽) 𝑞𝑞𝑖𝑖 > 𝑑𝑑 for any 𝑖𝑖, where 1 ≤ 𝑖𝑖 ≤ 𝑟𝑟. let us consider the case (𝛼𝛼).we denote the number 𝑞𝑞𝑖𝑖 such that 𝑞𝑞𝑖𝑖 ≤ 𝑑𝑑 by 𝑧𝑧. the pair (𝑥𝑥0, 𝑧𝑧), where 𝑥𝑥0 > 𝑑𝑑 belongs to 𝜋𝜋∗, therefore, using lemma 3.2 we conclude that any pair (𝑥𝑥, 𝑧𝑧), where 𝑥𝑥 > 𝑑𝑑, belongs to 𝜋𝜋∗. the pair (𝑧𝑧, 𝑦𝑦0), where 𝑦𝑦0 > 𝑑𝑑, belongs to 𝜋𝜋∗, therefore, using lemma 3.3 we conclude that any pair (𝑧𝑧, 𝑦𝑦), where 𝑦𝑦 > 𝑑𝑑 belongs to 𝜋𝜋∗. hence, any pair (𝑥𝑥, 𝑦𝑦), where 𝑥𝑥 > 𝑑𝑑, 𝑦𝑦 > 𝑑𝑑, belongs to 𝜋𝜋∗. so in the case (𝛼𝛼) the set 𝑆𝑆4∗ is defined by the formula (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑). let us note that similar conclusion concerning the set 𝑆𝑆4∗ can be made if there exists any etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) such that 𝑞𝑞1 > 𝑑𝑑, 𝑞𝑞𝑟𝑟 > 𝑑𝑑 and 𝑞𝑞𝑖𝑖 ≤ 𝑑𝑑 for some 𝑖𝑖, 1 < 𝑖𝑖 < 𝑟𝑟. now let us consider the case (𝛽𝛽). we will investigate the properties of all etc-sequences (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) such that 𝑞𝑞𝑖𝑖 > 𝑑𝑑 for 1 ≤ 𝑖𝑖 ≤ 𝑟𝑟, and (𝑞𝑞𝑖𝑖, 𝑞𝑞𝑖𝑖+1) ∈ 𝜋𝜋 for 1 ≤ 𝑖𝑖 < 𝑟𝑟. we distingmish the following subcases: (𝛽𝛽1), (𝛽𝛽2), (𝛽𝛽3), (𝛽𝛽4). (𝛽𝛽1) in some etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) of the mentioned kind there exists an index 𝑖𝑖 such that 1 ≤ 𝑖𝑖 < 𝑟𝑟, 𝑞𝑞𝑖𝑖+1 = 𝑞𝑞𝑖𝑖 + 1, but there is no etc-sequence of the mentioned kind containing an index 𝑗𝑗 such that 𝑞𝑞𝑗𝑗+1 = 𝑞𝑞𝑗𝑗 − 1. (𝛽𝛽2) in some etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) of the mentioned kind there exists an index 𝑖𝑖 such that 1 ≤ 𝑖𝑖 < 𝑟𝑟, 𝑞𝑞𝑖𝑖+1 = 𝑞𝑞𝑖𝑖 − 1, but there is no etc-sequence of the mentioned kind containing an index 𝑗𝑗 such that 𝑞𝑞𝑗𝑗+1 = 𝑞𝑞𝑗𝑗 + 1. (𝛽𝛽3) in some etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) of the mentioned kind there exists an index 𝑖𝑖 such that 1 ≤ 𝑖𝑖 < 𝑟𝑟, 𝑞𝑞𝑖𝑖+1 = 𝑞𝑞𝑖𝑖 + 1; besides, in some etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑡𝑡) of the mentioned kind there exists an index 𝑗𝑗 such that 1 ≤ 𝑗𝑗 < 𝑡𝑡, 𝑞𝑞𝑗𝑗+1 = 𝑞𝑞𝑗𝑗 − 1. (𝛽𝛽4) there is no etc-sequence of the mentioned kind satisfying the conditions described in the subcases (𝛽𝛽1)-( 𝛽𝛽3). clearly, the subcase (𝛽𝛽1) takes place if some 𝐹𝐹𝑖𝑖 in the structure of 𝐹𝐹 has the form (𝑦𝑦 = 𝑆𝑆(𝑥𝑥)), but there is no 𝐹𝐹𝑗𝑗 having the form (𝑥𝑥 = 𝑆𝑆(𝑦𝑦)). similarly, the subcase (𝛽𝛽2) takes place if some 𝐹𝐹𝑖𝑖 in the structure of 𝐹𝐹 has the form (𝑥𝑥 = 𝑆𝑆(𝑦𝑦)), but there is no 𝐹𝐹𝑗𝑗 having the form (𝑦𝑦 = 𝑆𝑆(𝑥𝑥)). the subcase (𝛽𝛽3) takes place if some 𝐹𝐹𝑖𝑖 and 𝐹𝐹𝑗𝑗 in the structure of 𝐹𝐹 have the forms, correspondingly 74 on transitive closures of two-dimensional strongly positive arithmetical sets (𝑦𝑦 = 𝑆𝑆(𝑥𝑥)) and (𝑥𝑥 = 𝑆𝑆(𝑦𝑦)). the subcase (𝛽𝛽4) takes place if the formula 𝐹𝐹 contains no 𝐹𝐹𝑖𝑖 having one of the mentioned forms. it is easily seen that in the subcase (𝛽𝛽1) the set 𝑆𝑆4∗ is defined by the formula (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)&(𝑥𝑥 < 𝑦𝑦) or by the formula (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)&(𝑥𝑥 ≤ 𝑦𝑦) (see lemma 3.4). in the subcase (𝛽𝛽2) the set 𝑆𝑆4∗ is defined by the formula (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)&(𝑥𝑥 > 𝑦𝑦) or by the formula (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)&(𝑥𝑥 ≥ 𝑦𝑦) (see lemma 3.5). let us note that the inequalities 𝑥𝑥 ≤ 𝑦𝑦 and 𝑥𝑥 ≥ 𝑦𝑦 are obtained in the subcases (𝛽𝛽1) and (𝛽𝛽2) if some 𝐹𝐹𝑖𝑖 in the structure of 𝐹𝐹 has the form 𝑥𝑥 = 𝑦𝑦 (see case 12 mentioned above). in the subcase (𝛽𝛽3) the set 𝑆𝑆4∗ is defined by the formula (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑) (see lemma 3.6). in the subcase (𝛽𝛽4) the set 𝑆𝑆4∗ is either empty or is defined by the formula (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)&(𝑥𝑥 = 𝑦𝑦). this completes the proof. proof of theorem 1. as it is established in lemmas 3.1-3.7, the sets 𝑆𝑆1∗, 𝑆𝑆2∗, 𝑆𝑆3∗, 𝑆𝑆4∗, are defined by formulas in the signature {0, =, <, 𝑆𝑆}. hence, the set 𝜋𝜋∗ = 𝑆𝑆1∗ ∪ 𝑆𝑆2∗ ∪ 𝑆𝑆3∗ ∪ 𝑆𝑆4∗ is defined by the disjunction of the mentioned formulas. this completes the proof. proof of theorem 2. let 𝐴𝐴 be the set {(𝑥𝑥, 𝑦𝑦)/𝑦𝑦 = 𝑆𝑆(𝑆𝑆(𝑥𝑥))}, let 𝐵𝐵 be any 2-dimensional strongly positive set, let 𝐵𝐵∗ be the transitive closure of 𝐵𝐵. we define the number 𝑑𝑑 for the set 𝐵𝐵 by the method given above. by 𝐷𝐷 we denote the set {(𝑥𝑥, 𝑦𝑦)/(𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)}. using lemma 3.7 we conclude that the set 𝐵𝐵∗ ∩ 𝐷𝐷 either is empty or is defined by one of the following formulas: (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)&(𝑥𝑥 > 𝑦𝑦), (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)&(𝑥𝑥 < 𝑦𝑦), (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)&(𝑥𝑥 = 𝑦𝑦), or by the disjunction of some of these formulas. similarly, using lemmas 3.4-3.6 we conclude that the set 𝐵𝐵 ∩ 𝐷𝐷 either is empty or is defined by one of the following formulas: (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)&(𝑦𝑦 = 𝑆𝑆(𝑥𝑥)), (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)&(𝑥𝑥 = 𝑆𝑆(𝑦𝑦)), (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)&(𝑥𝑥 = 𝑦𝑦), or by the disjuction of some of these formulas. therefore, in all the cases the set 𝐴𝐴 ∩ 𝐷𝐷 is different form 𝐵𝐵 ∩ 𝐷𝐷 and 𝐵𝐵∗ ∩ 𝐷𝐷. hence, 𝐴𝐴 ≠ 𝐵𝐵 and 𝐴𝐴 ≠ 𝐵𝐵∗. this completes the proof. note 1. the statement of theorem 2 is true also for any set defined by the formula 𝑦𝑦 = 𝑆𝑆(𝑆𝑆 … 𝑆𝑆(𝑥𝑥) … ), where the symbol 𝑆𝑆 is repeated 𝑛𝑛 ≥ 2 times. the proof is similar to that of theorem 2. note 2. obviously, any set defined by a formula in the signature (0, =, <, 𝑆𝑆) is primitive recursive, however, the reverse is not true (for example, the set of even numbers is primitive recurcive, but it cannot be defined by arithmetical formula in the signature (0, =, <, 𝑆𝑆) (see [6])). so the statement of theorem 1 is stronger than the statement of theorem 2 in [3]. references [1] s. n. manukian, “on the representation of recursively enumerable sets in weak arithmetics”, transactions of the iiap of nas of ra, mathematical problems of computer science, vol. 27, pp. 90-110, 2006. [2] s. n. manukian, “on an algebraic classification of multidimensional recursively enumerable sets expressible in formal arithmetical systems”, transactions of the iiap of nas of ra, mathematical problems of computer science, vol. 41, pp. 103-113, 20014. s. manukian 75 [3] s. n.manukian, “on strongly positive multidimensional arithmetical sets”, transactions of the iiap of nas of ra, mathematical problems of computer science, vol. 43, pp. 3241, 2015. [4] s. c. kleene, introduction to metamathematics, d.van nostrand comp., inc., new york – toronto, 1952. [5] e. mendelson, introduction to mathematical logic, d.van nostrand comp., inc., princeton – toronto – new york – london, 1964. [6] h. b. enderton, a mathematical introduction to logic, 2nd edition, san diego, harcourt, academic press, 2001. [7] g. s. tseytin, “one method of representation for the theory of algorithms and enumerable sets”, transactions of steklov institute of the acad. sci. ussr (in russian), vol. 72, pp. 6998, 1964. [8] a. i. maltsev, algorithms and recursive functions, 2nd edition (in russian), m.,”nauka”, 1986. submitted 04.10.2015, accepted 15.01.2016 խիստ պոզիտիվ երկչափ թվաբանական բազմությունների տրանզիտիվ փակումների մասին ս. մանուկյան ամփոփում պոզիտիվ և խիստ պոզիտիվ թվաբանական բազմությունների գաղափարները սահմանված են [1]-[3] հոդվածներում: [3] հոդվածում նշված է, որ ցանկացած երկչափ խիստ պոզիտիվ բազմության տրանզիտիվ փակումը պարզագույն անդրադարձ է: այս հոդվածում ապացուցվում է ավելի ուժեղ պնդում, այսինքն՝ ցանկացած երկչափ խիստ պոզիտիվ բազմության տրանզիտիվ փակումը նկարագրվում է թվաբանական բանաձևի միջոցով (0, =, <, 𝑆𝑆) սիգնատուրայում (որտեղ 𝑆𝑆(𝑥𝑥) = 𝑥𝑥 + 1): բացի դրանից ապացուցվում է, որ երկչափ խիստ պոզիտիվ բազմությունների դասը և այդ բազմությունների տրանզիտիվ փակումների դասը չեն համընկնում (0, =, <, 𝑆𝑆) սիգնատուրայում արտահայտվող թվաբանական բազմությունների դասի հետ: 76 on transitive closures of two-dimensional strongly positive arithmetical sets о транзитивных замыканиях строго позитивных арифметических множеств размерности 2 с. манукян аннотация понятия позитивного и строго позитивного множества рассматриваются в [1]-[3]. в [3] указано, что транзитивное замыкание всякого строго позитивного множества размерности 2 примитивно рекурсивно. в этой статье доказывается более сильное утверждение: транзитивное замыкание всякого строго позитивного множества размерности 2 задается арифметической формулой в сигнатуре (0, =, <, 𝑆𝑆), где 𝑆𝑆(𝑥𝑥) = 𝑥𝑥 + 1. доказывается также, что класс строго позитивных множеств размерности 2 и класс транзитивных замыканий таких множеств не совпадают с классом арифметических множеств размерности 2, задаваемых посредством арифметических формул в сигнатуре (0, =, <, 𝑆𝑆). начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 46, 87--91, 2016. linear orderings of tridimensional grids david h. muradian institute for informatics and automation problems of nas ra e-mail: david.h.muradian@gmail.com abstract the minimal linear arrangement problem (minla) is defined as follows: given a graph g, find a linear ordering (layout) 𝜑𝜑 for the vertices of g on a line such that the sum of the edge lengths is minimized over all orderings. edge length for an edge (x, y) is defined as | 𝜑𝜑(𝑥𝑥) − 𝜑𝜑(𝑦𝑦)|. in this paper we describe the class of minimal orderings of the special case of tridimensional grids – cartesian product of three simple paths, when one of them consists of two vertices. keywords: linear ordering, minimal linear arrangement problem, grids, wirelength. 1. introduction given a graph g=(x,u), a layout 𝜑𝜑 is a one-to-one mapping 𝜑𝜑 : x →{1,…,|x|}. for a given graph g=( x,u) and a layout 𝜑𝜑, we define 𝐸𝐸𝜑𝜑(𝐺𝐺) = � |𝜑𝜑(𝑥𝑥) − 𝜑𝜑(𝑦𝑦)| (𝑥𝑥,𝑦𝑦)∈𝑈𝑈 , as a wirelength of 𝜑𝜑. we define also wirelength of g as 𝐸𝐸(𝐺𝐺) = min 𝜑𝜑 𝐸𝐸𝜑𝜑(𝐺𝐺), where 𝜑𝜑 ranges over all layouts of g, and a layout 𝜑𝜑0 is called minimal if 𝐸𝐸𝜑𝜑0(𝐺𝐺)= 𝐸𝐸(𝐺𝐺). let’s denote by 𝜙𝜙𝐺𝐺 𝐸𝐸 the class of minimal layouts of g. let 𝑋𝑋′, 𝑋𝑋′′ ⊂ 𝑋𝑋 be nonempty disjoint sets, 𝑘𝑘 ∈ 1, 𝑁𝑁�� and 𝜑𝜑 be some layout of g. let’s denote: 𝑋𝑋𝜑𝜑𝑘𝑘 = {𝜑𝜑−1(1), 𝜑𝜑−1(2), … , 𝜑𝜑−1(𝑘𝑘), } 𝜔𝜔(𝑋𝑋′, 𝑋𝑋′′) = |{(𝑥𝑥, 𝑦𝑦) ∈ 𝑈𝑈 ⁄ 𝑥𝑥 ∈ 𝑋𝑋′; 𝑦𝑦 ∈ 𝑋𝑋′′ }| δ𝜑𝜑(𝑋𝑋′) = 1 |𝑋𝑋′| (|{(𝑥𝑥, 𝑦𝑦) ∈ 𝑈𝑈 ⁄ 𝑥𝑥 ∈ 𝑋𝑋′; 𝑦𝑦 ∉ 𝑋𝑋′; 𝜑𝜑(𝑥𝑥) < 𝜑𝜑(𝑦𝑦) }| − |{(𝑥𝑥, 𝑦𝑦) ∈ 𝑈𝑈 ⁄ 𝑥𝑥 ∈ 𝑋𝑋′; 𝑦𝑦 ∉ 𝑋𝑋′; 𝜑𝜑(𝑥𝑥) > 𝜑𝜑(𝑦𝑦) }|) 87 linear orderings of tridimensional grids 88 definition: we say that a set 𝑋𝑋′ (𝑋𝑋′ ⊂ 𝑋𝑋) is compact with respect to layout 𝜑𝜑, if max 𝑥𝑥∈𝑋𝑋′ 𝜑𝜑(𝑥𝑥) − min 𝑥𝑥∈𝑋𝑋′ 𝜑𝜑(𝑥𝑥) = |𝑋𝑋′| − 1. definition: we say that a set 𝑋𝑋′ (𝑋𝑋′ ⊂ 𝑋𝑋) directly goes behind the set 𝑋𝑋′′ (𝑋𝑋′′ ⊂ 𝑋𝑋) (this is denoted by 𝑋𝑋′ 𝜑𝜑 ← 𝑋𝑋′′), if 𝑋𝑋′, 𝑋𝑋′′ are compact and 𝑚𝑚𝑚𝑚𝑥𝑥 𝑥𝑥∈𝑋𝑋′ 𝜑𝜑(𝑥𝑥) = 𝑚𝑚𝑚𝑚𝑚𝑚 𝑥𝑥∈𝑋𝑋′′ 𝜑𝜑(𝑥𝑥) + 1. definition: we say that the sets 𝑋𝑋′, 𝑋𝑋′′ are independent of one another, if 𝜔𝜔(𝑋𝑋′, 𝑋𝑋′′) = 0. in the present paper the following lemma from [1] will play an essential role. lemma: if 𝜔𝜔(𝑋𝑋′, 𝑋𝑋′′) = 0, 𝑋𝑋′ 𝜑𝜑 ← 𝑋𝑋′′ and 𝜑𝜑 is a minimal layout, then . δ𝜑𝜑(𝑋𝑋′) ≤ δ𝜑𝜑(𝑋𝑋′′) let 𝜑𝜑 be some layout of g=(x,u) and 𝐺𝐺′ be an induced subgraph with vertex set 𝑋𝑋′ ⊂ 𝑋𝑋. let the vertices of 𝑋𝑋′ have the following numbers at the layout 𝜑𝜑: 𝑚𝑚1 < 𝑚𝑚2 < ⋯ < 𝑚𝑚�𝑋𝑋′�. consider the following layout 𝜑𝜑′: 𝜑𝜑′�𝜑𝜑−1(𝑚𝑚𝑖𝑖)� = 𝑚𝑚 �𝑚𝑚 = 1, |𝑋𝑋′|��. definition: we say that a subgraph 𝐺𝐺′ is ordered minimally at 𝜑𝜑, if 𝜑𝜑′ is a minimal layout for 𝐺𝐺′. consider the graph p2,m,n with the vertex set 𝛱𝛱2,𝑚𝑚,𝑛𝑛 = �𝑥𝑥𝑖𝑖,𝑗𝑗,𝑘𝑘/ 𝑚𝑚 = 1,2��; 𝑗𝑗 = 1, 𝑚𝑚��; 𝑘𝑘 = 1, 𝑚𝑚�� and the edge set u, where �𝑥𝑥𝑖𝑖,𝑗𝑗,𝑘𝑘, 𝑥𝑥𝑖𝑖′,𝑗𝑗′,𝑘𝑘′� ∈ 𝑈𝑈 if and only if |𝑚𝑚 − 𝑚𝑚 ′| + |𝑗𝑗 − 𝑗𝑗′| + |𝑘𝑘 − 𝑘𝑘′| = 1. let’s denote 𝛱𝛱𝑖𝑖1,𝑗𝑗1,𝑘𝑘1 𝑖𝑖2,𝑗𝑗2,𝑘𝑘2=�𝑥𝑥𝑖𝑖,𝑗𝑗,𝑘𝑘 / 𝑚𝑚1 ≤ 𝑚𝑚 ≤ 𝑚𝑚2; 𝑗𝑗1 ≤ 𝑗𝑗 ≤ 𝑗𝑗2; 𝑘𝑘1 ≤ 𝑘𝑘 ≤ 𝑘𝑘2�, ω0 = �𝑥𝑥1,1,1, 𝑥𝑥1,𝑚𝑚,1, 𝑥𝑥1,1,𝑛𝑛, 𝑥𝑥1,𝑚𝑚,𝑛𝑛, 𝑥𝑥2,1,1, 𝑥𝑥2,𝑚𝑚,1, 𝑥𝑥2,1,𝑛𝑛, 𝑥𝑥2,𝑚𝑚,𝑛𝑛� where 1 ≤ 𝑚𝑚1 ≤ 𝑚𝑚2 ≤ 2; 1 ≤ 𝑗𝑗1 ≤ 𝑗𝑗2 ≤ 𝑚𝑚; 1 ≤ 𝑘𝑘1 ≤ 𝑘𝑘2 ≤ 𝑚𝑚. definition: we say that the set 𝑋𝑋′ ⊂ 𝛱𝛱2,𝑚𝑚,𝑛𝑛 is concise with respect to 𝑥𝑥1,1,1, if for every 𝑥𝑥𝑖𝑖,𝑗𝑗,𝑘𝑘 ∈ 𝑋𝑋′ we have 𝛱𝛱1,1,1 𝑖𝑖,𝑗𝑗,𝑘𝑘 ⊆ 𝑋𝑋′. definition: we say that a layout φ is concise with respect to 𝑥𝑥1,1,1, if for every 𝑘𝑘 ∈ 1,2𝑚𝑚𝑚𝑚�� the set 𝑋𝑋𝜑𝜑𝑘𝑘 is concise with respect to 𝑥𝑥1,1,1. similarly one can define conciseness of sets and layouts with respect to other vertices from ω0. the following statements are valid. 1. if 𝜑𝜑 ∈ 𝜙𝜙p2,m,n 𝐸𝐸 , then for every 𝑘𝑘 ∈ 1,2𝑚𝑚𝑚𝑚�� the set 𝑋𝑋𝜑𝜑𝑘𝑘 is concise with respect to at least one vertex from ω0. 2. for each vertex from ω0, there is a minimal, concise with respect to its layout. we will leave out the proofs of the above statements as they are very similar to analogous statements from [1] d. muradian 89 the following theorem is a main result of this paper. theorem: let 𝜑𝜑 be concise with respect to 𝑥𝑥1,1,1. then 𝜑𝜑 is minimal if and only if for each i,j (𝑚𝑚 ∈ 1, 𝑚𝑚��; 𝑗𝑗 ∈ 1, 𝑚𝑚��) 𝑥𝑥1,𝑖𝑖,𝑗𝑗 𝜑𝜑 ← 𝑥𝑥2,𝑖𝑖,𝑗𝑗 and the subgraphs induced by the sets 𝛱𝛱1,1,1 1,𝑚𝑚,𝑛𝑛, 𝛱𝛱2,1,1 2,𝑚𝑚,𝑛𝑛 are ordered minimally at 𝜑𝜑. proof: only taking into consideration conciseness of 𝜑𝜑 with respect to 𝑥𝑥1,1,1, the set 𝛱𝛱2,𝑚𝑚,𝑛𝑛 is divided into subsets 𝛱𝛱𝑖𝑖 (regarding δ𝜑𝜑(𝑥𝑥)): 𝛱𝛱0 = �𝑥𝑥1,1,1� 𝛱𝛱1 = 𝛱𝛱1,2,1 1,𝑚𝑚−1,1 ∪ 𝛱𝛱1,1,2 1,1,𝑛𝑛−1; 𝛱𝛱2 = 𝛱𝛱1,2,2 1,𝑚𝑚−1,𝑛𝑛−1 ∪ �𝑥𝑥1,𝑚𝑚,1, 𝑥𝑥1,1,𝑛𝑛, 𝑥𝑥2,1,1�; 𝛱𝛱3 = 𝛱𝛱1,𝑚𝑚,2 1,𝑚𝑚,𝑛𝑛−1 ∪ 𝛱𝛱1,2,𝑛𝑛 1,𝑚𝑚−1,𝑛𝑛 ∪ 𝛱𝛱2,2,1 2,𝑚𝑚−1,1 ∪ 𝛱𝛱2,1,2 2,1,𝑛𝑛−1; 𝛱𝛱4 = 𝛱𝛱2,2,2 2,𝑚𝑚−1,𝑛𝑛−1 ∪ �𝑥𝑥1,𝑚𝑚,𝑛𝑛, 𝑥𝑥2,𝑚𝑚,1, 𝑥𝑥2,1,𝑛𝑛�; 𝛱𝛱5 = 𝛱𝛱2,𝑚𝑚,2 2,𝑚𝑚,𝑛𝑛−1 ∪ 𝛱𝛱2,2,𝑛𝑛 2,𝑚𝑚−1,𝑛𝑛; 𝛱𝛱6 = �𝑥𝑥2,𝑚𝑚,𝑛𝑛�. and δ𝜑𝜑(𝑥𝑥) = 3 − 𝑚𝑚 at 𝑥𝑥 ∈ 𝛱𝛱𝑖𝑖. at first we will prove that 𝑥𝑥1,1,1 𝜑𝜑 ← 𝑥𝑥2,1,1, i.e., φ(𝑥𝑥2,1,1)=2. let’s assume the reverse: 𝑥𝑥1,1,1 𝜑𝜑 ← 𝑆𝑆 𝜑𝜑 ← 𝑥𝑥2,1,1, and 𝑆𝑆 ≠ ∅. consider a case 𝑥𝑥1,𝑚𝑚,𝑛𝑛 ∉ 𝑆𝑆. we have δ𝜑𝜑(𝑆𝑆) ≤ δ𝜑𝜑�𝑥𝑥2,1,1� = 1 by the lemma. it is easy to see that for every set 𝑋𝑋′: δ𝜑𝜑(𝑋𝑋′) = 1 |𝑋𝑋′| � δ𝜑𝜑(𝑥𝑥). 𝑥𝑥∈𝑋𝑋′ as 𝜑𝜑 is concise with respect to 𝑥𝑥1,1,1, then from 𝑥𝑥1,𝑚𝑚,𝑖𝑖 ∈ 𝑆𝑆 follows 𝑥𝑥1,1,𝑖𝑖 ∈ 𝑆𝑆, where 𝑚𝑚 ∈ 2, 𝑚𝑚 − 1�� (and from 𝑥𝑥1,𝑗𝑗,𝑛𝑛 ∈ 𝑆𝑆 follows 𝑥𝑥1,𝑗𝑗,1 ∈ 𝑆𝑆, where 𝑗𝑗 ∈ 2, 𝑚𝑚 − 1��). therefore, δ𝜑𝜑(𝑆𝑆) ≥ 1 and δ𝜑𝜑(𝑆𝑆) = 1 if and only if 𝑆𝑆 = 𝛱𝛱1,1,1 1,𝑚𝑚,𝑛𝑛\�𝑥𝑥1,1,1, 𝑥𝑥1,𝑚𝑚,𝑛𝑛�. let 𝑆𝑆 𝜑𝜑 ← 𝑅𝑅 𝜑𝜑 ← 𝑥𝑥1,𝑚𝑚,𝑛𝑛 (obviously 𝑥𝑥2,1,1 ∈ 𝑅𝑅). easy to see that 𝜔𝜔�𝑅𝑅, 𝑥𝑥1,𝑚𝑚,𝑛𝑛� = 0, δ𝜑𝜑�𝑥𝑥1,𝑚𝑚,𝑛𝑛� = −1, and from the conciseness of 𝜑𝜑 we have δ𝜑𝜑(𝑅𝑅) > −1, which contradicts the lemma. let’s now consider the case 𝑥𝑥1,𝑚𝑚,𝑛𝑛 ∈ 𝑆𝑆. then 𝛱𝛱1,1,1 1,𝑚𝑚,𝑛𝑛 𝜑𝜑 ← 𝛱𝛱2,1,1 2,𝑚𝑚,𝑛𝑛 and the subgraphs 𝐺𝐺1, 𝐺𝐺2 induced with them are ordered minimally at 𝜑𝜑. really, it is easy to see that for every ordering ψ, for which 𝛱𝛱1,1,1 1,𝑚𝑚,𝑛𝑛 𝜓𝜓 ← 𝛱𝛱2,1,1 2,𝑚𝑚,𝑛𝑛, we will have 𝐸𝐸𝜓𝜓(𝛱𝛱2,𝑚𝑚,𝑛𝑛) = 𝑚𝑚2𝑚𝑚2 + 𝐸𝐸𝜓𝜓1(𝐺𝐺1) + 𝐸𝐸𝜓𝜓2(𝐺𝐺2), where 𝜓𝜓1(𝑥𝑥) = 𝜓𝜓(𝑥𝑥) when 𝑥𝑥 ∈ 𝛱𝛱1,1,1 1,𝑚𝑚,𝑛𝑛 and 𝜓𝜓2(𝑥𝑥) = 𝜓𝜓(𝑥𝑥) − 𝑚𝑚𝑚𝑚 when 𝑥𝑥 ∈ 𝛱𝛱2,1,1 2,𝑚𝑚,𝑛𝑛. therefore, 𝜑𝜑 ∈ 𝜙𝜙p2,m,n 𝐸𝐸 if and only if 𝜓𝜓1 ∈ 𝜙𝜙𝐺𝐺1 𝐸𝐸 , 𝜓𝜓2 ∈ 𝜙𝜙𝐺𝐺2 𝐸𝐸 . so 𝐺𝐺1, 𝐺𝐺2 at 𝜑𝜑 are ordered minimally. then from [1] we will have the following. if m ≤ n, then a) at m > 4: 𝛱𝛱1,1,1 1,𝜆𝜆0,𝜆𝜆0 𝜑𝜑 ← 𝛱𝛱1,1,1 1,𝑚𝑚,𝑛𝑛\𝛱𝛱1,1,1 1,𝜆𝜆0,𝜆𝜆0 𝜑𝜑 ← 𝛱𝛱2,1,1 2,𝜆𝜆0,𝜆𝜆0, where 2 ≤ 𝜆𝜆0 < 1 2 m; b) at m < 4: 𝛱𝛱1,1,1 1,𝑚𝑚,1 𝜑𝜑 ← 𝛱𝛱1,1,1 1,𝑚𝑚,𝑛𝑛\𝛱𝛱1,1,1 1,𝑚𝑚,1 𝜑𝜑 ← 𝛱𝛱2,1,1 2,𝑚𝑚,1; c) at m = 4: the case а) or b) is happened. it is not difficult to compute: δ𝜑𝜑 �𝛱𝛱1,1,1 1,𝑚𝑚,𝑛𝑛\𝛱𝛱1,1,1 1,𝜆𝜆0,𝜆𝜆0� = 𝑚𝑚𝑚𝑚 − 𝜆𝜆02 − 2𝜆𝜆0 𝑚𝑚𝑚𝑚 − 𝜆𝜆0 2 = 1 − 2𝜆𝜆0 𝑚𝑚𝑚𝑚 − 𝜆𝜆0 2 > 0; linear orderings of tridimensional grids 90 δ𝜑𝜑 �𝛱𝛱2,1,1 2,𝜆𝜆0,𝜆𝜆0� = 2𝜆𝜆0 − 𝜆𝜆02 𝜆𝜆0 2 = 2 𝜆𝜆0 − 1 ≤ 0; δ𝜑𝜑�𝛱𝛱1,1,1 1,𝑚𝑚,𝑛𝑛\𝛱𝛱1,1,1 1,𝑚𝑚,1� = 𝑚𝑚(𝑚𝑚 − 1) − 𝑚𝑚 𝑚𝑚(𝑚𝑚 − 1) = 𝑚𝑚 − 2 𝑚𝑚 − 1 > 0; δ𝜑𝜑�𝛱𝛱2,1,1 2,𝑚𝑚,1� = 0. the last relations obviously contradict the lemma. therefore, 𝑥𝑥1,1,1 𝜑𝜑 ← 𝑥𝑥2,1,1. now let’s show, that 𝑥𝑥1,𝑖𝑖,𝑗𝑗 𝜑𝜑 ← 𝑥𝑥2,𝑖𝑖,𝑗𝑗 for each i,j (𝑚𝑚 ∈ 1, 𝑚𝑚��; 𝑗𝑗 ∈ 1, 𝑚𝑚��). we will say that the vertices 𝑥𝑥1,𝑖𝑖,𝑗𝑗, 𝑥𝑥2,𝑖𝑖,𝑗𝑗 are neighbors. let’s assume the reverse. let z be a vertex with the smallest number, which does not directly goes behind its neighbor (denote the latter by y). so we have 𝑦𝑦 𝜑𝜑 ← 𝑆𝑆 𝜑𝜑 ← 𝑧𝑧; s ≠ ∅; δ𝜑𝜑(𝑦𝑦) = δ𝜑𝜑(𝑧𝑧) + 2. by the definition of z every vertex from 𝛱𝛱2,1,1 2,𝑚𝑚,𝑛𝑛⋂𝑆𝑆 directly goes behind its neighbor. let �𝛱𝛱2,1,1 2,𝑚𝑚,𝑛𝑛⋂𝑆𝑆 � = 𝑘𝑘 and 𝑦𝑦 𝜑𝜑 ← 𝑀𝑀1 𝜑𝜑 ← 𝑁𝑁1 𝜑𝜑 ← … 𝜑𝜑 ← 𝑀𝑀𝑘𝑘 𝜑𝜑 ← 𝑁𝑁𝑘𝑘 𝜑𝜑 ← 𝑀𝑀𝑘𝑘+1 𝜑𝜑 ← 𝑧𝑧, where 𝑁𝑁𝑖𝑖 − one pair of neighbors, and 𝑀𝑀𝑗𝑗 ⊂ 𝛱𝛱1,1,1 1,𝑚𝑚,𝑛𝑛. then as 𝜑𝜑 is concise, we have 𝜔𝜔(𝑆𝑆, 𝑧𝑧) = 0; 𝜔𝜔(𝑦𝑦, 𝑁𝑁𝑖𝑖) = 0; 𝜔𝜔( 𝑀𝑀𝑖𝑖, 𝑁𝑁𝑖𝑖) = 0, (1) at 1 ≤i ≤ j ≤ k. notice, that y and s cannot be independent of one another. otherwise, by the lemma we would have δ𝜑𝜑(𝑦𝑦) ≤ δ𝜑𝜑(𝑆𝑆) ≤ δ𝜑𝜑(𝑧𝑧) which would contradict the relation δ𝜑𝜑(𝑦𝑦) = δ𝜑𝜑(𝑧𝑧) + 2. therefore: ⋃𝑀𝑀𝑖𝑖≠ ∅. let’s show, that δ𝜑𝜑(𝑆𝑆) > −1. let’s assume the reverse: δ𝜑𝜑(𝑆𝑆) ≤ −1 . then, as δ𝜑𝜑(𝑁𝑁𝑖𝑖) ≥ −1 for every 𝑚𝑚 ∈ 1, 𝑘𝑘��, then ⋃𝑀𝑀𝑖𝑖 consists of a unique vertex 𝑥𝑥1,𝑚𝑚,𝑛𝑛. since 𝜔𝜔(𝑦𝑦, 𝑆𝑆) ≠ 0 , then by (1) we will have 𝑦𝑦 ∈ 𝛱𝛱3. therefore, δ𝜑𝜑(𝑧𝑧) = −2. but from the conciseness of 𝜑𝜑 we can conclude, that 𝑥𝑥1,𝑚𝑚,𝑛𝑛 𝜑𝜑 ←𝑧𝑧 , which contradicts the lemma. from δ𝜑𝜑(𝑆𝑆) > −1 we have δ𝜑𝜑(𝑧𝑧) = 0 (δ𝜑𝜑(𝑦𝑦) = 2). notice, that δ𝜑𝜑(𝑁𝑁𝑖𝑖) takes values from {-1;0;1}. let’s assume, that δ𝜑𝜑(𝑁𝑁𝑖𝑖) ≥ 0 for each 𝑚𝑚 ∈ 1, 𝑘𝑘��. then it is easy to see, that δ𝜑𝜑(𝑆𝑆) > 0, which is not possible by the lemma. therefore, there would be 𝑁𝑁𝑖𝑖, for which δ𝜑𝜑(𝑁𝑁𝑖𝑖) = −1. let 𝑁𝑁𝑝𝑝 be a pair with the smallest index from {𝑁𝑁𝑖𝑖}𝑖𝑖=1,𝑘𝑘�� , for which δ𝜑𝜑(𝑁𝑁𝑖𝑖) = −1. we have p ≥ 1 . we will prove by induction that 𝑀𝑀𝑖𝑖 = ∅ for all 𝑚𝑚 ∈ 2, 𝑝𝑝��. really, 𝑀𝑀𝑝𝑝 = ∅ by the lemma and (1). let the sets 𝑀𝑀𝑖𝑖+1, 𝑀𝑀𝑖𝑖+2, … , 𝑀𝑀𝑝𝑝 be empty. we have 𝑀𝑀𝑖𝑖 𝜑𝜑 ← ⋃ 𝑁𝑁𝑗𝑗 𝑝𝑝 𝑗𝑗=𝑖𝑖+1 ; δ𝜑𝜑(𝑀𝑀𝑖𝑖)≥0; δ𝜑𝜑�⋃ 𝑁𝑁𝑗𝑗 𝑝𝑝 𝑗𝑗=𝑖𝑖+1 � < 0, and by the lemma we will have 𝑀𝑀𝑖𝑖 = ∅ . then 𝑦𝑦 𝜑𝜑 ← 𝑀𝑀1 𝜑𝜑 ←⋃ 𝑁𝑁𝑗𝑗 𝑝𝑝 𝑗𝑗=1 . but δ𝜑𝜑(𝑦𝑦⋃𝑀𝑀1) > 0 , δ𝜑𝜑�⋃ 𝑁𝑁𝑗𝑗 𝑝𝑝 𝑗𝑗=1 � < 0, which contradicts the lemma. thus, we obtained that 𝑥𝑥1,𝑖𝑖,𝑗𝑗 𝜑𝜑 ← 𝑥𝑥2,𝑖𝑖,𝑗𝑗 for each i,j (𝑚𝑚 ∈ 1, 𝑚𝑚��; 𝑗𝑗 ∈ 1, 𝑚𝑚��), i.e., vertices of 𝐺𝐺1 got odd numbers, while the vertices of 𝐺𝐺2 – even numbers. let’s define layouts 𝜑𝜑1 and 𝜑𝜑2 for the graphs 𝐺𝐺1, 𝐺𝐺2: 𝜑𝜑1�𝜑𝜑−1(2𝑘𝑘 − 1)� = 𝑘𝑘; 𝜑𝜑2�𝜑𝜑−1(2𝑘𝑘)� = 𝑘𝑘; for each ∈ 1, 𝑚𝑚𝑚𝑚�� . then it is easy to see, that 𝐸𝐸𝜑𝜑(𝑃𝑃2,𝑚𝑚,𝑛𝑛) = 𝑚𝑚𝑚𝑚 + 2𝐸𝐸𝜑𝜑1(𝐺𝐺1) + 2𝐸𝐸𝜑𝜑2(𝐺𝐺2). (2) therefore, 𝜑𝜑 is minimal if and only if 𝜑𝜑 ∈ 𝜙𝜙𝐺𝐺𝑖𝑖 𝐸𝐸 (𝑚𝑚 ∈ 1,2��). this proves the theorem. substituting the formula of 𝐸𝐸(𝐺𝐺𝑖𝑖) from [1] into (2), at m ≤ n we will have: d. muradian 91 𝐸𝐸𝜑𝜑(𝑃𝑃2,𝑚𝑚,𝑛𝑛) = 𝑚𝑚𝑚𝑚 + 4 �− 2 3 𝜆𝜆0 3 + 2𝑚𝑚𝜆𝜆02 − �𝑚𝑚2 + 𝑚𝑚 − 2 3 �𝜆𝜆0 + 𝑚𝑚(𝑚𝑚2 + 𝑚𝑚 − 1) − 𝑚𝑚�, where �𝑚𝑚 − �𝑚𝑚 2 2 − 𝑚𝑚 2 + 1 4 � ≤ 𝜆𝜆0 ≤ �𝑚𝑚 + 1 2 − �𝑚𝑚 2 2 − 𝑚𝑚 2 + 1 4 �. references 1. d. o. muradian and t. e. piliposyan, “minimal numberings of vertices of a rectangular lattice”, akad. nauk. armjan.ssr 1, in russian, vol.70, pp. 21-27, 1980. submitted 07.07.2016, accepted 26.10.2016. գծային համարակալումներ եռաչափ ցանցերի համար դ. մուրադյան ամփոփում գրաֆի մինիմալ համարակալում գտնելու խնդիրը սահմանվում է հետևյալ կերպ: պահանջվում է գտնել տրված գրաֆի գագաթների այնպիսի տեղաբաշխում թվային առանցքի վրա, որ էջերի երկարությունների գումարը լինի նվազագույն, որտեղ էջի երկարությունը նրան կից գագաթների համարների տարբերության բացարձակ արժեքն է: այս աշխատանքում նկարագրվում է մինիմալ համարակալումների դասը եռաչափ ցանցերի մի մասնավոր դեպքի՝ երեք պարզ շղթաների դեկարտյան արտադրյալի համար, որոնցից մեկն ունի երկու գագաթ: линейные нумерации трехмерных решеток д. мурадян аннотация в работе описывается класс минимальных по длине нумераций частного случая трехмерных решеток – декартого произведения трех простых цепей, когда один из них состоит из двух вершин. минимальная длина нумерациия графа определяется следующим образом: для данного графа g требуется найти такую линейную нумерацию его вершин, чтобы сумма длин ребер (абсолютное число разности номеров инцидентных ей вершин) была минимальна относительно всевозможных нумераций графа. 1. introduction references начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 48, 105--111, 2017. further results for encoding and decoding procedures of asymmetric low magnitude error correcting codes hamlet k. khachatrian institute for informatics and automation problems of nas ra e-mail: hamletkh@ipia.sci.am abstract in this paper an implementation of encoding and decoding procedures for double ±1 error correcting optimal linear codes over rings 𝑍𝑍7 and 𝑍𝑍9 is presented. keywords: error correcting codes, asymmetric low magnitude error correcting codes, encoding and decoding procedures. 1. introduction codes over finite rings, particularly over integer residue rings and their applications in coding theory, have been studied for a long time. errors happening in the channel are basically asymmetrical; they also have a limited magnitude, and this effect is particularly applicable to flash memories. there have been a couple of papers regarding the optimal ±1 single error correcting codes over the alphabet 𝑍𝑍𝑚𝑚 [1, 2]. also there are some papers regarding the construction of optimal double ±1 error correcting codes [3, 4]. here, we propose to construct encoding and decoding algorithms for the optimal codes correcting double ±1 errors. in [5] you can see the construction of encoding and decoding procedures for the optimal linear code (12, 8) over ring 𝑍𝑍5, which was given by parity check matrix 𝐻𝐻5: 𝐻𝐻5 = � 1 1 1 1 1 0 1 2 3 4 1 1 0 1 2 3 4 2 2 2 2 2 1 1 3 2 4 4 2 3 2 4 4 2 1 1 1 1 1 1 1 3 2 4 4 2 0 4 �. in this case the number of combinations for each code word that can be corrected is: (1 + 12 ∗ 2 + (12 𝑐𝑐ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 2) ∗ 4) = 289. 105 further results for encoding and decoding procedures of asymmetric low error correcting codes 106 implementation of codes over large alphabets is much more difficult compared with small alphabets. in this paper we construct encoding and decoding procedures for the codes (16, 12) and (20, 16) over rings 𝑍𝑍7 and 𝑍𝑍9, which are developed in [4]. using this codes we can correct consequently 512 and 800 errors of type ±1 in any vectors from 𝑍𝑍7 and 𝑍𝑍9 with lengths 12 and 16 by adding only 4 parity check symbols. 2. presentation of the codes (16, 12) and (20, 16) over rings 𝒁𝒁𝟕𝟕 and 𝒁𝒁𝟗𝟗 in [4] you can see the construction of optimal linear codes over rings z7 and z9 correcting double ±1 errors. 2.1 code (16, 12) over ring 𝒁𝒁𝟕𝟕 let a linear (16, 12) code over ring z7 be given by the following parity check matrix 𝐻𝐻7: 𝐻𝐻7 = � 1 1 1 1 1 1 1 0 1 2 3 4 5 6 1 1 6 5 4 3 2 1 0 2 2 2 2 2 2 2 1 1 4 3 6 6 3 4 2 4 3 6 6 3 4 2 1 6 1 1 1 1 1 1 1 4 3 6 6 3 4 2 0 0 � . a linear code over ring z7 , with 12 information and 4 parity check symbols, given by the parity check matrix 𝐻𝐻7 can correct up to two errors of the type ±1, because 𝐻𝐻7 has a property according to which all the syndromes resulting from adding and subtracting operations between any two columns of the matrix 𝐻𝐻7 are different (±hi ± hj and hi ≠ hj). this code is optimal in the sense that it has a minimal possible number of parity check symbols. in this case the number of combinations for each code word that can be corrected is: 16 ∗ 2 + (16 𝑐𝑐ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 2) ∗ 4 = 512. 2.2 code (20, 16) over ring 𝒁𝒁𝟗𝟗 the parity check matrix 𝐻𝐻9 for an optimal linear code (20, 16) correcting double errors of the type ±1 over ring z9 has the following form: 𝐻𝐻9 = � 1 1 1 1 1 1 1 1 8 7 6 5 4 3 2 1 1 1 2 4 7 6 5 4 3 2 1 0 2 2 2 2 2 2 2 2 1 1 2 4 7 3 2 4 4 2 3 7 7 3 2 4 4 2 3 7 1 1 2 4 1 1 1 1 1 1 1 1 7 3 2 4 4 2 3 7 6 3 7 2 �. a linear code over ring z9 , with 16 information and 4 parity check symbols, given by the parity check matrix 𝐻𝐻9 can correct up to two errors of the type ±1. this code is optimal too in the sense that it has a minimal possible number of parity check symbols. in this case the number of combinations for each code word that can be corrected is: ( 20 ∗ 2 + (20 𝑐𝑐ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 2) ∗ 4) = 800 . in the next chapter we will construct encoding and decoding procedures for these two optimal linear codes. h. khachatrian 107 3. encoding and decoding procedures 3.1 code (16, 12) for encoding every message in z7 we must have the generator matrix 𝐺𝐺7. for this we should construct a combinatorial equivalent matrix 𝐻𝐻′7 from parity check matrix 𝐻𝐻7 of the code (16, 12): 𝐻𝐻′7 = � 1 0 0 0 5 2 5 1 5 2 5 0 1 1 6 1 0 1 0 0 2 1 5 5 0 6 4 1 4 6 0 4 0 0 1 0 0 0 0 0 4 1 6 5 5 6 5 5 0 0 0 1 1 5 5 2 3 1 1 3 0 6 2 3 �. here all syndromes will be different, too. we know the theorem, which says, that if h′ = [−pt|in−k], then g = [ik|p] (the reverse statement is also true), where ik is a 𝑘𝑘 ∗ 𝑘𝑘 identity matrix and p is a 𝑘𝑘 ∗ (𝑛𝑛 − 𝑘𝑘) matrix, 𝐺𝐺𝐻𝐻′𝑇𝑇 = 0. (1) thus, we can construct the generator matrix 𝐺𝐺7: 𝐺𝐺7 = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 2 5 0 6 1 0 0 0 0 0 0 0 0 0 0 0 5 6 0 2 0 1 0 0 0 0 0 0 0 0 0 0 2 2 0 2 0 0 1 0 0 0 0 0 0 0 0 0 6 2 0 5 0 0 0 1 0 0 0 0 0 0 0 0 2 0 3 4 0 0 0 0 1 0 0 0 0 0 0 0 5 1 6 6 0 0 0 0 0 1 0 0 0 0 0 0 2 3 1 6 0 0 0 0 0 0 1 0 0 0 0 0 0 6 2 4 0 0 0 0 0 0 0 1 0 0 0 0 6 3 2 0 0 0 0 0 0 0 0 0 1 0 0 0 6 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 2 5 0 0 0 0 0 0 0 0 0 0 1 0 6 3 2 4 0 0 0 0 0 0 0 0 0 0 0 1⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ . encoding procedure: in our scheme the message was presented by 12-tuples in z7. 𝑣𝑣 = (𝑎𝑎1, 𝑎𝑎2, 𝑎𝑎3, … , 𝑎𝑎12) is an arbitrary 12-tuple, and consider the vector 𝑢𝑢 that is the linear combination of columns 𝐺𝐺7 with 𝑎𝑎𝑖𝑖 is the 𝑖𝑖𝑡𝑡ℎ coefficient. 𝑢𝑢 = 𝑣𝑣𝐺𝐺 = (𝑐𝑐1, 𝑐𝑐2, 𝑐𝑐3, 𝑐𝑐4, 𝑎𝑎1, 𝑎𝑎2, 𝑎𝑎3, … , 𝑎𝑎12), where the first 4 components of the code vector are the parity check symbols and the next 12 components are information symbols, where 𝑐𝑐𝑗𝑗 = �∑ 𝑎𝑎𝑖𝑖𝑝𝑝𝑖𝑖𝑗𝑗 𝑘𝑘 𝑖𝑖=1 �𝑚𝑚𝑜𝑜𝑚𝑚7. (2) let us show the example to describe how we do these procedures. further results for encoding and decoding procedures of asymmetric low error correcting codes 108 example. let (0 1 2 6 4 0 6 5 4 1 2 2) be the message vector in z7. from (2) we can obtain parity check symbols by multiplying this message vector with the columns of the matrix 𝐺𝐺7. for example, the first parity check symbol is c1: 𝑐𝑐1 = (0 ∗ 2) + (1 ∗ 5) + (2 ∗ 2) + (6 ∗ 6) + (4 ∗ 2) + (0 ∗ 5) + (6 ∗ 2) + (5 ∗ 0) + (4 ∗ 6) + (1 ∗ 6) + (2 ∗ 1) + (2 ∗ 6) = 0 + 5 + 4 + 1 + 1 + 0 + 5 + 0 + 3 + 6 + 2 + 5 = 4(𝑚𝑚𝑜𝑜𝑚𝑚7). (all operations are in z7.) similarly, we can find other 3 parity check symbols: 𝑐𝑐2 = 5, 𝑐𝑐3 = 3, 𝑐𝑐4 = 1. after performing other multiple operations with matrix 𝐺𝐺7 we obtain this encoded vector: (4 5 3 1 0 1 2 6 4 0 6 5 4 1 2 2). as we can see in this code, the encoded message (codeword) has the length 16, from which the first 4 are parity check symbols, and the last 12 are information symbols. decoding procedure: now we will show how a decoding procedure will be implemented using the parity check matrix 𝐻𝐻′7, if during the message sending process the errors occured in the codewords. we will describe the decoding procedure by 3 steps: 1. receiver multiplies the vector with every row of matrix h′7 and gets the syndrome s = 𝐯𝐯h′. if s = (0,0,0,0) then there were not any errors in the received vector. 2. if the calculated syndrome s is a nonzero vector, then there are some occurred errors. these codes can correct only up to two errors with magnitude ±1. we know that all possible syndromes of matrix h′7 are different (±hi ± hj and hi ≠ hj). after calculating the syndrome the receiver knows from which two columns of the matrix h′7 the syndrome was resulted, consequently, it can find the two corresponding components of the vector, where the error was occurred and the direction of the error (if +hi, then upward direction or if −hi downward direction). on the other hand, if in the table of syndromes we do not have the resulted syndrome, then we cannot correct this kind of errors. 3. after finding the error components the receiver adds or subtracts 1 from them and obtains the sent code vector (c1, c2, c3, c4, a1, a2, a3, … , a12). so (a1, a2, a3, … , a12) is our message vector. an example. (4 5 3 1 0 1 2 6 4 0 6 5 4 1 2 2) is an encoded vector from the previous example. let 2 errors occur in the channel, and the receiver gets the vector (4 5 2 1 0 1 2 6 4 0 6 5 4 1 2 1). after performing multiple operations with rows of matrix 𝐻𝐻′7 the receiver obtains the syndrome(6 3 1 4). next from the table of syndromes it finds the corresponding columns, now they are 3 and 16. hence, the syndrome (6 3 1 4) was resulted from adding a negated column 3 of matrix 𝐻𝐻′7 to the negated column 16: 0 −1 0 −4 −1 −5 0 −3 = −1 −4 −6 −3 (𝑚𝑚𝑜𝑜𝑚𝑚7) = (6 3 1 4) (because in 𝑍𝑍7 0 = 7, −1 = 6, −2 = 5 , −3 = 4, −5 = 2, −6 = 1). hence, the error positions of encoded vector are 3 and 16 (both have a downward direction). h. khachatrian 109 so, it adds 1 to 𝑡𝑡ℎ𝑜𝑜 3rd component and 1 to 16𝑡𝑡ℎ of vector (4 5 𝟐𝟐 1 0 1 2 6 4 0 6 5 4 1 2 𝟏𝟏) and obtains the sent encoded vector (4 5 3 1 0 1 2 6 4 0 6 5 4 1 2 2). consequently, the message vector (code word) is (0 1 2 6 4 0 6 5 4 1 2 2) as we have in the example of the encoding procedure. using this code we can find and correct all possible 512 errors of the type ±1 in every vector over ring 𝑍𝑍7. 3.2 encoding and decoding for the code (20, 16) for the code (20, 16) over the ring z9 correcting double errors of the type ±1 we can do the same encoding and decoding processes as we did for the code (16, 12). in this case, the parity check matrix 𝐻𝐻′9 and the generator matrix 𝐺𝐺9 will have the following form: 𝐻𝐻′9 = � 1 0 0 0 6 7 8 5 0 6 6 0 7 3 5 4 7 4 7 4 0 1 0 0 0 6 0 2 4 7 1 4 1 1 1 1 7 4 3 3 0 0 1 0 6 4 5 6 6 3 0 6 3 5 6 3 7 1 4 5 0 0 0 1 1 4 1 5 2 1 6 8 6 3 0 6 4 1 7 0 �, 𝐺𝐺9= ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 3 0 3 8 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 3 5 5 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 4 8 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 7 3 4 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 5 3 7 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 3 2 6 8 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 3 8 0 3 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 5 3 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 8 6 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 6 8 4 6 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 4 8 3 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 5 8 6 3 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 2 2 2 5 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 5 5 8 8 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 2 6 5 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 5 6 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ unlike the previous case for the code (16, 12) over ring 𝑍𝑍7, in this case the message was presented by 16-tuples in 𝑍𝑍9. the encoded vector 𝑢𝑢 (codeword) has a length 20: 𝑢𝑢 = 𝑣𝑣𝐺𝐺 = (𝑐𝑐1, 𝑐𝑐2, 𝑐𝑐3, 𝑐𝑐4, 𝑎𝑎1, 𝑎𝑎2, 𝑎𝑎3, … , 𝑎𝑎16), where the first four are the parity check symbols: 𝑐𝑐𝑗𝑗 = �∑ 𝑎𝑎𝑖𝑖𝑝𝑝𝑖𝑖𝑗𝑗 𝑘𝑘 𝑖𝑖=1 �𝑚𝑚𝑜𝑜𝑚𝑚9 and the next 16 are information symbols. using this code we can find and correct all possible 800 errors of the type ±1 in every vector over ring 𝑍𝑍9. further results for encoding and decoding procedures of asymmetric low error correcting codes 110 4. conclusion in this paper an implementation of encoding and decoding procedures of optimal (16, 12) and (20, 16) linear codes over ring 𝑍𝑍7 and 𝑍𝑍9 correcting double ±1 errors is presented. we propose that this approach can be extended for implementation of similar procedures for the optimal codes over other rings 𝑍𝑍𝑛𝑛 and the research in this direction will follow. references [1] s. martirossian, “single error correcting close packed and perfect codes”, proc.1st intas int. seminar coding theory and combinatorics, armenia, pp. 90-115, 1996. [2] h. kostadinov, n.manev and h.morita, “on ±1 error correctable codes”, ieice trans.fundamentals, vol. e93-a, pp. 2578-2761, 2010. [3] g. khachatrian and h. morita, “construction of optimal 1 double error correcting linear codes over ring z5 ”, 3rd international workshop on advances in communications, boppard, germany, pp. 10-12, may 2014. [4] g. khachatryan and h. khachatryan, “construction of double ±1 error correcting linear optimal codes over rings 𝑍𝑍7 and 𝑍𝑍9” , mathematical problems of computer science, vol. 45, pp. 106—110, 2016. [5] h. khachatryan, “encoding and decoding procedures for double ±1 error correcting linear code over ring 𝑍𝑍5”, mathematical problems of computer science, vol. 43, pp. 57—61, 2015. submitted 17.09.2017, accepted 04.12.2017. այլ արդյունքներ ասիմետրիկ փոքր մեծությամբ սխալներ ուղղող կոդերով կոդավորման և ապակոդավորման ալգորիթմների համար հ. խաչատրյան ամփոփում այս հոդվածի շրջանակներում ներկայացված են կոդավորման և ապակոդավորման ալգորիթմները 𝑍𝑍7 և 𝑍𝑍9 օղակներում կառուցված ասիմետրիկ փոքր ամպլիտուդայով սխալներ ուղղող կոդերի համար: h. khachatrian 111 алгоритмы кодирования и декодирования для кодов исправляющих асимметричные двойные ошибки г. хачатрян аннотация в данной статье представлены алгоритмы кодирования и декодирования для кодов в кольцах 𝑍𝑍7 и 𝑍𝑍9 исправляющих двойные асимметричные ошибки. hamlet k. khachatrian начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 48, 64--73, 2017. effective and accurate binary clone detection hayk k. aslanyan ivannikov institute for system programming of the ras e-mail: hayk@ispras.ru abstract software developers usually copy and paste a particular piece of code as they prefer to use a pre-written or a partial solution as a basis for solving their problem. however, it can lead to various errors, as well as increase the size of the source and binary code. finding similar parts of code (clones) in binary code becomes more applicable when the source code is not available. additionally, a compiler can copy some parts of code during various transformations and create code clones, which do not exist in the source code. detection of binary code clones is used for malware analysis, finding semantic errors, detecting copyright violation, etc. this article discusses the existing methods of binary code clones detection and introduces a new method for binary clone detection. it consists of three main stages. the first stage is based on binnavi platform [1] and generates program dependence graphs for each binary function. graphs are generated based on reil [2] (reverse engineering intermediate language) platform-independent language. reil representation is supported for several architectures (x86, x86-64, arm, mips, ppc), thus ensuring the independence of the tool from the target architecture. the second stage detects clones based on previously created graphs. a polynomial heuristic algorithm is suggested for finding the maximum common subgraph of two program dependence graphs. at the third stage, the obtained clones are visualized for manual analysis. keywords: binary code clone, program static analysis, program dependence graph. 1. introduction the reuse of code fragments (a continuous sequence of code lines) is often encountered in software development process. there are several approaches for finding code clones, which are based on text [3], lexical [4], syntactic [5]—[7] and semantic [8]—[14] analysis of the program. however, all these methods analyze source code and the task of finding clones in binary code is less studied. these approaches are not applicable for binary code as binary code is platform dependent and registers and direct memory access are used, while in source code only variables are considered. detection of binary code clones has many practical applications such as finding functionally similar parts of the program, malware, semantic errors and copyright violations. binary code clones are divided into three main types. the first type of binary code clones are code fragments that completely match. the second type of binary code clones are code fragments, 64 h. aslanyan 65 which can differ in types, values, names of data and registers. the third type of binary code clones are code fragments that can differ in types, names of data and registers, and may also differ in some instructions (in a particular fragment some instructions may be added or removed). examples of assembly code clones (for x86 architecture) are shown in fig.1. the clone of the first type is exactly matched fragments. the clone of the second type uses ecx register instead of eax. the clone of the third type uses ecx register instead of eax and has one deleted instruction (imul eax, ebp+var_4]). fig.1. examples of code clones (x86 assembler). 2. binary code clone detection approaches text-based approach. jang et al. [15] proposed a fingerprinting algorithm called bitshred based on bloom filters to cluster malware samples. bitshred consists of three phases: shredding a file, creating a fingerprint, and comparing fingerprints. in the shredding phase, bitshred divides all executable code sections into fragments. then for each fragment fingerprint is calculated based on bloom filters [16]. at the last stage it compares fingerprints by the following ratio: j(a,b) = s(bfa∧bfb) / s(bfa∨bfb), where s(bf) is the count of set bits in the bf. finally, fragments, having a higher similarity score, are clustered together. the algorithm finds clones of only the first type. token-based approach. a. schulman [17] proposed a system that creates a hash for each function in a binary file. matched hashes that occur in more than one file indicate a clone of the code. hashes are based on opcodes and location labels of the opcodes. karim et al. [18] algorithm also considers hashes on instructions opcodes, but they are calculated using n-grams. the algorithms of the approach allow finding clones of the first and second types. metrics-based approach. the system, created by d. bruschi et al. [19], finds clones in binary files to detect malicious programs. first, the binary file is disassembled and normalized, then the dead code is deleted and the code is split into fragments. for each fragment, a metric is constructed based on the control flow graph. at the last stage, clones are detected by comparing the obtained metrics. effective and accurate binary clone detection 66 sæbjørnsen et al. [20] after obtaining the assembler from the binary file, create intermediate representations of the assembly code. then, a binary code is partitioned into overlapping code segments that consist of a block of contiguous assembly from a function. then they create a normalized instruction sequence, abstracting the information of memory location and registers. next, it performs clone detection on the normalized sequence. they define two methods for creating clone clusters. the first method is an exact match that uses a hash for each code region, and a clone exists if there are any repeated hash values. the second method is an inexact match, which extracts a set of features from a code region and looks for other code regions with the same feature set. they count the number of occurrences of each feature to create a feature vector for each region. next, they use locality-sensitive hashing (lsh) (andoni and indyk, [21]) on each region and perform a distance calculation for clustering based on features for inexact matching. based on this work, m. farhadi et al. [22] created a system for detecting clones of malicious code in programs. algorithms of the approach detect all three types of clones. structural-based approach. t. dullien et al. [23] proposed a system for comparing binary files based on structural analysis, to search for malicious code. the algorithm consists of two stages: the generation of several hashes of malicious code and the recognition of similarity between different sections of code based on the control flow graph. y. david and e.yahav [24] detect similarity among functions based on decomposition of functions into tracelets, which are continuous, partial traces of execution and are obtained from control flow graph. to measure similarity between two tracelets, they define a set of simple rewrite rules and measure how many rewrites are required to reach from one tracelet to another. they do this step by encoding the problem as a constraint-solving problem and measure distance using the number of constraints that have to be violated to reach a match. algorithms of the approach detect all three types of clones. behavior-based approach. in [25] d.e. krutz and e.shahab propose a new approach for code clone detecting, which also detects semantic clones. they use concolic analysis, which combines concrete and symbolic values in order to traverse all possible paths. 3. tool architecture for binary code clone detection the proposed model takes into account the following requirements:  finding all types of clones;  independence from the target architecture;  scalability: the size of the analyzed programs can reach hundreds of mb;  a large percentage of true positives (> 90%). it is allowed to specify values of two variables: the minimum number of instructions for clones (mn) and the minimum percentage of similarity (mp) of clones. the work of the tool is divided into three main stages: 1. the first stage is based on ida pro disassembler [26] and binnavi static analysis platform [1]. the ida pro disassembler is used as a tool for restoring structures and the control flow of the program. at this stage, machine code is translated to reil representation, then, pdg (program dependence graph) is generated for each function. 2. at the second stage, the assembler code clones are detected taking into account the parameters of the mn and mp. 3. at the third stage code clones and their corresponding pdgs are visualized. h. aslanyan 67 the main advantage of the proposed tool is that it is based on a semantic approach, which is more correct, than other approaches. 3.1 generation of pdgs binnavi platform is used to generate program dependence graphs for each function. binnavi provides an interface for generating and using various intermediate representations of the program based on reil, including the generation of the control flow graph, the generation of the call graph and data dependency graph. as a part of the tool, a new functionality has been added to the binnavi platform, which allows each function to automatically generate a control flow graph and a data flow graph and merge them into program dependence graph (fig. 2). the vertices of pdg correspond to reil instructions, and the edges are data and control dependencies between instructions. each vertex has id, which is the opcode of its reil instruction. fig. 2. example of pdg. 3.2 heuristic algorithm for maximum common subgraph detection of two pdgs at this stage a maximum common subgraph is calculated for each pair of pdgs with heuristic algorithm. this algorithm is named tracebasedslice. it uses several procedures, which are defined below. let 𝐺𝐺(𝑉𝑉, 𝐸𝐸) be a pdg and 𝑋𝑋 and 𝑌𝑌 be any sets of pdg vertices, such that 𝑋𝑋 ⊆ 𝑉𝑉. two vertices can be matching candidates, if the first vertex’s id, predecessors’ and successors’ amounts are equal to the second vertex’s id, predecessors’ and successors’ amount. definition 1: getpredecessors(x,y) procedure returns empty set if y is empty, otherwise returns vertices from x, which are not in y and are predecessors for y’s vertices. effective and accurate binary clone detection 68 definition 2: getsuccessors(x,y) procedure returns empty set if y is empty, otherwise returns vertices from x, which are not in y and are successors for y’s vertices. definition 3: sortvertices(x) procedure sorts vertices of x by their ids, predecessors count, successors count and binary address. definition 4: makecorrespondence (x, y) procedure returns pairs of vertices from sorted x and y sets, which are matching candidates. it considers vertices’ ids, predecessors, successors count and based on merging algorithm. definition 5: makeonecorrespondence(x, y) procedure returns a pair of vertices from sorted x and y sets, which are matching candidates. definition 6: for any m∈x and n∈y checkpredecessors(x,y,m,n) condition is satisfied, if predecessors of m from x and predecessors of n from y have the same set of ids. definition 7: for any m∈x and n∈y checksuccessors(x,y,m,n) condition is satisfied, if successors of m from x and successors of n from y have the same set of ids. definition 8: inducedsubgraph(x, g) procedure returns induces subgraph of x in g. procedure tracebasedslice input: pair of pdgs g1 (v1, e1), g2 (v2, e2) output: maximum common subgraph of g1 and g2 1. matchednodes1 ⊆ 𝑉𝑉1, matchednodes2 ⊆ 𝑉𝑉2 2. matchednodes1 ← ∅, matchednodes2 ← ∅ 3. nopredecessor1←{n∈v1:n hasn't predecessor} 4. nopredecessor2←{n∈v2:n hasn't predecessor} 5. continuematching ← true 6. while (continuematching) 7. continuematching←false 8. tempmatching⊆ 𝑉𝑉1 × 𝑉𝑉2 9. tempmatching← ∅ 10. sortedneighbours1← sortvertices (getpredecessors (v1, matchednodes1)) 11. sortedneighbours2← sortvertices (getpredecessors (v2, matchednodes2)) 12. tempmatching←makecorrespondence(sortedneighbours1, sortedneighbours2) 13. sortedneighbours1← sortvertices (getsuccessors (v1, matchednodes1)) 14. sortedneighbours2← sortvertices (getsuccessors (v2, matchednodes2)) 15. tempmatching←tempmatching ∪ makecorrespondence(sortedneighbours1, sortedneighbours2) 16. if tempmatching is empty 17. tempmatching←makeonecorrespondence (nopredecessor1, nopredecessor2) 18. if tempmatching is not empty 19. continuematching←true 20. for all (v1, v2) ∈ tempmatching 21. if checkpredecessors(matchednodes1, matchednodes2, v1, v2) and checksuccessors(matchednodes1, matchednodes2, v1, v2) 22. matchednodes1 ←matchednodes1 ∪ {v1} 23. matchednodes2←matchednodes2 ∪ {v2} 24. return inducedsubgraph(matchednodes1, g1), inducedsubgraph(matchednodes2, g2) h. aslanyan 69 after detecting the maximum common subgraph, the tracebasedslice procedure returns the common part of two functions. if it satisfies mn and mp, then the obtained results are saved and visualized. 3.3 visualization of binary code clones the last stage of the tool is visualization of clones. the purpose of the created graphical interface is to demonstrate the assembler code of the obtained clones, the percentage of their similarity, the corresponding graphs and maximum common subgraph (fig. 3). fig. 3. visualization of detected clones 4. results to assess the effectiveness of the tool, it is tested on various real projects. two successive versions are analyzed for each project. the tool analyzes the pairs of functions, which have the same name in old and new versions of projects. as the same function most likely will change at a few instructions in new version, then they should be clones. if the function is not changed, then they should be clones of the second type. the target machine for testing is intel i5, 4 cores, ram 16 gb. the results on several programs are shown in table 1. effective and accurate binary clone detection 70 table 1: results project name version binaries sizes (mb) functions count with the same name analyze time detected clones count (mp = 50%) detected clones count (mp = 90%) old new old new grep 3.0 3.1 0.74 0.74 308 10s 307 299 gdb 8.0 8.1 49 66 12707 6m 14s 11894 10683 findutils 4.4.1 4.4.2 1.1 1.1 484 12s 484 482 gcc 4.9.0 5.4.0 3.2 3.4 1041 50s 956 662 git 2.6.0 2.9.5 9.4 9.8 3257 1m 15s 3098 2627 bison 2.3 2.4 1.3 1.5 498 19s 498 497 5. conclusion in this paper, the main 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[25] d. e. krutz and e. shihab, "cccd: concolic code clone detection," wcre, 2013. d. e. krutz and e. shihab, "cccd: concolic code clone detection," 2013 20th working conference on reverse engineering, koblenz, 2013, pp. 489-490. [26] https://www.hex-rays.com/products/ida. effective and accurate binary clone detection 72 [27] h. k. aslanyan, s. f. kurmangaleev, v. g. vardanyan, m. s. arutunian and s. s. sargsyan, "platform-independent and scalable tool for binary code clone detection," trudy ispran/proc. isp ras, vol. 1, no. 2, pp. 215-226, 2016. submitted 02.08.2017, accepted 23.11.2017. երկուական կոդի կլոնների արդյունավետ և ճշգրիտ որոնում հ․ ասլանյան ամփոփում ծրագրային ապահովման մշակողները հաճախ պատճենում և տեղադրում են որոշակի կոդ, քանի որ նրանք նախընտրում են օգտագործել նախապես եղած լուծումը կամ մասնակի լուծումը որպես այլ խնդրի լուծման հիմք: այնուամենայնիվ, դա կարող է հանգեցնել տարբեր սխալների, ինչպես նաև նախնական և երկուական կոդերի մեծացմանը: կոդի նմանատիպ մասերը (կլոններ) երկուական կոդում գտնելու խնդիրը դառնում է ավելի կիրառելի, երբ նախնական կոդը հասանելի չէ: բացի այդ, տարբեր ձևափոխությունների ընթացքում կոմպիլյատորը կարող է պատճենել կոդի որոշ մասեր և ստեղծել կոդի կլոններ, որոնք գոյություն չունեն նախնական կոդում: երկուական կոդի կլոնների հայտնաբերումը կարող է օգտագործվել վնասակար կոդի, սիմվոլիկ սխալների, հեղինակային իրավունքի խախտումների հայտնաբերման համար: հոդվածում քննարկվում է երկուական կոդերի հայտնաբերման գոյություն ունեցող մեթոդները և ներկայացվում է երկուական կոդի կլոնների հայտնաբերման նոր մեթոդ: այն բաղկացած է երեք հիմնական փուլերից: առաջին փուլը հիմնված է binnavi համակարգի վրա և կառուցում է ծրագրի կախվածության գրաֆներ յուրաքանչյուր երկուական ֆունկցիայի համար: գրաֆները ստեղծվում են` հիմնվելով reil (reverse engineering intermediate language) ճարտարապետությունից անկախ լեզվի վրա: reil ներկայացումը կարելի է ստանալ մի քանի ճարտարապետությունների ասեմբլերների համար (x86, x86-64, arm, mips, ppc), դրանով ապահովելով գործիքի անկախությունը կոնկրետ ճարտարապետությունից: երկրորդ փուլը հայտնաբերում է կլոններ՝ նախկինում ստեղծված գրաֆների հիման վրա: երկու ծրագրի կախվածությունների գրաֆների առավելագույն ընդհանուր ենթագրաֆի հայտնաբերման համար առաջարկվում է բազմանդամային մոտարկող ալգորիթմ: երրորդ փուլում ստացված կլոնները ցուցադրվում են հետագա վերլուծության համար: h. aslanyan 73 эффективное и точное обнаружение клонов бинарного кода а. асланян аннотация разработчики программного обеспечения обычно копируют и вставляют определенный фрагмент кода, поскольку предпочитают использовать предварительно написанное решение или частичное решение в качестве основы для решения своей проблемы. однако это может привести к различным ошибкам, а также увеличить размер исходного и бинарного кода. поиск похожих частей кода (клонов) в двоичном коде становится более применимым, если исходный код недоступен. кроме того, компилятор во время различных преобразований может скопировать некоторые части кода и создать клоны кода, которых нет в исходном коде. обнаружение клонов бинарного кода используется для нахождения вредоносного кода, поиска семантических ошибок, обнаружения нарушения авторских прав и т. д. в этой статье обсуждаются существующие методы обнаружения клонов бинарных кодов и вводится новый метод их обнаружения. он состоит из трех основных этапов. первый этап основан на платформе binnavi и генерирует графы зависимостей программы для каждой двоичной функции. графы создаются на основе независимого от платформы языка reil (reverse engineering intermediate language). представление reil поддерживается для нескольких архитектур (x86, x86-64, arm, mips, ppc), что обеспечивает независимость инструмента от целевой архитектуры. второй этап обнаруживает клоны на основе ранее созданных графов. предлагается полиномиальный эвристический алгоритм для нахождения максимального общего подграфа двух графов зависимостей программы. на третьем этапе полученные клоны визуализируются для ручного анализа. references microsoft word m. kyrexian_10.doc mathematical problems of computer science 34, 2010. 31 construction of irreducible, normal and primitive polynomials over finite fields melsik kyureghyan institute for informatics and automation problems cryptography has been experiencing a drastic development in contemporary times. this is highly conditioned by fast growth of network systems and progress of communication technologies for which information security has become a problem of special importance. secure, trustworthy and reliable delivery of information is a central problem in information theory and coding theory with applications in computer science and telecommunication. as a rule, error correcting/detecting codes used to provide reliable transmission of data over an unreliable communication channel are unstable and do not ensure required data protection. this has necessitated employment of joint coding: cryptographic coding for data encryption and error correcting/detecting coding to improve reliability of communication on a channel. both types of coding are applied sequentially, guaranteeing security of information during communications. modern cryptography intersects the disciplines of mathematics, computer science and telecommunication, and is a cornerstone of computer and communications security. it addresses a wide range of problems, such as: a) electronic signatures are commonly used for signing digital documents and ensuring that downloaded applications are provided by a trusted source, and are central to the operation of public key infrastructures and many network security schemes. development of easy-realizable fast algorithms for electronic signature is one of hard problems. b) modern block ciphers are widely used to provide encryption of quantities of information, and/or a cryptographic checksum to ensure the contents have not been altered. design of easy-realizable fast algorithms for block ciphers remains a problem of high importance. the problem of explicitly constructing irreducible, normal and primitive polynomials over galois fields is one of the challenging problems in computer algebra, coding theory, cryptography and theory of finite fields and plays a major role in modern engineering, primarily due to wide use of such polynomials in variety of coding, cryptographic and computational applications. moreover, recent advances in these areas have awakened an even more interest to the subject of such polynomials. researchers of data coding laboratory have conducted research on the theoretical foundations of cryptography, the application of cryptography to network and system security. some theoretic and practical problems in this area have been attacked, particularly: 32 1. several recurrent methods of constructing irreducible polynomials over finite fields have been proposed recently [1]. 2. some effective algorithms, employing recurrent methods to construct primitive polynomials over finite fields have been considered [1-2]. 3. security characteristics of block ciphers safer+ and safer++ of safer family have been investigated from theoretical and practical point of view. the coordinate permutation chosen for use in both ciphers safer+ and safer++ is the “armenian shuffle”, which is used in place of ‘hadamard shuffle’ employed in the previous ciphers of the safer family. ‘armenian shuffle’ not only provides even better diffusion of safer+ and safer++ and enhances strength of these ciphers against both differential and linear cryptanalysis, but also runs significantly faster as it allows fewer number of encryption rounds in the cipher. this property enables building new cryptosystems, equivalent to safer+ and safer++ by their crypto characteristics, which would exceed previous ciphers in speed and would be similarly strongly secure against cryptanalysis [4-5]. 4. new public-key encryption and public-key digital signature schemes have been proposed based on discrete logarithm problem. for the public-key encryption scheme it has been shown that the given algorithm has an advantage over well known el-gamal public-key encryption scheme in terms of complexity of implementation and bandwidth efficiency. the specific of presented digital signature scheme is that the signature is addressed from a given user with a given public key to another user with a different public key so that only the recipient will be able to verify the signature from a specified user. the complexity of implementation is similar to digital signature standard algorithm (dsa) [3]. references [1]m. k. kyuregyan and g. m kyureghyan. irreducible compositions of polynomials over finite fields, arxiv: 1008.18637v1 [math.nt] 11 august 2010. accepted for publication in designs, codes and cryptography, an international journal, editors-in-chief: d. jungnickel; j.d. key; p. wild. [2] m. k. kyuregyan. “iterated constructions of irreducible polynomials over finite fields with linearly independent roots”, j. finite fields and their applications, vol. 10, issue 3 (2004), pp. 323-341. [3] khachatrian g., kyuregyan m., new public key encryption and signature scheme. proceedings of russian-germanarmenian workshop applications of information theory, coding and security. yerevan, armenia april 14-16, 2010 pp. 31-34. http://ipia.sci.am/web/waitsc2010.htm [4] j. l. massey, g. h. khachatrian and m. k. kuregian, nomination of safer+ as candidate algorithm for the advanced encryption standard (aes), submission document from cylink corporation to nist, june 1998. [5] j. l. massey, g. h. khachatrian and m. k. kuregian, nomination of safer++ as candidate algorithm for the new european schemes for signatures, integrity, and encryption (nessie), submission document from cylink corporation, 2000. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 50, 67--75, 2018. feedback driven grammar-based fuzzing seryozha a. asryan institute for informatics and automation problems of nas ra e-mail: asryan@ispras.ru abstract in this paper, we present a method for grammar-based fuzzing, which improves its penetration power. it is based on input data generation using a fuzzer feedback. several other methods are prone to create an initial set of acceptable test cases before the actual fuzzing process, and hence are unable to use the runtime information to increase the generated input’s quality. the proposed method uses the coverage information gathered for each input sample and guides grammar-based input generation. this method uses more than 120 bnf (backus-naur form) grammar rules described in antlr (another tool for language recognition) platform. experimental results show that our method feedback driven random test generation, has higher code coverage capabilities compared with the existing methods. keywords: fuzzing, bnf grammars, structured data, automated test generation 1. introduction correctness of compilers is crucial to most software projects. random testing, or fuzzing, has emerged as an important tool for finding bugs in compilers and runtimes. the main idea behind fuzzing is to feed target application (program under test) with a large amount of mutated test inputs to trigger unintended program behavior like hangs and crashes. the existing fuzzing approaches can be classified in three basic categories – blackbox fuzzing, whitebox fuzzing and graybox fuzzing. blackbox fuzzers had no knowledge about the program’s internal structure and, hence, are less effective. sometimes they can use grammars to generate inputs with specific characteristics. the second type of fuzzers usually combine fuzzing with heavy-weighted symbolic execution to improve the effectiveness by applying a symbolic engine in cases where fuzzer is unable to explore a new execution path (i.e., increase the code coverage). graybox fuzzing is something in between. it uses a light-weighted program instrumentation to extract partial information of the program to generate guided input samples without sacrificing execution speed. unfortunately, when it comes to testing applications with complex structured-inputs, fuzzers have several limitations. examples of such applications can be compilers, translators and interpreters. these applications have a multi-pass design and process input in multiple stages 67 feedback driven grammar-based fuzzing 68 (lexer, parser etc.). because of complicated checks and a huge amount of possible execution paths at the first stage, fuzzers are mostly unable to generate inputs that could exercise code beyond the first stage. currently there are several methods and instruments [1-6, 8-11] that use whitebox fuzzing or grammar-based data generation approaches to address the challenge. csmith [1] has been successfully used to identify hundreds of bugs in c/c++ compilers, however this and other similar approaches have significant drawbacks. csmith couples input generation logic with target programming language specifications. producing inputs based on this strategy requires expert knowledge and a significant engineering effort, which needs to be repeated from scratch for each new language. for example, to support a new programming language, this method requires the definition of a corresponding grammar and manual implementation of language features. grammar-based whitebox fuzzing [2] enhances the whitebox fuzzing technique by using the input grammar specification to construct valid test cases. it presents a dynamic test generation algorithm that uses symbolic execution to directly generate grammar-based constraints, the satisfiability of which is checked using a custom grammar-based constraint solver. the two main disadvantages of this method are the long runtime and the small set of available grammars. another instrument syntesk (syntax testing kit) [3] implements unitesk [4] technology. using bnf grammars, syntesk generates two sets of programs. the first set contains test cases, which will be accepted by compiler. in the second set instruments collect invalid programs, which will be rejected by the compiler. gramfuzz [5] can automatically detect grammar rules based on the provided input samples. after that, those grammars are used for further data generation. this instrument is mainly used for web browsers fuzzing. in its first step gramfuzz considers a set of inputs of html, css and javascript. then it tries to extract bnf rules and construct corresponding ast (abstract syntax tree). during fuzzing, inputs are generated by replacing the ast nodes with available elements (duplications are also accepted). the main limitations of this instrument are: not all bnf rules can be extracted from the provided samples, not all generated inputs are valid programs for parser. another instrument ifuzzer [6] finds bugs in javascript interpreters. it uses evolutionary computing techniques, such as genetic algorithms [7], to guide the fuzzer in generating uncommon input code fragments that may trigger exceptional behavior in the interpreter. ifuzzer gets as an input the context-free grammar of a particular language and a test suite with valid inputs. based on that grammar, it generates parse trees and extracts code fragments (fragment pool) from a given testsuite. the initial population for a genetic algorithm consists of random selection of programs, from the input test samples. for each new generation, ifuzzer uses the fitness function on inputs from the previous generation to determine a set of inputs upon which the new generation should be constructed. elements of the next generation are created by selecting random code fragments of the appropriate input code for replacement. replacement was performed by choosing a random member from the fragment pool. fitness function consists of a fuzzer feedback (whether the program crashed or not) and input complexity. this method uses only the final state of program execution as feedback and doesn’t consider information about the program execution paths (code coverage information), as well as the correlation between program paths. instrument superion [8] proposes a grammar-aware coverage-based grey-box fuzzing. this instrument uses the grammar of its test inputs to parse each input into an ast. based on the constructed ast, superion performs a trimming operation to reduce the size of the inputs, iteratively removing each subtree in the ast of a test input and observing coverage differences. it uses also two types of mutation strategies. the first one is afl’s [9] standart dictionary mutation. the second one replaces one subtree in the ast of a test input with the subtree from itself or another test input in the queue. one of the limitations of superion is that it needs well-documented grammars, as well as an initial set of valid test cases. the paper [10] proposes the instrument blendfuzz that uses grammars to create syntactically valid test inputs and guide test generation. this approach s. asryan 69 consists of two stages. the first stage requires an initial set of valid inputs and a corresponding grammar. according to the provided grammar, blendfuzz breaks test cases into grammatical fragments, which will be used as basic building blocks in the next stage. in the second one, generated code fragments are used to mutate the existing test cases. more specifically, blendfuzz selects a grammatical fragment of one input and replaces it with another one in the pool. this procedure is repeated systematically to generate a large set of input samples. although blendfuzz is quite successful in practice, it is based on random testing technique and doesn’t incorporate results of program execution. the paper [11] describes a learn&fuzz algorithm that uses sample inputs and neural-network-based statistical machine-learning techniques to automatically generate input grammars for grammar-based fuzzing. this algorithm can also generate new inputs based on the probability distribution of the learnt model. learn&fuzz algorithm is trained over a corpus of pdf files to generate test inputs for the microsoft edge pdf parser. the results can vary depending on different input formats and training sets. after studying the variety of methods mentioned above, we came to the conclusion that these methods have several limitations:  input generation strategy requires expert knowledge and a significant engineering effort, which needs to be repeated from scratch for each new language [1]  input generation is based only on the usage of the final state of the program execution (whether the program crashed or not) and doesn’t consider information about the program execution paths (code coverage information), as well as correlation between program paths [6]  method needs well-documented grammars and an initial set of valid test cases [8]  test generation is based on random testing technique and doesn’t use feedback of program execution [10]. in this paper, we propose a method for generating input data based on target bnf grammars. we develop this method on top of our previous paper [12], which increases the overall effectiveness and percentage of correctly generated input samples that will successfully pass the parsing stage. our work makes the following contributions:  we use the push down automata representation of bnf rules and fuzzer edge coverage information to direct the input generation towards increasing coverage.  using the coverage information, we add weighted values to edges (transitions) of push down automata of each input sample. then we use that information at the input generation stage to produce inputs with higher chances to explore new paths of program under test. the rest of this paper is organized as follows: section 2 describes our approach to grammarbased data generation. section 3 provides a detailed description of the implemented model of interaction between a fuzzer and an input generation component (sd-gen – structured data generation). section 4 presents the results of the performed experiments, and comparison with other methods. finally, section 5 presents our conclusion. 2. feedback-driven data generation 2.1 antlr’s grammar structure antlr platform provides bnf grammars for more than 120 different languages. as we discussed in our previous paper [12], each of these grammars has its own set of pushdown automata representations. feedback driven grammar-based fuzzing 70 figure 1 shows an example of a rule from bnf grammar and its corresponding pushdown automata. the rule is described with its name (rule name), followed by a single alternative, terminated with a semicolon, or it can also have alternatives. alternatives are either a list of rule elements or an empty list. in the picture above, “object” is a rule, which has “pair”, “ ‘{’ ‘}’ ” as its alternatives. in antlr platform, each rule should have a pushdown automata representation. for “object” rule (1) antlr would generate pushdown automata (2). we use this representation to distinguish interesting rules, which were used to build inputs to exercise new paths in target program during its fuzzing. 2.2 guided test case generation it is difficult to generate test cases for compilers, because their inputs are highly structured. vast majority of existing compiler fuzzing systems generate a set of inputs before actually starting the testing process. hence, they are not able to change their data generation models based on runtime information. despite the fact that some instruments use bnf grammars, without instrumentation feedback from the testing system, the generated samples will cover random parts of the target program. our test generation system tries to overcome these problems by creating dependencies between the generated input structure and information based on the target code coverage. the method used in our previous version of sd-gen (paper [12]) to generate data was based on the following algorithm. it takes two main parameters as input – depth (d) for each rule (that is, the maximum number of recursive selection); length (l) of generated input. additionally, we use two counters cd=0 and cl=0 to store the current depth and length values. 1. select a random rule from the bnf grammar and mark it as a start rule. add that rule to rlist. set the value of cl to the number of nodes in automata of start rule. go to step 2. 2. walk through rlist and select the first non-terminal symbol (nt node). if there is no nt node go to step 6, otherwise continue to step 3. object: ‘{‘ pair ( ‘,‘ pair ) * ‘}‘ | ‘{‘ ‘}‘ ; pair: string ‘:‘ value ; fig. 1. grammar rule representation in antlr. s. asryan 71 3. on the corresponding automata of nt, the rule algorithm selects a random path using the depth first search algorithm (dfs). then it replaces the nt rule in rlist with the constructed path and updates cl and cd values. 4. if cl and cd values reached their limits (cl < l and cd < d values), go to step 5, if not, go to step 2. 5. for each non-terminal node in rlist, the algorithm constructs the shortest terminal path and replaces it in rlist. 6. store the content of rlist in output file. to implement guided input generation, we added weight values to each transition of existing automata (figure 2). the above figure shows the automata for “object” rule of some bnf grammar, with weights on its edges. due to those weights we are able to make modification in our algorithm:  instead of choosing a random path on the corresponding automata of nt rule in step 3, now we select a path with the highest weight value. this can be achieved by choosing each transition using its weight as measurement for probability, i.e., select “2512” rather than “2523” because the weight of the second edge is less than the weight of the first one. to prevent the algorithm from selecting the same transition on every iteration, we decrease its weight by the appropriate value. 3. implementing sd-gen as plugin for fuzzing tool the current implementation of sd-gen as a fuzzer mutation plugin supports the configuration file that can be used to manually specify the number of testcases to be generated and its iteration counts. the bnf data generation plugin is designed to manage the execution of sd-gen instrument based on the assigned parameters. this plugin has the following main responsibilities:  run sd-gen instrument, which generates a set of bnf-based inputs using information about the transition weights received from the fuzzer. if such information does not yet exist, a set of completely random inputs is generated  for every constructed input, the plugin sends its rule transition information (including weights, if they exist) to the fuzzer engine. on the other hand, after detecting an interesting input, the fuzzer retrieves its rule transition information and updates it based on the target execution coverage results. each time when sdgen is called, it checks whether there is information about transitions weights or not. if the 4 4 6 6 3 2 3 1 4 6 3 3 2 1 5 3 4 4 fig. 2. automata of bnf rule with weight values. feedback driven grammar-based fuzzing 72 required information is available, it will choose the rules and transitions of that rule based on the exited data. doing this iteratively, improves the generated data accuracy and makes it possible to change the test case generation strategies in runtime, hence significantly increasing the impact of the constructed inputs on the target code coverage. the bnf data generation plugin is called in two different phases. first, it acts as a mutation plugin to continuously generate new inputs, and second, it is activated whenever fuzzing detects an interesting input (increases coverage or finds some crashes/hangs) sample to update the corresponding weight values. 4. results this section provides comparison results of the proposed method with our previous method and also with the existing methods. for analysis purpose we use several well-known compilers and interpreters. one of the main parameters of measurement is the application code coverage, as well as the number of generated inputs to gain that coverage. as shown in table 1, almost in all cases, our method was able to achieve more code coverage with less inputs. in case of gcc/g++, csmith gets more coverage results due to its way of generating inputs sample, which, starting from the initial sample, are all valid program instances. application name our previous method feedback driven grammarbased fuzzing (our current method) execution count coverage info. (%) gcc-7.1 24959 26107 ~9800 +4.6 g++-7.1 26154 30103 ~8400 +15.1 python-2.7 7561 7962 ~41000 +5.3 php-v7.1.7 2036 2107 ~325000 +3.5 luac-5.3.4 13395 16274 ~9700 +21.5 gfortran-7.1 24060 24950 ~5300 +3.7 5. conclusion fuzzing is a powerful technique for testing an application by randomly mutating its input values. however, for the certain type of application (compiler, interpreters) it is hard to generate test cases that will not fail at first levels of execution. our proposed method implements guided data generation using bnf grammars. we are able to generate new input testcases based on target application feedback on each input sample. it allows us to make modifications in our testcase generation strategies while continuing the fuzzing process. this method results in improvement of target code coverage and increases the analyses effectiveness. table. 1. experimental results for testing of the proposed method s. asryan 73 references [1] xuejun yang, yang chen, eric eide and john regehr, “finding and understanding bugs in c compilers”, pldi '11 proceedings of the 32nd acm sigplan conference on programming language design and implementation, san jose, california, usa, pp. 283-294, 2011. [2] p. godefroid, m. y. levin and d. molnar, "grammar-based whitebox fuzzing", pldi '08 proceedings of the 29th acm sigplan conference on programming language design and implementation, tucson, az, usa, pp. 206-215, 2008. [3] с. в. зеленов и н. в. пакулин, “верификация компиляторов – систематический подход”, proceedings of the institute for system programming, russia, vol. 13, no. 1. стр. 47-64, 2007. [4] i. b. bourdonov, a. s. kossatchev, v. v. kuliamin and a. k. petrenko, "unitesk test suite", fme '02 proceedings of the international symposium of formal methods europe on formal methods getting it right, london, uk, pp. 77-88, 2002. [5] t. guo, p. zhang, x. wang and q. wei, "gramfuzz: fuzzing testing of web browsers based on grammar analysis and structural mutation", second international conference on informatics & applications (icia), pp. 212-215, 2013. [6] s. veggalam, s. rawat, i. haller and h. bos, “ifuzzer: an evolutionary interpreter fuzzer using genetic programming”, proceedings of computer security 21st european symposium on research in computer security, pp. 581-601, sep. 2016 [7] r. i. mckay, n. x. hoai, p. a. whigham, y. shan and m. o’neill, “grammar-based genetic programming: a survey,” genetic programming and evolvable machines, vol. 11, pp. 365–396, may 2010. [8] j. wang, b. chen, l. wei and y. liu, “superion: grammar-aware greybox fuzzing”, 2018. [9] m. zalewski, web-page american fuzzy lop [10] d. yang, y. zhang and q. liu, “blendfuzz: a model-based framework for fuzz testing programs with grammatical inputs”, ieee 11th international conference on trust, security and privacy in computing and communications, pp. 1070-1076, london, june 2012. [11] p. godefroid, h. peleg and r. singh, “learn&fuzz: machine learning for input fuzzing”, proceedings of the 32nd ieee/acm international conference on automated software engineering, pp. 50-59, usa, nov. 2017 [12] s. sargsyan, sh. kurmangaleev, m. mehrabyan, m. mishechkin, t. ghukasyan and s. asryan, “grammar-based fuzzing", proceedings of ivannikov memorial workshop, pp. 32-36, armenia, may 2018. submitted 05.08.2018, accepted 04. 12.2018. feedback driven grammar-based fuzzing 74 ֆազինգի մեթոդ` կոնտեքստից ազատ քերականությունների օգտագործմամբ. տվյալների գեներացիա ֆազերի հետ հետադարձ կապի միջոցով ս. ասրյան ամփոփում մեր օրերում ֆազինգը համարվում է ավտոմատ թեստավորման ամենաարդյունավետ և ամենահայտնի մեթոդներից մեկը։ սակայն գոյություն ունեցող մեթոդներն ունեն սահմանափակ հնարավորություններ հետազոտելու այնպիսի ծրագրային համակարգեր (օր.՝ կոմպիլյատոորներ), որոնք մշակում են բարդ կառուցվածք ունեցող տվյալներ։ հոդվածում ներկայացված է ֆազինգի մեթոդ, որն օգտագօրծում է կոնտեքստից ազատ քերականությունները մուտքային տվյալների կառուցման համար՝ հիմնվելով ֆազինգի ընթացքում ստացված տվյալների վրա։ գոյություն ունեցող մոթոդներից շատերը հակված են կառուցել տվյալների ամբողջական հավաքածուներ, մինչ ֆազինգի սկիզբը, որի հետևանքով անհնար է դառնում օգտագործել կատարման ընթացքում ստացված ինֆորմացիան կառուցվող տվյալների որակը բարձրացնելու համար։ առաջարկվող մեթոդի նպատակն է ուղղորդել քերականության վրա հիմնված մուտքային տվյալների կառուցումը՝ օգտագործելով ինֆորմացիա ուսումնասիրվող ծրագրային համակարգի կոդի ծածկույթի մասին։ այս մեթոդն օգտագործում է ավելի քան 120 քերականությունների bnf (bakus-naur form) ներկայացումներ, նկարագրված antlr ծրագրային համակարգում։ փորձերի արդյունքները ցույց են տալիս, որ ներկայացված մեթոդի միջոցով կառուցված տվյալների շնորհիվ հնարավոր է ստանալ հետազոտվող ծրագրային ապահովման կոդի ավելի մեծ ծածկույթ, քան մյուս հայտնի մեթոդները։ фаззинг с использованием грамматических правил: генерация данных на основе обратной связи с фаззером с. асрян аннотация в наши дни фаззинг является одним из наиболее эффективных и широко используемых методов автоматического динамического анализа. несмотря на это, существующие методы имеют ограничения при тестировании приложений (компиляторы и интерпретаторы) обрабатывающие входные данные, имеющие сложную структуру. s. asryan 75 в статье представлен метод фаззинга на основе грамматических правил. метод основан на генерации входных данных программы, используя обратную связь с фаззером (информацию в результате выполнения фаззинга). множество других методов склонны создавать начальный набор тестовых примеров до начала процесса фаззинга, и, следовательно, не могут использовать информацию, доступную во время выполнения фаззинга, для повышения качества генерируемых тестовых примеров. предлагаемый метод использует покрытие кода программы, собранный для каждого тестового примера и направляет процесс построения новых входных данных (на основе грамматик). данный метод использует бнф (форма бэкуса — наура) представления более 120 грамматик, описанных в платформе antlr (another tool for language recognition). проведенные тестирования показывают, что метод генерации случайных тестов учитывая обратную связь с фаззером, позволяет достичь большего покрытия кода, чем существующие методы. microsoft word e. haroutunian_6.doc mathematical problems of computer science 34, 2010. 20 on close interaction of information theory and mathematical statistics e. a. haroutunian iiap nas ra dedicated to the 70-th anniversary of academician yuri haik shoukourian the first publications of the author (being scientific secretary of the computing centre of the armenian academy of science) was [1], where the notion of the e -capacity of the discrete memoryless channel (dmc) was introduced. the paper [2], the most frequently cited by information theory specialists, presents method of derivation of sphere packing bound for error probability of the dmc. in 1987 the author came back to computing centre renamed as institute for informatics and automation problems and directed in course of long epoch of twenty years (from 1986) by yury shoukourian. in 1992 the author defended the thesis of the doctor of science in moscow. in recent years e. haroutunian prepared more than ten candidates and one doctor of science specialized in information theory and mathematical statistics. international publications of last 20 years of the author and of his disciples are presented below. they can be classified to three following clusters. publications in information theory [1, 2, 4, 6-14, 19, 20, 22, 29-35] are devoted to study of estimates of error probability of various complex information transmission and protection systems. the second group includes investigations of different statistical problems and applications [3, 5, 15-18, 21, 23-28, 37-39]. the third section puts together papers devoted to interaction of methods of information theory and mathematical statistics [36, 40-43]. except cited below international publications more than forty scientific works were published in transactions of iiap and other republican editions. the research in the noted fields will be continued by staff of the laboratory of information theory and applied statistics of iiap in near future. references 1. e. a. haroutunian, “upper estimate of transmission rate for memoryless channel with countable number of output signals under given error probability exponent,” (in russian), 3rd all union conf. on theory of information transmission and coding, uzhgorod, publishing hous of the uzbek ac.sc., tashkent, pp. 83--86, 1967. 2. e. a. haroutunian, “estimates of the error probability exponent for a semi-continuous memoryless channel,” (in russian), probl. inform. transm., vol. 4, no. 4, pp. 37—48, 1968. 3. e. a. haroutunian, “on asymptotically optimal criteria for markov chains,” (in russian), first world congress of bernoulli society, section 2, vol. 2, no. 3, pp. 153-156, 1990. 4. m. e. haroutunian, “e-capacity of arbitrarily varying channel with informed encoder,” probl. inform. transm., (in russian), vol. 26, no. 4, pp. 16-23, 1990. 5. e. a. haroutunian, “logarithmically asymptotically optimal testing of multiple statistical hypotheses,” probl. contr. and inform. theory, vol. 19, no. 5--6, pp. 413-421, 1990. 21 6. e. a. haroutunian, r. sh. maroutian, “  ,e  -achievable rates for multiple descriptions of random varying source,” probl. contr. and inform. theory, vol. 20, no. 2, pp. 165--178, 1991. 7. m. e. haroutunian, “bounds of e-capacity for the channel with random parameter,” probl. inform. transm., (in russian), vol. 27, no. 1, pp. 14--23, 1991. 8. e. a. haroutunian, m. e. haroutunian, “channel with random parameter," proc. of 12-th prague conf. on inform. theory, statis. decision func. random proc., p. 20, 1994. 9. a. n. harutyunyan, e. a. haroutunian, “on properties of rate-reliability-distortion function,” ieee trans. inform. theory, vol. 50, no. 11, pp. 2768—2769, 1996. 10. e. a. haroutunian, m. e. haroutunian, “bounds of e-capacity region for restricted two-way channel,” (in russian), probl. inform. transm., vol. 34, no. 3, pp. 7--16, 1998. 11. e. a. haroutunian, a. n. haroutunian, a. r. kazarian, (ghazaryan) “on rate-reliabilities-distortions function of source with many receivers,” proc. joint session 6-th prague symp. asymptotic statist. and 13-th prague conf. inform. theory, statist. decision func. random proc., prague, vol. 1, pp. 217-220, 1998. 12. e. a. haroutunian, a. n. harutyunyan, “successive refinement of information with reliability criterion," in proc. ieee intern. symp. inform. theory, sorrento, italy, p. 205, 2000. 13. e. a. haroutunian, a. n. harutyunyan, a. r. ghazaryan, “on rate-reliability-distortion function for robust descriptions system,” ieee trans. inform. theory, vol. 46, no. 7, pp. 2690--2697, 2000. 14. e. haroutunian, a. ghazaryan, “on the shannon cipher system with wiretapper guessing subject to distortion and reliability requirements”, ieee international symposium on information theory, losanna, switzerland, p.324, 2002. 15. h. shahumyan, e. haroutunian, “on the most informative indices selection method”, proceedings of the intern. congress of mathematicians, beijing, p. 189, 2002. 16. n. ajabyan, f. topsoe, e. haroutunian, “on application of entropy analysis to spatiotemporal evolution of ecological models”, proceedings of the international conference on csit, yerevan. pp. 190-195, 2003. 17. h. shahumyan, e. haroutunian, “armenian web site on applied statistics”, proceedings of the international conference csit, yerevan, pp. 449-451, 2003. 18. i. safaryan, e. haroutunian, “common approach to the distributions mixture identification and dependence models analysis”, proceedings of international conference csit, yerevan, pp. 184-186, 2003. 19. m. e. haroutunian, “on multiple-access channel with random parameter,” proceedings of international conference csit, yerevan, armenia, pp. 174--178, 2003. 20. a. n. harutyunyan, e. a. haroutunian, “on properties of rate-reliability-distortion function”, ieee trans. on inform. theory, vol. 11, pp. 2768–2773, 2004. 21. e. a. haroutunian, p. s. avetisyan, i. a. safaryan, a. v. manasyan, “analysis of the multidimensional distributions non-homogeneity in social-economic models of teaching”, proceedings of vi (jubilee) international school-seminar “multivariate statistical analysis and econometrics”, pp. 5354, tsahkadzor, june 2004. 22. m. e. haroutunian, s. a. tonoyan, “random coding bound of information hiding e-capacity,” trans. of ieee intern. symp. infrom. theory, chicago, usa, p. 536, 2004. 23. r. ahlswede, e. haroutunian, “testing of hypothesis and identification”, electronic notes in discrete mathematics, 21, pp. 185 -189, 2005. 24. r. ahlswede, e. aloyan, e. haroutunian, “on logarithmically asymptotically optimal hypothesis testing for arbitrarily varying source with side information”, electronic notes in discrete mathematics 21, pp. 91-95, 2005. 25. e. haroutunian, “reliability in multiple hypotheses testing and identification problem”, nato science series: computer and system sciences, vol. 198, ios press, pp. 189 – 201, 2005. 26. e. haroutunian, h. shahumyan, a. manasyan, “armenian statistical web lab”, proceedings international conference csit, pp. 591-594, yerevan, armenia, 2005. 27. r. f. ahlswede, e. aloyan, e. a. haroutunian, “on logarithmically asymptotically optimal hypothesis testing for arbitrarily varying source with side information,” lecture notes in computer 22 science, vol. 4123, “general theory of information transfer and combinatorics”, springer, pp. 457-461, 2006. 28. r. f. ahlswede, e. a. haroutunian, “on logarithmically asymptotically optimal testing of hypotheses and identification”, lecture notes in computer science, vol. 4123, “general theory of information transfer and combinatorics”, springer, pp. 462-478, 2006. 29. m. haroutunian, “bounds of e-capacity for multiple-access channel with random parameter”, lecture notes in computer science, vol. 4123, ``general theory of information transfer and combinatorics", springer, pp. 166--183, 2006. 30. a. n. harutyunyan , “notes on conditions for successive refinement of information", lecture notes in computer science, vol. 4123, “general theory of information transfer and combinatorics”, springer, pp. 130--138, 2006. 31. a. n. harutyunyan, a.j. han vinck, “error exponent in avs coding”, proc. of ieee int. symp. inform. theory, seattle, wa, july 9--14, pp. 2166--2170, 2006. 32. e. haroutunian, “on bounds for ecapacity of dmc”, ieee trans. on inform. theory, v. 53. no. 11, pp. 4210—4220, 2007. 33. e. haroutunian, “reliability approach in wiretapper guessing theory”, “aspects of network and information security”, nato science for peace and security series, information and communication security, vol. 17, pp. 248-260, ios press, 2008. 34. m. e. haroutunian, “e-capacity of information hiding systems”, “aspects of network and information security”, nato science for peace and security series, information and communication security, vol. 17, pp. 261-273, ios press, 2008. 35. a. n. harutunyan, “remarks on e -optimal rate function in dms coding”, “aspects of network and information security”, nato science for peace and security series, information and communication security, vol. 17, pp. 308-314, ios press, 2008. 36. e. haroutunian, m. haroutunian, a. harutyunyan, “reliability criteria in information theory and in statistical hypothesis testing”, foundations and trends in communications and information theory, vol. 4, no. 2-3, pp. 1-171, 2008. 37. e. haroutunian, n. grigoryan, “on arbitrarily varying markov source coding and hypothesis lao testing by non-informed statistician”, proceedings of ieee int. symp. inform. theory, seoul, south korea, june 28--july 3, pp. 981-985, 2009. 38. e. haroutunian, p. hakobyan, “multiple hypotheses lao testing for many independent object”, international journal “scholarly research exchange”, vol. 2009, pp. 1--6, 2009. 39. e. haroutunian, p. hakobyan, "on hypotheses optimal testing for many independent objects", proceedings of international conference csit, pp. 141-144, yerevan, 2009. 40. m. grigoryan, a. n. harutyunyan, “multiple hypothesis testing for arbitrarily varying sources", proceedings of international workshop on applications of information theory, coding and security, yerevan, armenia, pp. 67-70, 2010. 41. a. n. harutyunyan, n. m. grigoryan, “chernoff bounds of hypothesis testing for arbitrarily varying sources", proceedings of international workshop on applications of information theory, coding and security, yerevan, armenia, pp. 71-74, 2010. 42. n. m. grigoryan, a. n. harutyunyan, “error exponents in multiple hypothesis testing for arbitrarily varying sources”, ieee international workshop on information theory, dublin, ireland, pp. 1-5, 2010. 43. e. a. haroutunian, “information theory and statistics”, international encyclopedia of statistical science, springer-verlag, pp. 643-645, 2010. microsoft word article1.doc ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 31, 163—166, 2008. 163 о методике построения корпоративных сетей передачи данных на базе технологии mpls vpn нана д. григорян государственный педагогический университет им. х.абовяна zarasak@dolphin.am аннотация статья посвящена попытке анализа методики построения корпоративных сетей передачи данных на базе технологии mpls vpn (multiprotocol label switching virtual private network) организации вкс на базе многопротокольной коммутации ip-пакетов по меткам. технология mpls позволяет обеспечить управляемость, легкую наращиваемость, надежность и высокую готовность решений, необходимых для предоставления мультисервисных услуг организациям. данный метод целесообразно использовать при создании кспд единой информационной системы армении, целью которой является повышение эффективности контроля и надзора за безопасностью и качеством продукции и услуг при их создании, поступлении и обращении на потребительском рынке армении. литература [1] м. шестаков, принципы построения корпоративных сетей передачи данных, «компьютерра», № 256, 1997 г. [2] м. захватов, построение виртуальных частных сетей (vpn) на базе технологии mpls. руководство cisco., 2001 г. [3] edited by jeff doyle and matt kolon. mcgraw-hill/osborne // juniper networks routers: the complete reference, 2002. [4] ivan pepelnjak, jim guichard elnjak, jeff apcar. cisco systems // mpls and vpn architectures, vol. 2, 2004. [5] r. aggarwal et al., multicast in 2547 vpns and vpls, internet draft, work in progress, http://www.ietf.org/internet-drafts/draft-raggarwa-l3vpn-mvpn-vpls-mcast-01.txt november, 2004. [6] r. aggarwal, d. papadimitriou, s. yasukawa (editors) et al., extensions to rsvp-te for point to multipoint te lsps // internet draft, work in progress, http://www.ietf.org/internetdrafts/draft-ietf-mpls-rsvp-te-p2mp-01.txt november, 2004. [7] в. олвейк, структура и реализация современной технологии mpls., 2004 г. [8] y. serbest, ray qiu, rob nath. supporting ip multicast over vpls, february, 2005, http://www.ietf.org/ internet-drafts/draft-serbest-l2vpn-vpls-mcast-02.txt о методике построения корпоративных сетей передачи данных на базе mpls vpn 164 pls ï»ëýáéá·ç³ûç ñçù³ý íñ³ ïíû³éý»ñç ñ³õáñ¹ù³ý ïáñåáñ³ïçí ó³ýó»ñç ï³éáõóù³ý ù»ãá¹çï³ûç í»ñéáõíáõãûáõýá ü. ¶ñç·áñû³ý ²ù÷á÷áõù ðá¹í³íá mpls ï»ëýáéá·ç³ûç ñçù³ý íñ³ ïíû³éý»ñç ñ³õáñ¹ù³ý ïáñåáñ³ïçí ó³ýó»ñç ï³éáõóù³ý ù»ãá¹çï³ûç í»ñéáõíáõãû³ý ÷áñó ¿: mpls ï»ëýáéá·ç³ý ãáõûé ¿ ï³éçë ³å³ñáí»é ³ûý áñáßáõùý»ñç ï³ñ·³íáñ»éçáõãûáõýá, ³ñ³· ³×ý áõ ñáõë³éçáõãûáõýá, áñáýù ³ýññ³å»ßï »ý ï³½ù³ï»ñåáõãûáõýý»ñçý µ³½ù³½³ý í³é³ûáõãûáõýý»ñ ù³ïáõó»éáõ ñ³ù³ñ: îíû³é ù»ãá¹á ýå³ï³ï³ñ³ñù³ñ ¿ û·ï³·áñí»é ð³û³ëï³ýç ùç³ëý³ï³ý ï»õ»ï³ïí³ï³ý ñ³ù³ï³ñ·ç ïíû³éý»ñç ñ³õáñ¹ù³ý ïáñåáñ³ïçí ó³ýó ëï»õí»éáõ å³ù³ý³ï: начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 45, 35--43, 2016. analysis of experiments of a new approach for test quality evaluation mariam e. haroutunian, varazdat k. avetisyan institute for informatics and automation problems of nas ra e-mail: armar@ipia.sci.am, avetvarazdat@gmail.com abstract in the previous paper [1] we suggested a new model of test quality evaluation based on information measures such as shannon entropy and average mutual information. to establish the practical bounds of these measures and the required number of examinees, some experiments were conducted. in this paper the analysis of these experiments are provided. keywords: test quality, shannon entropy, average mutual information, classical test theory, item response theory. 1. introduction test developers are basically concerned about the quality of test items and how examinees respond to it when constructing tests. test theories and related models provide a frame of reference for doing test design work or solving other practical problems. a good test model might specify the precise relationships among test items and ability scores, so that careful design work can be done to produce desired test score distribution and errors of the size that can be allowed. a good test theory or model can also handle errors of measurements by helping understand the role that measurement errors play in estimating examinee’s ability and correlations between variables and true scores or ability scores. there are two currently popular statistical frameworks to address test data analysis and test quality evaluation: classical test theory (ctt) [2] and item response theory (irt) [3]. ctt is a theory about test scores that introduces three concepts test score, true score and error score. in the ctt, the notion of ability is expressed by the true score, which is defined as "the expected value of observed performance on the test of interest." an examinee's ability is defined only in terms of a particular test. when the test is "hard," the examinee will appear to have low ability; when the test is "easy," the examinee will appear to have higher ability. ctt was the dominant statistical approach for 35 mailto:armar@ipia.sci.am mailto:avetvarazdat@gmail.com analysis of the experiments of a new approach for test quality evaluation 36 testing data until lord and novick (1968) placed it in the context with several other statistical theories of mental test scores, notably irt. irt is a model-based measurement statistical theory in which the performance of an examinee on a test item can be predicted (or explained) by a set of factors called traits, latent traits, or abilities; and the relationship between the examinees' item performance and the set of traits underlying item performance can be described by a monotonically increasing function called an item characteristic function or item characteristic curve (icc). each of these approaches has its advantages and disadvantages [4]. for example, in ctt item parameters are dependent on the examinee sample from which they are obtained, but in irt these parameters are examinee group independent. but on the other hand, in case of ctt smaller examinee sample sizes are required for analysis and the methods are simpler compared to irt. besides the existing ctt and irt models, we have developed a new approach [1] based on information measures such as shannon entropy and average mutual information. the main idea of the new approach is the following. suppose that the test consists of n items, each item can be considered as a binary random variable (rv) x1, x2, .., xn with probabilities p for correct answers and 1 − p, for incorrect answers: 𝑋𝑋𝑖𝑖 = � 1 𝑤𝑤𝑤𝑤𝑤𝑤ℎ 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑤𝑤𝑤𝑤𝑝𝑝 𝑝𝑝𝑖𝑖, 0 𝑤𝑤𝑤𝑤𝑤𝑤ℎ 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑤𝑤𝑤𝑤𝑝𝑝 1 − 𝑝𝑝𝑖𝑖, 𝑤𝑤 = 1, 𝑁𝑁.�� we consider shannon entropy of rv 𝑋𝑋𝑖𝑖 𝐻𝐻(𝑋𝑋𝑖𝑖) = −�𝑝𝑝(𝑥𝑥𝑖𝑖 ) log 𝑝𝑝(𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 ) and the average mutual information of two items: 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗 � = � 𝑝𝑝(𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑗𝑗 ) log 𝑝𝑝(𝑥𝑥𝑖𝑖, 𝑥𝑥𝑗𝑗 ) 𝑝𝑝(𝑥𝑥𝑖𝑖) ∗ 𝑝𝑝(𝑥𝑥𝑗𝑗 )𝑥𝑥𝑖𝑖,𝑥𝑥𝑗𝑗 = 𝐻𝐻(𝑋𝑋𝑖𝑖) − 𝐻𝐻�𝑋𝑋𝑖𝑖 | 𝑋𝑋𝑗𝑗� = 𝐻𝐻�𝑋𝑋𝑗𝑗� − 𝐻𝐻�𝑋𝑋𝑗𝑗 | 𝑋𝑋𝑖𝑖�. our test quality evaluation model consists of the following methods: method 1. if the value of 𝐻𝐻(𝑋𝑋𝑖𝑖) is close to 0, it means that we have a bad test item, which can be very easy or very difficult. if the value of 𝐻𝐻(𝑋𝑋𝑖𝑖) is close to 1 we have a good test item. method 2. if the value of 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗 � is close to 0, it means that there is independency of test items xi and xj . in case of values close to min�𝐻𝐻(𝑋𝑋𝑖𝑖), 𝐻𝐻�𝑋𝑋𝑗𝑗�� xi and xj items repeat each other. method 3. if t h e value of conditional entropy �𝐻𝐻�𝑋𝑋𝑗𝑗 | 𝑋𝑋𝑖𝑖�� is close to 𝐻𝐻(𝑋𝑋𝑖𝑖 ), then xi and xj are independent. however, several questions remain open. 1. how precisely our model evaluates the quality of test items and how comparable is it to ctt and irt estimation methods? 2. which are the permissible limits of 𝐻𝐻(𝑋𝑋𝑖𝑖) and 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗 � ? m. haroutunian, v. avetisyan 37 3. which is the sufficient number of the examinee samples for precise evaluation? the answers to these questions can be found experimentally. 2. description of experiments the results of school final exams of armenian language and literature held in 2008 were selected for testing. the results were provided in encrypted form by the center for assessment and testing. four test-results are chosen to be analyzed. each test consists of 80 items, and the number of schoolchildren who participated in the examination process is 2000. the names of the first 50 𝑋𝑋𝑖𝑖 items are 𝐴𝐴1, 𝐴𝐴2, … 𝐴𝐴50 and the names of the last 30 𝑋𝑋𝑖𝑖 items are 𝐵𝐵1, 𝐵𝐵2, … 𝐵𝐵30. for analysis test quality evaluation system developed by us was used in [5]. for each item of four tests the 𝐻𝐻(𝑋𝑋𝑤𝑤), ctt difficulty index [2] and irt b parameter [3] values have been calculated, the comparability of the mentioned parameters observed and the permissible limits of 𝐻𝐻(𝑋𝑋𝑖𝑖)defined. difficulty is defined in both ctt and irt. in ctt the difficulty index p is the proportion of examinees who answer the item correctly. for multiple-choice, true/false, and other items that are scored as right (1 point) or wrong (0 points), item difficulty is the proportion of examinees who answered the item correctly. it ranges from 0 to 1. item difficulty for a polytomous item (an item scored in more than two ordinal categories) is simply the item mean or average item score. it ranges between the minimum and the maximum possible item scores. in irt the difficulty index b (irt b parameter) is on the same metric as the proficiencies or traits. this metric is arbitrary, but often it is anchored so that the proficiency distribution in a designated group has a mean of 0 and standard deviation of 1. the item difficulty identifies the proficiency at which about 50% of the examinees are expected to answer the item correctly. to observe the dependency of 𝐻𝐻(𝑋𝑋𝑖𝑖) and 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗� values on the examinee sample size and define the enough number of examinee samples five experiments are carried out for each test. the analysis was conducted by choosing the same test at random based on 500, 300, 200, 100, 50 participants’ results. for each test 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗� and ctt correlation coefficient 𝑅𝑅�𝑋𝑋𝑖𝑖 , 𝑋𝑋𝑗𝑗� between 𝑋𝑋𝑖𝑖 and 𝑋𝑋𝑗𝑗 items [2] was calculated, their compatibility was observed and the permissible limits of 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗� were defined. correlation coefficient ranges from -1 +1. coefficient value should be small or equal to 0.3. if coefficient value is close to +1, it means that test items repeat each other and one of that items should be removed from the test. the negative correlation means there is an independency of test items. 3. the analysis of results. based on the first experiment results comparison of 𝐻𝐻(𝑋𝑋𝑖𝑖), ctt difficulty index p and irt b parameter values of test1 are shown in figure 1. according to ctt test items for which difficulty values are between 0.3 and 0.74 interval are good items (not easy and not very difficult 34 items), and based on the analyzed data we can see that 𝐻𝐻(𝑋𝑋𝑖𝑖) values of test 1 for these items are between 0.82 and 1.0. for easy test items, difficulty values are between 0.75 and 0.9 (29 items), 𝐻𝐻(𝑋𝑋𝑖𝑖) values are between 0.48 and 0.81. for very easy test items difficulty values are between 0.9 and 1.0 (14 items), 𝐻𝐻(𝑋𝑋𝑖𝑖) values are between 0.12 and 0.47. approximately the same results were obtained for the tests 2, 3 and 4. as we can see in case of 𝐻𝐻(𝑋𝑋𝑖𝑖)’s large values close to 1 irt b parameter gets large values. analysis of the experiments of a new approach for test quality evaluation 38 fig. 1. 𝐻𝐻(𝑋𝑋𝑖𝑖), ctt difficulty parameters p and irt b parameters of test1. h(xi) irt b difficulty p -5 -4 -3 -2 -1 0 1 2 3 a11a21b19 a3 a48 b6 b9 a50 a4 b12a14a13a47b24a10 a8 a26a24b21a40a25b11b14a46b16b28 m. haroutunian, v. avetisyan 39 to analyze the dependency of 𝐻𝐻(𝑋𝑋𝑖𝑖) values on the number of examinee sample we draw 𝐻𝐻(𝑋𝑋𝑖𝑖) graphics of each test based on the results of five experiments. the graphics are shown in figure 2. fig. 2. 𝐻𝐻(𝑋𝑋𝑖𝑖) graphics for five experiments of test1. the maximum differences of 𝐻𝐻(𝑋𝑋𝑖𝑖) values are presented in table1. table 1. maximum difference of 𝐻𝐻(𝑋𝑋𝑖𝑖) values 500 300 200 100 50 500 0.089 0.07 0.135 0.2 300 0.089 0.13 0.15 0.19 200 0.07 0.13 0.16 100 0.135 0.15 0.16 0.3 50 0.2 0.19 0.23 0.3 while decreasing the examinee sample size until 100, it is obvious that the differences of 𝐻𝐻(𝑋𝑋𝑖𝑖) values are small and the maximum difference is 0.15. but when examinee sample size is decreased more than 100, the difference is close to 0.3, and in case of values equal to 50 the difference is close to 0.3. with the same principle for each test the mutual information 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗� was calculated and the dependency of 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗� values on the number of examinee sample was observed. the graphics are shown in figure 3. analysis of the experiments of a new approach for test quality evaluation 40 fig. 3. 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗� for five experiments of test1 a1 item. the average mutual information and correlation between test items also have been analyzed. the graphics based on some items’ data are presented in figure 4 and figure 5. fig. 4. correlation �𝑅𝑅�𝑋𝑋𝑖𝑖 , 𝑋𝑋𝑗𝑗�� and average mutual information �𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗�� between a1 item and other 80 items. a1 (500) a1 (300 a1 (200) a1 (100) a1 (50) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 a27b15a12a31 b8 b21a33b25a32 b1 b23a10a16a46 b4 a17a24b13a47b29a14b17 b9 b12 r(x,y) i(x^y)x10 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 a4 a7 a1 0 a1 3 a1 6 a1 9 a2 2 a2 5 a2 8 a3 1 a3 4 a3 7 a4 0 a4 3 a4 6 a4 9 b2 b5 b8 b1 1 b1 4 b1 7 b2 0 b2 3 b2 6 b2 9 m. haroutunian, v. avetisyan 41 fig. 5. correlation �𝑅𝑅�𝑋𝑋𝑖𝑖 , 𝑋𝑋𝑗𝑗�� and average mutual information �𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗�� between a11 item and other 80 items. table 2. a11 a5 a8 a15 a26 a31 b18 b19 b20 b23 r(x,y) -0.0083 -0.0186 -0.0581 -0.0195 -0.0385 -0.0005 -0.0151 0.0219 -0.0177 i(x∧y) 0.0011 0.0013 0.0049 0.0001 0.00001 0.0001 0.000004 0.0004 0.00006 for item a11 the negative correlation with other items and average mutual information values are presented in table 2. when we compare items’ correlations and average mutual information graphics, it is easy to see that the results are comparable. for example, from the correlation matrix and graphics shown in figure 5 we can see that a11 test item has negative correlation with other test items, for these items the values of mutual information are presented in table2. if correlation values are negative, average mutual information values are small enough, but from test we should remove those items, so the smallest permissible limit value of average mutual information should be 0.005. 4. conclusion in this research while analyzing the data, the following has been determined. 1. the methods suggested by us are correctly defining the quality of test items and the results are comparable with ctt and irt estimation methods. 2. simpler mathematical analysis is needed compared to irt. 3. 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗� describes the dependence of test items which does not have an equivalent in irt. 4. for test good items 𝐻𝐻(𝑋𝑋𝑖𝑖) values should be between 0.8 and 1.0, for fairly good items (easy) values are between 0.45 and 0.8, and for bad test items 𝐻𝐻(𝑋𝑋𝑖𝑖) values are between 0 and 0.45. r(x,y) i(x^y)x10 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 a1 a4 a7 a1 0 a1 3 a1 6 a1 9 a2 2 a2 5 a2 8 a3 1 a3 4 a3 7 a4 0 a4 3 a4 6 a4 9 b2 b5 b8 b1 1 b1 4 b1 7 b2 0 b2 3 b2 6 b2 9 analysis of the experiments of a new approach for test quality evaluation 42 5. for 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗 � preferable are the values smaller than 0.05 and greater than 0.005 0.005≤ 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗 � ≤ 0.05 6. smaller sample sizes are required in comparison with ctt and irt. the sample size should be more than 100. in ctt the sample size is between 200 and 500, and in irt it depends on the irt model, but samples over 500 are needed. references [1] m. haroutunian and v. avetisyan, “new approach for test quality evaluation based on shannon information measures”, transactions of ipia of nas ra, mathematical problems of computer science, vol. 44, pp. 7-21, 2015. [2] m. b. chelishkova, theory and practice of pedagogical tests constructing, moscow: logos, 2002. [3] c. demars, item response theory. oxford university press; 1 edition, 2010. [4] k. hambleton and w. jones, “comparison of classical test theory and item response theory and their applications to test development”, educational measurement: issues and practice, vol. 12, no. 3, pp. 38-47, 1993. [5] m. haroutunian and v. avetisyan, “development of the test quality evaluation system”, proceedings of the international conference on computer science and information technologies (csit 2015), yerevan, armenia, september 28-october 2, pp. 372--375, 2015. submitted 10.10.2015, accepted 20.01.2016 թեստի որակի գնահատման նոր մոտեցման փորձարկումների վերլուծություն մ․ հարությունյան,վ․ ավետիսյան ամփոփում նախորդ հոդվածում [1] հեղինակների կողմից առաջարկվել է թեստերի որակի գնահատման նոր մոդել՝ հիմնված շենոնի էնտրոպիայի և միջին փոխադարձ ինֆորմացիայի վրա։ այս մեծությունների սահմանային արժեքները և թեստավորման մասնակիցների բավարար քանակը որոշելու համար կատարվել են փորձարկումներ։ հոդվածում ներկայացված է այդ փորձարկումների վերլուծությունը, որից հետևում է, որ հաշվարկները ավելի պարզ են irt-ի համեմատությամբ, թեստավորման արդյունքների վերլուծության համար պահանջվում է մասնակիցների ավելի քիչ քանակ, քան` ctt-ում և irt-ում։ m. haroutunian, v. avetisyan 43 анализ экспериментов нового подхода для оценки качества теста м. арутюнян, в. аветисян аннотация в предыдущей статье [1] авторами была предложена новая модель оценки качества теста на основе энтропии шэннона и средней взаимной информации. для того, чтобы установить практические пределы этих величин и необходимое количество экзаменуемых, были проведены эксперименты. в данной статье представлен анализ этих экспериментов, из которого следует, что расчеты более просты по сравнению с irt, для анализа результатов тестирований требуется меньшее количество экзаменуемых, чем в стт и irt. analysis of experiments of a new approach for test quality evaluation mariam e. haroutunian, varazdat k. avetisyan 2. description of experiments d:\sbornik\...\rev.dvi mathematical problems of computer science 30, 5{17, 2008. an e ±cient m ethod for gener ation of m ar ch t ests b ased on for mulas gurgen harutunyany, davit melkumyany, hasmik elchyanz, valery vardaniany y virage logic e-mail: fgurgen, davit.melkumyan, valery.vardaniang@viragelogic.com zyerevan state university e-mail: hasmik.elchyan@yahoo.com abstract a general method for generation of minimal march tests to detect or diagnose any subclass of simple static or dynamic faults in static rams is described. the proposed method is shown to generate all possible march tests satisfying certain necessary conditions for detection of faults. a correspondence between march tests and natural numbers is established that allows construct a formula that enables generation of all march tests detecting certain faults. as an example, the method is applied for construction of new minimal march tests for detection of several subclasses of three-operation dynamic faults. the method can be generalized for detection/diagnosis of any subset of static or dynamic faults. refer ences [1 ] a .j. va n d e go o r , te s t in g s e m ic o n d u c t o r m e m o r ie s : th e o r y a n d p r a c t ic e , j ohn w iley & sons, 1 9 9 1 . [2 ] a . j. va n d e go o r , b . s m it , " th e a u t o m a t ic g e n e r a t io n o f m a r c h t e s t s " , ie e e international w orkshop m emory technology d esign and testing, p p . 1 3 1 -1 3 6 , 1 9 9 3 . [3 ] k . za r r in e h , s . j. u p a d h ya ya , a n d s . ch a kr a va r t y, " a n e w fr a m e wo r k fo r g e n e r a t in g o p t im a l m a r c h t e s t s fo r m e m o r y a r r a ys " , p roc. int. conf. (itc), p p . 7 3 -8 2 , 1 9 9 8 . [4 ] a . b e n s o , s . d i ca r lo , g. d i n a t a le , p . p r in e t o , " a n o p t im a l a lg o r it h m fo r t h e a u t o m a t ic g e n e r a t io n o f m a r c h t e s t s " , d ate 2002, ie e e d esign, automation and test in e urope conference and e xhibition, p p . 9 3 8 9 3 9 , 2 0 0 2 . [5 ] s . m. a l-h a r b i, s . k . gu p t a , " a n e ± c ie n t m e t h o d o lo g y fo r g e n e r a t in g o p t im a l a n d u n ifo r m m a r c h t e s t s " , ie e e vl si test symposium, p p . 2 3 1 -2 3 7 , 2 0 0 1 . [6 ] c.-f. w u , c.-t. h u a n g , a n d c.-w . w u , " r a ms e s : a fa s t m e m o r y fa u lt s im u la t o r " , p roc. int. symp. d efect and f ault tolerance in vl si systems (d f t), albuquerque, p p . 1 6 5 -1 7 3 , n o v. 1 9 9 9 . [7 ] j.-f. l i, k .-l . ch e n g , c.-t. h u a n g , a n d c.-w . w u , " ma r c h -b a s e d r a m d ia g n o s t ic a lg o r it h m s fo r s t u c k-a t a n d c o u p lin g fa u lt s " , p roc. ie e e itc, p p . 7 5 8 -7 6 7 , 2 0 0 1 . 5 6 an e±cient method for generation of march tests based on formulas [8 ] t. gyo n jya n , v . a . v a r d a n ia n , " a n e ± c ie n t a lg o r it h m fo r g e n e r a t in g m in im a l m a r c h t e s t s fo r fa u lt d e t e c t io n a n d d ia g n o s is in s t a t ic r a n d o m a c c e s s m e m o r ie s " , international d esign and test w orkshop, d ubai, p p . 1 9 -2 0 , 2 0 0 6 . [9 ] a . b e n s o , a . b o s io , s . d i ca r lo , g. d i n a t a le , p . p r in e t t o , " a u t o m a t ic m a r c h t e s t s g e n e r a t io n fo r s t a t ic a n d d yn a m ic fa u lt s in s r a ms " , [1 0 ] g. h a r u t u n ya n , v . a . v a r d a n ia n , y . zo r ia n , " min im a l m a r c h t e s t s fo r d yn a m ic fa u lt s in r a n d o m a c c e s s m e m o r ie s " , j ournal of e lectronic testing: theory and applications, v o l. 2 3 , n u m b e r 1 , p p . 5 5 -7 4 , 2 0 0 7 . [1 1 ] l . d ilillo , p . gir a r d , s . p r a vo s s o u d o vit c h , a . v ir a z e l, m. b a s t ia n , " r e s is t ive -o p e n d e fe c t in je c t io n in s r a m c o r e -c e ll: a n a lys is a n d c o m p a r is o n b e t we e n 0 .1 3 ¹m a n d 9 0 n m t e c h n o lo g ie s " , d esign automation conference, p p . 8 5 7 -8 6 2 , 2 0 0 5 . [1 2 ] l . d ilillo , p . gir a r d , s . p r a vo s s o u d o vit c h , a . v ir a z e l, s . b o r r i, m. h a g e -h a s s a n , " d yn a m ic r e a d d e s t r u c t ive a u lt in e m b e d d e d -s r a ms : a n a lys is a n d m a r c h t e s t s o lu t io n s " , p roc. ie e e e uropean test symposium, 2 0 0 4 . [1 3 ] s . h a m d io u i, a .j. va n d e go o r , m. r o d g e r s , " ma r c h s s : a t e s t fo r a ll s t a t ic s im p le fa u lt s " , r ecords of ie e e int. w orkshop m td t, p p . 9 5 -1 0 0 , 2 0 0 2 . [1 4 ] s . h a m d io u i, a .j. va n d e go o r , m. r o d g e r s , " l in ke d fa u lt s in r a n d o m a c c e s s m e m o r ie s : c o n c e p t , fa u lt m o d e ls , t e s t a lg o r it h m s , a n d in d u s t r ia l r e s u lt s " , ie e e trans. cad , vo l. 2 3 , n o . 5 , p p . 7 3 7 -7 5 6 , 2 0 0 4 . [1 5 ] g. h a r u t u n ya n , v . a . v a r d a n ia n , y . zo r ia n , " min im a l m a r c h t e s t s fo r u n lin ke d s t a t ic fa u lt s in r a n d o m a c c e s s m e m o r ie s " , p roc. 23rd ie e e vl si test symposium, p alm springs, ca, usa, p p . 5 3 -5 9 , 2 0 0 5 . [1 6 ] a . j. va n d e go o r , z. a l-a r s , " fu n c t io n a l m e m o r y fa u lt s a fo r m a l n o t a t io n a n d a t a xo n o m y" , p roc. ie e e vl si test symposium, m ontreal, canada, p p . 2 8 1 -2 9 0 , 2 0 0 0 . [1 7 ] a h o , s e t h i, u llm a n , co m p ile r s : p r in c ip le s , te c h n iqu e s , a n d to o ls , addison-w esley, is b n 0 -2 0 1 -1 0 0 8 8 -6 , 1 9 8 6 . ´³ý³ó¨»ñç ùççáóáí ù³ñß ï»ëï»ñç ï³éáõóù³ý ³ñ¹ûáõý³í»ï ù»ãá¹ ¶. ð³ñáõãûáõýû³ý, ¸. ø»éùáõùû³ý, ð. ¾éãû³ý, ì. ì³ñ¹³ýû³ý ²ù÷á÷áõù üï³ñ³·ñí³í ¿ ñçßíáõ ë³ñù»ñáõù ëï³ïçï ï³ù ¹çý³ùçï ³ýë³ñùáõãûáõýý»ñç ï³ù³û³ï³ý »ýã³µ³½ùáõãûáõý ñ³ûïý³µ»ñáõ ï³ù ³ëïáñáßáõ ùçýçù³é ù³ñß ï»ëï»ñç ï³éáõóù³ý ù»ãá¹: ²é³ç³¹ñí³í ù»ãá¹áõù ï³éáõóíáõù »ý áñáß³ïç ³ýññ³å»ßï å³ûù³ýý»ñçý µ³í³ñ³ñáõ µáéáñ ù³ñß ï»ëï»ñá: ø³ñß ï»ëï»ñç ¨ µý³ï³ý ãí»ñç ùçç¨ ïñí»é ¿ ³ñï³å³ïï»ñáõù, áñá ñý³ñ³íáñáõãûáõý ¿ ï³éçë ï³éáõó»é µ³ý³ó¨: ´³ý³ó¨ç ùççáóáí ëï³óíáõù »ý µáéáñ ³ûý ù³ñß ï»ëï»ñá, áñáýù ñ³ûïý³µ»ñáõù ï³ù ³ëïáñáßáõù »ý ïñí³í ³ýë³ñùáõãûáõýý»ñá: ø»ãá¹á ïçñ³éí»é ¿ »ñ»ù ·áñíáõáõãû³ùµ ¹çý³ùçï ³ýë³ñùáõãûáõýý»ñç áñáß³ïç »ýã³¹³ëá ñ³ûï³µ»ñáõ ùçýçù³é ù³ñß ï»ëï»ñ ëï³ý³éáõ ñ³ù³ñ: ø»ãá¹á ñý³ñ³íáñáõãûáõý ¿ ï³éçë ï³éáõó»é ï³ù³û³ï³ý ëï³ïçï ¨ ¹çý³ùçï ³ýë³ñùáõãûáõýý»ñ ñ³ûï³µ»ñáõ ï³ù ³ëïáñáßáõ ùçýçù³é ù³ñß ï»ëï»ñ: mathematical problems of computer science 59, 16–26, 2023. doi: 10.51408/1963-0098 udc 519.714 rdnf oriented analytics to random boolean functions levon h. aslanyan, irinaa. arsenyan, vilik m. karakhanyan and hasmik a. sahakyan institute for informatics and automation problems of nas ra,yerevan, armenia e-mail: kavilik@gmail.com abstract dominant areas of computer science and computation systems are intensively linked to the hypercube-related studies and interpretations. this article presents some transformations and analytics for some example algorithms and boolean domain problems. our focus is on the methodology of complexity evaluation and integration of several types of postulations concerning special hypercube structures. our primary goal is to demonstrate the usual formulas and analytics in this area, giving the necessary set of common formulas often used for complexity estimations and approximations. the basic example under considered is the boolean minimization problem, in terms of the average complexity of the so-called reduced disjunctive normal form (also referred to as complete, prime irredundant, or blake canonical form). in fact, combinatorial counterparts of the disjunctive normal form complexities are investigated in terms of sets of their maximal intervals. the results obtained compose the basis of logical separation classification algorithmic technology of pattern recognition. in fact, these considerations are not only general tools of minimization investigations of boolean functions, but they also prove useful structures, models, and analytics for constraint logic programming, machine learning, decision policy optimization and other domains of computer science. keywords: boolean function, hypercube, complexity, asymptotic, reduced disjunctive normal form. article info: received 14 february 2023; sent for review 10 march 2023; accepted 11 april 2023. 1. hypercube and related structures the metric theory of boolean functions (bf) [1], [2] arose in the 70’s, in parallel with the emergence of broader design and implementation ideas for mechanical and electronic computation devices. it was then that it turned out that the system of binary representation of numbers is the most optimal, both from the point of view of the algorithmic implementation of arithmetic calculations and also from the point of view of developing physical carriers of performing these calculations [3]. bf – functions with only binary variables, and also 16 l. aslanyan, i. arsenyan, v. karakhanyan and h. sahakyan 17 with values in the domain {0, 1}, although simple among the other similar mathematical concepts, they are quite complex in solving problems associated with their transformations and optimization. the metric theory of boolean functions provides the necessary knowledge for coding, transforming and implementing binary functions. although the way to minimal bf representations are and remains difficult, a rather complete picture of the main forms of function representation of functions has been obtained, and the basic role here takes the concept of disjunctive normal forms. successive steps of several transformations of functions are found to achieve minimal forms as a chain from the table or formula representation to the reduced d.n.f., then to the deadlock forms and finally – the minimal structures. the accompanying structures and bottlenecks of achieving acceptable optimization are investigated intensively [1], [4]–[7]. here we will not cover the whole theory but will pay attention to one fundamental construction, – to the concept of reduced disjunctive normal forms (r.d.n.f.) of boolean functions. r.d.n.f. is the collection of all minimal conjunctions and geometrically the system of all maximum intervals/sub-cubes of functions. these forms are a universal concept, and they also arise in problems such as circuit design from set of functional elements (logical part of chip design), in the theory of pattern recognition (logic separation algorithm, and generation of logical regularities) [8]–[11], in biological models of heredity and mutations (phylogeny, parsimony) [12, 13], etc. turning to the complexity characterization of structures associated with the reduced disjunctive normal form, where two types are usually considered: the largest and most typical characteristics, we will focus on the second component. in a concise survey of the domain, the initial studies of [5], [14], and [15], should be mentioned, that give the formulas of average numbers of maximal intervals in boolean functions. [16], [17] extended these results to the case of partially defined boolean functions. an alternative track of papers in these topics includes the articles [18], [19], [20]. current research on the topics of bf and complexities might be demonstrated through the papers [21]–[26]. methodologically, in studies in the area of bf, it should be taken into account that the function determination domain, as well as the number of functions itself, are finite, depending on the number of the variables – the dimensionality. so, considering the parameter π(f) over the functions, we get the split of these functions into finite classes by the values of this parameter. these are the rates and intensity of the accepted values of the parameter π(f). in some cases, it is convenient to refer to these valuations as probabilistic distributions, which is not obligatorily but is convenient in some contexts. in this concern, there appears a link to the model of random boolean functions and the combinatorial theories initiated by a. renyi and p. erdos [27], [28]. 1.1 concepts and definitions in the binary domain elementary conjunction, direction. let α̃ and β̃ – be arbitrary vertices of the ndimensional unite cube. and let ji, i = 1, 2, · · · , r be all coordinates, those where αji = βji. consider the formula k(x1, x2, · · · , xn) = r∧ i=1 x σji ji , with σji = αji, i = 1, 2, · · · , r. we say that k is an elementary conjunction stretched on the pair of vertices α̃ and β̃ of the n-dimensional unit cube en. the number of literals in k is the rank of k. the geometrical counterpart of k is a sub-cube defined as follows. assign 0 values to all but j1, j2, · · · , jr coordinates and denote this vertex by v0. similarly, assign 18 rdnf oriented analytics to random boolean functions these coordinates by the value 1, obtaining the vertex v1. these are the minimal and maximal vertices that belong to k, and they determine a unique sub-cube of all truth vertices of k. n − r, the number of variable coordinates of k is the size of its sub-cube. let λ = {j1, j2, · · · , jr} be a collection of r indices drawn up of variables x1, x2, · · · , xn, and let λ̄ be the complementary to the λ set of indices. conjunctions of the form ∧r i=1 x σji ji and the corresponding intervals will be called conjunctions and intervals of the direction λ. for a fixed r there are crn different directions, and each of them is determined by the appropriate selection of an r subset {j1, j2, · · · , jr} of the set {1, 2, ..., n}. the individual interval in the direction {j1, j2, · · · , jr} appears in result of assigning the values σ1, σ2, · · · , σr to the variables xj1, xj2, · · · , xjr. fig. 1. geometry of hypercube. this also means that there are 2n−r conjunctions and intervals in one of the r-directions. the collection λ̄ of indices defines another set of directions. let f be an arbitrary logical formula and m ⊆ bn. we say that f absorbs or covers m if on each tuple α̃ ∈ m the formula f accepts the unite (true) value. let α̃ ∈ en be an arbitrary vertex. call the value | α̃ |= ∑n i=1 αi the module or the weight of α̃. the set of all vertices β̃ ∈ en, with ρ(α̃, β̃) =| α̃ ⊕ β̃ |= k, call the k–the layer of en in relation to the vertex α̃ (⊕ – mentions mod2 summation). intervals nk1 and nk2, k1(x1, x2, · · · , xn) = r∧ i=1 x σ1 ji ji and k2(x1, x2, · · · , xn) = r∧ i=1 x σ2 ji ji of the same size and the same direction we call neighbors if ρ(σ̃1, σ̃2) = 1, where ρ – be the hamming distance, ρ(σ̃1, σ̃2) = ∑r i=1 | σ1ji − σ 2 ji | . let then ji0 is the number of that unique coordinate for which σ1ji0 ̸= σ2ji0 . then we say that the conjunctions k 1 and k2 (or the pair of neighbor intervals corresponding to them) joined by the coordinate xji0 , and, as a result, l. aslanyan, i. arsenyan, v. karakhanyan and h. sahakyan 19 a new conjunction (interval) appears: r∧ i ̸=i0,i=1 x σji ji . partition the variable set x1, x2, · · · , xn in an arbitrary manner into two nonempty groups: xi1, xi2, · · · , xk as the first group, and xik+1, xik+2, · · · , xin as the second. then, the n-dimensional unit cube en may be represented as the cartesian multiplication bk ×bn−k of two sub-cubes: bk and bn−k generated correspondingly by the sets of variables xi1, xi2, · · · , xik and xik+1, xik+2, · · · , xin. let us enumerate the vertices of b n−k by the layers relative to the vertex 0̃ of bn−k. enumeration among the vertices of a particular layer is arbitrary, but the first group that is enumerated by low numbers is layer zero, then the first layer, and so on. additional ordering among layer vertices may use lexicographic order, binary value based order, etc. consider an arbitrary k-dimensional sub-cube bk of en, the first k-dimensional interval bk1 in the direction of b k. list the neighbor intervals to the considered one, bk1, bk2, b k 3, · · · , bkn−k+1. let f be an arbitrary (partially defined) boolean function that satisfies the following conditions: α) bk1 doesn’t contain zero value vertices of f : (∀α̃ ∈ bk1, f(α̃) ̸= 0), β) each of the neighbor with bk1 interval contains at least one ‘unit’ value vertex f : (∀j, j = 2, 3, · · · , n − k + 1 ∃α̃ ∈ bkj , f(α̃) = 1), γ) bk1 contains at least one ‘unit’ vertex of f : (∃α̃ ∈ bk1, f(α̃) = 1). in conditions α), β), γ), we say that bk1 is a maximal interval of the function f. d.n.f., composed of all elementary conjunctions, corresponding to maximal intervals of function f is named the reduced disjunctive normal form of f. the number of disjunctive members of this formula is considered as its complexity. denoting by rk(f) the number of all maximal k–intervals of the function f we get the formula of complexity of the reduced disjunctive normal form of f: n∑ k=0 rk(f). 2. on the maximum number of k-dimensional maximal intervals of rbf consider the class p2(n) of all boolean functions of n variables x1, x2, · · · , xn. let p, 0 < p < 1 be a fixed number, and fp – the probability distribution on p2(n), generated in the following way. the function f ∈ p2(n) is induced as a result of a randomized experiment, where the values of the function on vertices of en are derived randomly. the value 1 appears with a probability p and the 0 value – with a complementary probability 1−p. the vertices of en take part in this experiment independently of each other, and the probabilistic distribution fp over the set of boolean functions is generated in this way. the probability of an individual boolean function f under the distribution fp depends on the balance between the 0 and 1 values of the function f (the volumes of the sets n{ and en−n{). for f ∈ p2(n), this probability is equal to p|n{|(1 − p)2 n−|n{|. when p = 1/2 this probability is simply 1/22 n and the corresponding distribution becomes the uniform distribution over 20 rdnf oriented analytics to random boolean functions the p2(n). we introduce the notation rk(f) for the number of k-dimensional maximal intervals of the function f ∈ p2(n). and let rk(n, p) be the average value of the number of k-dimensional maximal intervals of functions f ∈ p2(n) under the distribution fp. it is easy to make sure, that rk(n, p) = ∑ f∈p2(n) fp(f) ∗ rk(f) (1) the number rk(n, p) in the expression (1) is given by its definition as a sum over all functions of f ∈ p2(n), counting all their k-dimensional maximal intervals and taking into account the probabilities of f in the distribution fp. further evidence of these constructions is provided by the following scheme: fig. 2. this figure presents the bipartite graph of functions and k-dimensional maximal intervals. upper line functions are placed in order of the number of their ”true” values, from 0 to 2k. different functions include different numbers of k-dimensional maximal intervals and have different probabilities under the distribution fp. instead, each interval presented in the bottom line is connected to the same number of functions. this is because the sizes of intervals is the same. the order of intervals is by groups of intervals, that belong to the same direction. numeration inside the functions with the same number of ”ones” and inside the groups of intervals of the same direction is arbitrary. following [5], we change the order of counting in 1, first considering all k-dimensional intervals in en. we relay two events to these intervals: the one, about their maximality, and then the second, about the set of functions that accept the first event about maximality. in this regard, it is also convenient to split the en in parts: the current k-dimensional interval k and its all n−k neighboring k-dimensional intervals k1, k2, · · · , kn−k. this part, the current interval and its neighbors, covers an area e1 of 2k(n − k + 1) vertices of en. and the second part that we consider, consists of the complementary area e2 to e1 up to en. the probability of maximality of k for the function f becomes the product of probability of maximality of k together with the conditional probability of f when k is given to be maximal. the first probability equals p2 k (1 − p2k)n−k. the first and second parts consist of events, and their sums of probabilities are equal to 1 as a probabilistic distribution. now, when we sum up the mentioned conditional probabilities with all f, we get the probability 1, and the final probability of maximality of k, under the conditions of fp, becomes p2 k (1 − p2k)n−k. it reminds us to take this probability for all k-dimensional intervals, obtaining the following equivalent form for (1), l. aslanyan, i. arsenyan, v. karakhanyan and h. sahakyan 21 rk(n, p) = c k n2 n−kp2 k (1 − p2 k ) n−k . (2) theorem 1. rk(n, p) is a concave function of the parameter k in the interval [0, n]. it is important to know the behavior of the function rk(n, p defined on the interval [1, n]. initially, it is useful to calculate the values of the function at the boundary points of the domain of definition: k = 0, 1, ..., n − 1, n. we give these values both for the arbitrary p and the value 1/2. table 1: values of rk(n, p) on boundary points, such as k = 0, 1, ..., n − 1, n. boundary point values of rk(n, p) dimension k of maximal interval rk(n, p) rk(n, 1/2) k = 0 2np(1 − p)n 1/2 k = 1 n2n−1p2(1 − p2)n−1 (n/4)(3/2)n−1 ... ... ... k = n − 1 n2n−1p2n−1(1 − p2n−1) n2n−1(1 − 1/22n−1)/22n−1) k = n p2 n 1/22 n as we can see, both the left and right boundary point values of the interval (0, n) are small, but there is a noticeable increase from left to right at the left end, and a decrease from left to right at the right end. to get a complete picture of the behavior, consider a number of special intermediate point values of the function at: k1 = log 1 −logp , k0 = log logn −logp , and k1 = log n −logp . the technical element of choosing of these values is in simple evaluation of sub-formula ek = 2 2k, which is an important functional part of the 1. substituting k1, k0, and k2 into ek we get: ek1 = 1/2, ek0 = 1/n, ek2 = 1/2 n. (3) let us start the proof of postulations 1-3. for this, conduct a preliminary analysis of the expression (2) for rk(n, p). consider an arbitrary integer value function k(n) that obeys the restriction 0 ≤ k(n) ≤ n, and substitute it into the expression 2. we are interested in the behaviour of the received function rk(n)(n, p) depending on the parameter k(n) as n → ∞. first let’s make sure that with the increase of k the expression rk(n, p) increases monotonically by the k ≤ [k0], and then it decreases, when ]k0[≤ k. by doing this we compose the relation rk = rk+1(n, p) rk(n, p) = (n − k)p2k(1 + p2k)n−k 2(k + 1)(1 − p2k+1) . (4) this expression can be considered for an arbitrary (not only for the integer) assignment to the parameter k. we will follow by checking if this function is concave in the interval 0 < k < n for large n. the direct way of this is to derive the expression of the fraction 22 rdnf oriented analytics to random boolean functions rk and treat it for a possible constant/zero value of it. in such consideration, the most important role takes the part ak = (n − k)p2 k of the base expression 4. substituting k0 into ak we obtain that (n − k0)p2 k0 = (n − k0)p ( logn −logp ) = (n − k0)2 logp( −logn logp ) = n−k0 n , which is converging to 1 as n → ∞. with the help of formulas in section 3. we see that the part bk = (1 + p 2k)n−k of (4) is limited at the point k0: (6) gives (1 + p 2k0 )n → e as n → ∞, so that (1 + p2 k0 )n−k0 also tends to e. compose the fraction bk+1/bk in the following form: bk+1/bk = ( 1 + p2 k p2 k )n−k−1 ( 1 + p2 k )n−k = ( 1+p2 k p2 k 1+p2 k )n−k 1 + p2 k p2 k (5) fig. 3. differential of growing rk(n, p). note that the fraction 1+p 2k p2 k 1+p2 k is less than 1, so its n − k degree is also less than 1. and the denominator of (5) is greater than 1 so that, finally, the expression (5) is less than 1 for all k, which means a monotonic decrease of the expression rk in (5). in general, as k increases, all the factors of (4), other than bk, decrease monotonically and, besides this, as n → ∞ , this expression tends to zero at the point k0 and grows infinitely when k = k0 − 1. finally, we receive that with increasing k, for the beginning, ik(n, p) increases, achieving its maximal value at the point [k0] or ]k0[, and, then, it decreases. 3. on the dependency of number of k-dimensional maximal intervals on k consider the parameter k2 = log n −logp. since 0 < p < 1, we have k2 = logn + c, where c represents an absolute constant determined by the fixed value of p. we intend to obtain an asymptotic formula for ik(n, p) by the n → ∞ for the values of k of the form k2 + const. we make use of the following expressions ckn ∼ nk k! , (1 − p2k) ∼ 1, and n! ∼ nne−n √ 2πn as n → ∞, which are based on the formulas 1. if 0 ≤ x ≤ 1 and 0 ≤ y, then exp(x(1 − x 2 )y) ≤ (1 + x)y ≤ exp(xy). (6) l. aslanyan, i. arsenyan, v. karakhanyan and h. sahakyan 23 2. if 0 ≤ x ≤ 1 and 0 ≤ y, then (1 − x)y ≤ exp(−xy); and (7) exp(−x(1 − x)y) ≤ (1 − x)y, when additionally 0 ≤ x ≤ 1/2. 3. if x and y be natural numbers, and x ≤ y, then (1 − x y ) x−1 2 ≤ x−1∏ i=1 (1 − i y ) ≤ (1 − x 2y )x−1. (8) and are valid for the mentioned values of the parameter k, and for this reason ik(n, p) ∼ nkek2n−kp2 k kk √ 2πk = ĩk(n, p). (9) theorem 2. the probability, that functions of the class p2(n) under the distribution fp have maximal intervals of sizes k, k < [k1] or k > [k2], where k1 = log 1 −logp and k2 = log n −logp tends to zero with n → ∞. on the right side of (9) we have expression, that depends on the continuous argument k, and which is equivalent to the expression ik(n, p) for the integer values of the parameter k, of the form k2 + const. in the mentioned area, ĩk(n, p) decreases monotonically with the increase of k, ĩk2(n, p) tends to infinity, and ĩk2+1(n, p) tends to zero, when n → ∞, so that ik(n, p) → 0, for values k >]k2[ and ik(n, p) → ∞ for values k0 ≤ k ≤ [k2], by n → ∞. let us also denote, that we do not insist that i]k2[(n, p) as n → ∞ converges to any appropriate value. in what follows, we will use the first chebyshev inequality (1). the first inequality lets formulate an extension of a postulation from [29] for the case of the probability distribution fp. actually, if to consider the expression ik(f), as a parameter of π(f) then for the values k >]k2[ ik(n, p) → 0 by n → ∞, and taking into the force the first inequality for the arbitrary ϵ(n) ≥ 0 p(ik(f) ≥ ϵ(n)) → 0 when n → ∞. a similar situation takes place in the region of small values of the parameter k. for the value k = k1 and p = 1/2 by the (3) p 2k1 = 1/2 and rk1(n, p) → ∞ as n → ∞. for p > 1/2, already for the value k1 − 1, we observe that rk1−1(n, p) → 0 as n → ∞. this is just because 2n−k1+1 1−p2k1−1 is a decreasing exponent, which together with ckn tends to 0. 4. conclusion this article has two goals: first, it considers the set of formulas needed to analyze the complexity of structures associated with a multidimensional unit cube, providing the necessary transformations and approximations for these formulas. further, the paper considers a typical study for this field using these formulas. the problem under consideration estimates the complexity of the reduced disjunctive normal form of boolean functions on average, or, what is the same, for almost the entire class of functions. 24 rdnf oriented analytics to random boolean functions references [1] yu. i. zhuravlev, “set-theoretical methods of algebra of logic, problemi kibernetiki, vol. 8, pp. 544, 1962. 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[29] l. h. aslanyan, “on implementation of reduced disjunctive normal form in the problem of extension of partial boolean functions”, junior researcher, natural sciences, yerevan state university, vol. 20, no. 2, pp. 6575, 1974. 2 6 rdnf oriented analytics to random boolean functions ä³ï³ñ³ï³ý µáõéû³ý ýáõýïóç³ý»ñç î¸üò ïáõùýáñáßí³í í»ñéáõíáõãûáõý è¨áý ð. ²ëé³ýû³ý, æñçý³ ². ²ñë»ýû³ý, ìçéçï ø. î³ñ³ë³ýû³ý, ð³ëùçï ². ê³ñ³ïû³ý ðð ¶²² æýýáñù³ïçï³ûç ¨ ³íïáù³ï³óù³ý åñáµé»ùý»ñç çýëïçïáõï, ºñ¨³ý, ð³û³ëï³ý e-mail: kavilik@gmail.com ²ù÷á÷áõù àíàëèòèêà îðèåíòèðîâàííàÿ íà ñäíô ñëó÷àéíûõ áóëåâûõ ôóíêöèé ëåâîí à. àñëàíÿí, èðèíà à. àðñåíÿí, âèëèê ì. êàðàõàíÿí, àñìèê à. ñààêÿí èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà, åðåâàí, àðìåíèÿ e-mail: kavilik@gmail.com àííîòàöèÿ äàííàÿ ñòàòüÿ ïðåñëåäóåò äâå öåëè: âî-ïåðâûõ, â íåé ðàññìàòðèâàåòñÿ íàáîð ôîðìóë, íåîáõîäèìûõ äëÿ àíàëèçà ñëîæíîñòè ñòðóêòóð, ñâÿçàííûõ ñ ìíîãîìåðíûì åäèíè÷íûì êóáîì, ïðåäîñòàâëÿÿ íåîáõîäèìûå ïðåîáðàçîâàíèÿ è àïïðîêñèìàöèè äëÿ ýòèõ ôîðìóë. äàëåå, â ñòàòüå ðàññìàòðèâàåòñÿ òèïè÷íîå èññëåäîâàíèå äëÿ äàííîé îáëàñòè ñ èñïîëüçîâàíèåì ýòèõ ôîðìóë. ðàññìàòðèâàåìàÿ ïðîáëåìà îöåíèâàåò ñëîæíîñòü ñîêðàùåííîé äèçúþíêòèâíîé íîðìàëüíîé ôîðìû áóëåâûõ ôóíêöèé â ñðåäíåì, èëè, ÷òî òî æå ñàìîå, ïî÷òè äëÿ âñåãî êëàññà ôóíêöèé. êëþ÷åâûå ñëîâà: áóëåâà ôóíêöèÿ, ìíîãîìåðíûé åäèíè÷íûé êóá, ñëîæíîñòü, àñèìïòîòèêà, ñîêðàùåííàÿ äèçúþíêòèâíàÿ íîðìàëüíàÿ ôîðìàå. ²ûë ñá¹í³íý áõýç »ñïáõ ýå³ï³ï, ý³ë ³ûý ùýý³ñïáõù ¿ µ³ý³ó¨»ñç ùç ß³ñù, áñáýù ³ýññ³å»ßï »ý µ³½ù³ã³÷ ùç³íáñ ëáñ³ý³ñ¹ç ñ»ï ï³åí³í ï³éáõóí³íùý»ñç µ³ñ¹áõãûáõýá í»ñéáõí»éáõ ñ³ù³ñ` ³å³ñáí»éáí ³ýññ³å»ßï ÷áë³ï»ñåáõùý»ñ ¨ ùáï³ñïáõùý»ñ ³ûë µ³ý³ó¨»ñç ñ³ù³ñ: ²í»éçý, ñá¹í³íá ùýý³ñïáõù ¿ ³ûë áéáñïç ñ³ù³ñ áñáß µýáñáß áõëáõùý³ëçñáõãûáõý` û·ï³·áñí»éáí ³ûë µ³ý³ó¨»ñá: øýý³ñïíáõ áýã³ó³ï³ñ·á ·ý³ñ³ïáõù ¿ µáõéû³ý ýáõýïóç³ý»ñç ïñ×³ïí³í ¹ç½ûáõýïïçí ýáñù³é ó¨ç µ³ñ¹áõãûáõýá ùçççýáõù ï³ù, áñ ýáõûýý ¿, ¹³ëç ·ñ»ã» µáéáñ ýáõýïóç³ý»ñç ñ³ù³ñ: ´³ý³éç µ³é»ñ ´áõéû³ý ýáõýïóç³, µ³½ù³ã³÷ ùç³íáñ ëáñ³ý³ñ¹, µ³ñ¹áõãûáõý, ³ëçùåïáïçï³, ïñ×³ïí³í ¹ç½ûáõýïïçí ýáñù³é ó¨: 02_vilik_59_16_26 02 d:\sbornik\...\gen_haar_tr2.dvi mathematical problems of computer science 31, 79{89, 2008. fast gener alized h aar t r ansfor ms h a ko b g. s a r u kh a n ya n institue for informatics and automation problems of nas of ra e-mail: hakop@ipia.sci.am abstract the fast generalized haar transform algorithms of orders 4n; 3n; and 5n are presented. refer ences [1 ] a h m e d n ., r a o k .r . or t h o g o n a l tr a n s fo r m s fo r d ig it a l s ig n a l p r o c e s s in g . s p r in g e r v e r la g , n e w y o r k, 1 9 7 5 . [2 ] h a a r z. zu r th e o r ie d e r o r t h o g o n a le n fu n kt io n e n s ys t e m e , ma t h . a n n ., vo l 6 9 , p .3 3 1 3 7 1 , 1 9 1 4 . [3 ] r a o k .r ., n a r a s im h a m , r e vu lu r i k . a fa m ily o f d is c r e t e h a a r t r a n s fo r m s , co m p u t . e le c t r . e n g r g ., vo l. 2 , p .3 6 7 -3 6 8 , 1 9 7 5 . [4 ] s e b e r r y j., zh a n g x .m. s o m e o r t h o g o n a l d e s ig n s a n d c o m p le x h a d a m a r d m a t r ic e s b y u s in g t wo h a d a m a r d m a t r ic e s . a u s t r a l. j. o f co m b in . th e o r y., n o .4 , p .9 3 -1 0 2 , 1 9 9 1 . [5 ] co r in t h io s m.j. a n e w c la s s o f g e n e r a l-b a s e m a t r ic e s a n d a fo r m a lis m fo r o p t im a l p a r a lle l/ p ip e lin e d c o m p u t e r a r c h it e c t u r e . e le c t r ic a l a n d co m p u t e r e n g in e e r in g , 1 9 9 3 . ca n a d ia n co n fe r e n c e o n 1 4 -1 7 s e p t ., vo l.2 , p .8 5 1 -8 5 6 , 1 9 9 3 . [6 ] min g yo n g zh o u , zh o n g ka n l iu , h a m a h . a r e s o lu t io n -c o n t r o lla b le h a r m o n ic a l r e t r ie va l a p p r o a c h o n t h e ch r e s t e n s o n d is c r e t e s p a c e . ie e e tr a n s a c t io n s , s ig n a l p r o c e s s in g , vo l.4 2 , is s u e 5 , p .1 2 8 1 -1 2 8 4 , 1 9 9 4 . [7 ] mo r a g a , c., p o s wig j. p r o p e r t ie s o f t h e zh a n g -h a r t le y s p e c t r u m o f p a t t e r n s . p r o c e e d in g s o f t h e twe n t ie t h in t e r n a t io n a l s ym p o s iu m , mu lt ip le -v a lu e d l o g ic , p .6 2 -6 8 , 1 9 9 0 . [8 ] a g a ia n s ., b a ja d ia n h . ge n e r a liz e d o r t h o g o n a l h a a r s ys t e m s : s yn t h e s is , me t r ic a n d co m p u t in g p r o p e r t ie s . h a a r me m o r ia l co n fe r e n c e , vo l.1 , p .9 7 -1 1 3 , co llo g . ma t h . s o c . ja n o s b o lya i, 4 9 , n o r t h -h o lla n d , a m s t e r d a m , n e w y o r k, 1 9 8 7 . [9 ] ma lla t s . a w a ve le t to u r o f s ig n a l p r o c e s s in g . a c a d e m ic p r e s s , 2 0 0 1 . ð³³ñç áý¹ñ³ýñ³óí³í ³ñ³· ó¨³÷áëáõãûáõýý»ñ ð. ê³ñáõë³ýû³ý ²ù÷á÷áõù ðá¹í³íáõù ý»ñï³û³óí³í »ý 4 n; 3 n; ¨ 5 n ï³ñ·ç ð³³ñç áý¹ñ³ýñ³óí³í ³ñ³· ó¨³÷áëáõãûáõýý»ñç ³é·áñçãùý»ñá: 7 9 d:\sbornik\...\article.dvi mathematical problems of computer science 31, 122{129, 2008. i nter val t otal color ings of cer tain gr aphs p e t r o s a . p e t r o s ya n institute for informatics and automation problems of nas of ra, e-mail: pet petros@fipia.sci.am, yahoo.comg abstract an interval total t¡coloring of a graph g is a total coloring of g with colors 1; 2; : : : ; t such that at least one vertex or edge of g is colored by i; i = 1; 2; : : : ; t; and the edges incident with each vertex v together with v are colored by (dg(v) + 1) consecutive colors, where dg(v) is the degree of the vertex v in g. it is proved that complete graphs, complete bipartite graphs and n¡dimensional cubes have interval total colorings and bounds are found for the possible number of colors in such colorings. refer ences [1 ] a .s . a s r a t ia n , t.m.j. d e n le y, r . h a g g kvis t , b ip a r t it e gr a p h s a n d th e ir a p p lic a t io n s , ca m b r id g e tr a c t s in ma t h e m a t ic s , 1 3 1 , ca m b r id g e u n ive r s it y p r e s s , 1 9 9 8 . [2 ] a .s . a s r a t ia n , r .r . k a m a lia n , \ in t e r va l c o lo r in g s o f e d g e s o f a m u lt ig r a p h " , appl. m ath. 5, p p . 2 5 -3 4 , 1 9 8 7 . [3 ] a .s . a s r a t ia n , r .r . k a m a lia n , \ in ve s t ig a t io n o n in t e r va l e d g e -c o lo r in g s o f g r a p h s " , j ournal of combin. theory, ser. b , vo l. 6 2 , p p . 3 4 -4 3 , 1 9 9 4 . [4 ] m. b e h z a d , gr a p h s a n d th e ir ch r o m a t ic n u m b e r s , p h .d . t h e s is , mic h ig a n s t a t e u n ive r s it y, 1 9 6 5 . [5 ] m. b e h z a d , g. ch a r t r a n d , j.k . co o p e r , \ th e c o lo u r n u m b e r s o f c o m p le t e g r a p h s " , j ournal of l ondon m ath. soc. 42, p p . 2 2 6 -2 2 8 , 1 9 6 7 . [6 ] o.v . b o r o d in , \ on t h e t o t a l c o lo u r in g o f p la n a r g r a p h s " , j . r eine angew. m ath. 394, p p . 1 8 0 -1 8 5 , 1 9 8 9 . [7 ] o.v . b o r o d in , a .v . k o s t o c h ka , d .r . w o o d a ll, \ to t a l c o lo r in g s o f p la n a r g r a p h s wit h la r g e m a xim u m d e g r e e " , j ournal of graph theory 26, p p . 5 3 -5 9 , 1 9 9 7 . [8 ] k .h . ch e w, h .p . y a p , \ to t a l c h r o m a t ic n u m b e r o f c o m p le t e r¡ p a r t it e g r a p h s " , j ournal of graph theory 16, p p . 6 2 9 -6 3 4 , 1 9 9 2 . [9 ] t.r . je n s e n , b .to ft , gr a p h co lo r in g p r o b le m s , w ile y in t e r s c ie n c e s e r ie s in d is c r e t e ma t h e m a t ic s a n d op t im iz a t io n , 1 9 9 5 . [1 0 ] a .v . k o s t o c h ka , \ th e t o t a l c o lo r in g o f a m u lt ig r a p h s wit h m a xim a l d e g r e e 4 " , d iscrete m athematics 17, p p . 1 6 1 -1 6 3 , 1 9 7 7 . [1 1 ] a .v . k o s t o c h ka , \ th e t o t a l c o lo r in g o f a m u lt ig r a p h s wit h m a xim a l d e g r e e 5 is a t m o s t s e ve n " , d iscrete m athematics 162, p p . 1 9 9 -2 1 4 , 1 9 9 6 . [1 2 ] p .a . p e t r o s ya n , \ in t e r va l e d g e c o lo u r in g s o f c o m p le t e g r a p h s a n d n¡ c u b e s " , m athematical p roblems of computer science, vo l. 2 5 , p p . 5 -8 , 2 0 0 6 . [1 3 ] m. r o s e n fe ld , \ on t h e t o t a l c o lo r in g o f c e r t a in g r a p h s " , israel j . m ath. 9, p p . 3 9 6 -4 0 2 , 1 9 7 1 . 1 2 2 p. petrosyan 1 2 3 [1 4 ] d .p . s a n d e r s , y . zh a o , \ on t o t a l 9 -c o lo r in g p la n a r g r a p h s o f m a xim u m d e g r e e s e ve n " , j ournal of graph theory 31, p p . 6 7 -7 3 ,1 9 9 9 . [1 5 ] n . v ija ya d it ya , \ on t h e t o t a l c h r o m a t ic n u m b e r o f a g r a p h " , j ournal of l ondon m ath. soc. (2) 3, p p . 4 0 5 -4 0 8 , 1 9 7 1 . [1 6 ] v .g. v iz in g , \ on a n e s t im a t e o f t h e c h r o m a t ic c la s s o f a p¡ g r a p h " , d iskret. analiz 3, p p . 2 5 -3 0 , 1 9 6 4 . [1 7 ] d .b . w e s t , in t r o d u c t io n t o gr a p h th e o r y, p r e n t ic e -h a ll, n e w je r s e y, 1 9 9 6 . [1 8 ] h .p . y a p , to t a l co lo r in g s o f gr a p h s , l e c t u r e n o t e s in ma t h e m a t ic s 1 6 2 3 , s p r in g e r v e r la g , b e r lin , 1 9 9 6 . [1 9 ] z. zh a n g , j. zh a n d , j. w a n g , \ th e t o t a l c h r o m a t ic n u m b e r s o f s o m e g r a p h s " , scientia sinica a 31, p p . 1 4 3 4 -1 4 4 1 , 1 9 8 8 . àñáß ·ñ³ýý»ñç ùçç³ï³ûù³ûçý éç³ï³ï³ñ ý»ñïáõùý»ñ ä. ä»ïñáëû³ý ²ù÷á÷áõù g ·ñ³ýç éç³ï³ï³ñ ý»ñïáõùá 1 ; 2 ; :::t ·áõûý»ñáí ï³ýí³ý»ýù ùçç³ï³ûù³ûçý éç³ï³ï³ñ t-ý»ñïáõù, »ã» ³ù»ý ùç i ·áõûýáí, i = 1 ; 2 ; :::t ý»ñïí³í ¿ ³éýí³½ý ù»ï ·³·³ã ï³ù ïáõ ¨ ûáõñ³ù³ýãûáõñ ·³·³ãçý ïçó ïáõ»ñá ¨ ·³·³ãá ý»ñïí³í »ý ( dg ( v ) +1 ) ñ³çáñ¹³ï³ý ·áõûý»ñáí, áñï»õ dg ( v ) áí ýß³ý³ïí³í ¿ ·³·³ãç ³ëïç×³ýá g ·ñ³ýáõù: ²å³óáõóí³í ¿, áñ éñçí ·ñ³ýý»ñá, éñçí »ñïïáõù³ýç ·ñ³ýý»ñá ¨ ã³÷³ýç ëáñ³ý³ñ¹á áõý»ý ùçç³ï³ûù³ûçý éç³ï³ï³ñ ý»ñïáõùý»ñ ¨ ·ïýí³í »ý ·ý³ñ³ï³ï³ýý»ñ ³û¹ ý»ñïáõùý»ñç ù»ç ù³ëý³ïóáõ ·áõûý»ñç ñý³ñ³íáñ ãíç ñ³ù³ñ: d:\user\...\article_final.dvi mathematical problems of computer science 49, 103{109, 2018. on dependence of i nter pr etation algor ithms of t yped functional p r ogr ams on canonical n otion of ±-reduction d a vit a . gr ig o r ya n department of informatics and applied mathematics, ysu e-mail: david.grigoryan.a@gmail.com abstract in this paper the interpretation algorithms of typed functional programs are considered. the interpretation algorithm is based on substitutions, ¯-reduction and canonical ±-reduction. the basic semantics of typed functional program is a function with indeterminate values of arguments, which is the main component of its least solution. if the value of the basic semantics for some values of arguments is indeterminate, then the interpretation algorithm either stops with the value ?, or works endlessly. it is shown that seven known interpretation algorithms are ?-depend on canonical notion of ±-reduction. here are these algorithms: fs (of full substitution), pes (of parallel external substitution), les (of left external substitution), pis (of parallel inner substitution), lis (of left inner substitution), act (active algorithm), pas (passive algorithm). keywords: typed functional program, canonical ±-reduction, interpretation algorithm, ?-dependency. 1 . typ e d ¸-te r m s , ca n o n ic a l n o t io n o f ±-r e d u c t io n , typ e d fu n c t io n a l p r o g r a m s th e d e ¯ n it io n s o f t h is s e c t io n c a n b e fo u n d in [1 ; 2 ; 3 ]. l e t m b e a p a r t ia lly o r d e r e d s e t , wh ic h h a s a le a s t e le m e n t ?, wh ic h c o r r e s p o n d s t o t h e in d e t e r m in a t e va lu e . e a c h e le m e n t o f m is c o m p a r a b le o n ly wit h it s e lf a n d wit h ?, wh ic h is t h e le a s t e le m e n t o f m. l e t u s d e ¯ n e t h e s e t o f t yp e s ( d e n o t e d b y t ypes) . 1 . m 2 t ypes, 2 . if ¯,®1,...,®k 2 t ypes ( k > 0 ) , t h e n t h e s e t o f a ll m o n o t o n ic m a p p in g s fr o m ®1 £:::£®k in t o ¯ ( d e n o t e d b y [®1 £ ::: £ ®k ! ¯]) b e lo n g s t o t ypes. l e t ® 2 t ypes, t h e n t h e o r d e r o f t yp e ® ( d e n o t e d b y ord ( ®) ) will b e a n a t u r a l n u m b e r , wh ic h is d e ¯ n e d in t h e fo llo win g wa y: if ® = m t h e n ord ( ®) = 0 , if ® = [®1 £ ::: £ ®k ! ¯], wh e r e ¯; ®1; :::; ®k 2 t ypes; k > 0 , t h e n ord( ®) = max ( ord ( ®1 ) ; :::; ord( ®k ) ; ord( ¯ ) ) + 1 . if x is a va r ia b le o f t yp e ® a n d c o n s t a n t c 2 ®, t h e n ord( x ) = ord ( c) = ord( ® ) : l e t ® 2 typ e s a n d v® b e a c o u n t a b le s e t o f va r ia b le s o f t yp e ®, t h e n v = s ®2t ypes v® is t h e s e t o f a ll va r ia b le s . th e s e t o f a ll t e r m s , d e n o t e d b y ¤ = s ®2t ypes ¤ ®, wh e r e ¤ ® is t h e s e t o f t e r m s o f t yp e ®, is d e ¯ n e d in t h e fo llo win g wa y: 1 0 3 1 0 4 on dependence of interpretation algorithms of typed functional programs on canonical notion 1 . if c 2 ®; ® 2 typ e s , t h e n c 2 ¤ ®, 2 . if x 2 v®, ® 2 typ e s , t h e n x 2 ¤ ®, 3 . if ¿ 2 ¤ [®1£:::£®k!¯], ti 2 ¤ ®i , wh e r e ¯; ®i 2 t ypes; i = 1 ; :::; k; k ¸ 1 , t h e n ¿ ( t1; :::; tk ) 2 ¤ ¯ ( t h e o p e r a t io n o f a p p lic a t io n , ( t1; :::; tk ) is t h e s c o p e o f t h e a p p lic a t o r ¿ ) , 4 . if ¿ 2 ¤ ¯; xi 2 v®i wh e r e ¯; ®i 2 t ypes, i 6= j ) xi 6= xj; i; j = 1 ; :::; k; k ¸ 1 t h e n ¸x1:::xk[¿ ] 2 ¤ [®1£:::£®k!¯] ( t h e o p e r a t io n o f a b s t r a c t io n , ¿ is t h e s c o p e o f t h e a b s t r a c t o r ¸x1:::xk ) . th e n o t io n o f fr e e a n d b o u n d o c c u r r e n c e s o f va r ia b le s a s we ll a s fr e e a n d b o u n d va r ia b le a r e in t r o d u c e d in t h e c o n ve n t io n a l wa y. th e s e t o f a ll fr e e va r ia b le s in t h e t e r m t is d e n o t e d b y f v(t). te r m s t1 a n d t2 a r e s a id t o b e c o n g r u e n t ( wh ic h is d e n o t e d b y t1 ´ t2 ) if o n e t e r m c a n b e o b t a in e d fr o m t h e o t h e r b y r e n a m in g b o u n d va r ia b le s . th e o c c u r r e n c e o f fr e e va r ia b le in t h e t e r m is c a lle d in t e r n a l if it d o e s n o t e n t e r t h e a p p lic a t o r , t h e s c o p e o f wh ic h c o n t a in s a fr e e o c c u r r e n c e o f s o m e va r ia b le . th e o c c u r r e n c e o f fr e e va r ia b le in t h e t e r m is c a lle d e xt e r n a l if it d o e s n o t e n t e r t h e s c o p e o f t h e a p p lic a t o r t h a t c o n t a in s a fr e e o c c u r r e n c e o f s o m e va r ia b le . l e t t 2 ¤ ®; ® 2 t ypes a n d f v ( t ) ½ fy1; :::; yng; y0 = ( y01 ; :::; y0n ) , wh e r e yi 2 v¯i ; y0i 2 ¯i; ¯i 2 t ypes; i = 1 ; :::; n; n ¸ 0 : th e va lu e o f t h e t e r m t fo r t h e va lu e s o f t h e va r ia b le s y1; :::yn e qu a l t o y0 = ( y 0 1; :::; y 0 n ) , is d e n o t e d b y v aly0 ( t) a n d is d e ¯ n e d in t h e c o n ve n t io n a l wa y. l e t t e r m s t1; t2 2 ¤ ®; ® 2 t ypes, f v ( t1 ) [f v ( t2 ) = fy1; :::; yng; yi 2 v¯i ; ¯i 2 t ypes; i = 1 ; :::; n; n ¸ 0 , t h e n t e r m s t1 a n d t2 a r e c a lle d e qu iva le n t ( d e n o t e d b y t1 » t2 ) if fo r a n y y0 = ( y 0 1; :::; y 0 n ) , wh e r e y 0 i 2 v¯i ; i = 1 ; :::; n we h a ve t h e fo llo win g : v aly0 ( t1 ) = v aly0 ( t2 ) . a t e r m t 2 ¤ ®; ® 2 t ypes, is c a lle d a c o n s t a n t t e r m wit h va lu e a 2 ® if t » a. fu r t h e r , we a s s u m e t h a t m is a r e c u r s ive s e t a n d t h e c o n s id e r e d t e r m s u s e va r ia b le s o f a n y o r d e r a n d c o n s t a n t s o f o r d e r · 1 , wh e r e c o n s t a n t s o f o r d e r 1 a r e s t r o n g c o m p u t a b le , m o n o t o n ic fu n c t io n s wit h in d e t e r m in a t e va lu e s o f a r g u m e n t s . a fu n c t io n f : m k ! m; k ¸ 1 , wit h in d e t e r m in a t e va lu e s o f a r g u m e n t s , is s a id t o b e s t r o n g c o m p u t a b le if t h e r e e xis t s a n a lg o r it h m , wh ic h s t o p s wit h t h e va lu e f ( m1; :::; mk ) 2 m fo r a ll m1; :::; mk 2 m, s e e [2 ]. a t e r m t 2 ¤ wit h a ¯ xe d o c c u r r e n c e o f a s u b t e r m ¿1 2 ¤ ®, wh e r e ® 2 t ypes, is d e n o t e d b y t¿1 , a n d a t e r m wit h t h is o c c u r r e n c e o f ¿1 r e p la c e d b y ¿2, wh e r e ¿22 ¤ ®, is d e n o t e d b y t¿2. to s h o w m u t u a lly d i®e r e n t va r ia b le s o f in t e r e s t x1; :::; xk; k ¸ 1 , o f a t e r m t, t h e n o t a t io n t[x1; :::; xk] is u s e d . th e n o t a t io n t[t1; :::; tk] d e n o t e s t h e t e r m o b t a in e d b y t h e s im u lt a n e o u s s u b s t it u t io n o f t h e t e r m s t1; :::; tk fo r a ll fr e e o c c u r r e n c e s o f t h e va r ia b le s x1; :::; xk, r e s p e c t ive ly, wh e r e xi 2 v®i ; i 6= j ) xi 6´ xj, ti 2 ¤ ®i ; ®i 2 t ypes; i; j = 1 ; ::; k; k ¸ 1 . a t e r m o f t h e fo r m ¸x1:::xk[¿ [x1; :::; xk]]( t1; :::; tk ) , wh e r e xi 2 v®; i 6= j ) xi 6´ xj ; ¿ 2 ¤ ; ti 2 ¤ ®i ; ®i 2 t ypes; i; j = 1 ; :::; k; k ¸ 1 , is c a lle d a ¯-r e d e x, it s c o n vo lu t io n is t h e t e r m ¿ [t1; :::; tk]. th e s e t o f a ll p a ir s ( ¿0; ¿1 ) , wh e r e ¿0 is a ¯-r e d e x a n d ¿1 is it s c o n vo lu t io n , is c a lle d a n o t io n o f ¯-r e d u c t io n a n d is d e n o t e d b y ¯. a o n e -s t e p ¯-r e d u c t io n ( !¯ ) a n d ¯r e d u c t io n ( !!¯ ) a r e d e ¯ n e d in t h e c o n ve n t io n a l wa y. a t e r m c o n t a in in g n o ¯-r e d e xe s is c a lle d a ¯-n o r m a l fo r m . th e s e t o f a ll ¯-n o r m a l fo r m s is d e n o t e d b y ¯-n f . ±-r e d e x h a s a fo r m f ( t1; :::; tk ) , wh e r e f 2 [m k ! m ], ti 2 ¤ m ; i = 1 ; :::; k; k ¸ 1 , it s c o n vo lu t io n is e it h e r m 2 m a n d in t h is c a s e f ( t1; :::; tk ) » m o r a s u b t e r m ti a n d in t h is c a s e f ( t1; :::; tk ) » ti; i = 1 ; :::; k. a ¯ xe d s e t o f t e r m p a ir s ( ¿0; ¿1 ) , wh e r e ¿0 is a ±-r e d e x a n d ¿1 is it s c o n vo lu t io n , is c a lle d a n o t io n o f ±-r e d u c t io n a n d is d e n o t e d b y ±. a o n e -s t e p ±-r e d u c t io n ( !± ) a n d ±-r e d u c t io n ( !!± ) a r e d e ¯ n e d in t h e c o n ve n t io n a l wa y. d. grigoryna 1 0 5 a o n e -s t e p ¯±-r e d u c t io n ( !¯± ) a n d ¯±-r e d u c t io n ( !!¯± ) a r e d e ¯ n e d in t h e c o n ve n t io n a l wa y. a t e r m c o n t a in in g n o ¯±-r e d e xe s is c a lle d a n o r m a l fo r m . th e s e t o f a ll n o r m a l fo r m s is d e n o t e d b y nf . a n o t io n o f ±-r e d u c t io n is c a lle d a s in g le -va lu e d n o t io n o f ±-r e d u c t io n , if ± is a s in g le va lu e d r e la t io n , i.e ., if ( ¿0; ¿1 ) 2 ± a n d ( ¿0; ¿2 ) 2 ±, t h e n ¿1 ´ ¿2, wh e r e ¿0; ¿1; ¿2 2 ¤ m . a n o t io n o f ±-r e d u c t io n is c a lle d a n e ®e c t ive n o t io n o f ±-r e d u c t io n if t h e r e e xis t s a n a lg o r it h m , wh ic h fo r a n y t e r m f ( t1; :::; tk ) , wh e r e f 2 [m k ! m ], ti 2 ¤ m ; i = 1 ; :::; k; k ¸ 1 , g ive s it s c o n vo lu t io n if f ( t1; :::; tk ) is a ±-r e d e x a n d s t o p s wit h a n e g a t ive a n s we r o t h e r wis e . de¯nition 1. [3 ] an e®ective, single-valued notion of ±-reduction is called a canonical notion of ±-reduction if: 1. t 2 ¯-nf ,t » m; m 2 m n f?g ) t !! ±m, 2. t 2 ¯-nf ; f v ( t ) = ;; t » ? ) t !!± ?. t heor em 1. (on canonical notion of ±-reduction). f or every recursive set of strong computable, monotonic functions with indeterminate values of arguments there exists a canonical notion of ±-reduction. p r o ve d in [3 ]. typ e d fu n c t io n a l p r o g r a m p is t h e fo llo win g s ys t e m o f e qu a t io n s : p 8 >< >: f1 = t1[f1; :::; fn] ::: fn = tn[f1; :::; fn] ( 1 ) wh e r e fi 2 v®i ; i 6= j ) fi 6´ fj; ti[f1; :::; fn] 2 ¤ ®i , fv ( ti[f1; :::; fn]) ½ ff1; :::; fng, ®i 2 t ypes; i; j = 1 ; :::; n; n ¸ 1 ; ®1 = [m k ! m ]; k ¸ 1 . e ve r y t yp e d fu n c t io n a l p r o g r a m p h a s t h e le a s t s o lu t io n ( s e e [1 ]) . l e t ( f1; :::; fn ) 2 ®1 £ ::: £ ®n b e t h e le a s t s o lu t io n o f p , t h e n t h e ¯ r s t c o m p o n e n t f1 2 [m k ! m] o f t h e le a s t s o lu t io n is s a id t o b e t h e b a s ic s e m a n t ic s o f t h e p r o g r a m p a n d is d e n o t e d b y fp. f ix ( p ) = f ( m1; :::; mk; m ) j fp ( m1; :::; mk ) = m; wh e r e m; m1; :::; mk 2 m; k ¸ 1 g. 2 . in t e r p r e t a t io n a lg o r it h m s , ?-d e p e n d e n c e th e in p u t o f t h e in t e r p r e t a t io n a lg o r it h m a is a p r o g r a m p o f t h e fo r m ( 1 ) , a t e r m f1 ( m1; :::; mk ) , wh e r e mi 2 m , i = 1 ; :::; k; k ¸ 1 a n d a c a n o n ic a l n o t io n o f ±-r e d u c t io n . a lg o r it h m a s t o p s wit h t h e r e s u lt m 2 m o r wo r ks in ¯ n it e ly. a lg o r it h m a u s e s t h r e e kin d s o f o p e r a t io n s : s u b s t it u t io n o f t h e t e r m s t1; :::; tn in s t e a d o f s o m e fr e e o c c u r r e n c e s o f va r ia b le s f1; :::; fn, o n e -s t e p ¯-r e d u c t io n a n d o n e -s t e p ±-r e d u c t io n . p roca ( p ) = f ( m1; :::; mk; m ) j a lg o r it h m a s t o p s fo r t h e p r o g r a m p a n d t h e t e r m f1 ( m1; :::; mk ) wit h t h e r e s u lt m, wh e r e m; m1; :::; mk 2 m; k ¸ 1 g in t e r p r e t a t io n a lg o r it h m a is c o n s is t e n t if fo r a n y p r o g r a m p a n d fo r a n y c a n o n ic a l n o t io n o f ±-r e d u c t io n we h a ve : p roca ( p ) ½ f ix ( p ) . t heor em 2. e very interpretation algorithm a is consistent. 1 0 6 on dependence of interpretation algorithms of typed functional programs on canonical notion p r oof. fo llo ws fr o m t h e r e s u lt s o f [4 ]. de¯nition 2. an interpretation algorithm a ?-depends on canonical notion of ±-reduction if there exist ±1 and ±2 canonical notions of ±-reduction, program p and m1; :::; mk 2 m; k ¸ 1 such that: ( m1; :::; mk; ? ) 2 p roca ( p ) for ±1 and ( m1; :::; mk; ? ) 62 p roca ( p ) for ±2. to s h o w a s e qu e n c e fi1 ; :::; fis; s ¸ 1 o f s o m e fr e e o c c u r e n c e s o f va r ia b le s o f t h e s e t ff1; :::; fng in t h e t e r m t, t h e n o t io n t < fi1; :::; fis > is u s e d . th e n o t io n t < ti1; :::; tis > d e n o t e s t h e t e r m o b t a in e d b y t h e s im u lt a n e o u s s u b s t it u t io n o f t h e t e r m s ti1; :::; tis fo r fr e e o c c u r e n c e s fi1 ; :::; fis, r e s p e c t ive ly. de¯nition 3. (interpretation algorithms) f s, p e s, l e s, p is, l is, p as, act s te p 1 : f s, p e s, l e s, p is, l is, p as, act : if t 2 n f a n d f v ( t ) \ ff1; :::; fng = ; t h e n t. e ls e g o t o s t e p 2 . s te p 2 : f s, p e s, l e s, p is, l is : if t ´ t¿ wh e r e ¿ is a le ft m o s t r e d e x ( ±-r e d e x o r ¯-r e d e x) , t h e n a( p; t¿0 ) , wh e r e ¿ 0 is t h e c o n vo lu t io n o f t h e ¿ , a 2 ff s; p es; les; p is; lisg, e ls e g o t o s t e p 3 . act, p as : if t ´ tfi ; ( 0 · i · n) , wh e r e tfi is t h e t e r m t wit h a ¯ xe d le ft m o s t fr e e o c c u r r e n c e o f a va r ia b le o f t h e s e t ff1; :::; fng, wh ic h is lo c a t e d t o t h e le ft o f t h e le ft m o s t r e d e x, t h e n a ( p; tti ) , wh e r e a 2 fact; p asg, e ls e g o t o s t e p 3 . s te p 3 : f s : if t ´ t[f1; :::; fn], t h e n f s ( p; t[t1; :::; tn]) . p e s : if t ´ t < fi1 ; :::; fik >, wh e r e fi1 ; :::; fik ; k > 0 is t h e s e qu e n c e o f a ll e xt e r n a l fr e e o c c u r r e n c e s o f va r ia b le s o f t h e s e t ff1; :::; fng, t h e n p es ( p; t < ti1; :::; tik >) . l e s : if t ´ tfi , wh e r e fi is t h e le ft m o s t e xt e r n a l fr e e o c c u r r e n c e o f a va r ia b le o f t h e s e t ff1; :::; fng, t h e n les ( p; tti ) . p is : if t ´ t < fi1; :::; fik >, wh e r e fi1; :::; fik ; k > 0 is t h e s e qu e n c e o f a ll in t e r n a l fr e e o c c u r r e n c e s o f va r ia b le s o f t h e s e t ff1; :::; fng, t h e n p is ( p; t < ti1 ; :::; tik >) . l is : if t ´ tfi , wh e r e fi is t h e le ft m o s t in t e r n a l fr e e o c c u r r e n c e o f a va r ia b le o f t h e s e t ff1; :::; fng, t h e n lis ( p; tti ) . act : if t ´ t¿ wh e r e ¿ ´ ¸x1:::xr[¿[x1; :::; xr]]( ¿1; :::; ¿r ) a n d ¿ is a le ft m o s t r e d e x, t h e n act ( p; t¿ [act (p;¿1);:::;act (p;¿r)] ) , e ls e g o t o s t e p 4 . p as : if t ´ t¿ wh e r e ¿ ´ ¸x1:::xr[¿[x1; :::; xr]]( ¿1; :::; ¿r ) a n d ¿ is a le ft m o s t r e d e x, t h e n p as ( p; t¿ [¿1;:::;¿r] ) , e ls e g o t o s t e p 4 . s te p 4 : act, p as : if t ´ t¿ wh e r e ¿ is a le ft m o s t r e d e x, wh ic h is a ±-r e d e x, t h e n a ( p; t¿0 ) , wh e r e ¿ 0 is t h e c o n vo lu t io n o f t h e ¿, a 2 fact; p asg. t heor em 3. any interpretation algorithm f s, p e s, l e s, p is, l is, p as, act ?-depends on canonical notion of ±-reduction. p r oof. l e t u s ¯ x m = n [ f?g, wh e r e n = f 0 ; 1 ; 2 ; :::g a n d c = fnot eqg wh e r e not eq 2 [m 2 ! m] is a b u ilt -in fu n c t io n a n d fo r e ve r y m1; m2 2 m we h a ve : not eq ( m1; m2 ) = ( 1 ; if m1; m2 2 n and m1 6= m2 ?; otherwise d. grigoryna 1 0 7 it is e a s y t o s e e t h a t not eq is a s t r o n g c o m p u t a b le , n a t u r a lly e xt e n d e d fu n c t io n wit h in d e t e r m in a t e va lu e s o f a r g u m e n t s ( a fu n c t io n is s a id t o b e n a t u r a lly e xt e n d e d , if it s va lu e is ? wh e n e ve r t h e va lu e o f a t le a s t o n e o f t h e a r g u m e n t s is ? ) . th e r e fo r e , fr o m th e o r e m 1 it fo llo ws t h a t t h e r e e xis t s t h e fo llo win g c a n o n ic a l n o t io n o f ±-r e d u c t io n ± fo r t h e s e t c: ± is : ( not eq ( n1; n2 ) ; 1 ) 2 ±, wh e r e n1; n2 2 n a n d n1 6= n2 ( not eq ( n; n) ; ?) 2 ±, wh e r e n 2 n ( not eq ( n; ?) ; ? ) 2 ±, wh e r e n 2 n ( not eq ( ?; n) ; ? ) 2 ±, wh e r e n 2 n ( not eq ( ?; ? ) ; ? ) 2 ± l e t u s d e ¯ n e t wo c a n o n ic a l n o t io n s o f ±-r e d u c t io n ±1 a n d ±2. ±1 is : ( not eq ( n1; n2 ) ; 1 ) 2 ±1, wh e r e n1; n2 2 n a n d n1 6= n2 ( not eq ( t; t ) ; ?) 2 ±1, wh e r e t 2 ¤ m ( not eq ( t; ? ) ; ?) 2 ±1, wh e r e t 2 ¤ m ( not eq ( ?; t ) ; ?) 2 ±1, wh e r e t 2 ¤ m ±2 is : ( not eq ( n1; n2 ) ; 1 ) 2 ±2, wh e r e n1; n2 2 n a n d n1 6= n2 ( not eq ( m; m ) ; ?) 2 ±2, wh e r e m 2 m ( not eq ( t; ? ) ; ?) 2 ±2, wh e r e t 2 ¤ m ( not eq ( ?; t ) ; ?) 2 ±2, wh e r e t 2 ¤ m it is e a s y t o s e e t h a t ±1 is a n e ®e c t ive , s in g le -va lu e d n o t io n o f ±-r e d u c t io n . th e r e fo r e , t o s h o w t h a t ±1 is a c a n o n ic a l n o t io n o f ±-r e d u c t io n it s u ± c e s t o s h o w t h a t ± ½ ±1. l e t ( ¿1; ¿2 ) 2 ±, wh e r e ¿1; ¿2 2 m , t h e n t h e fo llo win g c a s e s a r e p o s s ib le : ¿1 ´ n1 a n d ¿2 ´ n2, wh e r e n1; n2 2 n a n d n1 6= n2, t h e n it is o b vio u s t h a t ( not eq ( n1; n2 ) ; 1 ) 2 ±1. ¿1 ´ ¿2 ´ n, wh e r e n 2 n , t h e n fr o m ( not eq ( t; t) ; ? ) 2 ±1, wh e r e t 2 ¤ m , fo llo ws t h a t ( not eq ( n; n) ; ?) 2 ±1. ¿1 ´ n a n d ¿2 ´ ?, wh e r e n 2 n , t h e n fr o m ( not eq ( t; ? ) ; ? ) 2 ±1, wh e r e t 2 ¤ m , fo llo ws t h a t ( not eq ( n; ? ) ; ? ) 2 ±1. ¿1 ´ ? a n d ¿2 ´ n, wh e r e n 2 n , t h e n fr o m ( not eq ( ?; t ) ; ? ) 2 ±1, wh e r e t 2 ¤ m , fo llo ws t h a t ( not eq ( ?; n) ; ? ) 2 ±1. ¿1 ´ ? a n d ¿2 ´ ?, t h e n fr o m ( not eq ( t; t) ; ? ) 2 ±1, wh e r e t 2 ¤ m , fo llo ws t h a t ( not eq ( ?; ? ) ; ? ) 2 ±1. it is e a s y t o s e e t h a t ±2 is a n e ®e c t ive , s in g le -va lu e d n o t io n o f ±-r e d u c t io n . th e r e fo r e , t o s h o w t h a t ±2 is a c a n o n ic a l n o t io n o f ±-r e d u c t io n it s u ± c e s t o s h o w t h a t ± ½ ±2. l e t ( ¿1; ¿2 ) 2 ±, wh e r e ¿1; ¿2 2 m , t h e n t h e fo llo win g c a s e s a r e p o s s ib le : ¿1 ´ n1 a n d ¿2 ´ n2, wh e r e n1; n2 2 n a n d n1 6= n2, t h e n it is o b vio u s t h a t ( not eq ( n1; n2 ) ; 1 ) 2 ±2. ¿1 ´ ¿2 ´ n, wh e r e n 2 n, t h e n fr o m ( not eq ( m; m ) ; ? ) 2 ±2, wh e r e m 2 m, fo llo ws t h a t ( not eq ( n; n) ; ? ) 2 ±2. ¿1 ´ n a n d ¿2 ´ ?, wh e r e n 2 n , t h e n fr o m ( not eq ( t; ? ) ; ? ) 2 ±2, wh e r e t 2 ¤ m , fo llo ws t h a t ( not eq ( n; ? ) ; ? ) 2 ±2. 1 0 8 on dependence of interpretation algorithms of typed functional programs on canonical notion ¿1 ´ ? a n d ¿2 ´ n, wh e r e n 2 n , t h e n fr o m ( not eq ( ?; t ) ; ?) 2 ±2, wh e r e t 2 ¤ m , fo llo ws t h a t ( not eq ( ?; n) ; ? ) 2 ±2. ¿1 ´ ? a n d ¿2 ´ ?, t h e n fr o m ( not eq ( m; m) ; ? ) 2 ±2, wh e r e m 2 m, fo llo ws t h a t ( not eq ( ?; ? ) ; ?) 2 ±2. l e t p b e t h e fo llo win g p r o g r a m , wh e r e f1; f2 2 v[m !m]; x 2 vm p ( f1 = ¸x[not eq ( f2 ( x) ; f2 ( x ) ) ] f2 = ¸x[f2 ( x ) ] fo r ±1, p r o g r a m p a n d f s; p es; les; p is; lis; p as; act we h a ve : f1 ( 0 ) ; ¸x[not eq ( f2 ( x) ; f2 ( x) ) ]( 0 ) !¯ not eq ( f2 ( 0 ) ; f2 ( 0 ) ) !±1?; th e r e fo r e ( 0 ; ? ) 2 p rocf s ( p ) , ( 0 ; ? ) 2 p rocp es ( p ) , ( 0 ; ?) 2 p rocles ( p ) , ( 0 ; ? ) 2 p rocp is ( p ) , ( 0 ; ?) 2 p roclis ( p ) , ( 0 ; ?) 2 p rocp as ( p ) , ( 0 ; ? ) 2 p rocact ( p ) . fo r ±2, p r o g r a m p a n d act , lis, p as, les we h a ve : f1 ( 0 ) ; ¸x[not eq ( f2 ( x) ; f2 ( x) ) ]( 0 ) !¯ not eq ( f2 ( 0 ) ; f2 ( 0 ) ) ; not eq ( ¸x[f2 ( x) ]( 0 ) ; f2 ( 0 ) ) !¯ not eq ( f2 ( 0 ) ; f2 ( 0 ) ) ; ... a n d s o o n . th e r e fo r e ( 0 ; ?) 62 p rocles ( p ) , ( 0 ; ? ) 62 p roclis ( p ) , ( 0 ; ? ) 62 p rocp as ( p ) , ( 0 ; ? ) 62 p rocact ( p ) . fo r ±2, p r o g r a m p a n d f s, p es, p is we h a ve : f1 ( 0 ) ; ¸x[not eq ( f2 ( x) ; f2 ( x) ) ]( 0 ) !¯ not eq ( f2 ( 0 ) ; f2 ( 0 ) ) ; not eq ( ¸x[f2 ( x) ]( 0 ) ; ¸x[f2 ( x ) ]( 0 ) ) !¯ not eq ( f2 ( 0 ) ; f2 ( 0 ) ) ) ; ... a n d s o o n . th e r e fo r e ( 0 ; ? ) 62 p rocf s ( p ) , ( 0 ; ? ) 62 p rocp es ( p ) , ( 0 ; ?) 62 p rocp is ( p ) . in c o n c lu s io n , fo r e a c h in t e r p r e t a t io n a lg o r it h m a 2 ff s; p es; les; p is; lis; p as; act g t h e r e e xis t ±1 a n d ±2 c a n o n c ia l n o t io n s o f ±-r e d u c t io n a n d p r o g r a m p s u c h t h a t ( 0 ; ? ) 2 p roca ( p ) fo r ±1 a n d ( 0 ; ? ) 62 p roca ( p ) fo r ±2, t h e r e fo r e a ?-d e p e n d s o n c a n o n ic a l n o t io n o f ±-r e d u c t io n . refer ences [1 ] s . a . n ig iya n \ fu n c t io n a l l a n g u a g e s " , p rogramming and computer software, vo l. 1 7 , n o . 5 , p p . 2 9 0 -2 9 7 , 1 9 9 2 . [2 ] s . a . n ig iya n , \ on n o n -c la s s ic a l t h e o r y o f c o m p u t a b ilit y" , p roceedings of the ysu, p hysical and m athematical sciences, n o .1 , p p .5 2 -6 0 , 2 0 1 5 . [3 ] s . a . n ig iya n a n d t.v .k h o n d ka r ya n , \ on c a n o n ic a l n o t io n o f ±-r e d u c t io n a n d o n t r a n s la t io n o f t yp e d ¸-t e r m s in t o u n t yp e d ¸-t e r m s " , p roceedings of the ysu, p hysical and m athematical sciences, n o . 1 , p p . 4 6 -5 2 , 2 0 1 7 . [4 ] r . y u . h a ko p ia n , \ on p r o c e d u r a l s e m a n t ic s o f s t r o n g t yp e d fu n c t io n a l p r o g r a m s " , p roceedings of ysu, natural sciences, ( in r u s s ia n ) , n o . 3 , p p .5 9 -6 9 , 2 0 0 8 . d. grigoryna 1 0 9 submitted 10.10.2017, accepted 18.01.2018. îçåç½³óí³í ýáõýïóçáý³é íñ³·ñ»ñç çýï»ñåñ»ï³óç³ûç ³é·áñçãùý»ñç ï³ëí³íáõãûáõýá ï³ýáýçï ±-é»¹áõïóç³ûç ·³õ³÷³ñçó ¸. ¶ñç·áñû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ¹çï³ñïí³í »ý ïçåç½³óí³í ýáõýïóçáý³é íñ³·ñ»ñç çýï»ñåñ»ï³óç³ûç ³é·áñçãùý»ñá: æýï»ñåñ»ï³óç³ûç ³é·áñçãùá ñçùýí³í ¿ ï»õ³¹ñù³ý, ¯é»¹áõïóç³ûç ¨ ï³ýáýçï ±-é»¹áõïóç³ûç ·áñíáõáõãûáõýý»ñç íñ³: îçåç½³óí³í ýáõýïóçáý³é íñ³·ñ»ñç ñçùý³ï³ý ë»ù³ýïçï³ý ³ýáñáß ³ñ·áõù»ýïý»ñáí ýáõýïóç³ ¿, áñá ÷áùñ³·áõûý éáõíù³ý ñçùý³ï³ý µ³õ³¹ñçãý ¿: ºã» ñçùý³ï³ý ë»ù³ýïçï³ûç ³ñå»ùá áñáß ³ñå»ùý»ñç ¹»åùáõù ³ýáñáß ¿, ³å³ çýï»ñåñ»ï³óç³ûç ³é·áñçãùá ï³ù ï³ý· ¿ ³éýáõù ? ³ñå»ùáí, ï³ù ³ßë³ïáõù ¿ ³ýí»ñç: òáõûó ¿ ïñí³í, áñ 7 ñ³ûïýç çýï»ñåñ»ï³óç³ûç ³é·áñçãùý»ñá ?-ï³ëí³í »ý ï³ýáýçï ±-é»¹áõïóç³ûç ·³õ³÷³ñçó: ²û¹ ³é·áñçãù»ñá ñ»ï¨û³éý »ý` fs (éçñí ï»õ³¹ñù³ý), pes (½áõ·³ñ»é ³ñï³ùçý ï»õ³¹ñù³ý), les (ó³ë ³ñï³ùçý ï»õ³¹ñù³ý), pis (½áõ·³ñ»é ý»ñùçý ï»õ³¹ñù³ý), lis (ó³ë ý»ñùçý ï»õ³¹ñù³ý), act (³ïïçí ³é·áñçãù), pas (å³ëçí ³é·áñçãù). î çàâèñèìîñòè àëãîðèòìîâ èíòåðïðåòàöèè òèïèçèðîâàííûõ ôóíêöèîíàëüíûõ ïðîãðàìì îò êàíîíè÷åñêîãî ïîíÿòèÿ ±-ðåäóêöèè ä. ãðèãîðÿí àííîòàöèÿ â äàííîé ðàáîòå ðàññìàòðèâàþòñÿ èíòåðïðåòàòîðû òèïèçèðîâàííûõ ôóíêöèîíàëüíûõ ïðîãðàìì. àëãîðèòì èíòåðïðåòàöèè îñíîâàí íà ïîäñòàíîâêàõ, ¯ðåäóêöèè è êàíîíè÷åñêîé ±-ðåäóêöèè. îñíîâíàÿ ñåìàíòèêà òèïèçèðîâàííûõ ôóíêöèîíàëüíûõ ïðîãðàìì åñòü ôóíêöèÿ ñ íåîïðåäåëåííûìè çíà÷åíèÿìè àðãóìåíòîâ, êîòîðàÿ ÿâëÿåòñÿ ãëàâíîé êîìïîíåíòîé åå íàèìåíüøåãî ðåøåíèÿ. åñëè çíà÷åíèå îñíîâíîé ñåìàíòèêè, äëÿ íåêîòîðûõ çíà÷åíèé àðãóìåíòîâ, åñòü íåîïðåäåëåííîñòü, òî àëãîðèòì èíòåðïðåòàöèè ëèáî îñòàíàâëèâàåòñÿ ñî çíà÷åíèåì ?, ëèáî ðàáîòàåò áåñêîíå÷íî. ïîêàçàíî, ÷òî ñåìü èçâåñòíûõ àëãîðèòìîâ èíòåðïðåòàöèè ?-çàâèñÿò îò êàíîíè÷åñêîãî ïîíÿòèÿ ±-ðåäóêöèè. âîò ýòè àëãîðèòìû: fs (ïîëíîé ïîäñòàíîâêè), pes (ïàðàëëåëüíîé âíåøíåé ïîäñòàíîâêè), les (ëåâîé âíåøíåé ïîäñòàíîâêè), pis (ïàðàëëåëüíîé âíóòðåííåé ïîäñòàíîâêè), lis (ëåâîé âíóòðåííåé ïîäñòàíîâêè), act (àêòèâíûé àëãîðèòì), pas (ïàññèâíûé àëãîðèòì). mathematical problems of computer science 53, 7–13, 2020. udc 510.6 polynomial bounded proof complexities for some classes of dnf-tautologies garik v. petrosyan yerevan state university e-mail: garik.petrosyan.1@gmail.com abstract in this paper, we present the results on frege proof complexities of some dnftautologies. at first we introduce the notion of complete dnfs and prove that complete dnfs are tautologies, we also show that if the proof complexities for the set of complete dnfs are polynomially bounded, then the set of dnf-tautologies d, for which the number of non-negated variables in every conjunct is o(log(d)), also has polynomially bounded proof complexities. later we show that the set of all balanced dnf-tautologies has polynomial proof complexities. keywords: frege systems, proof complexity, dnf, complete dnf, balanced formulas. 1. introduction one of the most fundamental problems of the proof complexity theory is to find an efficient proof system for classical propositional calculus. there is a widespread understanding that polynomial-time computability is the correct mathematical model of feasible computation. according to the opinion, a truly ”effective” system should have a polynomial-size p(n) proof for every tautology of size n. in [1], cook and reckhow named such a system a super system. they showed that np = conp iff there exists a super system. it is well known that many systems are not super. this question about the frege system, the most natural calculi for propositional logic, is still open. in many papers, some specific sets of tautologies are introduced, and it is shown that the question about polynomially bounded sizes for frege-proofs of all tautologies is reduced to an analogous question for a set of specific tautologies. in particular, lutz strasburger introduced in [2] the notion of balanced formulas and showed that if there are polynomially bounded frege proofs for the set of balanced tautologies, then the frege systems are super. an analogous result for some other class of tautologies is proved in [3]. one of the well-known classes of tautologies is the class of tautologies, given in disjunctive normal form (dnf-tautologies), and it is an open question if the frege-proof complexities for the set of dnf-tautologies have polynomial upper bounds. the frege-proof complexities of some dnf-tautology classes are investigated in this paper. at first the notion of complete 7 8 polynomial bounded proof complexities for some classes of dnf-tautologies dnf is introduced, and it is proved that if the proof complexities for the set of complete dnfs are polynomially bounded, then the set of dnf-tautologies d, for which the number of non-negated variables in every conjunct is o(log(d)), also has polynomially bounded proof complexities. then it is proved that the proof complexities of the set of balanced dnf-tautologies has polynomially bounded proof complexities as well. 2. main notions and notations here we give basic definitions, which are necessary to give main results. definition 1: a frege system f uses a denumerable set of propositional variables, a finite complete set of propositional connectives; f has a finite set of inference rules defined by a figure of the form a1a2...an b (the rules of inference with zero hypotheses are axiom schemes); f must be sound and complete, i.e., for each rule of inference a1a2...an b every truth-value assignment, satisfying a1a2...an, also satisfies b, and f must prove every tautology. the particular choice of a language for the presented propositional formulas is immaterial in this consideration. however, because of some technical reasons, we assume that the language contains the propositional variables pi (i ≥ 1) or pij (i ≥ 1; j ≥ 1) logical connectives ¬,∧,∨,⊃ and parentheses (, ). note that some parentheses and ∧ can be omitted in generally accepted cases. note that our convention for serial disjunction a1∨a2∨...∨an (conjunction a1 ∧a2 ∧ ...∧an) associated from left to right. by |ϕ| we denote the size of the formula ϕ defined as the number of all entries of logical signs in it. it is obvious that the full size of the formula, which is understood to be the number of all symbols, is bounded by some linear function in |ϕ|. in the theory of proof complexity, the four main characteristics of the proof are: tcomplexity (length), defined as the number of proof steps, l-complexity (size), defined as the sum of sizes for all formulas in the proof (size), s-complexity (space), informally defined as the maximum of minimal sum of sizes for formulas on blackboard needed to verify all steps in the proof (formal definitions are, for example, in [2]) and w-complexity (width), defined as the maximum of sizes of the proof formulas. definition 2: let φ be a proof system and ϕ be a tautology. we denote by tφϕ (l φ ϕ, s φ ϕ, w φ ϕ) the minimal possible value of t-complexity (l-complexity, s-complexity, w-complexity) for all -proofs of tautology ϕ. by analogy, we can define the mentioned proof complexity characteristics for the proof of any formula a from premises γ and denote them respectively by t φ γ`a (l φ γ`a, s φ γ`a, w φ γ`a). let m be some set of tautologies. definition 3: we call the φ-proofs of tautologies from the set m t-polynomially (lpolynomially, s-polynomially, w-polynomially) bounded if there is a polynomial p() such that tφϕ ≤ p(|ϕ|) (lφϕ ≤ p(|ϕ|) ,sφϕ ≤ p(|ϕ|) ,wφϕ ≤ p(|ϕ|)) for all tautologies ϕ from m. note that if φ-proofs of tautologies from the set m are l-polynomially bounded they are also t-polynomially, s-polynomially, w-polynomially bounded. following the usual terminology, we call the variables and negated variables literals for classical logic. the conjunct k can be represented simply as a set of literals (no conjunct contains a variable and its negation simultaneously). in [4], the notion of balanced formulas is introduced in the following way. g. petrosyan 9 definition 4: a propositional formula is called balanced if every variable has only two occurrences in it, one positive and one with negative. in [4], it is shown that the problem on l-polynomially bounded sizes of proofs for all tautologies is reduced to the problem on l-polynomially bounded sizes of proofs for all balanced tautologies. 3. main results in this part, frege-proof complexities for some classes of dnf-tautologies are investigated. a. here the notion of complete dnf-tautologies is introduced, and it is proved that if the set of complete dnf-tautologies has l-polynomially therefore also t-polynomially, spolynomially, w-polynomially bounded proofs, then the set of dnf-tautologies d, where the number of non-negated variable in each conjunct is o(log(|d|)), also has l-polynomially therefore also t-polynomially, s-polynomially, w-polynomially bounded proofs. let all variables of a dnf n1 ∨n2 ∨ ...∨nn be negated variables. conjunct k is called representative for d if k contains at least one variable (without negation) from each ni (1 ≤ i ≤ n), and every variable of k is from d. definition 5: dnf d1 = c1 ∨ c2 ∨ ... ∨ cm is called a completion of dnf d2 = n1 ∨ n2 ∨ ...∨nn iff for every representative k of d2 there is a ni (1 ≤ i ≤ n), which is a subset of k, and the expression d = (n1 ∨n2 ∨ ...∨nn)∨(c1 ∨c2 ∨ ...∨cm) is called a complete dnf. theorem 1: complete dnfs are tautologies. proof. let’s assume the opposite: there is a complete dnf d = (n1∨n2∨...∨nn)∨(c1∨ c2∨...∨cm) such that it is not a tautology. if d = 0 in any collection, then n1∨n2∨...∨nn should also be 0 in that collection. n1 ∨n2 ∨ ...∨nn = 0, only if each conjunct is equal to 0, therefore, we have that in each conjunct ni (1 ≤ i ≤ n) there is a literal equal to 0. if we consider the conjunct p constructed by the conjunction of the variables of these literals, as we have a complete dnf, there is a cj (1 ≤ j ≤ m) such that cj is a subset of p , hence cj = 1, but we have assumed that d = 0, which is a contradiction. from this proof it is easy to see that all dnf-tautologies, the conjuncts of which do not contain a variable and a negated variable simultaneously, are complete dnfs. to evaluate the proof complexities of dnf-tautologies by reducing the proof to the proof of complete dnftautologies, we must use some transformations of formulas, therefore we give the following auxiliary statements, which are used to perform those transformations. lemma 1: for all formulas a,b and c, the set of the following formulas 1. a∨b ⊃ c ≡ (a ⊃ c) ∧ (b ⊃ c) 2. a ⊃ a∨b 3. a ⊃ (b ∨c) ≡ (a ⊃ b) ∨ (a ⊃ c) 4. b ⊃ (a ⊃ a∧b) 5. (a ⊃ b) ⊃ ((a∨c) ⊃ (b ∨c) has l-polynomially bounded proofs. 10 polynomial bounded proof complexities for some classes of dnf-tautologies proof. the proof is obvious. most dnf-tautologies , the proof complexities of which are being investigated, have small conjuncts sizes compered to their size. using complete dnfs, we can construct polynomial proofs for such dnf-tautologies. the following theorem gives such a construction. theorem 2: if the set of all complete dnfs has l-polynomially bounded proofs then, the set of dnf-tautologies d, where the number of non-negated variables in each conjunct is o(log(|d|)), also has l-polynomially bounded proofs. proof. let’s consider d dnf-tautology, where the number of non-negated variables in each conjunct is o(log(|d|)). for each conjunct, we separate the sub-conjunct of variables with positive entrance and the set of all such conjuncts and its subsets we denote by p . since the number of non-negated variables in each conjunct is o(log(|d|)), there is such a polynomial p() function that |p| ≤ p(|d|). if dnf is a tautology, it should have at least one conjunct, where all the variables are negated, otherwise it does not cover the 0 point we denote the disjunction of all such conjuncts by n. if d is a tautology, then there is a completion of n such that all conjuncts are from p . if such a completion did not exist, we would take a conjunct constructed by taking one variable from each conjunct of n, and this conjunct would not be covered by p , therefore if we set the values of these variables 1 and the values of all other variables 1, the value of d will be 0. let’s denote that completion by c. n ∨ c is a completion dnf; to prove d, we need to prove n ∨c and n ∨c ⊃ d, the first part follows from our condition. let’s prove the second part n ⊃ d. from lemma 1.1 we get two tautologies to prove n ⊃ d and c ⊃ d. from lemma 1.2 we get the polynomial proof of n ⊃ d. now let’s prove c ⊃ d. suppose c = c1 ∨ c2 ∨ ... ∨ cm, where ci(1 ≤ i ≤ m) is a conjunct. using lemma 1.1, we can convert c ⊃ d into (c1 ⊃ d) ∧ (c2 ⊃ d) ∧ ...∧ (cm ⊃ d). as ci (1 ≤ i ≤ m) is a conjunct, we can prove each one alone and later join them with ∧. note that the number of such ci conjuncts is less than p , therefore it has a polynomial upper bound. using lemma 1.4 and 5, we can reduce the proof of each (ci ⊃ d) (1 ≤ i ≤ m) to a new dnf, which also satisfies the condition that the number of non-negated variables in each conjunct is o(log(|d|)). we can use the same technique to prove these new tautologies and note that their p sets are subsets of our initial p set, therefore we will use only the first p . if we continue the proof in this way, we will have several tautologies, the number of which is polynomial from |d|, and each reduction is performed using polynomial steps and polynomial formulas, therefore the proof is l-polynomially bounded. corollary 3: if the set of complete dnfs has l-polynomially bounded proofs, than the set of dnf-tautologies d, where the number of negated variables in each conjunct is o(log(|d|)), also has l-polynomially bounded proofs. proof. the proof can be obtained from the proof of theorem 2 with slight changes. b. here, the proof complexities of some subset of balanced dnf-tautologies are investigated. definition 6: a dnf-tautology is correct if dnf, obtained from it by removing any conjunct, is no longer a tautology. lemma 2: the number of conjuncts in a balanced correct dnf-tautology with n variables is n+1. g. petrosyan 11 proof. we prove the lemma by induction on number n of variables in a balanced dnftautology d. if n = 1, we have d = ¬p∨p. suppose the statement is valid for a balanced dnf-tautology, which has ≤ n variables. if the number of variables is n + 1, then the correctly balanced dnf should have at least one conjunct with at least two literals pα1i1 p α2 i2 . after assigning α1 to the variable pi1 everywhere in the given dnf, both the number of variables and the number of conjuncts decrease by one, hence the number of conjuncts in the primary dnf should be n + 2. corollary 4: every balanced correct dnf-tautology has at least one conjunct, consisting of one literal. theorem 5: all balanced correct dnf-tautologies have l-polynomially bounded proofs. proof. let us have a balanced correct dnf-tautology d = d1 ∨d2 ∨ ...∨dn+1, which depends on the variables p1,p2, ...,pn. we take ¬d and prove the a contradiction. ¬d = n1 ∧n2 ∧ ...∧nn+1, where ni = ¬di (1 ≤ i ≤ n + 1) and ni = pα1i1 ∨p α2 i2 ∨ ...∨pαrir , where αi ∈ {0, 1} (1 ≤ i ≤ n + 1). based on ni, we construct the formula ei by adding instead of every variable pj from p1,p2, ...,pn which has no occurrence in ni, the formula p1 ∧¬p1 on the j-th place with a disjunction. it is obvious that ni = ei, and this equivalence can be derived with polynomially bounded characteristics of proof complexities. by this notation we have that formula ¬d is equivalent to formula d′ = e1 ∧e2...∧en+1. now we introduce new propositional variables pij (1 ≤ i ≤ n + 1; 1 ≤ j ≤ n), where pij is true if the variable pj occurs in ni and is false for the opposite case, and construct on the base of ei new disjunctions d′i by replacing both the primary literals and the formulas p1 ∧¬p1 with the corresponding variables pij. now we consider the well-known formulas of pigeon hole principle phpn = ∧ni=0 ∨ n−1 k=0 pik ⊃∨ n−1 k=0 ∨0≤i 0, qi = sign(zi). (2) substituting (1) into the hamilton equations (see, for example, ([22])) we can find: −ẍi = ji−1, i(xi−1 − xizi−1zi−1) + ji, i+1(xi+1 − xizi−1zi+1), −ÿi = ji−1, i(yi−1 − yizi−1zi−1) + ji, i+1(yi+1 − yizi−1zi+1), (3) 24 computation statistical properties of disordered systems from the first principles of mechanics where the following notations are made, ξ̈i = ∂ 2ξi/∂t 2 and ξ = (x, y), in addition ”t” denotes the usual time. we will assume that near the nodes spins are localized and quasiperiodic movements commit, ξi(t) = ξ 0 i + δ ξ i (t), where ξ 0 i and δi(t) denote the position of the equilibrium and quasi-periodic function of the time, respectively. below we will study the statistical properties of the system, which are formed on time scales τ >> τ0, where τ0 is a characteristic time of spins oscillation and obviously, in this case; ⟨ẍ⟩τ0 = ⟨ÿ⟩τ0 ≈ 0. averaging equations (3) on the period τ0 can be found: ji−1, i(xi−1 − xizi−1zi−1) + ji, i+1(xi+1 − xizi−1zi+1) = 0, ji−1, i(yi−1 − yizi−1zi−1) + ji, i+1(yi+1 − yizi−1zi+1) = 0, (4) where for simplicity in equations the index ”0” over of variables are omitted, i.e., x0i → xi, y 0 i → yi and z0i → zi. as it is easy to verify these equations define the condition at which the hamiltonian (1) in the i-th node takes an extremal value. solving the system of equations (4) with respect to variables xi+1 and yi+1 may be found: xi+1 = cx/ji, i+1, yi+1 = cy/ji, i+1, (5) where the following notations are made: cx(y) = ax(y) − by(x)(c ± √ d) 1 + b2x + b 2 y , aη = ηizi −1zi−1 − ηi−1, bη = ηizi−1qi+1, d = (1 + b2x + b 2 y − a 2 x − a 2 y − c 2) > 0, c = axby − aybx, η = (x, y). now, for the hamiltonian (1) the conditions of a local minimum can be set. it is obvious i-th spin is in stable equilibrium if the following inequalities are satisfied: axixi(s 0 i ) > 0, axixi(s 0 i )ayiyi(s 0 i ) − a 2 xiyi (s0i ) > 0, (6) where aηiηi = ∂ 2h/∂η2i and axiyi = ∂ 2h/∂xi∂yi; in addition s 0 i denotes i-th spin which is in a stable equilibrium. using (2), (4) and (6), the explicit forms of the second order derivatives can be calculated: aηiηi = (η 2 i + z 2 i )z −3 i ∆i, axiyi = xiyiz −3 i ∆i, ∆i = (ji−1, izi−1 + ji+1, izi+1), (7) and taking into account (6) and (7) the conditions of a local minimum energy may be found: axixi = (1 − y 2 i )z −3 i ∆i > 0, axixiayiyi − a 2 xiyi = z−4i ∆ 2 i > 0. (8) since, at the zi > 0 both conditions in (8) are satisfied: ∆i = (ji−1, izi−1 + ji+1, izi+1) > 0. (9) thus, in each node the solutions determining the orientation of the spin in the state of the local equilibrium can be found. if there is such coupling constants ji, i+1, for which not only the conditions (8) or (9) are satisfied, but also the following inequality holds: j2i, i+1 ≥ c 2 x + c 2 y > 0. (10) a. gevorkyan and v. sahakyan 25 3. the statistical ensemble of 1d disordered spin-chains it is easy to show, that the solutions of equations (5) satisfying inequalities (8) can be of two types: a. if ji−1, isi−1 · si ≤ 0 and |ji, i+1| > |ji−1, i|, then there is only one solution, which we denote by; s+i+1 (queen), and respectively, b. if ji−1, isi−1 · si > 0 and |ji, i+1| ≥ |j0,1| · |s0 × s1|, then s+i+1 is the solution, in addition there is another solution, s−i+1 (drone) under the condition that, |ji, i+1| < |ji−1, i|. note that the solutions which are denoted with signs ”+” and ”−” are characterized as follows, if the previous solution is the queen ”+”, up to two different solutions may be found; s+i+1 and s − i+1, while after the drone ”−” the solution is only one s+i+1. taking into account this we can construct solutions graphically in the form of separate fibonacci subtrees (f̂st i) (see fig. 2). the mathematical expectation of branchings number depending on the height of f̂st i can be calculated as follows: m(n) = m(n − 1)⌊ (2ξn)⌋ = ⌊2nη(n)⌋, η(n) = 1 + n−1 n∑ k=1 log2(ξk) > 0, (11) where m(n − 1) number of branchings at the height (n − 1) and ξk denotes a random coefficient which belongs to the interval [1/2, 1]. for simplification in the expression (11) the subtree’s number i is omitted. since each f̂st i consists of the set of nodes and the set of edges (the set of constants {j} = [j1,2, j2,3, ...jn−1,n]), it can be represented as a graph gi(n) ∼= {gj(n), j ∈ m}, where gj(n) denotes a random string by length n, which is characterized by kolmogorov’s complexity [23, 24]. subtree1 subtree2 j6,7 j6,7 j7,8 j7,8 j3,4 j3,4 j4,5 j4,5 j6,7 j7,8 j5,6 j5,6 j5,6 j5,6 j6,7 j6,7 j1,2 j2,3 j7,8j7,8 j7,8 + s6 + s7 s7 + s8 + s3 + s4 s4 s6 + s7 + s8 + s5 + s5 + s6 s6 s1 + s2 + s7 + s8+ s8 + s7 + s8 s8 j6,7 j7,8 j7,8 j7,8 j5,6 j5,6 j6,7 j6,7j6,7 j6,7 j7,8 j7,8 j3,4 j3,4 j4,5 j4,5 j4,5 j5,6 j5,6 j1,2 j2,3 j7,8 j7,8 j6,7 s6 + s7 + s7 + s8 s7 + s8 + s5 + s6+ s6 + s7 s7 + s3 + s4 s4 s5 + s5 s1 + s2 + s8 + s7 s8 + s8 + s8 + s6 + s8 subtree3 subtree4 j7,8 j5,6 j6,7 j6,7 j7,8 j4,5 j4,5 j5,6 j6,7 j6,7 j7,8 j3,4 j1,2 j2,3 j7,8 s7 s8 s5 s6 s7 s8 s4 s5 s6 s7 s8 s3 s1 s2 s7 s8 j7,8 j6,7 j7,8 j7,8 j5,6 j5,6 j6,7 j6,7 j3,4 j3,4 j4,5 j4,5 j1,2 j2,3 j6,7 j7,8 j7,8 j5,6 j5,6 j7,8 j7,8 s7 s8 s6 s7 s5 s6 s6 s8 s8 s3 s4 s4 s1 s2 s6 s7 s5 s8 s8 s7 s8 s8 subtree5 subtree6 j6,7 j7,8 j7,8 j5,6 j6,7 j6,7 j7,8 j7,8 j4,5 j5,6 j5,6 j7,8 j7,8 j6,7 j7,8 j7,8 j4,5 j4,5 j5,6 j6,7 j6,7 j3,4 j3,4 j1,2 j2,3 s6 s7 s5 s6 s7 s7 s8 s4 s5 s7 s8 s7 s8 s6 s7 s4 s5 s6 s3 s1 s2 s8 s8 s8s8 s8 j7,8 j5,6 j5,6 j6,7 j6,7 j6,7 j7,8 j7,8 j7,8 j7,8 j7,8 j7,8 j1,2 j2,3 j6,7 j6,7 j3,4 j3,4 j4,5 j4,5 j5,6 s7 s8 s5 s6 s6 s7 s7 s7 s8 s8s8 s7 s8 s8s8 s1 s2 s6 s3 s4 s4 s5 fig. 1. two different fibonacci subtrees (graphs) each with the height 8. both of graphs grow from the same initial data (root) in the result of two independent numerical experiments. the same symbols si and ji,j on different graphs can have completely different values. thus, for calculations of different physical parameters of the statistical ensemble, it is necessary to take into account the contribution of all independent graphs {g(n)} = [g1(n), ...gi(n), ...]. 26 computation statistical properties of disordered systems from the first principles of mechanics with regard to the graph computation complexity, it is easy to prove that: kg(n) ∝ m(n)ks(n), (12) where m(n) ∝ 2n is the value of branching on the step n, ks(n) denotes kolmogorov’s complexity of the string gj(n), while kg(n) denote the complexity of the graph gi(n) ⊂ {g(n)}n. the computational complexity of {g(n)}n obviously will be kens ∝ nm(n)ks(n), where n is the total number of graphs in the ensemble. the mathematical expectation of random variable f characterizing the ensemble {g(n)}n can be calculated by the formula: e[f] = f̄ = ∑n i=1 wif̄i∑n i=1 wi , wi = ni/n̄, (13) where ni and n̄ denote the number of strings of the graph gi(n) and the total number of strings in the ensemble, respectively, in addition f̄i = ∑ gi(n) f denotes the expectation of random variable f in the gi(n), which is calculated similarly to formula (13). lemma. if statistical weights of all independent graphs gi(n) ⊂ {g(n)}n are approximately the same it can be shown that the statistical weights of all strings gj(n) ⊂ {g(n)}n are equal exactly. in this case we can use the law of large numbers and simplify the expression (13) writing it as: e[f] = f̄ = 1 n n∑ j=1 f̃j + o(n −1/3), (14) where f̃j = ∑ gj f denotes the expectation of the random variable f on a randomly selected string gj(n) ⊂ gi(n). thus, the computation of statistical parameters of the disordered spin system by the formula (13) is algorithmically equivalent to solving of np hard problem, while the simulation by the formula (14) is the p problem. 4. the numerical experiments as a rule the problems of spin glasses are studied in the framework of the partition function representation by methods of monte carlo which, however, do not allow to answer many important questions of the statistical ensemble. in particular an important problem is the fact that the spin glass in the state of a statistical equilibrium generally speaking is in a metastable state and has some distribution near the ground state. the system in this state, obviously, cannot be studied using monte carlo methods, because these methods are adapted only for calculating the ground state. at first let us consider one set of initial data ω1i (roots) which includes orientations of the first two spins of the chain and the coupling constant between them, which are generated randomly from the corresponding homogeneous distributions. using the system of recurrence equations (5), with consideration of inequality conditions (8), we perform successive calculations of spin-chain. recall that this system of equations connects three consecutive spins, so that knowing the configuration of two previous spins, we can generate from lognormal distribution ([25]) a random constant ji, i+1 and exactly to calculate the orientation of spin in the subsequent node. conducting the consecutive node-by-node calculations on a. gevorkyan and v. sahakyan 27 the n-th step, we generate a random graph gi(n) ⊂ {g(n)}n at internal nodes of which the spins are in local minima of energies. with regard to spins in the external nodes, then it is supposed that they are in local minima of energies on the basis of other considerations. we calculated the characteristic distributions and parameters of the 1d spin glass, which is in the state of the statistical equilibrium using two algorithms which are based on formulas (13) and (14), respectively. for the simulation of the problem, first of all we need min max μ σ -35. -2.9 -14.2 3. -34.4 -2.8 -14.2 3. chains trees -25 -20 -15 -10 -5 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 ¶ ph¶l 0.00 0.05 0.10 0.15 0.20 0.25 0.30 p hj l p min max μ σ -6.3 6.2 0. 1.5 -5.8 5.8 0. 1.5 chain tree -4 -2 0 2 4 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 j phjl fig. 2. the distributions of energies and spin-spin coupling constant. the beige curves denote the results of calculations using the algorithm with the complete enumeration of all branches of graphs (conditionally we will call np algorithm), while black curves are constructed in the result of calculations by p algorithm. to specify initial conditions in the form of a large number of independent configurations (roots), i.e., the large set of the first two spins and coupling constants between them; {ω11 = (s11, s11; j11,2)1, ...ω1n = (s 1 1, s 1 2; j 1 1,2)n} = ω̂. the steps of simulation using the algorithm np are as follows. using the initial data, ω̂ we perform parallel calculations of all graphs gi(n) of the ensemble gi(n) ⊂ {g(n)}n. note that each of these graphs in terms of classical mechanics represents the set of classical trajectories that go out from one initial value (root). the database which is obtained in the result of simulation using np algorithm allows to construct distributions of the main parameters of the statistically equilibrium ensemble. the simulation by p algorithm which is based on the formula (14), is performed in a similar way but with the difference that in this case instead of the set of graphs {g(n)}n we grow the set of strings {g(n)}n. in this case from each graph we choose only one string as a representative. note, that the string (branch) gj(n) ⊂ {g(n)}n we grow by way of randomly selecting only one solution in each node. in the result of parallel simulation of the set of strings, we get the database which allows to construct the distributions of main parameters of the statistical equilibrium ensemble {g(n)}n with accuracy o(n−1/3). we compared the results of numerical simulations on the example of the statistical ensemble, {g(20)}5·104 consisting from 5 · 104 graphs by heights 20 with the ensemble {g(20)}5·104 which consists of the 5 · 104 strings of lengths 20. as it can be seen from fig. 2, the distributions of various parameters of the statistical ensemble that have been calculated in the limit of statistical equilibrium using two np and p algorithms coincide ideally. 28 computation statistical properties of disordered systems from the first principles of mechanics thus, we have shown on the example of 1d heisenberg spin glass, that the np hard problem with the given accuracy may be reduced to the p problem. 5. partition function now we can return to the definition of the main object of statistical physics, i.e., the partition function. as known the multiparticle classical system in the state of the statistical equilibrium in the configuration space is described by the partition function of type: z(β) = ∫ ... ∫ exp{−β h({r})}⌈r∞..., ⌈rn , β = ∞/∥bt , {r} = (r∞, ..., rn ), (15) where h({r}) is the hamiltonian of the system in the configuration space, kb and t are the boltzmann constant and temperature of the system, respectively. for the considered model the partition function is calculated exactly and has the following form ([21]): z(β, {j}) = n∏ i=1 sinh(ai) ai , ai = βji, i+1, (16) where the coupling constant ji, i+1 ∈ {j} = (j1, 2, j2, 3, ...jn−1, n) is the random variable. the average value of the partition function for the ensemble, can be found by way of averaging over the distribution of the coupling constant. it is often assumed that the distribution is gaussian: w(j) = 1 σj √ 2π exp { − (j − j0)2 2σ2j } , (17) where σj is the variance and j0 is the average value of coupling constant. averaging of the partition function (16) by the distribution (17) can find: z̄(β) = ∫ +∞ −∞ z(β, {j})w(j)dj = k(β) √ 2π ∫ +∞ −∞ (sinh(σjβx) σjβx )n e− 1 2 (x−x0)2dx, (18) where x = j/σj and x0 = j0/σj, in addition k(β) is the normalization factor: k−1(β) = 1 2j̄ ∫ j̄ −j̄ (sinh(jβ) jβ )n dj = 1 ȳ ∫ ȳ 0 (sinh(y) y )n dy, j ∈ [j̄, −j̄], with j̄ > 0 and ȳ = j̄β. recall that the coefficient k(β) is constructed in such a way that the helmholtz free energy converges to zero in the limit of β → ∞. the helmholtz free energy per one spin in chain is calculated by the following formula: f(β) = − 1 nβ ln z̄(β). (19) since the integration in representation (15) is carried out over the complete configuration space, then obviously we take into account also the contributions of physically unrealizable spin configurations. recall that usually the measure of a set of such spin configurations is assumed to be equal to zero without any serious proof, that not only groundless, but in some cases may be wrong. taking into account the fact that a set of strings describing the statistical ensemble in configuration space formally can be represented as a trajectory a. gevorkyan and v. sahakyan 29 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0 1 2 3 4 5 fig. 3. the free energy of the ensemble. the red curve is obtained at using of the expression (20), while the blue curve is obtained in the result of calculation by the expression (19). note that parameters of ε0 and σε are found by the way of simulation of problem from the first principles, whereas parameters j0 and σj are chosen for reasons of the best approximation to red curve. of dynamical system, in the limit of ergodicity of system (see [26, 27]), for the partition function the following representation may be written: z⋆(β) = ∫ −n/β −nε0 p̄(ε)dε, p̄(ε) = c−1p(ε), c = ∫ 0 −nε0 p(ε)dε, (20) where ε < 0 and ε0 = µ < 0 (see fig. 2) denote the energy of 1d spin-chain and its average energy, respectively, p̄(ε) is the normalized energy distribution. if the energy distribution (see fig. 2) to approximate by gaussian function (see expression (17)), then using the representation (20), for the free energy attributable to a single spin we obtain the following analytical expression: f⋆(β) = − 1 nβ ln {1 2 [ 1 − erf (ε0 + n/β√ 2σε )]} , (21) where σε denotes the variance of spin-chains energy distribution. comparing helmholtz’s free energies f(β) and f⋆(β) for the ensemble {g(20)}5·105 shows that these curves diverge sharply already at finite temperatures (see fig. 3 ). as we can see, near the temperature β ≃ 0.3, the ensemble of spin-chains exhibits a critical behavior, since the free energy tends to infinity (the red curve) that is characteristic to the phase transitions of the first order. the latter obviously is connected with the fact that in the expression (21), only the physically realizable spin configurations are taken into account. 30 computation statistical properties of disordered systems from the first principles of mechanics 6. conclusion we studied 1d spin glass in the framework of heisenberg’s nearest-neighboring hamiltonian from the first principles of the classical mechanics. it was shown that in the framework of the considered approach, even this simple task, which in a sense, is exactly solvable, corresponds to the category of np-hard (see the estimation of complexity (11)). it was proved that in the limit of the statistical equilibrium the computational np-hard problem, with a given accuracy to the p problem is reduced. the developed method as opposed to standard approach allows to calculate the statistical distributions of all parameters of the ensemble, including the distribution of coupling constant (see fig. 2). in the paper, a new representation was suggested for the partition function in the form of one dimensional integral from the spin-chains’ energy distribution (see the expression (20)). we have compared the helmholtz’s free energies which were calculated by using the usual (19) and new (21) representations. as it is shown (see fig. 3), the corresponding curves are significantly different already at finite temperatures, moreover, near the value β ∼ 0.3 the ensemble of spin-chains demonstrates a critical property, that usually occurs at first-order phase transitions. this is obviously due to the fact that in the formula (21), only such spin configurations are presented which satisfy the basic principles of classical mechanics (see expressions (4)-(9). thus, the main advantages of the developed approach are that we have received clear answers to all the raised questions on the example of study 1d spin glass from the first principles of the classical mechanics without using any additional assumptions. the ideas lying in the base of the developed approach are enough universal and allow the generalization of model for a multidimensional case and at presence of external fields ([28]). lastly, note that the new formulation of the problem of spin glasses and disordered systems in general might become very useful for investigation of the more global problem: namely, the problem of reduction of np hard to the p problem. references [1] k. binder and a. young, “spin glasses, experimental facts, theoretical concepts and open questions”, rev. mod. phys., vol. 58, pp. 801-976, 1986. [2] m. mézard, g. parisi and m. virasoro, “spin glass theory and beyond (world scientific)”, singapore 1987. [3] a. young, spin glasses and random fields, world scientific, singapore, 1998. [4] r. fisch and a. harris, “spin-glass model in continuous dimensionality”, phys. rev. lett., vol. 47, page 620, 1981. [5] c. ancona-torres, d. silevitch, g. aeppli and t. rosenbaum, “quantum and classical glass transitions in lihoxy1-xf4”, phys. rev. lett., vol. 101, no. 5, 057201, 2008. [6] a. bovier, statistical mechanics of disordered systems, a mathematical perspective, cambridge series in statistical and probabilistic mathematics, 2006. [7] y. tu, j. tersoff and g. grinstein, “properties of a continuous-random-network model for amorphous systems”, phys. rev. let., vol. 81 ,page 2490, 1998. [8] k. chary and g. govil, “nmr in biological systems, from molecules to human”, springer, vol. 6, page 511, 2008. [9] e. baake, m. baake and h. wagner, “ising quantum chain is a equivalent to a model of biological evolution”, phys. rev. let., vol. 78, page 559, 1997. a. gevorkyan and v. sahakyan 31 [10] a. s. gevorkyan and h. g. abajyan, “a new parallel algorithm for simulation of spin glasses on scales of space-time periods of external fields with consideration of relaxation effects”, phys. of particles and nuclei letters, vol. 9, no. 6-7, page 530, 2012. [11] f. liers, m. palassini, a. k. hartmann and m. jünger,“ground state of the bethe lattice spin glass and running time of an exact optimization algorithm”,phys. rev. b, 68, 094406, 2003. [12] j. c. angles, d′auriac, m. preissmann and a. sebo leibniz-imag, “optimal cuts in graphs and statistical mechanics”, mathl. comput. modeling, 26, no. s-10, page 11, 1997. [13] c. papadimitriou, computational complexity, (1st ed.). addison-wesley. section 2.7: nondeterministic machines, 1993. [14] h. r. lewis and c. papadimitriou, elements of the theory of computation, (1st ed.). prentice-hall. section 4.6: nondeterministic turing machines, 1981. [15] n. metropolis, a. rosenbluth, m. rosenbluth, a. teller and e. teller, “equation of state calculations by fast computing machines”, j. chem. phys., vol. 21, no. 6, page 1087, 1953. [16] b. hayes, “can’t get no satisfaction”, am. scientist, vol. 85, page 108, 1997. [17] r. monasson et all, diameter of the world wide web, nature, london, 400, 133, 1999. [18] special issue of theor. comput. sci. 265, edited by o. dubois, r. monasson, b. selman, and r. zecchina, 2001. [19] m. j. alava, p. m. duxbury, c. f. moukarzel and h. rieger, phase transitions and critical phenomena, edited by c. domb and j. lebowitz, academic press, new york, 18, 2001. [20] a. k. hartmann and h. rieger, optimization algorithms in physics, while-vch, berlin, 2001. [21] c. j. thompson, “phase transitions and critical phenomena”, academic press, vol. 1, pp. 177-226, 1972. [22] h. goldstein, classical mechanics, reading, ma: addison-wesley, 2nd ed., 1980. [23] a. n. kolmogorov, “logical basis for information theory and probability theory”, ieee transactions on information theory, vol. 14, no. 5, pp. 662–664, 1968. [24] m. li and p. vitányi, an introduction to kolmogorov complexity and its applications, new york, springer-verlag, 1997. [25] s. y. park and a. k. bera, “maximum entropy autoregressive conditional heteroskedasticity model”, journal of econometrics, elsevier, vol. 50, no. 2, pp. 219–230, 2009. [26] g. d. birkhoff, “what is the ergodic theorem?”, the american mathematical monthly, vol. 49, no. 4, pp. 222–226, 1942. [27] v. i. arnol’d and a. avez, ergodic problems of classical mechanics, new york, w.a. benjamin, 1968. [28] e. a. ayryan, a. s. gevorkyan and v. v. sahakyan, “new algorithm for simulation of 3d classical spin glasses under the influence of external electromagnetic fields”, physics of particles and nuclei letters, vol. 2, no. 3, pp. 380–384, 2015. submitted 15.07.2015, accepted 27.11.2015 3 2 computation statistical properties of disordered systems from the first principles of mechanics âï³ñ·³íáñí³í ëåçý³ûçý ñ³ù³ï³ñ·»ñç íç×³ï³·ñ³ï³ý ñ³ïïáõãûáõýý»ñç ñ³ßí³ñïá »éý»éáí ¹³ë³ï³ý ù»ë³ýçï³ûç ñçùý³ñ³ñ ëï½µáõýùý»ñçó ². ¶¨áñ·û³ý ¨ ì. ê³ñ³ïû³ý ²ù÷á÷áõù ø»ýù ñ»ï³½áï»é »ýù ¹³ë³ï³ý 1d ëåçý³ûçý ³å³ïçý»ñá ð³û½»ýµ»ñ·ç ùá¹»éç ßñç³ý³ïý»ñáõù: ðçùýí»éáí ð³ùçéïáýç ñ³í³ë³ñáõùý»ç íñ³ ¹áõñë ¿ µ»ñí³í é»ïáõñ»ýï ñ³í³ë³ñáõùý»ñç ñ³ù³ï³ñ·, áñá ãáõûé ¿ï³éçë çñ³ï³ý³óý»é ëåçý³ûçý ßõã³ý»ñç ñ³ßí³ñïá ñ³ý·áõûó ³é ñ³ý·áõûó: òáõûó ¿ ïñí³í, áñá ¹³ë³ï³ý ù»ë³ýçï³ûç ñçùý³ñ³ñ ëï½µáõýùý»ñç ñçù³ý íñ³ ñ³ßí³ñïý»ñá µ»ñáõù »ý np µ³ñ¹ ëý¹ñç, áñá ë³ï³ûý íç×³ï³·ñ³ï³ý ñ³í³ë³ñ³ïßéáõãû³ý ë³ñù³ýáõù ïñí³í åßïáõãû³ùµ ñ³ßííáõù ¿ p ³é·áñçãùç ùççáóáí: ²ýë³ùµéç íç×³ï³·ñ³ï³ý ·áõù³ñç ñ³ù³ñ ³é³ç³ñïí³í ¿ ù»ïã³÷³ýç çýï»·ñ³é³ûçý ý»ñï³û³óáõù ëåçý³ûçý ßõã³ý»ñç ¿ý»ñ·ç³ûç µ³ßëù³ý ýáõýïóç³ûçó: âû÷èñëåíèå ñòàòèñòè÷åñêèå ñâîéñòâà íåóïîðÿäî÷åííûõ ñïèíîâûõ ñèñòåì èç ïåðâûõ ïðèíöèïîâ êëàññè÷åñêîé ìåõàíèêè à. ãåâîðêÿí è â. ñààêÿí àííîòàöèÿ ìû èññëåäîâàëè êëàññè÷åñêèå ñïèíîâûå ñòåêëà â ðàìêàõ 1d ìîäåëè ãåéçåíáåðãà. îñíîâûâàÿñü íà óðàâíåíèÿõ ãàìèëüòîíà, âûâåäåía ñèñòåìà ðåêóððåíòíûõ óðàâíåíèé, êîòîðàÿ ïîçâîëÿåò îñóùåñòâëÿòü âû÷èñëåíèå ñïèíîâîé öåïî÷êè óçåë çà óçåëîì. ïîêàçàíî, ÷òî ïðîâåäåííûå íà îñíîâå îñíîâíûõ ïðèíöèïîâ êëàññè÷åñêîé ìåõàíèêè ðàñ÷åòû ñâîäÿòñÿ ê np òðóäíîé çàäà÷å, êîòîðàÿ, òåì íå ìåíåå, â ïðåäåëå ñòàòèñòè÷åñêîãî ðàâíîâåñèÿ ñ çàäàííîé òî÷íîñòüþ âû÷èñëÿåòñÿ p àëãîðèòìîì. äëÿ ñòàòèñòè÷åñêîé ñóììû àíñàìáëÿ ïðåäëîæåíî îäíîìåðíîå èíòåãðàëüíîå ïðåäñòàâëåíèå îò ðàñïðåäåëåíèÿ ýíåðãèè ñïèíîâîé öåïî÷êè. article.pdf (p.1-10) 02.pdf (p.11) d:\sbornik\...\intel.dvi mathematical problems of computer science 23, 2004, 130{133. i ntellectual system of students' t esting s e r g e y v . b a r kh u d a r ya n institute for informatics and automation problems of nas of ra e-mail sergeybarkh@mail.ru abstract this paper presents a distance testing system that allows testing of students with dynamically created sequences of tests. the sequence of questions given to students is generated in the process of testing based on his/her answers to the consequent test. overall performance of developed system is tested by the creation of the corresponding mathematical model. refer ences [1 ] l a u r a th o m s o n , l u ke w e llin , p h p a n d mys ql w e b d e ve lo p m e n t . d ia s o ft p u b lis h in g , k ie v, 2 0 0 2 . [2 ] b ill b a ll, d a vid p it t s , r e d h a t l in u x 7 .x . d ia s o ft p u b lis h in g , k ie v-mo s c o w, 2 0 0 2 . [3 ] k .b e r g e , gr a p h t h e o r y a n d it s a p p lic a t io n . mir , mo s c o w, 1 9 6 2 . [4 ] n .ch r is t o ¯ d e s , gr a p h t h e o r y, a n a lg o r it h m ic a p p r o a c h . a c a d e m ic p r e s s , n e w y o r k, l o n d o n , 1 9 7 5 . [5 ] d is t a n c e l e a r n in g p o r t a l, www.d is t a n c e -le a r n in g .r u . r u s s ia , ma y 2 0 0 4 . [6 ] w e b s o ft , www.we b s o ft .r u . r u s s ia , ma y 2 0 0 4 . àõë³ýáõý»ñç ï»ëï³íáñù³ý çýï»é»ïïáõ³é ñ³ù³ï³ñ· ê. ì. ´³ñëáõ¹³ñû³ý ²ù÷á÷áõù ²é³ç³ñïíáõ ñá¹í³íáõù ýï³ñ³·ñí³í ¿ ñ³ù³ï³ñ·, áñá ñý³ñ³íáñáõãûáõý ¿ ï³éçë ï³½ù³ï»ñå»é ñ»é³ñ³ñ ï»ëï³íáñáõù ¹çý³ùçï ëï»õííáõ ñ³ñó³ß³ñ»ñç ùççáóáí: àõë³ýáõçý ïñíáõ ñ³ñó»ñç ñ³çáñ¹³ï³ýáõãûáõýá ·»ý»ñ³óíáõù ¿ ï»ëï³íáñù³ý áýã³óùáõù å³ï³ëë³ýç ñçù³ý íñ³: øß³ïí³í ñ³ù³ï³ñ·ç ·áñíáõý»áõãû³ý ³ñ¹ûáõý³í»ïáõãûáõýá ëïáõ·íáõù ¿ ñ³ù³å³ï³ëë³ý ù³ã»ù³ïçï³ï³ý ùá¹»éç ëï»õíù³ùµ: 1 3 0 microsoft word i. zaslavski_11.doc mathematical problems of computer science 34, 2010. 33 classical and non-classical logic, logical methods for electronic circuits testing i. d. zaslavsky institute for informatics and automation problems investigations in the domain of mathematical logic have been implemented in our institute since 1960. different works devoted to constructive mathematical analysis and constructive logic were published over 1960-1990. in particular, a new logical system—symmetric constructive logic, was created; systems of recursive realizability and logical-mathematical systems based on this logic were investigated in [1]. researches on many-valued logics have been implemented since 1970 and in fuzzy logic since 1990.theorems on compactness were proved for a large class of many-valued logics. criteria for completeness and solvability were established for logical-mathematical systems based on three-valued logics [2]. general forms of algebraic representations for recursively enumerable fuzzy sets and for operators on such sets were created, connections between such algebraic systems and the corresponding logicalmathematical systems were established [3]. formal logical languages for the representation of primitive recursive functions and primitive recursive string functions were created; connections between such languages were investigated [4]. last years new logical systems, namely, fuzzy constructive and fuzzy symmetric constructive logics were created, logical calculi, formalizing the logical deductions in these logics, were investigated [5]. logical-mathematical systems based on these logics, in particular, fuzzy formal arithmetics were defined. it may be supposed that investigations of such systems will give a possibility for constructing a mathematical theory of so called “fuzzy natural numbers”. the urgency of such theory was noted by n.n. luzin and p.k. rashevsky [6]. since 1980 logical methods of electronic circuits testing, in particular, combinational and sequential circuits as well as memory circuits testing have been developed. the methods of circuit testability improvement by the insertion of new points of observability and controllability were developed; possibilities of optimization of the structure of circuits obtained by such a way were investigated [7]. the methods for the improvement of the structure of sequential circuits by the correction of delays in such circuits and testing of obtained delays were created [8]. new testing algorithms were developed for random access memories; the possibilities of minimization of corresponding tests were investigated. the minimal volume of some tests obtained by this way was established. new methods of dynamic faults testing in memory circuits were created [9]. the results of mentioned investigations are used in practical engineering. 34 references 1. zaslavsky i.d. symmetric constructive logic (in russian). publ. house of acad. sci. armenia, 281p.,1978 2. zaslavsky i.d. formal axiomatic theories based on a three-valued logic., journal of mathematical sciences, vol.130, n 2, pp.4578-4597, 2005 3. manukian s.n. algebras of recursively enumerable sets and their applications to fuzzy logic. journal of mathematical sciences, vol.130, n 2, pp.4598-4606, 2005 4. khachatrian m.h. on the representation of arithmetical and string functions in formal languages. transactions of iapi of anas, vol. 27, pp. 37-53, 2006 5. zaslavsky i.d. fuzzy constructive logic (in russian). proceedings of scientific seminars of st.-petersburg dept. of math. inst. steklov, vol. 358, pp. 130-152, 2008 6. rashevsky p. k. on the dogma of natural numbers series. (in russian), uspekhi math nauk, vol. 28, n4(172), pp. 243-246, 1973 7. vardanian v.a., mirzoyan l.b. method for improvement of the error detection ability of concurrent checkers. proceedings of the international conference “computer science and information technologies”, csit-01, pp. 349-353, 2001 8. vardanian v.a. on completely robust path delay fault testable realization of logic functions., proceedings of the 14th ieee vlsi test symposium, princeton (usa), pp.302-307, 1996 9. harutunyan g., vardanian v.a., zoryan y. minimal march tests for dynamic faults in random access memories. journal of electronic testing, theory and applications, vol. 23, n1, pp. 65-74, 2007. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 46, 59--65, 2016. performances of methods for solving a linear system of equations in the architecture of gpu accelerator in case of small matrices hrachya v. astsatryan and edita e. gichunts institute for informatics and automation problems of nas ra e-mail: hrach@sci.am, editagich@ipia.sci.am abstract the algebraic operations with a large number of small matrices are a very important issue in science. the solutions for linear system of equations with lu factorization are specific of the mentioned operations that have numerous applications of algebraic operations with small matrices. in this work we consider the performances of methods for solving linear system of equations with batched lu factorization for small complex matrices on the graphic processor nvidia k40c.the versions with partial pivoting, without pivoting and random butterfly transformations of batched lu factorization for small matrices are presented and shown, which of these versions is the effective one, in which case we achieve high performance. keywords: gpu accelerator, magma , linear algebra operation, lu factorization, small matrix, performance, batched computation, random butterfly transformation. 1. introduction numerous scientific computations require a number of small independent solutions to the problems. since each individual problem size can fail to ensure the necessary parallelization for the given basic equipment, specifically in the case of accelerators, then these problems should be solved simultaneously so that to charge the device with the needed amount of work; hence the name of batched calculations came. the batched subprograms are designed to solve many problems independent of each other in a parallelization manner. such an approach is required in a number of problems of various spheres, for example, in astrophysics [1], metabolic networks [2], quantum chemistry [3], high-order fem schemes for hydrodynamics [4], direct-iterative preconditioned solvers [5], and some image [6] and signal processing [7]. as soon as a large number of heterogeneous systems with gpu accelerators appear, the linear algebra software is fully performed in small matrix operations. graphic processors (gpus) 59 performances of methods for solving a linear system of equations of gpu in case of small matrices 60 are high-performance processors that are capable of making hundreds of floating point operations in parallel. their large register files and the scratchpad memory of high performance are well suited for the model of streaming execution. in this paper, via the versions of batched lu factorization, we focus on solving of linear equation systems that are processed on the graphic processor. it should be noted that the experiments were carried out for general complex matrices. new functions are considered in the case of small matrices that are new in the magma 1.6.1 package [8] and are named as batched. in the previous article [9] the solving methods of linear equations via the versions of lu factorization are presented for large matrices as well as it is shown that the higher performances are achieved due to the solving method of random butterfly transformation. in this paper we will present the same problem for small matrices where the batched subprograms of the magma package are applied, and consider how high performance will be achieved for small matrices. 2. implementation of batched subprograms in the batched problem each matrix represents a separate problem which is solved independently. the batched subprograms in magma are called lapack in accordance with the subprograms of the library. magma 1.6.1 package contains four standard precision arithmetic of batched subprograms single real, double real, single complex and double complex. in this paper we present the implementation of batched solutions of the solving methods of linear system of equations for small matrices for single complex and double complex precision. as in the case of large matrices, in small matrices as well the batched solutions of linear system of equations are implemented with pivoting, without pivoting and with random butterfly transformations. the batched solution of linear system of equations, where the batched lu factorization is performed with partial pivoting, is implemented through the magma 1.6.1 library of the xgesv_batched.cpp function. the solution consists of the following steps:  a and b matrices are transferred from the cpu to the global memory of gpu.  batched lu factorization of the matrix a is performed through xgetrf_batched.cpp function of magma library using a partial pivoting with row interchanges.  a * x = b is solved through the function xgetrs_batched.cpp of magma library.  x derived solutions are transferred from the gpu to cpu. xgetrf_batched.cpp consists of xgetf2_batched, xlaswp_rowparallel_batched, xtrsm_outofplace_batchedandxgemm subprograms. a) xgetf2: dgetf2 computes an lu factorization of a general m-by-n matrix a using partial pivoting with row interchanges. the factorization has the form a = p * l * u, where p is a permutation matrix, l is a lower triangular with unit diagonal elements (lower trapezoidal if m > n), and u is an upper triangular (upper trapezoidal if m < n). in each step of lu factorization, dgetf2 subprogram is used while factorizing the m x nb panel. it consists of three subprograms idamax, dswap and dscal of level 1 blas library and dger subprogram of level 2. the only approach to the implementation of dgetf2 subprogram lies in the fact that all the panels are loaded in the common memory of gpu and then perform all calculations before the results are sent to the main memory. b) xlaswp: dlaswp performs a series of row interchanges on a general rectangular matrix. dlaswp performs a series of row interchanges on the matrix a. one row interchange is h.astsatryan and e.gichunts 61 initiated for each of rows k1 through k2 of a. to increase the digital stability, the transformation with pivoting is required. dlaswp subprogram is used in case of the transformation with pivoting, despite the fact that we get low performance. in any case, experiments show that the transformation with pivoting is quite expensive for the batched problem. c)xtrsm: dtrsm solves one of the matrix equations op( a )*x = alpha*b, or x*op( a ) = alpha*b, where alpha is a scalar, x and b are m by n matrices, a is a unit, or a non-unit, an upper or lower triangular matrix and op( a ) is one of op( a ) = a or op( a ) = a**t. the matrix x is overwritten on b. d) xgemm: dgemm performs one of the matrix-matrix operations c := alpha*op( a )*op( b ) + beta*c, where op( x ) is one of op( x ) = x or op( x ) = x**t, alpha and beta are scalars, and a, b and c are matrices, with op( a ) an m by k matrix, op( b ) a k by n matrix and c an m by n matrix. the aim of the batched lu factorization is to achieve batched dgemm high performance. a lot of effort has been concentrated for the optimization of dgemm subprogram. in particular, for our case, dgemm is not only used in the trailing matrix updates but also in the implementation of the triangular matrix solvers (xtrsm). the batched solution of linear system of equations where the batched lu factorization is performed without pivoting, is implemented through the magma 1.6.1 library of xgesv_nopiv_batched.cpp and xgerbt¬_batched.cpp functions. in case of xgesv_nopiv_batched.cpp function the solution sequence is as follows:  a and b matrices are transferred from the cpu to the global memory of gpu.  batched lu factorization of the matrix a is performed through xgetrf_nopiv_batched.cpp function of magma library without any pivoting.  a * x = b is solved through the function xgetrs_nopiv_batched.cpp of magma library.  x derived solutions are transferred from the gpu to cpu. to implement xgetrf_nopiv_batched.cpp function, the xgetrf_ panel_nopiv_batched subprogram of magma library, as well as xtrsm_ work_batched and xgemm_batched subprograms of blas library are called. to implement xgetrs_nopiv_batched.cpp function, only the xtrsm_outofplace_batched subprogram of blas library is called. the solution of linear system of equations, where the random butterfly transformation is applied on a and b matrices, and the batched lu factorization is performed without pivoting, is implemented via xgerbt_batched.cpp function of magma 1.6.1 library. this type of solution implementation is realized in the following sequence: • we generate the random matrices u and v in packed storage on the cpu. • the matrix a and the packed representation of u and v are sent from the host memory to the device memory. • randomization is performed on the gpu, updating a in the device memory. • perform partial random butterfly transformation on the gpu with magmablas_cprbt_batched () function. • we compute utbon the gpu, ary= utbis solved on the gpu, followed by the solution x = vy with magmablas_cprbt_mtv_batched () function. • the solution is sent to the host memory. performances of methods for solving a linear system of equations of gpu in case of small matrices 62 3. performance results the experiments were conducted on nvidia k40c gpu. the architecture of nvidia k40c consists of 2880 cuda processor cores. it is endowed with much higher bandwidth 288 gb/s of message transfer between cpu and gpu, having 12 gb of global memory, gddr5 memory interface, and cuda c programming environment. the operation system of k40c is ubuntu 14.04.2 lts. magma 1.6.1 package is installed. the code is compiled using the gnu gcc version 4.8, gfortran-4.8, g ++ 4.8 and the nvcc version 7.0 with the optimization flag -o3 and linked with the atlas library. figures 1 and 2 show the performance results of batched functions of linear solution equations for small matrices for complex single precision and complex double precision, respectively. fig. 1. complex single precision. fig. 2. complex double precision. in case of single precision the obtained results show that the performance of solutions determined by the lu factorization without pivoting is several times higher than the solutions with pivoting. thus, for example, the performance of cgesv_nopiv_batched function without 0 100 200 300 400 500 600 700 800 cgesv cgesv_nopiv cgerbt n g flo p/ s 0 100 200 300 400 500 600 700 zgesv zgesv_nopiv zgerbt n g flo p/ s h.astsatryan and e.gichunts 63 pivoting is twice higher than the performance of cgesv_batched function with pivoting, and the performance of cgerbt_batched function with random butterfly transformation without pivoting is 4.5 times higher than the performance of cgesv_batched function and 2 times higher than the performance of cgesv_nopiv_batched function. in case of double precision the obtained results show that the performance of zgesv_nopiv_batched function without pivoting is twice higher than the performance of zgesv_batched function with pivoting, and the performance of zgerbt_batched function with random butterfly transformation without pivoting is 6.5 times higher than the performance of zgesv_batched function and 3 times higher than the performance of zgesv_nopiv_batched function. also note that in case of single precision the performance is relatively higher than in the case of double precision for the corresponding functions with and without pivoting. 4. conclusion we presented the methods for solving of linear system of equations on gpu accelerator for small matrices with and without pivoting. for small matrices the batched functions of magma 1.6.1. package have been applied which are new and the batched function with random butterfly transformation was not applied in the tests. it is concluded from the experiment results that for small matrices a high performance in solutions of linear system of equations is achieved applying the method of random butterfly transformation. references [1] o.e.b. messer, j. a. harris, s. parete-koon and m. a. chertkow, “multicore and accelerator development for a leadership-class stellar astrophysics code”, in proceedings of "para2012: state-of-the-art in scientific and parallel computing", 2012. [2] j. c. liao khodayari, a. r. zomorrodi and c. d. maranas, “a kinetic model of escherichia coli core metabolism satisfying multiple sets of mutant flux data”, metabolic engineering, vol. 25, pp. 50–62, 2014. [3] a. a. auer, g. baumgartner, d. e. bernholdt, a. bibireata,v. choppella, d. cociorva, x. gao, r. harrison, s. krishnamoorthy, s. krishnan, c.-c. lam, q. luc, m. nooijene,r. itzerf, j. ramanujamg, p. sadayappan and a. sibiryakovc, “automatic code generation for many-body electronic structure methods: the tensor contraction engine’, molecular physics, vol. 104, no. 2, pp. 211–228, 2006. [4] t. dong, v. dobrev, t. kolev, r.rieben, s.tomov and j. dongarra, “a step towards energy efficient computing: redesigning a hydrodynamicapplication on cpu-gpu”, in ieee 28th international parallel distributed processingsymposium (ipdps), 2014. [5] eun-jin im, k. yelick and r. vuduc, “sparsity: optimization frameworkfor sparse matrix kernels”, int. j. high perform. comput. appl., vol. 18, no. 1, pp. 135–158, 2004. performances of methods for solving a linear system of equations of gpu in case of small matrices 64 [6] j. m. molero, e. m. garzón, i. garcía, e. s. quintana-ortí, and a. plaza, “poster: a batched cholesky solver for local rx anomaly detection on gpus”, pumps, 2013. [7] m. j. anderson, d. sheffield and k. keutzer, “a predictive model for solving smalllinear algebra problems in gpu registers’, in ieee 26th international parallel distributedprocessing symposium (ipdps), 2012. [8] matrix algebra on gpu and multicore architectures (magma), magma release 1.6.1, 2015. online. [available]: http://icl.cs.utk.edu/magma/ [9] h. v. astsatryan and e. e. gichunts, “performances of methods for solving a linear system of equations in the architecture of gpu accelerator”, transactions of iiap nas ra, mathematical problems of computer science, vol. 45, pp. 44—52, 2016. submitted 02.08.2016, accepted 25.10.2016. փոքր մատրիցների դեպքում գծային հավասարումների համակարգի լուծման մեթոդների արտադրողականությունները gpu արագագործչի ճարտարապետությունում ð. ²ëó³ïñû³ý ¨ ¾. ¶çãáõýó ամփոփում գիտության մեջ շատ կարևոր խնդիր է հանդիսանում բազմաքանակ փոքր մատրիցների հետ կապված հանրահաշվական գործողությունները: այդ գործողություններից առանձնահատուկ են lu վերլուծությամբ գծային հավասարումների համակարգի լուծումները, որոնք փոքր մատրիցների հետ հանրահաշվական գործողություններում բազմաթիվ կիրառություններ ունեն :այս աշխատանքում դիտարկվում են կոմպլեքս փոքր մատրիցների համար batched lu վերլուծությամբ գծային հավասարումների համակարգի լուծման մեթոդների արտադրողականությունները nvidia k40c գրաֆիկական պրոցեսորի վրա: փոքր մատրիցների համար ներկայացվում են batched lu վերլուծության պտույտով, առանց պտույտի և թիթեռնիկի պատահական ձևափոխությամբ տարբերակները և ցույց է տրվում, թե այդ տարբերակներից որն է արդյունավետը, որի դեպքում հասնում ենք բարձր արտադրողականության: http://mpcs.sci.am/filesimages/volumes/volume_45/06.pdf http://mpcs.sci.am/filesimages/volumes/volume_45/06.pdf h.astsatryan and e.gichunts 65 производительности методов решения систем линейных уравнений в архитектуре gpu ускорителя в случае небольших матриц г. асцатрян и э. гичунц аннотация алгебраические операции с большим числом малых матриц являются очень важными вопросами в науке. из упомянутых операций специфическими являются решения для линейной системы уравнений с lu факторизацией, которые имеют многочисленные применения алгебраических операций с небольшими матрицами. в этой работе мы рассмотрим производительности методов для решения системы линейных уравнений с batched lu факторизацией для малых комплексных матриц на графическом процессоре nvidia k40c. для небольших матриц представлены версии batched lu факторизации с поворотом, без поворота и со случайным преобразованием бабочки, а также показано, какой из этих вариантов является эффективным, и в каком случае мы достигаем высокой производительности. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 48, 74--81, 2017. graphical representations of e-achievability region for identification and secret-key generation system lilit a. ter-vardanyan institute for informatics and automation problems of nas ra e-mail: lilit@sci.am abstract in this paper we represent the computations of theoretical results, obtained in [1], on the example of binary symmetric channel. the computations and graphical representations are realized using the “advanced inftheo” module, created in 2015 [2, 3] for r environment. keywords: biometric identification system, secret key, rate-reliability, r environment. 1. introduction nowadays the research of different biometric systems is one of the most important parts of the development of safety measures. any biometric system may provide a certain degree of confidentiality in terms of information theory. depending on applications, different models of biometric identification and secret-key generation were considered [4]. for each model one of the most important tasks is to determine the identification rate or the achievable rate of a secret. the next most important task is the study of the rate-reliability or e-achievability region, which is not an easy task [5, 6]. some outer and inner bounds of that region for various models of biometric settings were obtained in [1, 7-10]. alongside with difficulties in theoretical research, difficulties in applying the theoretical results also arise. since the obtained mathematical formulas are complex and difficult to calculate, a module for r was created to perform such calculations. r is a programming language and software environment for statistical analysis, graphics representation and reporting. r was created by ross ihaka and robert gentleman at the university of auckland, new zealand, and is currently developed by the r development core team. the core of r is an interpreted computer language which allows branching and looping as well as modular programming using functions. r allows integration with procedures written in the c, c++, .net, python or fortran languages for efficiency. 74 l. ter-vardanyan 75 the use of r environment in modern society is growing very fast due to a number of advantages it has, compared to other statistical tools. the module “advanced inftheo” for r was developed for estimation and computation of complex formulas of information theory. in this paper we use this module for computation of outer and inner bounds of e-achievability region for identification and secret-key generation system. 2. the model of biometric identification and secret-key generation system the biometric identification and secret-key generation system was considered in [4, 1] (fig.1). in [1] the inner and outer bounds of e-achievable rates region of this model were obtained. to formulate that result we remind the notations and definitions. the model of biometric identification and secret-key generation system consists of enrollment and identification procedures. in an enrollment phase 𝒱𝒱 individuals are observed. for each individual 𝑣𝑣 = {1, 2, . . . , |𝒱𝒱|} in the system the biometric source produces a biometric enrollment sequence x(v) = (𝑥𝑥1, 𝑥𝑥2, … , 𝑥𝑥𝑁𝑁) ∈ 𝒳𝒳𝑁𝑁. all these sequences are supposed to be generated at random with a given probability distribution (pd) 𝑄𝑄𝑁𝑁(𝐱𝐱) = �𝑄𝑄(𝑥𝑥𝑛𝑛), 𝑁𝑁 𝑛𝑛=1 x ∈ 𝒳𝒳𝑁𝑁 . during the enrollment procedure the biometric sequence x(v) of individual 𝑣𝑣 = {1, 2, . . . , 𝒱𝒱} is encoded into helper data 𝑚𝑚(𝑣𝑣) ∈ {1, 2, . . . , |ℳ|} and a secret key 𝑠𝑠(𝑣𝑣) ∈ {1, 2, . . . , |𝒮𝒮|}, hence the encoder mapping is e(x(v)) = (𝑚𝑚(𝑣𝑣); 𝑠𝑠(𝑣𝑣)) for 𝑣𝑣 = {1, 2, . . . , |𝒱𝒱|}. fig. 1. the model of biometric identification and secret-key generation system. graphical representations of e-achievability region for identification and secret-key system 76 the helper data 𝑚𝑚(𝑣𝑣) is then stored in a (public) database at position 𝑣𝑣 and the generated secret key 𝑠𝑠(𝑣𝑣) is handed over to the individual. the helper data makes the reliable identification possible. during the identification procedure a biometric identification sequence 𝒚𝒚 = (y1, y2, . . . , yn) ∈ 𝒴𝒴𝑁𝑁 is observed. if individual 𝑣𝑣 was observed, the sequence 𝒚𝒚(𝑣𝑣) occurs with probability 𝑊𝑊𝑁𝑁(𝐲𝐲|𝐱𝐱) = �𝑊𝑊(𝑦𝑦𝑛𝑛| 𝑥𝑥𝑛𝑛) 𝑁𝑁 𝑛𝑛=1 , 𝐲𝐲 ∈ 𝒴𝒴𝒩𝒩 , 𝐱𝐱 ∈ 𝒳𝒳𝒩𝒩, since the biometric channel 𝑊𝑊𝑁𝑁(𝐲𝐲|𝐱𝐱) is memoryless. during identification, upon observing the biometric identification sequence 𝐲𝐲, the decoder forms an estimate 𝑣𝑣� of the identity of the observed individual as well as an estimate of his secret key s(𝑣𝑣�), �𝑣𝑣�, s(𝑣𝑣�)� = 𝑑𝑑(𝐲𝐲, 𝑚𝑚(1), 𝑚𝑚(2), … , 𝑚𝑚(𝒱𝒱)), where 𝑑𝑑 is the decoder mapping. definition (e-achievability): for 𝐸𝐸 > 0 an identification and secret-key rate pair 𝑅𝑅𝐼𝐼, 𝑅𝑅𝑆𝑆 with 𝑅𝑅𝐼𝐼 ≥ 0 and 𝑅𝑅𝐼𝐼 ≥ 0 is 𝐸𝐸-achievable in a biometric identification setting, if for all 𝛿𝛿 > 0 for all 𝑁𝑁 large enough, there exists an encoder and a decoder such that pr {(𝑣𝑣�, �̂�𝑠) ≠ (𝑣𝑣, 𝑠𝑠)} ≤ exp{−𝑁𝑁(𝐸𝐸 − 𝛿𝛿)}, 1 𝑁𝑁 log |𝒱𝒱| ≥𝑅𝑅𝐼𝐼 − 𝛿𝛿, 1 𝑁𝑁 log |𝒮𝒮| ≥𝑅𝑅𝑆𝑆 − 𝛿𝛿, 1 𝑁𝑁 𝐼𝐼(𝑆𝑆 ∧ 𝑀𝑀) ≤ 𝛿𝛿. we use the following pd in the formulation of result: 𝑃𝑃 = {𝑃𝑃(𝑥𝑥), 𝑥𝑥 ∈ 𝒳𝒳}; 𝑉𝑉 = {𝑉𝑉 (𝑦𝑦 | 𝑥𝑥), 𝑦𝑦 ∈ 𝒴𝒴, 𝑥𝑥 ∈ 𝒳𝒳}. for information-theoretic quantities, such as entropy 𝐻𝐻𝑃𝑃(𝑋𝑋), mutual information 𝐼𝐼𝑃𝑃,𝑉𝑉(𝑋𝑋 ∧ 𝑌𝑌), divergence 𝐷𝐷(𝑉𝑉||𝑊𝑊|𝑃𝑃) we refer to [5, 6, 11]. the main result of [1] is the following theorem. theorem 1: for the biometric identification and secret-key generation system with the given 𝑄𝑄, 𝑊𝑊 and for all 𝐸𝐸 > 0 ℛ𝑟𝑟(𝐸𝐸, 𝑄𝑄, 𝑊𝑊 ) ⊆ ℛ(𝐸𝐸, 𝑄𝑄, 𝑊𝑊) ⊆ ℛ𝑠𝑠𝑠𝑠(𝐸𝐸, 𝑄𝑄, 𝑊𝑊 ), where ℛ𝑟𝑟(𝐸𝐸, 𝑄𝑄, 𝑊𝑊) = {(𝑅𝑅𝐼𝐼, 𝑅𝑅𝑆𝑆): 𝑅𝑅𝐼𝐼 + 𝑅𝑅𝑆𝑆 ≤ 𝑚𝑚𝑚𝑚𝑚𝑚𝑃𝑃,𝑉𝑉:𝐷𝐷(𝑃𝑃∘𝑉𝑉||𝑄𝑄∘𝑊𝑊)≤𝐸𝐸 |𝐼𝐼𝑃𝑃,𝑉𝑉(𝑋𝑋 ∧ 𝑌𝑌) + 𝐷𝐷(𝑃𝑃 ∘ 𝑉𝑉||𝑄𝑄 ∘ 𝑊𝑊) − 𝐸𝐸|+} ℛ𝑠𝑠𝑠𝑠(𝐸𝐸, 𝑄𝑄, 𝑊𝑊) = {(𝑅𝑅𝐼𝐼, 𝑅𝑅𝑆𝑆): 𝑅𝑅𝐼𝐼 + 𝑅𝑅𝑆𝑆 ≤ 𝑚𝑚𝑚𝑚𝑚𝑚𝑃𝑃,𝑉𝑉:𝐷𝐷(𝑃𝑃∘𝑉𝑉||𝑄𝑄∘𝑊𝑊)≤𝐸𝐸 𝐼𝐼𝑃𝑃,𝑉𝑉(𝑋𝑋 ∧ 𝑌𝑌)}. the notation |𝑎𝑎|+ is used for 𝑚𝑚𝑎𝑎𝑥𝑥 (𝑎𝑎, 0). l. ter-vardanyan 77 3. graphical representation of computations let's consider a binary symmetric channel with the parameter d, it means that 𝑊𝑊(𝑦𝑦|𝑥𝑥) conditional distribution matrix is defined as 𝑊𝑊(1|1) = 𝑊𝑊(0|0) = 1 − 𝑑𝑑, 𝑊𝑊(1|0) = 𝑊𝑊(0|1) = 𝑑𝑑. we have performed computations of above formulas in the theorem using “advanced inftheo” module. figure 2 shows the dependence of the 𝑅𝑅𝐼𝐼+𝑅𝑅𝑆𝑆 from e for the parameter 𝑑𝑑 = 0.2. the calculations are realized with the step 0.003. fig.2. the bounds of e-achievable rate region, when d=0.2 with step 0.003 graphical representations of e-achievability region for identification and secret-key system 78 figure 3 shows the dependence of identification rate 𝑅𝑅𝐼𝐼 on e and figure 4 shows the dependence of secret-key rate 𝑅𝑅𝑆𝑆 on e for the same parameter d=0.2. fig. 3. the bounds of e-achievable identification rate fig. 4. the bounds of e-achievable secret key rate l. ter-vardanyan 79 in figure 5 the dependence of 𝑅𝑅𝐼𝐼 on𝑅𝑅𝑆𝑆 is represented for various e. in other words figure 5 shows the trade-off between identification and secret-key rates. it means that the more individuals would like to be able to reliably identify, the smaller secret keys can be assigned to individuals for identification purposes and the identification system becomes less secure. it means that it becomes easier to get access to the systems that deploy biometric identification, since smaller secrets are easier to guess and also require less biometric information for their reconstruction. 4. conclusion the outer and inner bounds for e-achievability region for the model of biometric identification system and secret-key generation is investigated by constructing graphics for binary symmetric channel. fig. 5. identification versus secret-key rate projection, d=0.2 graphical representations of e-achievability region for identification and secret-key system 80 references [1] m. haroutunian, l. ter-vardanyan, “investigation of e-achievability region for identification and secret-key generation system”, csit-2013 the 9th international conference on computer science and information technologies (dedicated to the 70th anniversary of the national academy of sciences of armenia), yerevan, armenia, pp. 107-110 , 2013. 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[6] e. haroutunian, "on bounds for e capacity of dmc," ieee transactions on information theory, vol. 53, no. 11, pp. 4210–4220, 2007. [7] m. haroutunian, a. muradyan and l. tervardanyan, “on the e-capacity of a biometric identification system”, csit-2011 the 8th international conference on computer science and information technologies, yerevan, armenia, pp. 132-134, 2011. [8] m. e. haroutunian, a. muradyan and l. ter-vardanyan, "upper and lower bounds of biometric identification e-capacity", transactions of iiap of nas ra, mathematical problems of computer science, vol. 36, pp .1-10, 2012. [9] m. haroutunian and n. pahlevanyan "information theoretical analysis of biometric secret key sharing model," transactions of iiap of nas ra, mathematical problems of computer science, vol. 42, pp. 17-27, 2014. 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[11] t. m. cover and j. a. thomas, elements of information theory, 2nd edition, new york, ny, usa: wiley-interscience, 2006. submitted 05.08.2017, accepted 04. 12.2017. l. ter-vardanyan 81 նույնականացման և գաղտնի բանալի գեներացման համակարգի համար e-հասանելի տիրույթի գրաֆիկական ներկայացումները լ. տեր-վարդանյան ամփոփոում հոդվածում ներկայացված են [1]-ում ստացված տեսական արդյունքների հաշվարկները երկուական սիմետրիկ կապուղու օրինակով: հաշվարկները և գրաֆիկները կատարվել են 2015թ.-ին r միջավայրի համար ստեղծված «advanced inftheo» մոդուլի միջոցով [2, 3]: графические представления области e-достижимости для системы идентификации и генерации секретного ключа л. тер-варданян аннотация в статье представлены расчеты теоретических результатов, полученных в [1], на примере двоичного симметричного канала. вычисления и графические представления реализованы с использованием модуля «advanced inftheo», созданного в 2015 году для среды r [2, 3]. d:\sbornik_45_tex\...\7_17.dvi mathematical problems of computer science 46, 7{17, 2016. on pr e-h amiltonian cycles in b alanced b ipar tite digr aphs s a m ve l k h . d a r b in ya n institute for informatics and automation problems of nas ra e-mail: samdarbin@ipia.sci.am abstract let d be a strongly connected balanced bipartite directed graph of order 2a ¸ 10. let x; y be distinct vertices in d. fx; yg dominates a vertex z if x ! z and y ! z; in this case, we call the pair fx; yg dominating. in this paper we prove: if maxfd(x); d(y)g ¸ 2a¡2 for every dominating pair of vertices fx; yg, then either the underlying graph of d is 2-connected or d contains a cycle of length 2a ¡ 2 or d is isomorphic to one digraph of order ten. keywords: digraphs, cycles, hamiltonian cycles, bipartite balanced digraph, pancyclic, even pancyclic. 1 . in t r o d u c t io n w e c o n s id e r d ir e c t e d g r a p h s ( d ig r a p h s ) in t h e s e n s e o f [1 ]. a c yc le o f a d ig r a p h d is c a lle d h a m ilt o n ia n if it c o n t a in s a ll t h e ve r t ic e s o f d. fo r c o n ve n ie n c e o f t h e r e a d e r t e r m in o lo g y a n d n o t a t io n s will b e g ive n in d e t a ils in s e c t io n 2 . a d ig r a p h d o f o r d e r n is h a m ilt o n ia n if it c o n t a in s a h a m ilt o n ia n c yc le a n d p a n c yc lic if it c o n t a in s c yc le s o f e ve r y le n g t h k, 3 · k · n. fo r g e n e r a l d ig r a p h s t h e r e a r e s e ve r a l s u ± c ie n t c o n d it io n s fo r e xis t e n c e o f h a m ilt o n ia n c yc le s in d ig r a p h s . in t h is p a p e r , we will b e c o n c e r n e d wit h t h e d e g r e e c o n d it io n s . th e we ll-kn o wn a n d c la s s ic a l a r e gh o u ila -h o u r i's , n a s h -w illia m s ', w o o d a ll's , me yn ie l's a n d th o m a s s e n 's t h e o r e m s ( s e e , e .g ., [2 ][6 ]) . th e r e a r e a n a lo g o u s r e s u lt s o f t h e a b o ve -m e n t io n e d t h e o r e m s fo r t h e p a n c yc lic it y o f d ig r a p h s ( s e e , e .g ., [7 -1 2 ]) . e a c h o f t h e o r e m s ( [2 ]-[6 ]) im p o s e s a d e g r e e c o n d it io n o n a ll p a ir s o f n o n a d ja c e n t ve r t ic e s ( o r o n a ll ve r t ic e s ) . in [1 3 ] a n d [1 4 ], s o m e s u ± c ie n t c o n d it io n s we r e d e s c r ib e d fo r a d ig r a p h t o b e h a m ilt o n ia n , in wh ic h a d e g r e e c o n d it io n is r e qu ir e d o n ly fo r s o m e p a ir s o f n o n a d ja c e n t ve r t ic e s . l e t u s r e c a ll o n ly t h e fo llo win g t h e o r e m o f t h e m . t heor em 1.1: ( b a n g -je n s e n , gu t in , h .l i [1 3 ]) . l et d be a strongly connected digraph of order n ¸ 2 . suppose that minfd( x ) ; d ( y ) g ¸ n ¡ 1 and d( x ) + d( y ) ¸ 2 n ¡ 1 for any pair of nonadjacent vertices x; y with a common in-neighbor. then d is hamiltonian. a c yc le o f a n o n -b ip a r t it e d ig r a p h d is c a lle d p r e -h a m ilt o n ia n if it c o n t a in s a ll t h e ve r t ic e s o f d e xc e p t o n e . th e c o n c e p t o f p r e -h a m ilt o n ia n c yc le fo r t h e b a la n c e d b ip a r t it e d ig r a p h s is t h e fo llo win g : 7 8 on pre-hamiltonian cycles in balanced bipartite digraphs a c yc le o f a b a la n c e d b ip a r t it e d ig r a p h d is c a lle d p r e -h a m ilt o n ia n if it c o n t a in s a ll t h e ve r t ic e s o f d e xc e p t t wo . a d ig r a p h d is c a lle d b ip a r t it e if t h e r e e xis t s a p a r t it io n x, y o f it s ve r t e x s e t in t o t wo p a r t it e s e t s s u c h t h a t e ve r y a r c o f d h a s it s e n d -ve r t ic e s in d i®e r e n t p a r t it e s e t s . it is c a lle d b a la n c e d if jxj = jy j. th e r e a r e r e s u lt s a n a lo g o u s t o t h e t h e o r e m s o f gh o u ila -h o u r i, n a s h -w illia m s , w o o d a ll, me yn ie l a n d th o m a s s e n fo r b a la n c e d b ip a r t it e d ig r a p h s ( s e e e .g ., [1 5 ]) a n d t h e p a p e r s c it e d t h e r e . l e t x; y b e a p a ir o f d is t in c t ve r t ic e s in a d ig r a p h d. w e c a ll t h e p a ir fx; yg d o m in a t in g , if t h e r e is a ve r t e x z in d s u c h t h a t x ! z a n d y ! z. a n a n a lo g u e o f th e o r e m 1 .1 fo r b ip a r t it e d ig r a p h s wa s g ive n b y r . w a n g [1 6 ] a n d r e c e n t ly s t r e n g t h e n e d b y t h e a u t h o r [1 7 ]. t heor em 1.2: ( r . w a n g [1 6 ]) . l et d be a strongly connected balanced bipartite digraph of order 2 a, where a ¸ 1 . suppose that, for every dominating pair of vertices fx; yg, either d( x ) ¸ 2 a ¡ 1 and d( y ) ¸ a + 1 or d ( y ) ¸ 2 a ¡ 1 and d( x ) ¸ a + 1 . then d is hamiltonian. l e t d b e a b a la n c e d b ip a r t it e d ig r a p h o f o r d e r 2 a ¸ 4 . fo r in t e g e r k ¸ 0 , we s a y t h a t d s a t is ¯ e s c o n d it io n bk wh e n maxfd( x ) ; d( y ) g ¸ 2 a ¡ 2 + k fo r e ve r y p a ir o f d o m in a t in g ve r t ic e s x a n d y. t heor em 1.3: ( d a r b in ya n [1 7 ]) . l et d be a strongly connected balanced bipartite digraph of order 2 a, where a ¸ 4 . suppose that d satis¯es condition b1, i.e., for every dominating pair of vertices fx; yg, either d ( x ) ¸ 2 a ¡ 1 or d( y ) ¸ 2 a ¡ 1 . then either d is hamiltonian or isomorphic to the digraph d ( 8 ) (for the de¯nition of d ( 8 ) , see e xample 1). a b a la n c e d b ip a r t it e d ig r a p h o f o r d e r 2 m is e ve n p a n c yc lic if it c o n t a in s a c yc le o f le n g t h 2 k fo r a n y 2 · k · m. a n e ve n p a n c yc lic ve r s io n o f th e o r e m 1 .3 wa s p r o ve d in [1 8 ]. t heor em 1.4: ( d a r b in ya n [1 8 ]) . l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 8 other than the directed cycle of length 2 a. if d satis¯es condition b1, i.e., maxfd( x ) ; d( y ) g ¸ 2 a¡ 1 for every dominating pair of vertices fx; yg, then either d contains cycles of all even lengths less than or equal to 2 a or d is isomorphic to digraph d ( 8 ) . t heor em 1.5: ( d a r b in ya n [1 8 ]) . l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 8 , which contains a pre-hmiltonian cycle (i.e., a cycle of length 2 a ¡ 2 ). if d satis¯es condition b0, i.e., maxfd( x ) ; d( y ) g ¸ 2 a ¡ 2 for every dominating pair of vertices fx; yg, then for any k, 1 · k · a ¡ 1 , d contains a cycle of length 2 k for every k, 1 · k · a ¡ 1 . in vie w o f th e o r e m 1 .5 it s e e m s qu it e n a t u r a l t o a s k wh e t h e r a b a la n c e d b ip a r t it e d ig r a p h o f o r d e r 2 a in wh ic h maxfd ( x) ; d ( y ) g ¸ 2 a ¡ 2 fo r e ve r y d o m in a t in g p a ir o f ve r t ic e s fx; yg c o n t a in s a p r e -h a m ilt o n ia n c yc le ( i.e ., a c yc le o f le n g t h 2 a ¡ 2 ) . th e u n d e r lyin g g r a p h o f a d ig r a p h d is t h e u n iqu e g r a p h s u c h t h a t it c o n t a in s a n e d g e xy if x ! y o r y ! x ( o r b o t h ) . in t h is p a p e r we p r o ve t h e fo llo win g t h e o r e m . t heor em 1.6: l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 1 0 with partite sets x and y . assume that d satis¯es condition b0. then either the underlying graph of d is 2-connected or d contains a cycle of length 2 a ¡ 2 unless d is isomorphic to the digraph d ( 1 0 ) (for the de¯nition of d ( 1 0 ) , see e xample 2). s. darbinyan 9 2 . te r m in o lo g y a n d n o t a t io n s te r m in o lo g y a n d n o t a t io n s n o t d e s c r ib e d b e lo w fo llo w [1 ]. in t h is p a p e r we c o n s id e r ¯ n it e d ig r a p h s wit h o u t lo o p s a n d m u lt ip le a r c s . fo r a d ig r a p h d, we d e n o t e b y v ( d ) t h e ve r t e x s e t o f d a n d b y a( d ) t h e s e t o f a r c s in d. th e o r d e r o f d is t h e n u m b e r o f it s ve r t ic e s . th e a r c o f a d ig r a p h d d ir e c t e d fr o m x t o y is d e n o t e d b y xy o r x ! y. th e n o t a t io n x $ y m e n a s t h a t x ! y a n d y ! x ( x $ y is c a lle d 2 -c yc le ) . w e d e n o t e b y a ( x; y ) t h e n u m b e r o f a r c s wit h e n d -ve r t ic e s x a n d y. fo r d is jo in t s u b s e t s a a n d b o f v ( d ) we d e ¯ n e a( a ! b ) a s t h e s e t fxy 2 a ( d ) =x 2 a; y 2 bg a n d a ( a; b ) = a( a ! b ) [ a ( b ! a) . if x 2 v ( d ) a n d a = fxg we s o m e t im e s wr it e x in s t e a d o f fxg. if a a n d b a r e t wo d is jo in t s u b s e t s o f v ( d ) s u c h t h a t e ve r y ve r t e x o f a d o m in a t e s e ve r y ve r t e x o f b, t h e n we s a y t h a t a d o m in a t e s b, d e n o t e d b y a ! b. th e n o t a t io n a $ b m e a n s t h a t a ! b a n d b ! a. th e o u t -n e ig h b o u r h o o d o f a ve r t e x x is t h e s e t n + ( x ) = fy 2 v ( d ) =xy 2 a( d ) g a n d n¡ ( x ) = fy 2 v ( d ) =yx 2 a( d ) g is t h e in -n e ig h b o u r h o o d o f x. s im ila r ly, if a µ v ( d ) , t h e n n + ( x; a ) = fy 2 a=xy 2 a( d ) g a n d n¡ ( x; a ) = fy 2 a=yx 2 a ( d ) g. th e o u t -d e g r e e o f x is d+ ( x ) = jn + ( x ) j a n d d¡ ( x ) = jn¡ ( x ) j is t h e in -d e g r e e o f x. s im ila r ly, d+ ( x; a) = jn + ( x; a) j a n d d¡ ( x; a) = jn¡ ( x; a ) j. th e d e g r e e o f t h e ve r t e x x in d is d e ¯ n e d a s d ( x) = d+ ( x ) + d¡ ( x ) ( s im ila r ly, d( x; a) = d+ ( x; a) + d¡ ( x; a ) ) . th e s u b d ig r a p h o f d in d u c e d b y a s u b s e t a o f v ( d ) is d e n o t e d b y dhai o r hai b r e vit y. th e p a t h ( r e s p e c t ive ly, t h e c yc le ) c o n s is t in g o f t h e d is t in c t ve r t ic e s x1; x2; : : : ; xm ( m ¸ 2 ) a n d t h e a r c s xixi+1, i 2 [1 ; m ¡ 1 ] ( r e s p e c t ive ly, xixi+1, i 2 [1 ; m ¡ 1 ], a n d xmx1 ) , is d e n o t e d b y x1x2 ¢ ¢ ¢ xm ( r e s p e c t ive ly, x1x2 ¢ ¢ ¢ xmx1 ) . w e s a y t h a t x1x2 ¢ ¢ ¢ xm is a p a t h fr o m x1 t o xm o r is a n ( x1; xm ) -p a t h . a c yc le t h a t c o n t a in s a ll t h e ve r t ic e s o f d is a h a m ilt o n ia n c yc le . a d ig r a p h d is s t r o n g ly c o n n e c t e d ( o r , ju s t , s t r o n g ) if t h e r e e xis t s a p a t h fr o m x t o y a n d a p a t h fr o m y t o x fo r e ve r y p a ir o f d is t in c t ve r t ic e s x; y. two d is t in c t ve r t ic e s x a n d y a r e a d ja c e n t if xy 2 a( d ) o r yx 2 a ( d ) ( o r b o t h ) . fo r in t e g e r s a a n d b, a · b, le t [a; b] d e n o t e t h e s e t o f a ll in t e g e r s wh ic h a r e n o t le s s t h a n a a n d a r e n o t g r e a t e r t h a n b. a d ig r a p h d is c a lle d a b ip a r t it e d ig r a p h if t h e r e e xis t s a p a r t it io n x, y o f v ( d ) in t o t wo p a r t it e s e t s s u c h t h a t e ve r y a r c o f d h a s it s e n d -ve r t ic e s in d i®e r e n t p a r t it e s e t s . it is c a lle d b a la n c e d if jxj = jy j. 3 . e xa m p le s e xample 1. l e t d ( 1 0 ) b e a b ip a r t it e d ig r a p h wit h p a r t it e s e t s x = fx0; x1; x2; x3; x4g a n d y = fy0; y1; y2; y3; y4g s a t is fyin g t h e fo llo win g c o n d it io n s : t h e in d u c e d s u b d ig r a p h hfx1; x2; x3; y0; y1gi is a c o m p le t e b ip a r t it e d ig r a p h wit h p a r t it e s e t s fx1; x2; x3g a n d fy0; y1g; fx1; x2; x3g ! fy2; y3; y4g; x4 $ y1; x0 $ y0 a n d xi $ yi+1 fo r a ll i 2 [1 ; 3 ]. d ( 1 0 ) c o n t a in s n o o t h e r a r c s . it is n o t d i± c u lt t o c h e c k t h a t t h e d ig r a p h d ( 1 0 ) is s t r o n g ly c o n n e c t e d a n d s a t is ¯ e s c o n d it io n b0, b u t t h e u n d e r lyin g g r a p h o f d ( 1 0 ) is n o t 2 -c o n n e c t e d a n d d ( 1 0 ) h a s n o c yc le o f le n g t h 8 . ( it fo llo ws fr o m t h e fa c t s t h a t d ( x0 ) = d( x4 ) = 2 a n d x0 ( x4 ) is o n 2 -c yc le ) . e xample 2. l e t d ( 8 ) b e a b ip a r t it e d ig r a p h wit h p a r t it e s e t s x = fx0; x1; x2; x3g a n d y = fy0; y1; y2; y3g s a t is fyin g t h e fo llo win g c o n d it io n s : t h e in d u c e d s u b d ig r a p h hfx1; x2; y0; y1; y3gi is a c o m p le t e b ip a r t it e d ig r a p h wit h p a r t it e s e t s fx1; x2g a n d fy0; y1; y3g; fx1; x2; x3g ! fy2; y3; y4g; x3 $ y3; x0 $ y0 a n d x0 $ y1 a n d d ( 8 ) c o n t a in s n o o t h e r a r c s . it is n o t d i± c u lt t o c h e c k t h a t t h e d ig r a p h d ( 8 ) is s t r o n g ly c o n n e c t e d a n d s a t is ¯ e s 1 0 on pre-hamiltonian cycles in balanced bipartite digraphs c o n d it io n b0, b u t t h e u n d e r lyin g g r a p h o f d ( 8 ) is n o t 2 -c o n n e c t e d a n d d ( 8 ) h a s n o c yc le o f le n g t h 6 . 4 . p r o o f o f t h e ma in r e s u lt p r oof of t heor em 1.6: s u p p o s e , o n t h e c o n t r a r y, t h a t t h e u n d e r lyin g g r a p h o f d is n o t 2 -c o n n e c t e d a n d d c o n t a in s n o c yc le o f le n g t h 2 a ¡ 2 . th e n v ( d ) = a [ b [ fug, wh e r e a a n d b a r e n o n e m p t y d is jo in t s u b s e t s o f ve r t ic e s o f d, t h e ve r t e x u is n o t in a [ b a n d t h e r e a r e n o a r c s b e t we e n a a n d b. s in c e d is s t r o n g , t h e r e a r e ve r t ic e s x 2 a a n d x0 2 b s u c h t h a t fx; x0g ! u, i.e ., fx; x0g is a d o m in a t in g p a ir . n o t e t h a t x a n d x0 b e lo n g t o t h e s a m e p a r t it e s e t , s a y x. th e n u 2 y . b y c o n d it io n b0 we h a ve maxfd( x ) ; d( x0 ) g ¸ 2 a ¡ 2 . w it h o u t lo s s o f g e n e r a lit y, we a s s u m e t h a t d ( x ) ¸ 2 a ¡ 2 . fr o m t h is a n d t h e fa c t t h a t t h e r e a r e n o a r c s b e t we e n a a n d b it fo llo ws t h a t a ¡ 2 · jy \ aj · a ¡ 1 . p u t y1 := y \ a. w e will c o n s id e r t h e c a s e s jy1j = a ¡ 2 a n d jy1j = a ¡ 1 s e p a r a t e ly. case 1: jy1j = a ¡ 2 . th e n jy \ bj = 1 . l e t y1 := fy1; y2; : : : ; ya¡2g a n d y \ b := fy0g. it is n o t d i± c u lt t o c h e c k t h a t t h e ve r t e x x a n d e ve r y ve r t e x o f y1 [ fug fo r m a 2 -c yc le , i.e ., x $ y1 [ fug. th e r e fo r e , e ve r y p a ir o f d is t in c t ve r t ic e s o f y1 [ fug is a d o m in a t in g p a ir . th is m e a n s t h a t y1 [ fug h a s a t le a s t a ¡ 2 ve r t ic e s o f d e g r e e a t le a s t 2 a ¡ 2 ( m a yb e e xc e p t , s a y ya¡2, o r u ) . th e n d ( y1 ) ¸ 2 a¡ 2 , s in c e a ¸ 5 . fr o m t h is it fo llo ws t h a t jx \aj = a¡ 1 a n d x \b = fx0g s in c e t h e r e a r e n o a r c s b e t we e n y1 a n d b. p u t x1 := fx1; x2; : : : ; xa¡1g, wh e r e x1 = x. th e r e fo r e , b = fx0; y0g. s in c e d is s t r o n g a n d s in c e y0 is n o t a d ja c e n t t o a n y ve r t e x o f x1, it fo llo ws t h a t y0 $ x0, u ! x0, d( x0 ) = 4 a n d d( y0 ) = 2 . b y c o n d it io n b0, we h a ve d ( u ) ¸ 2 a ¡ 2 s in c e fu; y0g ! x0. a s s u m e ¯ r s t t h a t d ( yi ) ¸ 2 a ¡ 2 fo r a ll i 2 [1 ; a ¡ 2 ]. th e n y1 $ x1, s in c e t h e r e a r e n o a r c s b e t we e n y1 a n d fx0g, i.e ., t h e in d u c e d s u b d ig r a p h dhy1 \ x1i is a c o m p le t e b ip a r t it e d ig r a p h wit h p a r t it e s e t s x1 a n d y1. s in c e d( u ) ¸ 2 a ¡ 2 , it fo llo ws t h a t t h e ve r t e x u a n d a t le a s t a ¡ 2 ve r t ic e s o f x1 fo r m a 2 c yc le . n o w we c a n c h o o s e a ve r t e x o f x1 o t h e r t h a n x, s a y x2, s u c h t h a t u $ x2. th e r e fo r e , x1ux2y2x3 : : : xa¡2ya¡2xa¡1y1x1 is a c yc le o f le n g t h 2 a ¡ 2 , wh ic h c o n t r a d ic t s t h e s u p p o s it io n t h a t d c o n t a in s n o c yc le o f le n g t h 2 a ¡ 2 . a s s u m e s e c o n d t h a t y1 h a s a ve r t e x, s a y ya¡2, o f d e g r e e a t m o s t 2 a ¡ 3 . th e n fr o m c o n d it io n b0 it fo llo ws t h a t d( yi ) ¸ 2 a ¡ 2 fo r a ll i 2 [1 ; a ¡ 3 ] s in c e x $ y1 [ fug. th is im p lie s t h a t t h e s u b d ig r a p h dhx1 [ fy1; y2; : : : ; ya¡3gi is a c o m p le t e b ip a r t it e d ig r a p h wit h p a r t it e s e t s x1 a n d fy1; y2; : : : ; ya¡3g. in p a r t ic u la r , y1 $ x1. th e n e ve r y p a ir o f d is t in c t ve r t ic e s o f x1 is a d o m in a t in g p a ir . co n d it io n b0 im p lie s t h a t x1 h a s a t le a s t a ¡ 2 ve r t ic e s , s a y x1; x2; : : : ; xa¡2 , o f d e g r e e a t le a s t 2 a ¡ 2 . th e n fx1; x2; : : : ; xa¡2g $ y1 [ fug; in p a r t ic u la r ya¡2 $ fx1; x2; : : : ; xa¡2g a n d u $ fx1; x2; : : : ; xa¡2g. th e r e fo r e , y1xa¡1y2x2y3x3 : : : ya¡2 xa¡2ux1y1 is a c yc le o f le n g t h 2 a ¡ 2 , wh ic h is a c o n t r a d ic t io n . case 2: jy1j = a ¡ 1 . l e t n o w y1 := fy1; y2; : : : ; ya¡1g. th e n y \ b = ;, i.e ., b µ x. s in c e d is s t r o n g , fr o m c o n d it io n b0 it fo llo ws t h a t b = fx0g, u $ x0 a n d jx \ aj = a ¡ 1 . l e t n o w x1 := x \ a = fx1; x2; : : : ; xa¡1g, wh e r e x1 = x ( r e c a ll t h a t x1 ! u ) . if d ( yi ) ¸ 2 a ¡ 2 fo r a ll i 2 [1 ; a ¡ 1 ] t h e n t h e s u b d ig r a p h dhx1 [ y1i is a c o m p le t e b ip a r t it e d ig r a p h wit h p a r t it e s e t s x1 a n d y1. th e r e fo r e , d c o n t a in s a c yc le o f le n g t h 2 a ¡ 2 , a c o n t r a d ic t io n . s. darbinyan 1 1 a s s u m e t h e r e fo r e t h a t y1 h a s a ve r t e x o f d e g r e e a t m o s t 2 a ¡ 3 . ob s e r ve t h a t y1 m a y h a s a t m o s t t h r e e ve r t ic e s o f d e g r e e le s s t h a n 2 a ¡ 2 s in c e d( x1 ) ¸ 2 a ¡ 2 ( fo r o t h e r wis e y1 c o n t a in s t wo ve r t ic e s , s a y v a n d z, s u c h t h a t fv; zg ! x1 a n d maxfd( v ) ; d ( z ) g · 2 a ¡ 3 , wh ic h c o n t r a d ic t s c o n d it io n b0 ) . w e will c o n s id e r t h e fo llo win g fo u r s u b c a s e s d e p e n d in g o n t h e n u m b e r o f ve r t ic e s o f y1, wh ic h h a ve d e g r e e a t m o s t 2 a ¡ 3 . subcase 2.1: y1 h a s e xa c t ly o n e ve r t e x o f d e g r e e le s s t h a n 2 a ¡ 2 . a s s u m e , wit h o u t lo s s o f g e n e r a lit y, t h a t d( ya¡1 ) · 2 a ¡ 3 a n d d( yi ) ¸ 2 a ¡ 2 fo r a ll i 2 [1 ; a ¡ 2 ]. th e n it is e a s y t o s e e t h a t t h e s u b d ig r a p h dhx1 [ y1 n fya¡1gi is a c o m p le t e b ip a r t it e d ig r a p h wit h p a r t it e s e t s x1 a n d y1 n fya¡1g s in c e d( x0; y1 ) = 0 . fr o m s t r o n g c o n n e c t e d n e s s o f d it fo llo ws t h a t d+ ( u; x1 ) ¸ 1 . if u ! xi fo r s o m e i 2 [2 ; a ¡ 1 ], t h e n b y s ym m e t r y b e t we e n t h e ve r t ic e s x2; x3; : : : ; xa¡1, we c a n a s s u m e t h a t u ! x2. th e n it is e a s y t o s e e t h a t ux2y2x3 : : : ya¡2xa¡1y1x1u is a c yc le o f le n g t h 2 a ¡ 2 , wh ic h is a c o n t r a d ic t io n . a s s u m e t h e r e fo r e t h a t d+ ( u; fx2; x3; : : : xa¡1g ) = 0 : ( 1 ) th e n u ! x1, d+ ( ya¡1 ) ¸ 1 a n d d¡ ( ya¡1 ) ¸ 1 , s in c e d is s t r o n g . if t h e r e e xis t t wo d is t in c t ve r t ic e s o f x1, s a y x1 a n d x2, s u c h t h a t x1 ! ya¡1 a n d ya¡1 ! x2, t h e n t h e c yc le x1ya¡1x2y2x3 : : : xa¡2ya¡2xa¡1y1x1 is a c yc le o f le n g t h 2 a ¡ 2 , a c o n t r a d ic t io n . a s s u m e t h e r e fo r e t h a t t h e r e a r e n o t wo d is t in c t ve r t ic e s xi a n d xj o f x1 s u c h t h a t xi ! ya¡1 a n d ya¡1 ! xj. th e n d+ ( ya¡1 ) = d¡ ( ya¡1 ) = 1 a n d ya¡1 $ xi fo r s o m e i 2 [1 ; a ¡ 1 ]. if i = 1 , i.e ., x1 $ ya¡1. th e n d( ya¡1 ) = 2 . n o w u s in g ( 1 ) a n d t h e fa c t t h a t d( u; fx0; x1g ) = 4 , we o b t a in d( u ) = d ( u; fx0; x1g ) + d¡ ( u; fx2; x3; : : : xa¡1g ) · a + 2 · 2 a ¡ 3 ; wh ic h c o n t r a d ic t s c o n d it io n b0 s in c e fu; ya¡1g ! x1 a n d a ¸ 5 . th e r e fo r e , i 2 [2 ; a ¡ 1 ]. a s s u m e , wit h o u t lo s s o f g e n e r a lit y, t h a t ya¡1 $ xa¡1. th e n a( xi; ya¡1 ) = 0 fo r a ll i 2 [1 ; a ¡ 2 ], in p a r t ic u la r , a ( x2; ya¡1 ) = a ( x3; ya¡1 ) = 0 . th is t o g e t h e r wit h ( 1 ) im p lie s t h a t maxfd( x2 ) ; d ( x3 ) g · 2 a ¡ 3 , wh ic h c o n t r a d ic t s c o n d it io n b0 s in c e fx2; x3g ! y1. th e d is c u s s io n o f s u b c a s e 2 .1 is c o m p le t e d . subcase 2.2: y1 h a s e xa c t ly t wo ve r t ic e s o f d e g r e e le s s t h a n 2 a ¡ 2 . a s s u m e , wit h o u t lo s s o f g e n e r a lit y, t h a t d( ya¡2 ) · 2 a ¡ 3 , d ( ya¡1 ) · 2 a ¡ 3 a n d d( yi ) ¸ 2 a¡ 2 fo r a ll i 2 [1 ; a¡ 3 ]. th e n it is e a s y t o s e e t h a t t h e s u b d ig r a p h dhx1 [y1 nfya¡2; ya¡1gi is a c o m p le t e b ip a r t it e d ig r a p h wit h p a r t it e s e t s x1 a n d y1 n fya¡2; ya¡1g s in c e d( x0; y1 ) = 0 . fo r t h e d is c u s s io n o f s u b c a s e 2 .2 it is c o n vie n t ¯ r s t t o p r o ve t h e fo llo win g cla im s 1 a n d 2 b e lo w. claim 1: if xj ! ya¡2 fo r s o m e j 2 [2 ; a ¡ 1 ], t h e n d+ ( ya¡2; fx1; x2; : : : ; xa¡1g n fxjg ) = 0 . p r oof of claim 1: a s s u m e , wit h o u t lo s s o f g e n e r a lit y, t h a t xa¡1 ! ya¡2, i.e ., j = a ¡ 1 . s u p p o s e t h a t t h e c la im is n o t t r u e , i.e ., ya¡2 ! xi fo r s o m e i 2 [1 ; a ¡ 2 ]. w e will c o n s id e r t h e c a s e s i = 1 a n d i 2 [2 ; a ¡ 2 ] s e p a r a t e ly. case. i = 1 , i.e., ya¡2 ! x1. fir s t we s h o w t h a t d+ ( u; fx2; x3; : : : ; xa¡1g ) = 0 : ( 2 ) p r oof of (2): s u p p o s e t h a t ( 2 ) is n o t t r u e , i.e ., t h e r e is a k 2 [2 ; a¡ 1 ] s u c h t h a t u ! xk. if k 2 [2 ; a ¡ 2 ], we m a y a s s u m e , wit h o u t lo s s o f g e n e r a lit y, t h a t u ! x2. th e n t h e c yc le xa¡1ya¡2x1ux2y1x3y2 : : : xa¡2ya¡3xa¡1 is a c yc le o f le n g t h 2 a ¡ 2 , c o n t r a d ic t io n . a s s u m e t h e r e fo r e t h a t k = a ¡ 1 . th e n u ! xa¡1 a n d d+ ( u; fx2; x3; : : : ; xa¡2g ) = 0 : ( 3 ) 1 2 on pre-hamiltonian cycles in balanced bipartite digraphs if ya¡2 ! xl, fo r s o m e l 2 [2 ; a ¡ 2 ] ( s a y ya¡2 ! x2 ) , t h e n t h e c yc le xa¡1ya¡2x2y1x3y2 : : : xa¡2ya¡3x1 uxa¡1 is a c yc le o f le n g t h 2 a ¡ 2 , a c o n t r a d ic t io n . a s s u m e t h e r e fo r e t h a t d+ ( ya¡2; fx2; x3; : : : ; xa¡2g) = 0 : ( 4 ) if xl ! u fo r s o m e l 2 [2 ; a ¡ 2 ] ( s a y x2 ! u ) , t h e n t h e c yc le xa¡1ya¡2x1y2x3y3 : : : ya¡3xa¡2y1x2uxa¡1 is a c yc le o f le n g t h 2 a ¡ 2 , a c o n t r a d ic t io n . a s s u m e t h e r e fo r e t h a t d¡ ( u; fx2; x3; : : : ; xa¡2g) = 0 : co m b in in g t h is t o g e t h e r wit h ( 3 ) a n d ( 4 ) , we o b t a in d( u; fx2; x3; : : : ; xa¡2g ) = d+ ( ya¡2; fx2; x3; : : : ; xa¡2g ) = 0 : th e r e fo r e , s in c e a ¸ 5 , we h a ve d( x2 ) a n d d ( x3 ) · 2 a ¡ 3 , wh ic h c o n t r a d ic t s c o n d it io n b0 s in c e fx2; x3g ! y1. th is c o n t r a d ic t io n p r o ve s ( 2 ) . s in c e d is s t r o n g , fr o m ( 2 ) it fo llo ws t h a t u ! x1. th e r e fo r e , fu; ya¡2g ! x1, i.e ., fu; ya¡2g is a d o m in a t in g p a ir . th is t o g e t h e r wit h c o n d it io n b0 im p lie s t h a t d( u ) ¸ 2 a ¡ 2 s in c e d ( ya¡2 ) · 2 a ¡ 3 ( b y o u r a s s u m p t io n ) . n o w u s in g ( 2 ) , we o b t a in 2 a ¡ 2 · d( u ) = d ( u; fx0; x1g ) + d¡ ( u; fx2; x3; : : : ; xa¡1g ) · 4 + a ¡ 2 = a + 2 : h e n c e , a · 4 , wh ic h c o n t r a d ic t s t h a t a ¸ 5 . th e d is c u s s io n o f t h e c a s e i = 1 is c o m p le t e d . case. i 2 [2 ; a ¡ 2 ], i.e., ya¡2 ! xi and ya¡2x1 =2 a ( d ) . a s s u m e , wit h o u t lo s s o f g e n e r a lit y, t h a t ya¡2 ! x2, i.e ., i = 2 . n o w we p r o ve t h a t d+ ( u; fx3; x4; : : : ; xa¡1g) = 0 : ( 5 ) p r oof of (5): s u p p o s e t h a t ( 5 ) is n o t t r u e , i.e ., t h e r e is a n l 2 [3 ; a ¡ 1 ] s u c h t h a t u ! xl. if l = a ¡ 1 , i.e ., u ! xa¡1, t h e n t h e c yc le xa¡1ya¡2x2y2x3 : : : ya¡3xa¡2y1x1uxa¡1 is a c yc le o f le n g t h 2 a ¡ 2 . a s s u m e t h e r e fo r e t h a t l 2 [3 ; a ¡ 2 ]. w it h o u t lo s s o f g e n e r a lit y, we m a y a s s u m e t h a t u ! x3. th e n t h e c yc le x1ux3y2x4 : : : ya¡4xa¡2ya¡3 xa¡1ya¡2x2y1x1 is a c yc le o f le n g t h 2 a ¡ 2 . in b o t h c a s e s we h a ve a c yc le o f le n g t h 2 a ¡ 2 , wh ic h is a c o n t r a d ic t io n . th e r e fo r e , ( 5 ) is t r u e . fr o m ( 5 ) a n d s t r o n g ly c o n n e c t e d n e s s o f d it fo llo ws t h a t u ! x1 o r u ! x2. a s s u m e ¯ r s t t h a t u ! x1. it is n o t d i± c u lt t o s h o w t h a t d¡ ( u; fx2; x3; : : : ; xa¡2g) = 0 : ( 6 ) in d e e d , if x2 ! u, t h e n t h e c yc le ya¡2x2ux1y1x3y2x4 : : : xa¡2ya¡3xa¡1ya¡2 h a s le n g t h 2 a ¡ 2 ; if xj ! u a n d j 2 [3 ; a ¡ 2 ], t h e n ( we m a y a s s u m e t h a t j = 3 , i.e ., x3 ! u ) t h e c yc le xa¡1ya¡2x2y1x3ux1y2x4 : : : ya¡4xa¡2ya¡3xa¡1 h a s le n g t h 2 a ¡ 2 . in b o t h c a s e s we h a ve a c o n t r a d ic t io n . th e r e fo r e , t h e e qu a lit y ( 6 ) is t r u e . if u ! x2, t h e n fr o m ya¡2 ! x2, d ( ya¡2 ) · 2 a ¡ 3 a n d c o n d it io n b0 it fo llo ws t h a t d( u ) ¸ 2 a ¡ 2 . on t h e o t h e r h a n d , u s in g ( 5 ) a n d ( 6 ) we o b t a in 2 a ¡ 2 · d( u ) = d( u; fx0; x1g ) + d+ ( u; fx2; x3; : : : ; xa¡1g ) + d¡ ( u; fx2; x3; : : : ; xa¡1g ) · 6 : s. darbinyan 1 3 th e r e fo r e , a · 4 , wh ic h c o n t r a d ic t s t h a t a ¸ 5 . a s s u m e t h e r e fo r e t h a t ux2 =2 a ( d ) . th e n b y ( 5 ) a n d ( 6 ) we h a ve d+ ( u; fx2; x3; : : : ; xa¡1g ) = d¡ ( u; fx2; x3; : : : ; xa¡2g ) = 0 : in p a r t ic u la r , a ( xj; u ) = 0 fo r a ll j 2 [2 ; a ¡ 2 ]. s in c e a ¸ 5 a n d s in c e fx2; x3g ! y1, it fo llo ws t h a t d ( x2 ) = 2 a ¡ 2 o r d( x3 ) = 2 a ¡ 2 . if d ( x2 ) = 2 a ¡ 2 , t h e n fya¡2; ya¡1g ! x2, a n d if d( x3 ) = 2 a ¡ 2 , t h e n fya¡2; ya¡1g ! x3. in e a c h c a s e we h a ve a c o n t r a d ic t io n t o c o n d it io n b0. a s s u m e s e c o n d t h a t ux1 =2 a( d ) a n d u ! x2. th e n b y c o n d it io n b0 we h a ve d( u ) ¸ 2 a¡ 2 s in c e fu; ya¡2g ! x2 a n d d( ya¡2 ) · 2 a ¡ 3 . n o w u s in g ( 5 ) , we o b t a in 2 a ¡ 2 · d( u ) = d( u; fx0; x1g ) + d+ ( u; fx2; x3; : : : ; xa¡1g ) + d¡ ( u; fx2; x3; : : : ; xa¡1g ) · a + 2 ; wh ic h is a c o n t r a d ic t io n , b e c a u s e o f a ¸ 5 . cla im 1 is p r o ve d . claim 2: if xj ! ya¡2 fo r s o m e j 2 [2 ; a ¡ 1 ], t h e n d¡ ( ya¡2; fx2; x3; : : : ; xa¡1g n fxjg ) = 0 . p r oof of claim 2: a s s u m e , wit h o u t lo s s o f g e n e r a lit y, t h a t xa¡1 ! ya¡2, i.e ., j = a ¡ 1 . s u p p o s e t h a t t h e c la im is n o t t r u e , i.e ., xl ! ya¡2 fo r s o m e l 2 [2 ; a ¡ 2 ]. fr o m cla im 1 a n d s t r o n g ly c o n n e c t e d n e s s o f d it fo llo ws t h a t ya¡2 ! xa¡1. th is t o g e t h e r wit h c o n d it io n b0 a n d maxfd( ya¡2 ) ; d( ya¡1 ) g · 2 a ¡ 3 im p lie s t h a t ya¡1xa¡1 =2 a( d ) . a s s u m e , wit h o u t lo s s o f g e n e r a lit y, t h a t x2 ! ya¡2, i.e ., l = 2 . if u ! x2, t h e n t h e c yc le ya¡2xa¡1y2x3y3 : : : xa¡3ya¡3xa¡2y1x1ux2ya¡2 h a s le n g t h 2 a¡ 2 , wh ic h is a c o n t r a d ic t io n . l e t u ! xk, wh e r e k 2 [3 ; a ¡ 2 ]. w e m a y a s s u m e t h a t k = 3 , i.e ., u ! x3. th e n ya¡2xa¡1y1x1ux3y2x4y3 : : : xa¡2ya¡3 x2ya¡2 is a c yc le o f le n g t h 2 a¡ 2 , wh ic h is a c o n t r a d ic t io n . th e r e fo r e , we m a y a s s u m e t h a t d+ ( u; fx2; x3; : : : ; xa¡2g ) = 0 : ( 7 ) fr o m ( 7 ) a n d s t r o n g ly c o n n e c t e d n e s s o f d it fo llo ws t h a t u ! x1 o r u ! xa¡1. a s s u m e ¯ r s t t h a t u ! x1. it is n o t d i± c u lt t o s e e t h a t if fo r s o m e j 2 [3 ; a ¡ 2 ], s a y j = 3 , xj ! u, t h e n t h e c yc le ya¡2xa¡1y1x3ux1y3x4 : : : ya¡3xa¡2y2x2ya¡2 h a s le n g t h 2 a2, a n d if xa¡1 ! u, t h e n t h e c yc le ya¡2xa¡1ux1y1x3y3 : : : xa¡3ya¡3xa¡2y2x2ya¡2 h a s le n g t h 2 a ¡ 2 , wh ic h is a c o n t r a d ic t io n . a s s u m e t h e r e fo r e t h a t d¡ ( u; fx3; x4; : : : ; xa¡1g ) = 0 : ( 8 ) n o w u s in g ( 7 ) a n d ( 8 ) , we o b t a in a( u; xj ) = 0 fo r a ll j 2 [3 ; a ¡ 2 ] a n d d( u ) = d ( u; fx0; x1g ) + d+ ( u; fx2; x3; : : : ; xa¡1g ) + d¡ ( u; fx2; x3; : : : ; xa¡1g) · 6 · 2 a ¡ 3 : fr o m ( 7 ) , ( 8 ) a n d cla im 1 it fo llo ws t h a t d ( xj ) · 2 a¡ 3 fo r a ll j 2 [3 ; a¡ 2 ]. h e n c e , a¡ 2 = 3 , i.e ., a = 5 a n d d ( x3 ) · 2 a ¡ 3 , a n d d ( x2 ) ; d( x4 ) ¸ 2 a ¡ 2 s in c e fx2; x3; : : : ; xa¡1g ! y1. fr o m ya¡1xa¡1 =2 a( d ) a n d xa¡1u =2 a( d ) ( a ¡ 1 = 4 ) it fo llo ws t h a t u ! xa¡1, wh ic h is a c o n t r a d ic t io n s in c e fu; ya¡2g ! xa¡1 a n d maxfd ( u ) ; d ( ya¡2g · 2 a ¡ 3 . a s s u m e s e c o n d t h a t u ! xa¡1 a n d ux1 =2 a( d ) . s in c e fu; ya¡2g ! xa¡1 a n d s in c e d( ya¡2 ) · 2 a¡ 3 it fo llo ws t h a t d( u ) ¸ 2 a¡ 2 . on t h e o t h e r h a n d , u s in g ( 7 ) a n d ux1 =2 a ( d ) , we o b t a in 2 a ¡ 2 · d( u ) = d( u; fx0; x1g ) + d+ ( u; fx2; x3; : : : ; xa¡1g ) + d¡ ( u; fx2; x3; : : : ; xa¡1g ) · a + 2 ; 1 4 on pre-hamiltonian cycles in balanced bipartite digraphs wh ic h c o n t r a d ic t s t h a t a ¸ 5 . cla im 2 is p r o ve d . n o w we a r e r e a d y t o c o m p le t e t h e d is c u s s io n o f s u b c a s e 2 .2 . a s s u m e t h a t d¡ ( yj; fx2; x3; : : : ; xa¡1g ) 6= 0 fo r j = a¡ 2 o r a¡ 1 ( s a y j = a ¡ 2 ) . a s s u m e , wit h o u t lo s s o f g e n e r a lit y, t h a t xa¡1 ! ya¡2. fr o m cla im s 1 a n d 2 it fo llo ws t h a t d+ ( ya¡2; fx1; x2; x3; : : : ; xa¡2g ) = d¡ ( ya¡2; fx2; x3; : : : ; xa¡2g ) = 0 : ( 9 ) th e r e fo r e , d( xi ) · 2 a ¡ 2 fo r a ll i 2 [2 ; a ¡ 2 ] s in c e a ( xi; ya¡2 ) = 0 . fr o m s t r o n g ly c o n n e c t e d n e s s o f d a n d ( 9 ) it fo llo ws t h a t ya¡2 ! xa¡1. th is t o g e t h e r wit h maxfd( ya¡2 ) ; d( ya¡1 ) g · 2 a ¡ 3 a n d c o n d it io n b0 im p lie s t h a t ya¡1xa¡1 =2 a( d ) . th e r e fo r e , d+ ( ya¡1; fx1; x2; x3; : : : ; xa¡2g ) 6= 0 s in c e d is s t r o n g . n o w we a p p ly cla im 1 t o ya¡1 we c o n c lu d e t h a t xa¡1ya¡1 =2 a( d ) . th e n a( xa¡1; ya¡1 ) = 0 a n d d( xa¡1 ) · 2 a ¡ 2 . s in c e fx2; x3; : : : ; xa¡1g ! y1, fr o m c o n d it io n b0 it fo llo ws t h a t fx2; x3; : : : ; xa¡1g h a s a t le a s t a ¡ 3 ve r t ic e s o f d e g r e e a t le a s t 2 a ¡ 2 . in p a r t ic u la r , d( x2 ) ¸ 2 a ¡ 2 o r d ( x3 ) ¸ 2 a ¡ 2 . w it h o u t lo s s o f g e n e r a lit y, we a s s u m e t h a t d( x2 ) ¸ 2 a ¡ 2 . th e n x2 ! fu; ya¡1g s in c e a( x2; ya¡2 ) = 0 . n o w u s in g ( 9 ) wit h r e s p e c t t o ya¡1, we o b t a in d+ ( ya¡1; fx1; x3; x4; : : : ; xa¡1g ) = d¡ ( ya¡1; fx3; x4; : : : ; xa¡1g ) = 0 : ( 1 0 ) in p a r t ic u la r , fr o m ( 9 ) a n d ( 1 0 ) we h a ve d¡ ( x1; fya¡2; ya¡1g ) = 0 . th e r e fo r e , x1 ! ya¡2 a n d u ! x1 s in c e d( x1 ) ¸ 2 a ¡ 2 . h e n c e , t h e c yc le x2ux1ya¡2xa¡1y1x3y2x4 : : : ya¡4xa¡2ya¡3x2 is a c yc le o f le n g t h 2 a ¡ 2 , wh ic h is a c o n t r a d ic t io n . a s s u m e n o w t h a t a ( fx2; x3; : : : ; xa¡1g ! fya¡2; ya¡1g ) = 0 : th e n , s in c e d is s t r o n g , it fo llo ws t h a t x1 ! fya¡2; ya¡1g. fr o m t h e la s t e qu a lit y we h a ve d( xj ) · 2 a ¡ 2 fo r a ll j 2 [2 ; a ¡ 1 ]. th is t o g e t h e r wit h fx2; x3; : : : ; xa¡1g ! y1 im p lie s t h a t fx2; x3; : : : ; xa¡1g h a s a t le a s t a ¡ 3 ve r t ic e s o f d e g r e e e qu a l t o 2 a ¡ 2 . a s s u m e , wit h o u t lo s s o f g e n e r a lit y, t h a t d ( x2 ) = 2 a ¡ 2 . th e n fya¡2; ya¡1g ! x2, wh ic h is a c o n t r a d ic t io n s in c e d( ya¡2 ) · 2 a ¡ 3 a n d d ( ya¡1 · 2 a ¡ 3 . in e a c h c a s e we o b t a in a c o n t r a d ic t io n , a n d h e n c e , t h e d is c u s s io n o f s u b c a s e 2 .2 is c o m p le t e d . subcase 2.3: y1 h a s e xa c t ly t h r e e ve r t ic e s o f d e g r e e le s s t h a n 2 a ¡ 2 . a s s u m e , wit h o u t lo s s o f g e n e r a lit y, t h a t d( yj ) · 2 a ¡ 3 fo r a ll j 2 [a ¡ 3 ; a ¡ 1 ] a n d d( yi ) ¸ 2 a ¡ 2 fo r a ll i 2 [1 ; a ¡ 4 ]. th e n it is e a s y t o s e e t h a t t h e s u b d ig r a p h dhfx1 [ fy1; y2; : : : ; ya¡4gi is a c o m p le t e b ip a r t it e d ig r a p h a n d d¡ ( xi; fya¡3; ya¡2; ya¡1g ) · 1 fo r a ll i 2 [1 ; a ¡ 1 ]. th is t o g e t h e r wit h c o n d it io n b0 im p lie s t h a t fx2; x3; : : : ; xa¡1g h a s a t le a s t a ¡ 3 ve r t ic e s o f , s a y x2; x3; : : : ; xa¡2, o f d e g r e e e qu a l t o 2 a ¡ 2 . th e n x1 $ u, xi ! fya¡3; ya¡2; ya¡1g if i 2 [1 ; a ¡ 2 ], a n d xj $ u if j 2 [2 ; a ¡ 2 ]. n o w it is n o t d i± c u lt t o s e e t h a t fo r e ve r y i 2 [1 ; a ¡ 2 ] t h e r e is a j 2 [a ¡ 3 ; a ¡ 1 ] s u c h t h a t xi $ yj. b e c a u s e o f t h e s ym m e t r y b e t we e n t h e ve r t ic e s x1; x2; : : : ; xa¡2, we c a n a s s u m e , x1 $ ya¡3. a s s u m e ¯ r s t t h a t a ( fya¡2; ya¡1g ! fx4; x5; : : : ; xa¡1g ) 6= ;: s. darbinyan 1 5 l e t ya¡2 ! xa¡1. th e n t h e c yc le x2ux3ya¡3 x1ya¡2xa¡1y1x4y2 : : : xa¡2ya¡4x2 h a s le n g t h 2 a ¡ 2 , if a ¸ 6 , a n d t h e c yc le x2ux3y2x1y3x4y1x2, if a = 5 h a s le n g t h 2 a ¡ 2 , wh ic h is a c o n t r a d ic t io n . a s s u m e s e c o n d t h a t a ( fya¡2; ya¡1g ! fx4; x5; : : : ; xa¡1g ) = ;: fr o m x1 $ ya¡3, maxfd ( ya¡3 ) ; d ( ya¡2 ) ; d( ya¡1 ) g · 2 a ¡ 3 a n d c o n d it io n b0 it fo llo ws t h a t d¡ ( x1; fya¡2; ya¡3g ) = 0 a n d minfd+ ( ya¡2; fx2; x3g ) ; d+ ( ya¡1; fx2; x3gg ) ¸ 1 : ( 1 1 ) w it h o u t lo s s o f g e n e r a lit y, we a s s u m e t h a t ya¡2 ! x2. if a ¸ 6 , t h e n t h e c yc le ya¡2x2ya¡3x1ux3y1xa¡1 y2x4y3 : : : xa¡3ya¡4xa¡2ya¡2 is a c yc le o f le n g t h 2 a ¡ 2 , wh ic h is a c o n t r a d ic t io n . a s s u m e t h e r e fo r e t h a t a = 5 . n o w u s in g ( 1 1 ) , y3 ! x2, y2 ! x1 a n d c o n d it io n b0, we o b t a in y3 ! x3 a n d d+ ( y4; fx2; x3g ) = 0 . th u s , we h a ve t h a t dhfx1; x2; x3; u; y1gi is a c o m p le t e b ip a r t it e d ig r a p h wit h p a r t it e s e t s fx1; x2; x3g a n d fu; y1g, fx1; x2; x3g ! fy2; y3; y4g, x4 $ y1, xi $ yi+1 fo r a ll i 2 [1 ; 3 ] a n d x0 $ u. it is e a s y t o c h e c k t h a t t h e o b t a in e d d ig r a p h is s t r o n g ly c o n n e c t e d a n d is o m o r p h ic t o d ( 1 0 ) , wh ic h s a t is ¯ e s c o n d it io n b0, b u t h a s n o c yc le o f le n g t h 8 . th e t h e o r e m is p r o ve d . fr o m th e o r e m s 1 .5 a n d 1 .6 fo llo ws t h e fo llo win g c o r o lla r y fo llo ws . cor ollar y: l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 1 0 . assume that d ( x) +d( y ) ¸ 4 a¡ 3 for every dominaiting pair of vertices x and y. then either the underlying graph of d is 2-connected or d contains a cycle of length k for every k 2 [1 ; a¡ 1 ] unless d is isomorphic to the digraph d ( 1 0 ) . refer ences [1 ] j. b a n g -je n s e n , g. gu t in , d igraphs: theory, algorithms and applications, s p r in g e r , 2 0 0 1 . 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[1 8 ] s .k h . d a r b in ya n , \ cyc le s o f e a c h e ve n le n g t h s in b a la n c e d b ip a r t it e d ig r a p h s " , a r x iv: 1 6 0 4 .0 8 7 3 3 v1 [m a t h .co] 1 4 ju l 2 0 1 6 . submitted 14.06.2016, accepted 28.10.2016. ð³í³ë³ñ³ïßéí³í »ñïù³ë ïáõùýáñáßí³í ·ñ³ýý»ñç ý³ë³ñ³ùçéïáýû³ý óçïé»ñç ù³ëçý ê. ¸³ñµçýû³ý ²ù÷á÷áõù ð³í³ë³ñ³ïßéí³í »ñïù³ë ïáõùýáñáßí³í ·ñ³ýç ïáõùýáñáßí³í óçïéá ïáãíáõù ¿ ý³ë³ñ³ùçéïáýû³ý, »ã» ³ûý å³ñáõý³ïáõù ¿ ³û¹ ·ñ³ýç µáéáñ ·³·³ãý»ñá ` µ³óç »ñïáõëçó: ü»ñï³ ³ßë³ï³ýùáõù óáõûó ¿ ïñíáõù ñ»ï¨û³é åý¹áõùá. â»áñ»ù: ¸çóáõù` d-ý 2 a ¸ 1 0 ·³·³ã³ýç ñ³í³ë³ñ³ïßéí³í »ñïù³ë ïáõùýáñáßí³í ·ñ³ý ¿: ºã» ³û¹ ·ñ³ýç ·³·³ãý»ñç ó³ýï³ó³í ñ³õãáõ ½áõû·ç ³éýí³½ý ù»ï ·³·³ãç éáï³é ³ëïç×³ýá ÷áùñ ã¿ 2 a ¡ 2 ãíçó, ³å³ d-ý å³ñáõý³ïáõù ¿ ý³ë³ñ³ùçéïáýû³ý óçïé ï³ù d-ç ãïáõùýáñáßí³í ñçùù ·ñ³ýá 2 -ï³å³ïóí³í ¿ ï³ù d-ý ç½áùáñý ¿ ù»ï 10 ·³·³ã³ýç ·ñ³ýçý: s. darbinyan 1 7 î ïðåäãàìèëüòîíîâûõ êîíòóðîâ â ñáàëàíñèðîâàííûõ äâóäîëüíûõ îðãðàôàõ ñ. äàðáèíÿí àííîòàöèÿ îðèåíòèðîâàííûé êîíòóð ïðîõîäÿùèé ÷åðåç âñå âåðøèíû ñáàëàíñèðîâàííîãî äâóäîëüíîãî îðãðàôà, êðîìå äâóõ âåðøèí, íàçûâàåòñÿ ïðåäãàìèëüòîíîâûì êîíòóðîì. â íàñòîÿùåé ñòàòüå äîêàçûâàåòñÿ: òåîðåìà: ïóñòü d 2 a-âåðøèííûé ( a ¸ 5 ) ñáàëàíñèðîâàííûé äâóäîëüíûé îðãðàô. åñëè äëÿ ëþáûõ äîìèíèðóþùèõ ïàð âåðøèí ïî êðàéíåé ìåðå îäíà âåðøèíà èìååò ëîêàëüíóþ ñòåïåíü íå ìåíüøå ÷åì 2 a ¡ 2 , òî d ñîäåðæèò ïðåäãàìèëüòîíîâûé êîíòóð èëè íåîðèåíòèðîâàííàÿ îñíîâà ãðàô îðãðàôà ÿâëÿåòñÿ d 2-ñâÿçíîé èëè d èçîìîðôåí îäíîìó îðãðàôó ñ äåñÿòüþ âåðøèíàìè. microsoft word w7.doc mathematical problems of computer science 37, 53--63, 2012. 53 least squares fitting with cubic splines mane khachatryan state engineering university of armenia e-mail: mane.khachatrian@gmail.com abstract in the paper cubic spline approximation of experimental data by least squares method is considered. an algorithm to construct a cubic spline with mixed-type boundary constraints is derived. references [1] ø. ¶. ê³ã³ïñû³ý, «ê³éá ïçåç »½ñ³ûçý ë³ñù³ý³÷³ïáõùý»ñáí ëáñ³ý³ñ¹³ûçý ë÷é³ûýç ï³éáõóáõùá», вестник гиуа. моделирование, оптимизация, управление. вып. 14, том 2. сс.33-40, 2011. [2] ф. р. гантмахер, теория матриц . москва, наука, 1967. [3] ю. с. завьялов, б. и. квасов, в. л. мирошниченко, методы сплайн-функций. москва, наука, 1980. [4] б. и. квасов. методы изогеометрической аппроксимации сплайнами. москва, физматлит, 2006. [5] дж. г. мэтьюз, к. д. финк. численные методы: использование matlab. м.: издательский дом “вильямс”, 2001. [6] р. хорн, ч. джонсон. матричный анализ. м.: мир, 1989. [7] d. kincaid and w. cheney. numerical analysis. brooks/cole, pacific grove, ca, 1991. [8] j. w. lewis. “inversion of tridiagonal matrices”, numer. math., vol. 38, pp. 333–345, 1982. êáñ³ý³ñ¹³ûçý ëåé³ûýý»ñáí ùáï³ñïáõùá ÷áùñ³·áõûý ù³é³ïáõëçý»ñç ù»ãá¹áí ø. ê³ã³ïñû³ý ²ù÷á÷áõù ðá¹í³íáõù ¹çï³ñïíáõù է ÷áñó³ñ³ñ³ï³ý ïíû³éý»ñç ùáï³ñïáõùá ëáñ³ý³ñ¹³ûçý ëåé³ûýý»ñáí` ÷áùñ³·áõûý ù³é³ïáõëçý»ñç ù»ãá¹áí: ²ñï³ííáõù ¿ ë³éá ïçåç »½ñ³ûçý ë³ñù³ý³÷³ïáõùý»ñáí ëáñ³ý³ñ¹³ûçý ëåé³ûýý»ñç ï³éáõóù³ý ³é·áñçãùá: least squares fitting with cubic splines 54 аппроксимация кубическими сплайнами по методу наименьших квадратов м. хачатрян аннотация в статье рассматривается аппроксимация экспериментальных данных методом наименьших квадратов с помощью кубических сплайнов. выводится алгоритм построения интерполяционных кубических сплайнов с краевыми ограничениями смешанного типа. . 2_mossine-myu_18-25.dvi mathematical problems of computer science 49, 18{25, 2018. t wo gener alized lower b ounds for the cir cumfer ence mo s s in e s . k o u la kz ia n a n d zh o r a g. n iko g h o s ya n institute for informatics and automation problems of nas ra e-mail: mossine@hotmail.com, zhora@ipia.sci.am abstract in 2013, the second author obtained two lower bounds for the length of a longest cycle c in a graph g in terms of the length of a longest path (a longest cycle) in g ¡c and the minimum degree of g (zh.g. nikoghosyan, "advanced lower bounds for the circumference", graphs and combinatorics 29, pp. 1531-1541, 2013). in this paper we present two analogous bounds based on the average of the ¯rst i smallest degrees in g ¡ c for appropriate i instead of the minimum degree. keywords: circumference, minimum degree, degree sums. 1 . in t r o d u c t io n l e t c b e t h e c ir c u m fe r e n c e t h e le n g t h o f a lo n g e s t c yc le o f a g r a p h g a n d ± t h e m in im u m d e g r e e in g. in t h is p a p e r we p r e s e n t t h e fo llo win g t wo r e s u lt s . t heor em 1. l et c be a longest cycle in a graph g, p̂ the order of a longest path in g ¡ c and ¹ the average of the ¯rst p̂ smallest degrees in g ¡ c. then c ¸ ( p̂ + 1 ) ( ¹ ¡ p̂ + 1 ) : t heor em 2. l et c be a longest cycle in a graph g, ĉ the order of a longest cycle in g ¡ c and ¹ the average of the ¯rst ĉ smallest degrees in g ¡ c. then c ¸ ( ĉ + 1 ) ( ¹ ¡ ĉ + 1 ) : ob s e r vin g t h a t ¹ ¸ ± in th e o r e m s 1 a n d 2 , we o b t a in t h e o r ig in a l lo we r b o u n d s [2 ] a s im m e d ia t e c o r o lla r ie s in t e r m s o f p̂, ĉ a n d ±. t heor em a [2 ]. l et c be a longest cycle in a graph g and p̂ the order of a longest path in g ¡ c. then c ¸ ( p̂ + 1 ) ( ± ¡ p̂ + 1 ) : t heor em b [2 ]. l et c be a longest cycle in a graph g and ĉ the order of a longest path in g ¡ c. then c ¸ ( ĉ + 1 ) ( ± ¡ ĉ + 1 ) : 1 8 m. koulakzian and zh. nikoghosyan 1 9 2 . d e ¯ n it io n s w e u s e b o n d y a n d mu r t y [1 ] fo r t e r m in o lo g y a n d n o t a t io n n o t d e ¯ n e d h e r e , a n d c o n s id e r o n ly ¯ n it e u n d ir e c t e d g r a p h s wit h o u t lo o p s a n d m u lt ip le e d g e s . th e ve r t e x s e t o f a g r a p h g is d e n o t e d b y v ( g) o r ju s t v ; t h e s e t o f e d g e s b y e ( g) o r ju s t e. fo r a s u b g r a p h h o f g we a ls o u s e g ¡ h s h o r t fo r g ¡ v ( h ) , a n d jhj s h o r t fo r jv ( h ) j. p a t h s a n d c yc le s in g c a n b e c o n s id e r e d a s c o n n e c t e d s u b g r a p h s o f g, h a vin g a m a xim u m d e g r e e 0 ,1 o r 2 . th e le n g t h o f a p a t h p a n d o f a c yc le q, d e n o t e d b y l ( p ) a n d l ( q ) , is jv ( p ) j ¡ 1 a n d jv ( q ) j, r e s p e c t ive ly. w e d e n o t e l ( p ) = ¡ 1 a n d l ( q ) = 0 if a n d o n ly if v ( p ) = v ( q ) = ;. a g r a p h is s a id t o b e h a m ilt o n ia n if it s lo n g e s t c yc le p a s s e s t h r o u g h a ll o f it s ve r t ic e s . th e ve r t ic e s a n d e d g e s in g c a n b e in t e r p r e t e d a s c yc le s o f le n g t h s 1 a n d 2 , r e s p e c t ive ly. a n ( x; y ) -p a t h is a p a t h wit h e n d ve r t ic e s x a n d y. give n a n ( x; y ) -p a t h l o f g we d e n o t e b y ¡! l t h e p a t h l wit h a n o r ie n t a t io n fr o m x t o y. if u; v 2 v ( l) t h e n u¡!l v d e n o t e s t h e c o n s e c u t ive ve r t ic e s o n ¡! l fr o m u t o v in t h e d ir e c t io n s p e c ī e d b y ¡! l . th e s a m e ve r t ic e s , in r e ve r s e o r d e r , a r e g ive n b y v ã¡ l u. fo r l = x ¡! l y a n d u 2 v ( l) , le t u+ ( ¡!l ) ( o r ju s t u+ ) d e n o t e t h e s u c c e s s o r o f u ( u 6= y ) o n ¡!l a n d u¡ d e n o t e it s p r e d e c e s s o r ( u 6= x ) . if a µ v ( l) ¡ y a n d b µ v ( l) ¡ x, t h e n we d e n o t e a+ = fv+jv 2 ag a n d b¡ = fv¡jv 2 bg. a s im ila r n o t a t io n is u s e d fo r t h e c yc le s . if q is a c yc le a n d u 2 v ( q ) , t h e n u¡!q u = u. fo r v 2 v , p u t n ( v ) = fu 2 v juv 2 eg, d( v ) = jn ( v ) j a n d ± = m in fd( u ) ju 2 v g. 3 . s p e c ia l d e ¯ n it io n s fo r t h e r e m a in d e r o f t h is s e c t io n , le t a s u b g r a p h f o f a g r a p h g a n d a p a t h ( o r a c yc le ) ¡! m in g ¡ f b e ¯ xe d . de¯nition 1. ( ¤i) -m in im a lit y, ( ¤i) -m a xim a lit y. w e u s e t h e n o t io n s o f ( ¤i ) -m in im a lit y a n d ( ¤i ) -m a xim a liy d e ¯ n e d wit h r e s p e c t t o c e r t a in o p e r a t io n s fo r i = 1 ; 2 ; :::; 1 0 . th e y will b e d e s c r ib e d in d e t a il c u r r e n t ly. de¯nition 2. mf -extension; ¡! t ( u) ; _u; äu. fo r e a c h u 2 v ( m ) , le t ¡!t ( u ) = u¡!t ( u ) äu b e a p a t h in g, h a vin g o n ly u in c o m m o n wit h v ( m ) . if v ( t ( u) ) \ v ( t ( v ) ) = ; a n d v ( t ( u ) ) µ v ( g ¡ f ) fo r a ll d is t in c t ve r t ic e s u; v 2 v ( m ) , t h e n t h e fo r e s t t , d e ¯ n e d b y ft ( u ) ju 2 v ( m ) g, is s a id t o b e mf -e xt e n s io n . if äu 6= u fo r s o m e u 2 v ( m ) , t h e n we u s e _u t o d e n o t e u+ ( ¡!t ( u) ) . de¯nition 3. ©u; '( u ) ; ª( u ) ; ã ( u ) . l e t t b e a n m f -e xt e n s io n . fo r e a c h u 2 v ( m ) , p u t ©u = n ( äu) \ v ( t ) ; 'u = j©uj; ªu = n ( äu ) \ v ( f ) ; ãu = jªuj: de¯nition 4. u0; u 0; u1; u ¤. 2 0 two generalized lower bounds for the circumference l e t t b e a n mf -e xt e n s io n . p u t u0 = fu 2 v ( m ) ju = äug; u 0 = v ( m ) ¡ u0; u ¤ = fu 2 u 0j©u µ v ( t ( u ) ) g; u1 = v ( m ) ¡ ( u0 [ u¤ ) : de¯nition 5. m aximal m f -extension. a n mf -e xt e n s io n t is s a id t o b e m a xim a l if it is e xt r e m a l wit h r e s p e c t t o t h e fo llo win g o p e r a t io n : if t h e r e e xis t s a n e d g e äuz s u c h t h a t u 2 v ( m ) a n d z 62 v ( t ) [ v ( f ) , t h e n r e p la c in g t ( u ) b y ut ( u) äuz, we o b t a in a n e w m f -e xt e n s io n t 0 wit h jv ( t 0 ) j > jv ( t ) j. de¯nition 6. ( u0 ) -minimal and ( u0; u ¤ ) -minimal m f -extensions. a n mf -e xt e n s io n t is s a id t o b e ( u0 ) -m in im a l, if it is c h o s e n s u c h t h a t u0 is ( ¤ 6 ) -m in im a l ( s e e t h e p r o o f o f th e o r e m 1 ) . a ( u0 ) -m in im a l mf -e xt e n s io n t is s a id t o b e ( u0; u ¤ ) -m in im a l if it is c h o s e n s u c h t h a t u ¤ is ( ¤1 0 ) -m in im a l ( s e e t h e p r o o f o f th e o r e m 2 ) . de¯nition 7. bu; b ¤ u; bu; b ¤ u. l e t t b e a n mf -e xt e n s io n a n d u 2 v ( m ) . p u t bu = fv 2 u0jv _u 2 eg a n d bu = jbuj. b y t h e d e ¯ n it io n , bu = ; fo r e a c h u 2 u0. fu r t h e r m o r e , fo r e a c h u 2 u0 , le t b¤u = fv 2 u 0ju _v 2 eg a n d jb¤uj = b¤u. 4 . p r e lim in a r ie s th e p r o o fs o f t h e fo llo win g le m m a s c a n b e ¯ n d in [2 ]. lemma 1. l et c be a cycle in a graph g and p a path in g¡c. l et ¡!p 0; :::; ¡! p p be pairwise disjoint paths in g ¡ c with ¡!p i = vi ¡! p iwi ( i = 0 ; 1 ; :::; p ) , having only v0; :::; vp in common with p . then either there is a cycle in g longer than c or jcj ¸ px i=0 jzij + ¯̄ ¯̄ ¯ p[ i=0 zi ¯̄ ¯̄ ¯ ; where zi = n ( wi ) \ v ( c ) ( i = 0 ; 1 ; :::; p ) . lemma 2. l et f be a subgraph of a graph g and r a longest cycle in g ¡ f with a ( u0 ) minimal rf -extension t . then either there is a cycle longer than r or l ( r) ¸ 'u + bu + 1 for each u 2 u1. lemma 3. l et f be a subgraph of a graph g and p a path in g¡f with a ( u0 ) -minimal p f extension t . then either there is a path longer than p or l ( p ) ¸ 'u+bu for each u 2 u1[u¤. m. koulakzian and zh. nikoghosyan 2 1 5 . p r o o fs p r oof of t heor em 1. l et q = u0:::uq be a path in g ¡ c with a ( u0 ) minimal qcextension t . assume without loss of generality that c is ( ¤ 1 ¡ ¤ 4 ) -extremal, and q is ( ¤ 7 ¡ ¤ 9 ) -extremal. since g is non-hamiltonian, we have q ¸ 0 . claim 1. if u 2 u0 a n d v 2 u 0, t h e n ©u \ v ( t ( v ) ) µ fv; _vg. p r oof. s u p p o s e o t h e r wis e . l e t z 2 v ( t ( v ) ) ¡ fv; _vg. th e n , r e p la c in g t ( u ) a n d t ( v ) b y uz ¡! t ( v ) äv a n d v ¡! t ( v ) z¡, r e s p e c t ive ly, we c a n fo r m ( d e n o t e t h is o p e r a t io n b y ( ¤ 6 ) ) a n e w qc-e xt e n s io n , c o n t r a d ic t in g t h e ( u0 ) m in im a lit y o f t . 2 claim 2. if u 2 u0, t h e n 'u · q + b¤u. p r oof. th e p r o o f fo llo ws im m e d ia t e ly fr o m d e ¯ n it io n s 3 , 7 a n d cla im 1 . 2 claim 3. if u 2 u 0, t h e n 'u · q ¡ bu. p r oof. u s in g l e m m a 3 wit h t h e fa c t t h a t q is ( ¤ 7 ¡ ¤9 ) -e xt r e m a l, we o b t a in q ¸ 'u + bu fo r e a c h u 2 u 0, a n d t h e r e s u lt fo llo ws . 2 ob s e r vin g t h a t x u2u0 b¤u = x u2u 0 bu ( b y t h e d e ¯ n it io n ) a n d u s in g cla im s 2 a n d 3 , we o b t a in qx i=0 'ui · q ( q + 1 ) + x u2u0 b¤u ¡ x u2u 0 bu = q ( q + 1 ) : s u p p o s e ¯ r s t t h a t 'ui + ãui 6= d( äui ) fo r s o m e i 2 0 ; q. th e n t h e r e e xis t s a n e d g e äuz s u c h t h a t z 62 v ( t ) [ v ( c ) . a d d in g äuz t o t we o b t a in a n e w qc-e xt e n s io n , c o n t r a d ic t in g t h e m a xim a lit y o f t ( d e ¯ n it io n 5 ) . n o w le t 'ui + ãui = d( äui ) ( i = 0 ; :::; q ) . th e n qx i=0 ãui = qx i=0 d( äui ) ¡ qx i=0 'ui ¸ qx i=0 d( äui ) ¡ q ( q + 1 ) : it fo llo ws , in p a r t ic u la r , t h a t m a x i fãuig ¸ 1 q + 1 qx i=0 ãui ¸ 1 q + 1 qx i=0 d ( äui ) ¡ q: b y l e m m a 1 , c ¸ qx i=0 ãui + m a x i fãuig ¸ ( q + 2 ) ã 1 q + 1 qx i=0 d( äui ) ¡ q ! ¸ ( q + 2 ) ( ¹q ¡ q ) : p r oof of t heor em 2. l e t h = u1:::uku1 b e a c yc le in g ¡ c wit h a n ( u0; u ¤ ) -m in im a l hc-e xt e n s io n t . l e t h b e ( ¤5 ) -e xt r e m a l. p u t u¤1 = ( u 2 u¤j'u · h 2 ) ; u¤2 = ( u 2 u¤j'u ¸ h + 1 2 ) : 2 2 two generalized lower bounds for the circumference claim 1. if u 2 u0 a n d v 2 u 0, t h e n ©u \ v ( t ( v ) ) µ fv; _vg. p r oof. th e p r o o f is ve r y s im ila r t o t h a t o f cla im 1 in th e o r e m 1 . 2 claim 2. if u 2 u0, t h e n 'u · h ¡ 1 + b¤u. p r oof. im m e d ia t e fr o m d e ¯ n it io n s 3 , 7 a n d cla im 1 . 2 claim 3. if u 2 u1, t h e n 'u · h ¡ 1 ¡ bu. p r oof. s in c e h is ( ¤ 5 ) -e xt r e m a l, b y l e m m a 2 , h ¸ 'u + bu + 1 fo r e a c h u 2 u1, a n d t h e r e s u lt fo llo ws . 2 claim 4. if u 2 u¤, t h e n 'u · h ¡ 1 ¡ bu + 'u ¡ h2 . p r oof. s in c e h is ( ¤ 5 ) -e xt r e m a l, b y t h e s t a n d a r d a r g u m e n t s , h ¸ 2 ( bu + 1 ) fo r e a c h u 2 u¤, a n d t h e r e s u lt fo llo ws im m e d ia t e ly. 2 claim 5. if u 2 u1, t h e n 'u · h ¡ 1 ¡ bu. p r oof. im m e d ia t e fr o m cla im s 3 a n d 4 . 2 if u¤2 = ;, t h e n b y cla im s 2 a n d 5 , p u 'u · h( h ¡ 1 ) . b u t t h e n , a s in th e o r e m 1 , c ¸ ( h + 1 ) ( ¸1 ¡ h + 1 ) , wh e r e ¸1 = 1h pk i=1 d( äui ) ¸ ¹h. n o w le t u¤2 6= ;. ch o o s e v 2 u ¤2 s u c h t h a t 'v = m a x u2u¤ 2 f'ug: ( 1 ) claim 6. if u 2 u¤2 , t h e n 'u · h ¡ 1 ¡ bu + 'v ¡ h2 . p r oof. im m e d ia t e fr o m ( 1 ) a n d cla im 4 . 2 u s in g cla im s 2 , 5 , 6 a n d r e c a llin g t h a t p u2u0 b ¤ u = p u2u 0 bu a n d ju0j+ju1[u ¤ 1 j+ju¤2 j = h, we g e t x u 'u = x u2u0 'u + x u2u1[u ¤1 'u + x u2u ¤2 'u · h( h ¡ 1 ) + ju ¤2 j ã 'v ¡ h 2 ! : ( 2 ) b y d e ¯ n it io n 3 , ©v µ v ( t ( v ) ) . l e t v1; :::; vt b e t h e e le m e n t s o f ©+v , o c c u r r in g o n ¡! t ( v ) in a c o n s e c u t ive o r d e r wit h vt = äv. cle a r ly t = j©vj = 'v. p u t n ( vi ) \ v ( t ) = ©0i; n ( vi ) \ v ( c ) = z0i ( i = 1 ; :::; t) : ( 3 ) if ©0i \ ( v ( t ) ¡ v ( t ( v ) ) ) 6= ; fo r s o m e i 2 1 ; t, t h e n r e p la c in g t ( v ) b y v ¡! t ( v ) v¡i äv ã¡ t ( v ) vi; we fo r m ( d e n o t e t h is o p e r a t io n b y ( ¤ 1 0 ) ) a n e w hc-e xt e n s io n , c o n t r a d ic t in g t h e m in im a lit y o f ju¤j. s o , we c a n a s s u m e ©0i µ v ( t ( v ) ) ( i = 1 ; :::; t) . a s s u m e w.l.o .g . t h a t m a xi j©0ij = j©0tj = 'v. s o , m a x i j©0ij = j©0tj = j©vj = 'v = t: ( 4 ) s in c e ãui = d ( ui ) ¡ 'ui ( i = 1 ; :::; h) a n d jz0ij = d ( vi ) ¡ j©0ij ( i = 1 ; :::; t ¡ 1 ) , we h a ve hx i=1 ãui + t¡1x i=1 jz0ij = hx i=1 ( d( ui ) ¡ 'ui ) + t¡1x i=1 ( d( vi ) ¡ j©0ij) m. koulakzian and zh. nikoghosyan 2 3 = hx i=1 d( ui ) + t¡1x i=1 d( vi ) ¡ hx i=1 'ui ¡ t¡1x i=1 j©0ij: ( 5 ) p u t ¸2 = 1 h + t ¡ 1 ã hx i=1 d( ui ) + t¡1x i=1 d ( vi ) ! ¸ ¸1 ¸ ¹h: case 1. ju ¤2 j = 1 . b y ( 2 ) , ( 4 ) a n d ( 5 ) , hx i=1 ãui + t¡1x i=1 jz0ij ¸ ( h + t ¡ 1 ) ¸2 ¡ h ( h ¡ 1 ) ¡ t + h 2 ¡ t¡1x i=1 t = ( h + t ¡ 1 ) ¸2 ¡ h2 ¡ t2 + 3 h 2 : it fo llo ws , in p a r t ic u la r , t h a t m a x i fãui ; jz0ijg ¸ ¸2 ¡ h2 + t2 ¡ 3h 2 h + t ¡ 1 ¸ ¸2 ¡ 3 h 2 + 2 : if ¸2 · h ¡ 1 , t h e n c le a r ly c ¸ ( h + 1 ) ( ¸2 ¡ h + 1 ) . l e t ¸2 ¸ h ¸ t + 1 . a p p lyin g l e m m a 1 t o q = äv ã¡ t ( v ) v ¡! h v¡, we g e t c ¸ hx i=1 ãui + t¡1x i=1 jz0ij + m a x i fãui ; jz0ijg ¸ ( h + 1 ) ( ¸2 ¡ h + 1 ) + ( t ¡ 1 ) ( ¸2 ¡ t ¡ 1 ) ¸ ( h + 1 ) ( ¸2 ¡ h + 1 ) : case 2. ju ¤2 j ¸ 2 . ch o o s e w 2 u ¤2 ¡ v s u c h t h a t 'v ¸ 'w ¸ 'u fo r e a c h u 2 u ¤2 ¡ fv; wg. d e ¯ n e wi; z00i ; ©00i ( i = 1 ; :::; r ) fo r t ( w ) in t h e s a m e wa y a s vi; z 0 i a n d © 0 i we r e d e ¯ n e d fo r t ( v ) . a s in ( 4 ) , we c a n a s s u m e w.l.o .g . t h a t m a xi j©00i j = j©00rj = j©wj = 'w = r. cle a r ly, t+r = 'v +'w ¸ h+ 1 . th e n tx i=1 jz0ij + rx i=1 jz00i j = tx i=1 ( d( vi ) ¡ j©0ij) + rx i=1 ( d ( wi ) ¡ j©00i j ) = tx i=1 d ( vi ) + rx i=1 d( wi ) ¡ tx i=1 j©0ij ¡ rx i=1 j©00i j ¸ ( t + r ) ¸3 ¡ t2 ¡ r2; wh e r e ¸3 = 1 t + r ã tx i=1 d( vi ) + rx i=1 d( wi ) ! ¸ ¹h: in p a r t ic u la r , m a x i fjz0ij; jz00i jg ¸ ¸3 ¡ t2 + r2 t + r : a p p lyin g l e m m a 1 t o q = äv ã¡ t ( v ) v ¡! h w ¡! t ( w ) äw, we g e t c ¸ tx i=1 jz0ij + rx i=1 jz00i j + m a x i fjz0ij; jz00i jg ¸ ( t + r ) ¸3 ¡ t2 ¡ r2 + ¸3 ¡ t2 + r2 t + r 2 4 two generalized lower bounds for the circumference ¸ ( h + 1 ) ( ¸3 ¡ h + 1 ) + ¸3 ( t + r ¡ h) + h2 ¡ 1 ¡ t2 ¡ r2 ¡ t2 + r2 t + r : if ¸3 · h ¡ 1 , t h e n c le a r ly, c ¸ ( h + 1 ) ( ¸3 ¡ h + 1 ) . ot h e r wis e , c ¸ ( h + 1 ) ( ¸3 ¡ h + 1 ) + h( t + r ) ¡ 1 ¡ t2 ¡ r2 ¡ t2 + r2 t + r ¸ ( h + 1 ) ( ¸3 ¡ h + 1 ) + ( h ¡ 1 ) ( t + r ) ¡ t2 ¡ r2: ob s e r vin g t h a t h ¡ 1 ¸ m a xft; rg ¸ t 2 + r2 t + r ; we o b t a in c ¸ ( h+1 ) ( ¸3¡h+1 ) . th u s , c ¸ ( h+1 ) ( ¸¡h+1 ) , wh e r e ¸ = m in f¸1; ¸2; ¸3g ¸ ¹h. refer ences [1 ] j.a . b o n d y a n d u .s .r . mu r t y, gr a p h th e o r y wit h a p p lic a t io n s . ma c m illa n , l o n d o n a n d e ls e vie r , n e w y o r k, 1 9 7 6 . [2 ] zh .g. n iko g h o s ya n , \ a d va n c e d lo we r b o u n d s fo r t h e c ir c u m fe r e n c e " , gr a p h s a n d co m b in a t o r ic s 2 9 , p p . 1 5 3 1 -1 5 4 1 , 2 0 1 3 . submitted 22.08.2017, accepted 24.12.2017. ¶ñ³ýç ³ù»ý³»ñï³ñ óçïéç »ñï³ñáõãû³ý »ñïáõ áý¹ñ³ýñ³óí³í ·ý³ñ³ï³ï³ýý»ñ ø. øáõé³ù½û³ý ¨ ä. üçïáõáëû³ý ²ù÷á÷áõù 2013-çý »ñïñáñá¹ ñ»õçý³ïá g ·ñ³ýç ³ù»ý³»ñï³ñ c óçïéç »ñï³ñáõãû³ý ñ³ù³ñ ëï³ó³í »ñïáõ ëïáñçý ·ý³ñ³ï³ï³ýý»ñ` ³ñï³ñ³ûïí³í g ¡ c-ç ³ù»ý³»ñï³ñ ßõã³ûç »ñï³ñáõãû³ý (³ù»ý³»ñï³ñ óçïéç »ñï³ñáõãû³ý) ¨ g ·ñ³ýç ýí³½³·áõûý ³ëïç×³ýç µýáõã³·ñçãý»ñáí (zh.g. nikoghosyan, advanced lower bounds for the circumference, graphs and combinatorics 29, pp. 1531-1541, 2013): ü»ñï³ ³ßë³ï³ýùáõù ý»ñï³û³óíáõù »ý »ñïáõ ñ³ù³ýù³ý ·ý³ñ³ï³ï³ýý»ñ, áñï»õ ýí³½³·áõûý ³ëïç×³ýç µýáõã³·ñçãá ÷áë³ñçýí³í ¿ g¡c-ç ·³·³ãý»ñç ³é³ççý i ³ù»ý³÷áùñ ³ëïç×³ýý»ñç ùçççý ãí³µ³ý³ï³ýáí g ¡ c-áí å³ûù³ý³íáñí³í áñáß i å³ñ³ù»ïñç ñ³ù³ñ: m. koulakzian and zh. nikoghosyan 2 5 äâå îáîáùåííûå íèæíèå îöåíêè äëÿ äëèíû äëèííåéøåãî öèêëà ãðàôà ì. êóëàêçÿí è æ. íèêîãîñÿí àííîòàöèÿ â 2013 ãîäó âòîðîé àâòîð ïîëó÷èë äâå íèæíèå îöåíêè äëÿ äëèíû äëèííåéøåãî öèêëà ãðàôà g âûðàæåííûå ÷åðåç äëèíó äëèííåéøåé öåïè (äëèííåéøåãî öèêëà) ïîäãðàôà g ¡ c è ìèíèìàëüíóþ ñòåïåíü ãðàôà g (zh.g. nikoghosyan, advanced lower bounds for the circumference, graphs and combinatorics 29, pp. 1531-1541, 2013)? â íàñòîÿùåé ðàáîòå ïðåäñòàâëÿþòñÿ äâå îáîáùåííûå àíàëîãè÷íûå îöåíêè, ãäå âìåñòî ìèíèìàëüíîé ñòåïåíè ðàññìàòðèâàåòñÿ ñðåäíÿÿ àðèôìåòè÷åñêàÿ ñòåïåíåé ïåðâûõ i íàèìåíüøèõ ñòåïåíåé âåðøèí ïîäãðàôà g ¡ c äëÿ ïîäõîäÿùåãî ïàðàìåòðà i. d:\sbornik\...\article_eng.dvi mathematical problems of computer science 30, 25{30, 2008. on e xistence of cer tain locally-balanced 2-par tition of a t r ee s u r e n v . b a likya n yerevan state university email: suren.balikyan@gmail.com abstract a necessary and su±cient condition is obtained for the problem of such partitioning of the set of vertices of a tree g into two disjoint sets v1 and v2, such that for given sets v 01 µ v (g) and v 02 µ v (g) (v 01 \ v 02 = ;) it satis¯es the conditions v 01 µ v1, v 02 µ v2 and jj¸(v) \ v1j ¡ j¸(v) \ v2jj · 1 for any vertex v of g, where ¸(v) is the set of all vertices of g adjacent to v. refer ences [1 ] s .v . b a likya n , r .r . k a m a lia n , " on n p -c o m p le t e n e s s o f t h e p r o b le m o f e xis t e n c e o f lo c a lly-b a la n c e d 2 -p a r t it io n fo r b ip a r t it e g r a p h s g wit h ¢ ( g ) = 3 " , r eports of nas r a, applied m athematics, vo l. 1 0 5 , n u m . 1 , p p . 2 1 { 2 7 , 2 0 0 5 . [2 ] s .v . b a likya n , r .r . k a m a lia n , " on n p -c o m p le t e n e s s o f t h e p r o b le m o f e xis t e n c e o f lo c a lly-b a la n c e d 2 -p a r t it io n fo r b ip a r t it e g r a p h s g wit h ¢ ( g) = 4 u n d e r t h e e xt e n d e d d e ¯ n it io n o f t h e n e ig h b o u r h o o d o f a v e r t e x" , r eports of nas r a, applied m athematics, vo l. 1 0 6 , n u m . 3 , p p . 2 1 8 { 2 2 6 , 2 0 0 6 . [3 ] s .v . b a likya n , " on lo c a lly-b a la n c e d 2 -p a r t it io n s o f s o m e b ip a r t it e g r a p h s " , abstracts of papers of 15th international conference "m athe m atics. com p uting. e d ucation.", vo l. 1 5 , p . 7 , d u b n a , r u s s ia , ja n u a r y 2 8 fe b r u a r y 0 2 2 0 0 8 . [4 ] f. h a r a r y, graph theory, a d d is o n -w e s le y, r e a d in g , ma , 1 9 6 9 . [5 ] c. b e r g e , graphs and hypergraphs, e ls e vie r s c ie n c e l t d , 1 9 8 5 . ì³éç éáï³é-ñ³í³ë³ñ³ïßéí³í áñáß³ïç 2-ïñáñù³ý ·áûáõãû³ý ù³ëçý ê.ì. ´³éçïû³ý ²ù÷á÷áõù êï³óí³í ¿ ³ýññ³å»ßï ¨ µ³í³ñ³ñ å³ûù³ýª g í³éç ·³·³ãý»ñç µ³½ùáõãû³ý v1 ¨ v2 ãñ³ïíáõ »ýã³µ³½ùáõãûáõýý»ñç ³ûýåçëç ïñáñù³ý ·áûáõãûáõýá å³ñ½»éáõ ñ³ù³ñ, 2 5 2 6 on existence of certain locally-balanced 2-partition of a tree áñ ïñí³í v 01 µ v ( g) ¨ v 02 µ v ( g ) ( v 01 \ v 02 = ;) µ³½ùáõãûáõýý»ñç ñ³ù³ñ µ³í³ñ³ñí»ý ñ»ï¨û³é å³ûù³ýý»ñá. v 01 µ v1, v 02 µ v2 ¨ í³éç ûáõñ³ù³ýãûáõñ v ·³·³ãç ñ³ù³ñ ï»õç áõý»ý³ ñ»ï¨û³é ³ýñ³í³ë³ñáõãûáõýá jj (̧ v ) \ v1j ¡ j (̧ v ) \ v2jj ¸ 1 , áñï»õ ¸ ( v ) -áí ýß³ý³ïí³í ¿ v-çý ïçó ·³·³ãý»ñç µ³½ùáõãûáõýá: microsoft word w12.doc mathematical problems of computer science 37, 96--101, 2012. 96 modification of fse method based on coefficients of homogeneity gevorg karapetyan institute for informatics and automation problems of nas of ra e-mail: gevorgka@gmail.com abstract in this paper a modification of frequency selective extrapolation (fse) method is described. the fse method is used for concealment of missing blocks in digital images. the modification of the method is based on missing block support area analysis with usage of canny edge detection algorithm. in the result of the analysis, for each missing block a suboptimal support area size is selected, which is used in fse method. the paper includes experimental results which are compared with the results of frequency fse method. references: [1]. t. stockhammer and m. m. hannuksela, “h.264/avc video for wireless transmission,” ieee wireless communications, vol. 12, no. 4, pp. 6–13, 2005. [2]. kaup, k. meisinger and t. aach, “frequency selective signal extrapolation with applications to error concealment in image communication,” international journal of electronics and communications, vol. 59, no. 3, pp. 147–156, 2005. [3]. e. o. brigham and r. e. morrow; “the fast fourier transform” , ieee, vol. 4 issue:12, dec. 1967. [4]. b. jahne. digital image processing, springer–verlag berlin heidelberg, 2005. [5]. s.agaian, h.sarukhanyan, k.egiazarian and j.astola. hadamard transforms, spie press, 2011, ðà¾ ù»ãá¹ç í»ñ³÷áëáõùá ñ³ù³ë»éáõãû³ý ·áñí³ïçóý»ñç ñçù³ý íñ³ ¶. î³ñ³å»ïû³ý ²ù÷á÷áõù ²ûë ³ßë³ï³ýùáõù ùß³ïí³í ¿ ñ³×³խաï³ý áýïñ³ýù³ûçý ¿ùëïñ³åáéû³óç³ûç (ðà¾) ù»ãá¹ç ùç ýáñ ï³ñµ»ñ³ï: üß»ýù, áñ ðà¾ ù»ãá¹ý û·ï³·áñííáõù ¿ ãí³ûçý å³ïï»ñý»ñáõù µéáï³ûçý ïáñáõëïý»ñç ùáõ³ñïù³ý ýå³ï³ïáí: ì»ñççýçë ùá¹çýçï³óç³ý ñçùýí³í ¿ ³ç³ïóáõ ïçñáõûãç í»ñéáõíáõãû³ý íñ³՝ û·ï³·áñí»éáí î³ýçç ³é·áñçãùá: ì»ñéáõíáõãû³ý ³ñ¹ûáõýùáõù ûáõñ³ù³ýãûáõñ µéáïç ñ³ù³ñ áýïրíáõù ¿ ³ç³ïóáõ ïçñáõûãç »ýã³ûåïçù³é (suboptimal) ã³÷, áñý ³ûýáõñ»ï¨ û·ï³·áñííáõù ¿ ðà¾ ³é·áñçãùáõù: ²ßë³ï³ýùáõù µ»ñí³í »ý ðà¾ ù»ãá¹ç ýáñ g. karapetyan 97 ï³ñµ»ñ³ïç ÷áñó³ñïù³ý ³ñ¹ûáõýùý»ñá, çýãå»ë ý³¨ ï³ï³ñí³í ¿ ñ³ù»ù³ï³ï³ý í»ñéáõíáõãûáõý ðà¾ ù»ãá¹ç ³ñ¹ûáõýùý»ñç ñ»ï: модификация метода чвэ на базе коэффициентов однородности г. карапетян аннотация работа посвящена модификации метода частотной выборочной экстраполяции (чвэ). метод используется для реконструкции блочных потерь в цифровых изображениях. модификация метода чвэ основана на анализе вспомогательной области с использованием алгоритма кани. вследствие анализа для каждого блока выбирается субоптимальный размер вспомогательной области, который далее используется в алгоритме чвэ. в работе приведены результаты экспериментов, которые сравнены с результатами метода чвэ. 11_lilit_mariam.dvi mathematical problems of computer science 39, 88{93, 2013. i nvestigation of e-capacity for b iometr ic i denti¯cation p r otocol with random p ar ameter ¤ ma r ia m e . h a r o u t u n ia n a n d l ilit a . te r -v a r d a n ya n institute for informatics and automation problems of nas ra e-mail: armar@ipia.sci.am, lilit@sci.am abstract in recent years biometrics is widely used in di®erent tasks in the ¯eld of security. in this paper we investigated the biometric identi¯cation system from an informationtheoretical point of view. we investigate the exponentially high reliability criterion in biometric identi¯cation systems. the biometric identi¯cation system with random parameter is considered, which is more realistic for application. the lower and upper bounds of identi¯cation e-capacity of the model with random parameter for maximal and average error probabilities are constructed. when e ! 0 we derive the corresponding bounds of the capacity of the biometric identi¯cation system with random parameter, which coincide and hence, as a corollary we obtain the identi¯cation capacity for this model. keywords: biometric identi¯cation system, identi¯cation capacity, e-capacity bounds, error exponents, channel with random parameter. 1 . in t r o d u c t io n in r e c e n t ye a r s b io m e t r ic s is wid e ly u s e d in d i®e r e n t t a s ks in t h e ¯ e ld o f s e c u r it y. b io m e t r ic s is b e in g u s e d fo r p h ys ic a l a c c e s s c o n t r o l, c o m p u t e r lo g -in , in t e r n a t io n a l b o r d e r c r o s s in g a n d id c a r d s , e -p a s s p o r t s , b a s e d o n b io lo g ic a l fe a t u r e s o f a n y p e r s o n , s u c h a s ¯ n g e r p r in t s , a n e ye ir is [2 ]. in t h is p a p e r we c o n s id e r t h e p r o t o c o l o f b io m e t r ic id e n t i¯ c a t io n . th e id e n t i¯ c a t io n s ys t e m wit h r a n d o m p a r a m e t e r is g ive n in fig .1 . a t yp ic a l p r o t o c o l fo r id e n t i¯ c a t io n c o n s is t s o f t wo s t e p s : e n r o llm e n t a n d id e n t i¯ c a t io n . d u r in g t h e e n r o llm e n t , t h e b io m e t r ic d a t a o f m s u b je c t s a r e c a p t u r e d a n d a n a lyz e d , a ft e r t h a t , fo r e a c h in d ivid u a l, a r e c o r d is a d d e d t o a d a t a b a s e . a p e r fe c t s ys t e m wo u ld a lwa ys r e c o g n iz e a n in d ivid u a l a n d r e je c t a n im p o s t o r . to b u ild a n id e a l c h a n n e l is im p o s s ib le b e c a u s e b io m e t r ic d a t a a r e g a t h e r e d fr o m in d ivid u a ls u n d e r e n vir o n m e n t a l c o n d it io n s a n d t h e c h a n n e ls a r e e xp o s e d t o n o is e . s u c h a s ys t e m is n o t p e r fe c t ly s e c u r e , it le a d s t o s o m e e r r o r s , a n d it c a n b e in fo r m a t io n -t h e o r e t ic a l s e c u r e u p t o a c e r t a in le ve l. in t h is p a p e r we in ve s t ig a t e d t h e b io m e t r ic id e n t ī c a t io n s ys t e m fr o m a n in fo r m a t io n -t h e o r e t ic a l p o in t o f vie w. th e e n r o llm e n t -d a t a in a d a t a b a s e is a n o is y ve r s io n o f t h e b io m e t r ic a l d a t a c o r r e s p o n d in g t o t h e in d ivid u a l. ¤research was supported by armenian national grant 11-1b255. 8 8 m. haroutunian and l. ter-vardanyan 8 9 fig. 1. model of biometric identi¯cation system with random parameter in t h e id e n t ī c a t io n p h a s e a n u n kn o wn in d ivid u a l is o b s e r ve d a g a in a n d a n o t h e r n o is y ve r s io n o f t h e b io m e t r ic d a t a is c o m p a r e d t o t h e e n r o llm e n t d a t a in t h e d a t a b a s e . th e s ys t e m h a s t o c o m e u p wit h a n e s t im a t e o f t h e in d ivid u a l. w ille m s e t a l [1 ,3 ] in ve s t ig a t e d t h e fu n d a m e n t a l p r o p e r t ie s o f b io m e r t ic id e n t i¯ c a t io n s ys t e m . it h a s b e e n s h o wn t h a t it is im p o s s ib le t o r e lia b ly id e n t ify m o r e p e r s o n s t h a n c a p a c it y wh ic h is a n in h e r e n t c h a r a c t e r is t ic o f a n y id e n t i¯ c a t io n s ys t e m . th e y d e r ive d t h e c a p a c it y c o f t h is m o d e l. in [4 ] t h e e-c a p a c it y n e w c o n c e p t fo r b io m e t r ic a l id e n t ī c a t io n s ys t e m wa s in t r o d u c e d . w e in ve s t ig a t e d t h e e xp o n e n t ia lly h ig h r e lia b ilit y c r it e r io n in b io m e t r ic id e n t ī c a t io n s ys t e m s . in o t h e r wo r d s , we in t r o d u c e d a n e w p e r fo r m a n c e c o n c e p t o f b io m e t r ic id e n t i¯ c a t io n e-c a p a c it y, wh ic h t a ke s in t o a c c o u n t a s t r o n g e r r e qu ir e m e n t o n id e n t i¯ c a t io n fa u lt e ve n t s wit h e xt r e m e ly s m a ll p r o b a b ilit y ( 2 ¡ne in s t e a d o f " ) . in t e r m s o f p r a c t ic a l a p p lic a t io n s a n e xp o n e n t ia l d e c r e a s e in e r r o r p r o b a b ilit y ( n a m e ly, in u n wa n t e d id e n t i¯ c a t io n fa u lt s ) is m o r e d e s ir a b le . in p r a c t ic e t h e in d ivid u a ls a r e o b s e r ve d in va r io u s p la c e s a n d a t va r io u s t im e s . th a t is wh y it is m o r e in t e r e s t in g fr o m t h e p r a c t ic a l p o in t o f vie w t o a s s u m e t h a t t h e c o n s id e r e d m o d e l c h a n n e ls d e p e n d o n a r a n d o m p a r a m e t e r o f t h e b io m e t r ic m o d e l. in t h is p a p e r we in ve s t ig a t e t h e b io m e t r ic id e n t i¯ c a t io n s ys t e m wit h r a n d o m p a r a m e t e r , wh ic h is m o r e r e a lis t ic fo r a p p lic a t io n . th e c h a n n e l wit h r a n d o m p a r a m e t e r wit h a d d it io n a l in fo r m a t io n o n t h e e n c o d e r wa s ¯ r s t c o n s id e r e d b y s h a n n o n [5 ] a n d s t u d ie d b y ge lfa n d s . i. a n d p in s ke r m. s . [6 ]. th e y fo u n d t h e c a p a c it y o f t h is c h a n n e l fo r t h e a ve r a g e e r r o r p r o b a b ilit y c in a s it u a t io n wh e n t h e s t a t e s e qu e n c e is kn o wn a t t h e e n c o d e r . a h ls we d e r . f. [7 ] s h o we d t h a t t h e c a p a c it y fo r t h e m a xim u m e r r o r p r o b a b ilit y c is t h e s a m e . th e e-c a p a c it y fo r t h e c h a n n e l wit h r a n d o m p a r a m e t e r , e-c a p a c it y a n d c a p a c it y fo r t h e m u lt ip le -a c c e s s c h a n n e l wit h r a n d o m p a r a m e t e r wa s in ve s t ig a t e d b y h a r o u t u n ia n e . a ., h a r o u t u n ia n m. e . [8 ,1 0 ]. th e c h a n n e l c a n b e c o n s id e r e d in fo u r c a s e s , wh e n t h e s t a t e s e qu e n c e is kn o wn o r u n kn o wn a t t h e e n c o d e r a n d d e c o d e r . p r o c e e d in g fr o m t h e a p p lic a t io n s in t h e id e n t ī c a t io n p r o t o c o l, t h e s it u a t io n , wh e n t h e s t a t e s e qu e n c e is u n kn o wn a t t h e e n c o d e r a n d d e c o d e r , is p o s s ib le . in t h is p a p e r t h e lo we r a n d u p p e r b o u n d s o f t h e id e n t ī c a t io n e-c a p a c it y o f t h e m o d e l wit h r a n d o m p a r a m e t e r fo r m a xim a l a n d a ve r a g e e r r o r p r o b a b ilit ie s a r e c o n s t r u c t e d . w h e n 9 0 investigation of e-capacity for biometric identi¯cation protocol with random parameter e ! 0 we d e r ive t h e c o r r e s p o n d in g b o u n d s o f t h e c a p a c it y o f t h e b io m e t r ic id e n t i¯ c a t io n s ys t e m wit h r a n d o m p a r a m e t e r , wh ic h c o in c id e a n d h e n c e , a s a c o r o lla r y we o b t a in t h e id e n t ī c a t io n c a p a c it y fo r t h is m o d e l. 2 . n o t a t io n s a n d d e ¯ n it io n s l e t x ; y; z; s b e ¯ n it e s e t s a n d w b e a fa m ily o f d is c r e t e m e m o r yle s s c h a n n e ls ws : x ! y, wit h a n in p u t a lp h a b e t x a n d a n o u t p u t a lp h a b e t y. th e s is t h e c h a n n e l s t a t e , va r yin g in d e p e n d e n t ly a t e a c h m o m e n t o f t h e c h a n n e l a c t io n wit h t h e s a m e kn o wn p d q( s) o n s. th e r e a r e m in d ivid u a ls a n d e a c h in d ivid u a l h a s a n in d e x m = f 1 ; 2 ; ¢ ¢ ¢ ; mg. a b io m e t r ic d a t a s e qu e n c e x( m ) = fx1; x2; ¢ ¢ ¢ ; xn g, wh e r e xn 2 x ; n = 1 ; n c o r r e s p o n d s t o e a c h in d ivid u a l m . a ll t h e s e s e qu e n c e s a r e s u p o s e d t o b e g e n e r a t e d a t r a n d o m wit h a g ive n p r o b a b ilit y d is t r ib u t io n p n ( x ) = ny n=1 p ( xn ) ; x 2 x n : e nr ollment phase. l e t u s h a ve t h e s t a t io n a r y a n d d is c r e t e m e m o r yle s s c h a n n e l w ( yjx; s) wit h r a n d o m p a r a m e t e r . in t h is p h a s e a ll b io m e t r ic d a t a s e qu e n c e s x ( m) a r e o b s e r ve d via t h is c h a n n e l. th e s t a t e o f t h e c h a n n e l is c h a n g e d b y t h e fo llo win g p r o b a b ilit y d is t r ib u t io n q( s) , it m e a n s w n ( yjx; s) = ny n=1 w ( ynjxn; sn ) ; qn ( s) = ny n=1 q ( sn ) x 2 x n ; y 2 yn ; s 2 sn : th e r e s u lt in g y ( m ) e n r o llm e n t o u t p u t s e qu e n c e s fo r a ll m = f 1 ; 2 ; ¢ ¢ ¢ ; mg a r e s t o r e d in t h e d a t a b a s e ( d e ¯ n e it a s ydb ) . i denti¯cation phase. in t h e id e n t ī c a t io n p h a s e t h e b io m e t r ic d a t a s e qu e n c e o f a n u n kn o wn in d ivid u a l is o b s e r ve d via t h e s a m e m e m o r yle s s c h a n n e l w ( zjx; s) wit h r a n d o m p a r a m e t e r . w n ( zjx; s) = ny n=1 w ( ynjxn; sn ) ; z 2 zn ; x 2 x n ; s 2 sn : th e r e s u lt in g id e n t i¯ c a t io n o u t p u t s e qu e n c e z is c o m p a r e d t o t h e s e qu e n c e s y ( m ) , m = 1 ; 2 ; ¢ ¢ ¢ ; m, fr o m t h e d a t a b a s e a n d t h e id e n t i¯ c a t io n fu n c t io n gn : zn ! f 0 ; 1 ; 2 ; ¢ ¢ ¢ ; mg p r o d u c e s t h e in d e x o f t h e u n kn o wn in d ivid u a l m0 = gn ( z) ; h e r e 0 s t a n d s fo r t h e c a s e , wh e n t h e u n kn o wn in d ivid u a l h a s n o t b e e n o b s e r ve d b y t h e e n r o llm e n t p h a s e . if t h e s t a t e s e qu e n c e is u n kn o wn a t t h e e n c o d e r a n d d e c o d e r , le t u s d e n o t e w ¤ ( yjx ) = x s2s q( s) w ( yjx; s) ; w ¤ ( zjx) = x s2s q( s) w ( zjx; s) : a n d p ¤ = fp ¤ ( y ) = x x w ¤ ( yjx ) p ( x) ; x 2 x; y 2 y g; w ¤ ( zjy ) = p x w ¤ ( yjx ) w ¤ ( zjx ) p ( x ) p ¤ ( y ) : m. haroutunian and l. ter-vardanyan 9 1 th e c h a n n e l w ¤ : y ! z is m e m o r yle s s : w n ( zjy ) = ny n=1 w ( znjyn ) ; z 2 zn ; y 2 yn : d e n o t e b y r t h e fo llo win g r a t e r = 1 n lo g 2 m: th e e r r o r p r o b a b ilit y o f t h e id e n t i¯ c a t io n o f t h e p e r s o n m is e ( n; m ) = w n ( zn ng¡1n ( m) jy ( m) ) ; wh e r e g¡1n ( m) = fz : gn ( z ) = mg: w e c o n s id e r t h e maximal a n d t h e aver age er r or pr obabilities e( n ) = m a x m2m e( m ) ; e ( n ) = 1 m x m2m e( m ) : th e e-c a p a c it y fu n c t io n fo r t h e g ive n e > 0 is d e ¯ n e d a s c ( e; p ¤; w ¤ ) = lim n!1 1 n lo g m ( e; p ¤; w ¤; n ) ; wh e r e m ( e; p ¤; w ¤; n ) = s u p gn fm : eq ( n ) · e xp ( ¡n e ) g: w e d e n o t e b y c ( e; p ¤; w ¤ ) t h e e-c a p a c it y fo r t h e a ve r a g e e r r o r p r o b a b ilit y. w e s h a ll u s e t h e fo llo win g p d in t h e fo r m u la t io n o f r e s u lt s : p = fp ( y ) ; y 2 yg; v = fv ( zjy ) ; z 2 z; y 2 yg: fo r in fo r m a t io n -t h e o r e t ic qu a n t it ie s , s u c h a s e n t r o p y hp ( y ) , m u t u a l in fo r m a t io n ip;v ( z ^ y ) , t h e d ive r g e n c e d ( v jjwjp ) a n d fo r t h e n o t io n o f t h e t yp e we r e fe r t o [9 ]{ [1 5 ]. 3 . fo r m u la t io n o f t h e r e s u lt to d e ¯ n e t h e lo we r b o u n d ( random coding bound ) o f t h e id e n t ī c a t io n o f e-c a p a c it y fo r t h e c h a n n e l wit h r a n d o m p a r a m e t e r le t u s d e n o t e : rr ( e; p ¤; w ¤ ) 4 = 4 = m in p;v :d(p ±v jjp ¤±w ¤)·e ¯̄ ¯̄ip;v ( z ^ y ) + d ( p ± v jjp ¤ ± w ¤ ) ¡ e ¯̄ ¯̄ + : ( 1 ) fo r t h e fo r m u la t io n o f t h e u p p e r b o u n d ( sphere packing bound ) o f t h e in d e n t i¯ c a t io n o f ec a p a c it y le t u s in t r o d u c e t h e fo llo win g fu n c t io n : rsp ( e; p ¤; w ¤ ) 4 = m in p;v :d(p ±v jjp ¤±w ¤)·e ip;v ( z ^ y ) : ( 2 ) 9 2 investigation of e-capacity for biometric identi¯cation protocol with random parameter t heor em. f or the biometric identi¯cation system with random parameter for the given p ¤; w ¤ and for all e > 0 rr ( e; p ¤; w ¤ ) · c ( e; p ¤; w ¤ ) · c ( e; p ¤; w ¤ ) · rsp ( e; p ¤; w ¤ ) : th e p r o o f o f t h e t h e o r e m is s im ila r t o t h e p r o o f e xp o s e d in [4 ]. cor ollar y. w hen e ! 0 we derive the lower and upper bounds capacity of the channel with random parameter, which coincide and hence, we obtain the capacity c = ip ¤;w ¤ ( z ^ y ) : refer ences [1 ] f. w ille m s , t. k a lke r , j. go s e lin g , a n d j.-p . l in n a r t z , \ on t h e c a p a c it y o f a b io m e t r ic a l id e n t i¯ c a t io n s ys t e m " , international symposium on information theory, y o ko h a m a , ja p a n , p . 8 2 , 2 0 0 3 . 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[9 ] e . a . h a r o u t u n ia n , m. e . h a r o u t u n ia n a n d a . n . h a r u t yu n ya n , " r e lia b ilit y c r it e r ia in in fo r m a t io n t h e o r y a n d in s t a t is t ic a l h yp o t h e s is t e s t in g " , f oundations and trends in communications and information theory, vo l. 4 , n o 2 -3 , p p . 9 7 -2 6 3 , 2 0 0 8 . [1 0 ] m. e . h a r o u t u n ia n , \ e s t im a t e s o f e-c a p a c it y a n d c a p a c it y r e g io n s fo r m u lt ip le -a c c e s s c h a n n e l wit h r a n d o m p a r a m e t e r " , l ecture notes in computer science, vo l. 4 1 2 3 , s p r in g e r v e r la g , p p . 1 9 6 -2 1 7 , 2 0 0 6 . [1 1 ] e . a . h a r o u t u n ia n , " on b o u n d s fo r e-c a p a c it y o f d mc" , ie e e transactions on information theory, v 5 3 , n 1 1 , p p . 4 2 1 0 -4 2 2 0 , 2 0 0 7 . [1 2 ] m. e . h a r o u t u n ia n , s . a . to n o ya n , \ r a n d o m c o d in g b o u n d o f in fo r m a t io n h id in g ec a p a c it y" , p roc. of ie e e international symposium on information theory, p . 5 3 6 , u s a , ch ic a g o , 2 0 0 4 . [1 3 ] t. m. co ve r a n d j. a . th o m a s , e lements of information theory, w ile y, n e w y o r k, 1 9 9 1 . m. haroutunian and l. ter-vardanyan 9 3 [1 4 ] i. cs is z ¶a r a n d j. k äo r n e r , information theory: coding theorems for d iscrete m emoryless systems, a c a d e m ic p r e s s , n e w y o r k, 1 9 8 1 . [1 5 ] i. cs is z ¶a r , \ th e m e t h o d o f t yp e s " , ie e e transactions on information theory, vo l. 4 4 , n o . 6 , p p . 2 5 0 5 -2 5 2 3 , 1 9 9 8 . submitted 22.11.2012, accepted 14.02.2013. e-áõý³ïáõãû³ý áõëáõùý³ëçñáõãûáõýá å³ï³ñ³ï³ý å³ñ³ù»ïñáí ï»ýë³ã³÷³ï³ý ýáõûý³ï³ý³óù³ý ³ñó³ý³·ñáõãû³ý ñ³ù³ñ ø. ð³ñáõãûáõýû³ý ¨ è. î»ñ-ì³ñ¹³ýû³ý ²ù÷á÷áõù ì»ñççý ï³ñçý»ñçý ï»ýë³ã³÷áõãûáõýá é³ûýáñ»ý ïçñ³éíáõù ¿ ³ýíï³ý·áõãû³ý áéáñïç ï³ñµ»ñ ëý¹çñý»ñáõù: ²ßë³ï³ýùáõù ñ»ï³½áïí»é ¿ ï»ýë³ã³÷³ï³ý ýáõûý³ï³ý³óù³ý ñ³ù³ï³ñ·á‘ çýýáñù³óçáý-ï»ë³ï³ý ï»ë³ýïûáõýçó: ø»ýù ñ»ï³½áïáõù »ýù óáõóã³ûçý µ³ñóñ ñáõë³éçáõãû³ý ã³÷³ýçßá‘ ï»ýë³ã³÷³ï³ý ýáõûý³ï³ý³óù³ý ñ³ù³ï³ñ·»ñáõù: ¸çï³ñïí³í ¿ å³ï³ñ³ï³ý å³ñ³ù»ïñáí ï»ýë³ã³÷³ï³ý ýáõûý³ï³ý³óù³ý ñ³ù³ï³ñ·á, áñá ïçñ³éáõãûáõýý»ñç ï»ë³ï»ïçó ³í»éç çñ³ï»ë³ï³ý ¿: î³éáõóí³í »ý eáõý³ïáõãû³ý í»ñçý ¨ ëïáñçý ·ý³ñ³ï³ï³ýý»ñá ³é³í»é³·áõûý ¨ ùçççý ëë³éç ñ³í³ý³ï³ýáõãûáõýý»ñç ¹»åùáõù å³ï³ñ³ï³ý å³ñ³ù»ïñáí ùá¹»éç ñ³ù³ñ: ºñµ e ! 0 , ëï³ýáõù »ýù å³ï³ñ³ï³ý å³ñ³ù»ïñáí ï»ýë³ã³÷³ï³ý ýáõûý³ï³ý³óù³ý ñ³ù³ï³ñ·ç áõý³ïáõãû³ý ñ³ù³å³ï³ëë³ý ·ý³ñ³ï³ï³ýý»ñá, áñáýù ñ³ùáýïýáõù »ý ¨, áñå»ë ñ»ï¨³ýù, ëï³ýáõù »ýù ³û¹ ùá¹»éç ýáõûý³ï³ý³óù³ý áõý³ïáõãûáõýá: èññëåäîâàíèå e-ïðîïóñêíîé ñïîñîáíîñòè äëÿ ïðîòîêîëà áèîìåòðè÷åñêîé èäåíòèôèêàöèè ñî ñëó÷àéíûì ïàðàìåòðîì ì. àðóòþíÿí è ë. òåð-âàðäàíÿí àííîòàöèÿ çà ïîñëåäíèå ãîäû áèîìåòðèêà øèðîêî èñïîëüçóåòñÿ â ðàçëè÷íûõ çàäà÷àõ â ñôåðå áåçîïàñíîñòè. â äàííîé ðàáîòå ìû èññëåäîâàëè ñèñòåìó áèîìåòðè÷åñêîé èäåíòèôèêàöèè ñ èíôîðìàöèîííî-òåîðåòè÷åñêîé òî÷êè çðåíèÿ. ìû èññëåäóåì êðèòåðèé ýêñïîíåíöèàëüíî âûñîêîé íàäåæíîñòè â áèîìåòðè÷åñêèõ ñèñòåìàõ èäåíòèôèêàöèè. ðàññìîòðåíà ñèñòåìà áèîìåòðè÷åñêîé èäåíòèôèêàöèè ñî ñëó÷àéíûì ïàðàìåòðîì, êîòîðàÿ áîëåå ðåàëèñòè÷íà ñ òî÷êè çðåíèÿ ïðèëîæåíèé. ïîñòðîåíû íèæíÿÿ è âåðõíÿÿ ãðàíèöû äëÿ e-ïðîïóñêíîé ñïîñîáíîñòè ìîäåëè ñî ñëó÷àéíûì ïàðàìåòðîì äëÿ ìàêñèìàëüíîé è ñðåäíåé âåðîÿòíîñòè îøèáêè. êîãäà e ! 0 ìû ïîëó÷àåì ñîîòâåòñòâóþùèå îöåíêè ïðîïóñêíîé ñïîñîáíîñòè äëÿ ñèñòåìû áèîìåòðè÷åñêîé èäåíòèôèêàöèè ñî ñëó÷àéíûì ïàðàìåòðîì, êîòîðûå ñîâïàäàþò è, êàê ñëåäñòâèå, ìû ïîëó÷àåì ïðîïóñêíóþ ñïîñîáíîñòü èäåíòèôèêàöèè äëÿ ýòîé ìîäåëè. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 47, 50--61, 2017. information-theoretic approach to community detection problem mariam e. haroutunian and karen k. mkhitaryan institute for informatics and automation problems of nas ra e-mail: armar@ipia.sci.am, karenmkhitaryan@gmail.com abstract real world complex networks possess hidden information called communities or clusters, which are composed of nodes that are tightly connected within communities and weakly connected between communities. investigation of communities proved to have countless applications in different sciences such as computer science and machine learning, biology, economics, and social networks. parallel to the development of various detection algorithms, probabilistic network models also gained more attention, particularly stochastic block model which is a generative model for random graphs generating networks with community structure. this paper explores the state of the art on the connections of stochastic block model with information theory. keywords: community detection, stochastic block model, network theory, clustering, information theory. 1. introduction in recent times, the computer revolution has provided specialists with massive data and sufficient computational resources to process and analyze these data. the size of real networks has also grown considerably, reaching millions or even billions of vertices and edges. the need to deal with such a large number of units has produced a deep change in the way graphs are approached. in a random graph, the distribution of edges among the vertices is highly homogeneous. real networks are not random graphs, the distribution of edges is not only globally, but also locally inhomogeneous, with high concentrations of edges within special groups of vertices, and low concentrations between these groups. this feature of real networks is called a community structure [1, 2]. the aim of community detection in graphs is to identify the modules and, possibly, their hierarchical organization, by only using the information encoded in the graph topology. network is a collection of entities called nodes or vertices which are connected through edges or links. complex network is a group of interacting entities with some nontrivial dynamical 50 mailto:armar@ipia.sci.am mailto:karenmkhitaryan@gmail.com m. haroutunian and k. mkhitaryan 51 behavior. there are many types of complex networks such as social networks, technological networks, informational networks, and biological networks. the study of complex networks is a young area of scientific research stimulated largely by the study of real-world networks like computer networks and social networks. identifying graph communities is a popular topic in computer science. in parallel computing, for instance, it is critical to know what the best way is to allocate tasks to processors so as to minimize the communications between them and enable a fast performance of the calculation. this can be accomplished by splitting the computer cluster into groups with roughly the same number of processors, so that the number of physical connections between processors of different groups is minimal. the mathematical formalization of this problem is called graph partitioning. the basic goal of community detection is similar to that of a graph partition: we want to separate the network into groups of vertices that have few connections between them. the main difference is that the size of the groups is not fixed. one of the most important features representing real networks is the community structure, i.e. distribution of vertices in communities with higher internal connectivity (nodes joined by edges inside the community) than external connectivity (nodes joined by edges between communities). detecting communities in networks proved to have many applications in various fields of science such as in protein-to-protein interactions from biology, recommendation systems from online product purchasing, social network analysis from network science, problems related to big data, etc. although it was introduced long ago and developed for decades it is hard to evaluate detection algorithms as network types may vary. this is the reason that this field still has many open problems. community detection is one of the central problems in network and data sciences. recent research has focused on developing community detection methods using various approaches. a recent review of existing approaches can be found in [3]. evaluating the performance of algorithms on models is non-trivial. in some cases, most algorithms may succeed, while in others, algorithms may fail due to computational barriers. thus, an important question is to characterize the situations where the clustering tasks can be solved efficiently or information-theoretically. in particular, models may benefit from asymptotic phase transition phenomena, which, in addition to being mathematically interesting, allow the location of hard cases to benchmark algorithms. probabilistic network models can be used to model real networks, to study the average-case complexity of np-hard problems on graphs, or to set benchmarks for clustering algorithms with well-defined ground truth. the latter is needed to show how the model fits the data sets, and is very important in community detection as a vast majority of algorithms are based on heuristics and no ground truth is available in applications. this is, in particular, a well-known challenge for big data problems where one cannot manually determine the quality of the clusters. parallel to community detection, probabilistic network models were also theoretically developed. the stochastic block model is one of the most popular network models exhibiting community structures. the model was first proposed in the 80s [4] and received significant attention in the mathematics and computer science literature, as well as in the statistics and machine learning literature [5 -9]. theoretical investigation brought new insights about connections of stochastic block model with information theory where decoding an information sent through channel is considered somewhat similar to community detection [5, 10, 11]. in this paper we focus on the connections between information theory and community detection. in the next section we consider some community detection algorithms. we focus on the methods based on statistical inference, mainly on the stochastic block model in the section 3.we define the weak, partial and exact recovery requirements. the last section analyzes the role of information-theoretic approach to community detection problem 52 information theory in the investigation of open problems, such as fundamental limits, comparing partitions, etc. 2. community detection communities or clusters are groups of vertices that play an important role in the network. in real world, networks can be towns, friendship circles, virtual groups on the web, etc. revealing communities and investigating how particular communities behave is an attractive problem resulting in countless practical applications. solving community detection problems on modern real world network datasets can sometimes be very much complicated because of computational complexity as graphs may contain billions of nodes and edges and complexity of exact detection is np-hard. even nowadays distinguishing between algorithms and questioning the fact which algorithm works best for specific network is impossible. moreover, there are several types of networks that make the process more challenging. these types of networks include directed and weighted networks, including those which may have overlapping communities. detection of such community structures in complex networks is not an easy task. in recent years many community detection algorithms are developed. although it's still tricky to distinguish between "good" and "bad" algorithms as one algorithm working well on one network can fail for another, there are traditional approaches used broadly. algorithms are classified into different categories. some of them try to maximize the given quality function such as modularity, some hierarchical clustering methods introduce a similarity measure such as cosine similarity and group similar nodes into communities, etc. in this section we give a short description of some of them. we are interested in the investigation of the stochastic block model, which is from the list of methods based on statistical inference. some advanced methods include dynamic algorithms, methods to find overlapping communities, multi-resolution methods and cluster hierarchy. these topics are not included in this paper. modularity optimization modularity is a community quality measure which measures the fraction of the edges in the network that connect vertices of the same type minus the expected value of the same quantity in a random network where community divisions are the same but connections between vertices are random. particular algorithms that maximize the modularity for finding the community structure in networks include the louvain and infomap algorithms, as well as the fast greedy modularity optimization algorithm. m. haroutunian and k. mkhitaryan 53 fig. 1. in the left stochastic block model is given with 60 vertices, within communities and between communities edge probabilities 0.9 and 0.3, respectively. in the right louvain algorithm is implemented partitioning the network into four communities with a modularity score of 0.23. minimum-cut method one of the commonly used algorithms is the minimum-cut method, which tries to find communities with a predefined size by separating groups of nodes that have minimum inter group connectivity. some of the popular algorithms related to minimum cut method are the kernighanlin and stoer-wagner algorithms. kernighan-lin algorithm inputs the undirected graph 𝐺𝐺 = (𝑉𝑉, 𝐸𝐸) and partitions the vertex set 𝑉𝑉 into two disjoint subsets 𝐴𝐴 and 𝐵𝐵 of equal size in a way that minimizes the number of edges crossing between 𝐴𝐴 and 𝐵𝐵. the algorithm works also for weighted graphs, and the task becomes to minimize the sum of edge weights. this algorithm is used in the layout of digital circuits where minimum connections are vital for increasing performance. for undirected graphs the stoer-wagner algorithm is used to calculate the minimum cut. the idea of the algorithm is to recursively merge the vertices which are tightly connected until the graph contains two vertex sets. after each step the weight of cut is listed and at the end the minimum cut will be the minimum of the graph. hierarchical clustering hierarchical clustering displays different levels of grouping vertices. the algorithms of hierarchical clustering are renowned for their applications in real world networks such as social network analysis, biology, engineering, etc., as these networks probably have a hierarchical structure. note that communities in graphs may not have a hierarchical structure at all but with its weaknesses it's still one of the popular methods for community detection. hierarchical clustering starts with the definition of similarity measure of nodes. after calculating similarities between each pair of nodes in the graph one will end up with the similarity matrix and group nodes in communities. hierarchical clustering algorithms are divided into agglomerative algorithms in which clusters are iteratively merged if their similarity is high, and divisive algorithms where clusters are iteratively split by removing edges connecting vertices with low similarity. information-theoretic approach to community detection problem 54 girvan-newman algorithm another popular method used for community detection and clustering is the girvan-newman algorithm. the algorithm is implemented by calculating betweenness values of all edges then the edge with the highest betweenness score is removed and after the iterative process when no edges remain the original network is split into communities. fig. 2. in the left the stochastic block model is given with 60 vertices, within communities and between communities edge probabilities 0.9 and 0.2, respectively. in the right girvan-newman algorithm is implemented partitioning the network into four communities with a modularity score of 0.488 3. stochastic block model block modeling is a common approach in statistics and social network analysis to decompose a graph in classes of vertices with common properties. in this way, a simpler description of the graph is attained. vertices are usually grouped in classes of equivalence. there are two main definitions of topological equivalence for vertices: structural equivalence in which vertices are equivalent if they have the same neighbors; regular equivalence, in which vertices of a class have similar connection patterns to vertices of the other classes. regular equivalence is a more general concept than structural equivalence. indeed, vertices which are structurally equivalent are also regularly equivalent, but the inverse is not true. the concept of structural equivalence can be generalized to probabilistic models, in which one compares classes of graphs, not single graphs, characterized by a set of linking probabilities between the vertices. in this case, vertices are organized in classes in such a way that the linking probabilities of a vertex with all other vertices of the graph are the same for vertices in the same class, which are called stochastically equivalent [4]. the model appeared independently in multiple scientific communities: the terminology stochastic block model comes from the machine learning and statistics literature, while the model is called a planted partition model in theoretical computer science, and an inhomogeneous random graphs model in the mathematics literature. the stochastic block model has recently come back to the center of attention at both the practical level, due to extensions allowing overlapping communities that have proved to fit well real data sets in massive networks, and at the theoretical level due to new phase transition phenomena discovered for the two-community case. the goal of community detection is to recover communities up to some level of accuracy. 1. weak recovery (also called detection). this only requires the algorithm to output a partition of the nodes which is positively correlated with the true partition. m. haroutunian and k. mkhitaryan 55 2. partial recovery. one may ask the question of how much can be recovered about the communities. 3. exact recovery (also called recovery or strong consistency.) finally, one may ask for the regimes for which an algorithm can recover the entire clusters. one can also study partial-exact-recovery, namely which communities can be exactly recovered. the investigation of these levels is interesting not only from the mathematical point of view, but they are also relevant for applications. definition 1 ([6]): let a positive integer 𝑛𝑛 indicates the number of vertices, 𝑚𝑚 be the number of communities, 𝑝𝑝 = (𝑝𝑝1, 𝑝𝑝2 … , 𝑝𝑝𝑚𝑚) be the probability vector on {1, … , 𝑚𝑚} and 𝑃𝑃 be a symmetric 𝑚𝑚 x 𝑚𝑚 matrix of connectivity probabilities. stochastic block model 𝑆𝑆𝐵𝐵𝑆𝑆(𝑛𝑛, 𝑝𝑝, 𝑃𝑃) is defined by the pair (𝑋𝑋, 𝐺𝐺), where 𝑋𝑋 is an 𝑛𝑛-dimensional random vector with components valued in {1, … , 𝑚𝑚} in proportions 𝑝𝑝 and 𝐺𝐺 is an 𝑛𝑛-vertex undirected graph where vertices 𝑖𝑖 and 𝑗𝑗 are connected with probability 𝑃𝑃𝑋𝑋𝑖𝑖,𝑋𝑋𝑗𝑗 independently of other pairs of vertices. fig. 3. stochastic block matrix (left) of stochastic block model (right) is given, where probabilities of edges between communities is 𝑃𝑃𝑖𝑖,𝑗𝑗 and inside communities 𝑃𝑃𝑖𝑖,𝑖𝑖 particularly if all 𝑃𝑃𝑖𝑖,𝑗𝑗 elements of matrix 𝑃𝑃 are the same, the model is equivalent to erdos-renyi random graph model which does not have a community structure. planted partition model is a special case where diagonal elements of matrix 𝑃𝑃 are constant 𝑝𝑝 and off diagonal elements are constant 𝑞𝑞. if 𝑝𝑝 > 𝑞𝑞 the model is called assortative and if 𝑝𝑝 < 𝑞𝑞 dissortative. definition 2 ([6]): an algorithm detects communities with accuracy 𝛼𝛼 ∈ [0,1], if it takes 𝐺𝐺 drawn from 𝑆𝑆𝐵𝐵𝑆𝑆(𝑛𝑛, 𝑝𝑝, 𝑃𝑃) and outputs a reconstruction 𝑋𝑋՛ of 𝑋𝑋 that has agreement 𝛼𝛼 with probability 1 − 𝑜𝑜𝑛𝑛(1). definition 3 ([6]): in stochastic block model exact recovery is possible, if there exists an algorithm with accuracy α = 1. strong recovery is possible, if there exists an algorithm with accuracy α = 1 − on(1). weak recovery is possible, if the algorithm detects communities with accuracy α = 1 k + ε for some ε > 0. communities 𝑪𝑪𝟏𝟏 𝑪𝑪𝟐𝟐 𝑪𝑪𝟑𝟑 𝑪𝑪𝟒𝟒 𝑪𝑪𝟏𝟏 𝑃𝑃11 𝑃𝑃21 𝑃𝑃31 𝑃𝑃41 𝑪𝑪𝟐𝟐 𝑃𝑃21 𝑃𝑃22 𝑃𝑃32 𝑃𝑃42 𝑪𝑪𝟑𝟑 𝑃𝑃31 𝑃𝑃23 𝑃𝑃33 𝑃𝑃43 𝑪𝑪𝟒𝟒 𝑃𝑃41 𝑃𝑃24 𝑃𝑃34 𝑃𝑃44 1c 4c 3c 2c 13p 14p 34p 23p 12p 22p 33p 44p 11p 24p information-theoretic approach to community detection problem 56 fig. 4. in the left erdos-renyi random graph with 40 vertices is shown, where probability of edges between any pairs is constant. in the right , stochastic block model is given where probability of edges inside the communities is greater than probabilities of edges between communities majority of work in this field is done by establishing fundamental thresholds illustrating when it is possible to detect communities. 4. the role of information theory while the sphere of community detection is developing for many years with the construction of various algorithms to solve detecting tasks, the biggest portion of it is still unsolved. main questions are how accurately a particular algorithm detects communities or which communities can be revealed. these issues need thorough analysis. the information theory plays an important role in solving different problems. community detection has natural connections with information theory at various levels. theory behind stochastic block model is similar to graph-based codes, fdivergences, broadcasting problems on trees which are renowned topics from information theory. fig. 5. an encoder sends a compressed information to a decoder about the topology of the graph on the left. the information gives a coarse description of the graph, which is used by the decoder to deduce the original graph structure. figure reproduced from [3, 12]. the modular structure of a graph can be considered as a compressed description of the graph to approximate the whole information contained in its matrix. based on this idea, rosvall and bergstrom [12] envisioned a communication process in which a partition of a graph in m. haroutunian and k. mkhitaryan 57 communities represents a synthesis y of the full structure that a signaler sends to a receiver, which tries to infer the original graph topology x from it. the best partition corresponds to the signal y that contains the most information about 𝑋𝑋. this can be quantitatively assessed by the minimization of the conditional entropy 𝐻𝐻(𝑋𝑋|𝑌𝑌) of 𝑋𝑋 given by 𝑌𝑌. one has to look for the ideal tradeoff between a good compression and a small enough conditional entropy 𝐻𝐻(𝑋𝑋|𝑌𝑌). rosvall and bergstrom used the same idea introducing an information-theoretic flow-based method to examine the multipartite organization of biological and social systems [13]. the method reveals the community structure in weighted and directed graphs. they used the probability flow of random walks on a network as a proxy for information flows and by compressing the description of the probability flow they decomposed the network into communities. the idea is in expressing the shannon entropy of the random walk within and between clusters. if clusters are well separated from each other, transitions of the random walker between clusters will be infrequent, so it is advantageous to use the map, with the clusters as regions, because in the description of the random walk the codewords of the clusters will not be repeated many times, while there is a considerable saving in the description due to the limited length of the codewords used to denote the vertices. instead, if there are no well-defined clusters and/or if the partition is not representative of the actual community structure of the graph, transitions between the clusters of the partition will be very frequent and there will be little or no gain by using the two-level description of the map. information theory has also been used to detect communities in graphs. ziv et al. [14] have designed a method in which the information contained in the graph topology is compressed in such a way as to preserve some predefined information. as a criterion the mutual information 𝐼𝐼(𝑋𝑋; 𝑌𝑌) of two random variables 𝑋𝑋 and 𝑌𝑌 is used [15]. if 𝑋𝑋 is the input variable, 𝑍𝑍 is the variable specifying the partition and 𝑌𝑌 is the variable encoding the information we want to keep relevant variable, the goal is to minimize the mutual information between 𝑋𝑋 and 𝑍𝑍 (to achieve the largest possible data compression), under the constraint that the information on 𝑌𝑌 extractable from 𝑍𝑍 be accurate. the optimal tradeoff between the values of 𝐼𝐼(𝑋𝑋; 𝑍𝑍) and 𝐼𝐼(𝑌𝑌; 𝑍𝑍) (i. e., compression versus accuracy) is expressed by the minimization of a functional. in the case of graph clustering, the question is what to choose as a relevant information variable. ziv et al. proposed to adopt the structural information encoded in the process of diffusion on the graph. information theory is also useful when comparing different partitions of the network. when a particular algorithm is implemented, to assess the quality of the partition, it must be compared with other partitions or with available ground truth. this can be done using several evaluation measures. most similarity measures can be divided into three categories: measures based on pair counting, cluster matching and information theory. the third class of similarity measures is based on reformulating the problem of comparing partitions as a problem of message decoding within the framework of information theory [15]. the idea is that, if two partitions are similar, one needs very little information to infer one partition given by the other. this extra information can be used as a measure of dissimilarity. the mutual information is not ideal as a similarity measure: in fact, given a partition 𝑋𝑋, all partitions derived from 𝑋𝑋 by further partitioning (some of) its clusters would all have the same mutual information with 𝑋𝑋, even though they could be very different from each other. in this case the mutual information would simply equal the entropy 𝐻𝐻(𝑋𝑋), because the conditional entropy would systematically be zero. to avoid that, danon et al. [16] adopted the normalized mutual information 𝐼𝐼𝑛𝑛𝑛𝑛𝑛𝑛𝑚𝑚(𝑋𝑋; 𝑌𝑌) = 2𝐼𝐼(𝑋𝑋; 𝑌𝑌) 𝐻𝐻(𝑋𝑋) + 𝐻𝐻(𝑌𝑌) , information-theoretic approach to community detection problem 58 which is currently very often used in tests of graph clustering algorithms. the normalized mutual information equals 1 if the partitions are identical, whereas it has an expected value of 0 if the partitions are independent. meila [17] introduced the variation of information 𝑉𝑉(𝑋𝑋; 𝑌𝑌) = 𝐻𝐻(𝑋𝑋|𝑌𝑌) + 𝐻𝐻(𝑌𝑌|𝑋𝑋) which has some desirable properties with respect to the normalized mutual information and other measures. in particular, it defines a metric in the space of partitions as it has the properties of distance. it is also a local measure, i.e., the similarity of partitions differing only in a small portion of a graph depends on the differences of the clusters in that region, and not on the partition of the rest of the graph. the maximum value of the variation of information is log n, so the similarity values for partitions of graphs with different sizes cannot be compared with each other. for meaningful comparisons one could divide 𝑉𝑉(𝑋𝑋; 𝑌𝑌) by log n, as suggested by karrer et al. [18]. in recent years fundamental limits were obtained using the new f-divergence function, which is called the ch-divergence in [5] as it generalizes both the chernoff and hellinger divergences. the definite characterization of the recovery threshold in the general stochastic block models provides an operational meaning to a divergence function analog to the kullback – leibler divergence (kl-divergence) in the channel coding theorem. at a high level, clustering the stochastic block model is similar to reliably decoding a codeword on a channel which is non-conventional in information theory. the channel inputs are the nodes' community assignments and the channel outputs are the network edges. it is shown [6] that this analogy is more than just high-level: reliable communication on this channel is equivalent to exact recovery, shows that the “clustering capacity" is obtained from the ch-divergence of channel-kernel pq, which is an f-divergence like the kl-divergence governing the communication capacity. interestingly recovering communities in stochastic block model have parallels with sending a codeword through discrete memoryless channel in coding theory. more generally, community detection pairs well with information theory as it can be viewed as a decoding problem on a noisy channel: the community labels are the input to a black-box channel that provides local and noisy interactions of the inputs. this view point was further developed in [19], with the notion of graphical channels. definition 4 ([6]): let 𝑉𝑉 = {1, … , 𝑛𝑛} and 𝐺𝐺 = �𝑉𝑉, 𝐸𝐸(𝐺𝐺)� be a hypergraph with 𝑁𝑁 = |𝐸𝐸(𝐺𝐺)|. let 𝒳𝒳 and 𝒴𝒴 be two finite sets called the input and output alphabets respectively and 𝑄𝑄 be a channel from 𝒳𝒳𝑘𝑘 to 𝒴𝒴 called the kernel. to each vertex in 𝑉𝑉, assign a vertex variable in 𝒳𝒳, and to each edge in 𝐸𝐸(𝐺𝐺), assign an edge variable in 𝒴𝒴. let 𝑦𝑦𝐼𝐼 denote the edge variable attached to edge 𝐼𝐼, and 𝑥𝑥[𝐼𝐼] denote the 𝑘𝑘 node variable adjacent to 𝐼𝐼. a graphical channel with graph 𝐺𝐺 and kernel 𝑄𝑄 is defined as the channel 𝑃𝑃 given by 𝑃𝑃(𝑦𝑦|𝑥𝑥) = ∏ 𝑄𝑄(𝑦𝑦𝐼𝐼|𝑥𝑥[𝐼𝐼])𝐼𝐼∈𝐸𝐸(𝐺𝐺) , where 𝑥𝑥 ∈ 𝒳𝒳𝑉𝑉,𝑦𝑦 ∈ 𝒴𝒴𝐸𝐸(𝐺𝐺). figure reproduced from [6] definition 4 strongly connects community detection in stochastic block model and information sent through channel in coding theory. m. haroutunian and k. mkhitaryan 59 figure reproduced from [5] reliable communication through noisy channel is possible if 𝑅𝑅 < 1 − 𝐻𝐻(𝜀𝜀) and this is close to exact recovery of communities in stochastic block model. community detection has a strong connection with information theory also because x is typically discrete. 5. conclusion. recent analysis of community detection problems and stochastic block model proved to have a strong perspective with information theory. we hope that our experience in solving coding problems for various noisy channels [20] will lead us to new results by investigating both community detection problems and stochastic block model by applying information-theoretical approach. references [1] m. newman, “networks: an introduction”, oxford university press, oxford, 2010. [2] m. girvan and m. newman, "finding and evaluating community structure in networks", physical review e, vol. 69, 026113, 2004. [3] s. fortunato, "community detection in graphs", physics reports 486, pp. 75-174, 2010. [4] p. holland, k. laskey and s. leinhardt, "stochastic block models: first steps ", social networks, vol. 5, no. 2, pp. 109-137, 1983. [5] e. abbe and c. sandon, "community detection in general stochastic block models: fundamental limits and efficient recovery algorithms", ieee 56th annual symp. on foundations of computer science, usa, pp. 670 – 688, 2015. [6] e. abbe, "community detection and the stochastic block model", ieee information theory society newsletter, vol. 66, no. 1, pp. 312, 2016. [7] e. abbe and c. sandon, "recovering communities in the general stochastic block model without knowing the parameters”, advances in neural information processing systems, vol. 28, pp. 676 – 684, 2015. [8] a. decelle, f. krzakala, c. moore and l. zdeborova, "asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications", physical review e, vol. 84, 066106, 2011. information-theoretic approach to community detection problem 60 [9] v. kanade, e. mossel and t. schramm, “global and local information in clustering labeled block models”, ieee transactions on information theory, vol. 62, no. 10, pp. 5906– 5917, 2016. [10] e. abbe and m. wainwright, “information theory meets machine learning”, tutorial, intern. symp. on information theory, 2015. [11] e. abbe, a. bandeira and g. hall, "exact recovery in the stochastic block model", ieee transactions on information theory, vol. 62, no. 1, pp. 471 – 487, 2016. [12] m. rosvall and c. t. bergstrom, “an information-theoretic framework for resolving community structure in complex networks”, proc. natl. acad. sci. usa, vol. 104, no. 18, pp. 7327 – 7331, 2007. [13] m. rosvall and c. t. bergstrom, “maps of random walks on complex networks reveal community structure”, proc. natl. acad. sci. usa, vol. 105, no. 4, pp. 1118 1123, 2008. [14] e. ziv, m. middendorf and c. h. wiggins, “information theoretic approach to network modularity”, physical review e, vol. 71, no. 4, 046117, 2005. [15] d. j. c. mackay, “information theory, inference and learning algorithms”, cambridge university press, fourth printing, 2005. [16] l. danon, a. diaz-guilera, j. duch and a. arenas, “comparing community stucture identification”, journal of statistical mechanics: theory and experiment, p09008, 2005. [17] m. meila, “comparing clusterings – an information based distance”, journal of multivar. anal., vol. 98, no. 5, pp. 873 – 895, 2007. [18] b. karrer, e. levina and m. newman, “robustness of community structure in networks”, physical review e, vol. 77, 046119, 2008. submitted 08.10.2016, accepted 20.02.2017. ինֆորմացիոն տեսական մոտեցում համայնքների հայտնաբերման խնդրին մ. հարությունյան և կ. մխիթարյան ամփոփում իրական աշխարհի բարդ ցանցերը օժտված են թաքնված ինֆորմացիայով, այսպես կոչված համայնքներով կամ խմբերով, որոնց ներսում հանգույցները ավելի խիստ են կապված, քան համայնքների միջև: համայնքների հետազոտությունը հիմնավորված է մեծ թվով կիրառություններով տարբեր գիտություններում, ինչպիսիք են` կոմպյուտերագիտությունը և մեքենայական ուսուցումը, կենսաբանությունը, տնտեսագիտությունը և սոցիալական ցանցերը: համայնքների հայտնաբերման տարբեր ալգորիթմների մշակման հետ զուգահեռ հետազոտողների ուշադրությունն են գրավում նաև հավանականային ցանցերի մոդելները, մասնավորապես, ստոխաստիկ m. haroutunian and k. mkhitaryan 61 բլոկ մոդելը, որը պատահական գրաֆների կառուցման մոդել է և ստեղծում է համայնքների կառուցվածքով ցանցեր: այս հոդվածում ուսումնասիրվում է ստոխաստիկ բլոկ մոդելի և ինֆորմացիայի տեսության միջև կապի վերաբերյալ գիտական արդյունքների վիճակը: информационно-теоретический подход к задаче обнаружения сообществ м. арутюнян и к. мхитарян аннотация реальные сложные сети обладают скрытой информацией под названием сообщества или кластеры, состоящих из узлов, тесно связаных в кластере и слабо связанных между сообществами. исследование сообществ подтвердило бесчисленное множество применений в различных науках, таких как компьютерные науки и машинное обучение, биология, экономика и социальные сети. параллельно с развитием различных алгоритмов обнаружения сообществ, модели вероятностных сетей также привлекают больше внимания, в частности стохастическая блочная модель, которая создает сети со структурой сообщества. в данной статье исследуется современное состояние науки о связях стохастической блок модели с теорией информации. 2. community detection william-final.dvi mathematical problems of computer science 46, 126{131, 2016. gossiping p r oper ties of the m odi¯ed knäodel gr aphs v ilya m h . h o vn a n ya n institute for informatics and automation problems of nas ra williamhovnanyan@gmail.com abstract in this paper we consider the gossiping process implemented on several modi¯cations of knäodel graphs. we show the ability of knäodel graphs to remain good network topology for gossiping even in case of cyclic permutation of its edge weights. the results shown in this paper could help us to construct edge-disjoint paths between any pairs of vertices of the knäodel graph. keywords: graphs, networks, telephone problem, gossip problem, knäodel graphs. 1 . in t r o d u c t io n go s s ip in g is o n e o f t h e b a s ic p r o b le m s o f in fo r m a t io n d is s e m in a t io n in c o m m u n ic a t io n n e t wo r ks . th e g o s s ip p r o b le m ( a ls o kn o wn a s a t e le p h o n e p r o b le m ) is a t t r ib u t e d t o a . b o yd ( s e e e .g ., [4 ] fo r r e vie w) , a lt h o u g h t o t h e b e s t kn o wle d g e o f t h e r e vie we r s , it wa s ¯ r s t fo r m u la t e d b y r . ch e s t e r s a n d s . s ilve r m a n ( u n iv. o f w it wa t e r s r a n d , u n p u b lis h e d , 1 9 7 0 ) . co n s id e r a s e t o f n p e r s o n s ( n o d e s ) e a c h o f wh ic h in it ia lly kn o ws s o m e u n iqu e p ie c e o f in fo r m a t io n t h a t is u n kn o wn t o t h e o t h e r s , a n d t h e y c a n m a ke a s e qu e n c e o f t e le p h o n e c a lls t o s p r e a d t h e in fo r m a t io n . d u r in g a c a ll b e t we e n t h e g ive n t wo n o d e s , t h e y e xc h a n g e t h e wh o le in fo r m a t io n kn o wn t o t h e m a t t h a t m o m e n t . th e p r o b le m is t o ¯ n d a s e qu e n c e o f c a lls wit h a m in im u m le n g t h ( m in im a l g o s s ip s c h e m e ) , b y wh ic h a ll t h e n o d e s will kn o w a ll p ie c e s o f t h e in fo r m a t io n ( c o m p le t e g o s s ip in g ) . it h a s b e e n s h o wn in n u m e r o u s wo r ks [1 ]{ [4 ] t h a t t h e m in im a l n u m b e r o f c a lls is 2 n ¡ 4 wh e n n ¸ 4 a n d 1 , 3 fo r n = 2 ; 3 , r e s p e c t ive ly. s in c e t h e n m a n y va r ia t io n s o f g o s s ip p r o b le m h a ve b e e n in t r o d u c e d a n d in ve s t ig a t e d ( s e e e .g ., [5 ]{ [1 0 ], [1 3 ]{ [1 7 ]) . a n o t h e r va r ia n t o f go s s ip p r o b le m c a n b e fo r m u la t e d b y c o n s id e r in g t h e m in im u m a m o u n t o f t im e r e qu ir e d t o c o m p le t e g o s s ip in g a m o n g n p e r s o n s , wh e r e t h e c a lls b e t we e n t h e n o n -o ve r la p p in g p a ir s o f n o d e s c a n t a ke p la c e s im u lt a n e o u s ly a n d e a c h c a ll r e qu ir e s o n e u n it o f t im e ( [1 1 ], [1 2 ],[1 8 ]) . ob vio u s ly, t h e g o s s ip p r o b le m c a n b e e a s ily m o d e le d a s a g r a p h , t h e ve r t ic e s o f wh ic h r e p r e s e n t p e o p le in g o s s ip s c h e m e a n d e d g e s r e p r e s e n t c a lls b e t we e n t h e m ( e a c h o f t h e m h a s we ig h t wh ic h r e p r e s e n t s t h e m o m e n t wh e n c o m m u n ic a t io n t o o k p la c e ) . s o t h e g r a p h is c a lle d a c o m p le t e g o s s ip g r a p h if t h e r e a r e a s c e n d in g p a t h s b e t we e n a ll t h e p a ir s o f ve r t ic e s o f t h is g r a p h . a p a t h b e t we e n t wo ve r t ic e s is a n a s c e n d in g p a t h if e a c h n e xt e d g e h a s h ig h e r we ig h t t h a n t h e p r e vio u s o n e . 1 2 6 v. hovnanyan 1 2 7 k n äo d e l g r a p h s a r e o n e o f t h e 3 we ll-kn o wn n e t wo r k t o p o lo g ie s fo r g o s s ip in g / b r o a d c a s t in g ( a lo n g s id e wit h h yp e r c u b e a n d r e c u r s ive c ir c u la n t g r a p h s ) . de¯nition 1: the k näodel graph on n ¸ 2 vertices (n even) and of maximum degree ¢ ¸ 1 is denoted by w¢ ;n. the vertices of w¢ ;n are the pairs ( i; j ) with i = 1 ; 2 and 0 · j · n=2 ¡ 1 . f or every j, 0 · j · n=2 ¡ 1 and l = 1 ; : : : ; ¢ , there is an edge with the label (weight) l between the vertex ( 1 ; j ) and ( 2 ; ( j + 2 l¡1 ¡ 1 ) m o d n=2 ) . the edges with the given label l are said to be in dimension l. n o t e t h a t w1;n c o n s is t s o f n= 2 d is c o n n e c t e d e d g e s . fo r ¢ ¸ 2 , w¢ ;n is c o n n e c t e d . in t h is p a p e r we g ive s o m e n e w o b s e r va t io n s , m o s t ly in t e r m s o f g o s s ip in g , c o n c e r n in g a s lig h t m o d ī c a t io n o f k n äo d e l g r a p h s , o b t a in e d b y a c yc lic p e r m u t a t io n o f t h e d im e n s io n s o f e d g e s . fo r t h e g r a p h u n d e r c o n s id e r a t io n , t h e m o d i¯ c a t io n lo o ks a s fo llo ws : d im e n s io n o f t h e e d g e c o n n e c t in g t h e ve r t ic e s ( 1 ; j ) a n d ( 2 ; ( j + 2 l¡1 ¡ 1 ) m o d n=2 ) , is r e p la c e d b y ( l + p ) m o d dlo g 2 ne wh e r e p = 1 ; : : : ; d¢ e. w e ¯ r s t c o n s id e r t h e c a s e o f n = 2 k ¡ 2 a n d s h o w t h e a b ilit y o f k n äo d e l g r a p h s t o p r e s e r ve g o s s ip in g p r o p e r t ie s fo r t h e o p e r a t io n d e s c r ib e d . th e n we t r y t o g e n e r a liz e t h e a b o ve s t a t e m e n t a b o u t k n äo d e l g r a p h s fo r a ll g e n e r ic n 6= 2 k. a t t h e e n d we s h o w t h e in a b ilit y o f t h e s e g r a p h s t o p r e s e r ve g o s s ip in g p r o p e r t ie s in t h e c a s e o f n = 2 k, e xc e p t fo r o n e s p e c ī c c a s e . 2 . mo d ī e d k n äo d e l gr a p h s w it h n = 2 k ¡ 2 v e r t ic e s in t h is s e c t io n we a r e g o in g t o c o n s id e r t h e m o d i¯ e d k n äo d e l g r a p h s wit h t h e n u m b e r o f ve r t ic e s e qu a l t o 2 k ¡ 2 , wh e r e k is a n y in t e g e r wit h k ¸ 3 . th e d e ¯ n it io n o f m o d i¯ e d k n äo d e l g r a p h is a s fo llo ws : de¯nition 2: the modi¯ed k näodel graph on n ¸ 2 vertices (n even) and of maximum degree ¢ ¸ 1 is denoted by m¢ ;n ( p) . the vertices of m¢ ;n ( p ) are the pairs ( i; j ) with i = 1 ; 2 and 0 · j · n= 2 ¡ 1 , respectively. f or every j, 0 · j · n=2 ¡ 1 and l = 1 ; : : : ; ¢ , there is an edge with the label ( l + p ) m o d dlo g 2 ne between the vertices ( 1 ; j ) and ( 2 ; ( j + 2 l¡1 ¡ 1 ) m o d n= 2 ) where p is a ¯xed integer for that graph with possible values p = 1 ; : : : ; ¢ . the edges with the given label l + p are said to be of dimension l + p. a c c o r d in g t o t h e a b o ve d e ¯ n it io n , t h e r e e xis t ¢ ¡ 1 m o d i¯ c a t io n s fo r e a c h w¢ ;n. in t h is p a p e r we c o n s id e r o n ly t h e c a s e , wh e n ¢ = dlo g 2 ne, a n d t h is va lu e is a s s u m e d fo r a ll r e fe r e n c e s o f ¢ o n wa r d s . de¯nition 3: consider two graphs g1 = ( v; e1 ) and g2 = ( v; e2 ) with the same set of vertices v and labeled edge sets e1 and e2, respectively. the edge sum of these graphs is a graph g1 + g2 = g = ( v; e ) with e = e1 [ e2, whose edges e 2 e are labeled by the following rules: tg ( e ) = 8 < : tg1 ( e) ; if e 2 e1; tg2 ( e) + m a x e02e1 tg1 ( e 0 ) ; if e 2 e2: ( 1 ) where tg ( e) is tha label of the edge e in the resulting graph g. it is kn o wn t h a t if g = w¢ ;n + w1;n, t h e n g is a c o m p le t e g o s s ip g r a p h . l e t u s ¯ r s t c o n s id e r t h e g r a p h g0 = m¢ ;n ( ¢ ) + m1;n ( ¢ ) . t heor em 1: the graph g0 is a complete gossip graph. 1 2 8 gossiping properties of the modi¯ed knäodel graphs 0 t h a t in c lu d e s t h e ve r t ic e s ( i; j ) , wh e r e i = 1 ; 2 a n d 1 · j · ¢ . l e t u s p a r t ic u la r ly fo c u s o n t h e ve r t ic e s , fo r wh ic h i = 0 a n d 1 · j · ¢ , a n d c o n s id e r t h e c a ll in d im e n s io n 1 . th e r e is a n e d g e in d im e n s io n 1 b e t we e n t h e s e a n d ( 1 ; j0 ) ve r t ic e s , wh e r e 2 ¢ · j0 · n=2 . th e s a m e is t r u e a b o u t t h e ve r t ic e s ( 1 ; j ) a n d ( 0 ; j0 ) , wh e r e 1 · j · 2 ¢ a n d 2 ¢ < j0 · n=2 . it is o b vio u s t h a t t h e s e e d g e s c o n n e c t a ll t h e r e m a in in g 2 k ¡ 2 ¡ 2 ¢ = 2 ¢ +1 ¡ 2 ¢ ¡ 2 = 2 ¢ ¡ 2 ve r t ic e s t o t h e subg0 . on t h e o t h e r h a n d , we h a ve t h a t subg0 is a t r e e r o o t e d a t t h e ve r t e x ( 0 ; 1 ) . th u s , t h e r e e xis t a s c e n d in g p a t h s b e t we e n e ve r y ve r t e x o f subg0 a n d t h e ve r t e x ( 0 ; 1 ) . co n s id e r in g t h e fa c t t h a t e a c h e d g e in t h e su bg0 h a s d im e n s io n g r e a t e r t h a n 1 , it fo llo ws t h a t t h e r e e xis t s a n a s c e n d in g p a t h b e t we e n a n y o t h e r ve r t e x a n d t h e ve r t e x ( 0 ; 1 ) . th e t wo fa c t s t h a t t h e c h o ic e o f t h e ve r t e x ( 0 ; 1 ) wa s a r b it r a r y, a ls o t h e m o d i¯ e d k n äo d e l g r a p h b e in g s ym m e t r ic b y c o n s t r u c t io n , p o in t o u t t o c o m p le t e g o s s ip in g o f g0. cor ollar y 2: since the graph g0 is a gossip graph, we can claim that m¢ ;n ( p) , where 0 · p · ¢ , is also a gossip graph. t o a d d w1;n in o r d e r t o o b t a in a c o m p le t e g o s s ip g r a p h . 3 . mo d ī e d k n äo d e l gr a p h s w it h n = 2 k v e r t ic e s in t h is s e c t io n we c o n s id e r m o d i¯ e d k n äo d e l g r a p h s wit h 2 k ve r t ic e s . k n äo d e l g r a p h s wit h 2 k ve r t ic e s a r e d i®e r e n t in t e r m s o f g o s s ip in g s in c e w¢ ;n is a c t u a lly a g o s s ip g r a p h . h e n c e , in t h is c a s e t h e r e is n o n e e d t o a d d w1;n t o g e t a c o m p le t e g o s s ip g r a p h . to s t a r t wit h , le t u s c o n s id e r m¢ ;n ( 2 ) . t heor em 3: the graph m¢ ;n ( 2 ) is a complete gossip graph. in c a s e j is e ve n , a n d jnew = n=2 + bj=2 c in c a s e o f o d d j. w e o b t a in t wo c o m p o n e n t s t h a t a r e k n äo d e l g r a p h s wit h t h e n u m b e r o f ve r t ic e s e qu a l t o n=2 , a n d t h e s e c o m p o n e n t s a r e c o n n e c t e d b y t h e e d g e s o f la b e l ¢ . th e r e fo r e , if we s im p ly r e m o ve t h e s e e d g e s , t h e n t h e r e will b e t wo d is jo in t c o m p o n e n t s wh ic h a r e k n äo d e l g r a p h s t h e m s e lve s . a n d t h e r e is a n o n e -t o -o n e c a ll m a p p in g b e t we e n t h e s e t wo c o m p o n e n t s wit h t h e c a lls la b e le d b y lo g 2 n ( h ig h e s t la b e le d e d g e s ) . ob vio u s ly, t h is g r a p h is a c o m p le t e g o s s ip g r a p h . t heor em 4: the graph m¢ ;n ( p ) , where p 6= 2 , is not a gossip graph. p r oof: s in c e t h is g r a p h is s ym m e t r ic , it is e n o u g h t o s h o w t h a t t h e r e e xis t s a n a s c e n d in g p a t h fr o m a n y o t h e r ve r t e x t o t h e ve r t e x ( 0 ; 1 ) . l e t u s c o n s id e r t h e s u b g r a p h o f g0, subg p r oof. it is a n e a s y e xe r c is e t o ve r ify t h a t g0 is is o m o r p h ic t o w¢ ;n ( fo r d e t a ile d p r o o f r e fe r t o [2 3 ]) . h e n c e , we c a n r e p e a t t h is p r o c e d u r e , a n d e a c h t im e we will o b t a in a n e w m¢ ;n ( p) , p = ¢ ¡ 1 ; ¢ ¡ 2 ; :::; 1 , t h a t is is o m o r p h ic t o in it ia l w¢ ;n. th u s , t h e r e is n o n e e d p r oof: l e t 's s t a r t o u r p r o o f b y d o in g t h e fo llo win g t r a n s fo r m a t io n o n t h e ve r t e x s e t o f t h is g r a p h v . th e e le m e n t s o f t h is s e t a r e t h e p a ir s ( i; j ) . in c a s e o f i = 1 , if j is o d d , t h e n jnew = dj= 2 e a n d jnew = n=2 + j=2 , o t h e r wis e . if i = 2 , t h e r u le s a r e a s fo llo ws : jnew = j=2 v. hovnanyan 1 2 9 l e t u s d e n o t e t h e s u b s e t o f t h e e d g e s e t o f g r a p h m wit h t h e we ig h t wfix = log2n¡i+2 b y e0, a n d it s e le m e n t s b y e0l, wh e r e l = 1 ; 2 ; : : : n= 2 . th e o p e r a t io n a ¡ e0 l ( s e e [2 2 ]) s h o u ld a p p ly t h e " p e r m u t e lo we r " o p e r a t io n o n a ll t h e e d g e s o f e0 r e s u lt in g in a c o m p le t ly n e w g r a p h . th e r e wo u ld o b vio u s ly b e d u p lic a t e ( d o u b le ) e d g e s in t h is g r a p h , s in c e t h e e d g e s wit h we ig h t s wf ix ¡ 1 a n d wf ix + 1 wo u ld c o n n e c t t h e s a m e ve r t ic e s . h e n c e , t a kin g in t o c o n s id e r a t io n t h e fa c t t h a t t h e g o s s ip in g t im e , i.e ., t h e n u m b e r o f r o u n d s r e qu ir e d t o c o m p le t e g o s s ip in g in w¢ ;n wa s log2n ( wh ic h is t h e m o s t o p t im a l va lu e o f t h is p r o p e r t y) , t h e g r a p h m¢ ;n ( p ) is p r o ve d t o b e a n o n -g o s s ip g r a p h . a c kn o wle d g e m e n t s th e a u t h o r is g r a t e fu l t o p r o f. y u .h . s h o u ko u r ia n , d r . s . p o g h o s ya n a n d d r . v . p o g h o s ya n fo r im p o r t a n t d is c u s s io n s a n d c r it ic a l r e m a r ks a t a ll s t a g e s o f t h e wo r k. refer ences [1 ] b .b a ke r a n d r .s h o s t a k, \ go s s ip s a n d t e le p h o n e s " , d iscrete m athematics, vo l. 2 , p p . 1 9 1 -1 9 3 , 1 9 7 2 . [2 ] r .t. b u m b y, \ a p r o b le m wit h t e le p h o n e s " , siam j ournal on algebraic d iscrete m ethods, vo l. 2 , p p . 1 3 -1 8 , 1 9 8 1 . [3 ] a . h a jn a l, e .c. miln e r a n d e . s z e m e r e d i, \ a c u r e fo r t h e t e le p h o n e d is e a s e " , canadian m athematical b ulletin., vo l. 1 5 , p p . 4 4 7 -4 5 0 , 1 9 7 2 . [4 ] t. tijd e m a n , \ on a t e le p h o n e p r o b le m " , nieuw archief voor w iskunde vo l. 3 , p p . 1 8 8 -1 9 2 , 1 9 7 1 . [5 ] a . s e r e s s , \ qu ic k g o s s ip in g b y c o n fe r e n c e c a lls " , siam j ournal on d iscrete m athematics, vo l. 1 , p p . 1 0 9 -1 2 0 , 1 9 8 8 . [6 ] d .b . w e s t , \ go s s ip in g wit h o u t d u p lic a t e t r a n s m is s io n s " , siam j ournal on algebraic d iscrete m ethods, vo l. 3 , p p . 4 1 8 -4 1 9 , 1 9 8 2 . 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[1 3 ] z. h o a n d m. s h ig e n o , \ n e w b o u n d s o n t h e m in im u m n u m b e r o f c a lls in fa ilu r e -t o le r a n t go s s ip in g " , networks, vo l. 5 3 , p p . 3 5 -3 8 , 2 0 0 9 . [1 4 ] k . a . b e r m a n a n d m. h a wr ylyc z , \ te le p h o n e p r o b le m s wit h fa ilu r e s " , siam j ournal on algebraic d iscrete m ethods, vo l. 7 , p p . 1 3 -1 7 , 1 9 8 7 . p r oof: to s t a r t wit h t h e p r o o f, le t u s r e c a ll o u r m e t h o d fo r m o d ifyin g g o s s ip g r a p h s , ¯ r s t ly in t r o d u c e d in [2 2 ]. th e o p e r a t io n s p e r m u t e h ig h e r a n d p e r m u t e lo we r we r e in t r o d u c e d t o m o d ify t h e t o p o lo g y o f t h e g r a p h wit h o u t c h a n g in g it s m a in g o s s ip in g p r o p e r t ie s ( n u m b e r o f e d g e s , n u m b e r o f r o u n d s , e t c .) . 1 3 0 gossiping properties of the modi¯ed knäodel graphs [1 5 ] r .w . h a d d a d , s . r o y a n d a .a . s c h a ®e r , \ on g o s s ip in g wit h fa u lt y t e le p h o n e lin e s " , siam j ournal on algebraic d iscrete m ethods, vo l. 8 , p p . 4 3 9 -4 4 5 , 1 9 8 7 . [1 6 ] t. h a s u n a m a a n d h . n a g a m o c h i, \ im p r o ve d b o u n d s fo r m in im u m fa u lt -t o le r a n t g o s s ip g r a p h s " , l ecture notes in computer science, vo l. 6 9 8 6 , p p . 2 0 3 -2 1 4 , 2 0 1 1 . [1 7 ] v . h . h o vn a n ya n , h . e . n a h a p e t ya n , s u . s . p o g h o s ya n a n d v .s . p o g h o s ya n , \ tig h t e r u p p e r b o u n d s fo r t h e m in im u m n u m b e r o f c a lls a n d r ig o r o u s m in im a l t im e in fa u lt t o le r a n t g o s s ip s c h e m e s " , a r x iv:1 3 0 4 .5 6 3 3 . [1 8 ] s a n d r a m. h e d e t n ie m i, s t e p h e n t. h e d e t n ie m i, a r t h u r l . l ie s t m a n , \ a s u r ve y o f g o s s ip in g a n d b r o a d c a s t in g in c o m m u n ic a t io n n e t wo r ks " , networks, vo l. 1 8 , p p . 3 1 9 3 4 9 , 1 9 8 8 . [1 9 ] d . b . w e s t , \ a c la s s o f s o lu t io n s t o t h e g o s s ip p r o b le m " , p a r t i, d iscrete m athematics, vo l. 3 9 , n o . 3 , p p . 3 0 7 -3 2 6 , 1 9 8 2 ; p a r t ii, d isc. m ath., vo l. 4 0 , n o . 1 , p p . 8 7 -1 1 3 , 1 9 8 2 ; p a r t iii, d iscrete m athematics, vo l. 4 0 , n o . 2 -3 , p p . 2 8 5 -3 1 0 , 1 9 8 2 . [2 0 ] a . s e r e s s , \ qu ic k go s s ip in g wit h o u t d u p lic a t e t r a n s m is s io n s " , graphs and combinatorics, vo l. 2 , p p . 3 6 3 -3 8 1 , 1 9 8 6 . [2 1 ] p . fr a ig n ia u d , a . l . l ie s t m a n a n d d . s o it e a u , \ op e n p r o b le m s " , p arallel p rocessing l etters, vo l. 3 , n o . 4 , p p . 5 0 7 -5 2 4 , 1 9 9 3 . [2 2 ] v . h . h o vn a n ya n , s u . s . p o g h o s ya n a n d v . s . p o g h o s ya n , \ me t h o d o f lo c a l in t e r c h a n g e t o in ve s t ig a t e go s s ip p r o b le m s " , transactions of iiap of nas r a, m athematical p roblems of computer science, vo l. 4 0 , p p . 5 -1 2 , 2 0 1 3 . [2 3 ] l .h . k h a c h a t r ia n , h .s . h a r u t o u n ia n , \ on op t im a l b r o a d c a s t gr a p h s " , p roc. of the fourth int. colloquium on coding theory, 6 5 -7 2 , a r m e n ia , 1 9 9 1 . submitted 28.07.2016, accepted 02.11.2016. øá¹çýçï³óí³í knäodel ·ñ³ýý»ñç gossiping (µ³ùµ³ë³ýù³ûçý) ñ³ïïáõãûáõýý»ñá ì. ðáíý³ýû³ý ²ù÷á÷áõù ²ûë ñá¹í³íáõù ù»ýù ¹çï³ñïáõù »ýù k n äo d e l ·ñ³ýý»ñç áñáß ùá¹çýçï³óç³ý»ñç íñ³ çñ³ï³ý³óíáõ gossiping åñáó»ëá: òáõûó ¿ ïñíáõù k n äo d e l ·ñ³ýç gossiping-ç ñ³ù³ñ é³í ó³ýó³ûçý ïáåáéá·ç³ ñ³ý¹çë³ý³éáõ ï³ñáõáõãûáõýá ýáõûýçëï ¹ñ³ ïáõ»ñç ïßçéý»ñç óçïéçï í»ñ³¹³ë³íáñù³ý å³ñ³·³ûáõù: ðá¹í³íáõù óáõûó ïñí³í ³ñ¹ûáõýùý»ñá ï³ñáõ »ý û·ï³ï³ñ éçý»é k n äo d e l ·ñ³ýç ï³ù³û³ï³ý »ñïáõ ·³·³ãý»ñç ùçç¨ ïáõ³ýï³ë ×³ý³å³ññý»ñç ï³éáõóù³ý ñ³ù³ñ: v. hovnanyan 1 3 1 ãîññèï ñâîéñòâà ìîäèôèöèðîâàííûõ êíåäåë ãðàôîâ â. îâíàíÿí àííîòàöèÿ â ýòîé ñòàòüå ìû ðàññìàòðèâàåì ïðîöåññ ãîññèïà ðåàëèçîâàííûé íà íåêîòîðûõ ìîäèôèêàöèÿõ êíåäåë ãðàôîâ. ïîêàçàíà ñïîñîáíîñòü êíåäåë ãðàôîâ îñòàâàòüñÿ õîðîøåé ñåòåâîé òîïîëîãèåé äëÿ ãîññèïà äàæå â ñëó÷àå öèêëè÷åñêîé ïåðåñòàíîâêè âåñîâ ðåáåð. ðåçóëüòàòû, ïðåäñòàâëåííûå â äàííîé ðàáîòå ñïîñîáñòâóþò ïîñòðîåíèþ ðåáåðíî-íåïåðåñåêàþùèõñÿ ïóòåé ìåæäó ëþáûìè ïàðàìè âåðøèí êíåäåë ãðàôîâ. d:\sbornik\...\article.dvi mathematical problems of computer science 37, 25{34, 2012. on t wo-stage logar ithmically asymptotically optimal t esting of m ultiple h ypotheses concer ning distr ibutions fr om the p air of families e vg u e n i h a r o u t u n ia n 1, p a r a n d z e m h a ko b ya n 1 a n d fa r s h in h o r m o z i n e ja d 2 institute for informatics and automation problems of nas of ra.1 islamic azad university, ahvaz branch, iran.2 evhar@ipia.sci.am, par h@ipia.sci.am, hormozi-nejad@iauahvaz.ac.ir abstract two-stage testing of multiple hypotheses for a model with two given families of hypothetical probability distributions is considered. the matrix of reliabilities of logarithmically asymptotically optimal hypotheses testing by a pair of stages is studied and compared with the case of similar one-stage testing. keywords: logarithmically asymptotically optimal (lao) test, multiple hypotheses testing, multistage tests, reliabilities matrix, error probability exponent. refer ences [1 ] r . a h ls we d e a n d e .a . h a r o u t u n ia n . \ on s t a t is t ic a l h yp o t h e s e s o p t im a l t e s t in g a n d id e n t ī c a t io n " , l ecture notes in computer science, vol. 4123, general theory of information transfer and combinatorics, s p r in g e r , p p . 4 6 2 -4 7 8 , 2 0 0 6 . [2 ] t.m co ve r a n d j.a . to m a s . e lements of information theory, s e c o n d e d it io n , w ile y, n e wy o r k, 2 0 0 6 . [3 ] i. cs is z ¶a r a n d j. k äo r n e r . information theory: coding theorems for d iscrete m emoryless systems, a c a d e m ic p r e s s , n e wy o r k, 1 9 8 1 . [4 ] i. cs is z ¶a r a n d g. l o n g o . \ on t h e e r r o r e xp o n e n t fo r s o u r c e c o d in g a n d fo r t e s t in g s im p le s t a t is t ic a l h yp o t h e s e s " , studia sc. m ath. hungarica, vo l. 6 , p p . 1 8 1 { 1 9 1 , 1 9 7 1 . [5 ] e .a . h a r o u t u n ia n . \ l o g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f m u lt ip le s t a t is t ic a l h yp o t h e s e s " , p roblems of control and information theory, vo l 1 9 , n o . 5 -6 , p p . 4 1 3 -4 2 1 , 1 9 9 0 . [6 ] e .a . h a r o u t u n ia n .\ r e lia b ilit y in m u lt ip le h yp o t h e s e s t e s t in g a n d id e n t ī c a t io n " , p roceedings of the nato-asi conference, vol. 198 of nato science series iii: computer and systems sciences, y e r e va n , a r m e n ia , p p . 1 8 9 -2 0 1 , ios p r e s s , 2 0 0 8 . [7 ] e .a . h a r o u t u n ia n , m.e . h a r o u t u n ia n a n d a .n . h a r o u t u n ia n . r e lia b ilit y c r it e r ia in in fo r m a t io n t h e o r y a n d in s t a t is t ic a l h yp o t h e s is t e s t in g , f oundations and trends in communications and information theory, vo l. 4 , n o . 2 -3 , 2 0 0 8 . 2 5 2 6 on two-stage logarithmically asymptotically optimal testing of multiple hypotheses [8 ] e .a . h a r o u t u n ia n a n d p .m. h a ko b ya n , \ mu lt ip le h yp o t h e s e s l a o t e s t in g fo r m a n y in d e p e n d e n t o b je c t s " , international j ournal \scholarly r esearch e xchange", p p . 1 { 6 , 2 0 0 9 . [9 ] w . h o e ®d in g . \ a s ym p t o t ic a lly o p t im a l t e s t s fo r m u lt in o m ia l d is t r ib u t io n s " , annals of m athematical statistics, vo l. 3 6 , p p . 3 6 9 -4 0 1 , 1 9 6 5 . [1 0 ] g. tu s n a d y. \ on a s ym p t o t ic a lly o p t im a l t e s t s " , annals of statistics, vo l. 5 , n o . 2 , p p . 3 8 5 -3 9 3 , 1 9 7 7 . ºñïáõ áýï³ýçùý»ñ ï³½ùáõ µ³ßëáõùý»ñç í»ñ³µ»ñû³é µ³½ù³ïç í³ñï³íý»ñç »ñï÷áõé éá·³ñçãùáñ»ý ³ëçùåïáïáñ»ý ûåïçù³é ï»ëï³íáñáõù º. ð³ñáõãûáõýû³ý, ö. ð³ïáµû³ý ¨ ü. ðáñùá½ç ý»å³¹ ²ù÷á÷áõù ¸çï³ñïí»é ¿ µ³ßëáõùý»ñç »ñïáõ áýï³ýçùý»ñçó ï³½ùí³í ùá¹»éç í»ñ³µ»ñû³é µ³½ù³ïç íç×³ï³·ñ³ï³ý í³ñï³íý»ñç ëïáõ·ù³ý ·áñíáýã³óá: ð»ï³½áïí»é »ý »ñïáõ ÷áõé»ñçó ûáõñ³ù³ýãûáõñç ëë³éý»ñç ½áõû·»ñç óáõóçãý»ñç (ñáõë³éçáõãûáõýý»ñç) ÷áëï³åí³íáõãûáõýý»ñç ù³ïñçóý»ñá: ð³ù»ù³ïí»é »ý »ñï÷áõé ï»ëïç ¨ ùç³÷áõé ï»ëïç ñáõë³éçáõãûáõýý»ñá ¨ ·áñíáýã³óý»ñç »ñï³ñáõãûáõýý»ñá: î äâóõ-ýòàïíîì ëîãàðèôìè÷åñêè àñèìïòîòè÷åñêè îïòèìàëüíîì òåñòèðîâàíèè ìíîãèõ ãèïîòåç îòíîñèòåëüíî ðàñïðåäåëåíèé èç äâóõ ñåìåéñòâ å. àðóòþíÿí, ï. àêîïÿí è ô. õîðìîçè íåæàä àííîòàöèÿ ðàññìîòðåíî òåñòèðîâàíèå ìîäåëè ñîñòîÿøåé èç äâóõ ñåìåéñòâ âîçìîæíûõ ðàñïðåäåëåíèé âåðîÿòíîñòåé. ìàòðèöà íàäåæíîñòåé ëîãîðèôìè÷åñêè àñèìïòîòè÷åñêè îïòèìàëüíîãî òåñòèðîâàíèÿ â äâà ýòàïà èçó÷åíà è ñðàâíåíà ñî ñëó÷àåì àíàëîãè÷íîãî îäíîýòàïíîãî òåñòèðîâàíèÿ. microsoft word tigran_hakob.doc mathematical problems of computer science 30, 60--70, 2008. 60 быстрое кпф и меры улучшения качества цветных изображений тигран манукян, акоп саруханян институт проблем информатики и автоматизации нан ра tigran@manukyan.am, hakop@ipia.sci.am аннотация в работе предложены метод улучшения качества цветного изображения и меры качества основанный на быстром кватернионном преобразовании фурье (кпф). приведены также примеры улучшения цветных изображений. литература [1]. a. k. jain, “fundamentals of digital image processing”, prentice hall, englewood cliffs, 2005. [2]. r. k. yarglagadda, e. j. hersey, “hadamard matrix analysis and synthesis with applications in signal/image processing”, springer, 1997. [3]. s. mallat, “a wavelet tour of signal processing”, academic press, london, new york, tokyo, 1998. [4]. h. sarukhanyan, a. petrosian, “construction and application of hybrid wavelet and other parametric orthogonal transform”. journal of mathematical imaging and vision, vol. 23, pp. 2546, 2005. [5]. s. agaian, k. panetta and a. m. grigoryan, “transform-based image enhancement algorithms with performance measure”, ieee trans. on image processing, vol. 10, no. 3, 2001. [6]. т. манукян, “об одном методе улучшения качества цветных изображений с использованием кватернионного фурье-преобразования”, российско-армянский (славянский) государственный университет. годичная научная конференция, c. 124-130, 2006. [7]. t. manukyan, “quaternionic fourier transform for enhancement and compression of color images”, mathematical problems of computer science, vol. 28, pp. 51-59, 2007. [8]. e. wharton, s. agaian , k. panetta, “comparative study of logarithmic enhancement algorithms with performance measure”, proceedings, electronic imaging, 2006. [9]. s. s. agaian, b. silver, k. a. panetta, “transform coefficient histogram-based image enhancement algorithms using contrast entropy”, ieee trans. on image processing, vol. 16, no. 3. 2007. т.манукян, а.саруханян 61 ֆուրյեի քվատերնիոն արագ ձևափոխություն և գունավոր պատկերների որակի լավացման չափի ֆունկցիաներ տ. մանուկյան, հ. սարուխանյան ամփոփում աշխատանքում առաջարկվում են ֆուրյեի քվատերնիոն արագ ձևափոխության միջոցով գունավոր պատկերների որակի լավացման մեթոդ, ինչպես նաև գունավոր պատկերների որակի լավացման չափի ֆունկցիաներ։ բերվում են գունավոր պատկերների լավացման օրինակներ։ mathematical problems of computer science 53, 21--28, 2020. udc 510.6 on some universal propositional proof systems for many-valued logic artur a. khamisyan yerevan state university e-mail: khamisyam@gmail.com abstract some uniform hilbert-like propositional proof system is suggested for all versions of many-valued logic to apply them to 3 versions of 3-valued logic, two of which have only one designated value, and the last one has two designated values. keywords: many-valued logic, hilbert-like proof system, uniform propositional proof system. 1. introduction it is known that many-valued logic (mvl) as a separate subject was first created and developed by łukasiewicz [1]. his intention was to use a third, additional truth value for “possible” (or “unknown”). in the meantime, many interesting applications of mvl were found in such fields as logic, mathematics, hardware design, artificial intelligence and some other areas of soft information technologies, therefore investigations in the area of many-valued logic are very actual. the main theoretical results concern several properties of formal systems, which can present different versions of mvl and, in particular, issues on logical completeness of such systems. the completeness of some types of proof systems is proved by hard, many-stepped operations of immersion into two-valued logic, and for the other systems itis proved by reducing tothe completeness of the first type. the generalization of kalmar’s proof of deducibility for two-valued tautologies in classical propositional logic [2] enables us to suggest 1) a new method of proving the completeness of a propositional proof system (pps) of some well-known mvl such that it is essentially simpler than other known proofs of completeness and can be easily modified even into a proof of completeness for fuzzy logic as well, 2) a method for defining of many traditional variants of pps for mvl, the completeness of which is easily proved directly, without the usual immersion into two-valued logic. several new pps for two versions of mvl are introduced in [3-7]. the current research refers to the problem of constructing some uniform hilbert-like pps for all possible versions of mvl. 21 on some universal propositional proof systems for many valued logic 22 2. main definitions of k-valued logics here we give the main notions and notations for different versions of mvl. let ek be the set �0, 1 k−1 , … , k−2 k−1 , 1�. we use the well-known notions of a propositional formula, which, as usual, are defined from propositional variables with values from ek, (may be also propositional constants), parentheses (,), and logical connectives & , ∨ , ⊃ ,¬, each of which can be defined by a different mode. additionally, we introduce several variants of the exponential function p .we use the well-known notion of a propositional formula and introduce an additional notion of a formula: for each formulas a and b, the expression 𝑨𝑨𝑩𝑩 is also a formula. in the considered logics, only 1 or each of the values 1 2 ≤ 𝑖𝑖 k−1 ≤ 1 can be fixed as designated values. definitions of main logical functions are: 𝒑𝒑 ∨ 𝒒𝒒 = 𝑚𝑚𝑚𝑚𝑚𝑚(𝑝𝑝, 𝑞𝑞) (1) disjunction, 𝒑𝒑 ∨ 𝒒𝒒 = 𝑚𝑚𝑚𝑚𝑚𝑚(𝑝𝑝 + 𝑞𝑞, 1) (2) disjunction, 𝒑𝒑&𝑞𝑞 = 𝑚𝑚𝑚𝑚𝑚𝑚(𝑝𝑝, 𝑞𝑞) (1) conjunction, 𝒑𝒑&𝑞𝑞 = 𝑝𝑝𝑞𝑞(𝑚𝑚𝑚𝑚𝑚𝑚 𝑘𝑘)/(𝑘𝑘 − 1) (2) conjunction for implication, we have the following two versions: 𝒑𝒑 ⊃ 𝒒𝒒 = � 1, 𝑓𝑓𝑚𝑚𝑓𝑓 𝑝𝑝 ≤ 𝑞𝑞 1 − 𝑝𝑝 + 𝑞𝑞, 𝑓𝑓𝑚𝑚𝑓𝑓 𝑝𝑝 > 𝑞𝑞 (1) łukasiewicz’s implication or p⊃ 𝒒𝒒 = � 1, 𝑓𝑓𝑚𝑚𝑓𝑓 𝑝𝑝 ≤ 𝑞𝑞 𝑞𝑞, 𝑓𝑓𝑚𝑚𝑓𝑓 𝑝𝑝 > 𝑞𝑞 (2) gödel’s implication and for negation, there are also two versions: ¬𝒑𝒑 = 1 − 𝑝𝑝 (1) łukasiewicz’s negation or ¬𝒑𝒑 = ((𝑘𝑘 − 1)𝑝𝑝 + 1)(𝑚𝑚𝑚𝑚𝑚𝑚 𝑘𝑘)/(𝑘𝑘 − 1) (2) cyclically permuting negation. sometimes we can use the notation 𝒑𝒑� instead of ¬𝒑𝒑. for the propositional variable p and 𝛅𝛅= 𝑖𝑖 k−1 (0≤i≤k-1), we additionally define “exponent” functions: p as (𝑝𝑝 ⊃ δ)& (δ ⊃ 𝑝𝑝) with (1) implication (1) exponent, p as p with (k-1)–i(2) negations. (2) exponent. note that both (1) exponent and (2) exponent are not new logical functions. if we fix “1” (each of the values 1 2 ≤ 𝑖𝑖 k−1 ≤ 1) as a designated value, then the formula φ with variables p1,p2,…pn is called a k-tautology if for every 𝛿𝛿 = (𝛿𝛿1, 𝛿𝛿2, … , 𝛿𝛿𝑛𝑛) ∈ 𝐸𝐸𝑘𝑘 𝑛𝑛assigning 𝛿𝛿j (1≤j≤n) to each pj gives the value 1 (or some value 1 2 ≤ 𝑖𝑖 k−1 ≤ 1) of φ. 3. uniform hilbert-like propositional proof system and its properties here we suggest some uniform hilbert-like pps for all possible versions of mvl. for all formulas a, b, c of mvl, each 𝜎𝜎1, 𝜎𝜎2 from the set ek and for ∗∈ {&,∨, ⊃} the following a. khamisyan 23 formulas are axiom schemes of some pps version: 1. 𝐴𝐴 ⊃ (𝐵𝐵 ⊃ 𝐴𝐴) 2. (𝐴𝐴 ⊃ 𝐵𝐵) ⊃ (�𝐴𝐴 ⊃ (𝐵𝐵 ⊃ 𝐶𝐶)� ⊃ �𝐴𝐴 ⊃ 𝐶𝐶𝜑𝜑m.p.(𝐵𝐵,𝐶𝐶)�) 3.𝐴𝐴𝜎𝜎 ⊃ (¬𝐴𝐴)𝜑𝜑¬(𝐴𝐴,𝜎𝜎) 4.-6.𝐴𝐴𝜎𝜎1 ⊃ (𝐵𝐵𝜎𝜎2 ⊃ (𝐴𝐴 ∗ 𝐵𝐵)𝜑𝜑∗(𝐴𝐴,𝐵𝐵,𝜎𝜎1,𝜎𝜎2)) 7.𝐴𝐴𝜎𝜎1 ⊃ (𝐵𝐵𝜎𝜎2 ⊃ (𝐴𝐴𝐵𝐵)𝜑𝜑exp(𝐴𝐴,𝐵𝐵,𝜎𝜎1,𝜎𝜎2)) 8.(𝐴𝐴1 ⊃ 𝐵𝐵) ⊃ (�𝐴𝐴 𝑘𝑘−1 𝑘𝑘−2� ⊃ 𝐵𝐵� ⊃ ⋯ ⊃ ��𝐴𝐴 1 𝑘𝑘−1 ⊃ 𝐵𝐵� ⊃ �(𝐴𝐴0 ⊃ 𝐵𝐵) ⊃ 𝐵𝐵�� … ), where many-valued functions 𝜑𝜑¬(𝐴𝐴, 𝜎𝜎), 𝜑𝜑∗(𝐴𝐴, 𝐵𝐵, 𝜎𝜎1, 𝜎𝜎2), 𝜑𝜑exp(𝐴𝐴, 𝐵𝐵, 𝜎𝜎1, 𝜎𝜎2) must be defined individually for each version of mvl, such that axioms 3-7 will be a k-tautology in this version. inference rule is a modification of modus ponens (m.m.p.) 𝐴𝐴,𝐴𝐴⊃𝐵𝐵 𝐵𝐵𝜑𝜑m.p.(𝐴𝐴,𝐵𝐵) , where many-valued functions 𝜑𝜑m.p.(𝐴𝐴, 𝐵𝐵) must be defined individually for each version of mvl, such that the formula 𝐴𝐴 ⊃ ((𝐴𝐴 ⊃ 𝐵𝐵) ⊃ 𝐵𝐵𝜑𝜑m.p.(𝐴𝐴,𝐵𝐵)) will also be a k-tautology in this version. we use the well-known notions of proof and proof from premises. for all defined systems, we first prove the well known deduction theorem using the first two axiom schemes, and then the generalization of kalmar’s proof of deducibility for two valued tautologies inside the classical propositional logic gives us a possibility to directly prove completeness for the mentioned proof systems, without the usual loading into two-valued logic. let pk be one of the defined pps for some version of mvl. the following statement can be proved as usual. deduction theorem. let γ be a set of some formulas and 𝐴𝐴 and 𝐵𝐵 be some formulas. if the formula b is derived in the system pk from the premises γ and 𝐴𝐴 (γ, 𝐴𝐴 ⊢𝑃𝑃𝑘𝑘 𝐵𝐵), then the formula𝐴𝐴 ⊃ 𝐵𝐵 is derived in the system pk from the premises γ(γ ⊢𝑃𝑃𝑘𝑘 𝐴𝐴 ⊃ 𝐵𝐵). lemma 1: let 1 2= { , , , }np p p p be the set of all variables of any formula a, then for every 𝛿𝛿 = (𝛿𝛿1, 𝛿𝛿2, … , 𝛿𝛿𝑛𝑛) ∈ 𝐸𝐸3 𝑛𝑛 𝑝𝑝1 𝛿𝛿1, 𝑝𝑝2 𝛿𝛿2, … , 𝑝𝑝𝑛𝑛 𝛿𝛿𝑛𝑛 ⊢𝐶𝐶𝐶𝐶3 𝐴𝐴𝐴𝐴(𝛿𝛿1,𝛿𝛿2,…,𝛿𝛿𝑛𝑛). to simplify the proof, we demonstrate them only for p3 logic. proof is given by induction on number n of logical connectives in the formula a. for n=1 we have: by 𝛿𝛿 = 0 𝑝𝑝0 ⊢ 𝑝𝑝0, by𝛿𝛿 = 1 2� 𝑝𝑝 1 2� ⊢ 𝑝𝑝 1 2� and by 𝛿𝛿 = 1 𝑝𝑝1 ⊢ 𝑝𝑝1. suppose that the statement is valid for the number of logical connectives ˂n. if the number of logical connectives is n, then the formula a can be in one of the following forms: 1. 𝐴𝐴 = 𝐴𝐴1 ∗ 𝐴𝐴2, where ∗∈ {&,∨, ⊃}, 2. a=𝐴𝐴1 𝐴𝐴2, 3. 𝐴𝐴 = ¬𝐴𝐴1. for the case 1. 𝐴𝐴1�𝛿𝛿� = 𝜎𝜎1, 𝐴𝐴2�𝛿𝛿� = 𝜎𝜎2 => 𝐴𝐴�𝛿𝛿� = 𝜎𝜎1 ∗ 𝜎𝜎2 on some universal propositional proof systems for many valued logic 24 by induction hypothesis 𝑝𝑝1 𝛿𝛿1, 𝑝𝑝2 𝛿𝛿2, … , 𝑝𝑝𝑛𝑛 𝛿𝛿𝑛𝑛 ⊢ 𝐴𝐴1 𝜎𝜎1 𝑝𝑝1 𝛿𝛿1, 𝑝𝑝2 𝛿𝛿2, … , 𝑝𝑝𝑛𝑛 𝛿𝛿𝑛𝑛 ⊢ 𝐴𝐴2 𝜎𝜎2 use one of the axiom schemes 4. – 6. we have 𝑝𝑝1 𝛿𝛿1, 𝑝𝑝2 𝛿𝛿2, … , 𝑝𝑝𝑛𝑛 𝛿𝛿𝑛𝑛 ⊢ 𝐴𝐴1 𝜎𝜎1 ⊃ (𝐴𝐴2 𝜎𝜎2 ⊃ (𝐴𝐴1 ∗ 𝐴𝐴2)𝜑𝜑∗(𝐴𝐴1,𝐴𝐴2,𝜎𝜎1,𝜎𝜎2)) and for 𝐴𝐴1 = 𝜎𝜎1, 𝐴𝐴2 = 𝜎𝜎2 𝑝𝑝1 𝛿𝛿1, 𝑝𝑝2 𝛿𝛿2, … , 𝑝𝑝𝑛𝑛 𝛿𝛿𝑛𝑛 ⊢ 𝐴𝐴1 𝜎𝜎1 ⊃ �𝐴𝐴2 𝜎𝜎2 ⊃ (𝐴𝐴1 ∗ 𝐴𝐴2)𝜑𝜑∗(𝐴𝐴1,𝐴𝐴2,𝜎𝜎1,𝜎𝜎2)� and (𝐴𝐴1 ∗ 𝐴𝐴2)𝜑𝜑∗(𝐴𝐴1,𝐴𝐴2,𝜎𝜎1,𝜎𝜎2) is derived after the double application of m.m.p. for the case 2. 𝐴𝐴1�𝛿𝛿� = 𝜎𝜎1, 𝐴𝐴2�𝛿𝛿� = 𝜎𝜎2 => 𝐴𝐴�𝛿𝛿�=�𝐴𝐴1 𝐴𝐴2� (𝝈𝝈𝟏𝟏 𝝈𝝈𝟐𝟐) and we must use the axiom scheme 7. for the case 3. 𝐴𝐴1�𝛿𝛿� = 𝜎𝜎 => 𝐴𝐴�𝛿𝛿� = ¬𝜎𝜎 we must use the axiom scheme 3. ⧠ corollary 1: if a is a 3-tautology, then for every 𝛿𝛿 = (𝛿𝛿1, 𝛿𝛿2, … , 𝛿𝛿𝑛𝑛) ∈ 𝐸𝐸3 𝑛𝑛 𝑝𝑝1 𝛿𝛿1, 𝑝𝑝2 𝛿𝛿2, … , 𝑝𝑝𝑛𝑛 𝛿𝛿𝑛𝑛 ⊢ 𝐴𝐴 theorem 1: any formula is derived in p3 iff it is a 3-tautology. proof: it is obvious that every formula, derived in p3, is a 3-tautology. let },,,{= 21 npppp  (n≥1) be the set of all variables of any tautology a. for every 𝛿𝛿 = (𝛿𝛿1, 𝛿𝛿2, … , 𝛿𝛿𝑛𝑛) ∈ 𝐸𝐸3 𝑛𝑛 by the above corollary, we have 𝑝𝑝1 𝛿𝛿1, 𝑝𝑝2 𝛿𝛿2, … , 𝑝𝑝𝑛𝑛 𝛿𝛿𝑛𝑛 ⊢ 𝐴𝐴. for every 𝛿𝛿1, 𝛿𝛿2, … , 𝛿𝛿𝑛𝑛−1 we take into consideration the following 3 truth values � 𝛿𝛿1, 𝛿𝛿2, … , 𝛿𝛿𝑛𝑛−1, 0 𝛿𝛿1, 𝛿𝛿2, … , 𝛿𝛿𝑛𝑛−1, 1 2� 𝛿𝛿1, 𝛿𝛿2, … , 𝛿𝛿𝑛𝑛−1, 1 , for which we have ⎩ ⎨ ⎧ 𝑝𝑝1 𝛿𝛿1, 𝑝𝑝2 𝛿𝛿2, … , 𝑝𝑝𝑛𝑛−1 𝛿𝛿𝑛𝑛−1, 𝑝𝑝𝑛𝑛0 ⊢ 𝐴𝐴 𝑝𝑝1 𝛿𝛿1, 𝑝𝑝2 𝛿𝛿2, … , 𝑝𝑝𝑛𝑛−1 𝛿𝛿𝑛𝑛−1, 𝑝𝑝𝑛𝑛 1 2� ⊢ 𝐴𝐴 𝑝𝑝1 𝛿𝛿1, 𝑝𝑝2 𝛿𝛿2, … , 𝑝𝑝𝑛𝑛−1 𝛿𝛿𝑛𝑛−1, 𝑝𝑝𝑛𝑛1 ⊢ 𝐴𝐴 . by deduction theorem we have 𝑝𝑝1 𝛿𝛿1, 𝑝𝑝2 𝛿𝛿2, … , 𝑝𝑝𝑛𝑛−1 𝛿𝛿𝑛𝑛−1 ⊢ 𝑝𝑝𝑛𝑛0 ⊃ 𝐴𝐴 𝑝𝑝1 𝛿𝛿1, 𝑝𝑝2 𝛿𝛿2, … , 𝑝𝑝𝑛𝑛−1 𝛿𝛿𝑛𝑛−1 ⊢ 𝑝𝑝𝑛𝑛 1 2� ⊃ 𝐴𝐴 𝑝𝑝1 𝛿𝛿1, 𝑝𝑝2 𝛿𝛿2, … , 𝑝𝑝𝑛𝑛−1 𝛿𝛿𝑛𝑛−1 ⊢ 𝑝𝑝𝑛𝑛1 ⊃ 𝐴𝐴. then 𝑝𝑝1 𝛿𝛿1, 𝑝𝑝2 𝛿𝛿2, … , 𝑝𝑝𝑛𝑛−1 𝛿𝛿𝑛𝑛−1 ⊢ (𝐴𝐴1 ⊃ 𝐵𝐵) ⊃ (�𝐴𝐴 1 2� ⊃ 𝐵𝐵� ⊃ �(𝐴𝐴0 ⊃ 𝐵𝐵) ⊃ 𝐵𝐵�) /axiom 8./ 𝑝𝑝1 𝛿𝛿1, 𝑝𝑝2 𝛿𝛿2, … , 𝑝𝑝𝑛𝑛−1 𝛿𝛿𝑛𝑛−1 ⊢ (𝑝𝑝𝑛𝑛 1 2� ⊃ 𝐴𝐴) ⊃ ((𝑝𝑝𝑛𝑛0 ⊃ 𝐴𝐴) ⊃ 𝐴𝐴) /m.m.p./ 𝑝𝑝1 𝛿𝛿1, 𝑝𝑝2 𝛿𝛿2, … , 𝑝𝑝𝑛𝑛−1 𝛿𝛿𝑛𝑛−1 ⊢ (𝑝𝑝𝑛𝑛0 ⊃ 𝐴𝐴) ⊃ 𝐴𝐴 /m.m.p./ 𝑝𝑝1 𝛿𝛿1, 𝑝𝑝2 𝛿𝛿2, … , 𝑝𝑝𝑛𝑛−1 𝛿𝛿𝑛𝑛−1 ⊢ 𝐴𝐴 /m.m.p./ a. khamisyan 25 so, the number of premises is now n-1. repeating the above steps, we finally obtain the derivation of tautology a in p3 ⧠ note that this proof is the complete analogy to proof of the corresponding theorem for the 2valued logic [2]. also note that after proving by analogy the corresponding lemma 1. for any kvalued logic for k≥4, the proof of the corresponding theorem above can be given by analogy as well. every such pps can be easily transformed into gentzen-like system as well. 4. examples of uniform pps for some versions of mvl 4.1 the first of the constructed systems lnk(łukasiewicz’s negation ) with fixed “1” as the designated value, uses conjunction, disjunction, (1) implication, (1) negation and (1) exponent, as well as constants 𝛅𝛅= 𝒊𝒊 𝒌𝒌−𝟏𝟏 (1≤i≤k-2) for using (1) exponent [3,5,7]. in particular, for all formulas a, b, c of 3-valued logic and each 𝜎𝜎1, 𝜎𝜎2 from the set {0,1/2,1}, the following formulas are axiom schemes of ln3. 1. 𝐴𝐴 ⊃ (𝐵𝐵 ⊃ 𝐴𝐴) 2. (𝐴𝐴 ⊃ 𝐵𝐵) ⊃ (�𝐴𝐴 ⊃ (𝐵𝐵 ⊃ 𝐶𝐶)� ⊃ (𝐴𝐴 ⊃ 𝐶𝐶)) 3. 𝐴𝐴𝜎𝜎1 ⊃ (𝐵𝐵𝜎𝜎2 ⊃ (𝐴𝐴 ⊃ 𝐵𝐵)𝜎𝜎1⊃𝜎𝜎2) 4. 𝐴𝐴𝜎𝜎1 ⊃ (𝐵𝐵𝜎𝜎2 ⊃ (𝐴𝐴 ∨ 𝐵𝐵)𝜎𝜎1∨𝜎𝜎2) 5. 𝐴𝐴𝜎𝜎1 ⊃ (𝐵𝐵𝜎𝜎2 ⊃ (𝐴𝐴&𝐵𝐵)𝜎𝜎1&𝜎𝜎2) 6. 𝐴𝐴𝜎𝜎 ⊃ (¬𝐴𝐴)¬𝜎𝜎 7. (𝐴𝐴1 ⊃ 𝐵𝐵) ⊃ (�𝐴𝐴 1 2� ⊃ 𝐵𝐵� ⊃ �(𝐴𝐴0 ⊃ 𝐵𝐵) ⊃ 𝐵𝐵�) inference rule is modus ponens /m.p./𝐴𝐴,𝐴𝐴⊃𝐵𝐵 𝐵𝐵 . 4.2. the second systems cnk (cyclically permuting negation) with fixed “1” as the designated value, use conjunction, disjunction, (2) implication, (2) negation and (2) exponent [4,6,7]. in particular, for all formulas a, b, c of 3-valued logic and each 𝜎𝜎1, 𝜎𝜎2 from the set {0,1/2,1,} the following formulas are axiom schemes of cn3 1. 𝐴𝐴 ⊃ (𝐵𝐵 ⊃ 𝐴𝐴) 2. (𝐴𝐴 ⊃ 𝐵𝐵) ⊃ (�𝐴𝐴 ⊃ (𝐵𝐵 ⊃ 𝐶𝐶)� ⊃ (𝐴𝐴 ⊃ 𝐶𝐶)) 3. 𝐴𝐴𝜎𝜎1 ⊃ (𝐵𝐵𝜎𝜎2 ⊃ (𝐴𝐴 ⊃ 𝐵𝐵)𝜑𝜑⊃(𝐴𝐴,𝐵𝐵,𝜎𝜎1,𝜎𝜎2)) 4. 𝐴𝐴𝜎𝜎1 ⊃ �𝐵𝐵𝜎𝜎2 ⊃ (𝐴𝐴 ∨ 𝐵𝐵)𝜑𝜑∨(𝐴𝐴,𝐵𝐵,𝜎𝜎1,𝜎𝜎2)� 5. 𝐴𝐴𝜎𝜎1 ⊃ (𝐵𝐵𝜎𝜎2 ⊃ (𝐴𝐴&𝐵𝐵)𝜑𝜑&(𝐴𝐴,𝐵𝐵,𝜎𝜎1,𝜎𝜎2)) 6.𝐴𝐴𝜎𝜎1 ⊃ (𝐵𝐵𝜎𝜎2 ⊃ (𝐴𝐴𝐵𝐵)𝜑𝜑exp(𝐴𝐴,𝐵𝐵,𝜎𝜎1,𝜎𝜎2)) 7. 𝐴𝐴𝜎𝜎 ⊃ (¬𝐴𝐴)𝜎𝜎� 8. (𝐴𝐴 ⊃ 𝐵𝐵) ⊃ ((�̅�𝐴 ⊃ 𝐵𝐵) ⊃ ��̿�𝐴 ⊃ 𝐵𝐵� ⊃ 𝐵𝐵�), where the main role is played by the following exponents: on some universal propositional proof systems for many valued logic 26 𝜑𝜑⊃(𝐴𝐴, 𝐵𝐵, 𝜎𝜎1, 𝜎𝜎2) = (𝜎𝜎1 ⊃ 𝜎𝜎2)&(¬(𝐴𝐴⋁�̅�𝐴)⋁(𝐵𝐵� ⊃ 𝐵𝐵))⋁(¬�𝐴𝐴⋁�̿�𝐴�&¬(𝐵𝐵⋁𝐵𝐵�)), 𝜑𝜑∨(𝐴𝐴, 𝐵𝐵, 𝜎𝜎1, 𝜎𝜎2) = (𝜎𝜎1⋁𝜎𝜎2)⋁((𝐴𝐴 ⊃ �̅�𝐴)&¬(𝐵𝐵�⋁𝐵𝐵�))⋁(¬��̅�𝐴⋁�̿�𝐴�&(𝐵𝐵 ⊃ 𝐵𝐵�)), 𝜑𝜑&(𝐴𝐴, 𝐵𝐵, 𝜎𝜎1, 𝜎𝜎2) = (𝜎𝜎1&𝜎𝜎2)⋁((𝐴𝐴&�̿�𝐴)⋁(𝐵𝐵&𝐵𝐵�))⋁((𝐴𝐴&�̅�𝐴)⋁(𝐵𝐵&𝐵𝐵�) 𝜑𝜑exp(𝐴𝐴, 𝐵𝐵, 𝜎𝜎1, 𝜎𝜎2) = 𝜎𝜎1𝜎𝜎2⋁(¬(𝜎𝜎1𝜎𝜎2)&¬(¬(𝐴𝐴𝜎𝜎1&𝐵𝐵�𝜎𝜎2)⋁¬¬(𝐴𝐴𝜎𝜎1&𝐵𝐵�𝜎𝜎2))) inference rule is modus ponens /m.p./𝐴𝐴,𝐴𝐴⊃𝐵𝐵 𝐵𝐵 . 4.3. for ln3,2 łukasiewicz’s logic with fixed “1/2” and “1” as the designated value, which use conjunction, disjunction, (1) implication, (1) negation and (1) exponent, as well as constants 0, ½ and 1 for using (1)exponent. in particular for all formulas a, b, c of 3-valued logic and each 𝝈𝝈𝟏𝟏, 𝝈𝝈𝟐𝟐 from the set {0,1/2,1}, the following formulas are axiom schemes of ln3,2: 1. 𝐴𝐴 ⊃ (𝐵𝐵 ⊃ 𝐴𝐴) 2. (𝐴𝐴 ⊃ 𝐵𝐵) ⊃ (�𝐴𝐴 ⊃ (𝐵𝐵 ⊃ 𝐶𝐶)� ⊃ �𝐴𝐴 ⊃ 𝐶𝐶 1 2� �) 3. 𝐴𝐴𝜎𝜎1 ⊃ (𝐵𝐵𝜎𝜎2 ⊃ (𝐴𝐴 ⊃ 𝐵𝐵)𝜑𝜑⊃(𝐴𝐴,𝐵𝐵,𝜎𝜎1,𝜎𝜎2)) 4. 𝐴𝐴𝜎𝜎1 ⊃ �𝐵𝐵𝜎𝜎2 ⊃ (𝐴𝐴 ∨ 𝐵𝐵)𝜑𝜑∨(𝐴𝐴,𝐵𝐵,𝜎𝜎1,𝜎𝜎2)� 5. 𝐴𝐴𝜎𝜎1 ⊃ (𝐵𝐵𝜎𝜎2 ⊃ (𝐴𝐴&𝐵𝐵)𝜑𝜑&(𝐴𝐴,𝐵𝐵,𝜎𝜎1,𝜎𝜎2)) 6.𝐴𝐴𝜎𝜎1 ⊃ (𝐵𝐵𝜎𝜎2 ⊃ (𝐴𝐴𝐵𝐵)𝜑𝜑exp(𝐴𝐴,𝐵𝐵,𝜎𝜎1,𝜎𝜎2)) 7. 𝐴𝐴𝜎𝜎 ⊃ (¬𝐴𝐴)𝜎𝜎� 8. (𝐴𝐴1 ⊃ 𝐵𝐵) ⊃ (�𝐴𝐴 1 2� ⊃ 𝐵𝐵� ⊃ �(𝐴𝐴0 ⊃ 𝐵𝐵) ⊃ 𝐵𝐵�), where 𝜑𝜑⊃(𝐴𝐴, 𝐵𝐵, 𝜎𝜎1, 𝜎𝜎2) = �̅�𝐴⋁𝐵𝐵⋁𝜎𝜎1��⋁𝜎𝜎2 𝜑𝜑⋁(𝐴𝐴, 𝐵𝐵, 𝜎𝜎1, 𝜎𝜎2) = 𝐴𝐴⋁𝐵𝐵⋁𝜎𝜎1⋁𝜎𝜎2 𝜑𝜑&(𝐴𝐴, 𝐵𝐵, 𝜎𝜎1, 𝜎𝜎2) = (𝐴𝐴⋁𝜎𝜎1⋁𝐵𝐵𝜎𝜎2��)&(𝐵𝐵⋁𝜎𝜎2⋁𝐴𝐴𝜎𝜎1��) 𝜑𝜑𝑒𝑒𝑒𝑒𝑒𝑒(𝐴𝐴, 𝐵𝐵, 𝜎𝜎1, 𝜎𝜎2) = 𝐴𝐴𝐵𝐵⋁𝜎𝜎1𝜎𝜎2 𝜑𝜑−(𝐴𝐴, 𝐵𝐵, 𝜎𝜎1, 𝜎𝜎2) = �̅�𝐴⋁𝜎𝜎1�� and inference rule is modification of modus ponens 𝑨𝑨,𝑨𝑨⊃𝑩𝑩 𝑩𝑩𝝋𝝋𝒎𝒎.𝒑𝒑.(𝑨𝑨,𝑩𝑩) , where 𝝋𝝋𝒎𝒎.𝒑𝒑.(𝑨𝑨, 𝑩𝑩)=1/2. the work with another version of mvl and for k =4 or k =8 is in progress. 5. conclusion we suggest some hilbert-like propositional proof system, which is universal for all versions of many-valued logic. here we give also the application of the suggested system to 3 versions of 3valued logic, two of which have only one designated value and the last one has two designated values. the investigation of the applications to the other versions of k-valued logics (k≥4) is in progress. a. khamisyan 27 6. acknowledgements this work was supported by the ra mes state committee of science, in the frames of the research project № 18t-1b034. i am grateful to my supervisor, professor of ysu anahit chubaryan for her encouragement and fruitful discussions, and also for helpful suggestions, which enabled to improve and expand the results. references [1] j. lukasiewicz, “o logice trójwartósciowej”, ruch filozeficzny english translation: on three valued-logic, vol. 5, pp. 169-171, 1920. [2] e. mendelson, introduction to mathematical logic, van nostrand, princeton, 1975. [3] a. a. chubaryan, a. s. tshitoyan and a. a. khamisyan, “on some proof systems for many-valued logics and on proof complexities in it”, (in russian) reports of nasa ra, vol. 116, no. 2, pp. 18-24, 2016. [4] a. chubaryan and a. khamisyan, “generalization of kalmar’s proof of deducibility in two valued propositional logic into many valued logic”, pure and applied mathematics journal, doi: 10.116448/j.pamj. 20170602.12, vol. 6, no. 2, pp. 71-75, 2017. [5] a. chubaryan, a. khamisyan and a. tshitoyan, “on some systems for łukasiewicz’s many-valued logic and its properties”,fundamentalisscientiam, vol. 8, no. 8, spain, pp. 74-79, 2017. [6] а. чубарян и а. хамисян, новый метод доказательства полноты пропозициональной системы трехзначной логики лукасевича и его приложения, evolutio, естественные науки, вып. 3, сс. 9-12, 2016. [7] a.chubaryan, a.khamisyan and g. petrosyan, on some systems for two versions of many-valued logics and its properties, lambert academic publishing (lap), 2017. submitted 18.12.2019, accepted 09.03.2020. on some universal propositional proof systems for many valued logic 28 բազմարժեք տրամաբանության ասույթային հաշվի որոշ համապիտանի համակարգերի մասին արթուր ա. խամիսյան երևանի պետական համալսարան e-mail: khamisyam@gmail.com ամփոփում սույն հոդվածում առաջարկվում է հիլբերտյան տիպի համապիտանի արտածման համակարգ բազմարժեք տրամաբանության ասույթային հաշվի տարատեսակների համար, ինչպես նաև տրվում է այդ համապիտանի համակարգի ներկայացումը երեք տարբեր եռարժեք տրամաբանությունների համար, որոնցից երկուսը մեկ առանձնացված արժեքով են, իսկ վերջինը՝ երկու: բանալի բառեր` բազմարժեք տրամաբանություն, հիլբերտյան տիպի արտածման համակարգեր, ասույթային հաշվի համապիտանի արտածման համակարգ: о некоторых универсальных пропозициональных системах выводов для многозначных логик артур а. хамисян ереванский государственный университет e-mail:khamisyam@gmail.com аннотация в данной статье предлагается некоторая универсальная пропозициональная система гильбертовского типа для всех версий многозначных логик и рассматриваются ее приложения для трех версий 3-значных логик, две из которых имеют одно выделенное значение, а последняя – два выделенных значения. ключевые слова: многозначная логика, системы выводов гильбертовского типа, универсальная пропозициональная система выводов. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 46, 92--102, 2016. a model of a high level chess concept "attack on a weak square" nairi p. hakobyan and vachagan g. vahradyan institute for informatics and automation problems of nas ra e-mail: hakobyannairi@gmail.com, vachaganv@yandex.ru abstract subjective chess concept “attack on a weak square” was formalized. the task is divided into three terms – “weak square”, “square under attack”, “pin”. to analyze these concepts the mathematical apparatus of fuzzy logic is applied and an acceptable comparison between computer and human estimations of a "weak square" for certain positions is demonstrated. keywords: fuzzy logic, chess concepts, subjective concepts. “chess is the art of analysis” (mikhail bottvinnik) 1. introduction in this paper we are going to formalize a subjective chess concept “attack on a weak square” to approach the one used by chess players. why is it important? along with theoretical interest to the model it has also a practical value, since it can be used for a semantic search in databases of chess games, when the request is made on a real chess jargon [1]. exploitation of this method during chess player’s warm up for a tournament may radically improve the quality of player’s readiness and deepen comprehension of opponent’s tactics. furthermore, it can be useful for teaching the beginners. in order to implement the conceived it is necessary to formalize a huge amount of chess concepts and terms so that the machine could “translate” the chess player’s request from human language to an apprehensible one. since the concept of “weakness” is an entirely subjective concept, it has been chosen to apply fuzzy logic. why fuzzy? fuzzy sets have been used successfully in previous model constructions of chess concepts such as “beautiful mate”.[5] 92 mailto:hakobyannairi@gmail.com n. hakobyan and v. vahradyan 93 2. methodology “we must avoid creating weaknesses, find small ways to improve our pieces, and think small but never stop thinking. “ (garry kasparov) we will review each of the terms separately. 2.1 “weak squares”. concept of “weakness” in chess the conception "weak" in the term "weak square" as well as in any general use is an exclusionary subjective notion. each chess player perceives "weakness of square” in his own way, depending on its character and mindset. “weak squares” (or “weak points”) are squares which cannot be defended by a pawn. it is one of the key concepts in chess strategy and tactics. point is called weak for the player, if the player cannot attack it by his pawn. in contrast, this square is considered strong for an opponent. [2][3] in addition, the squares, which although may be attackеd by a pawn, but that attack is associated with a cutting deterioration of position, are considered to be weak. in a more general sense the square is weak if it is available for the opponent’s invasion. even if the definition of conception is exact, but it is associated with the deterioration of the position, it can be perceived differently by different players. for this reason it was decided to use fuzzy sets in order to get rid of the subjectivity, аlso, it helped us to distinguish the conception “very weak square”. review and evaluation of the “weak square” were divided into two different cases: if the square is empty and if there is a piece on that square. this concept is one of the basic ones, as it is usеd in many other concepts (for example: “a pin”). 2.2 “square under attack” let us define some of the concepts associated with the term:  those fields, which are threatened by the opponent’s piece are called “squares under attack”.  the opponent’s pieces that are on the “squares under attack” are called “pieces under attack”.  those fields, which are protected from the opponent’s attack by some “piece under attack” are called “potential squares under attack”. thereafter, if that field is not empty, a piece on it is called “a potential piece under attack”. the above defined concepts help us to define the concept of “pin” (also – “strong pin”, “weak pin” ...). 2.3 “pin” now we pass to the third concept, which will complete the general concept “attack on a weak square”. so what is “a pin”? in chess, a pin is a situation brought on by an attacking piece in which a defending piece cannot move without exposing a more valuable defending piece on its other side to capture by the attacking piece. [4] a model of a high level chess concept "attack on a weak square" 94 pin occurs only in a straight line – horizontally, vertically, diagonally. it can be created by rook, bishop or queen. the situations, when a chess player makes an opponent to sacrifice any piece by using pin, are of great interest. what about the situations, when the covered piece is the king? this kind of pin is called absolute. we can also describe the conception “pin” by using the above defined conceptions (“square under attack”, “piece under attack”, etc.). “pin” will appear on a board if two conditions are met:  appearance of “piece under attack” and “potential fields under attack” on the same line (or row).  the value of “potential field under attack” should be higher than “field under attack” that's why it is very important to know the “weakness” of related fields and covered pieces. thus, we use the concept of “weak field” in the concept of “pin” and by the help of it we can also differ “strong” and “weak” pins. 3. construction of fuzzy models 3.1 actual cost of piece “the most important feature of the chess position is the activity of the pieces. this is absolutely fundamental in all phases of the game: opening, middlegame and especially endgame”. (michael stean) as mentioned before, the evaluation of “weakness” of the field/square is being carried out with the help of fuzzy sets. to do this, we need to calculate somehow “the value” of every piece on the board. how can we do that? it is based on our fuzzy approach to the concept of "the presence of pieces on the board" allowing, particularly defining the concept of "actual cost of piece" as follows: 𝑉𝑉𝑅𝑅�𝑓𝑓𝑗𝑗 𝑐𝑐� = 𝑉𝑉𝑁𝑁�𝑓𝑓𝑗𝑗 𝑐𝑐� ∗ µ�𝑓𝑓𝑗𝑗 𝑐𝑐�, where 𝑉𝑉𝑁𝑁�𝑓𝑓𝑗𝑗 𝑐𝑐� is the nominal cost of j piece of color c and µ�𝑓𝑓𝑗𝑗 𝑐𝑐� – the membership function of fuzzy set. "the presence of pieces on the board" was defined as follows: µ(𝑓𝑓) = mr(𝑓𝑓) mt(𝑓𝑓) , where mr(𝑓𝑓) is the power of set of fields covered by the piece f in a given position and mt(𝑓𝑓) – the power of usual set of fields covered by the piece f on the center of empty chessboard. [5] the nominal value of square vary in different systems (for example the queen can have a nominal value of 8 or 9). the nominal value of the pieces was chosen as follows: piece nominal cost pawn 1 bishop/knight 3 rook 5 queen 9 [6] n. hakobyan and v. vahradyan 95 3.2 fuzzy approaches in actual cost of piece as it can be seen, the actual cost of the piece would be in the range of [-9;9] membership functions of the basic terms are presented in the chart and have the following analytical form: [5] 𝜇𝜇𝑡𝑡𝑡𝑡(𝑢𝑢) = 1 1 + �|𝑢𝑢−9| 9 � 9, where 𝑡𝑡𝑡𝑡 ∈ 𝑇𝑇 is an element of the term-set, u – the actual cost of piece in a range [-9;9] function graphic 3.3 fuzzy approaches for square's “weakness” evaluation now we will turn to the evaluation of “weak square”. as mentioned before the evaluation of the “weakness” would be conducted by the help of fuzzy approaches. thus, the evaluation of “weak square” is being calculated by two different ways in two different cases. first, we consider the squares, where the piece is present. for each “piece under attack” the actual cost of attacking and defending pieces is being calculated. further, we check which player would turn out to be in a worse situation than his opponent after the exchange of pieces on that field. the difference of actual costs of attacking and defending pieces will be considered as the evaluation of “weak square”. our actual cost is already in the [0;1] range, so the obtained evaluation would also be in that range. if the evaluation of the “field” is greater than 0.7 – call it “weak”. if it is even higher than 0.9 – call it “very weak” (the “weakness” is regarded from the point of view of the defending player). in the second case, empty squares are being reviewed. we would call the empty field weak, if the following conditions are met:  the attacking player has at least one piece that can capture that field (move there) and not be caught under attack of the opponent’s piece with a less actual cost.  during the next turn, the defending player cannot protect that field (by moving one of his pieces in order to attack the initial field) with a piece that has less actual cost than the opponent’s piece in the initial field. a model of a high level chess concept "attack on a weak square" 96 the idea is simple – if we cannot capture some field without bearing more damage than our opponent, that field will not be considered as “weak”. moreover, at least one of the following conditions should be satisfied:  if after moving to an empty field the attack on a piece with a greater (or equal) value appears???. in this case we have a situation equal to the one, where we considered “non-empty” squares.  if the action is limited for the opponent king. it is accepted among chess players that the restriction of king’s moves is a crucial move, so in this case we assume the field as “very weak”.  if a passed pawn appears. as in the previous case, the passed pawn is also considered as a very strong piece. once capablanca even said: “a passed pawn increases in strength as the number of pieces on the board diminishes”. in this case we also consider the field as “very weak”.  opponent piece with a higher actual cost then the attacking piece is died out as a result. now we have to compare the actual costs of both pieces and depending on their difference decide the field is “weak” or “very weak”. the mentioned limits are set aiming to avoid situations that do not have a significant effect on the game, i.e., represent an intermediate move. 3.4 pin we will use the above defined concept as a base for the formalization of “pin”. for each “piece under attack”, if available, we will find the “potential pieces under attack” and will calculate their actual costs. we calculate the difference of the actual values. if this difference is greater than 0.2 then the pin is called "strong." there is an extra situation, when the pinned piece is the opponent’s king. in that case nothing is needed to be calculated and the pin is called “absolute”. 4. the experiments let us illustrate the viability of the above models for the following chess positions. 4.1.1 weak square (case of non-empty field) firstly, consider “weak squares” from “squares under attack”. n. hakobyan and v. vahradyan 97 descry e5 square  pxe5 ... white took the black pawn, having the presence of pieces of 0.5 and actual cost 0.626.  ... pf6xe5 black took the white pawn, having the presence of pieces 0.75 and actual cost 0.686  nхе5 ... white took the black pawn, having the presence of pieces 0 and actual cost 0. black pawn on d6 square cannot take the knight on e5 square, since it will open check and because of this limitation the field e5 is considered to be “weak” for defending (black) side. evaluation of weakness = 0.686 – 0.626 = 0.06 4.1.2 weak square (case of empty field) now let us discuss the case when an empty field is “weak”. 1. сf5! ... we can see that both of the initial conditions of the “weakness” of an empty field are met. firstly, the empty square f5, which is not protected by a piece with less actual cost (in this case it a model of a high level chess concept "attack on a weak square" 98 is not protected by any piece), is being captured. on the other hand, black does not have any piece which can attack the square f5 during the next move. the condition d) from the list of secondary conditions is also met. black rook cannot go on the field h8 and g7 because of bishop’s presence. the black pawn on f7 also restricts the rook’s action, while the pawn’s movement is limited by the presence of the bishop. 4.2 pin this example illustrates two types of pins. knight on c6 is pinned by bishop on b5. this pin is “absolute”. actual cost of f6’s knight – 2.625, while actual cost of black queen – 3.214 (as we see, the queen’s limitation from different sides influenced its actual cost to be so little, but still greater than the knight’s one). after a transition to [0;1] range we will respectively get 0.959 and 0.901. the difference is 0.058, which illustrates the evaluation of “pin”. 4.3 comparison between computer and human estimation of “weak square” and “pin” we have tested the program all in all for 10 cases of “weak square” and 2 cases of “pin”. the table shows the answers of 2 first grade players and one master. in the first 10 cases, the answers are a bunch of squares that chess players called “weak” (under the condition that has been named who will play next). the key field is also being mentioned by players (the square where they would prefer to attack (by bold)). in the other 2 cases the answer contains in which field the pin is present. the results were compared as follows. n. hakobyan and v. vahradyan 99 table i: pos itions che ss players a. martirosyan v. vahradyan p. hakobyan program results 1 d5, f5 d5,f5,d6 a6,d5, d6, f5 d5, f5 2 e5, e6 e5,d5 e5, d5 е5 3 f3, g2, h3 f3,h3 f3, h3 f3, h3 4 h1, f3, d1 h1,e4 h1, d2, e4 e4 5 f3, e2, h3, g2 f3,h3 f3, h3 f3, h3 6 a5,c5,h6 c5,a5 c5 c5 7 f6,b6,d6 f6,d6 f6 d6, d8,f6 8 f5,g7 f5 f5 f5 9 e7, g7 e7,f7 e7,a7,f7,e8 е7 10 b7 b7 b7,a6,b5,c4 b7 11 е7 (absolute pin) е7(absolute pin) е7(absolute pin) е7(absolute pin) 12 c6(absolute pin) f6 (pin) c6(absolute pin) f6 (pin) c6(absolute pin) f6 (pin) c6(absolute pin) f6 (pin) what does this table show us? firstly, in 9 cases out of 10 the key field selected by chess players is present in the program output. the only bad option (4) is a classic case of the “weakness” appearance during a few turns (in this case, mate to white king), but we defined weaknesses only within one stroke in current position. this means that our algorithm can find the key field to play against in most cases. secondly, in 9 cases out of 10, all of the program outputs appear in chess players’ answers. the only issue is d8 field in #7 example. this means that our program basically does not show unnecessary / abnormal cases. thirdly, a considerable amount of squares in chess players’ answers are not shown by our program. the reason for it is, as already mentioned, that we have defined “weakness” for only one stroke. most of not-shown cases represent 3-4 stroke “weakness”. to sum up the discussed points, we can confirm that the program shows the acceptable result, apart from cases where weakness is detected in a few strokes. such cases will be considered in the future, with the help of the already defined conception “weakness”. further examples showcast the combined. a model of a high level chess concept "attack on a weak square" 100 table 2: 1(white) 2(white) 3(white) 4(white) 1) nd5 (condition а) 2) nf5 (condition а) 1) pхе5 after the exchanging white would have advantage 1) bh3 (condition а) 2) bf3 (condition b) 1) схе4 table 2: (continue) 5(черные) 6(белые) 7(белые) 8(белые) 1) bf3 (condition b) 2) bh3 (condition b) 1) rс5 (condition c) note if black’s move c4 square is “weak” by same condition 1) bd6 (condition b) 2) bd8 (condition b) 3) bf6 (condition d) 1) nf5 (condition а) 9(white) 10(white) 10 11 1) ле7 (condition a or condition c) 1) сb7 (condition а) 1) ch4! 2) фc7 3) cd8! in this case pin is «absolute» double pin n. hakobyan and v. vahradyan 101 5. conclusion to analyze the chess concept “attack on a weak field”, the mathematical model of fuzzy logic has been used. the proof of the admissibility of the results has been brought with the help of experiments with some high-level chess players’ participation. experiments demonstrated the viability of fuzzy theory in modeling complex human concepts. at present we continue to develop fuzzy models for chess concepts focusing, particularly, on ones accumulated in the repository of chess vocabulary for about 300 concepts in [7]. the next step may be to define the concept of "weakness in several passages" and "weak line" by using the concept of ”weakness”. furthermore, we are now implementing the already developed chess concepts (attack on the king, attack on weak field ...) into solver16 program, which allows us to insert problems of rgt class and relevant to them knowledge, also search for strategies based on that knowledge. to do that, firstly, we need to implement all of chess conceptions/terms used during the development of those conceptions (actual cost, the presence of pieces on the board, passed pawn ...). references [1] э. погосян, адаптация комбинаторных алгоритмов, нан ра, армения, ереван, 294 с., 1983 [2] j.r. capablanca, chess fundamentals, new york, ny: mckay chess library, 2005. [3] a. soltis, pawn structure chess, new york, ny: david mckay company, 1995. [4] h. golombek, golombek’s encyclopedia of chess, crown publishing, 1977. [5] v. g. vahradyan, “a model of a high level chess concept “beautiful mate””, 2013. [6] j. r. capablanca and n. de firmian, chess fundamentals, (completely revised and updated for the 21st century), random house, 2006 [7] e. pogossian, m. hambartsumyan and y. harutunyan, “a repository of units of chess vocabulary ordered by complexity of their interpretations”, national academy of sciences of armenia, ipia, pp. 1-55, 1974-1980. submitted 04.08.2016, accepted 12.11.2016. «գրոհ թույլ դաշտի վրա» բարձր մակարդակի շախմատային հասկացության մոդել ն. հակոբյան եւ վ. վահրադյան ամփոփում «գրոհ թույլ դաշտի վրա» շախմատային հասկացության վերլուծության համար ստեղծվել է մաթեմատիկական մոդել՝ հիմնված ոչ հստակ տրամաբանության տեսության վրա: դրա ընդունելիության ապացույցը բերվել է փորձերի օգնությամբ՝ a model of a high level chess concept "attack on a weak square" 102 բարձր մակարդակի մի քանի շախմատիստի մասնակցությամբ: փորձերը ցույց են տվել ոչ հստակ տրամաբանության ընդունելի կիրառությունը բարդ մարդկային հասկացությունների մոդելավորման հարցում: ներկայումս մենք շարունակում ենք զարգացնել ոչ հստակ մոդելները շախմատային հասկացությունների/տերմինների մոդելավորման մեջ՝ դիտարկելով, մասնավորապես, շախմատային բառարանշտեմարանի մեջ առկա 300 հասկացությունները: հաջորդ քայլը կարող է իրենից ներկայացնել «դաշտի թուլության» մոդելավորումը՝ հաշվի առնելով մի քանի քայլի կատարման հնարավորությունը, ինչպես նաև՝ «թույլ գծի» մոդելավորումը՝ օգտագործելով արդեն սահմանված «դաշտի թուլությունը»: ավելին, ներկայումս մենք ներմուծում ենք արդեն սահմանված հասկացությունները («գրոհ արքայի վրա», «գրոհ թույլ դաշտի վրա») solver16 ծրագրի մեջ, որը մեզ թույլ է տալիս ներմուծել rgt դասի խնդիրներ ևնրանց համապատասխան գիտելիքներ, ինչպես նաևկատարել անհրաժեշտ ստրատեգիաների փնտրում՝ հիմնված ներմուծած գիտելիքների վրա: այդ անելու համար մենք նախևառաջ պետք է ներմուծենք այդ ծրագրի մեջ բոլոր նախկինում սահմանված հասկացությունները, որոնք օգտագործվել են մեր հիմնական հասկացությունների մոդելավորման մեջ (իրական արժեք, անցողիկ զինվոր...): модель шахматного понятия высокого уровня “атака на слабое поле”. н. акопян и в. ваградян аннотация для анализа шахматного понятия “атака на слабое поле” была применена математическая модель нечеткой логики. доказательство приемлемости результатов было доведено с помощью экспериментов с участием некоторых шахматистов высокого уровня. эксперименты продемонстрировали удачный выбор теории нечетких множеств при моделировании сложных человеческих понятий. в настоящее время мы продолжаем развивать нечеткие модели для шахматных понятий, сосредоточив внимание, в частности, на тех примерно 300 терминов, которые накоплены в шахматном словаре. следующим шагом может быть определение понятия "слабости поля в нескольких ходах" и "слабой линии", используя уже определенное понятие "слабости". кроме того, сейчас мы внедряем уже разработанные шахматные понятия (нападение на короля, нападение на слабое поле ...) в программу solver16, которая позволяет внедрять проблемы класса rgt и соответствующие им знания, а также поиск стратегий, основанные на этих знаниях. для этого, во-первых, мы должны реализовать все шахматные концепции / термины, используемые в процессе разработки основных концепций (реальная стоимость, наличие фигур на доске, проходная пешка ...). d:\sbornik\...\rafayel.dvi mathematical problems of computer science 33, 24{34, 2010. n ew appr oach to fft algor ithms r a fa ye l b a r s e g h ya n a n d h a ko b s a r u kh a n ya n institute for informatics and automation problems of nas ra abstract in this paper we present a new, e±cient modi¯cation of split-radix algorithm for computing a power of two discrete fourier transforms. the developed algorithm allows to 40% real arithmetic operations reduction in comparison with previous best results for 16-point discrete fourier transform. refer ences [1 ] r . b la h u t , fa s t a lg o r it h m s fo r d ig it a l s ig n a l p r o c e s s in g . a d d is o n -w e s le y, 1 9 8 5 . [2 ] p . d u h a m e l a n d m. v e t t e r li, fa s t fo u r ie r t r a n s fo r m s : a t u t o r ia l r e vie w a n d a s t a t e o f t h e a r t , s ig n a l p r o c e s s in g a p r ., vo l. 1 9 , p p . 2 5 9 2 9 9 , 1 9 9 0 . [3 ] m. fr ig o a n d s . g. jo h n s o n , \ a m o d i¯ e d s p lit -r a d ix fft wit h fe we r a r it h m e t ic o p e r a t io n s " ,äie e e tr a n s . s ig n a l p r o c c e s s in g , vo l. 5 5 , p p . 1 1 1 -1 1 9 , 2 0 0 7 . [4 ] h . s a r u kh a n ya n , s . a g a ia n , \ co n ve n t io n a l, in t e g e r t o in t e g e r a n d qu a n t iz e d fa s t fo u r ie r tr a n s fo r m s " , cs it, p p . 2 0 4 -2 0 7 , 2 0 0 7 . [5 ] i. g. p e t r o vs ki, l e c t u r e s o n t h e t h e o r y o f o r d in a r y d i®e r e n t ia l e qu a t io n s , ( in r u s s ia n ) , 1 9 8 4 . [6 ] m. v e t t e r li, h . j. n u s s b a u m e r , \ s im p le fft a n d d ct a lg o r it h m s wit h r e d u c e d n u m b e r o f o p e r a t io n s " , signal p rocessing , vo l. 6 , n r . 4 , p p . 2 6 7 -2 7 8 , 1 9 8 4 . [7 ] r .b a r s e g h ya n , \ fft wit h lift in g t r a n s fo r m s " , 3-th annual conference, r au , ( s u b m it t e d ) , 2 0 0 9 . [8 ] r .b a r s e g h ya n , \ s p lit -r a d ix fft a lg o r it h m wit h lift in g s t e p s " , vestnik r au, series: p hysics-mathematics and natural science, p p . 4 2 -5 0 , 2 0 0 9 . üáñ ùáï»óáõù ü²ò ³é·áñçãùý»ñç è. ´³ñë»õû³ý ¨ ð. ê³ñáõë³ýû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ùß³ïí³í ¿ ïïñïí³í ñçùùáí üáõñû»ç ³ñ³· ó¨³÷áëáõãû³ý ýáñ, ³ñ¹ûáõý³í»ï ó¨³÷áëí³í ³é·áñçãùá: ó¨³÷áëí³í ³é·áñçãùç ñçù³ý íñ³ ûåïçù³é³óí»é ¿ 16-ã³÷³ýç í»ïïáñç ó¨³÷áëáõãûáõýá, áñç ñ»ï¨³ýùáí å³ñ³ýçíáõ çñ³ï³ý ãí³µ³ý³ï³ý ·áñíáõáõãûáõýý»ñç ù³ý³ïá ýí³½»é ¿ 40% -áí: 2 4 microsoft word tpel.doc mathematical problems of computer science 36, 28—40, 2012. 28 analysis of characteristics for cyclic business processes lusine s. tarumyan it educational and research center, yerevan state university abstract a new method for evaluation of business process characteristics is proposed, which is applicable at design stage when probabilities of process transitions are unknown. among a variety of publications on the process analysis, only a few address processes under unknown probabilities of transitions. the latter consider only acyclic processes, which restricts their application scope. the suggested method is based on a transformation of unstructured cycles into loop cycles. it gives an opportunity to represent a process via a hierarchy of loops and to apply dynamic programming methodology for characteristics analysis. it allows to extend the algorithms, initially developed for acyclic processes, for the analysis of cyclic processes, preserving at the same time the polynomial complexity of source algorithms. the introduced method has been used to develop effective algorithms for determination of the main characteristics of a business process (including time, cost, revenue, and profit). references [1] w. van der aalst, m. van h. kees, workflow management, models, methods and systems, the mit press cambridge, 2002. [2] h. jonkers, h.m. franken, “quantitative modeling and analysis of business processes”, simulation in industry, 8th european simulation symposium, vol.i, pp.175-179, 1996. [3] l. tarumyan, “timing analysis for workflow processes”, 17th european simulation symposium and exhibition within i3m'05 international mediterranean modeling multiconference, pp. 51-57, 2005. [4] w. van der aalst, j. desel, e. kindler, “on the semantics of epcs: a vicious circle”, workshop on epk, pp 7-18, 2003. [5] j. koehler, r. hauser, “untangling unstructured cyclic flows a solution based on continuations”, 6th int. conference on cooperative information systems coopis, springer lncs 3290, pp. 121-138, 2004. [6] l. tarumyan, “a system of performance characteristics analysis for business processes”, csit2011 eighth international conference on computer science and information technologies, pp. 198-201, 2011. [7] f. dirk, c. favre, b. jobstmann, j. koehler, n. lohmann, h. völzer, k. wolf, “instantaneous soundness checking of industrial business process models”, 7th int. conference on business process management, springer lncs 5701, pp. 278-293, 2009. [8] j. cardoso, a. sheth, j. miller, “workflow quality of service”, international conference on enterprise integration and modeling technology and international enterprise modeling conference (iceimt/iemc’02), 2002. [9] j. h. son, m. h. kim, “analyzing the critical path for the well-formed workflow schema”, seventh international conference on database systems for advanced applications dasfaa, pp.146-147, 2001. [10] j. eder, e. panagos, “managing time in workflow systems”, workflow handbook 2001, layne fischer ed., isbn 0-9703509-0-2, pp. 109-132, 2001. l. tarumyan 29 [11] f.e. allen, j. cocke, “a program data flow analysis procedure”, ibm thomas j. watson research center, 1971. [12] j. vanhatalo, h. volzer, f. leymann, “faster and more focused control-flow analysis for business process models though sese decomposition”, icsoc, vol. 4749, pp. 43–55, 2007. [13] j. vanhatalo, h. völzer, j. koehler, “the refined process structure tree”, data & knowledge engineering, v.68 n.9, pp.793-818, 2009. [14] r. bellman. dynamic programming, princeton university press, 1957. [15] workflow management coalition (wfmc), workflow standard – process definition interchange, process model. wfmc tc-1016-p.version 1.1, 1999. [16] f. leymann, d. roller, production workflow: concepts and techniques. prentice hall, 2000. [17] m. a schaefer, mathematical theory of global program optimization. prentice-hall,1973. [18] p. raulefs, s. shoukourian, l. tarumyan, v matevosyan, “determination of critical paths in hammock type processes”, hpc‘2003, scs international advanced simulation technologies conference astc’2003, pp. 241-246, 2003. ´ýáõã³·ñçãý»ñç í»ñéáõíáõãûáõýá óçïéçï µç½ý»ë åñáó»ëý»ñç ñ³ù³ñ è. â³éáõùû³ý ²ù÷á÷áõù ²é³ç³ñïí³í ¿ µç½ý»ë åñáó»ëý»ñç µýáõã³·ñçãý»ñç ·ý³ñ³ïù³ý ýáñ ù»ãá¹, áñá ïçñ³é»éç ¿ ý³ë³·íù³ý ÷áõéáõù` »ñµ åñáó»ëç õ»ï³í³ñù³ý ³ýóáõùý»ñç ñ³í³ý³ï³ýáõãûáõýý»ñá ³ýñ³ûï »ý: äñáó»ëý»ñç í»ñéáõíáõãû³ýá í»ñ³µ»ñíáõ ³éï³ ññ³å³ñ³ïáõùý»ñçó áý¹³ù»ýá ùç ù³ýçëý »ý ³ý¹ñ³¹³éýáõù õ»ï³í³ñù³ý ³ýóáõùý»ñç ³ýñ³ûï ñ³í³ý³ï³ýáõãûáõýý»ñáí åñáó»ëý»ñçý: ì»ñççýý»ñë ¹çï³ñïáõù »ý ùç³ûý ³óçïéçï åñáó»ëý»ñá, áñá ë³ñù³ý³÷³ïáõù ¿ ýñ³ýó ïçñ³éù³ý áéáñïá: ²é³ç³ñïí³í ù»ãá¹á ñ»ýí³í ¿ óçïé»ñç áã ëïñáõïïáõñç½³óí³íçó loop ó¨³÷áëù³ý íñ³: ¸³ ñý³ñ³íáñáõãûáõý ¿ ï³éçë åñáó»ëá ý»ñï³û³óý»é loop-»ñç ñç»ñ³ñëç³ûç ï»ëùáí ¨ µýáõã³·ñçãý»ñç í»ñéáõíáõãû³ý ñ³ù³ñ ïçñ³é»é ¹çý³ùçï íñ³·ñ³íáñù³ý ù»ãá¹á: ¸³ ñý³ñ³íáñáõãûáõý ¿ ï³éçë ³óçïéçï åñáó»ëý»ñç ñ³ù³ñ ùß³ïí³í í»éáõíáõãû³ý ³é·áñçãùý»ñá áý¹é³ûý»é óçïéçï åñáó»ëý»ñç ñ³ù³ñ` ùç¨ýáõûý å³ù³ý³ï å³ñå³ý»éáí ý³ëý³ï³ý ³é·áñçãùý»ñç µ³½ù³ý¹³ù³ûçý µ³ñ¹áõãûáõýá: ü»ñï³û³óí³í ù»ãá¹á û·ï³·áñíí³í ¿ µç½ý»ë åñáó»ëý»ñç ³é³çý³ûçý µýáõã³·ñçãý»ñç áñáßù³ý ¿ý»ïïçí ³é·áñçãùý»ñç ùß³ïù³ý ñ³ù³ñ (å³ù³ý³ï, í³ëë, »ï³ùáõï, ß³ñáõûã): analysis of characteristics for cyclic business processes 30 àíàëèç õàðàêòåðèñòèê öèêëè÷åñêèõ áèçíåñ-ïðîöåññîâ ë. òàðóìÿí àííîòàöèÿ ïðåäëîæåí íîâûé ìåòîä îöåíêè õàðàêòåðèñòèê áèçíåñ-ïðîöåññîâ, êîòîðûé ïðèìåíèì íà ýòàïå ïðîåêòèðîâàíèÿ, êîãäà âåðîÿòíîñòè ïåðåõîäîâ ïðîöåññà íåèçâåñòíû. ñðåäè ìíîæåñòâà ïóáëèêàöèé, ïîñâÿùåííûõ àíàëèçó ïðîöåññîâ, ëèøü íåìíîãèå ðàññìàòðèâàþò ïðîöåññû ñ íåèçâåñòíûìè âåðîÿòíîñòÿìè ïåðåõîäîâ. íî â ýòèõ ðàáîòàõ ðàññìàòðèâàþòñÿ òîëüêî àöèêëè÷åñêèå ïðîöåññû, ÷òî îãðàíè÷èâàåò îáëàñòü èõ ïðèìåíåíèÿ. ïðåäëàãàåìûé ìåòîä îñíîâàí íà ïðåîáðàçîâàíèè íåñòðóêòóðèðîâàííûõ öèêëîâ â loop öèêëû. ýòî äàåò âîçìîæíîñòü ïðåäñòàâèòü ïðîöåññ â âèäå èåðàðõèè loop öèêëîâ è ïðèìåíèòü ìåòîä äèíàìè÷åñêîãî ïðîãðàììèðîâàíèÿ äëÿ àíàëèçà õàðàêòåðèñòèê. ìåòîä ïîçâîëÿåò ðàñøèðèòü àíàëèòè÷åñêèå àëãîðèòìû, êîòîðûå ïåðâîíà÷àëüíî ðàçðàáîòàíû äëÿ àöèêëè÷åñêèõ ïðîöåññîâ, äëÿ öèêëè÷åñêèõ áèçíåñ-ïðîöåññîâ, ïðè ýòîì ñîõðàíÿÿ ïîëèíîìèàëüíóþ ñëîæíîñòü íà÷àëüíûõ àëãîðèòìîâ. ïðåäëîæåííûé ìåòîä áûë èñïîëüçîâàí äëÿ ðàçðàáîòêè ýôôåêòèâíûõ àëãîðèòìîâ îïðåäåëåíèÿ êëþ÷åâûõ õàðàêòåðèñòèê áèçíåñïðîöåññîâ (âðåìÿ, çàòðàòû, äîõîäû, ïðèáûëü). d:\sbornik\...\tpel.dvi mathematical problems of computer science 36, 57{62, 2012. computation of the complexity of some recur sive constr ucted n or mal p olynomials ma h m o o d a liz a d e h islamic azad universityahvaz branch e-mail: alizadeh@iauahvaz.ac.ir abstract in this paper we give some algorithms for computing the complexity of some normal polynomials constructed by some recurrent methods. finally some results of our algorithms are given in a table. keywords: complexity, irreducible polynomial, normal polynomial. refer ences [1 ] s . a b r a h a m ya n , \ s o m e c o n s t r u c t io n o f n-p o lyn o m ia ls o ve r ¯ n it e ¯ e ld s " , national academy of sciences of armenia r eports, vo l. 1 1 1 , n o . 3 , 2 3 2 -2 3 9 , 2 0 1 1 . [2 ] m. a liz a d e h , \ s o m e a lg o r it h m s fo r n o r m a lit y t e s t in g ir r e d u c ib le p o lyn o m ia ls a n d c o m p u t in g c o m p le xit y o f t h e n o r m a l p o lyn o m ia ls o ve r ¯ n it e ¯ e ld s , applied m athematical sciences, vo l. 6 , n o . 4 0 , 1 9 9 7 2 0 0 3 , 2 0 1 2 . [3 ] s . ga o .\ n o r m a l b a s e s o ve r ¯ n it e ¯ e ld s " , p h .d . th e s is , w a t e r lo o , 1 9 9 3 . [4 ] m. k . k yu r e g ya n , \ it e r a t e d c o n s t r u c t io n o f ir r e d u c ib le p o lyn o m ia ls o ve r ¯ n it e ¯ e ld s wit h lin e a r ly in d e p e n d e n t r o o t s " ,f inite f ields appl. , vo l. 1 0 , p p . 3 2 3 -3 4 1 , 2 0 0 4 . [5 ] m. k . k yu r e g ya n ," r e c u r s ive c o n s t r u c t io n s o f n -p o lyn o m ia ls o ve r gf ( 2 s ) " , d iscrete applied m athematics, vo l. 1 5 6 , p p . 1 5 5 4 -1 5 5 9 , 2 0 0 8 . [6 ] r . l id l a n d h . n ie d e r r e it e r . f inite f ields. ca m b r id g e u n ive r s it y p r e s s , 1 9 8 7 . [7 ] a . j. me n e z e s , i. f.b la ke , x .ga o , r . c. mu llin , s . a . v a n s t o n e a n d t. y a g h o o b ia n , applications of ¯nite ¯elds, k lu we r a c a d e m ic p u b lis h e r s , b o s t o n , d o r d r e c h t , l a n c a s t e r , 1 9 9 3 . [8 ] h . me yn , " e xp lic it n -p o lyn o m ia l o f 2 -p o we r d e g r e e o ve r ¯ n it e ¯ e ld s " , d esigns, codes and cryptography, vo l. 6 , p p . 1 4 7 -1 5 8 , 1 9 9 5 . [9 ] r . mu llin , i. on ys z c h u k, s . v a n s t o n e a n d r . w ils o n ,\ op t im a l n o r m a l b a s e s in gf ( pn ) n " ,d iscreate applied math.,vo l. 2 2 , ,1 4 9 -1 6 1 , 1 9 8 8 / 1 9 8 9 . [1 0 ] j. om u r a a n d j. ma s s e y, \ co m p u t a t io n a l m e t h o d a n d a p p a r a t u s fo r ¯ n it e ¯ e ld a r it h m a t ic " ,u .s . p a t e n t 4 ,5 8 6 ,6 2 7 , 1 9 8 6 . [1 1 ] i. on ys z c h u k, r . mu llin , a n d s . v a n s t o n e , \ co m p u t a t io n a l m e t h o d a n d a p p a r a t u s fo r m u lt ip lic a t io n " ,u .s p a t e n t 4 ,7 4 5 ,5 6 8 , 1 9 8 8 . 5 7 5 8 computation of the complexity of some recursive constructed normal polynomials è»ïáõñëçí ï³éáõóí³í áñáß ýáñù³é µ³½ù³ý¹³ùý»ñç µ³ñ¹áõãû³ý ñ³ßíáõùá ø. ²ñéç½³¹»ñ ²ù÷á÷áõù ²ûë ³ßë³ï³ýùáõù ù»ýù ³é³ç³ñïáõù »ýù áñáß³ïç ³é·áñçãù ñ³ßí»éáõ ñ³ù³ñ áñáß ýáñù³é µ³½ù³ý¹³ùý»ñç µ³ñ¹áõãûáõýá, áñáýù ï³éáõóí»é »ý é»ïáõñ»ýï »õ³ý³ïý»ñáí: ´»ñí³í ¿ ³õûáõë³ï, áñáõù ýßí³í ¿ áñáß ³ñ¹ûáõýùý»ñ: âû÷èñëåíèå ñëîæíîñòè íåêîòîðûõ ðåêóðñèâíî ïîñòðîåííûõ ïîëèíîìîâ ì. àëèçàäå àííîòàöèÿ â ýòîé ñòàòüå ìû ïðåäëàãàåì àëãîðèòìû âû÷èñëåíèÿ ñëîæíîñòè íîðìàëüíûõ ïîëèíîìîâ, ïîñòðîåííûõ îïðåäåëåííûìè ðåêóððåíòíûìè ìåòîäàìè. â çàêëþ÷åíèè ïðèâåäåíà òàáëèöà, êîòîðàÿ ñîäåðæèò íåêîòîðûå ðåçóëüòàòû. microsoft word tpel.doc mathematical problems of computer science 36, 99--103, 2012. 99 a processing algorithm for separation of cardiac and pulmonic activities reflected in microwave doppler spectra hayk g. hayrapetyan institute of radiophysics and electronics, armenian national ac. sci. alikhanian 1, ashtarak, 0203, armenia e-mail: haykh@irphe.am abstract the problem of azimuthal scanning of the human body by cw doppler radar is considered. a mathematical method and processing algorithm for separation of oscillation activities of various internal organs in the microwave doppler spectra is proposed. references [1] r.l.yadava , “rf/microwaves in bio-medical applications”, proc. of 8th international conference on electromagnetic interference and compatibility (incemic-2003), pp. 81-85, 2003. [2] j. lin and c. li , “wireless non-contact detection of heartbeat and respiration using lowpower microwave radar sensor”, asia-pacific microwave conference, apmc-2007, pp.1-4, 2007. [3] a. hakhoumian, h. hayrapetyan, s. martirosyan, a.muzhikyan, n. poghosyan and t. zakaryan, “l-band doppler radar for heartbeat sensing”, proc. of the int. conf. on “the teqnique of microwave and thz waves and its application in biomedical and radar technologies and in remote sensing”, pp. 91-93, 2010. [4] d.calcultt, l.tetley, satellite communications: principles and applications, butterworthheinemann, 1994. գբհ դոպլերային սպեկտրներում արտապատկերված շնչառական և սրտի ակտիվության տարանջատման ալգորիթմ հ. հայրապետյան ամփոփում դիտարկված է մարդու մարմնի ազիմուտալ սկանավորումը անընդատ գործողության ռադիոլոկացիոն կայանի միջոցով: առաջարկված է անդրադարձած գբհ ազդանշանի դոպլերային սպեկտրներում զանազան ներքին օրգանների տատանողական ակտիվությունների տարանջատման մաթեմատիկական եղանակը և ալգորիթմը: a processing algorithm for separation of cardiac and pulmonic activities reflected 100 алгоритм обработки разделения дыхательной и сердечной отраженных активностей в микроволновых допплеровских спектрах а. айрапетян аннотация рассмотрено азимутальное сканирование организма человека допплеровской рлс непрерывного действия. предложена математическая методика и алгоритм обработки разделения колебательных активностей различных внутренних органов в свч допплеровских спектрах отраженных сигналов. microsoft word i. karapetyan_12+.doc mathematical problems of computer science 34, 2010. 35 on some problems in graph theory iskandar a. karapetyan institute for informatics and automation problems some existence problems are studied related to cycles and paths with given properties, and some algorithms are found solving these problems. in particular, the existence problems are studied concerning hamiltonian graphs, pancyclic directed and vertex pancyclic directed graphs. it is proved that in any directed graph g of order p (p ≥13) with semidegrees at least (p-3)/2 every vertex belongs to a cycle of length k, k [9, p-1], for each integer k, and if p≥30 then this vertex belongs to cycles of lengths 6,7,8. as a corollary, one can find a cycle with given length and passing through given vertex by a polynomial time algorithm. a number of upper bounds are obtained for transversal and chromatic numbers for intersection graphs of families of rectangles with sides parallel to coordinate axes in the plane. some effective algorithms are given for one-layer and single-row problem, as well as for vlsi global routing problem. it is proved that each 3-connected graph g with (n+2k)/4 has a dominating cycle, and each 4-connected graph g either has a cycle of length at least 4-2k or has a dominating cycle, where n denotes the order,  the minimum degree and k the connectivity of g. some problems are studied by graph theorists under heading of r.kamalyan, concerning interval edge colorings, generalized interval edge colorings, interval total colorings, cycliccontinued edge colorings and local-balanced vertex partitions. it is proved the existence of cyclic-continued edge coloring for each tree and all possible values of a number of colors in such colorings are found. some problems are studied concerning the existence (as well as constructing and estimating of some parameters) of interval edge colorings for bipartite graphs, cylinders and tores, complete graphs and n-cubes, complete bipartite and some regular graphs. the idea of interval edge colorings is generalized. а particular case of a problem (erdos, 1991) concerning bipartite graphs without interval edge colorings is solved. references 1. с.х. дарбинян и и.а. карапетян, о вершинной панцикличности направленных графов с большими полустепенями. мат. воп. киб. и выч. тех., №29, с. 66-84, 2007. 2. s.kh. darbinyan and i.a. karapetyan, a note on short paths in oriented graphs, mathematical problems of computer sciences 33 (2010) 35-40. 3. и.а. карапетян и с.х. дарбинян, однарядная трассировка. csit conference, september 2007, yerevan, armenia. 4. и.а. карапетян и с.х. дарбинян, о двух задачах трассировки, computer sciense and information technologies, 28 september – 2 october, p. 439-442, 2009, yerevan, armenia. 36 5. zh.g. nikoghosyan, dirac-type generalizations concerning large cycles in graphs, discrete mathematics 309 (2009) 1925-1930. 6. m.zh. nikoghosyan, zh.g. nikoghosyan, large cycles in 4-connected graphs, discrete mathematics 311, no.4 (2011) 302-306. 7. r.r. kamalian, v.v. mkrtchyan, on special maximum matchings constructing, discrete mathematics 308, 2008, pp 1792-1800. 8. r.r. kamalian, on a number of colors in cyclically interval edge colorings of trees, lith-mat-r-2010/09-se, linkoping university, 2010. 9. p.a. petrosyan, interval edge-colorings of complete graphs and n dimensional cubes, discrete mathematics 310, 2010, pp. 1580-1587. 10. p.a. petrosyan, h.z. arakelyan, v.m. baghdasaryan, a generalization of interval edgecolorings of graphs, discrete applied mathematics 158, 2010, pp. 1827-1837. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 45, 44--52, 2016. performances of methods for solving a linear system of equations in the architecture of gpu accelerator hrachya v. astsatryan, edita e. gichunts institute for informatics and automation problems of nas ra e-mail: hrach@sci.am, editagich@ipia.sci.am abstract we consider some important issues related to the solution of linear system of equations that arise in multi-processor and graphics processing unit architecture. a more effective method for solving a linear system of equations is considered through the lu factorization. investigations are conducted in case of general complex matrices, because for those matrices the random butterfly transformation is used. the paper presents performances of several ways of solving methods on the graphic processor nvidia k40c. keywords: lu factorization, linear system of equations, random butterfly transformation, gepp, genp, magma, gpu accelerator. 1. introduction similar to lapack, magma [1, 2, 3] is being built as a community effort, incorporating the newest developments in hybrid algorithms and scheduling, and aiming at minimizing synchronizations and communication in these algorithms. the goal of these efforts is to redesign the dense linear algebra algorithms in lapack to fully exploit the power of current heterogeneous systems of multi/manycore cpus and accelerators, and deliver the shortest possible time to an accurate solution within the given energy constraints. indeed, the algorithms included so far in magma 1.6 manage to overcome bottlenecks associated with just multicore or gpus, to significantly outperform the corresponding packages for any of these components taken separately. for the linear system solvers on current multicore or gpu architectures, a bottleneck in terms of communication cost and parallelism comes from the pivoting, a technique used to prevent divisions by too-small numbers in the gaussian elimination (ge) process. current libraries like lapack implement ge using a block algorithm, which factors the input matrix by iterating over its blocks of columns (panels). pivoting not only requires communication (or 44 h.astsatryan, e.gichunts 45 synchronization in a shared memory environment), but it also limits the exploitation of asynchronicity between the block operations. the solution of linear system of algebraic equations has the form ax = b, where a is a square matrix, b is either a right side vector or a matrix, consisting of columns of the right sides. x is the solutions of system equations and it is a vector if b is a vector and it is a matrix, if b is a matrix. special block algorithms are used to solve the system equations. the system solution is divided into two parts: 1. system matrix factorization, 2. system solution through factorization. depending on the matrix feature the case of factorization is different. the following two cases of lu factorization are used for the general matrix: 1. the lu factorization with partial pivoting and row interchanges is used to factor a as a=plu, where p is a permutation matrix, l is a lower triangular matrix, the main diagonal elements of which are 1, and u is the upper triangular matrix 2. the lu factorization with no pivoting is used to factor a as a = lu, where l is the lower triangular unit, and u is the upper triangular one. several ways of solutions of linear system of equations on nvidia k40c gpu accelerator are presented in the paper which are carried out through the two mentioned cases of lu factorization. they are presented only for general matrices, because to get high performance for them, random butterfly transformation (rbt) was used which was repeatedly increasing the solution performance of the system of equations. note that rbt has been studied in the systems with multicore [4] and distributed memory [5], but its performance has not been studied on gpu accelerator. section 2 of the paper describes the cases of solving the linear system of equations. section 3 presents the performances of solutions of the mentioned cases on gpu accelerator. section 4 presents the conclusion. 2. solution methods 2.1 lu factorization with partial pivoting the lu factorization (or decomposition) of a matrix a has the form a = plu, where l is a unit lower triangular matrix, u is an upper triangular matrix and p is a permutation matrix. the block lu factorization algorithm [6] proceeds in the following steps: initially, a set of nb columns (the panel) is factored and a pivoting pattern is produced. then the elementary transformations, resulting from the panel factorization, are applied in block fashion to the remaining part of the matrix (the trailing submatrix). first, nb rows of the trailing submatrix are swapped, according to the pivoting pattern. then a triangular solve is applied to the top nb rows of the trailing submatrix. finally, matrix multiplication of the form aij←aij −aik ×akj is performed, where aik is the panel without the top nb rows, akj is the top nb rows of the trailing submatrix and aij is the trailing submatrix without the top nb rows. then the procedure is applied repeatedly, descending down the diagonal of the matrix. the solution of linear system of equations, where the lu factorization is made with partial pivoting, is performed by magma 1.6.1 library through cgesv and cgesv_gpu functions. the difference between these two functions is as follows: in the first case the function itself carries the matrices from the cpu to gpu and vice versa, while in the second case it is realized by the user. the solution sequence is as follows: performances of methods for solving a linear system of equations in the architecture of gpu accelerator 46  a and b matrices are transferred from the cpu to the global memory of gpu.  lu factorization of the matrix a is performed through cgetrf_gpu function of magma library using a partial pivoting with row interchanges.  a * x = b is solved through the function cgetrs_gpu of magma library.  x derived solutions are transferred from the gpu to cpu. to implement cgetrs_gpu function, claswp subprojects of lapackf77 library and ctrsm subprojects of magma library are used. the claswp routine swaps rows of the trailing submatrix according to the pivoting pattern, established in the panel factorization. this operation only performs data motion and the gpus are very sensitive to the matrix layout in memory. in raw-major layout, threads in a warp can simultaneously access consecutive memory locations. the ctrsm routine uses the lower triangle of the nb × nb diagonal block to apply triangular solve to the block of right-hand-sides formed by the top nb rows of the trailing submatrix. an efficient implementation of this routine on a gpu is difficult due to the data-parallel nature of gpus and small size of the solve (32 ≤ nb ≤ 288) [7]. the claswp routines are implemented in lapack [8], while the ctrsm routines are the part of the basic linear algebra subroutines (blas [9]) standard. lapack is an academic project and, therefore, the source code is freely distributed online. blas is a set of standardized routines, and it is available in commercial packages (e.g., mkl [10] from intel, acml [11] from amd, essl [12] from ibm), in academic packages (e.g., atlas [13]) and also as a reference implementation in fortran 77 from the netlib software repository. 2.2 lu factorization without pivoting the lu factorization (or decomposition) of a matrix a consists of writing of that matrix as a matrix product a = lu, where l is the lower triangular and u is the upper triangular. it is a central kernel in linear algebra because it is commonly used in many important operations such as solving a nonsymmetric linear system, inverting a matrix, computing a determinant or an approximation of a condition number. lu decomposition is an algebraic process that transforms a matrix a into a product of a lower triangular matrix l the elements of which are only on the diagonal and below, and an upper triangular matrix u the elements of which are only on the diagonal and above determinant and the inverse of a matrix. the solution of linear system of equations, where the lu factorization is made without pivoting, is performed by magma 1.6.1 library through cgesv_rbt and cgesv_nopiv_gpu functions. here also in the first case the function itself carries the matrices from the cpu to gpu and vice versa, while in the second case it is realized by the user. in case of xgesv_nopiv_gpu.cpp functions the solution sequence is as follows:  a and b matrices are transferred from the cpu to the global memory of gpu.  lu factorization of the matrix a is performed through cgetrf_nopiv_gpu function of magma library without any pivoting.  a * x = b is solved through the function cgetrs_nopiv_gpu of magma library.  x derived solutions are transferred from the gpu to cpu. to implement cgetrs_nopiv_gpu function, only the ctrsm subproject of magma library is used. the cgesv_rbt function is the optimized version of the solution of linear system of equations. the genp algorithm can be unstable due to a potentially large growth factor. this is why we systematically perform iterative refinement on the computed solution of the randomized system. algorithm 1 describes how the iterative refinement is performed in our implementations. h.astsatryan, e.gichunts 47 we improve the computed solution until we reach the required accuracy or we reach a defined maximum number of iterations. algorithm 1 iterative refinement. input: a the original matrix. input:b the right hand side. input:x the computed solution. input:l and u the factorized form of a input:n size of the matrix a result:an improved solution x 1: eps = machine precision 2: itermax = 30 3: iter = 0 4: anrm = ||a||∞ 5: xnrm = max|x| 6: cte = anrm * eps * √n 7: r = b – ax 8: rnrm = max|r| 9: while rnrm > xnrm and iter < itermax do 10: solve: ly = r 11: solve:uz = y 12: x = x+r 13: r = b – ax 14: xnrm = max|x| 15: rnrm = max|r| 16: iter = iter + 1 17: end while the iterative refinement process is stopped if iter > itermax or for all the rhs we have: rnrm < sqrt(n)*xnrm*anrm*eps*bwdmax where  iter is the number of the current iteration in the iterative refinement process  rnrm is the infinity-norm of the residual  xnrm is the infinity-norm of the solution  anrm is the infinity-operator-norm of the matrix a  eps is the machine epsilon returned by slamch('epsilon')  the values itermax and bwdmax are fixed to 30 and 1.0d+00, respectively. note that eps is determined by xlamch("epsilon") function of the lapackf77 library, and anrm is determined by the xlange() function of the magmablas library. the following consecutive steps are made for the solution of this case:  a and b matrices are transferred from the cpu to the global memory of gpu.  lu factorization of the matrix a is performed through cgetrf_nopiv_gpu function of magma library without any pivoting.  the cgesv_rbt function of the magma library is called. performances of methods for solving a linear system of equations in the architecture of gpu accelerator 48 note that the cgesv_rbt function takes the factorized a matrix and by the cgetrs_nopiv_gpu function it gets the solutions of linear system of equations, afterwards for iterative refinement it improves the computed solution to a system of linear equations. random butterfly transformation (rbt) is a randomization technique initially described by parker and recently revisited for dense linear systems. the method for randomizing has been described in [14, 15]. it consists of a multiplicative preconditioning utav where the matrices u and v are chosen among a particular class of random matrices called recursive butterfly matrices. then gaussian elimination with no pivoting (genp) is performed on the matrix utav and, to solve ax = b, we instead solve (utav)y= utb followed by x = vy. the solution of linear system of equations, where the random butterfly transformation is applied on a and b matrices and the lu factorization is made without pivoting, is implemented through the cgerbt_gpu function of the magma 1.6.1 library. the implementation of this form of solution is realized in the following sequence:  we generate the random matrices u and v in packed storage on the cpu.  the matrix a and the packed representation of u and v are sent from the host memory to the device memory.  randomization is performed on the gpu, updating a in the device memory.  perform partial random butterfly transformation on the gpu with magmablas_cprbt() function.  we compute utb on the gpu, ary= utb is solved on the gpu, followed by the solution x = vy with magmablas_cprbt_mtv() function.  the solution is sent to the host memory. 3. results of experiments the experiments were conducted on nvidia k40c gpu. the architecture of nvidia k40c consists of 2880 cuda processor cores. it is endowed with much higher bandwidth 288 gb/s of message transfer between cpu and gpu, having 12 gb of global memory, gddr5 memory interface, and cuda c programming environment. the operation system of k40c is ubuntu 14.04.2 lts. magma 1.6.1 package is installed. the code is compiled using the gnu gcc version 4.8, gfortran-4.8, g ++ 4.8 and the nvcc version 7.0 with the optimization flag -o3 and linked with the atlas library. figures 1 and 2(a,b) show the time and performance schedules of methods for solving of linear system of equations. the obtained results show that the performance of solutions defined by lu factorization without pivoting is higher than that with pivoting, especially when the solutions of random butterfly transformation are repeatedly endowed with high performance. the results show that the performance of solutions defined by cgesv_rbt function is the lowest of the mentioned cases but it is the optimized version of solutions because for iterative refinement it improves the computed solution to a system of linear equations. for the solutions defined by lu factorization with pivoting the cgesv_gpu function performance is higher than that of cgesv function. this is because the data have been loaded into the global memory of gpu before the function appeal. h.astsatryan, e.gichunts 49 fig. 1. fig. 2(a) 0 20 40 60 80 100 120 140 160 180 200 0 5000 10000 15000 20000 25000 cgesv cgesv_gpu cgesv_rbt cgesv_nopiv cgerbt_gpu n ti m e (s ec on ds ) 0 20000 40000 60000 80000 100000 120000 140000 160000 180000 200000 0 10000 20000 30000 cgesv cgesv_gpu cgesv_rbt cgesv_nopiv cgerbt_gpu g flo p/ s n performances of methods for solving a linear system of equations in the architecture of gpu accelerator 50 fig. 2(b) without cgerbt_gpu. 3. conclusion we presented methods for solving of linear system of equations on gpu accelerator where the lu factorization is performed with and without pivoting. the received performance results lead to the following conclusion that to achieve high performance in solutions of linear system of equations, random butterfly transformation solution method is definitely in the first place. references [1] r. nath, s. tomov and j. dongarra, “an improved magma gemm for fermi gpus”, international journal of high performance computing applications, vol. 24, no. 4, pp. 511–515, 2010. [2] s. tomov, j. dongarra and m. baboulin, “towards dense linear algebra for hybrid gpu accelerated manycore systems”, parallel computing, vol. 36(5&6), pp. 232– 240, 2010. [3] s. tomov, r. nath and j. dongarra, “accelerating the reduction to upper hessenberg, tridiagonal, and bidiagonal forms through hybrid gpu-based computing”, parallel computing, vol. 36, no. 12, pp. 645–654, 2010. [4] m. baboulin, d. becker and j. j. dongarra, “a parallel tiled solver for dense symmetric indefinite systems on multicore architectures”, parallel & distributed processing symposium (ipdps), 2012. [5] m. baboulin, d. becker, g. bosilca, a. danalis and j. j. dongarra, “an efficient distributed randomized algorithm for solving large dense symmetric indefinite linear systems”, parallel computing, vol. 40, no. 7 , pp. 212--223, 2014. [6] j. w. demmel, applied numerical linear algebra, siam, 1997. isbn: 0898713897 [7] j. kurzak, p. luszczek, m. faverge, and j. dongarra, “lu factorization with partial pivoting for a multicore system with accelerators”, ieee transactions on parallel and distributed systems, vol. 24, no. 8, pp. 1613—1621, 2013. 0 200 400 600 800 1000 1200 0 10000 20000 30000 cgesv cgesv_gpu cgesv_rbt cgesv_nopiv g flo p/ s n h.astsatryan, e.gichunts 51 [8] e. anderson, z. bai, c. bischof, s. blackford, j. demmel, j. dongarra, j. du croz, a. greenbaum, s. hammarling, a. mckenney and d. sorensen, lapack user’s guide, siam, 1999, third edition. [9] k. goto, gotoblas. texas advanced computing center, university of texas at austin, usa. http: // www. otc. utexas. edu/ atdisplay. jsp, 2007. [10] intel. math kernel library (mkl). http://www.intel.com/software/products/mkl/. [11] amd. amd core math library (acml). [online]. available: http://developer. amd. com/acml. jsp, 2012. [12] ibm corporation. ibm parallel engineering and scientific subroutine library. guide and reference. (gc23-3836), 1995. [13] r. c. whaley and j. dongarra. automatically tuned linear algebra software. technical report ut-cs-97-366, university of tennessee, december 1997. [online]. available: http://www.netlib.org/lapack/lawns/lawn131.ps. [14] d. s. parker, “random butterfly transformations with applications in computational linear algebra”, technical report csd-950023, ucla computer science department, 1995. [15] d. s. parker and b. pierce, “the randomizing fft: an alternative to pivoting in gaussian elimination”, technical report csd-950037, computer science department, ucla, 1995. submitted 04.09.2015, accepted 12.01.2016 գծային հավասարումների համակարգի լուծման մեթոդների արտադրողականությունները gpu արագագործչի ճարտարապետությունում հ. ասցատրյան, է. գիչունց ամփոփում մենք դիտարկում ենք գծային հավասարումների համակարգի լուծմանը վերաբերող որոշ կարևորագույն հարցեր, որոնք առաջանում են բազմապրոցեսորային և գրաֆիկական պրոցեսորային ճարտարապետությունում: դիտարկվում է lu վերլուծության միջոցով գծային հավասարումների համակարգի լուծման մեթոդներից ավելի արդյունավետ եղանակը: ուսումնասիրությունները կատարվում են կոմպլեքս ընդհանուր մատրիցների դեպքում, քանի որ այդ մատրիցների համար կիրառվում է թիթեռնիկի պատահական ձևափոխությունը: աշխատանքում ներկայացվում են մի քանի մեթոդներով լուծման եղանակների արտադրողականությունները nvidia k40c գրաֆիկական պրոցեսորի վրա: performances of methods for solving a linear system of equations in the architecture of gpu accelerator 52 производительности методов решения систем линейных уравнений в архитектуре gpu ускорителя г. асцатрян, э. гичунц аннотация рассмотрeны некоторые важные вопросы, связанные с решением систем линейных уравнений, возникающих в многопроцессорных и графических процессорных архитектурах. с помощью lu факторизации рассматривается более эффективный метод для решения линейной системы уравнений. исследования проводятся для случая общих комплексных матриц, потому что для этих матриц используется случайное преобразование бабочки. в статье представлены производительности методов нескольких способов решения на графическом процессоре nvidia k40c. 14_samve_darbinyan.dvi mathematical problems of computer science 39, 106{118, 2013. on cycles t hr ough ver tices of lar ge semidegr ee in digr aphs s a m ve l k h . d a r b in ya n a n d is ka n d a r a . k a r a p e t ya n institute for informatics and automation problems of nas ra e-mail: samdarbin@ipia.sci.am, isko@ipia.sci.am abstract let d be a strong digraph on n = 2m + 1 ¸ 5 vertices. in this paper we show that if d contains a cycle of length n¡1, then d has also a cycle which contains all vertices with in-degree and out-degree at least m (unless some extremal cases). keywords: digraphs, cycles, hamiltonian cycles, cyclability. 1 . in t r o d u c t io n th e d ig r a p h d is h a m ilt o n ia n if it c o n t a in s a h a m ilt o n ia n c yc le , i.e . a c yc le o f le n g t h jv ( d ) j. a s e t s o f ve r t ic e s in a d ig r a p h d ( a n u n d ir e c t e d g r a p h g) is s a id t o b e c yc la b le in d ( in g) if d ( g ) c o n t a in s a c yc le t h r o u g h a ll ve r t ic e s o f s. th e r e a r e m a n y we ll-kn o wn c o n d it io n s wh ic h g u a r a n t e e t h e c yc la b ilit y o f a s e t o f ve r t ic e s in a n u n d ir e c t e d g r a p h . mo s t o f t h e m c a n b e s e e n a s r e s t r ic t io n s o f h a m ilt o n ia n c o n d it io n s t o t h e c o n s id e r e d s e t o f ve r t ic e s ( s e e [4 , 5 , 1 5 , 1 6 , 1 8 ]) . h o we ve r , fo r g e n e r a l d ig r a p h s , r e la t ive ly fe w d e g r e e c o n d it io n s a r e kn o wn t o g u a r a n t e e h a m ilt o n is it y in d ig r a p h s ( s e e [2 , 3 , 7 , 9 , 1 3 , 1 4 , 1 7 , 1 9 ]) . th e m o r e g e n e r a l a n d c la s s ic a l o n e is t h e fo llo win g t h e o r e m o f m. me yn ie l: t heor em a [1 3 ]. if d is a strong digraph of order n ¸ 2 and d( x ) + d( y ) ¸ 2 n ¡ 1 for all pairs of nonadjacent vertices in d, then d is hamiltonian . in [8 ], t h e ¯ r s t a u t h o r p r o ve d t h e fo llo win g : t heor em b [8 ]. l et d be a strong digraph of order n ¸ 3 . if d( x ) + d( y ) ¸ 2 n ¡ 1 for any two non-adjacent vertices x; y 2 v ( d ) ¡ fz0g, where z0 is some vertex of d, then d is hamiltonian or contains a cycle of length n ¡ 1 . the following result is immediately corollary of theorem b . cor ollar y [8 ]. l et d be a strong digraph of order n ¸ 3 . if d has n ¡ 1 vertices of degree at least n, then d is hamiltonian or contains a cycle of length n ¡ 1 . a me yn ie l s e t m is a s u b s e t o f v ( d ) s u c h t h a t d( x ) + d ( y ) ¸ 2 n ¡ 1 fo r e ve r y p a ir o f ve r t ic e s x, y in m wh ic h a r e n o n a d ja c e n t in d. in [4 ], k . a . b e r m a n a n d x . l iu im p r o ve d th e o r e m b p r o vin g t h e fo llo win g g e n e r a liz a t io n o f t h e we ll-kn o wn me yn ie l's t h e o r e m . t heor em c [4 ]. l et d be a digraph of order n. if d is strongly connected, then every m eyniel set m lies in a cycle. th e o r e m c a ls o g e n e r a liz e s t h e c la s s ic a l t h e o r e m s a . gh o u ila -h o u r i [1 1 ] a n d d .r . w o o d a ll [1 9 ]. 1 0 6 s. darbinyan and i. karapetyan 1 0 7 th e d ig r a p h d is s-s t r o n g ly c o n n e c t e d if fo r a n y p a ir x; y o f d is t in c t ve r t ic e s o f s t h e r e e xis t s a p a t h fr o m x t o y a n d a p a t h fr o m y t o x in d ( s e e [1 2 ]) . h . l i, e . fla n d r in a n d j. s h u [1 2 ] p r o ve d t h e fo llo win g g e n e r a liz a t io n o f th e o r e m c. t heor em d [1 2 ]. l et d be a digraph of order n and m be a m eyniel set in d. if d is m-strongly connected, then d contains a cycle through all vertices of m. c. th o m a s s e n [1 7 ] ( fo r n = 2 k + 1 ) a n d t h e ¯ r s t a u t h o r [7 ] ( fo r n = 2 k ) p r o ve d t h e fo llo win g : t heor em e [1 7 , 7 ]. if d is a digraph of order n ¸ 5 with minimum degree at least n ¡ 1 and with minimum semi-degree at least n= 2 ¡ 1 , then d is hamiltonian (unless some extremal cases which are characterized). w e p u t a qu e s t io n t o kn o w if t h is r e s u lt o f c. th o m a s s e n a n d t h e ¯ r s t a u t h o r h a s a c yc la b le ve r s io n . l e t d b e a d ig r a p h o f o r d e r n = 2 m + 1 . a th o m a s s e n s e t t is a s u b s e t o f v ( d ) s u c h t h a t d+ ( x ) ¸ m a n d d¡ ( x ) ¸ m fo r e ve r y x 2 t , we d e n o t e t h e ve r t ic e s o f t b y t -ve r t ic e s . th e c yc le c o n t a in in g a ll ve r t ic e s o f t is c a lle d a t -c yc le . in t h is p a p e r we p r o ve t h e fo llo win g t wo t h e o r e m s wh ic h p r o vid e s o m e s u p p o r t fo r t h e a b o ve qu e s t io n . t heor em 1. l et d be a strong digraph of order n = 2 m + 1 ¸ 3 and d contains a cycle of length n ¡ 1 . then one of the following holds: i. d contains a cycle containing all vertices with in-degree and out-degree at least m; ii. d is isomorphic to digraphs d5 or d7 or belongs to the set l1 [ l2; iii. k¤m;m+1 µ d µ [km + km+1]¤; iv. d contains a cycle c := x1x2 : : : x2mx1 of length n ¡ 1 , and if x =2 v ( c ) and x is not adjacent to the vertices xl1; xl2 ; : : : ; xlj , j ¸ 3 , then xli¡1x; xxli+1 2 d and n + ( x ) = n + ( xli ) and n ¡ ( x ) = n¡ ( xli ) for all i 2 [1 ; j]. in particular, fxl1; xl2; : : : ; xlj ; xg is an independent set of vertices. t heor em 2. l et d be a 2-strong digraph of order n = 2 m + 1 ¸ 3 . then any two t -vertices x and y are on a common cycle in d. ou r p r o o fs a r e b a s e d o n t h e a r g u m e n t o f [1 7 , 7 ]. 2 . te r m in o lo g y a n d n o t a t io n s w e s h a ll a s s u m e t h a t t h e r e a d e r is fa m ilia r wit h t h e s t a n d a r d t e r m in o lo g y o n t h e d ir e c t e d g r a p h s ( d ig r a p h s ) a n d r e fe r t h e r e a d e r t o m o n o g r a p h o f b a n g -je n s e n a n d gu t in [1 ] fo r t e r m in o lo g y n o t d is c u s s e d h e r e . in t h is p a p e r we c o n s id e r ¯ n it e d ig r a p h s wit h o u t lo o p s a n d m u lt ip le a r c s . fo r a d ig r a p h d, we d e n o t e b y v ( d ) t h e ve r t e x s e t o f d a n d b y a ( d ) t h e s e t o f a r c s in d. th e o r d e r jdj o f d is t h e n u m b e r o f it s ve r t ic e s . oft e n we will wr it e d in s t e a d o f a ( d ) a n d v ( d ) . th e a r c o f a d ig r a p h d d ir e c t e d fr o m x t o y is d e n o t e d b y xy. fo r d is jo in t s u b s e t s a a n d b o f v ( d ) we d e ¯ n e a ( a ! b ) a s t h e s e t fxy 2 a ( d ) =x 2 a; y 2 bg a n d a ( a; b ) = a ( a ! b ) [ a( b ! a) . if x 2 v ( d ) a n d a = fxg we wr it e x in s t e a d o f fxg. if a a n d b a r e t wo d is jo in t s u b s e t s o f v ( d ) s u c h t h a t e ve r y ve r t e x o f a d o m in a t e s e ve r y ve r t e x o f b, t h e n we s a y t h a t a d o m in a t e s b, d e n o t e d b y a ! b. th e s u b d ig r a p h o f d in d u c e d b y a s u b s e t a o f v ( d ) is d e n o t e d b y hai. th e p a t h ( r e s p e c t ive ly, t h e c yc le ) c o n s is t in g o f t h e d is t in c t ve r t ic e s x1; x2; : : : ; xm ( m ¸ 2 ) a n d t h e a r c s xixi+1, i 2 [1 ; m ¡ 1 ] ( r e s p e c t ive ly, xixi+1, i 2 [1 ; m ¡ 1 ], a n d xmx1 ) , is d e n o t e d x1x2 ¢ ¢ ¢ xm ( r e s p e c t ive ly, x1x2 ¢ ¢ ¢ xmx1 ) . fo r a c yc le ck = x1x2 ¢ ¢ ¢ xkx1, t h e s u b s c r ip t s c o n s id e r e d m o d u lo k, i.e . xi = xs fo r e ve r y s a n d i s u c h t h a t i ´ s ( m o d k ) . if p is a p a t h c o n t a in in g a s u b p a t h fr o m x t o y we le t p [x; y] 1 0 8 on cycles through vertices of large semidegree in digraphs d e n o t e t h a t s u b p a t h . s im ila r ly, if c is a c yc le c o n t a in in g ve r t ic e s x a n d y, c[x; y] d e n o t e s t h e s u b p a t h o f c fr o m x t o y. fo r a n u n d ir e c t e d g r a p h g, we d e n o t e b y g¤ s ym m e t r ic d ig r a p h o b t a in e d fr o m g b y r e p la c in g e ve r y e d g e xy wit h t h e p a ir xy, yx o f a r c s . kn ( r e s p e c t ive ly, kp;q ) d e n o t e s t h e c o m p le t e g r a p h o f o r d e r n ( r e s p e c t ive ly, c o m p le t e b ip a r t it e g r a p h wit h p a r t it e s e t s o f c a r d in a lit ie s p a n d q ) , a n d kn d e n o t e s t h e c o m p le m e n t o f c o m p le t e u n d ir e c t e d g r a p h o f o r d e r n. two d is t in c t ve r t ic e s x a n d y a r e a d ja c e n t if xy 2 a ( d ) o r yx 2 a ( d ) ( o r b o t h ) . w e d e n o t e b y a ( x; y ) t h e n u m b e r o f a r c s b e t we e n t h e ve r t ic e s x a n d y. in p a r t ic u la r , a( x; y ) = 0 ( r e s p e c t ive ly, a( x; y ) 6= 0 ) m e a n s t h a t x a n d y a r e n o t a d ja c e n t ( r e s p e c t ive ly, a r e a d ja c e n t ) . fo r in t e g e r s a a n d b, a · b, le t [a; b] d e n o t e t h e s e t o f a ll in t e g e r s wh ic h a r e n o t le s s t h a n a a n d a r e n o t g r e a t e r t h a n b. 3 . p r e lim in a r ie s th e fo llo win g we ll-kn o wn s im p le le m m a s a r e t h e b a s is o f o u r r e s u lt s a n d o t h e r t h e o r e m s o n d ir e c t e d c yc le s a n d p a t h s in d ig r a p h s . th e y will b e u s e d e xt e n s ive ly in t h e p r o o fs o f o u r r e s u lt s . lemma 1 [1 0 ]. l et d be a digraph on n ¸ 3 vertices containing a cycle cm, m 2 [2 ; n¡ 1 ]. l et x be a vertex not contained in this cycle. if d ( x; cm ) ¸ m + 1 , then d contains a cycle ck for all k 2 [2 ; m + 1 ]. lemma 2 [6 ]. l et d be a digraph on n ¸ 3 vertices containing a path p := x1x2 : : : xm, m 2 [2 ; n ¡ 1 ] and let x be a vertex not contained in this path. if one of the following conditions holds: (i) d ( x; p ) ¸ m + 2 ; (ii) d ( x; p ) ¸ m + 1 and xx1 =2 d or xmx1 =2 d; (iii) d ( x; p ) ¸ m, xx1 =2 d and xmx =2 d then there is an i 2 [1 ; m ¡ 1 ] such that xix; xxi+1 2 d, i.e., d contains a path x1x2 : : : xixxi+1 : : : xm of length m (we say that x can be inserted into p or the path x1x2 : : : xixxi+1 : : : xm is extended from p with x ). 4 . p r o o f o f th e o r e m 1 h e r e we p r o ve o n ly th e o r e m 1 a n d fo r it we n e e d t h e fo llo win g d e ¯ n it io n s . de¯nition 1. d7 is a digraph (see [1, 17]) with the vertex set v ( d7 ) = fx1; x2; x3; x4; x5; x; yg such that n + ( x1 ) = fx2; x5; yg, n + ( x2 ) = fx3; x4; yg, n + ( x3 ) = fx2; x4; xg, n + ( x4 ) = fx3; x5; xg, n + ( x5 ) = fx1; x; yg, n + ( x ) = fx1; x2; x3g and n + ( y ) = fx1; x4; x5g. de¯nition 2. d5 is a digraph (see [1, 17]) with the vertex set v ( d5 ) = fx1; x2; x3; x; yg such that n + ( x1 ) = fx2; yg, n + ( x2 ) = fx3; xg, n + ( x3 ) = fx; yg, n + ( x ) = fx1; x2g and n + ( y ) = fx1; x3g. w e d e n o t e b y l1 t h e s e t o f t h r e e d ig r a p h s o b t a in e d fr o m d5 b y a d d in g t h e a r c x1x3 o r x3x1 ( o r b o t h ) . de¯nition 3. b y l2 we denote the set of digraphs d with the vertex set v ( d ) = fx1; x2; : : : ; x2m; xg and with the following properties: i. d contains a cycle x1x2 : : : x2mx1 of length 2 m and the vertices x and x2m are not adjacent; ii. n + ( x ) =n + ( x2m ) =fx1; x2; : : : ; xmg and n¡ ( x) =n ¡ ( x2m ) = fxm; xm+1; : : : ; x2m¡1g; s. darbinyan and i. karapetyan 1 0 9 iii. a ( fx1; x2; : : : ; xm¡1g ! fxm+1; xm+2; : : : ; x2m¡1g ) = ;, the induced subdigraphs hfx1; x2; : : : ; xmgi and hfxm; xm+1; : : : ; x2m¡1gi are arbitrary and one may add any number of arcs that go from fxm+1; xm+2; : : : ; x2m¡1g to fx1; x2; : : : ; xmg. (note that the digraphs from l2 are not 2-strong and x, x2m are t -vertices which are not in the common cycle. p r oof of t heor em 1. th e p r o o f is b y c o n t r a d ic t io n . s u p p o s e t h a t th e o r e m 2 is fa ls e , in p a r t ic u la r , d is n o t h a m ilt o n ia n . l e t c := x1x2 : : : xn¡1x1 b e a n a r b it r a r y c yc le o f le n g t h n ¡ 1 in d a n d le t t h e ve r t e x x is n o t c o n t a in in g t h is c yc le c. in fu r t h e r , b y h we d e n o t e a h a m ilt o n ia n c yc le in d. th e n x is a t -ve r t e x. s in c e c is a lo n g e s t c yc le , u s in g l e m m a s 1 a n d 2 , we o b t a in t h e fo llo win g c la im : claim 1. ( i) . d( x ) = n ¡ 1 a n d t h e r e is a ve r t e x xl, l 2 [1 ; n ¡ 1 ] wh ic h is n o t a d ja c e n t t o x. ( ii) . if xix =2 d, t h e n xxi+1 2 d a n d if xxi =2 d, t h e n xi¡1x 2 d, wh e r e i 2 [1 ; n ¡ 1 ]. ( iii) . if t h e ve r t ic e s x a n d xi a r e n o t a d ja c e n t , t h e n xi¡1x; xxi+1 2 d a n d d ( xi ) = n ¡ 1 . b y cla im 1 ( i) , wit h o u t lo s s o f g e n e r a lit y, we m a y a s s u m e t h a t t h e ve r t ic e s x a n d xn¡1 a r e n o t a d ja c e n t . fo r c o n ve n ie n c e , le t p := n ¡ 2 a n d y := xn¡1. w e h a ve yx1; xpy 2 d a n d xpx, xx1 2 d b y cla im 1 ( iii) . th e r e fo r e , y is a t -ve r t e x a n d d( y ) = n ¡ 1 . claim 2. a t le a s t t wo ve r t ic e s o f c a r e n o t a d ja c e n t t o x u n le s s d is is o m o r p h ic t o d5 o r d7 o r b e lo n g s t o t h e s e t l1 [ l2. p r oof. w e p r o ve cla im 2 b y c o n t r a d ic t io n . l e t c := x1x2 : : : xn¡1x1. th e n , b y l e m m a 1 , d ( x ) = n ¡ 1 a n d d+ ( x ) = d¡ ( x ) = m, s in c e d is n o t h a m ilt o n ia n . it is e a s y t o s e e t h a t s o m e ve r t e x xi ( s a y, y := xn¡1 ) is n o t a d ja c e n t t o x. th e n , b y cla im 1 ( iii) , xpx, xx1 2 d. if y is n o t a t -ve r t e x, t h e n t h e c yc le x1x2 : : : xn¡2yx1 c o n t a in s a ll t -ve r t ic e s . s o , we c a n a s s u m e t h a t y is a t -ve r t e x. th e n d( y ) = n ¡ 1 ( b y l e m m a 1 ) a n d d+ ( y ) = d¡ ( y ) = m. fr o m o u r a s s u m p t io n it fo llo ws t h a t n + ( x ) = fx1; x2; : : : ; xmg a n d n ¡ ( x ) = fxm; xm+1; : : : ; xpg: ( 1 ) w e ¯ r s t p r o ve t h a t t h e r e is a ve r t e x xk, k 2 [2 ; p ¡ 1 ], wh ic h is n o t a d ja c e n t t o y. a s s u m e t h a t it is n o t t h e c a s e . th e n n + ( y ) = fx1; x2; : : : ; xmg a n d n ¡ ( y ) = fxm; xm+1; : : : ; xpg: ( 2 ) s in c e d is n o t h a m ilt o n ia n we h a ve a( fx1; : : : ; xm¡1g ! fxm+1; : : : ; xpg ) = ;; ( 3 ) fo r o t h e r wis e , if xixj 2 d, wh e r e i 2 [1 ; m ¡ 1 ] a n d j 2 [m + 1 ; n ¡ 2 ], t h e n b y ( 1 ) a n d ( 2 ) , h = x1 : : : xixj : : : xpxxi+1 : : : xj¡1yx1 is a h a m ilt o n ia n c yc le . th e r e fo r e a ( fx1; : : : ; xm¡1; x; yg ! fxm+1; : : : ; xn¡2g ) = ;; i.e ., d b e lo n g s t o t h e s e t l2 wh ic h is a c o n t r a d ic t io n . th u s , t h e r e is a ve r t e x xk wit h k 2 [2 ; p ¡ 1 ] wh ic h is n o t a d ja c e n t t o y. b y cla im 1 ( iii) , xk¡1y; yxk+1 2 d. ob s e r ve t h a t xk a ls o is a t -ve r t e x. if k 2 [m + 1 ; p ¡ 1 ], t h e n m ¸ 3 a n d fr o m d¡ ( xk; fx; yg) = 0 it fo llo ws t h a t t h e r e is a ve r t e x xi, i 2 [1 ; m¡ 1 ], s u c h t h a t xixk 2 d. th e r e fo r e h = x1 : : : xixk : : : xpxxi+1 : : : xk¡1yx1, a c o n t r a d ic t io n . s o , we c a n a s s u m e t h a t k · m. s im ila r ly, we c a n a s s u m e t h a t k ¸ m. th e r e fo r e it r e m a in s t o c o n s id e r t h e c a s e wh e n m = k a n d t h e ve r t e x y is a d ja c e n t t o a ll ve r t ic e s o f p n fxmg. if n = 5 , i.e ., m = 2 , t h e n x1y; yx3 2 d a n d x2x1 =2 d, x3x2 =2 d, i.e ., d is is o m o r p h ic t o t h e we ll-kn o wn d ig r a p h 1 1 0 on cycles through vertices of large semidegree in digraphs d5 o r d 2 l, s in c e if we a d d t h e a r c x1x3 o r x3x1 ( o r b o t h ) t o d5, t h e n t h e r e s u lt in g d ig r a p h a ls o is n o t h a m ilt o n ia n , i.e ., d 2 l1. a s s u m e t h a t m ¸ 3 . it is n o t d i± c u lt t o s e e t h a t d( xm; fx1; xpg ) = 0 a n d a ( fx1; : : : ; xm¡2g ! xm ) = a ( xm ! fxm+2; : : : ; xpg ) = ;; ( 4 ) in p a r t ic u la r , xm is n o t a d ja c e n t t o x1 a n d xp. th e r e fo r e fxm+1; : : : ; xp¡1g ! xm ! fx2; : : : ; xm¡1g: ( 5 ) th is im p lie s t h a t xp a n d x1 a r e t -ve r t ic e s , s in c e x1 : : : xm¡1yxm+1 : : : xp¡1xmxx1 ( r e s p e c t ive ly, x2 : : : xm¡1y xm+1 : : : xpxxmx2 ) is a c yc le o f le n g t h n ¡ 1 wh ic h d o e s n o t c o n t a in xp ( r e s p e c t ive ly, x1 ) . n o w we c o n s id e r t h e ve r t e x y. if xp¡1y 2 d, t h e n xxp =2 d a n d yxp =2 d im p ly t h a t xixp 2 d fo r s o m e i 2 [1 ; m¡ 1 ], a n d h e n c e h = x1 : : : xixpxxi+1 : : : xp¡1yx1, a c o n t r a d ic t io n . s o , we c a n a s s u m e t h a t xp¡1y =2 d a n d , s im ila r ly, yx2 =2 d, i.e ., yxp¡1; x2y 2 d. u s in g l e m m a 2 , we o b t a in t h a t fx1; x2; : : : ; xm¡1g ! y ! fxm+1; xm+2; : : : ; xpg: ( 6 ) it is n o t d i± c u lt t o s e e t h a t d+ ( x1; p [x3; xm+1]) = 0 , fo r o t h e r wis e , if x1xi 2 d, i 2 [3 ; m], t h e n b y ( 1 ) a n d ( 6 ) , h = x1xi : : : xpxx2 : : : xi¡1yx1, a n d if x1xm+1 2 d, t h e n b y ( 1 ) , ( 5 ) a n d ( 6 ) , h = x1xm+1 : : : xpxxmx2 : : : xm¡1yx1, wh ic h is a c o n t r a d ic t io n . s im ila r ly, we c a n s h o w t h a t d¡ ( xp; p [xm¡1; xp¡2]) = 0 . th e r e fo r e n + ( x1 ) = fx2; y; xm+2; xm+3; : : : ; xpg a n d n¡ ( xp ) = fxp¡1; y; x1; x2; : : : ; xm¡2g: ( 7 ) b y ( 7 ) , ( 5 ) a n d ( 6 ) it is e a s y t o s e e t h a t x1 : : : xm¡2xpyxm+1 : : : xp¡1xmxx1 ( r e s p e c t ive ly, x1xm+2 : : : xpxxm x2 : : : xm¡1yx1 ) is a c yc le o f le n g t h n ¡ 1 , wh ic h d o e s n o t c o n t a in xm¡1 ( r e s p e c t ive ly, xm+1 ) . th is m e a n s t h a t xm¡1 a n d xm+1 a r e t -ve r t ic e s . n o w we will c o n s id e r t h e ve r t e x xm¡1. th e n xm¡1xi =2 d fo r a ll i 2 [m + 2 ; p] ( fo r o t h e r wis e , b y ( 5 ) , h = x1 : : : xm¡1xi : : : xpyxm+1 : : : xi¡1xmxx1 ) a n d xm¡1x1 =2 d ( fo r o t h e r wis e , h = x1 : : : xm¡2yxm+1 : : : xpxxmxm¡1x1 b y ( 5 ) a n d ( 6 ) ) . th u s , we h a ve d+ ( xm¡1; fx1; x; xm+2; : : : ; xpg ) = 0 . th e r e fo r e xm¡1 ! fx2; : : : ; xm¡2; y; xm; xm+1g: ( 8 ) n o w, if m ¸ 4 , t h e n b y ( 7 ) , ( 1 ) , ( 8 ) a n d ( 5 ) we h a ve h = x1xpxxm¡1xm+1 : : : xp¡1xmx2 : : : xm¡2yx1, wh ic h is a c o n t r a d ic t io n . th e r e fo r e m = 3 , i.e ., n = 7 . fr o m ( 4 ) , ( 5 ) a n d ( 7 ) we o b t a in t h a t x4x3; x3x2; x1x5 2 d, x1 a n d x5 a r e t -ve r t ic e s a n d d ( x3; fx1; x5g ) = 0 . it is e a s y t o s e e t h a t d+ ( x2; fx1; x5g ) = d+ ( x5; fx2; x4g ) = 0 . fr o m t h is we c o n c lu d e t h a t x5x1 2 d. n o w we s e e t h a t x1x5yx4x3xx1 is a c yc le o f le n g t h n ¡ 1 wh ic h d o e s n o t c o n t a in x2. th is m e a n s t h a t x2 is a t -ve r t e x a n d d+ ( x2 ) = d ¡ ( x2 ) = 3 . s in c e d + ( x2; fx; x1; x5g ) = 0 , it fo llo ws t h a t x2x4 2 d. th e r e fo r e d is is o m o r p h ic t o t h e d ig r a p h d7. cla im 2 is p r o ve d . claim 3. l e t xp¡1x; yxp 2 d a n d fo r s o m e k 2 [2 ; p ¡ 2 ] xk a n d y a r e n o t a d ja c e n t . th e n xk a n d xp a ls o a r e n o t a d ja c e n t . p r oof. s in c e xk a n d y a r e n o t a d ja c e n t it fo llo ws t h a t xk¡1y; yxk+1 2 d ( b y cla im 1 ( iii) ) . n o w if xkxp 2 d, t h e n h = x1 : : : xkxpyxk+1 : : : xp¡1xx1; a n d if xpxk 2 d, t h e n h = x1 : : : xk¡1yxpxk : : : xp¡1xx1. in e a c h c a s e we h a ve o b t a in e d a h a m ilt o n ia n c yc le , wh ic h is a c o n t r a d ic t io n . s. darbinyan and i. karapetyan 1 1 1 claim 4. if xp¡1x a n d yxp 2 d, t h e n d( xi; fx; yg ) ¸ 1 fo r a ll i 2 [2 ; p ¡ 2 ]. p r oof. s u p p o s e , o n t h e c o n t r a r y, t h a t d( xi; fx; yg ) = 0 fo r s o m e i 2 [2 ; p ¡ 2 ]. th e n b y cla im 1 ( iii) , xi¡1 ! fx; yg ! xi+1, a n d b y cla im 3 t h e ve r t ic e s xi a n d xp a r e n o t a d ja c e n t . n o w, s in c e xi is a t -ve r t e x a n d c a n n o t b e in s e r t e d in t o p [x1; xi¡1] a n d in t o p [xi+1; xp¡1], u s in g l e m m a 2 , we o b t a in t h a t p + 1 = d( xi ) = d( xi; p [x1; xi¡1]) + d( xi; p [xi+1; xp¡1]) · i + p ¡ i = p; a c o n t r a d ic t io n . claim 5. if xp¡1x 2 d, t h e n t h e ve r t ic e s y a n d xp¡1 a r e a d ja c e n t . p r oof. s u p p o s e , o n t h e c o n t r a r y, t h a t y a n d xp¡1 a r e n o t a d ja c e n t . th e n b y cla im 1 ( iii) , xp¡2y; yxp 2 d. if xpxp¡1 2 d, t h e n h = x1 : : : xp¡2yxpxp¡1xx1, a c o n t r a d ic t io n . s o , we c a n a s s u m e t h a t xpxp¡1 =2 d. mo r e o ve r , if xxi 2 d wit h i 2 [2 ; p ¡ 2 ], t h e n xi¡1xp¡1 =2 d ( fo r o t h e r wis e , we wo u ld h a ve a h a m ilt o n ia n c yc le h = x1 : : : xi¡1xp¡1xpxxi : : : xp¡2yx1 ) . r e c a ll ( b y cla im 2 ) t h a t t h e r e is a ve r t e x xl wit h l 2 [2 ; p ¡ 2 ] wh ic h is n o t a d ja c e n t t o x. n o t e t h a t xl¡1x a n d xxl+1 2 d b y cla im 1 ( iii) . s in c e x is a t -ve r t e x, it fo llo ws t h a t d+ ( x; p [x2; xp¡2]) ¸ m ¡ 2 . if we c o n s id e r t h e ve r t e x xp¡1, t h e n fr o m d¡ ( xp¡1; fy; xpg ) = 0 a n d t h e a b o ve o b s e r va t io n it fo llo ws t h a t xxp¡1 a n d xl¡1xp¡1 2 d: ( 9 ) h e n c e xpxl =2 d ( fo r o t h e r wis e , if xpxl 2 d, t h e n h = x1 : : : xl¡1xxp¡1xpxl : : : xp¡2yx1 ) . case 5.1. l · p ¡ 3 . th e n it is n o t d i± c u lt t o s e e t h a t t h e ve r t ic e s xl a n d xp¡1 a r e n o t a d ja c e n t . in d e e d , if xp¡1xl 2 d, t h e n h = x1 : : : xl¡1xp¡1xl : : : xp¡2yxpxx1 b y ( 9 ) ; a n d if xlxp¡1 2 d, t h e n h = x1 : : : xlxp¡1xpxxl+1 : : : xp¡2yx1, wh ic h is a c o n t r a d ic t io n . fr o m t h is , p + 1 = d( xl ) = d( xl; p [x1; xl¡1]) + d( xl; p [xl+1; xp¡2]) + d ( xl; fy; xpg ) : ( 1 0 ) n o w we s h o w t h a t xlxp a n d xp¡2xl 2 d: ( 1 1 ) l e t ¯ r s t yxl 2 d. th e n xlx1 =2 d ( fo r o t h e r wis e , h = x1 : : : xl¡1xxl+1 : : : xpyxlx1 is a h a m ilt o n ia n c yc le , a c o n t r a d ic t io n ) . s in c e t h e ve r t e x xl c a n n o t b e in s e r t e d in t o p [x1; xl¡1] a n d p [xl+1; xp¡2], fr o m ( 1 0 ) , xpxl =2 d a n d l e m m a 2 it fo llo ws t h a t d ( xl; p [x1; xl¡1]) = l ¡ 1 , d( xl; p [xl+1; xp¡2]) = p ¡ l ¡ 1 a n d xlxp; xp¡2xl 2 d. l e t n e xt yxl =2 d. s im ila r ly a s in t h e c a s e yxl 2 d we d e d u c e t h a t d( xl; p [xl+1; xp¡2]) = p ¡ l ¡ 1 a n d xlxp; xp¡2xl 2 d. ( 1 1 ) is p r o ve d . n o w u s in g ( 9 ) a n d ( 1 1 ) , we o b t a in a h a m ilt o n ia n c yc le h = x1 : : : xl¡1xp¡1xxl+1 : : : xp¡2xlxpyx1, wh ic h is a c o n t r a d ic t io n . case 5.2. l = p ¡ 2 . th e n xpxp¡2 =2 d a n d d ( xp¡2; fxp¡1; xpg ) · 2 . b y t h e c o n s id e r e d c a s e l · p ¡ 3 , w.l.o .g . we c a n a s s u m e t h a t t h e ve r t e x x is a d ja c e n t t o a ll ve r t ic e s o f p [x1; xp¡3]. th e n n + ( x) = fx1; x2; : : : ; xm¡1; xp¡1g a n d n¡ ( x ) = fxm¡1; xm; : : : ; xp¡3; xp¡1; xpg: ( 1 2 ) th is t o g e t h e r wit h fxp¡3; xp¡1; xpg ! x im p lie s t h a t m ¸ 3 a n d xx2 2 d. subcase 5.2.1. yx2 2 d. a s s u m e t h a t yxp¡2 =2 d. th e n d+ ( y; p [x2; xp¡3]) = m ¡ 2 s in c e y a n d xp¡1 a r e n o t a d ja c e n t . fr o m t h is a n d d ¡ ( xp¡2; fx; y; xpg ) = 0 it fo llo ws t h a t xixp¡2; yxi+1 2 d fo r s o m e i 2 [1 ; p¡ 4 ]. th e r e fo r e h = x1 : : : xixp¡2xp¡1xpyxi+1 : : : xp¡3xx1, wh ic h is a c o n t r a d ic t io n . s o , we c a n a s s u m e t h a t yxp¡2 2 d. n o w it is e a s y t o s e e t h a t x1 1 1 2 on cycles through vertices of large semidegree in digraphs a n d xp¡2 a r e n o t a d ja c e n t . in d e e d , if x1xp¡2 2 d, t h e n h = x1xp¡2xp¡1xpyx2 : : : xp¡3xx1; a n d if xp¡2x1 2 d, t h e n h = x1 : : : xp¡3xxp¡1xpyxp¡2x1; wh ic h is a c o n t r a d ic t io n . s in c e xp¡2 c a n n o t b e in s e r t e d in t o p [x2; xp¡3], b y l e m m a 2 we h a ve d ( xp¡2; p [x2; xp¡3]) · p ¡ 3 . on t h e o t h e r h a n d , p + 1 = d( xp¡2 ) = d( xp¡2; p [x2; xp¡3]) + d ( xp¡2; fxp¡1; xpg ) + a ( xp¡2; y ) im p lie s t h a t d ( xp¡2; p [x2; xp¡3]) = p ¡ 3 . h e n c e , b y l e m m a 2 , xp¡2x2 2 d a n d x2 : : : xp¡3xxp¡1xpyxp¡2 x2 is a c yc le o f le n g t h n ¡ 1 wh ic h d o e s n o t c o n t a in x1. th e r e fo r e x1 is a t -ve r t e x. n o w we c o n s id e r t h e ve r t e x x1. ob s e r ve t h a t if x1xi 2 d, i 2 [m; p ¡ 2 ], t h e n b y ( 1 2 ) , h = x1xi : : : xpyx2 : : : xi¡1xx1; a n d if x1xp¡1 2 d, t h e n h = x1xp¡1xpyxp¡2x2 : : : xp¡3xx1, a c o n t r a d ic t io n . th e r e fo r e d + ( x1; fx; y; xm; xm+1; : : : ; xp¡1g ) = 0 wh ic h c o n t r a d ic t s t h a t x is a t -ve r t e x. subcase 5.2.2. th e ve r t ic e s x2 a n d y a r e n o t a d ja c e n t . th e n x1y; yx3 2 d b y cla im 1 ( iii) , a n d b y cla im 3 t h e ve r t ic e s x2 a n d xp a ls o a r e n o t a d ja c e n t . ob s e r ve t h a t if xix 2 d wit h i 2 [3 ; p ¡ 1 ], t h e n x2xi+1 =2 d ( fo r o t h e r wis e , h = x1x2xi+1 : : : xpyx3 : : : xixx1 ) . fr o m t h is we h a ve , if x2x =2 d, t h e n d¡ ( x; p [x3; xp¡1]) = m ¡ 1 a n d a t le a s t m + 2 ve r t ic e s a r e n o t d o m in a t e d b y x2 s in c e d + ( x2; fy; x; x1g ) = 0 , wh ic h c o n t r a d ic t s t h a t x2 is a t -ve r t e x. s o , we c a n a s s u m e t h a t x2x 2 d. s in c e t h e ve r t e x x is a d ja c e n t t o a ll ve r t ic e s o f p [x1; xp¡3] it fo llo ws t h a t m = 3 . n o t e t h a t x2x4 2 d b y ( 9 ) , a n d x2; x3; x4 a r e t -ve r t ic e s . it is e a s y t o s e e t h a t d+ ( x2; fx1; x5; yg ) = d+ ( x3; fx; x1; x2g ) = d+ ( x4; fy; x3; x1g ) = d¡ ( x3; fx; x4; x5g ) = 0 : th e r e fo r e x3x5; x4x2; x1x3 2 d. s in c e x1yx3x4x2xx1 ( r e s p e c t ive ly, x2x3yx5xx4x2 ) is a c yc le o f le n g t h n ¡ 1 = 6 , it fo llo ws t h a t x5 ( r e s p e c t ive ly, x1 ) is a t -ve r t e x. n o w fr o m d+ ( x5; fx2; x3; x4g ) = d¡ ( x1; fx2; x3; x4g ) = 0 we h a ve x5x1 2 d. th e r e fo r e , d is is o m o r p h ic t o t h e we ll-kn o wn d ig r a p h d7 o r is h a m ilt o n ia n , a c o n t r a d ic t io n t o o u r a s s u m p t io n . subcase 5.2.3. x2y 2 d a n d yx2 =2 d. th e n b y cla im 1 ( ii) we h a ve x1y 2 d a n d t h e r e is a ve r t e x xk t o k 2 [3 ; p ¡ 3 ] wh ic h is n o t a d ja c e n t t o y ( s in c e m ¸ 3 ) . th e n xk¡1y a n d yxk+1 2 d b y cla im 1 ( iii) . u s in g cla im 3 , we o b t a in t h a t xk is n o t a d ja c e n t t o x1 a n d xp. s in c e xk c a n n o t b e in s e r t e d in t o p [x2; xk¡1] a n d p [xk+1; xp¡1], a p p lyin g l e m m a 2 t o t h e s e p a t h s , we o b t a in t h a t d( xk; p [x2; xk¡1]) · k ¡ 1 ; d ( xk; p [xk+1; xp¡1]) · p ¡ k; p + 1 · d ( xk ) = d ( xk; p [x2; xk¡1]) + d ( xk; p [xk+1; xp¡1]) + a ( xk; x ) a n d a( xk; x ) = 2 ( in o t h e r wo r d s xxk; xkx 2 d ) a n d e a c h in e qu a lit y is , in fa c t , a n e qu a lit y. h e n c e , b y l e m m a 2 , xkx2; xp¡1xk 2 d. fr o m xxk; xkx 2 d we o b t a in t h a t n + ( x ) = fx1; x2; : : : ; xk; xp¡1g a n d n¡ ( x ) = fxk; xk+1; : : : ; xp¡3; xp¡1; xpg a n d x1 : : : xk¡1yxk+1 : : : xp¡1xkxx1 is a c yc le o f le n g t h n ¡ 1 . th e r e fo r e xp is a t -ve r t e x a n d k = m¡ 1 . n o w we will c o n s id e r t h e ve r t e x xp. th e n xpxi =2 d fo r a ll i 2 [k; p ¡ 1 ] [f2 g ( fo r o t h e r wis e , h = x1x2 : : : xi¡1xxp¡1xpxi : : : xp¡2yx1 wh e n i 2 [k + 1 ; p ¡ 2 ]; a n d h = x1 : : : xi¡1yxpxi : : : xp¡1xx1 wh e n i = 2 ; k; p ¡ 1 wh ic h is a c o n t r a d ic t io n ) . th u s , we h a ve t h a t t h e ve r t e x xp d o e s n o t d o m in a t e a t le a s t m + 1 ve r t ic e s , wh ic h is a c o n t r a d ic t io n , s in c e xp is a t -ve r t e x. th is c o n t r a d ic t io n c o m p le t e s t h e p r o o f o f cla im 5 . b y cla im 2 t h e r e is a ve r t e x xl, wh e r e l 2 [2 ; p ¡ 1 ], wh ic h is n o t a d ja c e n t t o x, a n d b y cla im 1 ( iii) , xl¡1x; xxl+1 2 d. remar k 1. l e t a ve r t e x xk, wh e r e k 2 [2 ; p¡ 1 ], is n o t a d ja c e n t t o t h e ve r t ic e s x a n d y ( in o t h e r wo r d s d ( xk; fx; yg ) = 0 ) . th e n xpxk; xkx1 2 d a n d n ¡ ( x ) = n ¡ ( y ) , n + ( x) = n + ( y ) . s. darbinyan and i. karapetyan 1 1 3 b y cla im 1 ( iii) , xk¡1 ! fx; yg ! xk+1, xk is a t -ve r t e x a n d xk c a n n o t b e in s e r t e d in t o p [x1; xk¡1] a n d p [xk+1; xp]. u s in g l e m m a 2 , we o b t a in t h a t d( xk; p [x1; xk¡1]) · k a n d d( xk; p [xk+1; xp]) · p ¡ k + 1 ; p + 1 = d ( xk ) = d ( xk; p [x1; xk¡1]) + d ( xk; p [xk+1; xp]) · p + 1 : th e r e fo r e , e a c h in e qu a lit y is , in fa c t , a n e qu a lit y. h e n c e , b y l e m m a 2 , xpxk; xkx1 2 d. n o w we s h o w t h a t n ¡ ( x ) = n¡ ( y ) a n d n + ( x) = n + ( y ) . a s s u m e t h a t t h is is n o t t h e c a s e . l e t xix 2 d a n d xiy =2 d. th e n xi =2 fxk¡1; xpg, a n d b y cla im 1 ( ii) , yxi+1 2 d. s in c e xkx1; xkxp 2 d, it is n o t d i± c u lt t o s e e t h a t h = x1x2 : : : xixxk+1 : : : xpyxi+1 : : : xkx1 wh e n i < k ¡ 1 a n d h = x1x2 : : : xk¡1yxi+1 : : : xpxk : : : xixx1 wh e n i > k, a c o n t r a d ic t io n . to s h o w t h a t n + ( x ) = n + ( y ) it s u ± c e s t o c o n s id e r t h e c o n ve r s e d ig r a p h o f d. claim 6. d+ ( xp¡1; fx; yg ) · 1 . p r oof. s u p p o s e , o n t h e c o n t r a r y, t h a t xp¡1x a n d xp¡1y 2 d. th e n l · p ¡ 2 . s in c e d is n o t h a m ilt o n ia n it fo llo ws t h a t if xxi+1 2 d o r yxi+1 2 d, t h e n xixp =2 d. th is t o g e t h e r wit h d¡ ( xp; fx; yg ) = 0 a n d d+ ( x; p [x2; xp¡1]) = m ¡ 1 , im p lie s t h a t a t le a s t m + 1 ve r t ic e s d o n o t d o m in a t e xp. cle a r ly, xp is n o t t -ve r t e x. w e will d is t in g u is h t h r e e c a s e s a c c o r d in g a s xly 2 d o r xly =2 d a n d yxl 2 d o r xl a n d y a r e n o t a d ja c e n t . case 6.1. xly 2 d. th e n d¡ ( xl; fxp; xp¡1g ) = 0 ( fo r o t h e r wis e , if xpxl 2 d, t h e n h = x1 : : : xl¡1xxl+1 : : : xpxlyx1; a n d if xp¡1xl 2 d, t h e n x1 : : : xl¡1xxl+1 : : : xp¡1xlyx1 is a t -c yc le , a c o n t r a d ic t io n ) . s o , b y t h e a b o ve o b s e r va t io n we h a ve t h a t xp a n d xl a r e n o t a d ja c e n t . s in c e xp¡1xl =2 d a n d t h e ve r t ic e s xl c a n n o t b e in s e r t e d in t o p [x1; xl¡1] a n d p [xl+1; xp¡1], u s in g l e m m a 2 , we o b t a in t h a t d ( xl; p [x1; xl¡1]) · l a n d d( xl; p [xl+1; xp¡1]) · p ¡ l ¡ 1 : th e r e fo r e p + 1 = d ( xl ) = d( xl; p [x1; xl¡1]) + d ( xl; p [xl+1; xp¡1]) + a( xl; y ) : fr o m t h is we c o n c lu d e t h a t yxl 2 d a n d e a c h in e qu a lit y is , in fa c t , a n e qu a lit y. h e n c e , b y l e m m a 2 , xlx1 2 d a n d h = x1 : : : xl¡1xxl+1 : : : xpyxlx1, wh ic h is a c o n t r a d ic t io n . case 6.2. xly =2 d a n d yxl 2 d. th e n xlx1 =2 d ( fo r o t h e r wis e , h = x1 : : : xl¡1xxl+1 : : : xpyxlx1 ) a n d fr o m d( y ) = n ¡ 1 b y cla im 1 ( ii) we h a ve , yxl+1 2 d. s in c e xl c a n n o t b e in s e r t e d in t o p [xl+1; xp] a n d in t o p [x1; xl¡1], u s in g l e m m a 2 , we o b t a in t h a t d ( xl; p [x1; xl¡1]) = l ¡ 1 a n d d( xl; p [xl+1; xp]) = p ¡ l + 1 ; a n d xpxl 2 d. b y cla im 2 t h e r e is a ve r t e x xk, wh e r e k 2 [2 ; p ¡ 2 ], wh ic h is n o t a d ja c e n t t o y. th e n xk¡1y; yxk+1 2 d ( b y cla im 1 ( iii) ) a n d xk is a t -ve r t e x. w e c a n a s s u m e t h a t xkx =2 d ( fo r o t h e r wis e , fo r t h e ve r t e x y we wo u ld h a ve ca s e 6 .1 ) . a s s u m e ¯ r s t t h a t k · l ¡ 1 . th e n fr o m xkx =2 d it fo llo ws t h a t k · l ¡ 2 . n o w we will c o n s id e r t h e ve r t e x xk. it is e a s y t o s e e t h a t xkxp =2 d s in c e d is n o t h a m ilt o n ia n . s in c e xp is n o t t -ve r t e x a n d yxl 2 d it fo llo ws t h a t if xpxk 2 d, t h e n h = x1 : : : xk¡1yxl : : : xpxk : : : xl¡1xx1 is a h a m ilt o n ia n c yc le , a n d if xp¡1xk 2 d, t h e n x1 : : : xk¡1yxl : : : xp¡1xk : : : xl¡1xx1 is a t -c yc le . in e a c h c a s e we h a ve a c o n t r a d ic t io n . th e r e fo r e t h e ve r t ic e s xk a n d xp a r e n o t a d ja c e n t a n d xp¡1xk =2 d. co n s e qu e n t ly, s in c e xk c a n n o t b e in s e r t e d in t o p [x1; xk¡1] a n d p [xk+1; xp¡1] b y l e m m a 2 we o b t a in d ( xk; p [x1; xk¡1]) · k a n d d( xk; p [xk+1; xp¡1]) · p ¡ k ¡ 1 : th e r e fo r e p + 1 = d ( xk ) = d ( xk; p [x1; xk¡1]) + d ( xk; p [xk+1; xp¡1]) + a( xk; x ) · k + p ¡ k ¡ 1 + 1 = p; wh ic h le a d s t o a c o n t r a d ic t io n , s in c e xkx =2 d ( a( xk; x ) · 1 ) . 1 1 4 on cycles through vertices of large semidegree in digraphs a s s u m e s e c o n d t h a t k ¸ l + 1 . fr o m xly =2 d it fo llo ws t h a t k ¸ l + 2 . w e m a y a s s u m e t h a t y is a d ja c e n t t o a ll ve r t ic e s o f p [x1; xl+1]. th e n fx1; x2; : : : ; xl+1g µ n + ( y ) a n d d¡ ( y; p [xl+1; xp¡1]) = m ¡ 1 : n o w c o n s id e r t h e ve r t e x xl. it is n o t d i± c u lt t o s e e t h a t if xiy 2 d, i 2 [l + 1 ; p ¡ 1 ], t h e n xlxi+1 =2 d ( fo r o t h e r wis e , h = x1 : : : xlxi+1 : : : xpxxl : : : xiyx1 ) . th e r e fo r e , s in c e xl is a t -ve r t e x a n d d+ ( xl; fx; yg ) = 0 , we o b t a in t h a t xl d o e s n o t d o m in a t e a t le a s t m + 1 ve r t ic e s , wh ic h is a c o n t r a d ic t io n a n d c o m p le t e s t h e p r o o f o f ca s e 6 .2 . l e t fxl1 ; xl2; : : : xlrg b e a s e t o f ve r t ic e s wh ic h a t t h e s a m e t im e a r e n o t a d ja c e n t t o x a n d y, wh e r e 2 · l1 < l2 < ¢ ¢ ¢ < lr · p ¡ 1 . n o t e t h a t ( b y cla im 1 ( iii) ) fo r a ll i 2 [1 ; r] we h a ve xli¡1x; xxli+1; xli¡1y a n d yxli+1 2 d. remar k 2. th e s e t fx; y; xl1 ; xl2; : : : ; xlrg is a n in d e p e n d e n t s e t o f ve r t ic e s . in d e e d , if xli xlj 2 d a n d li < lj , t h e n h = x1 : : : xli xlj : : : xpxxli+1 : : : xlj¡1yx1; a n d if xli xlj 2 d a n d li > lj , t h e n b y r e m a r k 1 , xpxli 2 d a n d h = x1 : : : xlj ¡1yxli+1 : : : xpxli xlj : : : xli¡1xx1. in e a c h c a s e we a r r ive a t a c o n t r a d ic t io n . case 6.3. th e ve r t ic e s xl a n d y a r e n o t a d ja c e n t . w e c a n a s s u m e t h a t fo r a ll j 2 [2 ; p¡ 2 ] t h e ve r t ic e s xj a n d x a r e n o t a d ja c e n t if a n d o n ly if xj a n d y a r e n o t a d ja c e n t . th e n b y r e m a r ks 1 a n d 2 fo r a ll i 2 [1 ; r] we h a ve n + ( x ) = n + ( y ) = n + ( xli ) a n d n ¡ ( x ) = n¡ ( y ) = n¡ ( xli ) ; a n d fx; y; xl1; xl2; : : : ; xlrg is a n in d e p e n d e n t s e t o f ve r t ic e s . n o t e t h a t if xxi+1 2 d, t h e n xixp =2 d ( fo r o t h e r wis e , h = x1 : : : xixpxxi+1 : : : xp¡1yx1 ) . fr o m t h is a n d d¡ ( xp; fx; yg ) = 0 it fo llo ws t h a t a t le a s t m + 1 ve r t ic e s d o n o t d o m in a t e xp. th e r e fo r e , xp is n o t a t -ve r t e x. s im ila r ly, we c a n s h o w t h a t if fxi; xi+1g ! x ( r e s p e c t ive ly, x ! fxj; xj+1g ) , t h e n xi+1 ( r e s p e c t ive ly, xj ) is n o t a t -ve r t e x; a n d if xxi 2 d a n d xjx 2 d, t h e n xi¡1xj+1 =2 d. th e p r o o f o f cla im 6 is c o m p le t e d . claim 7. xp¡1; x =2 d. p r oof. s u p p o s e , o n t h e c o n t r a r y, t h a t xp¡1x 2 d. th e n , b y cla im s 5 a n d 6 we h a ve xp¡1y =2 d a n d yxp¡1 2 d. h e n c e b y cla im 1 ( ii) , yxp 2 d. fr o m t h is a n d cla im 2 it fo llo ws t h a t m ¸ 3 . th e r e a r e t h r e e p o s s ib ilit ie s : xx2 2 d o r x a n d x2 a r e n o t a d ja c e n t o r x2x 2 d. case 7.1. xx2 2 d. if yx2 2 d o r y a n d x2 a r e n o t a d ja c e n t , t h e n fo r t h e c o n ve r s e d ig r a p h o f d we h a ve t h a t cla im 5 o r cla im 6 is n o t t r u e . th u s , we c a n a s s u m e t h a t x2y 2 d a n d yx2 =2 d. th e n x1y 2 d, b y cla im 1 ( ii) . r e c a ll t h a t t h e r e is a ve r t e x xk wit h k 2 [3 ; p ¡ 2 ] ( b y cla im 2 ) wh ic h is n o t a d ja c e n t t o t h e ve r t e x y a n d h e n c e , b y cla im 1 ( iii) , xk¡1y; yxk+1 2 d a n d xk is a t -ve r t e x. n o w we will p r o ve t h a t t h e ve r t e x xk is n o t a d ja c e n t t o t h e ve r t ic e s x1 a n d xp a n d xp¡1xk; xkx2; xkx; xxk 2 d: ( 1 3 ) s u p p o s e t h a t t h is is n o t t h e c a s e . if xkx1 2 d, t h e n h = x1yxk+1 : : : xpxx2 : : : xkx1; if x1xk 2 d, t h e n h = x1xk : : : xpxx2 : : : xk¡1yx1; if xkxp 2 d, t h e n h = x1 : : : xkxpyxk+1 : : : xp¡1xx1; a n d ¯ n a lly if xpxk 2 d, t h e n h = x1 : : : xk¡1yxpxk : : : xp¡1xx1. in e a c h c a s e we h a ve a c o n t r a d ic t io n . th e r e fo r e xk is n o t a d ja c e n t t o t h e ve r t ic e s x1 a n d xp. fr o m t h is it fo llo ws t h a t ( s in c e xk is a t -ve r t e x) p + 1 = d ( xk ) = d( xk; p [x2; xk¡1]) + d( xk; p [xk+1; xp¡1]) + a ( xk; x) : ( 1 4 ) s in c e t h e ve r t e x xk c a n n o t b e in s e r t e d in t o p [x2; xk¡1] a n d p [xk+1; xp¡1] b y l e m m a 2 , we h a ve d( xk; p [x2; xk¡1]) · k ¡ 1 a n d d( xk; p [xk+1; xp¡1]) · p ¡ k: s. darbinyan and i. karapetyan 1 1 5 th is t o g e t h e r wit h ( 1 4 ) im p lie s t h a t t h e a b o ve in e qu a lit ie s , in fa c t , a r e e qu a lit ie s a n d a ( x; xk ) = 2 ( in o t h e r wo r d s xkx; xxk 2 d ) . a g a in , u s in g l e m m a 2 , we o b t a in t h a t xp¡1xk; xkx2 2 d. ( 1 3 ) is p r o ve d . fr o m ( 1 3 ) a n d cla im 2 it fo llo ws t h a t m ¸ 4 . b y ( 1 3 ) , t h e c yc le x1 : : : xk¡1yxk+1 : : : xp¡1xkxx1 ( r e s p e c t ive ly, x2 : : : xk¡1yxk+1 : : : xpxxkx2 ) h a s le n g t h n ¡ 1 a n d d o e s n o t c o n t a in xp ( r e s p e c t ive ly, x1 ) . th e r e fo r e , xp a n d x1 a r e t -ve r t ic e s . it is e a s y t o s e e t h a t if yxi 2 d wit h i 2 [2 ; p ¡ 1 ]; t h e n xi¡1xp =2 d ( 1 5 ) ( o t h e r wis e , if yxi a n d xi¡1xp 2 d, t h e n x1 : : : xi¡1xpyxi : : : xp¡1xx1 is a h a m ilt o n ia n c yc le ) . n o t e t h a t xk¡1xp =2 d ( o t h e r wis e if xk¡1xp 2 d, t h e n b y ( 1 3 ) , x1 : : : xk¡1xpyxk+1 : : : xp¡1xkxx1 is a h a m ilt o n ia n c yc le , a c o n t r a d ic t io n ) . fr o m ( 1 5 ) , d+ ( y; p [x2; xp¡1]) = m ¡ 2 , xk¡1xp =2 d a n d xxp =2 d it fo llo ws t h a t a t le a s t m ve r t ic e s d o n o t d o m in a t e xp. co n s e qu e n t ly, t h e ve r t e x y is a d ja c e n t t o a ll ve r t ic e s o f p ¡ fxkg. h e n c e fx1; x2; : : : ; xk¡1g ! y ! fxk+1; xk+2; : : : ; xpg; ( 1 6 ) a n d k ¡ 1 = p ¡ k = m ¡ 1 . fr o m xk¡1xp =2 d a n d ( 1 5 ) ,( 1 6 ) we h a ve d¡ ( xp; p [xk¡1; xp¡2]) = 0 a n d fx1; x2; : : : ; xk¡2g ! xp: ( 1 7 ) fr o m t h is a n d ( 1 3 ) we h a ve t h a t x1 : : : xk¡2xpyxk+1 : : : xp¡1xkxx1 is a c yc le o f le n g t h n ¡ 1 wh ic h d o e s n o t c o n t a in xk¡1. th is m e a n s t h a t xk¡1 is a t -ve r t e x a n d xk¡1 c a n n o t b e in s e r t e d in t o p [x1; xk¡2] a n d p [xk+1; xp¡1]xk. n o w we will c o n s id e r t h e ve r t e x xk¡1 a n d c la im t h a t xk¡1 is n o t a d ja c e n t t o t h e ve r t ic e s x1 a n d xp. in d e e d , if x1xk¡1 2 d, t h e n b y ( 1 3 ) , h = x1xk¡1 : : : xpxx2 : : : xk¡2yx1; if xk¡1x1 2 d, t h e n b y ( 1 7 ) a n d ( 1 3 ) , h = x1xpyxk+1 : : : xp¡1xkxx2 : : : xk¡1x1; if xpxk¡1 2 d, t h e n b y ( 1 6 ) , h = x1 : : : xk¡2yxpxk¡1 : : : xp¡1xx1; if xk¡1xp 2 d, t h e n b y ( 1 3 ) a n d ( 1 6 ) , h = x1 : : : xk¡1xpyxk+1 : : : xp¡1xkxx1. in e a c h c a s e we h a ve o b t a in e d a c o n t r a d ic t io n . th e r e fo r e xk¡1 is n o t a d ja c e n t t o t h e ve r t ic e s x1 a n d xp. n o w b y l e m m a 2 we h a ve p + 1 = d( xk¡1 ) = d( xk¡1; p [x2; xk¡2]) + d( xk¡1; p [xk+1; xp¡1] [ fxkg) + a( xk¡1; fx; yg ) · p ¡ 1 + a ( xk¡1; fx; yg ) : it is p o s s ib le o n ly if a ( xk¡1; fx; yg ) = 2 ( i.e ., xk¡1y a n d xxk¡1 2 d s in c e yxk¡1 =2 d a n d xk¡1x =2 d ) . it is n o t d i± c u lt t o s e e t h a t d¡ ( x1; p [xk¡1; xp¡1]) = 0 ( o t h e r wis e if xix1 2 d, i 2 [k; p ¡ 1 ], t h e n h = x1yxi+1 : : : xpxx2 : : : xix1 ) . h e n c e xk¡2x1 2 d a n d b y ( 1 3 ) , h = x1yxk+1 : : : xpxxk¡1xkx2 : : : xk¡2x1, wh ic h is a c o n t r a d ic t io n . th e c o n t r a d ic t io n c o m p le t e s t h e p r o o f o f ca s e 7 .1 . case 7.2. th e ve r t ic e s x a n d x2 a r e n o t a d ja c e n t . th e n b y cla im 1 ( iii) , x1x a n d xx3 2 d. b y cla im 4 we h a ve t h a t t h e ve r t ic e s x2 a n d y a r e a d ja c e n t . if we c o n s id e r t h e c o n ve r s e d ig r a p h o f d, t h e n u s in g cla im 5 we s e e t h a t x2y 2 d a n d yx2 =2 d. th e r e fo r e , b y cla im 1 ( ii) , x1y 2 d s in c e y is a t -ve r t e x. n o w we will c o n s id e r t h e ve r t e x x2. n o t e t h a t x2 a ls o is a t -ve r t e x. if xpx2 2 d, t h e n h = x1yxpx2 : : : xp¡1xx1, a c o n t r a d ic t io n . s o , we c a n a s s u m e t h a t xpx2 =2 d. b y l e m m a 2 , d ( x2; p [x3; xp]) · p¡ 2 s in c e x2 c a n n o t b e in s e r t e d in t o p [x3; xp]. fr o m t h is , s in c e x a n d x2 a r e n o t a d ja c e n t , yx2 =2 d a n d x2 is a t -ve r t e x, we o b t a in t h a t x2x1 2 d. n o w it is e a s y t o s e e t h a t if yxi 2 d wit h i 2 [4 ; p], t h e n xi¡1x2 =2 d ( fo r o t h e r wis e , h = x1yxi : : : xpxx3 : : : xi¡1x2x1 ) . co n s e qu e n t ly, fr o m d + ( y; p [x4; xp]) = m ¡ 1 1 1 6 on cycles through vertices of large semidegree in digraphs a n d d¡ ( x2; fx; yg) = 0 it fo llo ws t h a t a t le a s t m + 1 ve r t ic e s d o n o t d o m in a t e x2, wh ic h is a c o n t r a d ic t io n . th e o b t a in e d c o n t r a d ic t io n c o m p le t e s t h e p r o o f o f ca s e 7 .2 . case 7.3. x2x 2 d. th e n x1x 2 d b y cla im 1 ( ii) . th e n fr o m d¡ ( x; fx1; x2; xp¡1; xpg ) = 4 we h a ve m ¸ 4 . it fo llo ws t h a t t h e r e is a l 2 [3 ; p ¡ 2 ] s u c h t h a t xl¡2x; xl¡1x; xxl+1 2 d a n d xl a n d x a r e n o t a d ja c e n t b y cla im 2 . n o t e t h a t wit h r e s p e c t t o t h e ve r t ic e s x2 a n d y t h e fo llo win g s u b c a s e s a r e p o s s ib le : yx2 2 d o r x2y 2 d o r t h e ve r t ic e s y a n d x2 a r e n o t a d ja c e n t . subcase 7.3.1. yx2 2 d. it is n o t d i± c u lt t o s e e t h a t t h e ve r t ic e s x1 a n d xl a r e n o t a d ja c e n t . in d e e d , if x1xl 2 d, t h e n h = x1xl : : : xpyx2 : : : xl¡1xx1; a n d if xlx1 2 d, t h e n h = x1xxl+1 : : : xpyx2 : : : xlx1, wh ic h is a c o n t r a d ic t io n . w e ¯ r s t p r o ve t h a t yxl; xlx2; xlxl¡1; xlxl¡2 2 d a n d xl¡2xl =2 d: ( 1 9 ) p r oof of (19). a s s u m e t h a t xpxl 2 d. th e n xly =2 d ( fo r o t h e r wis e , if xly 2 d, t h e n h = x1 : : : xl¡1xxl+1 : : : xpxlyx1 ) . s in c e x1 a n d xl a r e n o t a d ja c e n t a n d xl c a n n o t b e in s e r t e d in t o p [x2; xl¡1] a n d p [xl+1; xp], u s in g l e m m a 2 , we s e e t h a t p + 1 = d ( xl ) = d( xl; p [x2; xl¡1]) + d ( xl; p [xl+1; xp]) + a( xl; y ) · p + a ( xl; y ) : it fo llo ws t h a t d( xl; p [x2; xl¡1]) = l ¡ 1 a n d a( xl; y ) = 1 . th e r e fo r e yxl 2 d a n d xlx2 2 d b y l e m m a 2 . n o w a s s u m e t h a t xpxl =2 d. th e n , s im ila r ly, a s b e fo r e we o b t a in t h a t d( xl; p [x2; xl¡1]) = l ¡ 1 , d( xl; p [xl+1; xp]) = p ¡ l a n d a( xl; y ) = 2 ( i.e ., yxl; xly 2 d ) . b y l e m m a 2 , we h a ve t h a t xlx2 2 d. n o w we will c o n s id e r t h e p a t h xl+1xl+2 : : : xpyx1 : : : xl¡2xl¡1 a n d t h e ve r t e x xl in s t e a d o f y. th e n u s in g cla im s 6 a n d 5 we o b t a in t h a t xlxl¡1; xlxl¡2 2 d a n d xl¡2xl =2 d. s o , in d e e d , ( 1 9 ) is s a t is ¯ e d , a s d e s ir e d . w .l.o .g . we c a n a s s u m e t h a t xxl+2 =2 d a n d x a n d xl+2 a r e a d ja c e n t ( b e c a u s e o t h e r wis e fo r t h e p a t h xl+1xl+2 : : : xpyx1 : : : xl¡1 we wo u ld h a ve ca s e 7 .1 o r 7 .2 wh ic h we h a ve a lr e a d y d e a lt wit h ) . th e n b y cla im 1 ( ii) we h a ve , xl+1x; xl+2x 2 d. n o w we c o n s id e r t h e ve r t e x x1. if xix 2 d wit h i 2 [2 ; p ¡ 1 ], t h e n x1xi+1 =2 d ( fo r o t h e r wis e , h = x1xi+1 : : : xpyx2 : : : xixx1 ) . if x1xl+1 2 d, t h e n h = x1xl+1 : : : xpyxlx2 : : : xl¡1xx1 b y ( 1 9 ) . ob s e r ve t h a t x2 : : : xl¡1xxl+1 : : : xpyxlx2 is a c yc le o f le n g t h n ¡ 1 wh ic h d o e s n o t c o n t a in x1. th is m e a n s t h a t x1 is a t -ve r t e x. n o w fr o m d ¡ ( x; p [x2; xp¡1]) = m ¡ 2 a n d d+ ( x1; fy; xl+1g ) = 0 it fo llo ws t h a t t h e ve r t e x x is a d ja c e n t t o a ll ve r t ic e s o f p ¡ fxlg wh ic h is n o t p o s s ib le s in c e m ¸ 4 , xl+1x 2 d a n d d is n o t h a m ilt o n ia n . subcase 7.3.2. x2y 2 d. th e n b y cla im s 2 a n d 1 ( iii) t h e r e is a ve r t e x xk wit h k 2 [3 ; p ¡ 2 ] s u c h t h a t xk¡1y; yxk+1 2 d a n d y is n o t a d ja c e n t t o xk. it is e a s y t o s e e t h a t xp a n d xk a r e n o t a d ja c e n t ( i.e ., a ( xk; xp ) = 0 ) . in d e e d , if xkxp 2 d, t h e n h = x1 : : : xkxpyxk+1 : : : xp¡1xx1; a n d if xpxk 2 d, t h e n h = x1 : : : xk¡1yxpxk : : : xp¡1x x1, wh ic h is a c o n t r a d ic t io n . n o w we p r o ve t h a t xp¡1xk a n d xkx 2 d: ( 2 0 ) p r oof of (20). l e t xkx1 2 d. th e n xxk =2 d ( s in c e , o t h e r wis e , if xxk 2 d, t h e n h = x1 : : : xk¡1yxk+1 : : : xpxxkx1 ) a n d h e n c e , s in c e a( xk; xp ) = 0 a n d t h e p a t h s p [x1; xk¡1] a n d p [xk+1; xp¡1] c a n n o t b e e xt e n d e d wit h xk b y l e m m a 2 we h a ve d ( xk; p [x1; xk¡1]) · k, d( xk; p [xk+1; xp¡1]) · p ¡ k a n d p + 1 = d( xk ) = d( xk; p [x1; xk¡1]) + d( xk; p [xk+1; xp¡1]) + a ( xk; x ) = p + 1 : th e r e fo r e d ( xk; p [x1; xk¡1]) = k, d( xk; p [xk+1; xp¡1]) = p ¡ k a n d a ( xk; x ) = 1 ( i.e ., xkx 2 d ) . n o w, u s in g l e m m a 2 , we o b t a in t h a t xp¡1xk 2 d. s. darbinyan and i. karapetyan 1 1 7 l e t n o w xkx1 =2 d. th e n d( xk; p [x1; xk¡1]) · k ¡ 1 , a ( xk; x ) = 2 ( i.e ., xkx; xxk 2 d ) a n d d( xk; p [xk+1; xp¡1]) = p ¡ k. a g a in , u s in g l e m m a 2 , we o b t a in t h a t xp¡1xk 2 d. s o , in d e e d ( 2 0 ) is s a t is ¯ e d , a s d e s ir e d . n o w we will c o n s id e r t h e ve r t e x xp wh ic h is a t -ve r t e x s in c e x1 : : : xk¡1yxk+1 : : : xp¡1xkxx1 is a c yc le o f le n g t h n ¡ 1 . if xiy 2 d wit h i 2 [1 ; p ¡ 2 ], t h e n xpxi+1 =2 d ( fo r o t h e r wis e , h = x1 : : : xiyxpxi+1 : : : xp¡1xx1 ) . n o t e t h a t d ¡ ( y; p [x1; xp¡2]) = m ¡ 1 a n d xpxk+1 =2 d ( if xpxk+1 2 d, t h e n b y ( 2 0 ) , h = x1 : : : xk¡1yxpxk+1 : : : xp¡1xkxx1. it fo llo ws fr o m t h e o b s e r va t io n a b o ve t h a t t h e ve r t e x y is a d ja c e n t t o a ll ve r t ic e s o f p ¡ fxkg. th e r e fo r e n¡ ( y ) = fx1; x2; : : : ; xk¡1; xpg a n d n + ( y ) = fx1; xk+1; xk+2; : : : ; xpg: th e n fo r t h e p a t h xk+1xk+2 : : : xpxx1x2 : : : xk¡1 a n d fo r t h e ve r t e x y b y cla im s 5 a n d 6 we h a ve t h e c o n s id e r e d ca s e 7 .1 . subcase 7.3.3. th e ve r t ic e s y a n d x2 a r e n o t a d ja c e n t . th e n x1y; yx3 2 d ( b y cla im 1 ( iii) ) , x2 a n d xp a r e n o t a d ja c e n t ( b y cla im 3 ) a n d x2 is a t -ve r t e x. a s s u m e t h a t x2x1 2 d. th e n xix2 =2 d if xxi+1 2 d, i 2 [3 ; p ¡ 1 ] ( fo r o t h e r wis e , h = x1xxi+1 : : : xpyx3 : : : xix2x1 ) . n o w fr o m d + ( x; p [x4; xp¡1]) = m ¡ 1 a n d d¡ ( x2; fx; yg ) = 0 it fo llo ws t h a t d¡ ( x2 ) · m ¡ 1 , wh ic h is a c o n t r a d ic t io n . s o , we c a n a s s u m e t h a t x2x1 =2 d. th e r e fo r e p + 1 = d( x2 ) = d ( x2; p [x3; xp¡1]) + d( x2; fx1; xg) · d( x2; p [x3; xp¡1]) + 2 : h e n c e d( x2; p [x3; xp¡1]) = p ¡ 1 . b y l e m m a 2 , x2 c a n b e in s e r t e d in t o t h e p a t h p [x3; xp¡1], a c o n t r a d ic t io n wh ic h c o m p le t e s t h e p r o o f o f cla im 7 . l e t u s n o w c o m p le t e t h e p o o f o f t h e t h e o r e m . s in c e d is n o t h a m ilt o n ia n fr o m cla im 7 a n d r e m a r k 2 it fo llo ws t h a t fo r a n y c yc le c := x1x2 : : : x2mx1 o f le n g t h n ¡ 1 = 2 m if x =2 v ( c ) t h e n n + ( x ) = n¡ ( x ) = fx1; x3; : : : ; x2m¡1g a n d fx2; x4; : : : ; x2m; xg is a n in d e p e n d e n t s e t o f ve r t ic e s . th e r e fo r e k¤m;m+1 µ d µ [km + km+1]¤. th e p r o o f o f t h e th e o r e m is c o m p le t e . remar k 3. l e t d b e a d ig r a p h wit h t h e ve r t e x s e t v ( d ) = fx1; x2; x3; x4; x5; x; yg s u c h t h a t n + ( x1 ) = fx2; x4g, n + ( x2 ) = fx; y; x3; x5g, n + ( x3 ) = n + ( x ) = n + ( y ) = fx1; x2; x4; g, n + ( x4 ) = fx; y; x5g a n d n + ( x5 ) = fx; y; x3g. it is e a s y t o c h e c k t h a t t h e ve r t ic e s x; y; x2; x3 a n d x4 a r e t -ve r t ic e s a n d t h e ve r t ic e s x1 a n d x5 a r e n o t t -ve r t ic e s . mo r e o ve r , t h e d ig r a p h d is 2 -s t r o n g a n d c o n t a in s n o c yc le t h r o u g h x; y; x2; x3 a n d x4. refer ences [1 ] j. b a n g -je n s e n , g. gu t in , d igraphs: theory, algorithms and applications, s p r in g e r , 2 0 0 0 . 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[6 ] j.a . b o n d y, c. th o m a s s e n , \ a s h o r t p r o o f o f me yn ie l's t h e o r e m " , d iscrete m ath., vo l. 1 9 , n o . 1 , p p . 8 5 -9 2 , 1 9 7 7 . 1 1 8 on cycles through vertices of large semidegree in digraphs [7 ] s .k h . d a r b in ya n , \ a s u ± c ie n t c o n d it io n fo r t h e h a m ilt o n ia n p r o p e r t y o f d ig r a p h s wit h la r g e s e m id e g r e e s " , akad. nauk armyan. ssr d okl., vo l. 8 2 , n o . 1 ,p p . 6 -8 , 1 9 8 6 ( s e e a ls o a r x iv: 1 1 1 1 .1 8 4 3 v1 [m a t h .co] 8 n o v 2 0 1 1 ) . [8 ] k h . d a r b in ya n , \ h a m ilt o n ia n a n d s t r o n g ly h a m ilt o n -c o n n e c t e d d ig r a p h s " , akad. nauk armyan. ssr d okl., vo l. 9 1 , n o . 1 , p p . 3 -6 , 1 9 9 0 ( in r u s s ia n ) . [9 ] s .k h . d a r b in ya n , \ a s u ± c ie n t c o n d it io n fo r d ig r a p h s t o b e h a m ilt o n ia n " , akad. nauk armyan. ssr d okl., vo l. 9 1 , n o . 2 , p p . 5 7 -5 9 , 1 9 9 0 ( in r u s s ia n ) . [1 0 ] r . h äa g g kvis t , c. th o m a s s e n , \ on p a n c yc lic d ig r a p h s " , j . combin. theory ser. b , vo l. 2 0 , p p . 2 0 -4 0 , 1 9 7 6 . [1 1 ] a . gh o u ila -h o u r i, \ u n e c o n d it io n s u ± s a n t e d 'e xis t e n c e d 'u n c ir c u it h a m ilt o n ie n " , c. r . acad. sci. p aris ser. a-b , n o . 2 5 1 , p p . 4 9 5 -4 9 7 , 1 9 6 0 . [1 2 ] h . l i, e . fla n d r in , j. s h u , \ a s u ± c ie n t c o n d it io n fo r c yc la b ilit y in d ir e c t e d g r a p h s " , d iscrete m ath., vo l. 3 0 7 , p p . 1 2 9 1 -1 2 9 7 , 2 0 0 7 . [1 3 ] y . ma n o u s s a kis , \ d ir e c t e d h a m ilt o n ia n g r a p h s " , j . graph theory, vo l. 1 6 , n o . 1 , p p . 5 1 -5 9 , 1 9 9 2 . [1 4 ] m. me yn ie l, \ u n e c o n d it io n s u ± s a n t e d 'e xis t e n c e d 'u n c ir c u it h a m ilt o n ie n d a n s u n g r a p h e o r ie n t e " , j . combin. theory ser. b , vo l. 1 4 , p p . 1 3 7 -1 4 7 , 1 9 7 3 . [1 5 ] k . ot a , \ cyc le s t h r o u g h p r e s c r ib e d ve r t ic e s wit h la r g e d e g r e e s u m " , d iscrete m ath. vo l. 1 4 5 , p p . 2 0 1 -2 1 0 , 1 9 9 5 . [1 6 ] r . s h i, \ 2 -n e ig h b o r h o o d s a n d h a m ilt o n ia n c o n d it io n s " , j .graph theory, n o . 1 6 , p p . 2 6 7 -2 7 1 , 1 9 9 2 . [1 7 ] c. th o m a s s e n , \ l o n g c yc le s in d ig r a p h s " , p roc. l ondon m ath. soc., vo l. 3 , n o . 4 2 , p p . 2 3 1 -2 5 1 , 1 9 8 1 . [1 8 ] h .j. v e ld m a n , \ cyc le s c o n t a in in g m a n y ve r t ic e s o f la r g e d e g r e e " , d iscrete m ath., vo l. 1 0 1 , p p . 3 1 9 -3 2 5 , 1 9 9 2 . [1 9 ] d .r . w o o d a ll, \ s u ± c ie n t c o n d it io n s fo r c ir c u it s in g r a p h s " , p roc. l ondon m ath. soc., n o . 2 4 , p p . 7 3 9 -7 5 5 , 1 9 7 2 . submitted 20.12.2012, accepted 14.02.2013. ø»í ïçë³³ëïç×³ýý»ñ áõý»óáõ ·³·³ãý»ñáí ³ýóýáõ óçïé»ñç ù³ëçý ê. ¸³ñµçýû³ý ¨ æ. î³ñ³å»ïû³ý ²ù÷á÷áõù ü»ñï³ ³ßë³ï³ýùáõù óáõûó ¿ ïñí³í, áñ »ã» 2 m + 1 ·³·³ã³ýç ïáõùýáñáßí³í ·ñ³ýý áõýç 2 m »ñï³ñáõãû³ùµ óçïé, ³å³ ³û¹ ·ñ³ýá, µ³ó³éáõãû³ùµ ùç ù³ýç áñáß³ïç ·ñ³ýý»ñç, ýáõûýå»ë å³ñáõý³ïáõù ¿ ïáõùýáñáßí³í óçïé, áñý ³ýóýáõù ¿ µáéáñ ³ûý ·³·³ãý»ñáí, áñáýó ïçë³³ëïç×³ýý»ñá ÷áùñ ã»ý m-çó: î öèêëàõ ïðîõîäÿùèõ ÷åðåç âåðøèíû ñ áîëüøèìè ïîëóñòåïåíÿìè ñ. äàðáèíÿí è è. êàðàïåòÿí àííîòàöèÿ â íàñòîÿùåé ðàáîòå äîêàçàíî: åñëè ( 2 m + 1 ) -âåðøèííûé îðãðàô d ñîäåðæèò îðöèêë äëèíû 2 m, òî d òàêæå (êðîìå íåêîòîðûõ îðãðàôîâ) ñîäåðæèò îðöèêë ïðîõîäÿùèé ÷åðåç âñå âåðøèíû ñ ïîëóñòåïåíÿìè íåìåíüøèìè m. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 45, 77--89, 2016. two-party regular expression matching protocol without asymmetric cryptography operations gurgen h. khachatryan1, mihran m. hovsepyan2, aram h. jivanyan3 1 american university of armenia 2 russian-armenian (slavonic) university 3 onecryptor cjsc e-mail: gurgenkh@aua.am, hovsepyan.mihran@gmail.com, aram@skycryptor.com abstract in this paper we describe a new protocol for secure evaluation of deterministic finite automata (dfa) between two parties (client and server). the protocol has no restrictions on the dfa’s input alphabet and runs in a single client-server communication round. it uses 𝑂𝑂(𝑚𝑚𝑚𝑚) operations for client-side computations; 𝑂𝑂(𝑚𝑚𝑚𝑚|𝑄𝑄|) operations for server-side computations, and the network communication bandwidth is 𝑂𝑂(𝑚𝑚𝑚𝑚𝑚𝑚|𝑄𝑄|) bytes where 𝑚𝑚 is the security parameter of the protocol, 𝑚𝑚 is the size of the dfa’s input alphabet, 𝑚𝑚 is the length of the input text and |𝑄𝑄| is the number of the dfa states. as a building block our algorithm uses white-box based 1-out-of-n oblivious transfer protocol, which results that the protocol does no public-key operations. apart from the description of the protocol the paper also contains results of efficiency benchmarks done on our implementation of the protocol. keywords: cryptography, secure function evaluation, secure pattern matching, oblivious transfer. 1. introduction the problem of two-party oblivious (secure) evaluation of dfa is a subproblem of more generic two-party secure function evaluation problem. in two-party secure function evaluation problem there is a known for two parties function 𝐹𝐹: 𝑈𝑈 × 𝑉𝑉 → 𝑍𝑍, the first party has an input argument 𝑢𝑢 ∈ 𝑈𝑈 and the second party has an input argument 𝑣𝑣 ∈ 𝑉𝑉 and the objective of these two parties to calculate 𝐹𝐹(𝑢𝑢, 𝑣𝑣) allowing one or both parties to learn the result, but none of them should learn any additional information about the other’s input. this problem has various real life applications. one such example is searching a number of appearances of a specific dna pattern among people of some specific group (the fbi wants to 77 mailto:gurgenkh@aua.am mailto:hovsepyan.mihran@gmail.com mailto:aram@skycryptor.com two-party regular expression matching protocol without asymmetric cryptography operations 78 allow biogenetical researchers (clients) to find the number of people among the criminals who have a specific (featured by the researchers) pattern in their dnas without providing the entire list of dnas of such people, and without learning which patterns have been searched). in this example the input argument of the first party is the pattern and the input of the second party is the database of dnas. the next application is in banking: a person wants to take a credit from the bank. the bank needs to check whether he fits their requirements, particularly they want to check his credit history stored in credit report agencies (cra). but full credit report of a person may contain lots of private information as long as the criteria of the bank for giving a credit for the person also may be private. so the bank may want to check his criteria with the cra without learning the full credit history and without providing the criteria to the cra. the range of applications of the two-party secure function evaluation problem is not limited to the privacy-preserving genomic computations and credit checking, it can also be used in remote diagnostics, graph algorithms, data mining, medical diagnostics, face recognition, policy checking and in other areas. for more details see [3], it gives entire overview of the problem and its applications. here we consider a problem where the first party (client) has a dfa (regular expression) γ as long as the second party (server) has a text string 𝑋𝑋 which consists of the letters of the dfa’s input alphabet. their objective is to check whether the string 𝑋𝑋 belongs to the language generated by the γ or, in other words, whether the string 𝑋𝑋 matches γ allowing both parties to learn the answer so that neither the client nor the server learned any additional information about the input of each other. this problem may have some variations: a) the client wants to hide the answer from the server. b) the client or both parties want to learn whether the string 𝑋𝑋 has a substring 𝑦𝑦 which matches γ. c) the client or both parties want to learn positions of all substrings of the string 𝑋𝑋 which match γ. d) the client or both parties want to learn the number of substrings of the string 𝑋𝑋 which match γ. in this paper we describe a protocol which solves the initial problem and it is shown that after a little modification the protocol can be applied to these variations as well. similar to the “efficient protocol for oblivious dfa evaluation” [1] construction our protocol runs in a single client-server communication round and in both client and server sides it has the same as the protocol from [1] complexity of symmetric operations when the input alphabet of the dfa is binary. but unlike [1], our protocol has no restriction on the dfas input alphabet and it uses no asymmetric operations anymore. 2. preliminaries 2.1 notations we will assume that the client has a dfa γ(𝑄𝑄; σ; δ; 𝑠𝑠1; 𝐹𝐹) with input alphabet σ = �𝑏𝑏1, 𝑏𝑏2, … , 𝑏𝑏|σ|�; set of states 𝑄𝑄 = {𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞|𝑄𝑄|}; table of transitions δ|𝑄𝑄|×|σ|, where δ[𝑞𝑞, 𝑏𝑏] ∈ 𝑄𝑄 is the state following after state 𝑞𝑞 through letter 𝑏𝑏; initial state 𝑠𝑠1 and set of accepting (or final) states 𝐹𝐹, while the server has an 𝑚𝑚 length string 𝑋𝑋 ∈ σ𝑛𝑛. saying the result of evaluation of γ on the string 𝑋𝑋 (or simply γ(𝑋𝑋)) we mean the boolean value which is 1 (or 𝑡𝑡𝑡𝑡𝑢𝑢𝑡𝑡) if the state g. khachatryan , m. hovsepyan, a.jivanyan 79 δ�… δ�δ�𝑠𝑠1, 𝑋𝑋[1]�, 𝑋𝑋[2]� … , 𝑋𝑋[𝑚𝑚]�, is an accepting state (i.e., belongs to 𝐹𝐹) and 0 (or 𝑓𝑓𝑓𝑓𝑓𝑓𝑠𝑠𝑡𝑡), otherwise. by 𝑚𝑚 we denote the security parameter of the protocol. the pipe-sign | is used to denote concatenation of strings and the circled plus sign ⊕ is used to denote per-bit xor operation. by ⌈𝑡𝑡⌉ we denote the ceiling of a real number 𝑡𝑡. in matrix indexing sometimes we use letters of the input alphabet σ and states 𝑄𝑄 instead of their numbers (i.e., 𝑏𝑏𝑗𝑗 instead of 𝑗𝑗, 𝑞𝑞𝑗𝑗 instead of 𝑗𝑗). 2.2 definitions: we will use the following concepts for describing the protocol and its security. one round 1-out-of-n oblivious transfer protocol (later ot): a protocol is intended for solving the following problem. the client has a number 𝑖𝑖 from 1 to 𝑚𝑚 and the server has a set of 𝑚𝑚 secrets {𝑠𝑠1, 𝑠𝑠2, … , 𝑠𝑠𝑛𝑛}. they should communicate and after that the client should learn the 𝑖𝑖-th secret 𝑠𝑠𝑖𝑖, but the server should not learn 𝑖𝑖 and the client should not learn any other secret apart from the 𝑠𝑠𝑖𝑖. there are various implementations of ot protocols, here we will use white box based 1-out-of-n oblivious transfer described in [2]. secure two-party computation: let 𝑓𝑓 = (𝑓𝑓1, 𝑓𝑓2) of form 𝑓𝑓: {0,1}∗ × {0,1}∗ → {0,1}∗ × {0,1}∗ be a two-party computation and 𝜋𝜋 be a two-party protocol for computing 𝑓𝑓 between the parties 𝑝𝑝1 and 𝑝𝑝2. the input of 𝑝𝑝1 is 𝑥𝑥 and the input of 𝑝𝑝2 is 𝑦𝑦. here we will briefly define two notions of security. a) full security (simulation-based security) against malicious adversaries: this security level is defined by requiring indistinguishability (perfect, statistical or computational) between a real execution of the protocol and an ideal execution in which there is a ttp (trusted third party) who receives the parties input, evaluates the function and outputs the results to them. b) privacy against malicious adversaries: this level of security guarantees that a corrupted party will not learn any information about the honest parties input. however, this does not always guarantee that the joint outputs of the parties in the real world is simulatable in an ideal world. for more detailed discussion of these security notations refer to [12]. looking ahead we want to note that in our protocol, we prove full-secure against one malicious party and private against the other. 3. the protocol for binary dfas at first let us consider the case when the input alphabet of the dfa σ = {0,1} is binary, we call such dfas “binary dfa”. brief description of the protocol: the main steps of the protocol are the following: a) client: create a special evaluation matrix (dfa matrix) 𝑀𝑀γ of size 𝑚𝑚 × |𝑄𝑄| × 2 intended for evaluation of γ on 𝑚𝑚-length binary strings. b) client: create garbled dfa matrix 𝐺𝐺𝑀𝑀γ by permuting each row of the 𝑀𝑀γ matrix, then encrypting each cell of the matrix using one time pad. as a result, to calculate γ(𝑋𝑋) it will be enough for the server to have the garbled dfa matrix 𝐺𝐺𝑀𝑀γ and a key for each position 𝑖𝑖; 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚 of the string 𝑋𝑋 corresponding to the letter of that position. c) server: for each letter of string 𝑋𝑋 create an ot query token [2] and send them all along with the ot initialization data to the client. two-party regular expression matching protocol without asymmetric cryptography operations 80 d) client: for each ot query token create an ot response token for the corresponding key [2] and send them together with the garbled dfa matrix 𝐺𝐺𝑀𝑀γ to the server. e) server: from ot response tokens invoke the keys for positions of the string 𝑋𝑋, then compute γ(𝑋𝑋) using those keys and 𝐺𝐺𝑀𝑀γ garbled dfa matrix. now let us look at these steps in more detail. step a): for evaluating γ on any 𝑚𝑚-length binary string we create a dfa matrix 𝑀𝑀γ of size 𝑚𝑚 × |𝑄𝑄| × 2 such that for each state 𝑞𝑞 and letter 𝑏𝑏 𝑀𝑀γ[𝑖𝑖, 𝑞𝑞, 𝑏𝑏] = δ[𝑞𝑞, 𝑏𝑏] ∀𝑖𝑖; 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚 − 1 and 𝑀𝑀γ[𝑚𝑚, 𝑞𝑞, 𝑏𝑏] = 1 if δ[𝑞𝑞, 𝑏𝑏] is a final state and 𝑀𝑀γ[𝑚𝑚, 𝑞𝑞, 𝑏𝑏] = 0 if δ[𝑞𝑞, 𝑏𝑏] is not final. it is easy to see that in such notation γ(𝑋𝑋) is equal to 𝑀𝑀γ �𝑚𝑚, … 𝑀𝑀γ �2, 𝑀𝑀γ�1, 𝑠𝑠1, 𝑋𝑋[1]�, 𝑋𝑋[2]� … , 𝑋𝑋[𝑚𝑚]�. figure 1 (a, b) bellow illustrates an example of dfa with its dfa matrix. step b): here it is worth to note that for any permutation 𝑃𝑃: 𝑄𝑄 → 𝑄𝑄 the dfa γ is equal (i.e., accepts the same set of strings) to the dfa γ𝑃𝑃(𝑄𝑄; {0,1}; δ𝑃𝑃; 𝑃𝑃[𝑠𝑠1]; 𝐹𝐹𝑃𝑃) (where for each state 𝑞𝑞 and letter δ𝑃𝑃[𝑃𝑃[𝑞𝑞], 𝑏𝑏] = 𝑃𝑃�δ[𝑞𝑞, 𝑏𝑏]�; and 𝐹𝐹𝑃𝑃 = {𝑞𝑞: 𝑃𝑃−1[𝑞𝑞] ∈ 𝐹𝐹}). the first stage of dfa matrix garbling is based on this fact. namely, we first create 𝑚𝑚 random permutations of the dfa states 𝑄𝑄 and fill by them 𝑚𝑚 rows of an 𝑚𝑚 × |𝑄𝑄| permutations matrix 𝑃𝑃𝑃𝑃𝑃𝑃, then we create permuted dfa matrix 𝑃𝑃𝑀𝑀γ such that for each state 𝑞𝑞 and letter 𝑏𝑏 𝑃𝑃𝑀𝑀γ[𝑖𝑖, 𝑃𝑃𝑃𝑃𝑃𝑃[𝑖𝑖, 𝑞𝑞], 𝑏𝑏] = 𝑃𝑃𝑃𝑃𝑃𝑃�𝑖𝑖 + 1, 𝑀𝑀γ[𝑖𝑖, 𝑞𝑞, 𝑏𝑏]�, for each 𝑖𝑖; 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚 − 1 and 𝑃𝑃𝑀𝑀γ[𝑚𝑚, 𝑃𝑃𝑃𝑃𝑃𝑃[𝑚𝑚, 𝑞𝑞], 𝑏𝑏] = 𝑀𝑀γ[𝑚𝑚, 𝑞𝑞, 𝑏𝑏]. figure 1 (c) bellow illustrates an example of a 𝑃𝑃𝑀𝑀γ matrix. here it is not hard to observe that in terms of 𝑃𝑃𝑀𝑀γ matrix γ(𝑋𝑋) is equal to 𝑃𝑃𝑀𝑀γ �𝑚𝑚, … 𝑃𝑃𝑀𝑀γ �2, 𝑃𝑃𝑀𝑀γ�1, 𝑃𝑃𝑃𝑃𝑃𝑃[1, 𝑠𝑠1], 𝑋𝑋[1]�, 𝑋𝑋[2]� … , 𝑋𝑋[𝑚𝑚]�. for the second stage of dfa matrix garbling we generate an 𝑚𝑚 × 2 size matrix of random 𝑚𝑚-bit keys 𝐾𝐾 and an (𝑚𝑚 + 1) × |𝑄𝑄| size matrix of 𝑚𝑚′-bit pads 𝑃𝑃𝑃𝑃𝑃𝑃; where for each state 𝑞𝑞, and for each 𝑖𝑖; 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚 the 𝑃𝑃𝑃𝑃𝑃𝑃[𝑖𝑖, 𝑞𝑞] is a random 𝑚𝑚′-bit string and 𝑃𝑃𝑃𝑃𝑃𝑃[𝑚𝑚 + 1, 𝑞𝑞] = 00 … 0�� 𝑘𝑘′ . here by 𝑚𝑚′ we denote 𝑚𝑚 − ⌈log2|𝑄𝑄|⌉. for garbling the permuted dfa matrix we also need some csprng (cryptographically secure pseudo-random number generator) 𝑃𝑃𝑃𝑃𝐺𝐺: {0,1}𝑘𝑘 ′ → {0,1}2𝑘𝑘. since the 𝑃𝑃𝑃𝑃𝐺𝐺 receives a 𝑚𝑚′-bit string as an input and returns 2𝑚𝑚-bit string as an output by 𝑃𝑃𝑃𝑃𝐺𝐺(𝑌𝑌, 0) we denote the first 𝑚𝑚-bits of 𝑃𝑃𝑃𝑃𝐺𝐺(𝑌𝑌) and by 𝑃𝑃𝑃𝑃𝐺𝐺(𝑌𝑌, 1) we denote the second 𝑚𝑚-bits of 𝑃𝑃𝑃𝑃𝐺𝐺(𝑌𝑌). having all these, we create the garbled dfa matrix 𝐺𝐺𝑀𝑀γ such that 𝐺𝐺𝑀𝑀γ[𝑖𝑖, 𝑞𝑞, 𝑏𝑏] = �𝑃𝑃𝑀𝑀γ[𝑖𝑖, 𝑞𝑞, 𝑏𝑏] | 𝑃𝑃𝑃𝑃𝑃𝑃�𝑖𝑖 + 1, 𝑃𝑃𝑀𝑀γ[𝑖𝑖, 𝑞𝑞, 𝑏𝑏]�� ⊕ 𝐾𝐾[𝑖𝑖, 𝑏𝑏] ⊕ 𝑃𝑃𝑃𝑃𝐺𝐺(𝑃𝑃𝑃𝑃𝑃𝑃[𝑖𝑖, 𝑞𝑞], 𝑏𝑏) for each 𝑖𝑖; 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚, letter 𝑏𝑏 and state 𝑞𝑞. g. khachatryan , m. hovsepyan, a.jivanyan 81 𝑠𝑠1 𝑠𝑠2 𝑠𝑠3 𝑠𝑠4 𝑠𝑠5 … … … accepting state non-accepting state start 0 1 1 0 1 2 n n-1 (0, 0) (1, 0) (2, 3) (5, 4) (2, 3) (5, 4) (2, 3) (5, 4) 1 2 |q| … … … … … … … … … accepted . . . . . . (b) 𝑀𝑀γ (dfa matrix of γ) (a) a general binary dfa γ 1 2 n n-1 start accepted . . . . . . (c) permuted dfa matrix 𝑃𝑃𝑀𝑀γ … 1 … 𝐼𝐼2 − 1 … 1 … 𝐼𝐼1 − 1 … 1 … 𝐼𝐼𝑛𝑛−1 − 1 … 1 … 𝐼𝐼𝑛𝑛 − 1 (0, 0) 𝐼𝐼𝑛𝑛 = 𝑃𝑃𝑃𝑃𝑃𝑃[𝑚𝑚, 1] (𝐽𝐽𝑛𝑛, …) 𝐼𝐼𝑛𝑛−1 = 𝑃𝑃𝑃𝑃𝑃𝑃[𝑚𝑚 − 1,1] … 𝐼𝐼1 + 1 … 𝐽𝐽1 − 1 … 𝐽𝐽1 + 1 … |𝑄𝑄| … 𝐼𝐼2 + 1 … 𝐽𝐽2 − 1 … 𝐼𝐼𝑛𝑛 + 1 … 𝐽𝐽𝑛𝑛 − 1 … 𝐼𝐼𝑛𝑛−1 + 1 … 𝐽𝐽𝑛𝑛−1 − 1 … 𝐽𝐽𝑛𝑛 + 1 … |𝑄𝑄| … 𝐽𝐽𝑛𝑛−1 + 1 … |𝑄𝑄| … 𝐽𝐽2 + 1 … |𝑄𝑄| (𝐽𝐽3, …) 𝐼𝐼2 = 𝑃𝑃𝑃𝑃𝑃𝑃[2,1] (𝐽𝐽2, …) 𝐼𝐼1 = 𝑃𝑃𝑃𝑃𝑃𝑃[1,1] (1, 0) 𝐽𝐽𝑛𝑛 = 𝑃𝑃𝑃𝑃𝑃𝑃[𝑚𝑚, 2] (…, …) 𝐽𝐽𝑛𝑛−1 = 𝑃𝑃𝑃𝑃𝑃𝑃[𝑚𝑚 − 1,2] (…, …) 𝐽𝐽2 = 𝑃𝑃𝑃𝑃𝑃𝑃[2,2] (…, …) 𝐽𝐽1 = 𝑃𝑃𝑃𝑃𝑃𝑃[1,2] 𝑋𝑋 = 01 … 00 𝑋𝑋 = 01 … 00 start fig. 1. dfa γ, dfa matrix 𝑀𝑀γ, permuted dfa matrix 𝑃𝑃𝑀𝑀γ. two-party regular expression matching protocol without asymmetric cryptography operations 82 step c): having the garbled dfa matrix 𝐺𝐺𝑀𝑀γ and 𝐾𝐾�𝑖𝑖, 𝑋𝑋[𝑖𝑖]� for each 𝑖𝑖; 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚 the server will be able to calculate γ(𝑋𝑋) (we will see that in step e). hence, here for each letter 𝑋𝑋[𝑖𝑖] of its input string the server generates an ot query token ot𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄[𝑖𝑖] and along with ot initialization info ot𝐼𝐼𝑛𝑛𝑖𝑖𝐼𝐼 sends them to the client. for the details of ot initialization info and ot query token generation see [2]. step d): for each ot query the token ot𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄[𝑖𝑖] (1 ≤ 𝑖𝑖 ≤ 𝑚𝑚) the client, using ot initialization info ot𝐼𝐼𝑛𝑛𝑖𝑖𝐼𝐼 generates an ot response token ot𝑅𝑅𝑄𝑄𝑅𝑅𝑅𝑅𝑅𝑅𝑛𝑛𝑅𝑅𝑄𝑄[𝑖𝑖] which carries sufficient info to invoke 𝐾𝐾�𝑖𝑖, 𝑋𝑋[𝑖𝑖]� (see [2]). then the client sends those response tokens, garbled dfa matrix 𝐺𝐺𝑀𝑀γ, 𝑃𝑃𝑃𝑃𝑃𝑃[1, 𝑠𝑠1] and 𝑃𝑃𝑃𝑃𝑃𝑃�1, 𝑃𝑃𝑃𝑃𝑃𝑃[1, 𝑠𝑠1]� to the server. step e): at this step first the server for each 𝑖𝑖; 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚 invokes 𝐾𝐾�𝑖𝑖, 𝑋𝑋[𝑖𝑖]� key stored in the corresponding response token ot𝑅𝑅𝑄𝑄𝑅𝑅𝑅𝑅𝑅𝑅𝑛𝑛𝑅𝑅𝑄𝑄[𝑖𝑖] (see [2]). then having the keys, garbled dfa matrix 𝐺𝐺𝑀𝑀γ, 𝑃𝑃𝑃𝑃𝑃𝑃[1, 𝑠𝑠1] and 𝑃𝑃𝑃𝑃𝑃𝑃�1, 𝑃𝑃𝑃𝑃𝑃𝑃[1, 𝑠𝑠1]�, it runs the following algorithm to calculate γ(𝑋𝑋): evaluation of garbled dfa matrix 𝑐𝑐𝑢𝑢𝑡𝑡_𝑠𝑠𝑡𝑡𝑓𝑓𝑡𝑡𝑡𝑡_𝑖𝑖𝑖𝑖 ∶= 𝑃𝑃𝑃𝑃𝑃𝑃[1, 𝑠𝑠1]; 𝑐𝑐𝑢𝑢𝑡𝑡_𝑝𝑝𝑓𝑓𝑖𝑖 ∶= 𝑃𝑃𝑃𝑃𝑃𝑃[1, 𝑐𝑐𝑢𝑢𝑡𝑡_𝑠𝑠𝑡𝑡𝑓𝑓𝑡𝑡𝑡𝑡_𝑖𝑖𝑖𝑖]; for each row 𝑖𝑖 = 1 to 𝑚𝑚 do 𝑐𝑐𝑢𝑢𝑡𝑡_𝑠𝑠𝑡𝑡𝑓𝑓𝑡𝑡𝑡𝑡_𝑖𝑖𝑖𝑖|𝑐𝑐𝑢𝑢𝑡𝑡_𝑝𝑝𝑓𝑓𝑖𝑖 ∶= 𝐺𝐺𝑀𝑀γ �𝑖𝑖, 𝑐𝑐𝑢𝑢𝑡𝑡𝑅𝑅𝐼𝐼𝑠𝑠𝐼𝐼𝑄𝑄𝑖𝑖𝑖𝑖, 𝑋𝑋[𝑖𝑖]� ⊕ 𝐾𝐾�𝑖𝑖, 𝑋𝑋[𝑖𝑖]� ⊕ 𝑃𝑃𝑃𝑃𝐺𝐺(𝑐𝑐𝑢𝑢𝑡𝑡_𝑝𝑝𝑓𝑓𝑖𝑖, 𝑋𝑋[𝑖𝑖]); end for return 𝑐𝑐𝑢𝑢𝑡𝑡_𝑠𝑠𝑡𝑡𝑓𝑓𝑡𝑡𝑡𝑡_𝑖𝑖𝑖𝑖 the step by step construction of the garbled dfa matrix implies that this algorithm will give the same result as evaluation of the initial dfa matrix, so no additional proof is required here. these were all steps of the protocol for binary dfas, now let us move to the case where there is no restriction on the input alphabet of the dfa. 4. case of non-binary dfas here again the protocol consists of 5 steps and they are almost the same as in the case of binary dfas, the main difference is that the size of the third dimension of 𝑀𝑀γ, 𝑃𝑃𝑀𝑀γ and 𝐺𝐺𝑀𝑀γ matrixes is |σ| instead of 2. let us look at these steps in detail. step a): create a dfa matrix 𝑀𝑀γ of size 𝑚𝑚 × |𝑄𝑄| × |σ| such that for each state 𝑞𝑞 and letter 𝑏𝑏 ∈ σ 𝑀𝑀γ[𝑖𝑖, 𝑞𝑞, 𝑏𝑏] = δ[𝑞𝑞, 𝑏𝑏] ∀𝑖𝑖; 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚 − 1 and 𝑀𝑀γ[𝑚𝑚, 𝑞𝑞, 𝑏𝑏] is equal to 1 if δ[𝑞𝑞, 𝑏𝑏] is a final state and it is 0, otherwise. in such notation γ(𝑋𝑋) is equal to 𝑀𝑀γ �𝑚𝑚, … 𝑀𝑀γ �2, 𝑀𝑀γ�1, 𝑠𝑠1, 𝑋𝑋[1]�, 𝑋𝑋[2]� … , 𝑋𝑋[𝑚𝑚]�. figure 2 (a, b) bellow illustrates an example of dfa with its dfa matrix. step b): here again we create 𝑚𝑚 random permutations of the dfa states 𝑄𝑄 and fill by them 𝑚𝑚 rows of an 𝑚𝑚 × |𝑄𝑄| permutations matrix 𝑃𝑃𝑃𝑃𝑃𝑃, then we create permuted dfa matrix 𝑃𝑃𝑀𝑀γ of size 𝑚𝑚 × |𝑄𝑄| × |σ| such that for each state 𝑞𝑞 and letter 𝑏𝑏 ∈ σ g. khachatryan , m. hovsepyan, a.jivanyan 83 𝑃𝑃𝑀𝑀γ[𝑖𝑖, 𝑃𝑃𝑃𝑃𝑃𝑃[𝑖𝑖, 𝑞𝑞], 𝑏𝑏] = 𝑃𝑃𝑃𝑃𝑃𝑃�𝑖𝑖 + 1, 𝑀𝑀γ[𝑖𝑖, 𝑞𝑞, 𝑏𝑏]�, for each 𝑖𝑖; 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚 − 1 and 𝑃𝑃𝑀𝑀γ[𝑚𝑚, 𝑃𝑃𝑃𝑃𝑃𝑃[𝑚𝑚, 𝑞𝑞], 𝑏𝑏] = 𝑀𝑀γ[𝑚𝑚, 𝑞𝑞, 𝑏𝑏] figure 2 (c) bellow illustrates an example of a 𝑃𝑃𝑀𝑀γ matrix. in terms of 𝑃𝑃𝑀𝑀γ matrix γ(𝑋𝑋) is equal to 𝑃𝑃𝑀𝑀γ �𝑚𝑚, … 𝑃𝑃𝑀𝑀γ �2, 𝑃𝑃𝑀𝑀γ�1, 𝑃𝑃𝑃𝑃𝑃𝑃[1, 𝑠𝑠1], 𝑋𝑋[1]�, 𝑋𝑋[2]� … , 𝑋𝑋[𝑚𝑚]�. then we create an 𝑚𝑚 × |σ| size matrix of random 𝑚𝑚-bit keys 𝐾𝐾 and an (𝑚𝑚 + 1) × |𝑄𝑄| size matrix of 𝑚𝑚′-bit pads 𝑃𝑃𝑃𝑃𝑃𝑃; here again for each state 𝑞𝑞, and for each 𝑖𝑖; 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚 the 𝑃𝑃𝑃𝑃𝑃𝑃[𝑖𝑖, 𝑞𝑞] is a random 𝑚𝑚′-bit string and 𝑃𝑃𝑃𝑃𝑃𝑃[𝑚𝑚 + 1, 𝑞𝑞] = 00 … 0�� 𝑘𝑘′ , where 𝑚𝑚′ is 𝑚𝑚 − ⌈log2|𝑄𝑄|⌉. we also need some csprng 𝑃𝑃𝑃𝑃𝐺𝐺: {0,1}𝑘𝑘 ′ → {0,1}𝑘𝑘∙|σ|, and by 𝑃𝑃𝑃𝑃𝐺𝐺(𝑌𝑌, 𝑗𝑗) we denote the 𝑗𝑗-th 𝑚𝑚-bits of 𝑃𝑃𝑃𝑃𝐺𝐺(𝑌𝑌) for each 𝑗𝑗; 1 ≤ 𝑗𝑗 ≤ |σ|. after all, we create the garbled dfa matrix 𝐺𝐺𝑀𝑀γ such that 𝐺𝐺𝑀𝑀γ[𝑖𝑖, 𝑞𝑞, 𝑏𝑏] = �𝑃𝑃𝑀𝑀γ[𝑖𝑖, 𝑞𝑞, 𝑏𝑏] | 𝑃𝑃𝑃𝑃𝑃𝑃�𝑖𝑖 + 1, 𝑃𝑃𝑀𝑀γ[𝑖𝑖, 𝑞𝑞, 𝑏𝑏]�� ⊕ 𝐾𝐾[𝑖𝑖, 𝑏𝑏] ⊕ 𝑃𝑃𝑃𝑃𝐺𝐺(𝑃𝑃𝑃𝑃𝑃𝑃[𝑖𝑖, 𝑞𝑞], 𝑏𝑏) for each 𝑖𝑖; 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚, letter 𝑏𝑏 ∈ σ and state 𝑞𝑞 ∈ 𝑄𝑄. step c): and again having the garbled dfa matrix 𝐺𝐺𝑀𝑀γ and 𝐾𝐾�𝑖𝑖, 𝑋𝑋[𝑖𝑖]� for each 𝑖𝑖; 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚 the server will be able to calculate γ(𝑋𝑋). so here we start ot phase between the client and the server for transferring the corresponding keys and 𝐺𝐺𝑀𝑀γ. for each letter 𝑋𝑋[𝑖𝑖] of its input string the server generates an ot query token ot𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄[𝑖𝑖] and along with ot initialization info ot𝐼𝐼𝑛𝑛𝑖𝑖𝐼𝐼 sends them to the client [2]. step d): for each ot query the token ot𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄[𝑖𝑖] (1 ≤ 𝑖𝑖 ≤ 𝑚𝑚) the client, using ot initialization info ot𝐼𝐼𝑛𝑛𝑖𝑖𝐼𝐼 generates an ot response token ot𝑅𝑅𝑄𝑄𝑅𝑅𝑅𝑅𝑅𝑅𝑛𝑛𝑅𝑅𝑄𝑄[𝑖𝑖] which carries sufficient info to invoke 𝐾𝐾�𝑖𝑖, 𝑋𝑋[𝑖𝑖]� [2]. then the client sends those response tokens, garbled dfa matrix 𝐺𝐺𝑀𝑀γ, 𝑃𝑃𝑃𝑃𝑃𝑃[1, 𝑠𝑠1] and 𝑃𝑃𝑃𝑃𝑃𝑃�1, 𝑃𝑃𝑃𝑃𝑃𝑃[1, 𝑠𝑠1]� to the server. step e): at this step firstly the server invokes 𝐾𝐾�𝑖𝑖, 𝑋𝑋[𝑖𝑖]� keys stored in the corresponding response token ot𝑅𝑅𝑄𝑄𝑅𝑅𝑅𝑅𝑅𝑅𝑛𝑛𝑅𝑅𝑄𝑄[𝑖𝑖] for each 𝑖𝑖; 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚 [2]. then having the keys, garbled dfa matrix 𝐺𝐺𝑀𝑀γ, 𝑃𝑃𝑃𝑃𝑃𝑃[1, 𝑠𝑠1] and 𝑃𝑃𝑃𝑃𝑃𝑃�1, 𝑃𝑃𝑃𝑃𝑃𝑃[1, 𝑠𝑠1]�, it runs the following algorithm to calculate γ(𝑋𝑋): evaluation of garbled dfa matrix 𝑐𝑐𝑢𝑢𝑡𝑡_𝑠𝑠𝑡𝑡𝑓𝑓𝑡𝑡𝑡𝑡_𝑖𝑖𝑖𝑖 ∶= 𝑃𝑃𝑃𝑃𝑃𝑃[1, 𝑠𝑠1]; 𝑐𝑐𝑢𝑢𝑡𝑡_𝑝𝑝𝑓𝑓𝑖𝑖 ∶= 𝑃𝑃𝑃𝑃𝑃𝑃[1, 𝑐𝑐𝑢𝑢𝑡𝑡_𝑠𝑠𝑡𝑡𝑓𝑓𝑡𝑡𝑡𝑡_𝑖𝑖𝑖𝑖]; for each row 𝑖𝑖 = 1 to 𝑚𝑚 do 𝑐𝑐𝑢𝑢𝑡𝑡_𝑠𝑠𝑡𝑡𝑓𝑓𝑡𝑡𝑡𝑡_𝑖𝑖𝑖𝑖|𝑐𝑐𝑢𝑢𝑡𝑡_𝑝𝑝𝑓𝑓𝑖𝑖 ∶= 𝐺𝐺𝑀𝑀γ �𝑖𝑖, 𝑐𝑐𝑢𝑢𝑡𝑡𝑅𝑅𝐼𝐼𝑠𝑠𝐼𝐼𝑄𝑄𝑖𝑖𝑖𝑖, 𝑋𝑋[𝑖𝑖]� ⊕ 𝐾𝐾�𝑖𝑖, 𝑋𝑋[𝑖𝑖]� ⊕ 𝑃𝑃𝑃𝑃𝐺𝐺(𝑐𝑐𝑢𝑢𝑡𝑡_𝑝𝑝𝑓𝑓𝑖𝑖, 𝑋𝑋[𝑖𝑖]); end for return 𝑐𝑐𝑢𝑢𝑡𝑡_𝑠𝑠𝑡𝑡𝑓𝑓𝑡𝑡𝑡𝑡_𝑖𝑖𝑖𝑖 here again the step by step construction of the garbled dfa matrix implies that this algorithm will give the same result as evaluation of the initial dfa matrix. two-party regular expression matching protocol without asymmetric cryptography operations 84 𝑠𝑠1 𝑠𝑠2 𝑠𝑠3 𝑠𝑠4 𝑠𝑠5 … … … accepting state non-accepting state start 1 2 n n-1 (0,0,0,0…0) 1 2 |q| … … … … … … … … … accepted . . . . . . (b) 𝑀𝑀γ (dfa matrix of γ) (a) a general binary dfa γ 1 2 n n-1 start accepted . . . . . . (c) permuted dfa matrix 𝑃𝑃𝑀𝑀γ … 1 … 𝐼𝐼2 − 1 … 1 … 𝐼𝐼1 − 1 … 1 … 𝐼𝐼𝑛𝑛−1 − 1 … 1 … 𝐼𝐼𝑛𝑛 − 1 (0,0,0,0…0) 𝐼𝐼𝑛𝑛 = 𝑃𝑃𝑃𝑃𝑃𝑃[𝑚𝑚, 1] (𝐽𝐽𝑛𝑛, …) 𝐼𝐼𝑛𝑛−1 = 𝑃𝑃𝑃𝑃𝑃𝑃[𝑚𝑚 − 1,1] … 𝐼𝐼1 + 1 … 𝐽𝐽1 − 1 … 𝐽𝐽1 + 1 … |𝑄𝑄| … 𝐼𝐼2 + 1 … 𝐽𝐽2 − 1 … 𝐼𝐼𝑛𝑛 + 1 … 𝐽𝐽𝑛𝑛 − 1 … 𝐼𝐼𝑛𝑛−1 + 1 … 𝐽𝐽𝑛𝑛−1 − 1 … 𝐽𝐽𝑛𝑛 + 1 … |𝑄𝑄| … 𝐽𝐽𝑛𝑛−1 + 1 … |𝑄𝑄| … 𝐽𝐽2 + 1 … |𝑄𝑄| (𝐽𝐽3, …) 𝐼𝐼2 = 𝑃𝑃𝑃𝑃𝑃𝑃[2,1] (𝐽𝐽2, …) 𝐼𝐼1 = 𝑃𝑃𝑃𝑃𝑃𝑃[1,1] (1,0,0,0…0) 𝐽𝐽𝑛𝑛 = 𝑃𝑃𝑃𝑃𝑃𝑃[𝑚𝑚, 2] (…, …) 𝐽𝐽𝑛𝑛−1 = 𝑃𝑃𝑃𝑃𝑃𝑃[𝑚𝑚 − 1,2] (…, …) 𝐽𝐽2 = 𝑃𝑃𝑃𝑃𝑃𝑃[2,2] (…, …) 𝐽𝐽1 = 𝑃𝑃𝑃𝑃𝑃𝑃[1,2] 𝑋𝑋 = 𝑏𝑏1𝑏𝑏2 … 𝑏𝑏1𝑏𝑏1 𝑏𝑏2 𝑏𝑏3 𝑏𝑏1 𝑏𝑏3 𝑏𝑏4 … 𝑏𝑏|σ| 𝑏𝑏1 𝑏𝑏2 𝑏𝑏4 … 𝑏𝑏|σ| (2,3,4,1…1) (1,0,0,0…0) (5,4,3,2…2) (2,3,4,1…1) (2,3,4,1…1) (5,4,3,2…2) (5,4,3,2…2) 𝑋𝑋 = 𝑏𝑏1𝑏𝑏2 … 𝑏𝑏1𝑏𝑏1 start fig. 2. dfa γ, dfa matrix 𝑀𝑀γ, permuted dfa matrix 𝑃𝑃𝑀𝑀γ. g. khachatryan , m. hovsepyan, a.jivanyan 85 5. security in the protocol we use a white-box based 1-out-of-2 ot protocol. such ot protocol is considered to be secure if the underlying white-box encryption schema is secure. in our case we use white-box encryption schema based on safer+ encryption schema. white-box scheme can be considered secure if no computationally bounded adversary is able to extract the master encryption key from the white-box encryption tables and no computationally bounded adversary is able to make decryption functionality with the help of only white-box encryption tables. so white-box scheme is considered to be secure if it is secure against key-recovery and reverseengineering attacks. taking into account this and security definitions from chapter 2 we see that according to the theorem 1 from [1] our protocol is fully-secure when only one party is malicious and it is private when both parties are malicious. 6. implementation and benchmarks 6.1 implementation a c++ application has been created which implements the protocol. the implementation of the protocol is parameterized by the security parameter 𝑚𝑚 and by the csprng 𝑃𝑃𝑃𝑃𝐺𝐺. the application acts as both a client and a server. as a server, it receives a text string as an input, and as a client, its input is a regular expression which it converts to the equivalent dfa using thompson’s construction algorithm [8] to create ndfa equivalent to the regular expression, then using the subset construction algorithm [9] to convert the ndfa to the dfa. after all it runs the steps a) – e) of the protocol and calculates the result of evaluation of the dfa on the text string. during calculations it prints out time spent for different parts of computations. 6.2 benchmarks algorithm construction implies that overall number of computations depends on the security parameter 𝑚𝑚, the number of the dfa states |𝑄𝑄|, the length of the text string 𝑚𝑚 and the number of letters in the dfas input alphabet |σ|. besides that, in the algorithm we have generation of random pads and keys, as well as usage of csprng, and depending on the approach used for generation of random pads and keys and chosen csprng the efficiency of the algorithm may differ for the same input data (𝑚𝑚, |𝑄𝑄|, |σ| and |𝑋𝑋|). we did our benchmarks for 𝑚𝑚 = 128 and |σ| = 2 on a 64-bit windows 7 pc with intel® core™ 2 quad q6600 2.4 ghz processor and 4gb ram, using sha-256 as {0,1}𝑘𝑘 ′ → {0,1}𝑘𝑘∙|σ| csprng and c++ standard library’s rand() function for random pad/key generation. the method of garbled dfa matrix construction shows and our benchmarks confirmed that the number of operations, hence the spent time for garbled dfa matrix creation, is proportional to the |𝑄𝑄| ∙ |σ| ∙ 𝑚𝑚 multiplication. the first table shows performance of garbled dfa matrix generation of our implementation on inputs of different sizes. the table contains averaged results from 10 runs for each pair of inputs. two-party regular expression matching protocol without asymmetric cryptography operations 86 table 1. garbled dfa creation time when (sec). 𝑚𝑚 |𝑄𝑄| 10 100 1000 10000 100000 10 <0.001 0.002 0.017 0.17 1.7 100 0.002 0.017 0.17 1.7 17 1000 0.017 0.17 1.7 17 >100 10000 0.17 1.7 17 >100 >100 100000 1.7 17 >100 >100 >100 1000000 17 >100 >100 >100 >100 the second table shows the performance of the ot phase of the protocol and performance of the garbled dfa matrix evaluation by the server. table 2. time spent in ot phase when (sec). 𝑚𝑚 ot query generation and response extraction (server) ot response generation (client) garbled dfa evaluation (server) 10 <0.001 <0.001 <0.001 100 0.002 <0.001 <0.001 1000 0.021 0.007 <0.001 10000 0.21 0.07 0.002 100000 2.1 0.7 0.02 1000000 21 7 0.2 the table shows only the dependency of efficiencies of the operations from 𝑚𝑚 since the number of ot queries/responses and the number of steps for garbled dfa matrix evaluation is equal to 𝑚𝑚 and do not depend on structure of dfa. 7. conclusion unlike other protocols for oblivious dfa evaluation published in recent years [3], our protocol is totally free from public-key operations, and it makes it somewhat unique among others. table 3 below illustrates complexities of client and server computations and network communication bandwidth of our and other recent protocols. it is also worth to note that our protocol allows both parties to learn only γ(𝑋𝑋), but in some applications it may be inconvenient or insufficient. in [1] it is shown that for each of the following modifications of the problem it is possible to solve it after modifying their protocol a little, and that those modifications have no security leakage. all those modifications of their protocol work for our protocol as well. a) the client wants to hide the answer from the server. b) the client or both parties want to learn whether the string 𝑥𝑥 has a substring 𝑦𝑦 which matches γ. c) the client or both parties want to learn positions of all substrings of the string 𝑥𝑥 which match γ. d) the client or both parties want to learn the number of substrings of the string 𝑥𝑥 which match γ. g. khachatryan , m. hovsepyan, a.jivanyan 87 table 3. complexity of protocols. round complexity client computations server computations network bandwidth asymmetric symmetric asymmetric symmetric troncoso [4] 𝑂𝑂(𝑚𝑚) 𝑂𝑂(𝑚𝑚|𝑄𝑄|) none 𝑂𝑂(𝑚𝑚|𝑄𝑄|) 𝑂𝑂(𝑚𝑚|𝑄𝑄|) 𝑂𝑂(𝑚𝑚|𝑄𝑄|𝑚𝑚) frikken [5] 2 𝑂𝑂(𝑚𝑚 + |𝑄𝑄|) 𝑂𝑂(𝑚𝑚|𝑄𝑄|) 𝑂𝑂(𝑚𝑚 + |𝑄𝑄|) 𝑂𝑂(𝑚𝑚|𝑄𝑄|) 𝑂𝑂(𝑚𝑚|𝑄𝑄|𝑚𝑚) gennaro [6] 𝑂𝑂(min{|𝑄𝑄|, 𝑚𝑚} 𝑂𝑂(𝑚𝑚|𝑄𝑄|) none 𝑂𝑂(𝑚𝑚|𝑄𝑄|) none 𝑂𝑂(𝑚𝑚|𝑄𝑄|𝑚𝑚) yao [10] 1 𝑂𝑂(𝑚𝑚) 𝑂𝑂(𝑚𝑚|𝑄𝑄| log|𝑄𝑄 𝑂𝑂(𝑚𝑚) 𝑂𝑂(𝑚𝑚|𝑄𝑄| log|𝑄𝑄 𝑂𝑂(𝑚𝑚𝑚𝑚2) ishai [11] 1 𝑂𝑂(𝑚𝑚) none 𝑂𝑂(𝑚𝑚|𝑄𝑄|) none 𝑂𝑂(𝑚𝑚|𝑄𝑄|𝑚𝑚) mohassel [1] 1 𝑂𝑂(𝑚𝑚) 𝑂𝑂(𝑚𝑚) 𝑂𝑂(𝑚𝑚) 𝑂𝑂(𝑚𝑚|𝑄𝑄|) 𝑂𝑂(𝑚𝑚|𝑄𝑄|𝑚𝑚) this protocol |σ| = 2 1 none 𝑂𝑂(𝑚𝑚|𝑄𝑄|) none 𝑂𝑂(𝑚𝑚) 𝑂𝑂(𝑚𝑚|𝑄𝑄|𝑚𝑚) references [1] p. mohassel, s. niksefat, s. sadeghian and b. sadeghiyan, “an efficient protocol for oblivious dfa evaluation and applications”, in proceedings of cryptographers' track at the rsa conference, pp. 398-415, 2012. [2] g. khachatryan and a. jivanyan, “efficient oblivious transfer protocols based on white-box cryptography’, (submitted for publication). see more in http://cse.aua.am/applied-cryptography-laboratory/. [3] v. kolesnikov, a.-r. sadeghi and t. schneider, “from dust to dawn: practically efficient two-party secure function evaluation protocols and their modular design”, cryptology eprint archive, report 2010/079 (2010), https://eprint.iacr.org/2010/079.pdf. [4] j.r. troncoso-pastoriza, s. katzenbeisser and m. celik, “privacy preserving error resilient dna searching through oblivious automata”, in proceedings of the 14th acm conference on computer and communications security, pp. 519–528, 2007. [5] k. frikken, “practical private dna string searching and matching through efficient oblivious automata evaluation’, data and applications security xxiii, pp. 81–94, 2009. [6] r. gennaro, c. hazay and j. sorensen, “text search protocols with simulation based security”, public key cryptography–pkc 2010, pp. 332–350, 2010. [7] c. hazay and t. toft, “computationally secure pattern matching in the presence of malicious adversaries”, advances in cryptology-asiacrypt 2010, pp. 195–212, 2010. [8] k. thompson, “programming techniques: regular expression search algorithm”, communications of the acm , vol. 11, no. 6, pp. 419–422, 1968. [9] s. michael, introduction to the theory of computation, pws publishing company, 1997. [10] a. c. yao, “protocols for secure computations”, in proceedings of the 23rd annual symposium on foundations of computer science, citeseer, pp. 160–164, 1982. http://en.wikipedia.org/wiki/michael_sipser two-party regular expression matching protocol without asymmetric cryptography operations 88 [11] y. ishai, j. kilian, k. nissim and e. petrank, “extending oblivious transfers efficiently”, advances in cryptology-crypto 2003, pp. 145–161, 2003. [12] o. goldreich, foundations of cryptography: basic applications, cambridge univ pr, 2004. submitted 04.08.2015, accepted 15.02.2016 առանց գաղտնագրման ասիմետրիկ գործողությունների կիրառման կանոնավոր արտահայտության գաղտնի համապատասխանեցման երկկողմանի պրոտոկոլ գ. խաչատրյան, մ. հովսեփյան, ա. ջիվանյան ամփոփում այս հոդվածում նկարագրված է նոր երկկողմանի պրոտոկոլ, որը թույլ է տալիս առաջին կողմին՝ կլիենտին, գաղտնի կերպով ստուգել, արդյոք երկրորդ կողմի՝ սերվերի, մոտ եղած տողը համապատասխանում է իր մոտ եղած կանոնավոր արտահայտությանը: պրոտոկոլը նախատեսված է կամայական այբբենարանով կանոնավոր արտահայտությունների համար և աշխատում է մի փուլով: այն կատարում է 𝑂𝑂(𝑚𝑚𝑚𝑚) գործողություն կլիենտի կողմում և 𝑂𝑂(𝑚𝑚𝑚𝑚|𝑄𝑄|) գործողություն սերվերի կողմում, իսկ ցանցով տեղափոխվում է 𝑂𝑂(𝑚𝑚𝑚𝑚𝑚𝑚|𝑄𝑄|) բայթ ինֆորմացիա, որտեղ 𝑚𝑚-ն պրոտոկոլի անվտանգության պարամետրն է, 𝑚𝑚-ը կանոնավոր արտահայտության այբբենարանի տառերի քանակն է, 𝑚𝑚-ը սերվերի մոտ եղած տողի երկարությունն է՝ տառերի քանակը, իսկ |𝑄𝑄|-ն՝ կանոնավոր արտահայտությանը համապատասխանող մինիմալ դետերմինիստիկ վերջավոր ավտոմատի (դվա) վիճակների քանակը: g. khachatryan , m. hovsepyan, a.jivanyan 89 двухсторонний протокол тайного сопоставления регулярных выражений без применения операций асимметричного шифрования г. хачатрян, m. овсепян, а. дживанян аннотация в этой статье описан двухсторонний протокол, который позволяет первой стороне – клиенту, имеющего регулярное выражение, тайным образом проверять соответствует ли строка имеющееся у второй стороны – у сервера, его регулярному выражению. протокол предназначен для регулярных выражений с произвольным алфавитом и работает за один раунд коммуникаций между клиентом и сервером. протокол выполняет 𝑂𝑂(𝑚𝑚𝑚𝑚) операций на клиенте, 𝑂𝑂(𝑚𝑚𝑚𝑚|𝑄𝑄|) операций на сервере, а трафик сети 𝑂𝑂(𝑚𝑚𝑚𝑚𝑚𝑚|𝑄𝑄|) байтов, где 𝑚𝑚 это параметр безопасности протокола, 𝑚𝑚 количество букв в алфавите регулярного выражения, 𝑚𝑚 длина входной строки сервера, а |𝑄𝑄| количество состояний минимального детерминистического конечного автомата (дка) соответствующего регулярному выражению. mathematical problems of computer science 48, 5–13, 2017. an example of a multiplayer serious game on 3d sandpile model hayk e. nahapetyan institute for informatics and automation problems of nas ra e-mail: hayknahapetyan@yahoo.com abstract purpose of this paper is to consider cellular automata models as a base of a type of serious games.as an example of cellular automata, the abelian sandpile model on 3d graphs has been chosen. moreover, a networked multiplayer gaming environment (sandgame) has been developed using .net and c# language. keywords: serious games, ca, asm, .net, c#, multi-user, simulation, visualization. 1. introduction a brief survey of serious games reveals that there seems to be as many definitions available as there are actors involved, but most agree on a core meaning that serious games (digital) are games used for purposes other than mere entertainment. another question of interest concerns the claimed positive effects of such games, or of applications from related and sometimes overlapping areas, such as e-learning, edutainment, game-based learning, and digital game-based learning. besides, that serious games allow users to experience situations impossible in real life because of cost, time, safety etc., they can have positive impacts on the players’ development of a number of different skills. even so, it is not the case that all games are good for all learning outcomes. empirical studies on the use of serious games in science education here [1] are conducted. cellular automata (ca) are discrete models studied in computability theory, mathematics, physics, complexity theory, theoretical biology and microstructure modeling. the concept of self-organized criticality was first introduced by bak, tang and wiesenfeld in 1987 [2], and gave rise to growing interest in the study of self-organizing systems. bak et al. argued that in many natural phenomena, the dissipative dynamics of the system is such that it drives the system to a critical state, thereby leading to ubiquitous power law behaviors. this mechanism has been invoked to understand the power law distributions observed in turbulent fluids, earthquakes, distribution of visible matter in the universe, solar flares and surface roughening of growing interfaces. the sandpile models, being a class of cellular automata, are among the simplest theoretical models which exhibit self-organized criticality. a special subclass of interest consists of so called abelian sandpile models (asm). the abelian property means that the final stable state of the ca is independent of the order in which 5 6 an example of a multiplayer serious game on 3d sandpile model the updates of cells are carried out. this property plays a key role during the numerical, as well as analytical studies of the asm [[3], [4], [5]]. in the ca simulator, [6] a new module has been added as an example of a serious game, named sandgame. the sandgame, developed on the basis of abelian sandpile model, has a carefully thought-out educational purpose and is not intended to be played primarily for amusement, whereas at the same time, it is entertaining. the idea behind the developed model relies on various theorems regarded to sandpile model, which improved the concept of learning and increased attractiveness. undoubtedly, it is vital for universities and schools to have learning mechanisms equipped with scientific games. 2. concept of serious games today, the term “serious games”is becoming more and more popular. a google-search on “serious games” renders about 39,900,000 hits [2017-10-11]. nowadays, the term itself is established, but there is no current singleton definition of the concept. serious games usually refer to games used for training, advertising, simulation, or education that are designed to run on personal computers, mobile phones or even on video game consoles. according to corti (2006, p.1), game-based learning(serious games) is all about leveraging the power of computer games to captivate and engage end-users for a specific purpose, such as to develop new knowledge and skills. it could be argued that for purposes other than purely academic, there is no need to define serious games. however, while different groups of researchers use the same term, they also appear to refer to different things. for example, in [7] serious game definition is like “any piece of software that merges a non-entertaining purpose (serious) with a video game structure (game)”. in another paper [8], serious game is formally defined as an “interactive computer application, with or without significant hardware component, that has a challenging goal, is fun to play and engaging, incorporates some scoring mechanism, and supplies the user with skills, knowledge or attitudes useful in reality”. 3. sandpile model consider an undirected graph g = (v, e) described with the set of vertices v = {v1, v2, . . . , vn} and the set of edges e. each vertex vi ∈ v is assigned a variable hi which takes integer values and represents the height of the sand at that vertex. hmaxi denotes the maximal allowed height for the vertex vi in the graph g. for a d-dimensional lattice, we take hmaxi = 2d + 1. ct denotes the set of heights hi which determines the configuration of the system at a given discrete time t . a configuration is called stable, if all heights satisfy hi < h max i . the vertex vi is called closed, if h max i = deg(vi), where deg(vi) indicates the degree of vi. the dynamics of the system is defined by the following rules. consider a stable configuration ct at a given time t . we add a grain of sand at a random vertex vi ∈ v by setting hi to hi + 1 (we assume that the vertex is chosen randomly with a uniform distribution on the set v ). this new configuration, if stable, defines ct +1. if hi ≥ hmaxi , then the vi becomes unstable and topples losing h max i grains of sand, while all neighbors of vi receive one grain. note that if the vertex is open, then the system loses grains. during the toppling of the closed vertices, the number of grains is conserved. note also that toppling of a vertex may cause some of its neighboring vertices to become unstable. in this case those vertices also topple according to the same toppling rule. once all unstable vertices are h. nahapetyan 7 toppled, a new stable configuration ct +1 is obtained. if the finite connected graph g has at least one open vertex, then all vertices become stable after a finite number of topplings. moreover, the new stable configuration is independent of the toppling order. therefore, the dynamics is well defined. let âi be an operator, which acts on sandpile configurations and adds a grain at vertex i. it can be easily shown that âiâj = âjâi. this is the reason why the sandpile model is called abelian. 4. sandgame sandgame is based on the abelian sandpile model, which has been considered on 3 dimensional connected torus with (n ∗ n ∗ n) size and 2 dimensional connected lattice with (n ∗ n) size, where n is the length (nodes count) by any direction of torus or lattice. ”connected” attribute for both graph types makes each node to have equivalent parameters as for neighbors, as well as for critical height. besides, the model will not lose any grain, which grants the game to have a winner. consider a 2 dimensional lattice. every edge with random direction on the lattice is assigned an arrow parameter, also contours in the graph are excluded. let di be the number of arrows directed towards to the node i. model will become infinitely unstable if each node topples at least once during the overall toppling. as each node has 4 neighbors, we put 4 − di grains on every node i. total grains’ count will be 4 ∗ n2 − ∑ di = 2 ∗ n2, which is an indispensable number of grains that makes the model become infinitely unstable. with the statistical methods [9], it is shown that in order to provide an infinitely unstable state, the required average count of the grains equals 2.12588... ∗ n2. by toppling 2 ∗ n2 − 1 grains on a 2-dimensional lattice, users need to topple 0.1258 ∗ n2 grains in average in order to make the model become infinitely unstable. the same logic applied for a 3-dimensional torus, the indispensable count of the grains will equal 3 ∗ n3, meanwhile the number of toppled grains will be 3 ∗ n3 − 1. the sandgame (see fig. 2) supports 2d/3d visualization along with the model rotation and zooming in/out possibilities. microsoft .net implements a strategy for web services to connect information; people; systems, and devices via software tools, thus making easier sharing and using the information between multiple websites, programs, and computers. besides, client server architecture has been implemented to facilitate working on shared models. so, the game is the following. there is a model with sands on random generated vertices and there are players, that should play against each other. each player will topple a grain on the desired node of the same model by sequence, in order to make the model become infinitely unstable. the player who will succeed in that will be the winner. 4.1 sandgame guidline this subsection is about the playing flow for sandgame. here it is specified how the game is played, and all the interactions are indicated. the definition is divided into three sections: pre-construction, actions, winner. pre-construction: for starting a new game, the player selects “file” on top panel, then chooses the “new game” option, and then inserts a size for the parameter n. a new model will be created and initialized with 3∗n3−1 grains. to proceed with sharing the model/game, user starts broadcasting from the top panel ”broadcasting”, and then selects “start broadcasting” 8 an example of a multiplayer serious game on 3d sandpile model by typing a name for the game (e.g., harvard championship). meanwhile, the other players open the channels’ list on the top panel ”broadcasting”; select the ”connect to chanel (see fig. 4), and then choose the desired game channel from the list. a prominent advantage of the software is that it provides simulation of all changes made during the model exploration, even in case of players’ lateness. actions: after connecting to a channel players can start dropping grains by selecting ”edit”, and then ”add grain” on the top panel (see fig. 3) by giving node coordinates. players continue these steps until one of them becomes a winner. winner: the player, whose dropped grain destabilizes the model resulting in all nodes toppling at least once, becomes a winner of the game. in a word, the player wins when his dropped grain forces the model to switch his state into infinitely unstable. strong background and theory behind the sandgame, makes it harder to win when you play against professional players. each player can create his/her own strategy, but the winner will be the one, whose strategy will be better. for example, one of the simplest strategies could be toppling grains on the nodes with max heights, as we know that model will be in infinitely unstable state when each node topples at least once. besides, accounting of attributes is implemented for each change in state, such as: average/layer/critical solidities; the model stability/non stability; count of nodes of the same height, etc., (see fig. 5), in order to facilitate decision making. fig 1. sandgame on ca simulator. fig 2. dialog window for adding grain. h. nahapetyan 9 fig 3. dialog window for the list of games. fig 4. attributes. 5. sandgame from developer viewpoint sandgame was developed under ca simulator, using .net and c#. to facilitate the collaborative play in the global network, microsoft azure has been used. from the developers viewpoint, the sandgame may be divided into three modules: “visualization”; “local simulation” and “service-client architecture”. in order to visualize the model zooms in/out and provide rotation, .net’s native libraries have been used. within the “local simulation” module, a guihelper class has been designed to provide the models creation; saving and loading; grains adding and toppling, as well as attributes counting, as follows: public static class guihelper { event eventhandler grainadded; viewport3d mainviewport; int size; list points; list points // initialize 3d view 10 an example of a multiplayer serious game on 3d sandpile model public static void init(viewport3d vp); //creates model with given sizes public static void createmodel(size size); //draws sandpile model public static void drawsandpilemodel(); //adds grain on sandpile model public static void addgrain(position pos); //adds grain from visual aspects private static void addgrainonvertex (interactivesphere point); //returns color regarded to grains count private static brush getcolorbyweight (int weight); #region file/string io //saves model in file public static void writetofile() {} //loads model from file public static void loadfromfile(){} #endregion } within the “service-client architecture” module, we have a broadcastinghelper class which includes essential functions to enable the broadcaster-subscriber connection, and a seservice class which implements the iseservice interface. the logic behind is to provide for a broadcaster to subscribe itself the same channel in order to get changes from other users. the channel keeps the whole information about changes made by all subscribers. meanwhile ,when a new user starts to listen to that channel, he/she not only gets up-to-date knowledge of the model, but also he/she gets provided with all the changes made since the moment of broadcasting. for the channels’ database, sqlite has been chosen. public static class broadcastinghelper { public static long selfchannelid; public static long subscribedchannelid; private static long lastactionid; private static timer timer; private static actionmodel locker; public static eventhandler<> channelclosed; //starts to listen to the given channel public static void listenchannel (channelmodel channel); h. nahapetyan 11 //disconnects from channel if it’s closed static void timer elapsed (object sender, elapsedeventargs e) //ends broadcasting public static void endbroadcasting(); //disconnects from channel public static void disconnectfromchannel(); public interface iseservice { [operationcontract] long startbroadcasting(string name); [operationcontract] void endbroadcasting(long id); [operationcontract] void addaction(long channelid, actiontype type, string data); [operationcontract] actionmodel getnextaction(long channelid, long lastactionid); [operationcontract] list getactivechannels(); } as already mentioned, ca simulator has been created on the example of asm. there are two main asm related functions: the drawsandpilemodel() which provides visualization of changes in already created model for asm vision, and the addgrain(position pos) which supports changes performed by the user on an asm model. it is quite easy to generate another ca model simply by manipulating the visualization and model modification functions. in order to make it a new ca available within a global network, a few functions of the iseservice interface should be adapted to the new ca model described within the seservice class. the sources of the ca simulator can be found in bitbucket under https : //nhayk@bitbucket.org/nhayk/casimulator.git link. 6. conclusion in this paper, a serious game, namely, “sandgame”, has been presented. the goal of the sandgame is to provide joint study/research of sandpile on 2d/3d graphs. features developed currently, are: simulation of asm; visualization within 2d and 3d space; shared play on the same model at the same time within global networks; models’ attributes counting. sandgame has been developed on the concept of the multiuser simulator (ca simulator), which was introduced and implemented in a way to make the solution available and fitting to any other type of cellular automata. perspectives on the work will be outlined in the near 12 an example of a multiplayer serious game on 3d sandpile model future in order to make the game environment more user-friendly, as well as to increase its usability and scalability. enhancements in visualization techniques will be implemented to make the simulator applicable for larger graphs, for what the technique described here [10] can be used. acknowledgement the author is grateful to prof. yu. h. shoukourian, dr. s. poghosyan and dr. y. alaverdyan for important discussions and critical remarks at all stages of the work. this work was supported by the state committee of science mes ra, in the frames of the research project no. 16yr-1b008. references [1] ch. meng-tzu, ch. jhih-hao, c. sheng-ju and ch. shin-yen, “the use of serious games in science education: a review of selected empirical research from 2002 to 2013”, journal of computers in education, no. 01 pp. 353-375, 2015. [2] p. bak, c. tang and k. wiesenfeld,“self-organized criticality: an explanation of the 1/f noise”,phys. rev. lett., vol.59, no. 4, pp. 381–384, 1987. [3] v. s. poghosyan, s. y. grigorev, v. b. priezzhev and p. ruelle, , “pair correlations in the sandpile model: a check of logarithmic conformal field theory”, phys. lett. b, vol. 659, pp. 768–772, 2008. [4] su. s. poghosyan, v. s. poghosyan, v. b. priezzhev and p. ruelle, “numerical study of correspondence between the dissipative and fixed-energy abelian sandpile models”, phys.rev. e, 84, 066119, 2011. [5] v.s. poghosyan, s. s. poghosyan and h. e. nahapetyan, “the investigation of models of self-organized systems by parallel programming methods based on the example of an abelian sandpile model”, proc. csit conference 2013, yerevan armenia, sept. 23-27, pp. 260-262, 2013. [6] h. nahapetyan, j.-pierre jessel, s. poghosyan and yu. shoukourian,”a multi user and multi purpose ca simulator”,phys. rev. lett., vol.59, no. 4, pp. 381–384, 1987. proc. csit conference 2017, yerevan armenia, sept. 23-27, pp. 260-262. [7] djaouti damien; alvarez julian; jessel jean-pierre. ”classifying serious games: the g/p/s model” irit university of toulouse, france. [8] m. graafland, j. m. schraagen and m. p. schijven “systematic review of serious games for medical education and surgical skills training”, department of surgery, academic medical centre, amsterdam, and netherlands organization for applied scientific research, soesterberg, the netherlands. [9] a. fey, l. levine and d.b. wilson, “approach to criticality in sandpiles”, phys. rev. e 82, 031121 published 15 september 2010. [10] h. e. nahapetyan, s. s. poghosyan, v. s. poghosyan and yu. h. shoukourian, “the parallel simulation method for d-dimensional abelian sandpile automata”, mathematical problems of computer science, vol. 46, 117–125, 2016. submitted 14.08.2017, accepted 28.11.2017. h. nahapetyan 1 3 ´³½ù³û·ï³ï»ñ “èáõñç ë³õ”-ç ûñçý³ï ¹çï³ñïí³í ³í³½³ïáõûïç »é³ã³÷ ùá¹»éç íñ³ ð. ü³ñ³å»ïû³ý ²ù÷á÷áõù ²ûë ñá¹í³íç ýå³ï³ïý ¿ ¹çï³ñï»é µçç³ûçý ³íïáù³ïý»ñç ùá¹»éý»ñá áñå»ë ñçùù áñáß³ïç “èáõñç ë³õ»ñ”-ç ñ³ù³ñ: àñå»ë µçç³ûçý ³íïáù³ïç ûñçý³ï áýïñí»é ¿ ²µ»éû³ý ³í³½³ïáõûïç ùá¹»éá »é³ã³÷ ·ñ³ýç ñ³ù³ñ: êï»õíí»é ¿ ó³ýó³ûçý µ³½ù³û·ï³ï»ñ ë³õ³ûçý ñ³ù³ï³ñ· (sandgame) û·ï³·áñí»éáí .net and c # é»½áõý: ïðèìåð ìíîãîïîëüçîâàòåëüñêîé èãðû íà òðåõìåðíîé ìîäåëè ïåñ÷àíîé êó÷è ã. íàãàïåòÿí àííîòàöèÿ öåëü ýòîé ñòàòüè ðàññìîòðåòü ìîäåëè êëåòî÷íûõ àâòîìàòîâ êàê îñíîâó îïðåäåëåííûõ ñåðüåçíûõ èãð. â êà÷åñòâå ïðèìåðà êëåòî÷íûõ àâòîìàòîâ âûáðàíà ìîäåëü ïåñ÷àíîé êó÷è íà òðåõìåðíûõ ãðàôàõ. ðàçðàáîòàíà ñåòåâàÿ ìíîãîïîëüçîâàòåëüñêàÿ èãðîâàÿ ñðåäà (sandgame) ñ èñïîëüçîâàíèåì .net è ÿçûêà c # . hayk's article hayk microsoft word manukian-revised.doc mathematical problems of computer science 37, 64--74, 2012. 64 some algebraical and logical properties of twodimensional arithmetical sets representable in presburger’s system1 seda n. manukian institute for informatics and automation problems of nas of ra e-mail: zaslav@ipia.sci.am abstract a classification ...)2()1()0(  hhh of arithmetical sets representable in m.presburger’s system ([1]-[4]) and a classification ...)2()1()0(  hhh of twodimensional sets of the same kind are considered. it is proved that these classifications are strictly monotone and complete. the operations  ,  , ,  , 1 on twodimensional arithmetical sets ([5]-[7]) and the algebras 0 and 1 based on these operations ([5]-[7]) are considered. the relations of these operations and algebras to the mentioned classifications are investigated. references [1] m. presburger, “ über die vollstaendigkeit eines gewissen system der arithmetik ganzer zahlen, in welchem die addition als einige operation hervortritt”, comptes rendu du i congres des mathematiciens des pays slaves, warszawa, pp. 92-101, 1930. [2] d. hilbert und p. bernays. grundlagen der mathematik. band i. zweite auflage, berlin-heidelbergnew york, springer verlag, 1968. [3] r. stansifer. presburger's article on integer arithmetic: remarks and translation. department of computer science, cornell university, ithaca, new york, 1984. [4] h. enderton. a mathematical introduction to logic. 2nd ed., san diego, harcourt, academic press, 2001. [5] s. n. manukian, “algebras of recursively enumerable sets and their applications to fuzzy logic”, journal of mathematical sciences, vol. 130, №2, pp. 4598-4606, 2005. [6] s. n. manukian, “a classification of arithmetical sets expressible in presburger's system”, proceedings of the international conference «computer science and information technologies», csit05, yerevan, armenia, pp. 161-162, 2005. [7] s. n. manukian, “on the representation of recursively enumerable sets in weak arithmetic. mathematical problems of computer science, vol. 27, pp. 90-110, 2006. 1 this work is supported by the grant 11-1b 189 of the government of the republic of armenia. s. manukian 65 [8] s. n. manukian, “classification of many-dimensional arithmetical sets represented in m. presburger's system”, reports of the national academy of science of armenia, vol. 111, №2, pp. 114120, 2011 (in russian). [9] s. c. kleene. introduction to metamathematics. d. van nostard comp., inc., new york-toronto, 1952. մ. պրեսբուրգերի համակարգում ներկայացվող երկչափ թվաբանական բազմությունների որոշ հանրահաշվական և տրամաբանական հատկություններ ս. ն. մանուկյան ամփոփում դիտարկվում են ...)2()1()0(  hhh դասերը, որոնց հաջորդականությունը ընդգրկում է մ. պրեսբուրգերի համակարգում ներկայացվող թվաբանական բազմությունների դասը ([1]-[4]) և ...)2()1()0(  hhh դասերը, որոնց հաջորդականությունն ընդգրկում է նման տիպի երկչափ բազմությունների դասը: ապացուցվում է, որ նշված դասակարգումները խիստ մոնոտոն են և լրիվ: դիտարկվում են երկչափ թվաբանական բազմությունների վրա որոշված  ,  , ,  , 1 գործողությունները ([5]-[7]) և այդ գործողությունների վրա հիմնված 0 և 1 հանրահաշիվները ([5]-[7]): հետազոտվում են այդ գործողությունների և հանրահաշիվների փոխհարաբերությունները նշված դասակարգումների հետ: о некоторых алгебраических и логических особенностях двумерных арифметических множеств, представимых в системе пресбургера с. н. манукян аннотация рассматривается классификация ...)2()1()0(  hhh арифметических множеств, представимых в системе пресбургера ([1]-[4]), а также классификация ...)2()1()0(  hhh двумерных арифметических множеств аналогичного типа. доказывается полнота и строгая монотонность этих классификаций. рассматриваются операции  ,  ,  ,  , 1 на двумерных арифметических множествах ([5]-[7]), а также алгебры 0 и 1 , основанные на этих операциях ([5]-[7]). исследуются взаимоотношения рассматриваемых операций и алгебр с вышеупомянутыми классификациями. d:\sbornik\...\alizadeh2.dvi mathematical problems of computer science 35, 26{32, 2011. constr ucting m ethods for i r r educible p olynomials ma h m o o d a liz a d e h islamic azad universityahvaz branch e-mail: alizadeh@iauahvaz.ac.ir abstract in this paper we study the irreducibility of some composite polynomials, constructed with a polynomial composition method over ¯nite ¯elds. furthermore, a recurrent method for constructing families of irreducible polynomials of higher degree from given irreducible polynomials over ¯nite ¯elds is given. refer ences [1 ] e .r . b e r le ka m p , a lg e b r a ic c o d in g t h e o r y, mc gr a w-h ill, n e w y o r k, 1 9 6 8 . 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[1 1 ] a .j. me n e z e s , i. f. b la ke , x .ga o , r .c.mu llin , s .a .v a n s t o n e , t.y a g h o o b ia n , applications of ¯nite ¯elds, k lu we r a c a d e m ic p u b lis h e r s , b o s t o n , 1 9 9 3 . 2 6 m. alizadeh 2 7 âµ»ñíáõ µ³½ù³ý¹³ùý»ñç ï³éáõóù³ý »õ³ý³ï ø. ²éç½³¹»ñ ²ù÷á÷áõù ²ûë ³ßë³ï³ýùáõù ù»ýù áõëáõùý³ëçñáõù »ýù áñáß ïáùåá½çóçáý µ³½ù³ý¹³ùý»ñç ãµ»ñí»éçáõãûáõýá, áñáýù ï³éáõóí³í »ý µ³½ù³ùý¹³ù³ûçý ïáùåá½çóçáý ù»ãá¹áí, í»ñç³íáñ ¹³ßï»ñç íñ³: ²í»éçý ïñí»é ¿ é»ïáõñ»ýï ù»ãá¹, í»ñç³íáñ ¹³ßï»ñç íñ³ ïñí³í ãµ»ñíáõ µ³½ù³ý¹³ùçó µ³ñóñ ³ëïç×³ýç ãµ»ñíáõ µ³½ù³ý¹³ù ï³éáõó»éáõ ñ³ù³ñ: d:\sbornik\...\article.dvi mathematical problems of computer science 40, 23{30, 2013. on application of optimal m ultihypothesis t ests for the b ounds constr uction for the i ncr ease of gr owth rate of stock m ar ket e vg u e n i a . h a r o u t u n ia n , a r t a k r . ma r t ir o s ya n a n d a r a m o. y e s s a ya n institute for informatics and automation problems of nas ra yerevan, armenia e-mail: eghishe@ipia.sci.am abstract in the present paper new bounds for increase of the growth rate of stock market (when an erroneous probability distribution is used) are obtained applying the matrix of reliabilities of optimal tests. keywords: hypothesis testing, stock market, growth rate, portfolio. 1 . in t r o d u c t io n th e m o d e l o f s t o c k m a r ke t c o n s id e r e d in t h is p a p e r is p r e s e n t e d in [1 ], wh e r e t h e b o u n d fo r t h e in c r e a s e o f t h e g r o wt h r a t e o f s t o c k m a r ke t in c a s e o f c o n t in u o u s d is t r ib u t io n s is c o n s t r u c t e d . th is m o d e l a n d t h e c o r r e s p o n d in g o p t im a l p o r t fo lio p r o b le m s we r e d e e p ly in ve s t ig a t e d in [1 ] { [8 ]. th e a p p lic a t io n o f r e s u lt s o n t h e h yp o t h e s is o f lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l ( l a o) t e s t in g ( [9 ] { [1 3 ]) in p o r t fo lio t h e o r y is in t r o d u c e d in [1 1 ], wh e r e u s in g e r r o r p r o b a b ilit ie s a n d r a lia b ilit ie s , n e w, m o r e e xa c t b o u n d s a r e fo u n d fo r t h e in c r e a s e o f t h e g r o wt h r a t e o f s t o c k m a r ke t . u s in g t h e e xis t e n c e t h e o r e m o f l a o t e s t [1 0 ] b o u n d s o b t a in e d in [1 1 ] a r e r e ¯ n e d in t h e p r e s e n t p a p e r . w e s t u d y a s t o c k m a r ke t wit h k ( ¸ 1 ) s t o c ks d e ¯ n e d b y a r a n d o m ve c t o r x = ( x1; x2; :::; xk ) , xk ¸ 0 ; k = 1 ; k; wh e r e t h e p r ic e r e la t ive s ( t h e r a t io o f t h e p r ic e a t t h e e n d o f t h e d a y t o t h e p r ic e a t t h e b e g in n in g o f t h e d a y) xk, k = 1 ; k; a r e in d e p e n d e n t r a n d o m va r ia b le s ( r v s ) wit h u n kn o wn p r o b a b ilit y d is t r ib u t io n s ( p d s ) d e ¯ n e d o n t h e ¯ n it e s e t x . w e a s s u m e t h a t m d is t in c t p d s gm; m = 1 ; m , a r e g ive n o n x wh ic h a r e p o s s ib le p d s fo r r v s xk; k = 1 ; k. if t h e r ig h t p d o f t h e r v xk is gmk ; k = 1 ; k, t h e n t h e jo in t p d o f t h e s t o c k m a r ke t ve c t o r x will b e gm1;m2;:::;mk ( x) = kq k=1 gmk ( xk ) ; wh e r e x = ( x1; x2; :::; xk ) 2 x k : p o r t fo lio ve c t o r b = ( b1; b2; :::; bk ) , bk ¸ 0 , k = 1 ; k, p bk = 1 , is a n a llo c a t io n o f t h e we a lt h a c r o s s t h e s t o c ks , wh e r e bk is t h e fr a c t io n o f o n e 's we a lt h in ve s t e d in s t o c k k [1 ]. 2 3 2 4 on application of optimal multihypothesis tests for the bounds construction of stock market w h e n a n in ve s t o r u s e s a p o r t fo lio b fo r s t o c k m a r ke t x, t h e n t h e we a lt h r e la t ive will b e s = btx ( bt is t h e t r a n s p o s e d ve c t o r o f b) : a n d if t h e in ve s t o r in ve s t s in t h e s t o c k m a r ke t fo r n c o n s e c u t ive d a ys a n d u s e s t h e s a m e b e t t in g s t r a t e g y b e a c h d a y, t h e n t h e we a lt h a t t h e e n d o f n d a ys is g ive n b y sn = nq n=1 btxn; wh e r e t h e s t o c k m a r ke t ve c t o r s x1; x2; ::: a r e in d e p e n d e n t a n d id e n t ic a lly d is t r ib u t e d wit h p d gm1;m2;:::;mk ( x ) : th e o p t im a l g r o wt h r a t e is w ¤ ( g ) = m a x b w ( b; g ) , wh e r e w ( b; g ) = e ( lo g btx ) is t h e g r o wt h r a t e o f a s t o c k m a r ke t a t p o r t fo lio b = ( b1; b2; :::; bk ) wit h r e s p e c t t o a s t o c k p d gm1;m2;:::;mk ( if t h e lo g a r it h m is t o b a s e 2 , t h e g r o wt h r a t e is c a lle d a d o u b lin g r a t e [1 ]) . th e n a p o r t fo lio b¤ = a r g ( m a x b w ( b; g ) ) , is c a lle d a lo g { o p t im a l p o r t fo lio o r a g r o wt h o p t im a l p o r t fo lio . in t h is p a p e r we a s s u m e t h a t p d gm1;m2;:::;mk is u n kn o wn a n d m u s t b e d e t e r m in e d o n t h e b a s e o f s a m p le ( r e s u lt s o f n in d e p e n d e n t o b s e r va t io n s ) xn = ( x1;n; x2;n; : : : ; xk;n ) , n = 1 ; n; wh e r e xk;n is a r e s u lt o f t h e n { t h r e a liz a t io n o f t h e p r ic e r e la t ive xk; k = 1 ; k: w it h t h e s a m p le xn = ( x1; x2; :::; xn ) we c a n c o n s id e r t h e e m p ir ic a l p d o f t h e s t o c k m a r ke t a ft e r n d a ys : g(n)m1;m2;:::;mk ( x n ) = ny n=1 gm1;m2;:::;mk ( xn ) = ny n=1 ky k=1 gmk ( xk;n ) : w e c o n s id e r t h e c a s e , wh e n e a c h p d fr o m t h e g ive n p d s gm c a n b e a d o p t e d fo r s o m e p r ic e r e la t ive s s im u lt a n e o u s ly. th e p r o c e d u r e o f d e c is io n m a kin g is a n o n -r a n d o m iz e d t e s t '(n); wh ic h c a n b e d e ¯ n e d b y a d ivis io n o f t h e s a m p le s p a c e x kn in t o m k d is jo in t s u b s e t s a(n)l1;l2;:::;lk = fx n : '(n) ( xn ) = ( l1; l2; :::; lk ) g; lk = 1 ; m; k = 1 ; k: th e p r o b a b ilit y o f t h e e r r o n e o u s a c c e p t a n c e o f h yp o t h e s is hl1;l2;:::;lk p r o vid e d t h a t hm1;m2;:::;mk is t r u e , fo r ( l1; l2; :::; lk ) 6= ( m1; m2; :::; mk ) ; is [9 ] ® (n ) l1;l2;:::;lk jm1;m2;:::;mk = g (n) m1;m2;:::;mk ( a(n )l1;l2;:::;lk ) : fo r t h e p r o b a b ilit y o f r e je c t io n o f h yp o t h e s is hm1;m2;:::;mk , wh e n it is t r u e t h e fo llo win g d e ¯ n it io n is a d o p t e d : ® (n ) m1;m2;:::;mk jm1;m2;:::;mk = x (l1;l2;:::;lk ) 6=(m1;m2;:::;mk ) ® (n ) l1;l2;:::;lk jm1;m2;:::;mk : fo r t h e s e qu e n c e ' o f t e s t s we will c o n s id e r e r r o r p r o b a b ilit y e xp o n e n t s wh ic h a r e c a lle d r e lia b ilit ie s : ¡ lim n !1 1 n lo g ® (n ) l1;l2;:::;lk jm1;m2;:::;mk = el1;l2;:::;lkjm1;m2;:::;mk ¸ 0 ; mk; lk = 1 ; m: it is kn o wn , t h a t em1;m2;:::;mk jm1;m2;:::;mk = m in (l1;l2;:::;lk )6=(m1;m2;:::;mk ) el1;l2;:::;lkjm1;m2;:::;mk : th e m a t r ix e = fel1;l2;:::;lkjm1;m2;:::;mk g is a m a t r ix o f r e lia b ilit ie s o f t h e s e qu e n c e o f t e s t s ': w e c a ll t h e t e s t s e qu e n c e '¤ l a o if fo r t h e g ive n p o s it ive va lu e s o f c e r t a in p a r t o f e le m e n t s o f t h e m a t r ix o f r e lia b ilit ie s e ( '¤ ) t h e p r o c e d u r e p r o vid e s m a xim a l va lu e s fo r a ll t h e o t h e r e le m e n t s o f it . e. haroutunian, a. martirosyan, a. yessayan 2 5 2 . p r o b le m s t a t e m e n t l e t t h e t r u e p d o f t h e s t o c k m a r ke t x = ( x1; x2; :::; xk ) b e gm1;m2;:::;mk wit h t h e c o r r e s p o n d in g lo g { o p t im a l p o r t fo lio bm1;m2;:::;mk a n d le t bl1;l2;:::;lk ; ( l1; l2; :::; lk ) 6= ( m1; m2; :::; mk ) b e a lo g { o p t im a l p o r t fo lio c o r r e s p o n d in g t o s o m e o t h e r , wr o n g p d gl1;l2;:::;lk : w h e n t h e in c o r r e c t p o r t fo lio bl1;l2;:::;lk is u s e d in s t e a d o f bm1;m2;:::;mk , t h e n t h e in c r e a s e o f t h e g r o wt h r a t e is t h e fo llo win g : ¢ w = w ( bm1;m2;:::;mk ; gm1;m2;:::;mk ) ¡ w ( bl1;l2;:::;lk ; gm1;m2;:::;mk ) ¸ 0 : th e in ve s t o r c a n u s e t h e s a m e in c o r r e c t p o r t fo lio fo r t h e n e xt p e r io d a n d h a ve s o m e ¯ n a n c ia l lo s s e s . th e b o u n d s fo r ¢ w will s h o w t h e m a xim a l p o s s ib le a m o u n t o f lo s s e s . co n s id e r t h e fo llo win g qu a n t it ie s : ql1;l2;:::;lkm1;m2;:::;mk ( x ) = btm1;m2;:::;mk x btl1;l2;:::;lk x gl1;l2;:::;lk ( x ) ; x 2 x k gl1;l2;:::;lkm1;m2;:::;mk = x x2x k gm1;m2;:::;mk ( x) lo g ql1;l2;:::;lkm1;m2;:::;mk ( x ) gm1;m2;:::;mk ( x) : a s a d ir e c t c o n s e qu e n c e o f t h e k u h n { tu c ke r c o n d it io n s , it wa s p r o ve d in [1 ] a n d [3 ] t h a t e s s¤ · 1 ; fo r a ll s; wh e r e s ¤ = b¤tx is t h e r a n d o m we a lt h r e s u lt in g fr o m t h e lo g { o p t im a l p o r t fo lio b¤; a n d s = btx is t h e we a lt h r e s u lt in g fr o m a n y o t h e r p o r t fo lio b: fr o m in e qu a lit y e s s¤ · 1 a n d t h e fa c t t h a t bl1;l2;:::;lk is a lo g { o p t im a l p o r t fo lio fo r gl1;l2;:::;lk ; we h a ve ql1;l2;:::;lkm1;m2;:::;mk ( x ) · x x2x k ql1;l2;:::;lkm1;m2;:::;mk ( x) · 1 : l e t d ( rkp ) = p z2z r( z ) lo g r(z) p (z) < 1 b y ( k u llb a c k l e ib le r ) d ive r g e n c e fo r p d s b y r a n d p; z 2 z: t heor em 1 [11]: if ( l1; l2; :::; lk ) 6= ( m1; m2; :::; mk ) ; then 4w = d ( gm1;m2;:::;mk k gl1;l2;:::;lk ) + gl1;l2;:::;lkm1;m2;:::;mk ; where gl1;l2;:::;lkm1;m2;:::;mk · 0 with equality if and only if q l1;l2;:::;lk m1;m2;:::;mk ( x) = gm1;m2;:::;mk ( x) ; for all x 2 x k : cor ollar y 1: the analog of the bound exposed in [1] in the case of discrete p d s is 4w · d ( gm1;m2;:::;mk k gl1;l2;:::;lk ) : fr o m th e o r e m 1 it fo llo ws t h a t fo r e a c h p a ir ( l1; l2; :::; lk ) 6= ( m1; m2; :::; mk ) t h e in ve s t o r ¯ r s t m u s t c h e c k t h e c o n d it io n ql1;l2;:::;lkm1;m2;:::;mk ( x) = gm1;m2;:::;mk ( x) ; fo r a ll x 2 x k : in s u c h c a s e s h is lo s s e s will b e m a xim a l. a ll o t h e r c a s e s ( l1; l2; :::; lk ) 6= ( m1; m2; :::; mk ) ( a n d t h e c o r r e s p o n d in g p o r t fo lio s bl1;l2;:::;lk 6= bm1;m2;:::;mk ) we will c a ll c a s e s ( p o r t fo lio s ) o f \ nonmaximal losses " a n d will a s s u m e t h a t t h e fo llo win g c o n d it io n is fu l¯ lle d : gl1;l2;:::;lkm1;m2;:::;mk ¸ ¡d ( gl1;l2;:::;lk k gm1;m2;:::;mk ) : ( 1 ) 2 6 on application of optimal multihypothesis tests for the bounds construction of stock market 3 . n e w b o u n d fo r gr o wt h r a t e l e t n ( xjx ) b e t h e n u m b e r o f r e p e t it io n s o f t h e e le m e n t x 2 x in t h e ve c t o r x 2 x n ; a n d le t q = fq( x ) = n ( xjx ) =n; x 2 x g b e t h e e m p ir ic a l d is t r ib u t io n ( t yp e ) o f t h e s a m p le x: fo r t h e g ive n p o s it ive d ia g o n a l e le m e n t s e1j1; e2j2; : : : ; em¡1jm ¡1 o f t h e r e lia b ilit y m a t r ix we c o n s id e r s e t s o f p d s rl = fq : d ( qjjgl ) · eljlg; l = 1 ; m ¡ 1 ; ( 2 ) rm = fq : d ( qjjgl ) > eljl; l = 1 ; m ¡ 1 g; ( 3 ) a n d d e ¯ n e t h e va lu e s fo r t h e e le m e n t s o f t h e fu t u r e r e lia b ilit y m a t r ix o f t h e l a o t e s t s s e qu e n c e a s fo llo ws : e¤ljl = e ¤ ljl ( eljl ) = eljl; l = m ¡ 1 ; ( 4 ) e¤ljm = e ¤ ljm ( eljl ) = in f q2r l d ( qjjgm ) ; m = 1 ; m ; m 6= l; l = 1 ; m ¡ 1 ; ( 5 ) e¤mjm = e ¤ mjm ( e1j1; : : : ; em¡1jm ¡1 ) = in f q2r m d ( qjjgm ) ; m = 1 ; m ¡ 1 ; ( 6 ) e¤mjm = e ¤ m jm ( e1j1; : : : ; em¡1jm¡1 ) = m in l=1;m¡1 e¤mjl: ( 7 ) th e l a o t e s t e xis t in g t h e o r e m c o n c e r n in g o n e o b je c t is t h e fo llo win g [1 0 ]: t heor em 2 [10]: if the distributions gm are di®erent, that is all divergences d ( gljjgm ) , l 6= m, l; m = 1 ; m , are strictly positive, then two statements hold: a) when the given numbers e1j1; e2j2; : : : ; em¡1jm ¡1 satisfy the conditions 0 < e1j1 < m in l=2;m d ( gljjg1 ) ; ( 8 ) 0 < emjm < m in [ m in l=1;m¡1 e¤ljm ( eljl ) ; m in l=m+1;m d ( gljjgm ) ]; m = 2 ; m ¡ 1 ; ( 9 ) then there exists a l ao sequence of tests '¤, the elements of the reliability matrix of which e ( '¤ ) = fe¤ljmg are de¯ned in (4){(7) and all of them are strictly positive; b) even if one of the conditions (8) or (9) is violated, then the reliability matrix of any such test includes at least one element equal to zero. t heor em 3: w hen the ¯rst m -1 diagonal elements of reliability matrix are ¯xed so that the numbers e1j1; e2j2; : : : ; em¡1jm ¡1 satisfy the conditions (8)-(9), then in case of condition (1), for portfolios bl1;l2;:::;lk 6= bm1;m2;:::;mk ; ( l1; l2; :::; lk ) 6= ( m1; m2; :::; mk ) ; of non maximal losses the following bound is true: ¢ w · d ( gm1;m2;:::;mk k gl1;l2;:::;lk ) ¡e¤l1;l2;:::;lkjm1;m2;:::;mk ; wh e r e e¤l1;l2;:::;lkjm1;m2;:::;mk = x k=1;k: mk 6=lk e¤lkjmk ( 'k ) : p r oof: b y co r o lla r y 1 fr o m [1 1 ] in c a s e o f c o n d it io n ( 1 ) , fo r p o r t fo lio s bl1;l2;:::;lk 6= bm1;m2;:::;mk ; ( l1; l2; :::; lk ) 6= ( m1; m2; :::; mk ) ; o f n o n -m a xim a l lo s s e s we h a ve ¢ w · d ( gm1;m2;:::;mk k gl1;l2;:::;lk ) ¡el1;l2;:::;lkjm1;m2;:::;mk : e. haroutunian, a. martirosyan, a. yessayan 2 7 th e r e fo r e t h e b e s t b o u n d fo r t h e in c r e a s e o f t h e g r o wt h r a t e will b e in c a s e o f m a xim a l va lu e s o f r e lia b ilit ie s . w h e n t h e ¯ r s t m-1 d ia g o n a l e le m e n t s o f o n e d im e n s io n a l r e lia b ilit y m a t r ix a r e ¯ xe d c o r r e s p o n d in g t o t h e c o n d it io n s ( 8 ) -( 9 ) t h e n fr o m th e o r e m 2 it fo llo ws t h a t t h e r e e xis t s a l a o s e qu e n c e o f t e s t s '¤, t h e e le m e n t s e ( '¤ ) = fe¤ljmg o f o n e d im e n s io n a l r e lia b ilit y m a t r ix o f wh ic h a r e d e ¯ n e d in ( 4 ) { ( 7 ) wit h m a xim a l va lu e s . th e c o m p o u n d t e s t '(n) fo r k s t o c ks c a n b e r e p r e s e n t e d a s a c o lle c t io n o f k in d ivid u a l t e s t s ' (n) 1 , ' (n ) 2 , ..., ' (n ) k fo r e a c h o f k s t o c ks [1 2 ], t h e n t h e in ¯ n it e c o m p o u n d t e s t ' is a c o lle c t io n o f in ¯ n it e t e s t s '1, '2,..., 'k : b y t h e l e m m a fr o m [1 2 ] if e le m e n t s elijmi ( ' i ) , mi; li = 1 ; m , i = 1 ; k; a r e s t r ic t ly p o s it ive , t h e n fo r ' = ( '1; '2; :::; 'k ) el1;l2;:::;lkjm1;m2;:::;mk ( ') = x i=1;k: mi 6=li elijmi ( 'i ) : s in c e t h e e le m e n t s e¤lijmi ( 'i ) a r e m a xim a l, t h e r e fo r e t h e s u m e ¤ l1;l2;:::;lkjm1;m2;:::;mk ( ') will a ls o b e m a xim a l. t heor em is pr oved. th e fo llo win g e xa m p le illu s t r a t e s t h e r e s u lt o f t h e th e o r e m 3 . e xample: co n s id e r t h e s e t x = f 1 ; 2 g a n d t wo p d s g1 = f 1 =2 ; 1 =2 g a n d g2 = f 1 = 3 ; 2 = 3 g d e ¯ n e d o n x : co r o lla r y 1 g ive s t h e fo llo win g b o u n d s : ¢ w = ¢ w ( g1 k g2 ) · d ( g1kg2 ) ¼ 0 :0 5 8 8 9 a n d ¢ w = ¢ w ( g2 k g1 ) · d ( g2kg1 ) ¼ 0 :0 5 6 6 3 : a p p lyin g th e o r e m 3 we s h a ll im p r o ve t h e b o u n d s . fo r t wo h yp o t h e s is we c a n t a ke e1j1 = 0 :0 5 : l e t q = fq1; q2g b e t h e t yp e o f t h e ve c t o r x 2 x n ; wh e r e q1 = n ( 1 jx ) =n a n d q2 = n ( 2 jx ) =n: fig. 1. q log 2q + (1 ¡ q) log 2(1 ¡ q) 2 8 on application of optimal multihypothesis tests for the bounds construction of stock market fig. 2. q log 3q + (1 ¡ q) log( 3 2 (1 ¡ q)) w e n e e d t o c a lc u la t e ( 5 ) a n d ( 6 ) e¤1j2 = in f d(qjjg1)·e1j1 d ( qjjg2 ) = in f q1 log 2q1+q2 log 2q2·e1j1 ( q1 lo g 3 q1 + q2 lo g ( 3 2 q2 ) ) a n d e¤2j1 = in f d(qjjg1)>e1j1 d ( qjjg1 ) = in f q1 log 2q1+q2 log 2q2>e1j1 ( q1 lo g 2 q1 + q2 lo g ( 2 q2 ) ) : ta kin g in t o a c c o u n t t h a t q1 + q2 = 1 ; fr o m t h e e xp r e s s io n q1 lo g 2 q1 + q2 lo g 2 q2 = 0 : 0 5 we o b t a in t h e e qu a t io n q lo g 2 q + ( 1 ¡ q ) lo g 2 ( 1 ¡ q ) = 0 : 0 5 : th e a p p r o xim a t e s o lu t io n s o f t h is e qu a t io n a r e q¡ ¼ 0 : 3 4 3 2 2 a n d q¡ ¼ 0 : 6 5 6 7 8 : in fig . 1 a n d fig . 2 t h e g r a p h s o f t h e fu n c t io n s q lo g 2 q +( 1 ¡q ) lo g 2 ( 1 ¡q ) a n d q lo g 3 q+( 1 ¡q ) lo g ( 3 2 ( 1 ¡q ) ) a r e r e p r e s e n t e d . th e s o lu t io n o f t h e in e qu a lit y q lo g 2 q + ( 1 ¡ q ) lo g 2 ( 1 ¡ q ) · e1j1 = 0 : 0 5 is q 2 [q¡; q¡]: th e r e fo r e , n o w it is e a s y t o s e e t h a t e¤1j2 = in f [q¡;q¡] ( q lo g 3 q + ( 1 ¡ q ) lo g ( 3 2 ( 1 ¡ q ) ) ) ¼ 0 :0 0 0 2 2 : it is a ls o o b vio u s t h a t e¤2j1 = in f [0;q¡)[(q¡;1] ( q lo g 2 q + ( 1 ¡ q ) log ( 2 ( 1 ¡ q ) ) ) = 0 : 0 5 : th u s , n e w b o u n d s a r e ¢ w ( g1 k g2 ) · 0 : 0 0 8 8 9 ; ¢ w ( g2 k g1 ) · 0 : 0 5 6 4 1 : 4 . co n c lu s io n th is p a p e r r e p r e s e n t s t h e a p p lic a t io n o f c o e ± c ie n t s o f t h e r e lia b ilit y m a t r ix o f l a o t e s t s in s t o c k ma r ke t p o r t fo lio t h e o r y. in ve s t o r s c a n u s e t h e o b t a in e d fo r m u la fo r b o u n d t o e va lu a t e t h e fu t u r e r is ks , m in im a l in c o m e s a n d d o n e c e s s a r y c o r r e c t io n s in c u r r e n t p o r t fo lio . ch o o s in g s o m e p d fr o m t h e g ive n s e t o f p d s , t h e in ve s t o r will h a ve s o m e e r r o r s . th e e xis t in g e m p ir ic a l d a t a a llo w t o e s t im a t e s t a t is t ic a lly t h e p r o b a b ilit ie s o f t h o s e e r r o r s . a n d t h e r e lia b ilit y c o e ± c ie n t s o f t h e c o r r e s p o n d in g e r r o r p r o b a b ilit ie s a r e u s e d fo r o b t a in in g o f n e w b o u n d s . it is s h o wn t h a t t h e b o u n d s c a n b e m in im iz e d u s in g l a o t e s t s . fixin g va lu e s e. haroutunian, a. martirosyan, a. yessayan 2 9 fo r d ia g o n a l e le m e n t s o f r e lia b ilit y m a t r ix t h e l a o t e s t a llo ws t o o b t a in m a xim a l va lu e s fo r a ll o t h e r e le m e n t s wh ic h e n a b le s o b t a in in g o f le s s e r b o u n d s fo r in c r e a s e o f g r o wt h r a t e o f t h e s t o c k m a r ke t . th e d e r ive d b o u n d c a n b e e ®e c t ive in m o d e lin g o f s t o c k m a r ke t . e va lu a t in g wit h n e w b o u n d t h e in ve s t o r h a vin g h is t o r ic a l d a t a c a n m in im iz e h is lo s s e s m o r e t h a n t h e in ve s t o r wit h o u t u s in g t h e h is t o r ic a l d a t a . a c kn o wle d g e m e n t th is wo r k wa s s u p p o r t e d in p a r t b y s cs o f me s o f r a u n d e r th e m a t ic p r o g r a m n o s cs 1 3 { 1 a 2 9 5 . refer ences [1 ] t. m. co ve r a n d j. a . th o m a s , e lements of information theory, s e c o n d e d it io n , n e w je r s e y, 2 0 0 6 . [2 ] t. m. co ve r , \ a n a lg o r it h m fo r m a xim iz in g e xp e c t e d lo g in ve s t m e n t r e t u r n " , ie e e transections on information theory, vo l. it { 3 0 , n o . 2 , p p . 3 6 9 { 3 7 3 , ma r c h 1 9 8 4 . [3 ] a . r . b a r r o n a n d t. m. co ve r , \ a b o u n d o n t h e ¯ n a n c ia l va lu e o f in fo r m a t io n " , ie e e transections on information theory, vo l. 3 4 , n o . 5 , p p . 1 0 9 7 { 1 1 0 0 , 1 9 8 8 . [4 ] e . e r kip a n d t. m. co ve r , \ th e e ± c ie n c y o f in ve s t m e n t in fo r m a t io n " , ie e e transections on information theory, vo l. 4 4 , n o . 3 , p p . 1 0 2 6 { 1 0 4 0 , 1 9 9 8 . [5 ] e . or d e n t lic h a n d t. m. co ve r , \ th e c o s t o f a c h ie vin g t h e b e s t p o r t fo lio in h in d s ig h t " , m athematics of operations r esearch, vo l. 2 3 , n o . 4 , p p . 9 6 0 { 9 8 2 , 1 9 9 8 . [6 ] g. n . iye n g a r a n d t. m. co ve r , \ gr o wt h o p t im a l in ve s t m e n t in h o r s e r a c e m a r ke t s wit h c o s t s " , ie e e transections on information theory, vo l. 4 6 , n o . 7 , p p . 2 6 7 5 { 2 6 8 3 , 2 0 0 0 . [7 ] p . h . a lg o e t a n d t. m. co ve r , \ a s ym p t o t ic o p t im a lit y a n d a s ym p t o t ic e qu ip a r t it io n p r o p e r t ie s o f lo g { o p t im a l in ve s t m e n t " , the annals of p robability, vo l. 1 6 , n o . 2 , p p . 8 7 6 { 8 9 8 , 1 9 8 8 . [8 ] r . b e ll a n d t. m. co ve r , \ ga m e { t h e o r e t ic o p t im a l p o r t fo lio s " , m anagment science, vo l. 3 4 , n o . 6 , p p . 7 2 4 { 7 3 3 , 1 9 8 8 . [9 ] e . h a r o u t u n ia n , m. h a r o u t u n ia n a n d a . h a r u t yu n ya n , \ r e lia b ilit y c r it e r ia in in fo r m a t io n t h e o r y a n d in s t a t is t ic a l h yp o t h e s is t e s t in g " , f oundations and trends in communications and information theory, vo l. 4 , n o . 2 , p p . 9 7 -2 6 3 , 2 0 0 8 . [1 0 ] e . h a r o u t u n ia n , \ l o g a r it h m ic a lly a s ym p t o t ic a ly o p t im a l t e s t in g o f m u lt ip le s t a t is t ic a l h yp o t h e s e s " , p roblems of control and information theory, vo l. 1 9 , n o . 5 -6 , p p . 4 1 3 -4 2 1 , 1 9 9 0 . [1 1 ] a . r . ma r t ir o s ya n , \ n e w b o u n d s c o n s t r u c t io n b y m e a n s o f r e lia b ilit ie s fo r t h e in c r e a s e o f g r o wt h r a t e o f s t o c k" , p roceedings of international conference csit 2011, p p . 1 8 6 { 1 8 9 , y e r e va n , 2 0 1 1 . [1 2 ] e . h a r o u t u n ia n a n d p . h a ko b ya n , \ mu lt ip le h yp o t h e s is l a o t e s t in g fo r m a n y in d e p e n d e n t o b je c t s " , scholarly r esearch e xchange, 2 0 0 9 . [1 3 ] r . a h ls we d e a n d e . h a r o u t u n ia n , \ on lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f h yp o t h e s e s a n d id e n t ī c a t io n " , information transfer and combinatorics, l ecture 3 0 on application of optimal multihypothesis tests for the bounds construction of stock market notes in computer science, vo l. 4 1 2 3 , p p . 5 5 3 { 5 7 1 , s p r in g e r , n e w y o r k, n y , u s a , 2 0 0 6 . submitted 16.08.2013, accepted 01.10.2013. ²ñå»ãõã»ñç ßáõï³ûç ù»í³óù³ý ³ñ³·áõãû³ý ³×ç ·ý³ñ³ï³ï³ýç ï³éáõóù³ýá µ³½ù³ïç í³ñï³íý»ñç ýï³ïù³ùµ ûåïçù³é ï»ëï»ñç ïçñ³éáõãû³ý ù³ëçý º. ð³ñáõãûáõýû³ý, ². ø³ñïçñáëû³ý ¨ ². ºë³û³ý ²ù÷á÷áõù úåïçù³é ï»ëï»ñç ñáõë³éçáõãûáõýý»ñç ù³ïñçóç ïçñ³éù³ùµ ëï³óí³í ¿ ëë³é ñ³í³ý³ï³ý³ûçý µ³ßëù³ý ïçñ³éù³ý ¹»åùáõù ³ñå»ãõã»ñç ßáõï³ûç ïáïáë³ûçý ³×ç ýáñ ·ý³ñ³ï³ï³ý: î ïðèìåíåíèè îïòèìàëüíûõ òåñòîâ îòíîñèòåëüíî ìíîãèõ ãèïîòåç äëÿ ïîñòðîåíèÿ îöåíêè óâåëè÷åíèÿ ñêîðîñòè ðîñòà ðûíêà öåííûõ áóìàã å. àðóòþíÿí, à. ìàðòèðîñÿí è à. åñàÿí àííîòàöèÿ ïóòåì èñïîëüçîâàíèÿ ìàòðèöû íàäåæíîñòåé îïòèìàëüíûõ òåñòîâ ïîëó÷åíà íîâàÿ îöåíêà óâåëè÷åíèÿ ñêîðîñòè ðîñòà ðûíêà öåííûõ áóìàã ïðè ïðèìåíåíèè îøèáî÷íîãî ðàñïðåäåëåíèÿ âåðîÿòíîñòåé. microsoft word ruben_vardanyan_1.doc mathematical problems of computer science 36, 133--139, 2012. 133 interactive multimedia publication system for building web services ruben vardanyan it educational and research center e-mail: ruben@emagy.com abstract interactive multimedia publications are of high importance in many application areas including education, training and entertainment. it explains the high demand in web services providing the effective tools of authoring and interacting with this kind of publications. this paper is aimed to present the interactive multimedia publication system, which can be used for building the above mentioned web services and having capabilities significantly exceeding those of the existing ones. the paper gives the high-level architectural view of the systems and outlines the communication protocol used between the client and server components of the system. references 1. t. beckers and n. oorts, f. hendrickx and r. van de walle, “multichannel publication of interactive media documents in a news environment”,14th international www conference, 2005. 2. j. p. jarmasz and n. d. georganas, “designing a distributed multimedia synchronization scheduler”, international conference on multimedia computing and systems, 1997. 3. s. wirag. adaptive scheduling of multimedia documents. fakultätsbericht 1997/12, universität stuttgart, 1997. 4. s. wirag. “modeling of adaptable multimedia documents”, in proc. interactive distributed multimedia systems and telecommunication services; international workshop, idms’97, darmstadt, germany, 1997. 5. k. selguk candan, b. prabhakaran, and v. s. subrahmanian, “chimp: a framework for supporting distributed multimedia document authoring and presentation”, icde, 1998. 6. r. vardanyan, “an adaptive method for visualization of the interactive multimedia publications”, csit, 2011. 7. h. schulzrinne, a. rao and r. lanphier, “real time streaming protocol (rtsp)”, rfc 2326, 1998. 8. adobe systems incorporated, /adobe flash video file format specification. version 10.1/, 2010 9. l. shklar and r. rosen, web application architecture: principles, protocols and practices, 2nd edition/, john wiley & sons, 2009. 10. j. hauser, realization of an extensible multimedia document model, university of stuttgart, 1999. interactive multimedia publication system for building web services 134 æýï»ñ³ïïçí øáõéïçù»¹ç³ ðñ³å³ñ³ïáõùý»ñç ñ³ù³ï³ñ·՝ í»µ í³é³ûáõãûáõýý»ñ ëï»õí»éáõ ñ³ù³ñ è. ì³ñ¹³ýû³ý ամփոփում æýï»ñ³ïïçí øáõéïçù»¹ç³ ðñ³å³ñ³ïáõùý»ñá ù»í ýß³ý³ïáõãûáõý áõý»ý ³ñµ»ñ áéáñïý»ñáõù, ³û¹ ãíáõù, ïñãáõãû³ý, í»ñ³å³ïñ³ëïù³ý »õ å³ù³ýóç áéáñïý»ñáõù: ¸³ µ³ó³ïñáõù ¿ ³ûë ïçåç ññ³å³ñ³ïáõùý»ñç ñ»ï ³ßë³ïáõ í»µ í³é³ûáõãûáõýý»ñç ù»í å³ñ³ýç³ñï: ²ûë ñá¹í³íç ýå³ï³ïý ¿ ý»ñï³û³óý»é æýï»ñ³ïïçí øáõéïçù»¹ç³ ðñ³å³ñ³ïù³ý ð³ù³ï³ñ· (æøðð), áñá ï³ñáõ ¿ û·ï³·áñíí»é í»ñá ýßí³í í»µ í³é³ûáõãûáõýý»ñá ï³éáõó»éáõ ñ³ù³ñ, áñç ñý³ñ³íáñáõãûáõýý»ñá ¿³å»ë ï·»ñ³½³ýó»ý ý»ñï³ûçë ·áûáõãûáõý áõý»óáõý»ñçý: ðá¹í³íáõù ý»ñï³û³óí³í ¿ ñ³ù³ï³ñ·ç µ³ñóñ ù³ï³ñ¹³ïç ï³éáõóí³íùá »õ ùáõéïçù»¹ç³ûç ÷áë³ýóù³ý åñáïáïáéá, áñá ñ³ïáõï ùß³ïí»é ¿ æøðð-ç clientserver ñ³õáñ¹³ïóáõãû³ý ñ³ù³ñ: система интерактивных мультимедийных публикации для создания веб-служб р. варданян аннотация интерактивные мультимедийные публикации играют важную роль во многих сферах, включая сферы образования, обучения и развлечения. этим обоснован большой спрос на веб-сервисы, предоставляющие эффективные инструменты для создания и взаимодействия с публикациями данного типа. целью данной статьи является представление системы опубликования интерактивной мультимедиа, которая может быть использована для создания вышеупомянутых веб-сервисов и иметь возможности намного превосходящие возможности ныне существующих систем. данная статья предлагает архитектурный вид системы и описывает протокол используемый для коммуникации между клиентскими и серверными частями системы. d:\sbornik\...\article.dvi mathematical problems of computer science 37, 35{38, 2012. on m aximal dead-end recognizing systems in the class of t wo-e lement subsets concer ning oper ations of i nter section and complement s e yr a n m. v a r d a n ya n institute for informatics and automation problems of nas of ra e-mail: seyranv@ipia.sci.am abstract the structure of dead-end n{recognizing systems having a maximal possible length in the class of two-element subsets concerning the operations of intersection and complement is investigated. the quantity of such systems is estimated. refer ences [1 ] ï. ýðäåø, äæ. ñïåíñåð, âåðîÿòíîñòíûå ìåòîäû â êîìáèíàòîðèêå. ìîñêâà, ”ìèð”, 1976. [2 ] ñ. ì. âàðäàíÿí, “îá îäíîé çàäà÷å ðàñïîçíàâàíèÿ ìíîæåñòâ”, äàí àðì. ññð, òîì 72, ññ. 141143, 1981. [3 ] ñ. ì. âàðäàíÿí, “î äëèíàõ òóïèêîâûõ ðàñïîçíàþùèõ ñèñòåì â êëàññå äâóõýëåìåíòíûõ ïîäìíîæåñòâ”, äíàí ðà, òîì 107, n 1, ññ. 37 43, 2007. [4 ] s . m. v a r d a n ya n , \ r e c o g n iz ig s e t s ( s ys t e m s ) " p roceedings of international conference computer science and information technologies csit05, p p .1 6 1 1 6 2 , y e r e va n , a r m e n ia 2 0 0 5 . [5 ] s . m. v a r d a n ya n , \ on t h e p o we r s o f d e a d -e n d r e c o g n iz in g s ys t e m s in t h e c la s s o f t wo e le m e n t s e t s c o n c e r n in g o p e r a t io n s o f in t e r s e c t io n a n d c o m p le m e n t " , m athematical p roblems of computer sciences, vo l. 3 5 , p p . 1 0 4 { 1 0 8 , 2 0 1 1 . [6 ] ñ. ì. âàðäàíÿí, “î ìèíèìàëüíîñòè íåêîòîðûõ ðàñïîçíàþùèõ ñèñòåì â êëàññå äâóõýëåìåíòíûõ ïîäìíîæåñòâ îòíîñèòåëüíî îïåðàöèé ïåðåñå÷åíèÿ è äîïîëíåíèÿ”, äíàí ðà, òîì 112, n 1, ññ. 57 66, 2012. [7 ] f. h a r a r y, graph theory, a d d is o n w e s le y. r e a d in g ma 1 9 6 9 . 3 5 3 6 on maximal dead-end recognizing systems in the class of two-element subsets ð³ïù³ý ¨ éñ³óù³ý ·áñíáõáõãûáõýý»ñç ýï³ïù³ùµ »ñïï³ññ »ýã³µ³½ùáõãûáõýý»ñç ¹³ëáõù ³é³í»é³·áõûý »ñï³ñáõãûáõý áõý»óáõ ×³ý³ãáõ ÷³ïáõõ³ûçý ñ³ù³ï³ñ·»ñç ù³ëçý ê. ì³ñ¹³ýû³ý ²ù÷á÷áõù ²é³í»é³·áõûý ñý³ñ³íáñ »ñï³ñáõãûáõý áõý»óáõ n-×³ý³ãáõ ÷³ïáõõ³ûçý ñ³ù³ï³ñ·»ñç ï³éáõóí³íùá ñ»ï³½áïíáõù ¿ »ñïï³ññ »ýã³µ³½ùáõãûáõýý»ñç ¹³ëáõù éñ³óù³ý ¨ ñ³ïù³ý ·áñíáõáõãûáõýý»ñç ýï³ïù³ùµ: ¶ý³ñ³ïíáõù ¿ ýù³ý ïçåç ñ³ù³ï³ñ·»ñç ù³ý³ïá: î ðàñïîçíàþùèõ òóïèêîâûõ ñèñòåìàõ, èìåþùèõ íàèáîëüøóþ âîçìîæíóþ äëèíó â êëàññå äâóõýëåìåíòíûõ ïîäìíîæåñòâ îòíîñèòåëüíî îïåðàöèé ïåðåñå÷åíèÿ è äîïîëíåíèÿ ñ. âàðäàíÿí àííîòàöèÿ èññëåäóåòñÿ ñòðóêòóðà n-ðàñïîçíàþùèõ òóïèêîâûõ ñèñòåì, èìåþùèõ íàèáîëüøóþ âîçìîæíóþ äëèíó â êëàññå äâóõýëåìåíòíûõ ïîäìíîæåñòâ îòíîñèòåëüíî îïåðàöèé ïåðåñå÷åíèÿ è äîïîëíåíèÿ. îïèñûâàþòñÿ âñåâîçìîæíûå òèïû n-ðàñïîçíàþùèõ ìàêñèìàëüíûõ òóïèêîâûõ ñèñòåì è äàåòñÿ îöåíêà êîëè÷åñòâà òàêèõ ñèñòåì. 43 mathematical problems of computer science 52, 43--53, 2019. udc 519.7 notes on monotone recognition in multi-valued grids levon h. aslanyan and hasmik a. sahakyan institute for informatics and automation problems of nas ra e-mail: lasl@sci.am, hsahakyan@sci.am abstract a novel method of monotone recognition based on the partitioning of the grid into discrete structures isomorphic to binary cubes (called “cube-split” technique) was proposed in our recent work, and a theoretical level description of two algorithms /algorithmic schemes/ solving this problem was also introduced. this paper provides implementation details of those algorithms, as well as focuses on the recognition of monotone binary functions with a small number of units. keywords: monotone function recognition, multi-valued grid, cube-splitting. 1. introduction let 𝛯𝛯𝑚𝑚+1 𝑛𝑛 denote the 𝑛𝑛-th cartesian degree of the set 𝛯𝛯𝑚𝑚+1 = {0,1, ⋯ , 𝑚𝑚}: 𝛯𝛯𝑚𝑚+1 𝑛𝑛 = {(𝑎𝑎1, ⋯ , 𝑎𝑎𝑛𝑛)|𝑎𝑎𝑖𝑖 ∈ 𝛯𝛯𝑚𝑚+1, 𝑖𝑖 = 1, ⋯ , 𝑛𝑛}, or, in other words, 𝛯𝛯𝑚𝑚+1 𝑛𝑛 is the set of vertices of the 𝑛𝑛-dimensional (𝑚𝑚 + 1)-valued discrete grid. the total number of vertices of 𝛯𝛯𝑚𝑚+1 𝑛𝑛 is equal to (𝑚𝑚 + 1)𝑛𝑛. we consider a component-wise partial order “≤” on 𝛯𝛯𝑚𝑚+1 𝑛𝑛 defined in the following way: for arbitrary vertices 𝑎𝑎 = (𝑎𝑎1, ⋯ , 𝑎𝑎𝑛𝑛) and 𝑏𝑏 = (𝑏𝑏1, ⋯ , 𝑏𝑏𝑛𝑛) of 𝛯𝛯𝑚𝑚+1 𝑛𝑛 , 𝑎𝑎 precedes 𝑏𝑏 (𝑎𝑎 ≤ 𝑏𝑏) if and only if 𝑎𝑎𝑖𝑖 ≤ 𝑏𝑏𝑖𝑖 for 𝑖𝑖 = 1, ⋯ , 𝑛𝑛. then, (𝛯𝛯𝑚𝑚+1 𝑛𝑛 , ≤) is a partially ordered set; we will use its hasse diagram for geometrical interpretations. 𝑓𝑓(𝑥𝑥1, 𝑥𝑥2, ⋯ , 𝑥𝑥𝑛𝑛): 𝛯𝛯𝑚𝑚+1 𝑛𝑛 → {0,1} is called a binary function defined on 𝛯𝛯𝑚𝑚+1 𝑛𝑛 . we say that 𝑓𝑓(𝑥𝑥1, 𝑥𝑥2, ⋯ , 𝑥𝑥𝑛𝑛) is a monotone function if for any two vertices 𝑎𝑎, 𝑏𝑏 of 𝛯𝛯𝑚𝑚+1 𝑛𝑛 , 𝑎𝑎 ≥ 𝑏𝑏 implies: 𝑓𝑓(𝑎𝑎) ≥ 𝑓𝑓(𝑏𝑏). the vertices of 𝛯𝛯𝑚𝑚+1 𝑛𝑛 , where 𝑓𝑓 takes the value “1”, are called units of the function. the set of units of 𝑓𝑓 is usually denoted by 𝑁𝑁𝑓𝑓. the vertices of 𝛯𝛯𝑚𝑚+1 𝑛𝑛 , where 𝑓𝑓 takes the value “0” are called zeros of the function. 𝑎𝑎1 ∈ 𝛯𝛯𝑚𝑚+1 𝑛𝑛 is called a lower unit of 𝑓𝑓, if 𝑓𝑓(𝑎𝑎1) = 1 and 𝑓𝑓(𝑏𝑏) = 0 for every 𝑏𝑏 ∈ 𝛯𝛯𝑚𝑚+1 𝑛𝑛 less than 𝑎𝑎1. 𝑎𝑎0 ∈ 𝛯𝛯𝑚𝑚+1 𝑛𝑛 is called an upper zero of 𝑓𝑓, if 𝑓𝑓(𝑎𝑎0) = 0 and 𝑓𝑓(𝑏𝑏) = 1 for every 𝑏𝑏 ∈ 𝛯𝛯𝑚𝑚+1 𝑛𝑛 greater than 𝑎𝑎0. mailto:lasl@sci.am mailto:hsahakyan@sci.am notes on monotone recognition in-multi-valued grids 44 fig. 1 demonstrates the hasse diagram of 𝛯𝛯5 3, and a monotone binary function 𝑓𝑓 defined on 𝛯𝛯5 3. highlighted vertices (4,4,4), (4,4,3), (4,3,4), (3,4,4), (4,4,2), (4,3,3), (3,4,3), (3,3,4), (2,4,4), (4,3,2), (3,3,3), (2,4,3), (1,4,4), (1,4,3) are units of the function, where (4,3,2), (3,3,3), and (1,4,3) are its lower units. the rest of vertices of 𝛯𝛯5 3 are zeros of the function, and (4,4,1), (4,2,4), (3,4,2),(2,3,4), and (0,4,4) are its upper zeros. we consider the problem of query-based algorithmic identification/recognition of monotone binary functions defined on 𝛯𝛯𝑚𝑚+1 𝑛𝑛 . this problem is initially investigated by v. korobkov and v. alekseev, but also, consecutively, by many other authors [1-4]. for 𝑚𝑚 = 1, this is the case of ordinary monotone boolean functions defined on the binary cube 𝐸𝐸𝑛𝑛, 𝐸𝐸𝑛𝑛 = {(𝛼𝛼1, ⋯ , 𝛼𝛼𝑛𝑛)|𝛼𝛼𝑖𝑖 ∈ {0,1}, 𝑖𝑖 = 1, ⋯ , 𝑛𝑛}. hansel’s chain-splitting technique of 𝐸𝐸𝑛𝑛 [5] is a wellknown effective tool for monotone boolean function recognition. the outline of the algorithm is as follows: the set of vertices of the binary cube is partitioned into disjoint chains of different lengths (there are a total of 𝐶𝐶𝑛𝑛 ⌊𝑛𝑛/2⌋ chains in the 𝑛𝑛-dimensional cube). a key property of the hansel’s chains is that once the function values are known for all the vertices in all the chains of length 𝑘𝑘, then the function values, inferable by monotonicity, are unknown for at most two vertices in each chain of the next length 𝑘𝑘 + 2. the maximum number of queries to recognize the monotone boolean function defined on the 𝑛𝑛-dimensional cube is 𝐶𝐶𝑛𝑛 ⌊𝑛𝑛/2⌋+𝐶𝐶𝑛𝑛 ⌊𝑛𝑛/2⌋+1. fig. 1. monotone function defined on 𝛯𝛯5 3. an extension of this technique to the case of multi-valued grids and monotone binary functions is obtained in [2-3]. in [2], v. alekseev developed the algorithm 𝑈𝑈0 for recognition of a monotone binary function defined on 𝛯𝛯𝑘𝑘1𝑘𝑘2⋯𝑘𝑘𝑛𝑛 = 𝛯𝛯𝑘𝑘1 × 𝛯𝛯𝑘𝑘2 × ⋯ × 𝛯𝛯𝑘𝑘𝑛𝑛, (𝛯𝛯𝑘𝑘𝑖𝑖 = {0,1, ⋯ , 𝑘𝑘 − 1}, 𝑖𝑖 = 1, ⋯ , 𝑛𝑛), which, in some sense, tries to generalize g.hansel’s algorithm. [2] proved that: 𝑇𝑇(𝑈𝑈0) 𝑇𝑇(𝑈𝑈𝑜𝑜𝑜𝑜𝑜𝑜) ≤ 1 2 ⌈𝑙𝑙𝑙𝑙𝑙𝑙2(𝑘𝑘 − 1)⌉, 444 443 434 344 442 433 424 343 334 244 432 423 342 324 243 234333 422 332 323 242 233 224 322 232 223 222 144 044134143314404413341431440 441 414 430 431 340 412 331 403 241 313 304 142 214 133 124 043 034 024033114042123204132213141303231312240321402330411420 410 400 401 320 311 230 302 221 140 212 131 203 122 041 113 032 104 023 014 004013022103031112040121202130211220301310 300 210 201 120 111 030 102 021 012 003 002011020101110200 100 010 001 000 l. aslanyan and h. sahakyan 45 where 𝑇𝑇(𝑈𝑈0) denotes the complexity of 𝑈𝑈0, and 𝑇𝑇(𝑈𝑈𝑜𝑜𝑜𝑜𝑜𝑜) is the complexity of the optimal algorithm 𝑈𝑈𝑜𝑜𝑜𝑜𝑜𝑜, 𝑘𝑘 = 𝑚𝑚𝑎𝑎𝑥𝑥𝑘𝑘𝑖𝑖. it is also found that: 𝑇𝑇(𝑈𝑈𝑜𝑜𝑜𝑜𝑜𝑜) ≥ |𝑀𝑀| + |𝑁𝑁| and 𝑇𝑇(𝑈𝑈0) ≤ |𝑀𝑀| + ⌊𝑙𝑙𝑙𝑙𝑙𝑙2𝑘𝑘⌋ ∙ |𝑁𝑁|, where 𝑀𝑀 and 𝑁𝑁 are the 2 sets of vertices in the middle layer area of 𝛯𝛯𝑘𝑘1𝑘𝑘2⋯𝑘𝑘𝑛𝑛: 𝑀𝑀 = �(𝑎𝑎1, ⋯ , 𝑎𝑎𝑛𝑛) ∈ 𝛯𝛯𝑘𝑘1𝑘𝑘2⋯𝑘𝑘𝑛𝑛 ∶ 𝑎𝑎1 + ⋯ + 𝑎𝑎𝑛𝑛 = � 1 2 ∑ (𝑘𝑘𝑖𝑖 − 1) 𝑛𝑛 𝑖𝑖=1 ��, 𝑁𝑁 = �(𝑎𝑎1, ⋯ , 𝑎𝑎𝑛𝑛) ∈ 𝛯𝛯𝑘𝑘1𝑘𝑘2⋯𝑘𝑘𝑛𝑛 ∶ 𝑎𝑎1 + ⋯ + 𝑎𝑎𝑛𝑛 = � 1 2 ∑ (𝑘𝑘𝑖𝑖 − 1) 𝑛𝑛 𝑖𝑖=1 � + 1�. in case of 𝛯𝛯𝑚𝑚+1 𝑛𝑛 the sets in the two middle layers are: 𝑀𝑀0 = �(𝑎𝑎1, ⋯ , 𝑎𝑎𝑛𝑛) ∈ 𝛯𝛯𝑚𝑚+1 𝑛𝑛 ∶ 𝑎𝑎1 + ⋯ + 𝑎𝑎𝑛𝑛 = � 𝑚𝑚∙𝑛𝑛 2 ��, 𝑁𝑁0 = �(𝑎𝑎1, ⋯ , 𝑎𝑎𝑛𝑛) ∈ 𝛯𝛯𝑚𝑚+1 𝑛𝑛 ∶ 𝑎𝑎1 + ⋯ + 𝑎𝑎𝑛𝑛 = � 𝑚𝑚∙𝑛𝑛 2 � + 1�, and consequently, the estimate of complexity of the algorithm 𝑈𝑈0 on 𝛯𝛯𝑚𝑚+1 𝑛𝑛 will be: 𝑇𝑇(𝑈𝑈0) ≤ |𝑀𝑀0| + ⌊𝑙𝑙𝑙𝑙𝑙𝑙2𝑚𝑚⌋ ∙ |𝑁𝑁0|. a recent novel method of the monotone recognition based on a partitioning of the grid into discrete structures isomorphic to binary cubes (called “cube-split” technique) is proposed in [6], and two algorithms /algorithmic schemes/ solving this problem are also introduced. this paper provides implementation details of these algorithms, as well as focuses on the recognition of monotone binary functions with a small number of units. the paper is organized as follows: section 2 introduces the cube-splitting technique. section 3 provides the implementation framework of these algorithmic schemes. section 4 addresses some particular cases with a small number of units that comes from applications. 2. the cube-splitting technique in this section we introduce the cube-splitting technique of recognition of monotone binary functions [6]. two homogeneous areas inside the 𝛯𝛯𝑚𝑚+1 𝑛𝑛 are defined in the following way: upper homogeneous area 𝐻𝐻�, this is the set of all “upper” elements of 𝛯𝛯𝑚𝑚+1𝑛𝑛 , i.e., elements with all-coordinate values ≥ 𝑚𝑚/2; lower homogeneous area 𝐻𝐻�, – this is the set of all “lower” elements, i.e., elements with all-coordinate values ≤ 𝑚𝑚/2. it is clear that: �𝐻𝐻�� = �𝐻𝐻�� = � (m+1 2 )nfor odd 𝑚𝑚, (𝑚𝑚 2 + 1)𝑛𝑛 for even 𝑚𝑚 . the following results were introduced in [6]. (1) 𝛯𝛯𝑚𝑚+1𝑛𝑛 can be split into �𝐻𝐻�� disjoint discrete structures isomorphic to binary cubes: notes on monotone recognition in-multi-valued grids 46 𝛯𝛯𝑚𝑚+1 𝑛𝑛 = ℰ1 ∪ ⋯∪ ℰ|𝐻𝐻�|, where every ℰ𝑖𝑖 contains exactly one vertex from 𝐻𝐻�, while the remaining vertices of ℰ𝑖𝑖 can be determined uniquely by this vertex through the complementarity interchanges of the coordinate values. ℰ𝑖𝑖 ∩ ℰ𝑗𝑗 = ∅, if 𝑖𝑖 ≠ 𝑗𝑗. the procedure is called “cube-splitting” of 𝛯𝛯𝑚𝑚+1 𝑛𝑛 . (2) the “cube-splitting” of 𝛯𝛯𝑚𝑚+1𝑛𝑛 keeps the monotonicity property in the following way: let 𝐹𝐹 be a monotone binary function defined on 𝛯𝛯𝑚𝑚+1 𝑛𝑛 , then either 𝑁𝑁𝐹𝐹 ∩ ℰ𝑖𝑖 is empty, or it satisfies the binary monotonicity property, i.e., for arbitrary vertex 𝑎𝑎 of 𝑁𝑁𝐹𝐹 ∩ ℰ𝑖𝑖, all vertices of ℰ𝑖𝑖 greater than 𝑎𝑎, also belong to 𝑁𝑁𝐹𝐹 ∩ ℰ𝑖𝑖 (for 𝑖𝑖 = 1, ⋯ , 𝑛𝑛). by integrating (1) and (2), a novel monotone recognition method has been proposed in [6]. 2.1 definitions/descriptions for each vertex 𝑉𝑉𝑖𝑖 = (𝑣𝑣𝑖𝑖1, ⋯ , 𝑣𝑣𝑖𝑖𝑛𝑛) of 𝐻𝐻� we compose the following set: ℰ𝑉𝑉𝑖𝑖 = {(𝑎𝑎1, ⋯ , 𝑎𝑎𝑛𝑛) ∈ 𝛯𝛯𝑚𝑚+1 𝑛𝑛 |𝑎𝑎𝑗𝑗 ∈ {𝑣𝑣𝑖𝑖𝑗𝑗, 𝑚𝑚 − 𝑣𝑣𝑖𝑖𝑗𝑗} for all 𝑗𝑗, 1 ≤ 𝑗𝑗 ≤ 𝑛𝑛}, and call ℰ𝑉𝑉𝑖𝑖 the vertical equivalence class of 𝑉𝑉𝑖𝑖. ℰ𝑉𝑉𝑖𝑖 contains a unique vertex from 𝐻𝐻�, this is the vertex with all coordinates ≥ 𝑚𝑚/2; and contains a unique vertex from 𝐻𝐻�, this is the vertex with all coordinates ≤ 𝑚𝑚/2. the remaining vertices of ℰ𝑉𝑉𝑖𝑖 can be obtained by component value inversions (with respect to 𝑚𝑚). ℰ𝑉𝑉𝑖𝑖 ∩ ℰ𝑉𝑉𝑗𝑗 = ∅ for different vertices 𝑉𝑉𝑖𝑖 and 𝑉𝑉𝑗𝑗 of 𝐻𝐻�. in this manner 𝛯𝛯𝑚𝑚+1 𝑛𝑛 can be split into �𝐻𝐻�� disjoint sets /equivalence classes/ uniquely defined by the elements of 𝐻𝐻� (or 𝐻𝐻�). the number of elements of ℰ𝑉𝑉𝑖𝑖 varies between 2 0 and 2𝑛𝑛 depending on the number of components of 𝑉𝑉𝑖𝑖 differing from 𝑚𝑚/2. indeed, if 𝑘𝑘 denotes the number of components of 𝑉𝑉𝑖𝑖 differing from 𝑚𝑚/2, i.e. 𝑘𝑘 = �{𝑣𝑣𝑖𝑖𝑗𝑗|𝑣𝑣𝑖𝑖𝑗𝑗 ≠ (𝑚𝑚 − 𝑣𝑣𝑖𝑖𝑗𝑗)}�, then �ℰ𝑉𝑉𝑖𝑖� = 2 𝑘𝑘. notice that 𝑘𝑘 = 𝑛𝑛 always for odd 𝑚𝑚. for example, fig. 2 demonstrates ℰ(3,4,3), ℰ(2,3,4) and ℰ(4,2,2) in 𝛯𝛯5 3. fig. 2. examples of cubes in a cube split of 𝛯𝛯5 3. 322 222 122 231 232 233 224 203302 241 311 321 331 111 121 131 102 142 113 123 133313 323 333 243 201 342 444 443 434 344 442 433 424 343 334 244 432 423 324 234 422 332 242 223 144 044134143314404413341431440 441 414 430 431 340 412 403 304 214 124 043 034 024033114042204132213141303312240402330411420 410 400 401 320 230 221 140 212 041 032 104 023 014 004013022103031112040202130211220301310 300 210 120 030 021 012 003 002011020101110200 100 010 001 000 l. aslanyan and h. sahakyan 47 for every vertex (𝑎𝑎1, ⋯ , 𝑎𝑎𝑛𝑛) of the equivalence class ℰ𝑉𝑉𝑖𝑖, we distinguish its sub-list of all coordinates accepting a value differing from 𝑚𝑚/2, let this be the list: (𝑎𝑎𝑠𝑠1, ⋯ , 𝑎𝑎𝑠𝑠𝑘𝑘). this list exactly fits the list of all coordinates of 𝑉𝑉𝑖𝑖 that are different from 𝑚𝑚/2. the reminder part of coordinates accepts the only value 𝑚𝑚/2 over the 𝑉𝑉𝑖𝑖, as well as over the whole set of vertices of ℰ𝑉𝑉𝑖𝑖. now, identifying (𝑎𝑎𝑠𝑠1, ⋯ , 𝑎𝑎𝑠𝑠𝑘𝑘) with the binary sequence 𝛽𝛽 = (𝛽𝛽1, ⋯ , 𝛽𝛽𝑘𝑘) of length 𝑘𝑘 such that 𝛽𝛽𝑗𝑗 = 1 if and only if 𝑎𝑎𝑠𝑠𝑗𝑗 > 𝑚𝑚/2, we map (𝑎𝑎1, ⋯ , 𝑎𝑎𝑛𝑛) into the vertex 𝛽𝛽 = (𝛽𝛽1, ⋯ , 𝛽𝛽𝑘𝑘) of the 𝑘𝑘-dimensional binary cube 𝐸𝐸𝑘𝑘. in this manner, we obtain a 1-1 mapping 𝑀𝑀: ℰ𝑉𝑉𝑖𝑖 ⟶ 𝐸𝐸 𝑘𝑘. the vertex of ℰ𝑉𝑉𝑖𝑖 with all coordinates < 𝑚𝑚/2 is mapped into the vertex (0, ⋯ ,0) of 𝐸𝐸 𝑘𝑘 (on the 0-th layer); the vertex of ℰ𝑉𝑉𝑖𝑖 with all coordinates > 𝑚𝑚/2 is mapped into the vertex (1, ⋯ ,1) of 𝐸𝐸 𝑘𝑘 (on the 𝑛𝑛-th layer); and, in general, all vertices of ℰ𝑉𝑉𝑖𝑖, which have 𝑙𝑙 coordinates > 𝑚𝑚/2 (consequently, 𝑚𝑚 − 𝑙𝑙 coordinates < 𝑚𝑚/2), are mapped into the vertices of 𝑙𝑙-th layer of 𝐸𝐸𝑘𝑘. hereafter, all structures (vertices, chains, cubes, functions, etc.) in 𝛯𝛯𝑚𝑚+1 𝑛𝑛 will be referred to as origin; and all structures (vertices, chains, cubes, functions, etc.) in binary cubes will be referred to as induced. for example, the induced binary cubes for ℰ(3,4,3), ℰ(2,3,4) and ℰ(4,2,2), are given in figure 3, (a), (b), and (c), correspondingly. (a) (b) (c) fig. 3. 2.2 the algorithmic framework let 𝐹𝐹 be a monotone binary function (which should be recognized with the help of an oracle), defined on 𝛯𝛯𝑚𝑚+1 𝑛𝑛 , and let 𝑁𝑁𝐹𝐹 denote its set of units of function 𝐹𝐹. algorithm 1 in a theoretical level description, the algorithm implements the following steps: 1. apply the cube-splitting on 𝛯𝛯𝑚𝑚+1𝑛𝑛 and let ℰ1, ⋯,.ℰ|𝐻𝐻�| be the equivalence classes of the upper homogeneous elements. 000 001010100 101110 111 011 01 00 10 11 0 1 notes on monotone recognition in-multi-valued grids 48 2. compose the corresponding induced binary cube 𝐸𝐸𝑖𝑖 for every ℰ𝑖𝑖. 3. apply the hansel’s algorithm to recognize the induced boolean function 𝑓𝑓𝑖𝑖, defined on 𝐸𝐸𝑖𝑖 as follows: for every 𝛽𝛽 ∈ 𝐸𝐸𝑖𝑖, 𝑓𝑓𝑖𝑖(𝛽𝛽) = 1 if and only if 𝐹𝐹(𝑏𝑏) = 1, where 𝑏𝑏 is the origin of 𝛽𝛽 in 𝛯𝛯𝑚𝑚+1 𝑛𝑛 (for 𝑖𝑖 = 1, ⋯ , �𝐻𝐻��). 4. transfer the recognition results into the 𝛯𝛯𝑚𝑚+1𝑛𝑛 . 3. implementation 3.1 implementation details of step 1 and step2 we consider the lexicographic ordering of the vertices of �𝐻𝐻��, where the smallest numerical values of coordinates are coming first. thus, the smallest vertex of 𝐻𝐻� in this ordering is (m+1 2 , m+1 2 , ⋯ , m+1 2 ) if 𝑚𝑚 is odd, and it is the vertex(𝑚𝑚 2 , 𝑚𝑚 2 , ⋯ , 𝑚𝑚 2 ), if 𝑚𝑚 is even; the greatest vertex is (𝑚𝑚, 𝑚𝑚, ⋯ , 𝑚𝑚). henceforth, we will assume that 𝐻𝐻� = {𝑉𝑉1, 𝑉𝑉2, ⋯ , 𝑉𝑉𝐻𝐻�} is the lexicographically ordered set of upper homogeneous elements. the cube splitting of 𝛯𝛯𝑚𝑚+1 𝑛𝑛 assumes that we compose for every vertex 𝑉𝑉𝑖𝑖 = (𝑣𝑣𝑖𝑖1, ⋯ , 𝑣𝑣𝑖𝑖𝑛𝑛) of 𝐻𝐻� its vertical equivalence class ℰ𝑉𝑉𝑖𝑖, and the corresponding induced binary cube to this. but at this point we do not need to compose and keep (and further to map to the binary cube) the whole set ℰ𝑉𝑉𝑖𝑖; instead, with every vertex 𝑉𝑉𝑖𝑖 = (𝑣𝑣𝑖𝑖1, ⋯ , 𝑣𝑣𝑖𝑖𝑛𝑛) of 𝐻𝐻� we will keep the following parameters: the number 𝜏𝜏𝑉𝑉𝑖𝑖 of coordinates of 𝑉𝑉𝑖𝑖 differing from 𝑚𝑚/2, this will determine the size of the induced binary cube 𝐸𝐸𝑖𝑖. when 𝑉𝑉𝑖𝑖, that is the issue, is evident, we will just use the notion 𝜏𝜏 for this, the positions of coordinates differing from 𝑚𝑚/2, we denote it by the vector 𝑉𝑉𝑖𝑖≠, and the values of coordinates differing from 𝑚𝑚/2, we denote as the vector 𝑉𝑉𝑖𝑖#. 𝜏𝜏, 𝑉𝑉𝑖𝑖≠, and 𝑉𝑉𝑖𝑖# will allow the easy reverse mapping, 𝑅𝑅𝑀𝑀 ∶ 𝐸𝐸𝜏𝜏 ⟶ ℰ𝑉𝑉𝑖𝑖, i.e., will allow to recover ℰ𝑉𝑉𝑖𝑖. for example, with the vertex (2,3,4) of 𝛯𝛯5 3 we keep: numerical value 2, this is the number of its coordinates differing from 𝑚𝑚/2, indexes 2,3 these are the coordinate indexes, where the values are differing from 𝑚𝑚/2, and 3,4 are the values at the coordinates 2 and 3. ℰ(2,3,4) is mapped into the 2-dimensional binary cube 𝐸𝐸2 according to 2nd and 3rd coordinates of (2,3,4), and the accompanying vectors are 𝑉𝑉𝑖𝑖≠ = (2,3) and 𝑉𝑉𝑖𝑖# = (3,4). the reverse mapping is as follows: given the pair (2,3,4) and 𝐸𝐸2 or alternatively, 𝑉𝑉𝑖𝑖≠, 𝑉𝑉𝑖𝑖# and 𝐸𝐸2. consider an arbitrary vertex of 𝐸𝐸2, for example, let 𝛽𝛽� = (1,0), then it follows that the origin of 𝛽𝛽� in ℰ(2,3,4) is 𝑎𝑎� = (2,3,0), because: the first component, missing at 𝑉𝑉𝑖𝑖≠, must be 𝑚𝑚/2, that is, 𝑎𝑎1 = 2, the second component should not be inverted in accord to 𝛽𝛽� = (1,0), and, thus, 𝑎𝑎2 = 3, the third component should be inverted, and thus, 𝑎𝑎3 = 4 − 4 = 0. in general, let (𝛽𝛽1, ⋯ , 𝛽𝛽𝑘𝑘) be an arbitrary vertex of the 𝑘𝑘-dimensional binary cube 𝐸𝐸𝑘𝑘, and let (𝑣𝑣1, ⋯ , 𝑣𝑣𝑛𝑛) be the upper homogeneous vector of the origin of 𝐸𝐸𝑘𝑘, and suppose that 𝑠𝑠1, ⋯ , 𝑠𝑠𝑘𝑘 are its coordinates differing from 𝑚𝑚/2. then, the origin of (𝛽𝛽1, ⋯ , 𝛽𝛽𝑘𝑘) is (𝑎𝑎1, ⋯ , 𝑎𝑎𝑛𝑛), where: l. aslanyan and h. sahakyan 49 𝑎𝑎𝑠𝑠𝑗𝑗 = � 𝑣𝑣𝑠𝑠𝑗𝑗 𝑖𝑖𝑓𝑓 𝛽𝛽𝑗𝑗 = 1 𝑚𝑚 − 𝑣𝑣𝑠𝑠𝑗𝑗𝑖𝑖𝑓𝑓 𝛽𝛽𝑗𝑗 = 0 for 𝑗𝑗 = 1, ⋯ , 𝑘𝑘 (1) 𝑎𝑎𝑖𝑖 = 𝑚𝑚/2 for 𝑖𝑖 ≠ 𝑠𝑠1, ⋯ , 𝑠𝑠𝑘𝑘. 3.2 implementation details of step 3 and step 4 in this part, algorithm 1 recognizes monotone boolean functions in the �𝐻𝐻�� number of binary cubes of different sizes (some of the functions might be identically 0, but we do not know this fact beforehand), by applying the hansel’s algorithm. also at this step, we will not deal with the binary cubes themselves, and we will be using the chain algebras, and therefore, we have to map (by the reverse mapping) all induced structures (vertices, functions, chains, etc.) into their origins in 𝛯𝛯𝑚𝑚+1 𝑛𝑛 . for example, consider some monotone boolean function 𝑓𝑓𝑖𝑖 on a 𝑘𝑘-dimensional binary cube. if we obtain the value 𝑓𝑓𝑖𝑖(𝛽𝛽�) on some vertex 𝛽𝛽� (in the process of the hansel’s algorithm), and know its origin upper homogeneous vertex (𝑣𝑣1, ⋯ , 𝑣𝑣𝑛𝑛) and also its coordinates differing from 𝑚𝑚/2 (let they be 𝑠𝑠1, ⋯ , 𝑠𝑠𝑘𝑘), then we can set that 𝐹𝐹(𝑎𝑎�) = 𝑓𝑓𝑖𝑖�𝛽𝛽��, where the coordinate values of 𝑎𝑎� are defined in accord to (1). similarly, we can map chains from the induced binary cubes into the 𝛯𝛯𝑚𝑚+1 𝑛𝑛 . for example, the maximum length chain <(000), (100), (110), (111)> in the cube in figure 2 (a) (which is induced to ℰ(3,4,3)), is mapped to the origin chain <(101), (301), (341), (343)> in 𝛯𝛯𝑚𝑚+1 𝑛𝑛 . upon receipt of the oracle’s response for a given vertex of the binary cube the response is mapped into the origin vertex of 𝛯𝛯𝑚𝑚+1 𝑛𝑛 . certainly, the response value could also be extended by the monotonicity property to the other relevant vertices of 𝛯𝛯𝑚𝑚+1 𝑛𝑛 . but in this research we prefer and emphasize the opportunity of the parallel implementation of the recognition algorithms in all the induced binary cubes, and so we keep them as separate nonintersecting processes. 4. small number of units note that algorithm 1 is worth applying when a large number of unit vertices of the monotone function appear in the upper homogeneous area 𝐻𝐻�. however, in some cases, mostly coming from applications, the function to be recognized has a small number of unit vertices in the upper area 𝐻𝐻�, or its complement has a small number of zero vertices in the lower homogeneous area 𝐻𝐻�. in the latter case we can recognize the complement of the monotone function. for this reason, the following points will be taken into account: ℰ𝑉𝑉𝑖𝑖 can be defined for each vertex of 𝐻𝐻� (obviously we will obtain the same set). in the example given in figure 2 the highlighted sets of 𝐻𝐻� will demonstrate ℰ(1,0,1), ℰ(2,1,0) and ℰ(0,2,2), as well. the mapping 𝑀𝑀: ℰ𝑉𝑉𝑖𝑖 ⟶ 𝐸𝐸 𝑘𝑘 will be defined as follows: for every vertex (𝑎𝑎1, ⋯ , 𝑎𝑎𝑛𝑛) of ℰ𝑉𝑉𝑖𝑖 let (𝑎𝑎𝑠𝑠1, ⋯ , 𝑎𝑎𝑠𝑠𝑘𝑘) denote its subsequence with all coordinates differing from 𝑚𝑚/2. identify (𝑎𝑎𝑠𝑠1, ⋯ , 𝑎𝑎𝑠𝑠𝑘𝑘) with the binary sequence 𝛽𝛽 = (𝛽𝛽1, ⋯ , 𝛽𝛽𝑘𝑘) of length 𝑘𝑘 such that 𝛽𝛽𝑗𝑗 = 1 if and only if 𝑎𝑎𝑠𝑠𝑗𝑗 < 𝑚𝑚/2. the vertex of ℰ𝑉𝑉𝑖𝑖 with all coordinate values < 𝑚𝑚/2 is mapped into the notes on monotone recognition in-multi-valued grids 50 vertex (1, ⋯ ,1) of 𝐸𝐸𝑘𝑘, this is on the 𝑛𝑛-th layer; the vertex of ℰ𝑉𝑉𝑖𝑖 with all coordinate values < 𝑚𝑚/2 is mapped into the vertex (0, ⋯ ,0) of 𝐸𝐸𝑘𝑘, this is on the 0-th layer; and, in general, all vertices of ℰ𝑉𝑉𝑖𝑖 which have 𝑙𝑙 coordinates < 𝑚𝑚/2 (consequently, 𝑚𝑚 − 𝑙𝑙 coordinates > 𝑚𝑚/2) are mapped into the vertices of 𝑙𝑙-th layer of 𝐸𝐸𝑘𝑘. the reverse mapping: 𝑅𝑅𝑀𝑀: 𝐸𝐸𝑖𝑖 ⟶ ℰ𝑉𝑉𝑖𝑖 will be implemented as follows: let (𝛽𝛽1, ⋯ , 𝛽𝛽𝑘𝑘) be an arbitrary vertex of the 𝑘𝑘-dimensional binary cube 𝐸𝐸𝑘𝑘, and let (𝑣𝑣1, ⋯ , 𝑣𝑣𝑛𝑛) be the lower homogeneous vector of the origin of 𝐸𝐸𝑘𝑘, where 𝑠𝑠1, ⋯ , 𝑠𝑠𝑘𝑘 are coordinates differing from 𝑚𝑚/ 2. then the origin of (𝛽𝛽1, ⋯ , 𝛽𝛽𝑘𝑘) is (𝑎𝑎1, ⋯ , 𝑎𝑎𝑛𝑛), where: 𝑎𝑎𝑠𝑠𝑗𝑗 = � 𝑣𝑣𝑠𝑠𝑗𝑗 𝑖𝑖𝑓𝑓 𝛽𝛽𝑗𝑗 = 0 𝑚𝑚 − 𝑣𝑣𝑠𝑠𝑗𝑗𝑖𝑖𝑓𝑓 𝛽𝛽𝑗𝑗 = 1 for 𝑗𝑗 = 1, ⋯ , 𝑘𝑘 (2) 𝑎𝑎𝑖𝑖 = 𝑚𝑚/2 for 𝑖𝑖 ≠ 𝑠𝑠1, ⋯ , 𝑠𝑠𝑘𝑘. 4.1 constraints consider the application, which is the generalized model of the known association rule mining in case where in addition to the presence or absence of elements in itemsets, the number of their repetitions is also included. the details are given in [7]. here we highlight the following constraints/restrictions that may appear with this problem. in terminology of supermarket basket analysis, here we distinguish two postulations: some item exists in the current basket, and second, which is the actual number of that item in the basket. let 𝑎𝑎𝑖𝑖 be the repetition number of the 𝑖𝑖-th element, for 𝑖𝑖 = 1, ⋯ , 𝑛𝑛. (1) the classic case is the (0,1) vector of item indicators in baskets, basket inventory. (2) 𝑎𝑎1 + ⋯ + 𝑎𝑎𝑛𝑛 ≤ 𝑟𝑟, the summary number of elements’ repetitions (the basket volume) is restricted by 𝑟𝑟, (3) 𝑎𝑎𝑖𝑖 ≤ 𝑟𝑟𝑖𝑖, -the repetition number of each 𝑎𝑎𝑖𝑖 is restricted by 𝑟𝑟𝑖𝑖, the item purchase restriction. in these cases, the problem deals with the recognition of monotone functions, where: (2) the zeros of the function appear in lower layers of the multivalued grid, (3) the zeros of the function appear in some homogeneous bottom area of the grid. in both cases it is more efficient to use the second algorithmic scheme of [6]. the idea is as follows. let 𝐹𝐹 be a monotone function defined on 𝛯𝛯𝑚𝑚+1 𝑛𝑛 . first we note that 𝑁𝑁𝐹𝐹 ∩ 𝐻𝐻� satisfies the monotonicity property, i.e., for arbitrary 𝑎𝑎, 𝑏𝑏 of 𝐻𝐻�, if 𝑎𝑎 ≥ 𝑏𝑏 then 𝐹𝐹(𝑎𝑎) ≥ 𝐹𝐹(𝑏𝑏). algorithm 2 1. firstly identify the part of the monotone function belonging to 𝐻𝐻� by one of the known resources of identification of monotone functions, and thus, reduce the size of the multivalued grid. as a result we obtain 𝑁𝑁𝐹𝐹 ∩ 𝐻𝐻�. 2. apply the cube-splitting according to 𝑁𝑁𝐹𝐹 ∩ 𝐻𝐻�, that considers the vertical equivalence classes ℰ1, ⋯,.ℰ|𝑁𝑁𝐹𝐹∩𝐻𝐻�| only for the vertices of 𝑁𝑁𝐹𝐹 ∩ 𝐻𝐻�. 3. implement 2-4 steps of algorithm 1. the algorithm can easily be adjusted for the identification of the complement of 𝐹𝐹 in 𝛯𝛯𝑚𝑚+1 𝑛𝑛 . l. aslanyan and h. sahakyan 51 4.2 resources to implement step 1 of algorithm 2 we have the following resources of identification of monotone functions in 𝐻𝐻�. a) the first resource is the known algorithm by v. alexeyev [2]. we notice that applying the algorithm to identification of monotone functions on 𝐻𝐻� (instead of 𝛯𝛯𝑚𝑚+1 𝑛𝑛 ) is a significant reduction of the work. the complexity of the algorithm 𝑈𝑈0 on 𝐻𝐻� will be: 𝑇𝑇(𝑈𝑈0) ≤ |𝑀𝑀1| + �𝑙𝑙𝑙𝑙𝑙𝑙2( 𝑚𝑚 2 )� ∙ |𝑁𝑁1|, where 𝑀𝑀1 and 𝑁𝑁1 are the middle layers of 𝐻𝐻�, defined accordingly. b) the second resource that we may use, is applying the cube-splitting itself for identifying the function on 𝐻𝐻�. we define the upper homogeneous area 𝐻𝐻�� of 𝐻𝐻�,: this is the set of all elements of 𝛯𝛯𝑚𝑚+1 𝑛𝑛 with all coordinate values ≥ 3𝑚𝑚/4; then 1. apply the cube-splitting on 𝐻𝐻� and find ℰ1, ⋯,.ℰ𝐻𝐻��, equivalence classes according to the elements of 𝐻𝐻��. 2. compose the corresponding induced binary cube 𝐸𝐸𝑖𝑖 for every ℰ𝑖𝑖. 3. apply the hansel’s algorithm to recognize the induced boolean function 𝑓𝑓𝑖𝑖, defined on 𝐸𝐸𝑖𝑖 as follows: for every 𝛽𝛽 ∈ 𝐸𝐸𝑖𝑖, 𝑓𝑓𝑖𝑖(𝛽𝛽) = 1 if and only if 𝐹𝐹(𝑏𝑏) = 1, where 𝑏𝑏 is the origin of 𝛽𝛽 in 𝐻𝐻� (for 𝑖𝑖 = 1, ⋯ , �𝐻𝐻��). 4. transfer the recognition results to 𝐻𝐻�. at this point we find 𝑁𝑁𝐹𝐹 ∩ 𝐻𝐻�, and then continue with: 5. apply the cube-splitting according to 𝑁𝑁𝐹𝐹 ∩ 𝐻𝐻�, that is find the vertical equivalence classes ℰ1, ⋯,.ℰ|𝑁𝑁𝐹𝐹∩𝐻𝐻�| only for the vertices of 𝑁𝑁𝐹𝐹 ∩ 𝐻𝐻�. 6. implement 2-4 steps of algorithm 1. c) the cube-splitting can be applied recursively. the following resource is also worth mentioning that can be used in all cases/algorithms. this is the growing technique [8] in monotone boolean function recognition and chain computation algebra [9]. 5. conclusion the problem of query based algorithmic recognition of monotone binary functions defined in multi-valued grids is known as a hard problem. it was investigated by v. korobkov, v. alekseev, a.serjantov, and others. it is known a chain-split type algorithm developed by v. alekseev, where the complexity estimates were given in terms of sizes of the middle layers of the grid. a novel method of identification of monotone binary functions based on the partitioning of the grid into discrete structures isomorphic to binary cubes was proposed in our recent work, where a theoretical level description of two algorithmic schemes was introduced. this paper provides the implementation details of the algorithms, as well as focuses on the recognition of monotone binary functions with a small number of units /or zeros/ distributed in homogeneous areas of the grid. notes on monotone recognition in-multi-valued grids 52 acknowledgement this work is partially supported by the grant № 18t-1b407 of the science committee of the ministry of education and science of armenia. references [1] v.korobkov, on monotone functions of algebra of logic, prob. cyb, 13, 1965. [2] v.alekseev, “on deciphering of some classes of monotone many valued functions”, ussr computational mathematics and mathematical physics, vol. 16, no. 1, pp. 189-198, 1976. [3] a.serjantov, “optimal algorithm for deciphering of some classes of monotone functions”,journal of computational mathematics and math. physics, vol. 23, no. 1, pp. 206-212, 1983. [4] n.zolotyk and a.chirkov, “on the complexity of deciphering of threshold functions of many-valued logic”, proceedings of the xi international seminar ”discrete mathematics and its applications”, commemorate on the 80th anniversary of academician lupanov, pp. 63-77, 2012. [5] g. hansel, “sur le nombre des fonctiones booleennes monotones de n variables”, c. r. acad. sci.p. 262, 1966. [6] l. aslanyan and h. sahakyan, “the splitting technique in monotone recognition”, discrete applied mathematics, vol. 216,pp. 502–512, 2017. [7] l. aslanyan and h. sahakyan, “cube-split technique in quantitative association rule mining”, information theories and applications, vol. 24, no. 3, pp. 3-20, 2017. [8] l. aslanyan and r. khachatryan, “association rule mining with n-dimensional unit cube chain split technique”, information theories and applications, vol. 1, no. 2, pp. 108-125, 2008. [9] g.tonoyan, “the successive splitting of vertices of an n-dimensional unit cube into chains and decoding problems of monotonic boolean functions”, ussr computational mathematics and mathematical physics, vol. 19, no. 6, pp. 179-191, 1979. submitted 24.07.2019, accepted 28.11.2019. մոնոտոն ֆունկցիայի ճանաչում բազմարժեք ցանցում լևոն հ. ասլանյան և հասմիկ ա. սահակյան հհ գաա ինֆորմատիկայի և ավտոմատացման պրոբլեմների ինստիտուտ e-mail: lasl@sci.am, hsahakyan@sci.am ամփոփում այս շարքի նախորդ աշխատանքներում առաջարկվել է մոնոտոն բինար ֆունկցիայի վերծանման նոր մոտեցում՝ հիմնված բազմարժեք բազմաչափ ցանցի խորանարդատիպ տրոհման մեթոդի վրա, որտեղ տեսական մակարդակում http://www.sciencedirect.com/science/journal/00415553 http://www.sciencedirect.com/science/journal/00415553 http://www.sciencedirect.com/science/journal/00415553 http://www.sciencedirect.com/science/journal/00415553 http://www.sciencedirect.com/science/journal/00415553/19/6 mailto:lasl@sci.am mailto:hsahakyan@sci.am l. aslanyan and h. sahakyan 53 առաջարկվել է խնդրի լուծման երկու ալգորիթմ /ալգորիթմական սխեմա/: ներկա աշխատանքում տրվում են այդ ալգորիթմների իրականացման մանրամասները, ինչպես նաև դիտարկվում է այն դեպքը, երբ մոնոտոն ֆունկցիան ունի փոքր թվով մեկ արժեքի գագաթներ: բանալի բառեր` մոնոտոն ֆունկցիայի ճանաչում, բազմարժեք ցանց, խորանարդատիպ տրոհում: заметки о монотоннoм распознавании в многозначных решетках левон а. асланян и асмик а. саакян институт проблем информатики и автоматизации нан ра e-mail: lasl@sci.am, hsahakyan@sci.am аннотация новый подход монотонного распознавания на основе разбиения многозначной решетки на дискретные структуры, изоморфные бинарным кубам (метод “кубического разбиения”) предложен в серии последних работ, где на теоретическом уровне дано описание двух алгоритмов /алгоритмических схем/ решения задачи. в данной статье приводится подробное описание деталей реализации этих алгоритмов, а также рассматривается случай распознавания монотонной функции с небольшим числом единиц, что связано с рядом практических приложений. ключевые слова: распознавание монотонной функции, многозначная решетка, кубическое разбиение. mailto:lasl@sci.am mailto:hsahakyan@sci.am 2. the cube-splitting technique 2.1 definitions/descriptions 2.2 the algorithmic framework 3. implementation 3.1 implementation details of step 1 and step2 3.2 implementation details of step 3 and step 4 4. small number of units 4.1 constraints 4.2 resources d:\sbornik\...\333.dvi mathematical problems of computer science 33, 35{40, 2010. a n ote on shor t p aths in or iented gr aphs s a m ve l k h . d a r b in ya n a n d is ka n d a r a . k a r a p e t ya n institute for informatics and automation problems of nas of ra e-mail: samdarbin@ipia.sci.am, isko@ipia.sci.am abstract let g be an oriented graph of order p ¸ 3 and minimum semi-degrees at least [p=2] ¡ k for a positive integer k. for a subset c of vertices g, we obtain su±cient conditions implying that for any pair of distinct vertices x; y 2 v (g) ¡ c there is a path from x to y of length less than a given integer which does not contain the vertices of c. refer ences [1 ] j. b a n g -je n s e n a n d g. gu t in , d ig r a p h s . th e o r y. a lg o r it h m s a n d a p p lic a t io n s , s p r in g e r , 2 0 0 0 . [2 ] j. d . u llm a n , co m p u t a t io n a l a s p e c t s o f v l s i,co m p u t e r s c ie n c e p r e s s , 1 9 8 4 . àõõõáñ¹í³í ·ñ³ýý»ñáõù ï³ñ× ×³ý³å³ññý»ñç ù³ëçý ê. ¸³ñµçýû³ý ¨ æ. î³ñ³å»ïû³ý ²ù÷á÷áõù ¸çóáõù g-ý p·³·³ã³ýç (p ¸ 3 ) áõõõáñ¹í³í ·ñ³ý ¿, áñç ·³·³ãý»ñç ïçë³³ëïç×³ýý»ñá ÷áùñ ã»ý [p= 2 ] ¡ k ãíçó (áñï»õ k ¸ 1 ¨ ³ùµáõç ãçí ¿), ¨ ¹çóáõù c-ý g ·ñ³ýç ·³·³ãý»ñç áñ¨> »ýã³µ³½ùáõãûáõý ¿: ü»ñï³ ³ßë³ï³ýùáõù ëï³óí³í »ý µ³í³ñ³ñ å³ûù³ýý»ñ, áñáýó ¹»åùáõù g ·ñ³ýç ó³ýï³ó³í »ñïáõ ï³ñµ»ñ x ¨ y ·³·³ãý»ñç ñ³ù³ñ ·áûáõãûáõý áõýç c µ³½ùáõãû³ý ·³·³ãý»ñáí ã³ýóýáõ ¨ ïñí³í ãíçó ÷áùñ »ñï³ñáõãû³ùµ x -çó ¹áõñë »ïáõ ¨ y-á ùïýáõ ïáõùýáñáßí³í ×³ý³å³ññ: 3 5 начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 48, 82-88, 2017. point clouds preprocessing for better registration aram v. gevorgyan russian-armenian university e-mail: aramgv@gmail.com abstract in this article, we review point clouds preprocessing algorithms that enhance the results and performance of point clouds registration. registration is a problem of merging two or more point clouds into one common. if a point cloud is obtained from a real physical object, it definitely has some noise and outliers. this noise can negatively affect the registration process and often prevent from good results. point clouds preprocessing reduces noise and helps to get better registration results. also we present several experiments on real dataset and results. keywords: point clouds, registration, icp, sor filter. 1. introduction 3d technologies are developing very fast in recent years and are used in many spheres. one of the most commonly used formats in which 3d data is stored is a point cloud. point cloud is a set of points in some coordinate system. point cloud can also store some additional information, such as color or surface normals. point clouds have many applications, and they can be converted to other formats. we can also get a surface from point clouds using one of the surface reconstruction algorithms. one of the fundamental problems in computer vision and computer graphics is 3d or point clouds registration problem. registration is the process of merging multiple partially overlapped point clouds from different scenes into one common point cloud. in other words, 3d registration transforms multiple point clouds from different coordinate systems into the same coordinate system so they can be aligned. process of getting a point cloud from a real physical object is known as reverse engineering or 3d scanning. there are active and passive methods of 3d scanning. active methods usually use laser or structured light for getting 3d information. while passive methods don't emit anything, 82 a. gevorgyan 83 they rely on the detection of reflected ambient radiation. examples of passive methods are stereovision, structure from motion, structure from shading. all these methods have their props and corns. but whatever method we use for getting a point cloud, it cannot be ideal. some noise and errors are presented. all these factors negatively affect the process of registration and can prevent for getting good results. also in case of big data, if point clouds have many points performance of the registration algorithms may be inacceptable. so in this article we review some point clouds preprocessing methods that can enhance the results and performance of registration algorithms. the article has the following structure: in section 2 we speak in details about point clouds registration and icp algorithm, in section 3 we review some preprocessing algorithms and in section 4 we present experiments and results. 2. related work suppose we want to align the source point cloud p }),...,,{( 2,1 npppp = with the target point cloud q }),...,,{( 2,1 nqqqq = . the aim of registration problem is to find such a transformation t (rotation matrix and translation vector) (1) applying which to source point cloud p will bring it to the same coordinate system with q.             == 1000 ),( 3333231 2232221 1131211 trrr trrr trrr trt . )1( the registration algorithms can be divided into 2 classes: coarse and fine registration. coarse registration algorithms are usually used for initial alignment and then results are refined with fine registration. the most popular 3d registration algorithm is iterative closest point (icp) [1-2] algorithm. icp is very simple for realization and has good performance. the main idea of icp algorithm is iteratively register point clouds, trying at each iteration to minimize distance between point clouds. icp algorithm consists of the following steps: 1) finding correspondent points between point clouds. the closest point between point clouds are defined as correspondent points. 2) transformation estimation applying which to source point cloud 𝑃𝑃 will minimize the error e. ( )∑ = −+= n i ii qtrpn e 1 21 , where 𝑅𝑅 rotation matrix, 𝑡𝑡 translation vector, 𝑝𝑝𝑖𝑖 and 𝑞𝑞𝑖𝑖 correspondent points, 𝑁𝑁 number of points. this error metric is called as point-to-point metric, there are also point-to-plane metrics defined in [2]. 3) apply the transformation to source point cloud 𝑃𝑃. the steps 1-3 are executed iteratively until the error is smaller than the given threshold. over the years many variants of icp algorithm are developed, more details about them in [3]. icp algorithm has a known local minimum issue: if the initial distance between the point clouds is big icp can give very wrong results. also noise and outliers can negatively affect the results of the registration. so it will be good to reduce noise before the registration process. point clouds preprocessing for better registration 84 3. point clouds preprocessing there are several point clouds preprocessing techniques that can be used for different purposes. let us review some of them. down sampling: down sampling filter decreases the number of points in a point cloud data set. the aim of down sampling is to reduce the number of points in point cloud while still preserving the object and scene it represents. down sampling filter is generally used on big data set to increase the performance of the algorithms working with point clouds. each point is described by a voxel and its centroid. voxel is a box with a specific size in the space, and centroid is a point the coordinates of which are the mean value of coordinates of all points in the voxel. there are two ways of down sampling: "voxel grid" and "uniform sampling". the first way replaces all points in the voxel with the centroid of that voxel, and the second one replaces with the nearest existing point to centroid. in the voxel grid approach a new point is generated, while in the uniform sampling no new point is inserted into the point cloud. the uniform sampling method is a bit slower than the voxel grid, but it represents the underlying surface more accurately. pass-through filter: pass through filter is used for cutting of values that are either inside or outside the given range. it can be used, for example, if we want to remove points that are deeper than some threshold. pass through filter checks for every point if its field (coordinate) is in the given interval or not. radius outlier removal filter: radius outlier removal filter for each point examines a circle with the given radius and with center in that point. if the number of points in that circle is less than the given threshold k, then this point is specified as an outlier and removed from the point cloud. in figure 1 we can see an example of radius outlier removal filter, if k = 1, the yellow point will be removed, if k =2, the green and yellow points will be removed. fig. 1. example of radius outlier removal filter [8]. statistical outlier removal filter (sor): statistical outlier removal filter removes points depending on their local neighborhood. for each point the mean distance from it to its k nearest neighbors is computed. by assuming that the resulted distribution is gaussian with a mean and a standard deviation, all points the mean distances of which are greater than the global distances mean plus standard deviation with some coefficient are defined as outliers and removed from dataset. point is specified as an outlier if its mean distance of its nearest k neighbors is bigger than m, where m is defined as: a. gevorgyan 85 σ*nsigmaavgm += , where 𝑎𝑎𝑎𝑎𝑎𝑎 is the average distance, 𝑛𝑛𝑛𝑛𝑛𝑛𝑎𝑎𝑛𝑛𝑎𝑎 – the number of times the standard deviation (userdefined parameter), 𝜎𝜎 – the standard deviation. an example of statistical outlier removal filter is shown in figure 2. fig. 2. example of statistical outlier removal filter. 4. experiments and results in this section we present some experiments and results with different datasets and preprocessing algorithms. for our realization we use point cloud library (pcl) [6] the biggest library for point clouds processing, viewing and manipulation. a registration with pcl is described in [7]. let us review some experiments of point clouds preprocessing steps. experiment 1: in this experiment we want to show how the downsample filter can affect the performance of registration algorithms, especially if point clouds have many points. we take two point clouds from stanford bunny dataset [4] each point cloud has 40256 points. we downsample these point clouds with the voxel of size 0.01. then we compare the performance of icp registration algorithm of original point clouds and these point clouds after down sample filter is applied. the results were shown in table 1. table 1: point clouds number of points icp (time) original point clouds 40256 32.91 downsampled point clouds 394 0.1 as it is seen downsampling greatly affects the performance, while results in this case are almost the same. experiment 2: in the second experiment we use our dataset of bear object. the point clouds in this dataset were generated from 2 web cameras by stereovision techniques. stereovision is a technique that estimates 3d information of a scene from two or more cameras. 3d information is obtained as a depth map, then the depth map can be converted into a point cloud. more about stereovision -in [5,10]. we scan the bear from different views and get a point cloud for each view. point clouds preprocessing for better registration 86 in this experiment we want to remove background to have a point cloud only of bear object. figure 3 shows the point cloud of whole scene. fig. 3. the whole scene. we segment our bear object from the scene using pass through filter, filtering by x, y and z coordinates. the result is shown in figure 4: fig. 4. segmented point cloud. experiment 3: in this experiment, we want to show how the noise can negatively affect the registration process. we take segmented point clouds from the previous experiment and merge these point clouds without applying any filter and after applying a statistical outlier removal filter. the results are shown in figures 5: a. gevorgyan 87 a) b) fig. 5. the result of merging 10 point clouds with icp algorithm. a) range point clouds, b) after applying statistical outlier removal filter. as we can see from the figures, the noise and outliers in original point clouds negatively affect the registration process bringing to an absolutely wrong result. but after applying the sor filter, we can get a good result. references [1] p. j. besl and h. d. mckay, “a method for registration of 3-dshapes”, ieee trans. pattern anal. mach. intell., vol. 14, no. 2, pp. 239–256, 1992. [2] y. chen and g. medioni, “object modeling by registration of multiple range images”, image and vision computing, vol. 3, no. 10, pp 145–155, april 1992. [3] s. rusinkiewicz and m. levoy, “efficient variants of the icp algorithm”, 3-d digital imaging and modeling, pp. 145–152, 2001. [4] stanford bunny dataset: [online]. available: https://graphics.stanford.edu/data/3dscanrep/ [5] p. j. bagga, “real time depth computation using stereo imaging”, journal electrical and electronic engineering, vol. 1, pp. 51-54, 2013. [6] point cloud library: [online]. available: http://www.pointclouds.org/ [7] d. holz, a.e. ichim, f. tombari, r.b. rusu and s. behnke, “registration with the point cloud library: a modular framework for aligning in 3-d”, ieee robotics & automation magazine, vol. 22, no. 4, pp. 110-124, 2015. [8] [online].available: http://www.pointclouds.org/documentation/tutorials/remove_outliers.php#remove-outliers [9] a. v. gevorgyan, “point clouds registration and generation from stereo images”, ithea journal, "information content and processing", vol. 3, no. 2, pp. 193-200, bulgaria, 2016. submitted 06.08.2017, accepted 04. 12.2017. point clouds preprocessing for better registration 88 կետերի ամպերի մշակում գրանցման արդյունքները բարելավելու նպատակով ա. գևորգյան ամփոփում սույն հոդվածում ներկայացված են կետերի ամպերի մշակման ալգորիթմներ, որոնք բարելավում են կետերի ամպերի գրանցման (registration) արդյունքները և արդյունավետությունը: գրանցման խնդիրը կայանում է երկու կամ ավելի կետերի ամպ միաձուլելու մեջ: եթե կետերի ամպը ստացված է իսկական ֆիզիկական օբյեկտից, այն պարունակում է աղմուկ, որը կարող է բացասական ազդեցություն ունենալ գրանցման գործընթացի վրա և խոչնդոտել ստանալ լավ արդյունքներ: կետերի ամպերի նախամշակումը նվազեցնում է աղմուկը և օգնում է ստանալ ավելի լավ գրանցման արդյունքներ: սույն հոդվածում ներկայացված են նաև մի քանի փորձերի արդյունքներ իրական տվյալների վրա: обработка облаков точек для лучшей регистрации а. геворкян аннотация в данной статье представлены алгоритмы обработки облаков точек, которые улучшают результаты и эффективность регистрации облаков точек. регистрация это задача соединения двух или более облаков точек. если облако точек получено от реального физического объекта, то в нем присутствует шум. данный шум отрицательно влияет на процесс регистрации и часто мешает получить хороший результат. предварительная обработка облаков точек уменьшает шум и помогает получить лучший результат регистрации. в данной статье, также представлены результаты нескольких экспериментов на реальных данных. microsoft word 13.doc mathematical problems of computer science 37, 102--107, 2012. 102 necessary conditions for optimal permissible placement by the height of the transitive directed tree with one root (part second) armen khachaturyan yerevan state university e-mail: khachaturyanarmen@gmail.com abstract the present paper is the second part of the paper [1]. here we have introduced a couple of additional concepts and have obtained some additional necessary conditions for the solution of the problem formulated in paper [1]. keywords: transitive directed graph, optimal placement. references 1. a.h. khachaturyan, “necessary conditions for optimal permissible placement by the height of the transitive directed tree with one root”, mathematical problems of computer science, vol. 36, pp. 104-114, 2012. 2. a.h. khachaturyan, “the optimal permissible placement by the height of the transitive oriented tree containing one vertex of branching”, mathematical problems of computer science, vol. 30, pp. 71-75, 2008. 3. m.r. garey, d.s. johnson, computers and intractability: a guide to the theory of npcompleteness. san francisco, ca: w.h. freeman, 1979. 4. f. gavril, “some np-complete problems on graphs,” proc.11th conf. on information sciences and systems, johns hopkins university, baltimore, md, pp. 91-95, 1977. 5. m.r. garey, r.l. graham, d.s. johnson and d.e. knuth, “complexity results for bandwidth minimization”, siam j. appl. math., vol. 34, pp. 477–495. 1978. 6. m.r. garey, d.s. johnson and l. stockmeyer, “some simplified np-complete graph problems”, theor. comput. sci., vol. 1, pp. 237–267. 1976. 7. ch.h. papadimitriou, “the np-copleteness of the bandwidth minimization problem”, computing, v. 16, pp. 263–270. 1976. 8. a.v. petrosyan, s.e. markosyan, yu.g. shukuryan, mathematical problems of automation and projection of calculating-machine. yer., (in russian). 1977. 9. g.g. geoletsyan, “flat placement of the vertices of tree with minimization of width”, dan arm. ssr, issue 56, no. 4, pp. 202–207 (in russian). 1973. a. khachaturyan 103 10. l. m. goldberg and i. a. klipker, “minimum placement of trees on a line,” technical report, physico-technical institute of low termeperatures, academy of sciences of ukraina ssr, ussr. 1976. 11. y. shiloach, “a minimum linear arrangement algorithm for undirected trees” report, dept. of applied mathematics, weizmann institute, rehovot, israel. 1976. 12. d. adolphson and t.c. hu, “optimal linear ordering”, siam j. appl. math., vol. 25, no. 3, pp. 403–423. 1973. 13. c. berge, the theory of graphs and its applications. new york: wiley, 1962. ø»ï ³ñù³ïáí ïñ³ý½çïçí ûñç»ýï³óí³í í³éç áëï µ³ñóñáõãû³ý ûåïçù³é ãáõûé³ïñ»éç ï»õ³¹ñù³ý ³ýññ³å»ßï å³ûù³ýý»ñ (ù³ë »ñïñáñ¹) ². ê³ã³ïáõñû³ý ²ù÷á÷áõù êáõûý ñá¹í³íá ñ³ý¹çë³ýáõù ¿ [1] ñá¹í³íç ß³ñáõý³ïáõãûáõýá: ²ûëï»õ ù»ýù ý»ñùáõí»é »ýù áñáß ýáñ ñ³ëï³óáõãûáõýý»ñ ¨ ëï³ó»é [1] ñá¹í³íáõù ó¨³ï»ñåí³í ëý¹ñç éáõíù³ý ñ³ù³ñ ¨ë ùç ù³ýç ³ýññ³å»ßï å³ûù³ýý»ñ: íåîáõîäèìûå óñëîâèÿ îïòèìàëüíîé äîïóñòèìîé ðàññòàíîâêè ïî âûñîòå òðàíçèòèâíî îðèåíòèðîâàííîãî äåðåâà ñ îäíèì êîðíåì (÷àñòü âòîðàÿ) à. õà÷àòóðÿí аннотация íàñòîÿùàÿ ñòàòüÿ ÿâëÿåòñÿ ïðîäîëæåíèåì ñòàòüè [1]. çäåñü ìû ïðèâåëè íåêîòîðûå íîâûå êîíöåïèè è ïîëó÷èëè åùå íåñêîëüêî íåîáõîäèìûõ óñëîâèé äëÿ ðåøåíèÿ çàäà÷è ñôîðìóëóðîâàííîé â ñòàòüå [1]. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 46, 66-72, 2016. development and implementation of some advanced web server protection methods arthur s. petrosyan and gurgen s. petrosyan institute for informatics and automation problems of nas ra e-mail: arthur@sci.am, gurgen@sci.am abstract article describes the development work done in the academic scientific research computer network of armenia (asnet-am) managed by the institute for informatics and automation problems (iiap) of the national academy of sciences of the republic of armenia (nas ra) regarding the implementation of web server protection and hardening. some advanced methods and solutions implemented recently within asnet-am web hosting services are being described. based on the experience of asnet-am web hosting services the best practice recommendations about secure web hosting environment implementation are given. keywords: www, web hosting environment, web server, apache, fail2ban 1. introduction increased capabilities in the age of modern-day information technology progress, makes the web server protection more and more urgent. new types of exploits, bots and brute force tools are being introduced. this article is a logical continuation of the previous research and deployment of the improved methods for web server protection [1] done in academic scientific research computer network of armenia (asnet-am) managed by the institute for informatics and automation problems (iiap) of the national academy of sciences of the republic of armenia (nas ra). work described in [1] was mainly focused on the methods of shared web server internal hardening and shared environment isolation. this paper is focused on several advanced methods to protect web server from external attacks. in this paper apache is considered as a web server example, because of being the most used web server on the internet [2]. 66 a. petrosyan and g. petrosyan 67 2. protection of writable directories web sites may require some files to be uploaded from web browsers. it means that a write permission to the user running the web server should be provided to be able to put the uploaded file into the destination directory. the common solution is to give all the users 777 (read/write/execute) access to the destination directory. such common solution becomes an easy way to upload a malicious program to the web server for later harmful execution. in case of apache 2 itk mpm [3] package, which is used within asnet-am web hosting services to deploy the improved web server protection methods, such write permissions are being limited only to a group level, without need of giving the 777 write access to all users. additionally it is recommended to put such writable directories out of virtual host area, i.e., on the upper level, than web site’s document root. this way no direct url-based access will be possible. example of such configuration follows: document root directory of ‘somesite.am’ website /home/somesite.am/www writable directory for uploaded files /home/somesite.am/uploadfiles (not /home/somesite.am/www/uploadfiles ) although the above configuration additionally hardens the web site, it is sometimes difficult to accomplish because of the specific web site structure. any writable directory should always be configured to prohibit execution of any scripts. the best way to do that is making appropriate static configuration for that web site in the web server’s main configuration [4]. if not possible (web site owner may not be granted permission to operate web server’s configuration) it should be done via the .htaccess file in that directory [5]. example of such configuration follows: php_flag engine 0 php_flag engine 0 php_flag engine 0 next protection method to be implemented is the limitation of file types and max file size to be allowed for uploading. this can be done by means of specific checks. following is the code example on php language to allow uploading only pdf files: if ($_files[file][tmp_name] <> "" && $_files[ file][tmp_name] <> "none" && $_files[file][size] <= $file_max_upl_size && is_uploaded_file($_files[file][tmp_name]) && $_files[file][type] == "application/pdf" || $_files[file][type] == "application/acrobat" || $_files[file][type] == "application/x-pdf" || $_files[file][type] == "applications/vnd.pdf" || $_files[file ][type] == "text/pdf" || $_files[file][type] == "text/x-pdf") { } development and implementation of some advanced web server protection methods 68 the above solutions are not complete panacea, because many web sites are based on the opensource solutions (like joomla, wordpress, drupal, etc.) and if not updated regularly, may contain known vulnerability/exploit. thus, an additional mechanism for writable directory contents checking is recommended to be used. it could be implemented by means of a specific shell script, regularly parsing writable directory contents in search of pre-defined prohibited file types (such as .php files). in case such file found the script moves it to quarantine location for further analysis and logs the incident. if such writable directory is protected with .htaccess file (since it is always located in the same directory) it could be replaced with some other one to allow the execution vulnerable scripts. thus, the script also checks for the presence of the .htaccess file, and if it does not find it, it recovers it from the backup copy and logs the incident. it also checks the contents of .htaccess file by comparing it with the backup version and if there is a difference detected recovers it from the backup copy and logs the incident. following is the shell script code example: function moveandlog { mv $1 $pgs_wrong_files_folder_path echo "$(date) phpmove::: $1" >> $pgs_log_file_path } export -f moveandlog /var/wrongfiles find $i -iregex '.*$php$' -exec bash -c 'moveandlog "{}"' \; find $i -iregex '.*$php3$' -exec bash -c 'moveandlog "{}"' \; find $i -iregex '.*$phtml$' -exec bash -c 'moveandlog "{}"' \; find $i -iregex '.*$phps$' -exec bash -c 'moveandlog "{}"' \; find $i -name ".htaccess" | while read fname; do if ! cmp -s $pgs_htaccess_path $fname then cp $pgs_htaccess_path $fname chown $pgs_chown_user:$pgs_chown_group $fname echo "$(date) overwrht:: $fname" >> $pgs_log_file_path fi done the script permanently runs as a daemon and makes checks every 3 seconds (time can be changed). thus, even if some web site vulnerability would enable an attacker to upload some malicious file to the writable directory, it will be detected and immediately moved out, and the incident will be logged. a. petrosyan and g. petrosyan 69 3. proper fail2ban configuration fail2ban [6] utility is available in most linux-based web server solutions. it can be used to detect and block certain ip addresses from which the attempts of unauthorized access to the server is done. these ip addresses are determined by the results of the monitoring of log files log-files (for example, /var/log/auth.log, /var/log/apache/error.log etc.). if any ip address in a certain period of time, makes too many unsuccessful log in attempts or any other suspicious activity, the host with the ip address is blocked (by adding iptables firewall rules) for a certain time interval specified by fail2ban configuration. the above-described default functionality of fail2ban allows to protect website not only from the so-called "brute force" attacks, but also from automatic scanning of web site by means of some bot scripts. below are given some best practice recommendations to protect web site using fail2ban, in order to increase the security of the web server. by scanning means of apache-auth filter unsuccessful attempts to gain access to the web server directories, that are protected by username and password (.htaccess / .htpasswd) can be identified. using apache-nohome filter attempts to scan the list of site scripting files can be detected. additional strict filter that scans and detects all types of files and folders was written by asnet-am team: failregex = ^%(_apache_error_client)s file does not exist while it adds web site protection, it should be used very carefully, because the web site developers or content editors may include broken links to different files, pictures, etc. and there is a chance to block normal users who just browse the site and occasionally click on the broken link, thus unintentionally giving the impression of improper access. it is probably better not to use this fail2ban filter for a web site that is not completely ready and tested. using apache-badbots filter enables to identify and prevent different bots crawl our web site to find email addresses to spam databases. several other filters (apache-botsearch, apache-fakegooglebot, apache-overflows) help to identify fake googlebot-s, long suspicious requests, etc. php-url-fopen filter can detect attempts to run php injection. apache-shellshock filter can detect attempts to exploit shellshock vulnerability. in spite of the above there are more intelligent bots that can identify the period of their blocking and produce scan or brute force in such a way, so as not to be blocked (slow brute force). to prevent such attacks a specific fajl2ban setting of different block time use can be configured. this is done by having a random time to be added for each blocking incident to some predetermined period of time. example of such settings follows: bantime = 86400 bantime.increment = true bantime.rndtime = 79m bantime.factor = 1 development and implementation of some advanced web server protection methods 70 fail2ban also has useful filter recidive that allows to scan own fail2ban’s log to identify the frequency of a particular ip address blocking (recidive), so as to re-block it for a longer period of time. it is advisable to set up fail2ban to all the services that will be active on the web server, such as ssh (after changing to a non-standard port), mail server, etc. 4. some additional techniques to increase web server security following additional methods of protecting web servers and web sites are also recommended for implementation as much as possible. many complex database-driven web sites have separate frontend (the site itself) and backend (content management system (cms)). in that case best practice would be to use different database users for frontend and backend. the frontend database user should have minimum rights (the best possible option is only to give read rights), while backend database user should have full rights to modify the database on which the web site is based. following are examples of mysql permissions for frontend and backend database users frontend user rights: grant select on somesitedb.* to somesitedbfrontenduser@localhost identified by 'somefrontendpassword'; backend user rights: grant all privileges on somesitedb.* to somesitedbbackenduser@localhost identified by 'somebackendpassword' with grant option; additionally it is best to separate the backend interface, by configuring it as a separate virtualhost and even putting it on a non-standard application port (different from 80). this will add a security layers to the backend. for some web sites based on the ready opensource solutions (like joomla, wordpress, drupal, etc.) it requires some tricks to separate backend and frontend parts. when the code is written the way that ties backend and frontend parts together, it could be easier, just to create two similar copies of website, but use different database users as shown above, so as to eliminate possibility of sql-injections or other database-related exploits to be run via public frontend. this is especially important for the ready opensource solutions, where some vulnerability may be found and not fixed yet, because of not being updated regularly. although present such vulnerability will have no effect, because database user for frontend doesn’t have enough privileges to modify the database. in case the web site backend is being used only from specific workstations, additional protection could be made by not registering the separate name of that backend virtual host in dns, but only statically adding that name in the ‘hosts’ file of such workstations. alternatively such backend protection could also be implemented through the special reverse proxy configuration. finally backend is good to have .htaccess / .htpasswd user/password protection in addition to any other protection methods. and of course backend connection should be made through the https protocol only. a. petrosyan and g. petrosyan 71 5. conclusion the use of the described methods to protect writable directories can prevent running the malicious scripts to harm the web site and the web server as a whole, as well as will help in timely detection and fixing of any vulnerabilities in the web site code, that could allow an attacker to upload undesirable files to the web server. using fal2ban utility can greatly increase the security of the web server and web sites, as it can identify and block attacks on services and sites such as brute force, vulnerability scans, etc. if also some additional non-standard protection methods described above are used, maximum protection could be achieved or at least we would make it more difficult to attack our web server. references [1] a. petrosyan and g. petrosyan, “research and deployment of improved web server protection methods”, transactions of iiap of nas ra mathematical problems of computer science, vol. 42, pp. 81-84, 2014. [2] wikipedia, the free encyclopedia, web server, market share, [online]. available: https://en.wikipedia.org/wiki/web_server#market_share [3] apache 2 itk mpm, [online]. available: http://mpm-itk.sesse.net/ [4] apache http server version 2.4, apache core features, directive, [online]. available: http://httpd.apache.org/docs/current/mod/core.html#directory [5] apache http server version 2.4, apache http server tutorial: .htaccess files, [online]. available: http://httpd.apache.org/docs/current/howto/htaccess.html [6] fail2ban. [online]. available: http://www.fail2ban.org/ submitted 19.07.2016, accepted 03.11.2016. վեբ սերվերների պաշտպանության որոշ առաջադեմ մեթոդների մշակումը և կիրառումը գ. պետրոսյան և ա. պետրոսյան ամփոփում հոդվածը նկարագրում է վեբ-սերվերների պաշտպանության բարձրացմանն ուղված հետազոտական աշխատանք, որը կատարվել է հայաստանի ակադեմիական գիտահետազոտական կոմպյուտերային ցանցում (asnet-am), որը ղեկավարում է հայաստանի հանրապետության գիտությունների ազգային ակադեմիայի (հհ գաա) ինֆորմատիկայի և ավտոմատացման պրոբլեմների ինստիտուտը (իապի): development and implementation of some advanced web server protection methods 72 նկարագրված են որոշ առաջադեմ մեթոդներ և լուծումներ, որոնք մշակվել և իրականացվել են վերջին ժամանակներում asnet-am ցանցի վեբ-հոսթինգի ծառայության համար: բերված են նաև asnet-am ցանցի վեբ-հոսթինգի ծառայության փորձի հիման վրա անվտանգ միջավայրի ստեղծման գործնական առաջարկություններ: разработка и реализация некоторых передовых методов защиты веб-серверов г. петросян и а. петросян аннотация статья описывает исследовательскую работу по усилению защиты веб-серверов, проведенную в академической научной исследовательской компьютерной сети армении (asnet-am), действующей под управлением института проблем информатики и автоматизации (ипиа) национальной академии наук республики армения (нан ра). описаны некоторые передовые методы и решения, разработанные и реализованные в последнее время для службы веб-хостинга сети asnet-am. также приведены практические рекомендации о реализации безопасной среды, на основе опыта работы службы веб-хостинга сети asnet-am. microsoft word tpel.doc mathematical problems of computer science 36, 104--114, 2012. 104 necessary conditions for optimal permissible placement by the height of the transitive directed tree with one root armen khachaturyan yerevan state university e-mail: khachaturyanarmen@gmail.com abstract in the graph theory the problem of the minimum placement of graph by the height, which is similarly formulated in [2] (the problem of minimum cut arrangement of graph), is known. the problem is np-complete [3]. in the present paper a partial case of this problem, i.e. the problem of optimal permissible placement by the height of the transitive directed tree with one root (which is a such transitive directed graph, the arc base of which forms a directed tree with one root), is formulated. in this paper some new concepts are introduced and necessary conditions for optimal solving of the new formulated problem are given. keywords: transitive directed graph, optimal placement. r e f e r e n c e s 1. a. h. khachaturyan, “the optimal permissible placement by the height of the transitive oriented tree containing one vertex of branching”, mathematical problems of computer science, vol. 33, pp183-186, 2010. 2. m.r. garey, d.s. johnson, computers and intractability: a guide to the theory of npcompleteness. san francisco, ca: w.h. freeman, 1979. 3. f. gavril, “some np-complete problems on graphs,” proc.11th conf. on information sciences and systems, johns hopkins university, baltimore, md, pp. 91-95, 1977. 4. m. r. garey, r. l. graham, d. s. johnson and d. e. knuth, “complexity results for bandwidth minimization”, siam j. appl. math., vol. 34, pp. 477–495. 1978. 5. m. r. garey, d. s. johnson and l. stockmeyer, “some simplified np-complete graph problems”, theor. comput. sci., vol. 1, pp. 237–267. 1976. 6. ch. h. papadimitriou, “the np-completeness of the bandwidth minimization problem”, computing, v. 16, pp. 263–270. 1976. 7. a.v. petrosyan, s. e. markosyan, yu. h. shukuryan, mathematical problems of automation and projection of calculating-machine. yer., (in russian). 1977. 8. g.g. geoletsyan, “flat placement of the vertices of tree with minimization of width”, dan arm. ssr, issue 56, no. 4, pp. 202–207 (in russian). 1973. 9. l. m. goldberg and i. a. klipker, “minimum placement of trees on a line,” technical report, physico-technical institute of low temperatures, academy of sciences of ukraine ssr, 1976. a. khachaturyan 105 10. y. shiloach, “a minimum linear arrangement algorithm for undirected trees” report, dept. of applied mathematics, weizmann institute, rehovot, israel. 1976. 11. d. adolphson and t.c. hu, “optimal linear ordering”, siam j. appl. math., vol. 25, no. 3, pp. 403–423. 1973. 12. c. berge, the theory of graphs and its applications. new york: wiley, 1962. ø»ï ³ñù³ïáí փոխանցական կողմնորոշ í³éç áëï µ³ñóñáõãû³ý ûåïçù³é ãáõûé³ïñ»éç ï»õ³¹ñáõãû³ý ³ýññ³å»ßï å³ûù³ýý»ñ ². ê³ã³ïáõñû³ý ²ù÷á÷áõù ¶ñ³ýý»ñç ï»ëáõãû³ý ù»ç ñ³ûïýç ¿ ·ñ³ýç áëï µ³ñóñáõãû³ý ûåïçù³é ï»õ³¹ñù³ý ëý¹çñá, áñá ñ³ù³ñå»ù ï»ñåáí ó¨³ï»ñåí³í ¿ [2]-áõù (·ñ³ýç ùçýçù³é ïïñí³íùáí ï³ñ·³íáñù³ý ëý¹çñá): êý¹çñá np-¹åí³ñ ¿: êáõûý ³ßë³ï³ýùáõù ó¨³ï»ñåí³í ¿ ³ûë ëý¹ñç ù³ëý³ïç ¹»åùá` ù»ï ³ñù³ïáí փոխանցական կողմնորոշ í³éç (¹³ ³ûý փոխանցական կողմնորոշ ·ñ³ýý ¿, áñç ³õ»õý»ñç µ³½³ý ï³½ùáõù ¿ ù»ï ³ñù³ïáí կողմնորոշ í³é) áëï µ³ñóñáõãû³ý ûåïçù³é ãáõûé³ïñ»éç ï»õ³¹ñù³ý ëý¹çñá: ²ûë ³ßë³ï³ýùáõù ý»ñï³û³óí»é »ý áñáß³ïç ýáñ ñ³ëï³óáõãûáõýý»ñ ¨ ïñí»é ó¨³ï»ñåí³í ëý¹ñç ûåïçù³é éáõíù³ý ³ýññ³í»ßï å³ûù³ýý»ñ: íåîáõîäèìûå óñëîâèÿ îïòèìàëüíîé äîïóñòèìîé ðàññòàíîâêè ïî âûñîòå òðàíçèòèâíî îðèåíòèðîâàííîãî äåðåâà ñ îäíèì êîðíåì à. õà÷àòóðÿí аннотация â òåîðèè ãðàôîâ èçâåñòíà ïðîáëåìà ìèíèìàëüíîé ðàññòàíîâêè ãðàôà ïî âûñîòå, êîòîðàÿ àíàëîãè÷íî ñôîðìóëèðîâàíà â [2] (ïðîáëåìà óïîðÿäî÷èâàíèÿ ãðàôà ñ ìèíèìàëüíûì ðàçðåçîì). ïðîáëåìà np-ïîëíà 3. â íàñòîÿùåé ñòàòüå ôîðìóëèðîâàí ÷àñòíûé ñëó÷àé ýòîé ïðîáëåìû: ïðîáëåìà îïòèìàëüíî äîíóñòèìîé ðàññòàíîâêè ïî âûñîòå òðàíçèòèâíî îðèåíòèðîâàííîãî äåðåâà ñ îäíèì êîðíåì (ýòî òàêîé òðàíçèòèâíî îðèåíòèðîâàííûé ãðàô, áàçà äóã êîòîðîãî ñîñòàâëÿåò îðèåíòèðîâàííîå äåðåâî ñ îäíèì êîðíåì). â ðàáîòå ââåäåíû íåêîòîðûå íîâûå ïîíÿòèÿ è äàíû íåîáõîäèìûå óñëîâèÿ äëÿ îïòèìàëüíîãî ðåøåíèÿ ñôîðìóëèðîâàííîé çàäà÷è. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 44, 85--92, 2015. combined digital methods for solving optimal resource allocation problems hasmik s. derdzyan vanadzor state university e-mail: jaslinna@mail.ru abstract this article is devoted to the development of new effective methods for solving optimal resource allocation problems. the simulated annealing and genetic methods of digital optimization are widely used for solving these kinds of problems. though these methods approach to the optimal solution of the problem, as usual, a long period of time is required for obtaining the exact solution. this article offers to combine the simulated annealing method with the modification of downhill simplex method to increase the convergence of the optimization method. keywords: simulated annealing, downhill simplex, modification, allocation, resources. 1. introduction the optimal usage of industrial, technological, financial and other resources is one of the important preconditions for further development of the society. the equal allocation of resources during the period of their usage is a special kind of optimal resource distribution. these kinds of problems arise during the parallel implementation of many industrial and technological processes as well as in the fields of construction engineering, financial investments, labor force distribution and so on. in such fields it is often required to work with stable number of employees, provide equal funding in a certain period of time, etc. usually these are the problems of digital optimization. according to the analysis of the recent scientific articles, researchers give preference to the meta-heuristic methods, particularly to the genetic, simulated annealing and other methods to solve these kinds of problems [1, 2]. these methods approach to the surroundings of global minimum, however, they require a great number of iterations for obtaining the exact global minimum of the objective function [3, 4]. that is why many researchers are satisfied with finding an approximate solution of optimization problems. so, it is necessary to develop new high performed and efficient methods and algorithms to find optimal solutions to the problems of equal distribution of resources. in this article the modification of downhill simplex method is combined with the simulated annealing method to speed up the search process for finding the optimal solution to these problems. 85 86 combined digital methods for solving optimal resource allocation problems 2. simulated annealing method the simulated annealing method is a meta-heuristic method used for finding the minimal value of the given objective function 𝑓𝑓(𝒙𝒙�), where 𝒙𝒙� ∈ 𝑅𝑅𝑛𝑛. the method allows to approach the global minimum of the given function in conditions of presence of several local minima. this method is based on crystallization processes of slow freezing thermodynamic systems, such as metals [5]. during the gradual decrease of temperature, the thermodynamic system shifts from the higher energy state to the lower energy state and comes to more stable structure. sometimes the system shifts towards the greater energy state, the result of which is more unstable metal structure. then the system changes its energy state again and comes to more stable condition. by the analogy of thermodynamic systems, the simulated annealing method does not permanently move towards the reduction of objective function values during the minimization process. the method sometimes allows moving to the direction of increment of the function values, depending on some probability. to determine the optimal value of objective function 𝑓𝑓(𝒙𝒙�) the following steps are done by this method. at first point 𝒙𝒙𝒄𝒄𝒄𝒄𝒄𝒄 ∈ 𝑅𝑅𝑛𝑛 is randomly selected from the search space of function 𝑓𝑓(𝒙𝒙�) and then initial temperature t is determined. then the temperature t is gradually decreased while it is higher than the already given temperature 𝑇𝑇𝑚𝑚𝑚𝑚𝑛𝑛 > 0. for each value of temperature t the following cycle is performed. the new point 𝒙𝒙𝒏𝒏𝒏𝒏𝒏𝒏 ∈ 𝑅𝑅𝑛𝑛 is chosen from the search space of the objective function. then ∆𝑓𝑓 = 𝑓𝑓(𝒙𝒙𝒏𝒏𝒏𝒏𝒏𝒏) − 𝑓𝑓(𝒙𝒙𝒄𝒄𝒄𝒄𝒄𝒄) difference is determined. 1. when ∆𝑓𝑓 < 0, in sense of minimization, the new value of function 𝑓𝑓(𝒙𝒙�) is “better” than the current value. in this case the new point 𝒙𝒙𝒏𝒏𝒏𝒏𝒏𝒏 is definitely accepted as the current point of the search space of the objective function. 2. when ∆𝑓𝑓 > 0, the objective function worsens its value on the new point. in this case the new point 𝒙𝒙𝒏𝒏𝒏𝒏𝒏𝒏 is accepted by probability 𝑝𝑝 = 𝑒𝑒−∆𝑓𝑓/𝑘𝑘𝑘𝑘, despite of the fact that the function takes higher value on that point (k is the constant of boltzmann). this step allows to observe wider areas. in the end the point is displayed on which the objective function takes minimal value. simulated annealing algorithm (algorithm 1) a. initialization step 1.1. generate a random point 𝒙𝒙𝒄𝒄𝒄𝒄𝒄𝒄 ∈ 𝑅𝑅𝑛𝑛 from the search space of the objective function 𝑓𝑓(𝒙𝒙�), 𝒙𝒙 ∈ 𝑅𝑅𝑛𝑛. choose 𝒙𝒙𝒐𝒐𝒐𝒐𝒐𝒐 as the best point. assign the point 𝒙𝒙𝒄𝒄𝒄𝒄𝒄𝒄 to the point 𝒙𝒙𝒐𝒐𝒐𝒐𝒐𝒐, 𝒙𝒙𝒐𝒐𝒐𝒐𝒐𝒐 ≔ 𝒙𝒙𝒄𝒄𝒄𝒄𝒄𝒄. step 1.2. select the minimal temperature 𝑇𝑇𝑚𝑚𝑚𝑚𝑛𝑛 > 0, the initial temperature 𝑇𝑇 > 0 and the value of parameter ℎ ∈ (0; 1). in [3] the following values are given: 𝑇𝑇𝑚𝑚𝑚𝑚𝑛𝑛 = 10−4, 𝑇𝑇 = � 𝑓𝑓(𝒙𝒙𝒄𝒄𝒄𝒄𝒄𝒄) ln (0.9) � , ℎ = 0.95. b. main cycle while 𝑇𝑇 > 𝑇𝑇𝑚𝑚𝑚𝑚𝑛𝑛, repeat steps 2.1-2.5. step 2.1. generate the new random point 𝑥𝑥𝑛𝑛𝑛𝑛𝑛𝑛 ∈ 𝑅𝑅𝑛𝑛 from the search space of the function 𝑓𝑓(𝒙𝒙�). step 2.2. determine the difference of values of the function 𝑓𝑓(𝒙𝒙�) on points 𝒙𝒙𝒏𝒏𝒏𝒏𝒏𝒏 and 𝒙𝒙𝒄𝒄𝒄𝒄𝒄𝒄. ∆𝑓𝑓 ≔ 𝑓𝑓(𝒙𝒙𝒏𝒏𝒏𝒏𝒏𝒏) − 𝑓𝑓(𝒙𝒙𝒄𝒄𝒄𝒄𝒄𝒄) step 2.3. if ∆𝑓𝑓 ≤ 0, accept the point 𝒙𝒙𝒏𝒏𝒏𝒏𝒏𝒏 as a current point of the search space, 𝒙𝒙𝒄𝒄𝒄𝒄𝒄𝒄 ≔ 𝒙𝒙𝒏𝒏𝒏𝒏𝒏𝒏. h. derdzyan 87 moreover, if 𝑓𝑓(𝒙𝒙𝒏𝒏𝒏𝒏𝒏𝒏) < 𝑓𝑓�𝒙𝒙𝒐𝒐𝒐𝒐𝒐𝒐�, accept the point 𝑥𝑥𝑛𝑛𝑛𝑛𝑛𝑛 as the optimum point of that moment. 𝒙𝒙𝒐𝒐𝒐𝒐𝒐𝒐 ≔ 𝒙𝒙𝒏𝒏𝒏𝒏𝒏𝒏. go to step 2.5. step 2.4. if ∆𝑓𝑓 > 0, decide whether to accept the new worse point as a current point of the search space or not depending on some probability. do steps 2.4.1-2.4.3. 2.4.1. generate the random number 𝑟𝑟 ∈ [0; 1]. 2.4.2. calculate 𝑝𝑝 = 𝑒𝑒− ∆𝑓𝑓 𝑘𝑘𝑘𝑘 probability, where 𝑘𝑘 = 1.380650524 ∗ 10−23. 2.4.3. if 𝑟𝑟 ≤ 𝑝𝑝, accept 𝑥𝑥𝑛𝑛𝑛𝑛𝑛𝑛 as a current point, 𝒙𝒙𝒄𝒄𝒄𝒄𝒄𝒄 ≔ 𝒙𝒙𝒏𝒏𝒏𝒏𝒏𝒏. step 2.5. 𝑇𝑇: = ℎ ∙ 𝑇𝑇 (decrease the current temperature of the system). c. final stage step 3.1. display the vector 𝒙𝒙𝒐𝒐𝒐𝒐𝒐𝒐 and the value 𝑓𝑓(𝒙𝒙𝒐𝒐𝒐𝒐𝒐𝒐) as the best value of the objective function. terminate. in this case there is a high probability that the point 𝒙𝒙𝒐𝒐𝒐𝒐𝒐𝒐 is quite close to the global minimum of the objective function. it is necessary to take higher initial temperature t, and lower 𝑇𝑇𝑚𝑚𝑚𝑚𝑛𝑛 in order to get exact results. this step significantly increases the number of calculations. to overcome this disadvantage, the simulated annealing method is combined with downhill simplex method in [3]. in this article the simulated annealing method is combined with the modification of downhill simplex method in order to increase the convergence of the proposed method. 3. the modification of downhill simplex method the downhill simplex method is a method of local optimization for finding minimal value of non-linear objective function 𝑓𝑓(𝒙𝒙�), where 𝒙𝒙� ∈ 𝑅𝑅𝑛𝑛. the method is initially proposed by nelder and mead [6]. simplex is a convex set with n+1 vertices in n-dimensional space. at first, the regular simplex s={𝒙𝒙�𝒊𝒊} is constructed in the search space of the objective function, 𝒙𝒙�𝒊𝒊 ∈ 𝑅𝑅𝑛𝑛, i = 1, n + 1��. then, the center 𝒙𝒙�𝒄𝒄 of simplex s is determined, by formula 𝒙𝒙�𝒄𝒄 = 1 𝑛𝑛 ∑ 𝒙𝒙�𝒊𝒊 𝑛𝑛 𝑚𝑚=1 . the center 𝒙𝒙�𝒄𝒄 is disposed equally from the first n vertices in the initial method. the 𝒙𝒙�𝒏𝒏+𝟏𝟏 vertex of simplex is replaced with the point which decreases the value of the objective function and belongs to the line connecting 𝒙𝒙�𝒄𝒄 and 𝒙𝒙�𝒏𝒏+𝟏𝟏. the suggested modification is based on the idea of weighting coefficients. in this case, the search process of the minimum value of the objective function occurs towards the direction in which the values of the given function decrease at high speed. in this case the weighted center 𝒙𝒙�𝒄𝒄′ of simplex is deteremined by formula 𝒙𝒙�𝒄𝒄′ = 𝜆𝜆1𝐱𝐱�𝟏𝟏 + 𝜆𝜆2𝐱𝐱�𝟐𝟐 + ⋯ + 𝜆𝜆𝑛𝑛𝐱𝐱�𝒏𝒏 , where ∑ 𝜆𝜆𝑚𝑚 = 1 𝑛𝑛 𝑚𝑚=1 , 𝜆𝜆𝑚𝑚 > 0, 𝑖𝑖 = 1, 𝑛𝑛��. it’s proposed to choose such values for coefficients 𝜆𝜆𝑚𝑚 which will make the weighted center 𝒙𝒙�𝒄𝒄′ tilt to the vertex of simplex, towards which the function values decrease at high speed. then, the vertex 𝒙𝒙�𝒏𝒏+𝟏𝟏 is replaced with the point which decreases the value of objective function and belongs to the line connecting the points 𝒙𝒙�𝒄𝒄′ and 𝒙𝒙�𝒏𝒏+𝟏𝟏. the detailed description of the suggested modification of the downhill simplex method is given in the publication [7]. 88 combined digital methods for solving optimal resource allocation problems modified downhill simplex algorithm (algorithm 2) a. initialization step 1.1. generate the random point 𝒙𝒙�𝟏𝟏 ∈ rn from the search space of the objective function 𝑓𝑓(𝒙𝒙�). step 1.2. around the point 𝒙𝒙�𝟏𝟏, construct the regular simplex s={𝒙𝒙�𝒊𝒊}, 𝑖𝑖 = 1, n + 1�� with the given rib h>0. choose the sufficiently small number 𝜀𝜀 > 0, for example 𝜀𝜀 = 10−8. b. main cycle while |𝑓𝑓(𝒙𝒙�𝒏𝒏+𝟏𝟏) − 𝑓𝑓(𝒙𝒙�𝟏𝟏)| > 𝜀𝜀 or 𝜌𝜌(𝒙𝒙�𝟏𝟏, 𝒙𝒙�𝒏𝒏+𝟏𝟏) = �∑ (𝑥𝑥𝑛𝑛+1,𝑗𝑗 − 𝑥𝑥1,𝑗𝑗)2𝑛𝑛𝑗𝑗=1 > 𝜀𝜀, the steps 2-7 are done, otherwise the step 8 is done. step 2. the vertices of the simplex are sorted to meet condition 𝑓𝑓(𝒙𝒙�𝟏𝟏) ≤ 𝑓𝑓(𝒙𝒙�𝟐𝟐) ≤ ⋯ ≤ 𝑓𝑓(𝒙𝒙�𝒏𝒏+𝟏𝟏). step 3. determine the coefficients 𝜇𝜇𝑚𝑚 = �𝑓𝑓(𝒙𝒙�𝒏𝒏+𝟏𝟏) − 𝑓𝑓(𝒙𝒙�𝑚𝑚)� 𝜌𝜌(𝒙𝒙�𝒏𝒏+𝟏𝟏, 𝒙𝒙�𝑚𝑚)⁄ , where 𝜌𝜌(𝒙𝒙�𝒏𝒏+𝟏𝟏, 𝒙𝒙�𝑚𝑚) = �∑ (x𝑛𝑛+1,𝑗𝑗 − x𝑚𝑚,𝑗𝑗)2nj=1 is the distance between vertices 𝒙𝒙�𝒏𝒏+𝟏𝟏 and 𝒙𝒙�𝒊𝒊 = �x𝑚𝑚,1, x𝑚𝑚,2, … , x𝑚𝑚,𝑛𝑛� ∈ rn, where 𝜇𝜇𝑚𝑚 shows the relative change of function 𝑓𝑓(𝒙𝒙�) on the vertices 𝒙𝒙�𝒏𝒏+𝟏𝟏 and 𝒙𝒙�𝑚𝑚, 𝑖𝑖 = 1, 𝑛𝑛��. step 4. determine new coefficients 𝜆𝜆𝑚𝑚 by normalizing weighted coefficients 𝜇𝜇𝑚𝑚, 𝑖𝑖 = 1, 𝑛𝑛��. choose 𝜆𝜆𝑚𝑚 = 𝜇𝜇𝑚𝑚 𝜇𝜇⁄ , 𝑖𝑖 = 1, 𝑛𝑛��, where 𝜇𝜇 = ∑ 𝜇𝜇𝑚𝑚𝑛𝑛𝑚𝑚=1 . coefficients 𝜆𝜆𝑚𝑚 meet the condition ∑ 𝜆𝜆𝑚𝑚 = 1 𝑛𝑛 𝑚𝑚=1 . step 5. determine the weighted center of simplex by formula 𝒙𝒙�𝒄𝒄′ = 𝜆𝜆1𝒙𝒙�𝟏𝟏 + 𝜆𝜆2𝒙𝒙�𝟐𝟐 + ⋯ + 𝜆𝜆𝑛𝑛𝒙𝒙�𝒏𝒏. step 6. a new point, which decreases the value of the objective function 𝑓𝑓(𝒙𝒙�), is being searched on the line 𝑙𝑙2 = {𝒙𝒙�𝒄𝒄′ + 𝛼𝛼 (𝒙𝒙�𝒄𝒄′ − 𝒙𝒙�𝒏𝒏+𝟏𝟏)}. step 6.1. for this purpose, on the line 𝑙𝑙2 the points 𝒙𝒙�𝒄𝒄, 𝒙𝒙�𝒏𝒏, 𝒙𝒙�𝒐𝒐𝒄𝒄, 𝒙𝒙�𝒊𝒊𝒄𝒄 are observed, corresponding to the values {1; 2; 0.5; -0.5} of 𝛼𝛼. step 6.2. for the mentioned points the following condition is checked by order: if on the new point the value of the function 𝑓𝑓(𝒙𝒙�) is lower, than on 𝒙𝒙�𝒏𝒏+𝟏𝟏 point, replace vertex 𝒙𝒙�𝒏𝒏+𝟏𝟏 of the simplex with that point, go back to the step 2. if there is no replacement move to the step 7. step 7. shrink the simplex to the vertex 𝒙𝒙�𝟏𝟏 by formula 𝒙𝒙�𝒊𝒊 = 𝒙𝒙�𝟏𝟏 + σ(𝒙𝒙�𝒊𝒊 − 𝒙𝒙�𝟏𝟏), i=2,…,n+1, σ = 0.5. c. final stage step 8. display the vector 𝑥𝑥1 and the optimal value 𝑓𝑓(𝑥𝑥1). end the algorithm. 4. the proposed combined algorithm (algorithm 3) in order to reach the neighborhood of the objective function 𝑓𝑓(�̅�𝑥), 𝑥𝑥 ∈ 𝑅𝑅𝑛𝑛, a certain number of steps of simulated annealing algorithm are done, by choosing 𝑇𝑇𝑚𝑚𝑚𝑚𝑛𝑛 = 10−2. in the result the vector 𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 ∈ 𝑅𝑅𝑛𝑛 is obtained, which is quite close to the global minimum of the objective function. then, a regular simplex is constructed around the point 𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 ∈ 𝑅𝑅𝑛𝑛 and the proposed modification of downhill simplex method is applied. a. description of the proposed combined algorithm the following steps are done to get the minimum value of the objective function 𝑓𝑓(�̅�𝑥), 𝑥𝑥 ∈ 𝑅𝑅𝑛𝑛. h. derdzyan 89 1. perform a certain number of steps of simulated annealing algorithm (algorithm 1), by choosing 𝑇𝑇𝑚𝑚𝑚𝑚𝑛𝑛 = 10−2. obtain 𝒙𝒙𝒐𝒐𝒐𝒐𝒐𝒐 ∈ 𝑅𝑅𝑛𝑛 vector in the result of this algorithm. 2. assign 𝒙𝒙1 ≔ 𝒙𝒙𝑜𝑜𝑜𝑜𝑜𝑜. by this step the vector 𝒙𝒙𝒐𝒐𝒐𝒐𝒐𝒐, obtained by the simulated annealing method, is chosen as the first 𝒙𝒙1 ∈ 𝑅𝑅𝑛𝑛 vertex of the constructed initial simplex. 3. apply the modified downhill simplex algorithm (algorithm 2), starting from the step 1.2. 4. in the end of the modified downhill simplex method, display the obtained vector 𝒙𝒙1 and the value 𝑓𝑓(𝒙𝒙1) as the minimal value of the objective function. this combined method has been tested on standard test functions of global optimization listed in [3]. a hundred independent tests have been done to solve each task and attempt the optimal value. every time the algorithm was started from different points of the search space of test function. the proposed combined method was compared with the combination of the simulated annealing method and nelder-mead’s downhill simplex method. table 1 shows the comparative evaluations between the two combined methods where the success rate shows the number of the successful cases of determining the global minimum of each standard function during the 100 independent tests. the test is considered successful and the minimal value of the test functions is achieved if condition �𝑓𝑓∗ − 𝑓𝑓� < 10−4 ∙ |𝑓𝑓∗| + 10−6 is satisfied [3]. here 𝑓𝑓∗ is the exact global minimum of the test function and 𝑓𝑓 is the optimal value obtained by the combined methods. table 1. combination of the simulated annealing method with the initial and modified downhill simplex methods. n test function n (number of variables) combination with the initial downhill simplex method combination with the modified downhill simplex method success rate success rate 1 rosenbrock's valley 4 80% 85% 2 freudenstein and roth 2 82% 87% 3 shubert 2 86% 89% 4 griewank 2 100% 100% 5 griewank 4 100% 100% 6 griewank 6 100% 100% 7 goldstein and price 2 100% 100% 8 bohachevsky 2 2 82% 85% 9 bohachevsky 3 2 88% 91% 10 schwefel 2 78% 82% 11 six-hump camel back 2 100% 100% table 1 shows that the combination of the simulated annealing method with the modified downhill simplex method has better results than the combination with nelder-mead initial method. during the hundred independent tests the proposed modified combined method in the algorithm 3 has obtained the global minimum of the test function more often than the initial combined method. for example, the number of the cases of obtaining the global minimum of rosenbrock's valley, freudenstein and roth test functions is increased by 5%, while for shubert, bohachevsky, schwefel functions it is increased by 3-4%. 90 combined digital methods for solving optimal resource allocation problems 5. the application of the proposed combined method for solving problems of equal allocation of resources in time the proposed combined method is applicable for the solution of the following problem of the optimal resource distribution. problem 1: n number of industrial, technological or other type of projects are accomplished. the given 𝑓𝑓𝑚𝑚(𝑡𝑡), 𝑡𝑡 ∈ [0, 𝑑𝑑𝑚𝑚], 𝑖𝑖 = 1, 𝑛𝑛�� functions show the number of the required resources for execution of the project i at time t . for each project the duration di is given. it is allowed to postpone the implementation of the given projects. ci is the earliest and vi is the latest start allowed for the project i. if the project i starts with the delay 𝑡𝑡𝑚𝑚 ∈ [𝑐𝑐𝑚𝑚, 𝑣𝑣𝑚𝑚] the function describing the allocation of resources at time t is 𝑓𝑓𝑚𝑚(𝑡𝑡 − 𝑡𝑡𝑚𝑚) where 𝑡𝑡 ∈ [𝑡𝑡𝑚𝑚, 𝑡𝑡𝑚𝑚 + 𝑑𝑑𝑚𝑚], 𝑖𝑖 = 1, 𝑛𝑛��. in this case, the sum function ∑ 𝑓𝑓𝑚𝑚(𝑡𝑡 − 𝑡𝑡𝑚𝑚) 𝑛𝑛 𝑚𝑚=1 will present the total number of resources required at time t. it is required to select such values of 𝑡𝑡1, 𝑡𝑡2, … , 𝑡𝑡𝑛𝑛 start points which allow the equal allocation of the total resources during the given time. a. the selection of criteria for equal allocation of resources we take the following criteria for the equal allocation of resources: 𝐼𝐼1(𝑡𝑡1, 𝑡𝑡2, … , 𝑡𝑡𝑛𝑛) and 𝐼𝐼2(𝑡𝑡1, 𝑡𝑡2, … , 𝑡𝑡𝑛𝑛), where 𝐼𝐼1(𝑡𝑡1, 𝑡𝑡2, … , 𝑡𝑡𝑛𝑛) = 𝑚𝑚𝑚𝑚𝑥𝑥𝑎𝑎≤𝑜𝑜≤𝑏𝑏 |∑ 𝑓𝑓𝑚𝑚(𝑡𝑡 − 𝑡𝑡𝑚𝑚) − 𝑀𝑀 𝑛𝑛 𝑚𝑚=1 | and 𝐼𝐼2(𝑡𝑡1, 𝑡𝑡2, … , 𝑡𝑡𝑛𝑛) = � 1 𝑏𝑏−𝑎𝑎 ∙ ∫ [∑ 𝑓𝑓𝑚𝑚(𝑡𝑡 − 𝑡𝑡𝑚𝑚) − 𝑀𝑀 𝑛𝑛𝑚𝑚=1 ]2𝑑𝑑𝑡𝑡 𝑏𝑏 𝑎𝑎 . these criteria represent the maximal deviation of sum function ∑ 𝑓𝑓𝑚𝑚(𝑡𝑡 − 𝑡𝑡𝑚𝑚) 𝑛𝑛 𝑚𝑚=1 from its mean value m and mean square deviation, correspondingly. the mean value m of required resources during the implementation period of projects is determined by 𝑀𝑀 = 1 𝑏𝑏−𝑎𝑎 ∙ ∫ ∑ 𝑓𝑓𝑚𝑚(𝑡𝑡 − 𝑡𝑡𝑚𝑚) 𝑛𝑛 𝑚𝑚=1 𝑑𝑑𝑡𝑡, 𝑚𝑚 = 𝑚𝑚𝑖𝑖𝑛𝑛𝑚𝑚 𝑡𝑡𝑚𝑚 = 0 , 𝑏𝑏 = 𝑚𝑚𝑚𝑚𝑥𝑥𝑚𝑚 𝑡𝑡𝑚𝑚 + 𝑑𝑑𝑚𝑚 , 𝑏𝑏 𝑎𝑎 where (b-a) is the implementation period of the projects. equal allocation of sum resources will be obtained in such values of parameters 𝑡𝑡1, 𝑡𝑡2, … , 𝑡𝑡𝑛𝑛 for which criteria 𝐼𝐼1(𝑡𝑡1, 𝑡𝑡2, … , 𝑡𝑡𝑛𝑛) or 𝐼𝐼2(𝑡𝑡1, 𝑡𝑡2, … , 𝑡𝑡𝑛𝑛) will accept their minimal value. so the given problem comes to the following problems (k=1, 2). 𝒎𝒎𝒊𝒊𝒏𝒏 𝑜𝑜1,𝑜𝑜2,…,𝑜𝑜𝑛𝑛 𝐼𝐼𝑘𝑘(𝑡𝑡1, 𝑡𝑡2, … , 𝑡𝑡𝑛𝑛), for 𝑐𝑐𝑚𝑚 ≤ 𝑡𝑡𝑚𝑚 ≤ 𝑣𝑣𝑚𝑚, 𝑖𝑖 = 1, 𝑛𝑛�� restrictions. the general formulation of the given problem is applicable for the following particular problems. problem 1: the system of n parallel operating aggregates is given. given functions 𝑓𝑓𝑚𝑚(𝑡𝑡), 𝑡𝑡 ∈ [0, 𝑑𝑑𝑚𝑚], 𝑖𝑖 = 1, 𝑛𝑛�� show the workload of aggregate i at time t. it is required to select such start points 𝑡𝑡𝑚𝑚 ∈ [𝑐𝑐𝑚𝑚, 𝑣𝑣𝑚𝑚] for operating the given aggregates which will make the total workload of the system more equal during the time. problem 2: n number of parallel implenting projects are given. given functions 𝑓𝑓𝑚𝑚(𝑡𝑡), 𝑡𝑡 ∈ [0, 𝑑𝑑𝑚𝑚], 𝑖𝑖 = 1, 𝑛𝑛�� show the labor force required for the work i at time t. it is required to select such start points 𝑡𝑡𝑚𝑚 ∈ [𝑐𝑐𝑚𝑚, 𝑣𝑣𝑚𝑚] for the implementation of the projects which will make the distribution of the total labor force more equal during the time. h. derdzyan 91 references [1] s. h. h. doulabi, a. seifi and s.y. shariat, “efficient hybrid genetic algorithm for resource leveling via activity splitting”, journal of construction engineering and management, vol.137, no. 2, pp. 137–146, 2011. [2] a. r. mushi, algorithms for the resource leveling problem, phd thesis, dublin city university, 1997. [3] a. r. hedar and m. fukushima, “hybrid simulated annealing and direct search method for nonlinear unconstrained global optimization”, optimization methods and software, vol. 17, no. 5, pp. 891-912, 2002. [4] a. r. hedar and m. fukushima, “simplex coding genetic algorithm for the global optimization of nonlinear functions”, multi-objective programming and goal programming, part 2, pp. 135-140, 2003. [5] s. kirkpatrick, c. d. gelatt and m. p. vecchi, “optimization by simulated annealing”, science, new series, vol. 220, no. 4598, pp. 671-680, 1983. [6] j. a. nelder and r. mead, “a simplex method for function minimization”, computer journal, vol. 7,pp. 308-313, 1965. [7] в. с. овсепян и а. с. дерцян, “модификация метода последовательного симплексного планирования и ее применение к решению задач оптимизации”, известия мгту “мами”, москва, т. 4, стр. 40-48, 2013 г. submitted 27.08.2015, accepted 25.11.2015 համակցված թվային մեթոդներ՝ ռեսուրսների օպտիմալ բաշխման խնդիրների լուծման համար հ. դերձյան ամփոփում հոդվածը նվիրված է ըստ ժամանակի ռեսուրսների օպտիմալ բաշխման խնդիրների լուծման արդյունավետ մեթոդների մշակմանը: նշված խնդիրների լուծման տարածված մեթոդներից են թրծման մոդելավորման, գենետիկական մեթոդները: այս մեթոդները սովորաբար մոտենում են խնդրի օպտիմալ լուծմանը, բայց ճշգրիտ լուծում գտնելու համար նրանցից պահանջվում է բավական երկար ժամանակ: սույն հոդվածի մեջ առաջակվում է թրծման մոդելավորման մեթոդը համակցել սիմպլեքս պլանավորման մեթոդի վերափոխման հետ՝ նշված խնդիրների լուծումը ճշգրտելու նպատակով: 92 combined digital methods for solving optimal resource allocation problems комбинированные численные методы для решения проблем оптимального распределения ресурсов а. дерцян аннотация статья посвящена разработке эффективных методов для решения проблем оптимального распределения ресурсов по времени. для решения этих проблем широко используются метод отжига, генетический метод и другие. эти методы приближаются к решению данных задач, но для нахождения точного решения требуется много времени. в этой статье предлагается скомбинировать метод отжига с модифицированным методом последовательного симплексного планирования, с целью нахождения более точного решения данных задач. d:\sbornik\...\said_9\tpel.dvi mathematical problems of computer science 35, 63{69, 2011. i r r educible compositions of p olynomials over finite fields of e ven char acter istic¤ s a e id m. me h r a b i a n d me ls ik k . k yu r e g ya n institue for informatics and automation problems of nas of ra e-mail smbatt@ipia.sci.am abstract this note presents some results with the constructive theory of synthesis of irreducible polynomials over a galois ¯eld with even characteristic. we prove a theorem that plays an important role for constructing irreducible polynomials. by this theorem a recurrent method for constructing families of irreducible polynomials of degree n2k (k = 1; 2; :::) over f2s is proposed. refer ences [1 ] e .r . b e r le ka m p , a lg e b r a ic co d in g th e o r y, mc gr a w-h ill, n e w y o r k, 1 9 6 8 . [2 ] i.f. b la ke , g.s e r o u s s i a n d n .p .s m a r t , e llip t ic c u r ve s in cr yp t o g r a p h y, ca m b r id g e u n ive r s it y p r e s s , ca m b r id g e , r e p r in t e d 2 0 0 0 . [3 ] b . ch o r , r . r ive s t , \ a kn a p s a c k-t yp e p u b lic ke y c r yp t o s ys t e m b a s e d o n a r it h m e t ic in ¯ n it e ¯ e ld s " , ie e e trans.inform.theory, vo l. 3 4 , p p . 9 0 1 -9 0 9 , 1 9 8 8 . [4 ] j. ca lm e t , \ a lg e b r a ic a lg o r it h m s in gf ( q) " ,d iscrete m ath, vo l. 5 6 p p . 1 0 1 -1 0 9 , 1 9 8 5 . [5 ] r . ch a p m a n , \ co m p le t e ly n o r m a l e le m e n t s in it e r a t e d qu a d r a t ic e xt e n s io n s of ¯ n it e ¯ e ld s " ,f inite f ields appl, vo l. 3 , p p . 3 -1 0 , 1 9 9 7 . [6 ] s . d . co h e n ,\ on ir r e d u c ib le p o lyn o m ia ls o f c e r t a in t yp e s in ¯ n it e ¯ e ld s " , p roc. cambridge p hilos. s o c , vo l. 6 6 , p p . 3 3 5 -3 4 4 , 1 9 6 9 . [7 ] s . d . co h e n ,\ th e e xp lic it c o n s t r u c t io n o f ir r e d u c ib le p o lyn o m ia ls o ve r ¯ n it e ¯ e ld s " , d e. codes cryptogr, vo l. 2 , p p . 1 6 9 -1 7 3 , 1 9 9 2 . [8 ] w . e b e r ly,\ v e r y fa s t p a r a lle l m a t r ix a n d p o lyn o m ia l a r it h m e t ic " , 25th annual symposium on f oundations of computer science, p p . 2 1 -3 0 , 1 9 8 4 . [9 ] s . ga o , n o r m a l b a s e s o ve r ¯ n it e ¯ e ld s , p h .d th e s is , w a t e r lo o , 1 9 9 3 . [1 0 ] m.k . k yu r e g ya n ,\ r e c u r r e n t me t h o d s fo r co n s t r u c t in g ir r e d u c ib le p o lyn o m ia ls o ve r " , f inite f ields apple, vo l. 8 , p p . 5 2 -6 8 , 2 0 0 2 . [1 1 ] m. k . k yu r e g ya n ,\ it e r a t e d c o n s t r u c t io n s o f ir r e d u c ib le p o lyn o m ia ls o ve r ¯ n it e ¯ e ld s wit h lin e a r ly in d e p e n d e n t r o o t s " ,f inite f ields apple, vo l. 1 0 , p p . 3 2 3 -4 3 1 , 2 0 0 4 . [1 2 ] m. k . k yu r e g ya n ,\ r e c u r r e n t m e t h o d s fo r c o n s t r u c t in g ir r e d u c ib le p o lyn o m ia ls o ve r fq o f o d d c h a r a c t e r is t ic s " , f inite f ields appl, vo l. 9 , p p . 3 9 -5 8 , 2 0 0 3 . ¤2000 mathematics subject classi¯cations: 47a68, 47a70 6 3 6 4 irreducible compositions of polynomials over finite fields of even characteristic [1 3 ] m. k . k yu r e g ya n , \ r e c u r r e n t m e t h o d s fo r c o n s t r u c t in g ir r e d u c ib le p o lyn o m ia ls o ve r fq o f o d d c h a r a c t e r is t ic s ii" , f inite f ields appl, vo l. 1 2 , p p . 3 5 7 -3 7 8 , 2 0 0 6 . [1 4 ] m.k . k yu r e g ya n , \ qu a d r a t ic t r a n s fo r m a t io n s a n d s yn t h e s is o f ir r e d u c ib le p o lyn o m ia ls o ve r ¯ n it e ¯ e ld s " , d okl, akad, nauk arm, ssr , vo l. 8 4 ( 2 ) , p p . 6 7 -7 1 , 1 9 8 7 . ( in r u s s ia n ) . [1 5 ] n .k o b lit z , a lg e b r a ic a s p e c t s o f cr yp t o g r a p h y, s p r in g e r , b e r lin , 1 9 9 8 . [1 6 ] r . l id l a n d h . n ie d e r r e it e r . fin it e fie ld s . ca m b r id g e u n ive r s it y p r e s s , 1 9 8 7 . [1 7 ] a . me n e z e s , i.f. b la ke , x . ga o, r . c. mu llin , s . a . v a n s t o n e , t.y a g h o o b ia n . a p p lic a t io n s o f fin it e fie ld s , k lu we r a c a d e m ic p u b lis h e r s , b o s t o n d o r d r e c h t l a n c a s t e r , 1 9 9 3 . [1 8 ] f.j.ma c w illia m s , n .j.a , s lo a n e . th e t h e o r y o f e r r o r -c o r r e c t in g c o d e s , p a r t , b e ll l a b o r a t o r ie s mu r r a y h ill, n j, u s a , n o r t h -h o lla n d p u b lis h in g co m p a n y, a m s t e r d a m , n e w y o r k, oxfo r d . [1 9 ] g. mc n a y, to p ic s in ¯ n it e ¯ e ld s , p h .d . th e s is , u n ive r s it y o f gla s g o w, 1 9 9 5 . [2 0 ] h . me yn ,\ e xp lic it n -p o lyn o m ia ls o f 2 -p o we r d e g r e e o ve r ¯ n it e ¯ e ld s " , d esigns codes cryptogr, vo l. 6 , p p . 1 0 7 -1 1 6 , 1 9 9 5 . [2 1 ] s . p e r lis , \ n o r m a l b a s e s o f c yc lic ¯ e ld s o f p r im e -p o we r d e g r e e " ,d uke m ath. j ., vo l. 9 , p p . 5 0 7 -5 1 7 , 1 9 4 2 . [2 2 ] j.v o n z u r ga t h e n , e . k a lt o fe n ,\ fa c t o r iz a t io n o f m u lt iva r ia t e p o lyn o m ia ls o ve r ¯ n it e ¯ e ld s " , m ath.comput, vo l.4 5 , p p . 2 5 1 -2 6 1 , 1 9 8 5 . [2 3 ] r . r . v a r s h a m o v,\ a g e n e r a l m e t h o d o f s yn t h e s iz in g ir r e d u c ib le p o lyn o m ia ls o ve r ga lo is ¯ e ld s " , soviet m ath. d okl, vo l. 2 9 , p p . 3 3 4 3 3 6 , 1 9 8 4 . ¼áõû· µýáõã³·ñçãáí ãµ»ñíáõ µ³½ù³ý¹³ùý»ñç µ³õ³¹ñáõãûáõýý»ñ í»ñç³íáñ ¹³ßï»ñç íñ³ ê. ø»ññ³µç ¨ ø. îûáõñ»õû³ý ²ù÷á÷áõù ²ûë ³ßë³ï³ýùá ý»ñï³û³óýáõù ¿ í»ñç³íáñ ¹³ßï»ñç íñ³ ½áõû· µýáõã³·ñçãáí ãµ»ñíáõ µ³½ù³ý¹³ùý»ñç ëçýï»½ù³ý ïáýëïñáõïïçí ï»ëáõãû³ý áñáß ³ñ¹ûáõýùý»ñ: ø»ýù ³å³óáõó»é »ýù ã»áñ»ù, áñá ù»í ¹»ñ ¿ ë³õáõù ãµ»ñíáõ µ³½ù³ý¹³ùý»ñ ï³éáõó»éáõó: ²ûë ã»áñ»ùç û·ýáõãû³ùµ ³é³ç³ñïíáõù ¿ n2 k ( k = 1 ; 2 :::) ³ëïç×³ýç ãµ»ñíáõ µ³½ù³ý¹³ù ï³éáõó»éáõ é»ïáõé»ýï ù»ãá¹: d:\sbornik\...\tpel.dvi mathematical problems of computer science 36, 70{78, 2012. n ew repr esentation for n on ruin p r obability of i nsur ance m odel with rent contr acts and i ts application for assessment of cr itical risks a r t a k r . ma r t ir o s ya n institute for informatics and automation problems of nas of ra e-mail: artakm81@inbox.ru abstract the paper considers the particular solution of one integro-di®erential equation of insurance risk theory. new representation for that solution is found, which is used for assessment of critical risks of the insurance companies that conduct purely rent operations. the critical risks are found in the case of a heavy tra±c and regular tail variation of the insurance premiums distribution function. keywords: insurance, rent, regular variation, heavy tra±c, critical risks, integrodi®erential equation. refer ences [1 ] l . ta ka c h , combinatorial methods in the random process theory ( mir , mo s c o w, 1 9 7 1 ) . [2 ] t. s a xe n , on the probability of ruin in the collective risk theory for insurance enterprises with only negative risk sums, ( s ka n d . a kt ., 3 1 , p p . 1 9 9 2 2 8 , 1 9 4 8 ) . [3 ] t. s a xe n , sur les mouvements aleatoires et le probleme de ruine de la theorie du risque collective, ( s o c . s c i. fe n n . co m m . p h ys . ma t h ., 1 6 , p p . 1 5 5 , 1 9 5 1 ) . [4 ] a . r . ma r t ir o s ya n , " cr it ic a l r is ks in m o d e ls o f c o lle c t ive in s u r a n c e " , actuary, n 3 , 2 0 0 9 ( in r u s s ia n , s e e www:actuaries:ru=magazine=?sect ioni d = 4 1 1 ) . [5 ] a . g. s h o lo m ic kiy, " r is k t h e o r y: c h o ic e a t t h e a n c e r t a in t y a n d r is k m o d e lin g " , m .:, 2 0 0 5 . [6 ] e . s e n e t a , r egularly varying functions ( n a u ka , mo s c o w, 1 9 8 5 ) . [7 ] a . m. s e d le t s kii, analytic fourier transforms and exponential approximations, s e r ie s : co n t e m p o r a r y ma t h e m a t ic s , fu n d a m e n t a l d ir e c t io n s 5 ( mo s c o w, 2 0 0 3 ) . [8 ] v . fe lle r , introduction to probability theory and its applications 2 ( mir , mo s c o w, 1 9 8 4 ) . [9 ] a . r . ma r t ir o s ya n , \ th e a s ym p t o t ic a n a lys is o f t h e c h a r a c t e r is t ic s o f t h e in s u r a n c e m o d e ls in c r it ic a l s it u a t io n s " , d is s e r t a t io n o f c a n d id a t e o f p h ys ic a l a n d m a t h e m a t ic a l s c ie n c e s , y e r e va n , 2 0 0 8 ( s e e e le c t r o n ic jo u r n a l d i®e r e n t ia l e qu a t io n s a n d c o n t r o l p r o c e s s e s , h t t p :/ / www.m a t h .s p b u .r u / d i®jo u r n a l/ r u / n u m b e r s / 2 0 0 8 .3 / is s u e .h t m l, o r h t t p :/ / www.p r o r e c t o r .o r g / a vt o r s / m a r t ir o s ya n / d is s .p d f ) . [1 0 ] e . a . d a n ie lia n , " l im it t h e o r e m s fo r wa it in g t im e in s in g le -c h a n n e l s is t e m s " , r eports of academy of sciences of armenia, vo l. l x x i, n 3 , p p . 1 2 9 -1 3 5 , 1 9 8 0 . 7 0 a. martirosyan 7 1 [1 1 ] v . g. s a a kya n , " r a n d o m c h o ic e d is c ip lin e in m o d e ls ¡! mrj ¡! grj 1 j1 " , m .: d issertation for physical and mathematical sciences, 1 9 8 5 . [1 2 ] e . a . d a n ie lia n , r . n . ch it c h ya n , " mu lt id im e n s io n a l lim it t h e o r e m s fo r t h e wa it in g t im e in p r io r it y qu e u e s o f ¡! mrj ¡! grj1 j1 t yp e " , acta cybernetica, vo l. 5 , fa s c 3 , p p . 3 2 5 -3 4 3 , s z e g e d , 1 9 8 1 . è»ýï³ý»ñç å³ûù³ý³·ñ»ñáí ³å³ñáí³·ñ³ï³ý ùá¹»éç ãëý³ýï³óù³ý ñ³í³ý³ï³ýáõãû³ý ýáñ ý»ñï³û³óáõù ¨ ýñ³ ïçñ³éáõãûáõýá ïñçïçï³ï³ý éçëï»ñç ·ý³ñ³ïù³ý ýå³ï³ïáí ². ø³ñïçñáëû³ý ²ù÷á÷áõù ðá¹í³íáõù ¹çï³ñïíáõù ¿ ³å³ñáí³·ñ³ï³ý éçëï»ñç ï»ëáõãû³ý ù»ï çýï»·ñá¹çý»ñ»ýóç³é ñ³í³ë³ñù³ý ù³ëý³ïç éáõíáõùá: ²û¹ éáõíù³ý ñ³ù³ñ ·ïýí³í ¿ ýáñ ý»ñï³û³óáõù, áñý û·ï³·áñíí»é ¿ ùç³ûý é»ýï³ý»ñç ·áñí³ñùý»ñáí ½µ³õíáõ ³å³ñáí³·ñ³ï³ý áýï»ñáõãû³ý ïñçïçï³ï³ý éçëï»ñç ·ý³ñ³ïù³ý ñ³ù³ñ: îñçïçï³ï³ý éçëï»ñá ·ïýí³í »ý í³ýñ³µ»éýí³íáõãû³ý ¨ ï³ýáý³í³ñ ÷á÷áëíáõ åáã»ñáí ³å³ñáí³·ñ³ï³ý í×³ñý»ñç µ³ßëù³ý ¹»åùáõù: íîâîå ïðåäñòàâëåíèå äëÿ âåðîÿòíîñòè íåðàçîðåíèÿ ñòðàõîâîé ìîäåëè ñ äîãîâîðàìè ñâÿçàííûìè ñ ðåíòàìè è åå ïðèìåíåíèåì ïðè îöåíêå êðèòè÷åñêèõ ðèñêîâ à. ìàðòèðîñÿí àííîòàöèÿ â ðàáîòå ðàññìîòðåíî ÷àñòíîå ðåøåíèå îäíîãî èíòåãðîäèôôåðåíöèàëíîãî óðàâíåíèÿ òåîðèè ñòðàõîâîãî ðèñêà. äëÿ ýòîãî ðåøåíèÿ íàéäåíî íîâîå ïðåäñòàâëåíèå, êîòîðîå ïðèíèìàåòñÿ ïðè îöåíêå êðèòè÷åñêèõ ðèñêîâ ñòðàõîâûõ êîìïàíèé, çàíèìàþøèõñÿ îïåðàöèÿìè, ñâÿçàííûìè ñ îáû÷íîé ðåíòîé. êðèòè÷åñêèå ðèñêè íàéäåíû â ñëó÷àÿõ êðèòè÷åñêîé çàãðóçêè è ïðè ïðàâèëüíîì èçìåíåíèè õâîñòîâ ðàñïðåäåëåíèé ñòðàõîâûõ âûïëàò. d:\final\...\4_armen_33--41.dvi mathematical problems of computer science 48, 33{41, 2017. a n ew m ethod of solving diophantine e quation a3 + b3 + c 3 = d a r m e n a . a va g ya n armenian state pedagogical university e-mail: avagyana73@gmail.com abstract the article is dedicated to the famous diophantine equation of the form a3+b3+c3 = d. we solve this problem in some particular cases. using the package mathematica 11 we ¯nd an e±cient algorithm to solve this problem. this algorithm is simpler and uses a signi¯cantly smaller number of operations than the other known algorithms for solving these equations. keywords: diophantine equations, the sum of three cubes, parametric solutions, polynomial identities. 1 . in t r o d u c t io n fe r m a t 's e qu a t io n fo r o d d e xp o n e n t s n a s ks fo r t h r e e in t e g e r s , e a c h wit h a n a b s o lu t e va lu e g r e a t e r t h a n 0 , s u c h t h a t t h e s u m o f t h e ir n¡ t h p o we r s is z e r o . a r e la t e d p r o b le m is t o ¯ n d t h r e e in t e g e r s , e a c h wit h a n a b s o lu t e va lu e g r e a t e r t h a n t h e n¡ t h r o o t o f k, s u c h t h a t t h e s u m o f t h e ir n¡ t h p o we r s e qu a ls k. fo r e xa m p le , d e t e r m in e t h e in t e g e r s a; b; c; wit h jaj ; jbj ; jcj > 1 s u c h t h a t a3 + b3 + c3 = d: ( 1 ) th is h a s in ¯ n it e ly m a n y s o lu t io n s b e c a u s e o f t h e id e n t it y ³ 1 ¡ 9 m3 ´3 + ³ 9 m4 ´3 + ³ ¡ 9 m4 + 3 m ´3 = 1 : ( 2 ) h o we ve r , t h e r e a r e o t h e r s o lu t io n s a s we ll. a r e t h e r e a n y o t h e r id e n t it ie s t h a t g ive a d i®e r e n t 1 -p a r a m e t e r fa m ily o f s o lu t io n s ? is e ve r y s o lu t io n o f ( 1 ) a m e m b e r o f a fa m ily like t h is ? in g e n e r a l, it is kn o wn t h a t t h e r e is n o ¯ n it e m e t h o d fo r d e t e r m in in g wh e t h e r a g ive n d io p h a n t in e e qu a t io n h a s s o lu t io n s . h o we ve r , it is a n o p e n p r o b le m , wh e t h e r t h e r e is a g e n e r a l m e t h o d fo r d e t e r m in in g if a g ive n d io p h a n t in e e qu a t io n h a s " a lg e b r a ic " s o lu t io n s , i.e ., a n a lg e b r a ic id e n t it y like t h e o n e a b o ve t h a t g ive s a n in ¯ n it e fa m ily o f s o lu t io n s . mo r e s p e c ī c a lly, is t h e r e a p r o p o s it io n , t h a t o n ly e qu a t io n s o f genus < 2 c a n h a ve a n a lg e b r a ic s o lu t io n ? it m a y b e wo r t h m e n t io n in g t h a t t h e c o m p le t e r a t io n a l-s o lu t io n o f t h e e qu a t io n a3 + b3 + c3 = t3 is kn o wn , a n d is g ive n b y a = q h 1 ¡ ( x ¡ 3 y ) ³ x2 + 3 y2 ´i ; 3 3 3 4 a new method of solving diophantine equation a3 + b3 + c3 = d b = ¡q h 1 ¡ ( x + 3 y ) ³ x2 + 3 y2 ´i ; c = q ·³ x2 + 3 y2 ´2 ¡ ( x + 3 y ) ¸ ; t = q ·³ x2 + 3 y2 ´2 ¡ ( x ¡ 3 y ) ¸ ; wh e r e q; x; y a r e a n y r a t io n a l n u m b e r s . th u s , if we s e t q e qu a l t o t h e in ve r s e o fh ( x2 + 3 y2 ) 2 ¡ ( x ¡ 3 y ) i we h a ve r a t io n a l s o lu t io n s o f ( 1 ) . h o we ve r , i t h in k t h e p r o b le m o f ¯ n d in g t h e in t e g e r -s o lu t io n s is m o r e d i± c u lt . if t is a llo we d t o b e a n y in t e g e r ( n o t ju s t 1 ) t h e n r a m a n u ja n g a ve t h e in t e g e r s o lu t io n s a s a = 3 n2 ¡ 5 n m ¡ 5 m2 b = 4 n2 ¡ 4 n m + 6 m2 c = 5 n2 ¡ 5 n m ¡ 3 m2 t = 6 n2 ¡ 4 n m ¡ 4 m2 th is o c c a s io n a lly g ive s a s o lu t io n o f e qu a t io n ( 1 ) ( wit h a p p r o p r ia t e c h a n g e s in s ig n ) , a s in t h e fo llo win g c a s e s n m a b c | | { | | | | { | | { 1 -1 1 2 -2 1 -2 9 1 0 -1 2 5 -1 2 -1 3 5 -1 3 8 1 7 2 1 9 -8 -7 9 1 -8 1 2 1 0 1 0 4 6 -1 0 9 1 1 1 6 1 1 1 4 6 8 -1 4 2 5 8 7 3 -1 7 3 6 5 6 0 1 6 7 4 0 2 -8 3 8 0 2 4 1 9 -9 9 3 -9 5 1 6 9 0 -9 2 6 2 7 1 1 1 8 3 2 5 8 h o we ve r , t h is d o e s n o t c o ve r a ll t h e s o lu t io n s g ive n b y ( 2 ) . b y t h e wa y, t h e e qu a t io n a3 +b3 +c3 = 1 h a s a lg e b r a ic s o lu t io n s [1 ], [2 ] o t h e r t h a n ( 2 ) . th e r e a r e kn o wn t o b e in ¯ n it e ly m a n y a lg e b r a ic s o lu t io n s , fo r in s t a n c e : ( 1 ¡ 9 t3 + 6 4 8 t6 + 3 8 8 8 t9 ) 3 + ( ¡1 3 5 t4 + 3 8 8 8 t10 ) 3 + ( 3 t ¡ 8 1 t4 ¡ 1 2 9 6 t7 ¡ 3 8 8 8 t10 ) 3 = 1 h o we ve r , it is n o t kn o wn wh e t h e r e ve r y s o lu t io n o f t h e e qu a t io n lie s in s o m e fa m ily o f s o lu t io n s wit h a n a lg e b r a ic p a r a m e t e r iz a t io n . in t e r e s t in g ly, n o t e , t h a t if yo u r e p la c e 1 b y 2 , t h e n a g a in t h e r e is a p a r a m e t r ic s o lu t io n : ³ 6 t3 + 1 ´3 ¡ ³ 6 t3 ¡ 1 ´3 + ³ 6 t2 ´3 = 2 ( 3 ) a n d a g a in t h is d o e s n o t c o ve r a ll t h e kn o wn in t e g e r s o lu t io n s . n o t e , t h a t p r e c is e ly o n e s o lu t io n is kn o wn t h a t is n o t g ive n b y ( 3 ) ( s e e [1 ]) : 1 2 1 4 9 2 8 3 + 3 4 8 0 2 0 5 3 ¡ 3 5 2 8 8 7 5 3 = 2 . it is e vid e n t ly n o t kn o wn u n t il t o d a y, if t h e r e a r e a n y o t h e r a lg e b r a ic s o lu t io n s b e s id e s t h e o n e n o t e d a b o ve . fo r d > 2 k e n ji k o ya m a [3 ] h a s g e n e r a t e d a la r g e t a b le o f in t e g e r s o lu t io n s o f a3 + b3 + c3 = d fo r n o n c u b e s d in t h e r a n g e 1 · d · 1 0 0 0 a n d jaj · jbj · jcj · 2 21 ¡ 1 a. avagyan 3 5 c o n s is t s o f t wo t a b le s : ta b le 1 ( 5 5 p a g e s ) c o n t a in s t h e in t e g e r s o lu t io n s , s o r t e d b y d, a n d ta b le 2 ( 2 p a g e s ) lis t s t h e n u m b e r o f p r im it ive s o lu t io n s fo u n d fo r e a c h d in t h e s e a r c h r a n g e . in g e n e r a l, it s e e m s t o b e a d i± c u lt p r o b le m t o c h a r a c t e r iz e a ll t h e s o lu t io n s o f a3 + b3 + c3 = d fo r s o m e a r b it r a r y in t e g e r d > 2 . in p a r t ic u la r , t h e qu e s t io n o f wh e t h e r a ll in t e g e r s o lu t io n s a r e g ive n b y a n a lg e b r a ic id e n t it y s e e m s b o t h d i± c u lt a n d in t e r e s t in g . n e ve r t h e le s s , fo r in s t a n c e , in t h e c a s e o f d = 3 , t h e r e is s t ill n o s o lu t io n kn o wn a p a r t fr o m t h e o b vio u s o n e s : ( 1 , 1 , 1 ) , ( 4 , 4 ,5 ) , ( 4 ,5 , 4 ) , a n d ( 5 , 4 , 4 ) . fo r d = 3 0 , t h e ¯ r s t s o lu t io n wa s fo u n d b y n . e lkie s a n d h is c o wo r ke r s in 2 0 0 0 [5 ]. it is in t e r e s t in g , t h a t in 1 9 9 2 , d . r . h e a t h b r o wn [6 ] m a d e a p r e d ic t io n o n t h e d e n s it y o f t h e s o lu t io n s fo r d = 3 0 wit h o u t kn o win g a n y s o lu t io n e xp lic it ly. ove r t h e ye a r s , a n u m b e r o f a lg o r it h m s h a ve b e e n d e ve lo p e d in o r d e r t o a t t a c k t h e g e n e r a l p r o b le m . co n c e r n in g t h e va r io u s a p p r o a c h e s , a n e xc e lle n t o ve r vie w, in ve n t e d b e fo r e 2 0 0 0 , wa s g ive n in [7 ], wh ic h wa s p u b lis h e d in 2 0 0 7 . h is t o r ic a lly, t h e ¯ r s t a lg o r it h m , wh ic h h a s a c o m p le xit y o f o ( b1+² ) fo r a s e a r c h b o u n d o f b is t h e m e t h o d b y r . h e a t h -b r o wn [8 ]. r e t u r n t h e in t e r e s t e d r e p r e s e n t a t io n s d = a3 + b3 + c3 ( 4 ) o f va r io u s in t e g e r s d a s s u m s o f t h r e e c u b e s . th e c u b ic r e s id u e s wit h r e s p e c t t o m o d u le 9 a r e : 0 , 1 , 8 , t h u s , it fo llo ws b y in s p e c t io n o f c a s e s t h a t fo r e ve r y in t e g e r s o lu t io n t o ( 4 ) we o b t a in d 6= 6 § 4 ( m o d 9 ) . a n y g ive n s o lu t io n c a n b e wr it t e n in o n e o f t h e fo llo win g fo r m s fo r n o n -n e g a t ive a; b; c : jdj = a3 + b3 + c3 o r jdj = a3 + b3 ¡ c3 o r jdj = a3 ¡ b3 ¡ c3 th e r e fo r e , it s u ± c e s t o c o n s id e r n o n -n e g a t ive s o lu t io n s t o t h e e qu a t io n s a3 + b3 = c3 § d a n d a3 + b3 + c3 = d ( fo r d = 0 it is a c a s e o f fe r m a t s t h e o r e m t h a t t h e r e a r e n o in t e g e r s o lu t io n s ) . in p r a c t ic e , we n e e d t o s e a r c h fo r p r im it ive s o lu t io n s , i.e ., gcd ( a; b; c) is n o t d ivis ib le b y d, s in c e t h e n o n -p r im it ive s o lu t io n s fo r ¯ xe d n a r e r o u t in e ly o b t a in e d fr o m t h e p r im it ive s o lu t io n s fo r t h e ir d ivis o r s . co n s id e r in g d = m3; d = m12 a n d d = 2 m9 t yp e va lu e s o f d, a n d m u lt ip lyin g b o t h s id e s o f ( 3 ) b y m9, a ft e r a p p lyin g t h e c h a n g e o f va r ia b le tt=m, o n e c a n o b t a in t h e m o r e g e n e r a l s o lu t io n ³ 6 t3 + m3 ´3 ¡ ³ 6 t3 ¡ m3 ´3 ¡ ³ 6 t2 ´3 = 2 m9 ( 5 ) wh ic h is p r im it ive fo r gcd ( 6 t; m ) = 1 . if gcd ( 6 t; m) > 1 , t h e n d ivid in g ( 5 ) b y ( gcd ( 6 t3; m3 ) ) 3 g ive s a p r im it ive s o lu t io n . fo r l; k ¸ 1 t h e s o lu t io n s a r e : ( 3 t3 + 2 3l¡1m3 ) 3 ¡ ( 3 t3 ¡ 2 3l¡1m3 ) 3 ¡ ( 2 l 3 mt2 ) 3 = 2 9l¡2m9 ( 6 ) ( 2 t3 + 3 3k¡1m3 ) 3 ¡ ( 2 t3 ¡ 3 3k¡1m3 ) 3 ¡ ( 2 3 kmt2 ) 3 = 2 3 9k¡3m9 ( 7 ) ( t3 + 2 3l¡1 3 3k¡1m3 ) 3 ¡ ( t3 ¡ 2 3l¡1 3 3k¡1m3 ) 3 ¡ ( 2 l 3 kmt2 ) 3 = 2 9l¡2 3 9k¡3m9 ( 8 ) n o t e , t h a t t h e y a r e p r im it ive fo r gcd ( 3 t; 2 m ) = 1 , gcd ( 2 t; 3 m ) = 1 a n d gcd ( t; 6 m) = 1 , r e s p e c t ive ly. th e la s t e qu a t io n s g ive p o lyn o m ia l fa m ilie s fo r n = 2 , 1 2 8 , 1 4 5 8 , 6 5 5 3 6 , 9 3 3 1 2 , 3 9 0 6 2 5 0 , 2 8 6 9 7 8 1 4 , e t c . 3 6 a new method of solving diophantine equation a3 + b3 + c3 = d a n a n a lo g o u s p r o c e d u r e m a y b e a p p lie d fo r 3 t o o b t a in fa m ilie s o f s o lu t io n s fo r n u m b e r s o f t h e fo r m m12. mu lt ip lyin g b o t h s id e s b y m12 a n d a p p lyin g t h e t r a n s fo r m a t io n t=m, we will g e t ³ 9 mt3 + m4 ´3 ¡ ³ 9 t4 + 3 mt ´3 + ³ 9 t4 ´3 = m12 ( 9 ) wh ic h is p r im it ive fo r gcd ( t; 3 m ) = 1 . in p a r t ic u la r , fo r 3 ¡ m a n d k 1 , ³ 3 kmt3 + 3 4k¡2m4 ´3 ¡ ³ t4 + 3 3k¡1lm3t ´3 + ³ t4 ´3 = 3 12k¡6m12 ( 1 0 ) is p r im it ive fo r gcd ( t; 3 m ) = 1 . e qu a t io n s ( 9 ) a n d ( 1 0 ) g ive fa m ilie s o f s o lu t io n s fo r n = 1 , 7 2 9 , 4 0 9 6 , 2 9 8 5 9 8 4 , 1 6 7 7 7 2 1 6 , 2 4 4 1 4 0 6 2 5 , 3 8 7 4 2 0 4 8 9 ,e t c . 2 . n e w me t h o d a n d r e s u lt s co n s id e r in g t h e m o r e g e n e r a liz e d p r o b le m o f t h e s u m o f t h r e e c u b e s , we a r e s e e kin g p1 ( y ) ; p2 ( y ) ; p3 ( y ) wit h t h e h ig h e s t p o s s ib le d e g r e e p o lyn o m ia ls a n d q ( y ) wit h t h e lo we s t p o s s ib le d e g r e e p o lyn o m ia l, s u c h a s p 31 ( y ) + p 3 2 ( y ) + p 3 3 ( y ) = q ( y ) a c t u a lly, t h e s o lu t io n o f t h is p r o b le m h a s a c lo s e r e la t io n wit h t h e a b o ve t r ivia l p r o b le m , s in c e t h e c a s e o f degq ( y ) = 0 c o in c id e s wit h o u r p r o b le m . n e ve r t h e le s s , t h e e s t im a t io n o f p o s s ib ilit y o f m in im iz a t io n o f degq( y ) it s e lf is a ls o a n in t e r e s t in g p r o b le m . result 1: th e ¯ r s t r e s u lt o f t h is p a p e r is d e vo t e d t o t h e c a s e o f d e g r e e s ( 8 , 8 , 6 ) . w e s e a r c h t h e d e s ir e d p o lyn o m ia ls wit h in t h e c la s s o f p o lyn o m ia ls o f t h e fo r m ³ ax8 + bx5 + cx2 ´3 ¡ ³ ax8 + b1x 5 + c1x 2 ´3 ¡ ³ ax6 + bx3 + c ´3 ( 1 1 ) fir s t o f a ll, we e xp a n d it ¡c3 ¡ 3 bc2x3 + ³ c3 ¡ 3 b2c ¡ 3 ac2 ¡ c31 ´ x6 + ³ ¡b3 + 3 bc2 ¡ 6 abc ¡ 3 b1c21 ´ x9 + ³ ¡ 3 ab2 + 3 b2c + 3 ac2 ¡ 3 a2c ¡ 3 b21c1 ¡ 3 ac21 ´ x12+ ³ b3 ¡ 3 a2b ¡ b31 + 6 abc ¡ 6 ab1c1 ´ x15 + ³ ¡a3 + 3 ab2 ¡ 3 ab21 + 3 a2c ¡ 3 a2c1 ´ x18 + ³ 3 a2b ¡ 3 a2b1 ´ x21 fu r t h e r ,we t a ke b1 = b, c1 = ¡a3+3a2c 3a2 , b = 2ab 3a , c = ¡a 4¡aab2+6a2ac 9a3 a n d g e t t h e fo llo win g fo r m : a12 7 2 9 a9 + a9b2 2 4 3 a8 + a6b4 2 4 3 a7 + a3b6 7 2 9 a6 ¡ 2 a 9c 8 1 a7 ¡ 4 a 6b2c 8 1 a6 ¡ 2 a 3b4c 8 1 a5 + 4 a6c2 2 7 a5 + 4 a3b2c2 2 7 a4 ¡ 8 a 3c3 2 7 a3 + + ã ¡ 2 a 9b 8 1 a7 ¡ 4 a 6b3 8 1 a6 ¡ 2 a 3b5 8 1 a5 + 8 a6bc 2 7 a5 + 8 a3b3c 2 7 a4 ¡ 8 a 3bc2 9 a3 ! x3+ + ã 2 a6b2 2 7 a5 + a3b4 9 a4 + a6c 9 a4 ¡ 4 a 3b2c 9 a3 ¡ a 3c2 3 a2 ! x6 + ã a6b 9 a4 + 4 a3b3 2 7 a3 ¡ 2 a 3bc 3 a2 ! x9 th u s , fu r t h e r c o n s id e r a t io n s a r e d e vo t e d t o ¯ n d in g t h e c a s e s , wh ic h a r e in t e r e s t in g fo r u s : a. avagyan 3 7 case 1: b = 0 . th e r e s u lt h a s t h e fo r m a12 7 2 9 a9 ¡ 2 a 9c 8 1 a7 + 4 a6c2 2 7 a5 ¡ 8 a 3c3 2 7 a3 + ã a6c 9 a4 ¡ a 3c2 3 a2 ! x6 subcase 1.1: c = 0 . th e r e s u lt g e t s t h e fo r m a 12 729a9 , wh ic h is a c u b e o f a n in t e g e r , t h u s , it is p r im it ive a n d n o t in t e r e s t in g . subcase 1.2: c = a 3 3a2 , t h e r e s u lt is ¡ a12 729a9 , a g a in p r im it ive . case 2: c = 3a 3+4ab2 18a2 . th e r e s u lt : ¡ a 3b6 1 9 6 8 3 a6 ¡ 2 a 3b5x3 7 2 9 a5 + ã a9 1 0 8 a6 ¡ a 3b4 2 4 3 a4 ! x6 fa c t o r iz in g t h e c o e ± c ie n t o f t h e la s t t e r m , o n e c a n o b t a in : ¡a 3 ( ¡ 3 a3 + 2 ab2 ) ( 3 a3 + 2 ab2 ) 9 7 2 a6 s u b s t it u t in g a = 3a 3 2b2 . th e n t h e r e s u lt will g e t t h e fo r m : ¡ 6 4 b 18 1 4 3 4 8 9 0 7 a6 ¡ 6 4 b 15x3 1 7 7 1 4 7 a3 fin a lly, t a kin g x = 2yb a we o b t a in t h e r e s u lt , wh ic h is in t e r e s t in g fo r u s : ¡1 ¡ 6 4 8 y3. p r actical consider ations: n o w we in ve s t ig a t e t h is r e s u lt fo r a p p lic a t io n s t o s o lve t h e p r o c e s s o f t h e e qu a t io n ( 4 ) . s in c e fo r max[abs[a; b; c]] · 1 0 14 t h e r e a r e we ll-kn o wn t a b le s in [4 ], t h u s , we s e e k s o lu t io n s o f ( 4 ) s a t is fyin g t h e c o n d it io n max[abs[a; b; c]] ¸ 1 0 15 wit h p o s s ib le s m a ll va lu e s o f abs[d] ( d e s ir a b ly le s s t h a n 1 0 0 0 ) . th u s , t h e r e s u lt is : ³ 5 4 y2 ³ 1 + 3 6 y3 + 4 3 2 y6 ´´3 ¡ ³ 1 8 y2 ³ 1 + 1 0 8 y3 + 1 2 9 6 y6 ´´3 ¡ ³ 1 + 2 1 6 y3 + 3 8 8 8 y6 ´3 = ¡ 1 ¡ 6 4 8 y3 s u r e , c a lc u la t io n s we r e e xp e c t e d t o b e s ig n i¯ c a n t ly h a r d , fo r wh ic h we will u s e ma t h e m a t ic a 1 1 .0 c o d e : g8[y ] := 1 . gcd h³ 5 4 y2 ³ 1 + 3 6 y3 + 4 3 2 y6 ´´ ; ³ 1 8 y2 ³ 1 + 1 0 8 y3 + 1 2 9 6 y6 ´´ ; ³ 1 + 2 1 6 y3 + 3 8 8 8 y6 ´i f8[y ]:=g8[y] n³ 5 4 y2 ³ 1 + 3 6 y3 + 4 3 2 y6 ´´ ; ³ 1 8 y2 ³ 1 + 1 0 8 y3 + 1 2 9 6 y6 ´´ ; ³ 1 + 2 1 6 y3 + 3 8 8 8 y6 ´o v8[y ] := g8[y] 3 ³ ¡1 ¡ 6 4 8 y3 ´ f or[i = ¡ 5 0 ; i · 5 0 ; i + + if[abs[v8[i]] < 1 0 0 0 0 0 0 ; if[max[abs[f8[i]]] > 1 0 0 0 0 0 0 0 0 0 0 0 0 ; p rint[fi; f8[i]; v8[i]g]]]] th e r e s u lt is : f¡ 1 1 ; f 5 0 0 0 2 5 0 8 9 9 3 5 8 ; 5 0 0 0 2 5 0 8 9 5 0 0 2 ; 6 8 8 7 5 4 1 6 7 3 g; 8 6 2 4 8 7 g 3 8 a new method of solving diophantine equation a3 + b3 + c3 = d f¡ 1 0 ; f 2 3 3 2 6 0 5 6 0 5 4 0 0 ; 2 3 3 2 6 0 5 6 0 1 8 0 0 ; 3 8 8 7 7 8 4 0 0 1 g; 6 4 7 9 9 9 g f¡ 9 ; f 1 0 0 4 0 7 9 1 2 0 6 0 6 ; 1 0 0 4 0 7 9 1 1 7 6 9 0 ; 2 0 6 6 0 8 5 1 4 5 g; 4 7 2 3 9 1 g f 9 ; f 1 0 0 4 3 0 8 7 0 3 1 1 8 ; 1 0 0 4 3 0 8 7 0 0 2 0 2 ; 2 0 6 6 4 0 0 0 7 3 g; ¡4 7 2 3 9 3 g f 1 0 ; f 2 3 3 2 9 9 4 4 0 5 4 0 0 ; 2 3 3 2 9 9 4 4 0 1 8 0 0 ; 3 8 8 8 2 1 6 0 0 1 g; ¡6 4 8 0 0 1 g f 1 1 ; f 5 0 0 0 8 7 7 0 6 5 6 4 6 ; 5 0 0 0 8 7 7 0 6 1 2 9 0 ; 6 8 8 8 1 1 6 6 6 5 g; ¡8 6 2 4 8 9 g th is m e a n s t h a t , fo r in s t a n c e : 1 0 0 4 0 7 9 1 2 0 6 0 6 3 ¡ 1 0 0 4 0 7 9 1 1 7 6 9 0 3 ¡ 2 0 6 6 0 8 5 1 4 5 3 = 4 7 2 3 9 1 : result 2: mo r e e n h a n c e d r e s u lt is o b t a in e d fo r t h e c a s e ( 9 , 9 , 7 ) : ( 3 + 3 6 0 y3 + 1 0 3 6 8 y6 + 9 3 3 1 2 y9 ) 3 ¡ ( ¡ 1 + 2 1 6 y3 + 1 0 3 6 8 y6 + 9 3 3 1 2 y9 ) 3 ¡( 4 y ( 5 + 3 2 4 y3 + 3 8 8 8 y6 ) ) 3 = 2 8 + 1 0 7 2 y3 g9[y¡] := 1 =gcd[( 3 + 3 6 0 y 3 + 1 0 3 6 8 y6 + 9 3 3 1 2 y9 ) ; ( ¡ 1 + 2 1 6 y3 + 1 0 3 6 8 y6 + 9 3 3 1 2 y9 ) ; ( 4 y ( 5 + 3 2 4 y3 + 3 8 8 8 y6 ) ) ] f9[y¡] := g9[y] ¤ f ( 3 ( 1 + 1 2 0 y3 + 3 4 5 6 y6 + 3 1 1 0 4 y9 ) ) ; ( ( ¡ 1 + 2 1 6 y3 + 1 0 3 6 8 y6 + 9 3 3 1 2 y9 ) ) ; ( 4 y ( 5 + 3 2 4 y3 + 3 8 8 8 y6 ) ) g v9[y¡] := g9[y] 3 ¤ ( 2 8 + 1 0 7 2 y3 ) f or[i = ¡ 5 0 ; i · 5 0 ; i + +; if[abs[v9[i]] < 1 0 0 0 0 0 0 0 ; if[max[abs[f9[i]]] > 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ; p rint[fi; f9[i]; v9[i]g]]] th e r e s u lt is : f¡ 2 1 ; f¡ 7 4 1 1 4 9 7 0 4 8 6 7 5 7 7 0 1 ; ¡ 7 4 1 1 4 9 7 0 4 8 5 4 2 4 1 2 1 ; ¡ 2 8 0 1 0 2 7 6 9 4 2 6 7 6 g; ¡ 9 9 2 7 7 6 4 g f¡ 2 0 ; f¡ 4 7 7 7 5 0 8 0 4 5 0 8 7 9 9 9 7 ; ¡ 4 7 7 7 5 0 8 0 4 4 9 7 2 8 0 0 1 ; ¡ 1 9 9 0 6 3 5 2 6 4 0 4 0 0 g; ¡ 8 5 7 5 9 7 2 g f¡ 1 9 ; f¡ 3 0 1 1 0 1 4 6 6 8 5 9 2 9 0 7 7 ; ¡ 3 0 1 1 0 1 4 6 6 8 4 9 4 1 3 8 5 ; ¡ 1 3 9 0 1 3 2 4 3 8 9 2 9 2 g; ¡ 7 3 5 2 8 2 0 g f¡ 1 8 ; f¡ 1 8 5 0 8 9 4 9 4 6 6 1 7 9 9 0 1 ; ¡ 1 8 5 0 8 9 4 9 4 6 5 3 4 0 0 9 7 ; ¡9 5 2 1 1 0 9 8 8 9 1 2 8 g; ¡ 6 2 5 1 8 7 6 g f¡ 1 7 ; f¡ 1 1 0 6 5 4 2 1 6 7 5 1 4 1 3 4 9 ; ¡ 1 1 0 6 5 4 2 1 6 7 4 4 3 3 8 8 1 ; ¡6 3 8 1 4 7 8 7 9 9 6 2 0 g; ¡ 5 2 6 6 7 0 8 g f¡ 1 6 ; f¡6 4 1 2 1 7 7 8 6 8 4 8 8 7 0 1 ; ¡ 6 4 1 2 1 7 7 8 6 7 8 9 8 8 8 1 ; ¡ 4 1 7 4 6 2 3 2 7 7 3 7 6 g; ¡ 4 3 9 0 8 8 4 g f¡ 1 5 ; f¡3 5 8 7 1 0 8 6 5 3 2 1 4 9 9 7 ; ¡ 3 5 8 7 1 0 8 6 5 2 7 2 9 0 0 1 ; ¡ 2 6 5 7 1 3 9 3 9 0 3 0 0 g; ¡ 3 6 1 7 9 7 2 g f¡ 1 4 ; f¡1 9 2 7 8 4 5 5 3 2 2 6 7 1 9 7 ; ¡ 1 9 2 7 8 4 5 5 3 1 8 7 2 0 6 5 ; ¡ 1 6 3 9 3 4 1 0 2 7 3 5 2 g; ¡ 2 9 4 1 5 4 0 g f 1 4 ; f 1 9 2 8 0 0 1 6 6 4 7 2 5 6 9 9 ; 1 9 2 8 0 0 1 6 6 4 3 3 0 5 5 9 ; 1 6 3 9 4 4 0 6 0 1 6 2 4 g; 2 9 4 1 5 9 6 g f 1 5 ; f 3 5 8 7 3 4 4 8 4 9 2 1 5 0 0 3 ; 3 5 8 7 3 4 4 8 4 8 7 2 8 9 9 9 ; 2 6 5 7 2 7 0 6 1 0 3 0 0 g; 3 6 1 8 0 2 8 g f 1 6 ; f 6 4 1 2 5 2 5 7 6 0 8 3 9 6 8 3 ; 6 4 1 2 5 2 5 7 6 0 2 4 9 8 5 5 ; 4 1 7 4 7 9 3 1 4 6 6 8 8 g; 4 3 9 0 9 4 0 g f 1 7 ; f 1 1 0 6 5 9 2 2 1 9 1 7 7 2 1 3 9 ; 1 1 0 6 5 9 2 2 1 9 1 0 6 4 6 6 3 ; 6 3 8 1 6 9 5 2 8 6 0 5 2 g; 5 2 6 6 7 6 4 g f 1 8 ; f 1 8 5 0 9 6 5 4 7 4 3 6 5 6 7 7 1 ; 1 8 5 0 9 6 5 4 7 4 2 8 1 6 9 5 9 ; 9 5 2 1 3 8 1 9 8 6 9 2 0 g; 6 2 5 1 9 3 2 g f 1 9 ; f 3 0 1 1 1 1 2 2 2 2 9 3 1 7 4 9 9 ; 3 0 1 1 1 1 2 2 2 2 8 3 2 9 7 9 9 ; 1 3 9 0 1 6 6 2 1 8 1 3 2 4 g; 7 3 5 2 8 7 6 g a. avagyan 3 9 f 2 0 ; f 4 7 7 7 6 4 0 7 5 5 4 8 8 0 0 0 3 ; 4 7 7 7 6 4 0 7 5 5 3 7 2 7 9 9 9 ; 1 9 9 0 6 7 6 7 3 6 0 4 0 0 g; 8 5 7 6 0 2 8 g f 2 1 ; f 7 4 1 1 6 7 4 8 9 3 3 0 4 2 7 6 3 ; 7 4 1 1 6 7 4 8 9 3 1 7 0 9 1 7 5 ; 2 8 0 1 0 7 8 1 0 3 7 4 2 8 g; 9 9 2 7 8 2 0 g h e r e t h e m o s t in t e r e s t in g t r ip le t is : 1 9 2 8 0 0 1 6 6 4 7 2 5 6 9 9 3 ¡ 1 9 2 8 0 0 1 6 6 4 3 3 0 5 5 9 3 ¡ 1 6 3 9 4 4 0 6 0 1 6 2 4 3 = 2 9 4 1 5 9 6 result 3: n o w, c o n s id e r in g t h e c a s e wh e n ( 2 5 ; 2 5 ; 1 8 ) , u s in g t h e s a m e a p p r o a c h we o b t a in : µ 1 1 8 y ( 6 3 + 3 6 y3 + 2 8 0 y6 + 6 7 2 y12 + 7 6 8 y18 + 5 1 2 y24 ) ¶3 ¡ µ 1 1 8 y ( 6 3 ¡ 3 6 y3 + 2 8 0 y6 + 6 7 2 y12 + 7 6 8 y18 + 5 1 2 y24 ) ¶3 ¡ µ 2 3 ( 3 + 2 0 y6 + 3 2 y12 + 3 2 y18 ) ¶3 = ¡ 8 ¡ 1 3 y6 th u s , o n e u s e s t h e fo llo win g c o d e : g25[y¡] := 1 =gcd[ µ 1 1 8 y ³ 6 3 + 3 6 y3 + 2 8 0 y6 + 6 7 2 y12 + 7 6 8 y18 + 5 1 2 y24 ´¶ ; µ 1 1 8 y ³ 6 3 ¡ 3 6 y3 + 2 8 0 y6 + 6 7 2 y12 + 7 6 8 y18 + 5 1 2 y24 ´¶ ; µ 2 3 ³ 3 + 2 0 y6 + 3 2 y12 + 3 2 y18 ´¶ ] f25[y¡] := g25[y] ¤ f µ 1 1 8 y ³ 6 3 + 3 6 y3 + 2 8 0 y6 + 6 7 2 y12 + 7 6 8 y18 + 5 1 2 y24 ´¶ ; µ 1 1 8 y ³ 6 3 ¡ 3 6 y3 + 2 8 0 y6 + 6 7 2 y12 + 7 6 8 y18 + 5 1 2 y24 ´¶ ; µ 2 3 ³ 3 + 2 0 y6 + 3 2 y12 + 3 2 y18 ´¶ g v25[y¡] := g25[y] 3 ¤ ( ¡8 ¡ 1 3 y6 ) f or[i = ¡ 5 0 ; i · 1 3 ; i + +; if[abs[v25[i]] < 1 0 0 0 0 0 0 0 0 0 ; if[max[abs[f25[i]]] > 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ; p rint[fi; f25[i]; v25[i]g]]]] th e r e s u lt is : f¡ 2 8 ; f¡2 1 4 7 4 2 6 1 8 8 3 0 1 0 5 7 5 9 5 1 0 7 2 1 8 8 8 9 0 0 7 3 0 7 9 6 0 1 ; ¡ 2 1 4 7 4 2 6 1 8 8 3 0 1 0 5 7 5 9 5 1 0 7 2 1 8 8 8 9 0 0 7 4 3 0 8 9 1 3 ; 1 1 9 3 6 3 9 7 9 2 9 6 4 3 8 8 5 1 9 0 1 0 2 2 2 0 8 1 g; ¡ 7 8 3 0 7 1 7 4 5 g f¡ 2 4 ; f¡ 4 5 5 2 4 8 4 8 2 0 7 1 5 5 3 6 3 5 5 8 6 9 3 8 7 6 1 9 4 3 6 4 2 1 5 4 ; ¡ 4 5 5 2 4 8 4 8 2 0 7 1 5 5 3 6 3 5 5 8 6 9 3 8 7 6 1 9 4 4 3 0 5 7 0 6 ; 7 4 4 4 4 2 3 5 9 0 5 1 1 7 1 0 8 6 2 3 6 3 8 5 2 9 g; ¡ 3 1 0 5 4 2 3 3 7 g f¡ 2 0 ; f¡ 4 7 7 2 1 8 5 9 9 6 2 9 2 5 5 2 6 4 0 2 8 4 4 5 4 3 9 9 8 4 0 0 3 5 ; ¡ 4 7 7 2 1 8 5 9 9 6 2 9 2 5 5 2 6 4 0 2 8 4 4 5 4 4 0 0 1 6 0 0 3 5 ; 2 7 9 6 2 0 2 7 1 0 3 5 7 3 3 3 7 6 0 0 0 0 0 0 1 g; ¡ 1 0 4 0 0 0 0 0 1 g f¡1 8 ; f¡ 6 8 5 1 8 8 5 5 8 8 8 4 8 6 8 2 6 6 8 6 7 5 8 4 5 4 6 8 1 2 4 4 7 ; ¡ 6 8 5 1 8 8 5 5 8 8 8 4 8 6 8 2 6 6 8 6 7 5 8 4 5 4 7 2 3 2 3 5 1 ; 8 3 9 3 9 0 0 6 3 6 1 8 7 2 9 3 9 2 1 9 7 1 2 2 g; ¡ 4 4 2 1 5 8 9 2 0 g f¡ 1 6 ; f¡ 1 8 0 2 8 8 1 0 1 4 8 4 8 0 4 3 9 4 8 5 7 6 4 9 2 1 6 5 5 3 2 4 ; ¡ 1 8 0 2 8 8 1 0 1 4 8 4 8 0 4 3 9 4 8 5 7 6 4 9 2 1 7 8 6 3 9 6 ; 5 0 3 7 1 9 1 2 1 5 3 0 0 9 4 1 2 3 7 4 5 2 9 g; ¡ 2 7 2 6 2 9 7 7 g f¡ 1 4 ; f¡1 2 7 9 9 6 6 0 0 2 3 5 5 3 5 2 2 7 1 3 1 9 7 3 3 0 1 4 0 3 3 ; ¡ 1 2 7 9 9 6 6 0 0 2 3 5 5 3 5 2 2 7 1 3 1 9 7 3 3 1 6 7 6 9 7 ; 4 0 a new method of solving diophantine equation a3 + b3 + c3 = d 9 1 0 6 7 5 0 0 9 9 2 9 7 1 4 8 2 4 3 4 5 8 g; ¡9 7 8 8 3 9 7 6 g f¡1 3 ; f¡ 4 0 1 4 3 1 4 5 0 0 4 0 7 5 5 1 8 5 9 3 7 7 2 7 8 0 8 2 3 9 ; ¡ 4 0 1 4 3 1 4 5 0 0 4 0 7 5 5 1 8 5 9 3 7 7 2 8 0 3 6 7 2 7 ; 4 7 9 8 0 9 8 3 5 7 3 3 5 2 7 6 0 4 7 6 0 4 g; ¡ 5 0 1 9 8 8 2 0 0 g f¡ 1 2 ; f¡ 1 3 5 6 7 4 6 8 7 3 8 2 8 7 3 5 4 5 1 2 1 7 5 5 4 2 0 3 7 ; ¡ 1 3 5 6 7 4 6 8 7 3 8 2 8 7 3 5 4 5 1 2 1 7 5 5 8 3 5 0 9 ; 2 8 3 9 8 2 3 1 6 7 6 7 8 6 7 2 8 9 6 0 1 g; ¡4 8 5 2 2 2 5 g f¡ 1 1 ; f¡6 1 6 3 7 4 9 0 4 4 4 8 3 6 8 1 0 3 7 3 1 1 6 9 3 6 8 1 ; ¡ 6 1 6 3 7 4 9 0 4 4 4 8 3 6 8 1 0 3 7 3 1 1 8 1 0 8 0 9 ; 2 3 7 2 2 3 2 7 2 6 1 5 3 2 6 5 2 4 9 1 6 g; ¡ 1 8 4 2 4 2 4 0 8 g f¡1 0 ; f¡ 2 8 4 4 4 4 8 7 1 1 1 1 4 8 4 4 4 4 5 9 9 9 8 0 0 3 5 ; ¡ 2 8 4 4 4 4 8 7 1 1 1 1 4 8 4 4 4 4 6 0 0 0 2 0 0 3 5 ; 2 1 3 3 3 3 5 4 6 6 6 6 8 0 0 0 0 0 0 2 g; ¡ 1 3 0 0 0 0 0 8 g f¡ 9 ; f¡ 4 0 8 4 0 5 3 4 1 2 8 2 2 8 4 2 5 9 4 2 6 5 8 2 9 1 ; ¡ 4 0 8 4 0 5 3 4 1 2 8 2 2 8 4 2 5 9 4 2 7 1 0 7 7 9 ; 6 4 0 4 0 4 9 8 2 3 0 1 3 0 2 4 1 1 6 g; ¡ 5 5 2 6 9 9 2 8 g f¡ 8 ; f¡ 5 3 7 3 0 3 4 3 8 7 4 0 7 7 6 2 6 5 1 8 3 2 4 6 ; ¡ 5 3 7 3 0 3 4 3 8 7 4 0 7 7 6 2 6 5 1 9 1 4 3 8 ; 1 9 2 1 5 4 3 1 7 1 1 0 6 4 0 6 4 1 g; ¡ 4 2 5 9 8 5 g f¡ 7 ; f¡ 7 6 2 9 2 8 7 6 4 0 9 3 9 5 7 8 2 3 6 5 1 7 3 ; ¡ 7 6 2 9 2 8 7 6 4 0 9 3 9 5 7 8 2 3 8 4 3 8 1 ; 6 9 4 7 9 5 7 0 7 4 2 2 3 7 0 4 4 g; ¡1 2 2 3 5 5 6 0 g f¡ 6 ; f¡ 8 0 8 7 0 9 7 4 8 2 4 3 3 9 9 9 9 3 3 3 3 ; ¡ 8 0 8 7 0 9 7 4 8 2 4 3 3 9 9 9 9 8 5 1 7 ; 2 1 6 6 6 5 8 8 4 7 5 7 1 4 5 8 g; ¡ 6 0 6 5 3 6 g f 6 ; f8 0 8 7 0 9 7 4 8 2 4 3 3 9 9 9 9 8 5 1 7 ; 8 0 8 7 0 9 7 4 8 2 4 3 3 9 9 9 9 3 3 3 3 ; 2 1 6 6 6 5 8 8 4 7 5 7 1 4 5 8 g; ¡ 6 0 6 5 3 6 g f 7 ; f 7 6 2 9 2 8 7 6 4 0 9 3 9 5 7 8 2 3 8 4 3 8 1 ; 7 6 2 9 2 8 7 6 4 0 9 3 9 5 7 8 2 3 6 5 1 7 3 ; 6 9 4 7 9 5 7 0 7 4 2 2 3 7 0 4 4 g; ¡1 2 2 3 5 5 6 0 g f 8 ; f 5 3 7 3 0 3 4 3 8 7 4 0 7 7 6 2 6 5 1 9 1 4 3 8 ; 5 3 7 3 0 3 4 3 8 7 4 0 7 7 6 2 6 5 1 8 3 2 4 6 ; 1 9 2 1 5 4 3 1 7 1 1 0 6 4 0 6 4 1 g; ¡ 4 2 5 9 8 5 g f 9 ; f 4 0 8 4 0 5 3 4 1 2 8 2 2 8 4 2 5 9 4 2 7 1 0 7 7 9 ; 4 0 8 4 0 5 3 4 1 2 8 2 2 8 4 2 5 9 4 2 6 5 8 2 9 1 ; 6 4 0 4 0 4 9 8 2 3 0 1 3 0 2 4 1 1 6 g; ¡ 5 5 2 6 9 9 2 8 g f 1 0 ; f 2 8 4 4 4 4 8 7 1 1 1 1 4 8 4 4 4 4 6 0 0 0 2 0 0 3 5 ; 2 8 4 4 4 4 8 7 1 1 1 1 4 8 4 4 4 4 5 9 9 9 8 0 0 3 5 ; 2 1 3 3 3 3 5 4 6 6 6 6 8 0 0 0 0 0 0 2 g; ¡ 1 3 0 0 0 0 0 8 g f1 1 ; f 6 1 6 3 7 4 9 0 4 4 4 8 3 6 8 1 0 3 7 3 1 1 8 1 0 8 0 9 ; 6 1 6 3 7 4 9 0 4 4 4 8 3 6 8 1 0 3 7 3 1 1 6 9 3 6 8 1 ; 2 3 7 2 2 3 2 7 2 6 1 5 3 2 6 5 2 4 9 1 6 g; ¡ 1 8 4 2 4 2 4 0 8 g f 1 2 ; f 1 3 5 6 7 4 6 8 7 3 8 2 8 7 3 5 4 5 1 2 1 7 5 5 8 3 5 0 9 ; 1 3 5 6 7 4 6 8 7 3 8 2 8 7 3 5 4 5 1 2 1 7 5 5 4 2 0 3 7 ; 2 8 3 9 8 2 3 1 6 7 6 7 8 6 7 2 8 9 6 0 1 g; ¡4 8 5 2 2 2 5 g f 1 3 ; f 4 0 1 4 3 1 4 5 0 0 4 0 7 5 5 1 8 5 9 3 7 7 2 8 0 3 6 7 2 7 ; 4 0 1 4 3 1 4 5 0 0 4 0 7 5 5 1 8 5 9 3 7 7 2 7 8 0 8 2 3 9 ; 4 7 9 8 0 9 8 3 5 7 3 3 5 2 7 6 0 4 7 6 0 4 g; ¡ 5 0 1 9 8 8 2 0 0 g h e r e t h e m o s t in t e r e s t in g t r ip le is : ( ¡ 2 1 4 7 4 2 6 1 8 8 3 0 1 0 5 7 5 9 5 1 0 7 2 1 8 8 8 9 0 0 7 3 0 7 9 6 0 1 ) 3+( 2 1 4 7 4 2 6 1 8 8 3 0 1 0 5 7 5 9 5 1 0 7 2 1 8 8 8 9 0 0 7 4 3 0 8 9 1 3 ) 3 ¡( 1 1 9 3 6 3 9 7 9 2 9 6 4 3 8 8 5 1 9 0 1 0 2 2 2 0 8 1 ) 3 = ¡7 8 3 0 7 1 7 4 5 a. avagyan 4 1 refer ences [1 ] g. p a yn e a n d l . n . v a s e r s t e in , \ s u m s o f t h r e e c u b e s " , the arithmetic of f unction f ields, d e gr u yt e r , p p . 4 4 3 { 4 5 4 1 9 9 2 . [2 ] d . h .l e h m e r , \ on t h e d io p h a n t in e e qu a t io n " ,j . l ondon m ath. soc., vo l. 3 1 , p p . 2 7 5 { 2 8 0 , 1 9 5 6 . [3 ] k . k o ya m a , \ ta b le s o f s o lu t io n s o f t h e d io p h a n t in e e qu a t io n " , m athematics of computation, vo l. 6 2 , p p . 9 4 1 { 9 4 2 , 1 9 9 4 . [4 ] d . b e r n s t e in , \ th r e e c u b e s " , on lin e . a va ila b le : h t t p :/ / c r .yp .t o / t h r e e c u b e s .h t m l. [5 ] n . d .e lkie s , \ r a t io n a l p o in t s n e a r c u r ve s a n d s m a ll n o n z e r o jx3y2j via la t t ic e r e d u c t io n " , algorithmic number theory (l eiden 2000), l e c t u r e n o t e s in co m p u t e r s c ie n c e 1 8 3 8 , s p r in g e r , b e r lin , p p . 3 3 { 6 3 , 2 0 0 0 . [6 ] d . r . h e a t h -b r o wn , \ th e d e n s it y o f z e r o s o f fo r m s fo r wh ic h we a k a p p r o xim a t io n fa ils " , m ath. comp., vo l. 5 9 , p p . 6 1 3 { 6 2 3 , 1 9 9 2 . [7 ] m. b e c k, e . p in e , w . ta r r a n t a n d k . y a r b r o u g h je n s e n , \ n e w in t e g e r r e p r e s e n t a t io n s a s t h e s u m o f t h r e e c u b e s " , m ath. comp., vo l. 7 6 , p p . 1 6 8 3 { 1 6 9 0 , 2 0 0 7 . [8 ] d . r . h e a t h -b r o wn , w . m. l io e n a n d h . j. j. t e r ie le , \ on s o lvin g t h e d io p h a n t in e e qu a t io n o n a ve c t o r c o m p u t e r " , m ath. comp., vo l. 6 1 , p p . 2 3 5 -2 4 4 , 1 9 9 3 . submitted 12.04.2017, accepted 10.11.2017. a3 + b3 + c3 = d ï»ëùç ¹çáý³ýïû³ý ñ³í³ë³ñù³ý éáõíù³ý ýáñ ù»ãá¹ ². ²í³·û³ý ²ù÷á÷áõù ðá¹í³íá ýíçñí³í ¿ ñ³ûïýç a3 + b3 + c3 = dï»ëùç ¹çáý³ýïû³ý ñ³í³ë³ñáõùý»ñç áñáß ù³ëý³íáñ ¹»åù»ñç éáõíù³ýá:îçñ³é»éáí “mathematica 11” ÷³ã»ãá, ñ³çáõí»é ¿ ·ïý»é ýßí³í ëý¹ñç éáõíù³ý ³ñ¹ûáõý³í»ï ³é·áñçãù, áñý ³ûë ëý¹ñç éáõíù³ý ³ûé ³é·áñãùý»ñç ñ³ù»ù³ï ³í»éç å³ñ½ ¿ ¨ û·ï³·áñíáõù ¿ ½·³éçáñ»ý ÷áùñ ù³ý³ïáõãû³ùµ ·áñíáõáõãûáõýý»ñ: íîâûé ìåòîä ðåøåíèÿ äèîôàíòîãî óðàâíåíèÿ a3 + b3 + c3 = d à. àâàãÿí àííîòàöèÿ ñòàòüÿ ïîñâÿùåíà ðåøåíèþ èçâåñòíîãî äèîôàíòîâîãî óðàâíåíèÿ âèäà a3 +b3 + c3 = d â íåêîòîðûõ ÷àñòíûõ ñëó÷àÿõ. èñïîëüçóÿ ïàêåò ”mathematica 11” óäàëîñü íàéòè ýôôåêòèâíûé àëãîðèòì íàõîæäåíèÿ ðåøåíèé ýòîé çàäà÷è, êîòîðûé ïî ñðàâíåíèþ ñ äðóãèìè èçâåñòíûìè àëãîðèòìàìè, äàþùèìè ðåøåíèÿ ýòîé çàäà÷è, ÿâëÿåòñÿ áîëåå ïðîñòûì è èñïîëüçóåò çíà÷èòåëüíî ìåíüøåå êîëè÷åñòâî îïåðàöèé. d:\sbornik\...\hakob2012.dvi mathematical problems of computer science 36, 17{27, 2012. classical spin glasses with consider ation of relaxation e ®ects a s h o t s . ge vo r kya n a n d h a ko b g. a b a jya n institute for informatics and automation problems of nas of ra e-mail g ashot@sci.am, habajyan@ipia.sci.am abstract the complex-classical short-range interaction hamiltonian is used for the ¯rst time for solving spin glasses with consideration of relaxation e®ects. a system of recurrent equations is obtained on the nodes of the 1d lattice. an e±cient mathematical algorithm is developed on the basis of these equations with consideration of extended sylvester conditions which allows node-by-node construct a huge number of stable spin chains in parallel. as a result of the simulation, distribution functions of di®erent parameters of a spin glass are constructed from the ¯rst principles of complexclassical mechanics. also, the critical properties of spin glass such as catastrophes in the clausius-mossotti equation are studied depending on the external ¯eld. it is shown that the developed approach excludes these catastrophes, which allows to organize continuous parallel computation based on the whole-range values of the external ¯eld. a new representation of the partition function is suggested which, opposite to the usual de¯nition, is a complex function with the derivatives de¯ned everywhere, including at critical points. refer ences [1 ] k . b in d e r , a . p . y o u n g , spin glasses: e xperimental facts, theoretical concepts, and open questions, r e vie ws o f mo d e r n p h ys ic s , vo l. 5 8 , n o . 4 , p p . 8 0 1 -9 7 6 , 1 9 8 6 . [2 ] m. m¶e z a r d , g. p a r is i, m. a . v ir a s o r o , spin glass theory and b eyond, w o r ld s c ie n t i¯ c , vo l. 9 , 1 9 8 7 . [3 ] a . p . y o u n g ( e d .) , spin glasses and r andom f ields, w o r ld s c ie n t ī c , 1 9 9 8 . [4 ] s . f. e d wa r d s , p . w . a n d e r s o n , theory of spin glasses, jo u r n a l o f p h ys ic s f, vo l. 9 , p . 9 6 5 , 1 9 7 5 . [5 ] r . fis c h , a . b . h a r r is , spin-glass model in continuous dimensionality, p h ys ic a l r e vie w l e t t e r s , vo l. 4 7 , n o . 8 , p . 6 2 0 , 1 9 8 1 . [6 ] a . b o vie r , statistical m echanics of d isordered systems: a m athematical p erspective, ca m b r id g e s e r ie s in s t a t is t ic a l a n d p r o b a b ilis t ic ma t h e m a t ic s , n o . 1 8 , p . 3 0 8 , 2 0 0 6 . [7 ] y . tu , j. te r s o ®, g. gr in s t e in , p roperties of a continuous-r andom-network m odel for amorphous systems, p h ys ic a l r e vie w l e t t e r s , vo l. 8 1 , n o . 2 2 , p p . 4 8 9 9 -4 9 0 2 , 1 9 9 8 . [8 ] k . v . r . ch a r y, g. go vil, nm r in b iological systems: f rom m olecules to human, s p r in g e r , vo l. 6 , p . 5 1 1 , 2 0 0 8 . 1 7 1 8 classical spin glasses with consideration of relaxation e®ects [9 ] e . b a a ke , m. b a a ke , h . w a g n e r , ising quantum chain is a e quivalent to a m odel of b iological e volution, p h ys ic a l r e vie w l e t t e r s , vo l. 7 8 , n o . 3 , p p . 5 5 9 -5 6 2 , 1 9 9 7 . [1 0 ] d . s h e r r in g t o n , s . k ir kp a t r ic k, a solvable m odel of a spin-glass, p h ys ic a l r e vie w l e t t e r s , vo l. 3 5 . p . 1 9 7 2 . [1 1 ] b . d e r r id a , r andom-e nergy m odel: an e xactly solvable m odel of d isordered systems, p h ys ic a l r e vie w b , vo l. 2 4 , p p . 2 6 1 3 -2 6 2 6 , 1 9 8 1 . [1 2 ] g. p a r is i, in¯nite number of order p arameters for spin-glasses, p h ys ic a l r e vie w l e t t e r s , vo l.4 3 . p p .1 7 5 4 -1 7 5 6 , 1 9 7 9 . [1 3 ] a . j. b r a y, m. a . mo o r e , r eplica-symmetry b reaking in spin-glass theories, p h ys ic a l r e vie w l e t t e r s , vo l. 4 1 , p p . 1 0 6 8 -1 0 7 2 , 1 9 7 8 . [1 4 ] j. f. fe r n a n d e z , d . s h e r r in g t o n , r andomly l ocated spins with oscillatory interactions, p h ys ic a l r e vie w b , vo l. 1 8 , p p . 6 2 7 0 -6 2 7 4 , 1 9 7 8 . [1 5 ] f. b e n a m ir a , j. p . p r o vo s t , g. j. v a ll¶e e , separable and non-separable spin glass m odels, jo u r n a l d e p h ys iqu e , vo l.4 6 , n o .8 , p p .1 2 6 9 -1 2 7 5 , 1 9 8 5 . [1 6 ] d . gr e n s in g , r . k äu h n , on classical spin-glass m odels, jo u r n a l d e p h ys iqu e , vo l. 4 8 , n o .5 , p p . 7 1 3 -7 2 1 , 1 9 8 7 . [1 7 ] a . s ge vo r kya n , quantum 3d spin-glass system on the scales of space-time p eriods of e xternal e lectromagnetic f ields, in p r e s s , p h ys ic s o f a t o m ic n u c le i. [1 8 ] c. m. b e n d e r , j. h . ch e n , d . w . d a r g , k . a . milt o n , classical trajectories for complex hamiltonians, jo r n a l o f p h ys ic s a : ma t h e m a t ic a l a n d ge n e r a l, vo l. 3 9 , p . 4 2 1 9 , 2 0 0 6 . [1 9 ] c. m. b e n d e r , d . w . d a r g , spontaneous b reaking of classical p t symmetry, jo u r n a l o f ma t h e m a t ic a l p h ys ic s , vo l. 4 8 , p . 2 7 0 3 , 2 0 0 7 . [2 0 ] c. m. b e n d e r , d . w . h o o k, e xact isospectral pairs of p t symmetric hamiltonians, jo u r n a l o f p h ys ic s a : ma t h e m a t ic a l a n d th e o r e t ic a l, vo l. 4 1 , p p . 1 7 5 1 -8 1 1 3 , 2 0 0 8 . [2 1 ] a . s . ge vo r kya n , e t a l., r egular and chaotic quantum dynamic in atom-diatom reactive collisions, p h ys ic s o f a t o m ic n u c le i, vo l.7 1 , p p . 8 7 6 -8 8 3 , 2 0 0 8 . [2 2 ] a . v . j. s m ilg a , cryptogauge symmetry and cryptoghosts for crypto-hermitian hamiltonians, jo u r n a l o f p h ys ic s a : ma t h e m a t ic a l a n d th e o r e t ic a l, vo l. 4 1 , p . 4 0 2 6 , 2 0 0 8 . [2 3 ] c. it z yks o n , j. m. d r o u ®e , statistical f ield theory: f rom b rownian motion to renormalization and lattice gauge theory, ca m b r id g e u n ive r s it y p r e s s , vo l. 2 , p . 4 2 8 , 1 9 9 1 . [2 4 ] a . s . ge vo r kya n , h . g. a b a jya n , h . s . s u kia s ya n , a new parallel algorithm for simulation of spin-glass systems on scales of space-time periods of an external ¯eld, jo u r n a l o f mo d e r n p h ys ic s , vo l.2 , p p . 4 8 8 -4 9 7 , 2 0 1 1 . [2 5 ] a . v . b o g d a n o v, a . s . ge vo r kya n , g. v . d u b r o vs kiy, on mechanisms of protonhydrogen resonance recharge at moderate energies, p is m a v zh .t.f., vo l. 9 , p p . 3 4 3 -3 4 8 , 1 9 8 3 . [2 6 ] i. ib r a g im o v, y u . l in n ik, independent and stationary sequences of r andom variebles, w o lt e r s -n o o r d h o ® p u b lis h in g gr o n in g e n , vo l. 4 8 , p p . 1 2 8 7 -1 7 3 0 , 1 9 7 1 . [2 7 ] e . b o lt h a u s e n , a . b o vie r ( e d s .) , spin glasses, s p r in g e r , vo l. 1 6 3 , p p . 1 9 0 0 -2 0 7 5 , 2 0 0 7 . [2 8 ] g. w a n n ie r , statistical p hysics, d o ve r p u b lic a t io n s , p . 5 3 2 , 1 9 8 7 . a. gevorkyan and h. abajyan 1 9 ¸³ë³ï³ý ëåçý³ûçý ³å³ïçý»ñá ñ³ßíç ³éýí³í é»é³ùë³óçáý »ñ¨áõûãý»ñá ². ¶¨áñ·û³ý ¨ ð. ²µ³çû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù áõëáõùý³ëçñí³í ¿ ³ñï³ùçý ¹³ßïç ³éï³ûáõãû³ùµ ï³ñµñ »ñï³ñáõãû³ùµ 1 d ãï³ñ·³íáñí³í ï³ñ³í³ï³ý ëåçý³ûçý ßõã³ý»ñç (îêþ) ñ³ùáõûãç íç×³ï³·ñ³ï³ý ñ³ïïáõãûáõýý»ñá‘ ñ³ßíç ³éý»éáí é»é³ùë³óçáý »ñ¨áõûãý»ñá: ²é³ççý ³ý·³ù û·ï³·áñíí»é ¿ ïáùåé»ùë-¹³ë³ï³ý ð³ùçéïáýç³ýá: ä³ñµ»ñ³ï³ý 1 d ó³ýóç ñ³ý·áõûóý»ñáõù ëï³óí»é ¿ é»ïáõñ»ýï »é³ýïûáõý³ã³÷³ï³ý ñ³í³ë³ñáõùý»ñç ñ³ù³ï³ñ·á, áñáýù êçéí»ëïñç å³ûù³ýý»ñç ñ»ï ùç³ëçý ³ý³éçïçïáñ»ý ß³ñáõý³ïíáõù »ý ïáùåé»ùë ï³ñ³íáõãû³ý ù»ç ¨ ñý³ñ³íáñáõãûáõý »ý ï³éçë ñ³ý·áõûó ³é ñ³ý·áõûó ñ³ßí»é ëåçýç áõõõáñ¹í³íáõãûáõýá‘ ñ³ßíç ³éý»éáí ëåçý³ûçý ßõã³ý»ñáõù é»é³ùë³óçáý »ñ¨áõûãý»ñá: àõëáõùý³ëçñí³í »ý ý³¨ ëåçý³ûçý ñ³ùáõûãáõù ï»õç áõý»óáõ áñáß³ïç ïñçïçï³ï³ý »ñ¨áõûãý»ñ, çýãåçëçù »ý îé³áõ½çáõë-øáëëáïçç (î-ø) ñ³í³ë³ñù³ý ù»ç ³õ»ïý»ñá‘ ï³ëí³í ³ñï³ùçý ¹³ßïç ù»íáõãûáõýçó: ²é³ç³ñïí³í ¿ íç×³ï³·ñ³ï³ý ·áõù³ñç ýáñ ý»ñï³û³óáõù í»ñç³íáñ ãíáí çýï»·ñ³é³ûçý ³ñï³ñ³ûïáõãû³ý ï»ëùáí‘ ¿ý»ñ·ç³ûç ¨ µ¨»é³óí³íáõãû³ý ï³ñ³íáõãûáõýáõù: êëàññè÷åñêèå ñïèíîâûå ñòåêëà ñ ó÷åòîì ðåëàêñàöèîííûõ ýôôåêòîâ à. ñ. ãåâîðêÿí à. ã. àáàäæÿí àííîòàöèÿ â äàííîé ðàáîòå èññëåäîâàíû ñòàòèñòè÷åñêèå ñâîéñòâà àíñàìáëÿ íåóïîðÿäî÷åííûõ 1 d ïðîñòðàíñòâåííûõ ñïèí öåïî÷åê (ïñö) ñ îïðåäåëåííîé äëèíîé âî âíåøíåì ïîëå ñ ó÷åòîì ðåëàêñàöèîííûõ ýôôåêòîâ. äëÿ ðåøåíèÿ ýòîé ïðîáëåìû âïåðâûå áûë èñïîëüçîâàí êîðîòêîäåéñòâóþùèé êîìïëåêñíî-êëàññè÷åñêèé ãàìèëüòîíèàí è ðàçðàáîòàí ýôôåêòèâíûé ìàòåìàòè÷åñêèé àëãîðèòì, êîòîðûé, ñ ó÷åòîì ðàñøèðåííûõ óñëîâèé ñèëüâåñòðà, ïîçâîëÿåò ïàðàëëåëüíî, øàã çà øàãîì ïîñòðîèòü áîëüøîå êîëè÷åñòâî ñòàáèëüíûõ 1 d ïñö. ôóíêöèè ðàñïðåäåëåíèÿ ðàçëè÷íûõ ïàðàìåòðîâ ñïèíîâîãî ñòåêëà ïîñòðîåíû íà îñíîâå àíàëèçà ðåçóëüòàòîâ ðàñ÷åòà 1 d ïñöàíñàìáëÿ. ïîêàçàíî, ÷òî ðàñïðåäåëåíèÿ ðàçíûõ ïàðàìåòðîâ ñïèíîâîãî ñòåêëà ïî-ðàçíîìó âåäóò ñåáÿ â çàâèñèìîñòè îò âíåøíåãî ïîëÿ. ïîêàçàíî, ÷òî îáîáùåííûé êîìïëåêñíî-êëàññè÷åñêèé ïîäõîä èñêëþ÷àåò âîçìîæíîñòü âîçíèêíîâåíèÿ êàòàñòðîô â óðàâíåíèè êëàóçèóñàìîññîòòè, ÷òî ïîçâîëÿåò îðãàíèçîâàòü íåïðåðûâíûå âû÷èñëåíèÿ íà âñåì èíòåðâàëå çíà÷åíèé âíåøíåãî ïîëÿ, âêëþ÷àÿ êðèòè÷åñêèå òî÷êè. íà îñíîâå ïðîâåäåííûõ èññëåäîâàíèé ïðåäëîæåí íîâûé, áîëåå òî÷íûé ñïîñîá ïîñòðîåíèÿ ñòàòèñòè÷åñêîé ñóììû ñèñòåìû, êîòîðàÿ â îòëè÷èå îò îáû÷íûõ ïðåäñòàâëåíèé, ÿâëÿåòñÿ êîìïëåêñíîé ôóíêöèåé. ñòàòèñòè÷åñêàÿ ñóììà, è åå ïðîèçâîäíûå àíàëèòè÷íû ïîâñþäó âêëþ÷àÿ êðèòè÷åñêèå òî÷êè. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 47, 62--67, 2017. design and implementation of "handipum" webrtcbased conferencing service in asnet-am network arthur s. petrosyan, gurgen s. petrosyan and robert n. tadevosyan institute for informatics and automation problems of nas ra e-mail: arthur@sci.am, gurgen@sci.am, robert@sci.am abstract this paper presents the results of research and deployment experience with design and implementation of the asnet-am webrtc-based audio/video conferencing service, called "handipum" (handipum.asnet.am). since the webrtcservices become widely used, this service can provide a modern browser-based conferencing environment for asnet-am users. keywords: real-time communications, rtc, webrtc, selective forwarding unit, sfu, conferencing, multimedia, middleware 1. introduction webrtc is a modern open source technology that enables web browsers with real-time communications (rtc) capabilities via javascript apis. it has been initially developed as a peer-to-peer technology, where clients (web browsers) can directly communicate without any intermediate infrastructure. this model is enough for creating basic applications but features such as group communications, media stream recording, media broadcasting or media transcoding are difficult to implement on top of it. for this reason, many applications require using some kind of intermediate media server. conceptually, the webrtc media server is just a kind of “multimedia middleware” (it is in the middle of the communicating peers) where media traffic passes through when moving from source to destinations. media servers are capable of processing media streams and offering different types including group communications (distributing the media stream that one peer generates among several receivers), mixing (transforming several incoming streams into one single composite stream), transcoding (adapting codecs and formats between incompatible clients), recording (storing in a persistent way the media exchanged among peers), etc. 62 mailto:arthur@sci.am mailto:gurgen@sci.am mailto:robert@sci.am a. petrosyan, g. petrosyan and r. tadevosyan 63 2. architecture models there are in general 3 main architecture models of deploying a multiparty webrtc-based video conferences: mesh, multipoint conferencing unit (mcu) & selective forwarding unit (sfu). in mesh multipoint architecture every participant sends and receives his media to all other participants. this variant is a very common technique used in webrtc to build multipoint conferences. advantages of mesh architecture are that it’s simple to implement in webrtc and requires very little backend infrastructure, keeping the resulting service cheap to operate. but it requires a lot of uplink bandwidth from each participant and thus can’t scale to a large number of participants. an mcu offers the ability to connect multiple participants in a single voice or video session. in mcu participants “speak” to a central entity, which mixes all inputs and sends out a single stream towards each participant. mcus generally implement a mixing architecture and are expensive due to their need for a lot of processing power per session. an sfu is capable of receiving multiple media streams and then decide which of these media streams should be sent to which participants. in sfu the participant sends his media to a central entity, which routes all incoming media as it sees fit to participants each of them receiving usually more than a single stream. so sfu can be treated as a lightweight webrtc implementation, allowing for multiuser video communication by simply relaying the received multimedia flows to all call participants, without mixing into a composite stream. one of the effective open-source sfu solutions is jitsi videobridge. together with other components from jitsi project [1] it proved to be an interesting solution that was used as a basis for creating a webrtc-based audio/video conferencing service in asnet-am network, called "handipum" [2]. 3. "handipum" conferencing service infrastructure there are lots of websites offering online video conferencing solutions on the internet today. many have a user friendly interface and very good video and audio quality. some even are free for use after registration. however, the downside of such solutions is that the data moves through a foreign server. plus no armenian interface is available anywhere in conferencing solutions. thus, it was decided to design and implement an own conferencing solution for asnetam users, based on freely available solutions. after some comparative research, one of such solutions was found to be the jitsi project, which provides an open source, encrypted, and sustainable interface. it works via web front end application called jitsi meet [3], thus making it possible to use any webrtc-compatible web browser as a client. such a modern flexible browser-based solution eliminates the need for a specific client and at the same time provides rich conferencing capabilities. two main components of "handipum" conferencing service are – jitsi videobridge and jitsi meet. they work together to transfer video and audio between the users. videobridge acts as a web real-time communication webrtc-compatible sfu and allows for multiuser communication. jitsi meet is a front end to videobridge and is implemented in javascript. jitsi meet can be configured to be hosted on some http server and in our case the nginx web server [4] was used for that. in addition some implementation of xmpp server is required on the server side. the prosody [5] lightweight, open source jabber/xmpp server is the basic solution used as an xmpp server. design and implementation of "handipum" webrtc-based conferencing service in asnet-am 64 4. "handipum" conferencing service interface "handipum" conferencing service currently offers video and audio conferences in web browsers with real-time text chat, youtube video sharing. other additional services that are under development include: presentation sharing, screen sharing, shared document editing, voip integration. we have done the localization of jitsi meet web interface and "handipum" conferencing service is now available in armenian. we have also provided the armenian translation to the public [6]. so jitsi meet is now translated into 16 languages, among which armenian is proudly presented, thanks to the asnet-am team efforts. users can take part in meetings with webrtc-enabled browsers. the current versions of opera, google chrome, and mozilla firefox already support standard; additional functions such as desktop sharing require browser add-ons. microsoft's internet explorer/edge and apple's safari browsers do not have proper webrtc support yet. all conferencing communication is secure enough thanks to the https encrypted connections. strong ssl certificate with 4096-bit keys sha-256 signature is used in "handipum" conferencing service. additionally each conference room created, can be locked by a password, to ensure limited access. 5. conference room creation users can open a new room in "handipum" conferencing service by opening its start page https://handipum.asnet.am (figure 1) and typing a unique name for conference room. the room name should not contain spaces and should consist of only latin letters and numbers. fig. 1. "handipum" conferencing service start page after clicking the “go” button the system creates a unique room interface as shown on figure 2. https://handipum.asnet.am/ a. petrosyan, g. petrosyan and r. tadevosyan 65 fig. 2. "handipum" conferencing service room interface while entering the conference room, participants should confirm that they allow the "handipum" conferencing service to access their camera and microphone. the initiator of the conference (the user, who initially created a new non-existing room) automatically receives moderator rights, can then protect the room with a password and mute or remove participants. currently it is not possible to provide moderator rights to any other participant. all current room participants are displayed in separate mini-windows at the bottom, and the moderator is marked with an asterisk. 6. sip integration "handipum" service is configured to make instant calls from a conference room to some sip number within asnet-am internal ip telephony service. sip integration is implemented with jigasi (jitsi gateway to sip) [7]. jigasi is a server-side application that allows regular sip clients to join the "handipum" conference. integration is made by setting a specific asnet-am sip server account in the jigasi configuration. a small phone icon appears in conference rooms in the web interface. any participant can call someone at a sip number and add him to current room conversation. clicking on phone icon opens a box in which the inviting participant needs to enter the sip number to call (figure 3). if the individual being called picks up, he/she can take part in the audio conversation within that room. by this, users with sip phones can take part in "handipum" conferences. fig. 3. inviting participant needs to enter the sip number to call design and implementation of "handipum" webrtc-based conferencing service in asnet-am 66 7. conclusion conferencing solution presented above is a modern webrtc-based browser-driven system. it is very efficient and scalable. it is able to run thousands of simultaneous video streams at small bandwidth and cpu usage [8]. before "handipum" many of our users scientists, scientific and technical associates, postgraduates and students used other tools such as microsoft skype or google hangouts for conferencing. since the "handipum" conferencing service is now open to our community and since it works directly from web browser and does not require any specific client installation, we hope it to become a viable alternative that perfectly matches our users’ needs. asnet-am members were informed about the "handipum" conferencing service available, by publishing the appropriate news at asnet-am website. references [1] jitsi project, [online]. available: http://jitsi.org [2] "handipum" asnet-am conferencing service, [online]. available: https://handipum.asnet.am [3] jitsi meet, [online]. available: https://jitsi.org/projects/jitsimeet [4] nginx http server, [online]. available: https://nginx.org/ [5] prosody xmpp communication server, [online]. available: https://prosody.im/ [6] jitsi meet armenian interface, [online]. available: http://translate.jitsi.org/hy/jitsi-meet/ [7] jigasi, [online]. available: https://github.com/jitsi/jigasi [8] jitsi videobridge performance evaluation, [online]. available: https://jitsi.org/projects/jitsivideobridgeperformance submitted 08.10.2016, accepted 22.02.2017. webrtc միջավայրում "handipum" վեբ կոնֆերանս համակարգի նախագծում և իրականացում asnet-am ցանցում ա. պետրոսյան, գ. պետրոսյան և ռ․ թադևոսյան ամփոփում հոդվածում նկարագրված է նոր սերնդի վեբ կոնֆերանս համակարգի նախագծումն ու իրականացումը asnet-am ցանցում, որն իրականացված է webrtc միջավայրում։ webrtc համակարգերը լայն տարածում են ստացել այն պատճառով, որ օգտագործողի համակարգչի վրա չեն պահանջում հատուկ ծրագրերի տեղադրում և http://jitsi.org/ https://handipum.asnet.am/ https://jitsi.org/projects/jitsimeet https://nginx.org/ https://prosody.im/ http://translate.jitsi.org/hy/jitsi-meet/ https://github.com/jitsi/jigasi a. petrosyan, g. petrosyan and r. tadevosyan 67 աշխատում են անմիջապես վեբ բրաուզերներում։ asnet-am ցանցում ներկայումս գործող այդպիսի համակարգը, որի մասին խոսվում է հոդվածում, հիմնված է jitsi բաց կոդով ծրագրային փաթեթի վրա։ նկարագրված են համակարգի հիմնական հնարավորությունները։ վեբ կոնֆերանս համակարգը անվանվել է "handipum" և հասանելի է https://handipum.asnet.am հասցեով։ մինչև "handipum" համակարգի ներդրումը, մեր օգտագործողներից շատերը, որոնց թվում են՝ գիտնականները, գիտական և տեխնիկական աշխատողները, ասպիրանտները, ուսանողները և այլ մասնագետները, օգտվում էին տարբեր այլ արտաքին կոնֆերանս համակարգերից։ այժմ, երբ "handipum" համակարգը հասանելի է ողջ հայկական գիտակրթական հանրության համար և աշխատում է անմիջապես բրաուզերից, հույս ունենք, որ այն արժանի տեղական այլընտրանք կլինի մեր օգտագործողների համար։ разработка и реализация сервиса конференций "handipum" на основе протоколов webrtc в сети asnet-am а. петросян, г. петросян и р․ тадевосян аннотация в статье описана разработка и реализация нового поколения сервиса веб конференций в сети asnet-am на основе протоколов webrtc․ реализация сервисов веб конференций в среде webrtc в настоящее время пользуется большой популярностью по причине отсутствия необходимости установки специального клиентского приложения, в качестве которого выступает обычный веб браузер. действующая в настоящее время в сети asnet-am услуга веб конференций на основе протоколов webrtc базируется на активно развивающемся проекте с открытым кодом jitsi․ описаны основные возможности сервиса․ сервис назван "handipum" и функционирует по адресу https://handipum.asnet.am․ до внедрения сервиса "handipum", многие из наших пользователей ученые, аспиранты и студенты, пользовались прочими внешними возможностями для конференций․ поскольку сервис "handipum" в настоящее время открыт для армянской научно-образовательной среды и работает прямо из браузера, мы надеемся, что он станет достойной альтернативой для наших пользователей․ https://handipum.asnet.am/ mathematical problems of computer science 45, 5–13, 2016. results of performance analysis of advanced inftheo new package for r narek s. pahlevanyan, mariam e. haroutunian gyumri state pedagogical institute institute for informatics and automation problems of nas ra e-mail: narek@ravcap.com, armar@ipia.sci.am abstract the authors have developed a new package “advanced inftheo” [1] for r to perform computations of complex information-theoretical results. for achieving higher computational speed, the package includes different types of parallelization. for evaluation of package performance various experiments were conducted. the analysis of these experiments are presented in this paper. moreover, the advantages of advacned inftheo package compared with the existing infotheo package are introduced. keywords: r language, r package, multithreading, e-capacity bounds. 1. introduction the popularity of r has grown significantly in recent years. created in 1993 as an alternative to proprietary statistical software, r is an integrated suite of software tools for data manipulation, calculation and graphical display. this open source language has become an important part of the it arsenal for analysts and data scientists performing statistical analysis on big data. currently, there are about 3 million r users worldwide who operate thousands of open sources packages within the r environment to increase productivity in such domains as spatial statistics, bioinformatics, financial market analysis and linear/nonlinear modeling, etc. large companies are using r, too. for example, facebook uses r for exploratory data analysis, big-data visualization, human resources and user behaviour analysis related to status updates and profile pictures. recent concerns in the financial sector has stimulated oracle to support r, for instance, oracle is now bundling it as part of its big data appliance product. google uses r for advertising effectiveness, economic forecasting, and big-data statistical modeling. twitter uses r for data visualization and semantic clustering. it contains a number of built-in functions for organizing data, running calculations on the information and creating elegant graphical representations of data sets which are very useful for data scientists. r provides a lot of different techniques for time-series analysis, classification, clustering as well as graphical packages for creating high quality, and sophisticated, customized plots with very simple syntax. the facilities of r can be extended through user-created packages. packages (also known as modules) are libraries developed in c++, that include specific functions set for 5 6 results of performance analysis of advanced inftheo new package for r certain applications. r comes with core set of packages, in addition to that there are more than 5,800 various packages and 120,000 functions free available for download [2] . we believe that r can be beneficial and helpful for calculations of complex formulas of information theory. in practice many information-theoretical results are difficult for computing because of the large volume of distributions. for example, the investigation of rate-reliability function [3] for various applications [4], [5], [6] is time consuming, and computational results are hard to obtain. r already had a package for calculating various measures of information theory called infotheo, but there was a need in creation of a new package for estimation and computation of more complicated formulas mentioned above. to perform computations of complex information-theoretical results in information theory the authors have developed a new package for r, called advanced inftheo. it was developed in c++; allows computation of more complicated formulas of information theory, such as functionality for computation of the lower and upper bounds of rate-reliability function, etc. functions inside advanced inftheo are parallelized. it allows three types of parallelization (cpu-based, gpu-based and cluster-based) to a consumer. moreover, it has a core set of functions included in infotheo package but with different technical implementation. the reason of reimplementation of the existing functions of infotheo inside advanced inftheo package was the investigation results of infotheo source code that revealed that the functions inside package are single threaded. technical solutions used inside advanced inftheo package for overcoming the existing limitations [7], [8] of r language as well as for solving the issues arising with multi-threaded approach are discussed in [1]. the advanced inftheo package experimentation results are published in [9]. specifically, in [9] the authors have computed the lower and upper bounds of e-achievable secret key rate of the biometric generated secret key sharing system obtained in [6] for various distributions. moreover, they provide graphical representations of the experimentation results to simplify the solutions in building of applications. in this paper we present the results of performance analysis of advanced inftheo package functions for various parallelization modes. moreover, we introduce the main differences of advanced inftheo package from the existing infotheo package and provide performance analysis results that show advantages in computational speed for the same functions of 2 packages. 2. description of infotheo package this package implements various measures of information theory based on several entropy estimators. the package was developed by patrick e. meyer, at the moment latest available version is 1.2.0 (published on july 2014). package includes the following primary functions available for usage [10]: • entropy() computes entropy, takes the dataset as input and computes the entropy in nats (base e). • condentropy() computes conditional entropy, takes two random vectors x and y as input and returns the conditional entropy h(x|y ) in nats. • condinformation() computes conditional mutual information, takes three random variables as input and computes the conditional mutual information in nats. n. pahlevanyan, m. haroutunian 7 • mutinformation() computes mutual information, takes two random variables as input and computes the mutual information in nats. • multiinformation() computes multiinformation, takes a dataset as input and computes the multiinformation (also called total correlation) among the random variables in the dataset. returned value is in nats. • interinformation() computes interaction information, takes a dataset as input and computes the interaction information among the random variables in the dataset. this measure is also known as synergy or complementarity. • discretize() performs unsupervized data discretization, discretizes data using the equal frequencies or equal width binning algorithm. from the descriptions of available functions inside infotheo we reveal that this package does not provide options for computation of more complicated formulas rate-reliability function, error exponent and so on. furthermore, analyzing the source code of infotheo showed that all functions inside the package are single threaded, which can lead to loss of productivity and efficiency on big datasets. 3. description of advanced inftheo package advanced inftheo package provides functionality for computation of various complex formulas of information theory as well as it implements the main functions of infotheo package. the main feature of advanced inftheo package is that it is using multithreaded computations for faster performance [11]. it means that advanced inftheo can use the advantage of multiprocessor hardware. moreover, it allows 3 types of parallelization: • cpu-based parallelization, • gpu-based parallelization, • cluster-based parallelization (mpi technology). inside cpu-based parallelization of advanced inftheo package is the idea of multi-core systems, that are now very common, and the number of processors per chip is growing. there is a need for integration of r code into multi-core environments. this approach has the potential to speed up a numerically intensive code. to design a package to support a more abstract use of multi-core systems from package development point of view seems to be a very difficult task, because the interactive nature of r and the existing multi-core technology implies runtime compilation. in terms of hardware, the parallel computing power of graphic processing units (gpus) might provide significant performance benefits to users of advanced inftheo. nvidia cuda technology is being used inside advanced inftheo package for gpu-based parallelization, cuda offers a programming model that is designed to allow direct access to the specific graphics hardware, with the graphics hardware running a very high number of threads in parallel. from user perspective, computer clusters are often unavailable, the physical costs (maintenance, etc.) are high, and effective use requires a various skill set (e.g, advanced user-level 8 results of performance analysis of advanced inftheo new package for r system configuration, mastery of batch job submission, careful formulation of problems to effectively manage communication costs). therefore, it is difficult to use this technology. in cluster-based parallelization type of advanced inftheo we consider that clusterized environment is already set up and r is properly installed there. the integration with cluster is done with openmpi library, which allows advanced inftheo package to take default settings (processors count, ram limit etc) for performing computations. in addition to functions described above for infotheo package, advanced inftheo includes the following functions : • setcomputationmode() allows selection of parallelization type (cpu, gpu or mpi) that’s used in rest of functions. • getcomputationmode() retrieves current parallelization type. • setverbosemode() sets package in debug mode. in this mode debug messages will be shown on every step of computations. • setthreadscount() sets threads count used in package functions. if this value is not set, the package will find and use the number of concurrent threads supported by hardware. • getthreadscount() retrieves currently used threads count. • setgpucorescount() sets gpu cores count used in package functions. this value can’t be bigger than the count of available physical multiprocessors. • calcentropy() calculates entropy (in bits) of discrete random variable using probability distribution vector. • calcjointentropy() calculates joint entropy of two discrete random variables using joint distribution matrix. • calcrelativeentropy() calculates kullback-leibler divergence (information gain in bits). • calcmutualinformation() calculates mutual information of two discrete random variables using joint distribution matrix. • calcconditionalentropy() calculates conditional entropy of two discrete random variables using joint distribution matrix. • calcmarginaldistribution() calculates marginal probability distribution of n×m sized probability matrix. • calclowerboundofecapacity() calculates lower bound of e-capacity for provided reliability and distribution matrix, based on computation of minimum mutual information for all distributions in the condition of limited divergence. • calcupperboundofecapacity() calculates upper bound of e-capacity for provided reliability and distribution matrix, based on computation of minimum mutual information for all distributions in the condition of limited divergence. for evaluation of advanced inftheo package performance some experiments were conducted: load testing, speed testing, configuration testing, isolation testing, etc. n. pahlevanyan, m. haroutunian 9 4. performance analysis results here we provide performance analysis of advanced inftheo package functions and give general recommendations for setting optimal thread count for various n (count of input arguments) values. also we compare performance of common functions in both packages. computations in fig.1 have been performed on machine with medium parameters (intel core 2 duo 2 x 2.00ghz, with 2.5gb ram) with default hardware concurrency (in our case it was 2). fig. 1. comparison of common functions when advanced inftheo is in cpu mode with thread count set to 2 and for n = 106. from fig. 1 we can see that in cpu mode with 2 threads advanced inftheo is in average 1.85 times faster than infotheo. we will have different results for smaller n if we use gpu mode with higher number of cores. fig. 2 shows that when advanced inftheo is in gpu mode and the lower n is used we have downgrade in computational speed compared with infotheo. 10 results of performance analysis of advanced inftheo new package for r fig. 2. comparison of common functions when advanced inftheo is in gpu mode with core count set to 32 and for n = 500. next figures (fig. 3 fig. 5) are demonstrating performance of calcupperboundofecapacity() function for various n and for various modes of advanced inftheo. fig. 3. performance of calcupperboundofecapacity() function for various n when advanced inftheo is in cpu mode and threads count is set to 8. n. pahlevanyan, m. haroutunian 11 fig. 4. performance of calcupperboundofecapacity() function for various n when advanced inftheo is in gpu mode with core count set to 32. fig. 5. performance of calcupperboundofecapacity() function for various n when advanced inftheo is in mpi mode with usage of 40 processors. 12 results of performance analysis of advanced inftheo new package for r in gpu mode tests were performed on nvidia geforce gt 440 graphical adapter. experiments in mpi mode (cluster-based parallelization) are depicted in fig. 5, those tests were been executed on armcluster with usage of 40 processors. from fig. 4 and fig. 5 we can see that we will have time gain if we use advanced inftheo in gpu or mpi modes. the thread count for gpu mode should be selected wisely, otherwise downgrade in performance is imminent. our other experiments with gpu mode showed that the number of threads should be set as multiplier of gpu multiprocessors warp size for better performance. 5. conclusion parallelism inside advanced inftheo can give huge time gain in computations of informationtheoretical results for practical applications. different modes inside advanced inftheo package will provide various performance results, to achieve higher performance results the thread count in cpu and gpu modes should be calculated precisley based on computational problem. for computation of non-complex formulas (such as entropy, mutual information etc.) when n < 103 we recommend the usage of smaller threads count (2-4). for computation of complex formulas (such as calculation of upper bound of e-capacity for provided reliability and distribution matrix) gpu and mpi modes can give big time gain compared with cpu mode, thread count for gpu mode should be set as multiply of multiprocessor warp size. references [1] n. pahlevanyan and m. haroutunian, “technical solutions of developing advanced inftheo new module for r,” in proceedings of the 10th international conference on computer science and information technologies, yerevan, armenia, 2015, pp. 306–309. [2] w. n. venables, d. m. smith and the r core team, “an introduction to r,” version 3.1.1, pp. 51–77, 2014. [3] e. a. haroutunian, m. e. haroutunian and a. n. harutyunyan, “reliability criteria in information theory and in statistical hypothesis testing,” foundations and trends in communications and information theory, vol. 4, no. 2-3, 2008. [4] t. ignatenko and f. m. willems, “biometric security from an information-theoretical perspective,” foundations and trends in communications and information theory, vol. 7, no. 2-3, 2012. [5] r. ahlswede and i. csiszar, “common randomness in information theory and cryptography. i secret sharing,” ieee transactions on information theory, vol. 39, no. 4, pp. 1121–1132, 1993. [6] m. haroutunian and n. pahlevanyan, “information theoretical analysis of biometric secret key sharing model,” transactions of iiap of nas of ra, mathematical problems of computer science, vol.42, pp. 17-27, 2014. [7] r development core team, “r: a language and environment for statistical computing,” r foundation for statistical computing, 2011. [8] d. smith, “the r ecosystem,” user! conference, coventry, united kingdom, august 2011. [9] m. haroutunian and n. pahlevanyan, “experimentation of advanced inftheo module for r on the example of biometric generated secret key sharing system,” international journal information content and processing, vol. 2, no. 1, pp. 62–70, 2015. n. pahlevanyan, m. haroutunian 1 3 [1 0 ] p . e . me ye r , \ p a c ka g e in fo t h e o ," the comprehensive r archive network, p p . 1 { 1 2 , ju ly 2 0 1 4 . [1 1 ] c. h u g h e s a n d t. h u g h e s , p rofessional m ulticore p rogramming: d esign and implementation for c++ d evelopers. b ir m in g h a m , u k , u k : w r o x p r e s s l t d ., 2 0 0 8 . submitted 12.09.2015, accepted 26.01.2016 r-ç ýáñ advanced inftheo ÷³ã»ãç ³ñï³¹ñáõ³ï³ýáõãû³ý í»ñéáõíáõãû³ý ³ñ¹ûáõýùý»ñá ü. ö³ñé¨³ýû³ý, ø. ð³ñáõãûáõýû³ý ²ù÷á÷áõù æýýáñù³óç³ûç ï»ëáõãû³ý µ³ñ¹ µ³ý³ó¨»ñç ñ³ßíù³ý ñ³ù³ñ ùß³ïí»é ¿ ýáñ ÷³ã»ã advanced inftheo [1] r é»½íç ñ³ù³ñ: ð³ßíáõ³ï³ý ù»í ³ñ³·áõãû³ý ñ³ëý»éáõ ñ³ù³ñ ÷³ã»ãý çñ ù»ç ý»ñ³éáõù ¿ ï³ñµ»ñ ïçåç ½áõ·³ñ»é³óù³ý ù»ë³ýç½ùý»ñ: ö³ã»ãç ³ñï³¹ñáõ³ï³ýáõãû³ý ·ý³ñ³ïù³ý ñ³ù³ñ ï³ï³ñí»é »ý ï³ñµ»ñ ÷áñó»ñ: ²û¹ ÷áñó»ñç í»ñéáõíáõãûáõýá ý»ñï³û³óí³í ¿ ³ûë ñá¹í³íáõù: øýý³ñïí³í »ý ý³¨ advanced inftheo ÷³ã»ãç ³é³í»éáõãûáõýý»ñá ·áûáõãûáõý áõý»óáõ infotheo ÷³ã»ãç ñ³ù»ù³ïáõãû³ùµ: ðåçóëüòàòû àíàëèçà ïðîèçâîäèòåëüíîñòè íîâîãî ïàêåòà advanced inftheo äëÿ r í. ïàéëåâàíÿí, ì. àðóòþíÿí àííîòàöèÿ äëÿ âû÷èñëåíèé ðåçóëüòàòîâ ñëîæíûõ ôóíêöèé òåîðèè èíôîðìàöèè ðàçðàáîòàí íîâûé ïàêåò advanced inftheo [1] äëÿ ÿçûêà r. â ìîäóëå äëÿ äîñòèæåíèé âûñîêîé ýôôåêòèâíîñòè âñòðîåíû ðàçíûå ìåõàíèçìû ïàðàëëåëèçàöèé. ïðîâåäåíû ðàçëè÷íûå ýêñïåðåìåíòû äëÿ îöåíêè ïðîèçâîäèòåëüíîñòè ïàêåòà. â äàííîé ñòàòüå ïðèâîäèòñÿ àíàëèç ýòèõ ýêñïåðèìåíòîâ. êðîìå òîãî, ïðåäñòàâëåíû ïðåèìóùåñòâà ïàêåòà advanced inftheo â ñðàâíåíèè ñ ñóùåòâóþùèì ìîäóëåì infotheo. 01.pdf (p.1-8) abstract_mariam.pdf (p.9) microsoft word e. pogossian_13.doc mathematical problems of computer science 34, 2010. 37 developing models of mental behavior edward pogossian cognitive algorithms and models laboratory at the academy of science of armenia (ipia), state engineering university of armenia the research is aimed to advance in answer to the fundamental question whether models of mind can be mental not being living realties (lr), assembled from lr or developed from the springs of lr? in other words, whether are models of mind which do not depend from lr but are classified as mind possible if mind uses the same criteria when forms the class mind? to answer the question constructive models of mind and criteria of measuring their mentality as well as the exhaustive experiments on revealing the truth are needed. a measurable approach to the models of mind, cognizers, is studied [pogossian 2010] and the criteria and experiments of testing of mentality of cognizers are questioned. the approach roots in fundamental works by [flavell,1962, neuman,1966, botvinnik,1984, atkinson1993, pylyshin,2004, roy,2005, shrodinger,1956, winograd, 986, mendler,2004] and develops the ideas in [pogossian, 1983,2005-7] on interpretation of the recognized views on mind by models having unanimous communalized meanings followed by experiments on validity of those models. valid cogs, if constructed, confirm the assertion that mind is a modeling based problem formation and solving procedure able to use knowledge gained from the solutions to promote the utilities of lr in their negentropic games. synchronously, mental cogs provide a constructive model of mind as the ultimate instrument for cognition. knowledge on the nature of instruments for revealing new knowledge gives a new look on the knowledge already gained or expected and raise new consequent questions. therefore, revealing by cogs the new knowledge on the instruments of cognition it is worth to question the new aspects of relationships between mind and the overall knowledge mind creates and uses. ongoing experiments on study of cogs are based on the technique of evaluating adaptive programs and their parts by local tournaments and use the game solving package fig1,2. with its kernel personalized planning and integrated testing (ppit) and strategy evaluation units specified for the class of problems where space of solutions can be represented by reproducible game trees (ssrgt class, fig.2) [pogossian,1983, 2005-7]. common structures for problems, the space solutions and representation of expert knowledge allows to vary the problems in focusing particular research questions followed by natural generalization for the ssrgt class. particularly, the studying of chess vocabulary and winning by zermelo classes of chess positions and strategies [pogossian, 2006] argues that real implementation of the models of human expertise for ssrgt problems , in principle, can only approximate game tree structures due to irresistible complexity of computations to prove correctness of the models. thus, for the same 38 requests both game players and computers will, as a rule, use different models of realities essentially based on their individual experience, any preferences between those models include uncertainty and interpretations of units of vocabulary along with communalized dimension have to essentially relay on certain personalized experiences of the communication parties. viability of the package was demonstrated for the intrusion protection ssrgt problems and the intermediate goals at first (igaf) algorithms [pogossian2005]. the igaf algorithms as well as [botvinnik,1984] are based on a common knowledge planning and dynamic testing of the plans in the corresponding game trees. it was proven that the igaf2 algorithms are able to acquire a range of expert knowledge in the form of goals or rules and to increase the efficiency of strategy formation with an increase in the amount of expert knowledge available to the algorithm. the effectiveness of the igaf2 algorithms was successfully tested for the network intrusion protection problems against representatives of four classes of attacks: syn-flood, fraggle, smurf and login-bomb, allowing to formulate, in particular, the following statements:  number of nodes searched by the igaf2 algorithm for making decisions with all expert rules and subgoals is the smallest compared with the igaf1 algorithm or with the minimax algorithm in which the depth of the search is increasing up to 13  the recommended version of the igaf2 algorithm with all expert rules and subgoals, for the depth of search 5 and 200 defending steps, is outperforming the productivity of the minmax algorithm by 14% while using 6 times less computing time and searching 27 times less nodes of the tree. knowledge based strategies for another ssrgt problem where supply chain management agents compete in a market to gain max profit for different customer requests were studied in [bagdasaryan, 2005]. for each request the agents generate possible offers and for each offer a game tree is constructed to allow simulation and tracing further actions with suppliers, production and delivery. in the mean time, strategic plans based on the handbook common expert knowledge are generated followed by the quantification of the plans for strategy analysis and their evaluation for final decision making. publications 1. baghdasaryan t., danielyan e, pogossian e. testing oligopoly strategy plans by their on the job performance simulation. //proceedings of the international conference csit2005, 2. [pogossian,2007] e.pogossian. on measures of performance of functions of human mind. 6th international conference in computer science and information technologies, csit2007 3. [pogossian ,2006] e.pogossian. specifying personalized expertise. international association for development of the information society (iadis): international conference cognition and exploratory learning in digital age (celda 2006), 8-10 dec., barcelona, spain, 151-159 4. pogossian e., javadyan a., ivanyan e.: effective discovery of intrusion protection strategies. the intern. workshop on agents and data mining, st. petersburg, russia, lecture notes in computer science, vol. 3505 (2005) 263-274 5. [pogossian,2007] e.pogossian, v. vahradyan, a. grigoryan. on competing agents consistent with expert knowledge", lecture notes in computer science, ais-adm-07: int. workshop on autonomous intelligent agents and data mining, june 5 -7, st. petersburg, 11pp. 6. [pogossian,1983] e.pogossian. adaptation of combinatorial algorithms (a monograph in russian), 293 pp. 1983. yerevan., 7. pogossian e. on measurable models of promotion of negentropic strategies by cognition, proceedings of international conference “information-interaction-intellect”, varna, bulgaria, june 21-27, 2010, p. 161-168. microsoft word article.doc mathematical problems of computer science 37, 75--82, 2012. 75 a new approach to receipt-free e-voting a. jivanyan russian-armenian(slavonic) university e-mail: jivanyan@gmail.com abstract in this paper we introduce a new electronic voting protocol with provable security properties. the generic method of voting presented here allows choosing different approaches for ballot casting to provide the most user-friendly interface. references [1] t. el-gamal, “a public key cryptosystem and a signature scheme based on discrete logarithms”, advances in cryptology crypto '84, lncs 196, pp 10-18, 1984 [2] a. fiat and a. shamir, “how to prove yourself: practical solutions to identification and signature problems”, advances in cryptology-crypto '86, vol. 263 of lncs [3] d. chaum and t. pedersen, “wallet databases with observers”. in proc. of crypto'92, springer-verlag, pp. 89-105, 1993.lncs 740. [4] d. chaum, “secret ballot receipts and transparent integrity better and less-costly electronic voting at polling places”, http://vreceipt.com/article.pdf, 2004. [5] d. chaum, “untraceable electronic mail, return addresses, and digital pseudonyms,” in communications of the acm, 24(2), pp. 84-88, 1981. [6] a. shamir,“how to share a secret”, communications of the acm, 22(11), pp. 612-613, 1979. [7] d. boneh, p. golle, “almost entirely correct mixing with application to voting.” [8] a. neff. “a verifiable secret shuffle and its application to e-voting”, in proc. of acm ccs'01,acm press, pp. 116-125. 2001. [9] j. furukawa and k. sako, “an efficient scheme for proving a shuffle”, proc. ofcrypto '01,springer-verlag, lncs 2139, 2001. [10] p. golle, s. zhong, d. boneh, m. jakobsson, a. juels, “optimsitic mixing for exit-polls”, advances in cryptology (asiacrypt 2002)', vol. 2501 oflncs, springer-verlag, 2002. [11] m. jakobsson, a. juels and r. rivest. “making mix nets robust for electronic voting by randomized partial checking”, in proc. of usenix'02. [12] d. wikstrom. “four practical attacks for "optimistic mixing for exit-polls"”, technical report t2003-04, swedish institute of computer science, 2003. [13] c. park, k. itoh and k. kurosawa. “efficient anonymous channel and all/nothingelectionscheme.”in proc. of eurocrypt 93, springer-verlag, lncs765, pp. 248259, 1993. [14] w. ogata, k. kurosawa, k. sako and k. takatani. “fault tolerant anonymouschannel”. in proc. of icics 97, lncs 1334, pp. 440-444, 1997. [15] k. sako and j. kilian.“receipt-free mix-type voting scheme”.in proc. of eurocrypt95.springer-verlag, lncs 921, 1995. a. jivanyan 76 [16] m. abe. “universally verifiable mix-net with verification work independent of thenumber of mix-servers”, in proc. of eurocrypt 98, springer-verlag, pp. 437-447, 1998. [17] t. moran, m. naor. “receipt-free universally-verifiable voting with everlasting privacy.”in advances in cryptology crypto 2006. [18] b. adida and c. a. neff.“efficient receipt-free ballot casting resistant to covertchannels.”in evt/wote, 2009. [19] m. stadler “publicly verifiable secret sharing”. proc. eurocrypt '96, pp. 190-199. 1996. [20] m. naor and a. shamir.“visual cryptography”.in eurocrypt, pp 112,1994. գաղտնիություն ապահովող էլեկտրոնային քվեարկության նոր մոտեցում ա. ջիվանյան ամփոփում մենք ներկայացնում ենք էլեկտրոնային քվեարկության նոր համակարգ՝ անվտանգության ձևական ապացուցվող հատկություններով: առաջարկված է քվեարկության ընդհանուր մոտեցում, որը թույլ է տալիս ընտրել վերջնական քվեարկության տարբեր մեթոդներ, ինչը հնարավորություն է տալիս ընտրել օգտագործողի տեսակետից ամենահարմար ինտերֆեյսը: новый подход к обеспечению секретного электронного голосования а. дживанян аннотация мы представляем новую систему электронного голосования с формально доказуемыми свойствами безопасности. предложен общий подход, позволяющий использовать различные конечные методы, что дает пользователям свободу выбора наиболее подходящего для них интерфейса. d:\sbornik\...\tpel.dvi mathematical problems of computer science 35, 33{36, 2011. e r r or p r obability e xponents and achievable region in t esting of m any h ypotheses for t wo i ndependent objects a r a m o. y e s s a ya n , e vg u e n i a . h a r o u t u n ia n a n d p a r a n d z e m m. h a ko b ya n institute for informatics and automation problems of nas of ra e-mail: evhar@ipia.sci.am abstract the model of many hypotheses testing for one objects was examined by e. tuncel. in the present work it is supposed that l hypothetical probability distributions are known and two objects independently each from other follow to one of them. n -vectors of values of discrete independent random variables represent results of n observations for each object. decisions concerning realized probability distributions of the objects must be made on the base of such samples. it is proved that de¯ned region for vector of error probability exponents \reliabilities for two objects completly characterizes set of all achivable vectors. refer ences [1 ] i. cs is z ¶a r a n d p . c. s h ie ld s , information theory and statistics: a tutorial. f oundations and trends in communications and information theory, vo lu m e 1 , n o . 4 , 2 0 0 4 . [2 ] t. m. co ve r a n d j. a . th o m a s , e lements of information theory, second e dition, w ile y, n e w y o r k, 2 0 0 6 . [3 ] r . e . b e c h h o fe r , j. k ie fe r a n d m. s o b e l, sequential identi¯cation and r anking p rocedures. th e u n ive r s it y o f ch ic a g o , p r e s s , 1 9 6 8 . [4 ] r . e . b la h u t , \ h yp o t h e s e s t e s t in g a n d in fo r m t io n t h e o r y," ie e e transaction on information theory, vo l 2 0 , p p . 4 0 5 -4 1 7 , 1 9 7 4 . [5 ] r . e . b la h u t , p rinciples and p ractice of information theory. r e a d in g , ma : a d d is o n w e s le y, 1 9 8 7 . [6 ] e .tu n c e l, \ on e r r o r e xp o n e n t s in h yp o t h e s is t e s t in g " , ie e e trans. inf. theory, vo l. 5 1 , n o . 8 , p p . 2 9 4 5 -2 9 5 0 , 2 0 0 5 . [7 ] e . h a r o u t u n ia n , \ l o g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f m u lt ip le s t a t is t ic a l h yp o t h e s e s " , p roblems of control and information theory, vo l. 1 9 ( 5 -6 ) , p p . 4 1 3 { 4 2 1 , 1 9 9 0 . [8 ] r . f. a h ls we d e a n d e . a . h a r o u t u n ia n , \ on lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f h yp o t h e s e s a n d id e n t i¯ c a t io n " . l ecture notes in computer science, vol. 4123, \general theory of information transfer and combinatorics", springer, p p . 4 6 2 { 4 7 8 , 2 0 0 6 . 3 3 3 4 error probability exponents and achievable region in testing of many hypotheses for two independent objects [9 ] e . h a r o u t u n ia n , \ r e lia b ilit y in m u lt ip le h yp o t h e s e s t e s t in g a n d id e n t i¯ c a t io n " , p roceedings of nato asi, yerevan, 2003, nato science series iii: computer and system sciences, vo l. 1 9 8 , p p . 1 8 9 -2 0 1 , ios p r e s s , 2 0 0 5 . [1 0 ] e . h a r o u t u n ia n a n d p . h a ko b ya n , \ mu lt ip le h yp o t h e s e s l a o t e s t in g fo r m a n y in d e p e n d e n t o b je c t s " , international j ournal scholarly r esearch e xchange, vo l. 2 0 0 9 , p p . 1 -6 , 2 0 0 9 . [1 1 ] e . h a r o u t u n ia n a n d a . y e s s a ya n , \ on o p t im a l t e s t in g o f t h r e e h yp o t h e s e s fo r t wo d e p e n d e n t o b je c t s " , m athematical p roblems of computer sciences, vo l. x x v i, p p . 8 9 -9 4 , 2 0 0 6 . ºñïáõ ³ýï³ë ûµû»ïïý»ñç í»ñ³µ»ñû³é µ³½ù³ãçí í³ñï³íý»ñç ï»ëï³íáñù³ý ëë³éý»ñç ñ³í³ý³ï³ýáõãûáõýý»ñç óáõóçãý»ñá ². ºë³û³ý, º. ð³ñáõãûáõýû³ý ¨ ö. ð³ïáµû³ý ²ù÷á÷áõù ðá¹í³íáõù »ýã³¹ñíáõù ¿, áñ l ñ³í³ý³ï³ý³ûçý µ³ßëáõùý»ñá ñ³ûïýç »ý, çëï ûµû»ïïý»ñçó ûáõñ³ù³ýãûáõñá ³ýï³ë ù»ïá ùûáõëçó ï³ñáõ »ý µ³ßëí³í éçý»é ïñí³íý»ñçó ûáõñ³ù³ýãûáõñáí: úµû»ïïý»ñç µ³ßëí³íáõãû³ý í»ñ³µ»ñû³é áñáßáõùý»ñý áý¹áõýíáõù »ý »ñïáõ ûµû»ïïý»ñç n -³ï³ý ³ýï³ë ¹çï³ñïáõùý»ñç ³ñ¹ûáõýùý»ñç ñçù³ý íñ³: ðá¹í³íáõù ³å³óáõóí»é ¿, áñ ³ýï³ë ûµû»ïïý»ñç ñáõë³éçáõãûáõýý»ñç (ëë³éý»ñç ñ³í³ý³ï³ýáõãûáõýý»ñç óáõóçãý»ñç) í»ïïáñý ³ùµáõçáõãû³ùµ µýáõã³·ñáõù ¿ ñ³ë³ý»éç ïáãíáõ í»ïïáñý»ñç µ³½ùáõãûáõýá: d:\sbornik\...\tpel.dvi mathematical problems of computer science 36, 140{145, 2012. n umer ical solution of 1d schr äodinger e quation at adiabatically changing p otential a s h o t s . ge vo r kya n 1;2 a n d mis a k g. n a lb a n d ya n 1 1institute for informatics and automation problems, nas of armenia 2joint institute of nuclear research, 141980 dubna, moscow reg., russia e-mail g ashot@sci.am, habajyan@ipia.sci.am abstract we study the eigenfunction and eigenvalue problem of 1d schräodinger equation with adiabatically changing along the reaction coordinate (external parameter) mors potential. as an example the 2d interaction potential of the collinear reactive collision h ¡ h ¡ h is calculated which later is ¯tted by the generalized 2d mors potential. it is shown that the vibration state of body system are characterized by the set of ¯ve orthonormalized wavefunctions and corresponding energies which are slowly changed along the curve of the reaction coordinate. the mentioned problem is solved also with taking into account rotation motion of bodies system. it is shown that the solution of previous problem in this case is insigni¯cantly modi¯ed. keywords: bodies system, eigenfunction and eigenvalue problem of schräodinger equation, ¯tting, mors potential. refer ences [1 ] a . s . ge vo r kya n , g. g. b a lin t -k u r t i a n d g. n ym a n : n o ve l a lg o r it h m fo r s im u la t io n o f 3 d qu a n t u m r e a c t ive a t o m -d ia t o m s c a t t e r in g . p r o c e d ia cs 1 ( 1 ) , p p . 1 1 9 5 -1 2 0 , 2 0 1 0 . 1 0 .1 0 1 6 / j.p r o c s .2 0 1 0 .0 4 .1 3 3 [2 ] s . flg g e , p ractical quantum mechanics i,. b e r lin , s p r in g e r , 1 9 7 4 . ²¹ç³µ³ïçï ÷á÷áëíáõ åáï»ýóç³éáí 1 d þñ»¹çý·»ñç ñ³í³ë³ñù³ý ãí³ûçý éáõíáõù ². ¶¨áñ·û³ý ¨ ø. ü³éµ³ý¹û³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù áõëáõùý³ëçñí³í ¿ ³ñï³ùçý ¹³ßïç ³éï³ûáõãû³ùµ ï³ñµ»ñ »ñï³ñáõãû³ùµ 1 d ãï³ñ·³íáñí³í ï³ñ³í³ï³ý ëåçý³ûçý ßõã³ý»ñç (îêþ) ñ³ùáõûãç íç×³ï³·ñ³ï³ý ñ³ïïáõãûáõýý»ñá‘ ñ³ßíç ³éý»éáí é»é³ùë³óçáý »ñ¨áõûãý»ñá: ²é³ççý ³ý·³ù û·ï³·áñíí»é ¿ ïáùåé»ùë-¹³ë³ï³ý ð³ùçéïáýç³ýá: ä³ñµ»ñ³ï³ý 1 d ó³ýóç ñ³ý·áõûóý»ñáõù ëï³óí»é »ý é»ïáõñ»ýï »é³ýïûáõý³ã³÷³ï³ý ñ³í³ë³ñáõùý»ñ, áñáýù 1 4 0 a. gevorkyan and m. nalbandyan 1 4 1 êçéí»ëïñç å³ûù³ýý»ñç ñ»ï ùç³ëçý ³ý³éçïçïáñ»ý ß³ñáõý³ïíáõù »ý ïáùåé»ùë ï³ñ³íáõãû³ý ù»ç ¨ ñý³ñ³íáñáõãûáõý »ý ï³éçë ñ³ý·áõûó ³é ñ³ý·áõûó ñ³ßí»é ëåçýç áõõõáñ¹í³íáõãûáõýá‘ ñ³ßíç ³éý»éáí ëåçý³ûçý ßõã³ý»ñáõù é»é³ùë³óçáý »ñ¨áõûãý»ñá: àõëáõùý³ëçñí³í »ý ý³¨ ëåçý³ûçý ñ³ùáõûãáõù ï»õç áõý»óáõ áñáß³ïç ïñçïçï³ï³ý »ñ¨áõûãý»ñ, çýãåçëçù »ý îé³áõ½çáõë-øáëëáïçç (î-ø) ñ³í³ë³ñù³ý ù»ç ³õ»ïý»ñá‘ ï³ëí³í ³ñï³ùçý ¹³ßïç ù»íáõãûáõýçó: ²é³ç³ñïí³í ¿ íç×³ï³·ñ³ï³ý ·áõù³ñç ýáñ ý»ñï³û³óáõù í»ñç³íáñ ãíáí çýï»·ñ³é³ûçý ³ñï³ñ³ûïáõãû³ý ï»ëùáí‘ ¿ý»ñ·ç³ûç ¨ µ¨»é³óí³íáõãû³ý ï³ñ³íáõãûáõýáõù: ×èñëåííîå ðåøåíèå 1 d óðàâíåíèÿ øðåäèíãåðà ñ àäèàáàòè÷åñêè èçìåíÿþùèìñÿ ïîòåíöèàëîì à. ãåâîðêÿí è ì. íàëáàíäÿí àííîòàöèÿ ìû èññëåäóåì ïðîáëåìó ñîáñòâåííûõ ôóíêöèé è ñîáñòâåííûõ çíà÷åíèé äëÿ 1 d óðàâíåíèÿ øðåäèíãåðà ñ àäèàáàòè÷åñêè èçìåíÿþùèìñÿ âäîëü êîîðäèíàòû ðåàêöèè (âíåøíûé ïàðàìåòåð) ïîòåíöèàëîì ìîðñà. â êà÷åñòâå ïðèìåðà âû÷èñëåí 2 d ïîòåíöèàë âçàèìîäåéñòâèÿ êîëëèíåàðíîãî ðåàêòèâíîãî ñòîëêíîâåíèÿ h ¡ h ¡ h, êîòîðûé äàëåå àïïðîêñèìèðîâàí îáîáùåííûì 2 d ïîòåíöèàëîì ìîðñà. ïîêàçàíî, ÷òî êîëåáàòåëüíîå ñîñòîÿíèå ñèñòåìû òåë õàðàêòåðèçóåòñÿ ìíîæåñòâîì èç ïÿòè îðòîíîðìèðîâàííûõ âîëíîâûõ ôóíêöèé è ñîîòâåòñòâóþùèõ ýíåðãèé, êîòîðûå ìåäëåííî ìåíÿþòñÿ âäîëü êðèâîé êîîðäèíàòû ðåàêöèè. óêàçàííàÿ ïðîáëåìà ðåøàåòñÿ òàê æå ñ ó÷åòîì âðàùàòåëüíîãî äâèæåíèÿ ñèñòåìû òåë. ïîêàçàíî, ÷òî â ýòîì ñëó÷àå ðåøåíèå ïðåäûäóùåé ïðîáëåìû íåçíà÷èòåëüíî ìîäèôèöèðóåòñÿ. nikoghosyan_article.dvi mathematical problems of computer science 50, 15{34, 2018. a shar p i mpr ovement of a t heor em of b auer and schmeichel zh o r a g. n iko g h o s ya n institute for informatics and automation problems of nas ra e-mail: zhora@ipia.sci.am abstract let g be a graph on n vertices with minimum degree ±. the earliest nontrivial lower bound for the circumference c (the length of a longest cycle in g) was established in 1952 due to dirac in terms of n and ±: (i) if g is a 2-connected graph, then c ¸ minfn; 2±g. the bound in theorem (i) is sharp. in 1986, bauer and schmeichel gave a version of this classical result for 1-tough graphs: (ii) if g is a 1-tough graph, then c ¸ minfn; 2±+2g. in this paper we present an improvement of (ii), which is sharp for each n: (iii) if g is a 1-tough graph, then c ¸ minfn; 2± + 2g when n ´ 1(mod 3); c ¸ minfn; 2± + 3g when n ´ 2(mod 3) or n ´ 1(mod 4); and c ¸ minfn; 2± + 4g otherwise. keywords: hamilton cycle; circumference; minimum degree; 1-tough graphs. 1 . in t r o d u c t io n th r o u g h o u t t h is a r t ic le we c o n s id e r o n ly ¯ n it e u n d ir e c t e d g r a p h s wit h o u t lo o p s o r m u lt ip le e d g e s . th e s e t o f ve r t ic e s o f a g r a p h g is d e n o t e d b y v ( g ) a n d t h e s e t o f e d g e s b y e ( g) . w e u s e n, ± a n d c t o d e n o t e t h e o r d e r o f g, t h e m in im u m d e g r e e a n d t h e c ir c u m fe r e n c e t h e le n g t h o f a lo n g e s t c yc le in g, r e s p e c t ive ly. a g o o d r e fe r e n c e fo r a n y u n d e ¯ n e d t e r m s is [2 ]. th e e a r lie s t n o n t r ivia l lo we r b o u n d fo r t h e c ir c u m fe r e n c e wa s e s t a b lis h e d in 1 9 5 2 d u e t o d ir a c [4 ] in t e r m s o f n a n d ±: t heor em a [4]: if g is a 2-connected graph, then c ¸ m in fn; 2 ±g. th e b o u n d 2 ± in th e o r e m a is s h a r p . in 1 9 7 3 , ch v¶a t a l [3 ] in t r o d u c e d t h e c o n c e p t o f t o u g h n e s s . s in c e t h e n a lo t o f r e s e a r c h h a s b e e n d o n e t o wa r d s ¯ n d in g t h e e xa c t a n a lo g s o f c la s s ic a l h a m ilt o n ia n r e s u lt s u n d e r a d d it io n a l 1 -t o u g h c o n d it io n in s t e a d o f 2 -c o n n e c t ivit y a n a lt e r n a t ive a n d s t r o n g e r n e c e s s a r y c o n d it io n fo r a g r a p h t o b e h a m ilt o n ia n . th e a n a lo g o f t h e c la s s ic a l th e o r e m a fo r 1 -t o u g h g r a p h s wa s e s t a b lis h e d b y b a u e r a n d s c h m e ic h e l ( [1 ], 1 9 8 6 ) . 1 5 1 6 a sharp improvement of a theorem of bauer and schmeichel t heor em b [1]: if g is a 1-tough graph, then c ¸ m in fn; 2 ± + 2 g. th e b o u n d 2 ± + 2 in th e o r e m b is s h o wn [1 ] t o b e s h a r p b y c o n s t r u c t in g g r a p h s o f o r d e r n ´ 1 ( mod 3 ) wit h c = 2 ± + 2 . in t h is p a p e r we s h o w t h a t t h e b o u n d 2 ± + 2 in th e o r e m b is s h a r p if a n d o n ly if n ´ 1 ( mod 3 ) . fu r t h e r m o r e , we p r e s e n t a s h a r p r e ¯ n e m e n t o f th e o r e m b , wh ic h is s h a r p fo r e a c h n. t heor em 1: e very 1-tough graph is either hamiltonian, or c ¸ 8 >< >: 2 ± + 2 when n ´ 1 ( mod 3 ) ; 2 ± + 3 when n ´ 2 ( mod 3 ) or n ´ 1 ( mod 4 ) ; 2 ± + 4 otherwise: to s e e t h a t th e o r e m 1 is s h a r p fo r e a c h n, le t h1; h2; :::; hh b e d is jo in t c o m p le t e g r a p h s wit h d is t in c t ve r t ic e s xi; yi 2 v ( hi ) ( i = 1 ; 2 ; :::; h) . fo r m a n e w g r a p h h ( t1; t2; :::; th ) b y id e n t ifyin g t h e ve r t ic e s x1; x2; :::; xh a n d a d d in g a ll p o s s ib le e d g e s b e t we e n y1; y2; :::; yh, wh e r e ti = jv ( hi ) j ( i = 1 ; 2 ; :::; h) . th e g r a p h h ( ± + 1 ; ± + 1 ; ± + 1 ) s h o ws t h a t t h e b o u n d 2 ± + 2 in th e o r e m 1 c a n n o t b e r e p la c e d b y 2 ± + 3 wh e n n ´ 1 ( mod 3 ) . n e xt , t h e g r a p h s h ( ± + 2 ; ± + 1 ; ± + 1 ) a n d h ( ± + 1 ; ± + 1 ; ± + 1 ; ± + 1 ) s h o w t h a t t h e b o u n d 2 ± + 3 c a n n o t b e r e p la c e d b y 2 ±+4 wh e n n ´ 2 ( mod 3 ) o r n ´ 1 ( mod 4 ) . fin a lly, t h e g r a p h h ( ±+2 ; ±+2 ; ±+1 ) s h o ws t h a t t h e b o u n d 2 ± + 4 c a n n o t b e r e p la c e d b y 2 ± + 5 . 2 . n o t a t io n s a n d p r e lim in a r ie s l e t g b e a g r a p h . fo r s a s u b s e t o f v ( g) , we d e n o t e b y gns t h e m a xim u m s u b g r a p h o f g wit h ve r t e x s e t v ( g) ns. w e wr it e hsi fo r t h e s u b g r a p h o f g in d u c e d b y s. fo r a s u b g r a p h h o f g we u s e gnh s h o r t fo r gnv ( h ) . th e n e ig h b o r h o o d a n d t h e d e g r e e o f a ve r t e x x 2 v ( g) will b e d e n o t e d b y n ( x) a n d d( x ) , r e s p e c t ive ly. fu r t h e r m o r e , fo r a s u b g r a p h h o f g a n d x 2 v ( g) , we d e ¯ n e nh ( x ) = n ( x ) \ v ( h ) a n d dh ( x ) = jnh ( x ) j. l e t s( g ) d e n o t e t h e n u m b e r o f c o m p o n e n t s o f a g r a p h g. a g r a p h g is 1 -t o u g h if jsj ¸ s( gns ) fo r e ve r y s u b s e t s o f t h e ve r t e x s e t v ( g) wit h s( gns ) > 1 . a g r a p h g o n n ve r t ic e s is h a m ilt o n ia n if g c o n t a in s a h a m ilt o n c yc le , i.e ., a c yc le o f le n g t h n. p a t h s a n d c yc le s in a g r a p h g a r e c o n s id e r e d a s s u b g r a p h s o f g. if q is a p a t h o r a c yc le , t h e n t h e le n g t h o f q, d e n o t e d b y jqj, is je ( q ) j. w e wr it e q wit h a g ive n o r ie n t a t io n b y ¡! q . fo r x; y 2 v ( q ) , we d e n o t e b y x¡!q y t h e s u b p a t h o f q in t h e c h o s e n d ir e c t io n fr o m x t o y. fo r x 2 v ( c ) , we d e n o t e t h e h-t h s u c c e s s o r a n d t h e h-t h p r e d e c e s s o r o f x o n ¡!c b y x+h a n d x¡h, r e s p e c t ive ly. w e a b b r e via t e x+1 a n d x¡1 b y x+ a n d x¡, r e s p e c t ive ly. fo r e a c h x ½ v ( c ) , we d e ¯ n e x+ = fx+jx 2 xg a n d x¡ = fx¡jx 2 xg. special de¯nitions. l e t g b e a g r a p h , c a lo n g e s t c yc le in g a n d p = x ¡! p y a lo n g e s t p a t h in gnc o f le n g t h p ¸ 0 . l e t »1; »2; :::; »s b e t h e e le m e n t s o f nc ( x ) [ nc ( y ) o c c u r r in g o n ¡! c in a c o n s e c u t ive o r d e r . s e t ii = »i ¡! c »i+1; i ¤ i = » + i ¡! c »¡i+1 ( i = 1 ; 2 ; :::; s) ; wh e r e »s+1 = »1. zh. nikoghosyan 1 7 ( 1 ) th e s e g m e n t s i1; i2; :::; is a r e c a lle d e le m e n t a r y s e g m e n t s o n c in d u c e d b y nc ( x ) [ nc ( y ) . ( 2 ) w e c a ll a p a t h l = z ¡! l w a n in t e r m e d ia t e p a t h b e t we e n t wo d is t in c t e le m e n t a r y s e g m e n t s ia a n d ib, if z 2 v ( i¤a ) ; w 2 v ( i¤b ) ; v ( l) \ v ( c [ p ) = fz; wg: ( 3 ) d e ¯ n e (̈ ii1; ii2; :::; iit ) t o b e t h e s e t o f a ll in t e r m e d ia t e p a t h s b e t we e n e le m e n t a r y s e g m e n t s ii1 ; ii2 ; :::; iit . ( 4 ) if (̈ i1; :::; is ) µ e, t h e n t h e m a xim u m n u m b e r o f in t e r m e d ia t e in d e p e n d e n t e d g e s ( n o t h a vin g a c o m m o n ve r t e x) in (̈ i1; :::; is ) will b e d e n o t e d b y ¹ ( )̈ . ( 5 ) w e s a y t h a t t wo in t e r m e d ia t e in d e p e n d e n t e d g e s w1w2; w3w4 h a ve a c r o s s in g , if e it h e r w1; w3; w2; w4 o r w1; w4; w2; w3 o c c u r o n ¡! c in a c o n s e c u t ive o r d e r . lemma 1: l et g be a graph, c a longest cycle in g and p = x ¡! p y a longest path in gnc of length p ¸ 1 . if jnc ( x ) j ¸ 2 , jnc ( y ) j ¸ 2 and nc ( x) 6= nc ( y ) , then c ¸ ( 3 ± + m a xf¾1; ¾2g ¡ 1 ¸ 3 ± if p = 1 ; 4 ± ¡ 2 p if p ¸ 2 ; where ¾1 = jnc ( x ) nnc ( y ) j and ¾2 = jnc ( y ) nnc ( x ) j. lemma 2: l et g be a graph, c a longest cycle in g and p = x ¡! p y a longest path in gnc of length p ¸ 0 . l et nc ( x ) = nc ( y ) , jnc ( x ) j ¸ 2 and f; g 2 f 1 ; :::; sg. (a1) if l 2 (̈ if ; ig ) , then jif j + jigj ¸ 2 p + 2 jlj + 4 : (a2) if (̈ if ; ig ) µ e ( g ) and j (̈ if ; ig ) j = " for some " 2 f 1 ; 2 ; 3 g, then jif j + jigj ¸ 2 p + " + 5 ; (a3) if (̈ if ; ig ) µ e ( g ) and (̈ if ; ig ) contains two independent intermediate edges, then jifj + jigj ¸ 2 p + 8 : th e fo llo win g r e s u lt is d u e t o v o s s [5 ]. lemma 3 [5]: l et g be a hamiltonian graph, fv1; v2; :::; vtg µ v ( g ) and d( vi ) ¸ t ( i = 1 ; 2 ; :::; t ) . then each pair x; y of vertices of g is connected in g by a path of length at least t. 1 8 a sharp improvement of a theorem of bauer and schmeichel 3 . p r o o fs p r oof of lemma 1. p u t a1 = nc ( x ) nnc ( y ) ; a2 = nc ( y ) nnc ( x ) ; m = nc ( x) \ nc ( y ) : b y t h e h yp o t h e s is , nc ( x ) 6= nc ( y ) , im p lyin g t h a t m a xfja1j; ja2jg ¸ 1 : l e t »1; »2; :::; »s b e t h e e le m e n t s o f nc ( x ) [ nc ( y ) o c c u r r in g o n ¡! c in a c o n s e c u t ive o r d e r . p u t ii = »i ¡! c »i+1 ( i = 1 ; 2 ; :::; s) , wh e r e »s+1 = »1. cle a r ly, s = ja1j + ja2j + jmj. s in c e c is e xt r e m e , we h a ve jiij ¸ 2 ( i = 1 ; 2 ; :::; s) . n e xt , if f»i; »i+1g\m 6= ; fo r s o m e i 2 f 1 ; 2 ; :::; sg, t h e n jiij ¸ p + 2 . fu r t h e r , if e it h e r »i 2 a1, »i+1 2 a2 o r »i 2 a2, »i+1 2 a1, t h e n a g a in jiij ¸ p + 2 . case 1. p = 1 . case 1.1. jaij ¸ 1 ( i = 1 ; 2 ) . it fo llo ws t h a t a m o n g i1; i2; :::; is t h e r e a r e jmj + 2 s e g m e n t s o f le n g t h a t le a s t p + 2 . ob s e r vin g a ls o t h a t e a c h o f t h e r e m a in in g s ¡ ( jmj + 2 ) s e g m e n t s h a s a le n g t h a t le a s t 2 , we h a ve c ¸ ( p + 2 ) ( jmj + 2 ) + 2 ( s ¡ jmj ¡ 2 ) = 3 ( jmj + 2 ) + 2 ( ja1j + ja2j ¡ 2 ) = 2 ja1j + 2 ja2j + 3 jmj + 2 : s in c e ja1j = d( x ) ¡ jmj ¡ 1 a n d ja2j = d( y ) ¡ jmj ¡ 1 , we h a ve c ¸ 2 d ( x) + 2 d( y ) ¡ jmj ¡ 2 ¸ 3 ± + d ( x) ¡ jmj ¡ 2 : r e c a llin g t h a t d( x ) = jmj + ja1j + 1 , we g e t c ¸ 3 ± + ja1j ¡ 1 = 3 ± + ¾1 ¡ 1 : a n a lo g o u s ly, c ¸ 3 ± + ¾2 ¡ 1 . s o , c ¸ 3 ± + m a xf¾1; ¾2g ¡ 1 ¸ 3 ±: case 1.2. e it h e r ja1j ¸ 1 ; ja2j = 0 o r ja1j = 0 ; ja2j ¸ 1 . a s s u m e w.l.o .g . t h a t ja1j ¸ 1 a n d ja2j = 0 , i.e . jnc ( y ) j = jmj ¸ 2 a n d s = ja1j + jmj. h e n c e , a m o n g i1; i2; :::; is t h e r e a r e jmj + 1 s e g m e n t s o f le n g t h a t le a s t p + 2 = 3 . ta kin g in t o a c c o u n t t h a t jmj + 1 = d( y ) a n d e a c h o f t h e r e m a in in g s ¡ ( jmj + 1 ) s e g m e n t s h a s a le n g t h a t le a s t 2 , we g e t c ¸ 3 ( jmj + 1 ) + 2 ( s ¡ jmj ¡ 1 ) = 3 d( y ) + 2 ( ja1j ¡ 1 ) ¸ 3 ± + ja1j ¡ 1 = 3 ± + m a xf¾1; ¾2g ¡ 1 ¸ 3 ±: case 2. p ¸ 2 . case 2.1. jaij ¸ 1 ( i = 1 ; 2 ) . it fo llo ws t h a t a m o n g i1; i2; :::; is t h e r e a r e jmj + 2 s e g m e n t s o f le n g t h a t le a s t p + 2 . fu r t h e r , s in c e e a c h o f t h e r e m a in in g s ¡ ( jmj + 2 ) s e g m e n t s h a s a le n g t h a t le a s t 2 , we g e t c ¸ ( p + 2 ) ( jmj + 2 ) + 2 ( s ¡ jmj ¡ 2 ) zh. nikoghosyan 1 9 = ( p ¡ 2 ) jmj + ( 2 p + 4 jmj + 4 ) + 2 ( ja1j + ja2j ¡ 2 ) ¸ 2 ja1j + 2 ja2j + 4 jmj + 2 p: ob s e r vin g a ls o t h a t ja1j + jmj + p ¸ d ( x) ; ja2j + jmj + p ¸ d( y ) ; we h a ve 2 ja1j + 2 ja2j + 4 jmj + 2 p ¸ 2 d( x ) + 2 d( y ) ¡ 2 p ¸ 4 ± ¡ 2 p; im p lyin g t h a t c ¸ 4 ± ¡ 2 p. case 2.2. e it h e r ja1j ¸ 1 ; ja2j = 0 o r ja1j = 0 ; ja2j ¸ 1 . a s s u m e w.l.o .g . t h a t ja1j ¸ 1 a n d ja2j = 0 , t h a t is jnc ( y ) j = jmj ¸ 2 a n d s = ja1j+jmj. it fo llo ws t h a t a m o n g i1; i2; :::; is t h e r e a r e jmj+1 s e g m e n t s o f le n g t h a t le a s t p+2 . ob s e r vin g a ls o t h a t jmj + p ¸ d( y ) ¸ ±, i.e ., 2 p + 4 jmj ¸ 4 ± ¡ 2 p, we g e t c ¸ ( p + 2 ) ( jmj + 1 ) ¸ ( p ¡ 2 ) ( jmj ¡ 1 ) + 2 p + 4 jmj ¸ 2 p + 4 jmj ¸ 4 ± ¡ 2 p: p r oof of lemma 2. l e t »1; »2; :::; »s b e t h e e le m e n t s o f nc ( x ) o c c u r r in g o n ¡! c in a c o n s e c u t ive o r d e r . p u t ii = »i ¡! c »i+1 ( i = 1 ; 2 ; :::; s) , wh e r e »s+1 = »1: to p r o ve ( a1 ) , le t l 2 (̈ if ; ig ) . fu r t h e r , le t l = z ¡! l w wit h z 2 v ( i¤f ) a n d w 2 v ( i¤g ) . p u t j»f ¡! c zj = d1; jz ¡! c »f+1j = d2; j»g ¡! c wj = d3; jw ¡! c »g+1j = d4; c0 = »f x ¡! p y»g ã¡ c z ¡! l w ¡! c »f : cle a r ly, jc0j = jcj ¡ d1 ¡ d3 + jlj + jp j + 2 : s in c e c is e xt r e m e , we h a ve jcj ¸ jc0j, im p lyin g t h a t d1 + d3 ¸ p + jlj + 2 . b y a s ym m e t r ic a r g u m e n t , d2 + d4 ¸ p + jlj + 2 . h e n c e jif j + jigj = 4x i=1 di ¸ 2 p + 2 jlj + 4 : th e p r o o f o f ( a1 ) is c o m p le t e . to p r o ve ( a 2 ) a n d ( a 3 ) , le t (̈ if ; ig ) µ e ( g) a n d j (̈ if ; ig ) j = " fo r s o m e " 2 f 1 ; 2 ; 3 g. case 1. " = 1 . l e t l 2 (̈ if ; ig ) , wh e r e jlj = 1 . b y ( a 1 ) , jif j + jigj ¸ 2 p + 2 jlj + 4 = 2 p + 6 : case 2. " = 2 . 2 0 a sharp improvement of a theorem of bauer and schmeichel it fo llo ws t h a t (̈ if ; ig ) c o n s is t s o f t wo e d g e s e1; e2. p u t e1 = z1w1 a n d e2 = z2w2, wh e r e fz1; z2g µ v ( i¤f ) a n d fw1; w2g µ v ( i¤g ) . case 2.1. z1 6= z2 a n d w1 6= w2. a s s u m e w.l.o .g . t h a t z1 a n d z2 o c c u r in t h is o r d e r o n if . case 2.1.1. w2 a n d w1 o c c u r in t h is o r d e r o n ig. p u t j»f ¡! c z1j = d1; jz1 ¡! c z2j = d2; jz2 ¡! c »f +1j = d3; j»g ¡! c w2j = d4; jw2 ¡! c w1j = d5; jw1 ¡! c »g+1j = d6; c0 = »f ¡! c z1w1 ã¡ c w2z2 ¡! c »gx ¡! p y»g+1 ¡! c »f : cle a r ly, jc0j = jcj ¡ d2 ¡ d4 ¡ d6 + jfe1gj + jfe2gj + jp j + 2 = jcj ¡ d2 ¡ d4 ¡ d6 + p + 4 : s in c e c is e xt r e m e , we h a ve jcj ¸ jc0j, im p lyin g t h a t d2 + d4 + d6 ¸ p + 4 . b y a s ym m e t r ic a r g u m e n t , d1 + d3 + d5 ¸ p + 4 . h e n c e jifj + jigj = 6x i=1 di ¸ 2 p + 8 : case 2.1.2. w1 a n d w2 o c c u r in t h is o r d e r o n ig. p u t t in g c0 = »f ¡! c z1w1 ¡! c w2z2 ¡! c »gx ¡! p y»g+1 ¡! c »f ; we c a n a r g u e a s in ca s e 2 .1 .1 . case 2.2. e it h e r z1 = z2, w1 6= w2 o r z1 6= z2, w1 = w2. a s s u m e w.l.o .g . t h a t z1 6= z2, w1 = w2 a n d z1; z2 o c c u r in t h is o r d e r o n if . p u t j»f ¡! c z1j = d1; jz1 ¡! c z2j = d2; jz2 ¡! c »f +1j = d3; j»g ¡! c w1j = d4; jw1 ¡! c »g+1j = d5; c0 = »f x ¡! p y»g ã¡ c z1w1 ¡! c »f ; c00 = »f ¡! c z2w1 ã¡ c »f+1x ¡! p y»g+1 ¡! c »f : cle a r ly, jc0j = jcj ¡ d1 ¡ d4 + jfe1gj + jp j + 2 = jcj ¡ d1 ¡ d4 + p + 3 ; jc00j = jcj ¡ d3 ¡ d5 + jfe2gj + jp j + 2 = jcj ¡ d3 ¡ d5 + p + 3 : s in c e c is e xt r e m e , jcj ¸ jc0j a n d jcj ¸ jc00j, im p lyin g t h a t d1 + d4 ¸ p + 3 ; d3 + d5 ¸ p + 3 : zh. nikoghosyan 2 1 h e n c e , jif j + jigj = 5x i=1 di ¸ d1 + d3 + d4 + d5 + 1 ¸ 2 p + 7 : case 3. " = 3 . it fo llo ws t h a t (̈ if ; ig ) c o n s is t s o f t h r e e e d g e s e1; e2; e3. l e t ei = ziwi ( i = 1 ; 2 ; 3 ) , wh e r e fz1; z2; z3g µ v ( i¤f ) a n d fw1; w2; w3g µ v ( i¤g ) . if t h e r e a r e t wo in d e p e n d e n t e d g e s a m o n g e1; e2; e3, t h e n we c a n a r g u e a s in ca s e 2 .1 . ot h e r wis e , we c a n a s s u m e w.l.o .g . t h a t w1 = w2 = w3 a n d z1; z2; z3 o c c u r in t h is o r d e r o n if . p u t j»f ¡! c z1j = d1; jz1 ¡! c z2j = d2; jz2 ¡! c z3j = d3; jz3 ¡! c »f+1j = d4; j»g ¡! c w1j = d5; jw1 ¡! c »g+1j = d6; c0 = »f x ¡! p y»g ã¡ c z1w1 ¡! c »f ; c00 = »f ¡! c z3w1 ã¡ c »f+1x ¡! p y»g+1 ¡! c »f : cle a r ly, jc0j = jcj ¡ d1 ¡ d5 + jfe1gj + p + 2 ; jc00j = jcj ¡ d4 ¡ d6 + jfe3gj + p + 2 : s in c e c is e xt r e m e , we h a ve jcj ¸ jc0j a n d jcj ¸ jc00j, im p lyin g t h a t d1 + d5 ¸ p + 3 ; d4 + d6 ¸ p + 3 : h e n c e , jif j + jigj = 6x i=1 di ¸ d1 + d4 + d5 + d6 + 2 ¸ 2 p + 8 : p r oof of t heor em 1. l e t g b e a 1 -t o u g h g r a p h . if c ¸ 2 ± + 4 , t h e n we a r e d o n e . h e n c e , we c a n a s s u m e t h a t c · 2 ± + 3 : ( 1 ) l e t c b e a lo n g e s t c yc le in g a n d p = x1 ¡! p x2 a lo n g e s t p a t h in gnc. p u t jp j = jv ( p ) j ¡ 1 = p. if jv ( p ) j = 0 , t h e n c is a h a m ilt o n c yc le a n d we a r e d o n e . l e t jv ( p ) j ¸ 1 , t h a t is p ¸ 0 . p u t x = nc ( x1 ) [ nc ( x2 ) a n d le t »1; :::; »s b e t h e e le m e n t s o f x o c c u r r in g o n c in a c o n s e c u t ive o r d e r . p u t ii = »i ¡! c »i+1; i ¤ i = » + i ¡! c »¡i+1 ( i = 1 ; :::; s) ; wh e r e »s+1 = »1: s in c e g is a 1 -t o u g h g r a p h , we h a ve ± ¸ 2 . case 1. p · ± ¡ 2 . it fo llo ws t h a t s ¸ jnc ( xi ) j ¸ ± ¡ p ¸ 2 ( i = 1 ; 2 ) . a s s u m e ¯ r s t t h a t nc ( x1 ) 6= nc ( x2 ) , im p lyin g t h a t p ¸ 1 . if p ¸ 2 , t h e n b y l e m m a 1 , c ¸ 4 ± ¡ 2 p ¸ 2 ± + 4 , c o n t r a d ic t in g ( 1 ) . h e n c e p = 1 , wh ic h yie ld s ± ¸ p + 2 = 3 . b y l e m m a 1 , c ¸ 3 ± ¸ 9 . if ± ¸ 4 , t h e n c ¸ 3 ± ¸ 2 ± + 4 , c o n t r a d ic t in g ( 1 ) . l e t ± = 3 . n e xt , we c a n s u p p o s e t h a t c = 9 , s in c e o t h e r wis e c ¸ 1 0 = 3 ± + 1 = 2 ± + 4 , c o n t r a d ic t in g ( 1 ) . fu r t h e r , we c a n s u p p o s e t h a t s ¸ 3 , s in c e nc ( x1 ) = nc ( x2 ) wh e n s = 2 , c o n t r a d ic t in g t h e h yp o t h e s is . fin a lly, we c a n s u p p o s e 2 2 a sharp improvement of a theorem of bauer and schmeichel t h a t s = 3 , s in c e c le a r ly c ¸ 1 0 wh e n s ¸ 4 , a c o n t r a d ic t io n . th u s , ji1j = ji2j = ji3j = 3 a n d it is n o t h a r d t o s e e t h a t gnf»1; »2; »3g h a s a t le a s t fo u r c o m p o n e n t s , c o n t r a d ic t in g ¿ ¸ 1 . n o w a s s u m e t h a t nc ( x1 ) = nc ( x2 ) . s in c e c is e xt r e m e , we h a ve jiij ¸ j»ix1 ¡! p x2»i+1j ¸ p + 2 ( i = 1 ; :::; s) : case 1.1. s ¸ ± ¡ p + 1 . cle a r ly, c = sx i=1 jiij ¸ s( p + 2 ) ¸ ( ± ¡ p + 1 ) ( p + 2 ) = ( ± ¡ p ¡ 2 ) p + 2 ± + p + 2 : ( 2 ) if p ¸ 2 , t h e n b y ( 2 ) , c ¸ 2 ± + 4 , c o n t r a d ic t in g ( 1 ) . l e t p · 1 . case 1.1.1. p = 0 . if (̈ i1; :::; is ) = ;, t h e n gnf»1; :::; »sg h a s a t le a s t s + 1 c o m p o n e n t s , c o n t r a d ic t in g t h e fa c t t h a t ¿ ¸ 1 . ot h e r wis e (̈ ia; ib ) 6= ; fo r s o m e d is t in c t a; b 2 f 1 ; :::; sg. l e t l 2 (̈ ia; ib ) . b y l e m m a 2 ( a 1 ) , jiaj + jibj ¸ 2 p + 2 jlj + 4 ¸ 6 : r e c a llin g a ls o t h a t s ¸ ± ¡ p + 1 = ± + 1 , we g e t c = sx i=1 jiij ¸ jiaj + jibj + 2 ( s ¡ 2 ) = 2 s + 2 ¸ 2 ± + 4 ; c o n t r a d ic t in g ( 1 ) . case 1.1.2. p = 1 . b y ( 2 ) , c ¸ 3 ±. w e c a n s u p p o s e t h a t ± · 3 , s in c e c ¸ 3 ± ¸ 2 ± + 4 wh e n ± ¸ 4 , c o n t r a d ic t in g ( 1 ) . on t h e o t h e r h a n d , b y t h e h yp o t h e s is , ± ¸ p + 2 = 3 , im p lyin g t h a t ± = 3 . b y t h e h yp o t h e s is , s ¸ ± ¡ p + 1 = 3 . n e xt , we c a n s u p p o s e t h a t s = 3 , s in c e c ¸ s( p + 2 ) ¸ 1 2 = 2 ± + 6 wh e n s ¸ 4 , c o n t r a d ic t in g ( 1 ) . fu r t h e r , if (̈ i1; i2; i3 ) = ;, t h e n gnf»1; »2; »3g h a s a t le a s t fo u r c o m p o n e n t s , c o n t r a d ic t in g ¿ ¸ 1 . ot h e r wis e (̈ ia; ib ) 6= ; fo r s o m e d is t in c t a; b 2 f1 ; 2 ; 3 g, s a y a = 1 a n d b = 2 . l e t l 2 (̈ i1; i2 ) . b y l e m m a 2 ( a 1 ) , ji1j + ji2j ¸ 2 p + 2 jlj + 4 = 8 ; wh ic h yie ld s c ¸ ji1j + ji2j + ji3j ¸ 1 1 = 2 ± + 5 , c o n t r a d ic t in g ( 1 ) . case 1.2. s = ± ¡ p. it fo llo ws t h a t x1x2 2 e. th e n x1x2 ã¡ p x+1 is a n o t h e r lo n g e s t p a t h in gnc. w e c a n s u p p o s e t h a t nc ( x1 ) = nc ( x + 1 ) , s in c e o t h e r wis e we c a n a r g u e a s in ca s e 1 . b y t h e s a m e r e a s o n , nc ( x1 ) = nc ( x + 1 ) = nc ( x +2 1 ) = ::: = nc ( x2 ) : s in c e c is e xt r e m e , we h a ve jiij ¸ j»ix1 ¡! p x2»i+1j = p + 2 ( i = 1 ; :::; s) . if (̈ i1; :::; is ) = ;, t h e n gnf»1; :::; »sg h a s a t le a s t s + 1 c o m p o n e n t s , c o n t r a d ic t in g ¿ ¸ 1 . ot h e r wis e (̈ ia; ib ) 6= zh. nikoghosyan 2 3 ; fo r s o m e d is t in c t a; b 2 f 1 ; :::; sg. l e t l 2 (̈ ia; ib ) wit h l = z1 ¡! l z2, wh e r e z1 2 v ( i¤a ) a n d z2 2 v ( i¤b ) . b y l e m m a 2 ( a 1 ) , jiaj + jibj ¸ 2 p + 6 . h e n c e c = sx i=1 jiij ¸ jiaj + jibj + ( s ¡ 2 ) ( p + 2 ) ¸ s( p + 2 ) + 2 = ( ± ¡ p) ( p + 2 ) + 2 = 2 ± + 2 + p ( ± ¡ p ¡ 2 ) : ( 3 ) claim 1. ( a1 ) 2 p + 6 · jiaj + jibj · 2 p + 7 a n d jiij · p + 5 ( i = 1 ; :::; s) . ( a 2 ) if jiaj + jibj = 2 p + 7 , t h e n jiij = p + 2 fo r e a c h i 2 f 1 ; :::; sgnfa; bg. ( a 3 ) if jiaj + jibj = 2 p + 6 , t h e n jif j · p + 3 fo r s o m e f 2 f1 ; :::; sgnfa; bg a n d jiij = p + 2 fo r e a c h i 2 f 1 ; :::; sgnfa; b; fg. ( a 4 ) if jifj = p + 5 fo r s o m e f 2 fa; bg, t h e n jiij = p + 2 fo r e a c h i 2 f1 ; :::; sgnffg. ( a 5 ) fo r e a c h d is t in c t f; g; h 2 f 1 ; :::; sg, jif j + jigj + jihj · 3 p + 9 . ( a 6 ) (̈ i1; :::; is ) µ e. p r oof. if jif j ¸ p + 6 fo r s o m e f 2 f1 ; :::; sg, t h e n c = sx i=1 jiij ¸ jif j + ( s ¡ 1 ) ( p + 2 ) ¸ s( p + 2 ) + 4 = 2 ± + 4 + p ( ± ¡ p ¡ 2 ) ¸ 2 ± + 4 ; c o n t r a d ic t in g ( 1 ) . n e xt , if jiaj + jibj ¸ 2 p + 8 , t h e n c ¸ jiaj + jibj + ( s ¡ 2 ) ( p + 2 ) ¸ s( p + 2 ) + 4 ¸ 2 ± + 4 ; a g a in c o n t r a d ic t in g ( 1 ) . h e n c e ( a1 ) h o ld s . s t a t e m e n t s ( a2 ) ¡ ( a 4 ) c a n b e p r o ve d b y a s im ila r wa y. to p r o ve ( a 5 ) , a s s u m e t h e c o n t r a r y, t h a t is jif j + jigj + jihj ¸ 3 p + 1 0 fo r s o m e d is t in c t f; g; h 2 f1 ; :::; sg. th e n c = sx i=1 jiij ¸ jif j + jigj + jihj + ( s ¡ 3 ) ( p + 2 ) ¸ 3 ( p + 2 ) + 4 + ( s ¡ 3 ) ( p + 2 ) = 2 ± + 4 + p( s ¡ 2 ) ¸ 2 ± + 4 ; c o n t r a d ic t in g ( 1 ) . s t a t e m e n t ( a6 ) fo llo ws fr o m l e m m a 2 ( a 1 ) a n d cla im 1 ( a 1 ) . cla im 1 is p r o ve d . claim 2. p + 3 · d1 · p + 4 a n d p + 3 · d2 · p + 4 , wh e r e d1 = j»a ¡! c z1j + j»b ¡! c z2j; d2 = jz1 ¡! c »a+1j + jz2 ¡! c »b+1j: p r oof. p u t q = »ax1 ¡! p x2»b ã¡ c z1z2 ¡! c »a: cle a r ly, jqj = jcj¡d1+p+3 . s in c e c is e xt r e m e , we h a ve jcj ¸ jqj, im p lyin g t h a t d1 ¸ p+3 . b y a s ym m e t r ic a r g u m e n t , d2 ¸ p + 3 . b y cla im 1 ( a 1 ) , jiaj + jibj = d1 + d2 · 2 p + 7 . if d1 ¸ p + 5 , t h e n 2 p + 7 ¸ d1 + d2 ¸ p + 5 + d2, im p lyin g t h a t d2 · p + 2 , a c o n t r a d ic t io n . h e n c e , d1 · p + 4 . b y a s ym m e t r ic a r g u m e n t , d2 · p + 4 . cla im 2 is p r o ve d . claim 3. if v1 2 v ( »+a ¡! c z¡1 ) a n d v2 2 v ( z+1 ¡! c »¡a+1 ) , t h e n v1v2 62 e. 2 4 a sharp improvement of a theorem of bauer and schmeichel p r oof. a s s u m e t h e c o n t r a r y, t h a t is v1v2 2 e. p u t q = »a ¡! c v1v2 ã¡ c z1z2 ã¡ c »a+1x1 ¡! p x2»b+1 ¡! c »a; j»a ¡! c v1j = d1; jv1 ¡! c z1j = d2; jz1 ¡! c v2j = d3; jv2 ¡! c »a+1j = d4; j»b ¡! c z2j = d5; jz2 ¡! c »b+1j = d6: cle a r ly, jqj = jcj ¡ d2 ¡ d4 ¡ d6 + p + 4 . s in c e c is e xt r e m e , we h a ve jqj · jcj, im p lyin g t h a t d2 + d4 + d6 ¸ p + 4 . b y a s ym m e t r ic a r g u m e n t , d1 + d3 + d5 ¸ p + 4 . b y s u m m in g , we g e t 6x i=1 di = jiaj + jibj ¸ 2 p + 8 ; c o n t r a d ic t in g cla im 1 ( a 1 ) . th u s , v1v2 62 e. cla im 3 is p r o ve d . claim 4. l e t »f ; »g; »h o c c u r o n ¡! c in a c o n s e c u t ive o r d e r fo r s o m e f; g; h 2 f 1 ; :::; sg a n d w1w2 2 e fo r s o m e w1 2 v ( i¤f ) a n d w2 2 v ( i¤g ) . if n ( w3 ) \ f»f+1; »gg 6= ; fo r s o m e w3 2 v ( i¤h ) , t h e n jw1 ¡! c »f +1j + j»g ¡! c w2j + j»h ¡! c w3j ¸ p + 4 : fu r t h e r , if n ( w4 ) \ f»f +1; »gg 6= ; fo r s o m e w4 2 v ( i¤h¡1 ) , t h e n jw1 ¡! c »f +1j + j»g ¡! c w2j + jw4 ¡! c »hj ¸ p + 4 : p r oof. a s s u m e ¯ r s t t h a t w3»f+1 2 e. p u t q = »f ¡! c w1w2 ¡! c »hx1 ¡! p x2»g ã¡ c »f +1w3»f : cle a r ly, jqj = jcj ¡ jw1 ¡! c »f +1j ¡ j»g ¡! c w2j ¡ j»h ¡! c w3j + p + 4 : s in c e jqj · jcj, t h e d e s ir e d r e s u lt h o ld s im m e d ia t e ly. if w4»f +1 2 e, t h e n we c a n u s e t h e fo llo win g c yc le q0 = »f ¡! c w1w2 ¡! c w4»f +1 ¡! c »gx2 ã¡ p x1»h ¡! c »f in s t e a d o f q. b y a s ym m e t r ic a r g u m e n t , t h e d e s ir e d r e s u lt h o ld s wh e n e it h e r w3»g 2 e o r w4»g 2 e. cla im 4 is p r o ve d . claim 5. e ve r y t wo in t e r m e d ia t e in d e p e n d e n t e d g e s e1; e2 in (̈ i1; :::; is ) h a ve a c r o s s in g wit h e1; e2 2 (̈ if ; ig; ih ) fo r s o m e d is t in c t f; g; h 2 f 1 ; :::; sg. p r oof. l e t e1 = w1w2 a n d e2 = w3w4. w e d is t in g u is h t h r e e d i®e r e n t c a s e s . fir s t , if e1; e2 2 (̈ if ; ig ) fo r s o m e d is t in c t f; g, t h e n b y l e m m a 2 ( a 3 ) , jif j + jigj ¸ 2 p + 8 , c o n t r a d ic t in g cla im 1 ( a 1 ) . n e xt , if e1 2 (̈ if ; ig ) a n d e2 2 (̈ ih; ir ) fo r s o m e d is t in c t f; g; h; r, t h e n b y l e m m a 2 ( a 1 ) , jifj + jigj ¸ 2 p + 6 a n d jihj + jirj ¸ 2 p + 6 , im p lyin g t h a t c ¸ jif j + jigj + jihj + jirj + ( s ¡ 4 ) ( p + 2 ) = 4 p + 1 2 + ( s ¡ 4 ) ( p + 2 ) = s( p + 2 ) + 4 = 2 ± + 4 + p( ± ¡ p ¡ 2 ) ¸ 2 ± + 4 ; wh ic h a g a in c o n t r a d ic t s ( 1 ) . fin a lly, le t e1 2 (̈ if ; ig ) a n d e2 2 (̈ if ; ih ) fo r s o m e d is t in c t f; g; h. a s s u m e w.l.o .g . t h a t »f ; »g; »h o c c u r o n ¡! c in a c o n s e c u t ive o r d e r a n d w1; w3 2 v ( i¤f ) , zh. nikoghosyan 2 5 w2 2 v ( i¤g ) , w4 2 v ( i¤h ) . w e c a n a s s u m e a ls o t h a t w3 a n d w1 o c c u r o n if in a c o n s e c u t ive o r d e r , s in c e o t h e r wis e e1 a n d e2 h a ve a c r o s s in g a n d we a r e d o n e . p u t q = »f ¡! c w3w4 ã¡ c w2w1 ¡! c »gx2 ã¡ p x1»h+1 ¡! c »f ; j»f ¡! c w3j = d1; jw3 ¡! c w1j = d2; jw1 ¡! c »f+1j = d3; j»g ¡! c w2j = d4; jw2 ¡! c »g+1j = d5; j»h ¡! c w4j = d6; jw4 ¡! c »h+1j = d7: cle a r ly, jqj = jcj¡d2 ¡d4 ¡d7 +p + 4 . s in c e c is e xt r e m e , we h a ve jqj · jcj, im p lyin g t h a t d2 + d4 + d7 ¸ p + 4 . on t h e o t h e r h a n d , b y l e m m a 2 , d3 + d5 ¸ p + 3 a n d d1 + d6 ¸ p + 3 . b y s u m m in g , we g e t p7 i=1 di = jif j + jigj + jihj ¸ 3 p + 1 0 . th e n jcj ¸ jif j + jigj + jihj + ( s ¡ 3 ) ( p + 2 ) = s( p + 2 ) + 4 ¸ 2 ± + 4 ; c o n t r a d ic t in g ( 1 ) . cla im 5 is p r o ve d . claim 6. if ¹( )̈ = 1 , t h e n s · 3 a n d e it h e r »+a »¡b+1 2 e wit h »a = »b+1 o r »¡a+1»+b 2 e wit h »a+1 = »b. if ¹ ( )̈ = 1 a n d s = 3 , t h e n ji1j = ji2j = ji3j = p + 3 . p r oof. s in c e ¹( )̈ = 1 , e it h e r o n e o f t h e ve r t ic e s z1; z2, s a y z1, is a c o m m o n ve r t e x fo r a ll e d g e s in (̈ i1; :::; is ) o r z1z3; z2z3 2 (̈ i1; :::; is ) fo r s o m e z3 2 v ( i¤f ) a n d f 2 f 1 ; :::; sgnfa; bg. case a1. z1 is a c o m m o n ve r t e x fo r a ll e d g e s in (̈ i1; :::; is ) . if z1 62 f»+a ; »¡a+1g, t h e n b y cla im 3 , gnf»1; :::; »s; z1g h a s a t le a s t s + 2 c o m p o n e n t s , c o n t r a d ic t in g ¿ ¸ 1 . l e t z1 2 f»+a ; »¡a+1g, s a y z1 = »+a . case a1.1. z1» ¡ b+1 62 e. it fo llo ws t h a t z2 6= »¡b+1. b y cla im 2 , j»b ¡! c z2j ¸ p + 2 . case a1.1.1. z1» ¡2 b+1 62 e. it fo llo ws t h a t jibj ¸ p + 5 . b y cla im 1 ( a 1 ) , jiaj = p + 2 . mo r e o ve r , we h a ve jibj = p + 5 , j»b ¡! c z2j = p + 2 , z2 = »¡3b+1 a n d n ( z1 ) \ v ( i¤b ) = fz2g. b y cla im 1 ( a 4 ) , jiij = p + 2 fo r e a c h i 2 f 1 ; :::; sgnfbg. n e xt , b y l e m m a 2 ( a 1 ) , (̈ ia; ii ) = ; fo r e a c h i 2 f 1 ; :::; sgnfa; bg. th u s , if z1y 2 (̈ i1; :::; is ) , t h e n y = z2, im p lyin g t h a t (̈ i1; :::; is ) = fz1z2g. b e s id e s , s in c e j»b ¡! c z2j = p + 2 ¸ 2 , we h a ve z2 62 f»+b ; »¡b+1g. th e r e fo r e , b y cla im 3 , gnf»1; :::; »s; z2g h a s a t le a s t s + 2 c o m p o n e n t s , c o n t r a d ic t in g ¿ ¸ 1 . case a1.1.2. z1» ¡2 b+1 2 e. it fo llo ws t h a t jibj ¸ p + 4 . a s s u m e ¯ r s t t h a t jibj = p + 5 . if z1»¡3b+1 62 e, t h e n c le a r ly z2 = » ¡2 b+1 a n d we c a n a r g u e a s in ca s e a 1 .1 .1 . ot h e r wis e t h e fo llo win g c yc le »ax1 ¡! p x2»a+1 ¡! c »¡3b+1z1» ¡2 b+1 ¡! c »a is lo n g e r t h a n c, a c o n t r a d ic t io n . n o w a s s u m e t h a t jibj = p + 4 , t h a t is j»b ¡! c »¡2b+1j = p + 2 . if z1y 2 e fo r s o m e y 2 v ( »b ¡! c »¡3b+1 ) , t h e n b y cla im 2 , j»b ¡! c yj ¸ p+2 , im p lyin g t h a t jibj ¸ p+5 , a c o n t r a d ic t io n . h e n c e , if z1y 2 (̈ ia; ib ) , t h e n c le a r ly y = »¡2b+1. in p a r t ic u la r , we h a ve z2 = »¡2b+1. fu r t h e r , if z1y 2 (̈ ia; if ) fo r s o m e f 2 f 1 ; :::; sgnfbg, t h e n b y l e m m a 2 ( a 1 ) , jiaj + jifj ¸ 2 p + 6 , 2 6 a sharp improvement of a theorem of bauer and schmeichel t h a t is jiaj + jibj + jif j ¸ 3 p + 1 0 , c o n t r a d ic t in g cla im 1 ( a 5 ) . th u s , z2 is a c o m m o n ve r t e x fo r a ll e d g e s in (̈ i1; :::; is ) . b y cla im 3 , gnf»1; :::; »s; z2g h a s a t le a s t s + 2 c o m p o n e n t s , c o n t r a d ic t in g ¿ ¸ 1 . case a1.2. »+a » ¡ b+1 2 e. b y cla im 2 , j»+a ¡! c »a+1j ¸ p + 2 a n d j»b ¡! c »¡b+1j ¸ p + 2 . if j»+a ¡! c »a+1j ¸ p + 3 a n d j»b ¡! c »¡b+1j ¸ p + 3 , t h e n jiaj + jibj ¸ 2 p + 8 , c o n t r a d ic t in g cla im 1 ( a 1 ) . h e n c e , we c a n a s s u m e w.l.o .g . t h a t j»b ¡! c »¡b+1j = p + 2 , t h a t is jibj = p + 3 a n d jiaj ¸ p + 3 . fu r t h e r , we h a ve »+b »a; » + b »b+1 62 e ( b y cla im 4 ) a n d »+b »+a 62 e ( b y cla im 2 ) . case a1.2.1. n ( »+b ) 6µ v ( c ) . l e t q = »+b ¡! q v b e a lo n g e s t p a t h in g wit h v ( q ) \v ( c ) = f»+b g. s in c e c is e xt r e m e , we h a ve v ( q ) \v ( p ) = ;. n e xt , s in c e p is a lo n g e s t p a t h in gnc, we h a ve jqj · p+1 . fu r t h e r , r e c a llin g t h a t »+b »a; » + b »b+1; » + b » + a 62 e ( s e e ca s e a 1 .2 ) , we c o n c lu d e t h a t v»a; v»b+1; v»+a 62 e, a s we ll. if vy 62 e fo r e a c h y 2 ( »+2b ¡! c »¡b+1 ) , t h e n c le a r ly n ( v ) µ ( v ( q ) [ f»1; :::; »sg ) nf»a; »b+1»+a g; t h a t is d( v ) · jqj + s ¡ 2 · p + s ¡ 1 = ± ¡ 1 , a c o n t r a d ic t io n . n o w le t vy 2 e fo r s o m e y 2 v ( »+2b ¡! c »¡b+1 ) . a s s u m e t h a t y is c h o s e n s o a s t o m in im iz e j»+b ¡! c yj. s in c e c is e xt r e m e , we h a ve j»+b ¡! c yj ¸ jqj + 1 . fu r t h e r , s in c e jn ( v ) \ v ( y¡!c »¡b+1 ) j ¸ ± ¡ ( s ¡ 2 ) ¡ jqj; we h a ve j»+b ¡! c »¡b+1j ¸ jqj + 1 + 2 ( ± ¡ s + 1 ¡ jqj ) = 2 ± ¡ jqj ¡ 2 s + 3 ¸ 2 ± ¡ p ¡ 2 s + 2 = p + 2 : b u t t h e n jibj ¸ p + 4 , a c o n t r a d ic t io n . case a1.2.2. n ( »+b ) µ v ( c ) . s in c e ¹ ( )̈ = 1 a n d »+b » + a 62 e, we h a ve n ( »+b ) µ v ( »+2b ¡! c »¡b+1 ) [ f»1; :::; »sgnf»a; »b+1g: if »a 6= »b+1, t h e n d( »+b ) · p + s ¡ 1 = ± ¡ 1 , a c o n t r a d ic t io n . h e n c e »a = »b+1. case a1.2.2.1. jifj = p + 2 fo r s o m e f 2 f 1 ; :::; sgnfa; bg. if n ( »+f ) µ v ( c ) , t h e n a s in d ic a t e d a b o ve , d( »+f ) · s ¡ 1 + j»+f ¡! c »¡f+1j = p + s ¡ 1 = ± ¡ 1 ; a c o n t r a d ic t io n . if n ( »+f ) 6µ v ( c ) , t h e n we c a n a r g u e a s in ca s e a 1 .2 .1 . case a1.2.2.2. jiij ¸ p + 3 fo r e a c h i 2 f 1 ; :::; sgnfa; bg. if s ¸ 4 , t h e n jcj = sx i=1 jiij ¸ s( p + 3 ) = ( ± ¡ p ) ( p + 3 ) zh. nikoghosyan 2 7 = 2 ± + 2 p + 4 + ( ± ¡ p ¡ 4 ) ( p + 1 ) ¸ 2 ± + 4 ; c o n t r a d ic t in g ( 1 ) . h e n c e , s · 3 . mo r e o ve r , if s = 3 , t h e n b y cla im 1 ( a 5 ) , ji1j = ji2j = ji3j = p + 3 . case a2. z1z3; z2z3 2 (̈ i1; :::; is ) , wh e r e z3 2 v ( i¤f ) a n d f 2 f 1 ; :::; sgnfa; bg. a s s u m e w.l.o .g . t h a t »a; »b; »f o c c u r o n ¡! c in a c o n s e c u t ive o r d e r . p u t j»a ¡! c z1j = d1; jz1 ¡! c »a+1j = d2; j»b ¡! c z2j = d3; jz2 ¡! c »b+1j = d4; j»f ¡! c z3j = d5; jz3 ¡! c »f +1j = d6: b y cla im 2 , d1 + d3 ¸ p + 3 ; d1 + d5 ¸ p + 3 ; d2 + d4 ¸ p + 3 ; d2 + d6 ¸ p + 3 ; d3 + d5 ¸ p + 3 ; d4 + d6 ¸ p + 3 : s u m m in g u p , we g e t 2 6x i=1 di = 2 ( jiaj + jibj + jifj ) ¸ 6 ( p + 3 ) : on t h e o t h e r h a n d , b y cla im 1 ( a 5 ) , jiaj + jibj + jif j · 3 ( p + 3 ) , im p lyin g t h a t d1 = d2 = ::: = d6 = ( p + 3 ) = 2 a n d p is o d d . h e n c e di ¸ 2 a n d u s in g cla im 3 , we c a n s t a t e t h a t gnf»1; :::; »s; z1; z2g h a s a t le a s t s + 3 c o m p o n e n t s , c o n t r a d ic t in g ¿ ¸ 1 . cla im 6 is p r o ve d . claim 7. e it h e r ¹ ( )̈ = 1 o r ¹ ( )̈ = 3 . p r oof. th e p r o o f is b y c o n t r a d ic t io n . if ¹( )̈ = 0 , t h e n gnf»1; :::; »sg h a s a t le a s t s + 1 c o m p o n e n t s , c o n t r a d ic t in g ¿ ¸ 1 . l e t ¹( )̈ ¸ 1 . case a1. ¹ = 2 . b y cla im 5 , (̈ i1; :::; is ) c o n s is t s o f t wo c r o s s in g in t e r m e d ia t e in d e p e n d e n t e d g e s w1w2 2 (̈ if ; ig ) a n d w3w4 2 (̈ if ; ih ) fo r s o m e d is t in c t f; g; h. a s s u m e t h a t b o t h »f ; »g ; »h a n d w1; w3; w2; w4 o c c u r o n ¡! c in a c o n s e c u t ive o r d e r . p u t q = »f ¡! c w1w2 ¡! c w4w3 ¡! c »gx2 ã¡ p x1»h+1 ¡! c »f ; j»f ¡! c w1j = d1; jw1 ¡! c w3j = d2; jw3 ¡! c »f+1j = d3; j»g ¡! c w2j = d4; jw2 ¡! c »g+1j = d5; j»h ¡! c w4j = d6; jw4 ¡! c »h+1j = d7: cle a r ly, jqj = jcj ¡ d2 ¡ d4 ¡ d7 + p + 4 . s in c e jqj · jcj, we h a ve d2 + d4 + d7 ¸ p + 4 . if d3 + d6 ¸ p + 3 a n d d1 + d5 ¸ p + 3 , t h e n p7 i=1 di = jif j + jigj + jihj ¸ 3 p + 1 0 , c o n t r a d ic t in g cla im 1 ( a 5 ) . ot h e r wis e , e it h e r d3 + d6 · p + 2 o r d1 + d5 · p + 2 , s a y d3 + d6 · p + 2 . fu r t h e r , if e it h e r d7 = 1 o r » ¡ h+1w3 2 e, t h e n b y cla im 2 , d3 ¸ p + 2 , t h a t is d3 + d6 ¸ p + 3 , a c o n t r a d ic t io n . h e n c e , d7 ¸ 2 a n d »¡h+1w3 62 e. b y cla im 4 , »¡h+1»f +1; »¡h+1»h 62 e. if jihj ¸ p + 4 , t h e n t a kin g in t o a c c o u n t t h a t jifj + jigj ¸ 2 p + 6 ( b y cla im 1 ( a 1 ) ) , we g e t jif j + jigj + jihj ¸ 3 p + 1 0 , c o n t r a d ic t in g cla im 1 ( a 5 ) . h e n c e , jihj · p + 3 . b y a s ym m e t r ic a r g u m e n t , jigj · p + 3 . case a1.1. n ( »¡h+1 ) µ v ( c ) . 2 8 a sharp improvement of a theorem of bauer and schmeichel if »¡h+1w2 62 e, t h e n r e c a llin g t h a t ¹( )̈ = 2 , we g e t n ( »¡h+1 ) µ v ( w4 ¡! c »¡2h+1 ) [ f»1; :::; »sgnf»f+1; »hg; im p lyin g t h a t jn ( »¡h+1 ) j · p + s ¡ 1 = ± ¡ 1 , a c o n t r a d ic t io n . n o w le t »¡h+1w2 2 e. b y cla im 1 ( a 1 a n d a 5 ) , jif j = jigj = jihj = p + 3 . mo r e o ve r , b y cla im 2 , d5 = p + 2 a n d d4 = 1 . th e n , fo r t h e s a m e r e a s o n , d1 = p + 2 , im p lyin g t h a t jiaj ¸ p + 4 , a c o n t r a d ic t io n . case a1.2. n ( »¡h+1 ) 6µ v ( c ) . w e c a n a r g u e a s in t h e p r o o f o f cla im 6 ( ca s e a 1 .2 .1 ) . case a2. ¹( )̈ ¸ 4 . b y cla im 5 , t h e r e a r e a t le a s t fo u r p a ir wis e c r o s s in g in t e r m e d ia t e in d e p e n d e n t e d g e s in (̈ i1; :::; is ) , wh ic h is im p o s s ib le . cla im 7 is p r o ve d . claim 8. if ¹ ( )̈ = 1 , t h e n e it h e r n ´ 1 ( mod 3 ) wit h c ¸ 2 ± + 2 o r n ´ 1 ( mod 4 ) wit h c ¸ 2 ± + 3 o r n ´ 2 ( mod 3 ) wit h c ¸ 2 ± + 3 . p r oof. b y cla im 6 , s · 3 a n d e it h e r »+a »¡b+1 2 e o r »¡a+1»+b 2 e, s a y »¡a+1»+b 2 e. case a1. s = 2 . it fo llo ws t h a t ± = p + s = p + 2 . l e t a = 1 a n d b = 2 . b y cla im 2 , j»1 ¡! c »¡2 j ¸ p + 2 a n d j»+2 ¡! c »1j ¸ p + 2 , im p lyin g t h a t jiij ¸ p + 3 ( i = 1 ; 2 ) . case a1.1. ji1j = p + 4 a n d ji2j = p + 3 . if v ( g ) = v ( c [ p ) , t h e n n = 3 p + 8 = 3 ± + 2 ´ 2 ( mod 3 ) wit h c = 2 p + 7 = 2 ± + 3 , a n d we a r e d o n e . ot h e r wis e n ( v1 ) 6µ v ( c [p ) fo r s o m e v1 2 v ( c [p ) . ob s e r vin g t h a t x1x2 2 e a n d r e c a llin g t h a t p is a lo n g e s t p a t h in v ( gnc ) , we c o n c lu d e t h a t v1 62 v ( p ) . ch o o s e a lo n g e s t p a t h q = v1 ¡! q v2 wit h v ( q ) \ v ( c ) = fv1g. cle a r ly, 1 · jqj · p + 1 = ± ¡ 1 a n d n ( v2 ) µ v ( c [ q) . case a1.1.1. v1 2 v ( »+22 ¡! c »¡1 ) . b y cla im 1 ( a 6 ) , n ( v2 ) \ v ( i¤1 ) = ;, t h a t is n ( v2 ) µ v ( i1 ) [ v ( q ) . a s s u m e t h a t v1 is c h o s e n s o a s t o m in im iz e jv1 ¡! c »1j, im p lyin g t h a t n ( v2 ) \ v ( v1 ¡! c »¡1 ) = ;. cle a r ly, jv1 ¡! c »1j · p + 1 . th e n b y cla im 4 , v1»2 62 e a n d t h e r e fo r e , v2»2 62 e, a s we ll. case a1.1.1.1. v2»1 2 e. it fo llo ws t h a t n ( v2 ) µ v ( q ) [ v ( »+2 ¡! c v¡1 ) [ f»1g. s in c e c is e xt r e m e a n d v2»1 2 e, we h a ve jv1 ¡! c »1j ¸ jqj+1 . if n ( v2 ) µ v ( q ) [f»1g, t h e n c le a r ly jqj ¸ ±¡ 1 = p+1 a n d t h e r e fo r e , jv1 ¡! c »1j ¸ p + 2 . b u t t h e n ji2j ¸ p + 4 , a c o n t r a d ic t io n . h e n c e , n ( v2 ) 6µ v ( q) [ f»1g, t h a t is v2y 2 e fo r s o m e y 2 v ( »+2 ¡! c v¡1 ) . a s s u m e t h a t y is c h o s e n s o a s t o m in im iz e jy ¡! c v1j. ob s e r vin g t h a t jy¡!c v1j ¸ jqj + 1 a n d ± = j»+2 ¡! c »1j ¸ 4 , we g e t j»+2 ¡! c »1j ¸ 2 ( jqj + 1 ) + 2 ( ± ¡ jqj ¡ 2 ) = 2 ± ¡ 2 ¸ ± + 2 = p + 4 ; a c o n t r a d ic t io n . case a1.1.1.2. v2»1 62 e. zh. nikoghosyan 2 9 it fo llo ws t h a t n ( v2 ) µ v ( q) [ v ( »+2 ¡! c v¡1 ) . if n ( v2 ) µ v ( q ) , t h e n jqj ¸ ± = p + 2 , a c o n t r a d ic t io n . ot h e r wis e v2y 2 e fo r s o m e y 2 v ( »+2 ¡! c v¡1 ) . a s s u m e t h a t y is c h o s e n s o a s t o m in im iz e jy¡!c v1j. s in c e jy ¡! c v1j ¸ jqj + 1 , we h a ve j»+2 ¡! c v1j ¸ jqj + 1 + 2 ( ± ¡ jqj ¡ 1 ) = 2 ± ¡ jqj ¡ 1 ¸ ± = p + 2 : b u t t h e n jibj ¸ 4 , a c o n t r a d ic t io n . case a1.1.2. v1 2 v ( »+1 ¡! c »¡32 ) . b y cla im 1 ( a 6 ) , n ( v2 ) \ v ( i¤2 ) = ;, t h a t is n ( v2 ) µ v ( q ) [ v ( i1 ) . a s s u m e t h a t v1 is c h o s e n s o a s t o m in im iz e j»1 ¡! c v1j, im p lyin g t h a t n ( v2 ) \ v ( »+1 ¡! c v¡1 ) = ;. cle a r ly, j»1 ¡! c v1j · p + 1 . th e n b y cla im 4 , v1»2 62 e a n d t h e r e fo r e , v2»2 62 e. case a1.1.2.1. »+2 » ¡2 2 2 e. b y cla im 3 , v1» ¡ 2 62 e, im p lyin g t h a t v2»¡2 62 e. case a1.1.2.1.1. v2»1 2 e. it fo llo ws t h a t n ( v2 ) µ v ( q ) [ v ( v1 ¡! c »¡22 ) [ f»1g. s in c e c is e xt r e m e a n d v2»1 2 e, we h a ve j»1 ¡! c v1j ¸ jqj + 1 . if n ( v2 ) µ v ( q) [ f»1g, t h e n jqj ¸ ± ¡ 1 = p + 1 a n d t h e r e fo r e , j»1 ¡! c v1j ¸ p + 2 . b u t t h e n ji1j ¸ p + 5 , a c o n t r a d ic t io n . h e n c e , n ( v2 ) 6µ v ( q) [ f»1g, t h a t is v2y 2 e fo r s o m e y 2 v ( v+1 ¡! c »¡22 ) . a s s u m e t h a t y is c h o s e n s o a s t o m in im iz e jv1 ¡! c yj. ob s e r vin g t h a t jv1 ¡! c yj ¸ jqj + 1 a n d ± = j»1 ¡! c »¡22 j ¸ 4 , we g e t j»1 ¡! c »¡22 j ¸ 2 ( jqj + 1 ) + 2 ( ± ¡ jqj ¡ 2 ) = 2 ± ¡ 2 ¸ ± + 2 = p + 4 ; a c o n t r a d ic t io n . case a1.1.2.1.2. v2»1 62 e. it fo llo ws t h a t n ( v2 ) µ v ( q) [ v ( v1 ¡! c »¡22 ) . if n ( v2 ) µ v ( q ) , t h e n jqj ¸ ± = p + 2 , a c o n t r a d ic t io n . ot h e r wis e v2y 2 e fo r s o m e y 2 v ( v+1 ¡! c »¡22 ) . b y c h o o s in g y s o a s t o m in im iz e jv1 ¡! c yj, we g e t jv1 ¡! c »¡22 j ¸ jqj + 1 + 2 ( ± ¡ jqj ¡ 1 ) = 2 ± ¡ jqj ¡ 1 ¸ ± = p + 2 : th is yie ld s jiaj ¸ p + 5 , a c o n t r a d ic t io n . case a1.1.2.2. »+2 » ¡2 2 62 e. if v2»1 2 e, t h e n a s in ca s e a 1 .1 .2 .1 .1 , j»1 ¡! c »¡2 j ¸ p +4 , c o n t r a d ic t in g t h e fa c t t h a t ji1j = p + 4 . ot h e r wis e , a s in ca s e a 1 .1 .2 .1 .2 , jv1 ¡! c »¡2 j ¸ p + 2 . s in c e ji1j = p + 4 , we h a ve v1 = »+1 , jqj = ± ¡ 1 = p + 1 a n d v3 = »¡2 . mo r e o ve r , we h a ve n ( v2 ) = ( v ( q ) [ f»¡2 g ) nfv2g. fu r t h e r , le t v b e a n a r b it r a r y ve r t e x in v ( q ) nfv1g. p u t q0 = v1 ¡! q v¡v2 ã¡ q v. s in c e q0 is a n o t h e r lo n g e s t p a t h wit h v ( q0 ) \ v ( c ) = fv1g, we c a n s u p p o s e t h a t n ( v ) = ( v ( q ) [ f»¡2 g ) nfvg fo r e a c h v 2 v ( q) nfv1g. fu r t h e r m o r e , if »1y 2 e fo r s o m e y 2 v ( »+21 ¡! c »¡22 ) , t h e n »1x1 ¡! p x2»2» + 2 » ¡ 2 v2 ã¡ q v1 ¡! c y»1 3 0 a sharp improvement of a theorem of bauer and schmeichel is lo n g e r t h a n c, a c o n t r a d ic t io n . h e n c e , »1y 62 e fo r e a c h y 2 v ( »+21 ¡! c »¡22 ) . a n a lo g o u s ly, if y»2 2 e fo r s o m e y 2 v ( »+1 ¡! c »¡22 ) , t h e n »1x1 ¡! p x2»2y ã¡ c »+1 ¡! q v2» ¡ 2 » + 2 ¡! c »1 is lo n g e r t h a n c, a c o n t r a d ic t io n . h e n c e , y»2 62 e fo r e a c h y 2 v ( »+1 ¡! c »¡22 ) . b u t t h e n gnf»+1 ; »¡2 g h a s a t le a s t t h r e e c o m p o n e n t s , c o n t r a d ic t in g ¿ ¸ 1 . case a1.1.3. v1 = » ¡2 2 . b y cla im 1 ( a 6 ) , n ( v2 ) µ v ( i1 ) . if v2y 2 e fo r s o m e y 2 v ( »+1 ¡! c v¡1 ) , t h e n we c a n a r g u e a s in ca s e a 1 .1 .2 . h e n c e , n ( v2 ) µ v ( q ) [ f»1; »2g. if v2»2 2 e, t h e n »1x1 ¡! p x2»2v2 ã¡ q v1» ¡ 2 » + 2 ¡! c »1 is lo n g e r t h a n c, a c o n t r a d ic t io n . th e n c le a r ly, v2»1 2 e a n d n ( v2 ) µ v ( q) [ f»1g. fu r t h e r m o r e , we h a ve jqj ¸ ± ¡ 1 , im p lyin g t h a t j»1 ¡! c v1j ¸ jqj + 1 ¸ ±. s in c e j»1 ¡! c v1j = ±, we h a ve jqj = ± ¡ 1 = p + 1 a n d n ( v2 ) = ( v ( q ) [ f»1g) nfv2g. mo r e o ve r , a s in ca s e 1 .1 .2 .2 , we h a ve n ( v ) = ( v ( q) [ f»1g ) nfvg fo r e a c h v 2 v ( q ) nfv1g. n o w c o n s id e r a n a r b it r a r y ve r t e x y 2 v ( »+1 ¡! c »¡32 ) . cle a r ly, j»1 ¡! c yj · p + 1 . b y cla im 2 , y»+2 62 e. n e xt , b y cla im 4 , y»2 62 e. fu r t h e r , if y»¡2 2 e, t h e n »1x1 ¡! p »2»2» + 2 » ¡ 2 y ¡! c »¡22 ¡! q v2»1 is lo n g e r t h a n c, a c o n t r a d ic t io n . fin a lly, s in c e ¹ ( )̈ = 1 , we h a ve yv 62 e fo r e a c h v 2 v ( »+22 ¡! c »¡1 ) . b u t t h e n gnf»1; »¡22 g h a s a t le a s t t h r e e c o m p o n e n t s , c o n t r a d ic t in g ¿ ¸ 1 . case a1.1.4. v1 = »1. if v2v3 2 e fo r s o m e v3 2 v ( »+22 ¡! c »¡1 ) [ v ( »+1 ¡! c »¡22 ) , t h e n we c a n a r g u e a s in ca s e s a 1 .1 .1 -a 1 .1 .3 . ot h e r wis e v2v3 2 e fo r s o m e v3 2 f»¡2 ; »+2 ; »2g. if v3 2 f»2; »+2 g, t h e n we c a n s h o w, a s in ca s e a 1 .1 .3 , t h a t gnf»1; v3g h a s a t le a s t t h r e e c o m p o n e n t s , c o n t r a d ic t in g ¿ ¸ 1 . n o w le t v3 = » ¡ 2 . co n s id e r a n a r b it r a r y ve r t e x v 2 v ( q) nfv1g. s in c e c is e xt r e m e , we h a ve n ( v ) \ f»2; »+2 g = ;. n e xt , if vy 2 e fo r s o m e y 2 v ( c ) nf»1; »2; »¡2 ; »+2 g, t h e n we c a n a r g u e a s in ca s e s a 1 .1 .1 -a 1 .1 .3 . th u s , we c a n a s s u m e t h a t n ( v ) µ v ( q) [ f»¡2 g, im p lyin g t h a t jqj ¸ ± ¡ 1 = p + 1 . l e t w 2 v ( »+1 ¡! c »¡32 ) . s in c e j»1 ¡! c wj · p + 1 , we h a ve w»+2 62 e ( b y cla im 2 ) a n d w»2 62 e ( b y cla im 4 ) . r e c a llin g a ls o t h a t ¹( )̈ = 1 , we c o n c lu d e t h a t n ( v ) µ v ( »1 ¡! c »¡2 ) . if » ¡2 2 »2; » ¡2 2 » + 2 62 e, t h e n c le a r ly gnf»1; »¡2 g h a s a t le a s t t h r e e c o m p o n e n t s , c o n t r a d ic t in g ¿ ¸ 1 . h e n c e , e it h e r »¡22 »2 2 e o r »¡22 »+2 2 e. case a1.1.4.1. »¡22 »2 2 e. if »¡22 » + 2 62 e, t h e n gnf»1; »2; »¡2 g h a s a t le a s t fo u r c o m p o n e n t s , c o n t r a d ic t in g ¿ ¸ 1 . h e n c e , »¡22 » + 2 2 e, t h a t is h»2; »¡2 ; »¡22 ; »+2 i is a c o m p le t e g r a p h . if v ( g) = v ( c [ p [ q) , t h e n n = 4 ± + 1 ´ 1 ( mod 4 ) wit h c = 2 ± + 3 , a n d we a r e d o n e . ot h e r wis e , a s in p r e vio u s c a s e s , we c a n s h o w t h a t ¿ < 1 , a c o n t r a d ic t io n . case a1.1.4.2. »¡22 » + 2 2 e. if »¡22 »2 62 e, t h e n gnf»1; »¡2 ; »+2 g h a s a t le a s t fo u r c o m p o n e n t s , c o n t r a d ic t in g ¿ ¸ 1 . ot h e r wis e h»2; »¡2 ; »¡22 ; »+2 i is a c o m p le t e g r a p h a n d we c a n a r g u e a s in ca s e a 1 .1 .4 .1 . zh. nikoghosyan 3 1 case a1.1.5. v1 2 f»2; »¡2 ; »+2 g. s in c e c is e xt r e m e , we h a ve v2 62 f»2; »¡2 ; »+2 g a n d t h e r e fo r e , we c a n a r g u e a s in ca s e s a 1 .1 .1 -1 .1 .4 . case a1.2. ji1j = ji2j = p + 3 . w e c a n s h o w t h a t n = 3 ± + 1 ´ 1 ( mod 3 ) wit h c = 2 ± + 2 , b y a r g u in g a s in ca s e a 1 .1 . case a2. s = 3 . b y cla im 6 , ji1j = ji2j = ji3j = p + 3 = ± a n d »¡2 »+2 2 e. if ± ¸ 4 , t h e n c = 3 ± ¸ 2 ± + 4 , c o n t r a d ic t in g ( 1 ) . h e n c e ± = 3 a n d t h e r e fo r e , p = 0 . p u t c = »1w1w2»2w3w4»3w5w6»1; wh e r e w2w3 2 e. u s in g cla im s 2 -5 , we c a n s h o w t h a t nc ( w1 ) = fw2; »1; »3g; nc ( w6 ) = fw5; »1; »3g: a n a lo g o u s r e la t io n s h o ld fo r w4; w5. if v ( gnc ) = fx1g, t h e n n = 1 0 ´ 1 ( mod 3 ) wit h c = 9 = 2 ± + 3 > 2 ± + 2 , a n d we a r e d o n e . ot h e r wis e n ( y ) = fv1; v2; v3g fo r s o m e y 2 v ( gnc ) nfx1g wit h n ( y ) µ v ( c ) . s in c e c is e xt r e m e , it is n o t h a r d t o s e e t h a t e it h e r n ( y ) = fw2; »1; »3g o r n ( y ) = fw3; »1; »3g o r n ( y ) = f»1; »2; »3g. b u t t h e n gnn ( y ) h a s a t le a s t fo u r c o m p o n e n t s , c o n t r a d ic t in g ¿ ¸ 1 . cla im 8 is p r o ve d . claim 9. if ¹ = 3 , t h e n g is t h e p e t e r s e n g r a p h , t h a t is n = 1 0 ´ 1 ( mod 3 ) wit h c ¸ 2 ± + 2 . p r oof. b y cla im 5 , (̈ i1; :::; is ) c o n t a in s t h r e e p a ir wis e c r o s s in g in t e r m e d ia t e in d e p e n d e n t e d g e s e1; e2; e3. l e t e1 = w1w2, e2 = w3w4 a n d e3 = w5w6. if w1; w3; w5 2 v ( i¤f ) fo r s o m e f 2 f 1 ; :::; sg, t h e n we c a n a r g u e a s in p r o o f o f cla im 7 . ot h e r wis e we c a n a s s u m e w.l.o .g . t h a t w1; w3 2 v ( i¤f ) , w2; w5 2 v ( i¤g ) a n d w4; w6 2 v ( i¤h ) fo r s o m e d is t in c t f; g; h 2 f 1 ; :::; sg, wh e r e b o t h »f ; »g; »h a n d w1; w3; w5; w2; w4; w6 o c c u r o n ¡! c in a c o n s e c u t ive o r d e r . b y cla im 1 ( a 1 a n d a 5 ) , jif j = jigj = jihj = p + 3 a n d jiij = p + 2 fo r e a c h i 2 f1 ; :::; sgnff; g; hg. p u t j»f ¡! c w1j = d1; jw1 ¡! c w3j = d2; jw3 ¡! c »f+1j = d3; j»g ¡! c w5j = d4; jw5 ¡! c w2j = d5; jw2 ¡! c »g+1j = d6; j»h ¡! c w4j = d7; jw4 ¡! c w6j = d8; jw6 ¡! c »h+1j = d9: if d3 + d7 ¸ p + 3 , d1 + d6 ¸ p + 3 a n d d4 + d9 ¸ p + 3 , t h e n c le a r ly jifj + jigj + jihj ¸ 3 p + 1 2 , a c o n t r a d ic t io n . ot h e r wis e we c a n a s s u m e w.l.o .g . t h a t d3 + d7 · p + 2 . fu r t h e r , if e it h e r d1 ¸ 2 o r d9 ¸ 2 , t h e n we c a n a r g u e a s in t h e p r o o f o f cla im 7 ( ca s e a 1 .1 ) . h e n c e , we c a n a s s u m e t h a t d1 = d9 = 1 . b y cla im 2 , d4 = d6 = 1 . fo r t h e s a m e r e a s o n , u s in g t h e fa c t t h a t d1 = d6 = 1 , we g e t d3 = d7 = 1 . case a1. e it h e r »h+1 6= »f o r »f +1 6= »g o r »g+1 6= »h. a s s u m e w.l.o .g . t h a t »h+1 6= »f , im p lyin g t h a t jif ¡1j = p + 2 . b y cla im 5 , »¡f y 62 e fo r e a c h y 2 v ( i¤i ) a n d i 2 f 1 ; :::; sgnff ¡ 1 g. mo r e o ve r , b y cla im 4 , »¡f y 62 e fo r e a c h y 2 f»f +1; »hg. if n ( »¡f ) µ v ( c ) , t h e n d( »¡f ) · ± ¡ 1 , a c o n t r a d ic t io n . ot h e r wis e we c a n 3 2 a sharp improvement of a theorem of bauer and schmeichel a r g u e a s in t h e p r o o f o f cla im 6 ( ca s e a 1 .2 .1 ) . case a2. »h+1 = »f , »f +1 = »g, »g+1 = »h. it fo llo ws t h a t s = 3 . a s s u m e w.l.o .g . t h a t f = 1 , g = 2 a n d h = 3 . case a2.1. e it h e r d2 ¸ 2 o r d5 ¸ 2 o r d8 ¸ 2 . a s s u m e w.l.o .g . t h a t d2 ¸ 2 , t h a t is w+1 6= w3. if p = 0 , t h e n ji1j = 3 , im p lyin g t h a t d2 = 1 , a c o n t r a d ic t io n . l e t p ¸ 1 . b y cla im 4 , w+1 »2; w+1 »3 62 e. if n ( w+1 ) µ v ( c ) , t h e n b y cla im 4 , n ( w+1 ) µ v ( w+21 ¡! c w3 ) [ f»1g. s in c e ji1j = p + 3 , we h a ve jw+1 ¡! c w3j · p. b u t t h e n d( w+1 ) · p + 1 = ± ¡ 2 , a c o n t r a d ic t io n . if n ( w+1 ) 6µ v ( c ) , t h e n we c a n a r g u e a s in t h e p r o o f o f cla im 6 ( ca s e a 1 .2 .1 ) . case a2.2. d2 = d5 = d8 = 1 . it fo llo ws t h a t jiij = 3 ( i = 1 ; 2 ; 3 ) , t h a t is p = 0 , ± = 3 a n d c = 9 . cle a r ly hv ( c ) [ fx1gi is t h e p e t e r s e n g r a p h . if v ( gnc ) 6= fx1g, t h e n it is n o t h a r d t o s e e t h a t c ¸ 1 0 , a c o n t r a d ic t io n . ot h e r wis e , n = 1 0 ´ 1 ( mod 3 ) wit h c = 9 = 2 ± + 3 > 2 ± + 2 . cla im 9 is p r o ve d . th u s , t h e r e s u lt h o ld s fr o m cla im s 7 ,8 ,9 . case 2. p = ± ¡ 1 . cle a r ly, jnc ( xi ) j ¸ 1 ( i = 1 ; 2 ) . case 2.1. x1y1; x2y2 2 e fo r s o m e d is t in c t y1; y2 2 v ( c ) . w e d is t in g u is h t h r e e m a in s u b c a s e s . case 2.1.1. th e r e e xis t s a p a t h q = z ¡! q y wit h z 2 v ( p ) , y 2 v ( c ) nfy1; y2g a n d v ( q ) \ v ( c [ p ) = fz; yg. a s s u m e w.l.o .g . t h a t y 2 v ( y+1 ¡! c y¡2 ) . s in c e c is e xt r e m e , we h a ve jy1 ¡! c yj ¸ jx1 ¡! p zj + 2 ; jy¡!c y2j ¸ jz ¡! p x2j + 2 ; jy2 ¡! c y1j ¸ ± + 1 : s u m m in g u p , we g e t jcj ¸ 2 ± + 4 , c o n t r a d ic t in g ( 1 ) . case 2.1.2. th e r e e xis t s a p a t h q = z ¡! q y wit h z 2 v ( y+1 ¡! c y¡2 ) , y 2 v ( y+2 ¡! c y¡1 ) a n d v ( q ) \ v ( c [ p ) = fz; yg. b y cla im 1 ( a 1 ) , jcj ¸ 2 p + 6 = 2 ± + 4 , c o n t r a d ic t in g ( 1 ) . case 2.1.3. gnfy1; y2g h a s a t le a s t t h r e e c o m p o n e n t s . it fo llo ws t h a t ¿ < 1 , c o n t r a d ic t in g t h e h yp o t h e s is . case 2.2. nc ( x1 ) = nc ( x2 ) = fyg fo r s o m e y 2 v ( c ) . it fo llo ws t h a t n ( x1 ) = ( v ( p ) [ fyg ) nfx1g; n ( x2 ) = ( v ( p ) [ fyg ) nfx2g: mo r e o ve r , x1 ¡! p v¡x2 ã¡ p v is a lo n g e s t p a t h in gnc fo r e a c h v 2 v ( x+1 ¡! p x2 ) . s in c e g is 2 c o n n e c t e d , we h a ve wz 2 e fo r s o m e w 2 v ( p ) a n d z 2 v ( c ) nfyg. if w = x1, t h e n u s in g t h e zh. nikoghosyan 3 3 p a t h zx1 ¡! p x2y, we c a n a r g u e a s in ca s e 2 .1 . ot h e r wis e we c a n u s e t h e p a t h yx1 ¡! p w¡x2 ã¡ p wz. case 3. p ¸ ±. case 3.1. x1y1; x2y2 2 e fo r s o m e d is t in c t y1; y2 2 v ( c ) . cle a r ly, jy1 ¡! c y2j ¸ ± +2 a n d jy2 ¡! c y1j ¸ ± +2 , wh ic h yie ld s jcj ¸ 2 ±+4 , c o n t r a d ic t in g ( 1 ) . case 3.2. nc ( x1 ) = nc ( x2 ) = fyg fo r s o m e y 2 v ( c ) . l e t y1; y2; :::; yt b e t h e e le m e n t s o f n + p ( x2 ) o c c u r r in g o n ¡! p in a c o n s e c u t ive o r d e r . p u t h = hv ( y¡1 ¡! p x2 ) i a n d pi = x1 ¡! p y¡i x2 ã¡ p yi ( i = 1 ; :::; t ) : s in c e pi is a lo n g e s t p a t h in gnc fo r e a c h i 2 f 1 ; :::; tg, we c a n a s s u m e w.l.o .g . t h a t p is c h o s e n s o a s t o m a xim iz e jv ( h ) j. if yiz 2 e fo r s o m e i 2 f1 ; :::; tg a n d z 2 v ( c ) nfyg, t h e n we c a n a r g u e a s in ca s e 3 .1 . ot h e r wis e n ( yi ) µ v ( h ) [ fyg ( i = 1 ; :::; t) , t h a t is jnh ( yi ) j ¸ ± ¡ 1 ( i = 1 ; :::; t) . b y l e m m a 3 , fo r e a c h d is t in c t u; v 2 v ( h ) , t h e r e is a p a t h in h o f le n g t h a t le a s t ± ¡ 1 , c o n n e c t in g u a n d v. s in c e g is 2 -c o n n e c t e d , h a n d c a r e c o n n e c t e d b y t wo ve r t e x d is jo in t p a t h s . th is m e a n s t h a t t h e r e is a p a t h q = y1 ¡! q y2 o f le n g t h a t le a s t ± + 1 wit h v ( q ) \ v ( c ) = fy1; y2g. fu r t h e r , we c a n a r g u e a s in ca s e 2 .1 . case 3.3. e it h e r nc ( x1 ) = ; o r nc ( x2 ) = ;. a s s u m e w.l.o .g . t h a t nc ( x1 ) = ;. b y a r g u in g a s in ca s e 3 .2 , we c a n ¯ n d a p a t h q = y1 ¡! q y2 o f le n g t h a t le a s t ± + 2 wit h v ( q ) \ v ( c ) = fy1; y2g, a n d t h e r e s u lt fo llo ws im m e d ia t e ly. th e o r e m 1 is p r o ve d . refer ences [1 ] d . b a u e r a n d e . s c h m e ic h e l, l o n g c yc le s in t o u g h g r a p h s , te c h n ic a l r e p o r t 8 6 1 2 , s t e ve n s in s t it u t e o f te c h n o lo g y, 1 9 8 6 . [2 ] j.a . b o n d y a n d u .s .r . mu r t y, graph theory with applications, ma c m illa n , l o n d o n a n d e ls e vie r , n e w y o r k 1 9 7 6 . [3 ] v . ch v¶a t a l, \ to u g h g r a p h s a n d h a m ilt o n ia n c ir c u it s , d iscrete m ath., vo l. 5 , p p . 2 1 5 -2 2 8 1 9 7 3 . [4 ] g. a . d ir a c , \ s o m e t h e o r e m s o n a b s t r a c t g r a p h s " , p roc. l ondon, m ath. soc., vo l. 2 , p p . 6 9 -8 1 , 1 9 5 2 . [5 ] h .-j. v o s s , \ b r id g e s o f lo n g e s t c ir c u it s a n d o f lo n g e s t p a t h s in g r a p h s " , b eitrage zur graphentheorie und deren anwendungen, vorgetr. auf dem. int. k olloq., ob e r h o f ( d d r ) , p p . 2 7 5 -2 8 6 , 1 9 7 7 . submitted 25.07.2018, accepted 20.11.2018. 3 4 a sharp improvement of a theorem of bauer and schmeichel ´³áõ»ñç ¨ þù³ûë»éç ã»áñ»ùç é³í³óáõù ä. üçïáõáëû³ý ²ù÷á÷áõù ¸çóáõù g-ý n ·³·³ã ¨ ± ýí³½³·áõûý ³ëïç×³ý áõý»óáõ ·ñ³ý ¿: ¶ñ³ýç ³ù»ý³»ñï³ñ óçïéç c »ñï³ñáõãû³ý ³é³ççý áã å³ñ½áõý³ï ·ý³ñ³ï³ï³ýá ëï³ó»é ¿ ¸çñ³ïá (1952). (i) î³ù³û³ï³ý 2-ï³å³ïóí³í ·ñ³ýáõù, c ¸ m in fn; 2 ±g: ²ûë ³ñ¹ûáõýùá 1986ã-çý ´³áõ»ñá ¨ þù³ûë»éá é³í³óñ»óçý 1-ïáßï ·ñ³ýý»ñç ñ³ù³ñ. (ii) î³ù³û³ï³ý 1-ïáßï ·ñ³ýáõù, c ¸ m in fn; 2 ±+2 g: êï³óí³í »ñïáõ ·ý³ñ³ï³ï³ýý»ñý ¿é ñ³ë³ý»éç »ý n å³ñ³ù»ïñç áñáß³ïç ³ñå»ùý»ñç ñ³ù³ñ: ü»ñï³ ³ßë³ï³ýùáõù µ»ñíáõù ¿ ´³áõ»ñç ¨ þù³ûë»éç ·ý³ñ³ï³ï³ýç ùç é³í³óáõù, áñá ñ³ë³ý»éç ¿ n å³ñ³ù»ïñç ó³ýï³ó³í ³ñå»ùç ¹»åùáõù: óëó÷øåíèå òåîðåìû áàóåðà è øìåéõåëÿ æ. íèêîãîñÿàí àííîòàöèÿ ïóñòü g ÿâëÿåòñÿ n âåðøèííûì ãðàôîì ñ ìèíèìàëüíîé ñòåïåíüþ ±. â 1952ã. äèðàê ïîëó÷èë ïåðâóþ íåòðèâèàëüíóþ îöåíêó äëÿ äëèíû c äëèííåéøåãî öèêëà ãðàôà g: (i) â ëþáîì 2-ñâÿçíîì ãðàôå, c ¸ m in fn; 2 ±g. ýòó îöåíêó â 1986ã. áàóåð è øìåéõåëü óëó÷øèëè äëÿ 1-æåñòêèõ ãðàôîâ: (ii) â ëþáîì 1-æåñòêîì ãðàôå, c ¸ m in fn; 2 ± + 2 g. ïîëó÷åííûå îöåíêè äîñòèãàåìû äëÿ îïðåäåëåííûõ çíà÷åíèé ïàðàìåòðà n. â íàñòîÿùåé ðàáîòå ïðåäëàãàåòñÿ óëó÷øåíèå îöåíêè áàóåðà è øìåéõåëüÿ, êîòîðîå íåóëó÷øàåìà äëÿ âñåõ çíà÷åíèé ïàðàìåòðà n. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 45, 53--58, 2016. method for analysis and classification of pavement based on quality davit. g. asatryan1, gurgen h. hakobyan2 1institute for informatics and automation problems of nas ra 2national polytechnic university of armenia e-mail: dasat@ipia.sci.am, hakobyan.gurgen@yahoo.com abstract the main objective of automated systems for pavement quality control is the detection of cracks and other surface defects, as well as the analysis of their corresponding parameters. on pavement areas the quality assessment is performed based on video surveys, followed by image processing with the corresponding mathematical methods. in this paper, a method for classification of video survey frames from pavement with and without cracks is proposed. each frame is processed with previously proposed methods of binarization and segmentation. as a result, two classes are formed for classification experiments based on the proposed procedures. the method of comparison with etalons is used, with application of proximity measure based on structural properties of images. on experimental data it is shown, that the proposed method of pavement classification gives acceptable results (the average of classification inaccuracy is about 26%). keywords: pavement, crack, classification, gradient magnitude, weibull distribution, proximity measure. 1. introduction for execution of regular works of pavement quality control, inspections and monitoring are essential. it will allow to have certain and updated information about deviations in pavement quality, such as cracks, potholes, roughness and other types of deviations from predefined criterion, which inevitably occur during road exploitation. depending on the type of the road and the rank of the observed pavement quality, corresponding works for collecting and processing information may be performed either manually or by using automated systems, which contain special sensors, high-speed cameras and other devices. manual surveys and processing of information require more time and financial expenses, they may be dangerous for surveyors 53 mailto:dasat@ipia.sci.am http://www.lingvo-online.ru/ru/search/translate/glossaryitemextrainfo?text=%d0%bc%d0%b5%d1%80%d1%8b%20%d0%b1%d0%bb%d0%b8%d0%b7%d0%be%d1%81%d1%82%d0%b8&translation=proximity%20measure&srclang=ru&destlang=en http://www.lingvo-online.ru/ru/search/translate/glossaryitemextrainfo?text=%d0%bc%d0%b5%d1%80%d1%8b%20%d0%b1%d0%bb%d0%b8%d0%b7%d0%be%d1%81%d1%82%d0%b8&translation=proximity%20measure&srclang=ru&destlang=en http://www.lingvo-online.ru/ru/search/translate/glossaryitemextrainfo?text=%d0%bd%d0%b5%d1%80%d0%be%d0%b2%d0%bd%d0%be%d1%81%d1%82%d1%8c&translation=roughness&srclang=ru&destlang=en method for analysis and classification of pavement based on quality 54 on active roads, also, there can be issues about objectivity of processing and achieved results [1]. therefore, in scientific literature the problem of automated analysis and pavement type classification is actual. during last decades, more attention is paid to analysis methods of pavement quality, which is based on techniques of computer vision and image processing. but a universal method for analysis and assessment of pavement important parameters does not exist. thus, the methods for solutions of this problem, which are proposed on literature are limited to certain, quite specific situations, which, however, are also many. in literature, a number of difficulties of automated crack detection are noted, which are related to low contrast difference between crack and background, intensity heterogeneity of crack itself, presence of shading, pollution and other types of artifacts [2, 3]. the most common method of crack detection uses methods of binarization, followed by analysis of the resulted image. in [4] a method of binarization and segmentation is proposed, which allows to distinguish forms of cracks, presented on the image alongside with background of artifacts. the visual comparison of images containing or not containing cracks, shows that there are significant statistical properties of parameters of these images, which allow to separate them into two types (classes) of situations. therefore, a procedure of classification can be developed, which, during automated pavement quality monitoring, can distinguish frames, which contain or do not contain cracks. this paper is dedicated to a simple method of classification, which is based on the usage of image structural properties, which are related to various behaviors of gradient fields of two types of examined images. to solve this problem, we followed the method of classification [5], which is applied to textured images. the reason for it is the possibility of considering images with cracks as textures. as it is known, texture classification is one of the important areas of image processing, computer vision and their applications. the goal of classification is to set the unknown sample to one of the predefined classes. usually, the class of objects is defined by an array of images, which were selected by a certain content based on formal features. the solution to this problem requires to set (or to find an appropriate) description (descriptor) and proximity measure of images, which correspond to this description. as stated above, suitable description of pavement images may be the distribution of segment sizes, which may be counted with the method described in [4]. 2. method for pavement type classification following [5], we consider images, the structure of which may be characterized by the aggregate of edges and borders contained inside the image. most procedures working with these types of structures, use descriptors, which use specific properties of gradient field of the image. in [5] a method for proximity measure of two images is proposed ),max(),max( ),min(),min( 2121 21212 σσηη σσηη =w , 10 2 ≤< w , (1) where the parameters 𝜂𝜂𝑖𝑖 and 𝜎𝜎𝑖𝑖 are statistical estimates of parameters of weibull distribution 𝑓𝑓(𝑥𝑥; 𝜂𝜂, 𝜎𝜎), which describe the gradient magnitude of the examined images. proposed classification procedure is based on the method of comparison with etalons and contains the following steps: http://www.lingvo-online.ru/ru/search/translate/glossaryitemextrainfo?text=%d0%bd%d0%b5%d0%be%d0%b4%d0%bd%d0%be%d1%80%d0%be%d0%b4%d0%bd%d0%be%d1%81%d1%82%d1%8c&translation=heterogenity&srclang=ru&destlang=en d. asatryan, g. hakobyan 55 step 1. for each image from training and testing samples, weibull distribution parameters η and σ are assessed; step 2. the mean values of iη and iσ are counted from parameter estimates iη ^ and iσ ^ for two specified classes by the corresponding items from the training samples. these values are stored as etalons for the next comparison with testing samples with them. also, mean values of 2w are counted and standard deviation is of proximity measure 2w over all pairs of samples inside each class. these values are used during assessment of proximity measure inside each class of training models. step 3. according to (1), for each testing sample the proximity measure is counted with each etalon. belonging of sample to one of the two classes is determined by the highest value of proximity measure. 3. experimental results experiments were made to test effectiveness of the proposed method of classification between two types of pavement. classification is done for classes, containing images with only artifacts or with cracks. these images were chosen manually by means of visual analysis of frames from video sequence, collected from real road with bad quality. 40 objects were included each inside each class, where 21 objects were chosen by random rule and used for training, and the rest for testing. inside the class of images with cracks were included images with various types of cracks, such as, longitudinal, transverse, block, alligator etc. [6], and in the class without cracks images not containing any obvious defects. fig 1. estimated parameters distribution of training samples. method for analysis and classification of pavement based on quality 56 described method was applied to the corresponding samples of images with results of binarization and segmentation. parameters η and σ of weibull distribution were estimated by the method of moments with data of gradient magnitude for each sample, which was counted by using of sobel operator [7]. figure 1 represents the scatters of specified estimates of training samples of two classes. it's visible, that classes with and without cracks are quite clearly separated from each other. table 1 contains the etalon values of iη and iσ for training samples with two classes (with and without cracks), 2w and standard deviations is of values of proximity measure inside each class. table 1. parameters of training samples. classes n iη iσ 2w is without cracks 21 80,40 0,56 0,50 0,284 with cracks 21 37,75 0,44 0,52 0,268 following the described method a classification was made from testing samples, consisting from the half of classes. results are shown in table 2. in the first row with class name, the quantities of values of testing images are shown, which were set to each class using the described classification procedure. it's visible, that classification was performed well enough, as the results of inaccuracy percents authenticate in table 2. table 2. classification inaccuracy distribution by classes. classes without cracks with cracks percent of inaccuracy without cracks 15 4 21,0 with cracks 6 13 31,6 4. conclusion for effective defect detection from pavement surface, it is advisable to apply methods of processing, which use structural properties of images by each frame of video survey, alongside with the known methods of data classification. proposed procedure of analysis and classification of pavement surface by quality allows to confidently distinguish frames from video survey containing or not containing defects of certain type. this procedure may be included in automated systems of pavement surface quality monitoring. d. asatryan, g. hakobyan 57 references [1] m. mustaffar, t. c. ling and o. c. puan, “automated pavement imaging program (apip) for pavement cracks classification and quantification a photogrammetric approach”, the international archives of the photogrammetry, remote sensing and spatial information sciences, vol. 37, part b4, pp. 367-372, 2008. [2] h. shimamura, k. oonuma and y. yasuda, “image analysis for automated pavement cracking evaluation”, mva'94 iapr workshop on machine vision applications dec. 13-15, kawasaki, pp. 83-86, 1994. [3] t. s. nguyen, m. avila and s. begot, “automatic detection and classification of defect on road pavement using anisotropy measure”, 17th european signal processing conference, pp. 617-624, 2009. [4] д. г. асатрян и г.o. акопян, “методика морфологического анализа изображения и оценивания качества дорожного покрытия”, в е с т н и к российско-армянского (славянского) университета, №1, сс. 36-44, 2015. [5] д. г. асатрян, в. в. куркчиян и л. р. харатян, “метод классификации текстур с использованием структурных характеристик изображения”, компьютерная оптика, том. 38, №3, сс. 574-579, 2014. [6] http://www.oregon.gov/odot/hwy/construction/docs/pavement/ /distress_survey_manual.pdf [7] d. g. asatryan and k. egiazarian, “quality assessment measure based on image structural properties”, proc. of the international workshop on local and non-local approximation in image processing, finland, helsinki, pp. 70-73, 2009. submitted 17.09.2015, accepted 20.01.2016 ճանապարհային ծածկույթի մակերևույթի որակի վերլուծության և դասակարգման մեթոդ դ. ասատրյան, գ. հակոբյան ամփոփում ճանապարհային ծածկույթների (ճծ) որակի հսկման համար նախատեսված ավտոմատացված համակարգերում հիմնական խնդիրը կայանում է մակերևույթի վրա առկա ճեղքերի և այլ արատների հայտնաբերումը, ինչպես նաև դրանց համապատասխան պարամետրերի վերլուծությունը: ճծ համապատասխան հատվածներում, որակի գնահատման նպատակով իրականացվում են տեսանկարահանումներ, այնուհետև method for analysis and classification of pavement based on quality 58 իրականացվում է պատկերների վերլուծություն՝ համապատասխան մաթեմատիկական մեթոդների կիրառմամբ: հոդվածում առաջարկվում է տեսանկարահանման կադրերի ճծ ճեղքեր պարունակող և չպարունակող պատկերների դասակարգման մեթոդ: առաջարկվող դասակարգման մեթոդի փորձարկման համար իրականացվում է յուրաքանչյուր կադրի անհատական մշակում՝ նախկինում առաջարկված երկակիացման և հատվածավորման մեթոդների հիման վրա, որի արդյունքում ձևավորվում են երկու դասի նմուշներ: օգտագործվում է էտալոնների հետ համեմատման մեթոդը՝ նմանության չափանիշի կիրառմամբ, որը հիմնված է պատկերի կառուցվածքային հատկությունների վրա: փորձարարական տվյալների վրա ցույց է տրված, որ ճծ դասակարգման առաջարկվող մեթոդը տալիս է ընդունելի արդյունքներ (դասակարգման սխալը կազմում է մոտ 26%): метод анализа и классификации поверхности дорожного покрытия по качеству д. асатрян, г. акопян аннотация в автоматизированных системах контроля качества дорожных покрытий (дп) основной задачей является выявление трещин и других дефектов поверхности, а также анализ их соответствующих параметров. для оценки качества производятся видеосъемки контролируемых участков дп с последующим анализом изображений соответствующими математическими методами. в настоящей работе предлагается процедура классификации кадров изображений видеосъемки, содержащих или не содержащих трещины дп. выполняется индивидуальная обработка каждого кадра в соответствии с предложенной ранее методикой бинаризации и сегментации, в результате чего образуются два класса образцов для испытания предлагемой процедуры классификации. используется метод сравнения с эталоном с применением меры близости, основанной на структурных свойствах изображений. на экспериментальном материале показано, что предложенная процедура классификации дп дает приемлемые результаты (ошибка классификации в среднем составляет около 26%.). mathematical problems of computer science 53, 63--66, 2020. udc 004 development of multi-eap radius configuration for eduroam service arthur s. petrosyan, gurgen s. petrosyan, robert n. tadevosyan and kevork kh. arsalanian institute for informatics and automation problems of nasra e-mail: arthur@sci.am, gurgen@sci.am, robert@sci.am, kevork.arsalanian@sci.am abstract this paper describes the mechanism of authenticating multiple-realm eduroam users via a single multi-eap radius configuration. the solution is based on the fact that some organizations, which are willing to join the eduroam community and use the service, especially in small communities, do not have a huge number of users, thus it will be costeffective to use a single radius server for them instead of having a separate radius server per each realm. the solution is unmatched in terms of practical open implementation. it has been implemented and tested in asnet-am network. keywords: eduroam, wifi, wireless, eap, authentication, radius 1. introduction in the original eduroam model [1], institutions that would like to join eduroam have to install and configure their own institutional radius server (irs), which should be registered at federation level radius server (flrs) within the national roaming operator (nro), meaning that every organization should follow these steps regardless of their number of users, which is not cost-effective for organizations with a few number of users. therefore, the solution of using a single radius server for multiple institutions (organizations, universities, etc.) has been thought up, developed and successfully implemented. 2. architecture generally, for each realm in eduroam, a separate radius server is being configured, which is normal for big organizations with thousands of active users. but for organizations with hundreds or tens of users having a separate radius server, it may not be so practical. in addition, some nros provide hosted radius solutions for connecting institutions, and in case of such relatively 63 mailto:arthur@sci.am mailto:gurgen@sci.am mailto:robert@sci.am development of multi-eap radius configuration for eduroam service 64 small ones it would be much more cost-effective for nro to use a single radius server for multiple supported realms instead of creating separate radius servers for each supported realm. with this configuration, it is now possible to use a central radius server with multiple extensible authentication protocol (eap) configuration for each supported realm [2]. the number of realms supported is unlimited. the authentication itself is performed at the institution site, as before, using the authentication method that is applicable to a particular institution (imap-based email authentication, ldap-based, etc.). when a user tries to authenticate, the request will be directed from the visited institution to the central radius server and from there to the user organization to get authenticated (fig. 1). fig. 1. authentication process. based on our previous boost concept [3], we mainly implemented this solution using the imap/email authentication method, but since the solution is based on freeradius [4], there is no limit for using any required authentication method. the solution described here has an appropriate ansible playbook [5] to automate setting up the process for multi-eap radius configuration. 3. advantages the main advantage of this solution is the creation of an automatically usable freeradius-based multi-eap radius configuration. the solution described in this paper will influence small organizations to use eduroam with minimal administrative overhead and minimal cost. while a. petrosyan, g. petrosyan, r. tadevosyan and k. arsalanian 65 carrying out investigations in this area, we found no such configuration described in public, except for the official general description [2], which lacks many important details about particular eap types. so, this solution is unmatched in terms of practical open implementation. 4. disadvantages although this solution of having multiple organizations on a single radius server is costefficient, it is not an ideal solution for large organizations with thousands of users. 5. conclusion this solution may be interesting in cases, where nros implement hosted radius solutions for relatively small connecting institutions (with hundreds or tens of users) to have a cost-effective single radius server for multiple supported realms instead of creating separate radius servers for each supported realm. references [1] [online]. available: https://www.eduroam.org/ [2] [online]. available: https://wiki.freeradius.org/modules/rlm_eap#i_want_to_enable_multiple_eaptypes_ho w_can_i_configure [3] a. petrosyan, g. petrosyan, r. tadevosyan and k. arsalanian, “identity infrastructure boost concept for eduroam service”, transactions of iiap nas ra, mathematical problems of computer science, vol. 52, pp. 61-65, yerevan 2019. [4] [online]. available: https://freeradius.org/ [5] [online]. available: https://github.com/asnet-am/eduroam-imap-playbook submitted 10.02.2020, accepted 26.05.2020. development of multi-eap radius configuration for eduroam service 66 բազմակի eap radius կարգավորումների մշակում eduroam ծառայության համար արթուր ս. պետրոսյան, գուրգեն ս. պետրոսյան, ռոբերտ ն․ թադևոսյան և գէորգ խ. արսալանյան հհ գաա ինֆորմատիկայի և ավտոմատացման պրոբլեմների ինստիտուտ e-mail: arthur@sci.am, gurgen@sci.am, robert@sci.am, kevork.arsalanian@sci.am ամփոփում հոդվածում նկարագրվում է eduroam-ի օգտագործողների վավերացման մեխանիզմ՝ բազմակի eap radius կարգավորումների միջոցով։ լուծումը հիմնված է այն փաստի վրա, որ որոշ կազմակերպություններ, որոնք ցանկանում են միանալ eduroam համայնքին և օգտվել ծառայությունից, չունեն հսկայական քանակությամբ օգտվողներ, ուստի ծախսարդյունավետ կլինի նմանատիպ կազմակերպություններից յուրաքանչյուրի համար առանձին radius սերվեր ունենալու փոխարեն, օգտագործել մեկ radius սերվեր մի քանի կազմակերպության համար: լուծումը եզակի է՝ գործնական բաց իրականացման առումով: այն իրականացվել և փորձարկվել է asnet-am ցանցում: բանալի բառեր` eduroam, wifi, անլար, eap, նույնականացում, radius. разработка конфигурации multi-eap radius для сервиса eduroam артур с. петросян, гурген с. петросян, роберт н. тадевосян и кеворк х. арсаланян институт проблем информатики и автоматизации нан ра e-mail: arthur@sci.am, gurgen@sci.am, robert@sci.am, kevork.arsalanian@sci.am аннотация в статье представлен механизм аутентификации пользователей eduroam с помощью конфигурации multi-eap radius для сервиса eduroam. решение основано на том факте, что некоторые, не имеющие большого количества пользователей, организации, желающие присоединиться к сообществу eduroam, могут использовать один общий radius сервер. данное решение не имеет аналогов с точки зрения практической открытой реализации. решение реализовано и опробовано в сети asnet-am. ключевые слова: eduroam, wifi, беспроводная связь, eap, аутентификация, radius. mailto:arthur@sci.am mailto:gurgen@sci.am mailto:robert@sci.am mailto:arthur@sci.am mailto:gurgen@sci.am mailto:robert@sci.am d:\sbornik\...\doc.dvi mathematical problems of computer science 33, 41{47, 2010. on computation of lower b ound of e capacity for the channel with t wo-sided state i nfor mation a r t h u r r . mu r a d ya n institute for informatics and automation problems of nas ra abstract in this paper a discrete memoryless channel (dmc) with 2 sided state information is examined in the sense of computations of capacity formulas and random-coding exponents. it is shown that computation of channel capacity and lower bound of e capacity requires a huge amount of computational resources and is rather slow when implemented with traditional methods. techniques are described how to massively parallelize the computations by means of gpgpu and obtain substantial acceleration. refer ences [1 ] t. m. co ve r a n d m. ch ia n g , \ d u a lit y b e t we e n c h a n n e l c a p a c it y a n d r a t e d is t o r t io n wit h t wo -s id e d s t a t e in fo r m a t io n " , ie e e transactions on information theory, vo l. 4 8 , n o . 6 , p p . 1 6 2 9 -1 6 3 8 , 2 0 0 2 . [2 ] m. h a r o u t u n ia n , a . mu r a d ya n , \ l o we r b o u n d fo r e c a p a c it y o f d is c r e t e m e m o r yle s s c h a n n e l wit h t wo -s id e d s t a t e in fo r m a t io n " , transactions of the institute for informatics and automation p roblems of the nas of r a, m athematical p roblems of computer sciences, vo l. 3 1 , p p . 2 8 -3 9 , 2 0 0 8 . [3 ] d . l u e b ke , g. h u m p h r e ys , \ h o w gp u s wo r k" , ie e e computer, 2 0 0 7 [4 ] r . w a n g , n . go o d n ig h t , g. h u m p h r e ys , \ co m p u t a t io n o n p r o g r a m m a b le g r a p h ic s h a r d wa r e " , ie e e computer graphics and applications, 2 0 0 5 . [5 ] j. h e n n e s s y; d . p a t t e r s o n , \ co m p u t e r a r c h it e c t u r e : a qu a n t it a t ive a p p r o a c h " , is b n 1 -5 5 8 6 0 -7 2 4 -2 , 3 r d e d it io n , 2 0 0 3 . [6 ] g. mo o r e , \ cr a m m in g m o r e c o m p o n e n t s o n t o in t e g r a t e d c ir c u it s " , e lectronics, vo l. 3 8 , n o 8 , 1 9 6 5 . [7 ] j. owe n s , d . l u e b ke , \ a s u r ve y o f g e n e r a l-p u r p o s e c o m p u t a t io n o n g r a p h ic s h a r d wa r e " , computer graphics f orum, vo l. 2 6 , n o 1 , p p . 8 0 -1 1 3 , 2 0 0 7 . [8 ] m. h a r r is , \ ma p p in g c o m p u t a t io n a l c o n c e p t s t o gp u s " , international conference on computer graphics and interactive techniques, n o 5 0 , 2 0 0 5 . [9 ] d . ib a r o u d e n e , \ p a r a lle l p r o c e s s in g " , ch a p t e r 1 , mo t iva t io n a n d h is t o r y. s t ma r y's u n ive r s it y, s a n a n t o n io , 2 0 0 8 . [1 0 ] t. co r m e n , c. l e is e r s o n , r . r ive s t , c. s t e in , \ in t r o d u c t io n t o a lg o r it h m s " , is b n 9 7 8 0 -2 6 2 -0 3 2 9 3 -3 , mit p r e s s , 2 0 0 1 4 1 4 2 on computation of lower bound of e capacity for the channel with two-sided state information e-áõý³ïáõãû³ý ëïáñçý ·ý³ñ³ï³ï³ýç ñ³ßí³ñïáõùá »ñïïáõù³ýç íç×³ïý»ñáí ï³åáõõçý»ñç ñ³ù³ñ ²ñãáõñ øáõñ³¹û³ý ²ù÷á÷áõù ²ßë³ï³ýùá ýíçñí³í ¿ áý¹ñ³ï ³é³ýó ñçßáõáõãû³ý »ñïïáõù³ýç íç×³ïý»ñáí ï³åáõõáõ ñ³ù³ñ ý³ëïçýáõù ëï³óí³í ³ñ¹ûáõýùý»ñç ñ³ßí³ñïù³ýá: ð³ûïýç áõý³ïáõãû³ý µ³ý³ó¨ç ¨ e-áõý³ïáõãû³ý å³ï³ñ³ï³ý ïá¹³íáñù³ý ·ý³ñ³ï³ï³ýç ñ³ßí³ñïù³ý ñ³ù³ñ å³ñ³ýçíáõù ¿ ï³ï³ñ»é ñëï³û³ï³ý ù³ý³ïáí ·áñíáõáõãûáõýý»ñ: ð³ßí³ñïý»ñç å³ù³ý³ïá ¿³å»ë ïñ×³ï»éáõ ýå³ï³ïáí ñ»ï³½áïí»é »ý ï³ñµ»ñ ùçç³í³ûñ»ñ ¨ ³é³ç³ñïí»é ¿ ù»ãá¹ ·áñíáõáõãûáõýý»ñá ñý³ñ³íáñçýë ½áõ·³ñ»é³óý»éáõ ñ³ù³ñ: òáõûó ¿ ïñí»é, áñ ýù³ý³ïçå ñ³ßí³ñïý»ñç ñ³ù³ñ ·ñ³ýçï³ï³ý ë³ñù»ñç û·ï³·áñíáõùá ï³éçë ¿ ³é³íáéáõãûáõý ³ûé ùççáóý»ñç ýï³ïù³ùµ: ð³ù»ù³ïáõãûáõýý»ñá ï³ï³ñí»é »ý ûñçý³ïç íñ³ ¨ ñ³ßí³ñïý»ñá ·ñ³ýçïáñ»ý å³ïï»ñí»é »ý: начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 39, 119--124, 2013. 119 on some properties of intersection and union of spheres in hamming metric haykaz e. danoyan yerevan state university e-mail: hdanoyan@yandex.ru abstract the problems of intersection and union of spheres of the same radius in hamming metric are considered. the formula for number of points in intersection is derived in case of two spheres. it is proved that three or more spheres of radius (covering radius of a code ) centered at points belonging to some quasi-perfect code intersect at most at one point. it is also proved that the increase of cardinality of union of spheres of the same radius, depending on radius, is a concave function and can have at most one or two maximum values depending on length. keywords: nearest neighbors, best-match, hamming spheres, concave function, quasiperfect code. 1. introduction let . denote by the set of vertices of the unit cube, that is each vector can be represented as , where , . for each pair of vectors denote by the hamming distance between the vectors and . for denote by the sphere with centre and radius , that is ⁄ . the carrier of the vector we define as ⁄ . denote by the weight of vector , that is . denote by the set of the first natural numbers, i.e. . a code will be called a subset of [1]. usually the codes are considered for which some other additional properties take place such as linearity, cyclicity, etc. in some cases a problem to find the intersection of spheres of the fixed radius can arise [2-4]. we will consider this problem for a simple case, namely when the centers of the spheres belong to a quasi-perfect code with the covering radius . recall that a code is called quasi perfect [1] if the following condition holds: , where and ⌊ ⁄ ⌋. the number is called a minimum distance of the code , i.e. . we also consider the problem of increase of union of spheres of some radius depending on the radius. more precisely let us have a set and let ⋃ ⋃ . it is important to check [2],[5] if the function is concave, with one or two maximal values depending on and . mailto:hdanoyan@yandex.ru on some properties of intersection and union of spheres in hamming metric 120 2. intersection of two spheres in the next section we will consider the problem of intersection of spheresin some cases, so it is useful at first to consider the case of two spheres.suppose we have two spheres with centers and and radii and respectively. we denote the intersection of the mentioned spheres by i.e. .it is easy to show that each case can be reduced to the case when , therefore furthermore we assume that the mentioned case takes place. without losing generality we assume, that . it is easy to see that when (where ) the considered intersection is empty, so we suppose that .we denote the cardinality of intersection by . let us take any vector and find out conditions, under which the vector belongs to the set . let and (figure1). β 0 … 0 0 … 1 … 1 1 … 1 d γ 1 … 1 0 … 0 … … 1 … 1 x y α 0 … 0 0 … 0 … … 0 … 0 fig. 1. in order for belongs to the intersection, and should satisfy the following system of inequalities: { let us denote [ ]. consider the following three cases. case a. . it is easy to see, that when then and when then (fig. 2). fig. 2. h. danoyan 121 therefore ∑ ∑ ( ) ( ) ∑ ∑ ( ) ( ) . in case when we can write (3) in a simpler form: ( ). case b. , . this case, in its turn, is divided into two separate cases: subcase b.1. . by straightforward verification we get that when then and when then (fig. 3). fig. 3. therefore ∑ ∑ ( ) ( ) ∑ ∑ ( ) ( ) . subcase b.2. . in this case we get that when . therefore ∑ ∑ ( ) ( ) . case c. . this case is also divided into two separate cases. subcase c.1. . we get that when then and when then (fig. 4). figure 4. fig. 4. on some properties of intersection and union of spheres in hamming metric 122 therefore ∑ ∑ ( ) ( ) ∑ ∑ ( ) ( ) . subcase c.2. . in this case we get that when . therefore ∑ ∑ ( ) ( ) . 3. intersection of spheres centered at codewords of quasi-perfect code suppose we have spheres with radii centered at points respectively. now we consider the intersection of spheres with the radius incase when the centers of spheres are codewords of any quasi-perfect code. as we mentioned, we can assume that . proposition 1. let vectors becodewords of the quasi-perfect code with an even minimum distance , then ⋂ if and only if i. ( ) , ii.⋂ . otherwise, the considered intersection is empty. proof. necessity is obvious let us prove thesufficiency. note that in this case we have , . from thecondition itfollows that the intersection of the spheres of radius centered at the points is nonempty only if ( ) , . so we get and ( ) , , . let . from the reasoning of the previous section it follows that , and ( ) therefore there is only belonging to the intersection. now let us have vectors which are codewords of any quasi-perfect code with an odd minimum distance, i.e . this means that the covering radius of the code is . suppose we want to know the intersection of spheres with the radius centered at .as we mentioned, consider that . partition the set into two nonintersecting subsets and in the following way { . from the condition follows that the considered intersection is not empty only when . without loosing generality assume that . let now we can formulate the following: proposition 2. let vectors be codewords of the quasi-perfect code with an odd minimum distance , then ⋂ if and only if i. ( ) ( ) ii.⋂ . otherwise the considered intersection is empty. proof. the schema of proof is analogous to proposition 1. 4. variation function of area size of sphere system is concave consider an arbitrary .then it is easy to check that for a fixed the function is concave. for odd there are two maximums of at and h. danoyan 123 . when is even, then there is one maximal value at the point .in a generalized model a number of packed spheres can be considered. given a subset and let ⋃ ⋃ . it is important to check if the function is also concave, with one or two maximal values depending on and . a harder generalization will be the case of different radii but our interest will be restricted to the basic case of one fixed . there can also be considered not the spheres but the spherical structures [6]. consider the case of two vertices, and . if one of these vectors is the negation of the other, then: if is odd, there is one maximum at and for ; when is even, then is increasing in interval and when it reminds to compare the following 2 values: ( ⁄ ) and ( ⁄ ). it is easy to check that the only maximum accepts at , just it needs to take into account that is even. now consider the case of and , not opposite in . choose an index and the variable so that . partition in the direction , then and belong to one of the parts of , let it be the subcube . denote by and the projections of and to . the areas of our consideration n, and , can be partitioned in the direction . in these two dimensional sub-cubes we receive the same pair of vertices: and , and the radius in , and and and the radius in . in theseterms . in fact, we have the same function and the same points for .if suppose that is concave, having 1 or 2 maximal values, then this shifted sum will have the same properties. now we consider the general case of arbitrary . denote by and the partition compounds of in direction . let and be projections of and in direction . in the first step, when we receive: in a union of ⋃ and , and similarly in a union of ⋃ and . in reality, in dimension we have to consider two sets of vertices – those are basic in that cube and these vertices draw spheres around, and the projection vertices that draw spheres around. the second set may add some new neighbours to the basic set produced by the first set. the difficulty to watch these sets concerns the radii difference. to use the induction hypotheses we have to transform the constructions so that a situation appears, that is standard in dimension . for this purpose by an evident note it is sufficient to consider not the basic sets of vertices but the ones appeared after the first step – ⋃ in and ⋃ in . in this case the basic will appear as the sum of 2 functions in dimension by the radius . at this stage we have 2 concave functions in , which have 1 or 2 maximums by the supposition of induction. it is evident that the sum of functions will increase when functions increase separately, and that it decreases in part in case of decrease in both of them. consider the point of the first decrease in . at least one of the sub-functions has its decrease at . let this be the function at . the sensitive part that initiates this decrease will be the set . the complementary part can’t be sensitive because its radius is higher in and its decrease must happen earlier and this will cause the summary decrease in . then, the projection of that appears with the radius in will have the same size in the next step when the radius is equalto . this will sufficiently initiate the decrease in the sub-function at . on some properties of intersection and union of spheres in hamming metric 124 references [1] f. j. mac williams, n. j. a. sloane, the theory of error-correcting codes, 1977. [2] i. honkala, “on the intersection and unions of hamming spheres”, the very knowledge of coding, pp. 70-81, 1987. [3] g. cohen, i. honkala, s. litsyn, a. lobstein, covering codes, 1997. [4] r.l. rivest, “on the optimality of elias's algorithm for performing best-match searches”, information processing, pp. 678-681, 1974. [5] yu.zhuravlev, selected scientific works, (in russian), 1998. [6] l. aslanyan, “metric decompositions and the discrete isoperimetry”, ifac symposium on manufacturing, modelling, management and control, pp. 433-438, 2000. submitted 24.12.2012, accepted 18.02.2013. հեմմինգի մետրիկայում սֆերաների հատման և միավորման որոշ հատկությունների մասին հ. դանոյան ամփոփում դիտարկվում է հեմմինգի մետրիկայում սֆերաների հատման և միավորման կետերի գտնելու խնդիրը: բերված է բանաձև` երկու տարբեր շառավիղներով սֆերաների հատման համար: ապացուցված է, որ երեք և ավելի r շառավղով սֆերաները, որոնց գագաթները պատկանում են որևէ քվազիկատարյալ կոդի, կարող են հատվել ամենաշատը մեկ կետով (r-ը նշված կոդի ծածկման շառավիղն է): ապացուցված է նաև, որ սֆերաների միավորման` շառավղից կախված ֆունկցիայի աճը ունի մեկ կամ երկու մաքսիմում` կախված տարածության չափողականությունից: о некоторых свойствах пересечения и объединения сфер в метрике хемминга а. даноян аннотация рассматривается проблема нахождения пересечения и объединения сфер в метрике хемминга. приведена формула для числа точек пересечения для двух сфер. доказано, что три и более сферы радиуса r,центры которых принадлежат некоторому квазисовершенному коду c, могут пересекаться лишь в одном точке (r радиус покрытия кода c). также доказано, что возрастание функции числа точек в объединении некоторых сфер от радиуса может иметь один или два максимума в зависимости от меры пространства. parandzem.dvi mathematical problems of computer science 46, 103{106, 2016. on n eyman-p ear son t esting for p air of i ndependent objects e vg u e n i a . h a r o u t u n ia n a n d p a r a n d z e m m. h a ko b ya n institute for informatics and automation problems of nas ra e-mail: eghishe@ipia.sci.am, par h@ipia.sci.am abstract the neyman-pearson principle for a model consisting of two independent objects is considered. it is supposed that two probability distributions are known and each object follows one of them independently. two approaches of neyman-pearson testing are considered for this model. the aim is to compare error probabilities to the corresponding two cases. keywords: independent objects, statistical hypotheses, neyman-pearson testing, error probability. 1 . in t r o d u c t io n in t h is p a p e r we d is c u s s n e ym a n -p e a r s o n h yp o t h e s e s t e s t in g p r o b le m fo r a m o d e l c o n s is t in g o f t wo in d e p e n d e n t o b je c t s . th is m o d e l wa s p r o p o s e d b y a h ls we d e a n d h a r o u t u n ia n [1 ]. th e c h a r a c t e r is t ic s o f t h e o b je c t s a r e x1 a n d x2 in d e p e n d e n t r a n d o m va r ia b le s ( r v s ) t a kin g va lu e s in t h e s a m e ¯ n it e s e t x . s o , t h a t c o n s id e r e d m o d e l is d e s c r ib e d b y t h e r a n d o m ve c t o r ( x1; x2 ) , wh ic h a s s u m e s va lu e s ( x 1; x2 ) 2 x £ x . th e n e ym a n -p e a r s o n le m m a is t h e fo u n d a t io n o f t h e m a t h e m a t ic a l t h e o r y o f s t a t is t ic a l h yp o t h e s is t e s t in g . th is p r in c ip le p la ys a c e n t r a l r o le in t h e t h e o r y a n d p r a c t ic e o f s t a t is t ic s . th e r e e xis t m a n y wo r ks wh e r e t h e n e ym a n -p e a r s o n le m m a is e xp o u n d e d fo r t h e c a s e o f t wo h yp o t h e s e s [2 ]{ [1 0 ]. th is p r in c ip le t o t h e c a s e o f m u lt ip le h yp o t h e s e s is s o lve d in [1 1 , 1 2 ]. 2 . p r o b le m s t a t e m e n t a n d r e s u lt fo r m u la t io n l e t x1 a n d x2 b e in d e p e n d e n t r a n d o m va r ia b le s t a kin g va lu e s in t h e s a m e ¯ n it e s e t x , e a c h o f t h e m wit h o n e o f t wo h yp o t h e t ic a l p r o b a b ilit y d is t r ib u t io n s gm = fgm ( x ) ; x 2 x , m = 1 ; 2 . l e t ( x1; x2 ) = ( ( x 1 1; x 2 1 ) ; :::; ( x 1 n; x 2 n ) ; :::; ( x 1 n ; x 2 n ) ) , x i n 2 x , i = 1 ; 2 , n = 1 ; n, b e a s e qu e n c e o f r e s u lt s o f n in d e p e n d e n t o b s e r va t io n s o f t h e ve c t o r ( x1; x2 ) . it is n e c e s s a r y t o d e ¯ n e u n kn o wn p d s o f t h e o b je c t s o n t h e b a s e o f t h e o b s e r ve d d a t a . th e d e c is io n fo r e a c h o b je c t m u s t b e m a d e fr o m t h e s a m e s e t o f h yp o t h e s e s : hm : g = gm, m = 1 ; 2 . th e r e a r e fo u r h yp o t h e t ic a l p r o b a b ilit y d is t r ib u t io n s fo r r a n d o m ve c t o r ( x1; x2 ) : 1 0 3 1 0 4 on neyman-pearson testing for pair of independent objects g1 ± g1 ( x1; x2 ) = fg1 ( x1 ) _g1 ( x2 ) ; ( x1; x2 ) 2 x £ x g, g1 ± g2 ( x1; x2 ) = fg1 ( x1 ) _g2 ( x2 ) ; ( x1; x2 ) 2 x £ x g, g2 ± g1 ( x1; x2 ) = fg2 ( x1 ) _g1 ( x2 ) ; ( x1; x2 ) 2 x £ x g a n d g2 ± g2 ( x1; x2 ) = fg2 ( x1 ) _g1 ( x2 ) ; ( x1; x2 ) 2 x £ x g. w e c a ll t h e p r o c e d u r e o f m a kin g d e c is io n o n t h e b a s e o f n o b s e r va t io n s t h e t e s t . it c a n b e d e ¯ n e d b y d ivis io n o f t h e s a m p le s p a c e x n £ x n o n 4 d is jo in t s u b s e t s bni;j, i; j = 1 ; 2 . th e s e t bni;j c o n s is t s o f a ll ve c t o r s ( x1; x2 ) fo r wh ic h t h e h yp o t h e s is gi ± gj is a d o p t e d . l e t ®l1;l2jm1;m2 = gm1 ±gnm2 ( b n l1;l2 ) b e t h e p r o b a b ilit y o f t h e e r r o n e o u s a c c e p t a n c e gl1 ±gl2 b y t h e t e s t p r o vid e d t h a t gm1 ± gm2 is t r u e , wh e r e ( l1; l2 ) 6= ( m1; m2 ) , li; mi = 1 ; 2 , i = 1 ; 2 . th e p r o b a b ilit y t o r e je c t a t r u e d is t r ib u t io n gm1 ± gm2 is a s fo llo ws : ®m1;m2jm1;m2 = x (l1;l2)6=(m1;m2) ®l1;l2jm1;m2 : th e m a t r ix o f e r r o r p r o b a b ilit ie s is a s fo llo ws : 0 bbb@ ®1;1j1;1 ®1;2j1;1 ®2;1j1;1 ®2;2j1;1 ®1;1j1;2 ®1;2j1;2 ®2;1j1;2 ®2;2j1;2 ®1;1j2;1 ®1;2j2;1 ®2;1j2;1 ®2;2j2;1 ®1;1j2;2 ®1;2j2;2 ®2;1j2;2 ®2;2j2;2 1 ccca n o w we will c o n s id e r n e ym a n -p e a r s o n t e s t in g fo r t h is m o d e l wit h t wo a p p r o a c h e s : a ) d ir e c t m e t h o d a n d b ) r e n u m b e r in g m e t h o d o f h yp o t h e s e s p a ir s . ou r a im is t o c o m p a r e t h e c o r r e s p o n d in g e r r o r p r o b a b ilit ie s o f t h o s e t wo a p p r o a c h e s . l e t u s d e s c r ib e t h o s e c a s e s . a ) d ir e c t a p p r o a c h l e t ®i1j1 a n d ® i 1j1 b e e r r o r p r o b a b ilit ie s o f t h e ¯ r s t a n d t h e s e c o n d o b je c t s , r e s p e c t ive ly. a c c o r d in g t o fu n d a m e n t a l n e ym a n -p e a r s o n le m m a , we c a n c h o o s e t i a n d t ii p o s it ive n u m b e r s a n d we c a n d ivid e t h e s a m p le s p a c e s o f t h e ¯ r s t a n d t h e s e c o n d o b je c t s a s fo llo ws : 1 ) fo r t h e ¯ r s t o b je c t : ai1 = ( x1 : gn1 ( x1 ) gn2 ( x1 ) > t i ) ; 1 ¡ gn1 ( ai1 ) = ®i1j1; a n d ai2 = x n =ai1: 2 ) fo r t h e s e c o n d o b je c t : aii1 = ( x2 : gn1 ( x2 ) gn2 ( x2 ) > t ii ) ; 1 ¡ gn1 ( aii1 ) = ®ii1j1; a n d aii2 = x n =aii1 : fin a lly, t h e n e ym a n -p e a r s o n d ivis io n o f t h e s a m p le s p a c e x n £ x n is t h e fo llo win g : a1;1 = ai1 £ aii1 ; a1;2 = ai1 £ aii2 ; a2;1 = ai2 £ aii1 ; a2;2 = ai2 £ aii2 : e. haroutunian and p. hakobyan 1 0 5 a c c o r d in g t o t h e d e ¯ n it io n o f e r r o r p r o b a b ilit ie s a n d in d e p e n d e n t s o f o b je c t s we c a n s e e t h a t t h e e r r o r p r o b a b ilit ie s o f t h is t e s t a r e fo r m u la t e d a s fo llo ws : ®l1;l2jm1;m2 = g n m1 ± gnm2 ( a n l1;l2 ) = x x 1 2a nl1 gm1 ( x1 ) £ x x2 2a nl2 gm2 ( x2 ) : b ) r e n u m b e r in g a p p r o a c h h e r e we h a ve t o r e n u m b e r t h e e a c h p a ir o f h yp o t h e s e s o n e a t a t im e a n d we h a ve t o a p p ly n e ym a n -p e a r s o n le m m a fo r m u lt ip le h yp o t h e s e s , wh ic h is in ve s t ig a t e d in [1 2 ]. in t h a t c a s e fo r t h e g ive n p o s it ive va lu e s ®¤1;1j1;1, ® ¤ 1;2j1;2 a n d ® ¤ 2;1j2;1 a n d c h o s e n n u m b e r s t1, t2 a n d t3 s e t s ai;j, i; j = 1 ; 2 , a r e t h e fo llo win g : a1;1 = ( ( x1; x2 ) : m in ã gn1 ( x1 ) gn2 ( x2 ) ; gn1 ( x2 ) gn2 ( x2 ) ; gn1 ( x1 ) ¢ gn1 ( x2 ) gn2 ( x1 ) ¢ gn2 ( x2 ) ! > t1 ) ; 1 ¡gn1 ±gn1 ( a1;1 ) = ®¤1;1j1;1; a1;2 = a1;1 ( ( x1; x2 ) : m in ã gn1 ( x1 ) gn2 ( x2 ) ; gn1 ( x1 ) ¢ gn2 ( x2 ) gn2 ( x1 ) ¢ gn1 ( x2 ) ! > t2 ) ; 1 ¡gn1 ±gn2 ( a1;2 ) = ®¤1;2j1;2 a2;1 = a1;1 \ a1;2 ( ( x1; x2 ) : gn1 ( x2 ) gn2 ( x2 ) > t3 ) ; 1 ¡ gn2 ± gn1 ( a2;1 ) = ®¤2;1j2;1 a n d a¤2;2 = a1;1 \ a1;2 \ a2;1: th e c o r r e s p o n d in g e r r o r p r o b a b ilit ie s a r e fo r m u la t e d a s a b o ve . l e t u s a s s u m e t h a t we h a ve fo u n d t h e e r r o r p r o b a b ilit ie s b y d ir e c t a p p r o a c h . b y c o n s id e r in g t h e fo llo win g ®1;1j1;1, ®1;2j1;2 a n d ®2;1j2;1 e r r o r p r o b a b ilit ie s a n d a p p lyin g t h e r e n u m b e r in g a p p r o a c h we will ¯ n d a ll t h e o t h e r e r r o r p r o b a b ilit ie s . b e c a u s e in [1 2 ] it is p r o ve d t h a t t h e s e e r r o r p r o b a b ilit ie s a r e t h e s m a lle s t . h e n c e , we c a n in s is t t h a t t h e r e n u m b e r in g a p p r o a c h o f n e ym a n -p e a r s o n t e s t in g is t h e b e s t m e t h o d . refer ences [1 ] r . f. a h ls we d e a n d e . a . h a r o u t u n ia n , " te s t in g o f h yp o t h e s e s a n d id e n t i¯ c a t io n " , e lectronic notes on d iscrete m athematics, vol. 21, p p . 1 8 5 { 1 8 9 , 2 0 0 5 . [2 ] j. n e ym a n a n d e . s . p e a r s o n , \ on t h e p r o b le m o f t h e m o s t e ± c ie n t t e s t s o f s t a t is t ic a l h yp o t h e s e s " , p h il. tr a n s . r o y. s o c . l o n d o n , s e r . a , 2 3 1 , p p . 2 8 9 -3 3 7 , 1 9 3 3 . [3 ] j. n e ym a n , f irst course in p robability and statistics, h o lt , r in e h a r t a n d w in s t o n , n e w y o r k, 1 9 5 0 . [4 ] e . l . l e h m a n a n d j.p . r o m a n o , testing statistical hypotheses, th ir d e d it io n . s p r in g e r , n e w y o r k, 2 0 0 5 . [5 ] a . a . b o r o vko v, m athematical statistics, in r u s s ia n , n a u ka , mo s c o w, 1 9 9 7 . [6 ] h . l . v a n tr e e s , d etection, e stimation and m odulation theory, p a r t 1 . n e w y o r k: w ile y, 1 9 6 8 . [7 ] m. h . d e gr o o t , p robability and statistics, 2 n d e d ., r e a d in g , ma , a d d is o n -w e s le y, 1 9 8 6 . [8 ] m. g. k e n d a ll a n d a . s t u a r t , the advanced theory of statistics, 2 , in fe r e n c e a n d r e la t io n s h ip , th ir d e d it io n . h a fn e r p u b lis h in g c o m p a n y, l o n d o n , 1 9 6 1 . 1 0 6 on neyman-pearson testing for pair of independent objects [9 ] a . k . b e r a , \ h yp o t h e s is t e s t in g in t h e 2 0 -t h c e n t u r y wit h a s p e c ia l r e fe r e n c e t o t e s t in g wit h m is s p e c i¯ e d m o d e ls " , in : \ s t a t is t ic s fo r t h e 2 1 -s t c e n t u r y. me t h o d o lo g ie s fo r a p p lic a t io n s o f t h e fit ir e " , ma r c e l d e kke r , in c ., n e w y o r k, b a s e l, p p . 3 3 -9 2 , 2 0 0 0 . [1 0 ] t. m. co ve r a n d j. a . th o m a s , e lements of information theory, s e c o n d e d it io n . w ile y, n e w y o r k, 2 0 0 6 . [1 1 ] e . a . h a r o u t u n ia n , \ n e ym a n -p e a r s o n p r in c ip le fo r m o r e t h a n t wo h yp o t h e s e s " , abstracts of armenian m athematical union annual session d edicated to 90 anniversary of r afael alexandrian, y e r e va n , p p .4 9 { 5 0 , 2 0 1 3 . [1 2 ] e . a . h a r o u t u n ia n a n d p . h a ko b ya n , \ n e ym a n -p e a r s o n p r in c ip le fo r m o r e t h a n t wo h yp o t h e s e s " , m athematical p roblems of computer science vo l. 4 0 , p p . 3 4 { 3 8 , 2 0 1 3 . submitted 24.08.2016, accepted 07.11.2016. ºñïáõ ³ýï³ë ûµû»ïïý»ñç ýï³ïù³ùµ ü»ûù³ýç-äçëáýç ï»ëï³íáñù³ý ù³ëçý º. ð³ñáõãûáõýû³ý ¨ ö. ð³ïáµû³ý ²ù÷á÷áõù ¸çïñïí»é ¿ »ñïáõ ³ýï³ë ûµû»ïïý»ñáí µýáõã³·ñíáõ ùá¹»éç í»ñ³µ»ñû³é ü»ûù³ýçäçñëáýç ëï½µáõýùá: ºýã³¹ñíáõù ¿, áñ ñ³ûïýç »ý »ñïáõ ñ³í³ý³ï³ý³ûçý µ³ßëáõùý»ñ ¨ ûµû»ïïý»ñçó ûáõñ³ù³ýãûáõñá ùûáõëçó ³ýï³ë µ³ßëí³í ¿ ¹ñ³ýóçó áñ¨¿ ù»ïáí: ²ûë ùá¹»éç ýï³ïù³ùµ ¹çï³ñïí»é ¿ ü»ûù³ýç-äçëáýç ï»ëï³íáñù³ý »ñïáõ ùáï»óáõù: üå³ï³ïý ¿ ñ³ù»ù³ï»é ³ûë »ñïáõ ùáï»óáõùý»ñç ñ³ù³å³ï³ëë³ý ëë³éý»ñç ñ³í³ý³ï³ýáõãûáõýý»ñá: î òåñòèðîâàíèè íåéìàíà-ïèðñîíà äëÿ ïàðû íåçàâèñèìûõ îáúåêòîâ å. àðóòþíÿí è ï. àêîïÿí àííîòàöèÿ ðàññìàòðèâàåòñÿ ïðèíöèï íåéìàíà-ïèðñîíà äëÿ ìîäåëè ñîñòîÿùåé èç äâóõ íåçàâèñèìûõ îáúåêòîâ. ïðåäïîëàãàåòñÿ, ÷òî äâà âåðîÿòíîñòíûõ ðàñïðåäåëåíèÿ èçâåñòíû, è êàæäûé îáúåêò ñëåäóåò îäíîìó èç íèõ íåçàâèñèìî îò äðóãîãî. ðàññìàòðèâàåòñÿ äâà ïîäõîäà òåñòèðîâàíèÿ íåéìàíà-ïèðñîíà äëÿ äàííîé ìîäåëè. öåëü ñîñòîèò â òîì, ÷òîáû ñðàâíèòü âåðîÿòíîñòè îøèáêè â äâóõ ñîîòâåòñòâóþùèõ ñëó÷àÿõ. hovnanyan_poghosyans.dvi mathematical problems of computer science 40, 5{12, 2013. m ethod of local i nter change to i nvestigate gossip p r oblems v ilya m h . h o vn a n ya n , s u r e n s . p o g h o s ya n a n d v a h a g n s . p o g h o s ya n institute for informatics and automation problems of nas ra e-mail: williamhovnanyan@gmail.com, psuren@yandex.ru, povahagn@gmail.com abstract in this paper the method of local interchange is introduced to investigate gossip problems. this method is based on a repetitive use of permute higher and permute lower operations, which map one gossip graph with n vertices to another by moving only its edges without changing the labels of edges (the moments of corresponding calls). using this method we obtain minimum time gossip graphs in which no one hears his own information (noho graphs) from gossip graphs based on knäodel graphs. the value of minimal time is t = dlog ne. this method also allowed us to ¯nd an alternative way of the proof that the number of minimal necessary calls in gossip schemes is 2n¡4, n ¸ 4. keywords: graphs, networks, telephone problem, gossip problem. 1 . in t r o d u c t io n go s s ip in g is o n e o f t h e b a s ic p r o b le m s o f in fo r m a t io n d is s e m in a t io n in c o m m u n ic a t io n n e t wo r ks . th e g o s s ip p r o b le m ( a ls o kn o wn a s a t e le p h o n e p r o b le m ) is a t t r ib u t e d t o a . b o yd ( s e e e x. [4 ] fo r r e vie w) , a lt h o u g h t o t h e b e s t kn o wle d g e o f t h e r e vie we r s , it wa s ¯ r s t fo r m u la t e d b y r . ch e s t e r s a n d s . s ilve r m a n ( u n iv. o f w it wa t e r s r a n d , u n p u b lis h e d , 1 9 7 0 ) . co n s id e r a s e t o f n p e r s o n s ( n o d e s ) e a c h o f wh ic h in it ia lly kn o ws s o m e u n iqu e p ie c e o f in fo r m a t io n t h a t is u n kn o wn t o t h e o t h e r s , a n d t h e y c a n m a ke a s e qu e n c e o f t e le p h o n e c a lls t o s p r e a d t h e in fo r m a t io n . d u r in g a c a ll b e t we e n t h e g ive n t wo n o d e s , t h e y e xc h a n g e t h e wh o le in fo r m a t io n kn o wn t o t h e m a t t h a t m o m e n t . th e p r o b le m is t o ¯ n d a s e qu e n c e o f c a lls wit h m in im u m le n g t h ( m in im a l g o s s ip s c h e m e ) , b y wh ic h a ll t h e n o d e s will kn o w a ll p ie c e s o f in fo r m a t io n ( c o m p le t e g o s s ip in g ) . it h a s b e e n s h o wn in n u m e r o u s wo r ks [1 ]-[4 ] t h a t t h e m in im a l n u m b e r o f c a lls is 2 n ¡ 4 , wh e n n ¸ 4 a n d 1 ; 3 , fo r n = 2 ; 3 , r e s p e c t ive ly. s in c e t h e n m a n y va r ia t io n s o f g o s s ip p r o b le m h a ve b e e n in t r o d u c e d a n d in ve s t ig a t e d [5 ]{ [1 0 ], [1 3 ]{ [1 7 ]. a n o t h e r va r ia n t o f t h e go s s ip p r o b le m c a n b e fo r m u la t e d b y c o n s id e r in g t h e m in im u m a m o u n t o f t im e r e qu ir e d t o c o m p le t e g o s s ip in g a m o n g n p e r s o n s , wh e r e t h e c a lls b e t we e n t h e n o n -o ve r la p p in g p a ir s o f n o d e s c a n t a ke p la c e s im u lt a n e o u s ly a n d e a c h c a ll r e qu ir e s o n e u n it o f t im e [1 1 , 1 2 , 1 8 ]. in s e c t io n 2 we in t r o d u c e t h e m e t h o d o f local interchange, wh ic h is b a s e d o n a r e p e t it ive u s e o f permute higher a n d permute lower o p e r a t io n s . th e s e o p e r a t io n s lo c a lly a c t o n t h e a d ja c e n t e d g e s t o a g ive n e d g e in a g o s s ip g r a p h ( a n o n -lo c a l interchange of two vertices from 5 6 method of local interchange to investigate gossip problems some point onwards in a list of telephone calls is d e ¯ n e d in [4 ]) . in s e c t io n 3 t h e m e t h o d o f local interchange is d e m o n s t r a t e d fo r t h e a lt e r n a t ive d e r iva t io n o f t h e m in im u m n u m b e r o f c a lls in g o s s ip s c h e m e . a n o t h e r a p p lic a t io n o f local interchange is d e s c r ib e d in s e c t io n 4 , wh e r e t h e p r o c e d u r e fo r o b t a in in g m in im u m t im e g o s s ip g r a p h s in wh ic h n o o n e h e a r s h is o wn in fo r m a t io n ( n oh o g r a p h s , [2 0 , 2 2 ]) fr o m g o s s ip g r a p h s b a s e d o n k n äo d e l g r a p h s is fo u n d . s in c e t h e m in im a l t im e o f g o s s ip g r a p h s is kn o wn [1 8 ], t h is p r o c e d u r e a llo we d u s t o e s t a b lis h t h e va lu e o f m in im a l t im e fo r n oh o g r a p h s t = dlo g ne. 2 . th e me t h o d a g o s s ip s c h e m e ( a s e qu e n c e o f c a lls b e t we e n n n o d e s ) c a n b e r e p r e s e n t e d b y a n u n d ir e c t e d e d g e -la b e le d g r a p h g = ( v; e ) wit h jv ( g ) j = n ve r t ic e s . th e ve r t ic e s a n d e d g e s o f g r e p r e s e n t c o r r e s p o n d in g ly t h e n o d e s a n d t h e c a lls b e t we e n t h e p a ir s o f n o d e s o f a g o s s ip s c h e m e . s u c h g r a p h s m a y h a ve m u lt ip le e d g e s , b u t n o t s e lf lo o p s . a n e d g e -la b e lin g o f g is a m a p p in g tg : e ( g) ! z+. th e la b e l tg ( e ) o f a g ive n e d g e e 2 e ( g) r e p r e s e n t s t h e m o m e n t o f t im e , wh e n t h e c o r r e s p o n d in g c a ll o c c u r s . a s e qu e n c e l = ( v0; e1; v1; e2; v2; : : : ; ek; vk ) wit h ve r t ic e s vi 2 v ( g) fo r 0 · i · k a n d e d g e s ei 2 e ( g) fo r 1 · i · k is c a lle d a wa lk o f le n g t h k fr o m a ve r t e x v0 t o a ve r t e x vk in g, if e a c h e d g e ei jo in s t wo ve r t ic e s vi¡1 a n d vi fo r 1 · i · k. a wa lk, in wh ic h a ll t h e ve r t ic e s a r e d is t in c t , is c a lle d a p a t h . if tg ( ei ) < tg ( ej ) fo r 1 · i < j · k, t h e n l is a n a s c e n d in g p a t h fr o m v0 t o vk in g. give n t wo ve r t ic e s u a n d v, if t h e r e is a n a s c e n d in g p a t h fr o m u t o v, t h e n v r e c e ive s t h e in fo r m a t io n o f u. n o t e t h a t t wo n o n -a d ja c e n t e d g e s c a n h a ve t h e s a m e la b e l. s in c e we c o n s id e r o n ly ( s t r ic t ly) a s c e n d in g p a t h s , t h e n s u c h e d g e s ( i.e . c a lls ) a r e in d e p e n d e n t , wh ic h m e a n s t h a t d u r in g t h e c o n s id e r a t io n o f m in im a l n u m b e r o f c a lls t h e e d g e s wit h t h e s a m e la b e l c a n b e r e o r d e r e d a r b it r a r ily b u t fo r a n y t1 < t2 a ll t h e e d g e s wit h t h e la b e l t1 a r e o r d e r e d b e fo r e a n y o f t h e e d g e s wit h t h e la b e l t2. th e r e fo r e , in t h is c a s e we c a n a s s u m e t h a t t h e r e a r e n o t a n y t wo e d g e s e1 a n d e2 s u c h t h a t tg ( e1 ) = tg ( e2 ) . d e n o t e t h e s e t o f e d g e s a d ja c e n t t o a g ive n ve r t e x v b y ev ( g) . give n a n e d g e e a n d o n e o f it s t wo e n d p o in t s v, we c o n s id e r t h e fo llo win g t wo s u b s e t s o f t h e s e t ev ( g ) : ½+v ( e; g) = fe0 2 ev ( g ) jtg ( e0 ) ¸ tg ( e) g; ( 1 ) ½¡v ( e; g ) = fe0 2 ev ( g) jtg ( e0 ) · tg ( e) g: ( 2 ) s o m e t im e s we o m it t h e a r g u m e n t g in n o t a t io n s . de¯nition 1: an identi¯cation of two vertices v1 and v2 [4] in a gossip graph g is a gossip graph g0, which is de¯ned as follows: the edges between the vertices v1 and v2 are deleted and these two vertices are replaced by a vertex u, whose set of incident edges is eu ( g 0 ) = ev1 ( g) [ ev2 ( g) a n in t e r c h a n g e o f t wo ve r t ic e s in a c a llin g s c h e m e is d e ¯ n e d a s fo llo ws : s t a r t e d fr o m t h e in d ic a t e d m o m e n t o f a t im e t o t h e e n d o f t h e c a llin g s c h e m e t wo ve r t ic e s ( a n d t h e e d g e s a d ja c e n t t o t h e m ) a r e r e p la c e d b y e a c h o t h e r . in t h is p a p e r we d e ¯ n e t h e c o n c e p t o f lo c a l in t e r c h a n g e , i.e . a n in t e r c h a n g e t h a t is d e ¯ n e d o n ly fo r a d ja c e n t ve r t ic e s . de¯nition 2: the permute higher operation p + ( e) on a selected edge e 2 e ( g) connecting vertices u and v is called a modi¯cation of g, which moves the edges of g adjacent to e as v. hovnanyan, su. poghosyan,v. poghosyan 7 follows: eu ( p + ( e ) g) = ½¡u ( e; g ) [ ½+v ( e; g) ; ( 3 ) and ev ( p + ( e ) g) = ½¡v ( e; g) [ ½+u ( e; g) : ( 4 ) correspondingly, the permute lower operation p ¡ ( e ) moves the edges of g adjacent to e as follows: eu ( p ¡ ( e ) g) = ½+u ( e; g ) [ ½¡v ( e; g) ; ( 5 ) and ev ( p ¡ ( e ) g) = ½+v ( e; g) [ ½¡u ( e; g) : ( 6 ) th e o p e r a t o r s p + a n d p ¡ a r e c a lle d t h e operators of local interchange. lemma 1: the result of the action of the operators of local interchange on a complete gossip graph is also a complete gossip graph. a c t u a lly, if a c a ll b e t we e n t wo p a r t ic ip a n t s t a ke s p la c e a t t h e m o m e n t t0, s t a r t e d fr o m t h a t p o in t t h e y b o t h h a ve t h e s a m e in fo r m a t io n a n d a r e e qu iva le n t . h e n c e , if we c h a n g e a ll c a lls wh ic h t o o k p la c e a ft e r t h a t m o m e n t ( p e r m u t e h ig h e r ) o u r n e w g o s s ip s c h e m e wo u ld a ls o b e c o m p le t e . a t t h e s a m e t im e , we c h a n g e d n e it h e r t h e n u m b e r o f e d g e s , n o r t h e n u m b e r o f r o u n d s r e qu ir e d t o p e r fo r m g o s s ip in g . th e s a m e a s s u m p t io n s c o u ld b e m a d e in c a s e o f p e r m u t e lo we r o p e r a t io n . l e t u s d e ¯ n e a n o p e r a t io n o n g o s s ip g r a p h s a+ = p + ( e1 ) p + ( e2 ) : : : p + ( ep ) ( a ¡ = p ¡ ( e1 ) p ¡ ( e2 ) : : : p ¡ ( ep ) ) is t h e s e qu e n c e o f p e r m u t e h ig h e r ( lo we r ) o p e r a t io n s o n e d g e s ei, i = 1 ; : : : ; p. lemma 2: the result of the operation a+ (a¡) does not depend on the order of the edges ei. p r oof. w it h o u t lo s s o f g e n e r a lit y we c a n o n ly c o n s id e r t h e p e r m u t e h ig h e r o p e r a t io n a+ wit h p = 2 . a s s u m e t h a t tg ( e1 ) < tg ( e2 ) , wh e r e tg is e d g e -o r d e r in g o f g o s s ip g r a p h g. th e r e a r e t wo p o s s ib le c a s e s : 1 ) e d g e s e1 a n d e2 a r e a d ja c e n t . in t h is c a s e , if a + = p + ( e1 ) p + ( e2 ) , a ft e r p + ( e1 ) a ll t h e e d g e s wh ic h a r e a d ja c e n t t o e1 ( in c lu d in g e2 ) will c h a n g e t h e ir e n d p o in t fr o m o n e e n d p o in t o f e1 t o a n o t h e r a n d a ft e r p + ( e2 ) a ll t h e e d g e s fo r wh ic h tg ( e ) > tg ( e2 ) a ls o will b e p e r m u t e d . if a+ = p + ( e2 ) p + ( e1 ) , t h e n a ft e r t h e ¯ r s t p h a s e o f a + a ll t h e e d g e s wit h tg ( e ) > tg ( e2 ) will b e p e r m u t e d a n d a ft e r t h e s e c o n d p h a s e a ll t h e e d g e s wh ic h a r e a d ja c e n t t o e1 a n d h a ve tg ( e ) > tg ( e1 ) ( in c lu d in g e2 ) will a ls o b e p e r m u t e d . th e s e o p e r a t io n s a r e d e m o n s t r a t e d in fig . 1 . it is o b vio u s t h a t t h e r e s u lt o f t h e s e t wo va r ia t io n s o f a+ is t h e s a m e . 2 ) e d g e s e1 a n d e2 a r e n o t a d ja c e n t . th is is a s im p le r c a s e in wh ic h t h e o p e r a t io n p + ( e ) o n o n e o f t h e m d o e s n o t a ®e c t d ir e c t ly t h e o t h e r . s o , it is o b vio u s t h a t a ll t h e va r ia t io n s o f a+ a r e e qu iva le n t . s o , in t h is s e c t io n t h e o p e r a t io n s o f lo c a l in t e r c h a n g e a n d t h e ir m a in p r o p e r t ie s we r e d e s c r ib e d . in t h e n e xt t wo s e c t io n we g ive s o m e a p p lic a t io n o f t h e o p e r a t io n s d e s c r ib e d in t h is s e c t io n . 8 method of local interchange to investigate gossip problems fig. 1. equivalence of di®erent variations of a+ 3 . min im u m go s s ip gr a p h s t heor em 1: the minimum required number of calls in complete gossip schemes is 2 n ¡ 4 , n ¸ 4 . p r oof. a s a lr e a d y m e n t io n e d in t h e in t r o d u c t io n it h a s b e e n s h o wn in n u m e r o u s wo r ks [1 , 2 , 3 , 4 ] t h a t t h e m in im a l n u m b e r o f c a lls in g o s s ip s c h e m e s is 2 n ¡ 4 , wh e r e n is t h e n u m b e r o f ve r t ic e s . in t h is s e c t io n we will p r o ve t h is s t a t e m e n t in a c o m p le c t ly n e w wa y, wh ic h is b a s e d o n t h e o p e r a t io n s o f lo c a l in t e r c h a n g e . l e t u s d e n o t e b y f ( n) t h e m in im u m n u m b e r o f e d g e s r e qu ir e d t o c o n s t r u c t a g o s s ip g r a p h o n n ve r t ic e s . th e r e a r e m a n y s o lu t io n s o f t h is p r o b le m t h a t g ive t h e n u m b e r o f e d g e s ( c a lls ) e qu a l t o 2 n ¡ 4 . th e r e fo r e , f ( n) · 2 n ¡ 4 is va lid . s o , t h e im p o r t a n t p a r t is t o p r o ve t h a t f ( n) ¸ 2 n ¡ 4 a ls o . co n s id e r a g o s s ip g r a p h wit h n ve r t ic e s a n d t h e n u m b e r o f e d g e s e qu a l t o f ( n) . ou r g o a l is t o p r o ve t h a t f ( n) ¸ f ( n ¡ 1 ) + 2 . a ft e r t h a t , t a kin g in t o a c c o u n t t h a t f ( 4 ) = 4 we will g e t t h a t f ( n) ¸ 2 n ¡ 4 . s u p p o s e we h a ve a g r a p h g wit h f ( n) c a lls . th e r e a r e 3 p o s s ib le kin d s o f g r a p h s d e p e n d in g o n t h e r e p e t it io n o f in fo r m a t io n . 1 ) it c o n t a in s a c yc le . s u p p o s e a c = ( e1; e2; : : : ; ep ) is a c yc le wit h le n g t h p a n d t h e e d g e s e1 a n d ep a r e a d ja c e n t t o t h e ve r t e x v ( s e e fig . 2 ) . fig. 2. cycle c in gossip graphs th e s im p le s t c a s e is if p = 2 . if t h is h o ld s , we c a n a p p ly a n o p e r a t io n o f t h e id e n t i¯ c a t io n o f t wo ve r t ic e s c o n n e c t e d b y t h is m u lt ip le e d g e wh ic h is d e s c r ib e d in t h e s e c t io n a b o ve . ob vio u s ly, a ft e r t h is o p e r a t io n o n c o m p le t e g o s s ip g r a p h s , t h e o b t a in e d n e w g o s s ip g r a p h s a r e a ls o c o m p le t e ( it d o e s n o t a ®e c t o n a s c e n d in g p a t h s b e t we e n t h e p a ir s o f ve r t ic e s ) . h e n c e , o u r n e w g r a p h g0 wit h n ¡ 1 ve r t ic e s h a s je ( g0 ) j ¸ f ( n ¡ 1 ) e d g e s . on t h e o t h e r h a n d g0 is v. hovnanyan, su. poghosyan,v. poghosyan 9 o b t a in e d fr o m g b y r e m o vin g t wo e d g e s a n d m e r g in g t wo ve r t ic e s , t h u s , je ( g0 ) j = f ( n) ¡ 2 . h e n c e , we o b t a in f ( n) ¸ f ( n ¡ 1 ) + 2 : ( 7 ) n o w c o n s id e r t h e c a s e o f p 6= 2 a n d a n o p e r a t io n a¡ = ( p ¡ ( e2 ) p ¡ ( e3 ) : : : p ¡ ( ep¡1 ) ) . a s it is p r o ve d in t h e p r e vio u s s e c t io n t h e r e s u lt o f a p p lic a t io n o f t h is o p e r a t io n o n a c o m p le t e g o s s ip g r a p h is a ls o a c o m p le t e g o s s ip g r a p h . in t h e r e s u lt t h e e d g e s e1 a n d ep will jo in a n d fo r m a d o u b le -e d g e . th u s , we c a m e t o t h e c a s e o f p = 2 . fu r t h e r s t e p s a r e d e s c r ib e d a b o ve . 2 ) th e r e a r e n o c yc le s ( n oh o g r a p h s ) . in t h is c a s e we c a n r e fe r t o [6 , 2 0 ], b u t le t u s a ls o c o n s id e r t h is c a s e . l e t l1 = ( e1; e2; : : : ; en ) a n d l2 = ( l1; l2; : : : ; lm ) b e a s c e n d in g p a t h s fr o m t h e ve r t e x v t o t h e ve r t e x u. l e t u s d e ¯ n e t h e o p e r a t io n s a¡e = ( p ¡ ( e2 ) p ¡ ( e3 ) : : : p ¡ ( en¡1 ) ) ( 8 ) a n d a¡l = ( p ¡ ( l2 ) p ¡ ( l3 ) : : : p ¡ ( lm¡1 ) ) ( 9 ) o n g o s s ip g r a p h . a ft e r a p p lic a t io n o f t h e s e o p e r a t io n s we will o b t a in a t e t r a g o n wh ic h in vo lve s t h e ve r t ic e s u a n d v a n d t h e e d g e s e1; l1; en; lm. th e n we c h o o s e t h e g r e a t e s t fr o m t h e p a ir e1; l1 a n d a p p ly o p e r a t io n " p e r m u t e lo we r " o n it . w e will o b t a in a t r ia n g le t h a t is a ls o a c yc le ( c a s e 1 ) . th is p r o c e s s is d e m o n s t r a t e d in fig . 3 . fig. 3. ascending paths in gossip graphs 3 ) th e r e a r e n o d u p lic a t e s o f in fo r m a t io n ( n od u p g r a p h s ) . in t h is c a s e t h e r e is e xa c t ly o n e a s c e n d in g p a t h b e t we e n a n y p a ir o f ve r t ic e s . it h a s b e e n s h o wn in [2 1 ], t h a t in t h is c a s e f 0 ( n) = 2 :2 5 n ¡ 6 , n = 4 k, k ¸ 2 , wh e r e f 0 ( n) is t h i m in im u m n u m b e r o f e d g e s o f n od u p g r a p h a n d it c o in c id e s wit h 2 n ¡ 4 o n ly wh e n n = 8 . s o , if it is a n od u p g r a p h , it a lwa ys h a s m o r e e d g e s t h a n 2 n ¡ 4 a n d t h e o n ly e xc e p t io n is t h e c a s e wh e n n = 8 . ca s e o f n = 4 wa s n o t c o n s id e r e d in [2 1 ], b u t in t h is c a s e f ( n) a n d f 0 ( n) a ls o c o in c id e . in t h e n e xt s e c t io n o t h e r a p p lic a t io n s o f lo c a l in t e r c h a n g e o p e r a t io n s a r e d e s c r ib e d . 4 . n oh o gr a p h s b a s e d o n k n äo d e l gr a p h s in [1 9 ] it is m e n t io n e d t h a t t h e m in im u m t im e n e e d e d t o c o m p le t e g o s s ip in g is dlo g ne. in [2 2 ] a p r o b le m o f ¯ n d in g m in im u m t im e o f g o s s ip in g in c a s e o f n oh o g r a p h s ( p r o b le m 2 6 ) is r a is e d . in t h is s e c t io n we will s h o w a n e w m e t h o d o f c o n s t r u c t io n o f n oh o g r a p h s wit h m in im u m g o s s ip in g t im e b y a p p lyin g a n o p e r a t io n o f lo c a l in t e r c h a n g e o n k n äo d e l g r a p h s ( [7 ]) . 1 0 method of local interchange to investigate gossip problems de¯nition 3: the k näodel graph on n ¸ 2 vertices (n even) and of degree ¢ ¸ 1 is denoted by w¢ ;n. the vertices of w¢ ;n are the pairs ( i; j ) with i = 1 ; 2 and 0 · j · n=2 ¡ 1 . f or every j, 0 · j · n= 2 ¡ 1 and l = 1 ; : : : ; ¢ , there is an edge with the label l between the vertex ( 1 ; j ) and ( 2 ; ( j + 2 l¡1 ¡ 1 ) m o d n= 2 ) . fr o m t h e d e ¯ n it io n it fo llo ws t h a t t h is g r a p h is c o n n e c t e d o n ly if ¢ ¸ 2 . w e will c o n s id e r t h e k n äo d e l g r a p h s wit h ¢ = dlo g 2 ne. fig. 4. gossip graph based on knäodel graph in [1 8 ] it wa s s h o wn t h a t t h e c o m b in a t io n o f t wo k n äo d e l g r a p h s wit h ¢ 1 = dlo g 2 ne a n d ¢ 2 = 1 is a c o m p le t e g o s s ip g r a p h ( s e e fig . 4 ) . l e t u s d e ¯ n e t h e o p e r a t io n a ¡ l = fp ¡ ( e ) jtg ( e ) < tg ( l ) g a s a n o p e r a t io n o f p e r m u t a t io n o f a ll e d g e s t h a t h a ve lo we r we ig h t va lu e ( o r la b e l) t h e n l. a ft e r a p p lic a t io n o f a¡2 o n k n äo d e l g r a p h s a ll e d g e s wit h la b e l 1 will b e p e r m u t e d ( s e e fig . 5 ) . fig. 5. noho graph based on knäodel graph lemma 3: the result of application of a¡2 on complete gossip graph based on k näodel graph is a noho gossip graph with minimum gossiping time t = dlo g 2 ne. p r oof. w it h o u t lo s s o f g e n e r a lit y we c a n c o n s id e r o n ly t h e p e r m u t a t io n o f a n e d g e t h a t c o n n e c t s ve r t ic e s ( 1 ; 0 ) a n d ( 2 ; 0 ) . th is e d g e h a s t wo a d ja c e n t e d g e s wit h la b e l 2 , h e n c e , a ft e r a¡2 it will c h a n g e it s p o s it io n t wic e . s o , a ft e r t h is o p e r a t io n t h e ve r t ic e s t h a t c o n n e c t t h is e d g e a r e ( 2 ; 1 ) a n d ( 1 ; n=2 ¡ 1 ) . th e s e ve r t ic e s we r e n o t c o n n e c t e d in o u r in it ia l g r a p h ( a c c o r d in g t o t h e d e ¯ n it io n o f k n äo d e l g r a p h ) , h e n c e , a ft e r a¡2 t h e r e will b e n o d u p lic a t e e d g e s . on t h e o t h e r h a n d , h e r e we a p p ly a n in ve r s e o p e r a t io n o f t h e o p e r a t io n d e s c r ib e d in t h e p r e vio u s s e c t io n o f c a s e 2 . r e a lly, a ft e r p e r m u t a t io n o f t h is e d g e t h e ve r t ic e s ( 1 ; 0 ) , ( 2 ; 0 ) , ( 2 ; 1 ) a n d ( 1 ; n=2 ¡ 1 ) a n d e d g e s wit h la b e ls tg ( e1 ) = 1 , tg ( e2 ) = 2 , tg ( e3 ) = 2 a n d tg ( e4 ) = dlo g 2 ne will fo r m a t e t r a g o n wh ic h is n o t a c yc le . th u s , we e xc lu d e t h e p o s s ib ilit y o f e xis t e n c e o f c yc le s a ft e r t h is o p e r a t io n . h e n c e , a ft e r a p p lic a t io n o f a¡2 we o b t a in a n oh o g r a p h wh ic h is e qu iva le n t t o o u r in it ia l c o m p le t e g o s s ip g r a p h . v. hovnanyan, su. poghosyan,v. poghosyan 1 1 th is m e t h o d s o f lo c a l in t e r c h a n g e a ls o h a ve s o m e o t h e r a p p lic a t io n s , wh ic h a llo we d u s t o in ve s t ig a t e m a n y va r ia t io n s o f b r o a d c a s t p r o b le m a n d im p r o ve s o m e kn o wn p a r a m e t e r s c o n c e r n in g t h e b r o a d c a s t s c h e m e s , b u t it is n o t t h e s u b je c t o f t h is p a p e r . a c kn o wle d g e m e n t s th is wo r k wa s s u p p o r t e d b y s t a t e co m m it t e e o f s c ie n c e ( s cs ) me s r a , in fr a m e o f t h e r e s e a r c h p r o je c t n o s cs 1 3 -1 b 1 7 0 . th e a u t h o r s a r e g r a t e fu l t o a ka d . y u .h . s h o u ko u r ia n fo r im p o r t a n t d is c u s s io n s a n d c r it ic a l r e m a r ks o n a ll s t a g e s o f t h e wo r k. refer ences [1 ] b .b a ke r a n d r .s h o s t a k, \ go s s ip s a n d t e le p h o n e s " , d iscrete m ath., vo l. 2 , p p . 1 9 1 -1 9 3 , 1 9 7 2 . [2 ] r .t.b u m b y, \ a p r o b le m wit h t e le p h o n e s " , siam j . alg. d isc. m ath., vo l. 2 , p p . 1 3 -1 8 , 1 9 8 1 . [3 ] a . h a jn a l, e .c. miln e r a n d e . s z e m e r e d i, \ a c u r e fo r t h e t e le p h o n e d is e a s e " , canad. m ath b ull., vo l. 1 5 , p p . 4 4 7 -4 5 0 , 1 9 7 2 . [4 ] t. tijd e m a n , \ on a t e le p h o n e p r o b le m " , nieuw arch. w isk., vo l. 3 , p p . 1 8 8 -1 9 2 , 1 9 7 1 . [5 ] a . s e r e s s , \ qu ic k g o s s ip in g b y c o n fe r e n c e c a lls " , siam j . d isc. m ath, vo l. 1 , p p . 1 0 9 1 2 0 , 1 9 8 8 . [6 ] d .b . w e s t , \ go s s ip in g wit h o u t d u p lic a t e t r a n s m is s io n s " , siam j . alg. d isc. m eth., vo l. 3 , p p . 4 1 8 -4 1 9 , 1 9 8 2 . [7 ] g. fe r t in a n d a . r e s p a u d , \ a s u r ve y o n k n äo d e l g r a p h s " , d iscrete applied m athematics, vo l. 1 3 7 ( 2 ) , p p . 1 7 3 -1 9 5 , 2 0 0 3 . [8 ] r . l a b a h n , \ k e r n e ls o f m in im u m s iz e g o s s ip s c h e m e s " , d iscr. m ath., vo l. 1 4 3 , p p . 9 9 -1 3 9 , 1 9 9 5 . [9 ] r . l a b a h n , \ s o m e m in im u m g o s s ip g r a p h s " , networks, vo l. 2 3 , p p . 3 3 3 -3 4 1 , 1 9 9 3 . [1 0 ] g. fe r t in a n d r . l a b a h n , \ co m p o u n d in g o f g o s s ip g r a p h s " , networks, vo l. 3 6 , p p . 1 2 6 -1 3 7 , 2 0 0 0 . [1 1 ] a . b a ve la s , \ co m m u n ic a t io n p a t t e r n in t a s k-o r ie n t e d g r o u p s " , j . acoust. soc. amer., vo l. 2 2 , p p . 7 2 5 -7 3 0 , 1 9 5 0 . [1 2 ] w . k n o d e l, \ n e w g o s s ip s a n d t e le p h o n e s " , d iscr. m ath., vo l. 1 3 , p p . 9 5 , 1 9 7 5 . [1 3 ] z. h o a n d m. s h ig e n o , \ n e w b o u n d s o n t h e m in im u m n u m b e r o f c a lls in fa ilu r e -t o le r a n t go s s ip in g " , networks, vo l. 5 3 , p p . 3 5 -3 8 , 2 0 0 9 . [1 4 ] k .a . b e r m a n a n d m. h a wr ylyc z , \ te le p h o n e p r o b le m s wit h fa ilu r e s " , siam j . alg. d isc. m eth., vo l. 7 , p p . 1 3 -1 7 , 1 9 8 7 . [1 5 ] r .w . h a d d a d , s . r o y a n d a .a . s c h a ®e r , \ on g o s s ip in g wit h fa u lt y t e le p h o n e lin e s " , siam j . alg. d isc. m eth., vo l. 8 , p p . 4 3 9 -4 4 5 , 1 9 8 7 . [1 6 ] t. h a s u n a m a a n d h . n a g a m o c h i, \ im p r o ve d b o u n d s fo r m in im u m fa u lt -t o le r a n t g o s s ip g r a p h s " , l ncs 6 9 8 6 , p p . 2 0 3 -2 1 4 , 2 0 1 1 . [1 7 ] r . a h ls we d e , l . ga r g a n o , h . s . h a r o u t u n ia n a n d l . h . k h a c h a t r ia n , \ fa u lt -t o le r a n t m in im u m b r o a d c a s t n e t wo r ks " , ne tw or k s, vo l. 2 7 , p p . 2 9 3 -3 0 7 , 1 9 9 6 . [1 8 ] v .h . h o vn a n ya n , h .e . n a h a p e t ya n , s u . s . p o g h o s ya n a n d v . s . p o g h o s ya n , \ tig h t e r u p p e r b o u n d s fo r t h e m in im u m n u m b e r o f c a lls a n d r ig o r o u s m in im a l t im e in fa u lt t o le r a n t g o s s ip s c h e m e s " , a r x iv:1 3 0 4 .5 6 3 3 . [1 9 ] h e d e t n ie m i, h e d e t n ie m i a n d l is t m a n , \ a s u r ve y o f g o s s ip in g a n d b r o a d c a s t in g in c o m m u n ic a t io n n e t wo r ks " , networks, vo l. 1 8 , p p . 3 1 9 -3 4 9 , 1 9 8 8 . 1 2 method of local interchange to investigate gossip problems [2 0 ] d . b . w e s t , \ a c la s s o f s o lu t io n s t o t h e go s s ip p r o b le m " , p a r t i, d isc. m ath. vo l. 3 9 , n o .3 , p p . 3 0 7 -3 2 6 , 1 9 8 2 ; p a r t ii, d isc. m ath. vo l. 4 0 , n o . 1 , p p . 8 7 -1 1 3 , 1 9 8 2 ; p a r t iii, d isc. m ath., vo l 4 0 2 -3 , p p . 2 8 5 -3 1 0 , 1 9 8 2 . [2 1 ] a . s e r e s s , \ qu ic k go s s ip in g wit h o u t d u p lic a t e t r a n s m is s io n s " , graphs and combinatorics, vo l. 2 , p p . 3 6 3 -3 8 1 , 1 9 8 6 . [2 2 ] p . fr a ig n ia u d , a .l . l ie s t m a n , d . s o it e a u , " op e n p r o b le m s " , p arallel p rocessing l etters, vo l. 3 4 , p p . 5 0 7 -5 2 4 , 1 9 9 3 . submitted 02.09.2013, accepted 18.10.2013. gossip ëý¹çñý»ñç ñ»ï³½áïáõùá \ èáï³é ÷áë³ý³ïù³ý \ ù»ãá¹ç ùççáóáí ì. ðáíý³ýû³ý, ê. äáõáëû³ý ¨ ì. äáõáëû³ý ²ù÷á÷áõù ²ûë ñá¹í³íáõù ýï³ñ³·ñí³í »ý éáï³é ÷áë³ý³ïù³ý ù»ãá¹á ¨ ýñ³ ïçñ³éáõãûáõýý»ñá gossip ëý¹çñý»ñç ñ»ï³½áïù³ý ñ³ù³ñ: ²ûë ù»ãá¹á ñçùýí³í ¿ ï»õ³÷áë»é ³í»éç ù»í»ñá ¨ ï»õ³÷áë»é ³í»éç ÷áùñ»ñá ·áñíáõáõãûáõýý»ñç µ³½ù³ïç ïçñ³éù³ý íñ³, áñáýù ³ñï³å³ïï»ñáõù »ý ùç n ·³·³ã³ýç gossip ·ñ³ýá ù»ï áõñçßç íñ³` ï»õ³÷áë»éáí ùç³ûý ïáõ»ñá, ã÷áë»éáí ïáõ»ñç ýçß»ñá (ñ³ù³å³ï³ëë³ý ½³ý·»ñç å³ù³ý³ïý»ñá): ú·ï³·áñí»éáí ³ûë ù»ãá¹á` knäodel ·ñ³ýý»ñç ñçù³ý íñ³ ù»ýù ëï³ó»é »ýù ùçýçù³é å³ù³ý³ï³ûçý noho gossip ·ñ³ýý»ñ (áã áù »ñµ»ù ãç éëáõù çñ çýýáñù³óç³ý): êï³óí³í ùçýçù³é å³ù³ý³ïç ×ß·ñçï ³ñå»ùý ¿` t = dlo g ne: ²ûë ù»ãá¹ç ùççáóáí ïñí»é ¿ gossip ëë»ù³ý»ñç ùçýçù³é ³ýññ³å»ßï ½³ý·»ñç ù³ý³ïç µ³ý³ó¨ç ( 2 n ¡ 4 ; n ¸ 4 ) ¨ë ùç ýáñ ³ñï³íáõù: èññëåäîâàíèå gossip ïðîáëåì ïðè ïîìîùè ìåòîäà ”ëîêàëüíûõ ïåðåñòàíîâîê” â. îâíàíÿí, ñ. ïîãîñÿí è â. ïîãîñÿí àííîòàöèÿ â ýòîé ñòàòüå ââåäåí ìåòîä ëîêàëüíûõ ïåðåñòàíîâîê äëÿ èçó÷åíèÿ gossip çàäà÷. ýòîò ìåòîä îñíîâàí íà ìíîãîêðàòíîì ïðèìåíåíèè îïåðàöèé ïåðåñòàâèòü áîëåå ñòàðøèå è ïåðåñòàâèòü ìåíåå ìëàäøèå, êîòîðûå îòîáðàæàþò îäèí ãðàô ñ n âåðøèíàìè íà äðóãîé, ïåðåìåùàÿ òîëüêî ðåáðà, íå ìåíÿÿ èõ ìåòîê (âðåìÿ ñîîòâåòñòâóþùåãî çâîíêà). èñïîëüçóÿ ýòîò ìåòîä, íà îñíîâå knäodel ãðàôîâ, ìû ïîëó÷èëè ìèíèìàëüíî âðåìåííûå noho gossip ãðàôû (íèêòî íèêîãäà íå óñëûøèò ñâîþ èíôîðìàöèþ). òî÷íîå çíà÷åíèå ïîëó÷åííîãî ìèíèìàëüíîãî âðåìåíè t = dlo g ne. ñ ïîìîùüþ ýòîãî ìåòîäà ïîëó÷åíî åùå îäíî íîâîå äîêàçàòåëüñòâî ôîðìóëû ( 2 n ¡ 4 ; n ¸ 4 ) äëÿ ìèíèìàëüíîãî íåîáõîäèìîãî êîëè÷åñòâà çâîíêîâ gossip ñõåì. d:\tex_522\04_pogh.dvi mathematical problems of computer science 52, 30{42, 2019. u d c 5 1 9 .1 rumor spr eading and i nvasion p er colation s u r e n s . p o g h o s ya n a n d v a h a g n s . p o g h o s ya n institute for informatics and automation problems of nas ra e-mail: psuren55@yandex.ru, povahagn@gmail.com abstract we study the models of rumor spreading and invasion bond percolation aimed at the revelation of possible connections between them. rumor spreading model describes the dissemination of a rumor due to the periodical repetition of sequential phone calls, whereas the invasion bond percolation refers to the spread of liquid in the porous environment. during a round of the rumor spreading, each node performs a call only once, meanwhile transmitting all the information that it knows at the moment of the call to its neighbors. the rumor reaches the receiver node during one round if there is a chain of successive calls between the source of the rumor and that node. the sequence of calls is taken uniformly at random from the set of all possible sequences (permutations of nodes). we compare the propagation of the rumor spreading with the invasion bond percolation in order to put forth necessary improvements of the percolation rules to map one model onto another, and vice versa. keywords: gossip problem, information dissemination, invasion percolation. 1 . in t r o d u c t io n a n d mo d e ls th e s p r e a d in g o f r u m o r s a n d in va s io n p e r c o la t io n a r e c o r e p r o b le m s in t wo d i®e r e n t ¯ e ld s : t h e t h e o r y o f d is s e m in a t io n o f in fo r m a t io n [1 ]{ [1 7 ] a n d t h e t h e o r y o f ° o w in a p o r o u s m e d ia [1 8 ]{ [2 3 ]. b o t h p r o b le m s c o n c e r n wit h t h e s t o c h a s t ic g r o wt h o f c lu s t e r s : t h e c lu s t e r o f in fo r m e d in d ivid u a ls in t h e ¯ r s t c a s e a n d t h e c lu s t e r o f in va d e d s it e s in t h e s e c o n d o n e . h e r e , we c o n s id e r t wo m o d e ls t yp ic a l fo r e a c h p r o b le m a n d e s t a b lis h a lin k b e t we e n t h e m . a m o n g t h e va r ie t y o f fo r m u la t io n s o f t h e ¯ r s t p r o b le m , we c h o o s e t h e r a n d o m p h o n e c a ll m o d e l [1 9 ] d e ¯ n e d o n t h e s o c a lle d c o m m u n ic a t io n g r a p h g ( v; e ) wit h t h e s e t o f ve r t ic e s v a n d t h e s e t o f e d g e s e. v e r t ic e s v d e n o t e t h e s e t o f p la ye r s , t h e e d g e s e d e n o t e p o s s ib le c o n n e c t io n s b e t we e n t h e p la ye r s b y p h o n e c a lls . w h e n e ve r a c o n n e c t io n is e s t a b lis h e d , t h e r u m o r c a n b e t r a n s m it t e d fr o m o n e p la ye r ( if s h e h o ld s t h e r u m o r ) t o a n o t h e r p la ye r ( if s h e d o e s n o t h o ld t h e r u m o r ye t ) . in [1 9 ], t h e a u t h o r s c o n s id e r e d t h e m o d e l wh e r e t h e p la ye r s c o m m u n ic a t e in parallel d u r in g c o m m u n ic a t io n r o u n d s . it m e a n s t h a t a n y in fo r m a t io n r e c e ive d in a r o u n d c a n n o t b e fo r wa r d e d t o a n o t h e r p la ye r in t h e s a m e r o u n d . th e a im is t o s p r e a d a ll r u m o r s fr o m a ll p la ye r s t o o t h e r s . th e p r o b le m wa s t o ¯ n d a n a lg o r it h m u s in g o n ly o ( ln jv j) r o u n d s a n d o ( jej ln ln jej ) c a lls . 3 0 su. poghosyan and v. poghosyan 3 1 in t h is p a p e r , we c o n c e n t r a t e o n kin e t ic s o f t h e r u m o r p r o p a g a t io n o n t h e c o m m u n ic a t io n g r a p h g ( v; e ) . to t h is e n d , we in ve s t ig a t e t h e e ®e c t o f sequential c o m m u n ic a t io n s wh e n a r u m o r c a n b e t r a n s la t e d t h r o u g h a c h a in o f r a n d o m c a lls d u r in g o n e r o u n d . co r r e s p o n d in g ly, e a c h r o u n d in o u r m o d e l is a ¯ n it e t im e in t e r va l c o n t a in in g a ll p o s s ib le c o n n e c t io n s ei 2 e; i = 1 ; 2 ; : : : ; jej, wh ic h h a p p e n e xa c t ly o n c e a t m o m e n t s t ( i) , wh e r e t( i ) ´ t( ei ) is t h e m o m e n t o f c o n n e c t io n ei in t h e c u r r e n t r o u n d . w e a s s u m e t h a t a ll m o m e n t s t( i ) ; i = 1 ; 2 ; : : : ; jej a r e d is t in c t a n d u n ifo r m ly d is t r ib u t e d b e t we e n t h e b e g in n in g a n d t h e e n d o f e a c h r o u n d . th e e ®e c t o f s e qu e n t ia l c o m m u n ic a t io n s le a d s t o a s p e c i¯ c p a t t e r n o f t h e c lu s t e r g r o wt h , wh ic h is s im ila r t o t h e in va s io n p e r c o la t io n . h o we ve r , t h e r u le s o f g r o wt h o f t h e c lu s t e r s o f in fo r m e d p la ye r s a n d in va d e d s it e s in t h e p e r c o la t io n p r o b le m a r e n o t c o m p le t e ly id e n t ic a l. l e t v¤ 2 v b e a ¯ xe d p la ye r wh o is t h e o r ig in o f t h e r u m o r a n d p = ( p1; p2; : : : ; pjej ) is a p e r m u t a t io n o f n u m b e r s ( 1 ; 2 ; : : : ; jej ) s u c h t h a t t( p1 ) < t ( p2 ) < ¢ ¢ ¢ < t( pjej ) in t h e ¯ r s t r o u n d . d e n o t in g b y eij 2 e t h e e d g e c o n n e c t in g a d ja c e n t ve r t ic e s i 2 v a n d 8 j 2 v , we d e ¯ n e a n o r d e r e d p a t h b e t we e n t h e ve r t ic e s i1 2 v a n d ik 2 v a s t h e s e qu e n c e o f c o n n e c t io n s ei1i2; ei2i3 ; : : : ; eik¡1ik . th e p a t h b e t we e n i1 a n d ik c o n d u c t s t h e r u m o r if t ( ei1i2 ) < t ( ei2i3 ) < ¢ ¢ ¢ < t( eik¡1ik ) . give n t h e p e r m u t a t io n o f c a lls p , we will s a y t h a t t h e r u m o r s t a r t e d fr o m v¤ r e a c h e s p la ye r v 2 v in t h e ¯ r s t r o u n d if t h e r e e xis t s a c o n d u c t in g p a t h b e t we e n v¤ a n d v. th e s u b s e t v1 ½ v o f ve r t ic e s r e a c h a b le fr o m v¤ d u r in g t h e ¯ r s t r o u n d r e p r e s e n t s t h e s e t o f p la ye r s g e t t in g in fo r m e d d u e t o p e r m u t a t io n o f c a lls p . th e qu a n t it y o f in t e r e s t is t h e n u m b e r o f in fo r m e d p la ye r s jv1jp a n d it s a ve r a g e t a ke n o ve r a ll p e r m u t a t io n s hjv1ji =p p jv1jp p robjv1jp . a c o n d u c t in g p a t h b e t we e n t wo a r b it r a r y ve r t ic e s is n o t n e c e s s a r ily s e lfa vo id in g . in d e e d , t h e p a t h m a y c o n t a in o n e o r m o r e lo o p s , wh ic h a p p e a r if b o t h a d ja c e n t ve r t ic e s i a n d j in c o n n e c t io n eij b e lo n g in g t o t h e p a t h a r e a lr e a d y in fo r m e d b e fo r e t h e m o m e n t t( eij ) . w e c a n c o in s u c h a c o n n e c t io n a s non-informative in c o n t r a s t t o a informative c o n n e c t io n t h a t c o n d u c t s t h e r u m o r . th e e xc lu s io n o f n o n -in fo r m a t ive c o n n e c t io n s fr o m a ll c o n d u c t in g p a t h s r e m o ve s a ll p o s s ib le lo o p s . th e n , t h e r e m a in in g in fo r m a t ive c o n n e c t io n s fo r m a c o n n e c t e d g r a p h , wh ic h is a t r e e s p a n n in g t h e s u b s e t o f in fo r m e d p la ye r s v1 a n d is r o o t e d a t v¤. b y d e ¯ n it io n , t h e ¯ r s t r o u n d is c o m p le t e if a ll p o s s ib le c o n n e c t io n s ei 2 e; i = 1 ; 2 ; : : : ; jej h a p p e n e d e xa c t ly o n c e . a lt e r n a t ive ly, we c a n s a y t h a t t h e r o u n d is c o m p le t e o n ly if s u b s e t v1 c o n t a in s a ll p o s s ib le ve r t ic e s r e a c h a b le fr o m v ¤ fo r t h e g ive n p e r m u t a t io n p . w h e n t h e ¯ r s t r o u n d is o ve r , t h e n e xt r o u n d s s t a r t c o n s e c u t ive ly wit h t h e s a m e s e qu e n c e o f c o n n e c t io n s t ( i ) ; i = 1 ; 2 ; : : : ; jej. p la ye r s r e c e ivin g t h e r u m o r in t h e ¯ r s t r o u n d c a n s e r ve a s o r ig in s o f t h e s a m e r u m o r fo r t h e s e c o n d r o u n d . a s a b o ve , t h e r u m o r is s p r e a d in g a lo n g t h e c o n d u c t in g p a t h s s t a r t e d a t t h e b o u n d a r y ve r t ic e s o f s u b s e t v1. ( a s u s u a l, we c a ll a ve r t e x v 2 v t h e b o u n d a r y ve r t e x o f a s u b s e t v1 ½ v if v 2 v1 a n d a t le a s t o n e o f t h e n e a r e s t n e ig h b o r s o f v in g r a p h g ( v; e ) d o e s n o t b e lo n g t o v1 ) . th e c o lle c t io n o f in fo r m a t ive c o n n e c t io n s in t h e s e c o n d r o u n d fo r m s a g r a p h , wh ic h is a fo r e s t wit h r o o t s a t t h e b o u n d a r y ve r t ic e s o f v1. th e fo r e s t s p a n s t h e s u b s e t o f p la ye r s v2 b e in g in fo r m e d d u r in g t h e s e c o n d c o m p le t e r o u n d . co n t in u in g t h e p r o c e s s r o u n d b y r o u n d , we o b t a in a s e qu e n c e o f s u b s e t s o f p la ye r s vi; i = 1 ; 2 ; : : : c h a r a c t e r iz e d b y t h e s iz e jvijp a n d t h e a ve r a g e s iz e hjviji fo r e a c h r o u n d a n d b y t h e a ve r a g e c o r r e la t io n s b e t we e n t h e r o u n d s . th e p r o c e s s is ¯ n is h e d wh e n a ll p la ye r s v b e c o m e in fo r m e d wit h t h e r u m o r . th e qu e s t io n is t h e a ve r a g e n u m b e r o f r o u n d s n e e d e d fo r t h e t o t a l s p r e a d in g o f t h e r u m o r . th e a s s o c ia t e d p r o b le m , in va s io n p e r c o la t io n , h a s m a n y fo r m u la t io n s a s we ll [2 0 ]. w e 3 2 rumor spreading and invasion percolation c h o o s e fr o m t h e m a p o p u la r ve r s io n , n a m e ly, t h e in va s io n b o n d p e r c o la t io n m o d e l [2 1 ]. s im ila r t o t h e ¯ r s t p r o b le m , we s t a r t wit h t h e g r a p h g ( v; e ) wit h s e t s o f ve r t ic e s a n d e d g e s v a n d e. w e a s s ig n r a n d o m n u m b e r s pij 2 [0 ; 1 ] t o e a c h e d g e eij 2 e. in va s io n p e r c o la t io n is a p r o c e s s o f s u c c e s s ive o c c u p a t io n o f t h e ve r t ic e s a n d e d g e s o f g ( v; e ) . a t t h e ¯ r s t s t e p , o n e o c c u p ie s a r a n d o m ly c h o s e n ve r t e x v¤ 2 v . th e n , o n e c o n s id e r s a ll e d g e s a d ja c e n t t o v¤ a n d o c c u p ie s t h e e d g e wit h t h e s m a lle s t pij t o g e t h e r wit h t h e s e c o n d ve r t e x b e lo n g in g t o it . a t s u b s e qu e n t s t e p s , o n e ¯ n d s t h e e m p t y e d g e wit h t h e s m a lle s t pij c o n n e c t e d t o t h e o c c u p ie d ve r t ic e s , a n d c h e c ks wh e t h e r t h is e d g e c o n n e c t s t wo o c c u p ie d ve r t ic e s o r n o t . in t h e fo r m e r c a s e , t h e e d g e is c a lle d trapped s in c e it is s it u a t e d b e t we e n t wo o c c u p ie d ve r t ic e s . if t h e e d g e is n o t t r a p p e d , it b e c o m e s o c c u p ie d t o g e t h e r wit h t h e e m p t y ve r t e x it c o n n e c t s wit h t h e o c c u p ie d c lu s t e r . in t h is wa y, o n e c o n s t r u c t s a g r a p h o f o c c u p ie d e d g e s a n d ve r t ic e s t ½ g( v; e ) , wh ic h is a t r e e s p a n n in g t h e c lu s t e r o f o c c u p ie d ve r t ic e s a t t h e g ive n s t a g e . th e d e s c r ib e d m o d e l wa s in t r o d u c e d b y b a r a b a s i [2 2 ] a n d c a lle d t h e in va s io n b o n d p e r c o la t io n m o d e l wit h a lo c a l t r a p p in g r u le t o d is t in g u is h it fr o m t h e o r d in a r y in va s io n b o n d p e r c o la t io n wh e r e t h e t r a p p e d e d g e s c a n b e o c c u p ie d a s we ll a s t h e n o n -t r a p p e d o n e s . b a r a b a s i p r o ve d t h a t t h e t r a p p in g r u le d o e s n o t a ®e c t t h e s c a lin g a n d d yn a m ic p r o p e r t ie s o f g r o win g c lu s t e r s o f o c c u p ie d ve r t ic e s a n d ,t h e r e fo r e , t h e t wo m o d e ls b e lo n g t o t h e s a m e u n ive r s a lit y c la s s . mo r e o ve r , t h e c o n s t r u c t io n o f t r e e t in t h e b a r a b a s i's m o d e l c o in c id e s wit h t h e p r im 's a lg o r it h m fo r ¯ n d in g t h e s h o r t e s t s p a n n in g t r e e o f a we ig h t e d r a n d o m g r a p h [2 3 ]. in t h e n e xt s e c t io n , we c o n s id e r a m o d ī c a t io n o f t h e p r im 's a lg o r it h m t o m a p it in t o t h e m o d e l o f t r a n s m is s io n o f r u m o r s . th e s u b s e qu e n t s e c t io n s will b e d e vo t e d t o t h e e xa c t s o lu t io n s o f t h e la t t e r m o d e l fo r s im p li¯ e d g r a p h s g ( v; e ) a n d t h e c o m p a r is o n o f c lu s t e r g r o wt h in m o d e ls o f t r a n s m is s io n o f r u m o r s a n d t h e in va s io n b o n d p e r c o la t io n . 2 . th e mo d i¯ e d mo d e l o f in va s io n p e r c o la t io n th e m a in d i®e r e n c e b e t we e n t h e m o d e ls o f t r a n s m is s io n o f r u m o r s a n d t h e in va s io n b o n d p e r c o la t io n is t h e m o n o t o n ic c h a r a c t e r o f t h e in va s io n p r o c e s s , wh ic h is n o t s u b d ivid e d in t o ¯ n it e t im e p e r io d s a p p e a r in g in t h e p r o p a g a t io n o f r u m o r s . th e in va s io n p r o c e s s c o n t in u e s p e r m a n e n t ly a n d s t o p s wh e n a ll ve r t ic e s o f g r a p h g( v; e ) b e c o m e o c c u p ie d . th e p r o p a g a t io n o f r u m o r s is in t e r m it t e d a t t h e e n d o f e a c h c o m p le t e r o u n d wh e n a ll p la ye r s r e a c h a b le fo r t h e g ive n p e r m u t a t io n o f c a lls b e c o m e in fo r m e d . to e lim in a t e t h e d i®e r e n c e , we in t r o d u c e a m o d i¯ e d m o d e l o f in va s io n p e r c o la t io n ( o r a m o d ī e d p r im 's a lg o r it h m ) . a s b e fo r e , we b e g in wit h t h e g r a p h g ( v; e ) a n d t h e s e t o f r a n d o m we ig h t s p ( i; j ) 2 [0 ; 1 ] a s s ig n e d t o e a c h e d g e eij 2 e. a g a in , we c h o o s e a ve r t e x v¤ 2 v a n d c o n s id e r it a s t h e s t a r t in g p o in t o f a g r o win g c lu s t e r . th e c lu s t e r g r o ws b y s u c c e s s ive a d d in g n e w o c c u p ie d e d g e s . th e o c c u p a t io n o f e d g e eij m e a n s t h e o c c u p a t io n o f a d ja c e n t ve r t ic e s i a n d j a s we ll. th e a d d it io n p r o c e s s yie ld s t wo r e s t r ic t io n s : ( a ) e a c h n e w e d g e we a r e g o in g t o o c c u p y c o n n e c t s t wo ve r t ic e s , o n e o f wh ic h b e lo n g s t o t h e c lu s t e r a n d t h e o t h e r d o e s n o t . th is r u le g u a r a n t ie s t h a t t h e g r o win g c lu s t e r is a t r e e . of t wo c o n n e c t e d e d g e s o f t h e t r e e , we c a ll a predecessor t h e e d g e t h a t is c lo s e r t o t h e s t a r t in g p o in t . th e o t h e r e d g e is t h e successor. ( b ) e a c h n e w o c c u p ie d e d g e b e in g a s u c c e s s o r , h a s t h e we ig h t la r g e r t h a n it s p r e d e c e s s o r . th e e d g e s yie ld in g b o t h r u le s ( a ) a n d ( b ) a r e c a lle d e lig ib le fo r t h e g r o wt h . th e m o d i¯ e d m o d e l o f in va s io n p e r c o la t io n is d e ¯ n e d b y t h e fo llo win g s t e p s . th e ¯ r s t s t e p is t h e o c c u p a t io n o f t h e s t a r t in g ve r t e x v¤. give n a c o n n e c t e d c lu s t e r o f e d g e s a n d ve r t ic e s c ( i ) o b t a in e d a ft e r su. poghosyan and v. poghosyan 3 3 i s t e p s , i = 1 ; 2 ; : : : , we s e le c t t h e s e t o f e d g e s e ( i ) e lig ib le fo r t h e g r o wt h ( fo r e d g e s a d ja c e n t t o t h e s t a r t in g p o in t , t h e r u le ( b ) is o m it t e d ) . ch o o s e in t h e s e t e ( i ) t h e e d g e eij wit h m in im a l we ig h t p ( i; j ) a n d a d d it t o t h e c lu s t e r o f o c c u p ie d e d g e s a n d ve r t ic e s . in t h is wa y, we o b t a in t h e c lu s t e r c ( i + 1 ) . if, a ft e r s o m e n u m b e r o f s t e p s i1, t h e r e a r e n o e d g e s e lig ib le fo r g r o wt h , we ¯ x t h e c lu s t e r c ( i1 ) a n d s t a r t t h e n e xt r o u n d o f g r o wt h . to t h is e n d , we t a ke t h e s a m e s e t o f r a n d o m we ig h t s p ( i; j ) 2 [0 ; 1 ] a n d c o n s id e r t h e b o u n d a r y ve r t ic e s o f t h e s e t c ( i1 ) a s t h e s t a r t in g p o in t s fo r t h e n e xt r o u n d . th e c lu s t e r o f e d g e s a n d ve r t ic e s o b t a in e d a ft e r i-t h s t e p o f t h e s e c o n d r o u n d is d e n o t e d b y c ( i1; i ) . fo r e a c h s t e p o f t h e s e c o n d r o u n d , we d e ¯ n e t h e s e t e ( i1; i) o f e d g e s e lig ib le fo r g r o wt h a n d c h o o s e a m o n g t h e m t h e e d g e eij wit h m in im a l we ig h t p( i; j ) . th e s e c o n d r o u n d c o n t in u e s d u r in g a n u m b e r o f s t e p s i2 u n t il o n e is a b le t o ¯ n d e d g e s e lig ib le fo r g r o wt h . th e s e t o f e d g e s a n d ve r t ic e s o c c u p ie d d u r in g t h e s e c o n d r o u n d is c ( i1; i2 ) . th e p r o c e s s p r o c e e d s r o u n d b y r o u n d u n t il t h e s e t o f ve r t ic e s in c ( i1; i2; : : : ; imax ) c o in c id e s wit h t h a t in g( v; e ) . w e g e t a m o d i¯ e d m o d e l o f in va s io n b o n d p e r c o la t io n d is p la yin g a n in t e r m it t e n c e o f t h e in va s io n p r o c e s s . in o r d e r t o g e t t h e c o r r e s p o n d e n c e b e t we e n t h e in t r o d u c e d m o d e l a n d t h e r a n d o m c a ll m o d e l, we r e s t r ic t t h e le n g t h o f e a c h r o u n d in t h e la t t e r m o d e l t o t h e u n it t im e in t e r va l. th e n , we c a n id e n t ify t h e m o m e n t s o f c a lls t ( i) ; i = 1 ; 2 ; : : : ; jej wit h t h e r a n d o m n u m b e r s pij 2 [0 ; 1 ] a s s ig n e d t o e a c h e d g e eij 2 e. th e c o n d it io n t h a t a ll pij a r e d is t in c t is n o t e s s e n t ia l s in c e pij a r e c o n t in u o u s va r ia b le s . th e c o n d it io n o f c o m p le t e n e s s o f e a c h r o u n d is id e n t ic a l fo r b o t h m o d e ls if o n e t a ke s in t o a c c o u n t t h e a b ilit y o f a c lu s t e r t o g r o w d u r in g t h e r o u n d . 3 . th e on e -d im e n s io n a l l a t t ic e in t h is s e c t io n , we illu s t r a t e t h e c o n s id e r e d m o d e ls wit h a s im p lī e d e xa m p le , wh e r e t h e c o m m u n ic a t io n g r a p h g( v; e ) is a ¯ n it e o n e -d im e n s io n a l la t t ic e . th e s e t o f ve r t ic e s v ½ z r e p r e s e n t in g t h e p la ye r s c o n s is t s o f l + 1 la t t ic e p o in t s i 2 v; i = 1 ; 2 ; : : : ; l + 1 . th e s e t o f e d g e s e r e p r e s e n t s l c o n n e c t io n s ej 2 e b e t we e n t h e n e ig h b o r in g ve r t ic e s j a n d j + 1 , j = 1 ; 2 ; : : : ; l. th e o r ig in o f t h e r u m o r is t h e ¯ r s t p la ye r i = 1 , a n d t ( j ) ´ t ( ej ) a r e t h e m o m e n t s o f c a lls u s e d b y t h e c o n n e c t io n s ej; j = 1 ; 2 ; : : : ; l. th e o r d e r o f c a lls c o r r e s p o n d s t o t h e p e r m u t a t io n p = ( p1; p2; : : : ; pl ) s u c h t h a t t( p1 ) < t( p2 ) < ¢ ¢ ¢ < t( pl ) . th e r u m o r r e a c h e s t h e p la ye r k1 + 1 d u r in g t h e ¯ r s t r o u n d a n d d o e s n o t p r o p a g a t e fu r t h e r if t ( 1 ) < t( 2 ) < ::: < t ( k1 ) ; t ( k1 ) > t ( k1 + 1 ) : ( 1 ) co n s id e r k1 + 1 n u m b e r s t ( 1 ) ; t ( 2 ) ; : : : ; t ( k1 + 1 ) . a m o n g ( k1 + 1 ) ! p e r m u t a t io n s o f t h e s e n u m b e r s , o n ly o n e is o r d e r e d a s t ( 1 ) < t( 2 ) < ::: < t ( k1 ) < t( k1 + 1 ) ( 2 ) wit h p r o b a b ilit y 1 =( k1 + 1 ) !. th e o r d e r ( 2 ) c a n b e b r o ke n a n d c o n ve r t e d in t o ( 1 ) in k1 wa ys b y r e p la c in g a n y o f k1 ¯ r s t n u m b e r s wit h t( k1 + 1 ) . th e r e fo r e , t h e p r o b a b ilit y o f o r d e r ( 1 ) is k1 t im e s g r e a t e r t h a n t h a t o f ( 2 ) a n d we g e t p rob ( k1 ) = k1 ( k1 + 1 ) ! : ( 3 ) 3 4 rumor spreading and invasion percolation th e r u m o r r e a c h e s t h e p la ye r k1 + 1 d u r in g t h e ¯ r s t r o u n d a n d t h e p la ye r k1 + k2 + 1 d u r in g t h e s e c o n d r o u n d b u t d o e s n o t p r o p a g a t e fu r t h e r if t( 1 ) < t ( 2 ) < ¢ ¢ ¢ < t( k1 ) ; t( k1 + 1 ) < t( k1 + 2 ) < ¢ ¢ ¢ < t ( k1 + k2 ) b u t t ( k1 ) > t( k1 + 1 ) ; t ( k1 + k2 ) > t( k1 + k2 + 1 ) to ¯ n d t h e p r o b a b ilit y p rob( k1; k2 ) , o n e h a s t o e xc lu d e p rob ( k1 + k2 ) fr o m t h e p r o d u c t o f t wo p r o b a b ilit ie s 1 =k1! a n d p rob ( k2 ) : p rob( k1; k2 ) = k2 k1!( k2 + 1 ) ! ¡ k1 + k2 ( k1 + k2 + 1 ) ! : th e c a lc u la t io n o f fu r t h e r p r o b a b ilit ie s p rob ( k1; k2; : : : ; kn ) is a d ir e c t a p p lic a t io n o f t h e in c lu s io n -e xc lu s io n p r in c ip le . fo r in s t a n c e , p rob ( k1; k2; k3 ) = k3= ( k1!k2!( 1 + k3 ) !) ¡ k3=( ( k1 + k2 ) !( 1 + k3 ) !) ¡ ( k2 + k3 ) = ( k1!( 1 + k2 + k3 ) !) + ( k1 + k2 + k3 ) =( 1 + k1 + k2 + k3 ) ! k n o win g t h e p r o b a b ilit ie s p rob( k1; k2; : : : ; ki ) , we c a n c a lc u la t e t h e e xp e c t e d va lu e s hkii in t h e lim it l ! 1: hkii = 1x k1=1;:::;ki=1 kip rob ( k1; k2; : : : ; ki ) th e ¯ r s t s e ve r a l r e s u lt s a r e : hk1i = e ¡ 1 hk2i = e2 ¡ 2 e hk3i = 3 e 2 ¡ 3 e2 + e3 hk4i = ¡ 2 e 3 + 4 e2 ¡ 4 e3 + e4 hk5i = 5 e 2 4 ¡ 1 0 e 2 3 + 1 5 e3 2 ¡ 5 e4 + e5 t h e n u m e r ic a l va lu e s o f wh ic h a r e : hk1i = 1 : 7 1 8 2 8 1 8 2 8 : : : hk2i = 1 : 9 5 2 4 9 2 4 4 2 : : : hk3i = 1 : 9 9 5 7 9 1 3 6 9 : : : hk4i = 2 : 0 0 0 0 3 8 8 5 0 : : : hk5i = 2 :0 0 0 0 5 7 5 7 8 : : : su. poghosyan and v. poghosyan 3 5 on e c a n e xp e c t t h a t hkni ! 2 wh e n n ! 1. to p r o ve t h is c o n je c t u r e , it is s u ± c ie n t t o ¯ n d t h e g e n e r a t in g fu n c t io n ( s e e a p p e n d ix ) r( x ) = 1x m=1 hkmixm = 1 ¡ x 1 ¡ x e xp ( 1 ¡ x) a n d ve r ify t h a t r ( x) » 2 =( 1 ¡ x) + o ( x ¡ 1 ) in t h e vic in it y o f t h e p o in t x = 1 . 4 . co n c lu s io n in o u r wo r ks [2 4 ]{ [2 9 ], t o p p lin g o f g r a in s a t n o d e s ( g r a p h ve r t ic e s in a b e lia n s a n d p ile m o d e l) wa s in t e r p r e t e d a s a t r a n s m is s io n o f t h e fu ll in fo r m a t io n a c c u m u la t e d in a ve r t e x, a c t iva t e d a t t h e m o m e n t , t o a ll it s n e ig h b o r s ( k-b r o a d c a s t ) . ta kin g in t o a c c o u n t t h a t a c t iva t io n o f ve r t ic e s is p e r fo r m e d a c c o r d in g t o a p r e d e ¯ n e d r a n d o m o r d e r , a n d a s in g le t im e t a c t is d e ¯ n e d t o b e a d is c r e t e t im e in t e r va l ( e qu a l t o t h e n u m b e r o f ve r t ic e s ) o f a c t iva t io n o f a ll t h e ve r t ic e s , it is n e c e s s a r y t o e s t im a t e t h e a ve r a g e n u m b e r o f t a c t s r e qu ir e d fo r t h e fu ll in fo r m a t io n e xc h a n g e / t r a n s m is s io n . th e e xa c t fo r m u la fo r t h e a ve r a g e n u m b e r o f t a c t s fo r ( n; n) la t t ic e s h a s n o t ye t b e e n d e r ive d . n e ve r t h e le s s , b a s e d o n t h e r e qu ir e d n u m b e r o f e xp e r im e n t a l d a t a , s t a t is t ic a l c u r ve s o f t h e m a in c h a r a c t e r is t ic s h a ve b e e n o b t a in e d , t h e s t u d y o f wh ic h m a d e it p o s s ib le t o s e t u p a h yp o t h e s is ( r e s e a r c h m e t h o d s a n d r e s u lt s a r e n o t in c lu d e d in t h is p a p e r ) t h a t t h e a ve r a g e n u m b e r o f t a c t s is 0 : 2 5 n. s t u d y o f t h e m o d e ls o f r u m o r s p r e a d in g a n d in va s io n b o n d p e r c o la t io n , a im e d a t t h e r e ve la t io n o f p o s s ib le c o n n e c t io n s b e t we e n t h e m , is p e r fo r m e d . it is s h o wn t h a t t h e r u m o r r e a c h e s t h e r e c e ive r n o d e d u r in g o n e r o u n d if t h e r e is a c h a in o f s u c c e s s ive c a lls b e t we e n t h e s o u r c e o f t h e r u m o r a n d t h a t n o d e . give n t h a t t h e s e qu e n c e o f c a lls is t a ke n u n ifo r m ly a t r a n d o m fr o m t h e s e t o f a ll p o s s ib le s e qu e n c e s , a n a lyt ic a l p r o p e r t ie s o f in fo r m a t io n d is s e m in a t io n h a ve b e e n in ve s t ig a t e d . th e r e s u lt s o b t a in e d m a ke it p o s s ib le t o im p r o ve t h e p e r c o la t io n r u le s a im e d a t m a p p in g o n e m o d e l o n t o a n o t h e r , a n d vic e ve r s a . fu t u r e wo r k will b e d e d ic a t e d t o t h e c o m p a r a t ive a n a lys is b e t we e n r u m o r s p r e a d in g a n d in va s io n b o n d p e r c o la t io n m o d e ls . 5 . a p p e n d ix w e d e ¯ n e a s e t o f a ll p a r t it io n s p( fk1; k2; : : : ; kng) o f a s e t o f n in t e g e r s fk1; k2; : : : kng a s a s e t o f t h e fo llo win g 2 n¡1 s u b s e t s p( fk1g ) = ffk1gg p ( fk1; k2g ) = f ffk1g; fk2gg; ffk1; k2gg g in g e n e r a l, r e c u r s ive ly, if t h e s e t o f m s u b s e t s o f fk1; k2; : : : ; kng si; i = 1 ; 2 ; : : : ; m, is a p a r t it io n fs1; s2; : : : ; smg 2 p( fk1; k2; : : : ; kng ) ( 4 ) 3 6 rumor spreading and invasion percolation t h e n fs1; s2; : : : ; sm; fkn+1gg 2 p( fk1; k2; : : : ; kn+1g ) a n d fs1; s2; : : : ; sm¡1; sm [ fkn+1gg 2 p( fk1; k2; : : : ; kn+1g ) ( 5 ) a r e a ls o p a r t it io n s . e a c h o f si; i = 1 ; 2 ; : : : ; m in ( 4 ) a n d ( 5 ) is a s e t o f k in t e g e r s s = fl1; l2; : : : ; lkg, wh e r e li 2 n, i = 1 ; 2 ; : : : ; k. w e d e n o t e jsj = k a n d ksk = l1 + l2 + ¢ ¢ ¢ + lk. th e n , ( u s in g t h e in c lu s io n -e xc lu s io n p r in c ip le ) , we o b t a in a s u m o ve r a ll e le m e n t s o f p( fk1; k2; : : : ; kng ) : p r o b ( k1; k2; : : : ; k n ) = nx m =1 x fs 1; s 2;::: ; s mg2p(fk1;k2;::: ;kng) ( ¡ 1 ) n +m k s 1k! k s 2k! ¢ ¢ ¢ k s m ¡1k! ks m k ( ks m k + 1 ) ! : th e r e fo r e , t h e a ve r a g e va lu e o f kn is hkni = 1x k1=1 1x k2=1 : : : 1x kn=1 kn p r o b ( k1; k2; : : : ; k n ) = nx m=1 x fs1;s2;::: ;smg2p(f1;2;::: ;ng) ( ¡ 1 ) n+m®js1j®js2j ¢ ¢ ¢ ®jsmj; ( 6 ) wh e r e ®n; n > 0 is t h e n-fo ld s u m ®n = 1x k1=1 1x k2=1 ¢ ¢ ¢ 1x kn=1 1 ( k1 + k2 + ¢ ¢ ¢ + kn ) ! ( 7 ) to c a lc u la t e ( 7 ) wit h n = 1 ; 2 ; : : : , we in t r o d u c e t h e g e n e r a t in g fu n c t io n fn ( x ) = 1x k1=1 1x k2=1 ¢ ¢ ¢ 1x kn=1 xk1+k2+¢¢¢+kn ( k1 + k2 + ¢ ¢ ¢ + kn ) ! : th e d e r iva t ive o f fn ( x) c a n b e e xp r e s s e d in t h e fo llo win g wa y d fn ( x ) d x = 1x k1=1 1x k2=1 ¢ ¢ ¢ 1x kn=1 xk1+k2+¢¢¢+kn¡1 ( k1 + k2 + ¢ ¢ ¢ + kn ¡ 1 ) ! = 1x k1=1 1x k2=1 ¢ ¢ ¢ 1x kn¡1=1 1x kn=0 xk1+k2+¢¢¢+kn ( k1 + k2 + ¢ ¢ ¢ + kn ) ! = fn ( x ) + fn¡1 ( x ) : ( 8 ) w e m u lt ip ly b o t h s id e s o f t h e r e c u r s ive d i®e r e n t ia l e qu a t io n ( 8 ) b y zn a n d t a ke t h e s u m o ve r n = 2 ; 3 ; : : : . in t r o d u c in g t h e g e n e r a t in g fu n c t io n o f t h e s e qu e n c e o f fu n c t io n s fn ( x ) : f ( z; x ) = 1x n=1 fn ( x ) z n; su. poghosyan and v. poghosyan 3 7 a n d t a kin g in t o a c c o u n t t h a t f1 ( x ) ´ 1x k1=1 xk1 k1! = ex ¡ 1 ; we o b t a in a d i®e r e n t ia l e qu a t io n fo r f ( z; x ) : @f ( z; x ) @x = ( z + 1 ) f ( z; x) + z ( 9 ) wit h in it ia l c o n d it io n s f ( z; 0 ) = 0 . th e s o lu t io n o f ( 9 ) is f ( z; x) = z z + 1 ¡ ex(z+1) ¡ 1 ¢ ; wh ic h c a n b e e xp a n d e d o ve r va r ia b le s x a n d z: f ( z; x) = 1x k=1 z ( z + 1 ) k¡1 k! xk = 1x k=1 kx n=1 xkzn ( n ¡ 1 ) !( k ¡ n) !k = 1x n=1 1x k=n xkzn ( n ¡ 1 ) !( k ¡ n ) !k : th e r e fo r e , fn ( x) = 1 ( n ¡ 1 ) ! 1x k=0 xn+k k!( n + k ) = 1 ( n ¡ 1 ) ! x 0 ¿ n¡1e¿ ¿: th u s , we o b t a in a n e xa c t e xp r e s s io n fo r ®n ®n = fn ( 1 ) = ( ¡1 ) n + e nx p=1 ( ¡ 1 ) n+p ( p ¡ 1 ) ! ; wh ic h a llo ws o n e t o c a lc u la t e t h e va lu e s o f ®n fo r va r io u s n: ®1 = e ¡ 1 ; ®2 = 1 ; ®3 = 12 ( e ¡ 2 ) ; ®4 = 3¡e 3 ; ®5 = 1 8 ( 3 e ¡ 8 ) ; ®6 = 130 ( 3 0 ¡ 1 1 e) : u s in g t h e fa c t t h a t 1 e = 1x p=0 ( ¡ 1 ) p p! ; we o b t a in a n a lt e r n a t ive r e p r e s e n t a t io n o f ®n: ®n = e 1x p=0 ( ¡ 1 ) p ( p + n) ! = e n! ¡ e ( n + 1 ) ! + e ( n + 2 ) ! ¡ ¢ ¢ ¢ : n o t e t h a t e n! ¡ e ( n + 1 ) ! < ®n < e n! ; i.e ., n n + 1 e n! < ®n < e n! : 3 8 rumor spreading and invasion percolation th e r e fo r e , t h e a s ym p t o t ic s o f ®n fo r la r g e n is ®n ' e n! ; n à 1 : n o w, we c a n r e wr it e t h e e xp r e s s io n ( 6 ) fo r hkni in t h e fo llo win g fo r m hkni = 1x ¾1=0 1x ¾2=0 ¢ ¢ ¢ 1x ¾n=0 ± ã nx i=1 i¾i; n ! ny i=1 ¡ ( ¡ 1 ) 1+i®i ¢¾i ( pn i=1 ¾i ) ! ¾1! ¾2! ¢ ¢ ¢ ¾n! : th e c o r r e s p o n d in g g e n e r a t in g fu n c t io n is r ( x) = 1 + 1x n=1 hknixn = 1x ¾1=0 1x ¾2=0 ¢ ¢ ¢ ã 1y i=1 ( ( ¡ 1 ) 1+i®i ) ¾i ¾i! ! ã 1x i=1 ¾i ! ! x p 1 i=1 i¾i : to c a lc u la t e r( x ) , we in t r o d u c e a n o t h e r g e n e r a t in g fu n c t io n d ( x; z ) = 1x ¾1=0 1x ¾2=0 ¢ ¢ ¢ ã 1y i=1 ( ( ¡ 1 ) 1+i®i ) ¾i ¾i! ! z p 1 i=1 ¾i x p 1 i=1 i¾i ; wh ic h c a n b e r e wr it t e n a s d ( x; z ) = 1y i=1 ã 1x ¾=0 ( ( ¡1 ) 1+i®i ) ¾ z¾xi¾ ¾! ! = 1y i=1 e xp © ( ¡ 1 ) 1+i®ixiz ª = e xp ( z 1x i=1 ( ¡ 1 ) 1+i®ixi ) th e fu n c t io n d ( x; z ) is t h e s e r ie s d ( x; z ) = 1x p=0 g ( x) p p! zp; wh e r e g( x ) = 1x i=1 ( ¡1 ) 1+i®ixi = 1x i=1 ã ¡ 1 + e ix j=1 ( ¡ 1 ) j¡1 ( j ¡ 1 ) ! ! xi = x x ¡ 1 + e 1x j=1 ã ( ¡ 1 ) j¡1 ( j ¡ 1 ) ! 1x i=j xi ! = x x ¡ 1 + e 1 ¡ x 1x j=1 ( ¡ 1 ) j¡1xj ( j ¡ 1 ) ! = x x ¡ 1 ¡ 1 + e1¡x ¢ su. poghosyan and v. poghosyan 3 9 on t h e o t h e r h a n d , t h e fu n c t io n r ( x) c a n b e e xp r e s s e d b y g( x ) : r ( x ) = 1x p=0 g( x ) p = 1 1 ¡ g ( x) = 1 ¡ x 1 ¡ xe1¡x : r ( x ) ' 2 1 ¡ x + o ( x ¡ 1 ) th e r e fo r e , lim n!1 hkni = 2 : references [1 ] b . b a ke r a n d r . s h o s t a k, \ go s s ip s a n d t e le p h o n e s " , d iscrete m ath., vo l. 2 , p p . 1 9 1 -1 9 3 , 1 9 7 2 . 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[2 4 ] v . h . h o vn a n ya n , s u . s . p o g h o s ya n a n d v . s . p o g h o s ya n , \ n e w m e t h o d s o f c o n s t r u c t io n o f fa u lt -t o le r a n t go s s ip g r a p h s " , ie e e p roceedings of the international conference on computer science and information technologies (csit'2013), d oi: 1 0 .1 1 0 9 / cs ite c h n o l.2 0 1 3 .6 7 1 0 3 4 1 . [2 5 ] v .h . h o vn a n ya n , s u .s . p o g h o s ya n a n d v .s . p o g h o s ya n , \ me t h o d o f l o c a l in t e r c h a n g e t o in ve s t ig a t e go s s ip p r o b le m s " , transactions of iiap of the nas of r a, m athematical p roblems of computer science, vo l. 4 0 , p p . 5 -1 2 , 2 0 1 3 . [2 6 ] v .h . h o vn a n ya n , s u .s . p o g h o s ya n a n d v .s . p o g h o s ya n , \ me t h o d o f l o c a l in t e r c h a n g e fo r t h e in ve s t ig a t io n o f go s s ip p r o b le m s : p a r t 2 " , transactions of iiap of the nas of r a, m athematical p roblems of computer science, vo l. 4 1 , p p . 1 5 -2 2 , 2 0 1 4 . [2 7 ] v . h . h o vn a n ya n , s u . s . p o g h o s ya n a n d v .s . p o g h o s ya n , \ gr a p h p lo t t e r : a s o ft wa r e to o l fo r t h e in ve s t ig a t io n o f fa u lt -t o le r a n t go s s ip gr a p h s " , p roceedings of international conference csit-2013, p p . 2 0 -2 2 . [2 8 ] v . h . h o vn a n ya n , s u . p o g h o s ya n a n d v . p o g h o s ya n , \ fa u lt -t o le r a n t go s s ip gr a p h s b a s e d o n w h e e l gr a p h s " , transactions of iiap of the nas of r a, m athematical p roblems of computer science, vo l. 4 2 , p p . 4 3 -5 3 , 2 0 1 4 . [2 9 ] v . h . h o vn a n ya n , s u . p o g h o s ya n a n d v .p o g h o s ya n , \ op e n p r o b le m s in g o s s ip / b r o a d c a s t s c h e m e s a n d t h e p o s s ib le a p p lic a t io n o f t h e m e t h o d o f lo c a l in t e r c h a n g e " , p roceedings of international conference csit 2015, p p . 7 3 -7 8 . submitted 19.06.2019, accepted 26.11.2019. su. poghosyan and v. poghosyan 4 1 ´³ùµ³ë³ýùç ï³ñ³íáõùá ¨ ý»ñã³÷³ýóáõ³ï³ý å»ñïáé³óç³ êáõñ»ý ê. äáõáëû³ý ¨ ì³ñ³·ý ê. äáõáëû³ý ðð ¶²² æýýáñù³ïçï³ûç ¨ ³íïáù³ï³óù³ý åñáµé»ùý»ñç çýëïçïáõï e-mail: psuren55@yandex.ru, povahagn@gmail.com ²ù÷á÷áõù ø»ýù áõëáõùý³ëçñáõù »ýù µ³ùµ³ë³ýùç ï³ñ³íù³ý ¨ ïáõ³ûçý ý»ñã³÷³ýóáõ³ï³ý å»ñïáé³óç³ý»ñç ùá¹»éý»ñá` ýå³ï³ï áõý»ý³éáí µ³ó³ñ³ûï»é ¹ñ³ýó ñ³í³ý³ï³ý ï³åí³íáõãûáõýá: ´³ùµ³ë³ýùç ï³ñ³íù³ý ùá¹»éá ýï³ñ³·ñáõù ¿ ¹ñ³ ï³ñ³íáõùá ñ³çáñ¹³ï³ý ñ»é³ëáë³½³ý·»ñç å³ñµ»ñ³µ³ñ ïñïýí»éáõ ³ñ¹ûáõýùáõù, çëï ïáõ³ûçý ý»ñã³÷³ýóáõ³ï³ý å»ñïáé³óç³ý»ñç ùá¹»éá í»ñ³µ»ñáõù ¿ í³ïáïï»ý ùçç³í³ûñáõù ñ»õáõïç ï³ñ³íù³ýá: ´³ùµ³ë³ýùç ï³ñ³íù³ý ùá¹»éáõù ù»ï ï³ïïç áýã³óùáõù ûáõñ³ù³ýãûáõñ ñ³ý·áõûó ï³ï³ñáõù ¿ ùç³ûý ù»ï ½³ý·` ½³ý·³ñ³ñ»éáõ å³ñçý çñ ïçñ³å»ï³í áõç ï»õ»ïáõãûáõýá ÷áë³ýó»éáí ñ³ñ¨³ýý»ñçý: àý¹ëùçý` µ³ùµ³ë³ýùá ù»ï ÷áõéç áýã³óùáõù ñ³ëýáõù ¿ ëï³óáõçý` µ³ùµ³ë³ýùç ³õµûáõñç ¨ ëï³óáõç ùçç¨ ñ³çáñ¹³ï³ý ½³ý·»ñç ßõã³ûç ³éï³ûáõãû³ý ¹»åùáõù: ¼³ý·»ñç ñ³çáñ¹³ï³ýáõãûáõýý áýïñíáõù ¿ µáéáñ ñý³ñ³íáñ ñ³çáñ¹³ï³ýáõãûáõýý»ñç ß³ñùçó` ñ³í³ë³ñ³ã³÷ å³ï³ñ³ï³ýáõãû³ý ëï½µáõýùáí (ñ³ý·áõûóý»ñç ï»õ³÷áëáõãûáõý): ø»ýù ñ³ù»ù³ïáõù »ýù µ³ùµ³ë³ýùç ï³ñ³íù³ý ùá¹»éá ïáõ³ûçý ý»ñã³÷³ýóáõ³ï³ý å»ñïáé³óç³ûç ñ»ï` ùá¹»éý»ñç ÷áë³¹³ñó ³ñï³å³ïï»ñáõùý»ñç çñ³ï³ý³óù³ý ýå³ï³ïáí å»ñïáé³óç³ûç ï³ýáýý»ñç ³ýññ³å»ßï µ³ñ»é³íáõùý»ñ ³é³ç³¹ñ»éáõ ñ³ù³ñ: ´³ý³éç µ³é»ñ` µ³ùµ³ë³ýùç ëý¹çñ, çýýáñù³óç³ûç ï³ñ³íáõù, ïáõ³ûçý ý»ñã³÷³ýóáõ³ï³ý å»ñïáé³óç³: ðàñïðîñòðàíåíèå ñïëåòåí è èíâàçèâíàÿ ïåðêîëÿöèÿ ñóðåí ñ. ïîãîñÿí è âààãí ñ. ïîãîñÿí èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà e-mail: psuren55@yandex.ru, povahagn@gmail.com àííîòàöèÿ ìû èçó÷àåì ìîäåëè ðàñïðîñòðàíåíèÿ ñïëåòåí è ðåáåðíîé èíâàçèâíîé ïåðêîëÿöèè ñ öåëüþ îáíàðóæåíèÿ âîçìîæíîé ñâÿçè ìåæäó íèìè. ìîäåëü ðàñïðîñòðàíåíèÿ ñïëåòåí îïèñûâàåò ðàñïðîñòðàíåíèå êàê âñëåäñòâèå èççà ïåðèîäè÷åñêîãî ïîâòîðåíèÿ ïîñëåäîâàòåëüíûõ òåëåôîííûõ çâîíêîâ, â òî âðåìÿ êàê ðåáåðíàÿ èíâàçèâíàÿ ïåðêîëÿöèÿ îòíîñèòñÿ ê ðàñïðîñòðàíåíèþ æèäêîñòè â ïîðèñòîé ñðåäå. âî âðåìÿ îäíîãî ðàóíäà ðàñïðîñòðàíåíèÿ ñëóõîâ êàæäûé óçåë âûïîëíÿåò âûçîâ òîëüêî îäèí ðàç, ïðè ýòîì ïåðåäàâàÿ âñþ èíôîðìàöèþ, äîñòóïíóþ â ìîìåíò âûçîâà, ñâîèì ñîñåäÿì. ñïëåòíÿ äîñòèãàåò óçëà-ïîëó÷àòåëÿ â òå÷åíèå îäíîãî ðàóíäà ïðè íàëè÷èè öåïî÷êè ïîñëåäîâàòåëüíûõ âûçîâîâ ìåæäó èñòî÷íèêîì ñïëåòíè è óçëîì-ïîëó÷àòåëåì. 4 2 rumor spreading and invasion percolation ïîñëåäîâàòåëüíîñòü âûçîâîâ âûáèðàåòñÿ ðàâíîìåðíî ñëó÷àéíûì îáðàçîì èç íàáîðà âñåõ âîçìîæíûõ ïîñëåäîâàòåëüíîñòåé (ïåðåñòàíîâîê óçëîâ). ìû ñðàâíèâàåì ìîäåëü ðàñïðîñòðàíåíèÿ ñïëåòåí ñ ðåáåðíîé èíâàçèâíîé ïåðêîëÿöèåé ñ öåëüþ âûäâèæåíèÿ íåîáõîäèìîé îïòèìèçàöèè ïðàâèë ïåðêîëÿöèè äëÿ äîñòèæåíèÿ îòîáðàæåíèÿ îäíîé ìîäåëè íà äðóãóþ, è, íàîáîðîò. êëþ÷åâûå ñëîâà: ïðîáëåìà ñïëåòíè, ðàñïðîñòðàíåíèå èíôîðìàöèè, ðåáåðíàÿ èíâàçèâíàÿ ïåðêîëÿöèÿ. microsoft word tpel_1.doc mathematical problems of computer science 36, 115--120, 2012. 115 off-line signature verification and recognition vahe khachaturyan institute for informatics and automation problems of nas of ra e-mail: vahe@7smarts.com abstract in this paper we propose a technique that can be used for signature recognition. this technique is a contour based technique. here we propose a simple and effective approach that can be easily implemented in a programming language. the paper deals with the recognition of the signature, as human operator generally makes the work of signature recognition. hence the algorithm simulates human behavior, to achieve perfection and skill through ai. the logic that decides the extent of validity of the signature must implement artificial intelligence pattern recognition is the science that concerns the description or classification of measurements, usually based on underlying model. since most pattern recognition tasks are first done by humans and automated later, the most fruitful source of features has been to ask the people who classify the objects how they tell them a part . signatures are a behavioral biometric that change over a period of time and are influenced by physical and emotional conditions of a subject. this technique gives acceptable results in a simple and fast way. key words: pattern recognition, signature recognition, image morphology references [1] m. sonka, v. hlavac and r. boyle, “image processing analysis, and machine vision”, thomson learning, singapore, pp. 563-64, 573, 583, 593, 2002. [2] s. loncaric, a. dhawan,” a morphological signature transform for shape description.” pattern recognition, 6: pp. 1029-1037. 1993. [3] evett and r. n. totty, ”study of the variation in the dimensions of genuine signatures”, journal of the forensic science society, vol. 25, pp. 207-215, 1985 [4] b. fang, c. h. leung, y. y. tang, k. w. tse, p. c. k. kwok, y. k. wong,”off-line signature verification by the tracking of feature and stroke positions”, pattern recognition, vol. 36, pp. 91-101, 2003. [5] y. mizukami, h. miike, m. yoshimura, and i. yoshimura, ”an off-line signature verification system using an extracted displacement function”, in proceedings of icdar, pp. 757-760, 1999. [6] w. f. nemcek and w. c. lin, ”experimental investigation of automatic signature verification” ieee transactions on systems, man and cybernetics, vol. 4 , pp. 121-126, 1974. off-line signature verification and recognition 116 [7] t. joachims, “text categorization with support vector machines: learning with many relevant features”. proceedings of the tenth european conference on machine learning. berlin: springer, pp. 137 -142, 1998. <<úýé³ûý>> ëïáñ³·ñáõãû³ý ×³ý³ãáõù ì. ê³ã³ïáõñû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ³é³ç³ñïíáõù ¿ <<úýé³ûý>> ëïáñ³·ñáõãû³ý ×³ý³ãù³ý »õ³ý³ï՝ ñçùýí³í »½ñ³·í³ûçý ý»ñïù³ý íñ³, ï³ëí³í ï³ñµ»ñ å³ñ³ù»ïñ»ñçó՝ ÷çùë»éý»ñç ù³ý³ï, ß»õù³ý ³ýïûáõý, é³ûýáõãûáõý, µ³ñóñáõãûáõý: ø»ýù û·ï³·áñíáõù»ýù ex-oring ëïáñ³·ñáõãû³ý çëïáõãûáõýý ëïáõ·»éáõ ñ³ù³ñ: ´»ñíáõù ¿ ëïáñ³·ñáõãû³ý ×³ý³ãù³ý ûñçý³ï: опознание “офлайн” подписи в. хачатурян аннотация в работе предлагается метод опазнания ,,офлайн” подписи основанный на контурное окрашивание в зависимости от многих параметров: количество пикселеи, угол наклона, широта, высота. мы используем ex-oring для проверки подлинности подписи. приводится пример опазнания подписи. mathematical problems of computer science 53, 67--71, 2020. udc 004 the concept of internal ip addresses system for nrens robert n. tadevosyan, arthur s. petrosyan and gurgen s. petrosyan institute for informatics and automation problems of nasra e-mail: robert@sci.am, arthur@sci.am, gurgen@sci.am abstract this paper presents the concept of a new system of internal ip addresses proposed for use by national research and education networks (nren). recently, various systems and services, have been actively developing, such as wi-fi, ip telephony, etc., for which it is necessary to ensure the security of streams, reducing the area of collisions, etc. the allocation of additional ip addresses in local networks makes it possible to realize all the abovementioned tasks and have the necessary reserve for future development. the concept is deployed in academic scientific research computer network of armenia (asnet-am). keywords: vlan, networking, eduroam, wi-fi, voip 1. introduction in general, the virtual local area network (vlan) technology is used to logically divide the network into different and independent parts. the proposed concept of ip addressing distribution system to subnets and methods of implementation with appropriate vlans technology for managing and controlling them, suggest a completely new vision to this technology. it takes into account many factors of the current development of national research and education networks (nrens) (the detailing is outside the scope of this work, since it may vary for different nrens), network infrastructure, installed devices and the specifics of the connected institutions. in addition, the presented concept allows flexible future growth of nren infrastructure. the proposed concept of ip addressing distribution promotes the development of virtually unlimited additional services in nren in the future without any big changes and reorganization of the network structure. as a rule of thumb, local area network (lan) internal ip addressing system has the following simple scheme without any logical division and vlans implementations. for each 67 mailto:robert@sci.am mailto:arthur@sci.am mailto:gurgen@sci.am the concept of internal ip addresses system for nrens 68 connected institute, one “c” class of internal ip addresses is allocated. addresses are distributed from the pool of addresses reserved for private use [1], such as prefixes 192.168/16, 172.16/12 and 10/8. as an example several lan addresses may be used according to the following system: “10.1.x.y”, where “x” is the allocated organization number in the ip space (in the third byte), and “y” is the space of 256 addresses, for use in the internal local networks. 2. concept architecture for each organization connected to nren, a new ip address space can be allocated for “b” class network and will be built according to the following scheme: “10.x.y.y”, where “x” is the allocated institution number in ip space, and “y.y” are spaces of 256 “c” class addresses (65536 ip addresses) for use in the internal local networks. to save the current ip numbering system of organizations, the organization numbers will be moved from the third byte to the second one: “10.1.x.y” “10.x.y.y”. networks started from 10.200.y.y will be used for addressing the globally controlled communication devices. each institution of “b” class network will be divided into logical and physical subspaces according to the following principle. 1. workstations 2. local servers 3. managed communication equipment within the organization, etc. 4. wi-fi devices 5. voice over internet protocol (voip) phones 6. … for future use at the moment, it is proposed to use the first 128 networks of institutions address spaces. the remaining 128 networks will be reserved for further expansion of network systems and services. the distribution of ip addressing will be carried out according to the following scheme: • 10.x.0-a.y desktops • 10.x.a+1-b.y local servers • 10.x.b+1-c.y local managed communication equipment • 10.x.c+1-g.y wi-fi devices • 10.x.g+1-h.y voip phones additionally, the subnet “10.x.c+1-g.y wi-fi devices” is proposed to be divided into subparts according to the type and area of use of the connected devices as follows: • 10.x.c+1-d.y the general academic wi-fi network devices (ssid: eduroam) • 10.x.d+1-e.y wi-fi network of current organization (for example, ssid: iiap) • 10.x.e+1-f.y guest wi-fi access by the organization (for example, ssid: iiapfreewifi) • 10.x.f+1-g.y will be implemented if necessary (for example: ssid: csit2019) the implementation of this new ip addressing system will be efficient through the creation of appropriate vlans for each of the subnets [2]. the following systems of standards are proposed for internal names and numbering of vlans:  10.x.0-a.y vlan name: vlanz00, vlan id: z00  10.x.a+1-b.y vlan name: vlanza+1, vlan id: za+1  10.x.b+1-c.y vlan name: vlan zb+1, vlan id: zb+1  10.x.c+1-d.y vlan name: vlan zc+1, vlan id: zc+1  10.x.d+1-e.y vlan name: vlan zd+1, vlan id: zd+1  10.x.e+1-f.y vlan name: vlan ze+1, vlan id: ze+1 r. tadevosyan, a. petrosyan and g. petrosyan 69  10.x.f+1-g.y vlan name: vlan zf+1, vlan id: zf+1  10.x.g+1-h.y vlan name: vlanzg+1, vlan id: zg+1 where x is the network number allocated to organizations in the ip subspace for the local network addresses of nren. fig. 1. internal ip address distribution system. for example, the vlan number for workstations will be z00, the vlan number for general academic wi-fi network devices will be zc+1, and the vlan number for the subnet of voip devices will be zg+1. additionally, if it is necessary to divide the subnets to subparts and subvlans, the numeration should be created according to the following scheme. an additional number from 0 to 9 should be added to the base vlan number. for example, when it is necessary to create additional subnets and vlans in “workstations” zone, the numbers will be z001, z002, z003, or za0, za1, etc. the first number of vlans for subnets started from 100, will be z+1, and for subnets started from 200, will be z+2. this is a general and theoretical vision of the new ip addressing system. practical implementation includes the creation of a real numeration standard and distribution, as default, which will be implemented for all connected institutions. implementation of this system as standard will facilitate easy configuring and managing the border routers of all connected institutions. since the proposed ip addressing system does not conflict with the standard addressing space, it can be implemented without any interruption of the network work. to start this process in the connected institution, it is necessary to configure new address spaces and vlans on the border router, and then, after starting to the step-by-step transition of all equipment and devices to the new addressing system. in the pan-european data network for the research and education community (géant) [3], which interconnects the nrens across europe, there are some proposals for internetworking vlans implementations [4], but this work covers another area of work scope. currently, the proposed numbering and distribution system with the appropriate vlans has already been created as a standard and enacted in production mode. this document with the default numbering system, is an internal document of the asnet-am network. the concept of internal ip addresses system for nrens 70 the implementation of this new internal ip addressing system on the border routers of nren-connected institutes will now allow the unimpeded expansion of network services and systems and create the basis for future growth. 3. conclusion the proposed concept can be used by any nren to make the internal infrastructure more efficient. currently, this new ip addressing system has already been implemented in the local networks of several academic scientific research computer network of armenia (asnetam)-connected organizations. in the mentioned institutions, the production of wi-fi network structure is already divided into subnets with the corresponding service set identifier (ssid) access. each subnet operates in its own dedicated vlan and receives, via the dynamic host configuration protocol (dhcp), ip addresses corresponding to its own vlan, which allows the general control and regulation of flows, as well as access control to resources of both asnetam and the internet [5]. in the future, this new addressing system will be expanded to all organizations connected to asnet-am. references [1] y. rekhter, b. moskowitz, d. karrenberg, g. j. de groot and e. lear, “address allocation for private internets”, silicon graphics, inc, february 1996, [online]. available: https://tools.ietf.org/html/rfc1918 [2] p. congdon, m. sanchez, b. aboba, “racius attibutes for virtual lan and priority support”, [online]. available: https://www.ietf.org/rfc/rfc4675.txt [3] [online]. available: https://www.geant.org/ [4] v. capone and m. usman , “the géant network: addressing current and future needs of the hep community”, journal of physics: conference series 664 052005, doi: 10.1088/1742-6596/664/5/052005, 2015, [online]. available: https://iopscience.iop.org/article/10.1088/1742-6596/664/5/052005/pdf [5] b. hartpence, routing and switching, o’reilly media, inc., isbn: 9781449306557, 2011, [online]. available: https://www.oreilly.com/library/view/packet-guide-to/9781449311315/ submitted 04.03.2020, accepted 18.06.2020. https://tools.ietf.org/html/rfc1918 https://www.ietf.org/rfc/rfc4675.txt https://www.geant.org/ https://iopscience.iop.org/article/10.1088/1742-6596/664/5/052005/pdf https://www.oreilly.com/library/view/packet-guide-to/9781449311315/ r. tadevosyan, a. petrosyan and g. petrosyan 71 հետազոտական և կրթության ազգային ցանցերի ներքին հասցեավորման համակարգի կոնցեպցիա ռոբերտ ն․ թադևոսյան, արթուր ս․ պետրոսյան և գուրգեն ս․ պետրոսյան հհ գաա ինֆորմատիկայի և ավտոմատացման պրոբլեմների ինստիտուտ էլ․ հասցե: robert@sci.am, arthur@sci.am, gurgen@sci.am ամփոփում այս հոդվածը ներկայացնում է ներքին ip հասցեների նոր համակարգի կոնցեպցիա, որն առաջարկվում է օգտագործել ազգային հետազոտական և կրթական ցանցերի (nren) կողմից: վերջերս տարբեր համակարգերի և ծառայությունների զարգացումը, ինչպիսիք են` wi-fi, ip հեռախոսակապ և այլն, անհրաժեշտություն են առաջացնում ապահովել հոսքերի անվտանգությունը, բախումների տարածքի կրճատում և այլն։ հաշվի առնելով դա, անհրաժեշտ են դառնում լոկալ ցանցերում լրացուցիչ ip հասցեների օգտագործումը և բաշխումը, ինչը հնարավորություն կտա լուծել վերոնշված խնդիրները և ունենալ անհրաժեշտ ռեզերվ ցանցի հետագա զարգացման համար: այս համակարգը ներդրված է հայաստանի ակադեմիական գիտահետազոտական կոմպյուտերային ցանցում (asnet-am)։ բանալի բառեր` vlan, networking, eduroam, wi-fi, voip концепция системы внутренних ip адресов в локальных сетях для национальных научно-образовательных сетей роберт. н. тадевосян, артур с. петросян и гурген с. петросян институт проблем информатики и автоматизации нан ра e-mail: robert@sci.am, arthur@sci.am, gurgen@sci.am аннотация в этом документе представлена концепция новой системы внутренних ip-адресов для использования национальными исследовательскими и образовательными сетями (nren). в последнее время развитие различных систем и сервисов, таких как wi-fi, ip телефония и др., требуют обеспечения безопасности потоков в выделенных каналах, уменьшения коллизий, что создает потребность в дополнительных ip адресах. эти ip адреса помогут устранить вышеперечисленные недостатки и обеспечат необходимый задел для будущего развития сети. данная концепция уже внедрена в академической научно-исследовательской компьютерной сети армении (asnet-am). ключевые слова: vlan, networking, eduroam, wi-fi, voip. mailto:robert@sci.am mailto:arthur@sci.am mailto:gurgen@sci.am mailto:robert@sci.am mailto:arthur@sci.am mailto:gurgen@sci.am d:\sbornik\...\article.dvi mathematical problems of computer science 40, 31{33, 2013. simple p r oofs of t wo dir ac-type t heor ems i nvolving connectivity ca r le n m. mo s e s ya n , mh e r zh . n iko g h o s ya n a n d zh o r a g. n iko g h o s ya n faculty of mathematics and informatics kh. abovyan armenian state university institute for informatics and automation problems of nas ra e-mail: mosesyan@list.ru, zhora@ipia.sci.am abstract in 1981, the third author proved that each 2-connected graph g with ± ¸ (n+·)=3 is hamiltonian and each 3-connected graph contains a cycle of length at least minfn; 3± ¡ ·g, where n denotes the order, ± the minimum degree and · the connectivity of g. short proofs of these two results were given by häaggkvist and yamashita, respectively, occupying more than three pages for actual proofs altogether. here we give much simpler and shorter proofs actually occupying the two-thirds of a page. keywords: minimum degree, connectivity, circumference. l e t g b e a ¯ n it e u n d ir e c t e d g r a p h wit h o u t lo o p s o r m u lt ip le e d g e s . a g o o d r e fe r e n c e fo r a n y u n d e ¯ n e d t e r m s is [1 ]. w e r e s e r ve n, ±, · a n d c t o d e n o t e t h e o r d e r ( t h e n u m b e r o f ve r t ic e s ) , m in im u m d e g r e e , c o n n e c t ivit y a n d c ir c u m fe r e n c e ( t h e le n g t h o f a lo n g e s t c yc le ) o f g. th e ve r t ic e s a n d e d g e s c a n b e in t e r p r e t e d a s s im p le c yc le s o f le n g t h s 1 a n d 2 , r e s p e c t ive ly. a c yc le c in g is c a lle d a h a m ilt o n c yc le if jcj = n a n d is c a lle d a d o m in a t in g if gnc is e d g e le s s . in 1 9 5 2 , d ir a c [2 ] p r o ve d t h a t e a c h g r a p h wit h ± ¸ n= 2 is h a m ilt o n ia n ( i.e . h a s a h a m ilt o n c yc le ) a n d in e a c h 2 -c o n n e c t e d g r a p h , c ¸ m in fn; 2 ±g. in 1 9 8 1 , t h e t h ir d a u t h o r [5 ] wa s a b le t o o b t a in t wo a n a lo g o u s d ir a c -t yp e r e s u lt s in vo lvin g c o n n e c t ivit y ·. t heor em 1 [5]: e very 2-connected graph with ± ¸ ( n + ·) =3 is hamiltonian. t heor em 2 [5]: in every 3-connected graph, c ¸ m in fn; 3 ± ¡ ·g. th e o r ig in a l p r o o fs [5 ] o f th e o r e m s 1 a n d 2 a r e s o m e wh a t le n g t h y a n d c o m p lic a t e d . s h o r t p r o o fs o f t h e s e t wo r e s u lt s we r e g ive n b y h äa g g kvis t [3 ] a n d y a m a s h it a [7 ], r e s p e c t ive ly, o c c u p yin g m o r e t h a n t h r e e p a g e s fo r a c t u a l p r o o fs a lt o g e t h e r . in t h is n o t e we p r e s e n t m u c h s im p le r a n d s h o r t e r p r o o fs o f t h e o r e m s 1 a n d 2 ( a c t u a l p r o o fs o n a b o u t 2 / 3 p a g e ) b a s e d m a in ly o n s t a n d a r d a r g u m e n t s a n d t h e fo llo win g t wo t h e o r e m s . t heor em 3 [4]: l et g be a 2-connected graph with ± ¸ ( n + 2 ) =3 . then every longest cycle in g is a dominating cycle. t heor em 4 [6]: l et g be a 3-connected graph. then either c ¸ 3 ± ¡ 3 or every longest cycle in g is a dominating cycle. th e s e t o f ve r t ic e s o f a g r a p h g is d e n o t e d b y v ( g ) a n d t h e s e t o f e d g e s b y e ( g) . fo r s a s u b s e t o f v ( g) , we d e n o t e b y gns t h e m a xim u m s u b g r a p h o f g wit h ve r t e x s e t 3 1 3 2 simple proofs of two dirac-type theorems involving connectivity v ( g ) ns. fo r a s u b g r a p h h o f g we u s e gnh s h o r t fo r gnv ( h ) . w e d e n o t e b y n ( x) t h e n e ig h b o r h o o d o f a ve r t e x x in a g r a p h g. w e wr it e a c yc le c o f g wit h a g ive n o r ie n t a t io n b y ¡! c . fo r x; y 2 v ( c ) , we d e n o t e b y x¡!c y t h e s u b p a t h o f c in t h e c h o s e n d ir e c t io n fr o m x t o y. fo r x 2 v ( c ) , we d e n o t e t h e s u c c e s s o r o f x o n ¡!c b y x+ a n d t h e p r e d e c e s s o r b y x¡. fo r x ½ v ( c ) , we d e ¯ n e x + = fx+jx 2 xg. lemma 1: l et g be a 2-connected graph and s a minimum cut-set in g. if every longest cycle in g is a dominating cycle, then either c ¸ 3 ± ¡ · + 1 or there exists a longest cycle c with s µ v ( c ) . p r oof: ch o o s e a lo n g e s t c yc le c in g s u c h t h a t jv ( c ) \ sj is a s g r e a t a s p o s s ib le . a s s u m e t h e c o n ve r s e , t h a t is s 6µ v ( c ) a n d x 2 snv ( c ) . s in c e c is d o m in a t in g , n ( x ) µ v ( c ) . l e t »1; :::; »t b e t h e e le m e n t s o f n ( x ) , o c c u r r in g o n ¡! c in a c o n s e c u t ive o r d e r . p u t m1 = f»ijv ( »+i ¡! c »¡i+1 ) \s 6= ;g a n d m2 = n ( x ) nm1. s in c e x 2 s, we h a ve jm1j · ·¡ 1 a n d jm2j = jn ( x ) j ¡ jm1j ¸ ± ¡ · + 1 . fu r t h e r , s in c e c is e xt r e m e a n d jv ( c ) \ sj is m a xim u m , n ( x ) \ n + ( x) \ m ++2 = ; a n d c ¸ jn ( x ) j + jn + ( x ) j + jm ++2 j = 2 jn ( x ) j + jm2j ¸ 3 ± ¡ · + 1 . p r oof of t heor em 2: l e t g b e a 3 -c o n n e c t e d g r a p h , s b e a m in im u m c u t -s e t in g a n d le t h1; :::; hh b e t h e c o n n e c t e d c o m p o n e n t s o f gns. th e r e s u lt h o ld s im m e d ia t e ly if c ¸ 3 ± ¡ 3 , s in c e 3 ± ¡ 3 ¸ 3 ± ¡ ·. ot h e r wis e , b y th e o r e m 4 , e ve r y lo n g e s t c yc le in g is a d o m in a t in g c yc le . if s 6µ v ( c ) fo r e a c h lo n g e s t c yc le c t h e n b y l e m m a 1 , c ¸ 3 ± ¡ · + 1 . l e t c b e a lo n g e s t c yc le wit h s µ v ( c ) . if v ( gnc ) = ; t h e n jcj = n a n d we a r e d o n e . l e t x 2 v ( gnc ) . a s s u m e w.l.o .g . t h a t x 2 v ( h1 ) . p u t y1 = n ( x ) [ n + ( x) . cle a r ly jy1j ¸ 2 ± a n d it r e m a in s t o ¯ n d a s u b s e t y2 in v ( c ) s u c h t h a t y1\y2 = ; a n d jy2j ¸ ±¡·. a b b r e via t e , v1 = v ( h1 ) [ s. s u p p o s e ¯ r s t t h a t y1 µ v1. if v ( h2 ) µ v ( c ) , t h e n t a ke y2 = v ( h2 ) s in c e jv ( h2 ) j ¸ ± ¡ · + 1 . ot h e r wis e , t h e r e e xis t y 2 v ( h2nc ) wit h n ( y ) µ v ( c ) ( s in c e c is d o m in a t in g ) a n d we c a n t a ke y2 = n ( y ) ns. n o w le t y1 6µ v1. a s s u m e w.l.o .g . t h a t y1 \ v ( h2 ) 6= ;. s in c e n ( x ) µ v1, we c a n c h o o s e z 2 n + ( x ) \ v ( h2 ) . if n ( z ) µ v ( c ) , t h e n t a ke y2 = n ( z ) ns, s in c e n + ( x) is a n in d e p e n d e n t s e t o f ve r t ic e s ( b y s t a n d a r d a r g u m e n t s ) a n d t h e r e fo r e , n ( z ) \ n + ( x ) = ;. ot h e r wis e , c h o o s e w 2 n ( z ) nv ( c ) . cle a r ly n ( w ) µ v ( c ) , w 2 v ( h2 ) a n d n ( w ) \ n + ( x) = fzg. th e n b y t a kin g y2 = ( n ( w ) nfzg ) n ( snfz¡g ) we c o m p le t e t h e p r o o f. p r oof of t heor em 1: a s s u m e t h e c o n ve r s e , t h a t is g is a n o n -h a m ilt o n ia n 2 -c o n n e c t e d g r a p h wit h ± ¸ ( n+·) =3 . l e t s b e a m in im u m c u t -s e t in g. s in c e ± ¸ ( n+·) =3 ¸ ( n+2 ) = 3 , b y th e o r e m 3 , e ve r y lo n g e s t c yc le in g is a d o m in a t in g c yc le . fu r t h e r , s in c e c · n · 3 ± ¡·, b y l e m m a 1 , g c o n t a in s a lo n g e s t c yc le c wit h s µ v ( c ) . a s in t h e p r o o f o f th e o r e m 2 , c ¸ 3 ± ¡ ·, c o n t r a d ic t in g t h e fa c t t h a t ± ¸ ( n + ·) / 3 . refer ences [1 ] j.a . b o n d y a n d u .s .r . mu r t y, graph theory with applications, ma c m illa n , l o n d o n a n d e ls e vie r , n e w y o r k, 1 9 7 6 . [2 ] g. a . d ir a c , \ s o m e t h e o r e m s o n a b s t r a c t g r a p h s " , p roc. l ondon, m ath. soc., vo l. 2 , p p . 6 9 { 8 1 , 1 9 5 2 . [3 ] r . h äa g g kvis t a n d g.g. n ic o g h o s s ia n , \ a r e m a r k o n h a m ilt o n ia n c yc le s " , j ournal combin. theory, s e r . b 3 0 , p p . 1 1 8 -1 2 0 , 1 9 8 1 . [4 ] c. s t . j. a . n a s h -w illia m s , studies in p ure m athematics, ( e d g e -d is jo in t h a m ilt o n ia n c yc le s in g r a p h s wit h ve r t ic e s o f la r g e va le n c y, p p . 1 5 7 { 1 8 3 ) , in : l . mir s ky ( e d ) , a c a d e m ic p r e s s , s a n d ie g o , l o n d o n , 1 9 7 1 . c. mosesyan, m. nikoghosyan, zh. nikoghosyan 3 3 [5 ] zh . g. n iko g h o s ya n , \ on m a xim a l c yc le o f a g r a p h " , d an arm.ssr , ( in r u s s ia n ) , vo l. l x x ii, n o . 2 , p p . 8 2 -8 7 , 1 9 8 1 . [6 ] h .-j. v o s s a n d c. zu lu a g a , \ ma xim a le g e r a d e u n d u n g e r a d e k r e is e in gr a p h e n i" , w iss. z. tech. hochschule ilmenau, vo l. 2 3 , p p . 5 7 -7 0 , 1 9 7 7 . ( m ath. soc., vo l. 2 p p . 6 9 { 8 1 , 1 9 5 2 .) [7 ] t. y a m a s h it a , \ a d e g r e e s u m c o n d it io n fo r lo n g e s t c yc le s in 3 -c o n n e c t e d g r a p h s " , j ournal of graph theory, vo l. 5 4 , n o . 4 , p p . 2 7 7 -2 8 3 , 2 0 0 7 . submitted 05.09.2013, accepted 17.10.2013. î³å³ïóí³íáõãû³ý å³ñ³ù»ïñáí ¸çñ³ïû³ý ï»ëùç »ñïáõ ã»áñ»ùý»ñç å³ñ½ ³å³óáõûóý»ñ î. øáë»ëû³ý, ø. üçïáõáëû³ý ¨ ä. üçïáõáëû³ý ²ù÷á÷áõù ºññáñ¹ ñ»õçý³ïá 1981-çý ³å³óáõó»é ¿, áñ ï³ù³û³ï³ý 2-ï³å³ïóí³í ·ñ³ý ± ¸ ( n+·) =3 å³ûù³ýç ¹»åùáõù ñ³ùçéïáýû³ý ¿, çëï ï³ù³û³ï³ý 3-ï³å³ïóí³í ·ñ³ý áõýç ³éýí³½ý m in fn; 3 ± ¡·g »ñï³ñáõãû³ý óçïé, áñï»õ n-á ·ñ³ýç ·³·³ãý»ñç ù³ý³ïý ¿, ± ý‘ ýí³½³·áõûý ³ëïç×³ýá, çëï ·-ý` ï³å³ïóí³íáõãûáõýá: ð³·íçëãá ¨ ú³ù³ßçï³ý, ñ³ù³å³ï³ëë³ý³µ³ñ, ½·³éçáñ»ý ïñ×³ï»é »ý ³ûë »ñïáõ ã»áñ»ùý»ñç ý³ëý³ï³ý ³å³óáõûóý»ñá‘ »ñ»ù ¿çç ë³ñù³ýý»ñáõù: ü»ñï³ ³ßë³ï³ýùáõù áý¹ñ³ýáõñá 2/3 ¿çç ë³ñù³ýý»ñáõù ý»ñï³û³óíáõù »ý ß³ï ³í»éç ï³ñ× ¨ å³ñ½ ³å³óáõûóý»ñ: ïðîñòûå äîêàçàòåëüñòâà äâóõ òåîðåì äèðàêñêîãî òèïà ñ ïàðàìåòðîì ñâÿçíîñòè ê. ìîñåñÿí, ì. íèêîãîñÿí è æ. íèêîãîñÿí àííîòàöèÿ òðåòèé àâòîð â 1981 ãîäó äîêàçàë, ÷òî êàæäûé 2-ñâÿçíûé ãðàô ïðè óñëîâèè ± ¸ ( n+·) =3 ãàìèëüòîíîâ, à êàæäûé 3-ñâÿçíûé ãðàô ëèáî ãàìèëüòîíîâ ëèáî ñîäåðæèò öèêë äëèíû ïî ìåíüøåé ìåðå 3 ± ¡ k , ãäå n îáîçíà÷àåò ÷èñëî âåðøèí ãðàôà, ±ìèíèìàëüíàÿ ñòåïåíü è k ñâÿçíîñòü. õàãâèñò è ßìàøèòà, ñîîòâåòñòâåííî, çíà÷èòåëüíî ñîêðàòèëè îðèãèíàëüíûå äîêàçàòåëüñòâà ýòèõ òåîðåì â ðàìêàõ òðåõ ñòðàíèö. â íàñòîÿøåé ñòàòüå â ðàìêàõ 2/3 ñòðàíèö ïðåäëàãàþòñÿ íîâûå áîëåå êîðîòêèå è ïðîñòûå äîêàçàòåëüñòâà. microsoft word 11_tigran_sokhakyans' paper_90--98 mathematical problems of computer science 45, 90--98, 2016. 90 optimization techniques for generic secure two-party computation platform tigran v. sokhakyan russian-armenian (slavonic) university e-mail: tigran.sokhakyan@gmail.com abstract in this article we present an implementation of general purpose secure twoparty computation framework offering security against semi-honest threat model. proposed framework implements yao’s garbled circuits protocol and incorporates novel oblivious transfer protocol based on white-box cryptography methods for the first time to avoid computationally expensive public key operations. also experimental results illustrating the efficiency of our framework compared with previous implementations are provided. keywords: secure two-party computation, yao’s garbled circuits protocol, white-box cryptography, oblivious transfer. 1. introduction as the world is getting more and more connected in many real world scenarios, parties with different and potentially conflicting interests have to interact. examples of this are citizens and governments (electronic passport and electronic id), patients and health insurers or medical institutions (electronic health card, e-health services), or companies and service providers (cloud computing). in the mentioned context questions of paramount importance are the architecture and the ability of an underlying communication system to fulfill varied security and privacy requirements of the involved parties. protocols for secure computations allow two or more mutually distrustful parties communicate and jointly compute some commonly agreed function on each other’s input without possessing a trusted party, having privacy and authenticity guarantees. andrew yao pointed out that secure two-party protocols can be constructed for computation of any computable function [1]. yao’s protocol remains one of the most actively studied methods for secure computations. although yao never published a precise protocol, the very first real world implementation of secure two-party computation (s2pc) [2] used yao’s basic garbled circuit approach, and it t. sokhakyan 91 remains the primary paradigm for the plenty of 2pc implementations that have been developed during the past eleven years [3 5]. yao’s protocol has a great practical significance. in many real-world situations, the inputs to a function may be too valuable or sensitive to share with other parties. efficient s2pc algorithms enable a variety of electronic transactions, previously impossible due to mutual mistrust of participants. bringer et al. presented recent advances in privacy-preserving biometric identification [6], when it is desirable for individual genetic data to be kept private but still checked against a specified list. the more general case of multiparty computation has already seen real-world use in computing market clearing prices in denmark [7]. this is not so forth full list of applications: auctions [8], contract signing [9], etc. organization of the paper. in section 2 necessary cryptographic background is covered. in section 3 we give implementation overview of our framework. section 4 provides high level description of framework usage and section 5 provides experimental results. 2. background in subsequent sections we briefly introduce the main cryptographic tools we have used in our framework: garbled circuits, white-box cryptography based oblivious transfer (ot) protocol and optimization techniques for yao’s protocol.yao’s garbled circuits protocol. 2.1. yao’s garbled circuits protocol yao’s garbled circuit protocol [1] allows two mutually distrustful parties holding inputs and to evaluate an arbitrary computable function , on their input values without leaking any side information about their inputs beyond what is explicitly implied by the function output. the main idea is that one party (called garbled circuit generator) generates an “encrypted” version of the boolean circuit computing the function , and the second party (called garbled circuit evaluator) obliviously computes the garbled circuit. note that reverse engineering techniques are not applicable to garbled circuit, thus the evaluator does not learn any intermediate value. suppose the generator has a boolean circuit with 2 fan-in gates computing the function . at the first step the generator fixes some integer and assigns two random looking bit strings and to each wire of circuit (label conceptually encodes value ∈ 0,1 for the wire ). then for the gate having output wire and input wires , , the generator prepares the garbled table with the following entries: , , , (1) where enc is an encryption scheme fixed by generator. the collection of all garbled gates is called a garbled circuit. then the generator passes the garbled circuit and mapping for the labels for the output wires to the evaluator. note that only the generator knows mapping between binary input bits for input wires and it can simply send the garbled circuit to the evaluator with label for input wire , where is the -th bit of its input. to obtain wire labels for its input the evaluator runs oblivious transfer protocol described next with the generator. optimization techniques for generic secure two-party computation platform 92 the evaluation of garbled circuit is done in a hierarchical way. given labels and of input wires of garbled gate the evaluator decrypts the appropriate entry of garbled table using the keys and . when labels of all output wires are computed the evaluator sends the function output value to the generator using the provided mapping for output wires. 2.2. white-box cryptography based ot protocol extension 1-out-of-2 oblivious protocol (ot) is an essential part of yao’s garbled circuit protocol. it involves two parties: sender holding two strings and , and receiver holding the selection bit . ot protocol allows the sender to transmit exactly one input string to receiver; the receiver learns nothing about ⊕ and the sender does not learn selection bit . currently several ot protocols are available. for implementation of ot we use novel white-box cryptography based ot protocol which is secure in the semi-honest setting and is introduced in [10], which is order of magnitude faster compared with previous ot protocol implementations. 2.3. efficient garbling schemes to get more practical use of yao’s garbled circuits protocol many optimizations are developed during the past decade, some of which are compatible with each other and are incorporated in our framework. kolesnikov and schneider [11] introduced a technique eliminating the need to garble xor gates (xor gates become “free”, involving no communication or cryptographic operations). also, we use the technique proposed by pinkas et al. [12] allowing to reduce the size of a garbled table from four to three ciphertexts, and saving 25% of network bandwidth for nonxor gates. another optimization to apply is the flexor technique by kolesnikov et al. [13] combined with two garbled row reduction [12] instead of one. we have added this option, as two garbled row reduction and freexor techniques are not compatible. the combination of optimization techniques is user configurable. 3. the structure of computation framework the framework implements yao’s garbled circuits protocol to perform the privacy-preserving computation of function , , where the actual values of arguments are provided by mutually distrustful participants. our framework is implemented in c++ programming language. it contains a compiler generating an equivalent boolean circuit from the algorithmic representation of given function , . popular and tools were used to generate the compiler. the details of compiler implementation and description of its’ input language are given in [16]. the second major part of our framework is the implementation of yao’s garbled circuits protocol. the implementation employees various state of the art optimizations together with usage of novel oblivious transfer protocol based on white-box cryptography operations. whitebox cryptography methods are used to avoid computationally expensive public key operations during oblivious transfer. in our implementation, we use the white-box implementation of safer+ [17]. t. sokhakyan 93 other optimizations, introduced earlier in this field, are incorporated for speedup of different aspects of practical yao’s garbled circuits protocol implementation. to avoid delays caused by transmission of small amounts of data blocks over the network, communication batching is employed. the usage of communication batching gives the opportunity of data compression sent over the network. this would be highly desirable especially for wide area networks, where the usage of the underlying network becomes prohibitively expensive. to the best of our knowledge, none of the previous implementations uses emphasized data compression tool. during the construction and the evaluation of garbled circuit encryption function fixed by underlying garbling scheme is used extensively. for performing this operation faster, whenever it is possible, we have made use of aes encryption instructions supported on popular intel processors. although, these instructions are not portable, they provide additional speedup whenever they are available. in the baseline yao’s garbled circuits protocol a garbled circuit is constructed and transmitted over the network for evaluation, which requires entire circuit to be stored in memory. we have incorporated pipelined construction and evaluation of the garbled circuit [16], to reduce the amount of memory needed during computations and enable the participants do the computations without delays. another optimization is employed to minimize the amount of memory needed for storing the intermediate wire labels during construction and evaluation of the garbled circuit. the compiler generates usage count for each gate, and it is decremented each time the value of a wire is accessed. the intermediate results are swept off the memory as the counter hits zero. the above-mentioned optimizations or their alternatives are used in the previous implementations of s2pc and none of the previous implementations incorporates all of them. the use of these techniques enabled us to automatically construct and evaluate circuits having more gates than previous implementations are capable. 4. usage of two-party computation framework for convenience suppose there are two parties called alice and bob wishing to compute the function , on their private values and . the higher level view of framework usage is presented in fig. 1, and the flow is described below. first the parties need to describe the function in using the source language of the compiler which is a part of the framework [16]. then an equivalent boolean circuit is generated implementing function . this circuit is then shared among alice and bob. note, that resourceintensive generation of the boolean circuit is only once and the circuit can be used multiple times. having circuit structure alice determines the number of input bits corresponding to bob’s input wires and generates garbled labels for these wires. after this alice and bob initiate 1-out-of-2 white-box oblivious transfer protocol extension where alice inputs generated garbled labels and bob inputs bits of his private input . after this step bob obliviously gets wire labels corresponding to his input value. then alice and bob perform pipelined evaluation (see section 4.1) of the circuit. after this step alice and bob communicate to get their respective private outputs without leaking anything. note, that alice knows the correspondence between plain boolean values and garbled ones for all wires. thus, she can restore her private output bits when actually garbled values are known to her. also, alice cannot reveal bob’s private output because she does not know actually garbled values for wires corresponding to bob’s private output. for these reasons bob sends to alice garbled labels corresponding to her private output wires and alice sends to bob garbling scheme for his private output wires. optimization techniques for generic secure two-party computation platform 94 fig. 1. the higher level view of steps performed by participants during computation of function f(a, b) using our framework. 5. experimental results this section presents comparison of running times between our framework and a previous implementation implementing yao’s garbled circuits protocol in semi-hones adversarial model t. sokhakyan 95 [18]. for this, we considered privacy-preserving evaluation of two popular functions, namely, secure computation of hamming and levenshtein distances between provided arguments. for convenience we call the participants of computations the client and the server. this naming convention comes from the practice, where these problems are considered in clientserver architecture. formally, the hamming distance between given pair of -bit binary strings and , denoted , , is defined the total number of correspondingly different bits between and . in a privacy-preserving setting, is the secret input of the client and is the secret input of the server. in this setting, the client and the server wish to jointly compute , , or use it as an intermediate result during subsequent computations, without revealing respective private inputs to another party. levenshtein distance problem between two strings has important applications in computational biology, comparing text files and various fields. the problem statement as follows; given a set ο consisting of basic operations applicable on individual characters of some string and two strings and , find the edit distance between them. the edit distance between strings and is denoted by levenstein(x, y) and is defined as the minimum number of basic operations from ο needed for transformation of string into . we considered the typical case, when the set ο consists of insertion, deletion and replacement of single character in certain position of the source string. fig. 2 presents comparison of overall running times of privacy-preserving hamming distance computation between the previous implementation and the implementation presented in this work. fig. 2. comparison of running times of secure hammaing distance computation between mightbeevil and the presented framework. optimization techniques for generic secure two-party computation platform 96 fig. 3 presents comparison of overall running times of privacy-preserving edit distance computation between the previous implementation and the implementation presented in this paper. fig. 3. comparison of running times of secure levenshtein distance computation between mightbeevil and the presented framework. as it was expectable, our framework outperforms the previous implementation. the difference of running times grows with the number of input bits of the function. the performance of previous implementations of s2pc suffers highly from extensive usage of public key operations during ot protocol execution, which is used to obliviously exchange the garbled values corresponding to the input bits of the server (for details see section 3). as the number of input bits grows, previous implementation spends more time on ot execution step compared with our implementation, which replaces the usage of computationally expensive public key operations with order of magnitude faster white-box cryptography counterparts. 6. conclusion in this article, we have presented a framework for secure two-party computation offering security in the semi-honest adversarial model. the framework incorporates various optimizations with usage of white-box cryptography methods, which are order of magnitude faster compared with public key operations, during oblivious transfer protocol execution step. experimental results given in the article prove that these optimizations really enhance the overall running time of the computations. the benefit of white-box cryptography methods usage is highly expressed when the length of the input arguments grows. t. sokhakyan 97 we plan to make further improvements in our system to enable safety against malicious adversaries, with comparable performance and experimentally measure the impact of white-box cryptography methods on used network bandwidth. references [1] a. c.-c. yao, "how to generate and exchange secrets (extended abstract)," 27th annual symposium on foundations of computer science, toronto, pp. 162 167, october 27-29, 1986. [2] d. malkhi, n. nisan, b. pinkas and y. sella, "fairplay secure two-party computation system," proceedings of the 13th usenix security symposium, pp. 287-302, 2004. [3] t. k. frederiksen, t. p. jakobsen, j. b. nielsen, p. s. nordholt and c. orlandi, "minilego: efficient secure two-party computation from general assumptions," eurocrypt, pp. 537-556, 2013. [4] y. huang, j. katz and d. evans, "quid-pro-quo-tocols: strengthening semi-honest protocols with dual execution," ieee symposium on security and privacy, sp 2012, may, pp. 21-23, 2012. [5] y. lindell, b. pinkas and n. p. smart, "implementing two-party computation efficiently with security against malicious adversaries," security and cryptography for networks, springer, pp. 2-20, 2008. [6] j. bringer, h. chabanne and a. patey, "privacy-preserving biometric identification using secure multiparty computation: an overview and recent trends," signal processing magazine, ieee, pp. 42-52, 2013. [7] p. bogetoft, d. l. christensen, i. damgard, m. geisler et al., "secure multiparty computation goes live," financial cryptography and data security, springer, pp. 325-343, 2009. [8] g. di crescenzo, "private selective payment protocols," financial cryptography, pp. 72-89, 2001. [9] s. even, o. goldreich and a. lempel, "a randomized protocol for signing contracts," commun. acm, vol. 28, no. 6, pp. 637-647, 1985. [10] a. jivanyan and g. khachatryan, "efficient oblivious transfer protocols based on whitebox cryptography," aua internal reports, 2013. [11] v. kolesnikov and t. schneider, "improved garbled circuit: free xor gates and applications," automata, languages and programming, 35th international colloquium, springer, pp. 486 498, 2008. [12] b. pinkas, t. schneider, n. p. smart and s. c. williams, "secure two-party computation is practical," advances in cryptology asiacrypt 2009, 15th international conference, pp. 250-267, 2009. [13] v. kolesnikov, p. mohassel and m. rosulek, "flexor: flexible garbling for xor gates that beats free-xor," advances in cryptology--crypto 2014, springer, pp. 440-457, 2014. [14] d. kozen , "lower bounds for natural proof systems," 18th annual symposium on foundations of computer science, ieee, pp. 254-266, sep 30, 1977. [15] c. s. geol, j. katz, r. kumaresan and h.-s. zhou. "on the security of the “free-xor” technique," theory of cryptography, springer berlin heidelberg, pp. 39-53, 2012. optimization techniques for generic secure two-party computation platform 98 [16] d. danoyan and t. sokhakyan, "a generic framework for secure computations", proceedings of russian-armenian (slavonic) university 2015 (physical, mathematical and natural sciences), vol. 2, pp. 14-21, 2015. [17] j. massey, g. khachatrian and m. kuregian. "nomination of safer+ as a candidate algorithm for advanced encryption standard (aes)," represented at the 1st aes conference, ventura, usa, august 20-25, 1998 [18] y. huang, d. evans, j. katz and l. malka, "faster secure two-party computation using garbled circuits," in usenix security symposium, vol. 201, no. 1, august 8, 2011. submitted 10.10.2015, accepted 25.01.2016 երկու մասնակցով ընդհանուր օգտագործման անվտանգ հաշվարկների համակարգի լավարկման մեթոդներ տ. սոխակյան ամփոփում այս հոդվածում ներկայացված է երկու մասնակցով անվտանգ հաշվարկների համակարգի իրականացում: առաջարկված համակարգը իրականացնում է յաոյի հաղորդակարգը և նմանատիպ համակարգերի իրականացման համար առաջին անգամ օգտագործում է սպիտակ արկղի գաղտնագրության մեթոդների վրա հիմնված անտեղյակ փոխանցման նոր հաղորդակարգը, որը խուսափում է հաշվողական տեսանկյունից դանդաղ բաց բանալիով գործողություններից: բերված են փորձնական արդյունքներ, որոնք ապացուցում են առաջարկված իրականացման արդյունավետությունը: методы оптимизации для обобщённой платформы конфиденциальных вычислений с двумя участниками т. сохакян аннотация в этой статье представлена реализация платформы конфиденциальных вычислений общего назначения с двумя участниками. предложенная платформа реализует протокол яо с искажёнными схемами, впервые используя протокол забывчивой передачи основанной на методах криптографии белого ящика, который не использует дорогостоящие операции с открытым ключом. также приводятся экспериментальные данные, показывающие эффективность реализованной системы. microsoft word nanasyan_15.doc математические вопросы кибернетики и вычислительной техники 34, 2010. 41 некоторые прикладные разработки ипиа нан ра последних лет в области информационных технологий а.нанасян в 2005-2010 гг. в ипиа выполнялись работы по разработке и внедрению новых новых сервисных услуг для почтовой службы asnet webmail (телефонная голосовая почта t-mail, информационная система mailinformer), созданию интерактивной экзаменационной системы i-тест, разработке концепции и эскизного проекта инновационного предложения ипиа «создание единой инфокоммуникационной и управляющей инфраструктуры г. еревана». телефонная голосовая почта t-mail предназначена для доступа к почтовым ресурсам интернет с абонентского телефона. являясь надстройкой над e-mail, голосовая почта t-mail позволяет посылать голосовые письма с телефона на любой e-mail адрес, также как и получать на телефон голосовые письма, посланные по e-mail. информационная система mail informer предназначен для оперативного уведомления пользователей электронной почты на мобильный телефон информации о поступлении в их почтовый ящик новых e-mail от выделенных отправителей, электронные адреса которых предварительно занесены пользователем в список разрешенных адресов . система, по заданному списку адресов, уведомит о поступлении корреспонденции от ожидаемых отправителей в виде sms. в интерактивной экзаменационной тестовой системе i-тест предусмотрены элементы, позволяющие в первом, грубом приближении, приблизить тестовую систему к традиционной модели взаимодействия «экзаменатор-экзаменуемый», когда при оценке знаний учитывается не только параметр «правильно-неправильно», но и возможные реакции, характер и поведение экзаменуемого во время прохождения экзамена. инновационное предложение ипиа «создание единой инфокоммуникационной и управляющей инфраструктуры г. еревана». в настоящее время в крупных городах создаются автоматизированные системы мониторинга и управления городским хозяйством. эти специализированные системы имеют узкую направленность и предназначены для решения конкретных задач. более того, подобные системы трудно адаптируются и, как правило, создаются для конкретных конфигураций. все эти системы в определенной мере функционально взаимосвязаны, при этом, очевидна целесообразность объединения существующих систем и вновь создаваемых интеллектуальных систем управления в единую инфкоммуникационную и управляющую инфраструктуру города для комплексного решения задач экологического и техногенного мониторинга, адаптивного управления городскими транспортными потоками, минимизации затрат на уличное освещение применением интеллектуальных систем освещения и т. п. в предлагаемой структуре будут использованы уже существующие высокопроизводительные вычислительные и коммуникационные ресурсы института проблем информатики и автоматизации нан ра, в том числе суперкомпьютер армкластер и распределенная вычислительная среда – армгрид. 42 предлагается решения трех актуальных проблем г. еревана на основе единой общегородской инфраструктуры  увеличение пропускной способности основных магистралей  контроль окружающей среды.  сокращение расходов на содержание городского хозяйства необходимость оптимизации управления городскими транспортными потоками , связана с резким увеличением количества автотранспорта в условиях применяемых методов регулирования (светофоры переключаются по заданной программе , без учета реальной обстановки на перекрестке) и сложившихся транспортных магистралей, пропускная способность которых не рассчитана на существующий в настоящее время трафик, соответственно, ограничивает движение транспорта и приводит к заторам. решение данной проблемы традиционными способами (расширение улиц, прокладка параллельных улиц, эстакады, многоуровневые развязки) крайне затруднительно в условиях существующей городской застройки. одним из принятых способов решения данной задачи является применение адаптивных (интеллектуальных ) систем регулирования уличным движением, когда длительность фаз переключения светофоров меняется в зависимости от количества машин на перекрестке, а при комплексировании светофоров всех перекрестков магистрали оптимизируется движение по всей магистрали. подобная организация управления движением позволяет увеличить пропускную способность магистрали на 2530 %, уменьшить вредные выбросы на 10-15 %. мониторинг окружающей среды с использованием спутниковых технологий для анализа и оптимизации управления транспортными потоками и сетями города контроль городской окружающей среды производится, как правило, специализированными организациями по конкретному направлению (метео, воздух, вода, земля) по определенному временному графику или, в случае непредвиденных ситуаций, с использованием специализированных мобильных (реже стационарных) измерительных станций. между тем, очевидно, что в крупном городе непрерывный экологический мониторинг должен осуществляться в реальном масштабе времени. прием и обработку оперативной информации о состоянии среды обитания целесообразнее, при этом, сосредоточить в едином городском информационно-аналитическом центре. в городском информационно-аналитическом центре предусматривается также использование спутниковых технологий (гис технологий), специально предназначенных для работы с информацией о городских транспортных объектах и сетях. технологии гис будут использоваться для решения задач моделирования, анализа, и оптимизации управления инфраструктурой и ее развития. программные средства гис адаптированы к вычислительной среде grid одна из составляющих минимизации расходов на содержание городского хозяйства – сокращение энергопотребления источников уличного освещения. данная задача решается применением интеллектуальных систем уличного освещения, которые изменяют уровень освещенности улицы светильниками в зависимости от времени суток, погоды (сумерки, ночь, туман, дождь) и интенсивности движения автотранспорта. применение подобных новых технологий (lon works, gsmcontrol и др.) позволяет снизить потребление электроэнергии на освещение улиц более, чем на 30-40 %. в рамках первого этапа реализации данного проекта предлагается начать работы по исследованию, разработке и реализации опытного сегмента системы, охватывающего проблемный с точки зрения транспортного трафика городской район еревана. microsoft word title-template_sbornik36.doc mathematical problems of computer science 36, 41--50, 2012. 41 digital mammogram segmentation and abnormal masses detection system armen sahakyan institute for informatics and automation problems of nas of ra e-mail: armensahakyan@gmail.com abstract digital mammogram has emerged as the most popular screening technique for early detection of breast cancer and other abnormalities. raw digital mammograms are medical images that are difficult to interpret so we need to develop computer aided diagnosis (cad) systems that will improve detection of abnormalities in mammogram images. extraction of the breast region by delineation of the breast contour allows the search for abnormalities to be limited to the region of the breast. we need to perform essential pre-processing steps to suppress artifacts, enhance the breast region and then extract breast region by the process of segmentation. in this paper we present an automated system for detection of abnormal masses by anatomical segmentation of breast region of interest (roi). references [1] l.-m. wun, r. m. merrill, and e. j. feuer, "estimating lifetime and age-conditional probabilities of developing cancer," lifetime data analysis, vol. 4, pp. 169-186, 1998. [2] "who cancer facts," http://www.who.int/cancer/en/, 2009. [3] l. shen, r. rangayyan, and j. desaultels, “detection and classification mammographic calcifications”, international journal of pattern recognition and artificial intelligence. singapore: world scientific, pp. 1403–1416, 1994. [4] f. aghdasi, r.ward, and b. palcic, “restoration of mammographic images in the presence of signal-dependent noise”, in state of the art in digital mammographic image analysis. singapore: world scientific, vol. 7, pp. 42–63, 1994. [5] y. chitre, a. dhawan, and m. moskowtz, “artificial neural network based classification of mammographic microcalcifications using image structure features”, in state of the art of digital mammographic image analysis. singapore: world scientific, vol. 7, pp. 167–197, 1994. [6] pisano and f. shtern,“image processing and computer-aided diagnosis in digital mammography,” in state of the art of digital mammographic image analysis. singapore: world scientific, vol. 7, pp. 280–291, 1994. digital mammogram segmentation and abnormal masses detection system 42 [7] a. sahakyan, h. sarukhanyan, "automatic segmentation of the breast region in digital mammograms", computer science and information technologies, proceedings of the conference, pp. 386 389, yerevan, armenia, september 26-30, 2011. [8] indra kanta maitra ; sanjay nag and prof. samir k. bandyopadhyay, "automated digital mammogram segmentation for detection of abnormal masses using binary homogeneity enhancement algorithm", indian journal of computer science and engineering, issue 3, vol. 2, pp. 416-427, 2011. [9] j. suckling et al., “the mammographic image analysis society digital mammogram database”, exerpta medica., vol. 1069, pp. 375– 378, 1994. âí³ûçý ù³ùá·ñ³ùý»ñç ë»·ù»ýï³íáñù³ý ¨ ³ýýáñù³é ½³ý·í³íý»ñç ñ³ûï³µ»ñù³ý ñ³ù³ï³ñ· ². ê³ñ³ïû³ý ²ù÷á÷áõù îñíù³·»õóç ù³õóï»õç ¨ ³ûé ³ýýáñù³é ½³ý·í³íý»ñç í³õ ñ³ûïý³µ»ñù³ý ù»ãá¹ý»ñçó ¿ ãí³ûçý ù³ùá·ñ³ýç³ý։ âí³ûçý ù³ùá·ñ³ùý»ñá µåßï³ï³ý å³ïï»ñý»ñ »ý, áñáýó í»ñ³ùß³ïù³ý ¨ í»ñéáõíáõãû³ý ñ³ù³ñ ³ýññ³å»ßï ¿ ùß³ï»é ñ³ù³ï³ñ·ã³ûçý ³ëïáñáßù³ý ¹ç³·ýáëïçï (cad) ñ³ù³ï³ñ·, áñá ïû·ýç ³ýýáñù³é ½³ý·í³íý»ñç ñ³ûïý³µ»ñù³ýá։ îñíùç »½ñ³·í»ñç ßñç³·íù³ý û·ýáõãû³ùµ, ïñíù³ûçý ßñç³ýç ³é³ýóý³óáõùá ãáõûé ¿ ï³éçë, áñ ³ýýáñù³é ½³ý·í³íý»ñç áñáýáõùá ë³ñù³ý³÷³ïíç ùç³ûý ïñíùç ßñç³·íáí։ ²ýññ³å»ßï ¿ çñ³ï³ý³óý»é ³õùáõïý»ñç ñ»é³óù³ý ý³ë³ùß³ïù³ý ù³ûé»ñ, µ³ñ»é³í»é ïñíù³·»õóç å³ïï»ñá ¨ ³ûýáõñ»ï¨ ë»·ù»ýï³íáñ»é։ ²ûë ñá¹í³íáõù ý»ñï³û³óí³í ¿ ³ýýáñù³é ½³ý·í³íý»ñç ñ³ûïý³µ»ñù³ý íñ³·ñ³ûçý ñ³ù³ï³ñ·, áñá çñ³ï³ý³óýáõù ¿ å³ïï»ñç ñ»ï³ùñùçñ ïçñáõûãý»ñç (roi) ë»·ù»ýï³íáñáõùá։ ñèñòåìà öèôðîâîé ñåãìåíòàöèè ìàììîãðàì è îáíàðóæåíèÿ àíîìàëüíûõ ìàññ à. ñààêÿí àííîòàöèÿ öèôðîâàÿ ìàììîãðàôèÿ ÿâëÿåòñÿ ñàìîé ïîïóëÿðíîé òåõíèêîé ñêðèíèíãà äëÿ ðàííåãî âûÿâëåíèÿ ðàêà ìîëî÷íîé æåëåçû è äðóãèõ íàðóøåíèé. öèôðîâûå ìàììîãðàìû ÿâëÿþòñÿ ìåäèöèíñêèìè èçîáðàæåíèÿìè è äëÿ èõ îáðàáîòêè íûæíî ðàçàðàáîòàòü âñïîìîãàòåëüíûå äèàãíîñòè÷åñêèå êîìïüþòåðíûå (cad) ñèñòåìû, êîòîðûå áóäóò ñïîñîáñòâîâàòü âûÿâëåíèþ íàðóøåíèé. îòìå÷åíèå îáëàñòè ãðóäè êîíòóðîì ïîçâîëÿåò ïîèñêû àíîìàëèè áûòü îðãàíèÿåííûì òîëüêî îáëàñòüþ ãðóäè. ìû äîëæíû âûïîëíèòü øàãè ïðåâàðèòåëüíîé îáðàáîòêè, ÷òîáû ïîäàâèòü øóìû, óëóòøèòü èçîáðàæåíèå îáëàñòè ãðóäè è çàòåì îòìåòèòü îáëàñòü ãðóäè ïðîöåññîì ñåôìåíòàöèè. â ýòîé ñòàòüå ìû ïðåäñòàâëÿåì àâòîìàòèçèðîâàííóþ ñèñòåìû äëÿ îáíàðóæåíèÿ àíîìàëüíûõ ìàññ àíàòîìè÷åñêîé ñåôìåíòàöèåé îáëàñòè èíòåðåñà (roi) ãðóäè. d:\tex_522\main.dvi mathematical problems of computer science 52, 21{29, 2019. u d c 5 1 9 .8 i nfor mation t heor etic m ethods for anomaly detection ma r ia m e . h a r o u t u n ia n a n d tig r a n s . b a d a s ya n institute for informatics and automation problems of nas ra e-mail: armar@ipia.sci.am; tigranbadasyan@gmail.com abstract maintaining the security of digital systems with a huge amount of data is one of the main concerns of it specialists in these times. anomaly detection in systems is one of the solutions to overcome this challenge. anomaly detection means ¯nding patterns that are not normal or deviate from normal behavior in a system. anomaly detection has various applications in bio-informatics, image processing, cyber security, security for databases, etc. there are many groups of methods that are used for anomaly detection including statistical methods, neural network methods and information theoretic methods. in this paper we survey pros and cons of anomaly detection based on information theoretic techniques. keywords: anomaly / outlier detection, entropy, relative entropy, local-search heuristic algorithm lsa. 1 . in t r o d u c t io n a n o m a ly o r o u t lie r is a n im p o r t a n t c o n c e p t o f t h e d a t a a n a lys is . a n o m a ly is d e ¯ n e d a s a d e via t io n fr o m n o r m a l d a t a . it m e a n s t h a t t h e d a t a o b je c t is n o t s im ila r t o t h e o t h e r o b s e r va t io n s in t h e d a t a s e t . it is ve r y im p o r t a n t t o d e t e c t t h e s e o b je c t s d u r in g t h e d a t a a n a lys is t o t r e a t t h e m d i®e r e n t ly fr o m t h e o t h e r d a t a . th e a n o m a ly d e t e c t io n m e t h o d s a r e wid e ly u s e d fo r t h e fo llo win g p u r p o s e s : cr e d it c a r d ( a n d m o b ile p h o n e ) fr a u d d e t e c t io n ; s u s p ic io u s w e b s it e d e t e c t io n ; w h o le -g e n o m e d n a m a t c h in g ; e cg-s ig n a l ¯ lt e r in g ; s u s p ic io u s t r a n s a c t io n d e t e c t io n a n d s o o n . th e in ve s t ig a t io n o f a n o m a ly d e t e c t io n t e c h n iqu e s is ve r y im p o r t a n t in s u c h ¯ e ld s a s me d ic a l a n d p u b lic h e a lt h ; d e c is io n m a kin g ; b u s in e s s in t e llig e n c e ; in d u s t r ia l d a m a g e d e t e c t io n a n d o t h e r s . 2 1 2 2 information theoretic methods for anomaly detection th e a n o m a ly d e t e c t io n p r o b le m h a s b e c o m e a r e c o g n iz e d r a p id ly-d e ve lo p in g t o p ic o f t h e d a t a a n a lys is . a s ig n ī c a n t n u m b e r o f m e t h o d s h a ve r e c e n t ly b e e n p r o p o s e d . ma n y s u r ve ys a n d s t u d ie s a r e d e vo t e d t o t h is p r o b le m , s e e fo r e xa m p le [1 ] [4 ]. th e y c o n c e r n d i®e r e n t a s p e c t s o f a n o m a ly d e t e c t io n . [1 ] r e vie ws t r a d it io n a l o u t lie r d e t e c t io n a lg o r it h m s , [2 ] o ve r vie ws t h e e xis t in g r e s e a r c h fo r t h e p r o b le m o f d e t e c t in g a n o m a lie s in d is c r e t e s e qu e n c e s , [3 ] is fo c u s e d o n e n s e m b le le a r n in g o n e s , wh ile [4 ] o n ly in vo lve s t h e la t e s t a n d p o p u la r a n o m a ly d e t e c t io n m e t h o d s fo r t h e d a t a wit h h ig h d im e n s io n a lit y a n d m ixe d t yp e s , o n wh ic h t h e c la s s ic a l d e t e c t io n m e t h o d s c a n n o t o p e r a t e ve r y we ll. th e r e is n o wid e ly a c c e p t a b le fo r m a l d e ¯ n it io n o f t h e a n o m a ly. th e p r e c is e d e ¯ n it io n o f t h e o u t lie r d e p e n d s o n t h e s p e c i¯ c p r o b le m a n d it s d a t a r e p r e s e n t a t io n . a t a n a b s t r a c t le ve l, a n a n o m a ly is d e ¯ n e d a s a p a t t e r n t h a t d o e s n o t c o n fo r m t o e xp e c t e d n o r m a l b e h a vio r . a s t r a ig h t fo r wa r d a n o m a ly d e t e c t io n a p p r o a c h , t h e r e fo r e , is t o d e ¯ n e a r e g io n r e p r e s e n t in g n o r m a l b e h a vio r a n d d e c la r e a n y o b s e r va t io n in t h e d a t a t h a t d o e s n o t b e lo n g t o t h is n o r m a l r e g io n a s a n a n o m a ly. b u t s e ve r a l fa c t o r s m a ke t h is a p p r o a c h ve r y c h a lle n g in g . s p e c i¯ c fo r m u la t io n o f t h e p r o b le m is d e t e r m in e d b y s e ve r a l d i®e r e n t fa c t o r s s u c h a s t h e n a t u r e o f t h e in p u t d a t a , t h e a va ila b ilit y o f la b e ls , t h e c o n s t r a in t s a n d r e qu ir e m e n t s in d u c e d b y t h e a p p lic a t io n d o m a in . th is ju s t ī e s t h e n e e d fo r t h e b r o a d s p e c t r u m o f a n o m a ly d e t e c t io n t e c h n iqu e s . b a s e d o n t h e r e s e a r c h a r e a t h e m a jo r it y o f a n o m a ly d e t e c t io n t e c h n iqu e s c a n b e c a t e g o r iz e d in t o c la s s i¯ c a t io n , n e a r e s t n e ig h b o r , c lu s t e r in g , s t a t is t ic a l a n d in fo r m a t io n t h e o r e t ic a l t e c h n iqu e s . e a c h o f t h e la r g e n u m b e r o f a n o m a ly d e t e c t io n t e c h n iqu e s h a s it 's o wn u n iqu e s t r e n g t h s a n d we a kn e s s e s . it is im p o r t a n t t o kn o w wh ic h a n o m a ly d e t e c t io n t e c h n iqu e is b e s t s u it e d fo r a g ive n p r o b le m . give n t h e c o m p le xit y o f t h e p r o b le m s p a c e , it is n o t fe a s ib le t o p r o vid e s u c h a n u n d e r s t a n d in g fo r e ve r y a n o m a ly d e t e c t io n p r o b le m . n o n e o f t h e p r e vio u s s u r ve ys , h o we ve r , fo c u s e s o n in fo r m a t io n t h e o r e t ic a l d e t e c t io n m e t h o d s . in t h is p a p e r we s u r ve y t h e s t a t e -o f-t h e -a r t t e c h n iqu e s o f a n o m a ly d e t e c t io n b a s e d o n in fo r m a t io n t h e o r y. th e a im is t o e xp lo r e t h e u s e o f t o o ls / p r o p o s a ls o f in fo r m a t io n th e o r y, wh ic h is c a p a b le o f b e in g u s e d t o s o lve t h e p r o b le m o f a n o m a ly d e t e c t io n fr o m n e w p e r s p e c t ive s . 2 . s o m e in fo r m a t io n th e o r e t ic me a s u r e s entropy is a n im p o r t a n t c o n c e p t in in fo r m a t io n t h e o r y a n d c o m m u n ic a t io n t h e o r y [5 ]. it m e a s u r e s t h e u n c e r t a in t y o f a c o lle c t io n o f d a t a it e m s : h ( x ) = ¡ x x p ( x ) lo g p ( x) ; fo r a d a t a s e t x o f it e m s x wit h p r o b a b ilit ie s p ( x) . e n t r o p y s p e c i¯ e s t h e n u m b e r o f b it s r e qu ir e d t o e n c o d e a n d t r a n s m it t h e c la s s ī c a t io n o f d a t a it e m . fo r a n o m a ly d e t e c t io n e n t r o p y c a n b e u s e d a s a m e a s u r e o f t h e r e g u la r it y o f a u d it d a t a . th e s m a lle r t h e e n t r o p y, t h e m o r e r e g u la r is t h e d a t a s e t . h ig h r e g u la r it y d a t a c a n p r e d ic t fu t u r e e ve n t s b e c a u s e e ve n t s r e p e a t e d m a n y t im e s in t h e c u r r e n t d a t a s e t a r e like ly t o a p p e a r in t h e fu t u r e . conditional entropy o f x g ive n y is h ( xjy ) = ¡ x x;y p ( x; y ) lo g p( xjy ) ; m. haroutunian and t. badasyan 2 3 wh e r e p ( x; y ) is t h e jo in t p r o b a b ilit y o f x a n d y a n d p ( xjy ) is t h e c o n d it io n a l p r o b a b ilit y o f x g ive n y. th is m e a s u r e c a n b e u s e d fo r a n o m a ly d e t e c t io n a s t h e va lu e o f d e p e n d e n c e r e g u la r it y. a s in c a s e o f e n t r o p y, t h e s m a lle r t h e c o n d it io n a l e n t r o p y, t h e b e t t e r . kullback-leibler divergence a ls o kn o wn a s relative entropy b e t we e n t wo p r o b a b ilit y d is t r ib u t io n s o n t h e s a m e x is d ( p k q) = x x p ( x) lo g p ( x ) q ( x ) : fo r a n o m a ly d e t e c t io n k l d ive r g e n c e c a n b e u s e d t o m e a s u r e t h e d is t a n c e o f r e g u la r it ie s b e t we e n t h e t r a in in g d a t a s e t a n d t h e t e s t d a t a s e t . a g a in , t h e s m a lle r t h e r e la t ive e n t r o p y, t h e b e t t e r . 3 . in fo r m a t io n th e o r y b a s e d a n o m a ly d e t e c t io n te c h n iqu e s in fo r m a t io n t h e o r e t ic t e c h n iqu e s a n a lyz e t h e in fo r m a t io n c o n t e n t o f a d a t a s e t u s in g d i®e r e n t in fo r m a t io n t h e o r e t ic m e a s u r e s s u c h a s e n t r o p y, r e la t ive e n t r o p y, a n d s o o n . s u c h t e c h n iqu e s a r e b a s e d o n t h e fo llo win g ke y a s s u m p t io n : a n o m a lie s in d a t a in d u c e ir r e g u la r it ie s in t h e in fo r m a t io n c o n t e n t o f t h e d a t a s e t . in [6 ] t h e in fo r m a t io n -t h e o r e t ic m e a s u r e s a r e p r o p o s e d fo r a n o m a ly d e t e c t io n in t h e fo llo win g g e n e r a l a p p r o a c h : me a s u r e t h e r e g u la r it y o f t h e d a t a a n d p e r fo r m t h e a p p r o p r ia t e d a t a t r a n s fo r m a t io n . it e r a t e t h is s t e p if n e c e s s a r y s o t h a t t h e d a t a s e t u s e d fo r m o d e lin g h a s h ig h r e g u la r it y. d e t e r m in e h o w t h e m o d e l s h o u ld b e b u ilt , i.e . h o w t o a c h ie ve t h e b e s t p e r fo r m a n c e o r t h e o p t im a l p e r fo r m a n c e / c o s t t r a d e -o ®, a c c o r d in g t o t h e r e g u la r it y m e a s u r e . u s e r e la t ive e n t r o p y t o d e t e r m in e wh e t h e r a m o d e l is s u it a b le fo r a n e w d a t a s e t ( e .g ., fr o m a n e w e n vir o n m e n t ) . d i®e r e n t s e qu e n c e le n g t h s a r e u s e d t o s h o w t h e r e la t io n s h ip b e t we e n r e g u la r it y a n d d e t e c t io n p e r fo r m a n c e . in p r a c t ic e , o n e c a n s im p ly c o m p u t e t h e r e g u la r it y o f a g ive n d a t a s e t a n d d e t e r m in e h o w t o b u ild a m o d e l, b e c a u s e c o m p u t in g r e g u la r it y, in g e n e r a l, is m u c h m o r e e ± c ie n t t h a n c o m p u t in g a m o d e l. th is is o n e o f t h e a d va n t a g e s o f t h is a p p r o a c h in c o m p a r is o n wit h t h e o t h e r s , wh e r e t h e r e is n o qu id e lin e fo r b u ild in g a m o d e l a n d e xp la in in g it s p e r fo r m a n c e . h o we ve r , t h e r e la t io n s h ip b e t we e n r e g u la r it y a n d d e t e c t io n p e r fo r m a n c e wa s s h o wn fo r t h e c la s s ī e r m o d e l, wh ile t h e r e a r e o t h e r p r o b a b ilis t ic a lg o r it h m s , t h a t c a n b e u s e d fo r a n o m a ly d e t e c t io n . h o w t h e in fo r m a t io n -t h e o r e t ic m e a s u r e s c a n b e u s e d fo r t h e s e a lg o r it h m s is a n o p e n qu e s t io n . s o m e r e a l-life a p p lic a t io n s c o n t a in categorical data, h o we ve r , m o s t o f t h e o u t lie r a lg o r it h m s a r e fo c u s e d o n n u m e r ic a l d a t a a n d d o n o t p e r fo r m we ll wh e n a p p lie d t o c a t e g o r ic a l d a t a . th e p r o b le m o f o u t lie r d e t e c t io n in c a t e g o r ic a l d a t a is c o n s id e r e d in [7 ]. h e r e e n t r o p y is u s e d t o m e a s u r e t h e d e g r e e o f d is o r d e r o f a d a t a s e t . th e o p t im iz a t io n p r o b le m is d e s c r ib e d a s fo llo ws : ¯ n d in g a s u b s e t o f k o b je c t s s u c h t h a t t h e e xp e c t e d e n t r o p y o f t h e r e s u lt a n t d a t a s e t a ft e r t h e r e m o va l o f t h is s u b s e t is m in im iz e d . th e g r e e d y o p t im iz a t io n s c h e m e t h a t u s e s lo c a l s e a r c h h e u r is t ic is s t u d ie d fo r qu a lit y t im e t r a d e o ®. a s a r e s u lt t h e local search algorithm (lsa) wa s p r e s e n t e d , wh ic h h a s b e e n s h o wn t o b e t h e b e s t in d e t e c t in g o u t lie r s b o t h in t e r m s o f a c c u r a c y a n d s p e e d wh e n c o m p a r e d wit h o t h e r t e c h n iqu e s . w o r kin g o f l s a c a n b e d e ¯ n e d a s fo llo ws . fo r a d a t a s e t d s , k o u t lie r s a r e t o b e d e t e c t e d u s in g e n t r o p y, th e n u m b e r o f o u t lie r s t o b e g e n e r a t e d ( k ) is u s e r d e ¯ n e d . 2 4 information theoretic methods for anomaly detection th e in it ia l s e t o f o u t lie r s ( s o) is e m p t y a n d a ll t h e d a t a s e t 's r e c o r d s a r e m a r ke d a s n o n o u t lie r s . k s c a n s a r e c a r r ie d o u t t o s e le c t k r e c o r d s a s o u t lie r s . d u r in g e a c h s c a n , e a c h r e c o r d t a g g e d a s a n o n -o u t lie r is t e m p o r a r ily r e m o ve d fr o m t h e d a t a s e t a n d t h e c h a n g e in e n t r o p y is c a lc u la t e d . th e r e c o r d t h a t a c h ie ve s t h e m a xim u m d e c r e a s e in e n t r o p y b y r e m o vin g t h a t r e c o r d , is s e le c t e d a s a n o u t lie r a n d a d d e d t o s o. th is c o n t in u e s fo r e a c h s c a n u n t il t h e s iz e o f os r e a c h e s t h e d e ¯ n e d va lu e o f k. a s t h e o p t im a l n u m b e r o f o u t lie r s va r ie s fr o m d a t a s e t t o d a t a s e t , it is n o t p o s s ib le t o p r e d ic t t h e n u m b e r o f o u t lie r s o r t o d e ¯ n e a s t a n d a r d o r ¯ xe d n u m b e r o f o u t lie r s t h a t c a n b e a p p lic a b le fo r e ve r y d a t a s e t . th is is t h e d is a d va n t a g e o f l s a . in [8 ] a n o u t lie r d e t e c t io n t e c h n iqu e fo r c a t e g o r ic a l d a t a is p r o p o s e d wh ic h is b a s e d o n e n t r o p y a n d c a lle d a u t o m a t e d e n t r o p y v a lu e fr e qu e n c y ( a e v f) . it r e qu ir e s n o u s e r in p u t a n d will a lwa ys g e n e r a t e t h e o p t im a l n u m b e r o f o u t lie r s . th is is t h e e xt e n d e d ve r s io n o f t h e l s a , wh ic h in t r o d u c e s t h e n e w t e r m s : e n t r o p y d i®e r e n c e g a p a n d m a x e n t r o p y g a p . th e entropy difference gap is t h e d i®e r e n c e in t h e va lu e s o f c h a n g e o f e n t r o p y b e t we e n o n e r e c o r d a n d t h e n e xt . th e max entropy gap is t h e m a xim u m e n t r o p y d i®e r e n c e g a p t h a t c a n e xis t a s d e ¯ n e d b y t h e u s e r . if t h e e n t r o p y d i®e r e n c e g a p b e c o m e s la r g e r t h a n t h e m a x e n t r o p y g a p , t h e a lg o r it h m t e r m in a t e s . th e a lg o r it h m is a t wo -s t e p p r o c e s s : ge n e r a t in g e n t r o p y c h a n g e va lu e s . th e c h a n g e in e n t r o p y va lu e fo r e a c h r e c o r d in a d a t a s e t is g e n e r a t e d a n d s t o r e d in a t a b le . on c e a ll r e c o r d s h a ve b e e n p r o c e s s e d , t h e t a b le is u p d a t e d s o t h a t t h e r e c o r d wit h t h e m a xim u m e n t r o p y c h a n g e va lu e is a t t h e t o p , fo llo we d b y t h e o t h e r r e c o r d s in d e s c e n d in g o r d e r o f e n t r o p y c h a n g e va lu e s . ge n e r a t in g o u t lie r s . th e e n t r o p y d i®e r e n c e g a p is d e t e r m in e d a n d t h e n c o m p a r e d wit h t h e m a x e n t r o p y g a p fo r e a c h va lu e fr o m t h e t o p o f t h e t a b le d o wn wa r d s . if t h e e n t r o p y d i®e r e n c e g a p is le s s t h a n o r e qu a l t o t h e m a x e n t r o p y g a p t h e n t h e a lg o r it h m c o n t in u e s c a r r yin g o u t c o m p a r is o n s d o wn t h e t a b le , o t h e r wis e t h e a lg o r it h m t e r m in a t e s a n d a ll t h e r e c o r d s u p t o t h a t p o in t a r e a d d e d t o os , wh ic h is t h e n d is p la ye d a s t h e o u t p u t . th e d is a d va n t a g e o f t h is a p p r o a c h is t h a t it is n o t p o s s ib le t o d e ¯ n e a s t a n d a r d m a x e n t r o p y g a p wh ic h c o u ld b e a p p lie d t o e ve r y d a t a s e t . in [8 ] a n a u t o m a t e d a lg o r it h m is p r o p o s e d b y c o m b in in g t h e t wo a b o ve a lg o r it h m s , wh ic h r e qu ir e s o n ly t h e d a t a s e t a s in p u t a n d d o e s n o t r e qu ir e e it h e r t h e n u m b e r o f o u t lie r s o r t h e m a x e n t r o p y g a p a s u s e r in p u t . a e v f a lg o r it h m is : fin d in g t h e o p t im a l m a x e n t r o p y g a p . th e c h a n g e in e n t r o p y va lu e fo r e a c h r e c o r d in t h e d a t a s e t is g e n e r a t e d a n d s t o r e d in a t a b le . th e a ve r a g e e n t r o p y c h a n g e is d e t e r m in e d , wh ic h is s e t a s t h e m a x e n t r o p y g a p . on c e a ll r e c o r d s h a ve b e e n p r o c e s s e d , t h e t a b le is u p d a t e d t o s h o w t h e r e c o r d wit h t h e m a xim u m e n t r o p y c h a n g e a t t h e t o p o f t h e t a b le , fo llo we d b y d e s c e n d in g va lu e s o f e n t r o p y c h a n g e . fin d in g t h e o p t im a l n u m b e r o f o u t lie r s . th e e n t r o p y d i®e r e n c e g a p is d e t e r m in e d a n d t h e n c o m p a r e d wit h t h e m a x e n t r o p y g a p . if t h e e n t r o p y d i®e r e n c e g a p is le s s t h a n o r e qu a l t o t h e m a x e n t r o p y g a p t h e n t h e a lg o r it h m c o n t in u e s wo r kin g d o wn t h e t a b le c a r r yin g o u t t h e c o m p a r is o n . ot h e r wis e t h e a lg o r it h m t e r m in a t e s a n d a ll t h e r e c o r d s u p t o t h a t p o in t a r e a d d e d t o os , wh ic h is t h e n d is p la ye d a s t h e o u t p u t . a n o ve l, p a r a m e t e r -fr e e m e t h o d , comprex, fo r id e n t ifyin g a n o m a lie s u s in g p a t t e r n b a s e d compression is in t r o d u c e d in [9 ]. co m p r e s s io n -b a s e d t e c h n iqu e s h a ve b e e n e xp lo r e d m o s t ly in c o m m u n ic a t io n s t h e o r y, fo r r e d u c e d t r a n s m is s io n c o s t a n d in c r e a s e d t h r o u g h p u t m. haroutunian and t. badasyan 2 5 a n d in d a t a b a s e s , fo r r e d u c e d s t o r a g e c o s t a n d in c r e a s e d qu e r y p e r fo r m a n c e . in t h e m e n t io n e d p a p e r t h e p r o p o s e d m e t h o d ¯ n d s a c o lle c t io n o f d ic t io n a r ie s t h a t d e s c r ib e t h e n o r m o f a d a t a b a s e s u c c in c t ly, a n d s u b s e qu e n t ly ° a g s t h o s e p o in t s d is s im ila r t o t h e n o r m wit h a h ig h c o m p r e s s io n c o s t a s a n o m a lie s . th is a p p r o a c h e xh ib it s fo u r ke y fe a t u r e s : 1 ) it is p a r a m e t e r fr e e ; it b u ild s d ic t io n a r ie s d ir e c t ly fr o m d a t a , a n d r e qu ir e s n o u s e r s p e c i¯ e d p a r a m e t e r s s u c h a s d is t a n c e fu n c t io n s o r d e n s it y a n d s im ila r it y t h r e s h o ld s , 2 ) it is g e n e r a l; it wo r ks fo r a b r o a d r a n g e o f c o m p le x d a t a b a s e s , in c lu d in g g r a p h , im a g e a n d r e la t io n a l d a t a b a s e s t h a t m a y c o n t a in b o t h c a t e g o r ic a l a n d n u m e r ic a l fe a t u r e s , 3 ) it is s c a la b le ; it s r u n n in g t im e g r o ws lin e a r ly wit h r e s p e c t t o b o t h d a t a b a s e s iz e a s we ll a s n u m b e r o f d im e n s io n s , a n d 4 ) it is e ®e c t ive ; e xp e r im e n t s o n a b r o a d r a n g e o f d a t a s e t s s h o w la r g e im p r o ve m e n t s in b o t h c o m p r e s s io n , a s we ll a s p r e c is io n in a n o m a ly d e t e c t io n , o u t p e r fo r m in g it s s t a t e -o f-t h e a r t c o m p e t it o r s . te c h n iqu e s fo c u s e d o n a n o m a ly d e t e c t io n in graph-based data h a ve r e c e n t ly b e e n t h e s u b je c t o f a t t e n t io n . a g e n e r a l, c o m p r e h e n s ive s u r ve y o f s t a t e -o f-t h e -a r t m e t h o d s fo r a n o m a ly d e t e c t io n in d a t a r e p r e s e n t e d a s g r a p h s is p r o vid e d in [1 0 ]. h e r e we fo c u s o n ly o n in fo r m a t io n -t h e o r e t ic a p p r o a c h e s . in [1 1 ] t h e a u t h o r s in t r o d u c e d t wo t e c h n iqu e s fo r g r a p h -b a s e d a n o m a ly d e t e c t io n . th e ¯ r s t , anomalous substructure detection, lo o ks fo r s p e c i¯ c , u n u s u a l s u b s t r u c t u r e s wit h in a g r a p h . in t h e s e c o n d m e t h o d , anomalous subgraph detection, t h e g r a p h is p a r t it io n e d in t o d is t in c t s e t s o f ve r t ic e s ( s u b g r a p h s ) , e a c h o f wh ic h is t e s t e d a g a in s t t h e o t h e r s fo r u n u s u a l p a t t e r n s . u s in g t h e c o n c e p t o f c o n d it io n a l e n t r o p y a m e a s u r e o f g r a p h r e g u la r it y c a lle d conditional substructure entropy h a s b e e n in t r o d u c e d t o d e ¯ n e t h e n u m b e r o f b it s n e e d e d t o d e s c r ib e a n a r b it r a r y s u b s t r u c t u r e s s u r r o u n d in g s . a s u b s t r u c t u r e is a c o n n e c t e d s u b g r a p h o f t h e o ve r a ll g r a p h . b y s u r r o u n d in g s , a u t h o r s a r e r e fe r r in g t o t h e e d g e s a n d ve r t ic e s a d ja c e n t t o t h e s u b s t r u c t u r e . th e s u r r o u n d in g s c a n b e t h o u g h t o f a s a s e t o f e xt e n s io n s t o t h e s u b s t r u c t u r e ; a n e xt e n s io n o f a s u b s t r u c t u r e is d e ¯ n e d t o b e t h e a d d it io n o f e it h e r a s in g le ve r t e x ( a lo n g wit h t h e e d g e c o n n e c t in g it t o t h e s u b s t r u c t u r e ) , o r a s in g le e d g e wit h in t h e s u b s t r u c t u r e . l e t y b e d e ¯ n e d t o c o n t a in a ll n-ve r t e x s u b s t r u c t u r e s wit h in t h e g r a p h , a n d x c o n t a in a ll e xt e n s io n s o f t h e s u b s t r u c t u r e s in y . fo r a g ive n s u b s t r u c t u r e y 2 y; p ( y ) is d e ¯ n e d a s t h e n u m b e r o f in s t a n c e s o f y in g, d ivid e d b y t h e t o t a l n u m b e r o f in s t a n c e s o f a ll n-ve r t e x s u b s t r u c t u r e s . fo r p a r t ic u la r s u b s t r u c t u r e s x 2 x; y 2 y , p ( xjy ) is d e ¯ n e d t o r e p r e s e n t t h e p e r c e n t a g e o f in s t a n c e s o f y t h a t e xt e n d t o a n in s t a n c e o f x. b y a n a lo g y o f t h e c o n d it io n a l e n t r o p y fo r s t r in g d a t a , t h e conditional substructure entropy is d e ¯ n e d a s h ( xjy ) = ¡ x x;y p( y ) [p ( xjy ) lo g p ( xjy ) + ( 1 ¡ p ( xjy ) ) lo g ( 1 ¡ p ( xjy ) ) ]: e n t r o p y a n d k l d ive r g e n c e m e t h o d s h a ve b e e n r e g a r d e d a s e ®e c t ive m e t h o d s fo r d e t e c t in g abnormal tra± c b a s e d o n ip a d d r e s s -d is t r ib u t io n s t a t is t ic s o r p a c ke t s iz e -d is t r ib u t io n s t a t is t ic s [1 2 ] [1 4 ]. in [1 5 ] t wo n e w a n d e ®e c t ive a n o m a ly-b a s e d d e t e c t io n m e t r ic s ( g e n e r iliz e d e n t r o p y a n d in fo r m a t io n d is t a n c e ) we r e p r o p o s e d , wh ic h id e n t ify d d o s a t t a c ks e a r ly a n d a c c u r a t e ly. a n e ®e c t ive ip t r a c e b a c k s c h e m e wa s p r o p o s e d b a s e d o n a n in fo r m a t io n d is t a n c e m e t r ic t h a t c a n t r a c e a ll a t t a c ks b a c k t o t h e ir o wn lo c a l a r e a n e t wo r ks ( l a n s ) in a s h o r t t im e . th e renyi or generalized entropy o f o r d e r ® is d e ¯ n e d a s fo llo ws : h® ( x) = 1 1 ¡ ® lo g 2 ( x i p®i ) ; ® ¸ 0 ; ® 6= 1 : 2 6 information theoretic methods for anomaly detection th e renyi divergence b e t we e n p r o b a b ilit y d is t r ib u t io n s p a n d q is : d® ( p jjq ) = 1 ® ¡ 1 lo g 2 ( x i p®i q 1¡® i ) ; ® ¸ 0 : th e r e n yi d ive r g e n c e , a s t h e g e n e r a liz e d d ive r g e n c e , c a n d e d u c e m a n y c o n c r e t e d ive r g e n c e fo r m s a c c o r d in g t o d i®e r e n t va lu e s o f o r d e r ®. th e information distance is t h e s ym m e t r ic iz e d m e t r ic d® ( p; q) = d® ( p jjq ) + d® ( qjjp ) : it is s ig n ī c a n t t h a t t h is m e t r ic s c a n im p r o ve t h e s ys t e m s ' d e t e c t io n s e n s it ivit y b y a d ju s t in g t h e va lu e o f o r d e r ® o f t h e g e n e r a liz e d e n t r o p y a n d in fo r m a t io n d is t a n c e m e t r ic s . a s t h e p r o p o s e d m e t r ic s c a n in c r e a s e t h e in fo r m a t io n d is t a n c e b e t we e n t h e a t t a c k t r a ± c a n d t h e le g it im a t e t r a ± c , t h e y c a n e ®e c t ive ly d e t e c t lo w-r a t e d d o s a t t a c ks e a r ly a n d r e d u c e t h e fa ls e p o s it ive r a t e c le a r ly. a s a r e s u lt [1 5 ] t h e p r o p o s e d in fo r m a t io n m e t r ic s im p r o ve t h e p e r fo r m a n c e o f lo w-r a t e d d o s a t t a c ks d e t e c t io n a n d ip t r a c e b a c k o ve r t h e t r a d it io n a l a p p r o a c h e s . a s in [9 ], u n ive r s a l s o u r c e c o d in g h a s b e e n u s e d fo r a n o m a ly d e t e c t io n in va r io u s p a p e r s m o s t ly b a s e d o n c o m p a r in g c o d e le n g t h , s e e fo r e xa m p le , [1 6 ] [2 0 ]. th e c o m p a r is o n is d o n e b y a m e a s u r e o f e n t r o p y r a t e . th e p r o b le m is t h a t t h e r e a r e m a n y d is s im ila r s o u r c e s t h a t h a ve t h e s a m e e n t r o p y r a t e . to o ve r c o m e t h is is s u e a n e w a p p r o a c h b a s e d o n t h e n o t io n o f atypicality is s u g g e s t e d in [2 1 ]. mo s t o f t h e va lu e in t h e in fo r m a t io n in s o m e a p p lic a t io n s is in t h e p a r t s t h a t d e via t e fr o m t h e a ve r a g e , t h a t a r e u n u s u a l, a t yp ic a l. a t yp ic a lit y is d e ¯ n e d a s d a t a t h a t c a n b e e n c o d e d wit h fe we r b it s in it s e lf r a t h e r t h a n u s in g t h e c o d e fo r t h e t yp ic a l d a t a . in [2 1 ] t h e a u t h o r s s h o w t h a t t h is d e ¯ n it io n h a s g o o d t h e o r e t ic a l p r o p e r t ie s a n d d e ve lo p a n im p le m e n t a t io n b a s e d o n u n ive r s a l s o u r c e c o d in g . th is id e a o f ¯ n d in g a lt e r n a t ive e xp la n a t io n s fo r d a t a r a t h e r t h a n m e a s u r in g s o m e kin d o f d i®e r e n c e fr o m t yp ic a l d a t a is wh a t s e p a r a t e s t h is m e t h o d fr o m u s u a l a p p r o a c h e s in o u t lie r d e t e c t io n a n d a n o m a ly d e t e c t io n . a t yp ic a lit y is p u r e ly a p r o p e r t y o f d a t a a n d t h e r e fo r e t h e r e a r e n o m is s e s o r fa ls e a la r m s : d a t a is a t yp ic a l o r n o t . 4 . co n c lu s io n th e p r e s e n t wo r k s t u d ie d m a in in fo r m a t io n t h e o r e t ic a p p r o a c h e s fo r a n o m a ly d e t e c t io n a p p lic a t io n s . in fo r m a t io n t h e o r y p r o vid e s a u n ive r s a l a p p r o a c h in s t e a d o f lo o kin g a t s p e c i¯ c s t a t is t ic s o f d a t a . th e a d va n t a g e s o f in fo 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[1 7 ] e . e . e ila n d a n d l . m. l ie b r o c k, \ a n a p p lic a t io n o f in fo r m a t io n t h e o r y t o in t r u s io n d e t e c t io n , p roc. f ourth ie e e int. w orkshop inf. assurance, 2 0 0 6 . [1 8 ] c.-k . h a n a n d h .-k . ch o i, \ e ®e c t ive d is c o ve r y o f a t t a c ks u s in g e n t r o p y o f p a c ke t d yn a m ic s , ie e e netw., vo l. 2 3 , n o . 5 , p p . 4 1 2 , 2 0 0 9 . [1 9 ] n . w a n g , j. h a n a n d j.fa n g , \ a n a n o m a ly d e t e c t io n a lg o r it h m b a s e d o n lo s s le s s c o m p r e s s io n , p roc. ie e e 7th int. conf. netw. archit. storage, p p . 3 1 3 8 , 2 0 1 2 . [2 0 ] h . s h a h r ia r a n d m. zu lke r n in e , \ in fo r m a t io n -t h e o r e t ic d e t e c t io n o f s ql in je c t io n a t t a c ks , p roc. ie e e 14th int. symp. high-assurance syst. e ng., p p . 4 0 4 7 , 2 0 1 2 . 2 8 information theoretic methods for anomaly detection [2 1 ] a . h o s t -ma d s e n , e . s a b e t i a n d c. w a lt o n , \ d a t a d is c o ve r y a n d a n o m a ly d e t e c t io n u s in g a t yp ic a lit y: t h e o r y" , ie e e trans. on inform. theory, vo l. 6 5 , n o . 9 , p p . 5 3 0 2 5 3 2 2 , 2 0 1 9 . submitted 10.06.2019, accepted 18.10.2019. ²ýáù³éç³ý»ñç ñ³ûïý³µ»ñù³ý çýýáñù³óçáý ï»ë³ï³ý ù»ãá¹ý»ñ ø³ñç³ù º. ð³ñáõãûáõýû³ý ¨ îç·ñ³ý ê. ´³¹³ëû³ý ðð ¶²² æýýáñù³ïçï³ûç ¨ ³íïáù³ï³óù³ý åñáµé»ùý»ñç çýëïçïáõï e-mail: armar@sci.am, tigranbadasyan@gmail.com ²ù÷á÷áõù ²ýáù³éç³ý»ñç ñ³ûïý³µ»ñáõùá ýß³ý³ïáõù ¿ ñ³½í³·ûáõï ïíû³éý»ñç ýáõûý³ï³ý³óáõù, áñáýù ýáñù³é ã»ý ï³ù ß»õíáõù »ý ñ³ù³ï³ñ·áõù ýáñù³é å³ñí³íùçó: ²ýáù³éç³ûç ñ³ûïý³µ»ñáõùá áõýç ï³ñµ»ñ ïçñ³éáõãûáõýý»ñ ï»ýë³çýýáñù³ïçï³ûç, å³ïï»ñý»ñç ùß³ïù³ý, ïçµ»ñ³ýíï³ý·áõãû³ý, ïíû³éý»ñç µ³½³ûç ³ýíï³ý·áõãû³ý ³å³ñáíù³ý ¨ ³ûé áéáñïý»ñáõù: î³ý µ³½ù³ãçí ù»ãá¹ý»ñç ëùµ»ñ, áñáýù û·ï³·áñííáõù »ý ³ýáù³éç³ý»ñç ñ³ûïý³µ»ñù³ý ñ³ù³ñ, ý»ñ³éû³é íç×³ï³·ñ³ï³ý ù»ãá¹ý»ñá, ý»ûñáý³ûçý ó³ýóç ù»ãá¹ý»ñá ¨ çýýáñù³óç³ûç ï»ëáõãû³ý ù»ãá¹ý»ñá: ²ûë ñá¹í³íáõù ùýý³ñïíáõù »ý ñçùý³ï³ý çýýáñù³óçáý-ï»ë³ï³ý ùáï»óáõùý»ñý ³ýáù³éç³ý»ñç ñ³ûïý³µ»ñù³ý ïçñ³éáõãûáõýý»ñç ñ³ù³ñ: æýýáñù³óç³ûç ï»ëáõãûáõýá ï³éçë ¿ ñ³ùáý¹ñ³ýáõñ ùáï»óáõù` ïíû³éý»ñç íç×³ï³·ñáõãûáõýá í»ñéáõí»éáõ ÷áë³ñ»ý: ´³ý³éç µ³é»ñ` ³ýáù³éç³ûç ñ³ûïý³µ»ñáõù, ¿ýïñáåç³, ñ³ñ³µ»ñ³ï³ý ¿ýïñáåç³, ï»õ³ûçý áñáýù³ý ¿íñçëïçï ³é·áñçãù lsa: èíôîðìàöèîííî-òåîðåòè÷åñêèå ìåòîäû äëÿ îáíàðóæåíèÿ àíîìàëèé ìàðèàì å. àðóòþíÿí è òèãðàí ñ. áàäàñÿ èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà e-mail: armar@sci.am, tigranbadasyan@gmail.com àííîòàöèÿ îáíàðóæåíèå àíîìàëèé îçíà÷àåò îïîçíàâàíèå ðåäêèõ äàííûõ, êîòîðûå íå ÿâëÿþòñÿ íîðìàëüíûìè èëè îòêëîíÿþòñÿ îò íîðìàëüíîãî ïîâåäåíèÿ â ñèñòåìå. îáíàðóæåíèå àíîìàëèé èìååò ðàçëè÷íûå ïðèìåíåíèÿ â áèîèíôîðìàòèêå, m. haroutunian and t. badasyan 2 9 îáðàáîòêå èçîáðàæåíèé, êèáåðáåçîïàñíîñòè, áåçîïàñíîñòè äëÿ áàç äàííûõ è ò. ä. ñóùåñòâóåò ìíîãî ãðóïï ìåòîäîâ, êîòîðûå èñïîëüçóþòñÿ äëÿ îáíàðóæåíèÿ àíîìàëèé, âêëþ÷àÿ ñòàòèñòè÷åñêèå ìåòîäû, ìåòîäû íåéðîííûõ ñåòåé è òåîðåòèêî-èíôîðìàöèîííûå ìåòîäû. â íàñòîÿùåé ðàáîòå ðàññìàòðèâàþòñÿ îñíîâíûå òåîðåòèêî-èíôîðìàöèîííûå ïîäõîäû äëÿ ïðèëîæåíèé îáíàðóæåíèÿ àíîìàëèé. òåîðèÿ èíôîðìàöèè ïðåäîñòàâëÿåò óíèâåðñàëüíûé ïîäõîä âìåñòî òîãî, ÷òîáû ñìîòðåòü íà êîíêðåòíûå ñòàòèñòè÷åñêèå äàííûå. êëþ÷åâûå ñëîâà: îáíàðóæåíèå àíîìàëèé, ýíòðîïèÿ, îòíîñèòåëüíàÿ ýíòðîïèÿ, ýâðèñòè÷åñêèé àëãîðèòì ëîêàëüíîãî ïîèñêà lsa. d:\sbornik\...\tpel.dvi mathematical problems of computer science 35, 5{13, 2011. development and application of i nter active algor ithms for n atur al language to un l and un l to n atur al language t r ansfor mations a r a m a . a ve t is ya n institute for informatics and automation problems of nas of ra e-mail: a.avetisyan@undlfoundation.org abstract the present article describes certain developments worked out for the creation of nl-unl transformation framework. in particular, new interactive algorithms aimed at analyzing and converting sentences of natural languages (nls) into unl and converting unl structures back into texts of natural languages, are described. in this article, we describe the research on the work done for the creation of self-learning conversion system, based on the analysis of the interactive user-de¯ned intervention. we also analyze the existing resources and describe certain algorithmic and technical solutions. refer ences [1 ] h . u c h id a , m. zh u , \ th e u n ive r s a l n e t wo r kin g l a n g u a g e ( u n l ) s p e c ī c a t io n s " , ve r s io n 7 , u n d l fo u n d a t io n , ju n e , 2 0 0 5 . [2 ] a . a ve t is ya n , \ s o m e a p p r o a c h e s t o t h e g e n e r a t io n o f s e n t e n c e s in n a t u r a l la n g u a g e fr o m u n l " , p roceedings of the conference csit, p . 2 6 8 ,y e r e va n , 2 0 0 9 . [3 ] \ gr a m m a r s p e c i¯ c a t io n s " , und l f oundation online resources, www.u n lwe b .n e t , 2 0 1 1 . [4 ] \ d ic t io n a r y s p e c ī c a t io n s " , und l f oundation online resources, www.u n lwe b .n e t , 2 0 1 1 . [5 ] i. za s la ws kiy, a . a ve t is ya n , v . ge vo r g ya n , \ im p le m e n t a t io n o f d ic t io n a r y l o o ku p a u t o m a t a fo r u n l a n a lys is a n d ge n e r a t io n " , international j ournal of information theories and applications, vo l. 1 7 , n . 4 , s o ¯ a , b u lg a r ia , 2 0 1 0 . 5 6 development and application of interactive algorithms for nl to unl and unl to nl transformations ´ý³ï³ý é»½íáí ý³ë³¹³ëáõãûáõýý»ñç ñ³ùáý¹ñ³ýáõñ ó³ýó³ûçý é»½íç ¨ ñ³ï³é³ïá ÷áñ³ï»ñåù³ý ñ³ù³ñ »ñïëáë³ï³ý ³é·áñçãùç ùß³ïáõùá ¨ ïçñ³éáõùá ². ²í»ïçëû³ý ²ù÷á÷áõù ø»ñ ³ñç¨ ¹ñí³í ¿ñ ëý¹çñª ùß³ï»é ýáñ íñ³·ñ³ûçý ñ³ù³ï³ñ· nl-unl »ñïïáõù³ýç ÷áë³ï»ñåáõùý»ñç ñ³ù³ñ, áñá ïï³ñáõõ³ý³ éáõí»é ý³ëáñ¹ íñ³·ñ³ûçý ùççáóý»ñç ïçñ³éù³ý ñ»ï ï³åí³í ëý¹çñý»ñá ¨ ï³é³ç³ñïç ýáñ ³é·áñçãù³ï³ý ¨ ï»ëýçï³ï³ý éáõíáõùý»ñ: üáñ ùß³ïí³í lily ñ³ù³ï³ñ·á éáõíáõù ¿ µáéáñ í»ñáñçßû³é ëý¹çñý»ñá, ³é³ç³ñï»éáí ï³ýáýý»ñç ý»ñï³û³óù³ý ýáñ áõ ×ïáõý³óí³í ï»ëù, µ³é³ñ³ý³ûçý ÷ýïñù³ý ¨ ï³ýáýý»ñç ñ³ù³å³ï³ëë³ýáõãû³ý ñ³ù³ñ ùß³ïí³í ³é·áñç÷ù»ñ: öáë³ï»ñåù³ý áýã³óùá ³ûåù ï³ñáõ ¿ éçý»é ³ùµáõçáíçý ³íïáù³ï³óí³í ï³ù ïçë³³íïáù³ï³óí³í (çýï»ñ³ïïçí), ãáõûé ï³éáí û·ï³·áñíáõçý áã ùçû³ûý ñ»ï¨»é ÷áë³ï»ñåù³ý áýã³óùçý, ³ûé ý³¨ ñ»ßïáõãû³ùµ ï³é³í³ñ»é ³ûý: ð³ù³ï³ñ·ç »ñïëáë³ï³ý éçý»éá, áý¹ó»éáõù ¿ ýáñ ñý³ñ³íáñáõãûáõý ùß³ï»éáõ ù»ë³ýç½ù, áñá ãáõûé ïï³ ñ³ù³ï³ñ·çý ëáíáñ»é ¨ µ³ñóñ³óý»é ÷áë³ï»ñåù³ý ³ñ¹ûáõýùý»ñç áñ³ïá: ²ûë ñý³ñ³íáñáõãûáõýá ï³ñ¨áñ ù³ûé ¿ ù»ñ³ï³ý³ï³ý ï³ýáýý»ñç ½³ñ·³óù³ý ñ³ù³ñ: d:\sbornik\...\tpel.dvi mathematical problems of computer science 35, 70{76, 2011. t he shannon cipher system with a guessing wir etapper e avesdr oping t hr ough a n oisy channel e vg u e n i a . h a r o u t u n ia n a n d tig r a n m. ma r g a r ya n institute for informatics and automation problems of nas of ra e-mail: eghishe@sci.am abstract in this paper we study the processes in the shannon cipher system with discrete memoryless source and a guessing wiretapper. the wiretapper observes a cryptogram of m -vector ciphered messages passed through the noisy channel and tries to guess the secret plaintext with length n. the security of encryption system is measured by the average number of guesses needed for the wiretapper to uncover the plaintext. the problem was suggested by arikan and merhav as a generalization of their result for noiseless channel to wairtapper. refer ences [1 ] e . a r ika n , \ a n in e qu a lit y o n g u e s s in g a n d it s a p p lic a t io n t o s e qu e n t ia l d e c o d in g " , ie e e trans. inform. theory, vo l. 4 2 , n o . 1 , p p . 9 9 -1 0 5 , 1 9 9 6 . [2 ] e . a r ika n a \ on t h e a ve r a g e n u b e r o f gu e s s e s r e qu ir e d t o d e t e r m in e t h e v a lu e o f a r a n d o m va r ia b le " , transactions of the 12th p rague conference on information theory, statistical d ecision f unction and r andom p rocesses, p p . 2 0 -2 3 , 1 9 9 4 . [3 ] e . a r ika n a n d n . me r h a v, \ gu e s s in g s u b je c t t o d is t o r t io n " , ie e e trans. inform. theory, vo l. 4 4 , n o . 3 , p p . 1 0 4 1 -1 0 5 6 , 1 9 9 8 . [4 ] e . a r ika n a n d n . me r h a v,\ jo in t s o u r c e -c h a n n e l c o d in g a n d g u e s s in g wit h a p p lic a t io n t o s e qu e n t ia l d e c o d in g " , ie e e trans. inform. theory, vo l. 4 4 , n o . 5 , p p . 1 7 5 6 -1 7 6 9 , 1 9 9 8 . [5 ] i. cs is z ¶a r a n d j. k äo r n e r , information theory: coding theorems for d iscrete m emoryless systems, n e w y o r k: a c a d e m ic , 1 9 8 1 . [6 ] t. m. co ve r a n d j. a . th o m a s , e lements of information theory, n e w y o r k: w ile y, 2 0 0 6 . [7 ] e . a . h a r o u t u n ia n a n d a . r . gh a z a r ya n , \ on c ip h e r s ys t e m wit h a wir e t a p p e r g u e s s in g wit h r e s p e c t t o ¯ d e lit y a n d r e lia b ilit y c r it e r ia " , p roceedings of the third conference on computer science and information technologies ( y e r e va n , a r m e n ia , 2 0 0 1 ) , p p . 2 1 5 -2 1 9 . [8 ] e . a . h a r o u t u n ia n a n d a . r . gh a z a r ya n , \ on t h e s h a n n o n c ip h e r s ys t e m wit h a wir e t a p p e r g u e s s in g s u b je c t t o d is t o r t io n a n d r e lia b ilit y r e qu ir e m e n t s " , p roceedings of the 2002 ie e e int. symp. inform. theory ( l a u s a n n a , s wit z e r la n d ) , p . 3 2 4 . 7 0 e. haroutunian, t. margaryan 7 1 [9 ] e . a . h a r o u t u n ia n , \ r e a lib ilit y a p p r o a c h in wir e t a p p e r g u e s s in g t h e o r y" , in \ a s p e c t s o f n e t wo r k a n d in fo r m a t io n s e c u r it y" , n a to s c ie n c e fo r p e a c e a n d s e c u r it y, s e r ie s d : in fo r m a t io n a n d co m m u n ic a t io n s e c u r it y, vo l. 1 7 , p p . 2 4 8 { 2 6 0 , ios p r e s s , 2 0 0 8 . [1 0 ] y . h a ya s h i a n d h . y a m a m o t o , \ co d in g t h e o r e m s fo r t h e s h a n n o n c ip h e r wit h a g u e s s in g wir e t a p p e r a n d c o r r e la t e d s o u r c e o u t p u t s " , ie e e trans. inform. theory, vo l. 5 4 , n o . 6 , p p . 2 8 0 8 -2 8 1 7 , ju n e 2 0 0 8 . [1 1 ] m. e . h e llm a n , \ a n e xt e n t io n o f t h e s h a n n o n t h e o r y a p p r o a c h t o c r yp t o g r a p h y" , ie e e trans. on inform. theory, vo l. 2 3 , n o . 3 , p p . 2 8 9 -2 9 9 , 1 9 7 7 . [1 2 ] d . ma lo n e a n d w . g. s u lliva n , \ gu e s s wo r k a n d e n t r o p y" , ie e e trans. inform. theory, vo l. 5 0 , n o . 3 , p p . 5 2 5 -5 2 6 , 2 0 0 4 . [1 3 ] j. l . ma s s e y, " gu e s s in g a n d e n t r o p y" , p roceedings of the 1994 ie e e international symp. inform. theory ( tr o n d h e im , n o r wa y, 1 9 9 4 ) , p . 2 0 4 . [1 4 ] n . me r h a v a n d e . a r ika n , " th e s h a n n o n c ip h e r s ys t e m wit h a g u e s s in g wir e t a p p e r " , ie e e trans. inform. theory, vo l. 4 5 , n o . 6 , p p . 1 8 6 0 -1 8 6 6 , 1 9 9 9 . [1 5 ] a . s g a r r o , " e r r o r p r o b a b ilit ie s fo r s im p le s u b s t it u t io n cip h e r s " , ie e e trans. inform. theory, vo l. 2 9 , n o . 2 , p p . 1 9 0 -1 9 7 , 1 9 8 3 . [1 6 ] a . s g a r r o , \ e xp o n e n t ia l-t yp e p a r a m e t e r s a n d s u b s t it u t io n cip h e r s " , p roblems of control and inform. theory, vo l. 1 4 ( 5 ) , p p . 3 9 3 -4 0 3 , 1 9 8 5 . þ»ýáýû³ý í³íï³·ñù³ý ñ³ù³ï³ñ·áõù ·³õïý³·áõç ·áõß³ïáõùá ³õùïáï ï³åáõõáí º. ð³ñáõãûáõýû³ý ¨ î. ø³ñ·³ñû³ý ²ù÷á÷áõù ðá¹í³íáõù éáõíí³í ¿ ø»ññ³íç ¨ ²ñçï³ýç ïáõùçó ³é³ç³¹ñí³í ëý¹çñá: ¸çï³ñïí³í ¿ ñ³ï¨û³é ñ³ù³ï³ñ·á. í³ýçãá µ³ý³éáõ û·ýáõãû³ùµ í³íï³·ñáõù ¿ ñ³õáñ¹³·ñáõãûáõýá ¨ ³ý³õùáõï ï³åáõõáí áõõ³ñïáõù ¿ ûñçý³ï³ý ñ³ë»³ïçñáçá, áñçý áõõ³ñïáõù ¿ ý³¨ µ³ý³éçý ù»ï ³ûé ³ý³õùáõï, å³ßïå³ýí³í ï³åáõõáí: ¶³õïý³·áõá, û·ïí»éáí ³õùïáï ï³åáõõáõó, ëï³ý³éáí í³íï³·çñá, ó·ïáõù ¿ í»ñí³ý»é ñ³õáñ¹³·ñáõãûáõýá ñ³çáñ¹³ï³ý ·áõß³ïáõùý»ñç ùççáóáí: ðá¹í³íáõù ëï³óí³í ¿ ïé³ñù³ý ³ñ³·áõãû³ý ·ýñ³ï³ï³ýý»ñá: mathematical problems of computer science 46, 18{25, 2016. spanning t r ees with few b r anch and end ver tices institute for informatics and automation problems of nas ra e-mail: zhora@ipia.sci.am abstract for a graph g, let ¾2 be the minimum degree sum of two nonadjacent vertices in g. a vertex of degree one in a tree is called an end vertex and a vertex of degree at least three is called a branch vertex. we consider: (¤) connected graphs on n vertices such that ¾2 ¸ n ¡ k + 1 for some positive integer k. in 1976, it was proved (by the author) that every graph satisfying (¤), has a spanning tree with at most k end vertices. in this paper we ¯rst show that every graph satisfying (¤), has a spanning tree with at most k + 1 branch and end vertices altogether. the next result states that every graph satisfying (¤), has a spanning tree with at most 1 2 (k ¡ 1) branch vertices. the third result states that every graph satisfying (¤), has a spanning tree with at most 3 2 (k ¡ 1) degree sum of branch vertices. all results are sharp. keywords: spanning tree, end vertex, k-ended tree, branch vertex, degree sum of the branch vertices, ore-type condition. 1 . in t r o d u c t io n th r o u g h o u t t h is a r t ic le we c o n s id e r o n ly ¯ n it e u n d ir e c t e d g r a p h s wit h o u t lo o p s o r m u lt ip le e d g e s . th e s e t o f ve r t ic e s o f a g r a p h g is d e n o t e d b y v ( g ) a n d t h e s e t o f e d g e s b y e ( g) . a g o o d r e fe r e n c e fo r a n y u n d e ¯ n e d t e r m s is [1 ]. fo r a g r a p h g, we u s e n a n d ® t o d e n o t e t h e o r d e r ( t h e n u m b e r o f ve r t ic e s ) a n d t h e in d e p e n d e n c e n u m b e r o f g, r e s p e c t ive ly. if ® ¸ k fo r s o m e in t e g e r k, le t ¾k b e t h e m in im u m d e g r e e s u m o f a n in d e p e n d e n t s e t o f k ve r t ic e s ; o t h e r wis e we le t ¾k = +1. fo r a s u b s e t s µ v ( g ) , we d e n o t e b y g[s] t h e s u b g r a p h o f g in d u c e d b y s. w e u s e dg ( v ) t o d e n o t e t h e n u m b e r o f n e ig h b o r s o f a ve r t e x v in g, c a lle d t h e d e g r e e o f v in g. th e m in im u m d e g r e e in g is d e n o t e d b y ±. if q is a p a t h o r a c yc le in a g r a p h g, t h e n t h e o r d e r o f q, d e n o t e d b y jqj, is jv ( q) j. th e g r a p h g is h a m ilt o n ia n if g c o n t a in s a h a m ilt o n c yc le , i.e ., a c yc le c o n t a in in g e ve r y ve r t e x o f g. w e wr it e a p a t h q wit h a g ive n o r ie n t a t io n b y ¡! q . fo r x; y 2 v ( q ) , we d e n o t e b y x¡!q y t h e s u b p a t h o f q in t h e c h o s e n d ir e c t io n fr o m x t o y. w e u s e x+ t o d e n o t e t h e s u c c e s s o r , a n d x¡ t h e p r e d e c e s s o r , o f a ve r t e x x 2 v ( q ) . fo r x µ v ( q) , we d e ¯ n e x + = fx+ : x 2 xg a n d x¡ = fx¡ : x 2 xg. ¤g.g. nicoghossian (up to 1997) 1 8 zh o r a g. n iko g h o s ya n ¤ zh. nikoghosyan 1 9 a ve r t e x o f d e g r e e o n e is c a lle d a n e n d -ve r t e x, a n d a n e n d -ve r t e x o f a t r e e is u s u a lly c a lle d a le a f. th e s e t o f e n d -ve r t ic e s o f g is d e n o t e d b y end ( g) . a b r a n c h ve r t e x o f a t r e e is a ve r t e x o f d e g r e e a t le a s t t h r e e . th e s e t o f b r a n c h ve r t ic e s o f a t r e e t will b e d e n o t e d b y b ( t ) . fo r a p o s it ive in t e g e r k, a t r e e t is s a id t o b e a k-e n d e d t r e e if jend( t ) j · k. a h a m ilt o n p a t h is a s p a n n in g 2 -e n d e d t r e e . a h a m ilt o n c yc le c a n b e in t e r p r e t e d a s a s p a n n in g 1 -e n d e d t r e e . in 1 9 5 2 , d ir a c [2 ] p r o ve d t h a t e ve r y g r a p h wit h ± ¸ n=2 h a s a h a m ilt o n c yc le . th e d e g r e e s u m ve r s io n o f t h is r e s u lt wa s p r o ve d in 1 9 6 0 d u e t o or e [3 ]: e ve r y g r a p h wit h ¾2 ¸ n h a s a h a m ilt o n c yc le . th e a n a lo g s o f t h e s e t wo c la s s ic a l r e s u lt s fo r h a m ilt o n p a t h s fo llo w e a s ily. t heor em a: [2 ]. e very graph with ± ¸ ( n ¡ 1 ) =2 has a hamilton path. t heor em b : [3 ]. e very graph with ¾2 ¸ n ¡ 1 has a hamilton path. th e n e xt r e s u lt t o o k a d i®e r e n t a p p r o a c h d u e t o ch v¶a t a l a n d e r d }o s [4 ] b a s e d o n c o n n e c t ivit y a n d in d e p e n d e n c e n u m b e r : e ve r y k-c o n n e c t e d ( k ¸ 1 ) g r a p h wit h ® · k h a s a h a m ilt o n c yc le . th e h a m ilt o n p a t h ve r s io n o f t h is r e s u lt c a n b e fo r m u la t e d a s fo llo ws . t heor em c: [4 ]. e very k-connected ( k ¸ 1 ) graph with ® · k + 1 has a hamilton path. a h a m ilt o n p a t h c a n b e r e g a r d e d a s a s p a n n in g t r e e wit h e xa c t ly t wo le a ve s , a s p a n n in g t r e e wit h n o b r a n c h ve r t e x, o r a s p a n n in g t r e e wit h m a xim u m d e g r e e t wo . th e r e fo r e , a s o n e o f g e n e r a liz e d p r o b le m s o f a h a m ilt o n p a t h p r o b le m , it is n a t u r a l t o lo o k fo r c o n d it io n s wh ic h e n s u r e t h e e xis t e n c e o f a s p a n n in g t r e e wit h fe w le a ve s , fe w b r a n c h ve r t ic e s o r b o u n d e d m a xim u m d e g r e e m o t iva t e d fr o m o p t im iz a t io n a s p e c t s wit h va r io u s a p p lic a t io n s . in t h is p a p e r we c o n s id e r t r e e p r o b le m s a r is in g in t h e c o n t e xt o f o p t ic a l a n d c e n t r a liz e d t e r m in a l n e t wo r ks : ( i) ¯ n d in g a s p a n n in g t r e e o f g wit h t h e m in im u m n u m b e r o f e n d ve r t ic e s , ( ii) ¯ n d in g a s p a n n in g t r e e wit h t h e m in im u m n u m b e r o f b r a n c h ve r t ic e s a n d ( iii) ¯ n d in g a s p a n n in g t r e e o f g s u c h t h a t t h e d e g r e e s u m o f t h e b r a n c h ve r t ic e s is m in im iz e d , m o t iva t e d b y n e t wo r k d e s ig n p r o b le m s wh e r e ju n c t io n s a r e s ig n i¯ c a n t ly m o r e e xp e n s ive t h a n s im p le e n d o r t h r o u g h -n o d e s , a n d a r e , t h u s , t o b e a vo id e d . a ll t h e s e p r o b le m s a r e n p -h a r d b e c a u s e t h e y c o n t a in t h e h a m ilt o n p a t h p r o b le m a s a p a r t ic u la r c a s e . th e c o n s t r a in t o n t h e n u m b e r o f e n d ve r t ic e s a r is e s b e c a u s e t h e s o ft wa r e a n d h a r d wa r e a s s o c ia t e d t o e a c h t e r m in a l d i®e r s a c c o r d in g ly wit h it s p o s it io n in t h e t r e e . u s u a lly, t h e s o ft wa r e a n d h a r d wa r e a s s o c ia t e d t o a \ d e g r e e -l" t e r m in a l is c h e a p e r t h a n t h e s o ft wa r e a n d h a r d wa r e u s e d in t h e r e m a in in g t e r m in a ls b e c a u s e fo r a n y in t e r m e d ia t e t e r m in a l j o n e n e e d s t o c h e c k if t h e a r r iva l m e s s a g e is d e s t in e d t o t h a t n o d e o r t o a n y o t h e r n o d e lo c a t e d a ft e r n o d e j. a s a c o n s e qu e n c e , t h a t p a r t ic u la r t e r m in a l n e e d s s o ft wa r e a n d h a r d wa r e fo r m e s s a g e r o u t in g . on t h e o t h e r h a n d , s u c h e qu ip m e n t is n o t n e e d e d in " d e g r e e -l" t e r m in a ls . a s s u m in g t h a t t h e h a r d wa r e a n d s o ft wa r e fo r m e s s a g e r o u t in g in t h e n o d e s is a lr e a d y a va ila b le , t h e a b o ve d is c u s s io n m o t iva t e s a c o n s t r a in t s t a t in g t h a t a t r e e s o lu t io n h a s t o c o n t a in e xa c t ly a c e r t a in n u m b e r o f \ d e g r e e -l" t e r m in a ls . a d i®e r e n t s it u a t io n , r e s u lt in g fr o m a n e w t e c h n o lo g y a llo win g a s wit c h t o r e p lic a t e t h e s ig n a l b y s p lit t in g lig h t . a lig h t -t r e e c o n n e c t s o n e n o d e t o a s e t o f o t h e r n o d e s in t h e n e t wo r k a llo win g m u lt ic a s t c o m m u n ic a t io n fr o m t h e s o u r c e t o a s e t o f d e s t in a t io n s ( in c lu d in g t h e p o s s ib ilit y o f t h e s e t o f d e s t in a t io n s c o n s is t in g o f a ll o t h e r n o d e s ) . th e s wit c h e s wh ic h c o r r e s p o n d t o t h e n o d e s o f d e g r e e g r e a t e r t h a n t wo h a ve t o b e a b le t o s p lit lig h t ( e xc e p t fo r t h e s o u r c e o f t h e m u lt ic a s t , wh ic h c a n t r a n s m it t o a n y n u m b e r o f n e ig h b o r s ) . typ ic a l o p t ic a l n e t wo r ks will h a ve a lim it e d n u m b e r o f t h e s e m o r e s o p h is t ic a t e d s wit c h e s , a n d o n e h a s t o p o s it io n t h e m in s u c h a wa y t h a t a ll p o s s ib le m u lt ic a s t s c a n b e p e r fo r m e d . th u s , we 2 0 spanning trees with few branch and end vertices a r e le a d t o t h e p r o b le m o f ¯ n d in g s p a n n in g t r e e s wit h a s fe w b r a n c h ve r t ic e s a s p o s s ib le . in 1 9 7 1 , l a s v e r g n a s [5 ] g a ve a d e g r e e c o n d it io n t h a t g u a r a n t e e s t h a t a n y fo r e s t in g o f lim it e d s iz e a n d wit h a lim it e d n u m b e r o f le a ve s c a n b e e xt e n d e d t o a s p a n n in g t r e e o f g wit h a lim it e d n u m b e r o f le a ve s in a n a p p r o p r ia t e s e n s e . th is r e s u lt im p lie s a s a c o r o lla r y a d e g r e e s u m c o n d it io n fo r t h e e xis t e n c e o f a t r e e wit h a t m o s t k le a ve s in c lu d in g th e o r e m b a s a s p e c ia l c a s e fo r k = 1 . t heor em d: [6 ], [5 ], [7 ]. l et g be a connected graph with ¾2 ¸ n ¡ k + 1 for some positive integer k. then g has a spanning k-ended tree. h o we ve r , th e o r e m d wa s ¯ r s t o p e n ly fo r m u la t e d a n d p r o ve d in 1 9 7 6 b y t h e a u t h o r [6 ]. l a t e r , it wa s r e p r o ve d in 1 9 9 8 b y b r o e r s m a a n d tu in s t r a [7 ]. th e n e xt g e n e r a liz a t io n c o n t a in s th e o r e m c a s a s p e c ia l c a s e ( k = 1 ) d u e t o w in [8 ]. t heor em e : [8 ]. l et g be an s-connected graph with ® · s + k ¡ 1 for some integer k ¸ 1 . then g has a spanning k-ended tree. on e o f t h e in t e r e s t in t h e e xis t e n c e o f s p a n n in g t r e e s wit h b o u n d e d n u m b e r o f b r a n c h ve r t ic e s a r is e s in t h e r e a lm o f m u lt ic a s t in g in o p t ic a l n e t wo r ks . ga r g a n o , h a m m a r , h e ll, s t a c h o a n d v a c c a r o [9 ] p r o ve d t h e fo llo win g . t heor em f: [9 ]. e very connected graph with ¾3 ¸ n ¡ 1 has a spanning tree with at most one branch vertex. fla n d r in e t a l. [1 0 ] p o s e d t h e fo llo win g c o n je c t u r e . conjectur e a: [1 0 ]. if g is a connected graph with ¾k+3 ¸ n ¡ k for some positive integer k, then g has a spanning tree with at most k branch vertices. r e c e n t ly, ma t s u d a , oz e ki a n d y a m a s h it a [1 1 ] e s t a b lis h e d a n u p p e r b o u n d fo r t h e in d e p e n d e n c e n u m b e r ® wh ic h im p lie s t h e e xis t e n c e o f a s p a n n in g t r e e wit h b o u n d e d n u m b e r o f b r a n c h ve r t ic e s in c o n n e c t e d c la w-fr e e g r a p h s . t heor em g: [1 1 ]. l et k be a non-negative integer. a connected claw-free graph g has a spanning tree with at most k branch vertices if ® · 2 k + 2 . in t h is p a p e r we p r e s e n t a s h a r p or e -t yp e c o n d it io n fo r t h e e xis t e n c e o f s p a n n in g t r e e s in c o n n e c t e d g r a p h s wit h b o u n d e d t o t a l n u m b e r o f b r a n c h a n d e n d ve r t ic e s im p r o vin g th e o r e m d b y in c o r p o r a t in g t h e n u m b e r o f b r a n c h ve r t ic e s a s a p a r a m e t e r . t heor em 1:. l et g be a connected graph of order n. if ¾2 ¸ n ¡ k + 1 for some positive integer k, then g has a spanning tree t with at most k ¡ jb ( t ) j + 1 end vertices. l e t g b e t h e c o m p le t e b ip a r t it e g r a p h k±;±+k¡1 o f o r d e r n = 2 ± + k ¡ 1 a n d m in im u m d e g r e e ±, wh e r e k ¸ 3 . cle a r ly, ¾2 ( g) = 2 ± = n¡k+1 . b y th e o r e m 1 , g h a s a s p a n n in g t r e e t wit h jend( t ) j · k ¡ b + 1 . ob s e r vin g t h a t t is n o t ( k ¡ 1 ) -e n d e d , t h a t is jend ( t ) j ¸ k, we h a ve b · 1 . on t h e o t h e r h a n d , we h a ve b ¸ 1 , s in c e jend( t ) j ¸ k ¸ 3 , wh ic h im p lie s b = 1 . th is m e a n s t h a t t is n o t ( k ¡ b ) -e n d e d a n d c o n s e qu e n t ly, th e o r e m 1 is s h a r p fo r e a c h k ¸ 3 . th e n e xt r e s u lt fo llo ws fr o m th e o r e m 1 p r o vid in g a s h a r p or e -t yp e c o n d it io n fo r t h e e xis t e n c e o f s p a n n in g t r e e s in c o n n e c t e d g r a p h s wit h fe w b r a n c h ve r t ic e s . t heor em 2: l et g be a connected graph of order n. if ¾2 ¸ n ¡ k + 1 for some positive integer k, then g has a spanning tree with at most ( k ¡ 1 ) =2 branch vertices. th e t h ir d r e s u lt p r o vid e s a n or e -t yp e c o n d it io n fo r t h e e xis t e n c e o f s p a n n in g t r e e s in c o n n e c t e d g r a p h s wit h b o u n d e d d e g r e e s u m o f t h e b r a n c h ve r t ic e s . t heor em 3: l et g be a connected graph of order n. if ¾2 ¸ n ¡ k + 1 for some positive integer k, then g has a spanning tree with at most 3 2 ( k ¡ 1 ) degree sum of the branch vertices. l e t g b e a g r a p h ( t r e e ) o b t a in e d fr o m t h e p a t h v0v1:::vbvb+1 b y a d d in g n e w ve r t ic e s zh. nikoghosyan 2 1 u1; :::; ub a n d t h e e d g e s uivi ( i = 1 ; :::; b) . cle a r ly, n = 2 b + 2 a n d ¾2 = 2 = n ¡ ( 2 b + 1 ) + 1 . s in c e jb ( g ) j = b, t h e b o u n d ( k ¡ 1 ) = 2 in th e o r e m 2 is s h a r p . fu r t h e r , s in c e pbi=1 d ( vi ) = 3 2 ( k ¡ 1 ) , t h e b o u n d 3 2 ( k ¡ 1 ) in th e o r e m 3 is s h a r p a s we ll. th e o r e m s 1 ,2 ,3 we r e a n n o u n c e d in 2 0 1 5 [1 2 ] a n d th e o r e m 1 wa s p r o ve d in d e p e n d e n t ly b y s a it o a n d s a n o [1 3 ]. 2 . p r o o f o f th e o r e m 1 p r oof of t heor em 1: l e t g b e a c o n n e c t e d g r a p h wit h ¾2 ¸ n ¡ k + 1 a n d le t t b e a s p a n n in g t r e e in g. a s s u m e t h a t ( a 1 ) t is c h o s e n s o t h a t jend( t ) j is a s s m a ll a s p o s s ib le . p u t end( t ) = f»1; :::; »f g. l e t ¡! p2 = »1 ¡! p2 »2 b e t h e u n iqu e p a t h in t wit h e n d ve r t ic e s »1 a n d »2. fu r t h e r , a s s u m e t h a t ( a 2 ) t is c h o s e n s o t h a t p2 is a s lo n g a s p o s s ib le , s u b je c t t o ( a1 ) . p u t jb ( t ) j = b. if f = 2 t h e n p2 is a 2 -e n d e d s p a n n in g t r e e ( h a m ilt o n p a t h ) in g wit h jb ( p2 ) j = b = 0 , im p lyin g t h a t f = 2 · k + 1 = k ¡ b + 1 . n o w le t f ¸ 3 , t h a t is b ¸ 1 . claim 1: if p is a hamilton path in g[v ( p2 ) ] with end vertices x; y, then n ( x) [ n ( y ) µ v ( p2 ) . p r oof: a s s u m e t h e c o n t r a r y a n d a s s u m e w.l.o .g . t h a t n ( x) 6µ v ( p2 ) . p u t t 0 = t ¡ e ( p2 ) + e ( p ) . cle a r ly, t 0 is a n f -e n d e d s p a n n in g t r e e in g a n d xv 2 e ( g ) fo r s o m e v 2 v ( g ¡ p ) . l e t c b e t h e u n iqu e c yc le in t 0 + xv a n d le t vv0 b e t h e u n iqu e e d g e o n c wit h v0 6= x. th e n t 0+xv¡vv0 is a n f-e n d e d s p a n n in g t r e e in g, c o n t r a d ic t in g ( a 2 ) . 4 b y cla im 1 , n ( »1 ) [ n ( »2 ) µ v ( p2 ) . if n ( »1 ) \ n + ( »2 ) 6= ; t h e n c le a r ly, g[v ( p2 ) ] h a s a h a m ilt o n c yc le . s in c e b ¸ 1 , g[v ( p2 ) ] h a s a h a m ilt o n p a t h wit h e n d ve r t e x x s u c h t h a t n ( x ) 6µ v ( p2 ) , c o n t r a d ic t in g cla im 1 . h e n c e , n ( »1 ) \ n + ( »2 ) = ;. ob s e r vin g a ls o t h a t »1 62 n ( »1 ) [ n + ( »2 ) a n d n + ( »2 ) µ v ( p2 ) , we g e t jp2j ¸ jn ( »1 ) j + jn + ( »2 ) j + jf»1gj = d( »1 ) + d ( »2 ) + 1 ¸ ¾2 + 1 : ( 1 ) fo r e a c h i 2 f 3 ; :::; fg, le t ¡!pi = »i ¡! pi zi b e t h e u n iqu e p a t h in t b e t we e n »i a n d t h e n e a r e s t ve r t e x zi o f p2. cle a r ly, zi 2 b ( t ) ( i = 3 ; :::; f ) . case 1: jpij = 2 ( i = 3 ; :::; f ) . it fo llo ws t h a t b ( t ) µ v ( p2 ) . if b = 1 , t h e n b y ( 1 ) , jp2j ¸ ¾2 + b a n d t h e r e fo r e , f = jf»3; :::; »f gj + 2 = n ¡ jp2j + 2 · n ¡ ¾2 ¡ b + 2 · k ¡ b + 1 : l e t b ¸ 2 a n d le t x1; :::; xb b e t h e e le m e n t s o f b ( t ) , o c c u r r in g o n ¡! p2 in a c o n s e c u t ive o r d e r . a s s u m e w.l.o .g . t h a t x1 = z3. fu r t h e r , a s s u m e t h a t ( a 3 ) t is c h o s e n s o t h a t dt ( x1 ) is a s s m a ll a s p o s s ib le , s u b je c t t o ( a1 ) a n d ( a2 ) . if »3v1 2 e ( g) fo r s o m e v1 2 v ( x+1 ¡! p2 »2 ) , t h e n t + »3v1 ¡ »3x1 is a n f-e n d e d t r e e , c o n t r a d ic t in g ( a 3 ) . h e n c e , we c a n a s s u m e t h a t n ( »3 ) µ v ( »1 ¡! p2 x1 ) , t h a t is ( n ( »3 ) ¡ z3 ) \ b ( t ) = ;: ( 2 ) n e xt , if n ¡ ( »1 ) \ ( n ( »3 ) ¡ z3 ) h a s a n e le m e n t v2, t h e n v2 ã¡ p2»1v + 2 ¡! p2 »2 is a h a m ilt o n p a t h in g[v ( p2 ) ] wit h e n d ve r t e x v2 s u c h t h a t n ( v2 ) 6µ v ( p2 ) , c o n t r a d ic t in g cla im 1 . h e n c e , n¡ ( »1 ) \ ( n ( »3 ) ¡ z3 ) = ;: ( 3 ) fin a lly, if n¡ ( »1 ) \ b ( t ) 6= ;, t h a t is »1z+i 2 e ( g ) fo r s o m e i 2 f 3 ; :::; fg, t h e n zi ã¡ p2»1z + i ¡! p2 »2 is a h a m ilt o n p a t h in g[v ( p2 ) ] wit h e n d ve r t e x zi s u c h t h a t n ( zi ) 6µ v ( p2 ) , a g a in c o n t r a d ic t in g cla im 1 . h e n c e , n¡ ( »1 ) \ b ( t ) = ;: ( 4 ) u s in g ( 2 ) , ( 3 ) , ( 4 ) a n d o b s e r vin g t h a t »2 62 n¡ ( »1 ) [ ( n ( »3 ) ¡ z3 ) [ b ( t ) , we g e t jv ( p2 ) j ¸ jn ¡ ( »1 ) j + jn ( »3 ) ¡ z3j + jb ( t ) j + jf»2gj ¸ d( »1 ) + d( »3 ) + b ¸ ¾2 + b; im p lyin g t h a t f = jf»3; :::; »f gj + 2 = n ¡ jv ( p2 ) j + 2 · n ¡ ¾2 ¡ b + 2 · k ¡ b + 1 : case 2: jpij ¸ 3 fo r s o m e i 2 f 3 ; :::; fg, s a y i = 3 . case 2.1: n¡ ( »1 ) \ n + ( »2 ) 6= ;. it fo llo ws t h a t »1w +; »2w ¡ 2 e ( g) fo r s o m e w 2 n ¡ ( »1 ) \ n + ( »2 ) . if z3 = w t h e n w ã¡ p2 »1w +¡!p2 »2 is a h a m ilt o n p a t h in g[v ( p2 ) ] wit h e n d ve r t e x w s u c h t h a t n ( w ) 6µ v ( p2 ) , c o n t r a d ic t in g cla im 1 . h e n c e z3 6= w. a s s u m e w.l.o .g . t h a t z3 2 v ( »1 ¡! p2 w ¡ ) . p u t t 0 = t + »1w + + »2w ¡ ¡ z3z¡3 ¡ ww¡: cle a r ly, t 0 is a s p a n n in g f-e n d e d t r e e in g a n d »3 ¡! p3z3 ¡! p2 w ¡»2 ã¡ p2 w +»1 ¡! p2 z ¡ 3 is a p a t h in t 0 lo n g e r t h a n p2, c o n t r a d ic t in g ( a 2 ) . case 2.2: n¡ ( »1 ) \ n + ( »2 ) = ;. p u t b1 = v ( p2 ) \ b ( t ) ; b2 = b ( t ) ¡ b1: 2 2 spanning trees with few branch and end vertices zh. nikoghosyan 2 3 u s in g cla im 1 , it is e a s y t o s e e t h a t n¡ ( »1 ) \ b1 = n + ( »2 ) \ b1 = ;: ob s e r vin g a ls o t h a t n¡ ( »1 ) [ n + ( »2 ) µ v ( p2 ) , we g e t jp2j ¸ jn¡ ( »1 ) j + jn + ( »2 ) j + jb1j = d( »1 ) + d( »2 ) + jb1j ¸ ¾2 + jb1j ¸ n ¡ k + 1 + jb1j: th e n n ¸ jp2j + jb2j + jf»3; :::; »f gj ¸ n ¡ k + 1 + jb1j + jb2j + f ¡ 2 = n ¡ k + b + f ¡ 1 ; im p lyin g t h a t f · k ¡ b + 1 . p r oof of t heor em 2: b y th e o r e m 1 , g h a s a s p a n n in g t r e e t wit h jend( t ) j · k ¡ b + 1 , wh e r e b = jb ( t ) j. on t h e o t h e r h a n d , it is n o t h a r d t o s e e t h a t jend ( t ) j ¸ b + 2 , im p lyin g t h a t b · ( k ¡ 1 ) = 2 . p r oof of t heor em 3: b y th e o r e m 1 a n d th e o r e m 2 , g h a s a s p a n n in g t r e e t wit h f = jend ( t ) j · k ¡ b + 1 a n d b · ( k ¡ 1 ) =2 , wh e r e b = jb ( t ) j. l e t d1; d2; :::; db b e t h e d e g r e e s o f b r a n c h ve r t ic e s o f t . ob s e r vin g t h a t f = bx i=1 ( di ¡ 2 ) + 2 ; we g e t bx i=1 di · k + b ¡ 1 · 3 2 ( k ¡ 1 ) : refer ences [1 ] j. a . b o n d y a n d u . s . r . mu r t y, graph theory with applications, ma c m illa n , l o n d o n a n d e ls e vie r , n e w y o r k, 1 9 7 6 . [2 ] g. a . d ir a c , \ s o m e t h e o r e m s o n a b s t r a c t g r a p h s " , p roc. l ondon, m ath. soc., vo l.2 , p p . 6 9 -8 1 , 1 9 5 2 . [3 ] o. or e , \ a n o t e o n h a m ilt o n ia n c ir c u it s " , am. m ath. m onth., vo l. 6 7 , p a g e 5 5 , 1 9 6 0 . [4 ] v . ch v¶a t a l a n d p . e r d }o s , \ a n o e o n h a m ilt o n ia n c ir c u it s " , d iscrete m ath., vo l. 2 , p p . 1 1 1 -1 1 3 , 1 9 7 2 . [5 ] m. l a s v e r g n a s , \ s u r u n e p r o p r i¶ e t ¶ e d e s a r b r e s m a xim a u x d a n s u n g r a p h e " , 1 9 7 2 c.r . a c a d .s c i.p a r is s ¶ e r . a , vo l. 2 7 2 , p p . 1 2 9 7 -1 3 0 0 , 1 9 7 1 . [6 ] zh .g. n iko g h o s ya n , \ two t h e o r e m s o n s p a n n in g t r e e s " , uchenie zapiski e gu (scienti¯c transactions of the yerevan state university), s e r . ma t e m a t ika , ( in r u s s ia n ) , n o . 3 , p p . 3 { 6 , 1 9 7 6 . [7 ] h . b r o e r s m a a n d h . tu in s t r a , \ in d e p e n d e n c e t r e e s a n d h a m ilt o n c yc le s " , j . graph theory, vo l. 2 9 , p p . 2 2 7 -2 3 7 , 1 9 9 8 . [8 ] s . w in , \ on a c o n je c t u r e o f l a s v e r g n a s c o n c e r n in g c e r t a in s p a n n in g t r e e s in g r a p h s " , r esultate m ath., vo l. 2 , p p . 2 1 5 -2 2 4 , 1 9 7 9 . [9 ] l . ga r g a n o , p . h e ll, l . s t a c h o a n d u . v a c c a r o , \ s p a n n in g t r e e s wit h b o u n d e d n u m b e r o f b r a n c h ve r t ic e s " , l ecture notes in comput. sci., vo l. 2 3 8 0 , p p . 3 5 5 -3 6 5 , 2 0 0 2 . [1 0 ] e . fla n d r in , t. k a is e r , r . k u · z e l, h . l i a n d z. r yj¶a · c e k, \ n e ig h b o r h o o d u n io n s a n d e xt r e m a l s p a n n in g t r e e s " , d iscrete m ath., vo l. 3 0 8 , p p . 2 3 4 3 -2 3 5 0 , 2 0 0 8 . [1 1 ] h . ma t s u d a , k . oz e ki a n d t. y a m a s h it a , \ s p a n n in g t r e e s wit h a b o u n d e d n u m b e r o f b r a n c h ve r t ic e s in a c la w-fr e e g r a p h " , graphs and combin. vo l. 3 0 , p p . 4 2 9 -4 3 7 , 2 0 1 4 . [1 2 ] zh .n iko g h o s ya n , \ on s p a n n in g t r e e p r o b le m s a r is in g in o p t ic a l a n d t e r m in a l n e t wo r ks " , csit conference 2015, y e r e va n , a r m e n ia , s e p t e m b e r 2 8 oc t o b e r 2 , p p . 9 4 -9 5 , 2 0 1 5 . [1 3 ] a . s a it o a n d k . s a n o , \ s p a n n in g t r e e s h o m e o m o r p h ic t o a s m a ll t r e e " , d iscrete m ath., vo l. 3 3 9 , p p . 6 7 7 -6 8 1 , 2 0 1 6 . submitted 01.07.2016, accepted 28.10.2016. îù³ëù³ûçý í³é»ñ ùçã ×ûáõõ³ûçý ¨ ï³ëí³í ·³·³ãý»ñáí ä. üçïáõáëû³ý ²ù÷á÷áõù g ·ñ³ýç ñ³ù³ñ ¾2-áí ýß³ý³ïíáõù ¿ g-ç áã ñ³ñ¨³ý ·³·³ãý»ñç ³ëïç×³ýý»ñç ýí³½³·áõûý ·áõù³ñá: ì³éç ù»ï ³ëïç×³ý áõý»óáõ ·³·³ãá ïáãíáõù ¿ ï³ëí³í ·³·³ã, çëï ³éýí³½ý »ñ»ù ³ëïç×³ý áõý»óáõ ·³·³ãá` ×ûáõõ³ûçý ·³·³ã: ²ßë³ï³ýùáõù ¹çï³ñïíáõù »ý ùç³ûý ( ¤ ) n ·³·³ã³ýç ï³å³ïóí³í ·ñ³ýý»ñ, áñáýù µ³í³ñ³ñáõù »ý ¾2 ¸ n ¡ k + 1 å³ûù³ýçý çýã-áñ ùç k µý³ï³ý ãíç ñ³ù³ñ: 1976-çý ³å³óáõóí»é ¿ (ñ»õçý³ïç ïáõùçó), áñ ( ¤ ) -çý µ³í³ñ³ñáõ ï³ù³û³ï³ý ·ñ³ý áõýç ³ù»ý³ß³ïá k ï³ëí³í ·³·³ã áõý»óáõ ïù³ëù³ûçý í³é: ü»ñï³ ³ßë³ï³ýùáõù ý³ë óáõûó ¿ ïñíáõù, áñ ( ¤ ) -çý µ³í³ñ³ñáõ ï³ù³û³ï³ý ·ñ³ý áõýç ïù³ëù³ûçý í³é, áñç ï³ëí³í ¨ ×ûáõõ³ûçý ·³·³ãý»ñç áý¹ñ³ýáõñ ù³ý³ïá ãç ·»ñ³½³ýóáõù ( k+1 ) -á: ºñïñáñ¹ ³ñ¹ûáõýùá åý¹áõù ¿, áñ ( ¤ ) -çý µ³í³ñ³ñáõ ï³ù³û³ï³ý ·ñ³ý áõýç ïù³ëù³ûçý í³é ³ù»ý³ß³ïá 1 2 ( k¡ 1 ) ×ûáõõ³ûçý ·³·³ãý»ñáí: àëï »ññáñ¹ ³ñ¹ûáõýùç, ( ¤ ) -çý µ³í³ñ³ñáõ ï³ù³û³ï³ý ·ñ³ý áõýç ïù³ëù³ûçý í³é, áñç ×ûáõõ³ûçý ·³·³ãý»ñç ³ëïç×³ýý»ñç ·áõù³ñá ãç ·»ñ³½³ýóáõù 3 2 ( k ¡ 1 ) -á: ´áéáñ »ñ»ù ·ý³ñ³ï³ï³ýý»ñá ñ³ë³ý»éç »ý: êàðêàñû ñ ìåíüøèì ÷èñëîì âèñÿ÷èõ è br-âåðøèí æ. íèêîãîñÿí àííîòàöèÿ äëÿ ãðàôà g ÷åðåç ¾2 îáîçíà÷àåòñÿ ìèíèìàëüíàÿ ñóììà ñòåïåíåé äâóõ íåñìåæíûõ âåðøèí. âåðøèíà â äåðåâå íàçûâàåòñÿ âèñÿ÷åé, åñëè èìååò ñòåïåíü 1; è íàçûâàåòñÿ br-âåðøèíîé, åñëè èìååò ñòåïåíü ïî ìåíüøåé ìåðå 3. â ðàáîòå ðàññìàòðèâàþòñÿ òîëüêî ( ¤ ) n-âåðøèííûå ñâÿçíûå ãðàôû, óäîâëåòâîðÿþùèå óñëîâèþ ¾2 ¸ n ¡ k + 1 äëÿ íåêîòîðîãî íàòóðàëüíîãî ÷èñëà k. â 1976 ãîäó 2 4 spanning trees with few branch and end vertices zh. nikoghosyan 2 5 áûëà äîêàçàíà (àâòîðîì), ÷òî ïðîèçâîëüíûé ãðàô óäîâëåòâîðÿþùèé óñëîâèþ ( ¤ ) , èìååò êàðêàñ ñ íå áîëåå ÷åì k âèñÿ÷èìè âåðøèíàìè. â íàñòîÿùåé ðàáîòå äîêàçûâàåòñÿ, ÷òî âñÿêèé ãðàô óäîâëåòâîðÿþùèé óñëîâèþ ( ¤ ) , èìååò êàðêàñ, ãäå îáùåå ÷èñëî âèñÿ÷èõ è br-âåðøèí íå ïðåâîñõîäèò k + 1 . âòîðîé ðåçóëüòàò óòâåðæäàåò, ÷òî âñÿêèé ãðàô, óäîâëåòâîðÿþùèé óñëîâèþ ( ¤ ) , èìååò êàðêàñ, ãäå ÷èñëî br-âåðøèí íå ïðåâîñõîäèò 1 2 ( k ¡ 1 ) . òðåòèé ðåçóëüòàò óòâåðæäàåò, ÷òî âñÿêèé ãðàô, óäîâëåòâîðÿþùèé óñëîâèþ ( ¤ ) , èìååò êàðêàñ, ãäå ñóììà ñòåïåíåé br-âåðøèí íå ïðåâîñõîäèò 3 2 ( k ¡ 1 ) . âñå òðè îöåíêè äîñòèãàåìû. 02.pdf (p.1-7) zhora_nikoghosyan_1.pdf (p.8) 2_nikoghosyan_14--18.dvi mathematical problems of computer science 48, 14{18, 2017. n ew e xtensions of dir ac's t heor ems mo s s in e s . k o u la kz ia n 1, ca r le n m. mo s e s ya n 2 a n d zh o r a g. n iko g h o s ya n 1 1institute for informatics and automation problems of nas ra e-mails: mossine@hotmail.com; zhora@ipia.sci.am 2faculty of mathematics, physics and informatics kh. abovyan armenian state university e-mail: mosesyan@list.ru abstract let g be a graph on n vertices with degree sequence ± = d1 · d2 · ::: · dn and let c be the circumference the length of a longest cycle in g. in 1952, dirac proved: (i) every graph with d1 ¸ n2 is hamiltonian; (ii) in every 2-connected graph, c ¸ minfn; 2d1g. in this paper we present the following two dirac-type extensions: (iii) every graph with d± ¸ n2 is hamiltonian; (iv) in every 2-connected graph, c ¸ minfn; 2d±g. the results are sharp. keywords: hamilton cycle, longest cycle, circumference, minimum degree, degree sequence. 1 . in t r o d u c t io n w e c o n s id e r o n ly ¯ n it e u n d ir e c t e d g r a p h s wit h n e it h e r lo o p s n o r m u lt ip le e d g e s . a g o o d r e fe r e n c e fo r a n y u n d e ¯ n e d t e r m s is [2 ]. th e s e t o f ve r t ic e s o f a g r a p h g is d e n o t e d b y v ( g ) a n d t h e s e t o f e d g e s b y e ( g ) . l e t n b e t h e o r d e r ( t h e n u m b e r o f ve r t ic e s ) o f g, c t h e o r d e r o f a lo n g e s t c yc le ( c a lle d c ir c u m fe r e n c e ) in g a n d p t h e o r d e r o f a lo n g e s t p a t h . w e u s e n ( v ) t o d e n o t e t h e s e t o f a ll n e ig h b o r s o f a ve r t e x v a n d d( v ) = jn ( v ) j t o d e n o t e t h e d e g r e e o f ve r t e x v. th e m in im u m d e g r e e in g is d e n o t e d b y ±. l e t d1; d2; :::; dn b e t h e d e g r e e s e qu e n c e in g wit h ± = d1 · d2 · ::: · dn. a p a t h ( s im p le p a t h ) o f o r d e r m is a s e qu e n c e o f d is t in c t ve r t ic e s v1; :::; vm, d e n o t e d b y v1v2:::vm, s u c h t h a t vi¡1vi is a n e d g e fo r a ll 2 · i · m. s im ila r ly. a c yc le o f o r d e r m is a s e qu e n c e o f d is t in c t ve r t ic e s v1; :::; vm, d e n o t e d b y v1v2:::vmv1, s u c h t h a t vi¡1vi a n d vmv1 a r e e d g e s fo r a ll 2 · i · m. in p a r t ic u la r , fo r m = 2 , v1v2v1 is a c yc le o f o r d e r 2 ; a n d fo r m = 1 , v1v1 is a c yc le o f o r d e r 1 . s o , b y t h e d e ¯ n it io n , e ve r y ve r t e x ( e d g e ) c a n b e c o n s id e r e d a s a c yc le o f o r d e r 1 ( 2 , r e s p e c t ive ly) . a g r a p h g is h a m ilt o n ia n if g c o n t a in s a h a m ilt o n c yc le , t h a t is a s im p le s p a n n in g c yc le . w e wr it e a c yc le ( a p a t h ) q wit h a g ive n o r ie n t a t io n b y ¡! q . th e r e ve r s e s e qu e n c e o f ve r t ic e s o f ¡! q is d e n o t e d b y ã¡ q . fo r x 2 v ( q) , we d e n o t e t h e s u c c e s s o r a n d t h e p r e d e c e s s o r o f x o n ¡! q ( if s u c h ve r t ic e s e xis t ) b y x+ a n d x¡, r e s p e c t ive ly. fo r u µ v ( q ) , we d e n o t e u + = fu+ju 2 ug a n d u¡ = fu¡ju 2 ug. w e u s e p = x¡!p y t o d e n o t e a p a t h wit h e n d 1 4 m. koulakzian, c. mosesyan and zh. nikoghosyan 1 5 ve r t ic e s x a n d y in t h e d ir e c t io n fr o m x t o y. w e s a y t h a t ve r t e x z1 p r e c e d e s ve r t e x z2 o n a p a t h ¡! q if z1, z2 o c c u r o n ¡! q in t h is o r d e r , a n d in d ic a t e t h is r e la t io n s h ip b y z1 á z2. in 1 9 5 2 , d ir a c [3 ] g a ve t h e ¯ r s t s u ± c ie n t c o n d it io n fo r a g r a p h t o b e h a m ilt o n ia n a n d t h e ¯ r s t n o n t r ivia l lo we r b o u n d fo r t h e c ir c u m fe r e n c e in t e r m s o f m in im u m d e g r e e d1 = ±. t heor em a: [3 ]. e very graph with d1 ¸ n2 is hamiltonian. t heor em b : [3 ]. in every 2-connected graph, c ¸ m in fn; 2 d1g. a g r e a t n u m b e r o f g e n e r a liz a t io n s a n d im p r o ve m e n t s o f th e o r e m s a a n d b a r e kn o wn u n d e r va r io u s c o n d it io n s . in t h is p a p e r we p r e s e n t t h e fo llo win g t wo d ir a c -t yp e e xt e n s io n s o f th e o r e m s a a n d b in t e r m s o f di ( fo r a p p r o p r ia t e i ) in s t e a d o f d1. t heor em 1: e very graph with d± ¸ n2 is hamiltonian. t heor em 2: in every 2-connected graph, c ¸ m in fn; 2 d±g. it is n o t h a r d t o s e e t h a t if g is a g r a p h wit h d± ¸ n2 o r g is a 2 -c o n n e c t e d g r a p h , t h e n ± ¸ 2 . to s e e t h a t th e o r e m 1 a n d th e o r e m 2 a r e s h a r p , le t g = k± + k±+1. s in c e d± = ± a n d c = 2 ± < n, we c o n c lu d e t h a t t h e b o u n d d± ¸ n2 in th e o r e m 1 c a n n o t b e r e p la c e d b y d± ¸ n¡12 . l e t g = k± + ( ±k1 [ k2 ) . cle a r ly, d± = ± a n d d±+1 = ± + 1 . ob s e r vin g a ls o t h a t c = 2 ± + 1 < n, we c o n c lu d e t h a t t h e b o u n d d± ¸ n2 in th e o r e m 1 c a n n o t b e r e p la c e d b y d±+1 ¸ n2 . a s fo r th e o r e m 2 , t h e g r a p h e xa m p le g = k± + k±+1 s h o ws t h a t t h e b o u n d c ¸ m in fn; 2 d±g in th e o r e m 2 c a n n o t b e r e p la c e d b y c ¸ m in fn; 2 d± + 1 g. fin a lly, t h e g r a p h e xa m p le g = k± + ( ±k1 [ k2 ) s h o ws t h a t t h e b o u n d c ¸ m in fn; 2 d±g c a n n o t b e r e p la c e d b y c ¸ m in fn; 2 d±+1g. th u s , th e o r e m 1 a n d th e o r e m 2 a r e s h a r p in a ll r e s p e c t s . l e t ¡! p = v1v2:::vp b e a lo n g e s t p a t h in g. cle a r ly, n ( v1 ) [ n ( vp ) µ v ( p ) . a vin e o f le n g t h m o n p is a s e t fli = wi ¡! l izi : 1 · i · mg o f in t e r n a lly-d is jo in t p a t h s s u c h t h a t ( a ) v ( li ) \ v ( p ) = fwi; zig ( i = 1 ; :::; m) , ( b ) v1 = w1 á w2 á z1 ¹ w3 á z2 ¹ w4 á ::: ¹ wm á zm¡1 á zm = vp o n p . lemma 1: [3 ]. if g is 2 -connected, then there is at least one vine on p . w e n e e d a ls o t h e fo llo win g le m m a . lemma 2: [1 ]. l et g be a 2-connected graph on n vertices. if v1v2:::vp is a longest path of g, then there exists a cycle of length at least m in fd( v1 ) + d( vp ) ; ng. 2 . p r o o fs p r oof of t heor em 1. a s s u m e ¯ r s t t h a t g is n o t c o n n e c t e d a n d le t h1 a n d h2 b e t wo c o n n e c t e d c o m p o n e n t s o f g. cle a r ly, jv ( hi ) j ¸ ± + 1 ( i = 1 ; 2 ) a n d m a xfd ( v ) : v 2 v ( hi ) g ¸ d±+1 ( i = 1 ; 2 ) : h e n c e 2 d± · 2 d±+1 · m a xfd( v ) : v 2 v ( h1 ) g + m a xfd( v ) : v 2 v ( h2 ) g 1 6 new extensions of dirac's theorems · jv ( h1 ) j + jv ( h2 ) j ¡ 2 · n ¡ 2 ; c o n t r a d ic t in g t h e h yp o t h e s is 2 d± ¸ n. s o , we c a n a s s u m e t h a t g is c o n n e c t e d . l e t¡! p = v1v2:::vp b e a lo n g e s t p a t h in g. cle a r ly, n ( v1 ) [ n ( vp ) µ v ( p ) . a s s u m e t h a t ( a 1 ) p is c h o s e n s o t h a t d ( v1 ) is m a xim u m . l e t x1; x2; :::; xt b e t h e e le m e n t s o f n ( v1 ) o c c u r r in g o n ¡! p in a c o n s e c u t ive o r d e r , wh e r e t = d( v1 ) ¸ ±. ob s e r ve t h a t fo r e a c h i 2 f2 ; :::; tg, x¡i ã¡ p v1xi ¡! p vp is a lo n g e s t p a t h in g. b y ( a 1 ) , d( v1 ) ¸ d( x¡i ) ( i = 1 ; 2 ; ::; t) : ( 1 ) a s s u m e ¯ r s t t h a t xt = vp, t h a t is c ¸ p. if c ¸ p + 1 , t h e n t h e c yc le o f o r d e r a t le a s t p + 1 c o n t a in s a p a t h o f o r d e r a t le a s t p + 1 , a c o n t r a d ic t io n . h e n c e , c = p. p u t ¡! c = v1v2:::vpv1. if p = n, t h e n c is a h a m ilt o n c yc le in g. ot h e r wis e , s in c e g is c o n n e c t e d , u1u2 2 e ( g ) fo r s o m e u1 2 v ( c ) a n d u2 62 v ( c ) . th e n u+1 ¡! c u1u2 is a p a t h o f o r d e r p + 1 , a c o n t r a d ic t io n . n o w a s s u m e t h a t xt 6= vp, t h a t is xt á vp. fu r t h e r , we c a n a s s u m e t h a t ( a 2 ) p is c h o s e n s o t h a t d ( vp ) is m a xim u m , s u b je c t t o ( a 1 ) . l e t y1; y2; :::; yf b e t h e e le m e n t s o f n ( vp ) o c c u r r in g o n ã¡ p in a c o n s e c u t ive o r d e r . b y ( a 2 ) , d( vp ) ¸ d( y+i ) ( i = 1 ; 2 ; ::; f ) : ( 2 ) b y ( 1 ) a n d ( 2 ) , d( v1 ) ¸ m a xfd ( x¡1 ) ; d( x¡2 ) ; :::; d( x¡t ) g ¸ m a xfd1; d2; :::; dtg = dt = dd(v1) ¸ d±: a n d d ( vp ) ¸ m a xfd( y+1 ) ; d ( y+2 ) ; :::; d( y+f g ¸ m a xfd1; d2; :::; dfg = df = dd(vp) ¸ d±; im p lyin g t h a t d( v1 ) + d ( vp ) ¸ 2 d± ¸ n: ( 3 ) b y l e m m a 2 , g is h a m ilt o n ia n . h o we ve r , we p r e s e n t a s h o r t p r o o f o f t h is fa c t a c c o r d in g t o t h e la t e s t t e r m in o lo g y. case 1. n ( v1 ) \ n + ( vp ) 6= ;. l e t v 2 n ( v1 ) \ n + ( vp ) , t h a t is v1v; vpv¡ 2 e ( g) . s in c e v1v ¡! p vpv ¡ã¡p v1 is a c yc le o f o r d e r p, we h a ve c ¸ p, im p lyin g t h a t c = p. s in c e g is c o n n e c t e d , we h a ve c = p = n, t h a t is g is h a m ilt o n ia n . case 2. n ( v1 ) \ n + ( vp ) = ;. m. koulakzian, c. mosesyan and zh. nikoghosyan 1 7 it fo llo ws t h a t n ¸ p ¸ jn ( v1 ) j + jn + ( vp ) j + jfv1gj ¸ ¸ jn ( v1 ) j + jn ( vp ) j + 1 ¸ d( v1 ) + d ( vp ) + 1 : b y ( 3 ) , n ¸ 2 d± + 1 , c o n t r a d ic t in g t h e h yp o t h e s is d± ¸ n2 . p r oof of t heor em 2. l e t ¡! p = v1v2:::vp b e a lo n g e s t p a t h in g. d e ¯ n e t h e ve r t ic e s x1; x2; :::; xt; y1; y2; :::; yf a s in p r o o f o f th e o r e m 1 , wh e r e we h a ve p r o ve d d ( v1 ) + d( vp ) ¸ 2 d±: ( 4 ) case 1. xt ¹ yf . l e t fli = wi ¡! l izi : 1 · i · mg b e a vin e o f m in im a l le n g t h m o n ¡! p . s in c e p is a lo n g e s t p a t h in g, we h a ve l1; lm 2 e ( g) . n e xt , s in c e m is m in im a l, we h a ve xt á z2, xt á w3 a n d wm¡1 á yf , zm¡2 á yf . ch o o s e z¤1 2 v ( p ) s u c h t h a t w2 á z¤1 a n d jv ( w2 ¡! p z¤1 ) j is m in im a l. a n a lo g o u s ly, c h o o s e w¤m 2 v ( p ) s u c h t h a t w¤m á zm¡1 a n d jv ( w¤m ¡! p zm¡1 ) j is m in im a l. p u t h = p [ m¡1[ i=2 li [ fv1z¤1 ; vpw¤mg: b y d e le t in g t h e fo llo win g p a t h s wi ¡! p zi¡1 ( i = 3 ; 4 ; :::; m ¡ 1 ) ; w2 ¡! p z¤1; w ¤ m ¡! p zm¡1 fr o m h ( e xc e p t fo r t h e ir e n d ve r t ic e s ) , we o b t a in a c yc le c wit h a t le a s t d( v1 ) + d( vp ) + 1 ve r t ic e s . b y ( 4 ) , c ¸ jv ( c ) j ¸ d ( v1 ) + d( vp ) + 1 > 2 d±: case 2. yf á xt. case 2.1. n ( v1 ) \ n + ( vp ) 6= ;. l e t v 2 n ( v1 ) \ n + ( vp ) , t h a t is v1v; vpv¡ 2 e ( g ) . s in c e v1v ¡! p vpv ¡ã¡p v1 is a c yc le o f o r d e r p a n d g is c o n n e c t e d , e it h e r p < jv ( g ) j a n d we c a n fo r m a p a t h lo n g e r t h a n p ( a c o n t r a d ic t io n ) o r p = jv ( g ) j, im p lyin g t h a t c = p = n. case 2.2. n ( v1 ) \ n + ( vp ) = ;. s in c e yf á xt, we c a n c h o o s e xi 2 n ( v1 ) a n d yj 2 n ( vp ) s u c h t h a t yj á xi a n d v1v; vpv 62 e ( g ) fo r e a c h ve r t e x v wit h yj á v á xi. p u t c = v1xi ¡! p vpyi ã¡ p v1: th e n c ¸ jv ( c ) j ¸ jn ( v1 ) j + jn + ( vp ) j + jfv1gj ¡ jfyjgj ¸ jn ( v1 ) j + jn ( vp ) j = d ( v1 ) + d( vp ) : b y ( 4 ) , c ¸ 2 d±. 1 8 new extensions of dirac's theorems 3 . a c kn o wle d g m e n t s w e wo u ld like t o t h a n k t h e r e fe r e e s fo r u s e fu l s u g g e s t io n s , wh ic h h a ve c o n s id e r a b ly im p r o ve d t h e p r e s e n t a t io n o f t h e p a p e r . refer ences [1 ] j. a . b o n d y, \ b a s ic g r a p h t h e o r y p a t h s a n d c yc le s " , h a n d b o o k o f co m b in a t o r ic s , vo l.1 , e ls e vie r , a m s t e r d a m , p p . 5 -1 1 0 , 1 9 9 5 . [2 ] j. a . b o n d y a n d u .s .r . mu r t y, graph theory with applications, ma c m illa n , l o n d o n a n d e ls e vie r , n e w y o r k 1 9 7 6 . [3 ] g. a . d ir a c , \ s o m e t h e o r e m s o n a b s t r a c t g r a p h s " , p roc. l ondon, m ath. soc., vo l. 2 , p p . 6 9 -8 1 , 1 9 5 2 . submitted 22.08.2017, accepted 04.12.2017. ¸çñ³ïç ã»áñ»ùý»ñç ýáñ áý¹ñ³ýñ³óáõùý»ñ ø. øáõé³ù½û³ý, î. øáë»ëû³ý ¨ ä. üçïáõáëû³ý ²ù÷á÷áõù ¸çóáõù g-ý ± ýí³½³·áõûý ³ëïç×³ý ¨ ± = d1 · d2 · ::: · dn ³ëïç×³ý³ûçý ñ³çáñ¹³ï³ýáõãûáõý áõý»óáõ n ·³·³ã³ýç ·ñ³ý ¿: g-ç ³ù»ý³»ñï³ñ óçïéç »ñï³ñáõãûáõýá ýß³ý³ïíáõù ¿ c-áí: 1952-çý ¸çñ³ïý ³å³óáõó»ó, áñ (i) d1 ¸ n2 å³ûù³ýçý µ³í³ñ³ñáõ ï³ù³û³ï³ý ·ñ³ý áõýç ð³ùçéïáýç óçïé. (ii) ï³ù³û³ï³ý 2-ï³å³ïóí³í ·ñ³ýáõù c ¸ m in fn; 2 d1g: ü»ñï³ ³ßë³ï³ýùáõù µ»ñíáõù »ý ýßí³í ³ñ¹ûáõýùý»ñç »ñïáõ áý¹é³ûýáõùý»ñ.(iii) d± ¸ n2 å³ûù³ýçý µ³í³ñ³ñáõ ï³ù³û³ï³ý ·ñ³ý áõýç ð³ùçéïáýç óçïé, (iv) ï³ù³û³ï³ý 2-ï³å³ïóí³í ·ñ³ýáõù c ¸ m in fn; 2 d±g: êï³óí³í ³ñ¹ûáõýùý»ñá »ýã³ï³ ã»ý µ³ñ»é³íù³ý: íîâûå îáîáùåíèÿ òåîðåì äèðàêà ì. êóëàêçÿí, ê. ìîñåñÿí è æ. íèêîãîñÿí àííîòàöèÿ ïóñòü ± = d1 · d2 · ::: · dn ïîñëåäîâàòåëüíîñòü ñòåïåíåé âåðøèí nâåðøèííîãî ãðàôà g ñ ìèíèìàëüíîé ñòåïåíüþ ±. äëèíà äëèííåéøåãî öèêëà ãðàôà îáîçíà÷àåòñÿ ÷åðåç c. â 1952 ãîäó äèðàê äîêàçàë, ÷òî (i) êàæäûé ãðàô óäîâëåòâîðÿþùèé óñëîâèþ d1 ¸ n2 , èìååò ãàìëüòîíîâ öèêë; (ii) åñëè g ÿâëÿåòñÿ 2-ñâÿçíûì ãðàôîì, òî c ¸ m in fn; 2 d1g. â íàñòîÿùåé ðàáîòå äîêàçûâàþòñÿ: (iii) êàæäûé ãðàô óäîâëåòâîðÿþùèé óñëîâèþ d± ¸ n2 , èìååò ãàìëüòîíîâ öèêë; (iv) åñëè g ÿâëÿåòñÿ 2-ñâÿçíûì ãðàôîì, òî c ¸ m in fn; 2 d±g. ïîëó÷åííûå ðåçóëüòàòû íåóëó÷øàåìû. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 44, 42--50, 2015. 3d scanner from two web cameras aram v. gevorgyan1 and hakob g. sarukhanyan2 1russian-armenian university, 2institute for informatics automation problems of nas ra e-mail: aramgv@gmail.com, hakop@ipia.sci.am abstract in this paper we present an approach how to build 3d scanner using two ordinary web cameras. this approach is based on stereovision and is beneficial from the other 3d scanners that it doesn’t need laser, structured light or other expensive sensors. we put the object in front of the cameras at a distance about 40-50 cm and rotate it at small degrees. for every rotation we compute the disparity map from two cameras and then get the depth map in the form of point cloud. then we filter out our object from the scene and then merge object point clouds from different views to get a full 3d model. keywords: stereovision, depth map, registration, icp, 3d model. 1. introduction the acquisition of 3d models from real objects is a very actual problem and has many applications in cinema, video games, education software and others. device that can automatically acquire 3d is called 3d scanner. there are different types of 3d scanners: contact 3d scanners that need physical touch to get 3d information, or non-contact, that usually use laser or structured lighting. but most of these devices are expensive. in this article, we represent a system for getting 3d model from 2 web cameras using stereovision. this approach is beneficial from the others that it’s enough to have two general web cameras and no other additional equipment. stereovision is a technique that gives 3d information ( x, y, z coordinates) of the scene using 2 or more cameras. it usually consists of the following steps: camera calibration, rectification, stereo correspondence (obtaining disparity map), triangulation (obtaining depth map from disparity map). we will speak about these steps in details in section 2. image with its depth (z coordinate) is usually called 2.5d. from the left and the right camera two images we get point cloud of the object from one scene of view, but it’s not enough to have an object 3d model. to build the object 3d model we need to take object images from different scenes and then merge these point clouds into one common, this process is known as registration and is covered in section 2. 42 mailto:gevorgka@gmail.com mailto:hakop@ipia.sci.am a. gevorgyan and h. sarukhanyan 43 2. related work 2.1 calibration camera calibration is the process of relating the ideal model of the camera to the actual physical device and of determining the position and orientation of the camera with respect to a world reference system. camera internal parameters include focal length, image format and principal point. external parameters contain information about camera position and orientation in a world, rotation and translation vectors. if we know the internal and the external parameters we can say that the camera is calibrated. cameras calibration is needed for rectification and depth calculation. camera calibration is usually done by using chessboard pattern. more detailed calibrations is described in [1],[2], [3]. 2.1 rectification if the two cameras aligned to be coplanar, the two images have the same image plane, so the corresponding points have the same row coordinates and the stereo correspondence problem is reduced from 2d to 1d. but in real world it is impossible to place two cameras ideally coplanar so rectification is needed. rectification is a process of reprojecting image planes onto the common plane parallel to the line between optical centers that is called baseline. rectification is done using epipolar geometry [4]. epipolar geometry is the intrinsic projective geometry used in stereovision. it is independent of scene structure, and only depends on the cameras' internal parameters and relative pose. main concepts of epipolar geometry: the epipole is the point of intersection of the baseline with the image plane. an epipolar plane is the plane defined by a 3d point (or equivalently image point) and the optical centres. an epipolar line is the intersection of an epipolar plane with the image plane. fig. 1. the epipolar constraint. epipolar constraint: we must search the correspondent point of the point in its correspondent epipolar line in the other image. 3d scanner from two web cameras 44 2.2 stereo correspondence stereo correspondence is a problem of finding matching pixels in the left and right images corresponding to the same points on the 3d surface. if images are rectified all epipolar lines are parallel and the corresponding points lie on the same row in both images. suppose (𝑥𝑥, 𝑦𝑦) are coordinates of the point in the left image and (𝑥𝑥′, 𝑦𝑦′) in the right image, where y= 𝑦𝑦′ because images are rectified. after finding the corresponding points we can compute the disparity map, where disparity for each pixel is 𝑥𝑥 − 𝑥𝑥′. the disparity measures the displacement of a point between the two images. correspondence problem is complicated by occlusion fact, some pixels are visible only in one of the images. there are plenty of stereo matching algorithms. they usually can be divided into two classes: local methods and global methods. in local methods disparity computation at a given point depends only on intensity values within a finite window. in contrast to local methods, in global methods disparity computation is done for all reference image pixels at once. this is achieved by minimizing energy function that consists of data term and smoothness term. local methods are fast enough but their accuracy is not good and may be inacceptable. while global methods have good accuracy but they are time consuming. most commonly used stereo correspondence algorithms are: block matching, semi-global matching [5] and graph cuts. block matching is a local method, graph cut is a global method that gives very good results but works slow and semi-global matching gives almost the same results as global methods but it works extremely faster. a good survey of stereo correspondence algorithms is in [6]. 2.3 triangulation if we know the geometric arrangement of the cameras, then we can calculate the depth map from the disparity map by triangulation. let us consider the simple case: suppose we have 2 cameras with parallel optical axes. let the baseline be perpendicular to the optical axes of the cameras and parallel to x-axis.the distance between the cameras is d and the cameras have equal focal length f. fig. 2. a simplified stereo imaging system. a. gevorgyan and h. sarukhanyan 45 consider (x, y, z) are coordinates of a point in three-dimensional world, where 𝑧𝑧 is a depth. let this point have (xl, yl) and (𝑥𝑥𝑥𝑥, 𝑦𝑦𝑥𝑥) coordinates in the left and right image planes of the respective cameras. then from the similar triangles we have: z f df z d xl xr xl xr = ⇒ = − − , where xl − xr is disparity. thus, the depth at various scene points may be recovered by knowing the disparities of the corresponding image points. also we can compute: ( ) ( ) ( ) ( ) d xl xr d yl yr x , y xl xr xl xr + + = = − −2 2 . as we can see depth is inversely proportional to disparity, so when disparity is near 0, small disparity differences make for large depth differences, and when disparity is large, small disparity differences do not change the depth by much. so the depth is computed more accurately for objects near the camera. having the depth map we can generate the point cloud. point cloud is a set of data points in some coordinate system. more about how to compute the depth is in [7], and how to compute the object distance from web cameras – in [8]. 2.4 registration registration is a process of merging point clouds from different scenes into one common model. in other words, registration transforms multiple 3d point clouds into the same coordinate system so as to align overlapping components of these sets. the most popular and commonly used algorithm of registration is the iterative closest point (icp). there are many variants of the algorithm, but all these variants affect one of six stages of the algorithm: 1. selection subsets of two point clouds 2. matching corresponding pairs 3. weighting the corresponding pairs appropriately 4. rejecting certain pairs based on looking at each pair individually or considering the entire set of pairs 5. compute an error metric based on the point pairs 6. minimizing the error metric more about icp and its variants are in [9], [10], [11]. 3d scanner from two web cameras 46 3. realization for our realization we use opencv the biggest image processing library [12], [13] and point cloud library [14] 2d/3d images and point clouds processing library. for data capture we use two logitech hd pro webcam c920 web cameras. we calibrate our cameras with 8x6 chessboard pattern printed on a2 paper and tightly attached on the glass. it is very important to have a flat and difficult stretching surface for calibration pattern. we also tried with a4 paper, but it didn't give the sufficient results. also illumination is very important for calibration. as it was mentioned above, stereo block matching (sbm) is one of the best stereo correspondence algorithms combining speed and accuracy, so we choose sbm for computing disparity map. we place an object at about 40-50 cm from the cameras and start rotating it about 1-3 degree angle. we capture the left and right camera images for every rotation and compute disparity maps for them. for every disparity map, having cameras parameters, by triangulation we compute the depth map in the form of a point cloud. then we segment our object from the point cloud by its depth values, to have point cloud only of the object with no background. a) b) a. gevorgyan and h. sarukhanyan 47 c) fig. 3. point clouds of the object from different positions. then the process of registration starts. at first we take the first point cloud as a global model. then at each next step we take the next consecutive point cloud and merge it to the global model, transforming it to the global models coordinates and adding to it. merging of two point clouds consists of the following steps: 1. finding correspondence between point clouds 2. rejecting bad correspondences 3. finding transformation matrix based on correspondences and transform point cloud 4. if convergence criteria is reached stop, else go to step 1 a) 3d scanner from two web cameras 48 b) c) fig. 4. constructed 3d model. 4. conclusion and future work as it was mentioned, our approach is based on stereovision and it is beneficial from the others that it was cheaper and more available. there are some other 3d scaners based on stereovision [15], [16], [17], but they need laser or sctructured light. benefits of our aproach that it is enough to have only two ordianry web cameras. our constructed 3d model is not ideal and its quality can be enhanced. for future work we plan to use color variant of icp algorithm, that is good for the objects that don't have good geometric features. due to many factors some holes appears at disparity map, so we also plan to try hole filling algorithms for disparity maps, that may enhance the registrations results. a. gevorgyan and h. sarukhanyan 49 references [1] p. hillman, “white paper: camera calibration and stereo vision”, lochrin terrace, edinburgh eh3 9ql, tech. rep, 2005. [2] z. zhang, “a flexible new technique for camera calibration”, ieee transactions on pattern analysis and machine intelligence, pp. 1330–1334, 2000. [3] z. zhang, "camera calibration with one-dimensional objects", ieee transactions on pattern analysis and machine intelligence, vol. 26, pp. 892-899, 2004. [4] epipolar geometry tutorial. [online]. available: http://homepages.inf.ed.ac.uk/rbf/cvonline/local_copies/owens/lect10/node3. html [5] h. hirschmuller, “accurate and efficient stereo processing by semi global matching and mutual information.” ieee computer vision and pattern recognition, vol. 2, pp. 807-814 2005. [6] d. scharstein and r. szeliski, “a taxonomy and evaluation of dense two-frame stereo correspondence algorithms”, international journal of computer vision, vol. 47, pp. 7-42, 2002. [7] p. j.bagga, “real time depth computation using stereo imaging”, journal electrical and electronic engineering, vol. 1, pp. 51-54, 2013. [8] m. a. mahammed, a. i. melhum and f. a. kochery, “object distance measurement by stereo vision”, international journal of science and applied information technology, vol. 2, pp. 05-08, march 2013. [9] p. j. besl and h. d. mckay, “a method for registration of 3-dshapes”, ieee trans. pattern anal. mach. intell., pp. 239–256, 1992. [10] s. rusinkiewicz and m. levoy, “efficient variants of the icp algorithm”, 3-d digital imaging and modeling, pp. 145–152, 2001. [11] a. johnson and s. kang, “registration and integration of textured 3d data”, in proc. 3dim ’97, ottawa, pp. 234–241, 1997. [12] g. bradski and a. kaehler, learning opencv: computer vision with the opencv library, 1st ed., o’reilly media, inc., 1005 gravenstein highway north, sebastopol, 2008. [13] the opencv website. [online]. available: http://opencv.org/ [14] the point cloud library website. [online]. available: http://pointclouds.org/ [15] z. lv and z. zhang, "build 3d scanner system based on binocular stereo vision", journal of computers, vol. 7., no 2, february 2012. [16] tzung-han lin, "resolution adjustable 3d scanner based on using stereo cameras", signal and information processing association annual summit and conference (apsipa), 2013 asia-pacific, 2013. [17] m. pashaei, s. m. mousavi, "implementation of a low cost structured light scanner", int. ch. photogramm. remote sens. spatial inf. sci., vol. xl-5/w2., pp.477-482, 2013. submitted 04.09.2015, accepted 20.11.2015 http://pointclouds.org/ 3d scanner from two web cameras 50 3d սկաներ երկու վեբ տեսախցիկից ա. գևորգյան և հ. սարուխանյան ամփոփում սույն հոդվածում ներկայացված է երկու սովորական տեսախցիկից 3d սկաներ կառուցելու եղանակ: այն հիմնված է ստերեոտեսողության վրա և տարբերվում է մնացած 3d սկաներներից նրանով, որ չի պահանջում լազեր, համակարգված լույս կամ այլ թանկարժեք սենսորներ: օբյեկտը տեղադրվում է տեսախցիկներից մոտ 40-50 սանտիմետր հեռավորության վրա և պտտվում փոքր անկյուններով: ամեն պտույտի համար հաշվարկվում է անհավասարությունների քարտեզը, որից ստացվում է խորությունների քարտեզը` կետերի ամպի տեսքով: այնուհետև առանձնացվում է օբյեկտը տեսարանից, միավորվում են տարբեր տեսարաններից կետերի ամպերը և ստացվում է ամբողջական 3d մոդել: 3d сканер из двух веб камер а. геворкян и а. саруханян аннотация в данной статье представлен метод построения 3d сканера при помощи двух обычных веб камер. этот метод основан на стереозрении и отличается от других 3d сканеров тем, что не требует лазера, структурированного света или других дорогих сенсоров. объект помещается на расстоянии 40-50 сантиметров перед камерами и вращается на маленькие градусы. для каждого вращения вычисляется карта смещений по двум камерам и строится карта глубины в виде облака точек. далее мы отделяем объект от сцены и соединяем облака точек объекта из разных ракурсов, чтобы получить полную 3d модель. 2. related work 2.1 calibration 2.1 rectification 2.2 stereo correspondence 2.3 triangulation 2.4 registration 3. realization 4. conclusion and future work references microsoft word w10.doc mathematical problems of computer science 37, 83--87, 2012. 83 overview of methods of biometric based key protection gurgen khachatrian and narek malkhasyan american university of armenia, institute for informatics and automation problems of nas of ra abstract the security of any modern cryptosystem relies on the assumption that secret keys used for the system such as secret keys for message encryption and authentication as well as private keys of public key cryptosystem are unknown. this assumption is not easy to satisfy in most practical applications. the most widely applicable method uses conventional passwords to encrypt secure keys stored on the computer device. however passwords are vulnerable against many kinds of attacks since they can be either guessed or stolen. another basic problem is the user authentication. it is well known that when using a traditional and widely used cryptographic methods the user authentication is achieved by challenge response protocols, the essence of which consists in verifying that the party which wants to confirm his authentication possesses a secret key. in this paper an overview of methods of password generation from biometric data is presented along with the discussion of the remaining challenges and possible directions of future research. references 1. m. maslennikov, practical cryptography, saint petersburg, 2003. 2. g. khachatrian and h. khasikyan “correlation based password generation from fingerprints”, proc. ita-2012 conference “classification, forecasting, data mining “. 3. a. juels and m. wattenberg, “a fuzzy commitment scheme”, in sixth acm conference computer and communication security”, pp. 28-36, 1999. 4. a. juels and m. sudan, “a fuzzy vault scheme” , proc. ieee international symposium on information theory” , pp.408, 2002. 5. u. uludag, s. pankanti and a. jein. “fuzzy vault for fingerprints”, lecture notes on computer science, pp. 55-71, 2005. overview of methods of biometric based key protection 84 բանալիների պաշտպանության բիոմետրիկ (կենսաչափական) մեթոդների դիտարկում գ. խաչատրյան և ն. մալխասյան ամփոփում ցանկացած ժամանակակից կրիպտոհամակարգի անվտանգությունը հիմնվում է այն ենթադրության վրա, որ համակարգում օրտագործվող բանալիները, ինչպես օրինակ՝ հաղորդագրությունների գաղտնագրման, նույնականացման և բաց բանալիով կրիպտոհամակարգերի բանալիները, անհայտ են: կիրառական համակարգերի մեծամասնությունում այս ենթադրությունը բավարարելը հեշտ չէ: ամենատարածված մեթոդը ավանդական գաղտնաբառերի օգտագործումն է համակարգչային սարքի վրա պահվող բանալին գաղտնագրելու համար: սակայն գաղտնաբառերը խոցելի են տարատեսակ հարձակումների նկատմամբ, քանի որ դրանք կարող են գուշակվել և գողացվել: մեկ այլ կարևոր խնդիր է օգտագործողների նույնականացման խնդիրը: հայտնի է, որ ավանդական և տարածված ծածկագրաբանական մեթոդներ օգտագործելիս օգտագործողի նույնականացումը իրականացվում է <<մարտահրավեր-պատասխան>> արձանագրությունների միջոցով, որոնց օգնությամբ օգտագործողը կարող է ապացուցել որևէ բանալու պատկանելությունը իրեն: սույն հոդվածում ներկայացված են բիոմետրիկ (կենսաչափական) տվյալներից գաղտնաբառերի ստացման մեթոդների, ինչպես նաև ոլորտի այլ մարտահրավերների և աշխատանքի ապագա հնարավոր ուղղությունների դիտարկումները: обзор биометрических методов защиты ключей г. хачатрян и н. малхасян аннотация безопасность всех современных криптосистем базируется на предположении, что секретные ключи используемые в системе, такие как ключи для шифрования сообщений, ключи аутентикации и ключи криптосистем с открытым ключом, неизвестны. в большинстве практических приложений удовлетворять это предположение непросто. самый распространенный метод это использование традиционных паролей для шифрования ключей хранящихся на некоторых компютерных устройствах. но пароли уязвимы множеством атак, так как они могут быть разгаданы и украдены. другая основная проблема заключается в аутентикации пользователей. хорошо известно, что при использовании традиционных и широко известных криптографических методов, аутентикация пользователей достигается при помощи протоколов “вызов-ответ”, при помощи которых пользователь может доказать принадлежность некоторого ключа ему. в этой статье представляется обзор методов генерирования паролей из биометрических данных, а также рассматриваются вызовы в этой сфере и возможные будущие пути исследования. d:\sbornik\...\tpel.dvi mathematical problems of computer science 36, 79{90, 2012. random coding b ound for secr ecy e-capacity region of the b r oadcast channel with t wo con¯dential m essages n a s r in a fs h a r , e vg u e n i h a r o u t u n ia n a n d ma r ia m h a r o u t u n ia n institute for informatics and automation problems of nas of ra e-mail: evhar@ipia.sci.am abstract we study secrecy e-capacity region of the discrete memoryless broadcast channel with two independent con¯dential messages sent to two receivers (bc-2cm). the system involves two sources, one encoder, two discrete memoryless channels and two receivers. each private message is sent to the corresponding receiver while keeping the other receiver in total ignorance of it. the level of ignorance is measured by the equivocation rate. e-capacity region is the set of rate pairs r1; r2 of codes with given error probability exponents (reliabilities) e1; e2 at respective receivers. we derived a random coding bound for secrecy e-capacity region of the bc-2cm. when error probability exponents are going to zero, this bound coincides with the inner bound of secrecy capacity region of the bc-2cm obtained by r. liu, i. maric, p. spasojevic and r. yates. key words: broadcast channel with con¯dential messages, secrecy e-capacity, equivocation rate, error probability exponent, method of types, random coding bound. refer ences [1 ] t. m. co ve r , \ b r o a d c a s t c h a n n e ls " , ie e e trans. inf. theory, vo l. it-1 8 , n o . 1 , p p . 2 -1 4 , 1 9 7 2 . [2 ] t. m. co ve r a n d j. a . th o m a s , \ e lements of information theory" , 2 n d e d it io n , a w ile y-in t e r s c ie n c e p u b lic a t io n , 2 0 0 6 . [3 ] i. cs is z ¶a r a n d j. k äo r n e r , \ b r o a d c a s t c h a n n e l wit h c o n ¯ d e n t ia l m e s s a g e s " , ie e e trans. inf. theory, vo l. it-2 4 , n o . 3 , p p . 3 3 9 -3 4 8 , 1 9 7 8 . [4 ] i. cs is z ¶a r a n d j. k äo r n e r , \ information theory: coding theorems for d iscrete m emoryless systems " , n e w y o r k, w ile y, 1 9 8 1 . [5 ] s . i. ge lfa n d a n d m. s . p in s ke r , \ ca p a c it y o f b r o a d c a s t c h a n n e l wit h o n e d e t e r m in is t ic c o m p o n e n t " , p robl. p ered. inf., vo l. 1 6 , n o . 1 , p p . 1 7 -2 5 , 1 9 8 0 . [6 ] e . a . h a r o u t u n ia n , \ u p p e r e s t im a t e o f t r a n s m is s io n r a t e fo r m e m o r yle s s c h a n n e l wit h c o u n t a b le n u m b e r o f o u t p u t s ig n a ls u n d e r g ive n e r r o r p r o b a b ilit y e xp o n e n t " , ( in r u s s ia n ) 3rd all union conference on theory of information transmission and coding, uzhgorod, p ublishing house of the uzbek academy of sciences, p p . 8 3 -8 6 , 1 9 6 7 . 7 9 8 0 random coding bound for secrecy e-capacity region of the broadcast channel [7 ] e . a . h a r o u t u n ia n , b . b e lb a s h ir , \ l o we r b o u n d o f t h e o p t im a l t r a n s m is s io n r a t e d e p e n d in g o n g ive n e r r o r p r o b a b ilit y e xp o n e n t fo r d is c r e t e m e m o r yle s s c h a n n e l a n d fo r a s ym m e t r ic b r o a d c a s t c h a n n e l" , ( in r u s s ia n ) , abstracts of p apers of 6th int. symp. inf. theory, tashkent, ussr , vo l. 1 , p p . 1 9 -2 1 , 1 9 8 4 . [8 ] e . a . h a r o u t u n ia n , \ on b o u n d s fo r e -ca p a c it y o f d mc" , ie e e trans. inf. theory, vo l. it-5 3 , n o . 1 1 , p p . 4 2 1 0 -4 2 2 0 , 2 0 0 7 . [9 ] e . a . h a r o u t u n ia n , m. e . h a r o u t u n ia n a n d a . n . h a r u t yu n ya n , \ r e lia b ilit y c r it e r ia in in fo r m a t io n t h e o r y a n d in s t a t is t ic a l h yp o t h e s is t e s t in g " , f oundations and trends in communications and information theory, vo l. 4 , n o s . 2 -3 , 2 0 0 8 . [1 0 ] e . a . h a r o u t u n ia n , m. e . h a r o u t u n ia n a n d n . a fs h a r , \ r a n d o m co d in g b o u n d fo r e-c a p a c it y r e g io n o f t h e w ir e t a p ch a n n e l" , 8th international conference of computer science and information technologies, yerevan, p p . 1 2 1 -1 2 4 , 2 0 1 1 . [1 1 ] m. e . h a r o u t u n ia n , \ r a n d o m c o d in g b o u n d fo r e-c a p a c it y r e g io n o f t h e b r o a d c a s t c h a n n e l" , m athematical p roblems of computer science, n o . 2 1 , p p . 5 0 -6 0 , 2 0 0 0 . [1 2 ] m. h a ya s h i , r . ma t s u m o t o , \ u n ive r s a lly a t t a in a b le e r r o r a n d in fo r m a t io n e xp o n e n t s fo r t h e b r o a d c a s t c h a n n e ls wit h c o n ¯ d e n t ia l m e s s a g e s " , arxiv:1104.4285v2 [cs.it], 2 4 a p r il 2 0 1 1 . [1 3 ] y . l ia n g , h . v . p o o r a n d s . s h a m a i, \ in fo r m a t io n t h e o r e t ic s e c u r it y" , f oundations and trends in communications and information theory, vo l. 5 , n o s . 4 -5 , 2 0 0 9 . [1 4 ] r . l iu , i. ma r ic , p . s p a s o je vic a n d r . y a t e s , \ d is c r e t e m e m o r yle s s in t e r fe r e n c e a n d b r o a d c a s t c h a n n e ls wit h c o n ¯ d e n t ia l m e s s a g e s : s e c r e c y r a t e r e g io n s " , ie e e trans. inf. theory, vo l. 5 4 , n o . 6 , p p . 2 4 9 3 -2 5 0 7 , 2 0 0 8 . [1 5 ] k . ma r t o n , \ a c o d in g t h e o r e m fo r t h e d is c r e t e m e m o r yle s s b r o a d c a s t c h a n n e l" , ie e e trans. inf. theory, vo l. it-2 5 , n o . 3 , p p . 3 0 6 -3 1 1 , 1 9 7 9 . [1 6 ] c. e . s h a n n o n , \ a m a t h e m a t ic a l t h e o r y o f c o m m u n ic a t io n " , b ell syst. tech. j .,| vo l. 2 7 , n o . 3 , p p . 3 7 9 -4 2 3 , 1 9 4 8 . 3 8 , n o . 5 , p p . 6 1 1 -6 5 9 , 1 9 5 9 . [1 7 ] a . d . w yn e r , \ th e wir e -t a p c h a n n e l" , b ell syst. tech. j ., vo l. 5 4 , n o . 8 , p p . 1 3 5 5 -1 3 8 7 , 1 9 7 5 . [1 8 ] j. x u , y . ca o a n d b . ch e n , \ ca p a c it y b o u n d fo r b r o a d c a s t c h a n n e ls wit h c o n ¯ d e n t ia l m e s s a g e s " , ie e e trans. inf. theory, vo l. 5 5 , n o . 1 0 , p p . 4 5 2 9 -4 5 4 2 , 2 0 0 9 . ºñïáõ ·³õïýç ñ³õáñ¹³·ñáõãûáõýý»ñáí é³ûý³ë÷ûáõé ï³åáõõáõ ·³õïýçáõãû³ý e-áõý³ïáõãû³ý ïçñáõûãç å³ï³ñ³ï³ý ïá¹³íáñù³ý ·ý³ñ³ï³ï³ýá ü. ²íß³ñ, º. ð³ñáõãûáõýû³ý ¨ ø. ð³ñáõãûáõýû³ý ²ù÷á÷áõù ø”ýù ñ»ï³½áïáõù »ýù »ñïáõ ³ýï³ë ·³õïýç ñ³õáñ¹³·ñáõãûáõýý»ñáí áý¹ñ³ï ³é³ýó ñçßáõáõãû³ý é³ûý³ë÷ûáõé ï³åáõõáõ ·³õïýçáõãû³ý e-áõý³ïáõãû³ý ïçñáõûãá: ³õïýçáõãû³ý ù³ï³ñ¹³ïá ã³÷íáõù ¿ ³ýáñáßáõãû³ý ³ñ³·áõãû³ùµ: î³éáõóí³í ¿ ·³õïýçáõãû³ý e-áõý³ïáõãû³ý ïçñáõûãç å³ï³ñ³ï³ý ïá¹³íáñù³ý ·ý³ñ³ï³ï³ýá: n. afshar, e. haroutunian and m. haroutunian 8 1 ãðàíèöà ñëó÷àéíîãî êîäèðîâàíèÿ îáëàñòè e-ïðîïóñêíîé ñïîñîáíîñòè ñåêðåòíîñòè øèðîêîâåùàòåëüíîãî êàíàëà ñ äâóìÿ ñåêðåòíûìè ñîîáùåíèÿìè í.àôøàð, å.àðóòþíÿí è ì. àðóòþíÿí àííîòàöèÿ ìû èçó÷àåì îáëàñòü ñåêðåòíîé -ïðîïóñêíîé ñïîñîáíîñòè äèñêðåòíîãî øèðîêîâåùàòåëüíîãî êàíàëà áåç ïàìÿòè ñ äâóìÿ íåçàâèñèìûìè ñåêðåòíûìè ñîîñáùåíèÿìè, ïîñûëàåìûìè äâóì àäðåñàòàì. óðîâåíü ñåêðåòíîñòè èçìåðÿåòñÿ ñêîðîñòüþ íåîïðåäåëåííîñòè. ìû ñòðîèì ãðàíèöó ñëó÷àéíîãî êîäèðîâàíèÿ äëÿ îáëàñòè ñåêðåòíîé å-ïðîïóñêíîé ñïîñîáíîñòè øèðîêîâåùàòåëüíîãî êàíàëà ñ äâóìÿ ñåêðåòíûìè ñîîáùåíèÿìè. d:\sbornik\...\12_g.dvi mathematical problems of computer science 39, 94{100, 2013. radicals and p r er adicals in the categor y of m odules over all rings gr ig o r g. e m in -te r ya n ( e m in ) institute for informatics and automation problems of nas ra e-mail: grigoremin@rambler.ru abstract let mod be a category whose objects are all possible pairs (a; u), where u is an associative ring, a is a right u-module (not unitary in the general case) and the set of morphisms of module (a; u) to module (b; v ) consists of pairs of mappings ('a; 'u ), where 'a or 'u , respectively, is a homomorphism of abelian group a to abelian group b (ring u to ring v ), where (a ¢ u)'a = a 'a ¢ u 'u , a 2 a, u 2 u. this pair of mappings is called a homomorphism of module (a; u) to module (b; v ). it is proved that strict radicals of mod in the sense of kurosh are described by means of systems of strict radicals of the category of associative rings as and categories of right u-modules mod ¡ u, u 2 as. it turned out that a wider classes of preradicals and radicals of mod can also be described by means of systems of preradicals and radicals of as and mod ¡ u, u 2 as, respectively, which satisfy some conditions. keywords: module, associative ring, abelian group, category, radical, preradical, torsion, ideally hereditary radical, strongly hereditary radical. l e t u s c o n s id e r t h e c a t e g o r y m od, wh o s e o b je c t s a r e a ll p o s s ib le p a ir s ( a; u ) , wh e r e a is a r ig h t u-m o d u le o ve r a n a s s o c ia t ive r in g u, a n d t h e m o r p h is m s a r e p a ir s ( 'a; 'u ) o f h o m o m o r p h is m s s a t is fyin g t h e c o n d it io n ( a ¢ u ) 'a = a 'a ¢ u 'u , a 2 a, u 2 u [1 ]. de¯nition 1: w e will say that a preradical r is de¯ned in the mod, if an ideal r ( a; u ) matches to each module ( a; u ) 2 mod such that r ( a; u ) ( 'a; 'u ) µ r ( b; v ) for each homomorphism ( 'a; 'u ) : ( a; u ) ! ( b; v ) of m od [2], [3]. in o t h e r wo r d s , t h e p r e r a d ic a l r is a n o r m a l s u b fu n c t o r o f t h e id e n t it y fu n c t o r idm od : mod ! m od. in e a c h m o d u le ( a; u ) 2 mod it s in g le s o u t a n id e a l r ( a; u ) s u c h t h a t fo r e ve r y h o m o m o r p h is m ( 'a; 'u ) : ( a; u ) ! ( b; v ) t h e d ia g r a m : ( a; u ) ¡! ( b; v ) " " r ( a; u ) ¡! r ( b; v ) is c o m m u t a t ive . a s s u m e t h a t in t h e c a t e g o r y mod s o m e p r e r a d ic a l r is s p e c ī e d . co n s id e r a fu ll s u b c a t e g o r y mod( as) o f m od wh o s e o b je c t s a r e a ll m o d u le s o f t h e fo r m ( 0 ; u ) . a n y id e a ls a n d 9 4 g. emin-teryan (emin) 9 5 a n y h o m o m o r p h ic im a g e s o f o b je c t s fr o m mod ( as) t h e m s e lve s lie in s u b c a t e g o r y mod( as) , a n d h e n c e , t h e p r e r a d ic a l r in d u c e s a c o m p le t e ly d e t e r m in a t e p r e r a d ic a l r in s u b c a t e g o r y mod( as) . th is m e a n s t h a t a p r e r a d ic a l r d e ¯ n e s a c o m p le t e ly d e t e r m in a t e p r e r a d ic a l r in c a t e g o r y as o f a s s o c ia t ive r in g a n d r ( 0 ; u ) = ( 0 ; r( u ) ) . lemma 1: l et r be any preradical in mod, (a,u) be any module and r ( a; u ) = ( a0; u 0 ) . then u 0 = r( u ) , where r is a preradical in as induced of preradical r of mod. p r oof. s in c e r is a p r e r a d ic a l a n d ( 0 ; u ) is a s u b m o d u le o f ( a; u ) t h e n ( 0 ; r ( u ) ) = r ( 0 ; u ) µ r ( a; u ) = ( a0; u 0 ) . h e n c e r( u ) µ u 0. b u t fo r h o m o m o r p h is m ( 0 ; 1 u ) : ( a; u ) ! ( 0 ; u ) we h a ve ( a0; u 0 ) ( 0 ; 1 u ) = r ( a; u ) ( 0 ; 1 u ) µ r ( 0 ; u ) = ( 0 ; r( u ) ) , i.e ., u 0 µ r ( u ) . l e m m a 1 : is p r o ve d . r e c a ll t h a t t h e s u b m o d u le ( a0; u 0 ) o f ( a; u ) will b e a n id e a l o f ( a; u ) if a n d o n ly if u 0 is a n id e a l o f t h e r in g u a n d t h e in c lu s io n s a ¢ u 0 µ a0 a n d a0 ¢ u µ a0 a r e o b s e r ve d [1 ]. a s s u m e r is a n y p r e r a d ic a l o f mod, ( a; u ) is a n a n y m o d u le a n d r ( a; u ) = ( a0; u 0 ) . s in c e ( a0; u 0 ) is a n id e a l o f ( a; u ) , t h e n a0 is a u -s u b m o d u le o f u-m o d u le a. n o w s h o w t h a t m a t c h in g s u b m o d u le a0 t o e a c h u -m o d u le a d e ¯ n e s t h e p r e r a d ic a l ru in t h e c a t e g o r y o f r ig h t u-m o d u le s mod ¡ u. a s s u m e a a n d b a r e u-m o d u le s , ' : a ! b is a n u-h o m o m o r p h is m a n d r ( a; u ) = ( a0; u 0 ) , r ( b; u ) = ( b0; u 0 ) , wh e r e u 0 = r( u ) b y l e m m a 1 :. w e 'll s h o w t h a t a0' µ b0. in d e e d , s in c e ( '; 1 u ) : ( a; u ) ! ( b; u ) is a h o m o m o r p h is m in mod, t h e n ( a0; u 0 ) ( '; 1 u ) = r ( a; u ) ( '; 1 u ) µ r ( b; u ) = ( b0; u 0 ) . h e n c e a0' µ b0. th u s , fo r a n y u -m o d u le a, if r ( a; u ) = ( a0; u 0 ) , m a t c h in g o f u-s u b m o d u le ru ( a ) = a 0 t o e a c h u-m o d u le a d e ¯ n e s t h e p r e r a d ic a l ru in t h e c a t e g o r y mod ¡ u. th u s , we h a ve t h e fo llo win g le m m a . lemma 2: e ach preradical r of the category m od induces a completely determinate preradical r in as and preradicals ru in the categories m od ¡ u ( u 2 as) such that r ( a; u ) = ( ru ( a ) ; r( u ) ) for any module ( a; u ) of the category m od. l e t u s c o n s id e r t h e a c t io n o f t h e r in g h o m o m o r p h is m ' : u ! v . a n y r ig h t v -m o d u le b b e c o m e s a r ig h t u -m o d u le if t h e a c t io n o f t h e o p e r a t o r s is d e ¯ n e d a s fo llo ws : a ± u = a ( u') , we will s a y t h a t t h e m o d u le b is c o n ve r t e d in t o t h e u-m o d u le b' b y wit h d r a wa l a lo n g '. a s s u m e r is a n y p r e r a d ic a l o f as a n d ru ( u 2 as) a r e p r e r a d ic a ls o f t h e c a t e g o r ie s mod ¡ u. co n s id e r t h e s ys t e m o f p r e r a d ic a ls fr; ru j u 2 asg. de¯nition 2: the system of preradicals fr; ru j u 2 asg will be called a matched system of preradicals, if it satis¯es the following conditions: (p 1) a ¢ r( u ) µ ru ( a ) for each u-module a (p 2) ru ( b' ) µ rv ( b ) ) for each v -module b and each homomorphism ' : u ! v of rings, where b' is the conversion of v -module b into an u-module by the withdrawal along '. t heor em 1: assume that the preradical r is speci¯ed in m od. then in as it induces the preradical r and in each category mod ¡ u ( u 2 as) it induces preradicals ru such that the system of preradicals fr; ru j u 2 asg is matched. conversely, every matched system of preradicals fr; ru j u 2 asg speci¯es a completely determinate preradical r in mod such that r ( a; u ) = ( ru ( a) ; r ( u ) ) for each ( a; u ) 2 mod. there exists a one-to-one correspondence between all the preradicals of m od and all the matched systems of preradicals. 9 6 radicals and preradicals in the category of modules over all rings p r oof. a s s u m e t h a t r is a p r e r a d ic a l o f mod. th e n , b y l e m m a 2 :, we h a ve a c o m p le t e ly d e t e r m in a t e s ys t e m o f p r e r a d ic a ls fr; ru j u 2 asg. s in c e fo r e a c h ( a; u ) 2 mod, r ( a; u ) = ( ru ( a ) ; r ( u ) ) is a n id e a l o f m o d u le ( a; u ) , h e n c e a ¢ r( u ) µ ru ( a ) . th u s , t h e c o n d it io n ( p 1 ) is s a t is ¯ e d . a s s u m e t h a t b is a v -m o d u le , ' : u ! v is a h o m o m o r p h is m o f r in g s , a n d b' is t h e c o n ve r s io n o f v -m o d u le b in t o a n u -m o d u le b y wit h d r a wa l a lo n g '. a s t h e p a ir ( 1 ; ') : ( b'; u ) ! ( b; v ) is a h o m o m o r p h is m in mod, t h e n ( ru ( b' ) ; r ( u ) ) ( 1 ; ') = r ( b'; u ) ( 1 ; ') µ r ( b; v ) = ( rv ( b ) ; r ( v ) ) . co n ve r s e ly, a s s u m e t h a t we a r e g ive n a m a t c h e d s ys t e m o f p r e r a d ic a ls fr; ru j u 2 asg a n d ( a; u ) 2 mod. s in c e r ( u ) is a n id e a l o f r in g u, ru ( a) is a s u b m o d u le o f u-m o d u le a a n d a ¢ r( u ) µ ru ( a) b y c o n d it io n ( p 1 ) , t h e n t h e s u b m o d u le ( ru ( a) ; r( u ) ) is a n id e a l o f t h e m o d u le ( a; u ) . l e t ( '; ã ) : ( a; u ) ! ( b; v ) b e a n y h o m o m o r p h is m . it is c le a r t h a t r ( u ) ã µ r ( v ) . on t h e o t h e r h a n d , it is e a s ily s e e n t h a t t h e h o m o m o r p h is m ( '; ã ) = ( '; 1 ) ¢ ( 1 ; ã ) , wh e r e ( '; 1 ) : ( a; u ) ! ( bã; u ) a n d ( 1 ; ã ) : ( bã; u ) ! ( b; v ) . s in c e ru is a p r e r a d ic a l in mod¡u a n d ( '; 1 ) is a n u -h o m o m o r p h is m , t h e n ru ( a) ' µ ru ( bã ) . a n d fr o m t h e c o n d it io n ( p 2 ) it fo llo ws t h a t ru ( bã ) µ rv ( b ) . h e n c e ru ( a) ' µ rv ( b ) . th e t h ir d a s s e r t io n o f t h is t h e o r e m fo llo ws fr o m t h e ¯ r s t t wo o n e s , s in c e t h e e xp r e s s io n s g ive n in t h e t h e o r e m a b o ve a r e m u t u a lly in ve r s ive . th u s , t h e p r o o f o f t h e t h e o r e m is c o m p le t e d . remar k 1 it is e a s y t o ve r ify [2 ] t h a t if r is a p r e r a d ic a l in m od, t h e n r ( a; u ) in c lu d e s a ll t h e r-s u b m o d u le s o f t h e m o d u le ( a; u ) fo r e a c h m o d u le ( a; u ) . a ls o t h e c la s s o f r-r a d ic a l m o d u le s is e n c lo s e d u n d e r t h e o p e r a t io n o f h o m o m o r p h ic im a g e s . de¯nition 3: a preradical r of mod is called a radical if r ( ( a; u ) =r ( a; u ) ) = ( 0 ; 0 ) for each module ( a; u ) [4]. s im ila r t o l e m m a s 1 : a n d 2 :, o n e m a y p r o ve t h e fo llo win g le m m a . lemma 3: l et r be a radical in mod, ( a; u ) be some module and let r ( a; u ) = ( a0; u 0 ) . then u 0 = r ( u ) , where r is the radical in as induced of the radical r of m od. lemma 4: e very radical r of the category mod induces a completely determinate radical r in as and radicals ru in categories mod ¡ u ( u 2 as) such that r ( a; u ) = ( ru ( a) ; r( u ) ) for any module ( a; u ) of the category mod. a s s u m e t h a t r is a r a d ic a l o f as a n d ru ( u 2 as) a r e r a d ic a ls o f c a t e g o r ie s m od ¡ u. co n s id e r t h e s ys t e m o f r a d ic a ls fr; ru j u 2 asg. de¯nition 4: the system of radicals fr; ru j u 2 asg is called a matched system of radicals if it satis¯es the following conditions: (p 1) a ¢ r ( u ) µ ru ( a ) for each u-module a (p 2) ru ( b' ) µ rv ( b ) ) for each v -module b and each homomorphism ' : u ! v of rings, where b' is the conversion of v -module b into an u-module by the withdrawal along '. g. emin-teryan (emin) 9 7 (p 3) ru=r(u) ( a) = ru ( a¼ ) ) for each u=r( u ) -module a, where ¼ : u ! u=r ( u ) is a natural epimorphism and a¼ is the conversion of u=r ( u ) module a into u-module by withdrawal along ¼. t heor em 2: assume that some radical r is speci¯ed in mod. then it induces the radical r in as and radicals ru in each category m od ¡ u, u 2 as such that the system of radicals fr; ru j u 2 asg is matched. conversely, every matched system of radicals fr; ru j u 2 asg speci¯es a completely determinate radical r in m od such that r ( a; u ) = ( ru ( a) ; r ( u ) ) for each ( a; u ) 2 mod. there exists a one-to-one correspondence between all the radicals of mod and all the matched systems of radicals. p r oof. a s s u m e t h a t r is a r a d ic a l in mod a n d fr; ru j u 2 asg is a c o m p le t e ly d e t e r m in e d s ys t e m o f p r e r a d ic a ls . ob vio u s ly, if r is a r a d ic a l in mod, r is a r a d ic a l in as. mo r e o ve r , ( ru=r(u) ( a=ru ( a ) ) ; r( u=r ( u ) ) ) = r ( ( a; u ) =r ( a; u ) ) = ( 0 ; 0 ) , i.e . ru=r(u ) ( a=ru ( a ) ) = 0 fo r e a c h ( a; u ) 2 mod. h e n c e , a s it fo llo ws fr o m c o n d it io n ( p 2 ) ru ( a=ru ( a ) ) = 0 fo r e ve r y u-m o d u le a. l e t ¼ : u ! u=r( u ) b e a n a t u r a l e p im o r p h is m a n d a a n a r b it r a r y u=r ( u ) -m o d u le . co n s id e r u -m o d u le a¼ in wh ic h a ± u = a ¤ ( u + r ( u ) ) , a 2 a, u 2 u b y d e ¯ n it io n . ob vio u s ly, e ve r y s u b m o d u le o f u-m o d u le a¼ is a s u b m o d u le o f u=r ( u ) -m o d u le a. s in c e a ± r ( u ) = 0 , t h e r e ve r s e c o n ve r s io n , i.e . c o n ve r s io n t o o p e r a t io n a ¤ ( u + r( u ) ) = a ± u, a 2 a, u 2 u a ls o h o ld s . th e r e fo r e u-m o d u le a¼ is t r a n s fo r m e d in t o a u=r( u ) m o d u le a, a n d t h e s u b m o d u le s a n d fa c t o r m o d u le s o f u-m o d u le a¼ a r e t r a n s fo r m e d in t o s u b m o d u le s a n d fa c t o r m o d u le s o f u=r ( u ) -m o d u le a. co n s id e r a n a t u r a l u -e p im o r p h is m µ : a¼ ! a¼=ru ( a¼ ) . ob vio u s ly, fo r t r a n s fe r t o ¤ o p e r a t io n t h e m a p p in g µ b e c o m e s a n u=r ( u ) -h o m o m o r p h is m . th u s , b e c a u s e ru=r(u) is a p r e r a d ic a l o f mod ¡ u=ru we h a ve ru=r(u ) ( a) µ µ ru=r(u) ( a¼=ru ( a¼ ) ) . on t h e o t h e r h a n d , a s r is a r a d ic a l o f m od, t h e n in vie w o f a s s e r t io n a b o ve ru=r(u ) ( a¼=ru ( a¼ ) ) = 0 . h e n c e ru=r(u ) ( a) µ = 0 , i.e . ru=r(u) ( a) µ ru ( a¼ ) fo r e a c h u=r ( u ) -m o d u le a. th e r e fo r e , b y c o n d it io n ( p 2 ) ru ( a¼ ) µ ru=r(u) ( a ) fo r e a c h u=r ( u ) -m o d u le a. co n ve r s e ly, le t ( a; u ) b e a r a n d o m m o d u le o f mod. s in c e r is a r a d ic a l o f as, t h e n r ( ( a; u ) =r ( a; u ) ) = ( ru=r(u) ( a=ru ( a ) ; 0 ) ) . a n a p p lic a t io n o f c o n d it io n ( p 3 ) t o u=r( u ) m o d u le a=ru ( a ) yie ld s ru=r(u) ( a=ru ( a ) ) = ru ( a=ru ( a) ) = 0 , a s ru is a r a d ic a l o f mod ¡ u . th e t h ir d a s s e r t io n in t h is t h e o r e m fo llo ws fr o m th e o r e m 1 :. th e t h e o r e m is p r o ve d . de¯nition 5: a preradical r of m od is called idempotent, if r ( r ( a; u ) ) = r ( a; u ) for each module ( a; u ) . t heor em 3: the matched system of preradicals fr; ru j u 2 asg de¯nes an idempotent preradical of the category mod if and only if all preradicals of that system are idempotent and the condition (p 4) ru ( a) = rr(u) ( a) holds for each u-module a. p r oof. a s s u m e t h a t r is a n id e m p o t e n t p r e r a d ic a l o f m od a n d fr; ru j u 2 asg is t h e c o r r e s p o n d in g m a t c h e d s ys t e m o f p r e r a d ic a ls . b e c a u s e r ( 0 ; u ) = ( 0 ; r ( u ) ) fo r a n y ( 0 ; u ) 2 mod( as) a n d r is id e m p o t e n t , r will b e a ls o id e m p o t e n t . th e r e fo r e , t h e id e m p o t e n c e o f r 9 8 radicals and preradicals in the category of modules over all rings m e a n s t h a t ( ru ( a) ; r ( u ) ) = ( rr(u) ( ru ( a ) ) ; r( u ) ) , i.e . ru ( a) = rr(u) ( ru ( a ) fo r e a c h m o d u le ( a; u ) o f m od. on t h e o t h e r h a n d , it fo llo ws fr o m ( p 2 ) t h a t rr(u ) ( ru ( a ) ) µ ru ( ru ( a ) ) . h e n c e ru ( a ) = rr(u) ( ru ( a) ) µ ru ( ru ( a ) ) µ ru ( a ) , i.e . ru ( ru ( a ) ) = ru ( a) fo r e a c h um o d u le a. a s we h a ve a lr e a d y s e e n if r is id e m p o t e n t , ru ( a) = rr(u) ( ru ( a) ) fo r e a c h u -m o d u le a. on t h e o t h e r h a n d fo r e a c h u -m o d u le a, ru ( a) is a r ( u ) -s u b m o d u le o f r( u ) -m o d u le a, b e c a u s e ru ( a ) ¢ r ( u ) µ a ¢ r( u ) µ ru ( a) b y c o n d it io n ( p 1 ) . h e n c e rr(u ) ( ru ( a ) ) µ rr(u) ( a) . w e o b t a in t h a t ru ( a ) µ rr(u ) ( a ) . b u t in vie w o f ( p 2 ) rr(u ) ( a) µ ru ( a) . h e n c e ru ( a ) = rr(u ) ( a ) fo r e a c h u -m o d u le a. co n ve r s e ly, a s s u m e t h a t fr; ru j u 2 asg is a m a t c h e d s ys t e m o f id e m p o t e n t p r e r a d ic a ls wh ic h s a t is fy t h e c o n d it io n ( p 4 ) a n d ( a; u ) is a m o d u le o f t h e c a t e g o r y m od. fr o m ( p 4 ) , wh ic h is u s e d fo r u -m o d u le ru ( a ) , it fo llo ws t h a t rr(u) ( ru ( a ) ) = ru ( ru ( a ) ) = ru ( a ) b e c a u s e ru is a n id e m p o t e n t p r e r a d ic a l in mod-u . th u s , t h e p r o o f o f t h e t h e o r e m is c o m p le t e d a s r( r ( u ) ) = r( u ) fo r e a c h u 2 as. de¯nition 6: the radical r of the category mod is called ideally hereditary if r ( a0; u 0 ) = r ( a; u ) \ ( a0; u 0 ) for all ideals ( a0; u 0 ) of ( a; u ) 2 mod [2], [5]. remar k 2 th e r a d ic a l r0 o f t h e s ys t e m fr; ru j u 2 asg is a r a d ic a l o f t h e c a t e g o r y o f a b e lia n g r o u p ab. s in c e r0 ( a ) ´ µ r0 ( a) fo r e ve r y a b e lia n g r o u p a a n d e ve r y e n d o m o r p h is m ´ o f a, t h e s u b g r o u p r0 ( a) will b e a n u-s u b m o d u le fo r e a c h u -m o d u le a. th u s , it is e a s y t o p r o ve t h a t r0 is a r a d ic a l in a ll t h e c a t e g o r ie s mod ¡ u ( u 2 as) . th e s ys t e m o f r a d ic a ls fr; ru j u 2 asgin wh ic h ru = r0 fo r a ll u 2 as t u r n s t o a p a ir o f fr; r0g. a s in s u c h s ys t e m s t h e c o n d it io n s ( p 2 ) a n d ( p 3 ) a r e e xe c u t e d a u t o m a t ic a lly, t h e n t h e y will b e m a t c h e d if t h e y s a t is fy t h e o n ly c o n d it io n : a ¢ r ( u ) µ r0 ( a ) fo r e a c h u-m o d u le a, i.e . t h e o n ly c o n d it io n ( p 1 ) . t heor em 4: assume that r is an ideally hereditary radical in mod. then in as it induces an ideally hereditary radical r and in ab it induces a torsion rab such that the pair fr; rabg of radicals is matched. conversely, every matched pair of radicals fr; rabg, where r is an ideally hereditary radical in as and rab is a torsion in ab, speci¯es a completely determinate ideally hereditary radical r in mod whose radical class includes all modules ( a; u ) such that u = r( u ) and a = rab ( a ) . there exists a one-to-one correspondence between all ideally hereditary radicals of mod and all such matched pairs. p r oof. l e t r b e a n id e a lly h e r e d it a r y r a d ic a l in mod a n d le t fr; ru ju 2 asg b e it s m a t c h e d s ys t e m o f r a d ic a ls . w e will p r o ve t h a t t h is m a t c h e d s ys t e m , wh ic h in p a r t ic u la r , s a t is ¯ e s t h e c o n d it io n ( p 1 ) will b e a p a ir o f fr; r0g, wh e r e r is a n id e a lly h e r e d it a r y r a d ic a l in as a n d r0 is a t o r s io n in ab. l e t u b e a n a s s o c ia t ive r in g a n d a b e a n y u -m o d u le . co n s id e r t h e m o d u le ( a; 0 ) . it is e a s y t o p r o ve t h a t ( a; 0 ) is a n id e a l o f m o d u le ( a; u ) . th e r e fo r e , a s r is a n id e a lly h e r e d it a r y r a d ic a l in mod, ( r0 ( a ) ; 0 ) = r ( a; 0 ) = ( a; 0 ) t r ( a; u ) = ( a t ru ( a ) ; 0 ) = ( ru ( a ) ; 0 ) , i.e . ru ( a ) = r0 ( a ) fo r e a c h u -m o d u le a, wh e r e u 2 as. th u s ru = r0 fo r e a c h u 2 as. n o w le t u b e a n a s s o c ia t ive r in g a n d v b e a n y id e a l o f u. it is o b vio u s t h a t t h e m o d u le ( 0 ; v ) is a n id e a l o f t h e m o d u le ( 0 ; u ) . th e r e fo r e , a s r is a n id e a lly h e r e d it a r y r a d ic a l in mod, ( 0 ; r ( v ) ) = r ( 0 ; v ) = ( 0 ; v ) t r ( 0 ; u ) ) = ( 0 ; v t r ( u ) ) . h e n c e r( v ) = v t r ( u ) , i.e . r is a n id e a lly h e r e d it a r y r a d ic a l in as. g. emin-teryan (emin) 9 9 s im ila r ly, c o n s id e r in g t h e m o d u le s ( a; 0 ) , we c a n p r o ve t h a t r0 is a t o r s io n in ab. co n ve r s e ly, le t t h e p a ir o f r a d ic a ls fr; r0g s a t is fy t h e c o n d it io n o f t h e t h e o r e m . a s t h is p a ir s a t is ¯ e s c o n d it io n ( p 1 ) , it will b e a m a t c h e d s ys t e m o f r a d ic a ls a c c o r d in g t o t h e r e m a r k a b o ve . th e r e fo r e , t h e c o n s id e r e d s ys t e m will d e ¯ n e a c o m p le t e ly d e t e r m in a t e r a d ic a l r in mod. n o w s h o w t h a t r is a n id e a lly h e r e d it a r y r a d ic a l. l e t ( a; u ) b e a m o d u le o f m od a n d ( b; v ) b e a n id e a l o f ( a; u ) . s in c e v is a n id e a l o f t h e r in g u , b is a s u b g r o u p o f a b e lia n g r o u p a, r is a n id e a lly h e r e d it a r y r a d ic a l in as a n d r0 is a t o r s io n in ab, r ( v ) = v t r ( u ) a n d r0 ( b ) = b t r0 ( a ) . th u s r ( b; v ) = ( r0 ( b ) ; r ( v ) ) = ( b t r0 ( a ) ; v t r( u ) ) = ( b; v ) t r ( a; u ) . th e t h ir d a s s e r t io n o f t h e t h e o r e m fo llo ws fr o m t h e ¯ r s t t wo s in c e t h e e xp r e s s io n s g ive n t h e r e a r e m u t u a lly in ve r s e . de¯nition 7: the radical r in the sense of k urosh [1] is called strict if r-radical r ( a; u ) of any module ( a; u ) contains all r-submodules of ( a; u ) . de¯nition 8: the radical r of the category mod is called s t r o n g ly h e r e d it a r y if r ( a1; u1 ) = r ( a; u ) t ( a1; u1 ) for all submodules ( a1; u1 ) of ( a; u ) 2 m od. it is e a s y t o p r o ve t h a t e ve r y s t r o n g ly h e r e d it a r y r a d ic a l is a s t r ic t r a d ic a l. in t h e s a m e wa y we m a y p r o ve t h e fo llo win g t h e o r e m . t heor em 5: assume that the strongly hereditary radical r is speci¯ed in mod. then in as it induces a strongly hereditary radical r and in ab it induces a torsion rab such that the pair fr; rabg of strongly hereditary radicals is matched. conversely, every matched pair of strongly hereditary radicals fr; rabg speci¯es a completely determinate strongly hereditary radical r in m od whose radical class includes all modules ( a; u ) such that u = r( u ) and a = rab ( a) . there exists a one-to-one correspondence between all strongly hereditary radicals of mod and all matched pairs of strongly hereditary radicals. refer ences [1 ] g.g. e m in , \ p r e m a n ifo ld s , g r o u p p o id o f m a n ifo ld s a n d s t r ic t r a d ic a ls in t h e c a t e g o r y o f m o d u le s o ve r a ll r in g s " , izv. an arm. ssr ser. matem., vo l. 1 4 , n o .3 , p p .2 1 1 -2 3 2 , 1 9 7 9 ; soviet j ournal of contemporary m athematical analysis (armenian academy of sciences), by allerton p ress, in c . n e w-y o r k, p p .5 2 -7 2 , 1 9 8 0 . [2 ] a . i. k a s h u , r a d ic a ls a n d to r s io n s in t h e m o d u le s , ( in r u s s ia n ) , iz d . s h t iin z a , k is h in e v, 1 9 8 3 . [3 ] g. g. e m in -te r ya n ( e m in ) , \ id e m p o t e n t p r e r a d ic a ls in t h e c a t e g o r y o f m o d u le s o ve r a ll r in g s " . p roceedings of csit, a r m e n ia , y e r e va n , p p . 6 1 -6 4 , 2 0 0 5 . [4 ] g. e m in -te r ya n ( e m in ) , \ r a d ic a ls in t h e c a t e g o r y o f m o d u le s o ve r a ll r in g s " p roceedings of csit, a r m e n ia , y e r e va n , p p . 3 5 -3 6 , 2 0 0 7 . [5 ] g. e m in -te r ya n ( e m in ) , \ h e r e d it a r y r a d ic a ls in t h e c a t e g o r y o f m o d u le s o ve r a ll r in g s " , p roceedings of csit, a r m e n ia , y e r e va n , p p . 4 4 -4 5 , 2 0 1 1 . submitted 23.11.2012, accepted 18.02.2013. 1 0 0 radicals and preradicals in the category of modules over all rings è³¹çï³éý»ñá ¨ ùçýãé³¹çï³éý»ñá µáéáñ ûõ³ïý»ñç íñ³ ùá¹áõéý»ñç ï³ï»·áñç³ûáõù ¶. ¾ùçý-î»ñû³ý ²ù÷á÷áõù ¸çï³ñïíáõù ¿ µáéáñ ûõ³ïý»ñç íñ³ ùá¹áõéý»ñç m od ï³ï»·áñç³ý: ²û¹ ï³ï»·áñç³ûç ûµû»ïïý»ñá µáéáñ ³ûýåçëç ( a; u ) ½áõû·»ñý »ý, áñï»õ u -ý ³ëáóç³ïçí ûõ³ï ¿ ¨ ³ýå³ûù³ý ã¿, áñ áõý»ý³ ùç³íáñ, çëï a-ý` ³ç u -ùá¹áõé : àý¹ëùçý, ( a; u ) ùá¹áõéçó ¹»åç ( b; v ) ùá¹áõé ùáñýç½ùý»ñç µ³½ùáõãûáõýá µ³õï³ó³í ¿ ( 'a; 'u ) ½áõû·»ñçó, áñï»õ 'a-ý ³µ»éû³ý a ëùµç ñáùáùáñýç½ù ¿ ¹»åç b ëáõùµ, 'u -ý u ûõ³ïç ñáùáùáñýç½ù ¿ ¹»åç v ûõ³ï ¨ ( a ¢ u) 'a = a'a ¢ u'u , a 2 a, u 2 u : ²µ»éû³ý ëùµ»ñç ab ï³ï»·áñç³ý ¨ ³ëáóç³ïçí ûõ³ïý»ñç as ï³ï»·áñç³ý modç éñçí »ýã³ï³ï»·áñç³ý»ñ »ý, ùçýã¹»é mod ¡ u ï³ï»·áñç³ý ýçùëí³í u ûõ³ïç íñ³ ³ç ùá¹áõéý»ñç ï³ï»·áñç³ý ¿, áñï»õ u -ý ó³ýï³ó³í ³ëáóç³ïçí ûõ³ï ¿` mod-ç »ýã³ï³ï»·áñç³: ú·ï³·áñí»éáí mod ï³ï»·áñç³ûç »ýã³ï³ï»·áñç³ý»ñç» ”áëï ß»ñï»ñç” ý»ñï³û³óù³ý ñ»õçý³ïç ïáõùçó ³é³ç³ñïí³í ù»ãá¹á, ñý³ñ³íáñ ¹³ñó³í ýï³ñ³·ñ»é ³û¹ ï³ï»·áñç³ûç é³¹çï³éý»ñá ¨ ùçýãé³¹çï³éý»ñá: ¸ñ³ýù ýï³ñ³·ñí³í »ý ³ëáóç³ïçí ûõ³ïý»ñç ¨ ýçùëí³í ûõ³ïç íñ³ ùá¹áõéý»ñç é³¹çï³éý»ñç, ñ³ù³å³ï³ëë³ý³µ³ñ, ùçýãé³¹çï³éý»ñç ñ³ù³ó³ûý»óí³í, ³ûëçýùý, ñ³ù³å³ï³ëë³ý å³ûù³ýý»ñçý µ³í³ñ³ñáõ ñ³ù³ï³ñ·»ñç û·ýáõãû³ùµ: ðàäèêàëû è ïðåäðàäèêàëû â êàòåãîðèè ìîäóëåé íàä âñåìè êîëüöàìè ã. ýìèí-òåðüÿí (ýìèí) àííîòàöèÿ ðàññìàòðèâàåòñÿ êàòåãîðèÿ mod ìîäóëåé íàä âñåìè êîëüöàìè. îáúåêòû ýòîé êàòåãîðèè âñåâîçìîæíûå ïàðû ( a; u ) , ãäå u àññîöèàòèâíîå êîëüöî, íå îáÿçàòåëüíî ñ åäèíèöåé, a-ïðàâûé u -ìîäóëü. ìíîæåñòâî ìîðôèçìîâ ìîäóëÿ ( a; u ) â ìîäóëü ( b; v ) ñîñòîèò èç ïàð ( 'a; 'u ) , ãäå 'a ãîìîìîðôèçì àáåëåâîé ãðóïïû a â àáåëåâó ãðóïïó b, à 'u ãîìîìîðôèçì êîëüöà u â êîëüöî v , ïðè÷åì ( a ¢ u ) 'a = a 'a ¢ u 'u , a 2 a, u 2 u. êàòåãîðèÿ àáåëåâûõ ãðóïï ab è êàòåãîðèÿ àññîöèàòèâíûõ êîëåö as ÿâëÿþòñÿ ïîëíûìè ïîäêàòåãîðèÿìè êàòåãîðèè m od. äëÿ ëþáîãî àññîöèàòèâíîãî êîëüöà u êàòåãîðèÿ m od ¡ u ïðàâûõ ìîäóëåé íàä ôèêñèðîâàííûì êîëüöîì u ÿâëÿåòñÿ ïîäêàòåãîðèåé êàòåãîðèè m od. ñ ïîìîùüþ ïðåäëîæåííîãî àâòîðîì ìåòîäà ”ïîñëîéíîãî” ïðåäñòàâëåíèÿ èññëåäóåìûõ ïîäêàòåãîðèé êàòåãîðèè m od îïèñàíû ðàäèêàëû è ïðåäðàäèêàëû ýòîé êàòåãîðèè. îíè îïèñàíû ñ ïîìîùüþ ñîãëàñîâàííûõ, òî åñòü, óäîâëåòâîðÿþùèõ íåêîòîðûì óñëîâèÿì ñèñòåì ðàäèêàëîâ, ñîîòâåòñòâåííî, ïðåäðàäèêàëîâ êàòåãîðèé àññîöèàòèâíûõ êîëåö è ìîäóëåé íàä ôèêñèðîâàííûì êîëüöîì. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 48, 89--97, 2017. modeling of compliance processes with specified criteria in complex dynamic systems ruben b. aghajanyan yerevan state university e-mail: ruboo1993@gmail.com abstract the urgency of research and modeling of complex systems lies in the fact that it is necessary to effectively manage such systems, to predict their development, to use corrective and preventive actions, to eliminate undesirable phenomena. it is necessary to ensure that the system parameters meet the specified requirements and criteria [1],[2],[3],[4]. the complexity of research and modeling of such systems is as follows: increase in the number of components and links in the system complication of internal random relationships existence of stochastic processes changes in the requirements for compliance of key indicators with specified criteria. the considered class of complex systems is encountered in practice in a number of industries in which strict requirements are imposed on the parameters of the system, to their estimates. for example, to indicators of the quality of products (aircraft construction, management of complex equipment, production of food and drugs, health). for such systems, the problem of identifying deviations, estimating the indices of individual parameters for the objects of the system and methods of applying corrective actions for their elimination is urgent. topical for the class of systems under consideration are the problems of self-regulation and self-development. keywords: dynamic network systems, deviations and inconsistencies, dynamic evaluation criteria, weight coefficients of groups of criteria, corrective and preventive actions (capa, corrective and preventive action). 1. introduction the main problem in the development and implementation of various methods for estimating parameters when deviations and inconsistencies are detected in complex systems is the duration and laboriousness of the corresponding processes. without developed technological 89 mailto:ruboo1993@gmail.com modeling of compliance processes with specified criteria in complex dynamic systems 90 tools, it is not always possible to quickly analyze a large number of dynamic estimates and make the right timely decision. in fig. 1, for the considered class of complex systems, a diagram of interacting processes for estimating parameters and monitoring their compliance with the established criteria is shown. fig. 1. sequence of control and analysis of key parameters. the main topic of this article is the study of the automation of the procedure for performing parameter estimation in dynamic complex systems when nonconformities are detected for making decisions about carrying out the necessary studies, obtaining estimates and performing corrective and preventive actions (capa) in order to eliminate the detected inconsistencies. 2. mathematical model for estimating parameters in complex systems first, we introduce the following notation in the language of set theory: 𝑆𝑆 = {𝑠𝑠1, 𝑠𝑠2 … , 𝑠𝑠𝑛𝑛} a finite set of components of the system to be evaluated, 𝐺𝐺 = {𝐺𝐺1, 𝐺𝐺2 … , 𝐺𝐺𝑚𝑚} a finite set of groups of criteria in which each element 𝐺𝐺𝑖𝑖 is in turn a set the elements of which are a collection of certain similar (with the same characteristic parameters) criteria , i.e., 𝐺𝐺1 = �𝑔𝑔11, 𝑔𝑔12, … �, 𝐺𝐺1 = �𝑔𝑔21, 𝑔𝑔22, … �, … 𝐺𝐺𝑖𝑖 = �𝑔𝑔𝑖𝑖1, 𝑔𝑔𝑖𝑖2, … �there are sets of different groups of criteria. to each 𝐺𝐺𝑖𝑖 we assign a certain weighting coefficient 𝜆𝜆(𝐺𝐺𝑖𝑖) ∈ 𝜆𝜆, where 𝜆𝜆(𝐺𝐺𝑖𝑖) = (0 ÷ 1). we define a binary function 𝐹𝐹 that can take the value 0 or 1 and is defined on the set 𝐺𝐺𝑖𝑖, 𝐹𝐹(𝐺𝐺𝑖𝑖) = 𝑁𝑁′ ∈ 𝑁𝑁 , where 𝑁𝑁′𝑖𝑖 = {0,1}. (1) we also introduce a set of standard estimates in the form of ordered subsets of natural numbers 𝐾𝐾 = {𝐾𝐾1,𝐾𝐾2,… 𝐾𝐾𝑧𝑧}, (2) where 𝐾𝐾𝑧𝑧 ⊂ n & (𝐾𝐾1 ∩ 𝐾𝐾2 ∩ … ∩ 𝐾𝐾𝑧𝑧 = {∅}&(𝐾𝐾1 ∪ 𝐾𝐾2 ∪ … ∪ 𝐾𝐾𝑧𝑧! = {∅}. the elements of each subset 𝐾𝐾𝑧𝑧 is the ordered sequence of natural numbers {𝑚𝑚1, 𝑚𝑚2 … , 𝑚𝑚𝑛𝑛} from the set of 𝑁𝑁, where 𝑚𝑚𝑗𝑗 = 𝑚𝑚𝑖𝑖 + 1, 𝑗𝑗 = 𝑖𝑖 + 1 . 𝐾𝐾𝑖𝑖 will be called the i-th correspondence class. for each element s we have a chain of averaged values for each group 𝐺𝐺𝑖𝑖: 𝐹𝐹(𝑠𝑠𝑚𝑚, 𝐺𝐺𝑖𝑖) = (𝑓𝑓(𝑔𝑔𝑖𝑖1, 𝑠𝑠𝑚𝑚) + (𝑓𝑓(𝑔𝑔𝑖𝑖2, 𝑠𝑠𝑚𝑚) … + (𝑓𝑓(𝑔𝑔𝑖𝑖𝑖𝑖, 𝑠𝑠𝑚𝑚) ), (3) • definition of the correspondence class. • expert opinion • calculation of new values of weight coefficients and evaluation criteria • calculation of estimates according to given criteria • calculation of estimates • system state analysis • selection of the sequence of criteria • defining assessment classes selection of evaluation criteria calculation of estimated values comparison with given values formation of new indicators r. aghajanyan 91 we define for sm an integral estimate with allowance for the weight coefficients 𝜆𝜆, 𝐼𝐼(𝑠𝑠𝑚𝑚) = � (𝐹𝐹(𝑠𝑠𝑚𝑚, 𝐺𝐺𝑖𝑖)𝜆𝜆𝑖𝑖) 𝑘𝑘 𝑖𝑖=1 , (4) where 𝑚𝑚 = {0, 1, … 𝑛𝑛} – the number of components in the system, 𝑘𝑘 = {0, 1, … 𝑧𝑧} – the number of set elements 𝐾𝐾 (2). finally, we get a couple of expressions that describe the procedure for obtaining a quantitative assessment and monitoring compliance with specified criteria։ 𝐼𝐼(𝑠𝑠𝑚𝑚) = � (𝐹𝐹(𝑠𝑠𝑚𝑚, 𝐺𝐺𝑖𝑖)𝜆𝜆𝑖𝑖) 𝑘𝑘 𝑖𝑖=1 𝑄𝑄(𝐼𝐼(𝑠𝑠𝑚𝑚)) = 𝐾𝐾′ ⊂ 𝐾𝐾𝑖𝑖 the problem is to determine for the integral estimate 𝐼𝐼(𝑠𝑠𝑚𝑚) belonging to one of the classes of correspondences 𝐾𝐾𝑖𝑖 . for each 𝑠𝑠𝑛𝑛 element, it is necessary to execute the sequence of the following 5 procedures: 1) generating a value chain 𝐹𝐹(𝐺𝐺𝑖𝑖1), 𝐹𝐹(𝐺𝐺2), … 𝐹𝐹(𝐺𝐺𝑖𝑖), (1) where 𝑖𝑖 – the number of criteria groups 2) calculation of the average values for each group, taking into account the weighting factors: 𝐹𝐹(𝑠𝑠𝑚𝑚, 𝐺𝐺1)𝜆𝜆1, (𝑠𝑠𝑚𝑚, 𝐺𝐺2)𝜆𝜆2,… 𝐹𝐹(𝑠𝑠𝑚𝑚, 𝐺𝐺𝑖𝑖)𝜆𝜆𝑖𝑖 , where 𝐹𝐹(𝑠𝑠𝑚𝑚, 𝐺𝐺𝑖𝑖) – the mean value according to expression (3) 3) calculation of the integral sum 𝐼𝐼(𝑠𝑠𝑚𝑚), according to expression (4) 4) the definition of 𝐾𝐾𝑖𝑖 (i=1,2…z) for which 𝐼𝐼(𝑠𝑠𝑚𝑚) ∈ 𝐾𝐾𝑖𝑖, i.e., the definition of the group that includes the given integral sum 𝐼𝐼(𝑠𝑠𝑚𝑚) (5) 5) generation of actions (expert opinion) according to the estimation of 𝐼𝐼(𝑠𝑠𝑚𝑚) and the corresponding group 𝐾𝐾𝑖𝑖 a practical example (pharmaceutical production) consider the foregoing with the example of evaluating and selecting a supplier of ingredients and packaging materials in a pharmaceutical company. the task is to obtain in a timely manner a possible objective integral evaluation with subsequent periodic reassessment that would help to make the right decision when choosing a supplier. the production of pharmaceuticals by pharmaceutical enterprises is a chain of interrelated processes from the choice of the supplier and the organization of the supply of ingredients to the production, packaging and supply of medicines to the consumer. all these processes must meet the quality requirements, according to the gmp standard. ingredients and packaging materials are the most important components of modern medicines and their manufacturers have to be cautious in choosing suppliers. among the most important aspects of supplier selection for manufacturers are the following: information about the supplier, its reputation and position in the market, the availability of quality certificates, financial statements, customer references and other available information, negative processes related to quality violations, deviations from specified parameters, etc. conditions for providing a guarantee for the delivered products, responsibility for the quality of the products supplied, readiness to assist in eliminating inconsistencies and compensating losses that arose, for example, during transportation terms of payment, product cost, cost and delivery time, quality and timeliness of registration of documentation for product registration and customs operations estimating the total cost of the cost of purchasing the ingredients, taking into account transportation costs, tax duties, the cost of excise taxes, registration certificates, insurance premiums and other payments (5) modeling of compliance processes with specified criteria in complex dynamic systems 92 types of packages of supplied ingredients and materials in terms of ease of transportation, storage and packaging compliance of the declared quality indicators with their actual estimated values during the period of interaction between the customer and the supplier estimation of profitability of production taking into account expenses for ingredients and packaging materials. the task is to obtain in a timely manner a possible objective integral evaluation with subsequent periodic reassessment that would help to make the right decision when choosing a supplier [5]. there are many vendor classification systems. such systems depend entirely on the competence and qualifications of relevant specialists in the pharmaceutical, food and other industries. each company needs to create its own acceptable classification system. one of the classification options for the pharmaceutical industry is presented in table 1. [5]. consider the application of the proposed algorithm for this classification option. table 1: vendor classification method. n topic (group of criteria) criteria (gij) values f(gi) (0,1) values 𝜆𝜆𝑖𝑖 (0-1) 1. quality level (𝐺𝐺1) • compliance with the specification parameters (𝑔𝑔11) • compliance with quality system requirements (𝑔𝑔12) • availability of certificates of conformity (𝑔𝑔13) • access to the product dossier (𝑔𝑔14) • product stability (𝑔𝑔15) • quality index by product series (𝑔𝑔16) 𝑓𝑓(𝑔𝑔11) 𝑓𝑓(𝑔𝑔12) 𝑓𝑓(𝑔𝑔13) 𝑓𝑓(𝑔𝑔14) 𝑓𝑓(𝑔𝑔15) 𝑓𝑓(𝑔𝑔16) 1 2. production level (𝐺𝐺2) • availability of full production cycle (𝑔𝑔21) • competence of the staff (𝑔𝑔22) • level and potential of development (𝑔𝑔23) 𝑓𝑓(𝑔𝑔21) 𝑓𝑓(𝑔𝑔22) 𝑓𝑓(𝑔𝑔23) 0.8 3. logistics (𝐺𝐺3) • types and dimensions of packaging (𝑔𝑔31) • offered service (𝑔𝑔32) • periodicity of delivery (𝑔𝑔33) • transport condition (𝑔𝑔34) • geographical location (𝑔𝑔35) • reliability of delivery (𝑔𝑔36) 𝑓𝑓(𝑔𝑔31) 𝑓𝑓(𝑔𝑔32) 𝑓𝑓(𝑔𝑔33) 𝑓𝑓(𝑔𝑔34) 𝑓𝑓(𝑔𝑔35) 𝑓𝑓(𝑔𝑔36) 0.7 4. marketing (𝐺𝐺4) • contract pric (𝑔𝑔41) • terms of payment (𝑔𝑔42) • proposed assortment today (𝑔𝑔43) • the claimed asortment of tomorrow (𝑔𝑔44) • image and reputation (𝑔𝑔45) • customer orientation (𝑔𝑔46) 𝑓𝑓(𝑔𝑔41) 𝑓𝑓(𝑔𝑔42) 𝑓𝑓(𝑔𝑔43) 𝑓𝑓(𝑔𝑔44) 𝑓𝑓(𝑔𝑔45) 𝑓𝑓(𝑔𝑔46) 0.3 r. aghajanyan 93 it is also necessary to specify several characteristic groups with given ranges of numbers that define the group to which the supplier in question can enter. for example, table 2: characteristic groups and their metrics. group metric (value range) characteristic 𝐾𝐾1 <5 does not meet the specified criteria 𝐾𝐾2 5-8 requires revision 𝐾𝐾3 8-10 partially matched 𝐾𝐾4 10-14.4 totally coincides then it is necessary to calculate the average value for all four groups of criteria: 1. 𝐴𝐴𝐴𝐴. (𝐺𝐺1) = (f(𝑔𝑔11) + f(𝑔𝑔12) + f(𝑔𝑔13) + f(𝑔𝑔14) + f(𝑔𝑔15) + f(𝑔𝑔16)) /6 2. 𝐴𝐴𝐴𝐴. (𝐺𝐺2) = (f(𝑔𝑔21) + f(𝑔𝑔22) + f(𝑔𝑔23))/3 3. 𝐴𝐴𝐴𝐴. (𝐺𝐺3) = (f(𝑔𝑔31) + f(𝑔𝑔32) + f(𝑔𝑔33) + f(𝑔𝑔34) + f(𝑔𝑔35) + f(𝑔𝑔36))/6 4. 𝐴𝐴𝐴𝐴. (𝐺𝐺4) = (f(𝑔𝑔41) + f(𝑔𝑔42) + f(𝑔𝑔43) + f(𝑔𝑔44) + f(𝑔𝑔45) + f(𝑔𝑔46))/6 further taking into account the given weight coefficients for each group of criteria, it is necessary to calculate the integral estimate according to a pair of expressions (5) : 𝐼𝐼𝑛𝑛𝐼𝐼. = 𝐴𝐴𝐴𝐴. (𝐺𝐺1) ∗ 1 + 𝐴𝐴𝐴𝐴. (𝐺𝐺2) ∗ 0.8 + 𝐴𝐴𝐴𝐴. (𝐺𝐺3) ∗ 0.7 + 𝐴𝐴𝐴𝐴. (𝐺𝐺4) ∗ 0.3 now it is necessary to check which group from the given set 𝐾𝐾 = {𝐾𝐾1, 𝐾𝐾2, 𝐾𝐾3,𝐾𝐾4 contains the obtained estimate to take appropriate action: 𝐼𝐼𝑛𝑛𝐼𝐼. ∈ 𝐾𝐾1 -> the supplier does not meet the specified criteria 𝐼𝐼𝑛𝑛𝐼𝐼. ∈ 𝐾𝐾2 -> data on the supplier need to be revised, it may be required to request additional materials 𝐼𝐼𝑛𝑛𝐼𝐼. ∈ 𝐾𝐾3 -> the supplier partially meets the specified criteria 𝐼𝐼𝑛𝑛𝐼𝐼. ∈ 𝐾𝐾4 -> the supplier fully meets the specified criteria let's pay attention to two basic singularities of similar systems of estimations: a. the risk of obtaining inaccurate data due to subjectivity in the development of a system of criteria and the designation of weighting factors. as noted above, you often have to rely on the professionalism of the staff involved in developing the criteria and the vendor evaluation system. b. when certain events occur, verification is required not at all but at certain criteria, i.е., the choice of the evaluation criteria system is dynamic. 3. model with dynamic definition of control criteria in order to take into account the above-mentioned peculiarities of the class of systems under consideration, we introduce an additional set of states 𝐶𝐶{𝑐𝑐1, 𝑐𝑐2 … } and the set of mappings 𝑊𝑊 = {𝑤𝑤1, 𝑤𝑤2 … , 𝑤𝑤𝑛𝑛} , where 𝑤𝑤𝑖𝑖(𝐶𝐶 → 𝐺𝐺1, 𝑖𝑖 = (1,2, … 𝑛𝑛) ), (6) modeling of compliance processes with specified criteria in complex dynamic systems 94 for which to a certain ci there corresponds a chain of the following subsets (𝐺𝐺1′ ⊂ 𝐺𝐺1) ∩ (𝐺𝐺2′ ⊂ 𝐺𝐺2) ∩ (𝐺𝐺3′ ⊂ 𝐺𝐺3) ∩ … ∩ (𝐺𝐺𝑖𝑖 ′ ⊂ 𝐺𝐺𝑖𝑖) , it is possible 𝐺𝐺𝑖𝑖 = {∅} for anyone (𝑖𝑖 = 1,2, … 𝑛𝑛). by the states ci we mean the appearance in the system of some deviations (inconsistencies), in the result of which it is required to test by groups of criteria according to (expressions 3,4,5). in practice, this may be, for example the inconsistency of the equipment's indicators with the specified permissible values or the breakdown of individual components, which may suspend some stages of the production process, etc. in this case, the problem (item "a") can be formulated as follows: determine the state of the system ci, choose the corresponding subsets of the criteria 𝐺𝐺1′, 𝐺𝐺2′ … and apply the above algorithm for a given chain of criteria. to determine the weighting coefficients (subsection "b"), we present the following procedure. suppose for a certain period n production processes were performed, during which, 𝑀𝑀(𝑐𝑐𝑖𝑖) times the deviation of 𝑐𝑐𝑖𝑖 from the set values was observed. at the same time, testing with the given criterion showed k times the discrepancy, i.e., 𝑓𝑓�𝑔𝑔𝑖𝑖𝑗𝑗� takes the value "0" (see expression 1), then 𝑀𝑀(𝑐𝑐𝑖𝑖)/𝑁𝑁 will be the probability of occurrence of the event 𝑐𝑐𝑚𝑚. and 𝑘𝑘(𝑐𝑐𝑖𝑖, 𝑔𝑔𝑖𝑖𝑗𝑗)/𝑀𝑀(𝑐𝑐𝑖𝑖) is the probability of occurrence of the discrepancy by the criterion 𝑔𝑔𝑖𝑖𝑗𝑗 at the occurrence of the event 𝑐𝑐𝑖𝑖. the total probability of non-compliance with the 𝑔𝑔𝑖𝑖𝑗𝑗 criterion during the n production processes will be: 𝑃𝑃(𝑐𝑐𝑖𝑖, 𝑔𝑔𝑖𝑖𝑗𝑗) = 𝑀𝑀(𝑐𝑐𝑖𝑖) 𝑁𝑁 ∗ 𝐾𝐾�𝑐𝑐𝑖𝑖,𝑔𝑔𝑖𝑖𝑖𝑖� 𝑀𝑀(𝑐𝑐𝑖𝑖) . (6) in this case, the value of the weighting coefficient for the given criteria 𝑔𝑔𝑖𝑖𝑗𝑗 from 𝐺𝐺𝑖𝑖 ′ at occurrence of the events ci can be calculated as follows: 𝜆𝜆′𝑖𝑖(𝑐𝑐𝑖𝑖) = � 𝑃𝑃�𝑔𝑔𝑖𝑖𝑖𝑖,𝑐𝑐𝑖𝑖� 𝑚𝑚 𝑚𝑚 𝑗𝑗=1 , (7) where 𝑗𝑗 = (1 ÷ 𝑚𝑚) , mthe number of criteria in a group 𝐺𝐺𝑖𝑖. similarly, calculating for the criteria of the group 𝐺𝐺𝑖𝑖 when other events 𝑐𝑐𝑗𝑗 ∈ 𝐶𝐶 occur, for which the condition 𝑤𝑤𝑖𝑖(𝐶𝐶 → 𝐺𝐺𝑖𝑖) is satisfied, we obtain the final value of the weight coefficient for the group 𝐺𝐺𝑖𝑖: 𝜆𝜆𝑖𝑖 = 𝑠𝑠𝑠𝑠𝑚𝑚(𝜆𝜆′𝑖𝑖(𝑐𝑐1𝑖𝑖) + 𝜆𝜆′𝑖𝑖(𝑐𝑐2𝑖𝑖) + ⋯ + 𝜆𝜆′𝑖𝑖(𝑐𝑐𝑚𝑚𝑖𝑖). (8) in this case, the expressions (5) describing the model of quantitative estimation and control of compliance with the given criteria for 𝑠𝑠𝑚𝑚 will take the following form: 𝐼𝐼(𝑠𝑠𝑚𝑚) = �(𝐹𝐹(𝑠𝑠𝑚𝑚, 𝐺𝐺𝑖𝑖)𝜆𝜆𝑖𝑖), 𝑘𝑘 𝑖𝑖=1 𝑄𝑄(𝐼𝐼(𝑠𝑠𝑚𝑚)) = 𝐾𝐾′ ⊂ 𝐾𝐾𝑖𝑖, 𝜆𝜆𝑖𝑖 = 𝑠𝑠𝑠𝑠𝑚𝑚(𝜆𝜆′𝑖𝑖(𝑐𝑐1𝑖𝑖) + 𝜆𝜆′𝑖𝑖(𝑐𝑐2𝑖𝑖) + ⋯ + 𝜆𝜆′𝑖𝑖(𝑐𝑐𝑚𝑚𝑖𝑖), 𝜆𝜆′𝑖𝑖(𝑐𝑐𝑖𝑖) = � 𝑃𝑃�𝑔𝑔𝑖𝑖𝑗𝑗, 𝑐𝑐𝑖𝑖� 𝑚𝑚 𝑚𝑚 𝑗𝑗=1 . the corresponding procedure can be specified in the form of the following 6 steps: r. aghajanyan 95 1. analysis of the state of the system and dynamic generation of the chain of criteria (𝐺𝐺1′ ⊂ 𝐺𝐺1) ∩ (𝐺𝐺2′ ⊂ 𝐺𝐺2) ∩ (𝐺𝐺3′ ⊂ 𝐺𝐺3) ∩ … ∩ (𝐺𝐺𝑖𝑖 ′ ⊂ 𝐺𝐺𝑖𝑖) , where “i” is the number of groups of evaluation criteria defined for a given state. 2. calculation of the average values for each group, taking into account the weighting factors: 𝐹𝐹1(𝑠𝑠𝑚𝑚, 𝐺𝐺1) ∗ 𝜆𝜆1, 𝐹𝐹2(𝑠𝑠𝑚𝑚, 𝐺𝐺2) ∗ 𝜆𝜆2 … 𝐹𝐹𝑖𝑖(𝑠𝑠𝑚𝑚, 𝐺𝐺𝑖𝑖) ∗ 𝜆𝜆𝑖𝑖 , where 𝐹𝐹𝑖𝑖(𝑠𝑠𝑚𝑚, 𝐺𝐺𝑖𝑖) – the mean value according to expression (1). 3. calculation of the sum 𝐼𝐼(𝑠𝑠𝑚𝑚) according to expression (2). 4. definition 𝐾𝐾𝑖𝑖, for which 𝐼𝐼(𝑠𝑠𝑚𝑚) ∈ 𝐾𝐾𝑖𝑖, where 𝑖𝑖 =1,2,3,4,5, 6, ie., the definition of the group that includes this integral sum 𝐼𝐼(𝑠𝑠𝑛𝑛). 5. generation of actions (expert opinion) according to the estimation of 𝐼𝐼(𝑠𝑠𝑚𝑚) and the corresponding group 𝐾𝐾𝑖𝑖. 6. formation of new probabilistic values for determining the chain of evaluation criteria and their weight coefficients. below is a structural and functional diagram of an information system (software) that implements interrelated processes of evaluation and control of the parameters of the class of complex systems under consideration. fig. 2. functional components of the information system. objects and evaluation criteria estimated values of parameters decision management generation of weighting coefficients registration of the onset of events creating lists of control objects generating lists of criteria and their classification by topic creating of weight coefficients for groups of criteria log valuation values calculating average values of object parameters calculation of the amount taking into account the average values and weights formation of evaluation groups for monitoring parameter values selection of the object of control selecting a list of criteria the definition of an evaluation group for each sum of values generation of corrective actions according to the evaluation group logging the results of the functioning of the system calculation of probabilistic values for the chain of criteria and weighting coefficients modeling of compliance processes with specified criteria in complex dynamic systems 96 4. conclusion in this paper we propose a model for estimating the parameters of complex systems and detecting nonconformities (deviations) to a given set of dynamically generated criteria. based on the described model, a method is proposed for determining corrective and preventive actions to eliminate the detected discrepancies. during the operation of the system, experimental data were obtained, which make it possible to calculate the probabilistic values of the interrelated evaluation criteria and their weight coefficients. the software that implements this method was introduced in pharmaceutical manufacturing companies. the main result obtained in the course of operation is the prompt elimination of inconsistencies and the reduction of current expenses by 45-50% due to 1) reducing the time for calculating the monitored parameters of the nonconformity detection system and taking decisions on the implementation of capa actions 2) reducing the risk of inconsistencies and deviations of the main parameters from the specified criteria. references [1] v. a. ostrejkovsky, theory of systems, m .: high school, 1997. [2] b.ya. sovetov, modeling of systems, moscow: higher school, 2001. [3] m. mesarovic, d. mako and i. tahakara, the theory of hierarchical multi-level systems, moscow: mir, 1978. [4] j. peterson, petri nets theory and systems modeling, moscow: mir, 1984. . [5] a. v. aleksandrov, “gmp: evaluation and audit of a pharmaceutical company supplier”, "industrial review", vol. 6, no. 11, pp. 42--45, 2008. submitted 08.10.2017, accepted 05. 12.2017. բարդ դինամիկ համակարգերում համապատասխանության վերահսկման պրոցեսների մոդելավորման մեթոդներ ռ. աղաջանյան ամփոփում բարդ համակարգերի հետազոտման և մոդելավորման կարևորությունը պայմանավորված է այդ համակարգերի արդյունավետ կառավարման անհրաժեշտությամբ, դրանց զարգացման կանխատեսման և գործունեության մեջ շեղումների վերացման ուղղությամբ։ համակարգի հիմնական խնդիրն է ապահովել համապատասխանությունը հիմնական ցուցանիշների և տրված չափորոշիչների միջև։ նման համակարգերի r. aghajanyan 97 հետազոտման և մոդելավորման բարդությունը պայմանավորված է մի շարք պատճառներով, որոնց թվում` համակարգում կոմպոնենտների քանակի ավելացումը մեծ թվով բարդ ներքին ստոխաստիկ կապերի առկայությունը պատահական պրոցեսների առկայությունը հիմնական ցուցանիշների համապատասխանության պահանջների փոփոխություն դիտարկվող բարդ համակարգերը հանդիպում են մի շարք ոլորտներում (բարդ տեխնիկայի կառավարում, սննդամթերքի և դեղերի արտադրություն, առողջապահություն), որոնց պարամետրերի նկատմամբ դրված են խիստ պահանջներ, օրինակ՝ արտադրանքի որակի չափանիշների նկատմամբ։ շեղումների հայտնաբերումը, համակարգի պարամետրերի ցուցանիշների գնահատումը և կանխարգելիչ մեթոդների կիրառումը համարվում են արդի նման համակարգերի ուսումնասիրման և մոդելավորման համար։ метод моделирования процессов контроля соответствия заданным критериям в сложных динамических системах р. агаджанян аннотация актуальность исследования и моделирования сложных систем обусловлена прежде всего необходимостью эффективного управления подобными системами, прогнозированием их развитием и использованием корректирующих и превентивных действий для устранения нежелательных явлений в их функционировании, обеспечении соответствия параметров системы заданным требованиям и критериям. сложность исследования и моделирования подобных систем обусловлена рядом причин, среди которых можно выделить следующие: увеличение числа компонент и звеньев системы наличие большого количества сложных внутренних случайных взаимосвязей наличие стохастических процессов изменения в требованиях на соответствие ключевых показателей заданным критериям. рассматриваемый класс сложных систем встречается на практике в ряде отраслях, в которых предъявляются строгие требования к параметрам системы, их оценкам, например к показателям качества выпускаемой продукции (авиастроение, управление сложным оборудованием, производство пищевых продуктов и медикаментов, здравоохранение). для подобных систем является актуальной задача выявления отклонений, оценка показателей отдельных параметров для объектов (звеньев) системы и методы применения корректирующих действий для их устранения. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 47, 68--75, 2017. performances of factorizations of complex hermitian matrices in the architecture of gpu accelerator hrachya v. astsatryan and edita e. gichunts institute for informatics and automation problems of nasra e-mail: hrach@sci.am, editagich@ipia.sci.am abstract linear algebra 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 factorizations are very important and widely used in scientific and engineering calculations. the factorizations with bunch-kaufmann and aasen’s algorithms, as well as the 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 factorization without pivoting also belong to this class of factorizations.the implementation of the mentioned three factorizations in hybrid architecture are presented in this work, and also performances are given on the nvidia tesla k40c graphic processor using the magma library for complex hermitian matrices. keywords: magma library, ldlt factorization, aasen's algorithm, bunchkaufman algorithm, hermitian matrix,triangular matrix. 1. introduction solutions of linear system of equations of hermitian matrices have repeatedly been used in physics. to calculate the solutions of linear system of equations of 𝐴𝐴𝐴𝐴 = 𝑏𝑏 form, the classical method turns 𝐴𝐴 matrix into the following 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 analysis form: 𝑃𝑃𝐴𝐴𝑃𝑃𝑇𝑇 = 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 ,where 𝐿𝐿 is the unit lower triangular, 𝐿𝐿 is the block diagonal with either 1-by-1 or 2-by-2 diagonal blocks, and 𝑃𝑃 is a permutation matrix to ensure the numerical stability of the factorization. the factorizations with bunch-kaufmann and aasen’s algorithms, as well as the 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 factorization without pivoting are very popular. in this paper we present the mentioned factorizations for complex hermitian matrices. novelty of this work is the realization of 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 factorization by the aasen’s algorithm on gpu with the application of magma library, as it has just been integrated into the magma 2.0.1 package. the realizations of 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 factorizations without pivoting and with bunch-kaufmann algorithm are also presented in this work, in order to have a complete understanding of performances of 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 factorizations of complex hermitian matrices. note also that this problem is presented in cases of complex single and complex double precisions. the work contains the following sections: the first sectionis introduction, the second one describes the steps of the above mentioned algorithms in cpu / gpu hybrid architecture. the 68 h.astsatryan and e. gichunts 69 third section contains the experiment results, where the graphs of performance and time of those algorithms are shown, as well as the analysis of comparisons of factorizations are given. the fourth section is the conclusion based on the experiment results. 2. ldlt factorizationin cpu/gpu hybrid architecture solutions of linear system of equations of the form 𝐴𝐴 ∗ 𝑍𝑍 = 𝐵𝐵 are being used in science on repeated occasions, where 𝐴𝐴 is a complex hermitian matrix, and 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 matrix factorization is used. describe three cases of these factorizations in hybrid architecture that are resolved with bunch-kaufmann and aasen’s algorithms, as well as the case without pivoting. lapack library is used in the systems with shared memory, where as in hybrid systems its parallelized magma library is used. 2.1aasen’s algorithm aasen’s algorithm [1] factorizes 𝐴𝐴 into an 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 decomposition of the form 𝑃𝑃𝐴𝐴𝑃𝑃𝑇𝑇 = 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 ,where 𝑃𝑃 is a permutation matrix, 𝐿𝐿 is a unit lower triangular matrix, and 𝐿𝐿 is a hermitian matrix. to exploit the memory hierarchy on a modern computer,a partitioned-version of aasen’s algorithm was recently proposed [2]. this algorithm first factorizes a panel in a left-looking fashion, and then uses blas-3 operations toupdate the trailing submatrix in a right-looking way. the blocked-version of aasen’s algorithm computes an 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 factorization of 𝐴𝐴, where 𝐿𝐿 is a banded matrix of theband width equal to the block size. aasen’s algorithm is a chetrf_aasen subprogram of magma library for complex hermitian matrices. chetrf_aasen computes the factorization of a complex hermitian matrix 𝐴𝐴 based on a communication-avoiding variant of the aasen's algorithm. the form of the factorization is as follows: 𝐴𝐴 = 𝑈𝑈 ∗ 𝐿𝐿 ∗ 𝑈𝑈 ∗∗ 𝐻𝐻 if uplo = magmaupper, or 𝐴𝐴 = 𝐿𝐿 ∗ 𝐿𝐿 ∗ 𝐿𝐿 ∗∗ 𝐻𝐻 if uplo = magmalower, where 𝑈𝑈 (or 𝐿𝐿) is a product of permutation and unit upper (lower) triangular matrices, and 𝐿𝐿 is hermitian and banded matrix of theband width equal to the block size. there are three cases of 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 factorization computing by the aasen’s algorithm which areright-and left-looking algorithms and a blocked left-looking version of the algorithm. factorization by the aasen’s algorithmis implemented with the left-looking algorithm in cpu / gpu hybrid architecture. the intermediate heisenberg matrix 𝐻𝐻 is used to calculate the 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 factorization that is defined by 𝐻𝐻 = 𝑇𝑇𝐿𝐿𝑇𝑇 . in the initial step the matrix 𝐴𝐴 is completely transferred from the cpu to the gpu memory through magma_csetmatrix_async () function. the algorithm consists of the following sequence: 𝐻𝐻(1: 𝑗𝑗 − 1, 𝑗𝑗) is calculated and 𝑇𝑇(𝑗𝑗, 𝑗𝑗) is updated: initially the aasen’s algorithm calculates the 𝑗𝑗−𝑡𝑡ℎ column of 𝐻𝐻, that is: 𝐻𝐻(𝑖𝑖, 𝑗𝑗) = 𝑇𝑇(𝑖𝑖, 𝑖𝑖 − 1) ∗ 𝐿𝐿(𝑗𝑗, 𝑖𝑖 − 1)′ + 𝑇𝑇(𝑖𝑖, 𝑖𝑖) ∗ 𝐿𝐿(𝑗𝑗, 𝑖𝑖)′ + 𝑇𝑇(𝑖𝑖, 𝑖𝑖 + 1) ∗ 𝐿𝐿(𝑗𝑗, 𝑖𝑖 + 1)′ , performanses of fuctorizations of complex hermitian matrices in the architecture of gpu accelerator 70 wherei is modified from 1 to 𝑗𝑗 − 1 . using magma_cgemm () function for matrix multiplication the (𝑗𝑗; 𝑗𝑗)𝑡𝑡ℎ element of 𝐻𝐻 will be the (𝑗𝑗; 𝑗𝑗)𝑡𝑡ℎ element of the equation 𝐴𝐴 = 𝐿𝐿𝐻𝐻 . in the next step 𝑇𝑇(𝑗𝑗, 𝑗𝑗) will be the (𝑗𝑗; 𝑗𝑗)𝑡𝑡ℎ element of the equation 𝐻𝐻 = 𝑇𝑇𝐿𝐿𝑇𝑇, namely 𝑇𝑇(𝑗𝑗, 𝑗𝑗) = 𝐴𝐴(𝑗𝑗, 𝑗𝑗) − 𝐿𝐿(𝑗𝑗, 1: 𝑗𝑗) ∗ 𝐻𝐻(1: 𝑗𝑗, 𝑗𝑗) is calculated using magma_cher2k () function, which performs the hermitian rank-2k update, and magmablas_csymmetrize () function, which copies the lower triangle to upper triangle, or vice-versa, to make 𝑑𝑑𝐴𝐴 a general representation of a symmetric matrix. if 𝑗𝑗 > 1 , then 𝑇𝑇(𝑗𝑗, 𝑗𝑗) = 𝑇𝑇(𝑗𝑗, 𝑗𝑗) − 𝐿𝐿(𝑗𝑗, 𝑗𝑗) ∗ 𝑇𝑇(𝑗𝑗, 𝑗𝑗 − 1) ∗ 𝐿𝐿(𝑗𝑗, 𝑗𝑗 − 1)′ is calculated. 2.2bunch-kaufman algorithm bunch-kaufmann algorithmhas a very wide application in the solution of linear system of equations of hermitian matrices through 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 factorization. the bunch-kaufman method performs the following decomposition: 𝑃𝑃𝐴𝐴𝑃𝑃𝑇𝑇 = 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 ,where 𝐿𝐿 is an n-by-n lower triangular matrix with a unit diagonal, and 𝐿𝐿 is a block diagonal matrix with either 1-by-1 or 2-by-2 diagonal blocks [3]. the pivoting strategies to compute the permutation matrix 𝑃𝑃 for the 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 factorization include complete pivoting (bunch-parlett algorithm) [4], partial pivoting (bunchkaufman algorithm) [5], rook pivoting (bounded bunch-kaufman) [6, p. 523]. in particular, the bunch-kaufman and rook pivoting are implemented in lapack [7]. a diagonal pivoting algorithm, the subprogram of which has the name_hetrf() in lapack library, is generally available by netlib [8]. a concise matrix-notation description of this algorithm can be found in [9]. bunch-kaufman algorithm is chetrf subprogram of magma library for complex hermitian matrices. chetrf computes the factorization of a complex hermitian matrix ausing the bunchkaufman diagonal pivoting method. the form of the factorization is 𝐴𝐴 = 𝑈𝑈 ∗ 𝐿𝐿 ∗ 𝑈𝑈𝐻𝐻 if uplo = magmaupper, or 𝐴𝐴 = 𝐿𝐿 ∗ 𝐿𝐿 ∗ 𝐿𝐿𝐻𝐻 if uplo = magmalower, where 𝑈𝑈 (or 𝐿𝐿) is a product of permutation and unit upper (lower) triangular matrices, and 𝐿𝐿 is hermitian and block diagonal with1-by-1 and 2-by-2 diagonal blocks. this is the blocked version of the algorithm, calling level 3 blas. in the first step of hybrid cpu / gpu programming the triangular matrix moves from cpu to gpu through magma_csetmatrix_async () function. if uplo = magmaupper, then the upper triangular matrix 𝐴𝐴 moves to gpu, and if uplo = magmalower, then the lower triangular matrix 𝐴𝐴 moves to gpu. afterwards 𝑈𝑈 ∗ 𝐿𝐿 ∗ 𝑈𝑈′ or 𝐿𝐿 ∗ 𝐿𝐿 ∗ 𝐿𝐿′ factorization of triangular matrix 𝐴𝐴 is carried out in the following way. in case ofthe upper triangular matrix each 𝑘𝑘 − 𝑘𝑘𝑏𝑏 + 1 ∶ 𝑘𝑘 column of the matrix is factorized, as well as uses a blocked code to update the columns 1 ∶ 𝑘𝑘 − 𝑘𝑘𝑏𝑏 through the following magma_clahef_gpu() function. magma_clahef_gpu() computes a partial factorization of a complex hermitian matrix 𝐴𝐴 using the bunch-kaufman diagonal pivoting method. 𝐾𝐾 is the main loop index, decreasing from 𝑁𝑁 to 1 in steps of 𝐾𝐾𝐵𝐵, where 𝐾𝐾𝐵𝐵 is the number of columns factorized by magma_clahef_gpu(). in case of the lower triangular matrix each 𝑘𝑘 − 𝑘𝑘𝑏𝑏 + 1: 𝑘𝑘 column of the matrix is factorized, as well as a blocked code to update the h.astsatryan and e. gichunts 71 columns 𝑘𝑘 + 𝑘𝑘𝑏𝑏 ∶ 𝑛𝑛 . 𝐾𝐾 is the main loop index, increasing from 1 to 𝑁𝑁 in steps of 𝐾𝐾𝐵𝐵, where 𝐾𝐾𝐵𝐵 is the number of columns factorized by magma_clahef_gpu( ). 2.3 ldlt factorizations with no pivoting this method of factorization for complex hermitian matrices is performed by chetrf_nopiv() function of magma library. chetrf_nopiv computes the 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 factorization of a complex hermitian matrix 𝐴𝐴. the factorization has the form 𝐴𝐴 = 𝑈𝑈𝐻𝐻 ∗ 𝐿𝐿 ∗ 𝑈𝑈 if uplo = magmaupper, or 𝐴𝐴 = 𝐿𝐿 ∗ 𝐿𝐿 ∗ 𝐿𝐿𝐻𝐻 if uplo = magmalower, where 𝑈𝑈 is an upper triangular matrix, 𝐿𝐿 is a lower triangular, and 𝐿𝐿 is a diagonal matrix. this is the block version of the algorithm, calling level 3 blas. in hybrid cpu/gpu programming 𝐴𝐴 = 𝑈𝑈′ ∗ 𝐿𝐿 ∗ 𝑈𝑈 factorization without pivoting will be performed in the following sequence. the matrix completely moves to gpu memory. after wards, all the 𝐴𝐴(𝑗𝑗, 𝑗𝑗) diagonal elements in the main cycle move to cpu memory. cpu is used to calculate the 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 factorization of the diagonal block. this diagonal block on cpu is factorized through magma_chetrf_nopiv_cpu () function. once the resulting 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 factors of the diagonal block are copied back to the gpu, the corresponding off-diagonal blocks of the 𝐿𝐿 -factor are computed by the triangular solve on the gpu. then the copy ofj column of 𝑈𝑈 is sent to cpu. compute the off-diagonal blocks of current block column by magma_ctrsm() function. after wards update the trailing submatrix with 𝐿𝐿 by magmablas_clascl_diag() function. finally, update each block column of the trailing submatrix, calling a matrix-matrix multiply on the gpu. 𝐴𝐴 = 𝐿𝐿 ∗ 𝐿𝐿 ∗ 𝐿𝐿′ factorization in similar sequence will be performed for the lower triangular matrix. 3. experimental results the experiments were conducted on nvidia k40c gpu. the architecture of tesla k40c consists of 2880 cuda processor cores. it is endowed with much higher bandwidth 288 gb/s of message transfer between cpu and gpu, having 12 gb of global memory, gddr5 memory interface, and cuda c programming environment. the operation system of tesla k40 is ubuntu 14.04.2 lts.cuda7 programming environment was used for the realization of programs. magma 2.0.1 package was installed in accordance with cuda7 environment. for the compilation of magma library the lapack-3.4.2, clapack-3.2.1 and atlas-3.10.0 packages were installed. gcc-4.8, gfortran-4.8, g ++ 4.8 and nvcc compilers were used. such references were made in make.inc file on libf77blas.a, libcblas.a, libf2c.a, libcublas.so, libcudart.so, libm.a, libstdc ++.so, libpthread.so, libdl.so, libcusparse.so static and dynamic libraries. magma 2.0.1 package contains libmagma.a and libmagma_sparse.a libraries. figures 1 and 2 show the time and performance graphs of factorizations with the aasen’s algorithm, bunch-kaufmann algorithm and 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 factorization without pivoting for complex single precision. performanses of fuctorizations of complex hermitian matrices in the architecture of gpu accelerator 72 fig. 1. complex single precision fig. 2. complex single precision in case of complex single precisionat most 𝑁𝑁 = 35840 -dimensional matrix can be sent to tesla k40c graphical processor. the experiment results show that in case of up to 𝑁𝑁 = 16384 dimensional incoming matrix the 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 factorization without pivoting, both in terms of time and performance, exceeds the aasen’s factorization algorithm for 5 times and the bunch-kaufmann factorization algorithm – for 3 times. where as the bunch-kaufmann algorithm exceeds the aasen’s algorithm for 3 times. increasing the incoming matrix up to 𝑁𝑁 = 35840 dimension the bunch-kaufmann algorithm is approaching the factorization without pivoting, and it turns out that they exceed the aasen’s factorization algorithm for 3 times both in terms of time and performance. figures 3 and 4 show the time and performance graphs of the mentioned three factorizations for complex double-precision. 0 20 40 60 80 100 120 140 0 10000 20000 30000 40000 aasen bunch-kauf trf_nopiv n ti m e (s ec on ds ) 0 200 400 600 800 1000 1200 1400 1600 1800 0 10000 20000 30000 40000 aasen bunch-kauf trf_nopiv n g flo p/ s h.astsatryan and e. gichunts 73 fig. 3. complex double precision fig. 4. complex double precision and in the case of complex double precision at most 𝑁𝑁 = 23552 -dimensional matrix can be sent to tesla k40c graphical processor. the experiment results show that in this casethe 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 factorization without pivoting in terms of time and performance exceeds the aasen’s factorization algorithm for 5 times and the bunch-kaufmann factorization algorithm – for 2 times, whereas the bunch-kaufmann algorithm, in terms of both time and performance, exceeds the aasen’s algorithm for 2.5 times. 4. conclusion we presented the performances of complex hermitian matrix 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 factorizations in cpu/gpu hybrid architecture. we also presented the performance results of factorizations with aasen’s algorithm, bunch-kaufmann algorithm and 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 factorization without pivoting using magma library. based on the obtained results we came to the conclusion that the 𝐿𝐿𝐿𝐿𝐿𝐿𝑇𝑇 factorization without pivotingis in a leading position by its high performance. in case of complex single 0 10 20 30 40 50 60 70 80 90 0 10000 20000 30000 aasen bunch-kauf trf_nopiv n ti m e (s ec on ds ) 0 200 400 600 800 1000 1200 0 10000 20000 30000 aasen bunch-kauf trf_nopiv n g flo p/ s performanses of fuctorizations of complex hermitian matrices in the architecture of gpu accelerator 74 precision it equalsthe bunch-kaufmann algorithm along with the increase of the incoming matrix, and in the case of complex double precision it exceeds for 2 times. it exceeds the aasen’s factorization algorithm for 5 times in complex single and double precisions cases. the latter concedes the bunch-kaufmann algorithm in both cases for 2.5-3 times. references [1] j. aasen, “on the reduction of a symmetric matrix to tridiagonal form”, bit 11, pp. 233242, 1971. [2] m. rozlo˘zn´ık, g. shklarski and s. toledo, “partitioned triangular tridiagonalization”, acm trans. math. softw.,vol. 37, no. 4, pp. 1–16, 2011. [3] g. golub and ch. van loan, matrix computations, john hopkins university press, baltimore, second edition, 1989. [4] j. r. bunch and b. n. parlett,“direct methods for solving symmetric indefinite systems of linear equations”, siam j. numerical analysis, vol. 8,pp. 639-655, 1971. [5] j. r. bunch and l. kaufman,“some stable methods for calculating inertia andsolving symmetric linear systems”, mathematics of computation, vol. 31,pp. 163-179, 1977. [6] c. ashcraft, r. g. grimes and j. g. lewis,“accurate symmetric indefinite linear equation solvers”, siam j. matrix anal. and appl., vol. 20, no. 2, pp. 513-561, 1998. [7] e. anderson, z. bai, j. j. dongarra, a. greenbaum, a. mckenney, j. ducroz, s. hammarling, j. w. demmel, c. bischof and d. sorensen, “lapack:a portable linear algebra library for high-performance computers”, proceedings of the1990 acm/ieee conference on supercomputing. [8] c. anderson et al.,lapack user's guide, siam press, philadelphia,1992. [9] p. e. strazdins. a dense complex symmetric indefinite solver for thefujitsu ap3000, technical report tr-cs-99-01, computer sciencedept, australian national university, may 1999. submitted 03.10.2016, accepted 27.01.2017. կոմպլեքս հերմիտյան մատրիցների ֆակտորիզացիաների արտադրողականությունները gpu արագագործչի ճարտարապետությունում հ. ասցատրյան և է. գիչունց ամփոփում գծային հանրահաշվի ldlt ֆակտորիզացիաները շատ կարևոր են և մեծ կիրառություն ունեն գիտական և ինժեներական հաշվարկներում: այդ ֆակտորիզացիաների դասին են պատկանում բանչ-կաուֆմանի և աասենի ալգորիթմներով ֆակտորիզացիաները, ինչպես h.astsatryan and e. gichunts 75 նաև առանց պտույտի ldlt ֆակտոիզացիան: այս աշխատանքում ներկայացված են նշված երեք ֆակտորիզացիաների իրականացումները հիբրիդային ճարտարապետությունում, և տրված են արտադրողականություններ nvidia tesla k40c գրաֆիկական պրոցեսորի վրա magma գրադարանի կիրառմամբ կոմպլեքս հերմիտյան մատրիցների համար: производительности факторизации комплексных эрмитовых матриц в архитектуре ускорителя gpu г. асцатрян и э. гичунц аннотация ldlt факторизации линейной алгебры очень важны и широко используются в научных и инженерных расчетах. к этому классу относятся факторизации с алгоритмами банча-кауфмана и аасена, а также факторизация ldlt без поворота. в этой работе представлены реализации упомянутых трех факторизаций в гибридной архитектуре, а также представлены производительности на графическом процессоре nvidia tesla k40c с использованием библиотеки magma для комплексных эрмитовых матриц. d:\sbornik\...\paper.dvi mathematical problems of computer science 37, 39{44, 2012. spher e p acking b ound for e-capacity of secr ecy leakage of the b r oadcast channel with con¯dential m essages n a s r in a fs h a r institute for informatics and automation problems of nas of ra abstract the discrete memoryless broadcast channel with con¯dential messages (bcc) involves two discrete memoryless channels with two sources and one encoder. a common message must be transmitted to two receivers and a private message to the intended receiver, while keeping the other receiver as ignorant of it as possible. secrecy leakage rate is the rate of information available to the unintended receiver about the private message. upper bound for e-capacity of secrecy leakage of the bcc is derived. key words: broadcast channel with con¯dential messages, e-capacity, error probability exponent, method of types, secrecy leakage rate, sphere packing bound. refer ences [1 ] i. cs is z ¶a r a n d j. k äo r n e r , \ b r o a d c a s t c h a n n e l wit h c o n ¯ d e n t ia l m e s s a g e s " , ie e e trans. inform. theory, vo l. it-2 4 , n o . 3 , p p . 3 3 9 -3 4 8 , 1 9 7 8 . [2 ] e . a . h a r o u t u n ia n , \ u p p e r e s t im a t e o f t r a n s m is s io n r a t e fo r m e m o r yle s s c h a n n e l wit h c o u n t a b le n u m b e r o f o u t p u t s ig n a ls u n d e r g ive n e r r o r p r o b a b ilit y e xp o n e n t " , ( in r u s s ia n ) 3rd all union conference on theory of information transmission and coding, uzhgorod, p ublishing house of the uzbek academy of sciences, p p . 8 3 -8 6 , 1 9 6 7 . [3 ] e . a . h a r o u t u n ia n , b . b e lb a s h ir , \ l o we r b o u n d o f t h e o p t im a l t r a n s m is s io n r a t e d e p e n d in g o n g ive n e r r o r p r o b a b ilit y e xp o n e n t fo r d is c r e t e m e m o r yle s s c h a n n e l a n d fo r a s ym m e t r ic b r o a d c a s t c h a n n e l" , ( in r u s s ia n ) , abstracts of p apers of 6th int. symp. inform. theory, tashkent, ussr , vo l. 1 , p p . 1 9 -2 1 , 1 9 8 4 . [4 ] e . a . h a r o u t u n ia n , \ on b o u n d s fo r e -ca p a c it y o f d mc" , ie e e trans. inform. theory, vo l. it-5 3 , n o . 1 1 , p p . 4 2 1 0 -4 2 2 0 , 2 0 0 7 . [5 ] e . a . h a r o u t u n ia n , m. e . h a r o u t u n ia n a n d a . n . h a r u t yu n ya n , \ r e lia b ilit y c r it e r ia in in fo r m a t io n t h e o r y a n d in s t a t is t ic a l h yp o t h e s is t e s t in g " , f oundations and trends in communications and information theory, vo l. 4 , n o s . 2 -3 , 2 0 0 8 . [6 ] e . a . h a r o u t u n ia n , m. e . h a r o u t u n ia n a n d n . a fs h a r , \ r a n d o m c o d in g b o u n d fo r e-c a p a c it y r e g io n o f t h e wir e t a p c h a n n e l" , 8th international conference of computer science and information technologies, y e r e va n , p p . 1 2 1 -1 2 4 , 2 0 1 1 . 3 9 4 0 sphere packing bound for e-capacity of secrecy leakage of the broadcast channel with con¯dential messages ¶³õïýç ñ³õáñ¹³·ñáõãûáõýý»ñáí é³ûý³ë÷ûáõé ï³åáõõáõ ·³õïýçùç ñáë³ïáñáõëïç ³ñ³·áõãû³ý ·ý¹»ñç ÷³ã»ã³íáñù³ý ·ý³ñ³ï³ï³ýá ü. ²íß³ñ ²ù÷á÷áõù ¶³õïýç ñ³õáñ¹³·ñáõãûáõýýññáí é³ûý³ë÷ûáõé ï³åáõõáõ áý¹ñ³ýáõñ ñ³õáñ¹³·ñáõãûáõýá ÷áë³ýóíáõù ¿ »ñïáõ ñ³ëó»³ï»ñ»ñçý, çëï ù³ëý³íáñ ñ³õáñ¹³·ñáõãûáõýáª ùç³ûý ù»ïçý. ³ûý å»ïù ¿ ñý³ñ³íáñçýë ·³õïýç éçýç ùûáõëç ýï³ïù³ùµ: î³éáõóí»é ¿ ·³õïýçùç ñáë³ïáñëïç ³ñ³·áõãû³ý e-áõý³ïáõãû³ý í»ñçý ·ý³ñ³ï³ï³ýá: ãðàíèöà ñôåðè÷åñêîé óïàêîâêè ñêîðîñòè óòå÷êè ñåêðåòà â øèðîêîâåùàòåëüíîì êàíàëå ñ ñåêðåòíûìè ñîîáùåíèÿìè í.àôøàð àííîòàöèÿ îáùåå ñîîáøåíèå øèðîêîâåùàòåëüíîãî êàíàëà ñ ñåêðåòíûìè ñîîáùåíèÿìè, ïåðåäàåòñÿ äâóì àäðåñàòàì, à ÷àñòíîå ñîîáùåíèå òîëüêî îäíîìó èç íèõ è äîëæíî áûòü íàñêîëüêî âîçìîæíî ñåêðåòíûì äëÿ âòîðîãî. ïîñòðîåíà âåðõíÿÿ ãðàíèöà e-ïðîïóñêíîé ñïîñîáíîñòè ñêîðîñòè óòå÷êè ñåêðåòà. d:\sbornik\...\tpel.dvi mathematical problems of computer science 33, 48{53, 2010. e ±cient computation of subset of output p oints of fft r a fa ye l b a r s e g h ya n institute for informatics and automation problems of nas ra abstract this paper presents e±cient pruning algorithms for computing the length q £ 2p dft for a subset of output points based on transform decomposition method and in new results in computation of fft. refer ences [1 ] j. w . co o le y a n d j. w . tu ke y, \ a n a lg o r it h m fo r t h e m a c h in e c o m p u t a t io n o f t h e c o m p le x fo u r ie r s e r ie s " , m ath. computation , p p . 2 9 7 { 3 0 1 , 1 9 6 5 . [2 ] r . b la h u t , fa s t a lg o r it h m s fo r d ig it a l s ig n a l p r o c e s s in g . a d d is o n -w e s le y, 1 9 8 5 . [3 ] r . y a vn e , \ a n e c o n o m ic a l m e t h o d fo r c a lc u la t in g t h e d is c r e t e fo u r ie r t r a n s fo r m ," p roc. af ip s f all j oint computer conf., p p . 1 1 5 1 2 5 , 1 9 6 8 . [4 ] m. fr ig o a n d s . g. jo h n s o n ,\ a m o d ī e d s p lit -r a d ix fft wit h fe we r a r it h m e t ic o p e r a t io n s " , ie e e trans. signal p roccessing, vo l. 5 5 , p p . 1 1 1 -1 1 9 , 2 2 0 7 . [5 ] h . s a r u kh a n ya n , r . b a r s e g h ya n , \ n e w e ± c ie n t fft wit h fe we r op e r a t io n s " , cs it, p p . 3 5 1 -3 5 4 , 2 0 0 9 . [6 ] g. go e r t z e l,\ a n a lg o r it h m fo r t h e e va lu a t io n o f ¯ n it e t r ig o n o m e t r ic s e r ie s " , amer. m ath. m onthly, p p . 3 4 -3 5 , 1 9 5 8 . [7 ] c. g. b o n c e le t , \ a r e a r r a n g e d d ft a lg o r it h m r e qu ir in g n 2=6 m u lt ip lic a t io n s ," ie e e trans. acoust., speech, signal p rocessing , p p . 1 6 5 8 -1 6 5 9 , 1 9 8 6 . [8 ] r . d e w ild , \ me t h o d fo r p a r t ia l s p e c t r u m c o m p u t a t io n ," p roc. inst. e lec. e ng., p p . 6 5 9 -6 6 6 , 1 9 8 7 . [9 ] j.d . ma r ke l, \ fft p r u n in g ," ie e e trans. audio e lectroacoust, p p . 3 0 5 -3 1 1 , 1 9 7 1 . [1 0 ] h . v . s o r e n s e n , c. s . b u r r u s , a n d d . l . jo n e s , \ a n e w e ± c ie n t a lg o r it h m fo r c o m p u t in g a fe w d ft p o in t s ," p roc. ie e e int., p p . 1 9 1 5 -1 9 1 8 , 1 9 8 8 . [1 1 ] r . b a r s e g h ya n ,\ s p lit -r a d ix fft a lg o r it h m wit h lift in g s t e p s ," vestnik r au, series: p hysics-mathematics and natural science p p . 4 1 -5 1 , 2 0 0 9 . [1 2 ] h . s a r u kh a n ya n , r . b a r s e g h ya n ,\ n e w a p p r o a c h t o fft a lg o r it h m s " , m athematical p roblems of computer sciencevo l. 3 3 , p p . 2 4 { 3 4 , 2 0 1 0 . [1 3 ] g. b i a n d y . q. ch e n , " fa s t d ft a lg o r it h m s fo r le n g t h ," ie e e trans. circuits and syst.ii: analog and d igital signal p rocessing, p p . 6 8 5 -6 9 0 , 1 9 9 8 . 4 8 r. barseghyan 4 9 ü²ò »éù³ûçý áñáß³ïç ïçñáõûãç ñ³ßíù³ý ³ñ¹ûáõý³í»ï ³é·áñçãù è. ´³ñë»õû³ý ²ù÷á÷áõù ò¨³÷áëáõãû³ý ¹»ïáùåá½çóç³ûç ñçù³ý íñ³ ³ßë³ï³ýùáõù ùß³ïí³í ¿ ëå»ïïñ³é ïçñáõûãç áñáß³ïç ñ³ïí³íç ñ³ßíù³ý ³ñ¹ûáõý³í»ï ³é·áñçãù: d:\sbornik\...\tpel.dvi mathematical problems of computer science 35, 37{45, 2011. dual laplace stieltjes t r ansfor mations of cr itical risks in case of n egative i nsur ance p ayments a r t a k ma r t ir o s ya n institute for informatics and automation problems of nas of ra e-mail: artakm81@inbox.ru abstract the present paper is devoted to critical risks of collective insurance models with negative insurance payments (connected with contracts with usual life rent). limit theorems arisen in critical situations are represented and the dual laplass-stiltjes transformations are found for critical risks arisen in collective insurance risks models with negative insurance payments. the speci¯cations of the considered collective risk model and the adaptive control strategy for multiperiodic insurance risk model introduced by malinovskii is illustrated. refer ences [1 ] l .ta ka c h , co m b in a t o r ia l m e t h o d s in t h e r a n d o m p r o c e s s e s t h e o r y, (in r ussian), m ir, m oscow, 1 9 7 1 . [2 ] t. s a xe n , \ on t h e p r o b a b ilit y o f r u in t h e c o lle c t ive r is k t h e o r y fo r in s u r a n c e e n t e r p r is e s wit h o n ly n e g a t ive r is k s u m s " , skand. akt., vo l. 3 1 , p p . 1 9 9 -2 2 8 , 1 9 4 8 . [3 ] t. s a xe n , \ s u r le s m o u ve m e n t s a le a t o ir e s e t le p r o b le m d e r u in e d e la t h e o r ie d u r is qu c o lle c t ive " , soc sci fenn comm phys math, n u m . 1 6 , p p . 1 -5 5 , 1 9 5 1 . [4 ] g. a r fwe d s o n , \ a s e m i-c o n ve r g e n t s e r ie s wit h a p p lic a t io n t o t h e c o lle c t ive t h e o r y o f r is k" , skand. akt., vo l. 3 5 , p p . 1 6 -3 5 , 1 9 5 2 . [5 ] v .k . ma lin o vs kii, \ on a n o n -lin e a r d yn a m ic s o lve n c y c o n t r o l m o d e l" , in: contribution to: xxxiv astin colloquium, b erlin, p p . 2 4 -2 7 , 2 0 0 3 . [6 ] v .k . ma lin o vs kii, \ a d a p t ive c o n t r o l s t r a t e g ie s a n d d e p e n d e n c e o f ¯ n it e t im e r u in o n t h e p r e m iu m lo a d in g " , insurance: m athematics and e conomics, vo l. 4 2 , p p . 8 1 -9 4 , 2 0 0 8 . [7 ] a . r . ma r t ir o s ya n , the asymptotic analysis of the characteristics of the insurance models in critical situations, www.n e va .r u / jo u r n a l/ j/ e n / n u m b e r s / 2 0 0 8 .3 / a r t ic le .8 0 .h t m l o r www.p r o r e c t o r .o r g / a vt o r s / m a r t ir o s ya n / d is s .p d f, ( in r u s s ia n ) , y e r e va n , d is s e r t a t io n , 2 0 0 8 . [8 ] n . u . p r a b g h u , th e m e t h o d s o f t h e o r y o f qu e u e a n d r e s e r ve s c o n t r o l, (in r ussian), m ., m ashinostroenie, 1 9 6 9 . [9 ] n . u . p r a b g h u , s t o c h a s t ic p r o c e s s e s o f t h e o r y o f r e s e r ve s , (in r ussian), m ., m ir, 1 9 8 4 . 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[1 4 ] v . co r o le v, i. s o ko lo v, a . go r d e e v, m. gr ig o r e va , s . p o p o v, n . ch e b o n e n ko , \ s o m e fo r e c a s t in g m e t h o d s o f in t e r im c h a r a c t e r is t ic o f r is ks c o n n e c t e d wit h c a t a s t r o p h ic e ve n t s " , (in r ussian), actuary: informational-analitical bulletin, n o . 1 , p p . 3 4 -4 0 , 2 0 0 7 . [1 5 ] a . n . s h ir ya e v, fo u n d a t io n s o f s t o c h a s t ic ¯ n e n c ia l m a t h e m a t ic s , (in r ussian), m ., f azis, vo l. 1 , 1 9 9 8 . [1 6 ] a . a . b o r o vko v, th e o r y o f p r o b a b ilit ie s , (in r ussian), m ., nauka, 1 9 8 6 . [1 7 ] e . a . d a n ie lia n , \ l im it t h e o r e m s fo r wa it in g t im e in s in g le -c h a n n e l s is t e m s " , (in r ussian), r eports of academy of sciences of armenia, vo l. l x x i, n o . 3 , p p . 1 2 9 -1 3 5 , 1 9 8 0 . [1 8 ] v . g. s a a kya n , \ r a n d o m c h o ic e d is c ip lin e in m o d e ls ¡! mrj ¡! grj 1 j1 " , (in r ussian), m ., d issertation, 1 9 8 5 . [1 9 ] e . a . d a n ie lia n , r . n . ch it c h ya n , \ mu lt id im e n s io n a l lim it t h e o r e m s fo r t h e wa it in g t im e in p r io r it y qu e u e s o f ¡! mrj ¡! grj 1 j1 t yp e " , acta cybernetica, vo l. 5 , fa s c 3 , p p . 3 2 5 3 4 3 , s z e g e d , 1 9 8 1 . [2 0 ] v . m. zo lo t a r e v, on e -d im e n s io n a l s t a b le d is t r ib u t io n s , (in r ussian), m ., nauka, 1 9 8 0 . îñçïçï³ï³ý éçëï»ñç ïñïý³ïç è³åé³ë-êïçéï»ëç ó¨³÷áëáõãûáõýý»ñá µ³ó³ë³ï³ý ³å³ñáí³·ñ³ï³ý í×³ñý»ñç ¹»åùáõù ². ø³ñïçñáëû³ý ²ù÷á÷áõù êáõûý ³ßë³ï³ýùá ýíçñí³í ¿ µ³ó³ë³ï³ý í×³ñý»ñáí (ëáíáñ³ï³ý ïû³ýùç é»ýï³ý»ñç å³ûù³ý³·ñ»ñáí) ³å³ñáí³·ñ³ï³ý ùá¹»éç ïñçïçï³ï³ý éçëï»ñçý: ü»ñï³û³óí³í »ý ïñçïçï³ï³ý çñ³íç×³ïý»ñáõù ³é³ç³óáõ ë³ñù³ý³ûçý ã»áñ»ùý»ñ ¨ ·ïýí³í »ý µ³ó³ë³ï³ý í×³ñý»ñáí ïáéé»ïïçí ³å³ñáí³·ñáõãû³ý éçëïç ùá¹»éáõù ³é³ç³óáõ ïñçïçï³ï³ý éçëï»ñç ïñïý³ïç è³åé³ë-êïçéï»ëç ó¨³÷áëáõãûáõýý»ñá: èáõë³µ³ýí³í »ý ¹çï³ñïí³í ïáéé»ïïçí éçëïç ùá¹»éç ¨ ø³éçýáíëïáõ ïáõùçó µ³½ù³å³ñµ»ñ³ï³ý ³å³ñáí³·ñ³ï³ý éçëïç ùá¹»éç ñ³ù³ñ ý»ñ¹ñí³í ³¹³åïçí ï³é³í³ñù³ý ·áñíáýã³óç ù³ýñ³ù³ëý»ñá: microsoft word y.movsisyan_16.doc математические вопросы кибернетики и вычислительной техники 34, 2010. 43 бирешетки ю. м. мовсисян бирешетки, как алгебры с двумя решеточными структурами, введены м. гинсбергом и м. фиттингом в 1986-90гг. они имеют широкое приложение в исследованиях по логическому программированию, многозначной логике и нтелектуальным системам. доказывается, что описание бирешеток гинсберга с условиями сплетенности и ограниченности, полученное в работах разных авторов, остается в силе без условия ограниченности, а вместо условия сплетенности оказывается достаточно взять ослабленную форму сплетенности, которую мы называем “слабой сплетенностью”. бирешетки с этим свойством называем слабосплетенными. также доказывается, что любая слабо-сплетенная бирешетка изоморфна суперпроизведению двух решеток. бирешетки из различных многообразий характеризуются с помощью сверхтождеств. список литературы [1] m.l. ginsberg, multi-valued logics: a uniform approach to reasoning in ortificial intelligence, computational intelligence 4(1988), 265-316. [2] m.c. fitting, bilattices in logic programming, in: g. epstain ed., proc 20-th internat. symp. on multiple-valued logic, ieee, new-york, 1990, 63-70. [3] m.c. fitting, logic programming on a topological bilattice, found. inform. 11(1988), 209218 [4] m.c. fitting, bilattices and semantics of logic programming, journal of logic programming, 11(1991), 91-116. [5] b. jonsson, distributive bilattices, vanderbilt university. [6] b. mobasher, d. pigozzi, g. slutzki, multi-valued logic programming semantics: an algebraic approach, teoretical computer science, 171(1997), 77-109, [7] b. mobasher, d.pigozzi, g. slutzki, g. voutsadakis, a duality theory for bilattices, algebra univers., 43(2000), 109-125. [8] a.b. romanowska, a. trakul, on the structure of some bilattices, universal and applied algebra, world scientific, 1989, 235-253. [9] a. avron, the structure of interlaced bilattices, math. struct. in comp. science, cambridge university press, 6(1996), 287-299. [10] yu.m. movsisyan, a.b. romanowska, j.d.h. smith., superproducts, hyperidentities, and algebraic structures of logic programming, comb. math. and. comb. comp., 58(2006), 101-111. [11] yu.m. movsisyan, interlaced, modular, distributive and boolean bilattices, armenian journal of mathematics, 1, 7-13, 2008. 44 [12] ю.м. мовсисян, введение в теорию алгебр со сверхтождествами, изд-во ереванского госуниверситета, 1986. [13] ю.м. мовсисян, сверхтождества в алгрбрах и многообразиях, умн, 1998, т.53, n1(319), 61-114. [14] ю.м. мовсисян, алгебры со сверхтождествами многообразия булевых алгебр, изв. ран. сер. матем. 1996, т.60, n6, 127-168. [15] ю.м. мовсисян, сверхтождества булевых алгебр, изв. ран. сер. матем. 1992, т.56, n3, 654-672. [16] yu.movsisyan, hyperidentities and hypervarieties, scientiae mathematicae japonicae, 54,3(2001,595-640. [17] r. padmanabhan, p. penner, binary hyperidentities of lattices, aequations math., 1992, v.44, 154-167. [18] r. padmanabhan, p. penner, a hyperbase for binary lattice hyperidentities, journal of automated reasoning, 24(2000), 365-370. [19] k. denecke and sh.l.wismath, hyperidentities and clones, gordon and breach science publishers, 2000. [20] j. koppitz and k. denecke, m-solid varieties of algebras, springer, 2006. [21] в.д. аносов, о гомоморфизмах многоосновных алгебраических систем в связи с криптографическими применениями, дискретная математика, 2007, т.19, n2, 27-44. [22] j.m. font, m.moussavi, notes on a six-valued extension of tree-valued logic, jour. of applied non-classical logic, vol. 3(1993), 173-187. [23] m. kondo, on the structures of weak interlaced bilattices, in 32nd ieee international symposium on multiple-valued logic (ismvl 2002), may 15-18, 2002, boston, massachusetts, usa, pages 23, ieee computer society, 2002. [24] г. гретцер, общая теория решеток, м., “ мир”, 1982. [25] yu.m. movsisyan, l.m. budaghyan, the elementary characterization of algebras with hyperidentities of boolean algebras, intrnational conference mathematical logic, algebra and set theory dedicated to the 100-th anniversary of p.s. novikov, august 27-31, 2001, moscow 2001, p. 32. [26] yu.m. movsisyan, l.m. budaghyan, on elementary decidability of quasi-boolean algebras, intrnational conference mathematics in armenia, advanced and perspectives, september 30-october 7, 2003, yerevan, armenia, p. 55-57. [27] б.и. плоткин, универсальная алгебра, алгебраическая логика и базы данных, м., наука, 1991. microsoft word sergey_tpel_15.doc mathematical problems of computer science 36, 121--127, 2012. 121 workload management for grid environment with the restriction on the waiting time vladimir sahakyan and sergey petrosyan institute for informatics and automation problems of nas of ra vladimir.sahakyan@sci.am sergpet@ipia.sci.am abstract resource management and job scheduling in multiprocessor computing system and grid environment are challenging problems. although significant results were achieved in the past, there are some problems that still exist, and need to be completely solved. more restrictions in job will make queue management run efficiently. one of the main parameters in the job scheduling is a waiting time. waiting time is the time period, that job is ready to wait until it runs. in this article one approach to organize workload management is considered, it gives an overall solution for problem, and may be upgraded to support nonhomogeneous systems or adding some new fetchers. the article offers two parametric models of queue service discipline fifo with optimizations and restriction on waiting time. the models will be compared and underlined within the main usage of two models. keywords: workload management, queuing theory, multiprocessor system, high performance computing system, computing grid. references [1] official web-site of the armeniannational grid initiative foundation, http://www.grid.am [2] v. sahakyan and s. petrosyan, “ simulation of the queue with the restriction on the waiting time for multiprocessor systems”, proceedings of conference computer science and information technologies, pp. 272-273, 2011. [3] v.sahakyan, “about the queue organization in the multiprocessor computing systems”, mathematical problems of computer science, vol. 34, pp.18-19, 2010. [4] s. petrosyan, “simulation of the queue with the restriction on the waiting time for multiprocessor systems”, proceedings of conference computer science and information technologies, pp. 263-265, 2011. [5] t. grigoryan and v. sahakyan, “dynamic resource manager for clusters”, proceedings of conference. computer science and information technologies, 2005. [6] g. avellino et al., “the first deployment of workload management services on the eu datagridtestbed: feedbackon design and implementation”, in proceedings of the 2003 computing in high energy and nuclear physics conference (chep03), la jolla, ca, usa, march 2003. [7] data grid jdl attributes datagrid-01-ten—0142-0_2 http://www.grid.org.tr/servisler/dokumanlar/jdl_atributes_datagrid.pdf workload management for grid environment with the restriction on the waiting time 122 [8] t. xy, целочисленное программирование и потоки в сетях, москва: мир, 1974. հերթիկազմակերպման մոտեցում գրիդ միջավայրում, սպասման ժամանակի սահմանափակմամբ վ. սահակյան և ս. պետրոսյան ամփոփում ռեսուրսների կառավարման և առաջադրանքների պլանավորումը բազմապրոցեսորային հաշվողական գրիդ միջավայրերում դժվարին խնդիրներից են: չնայած նախկինում ստացված էական արդյունքներին, դեռևս առկա են խնդիրներ, որոնց լուծումը կհանգեցնի առավել օպտիմալ ռեսուրսների կառավարման: ավելացնելով սահմանափակումներ առաջադրանքներին, հնարավոր է ավելի արդյունավետ դարձնել առաջադրանքների հերթի ղեկավարումը և հասնել էֆեկտիվության բարձրացման: հիմնական սահմանափակումներից մեկն է սպասման ժամանակի սահմանափակումը: սպասման ժամանակը դա այն ժամանակն է, երբ առաջադրանքը կարող է սպասել մինչև կատարման անցնելը: սույն հոդվածում դիտարկվում է առաջադրանքների հերթի կառավարման նոր մոտեցում։ այն տալիս է ընդհանրացված լուծման մեխանիզմ, որը կարելի է զարգացնել, հետերոգեն համակարգերի համար ավելացնելով նոր ֆունկցիոնալություն: հոդվածում առաջարկվում է հերթի կազմակերպման պարամետրացված երկու մոտեցում ֆիֆո վարքով` սպասման ժամանակի սահմանափակմամբ: կատարվում է մոդելների համեմատություն և նշվում են մոտեցումների հիմնական կիրառման պայմանները: îðãàíèçàöèÿ î÷åðåäè â ãðèä ñðåäå ñ îãðàíè÷åíèåì âðåìåíиì îæèäàíèÿ â. ñààêÿí è ñ. ïåòðîñÿí àííîòàöèÿ óïðàâëåíèå ðåñóðñàìè è ïëàíèðîâàíием çàäà÷ â ìíîãîïðîöåññîðíîé ñðåäå âû÷èñëèòåëüíîé ñèñòåìû ãðèä ÿâëÿåòñÿ îäíîé èç òðóäíîðåøàåìûõ çàäà÷. íåñìîòðÿ íà çíà÷èòåëüíûå ðåçóëüòàòû, ïîëó÷åííûå ðàíåå,îñòàþòñÿ ïðîáëåìû, ðåøåíèå êîòîðûõ ïðèâîäèò ê îïòèìàëüíîìó óïðàâëåíèþ ðåñóðñàìè.ïóòåì äîáàâëåíèÿ îãðàíè÷åíèé íà ïîñòàâëåííûå çàäà÷è âîçìîæíî óâåëè÷èòü ýôôåêòèâíîñòü óïðàâëåíèÿ î÷åðåäüþ çàäà÷ è â èòîãå äîñòè÷ü ïîâûøåíèÿ ýôôåêòèâíîñòè.îäíèì èç îñíîâíûõ îãðàíè÷åíèé ÿâëÿåòñÿ îãðàíè÷åíèå âðåìåíиì îæèäàíèÿ.âðåìÿ îæèäàíèÿ, ýòî âðåìÿ, â òå÷åíèå êîòîðîãî çàäà÷à ìîæåò æäàòü âûïîëíåíèÿ. â äàííîé ñòàòüå ðàññìàòðèâàåòñÿ íîâûé ïîäõîä ê óïðàâëåíèþ î÷åðåäüþ çàäà÷. îí äàåò îáîáùåííûé ìåõàíèçì ðåøåíèÿ, êîòîðûé ìîæíî ðàçâèâàòü, äîáàâëÿÿ íîâûå ôóíêöèîíàëüíûå âîçìîæíîñòè äëÿ ãåòåðîãåííûõ ñèñòåì. â ñòàòüå ïðåäëàãàюòñÿ äâå ïàðàìåòðèçîâàííûå ìîäåëè óïðàâëåíèÿ î÷åðåäüþ ñ îãðàíè÷åíèåì âðåìåíè è ñfifo ïîâåäåíèåì. ïðåäîñòаâëÿåòñÿ ñðàâíåíèå ìîäåëåé è îòìå÷àþòñÿ îñíîâíûå óñëîâèÿ ïðèìåíåíèé. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 46, 73--80, 2016. a polynomial algorithm for the minimum bandwidth of interval graphs david h. muradian institute for informatics and automation problems of nas ra e-mail: david.h.muradian@gmail.com abstract let g be a connected graph with vertex set x and edge set u. a layout of g is a one-to-one map 𝜑𝜑 from x onto {1, 2, . . . , |x|}. the bandwidth of 𝜑𝜑 is 𝐵𝐵𝜑𝜑(𝐺𝐺) = max | 𝜑𝜑(𝑢𝑢) − 𝜑𝜑(𝑣𝑣)|, where (u, v) ranges over all edges of g. the bandwidth of g, denoted by b(g), is defined as b(g) = min𝐵𝐵𝜑𝜑(𝐺𝐺) where 𝜑𝜑 ranges over all layouts of g. interval graphs are the intersection graphs of a family of intervals over the real line. in this paper we show that the bandwidth minimization problem for interval graphs can be solved in time o�n∆2log(∆)�, where n is the vertex number and δ is the maximal degree of vertex of the interval graph. keywords: graph layout, bandwidth, interval graphs. 1. introduction the bandwidth minimization problem for graphs was first stated in 1966 by harper ([1]), where the problem was solved for hypercubes. finding the bandwidth of an arbitrary graph is an np-complete problem ([2]) and it remains np-complete for many simple structures, e.g., for cyclic caterpillars with hair length 1, graphs in which the removal of all pendant vertices results in a simple cycle ([3]). there are only few classes of graphs for which an efficient solution (i.e., a polynomial algorithm or analytic result) to the bandwidth problem is known. classes of graphs the bandwidth of which can be computed efficiently are butterflies ([4]), chain graphs ([5]), caterpillars with hair length at most 2 ([6]). another nontrivial class, for which the problem was solved efficiently, is the class of interval graphs, graphs which are the intersection graphs of a family of intervals over the real line. the first polynomial algorithm for interval graphs was given in 1986 by the author ([7]). it was published in the reports of nas ra, where the algorithm is described in detail, and besides a brief proof of its correctness is given. since this result was obtained independently and published in the following years ([8], [9], [10]), we consider it reasonable to publish the full proof of our algorithm’s correctness. 73 a polynomial algorithm for the minimum bandwidth of interval graphs 74 2. preliminaries let 𝐺𝐺 be a connected graph with vertex set 𝑋𝑋 and edge set 𝑈𝑈. a numbering of 𝐺𝐺 is a one-to-one map 𝜑𝜑 from 𝑋𝑋 onto {1, 2, . . . , | 𝑋𝑋|}. the bandwidth of 𝜑𝜑 is 𝐵𝐵𝜑𝜑(𝐺𝐺)= max | 𝜑𝜑(𝑢𝑢) − 𝜑𝜑(𝑣𝑣)|, where (𝑢𝑢, 𝑣𝑣) ranges over all edges of 𝐺𝐺. the bandwidth of 𝐺𝐺, denoted by 𝐵𝐵(𝐺𝐺), is defined as 𝐵𝐵(𝐺𝐺) = min 𝐵𝐵𝜑𝜑(𝐺𝐺), where 𝜑𝜑 ranges over all numberings of 𝐺𝐺. length of an edge (𝑢𝑢, 𝑣𝑣) is defined as | 𝜑𝜑(𝑢𝑢) − 𝜑𝜑(𝑣𝑣)|. given a graph 𝐺𝐺(𝑋𝑋, 𝑈𝑈) and its some layout 𝜑𝜑. let’s define a new layout 𝜑𝜑𝐴𝐴,𝐵𝐵 – swap of two disjoint subsets 𝐴𝐴 and 𝐵𝐵 of 𝑋𝑋 as follows. if 𝐴𝐴 and 𝐵𝐵 are disjoint subsets of vertices of 𝐺𝐺, at that for any 𝑥𝑥 ∈ 𝐴𝐴 and 𝑦𝑦 ∈ 𝐵𝐵 we have 𝜑𝜑(𝑥𝑥) < 𝜑𝜑(𝑦𝑦) and max 𝑧𝑧∈𝐵𝐵 𝜑𝜑(𝑧𝑧) − min 𝑧𝑧∈𝐴𝐴 𝜑𝜑(𝑧𝑧) = |𝐴𝐴| + |𝐵𝐵| − 1, then 𝜑𝜑(𝑧𝑧)/ 𝑧𝑧 ∈ 𝑋𝑋\(𝐴𝐴 ∪ 𝐵𝐵) 𝜑𝜑𝐴𝐴,𝐵𝐵(𝑧𝑧) = 𝜑𝜑(𝑧𝑧) + |𝐵𝐵|/𝑧𝑧 ∈ 𝐴𝐴 𝜑𝜑(𝑧𝑧) − |𝐴𝐴|/𝑧𝑧 ∈ 𝐵𝐵 let’s denote 𝑋𝑋𝜑𝜑[𝑘𝑘, 𝑙𝑙] = {𝑥𝑥 ∈ 𝑋𝑋/𝑘𝑘 ≤ 𝜑𝜑(𝑥𝑥) ≤ 𝑙𝑙} for 1 ≤ k ≤ l ≤ n. let’s consider an interval graph 𝐺𝐺 = (𝑋𝑋, 𝑈𝑈). let’s denote by 𝑥𝑥� an interval corresponding to the vertex 𝑥𝑥 ∈ 𝑋𝑋. we say that an interval 𝑥𝑥� = (𝑎𝑎, 𝑏𝑏) is entirely on the left side (right side) of an interval 𝑦𝑦� = (𝑐𝑐, 𝑑𝑑) if 𝑏𝑏 < 𝑐𝑐 (correspondingly 𝑎𝑎. 𝑑𝑑. let’s denote by 𝛤𝛤−(𝑥𝑥�) (𝛤𝛤+(𝑥𝑥�)) the set of intervals which entirely are on the left side (correspondingly on the right side) of 𝑥𝑥�. let’s define a layout 𝜑𝜑0 for the graph 𝐺𝐺. for any vertices 𝑥𝑥, 𝑦𝑦 ∈ 𝑋𝑋 if 𝛤𝛤−(𝑥𝑥�) = 𝛤𝛤−(𝑦𝑦�) and 𝛤𝛤+(𝑥𝑥�) = 𝛤𝛤+(𝑦𝑦�), then 𝜑𝜑0(𝑥𝑥) < 𝜑𝜑0(𝑦𝑦) or 𝜑𝜑0(𝑥𝑥) > 𝜑𝜑0(𝑦𝑦). otherwise, 𝜑𝜑0(𝑥𝑥) < 𝜑𝜑0(𝑦𝑦) if and only if 𝛤𝛤−(𝑥𝑥�) ⊂ 𝛤𝛤−(𝑦𝑦�) or 𝛤𝛤−(𝑥𝑥�) = 𝛤𝛤−(𝑦𝑦�) but 𝛤𝛤+(𝑥𝑥�) ⊂ 𝛤𝛤+(𝑦𝑦�). it is easy to check that 𝜑𝜑0 is well-defined by the above conditions and has the following properties. 1. if 𝑥𝑥� is entirely on the left side of 𝑦𝑦�, then 𝜑𝜑0(𝑥𝑥) < 𝜑𝜑0(𝑦𝑦). it is obvious by the definition. 2. if 1 ≤ i < j ≤ n and the vertices 𝑥𝑥 = 𝜑𝜑0−1(𝑖𝑖), 𝑦𝑦 = 𝜑𝜑0−1(𝑗𝑗) are adjacent, then 𝑥𝑥 is adjacent to any vertex 𝑧𝑧 = 𝜑𝜑0−1(𝑘𝑘), where i < k ≤ j. really, let’s assume that 𝑥𝑥, 𝑧𝑧 are not adjacent. then 𝑥𝑥� is entirely on the left side of �̂�𝑧. the vertices 𝑧𝑧, 𝑦𝑦 should be adjacent, otherwise as 𝑥𝑥, 𝑦𝑦 are adjacent, then 𝑦𝑦� should be entirely on the left side of �̂�𝑧 and therefore should have a smaller number in the layout 𝜑𝜑0. if 𝑧𝑧, 𝑦𝑦 are adjacent, then again 𝑦𝑦� should have a smaller number than �̂�𝑧 because 𝛤𝛤−(𝑦𝑦�) ⊂ 𝛤𝛤−(�̂�𝑧) (at least on account of 𝑥𝑥�), which leads to a contradiction. 3. let x and y be vertices for which we have 𝛤𝛤−(𝑦𝑦�) ⊂ 𝛤𝛤−(𝑥𝑥�) and 𝛤𝛤+(𝑦𝑦�) ⊂ 𝛤𝛤+(𝑥𝑥�). then two disjoint intervals exist, one of which is entirely on the left side and the other – entirely on the right side of 𝑥𝑥� and both are overlaps with 𝑦𝑦�. it is obvious by definition. let 𝑥𝑥�, 𝑦𝑦�, 𝑧𝑧1� and 𝑧𝑧2� be intervals where 𝑧𝑧1� is entirely on the left side of 𝑥𝑥� and 𝑧𝑧2� entirely on the right side of 𝑥𝑥� and all of them overlap with 𝑦𝑦�. then we will say that 𝑥𝑥� is a proper interval for 𝑦𝑦� and record this fact as 𝑥𝑥 ⊂̇ 𝑦𝑦. for any vertex 𝑥𝑥 ∈ 𝑋𝑋, we will name 𝜑𝜑0(𝑥𝑥) as its index. d. muradian 75 3. algorithm step i (i ≥ 1). at step i the algorithm having as an input the layout 𝜑𝜑 = 𝜑𝜑𝑖𝑖−1, creates a layout 𝜑𝜑𝑖𝑖 or stops, claiming that 𝐵𝐵(𝐺𝐺) > 𝐾𝐾. let 𝑦𝑦 be a vertex with the greatest number at 𝜑𝜑, which is incident to an edge with a length above 𝐾𝐾, and let (𝑥𝑥, 𝑦𝑦) have the greatest length among them (i.e., 𝑦𝑦 is not adjacent to vertices, the numbers of which are less than 𝜑𝜑(𝑥𝑥)). then the algorithm tries to find a vertex with the smallest number from 𝑋𝑋𝜑𝜑[𝜑𝜑(𝑥𝑥) + 1, 𝜑𝜑(𝑦𝑦) − 1], which is not adjacent to y. if such vertex does not exist, then it claims that 𝐵𝐵(𝐺𝐺) > 𝐾𝐾. let 𝑧𝑧 be the sought-for vertex. let’s denote 𝑀𝑀 = 𝑋𝑋𝜑𝜑[𝜑𝜑(𝑥𝑥), 𝜑𝜑(𝑧𝑧) − 1]. let 𝑎𝑎𝑗𝑗 be a vertex from the set 𝑀𝑀\𝑋𝑋𝜑𝜑�𝜑𝜑�𝑎𝑎𝑗𝑗−1�, 𝜑𝜑(𝑧𝑧) − 1� having the greatest index there (with an agreement, that 𝑋𝑋𝜑𝜑[𝜑𝜑(𝑎𝑎0), 𝜑𝜑(𝑧𝑧) − 1] = ∅). denote 𝑆𝑆𝑗𝑗 = 𝑋𝑋𝜑𝜑�𝜑𝜑�𝑎𝑎𝑗𝑗� + 1, 𝜑𝜑�𝑎𝑎𝑗𝑗−1� − 1�. note that for some j-s sets 𝑆𝑆𝑗𝑗 may be empty. the layout 𝜑𝜑𝑖𝑖 is defined as follows: 𝜑𝜑𝑖𝑖−1(𝑥𝑥)/ 𝑎𝑎 = 𝑧𝑧 𝜑𝜑𝑖𝑖(𝑎𝑎) = 𝜑𝜑𝑖𝑖−1(𝑎𝑎)/𝑎𝑎 ∈ ⋃ 𝑆𝑆𝑗𝑗 ∪ �𝑋𝑋\(𝑀𝑀 ∪ {𝑧𝑧})�𝑗𝑗 𝜑𝜑𝑖𝑖−1(𝑎𝑎) + 1 + �𝑆𝑆𝑗𝑗�/𝑎𝑎 = 𝑎𝑎𝑗𝑗 in other words 𝜑𝜑𝑖𝑖 is obtained from 𝜑𝜑𝑖𝑖−1 via successive swaps: (m,{𝑧𝑧}), ({𝑎𝑎1}, 𝑆𝑆1), ({𝑎𝑎2}, 𝑆𝑆2), … then the algorithm checks if there is an edge with the length above 𝐾𝐾 at 𝜑𝜑𝑖𝑖. if not, then it stops, claiming that 𝜑𝜑𝑖𝑖 is the sought-for layout with the bandwidth no more than 𝐾𝐾. if yes, then the algorithm turns to the step 𝑖𝑖 + 1. 4. proof of the algorithm’s correctness let’s define a class of layouts φ0 for the graph 𝐺𝐺: 𝜑𝜑 ∈ φ0 if and only if for any vertices a and b, for which 𝜑𝜑(𝑎𝑎) < 𝜑𝜑(𝑏𝑏) and 𝜑𝜑0(𝑎𝑎) > 𝜑𝜑0(𝑏𝑏), will take place 𝑎𝑎 ⊂̇ 𝑏𝑏. from the definition of φ0 it follows immediately that for any 𝜑𝜑 ∈ φ0 and two nonadjacent vertices x and y, if 𝜑𝜑0(𝑥𝑥) < 𝜑𝜑0(𝑦𝑦), then 𝜑𝜑(𝑥𝑥) < 𝜑𝜑(𝑦𝑦). it is easy to see that 𝜑𝜑0 ∈ φ0. moreover, the algorithm, beginning with 𝜑𝜑0, never leaves the class φ0. lemma 1: 𝜑𝜑𝑖𝑖 ∈ φ0 in each step 𝑖𝑖. proof: we will prove the statement by induction. for 𝑖𝑖 = 0 the statement obviously is true, let it be true for each step until 𝑖𝑖 − 1. let’s show that 𝜑𝜑𝑖𝑖 ∈ φ0. then it is sufficient to show that 𝑧𝑧 ⊂̇ 𝑞𝑞 any 𝑞𝑞 ∈ 𝑀𝑀, and 𝑎𝑎𝑡𝑡 ⊂̇ 𝑞𝑞𝑡𝑡 for all 𝑞𝑞𝑡𝑡 ∈ 𝑆𝑆𝑡𝑡. since (𝑞𝑞, 𝑦𝑦) ∈ 𝑈𝑈 and (𝑧𝑧, 𝑦𝑦) ∉ 𝑈𝑈, then naturally q cannot be a proper interval for z, and by the fact that 𝜑𝜑𝑖𝑖−1 ∈ φ0, we will have 𝜑𝜑0(𝑞𝑞) < 𝜑𝜑0(𝑧𝑧). but as 𝛤𝛤+(𝑞𝑞�) ⊂ 𝛤𝛤+(�̂�𝑧), it follows immediately that 𝛤𝛤−(𝑞𝑞�) ⊂ 𝛤𝛤−(�̂�𝑧), 𝑖𝑖.e., 𝑧𝑧 ⊂̇ 𝑞𝑞. then the layout obtained by the swap of m and {z} certainly belongs to φ0, and therefore, 𝑎𝑎𝑡𝑡 ⊂̇ 𝑞𝑞𝑡𝑡 for all 𝑞𝑞𝑡𝑡 ∈ 𝑆𝑆𝑡𝑡. this proves the lemma. in the next two lemmas several new properties of the layout 𝜑𝜑𝑖𝑖 are proved. lemma 2: in the step 𝑖𝑖, the length of the edge (𝑥𝑥, 𝑦𝑦) decreases by 1 and none of vertices from 𝑋𝑋𝜑𝜑𝑖𝑖[𝜑𝜑𝑖𝑖(𝑦𝑦) + 1, 𝑛𝑛] is incident to an edge with the length above k. proof: by lemma 1 we have 𝑧𝑧 ⊂̇ 𝑥𝑥, and z together with x don’t have adjacent vertices in 𝑋𝑋𝜑𝜑𝑖𝑖[𝜑𝜑𝑖𝑖(𝑦𝑦) + 1, 𝑛𝑛]. therefore, there is no vertex from 𝑋𝑋𝜑𝜑𝑖𝑖[𝜑𝜑𝑖𝑖(𝑦𝑦) + 1, 𝑛𝑛], incident to an edge with length above k. now we will show that the length of the edge (x, y) decreases exactly by 1. if a polynomial algorithm for the minimum bandwidth of interval graphs 76 not, then denoting = 𝜑𝜑𝑖𝑖−1 −1 (𝜑𝜑𝑖𝑖−1(𝑥𝑥) + 1) , we will have 𝑥𝑥 ⊂̇ 𝑢𝑢. by definition u doesn’t have adjacent vertices in 𝑋𝑋𝜑𝜑𝑖𝑖−1[𝜑𝜑𝑖𝑖−1(𝑦𝑦) + 1, 𝑛𝑛]. let j be the greatest number, for which 𝜑𝜑𝑗𝑗(𝑢𝑢) < 𝜑𝜑𝑗𝑗(𝑥𝑥) and 𝜑𝜑𝑗𝑗+1(𝑢𝑢) > 𝜑𝜑𝑗𝑗+1(𝑥𝑥). it is clear that j ≤ i-2. in the sub-step j+1 some set u swaps with {x}, where uϵu, at that there exists a vertex v, which is not adjacent to x, but is adjacent to all vertices from u, and therefore, vϵ𝑋𝑋𝜑𝜑𝑖𝑖−1[𝜑𝜑𝑖𝑖−1(𝑦𝑦) + 1, 𝑛𝑛]. obtained contradiction proves the lemma. lemma 3: let vertices 𝑎𝑎 and 𝑏𝑏 satisfy the conditions: 𝜑𝜑𝑖𝑖(𝑎𝑎) < 𝜑𝜑𝑖𝑖(𝑏𝑏) and 𝜑𝜑0(𝑎𝑎) > 𝜑𝜑0(𝑏𝑏). (1) then there are vertices 𝐷𝐷 and 𝑑𝑑, such that: 𝐷𝐷𝐷𝐷𝑋𝑋𝜑𝜑𝑖𝑖[𝜑𝜑𝑖𝑖(𝑎𝑎) + 1, 𝜑𝜑𝑖𝑖(𝑏𝑏)] and 𝜑𝜑𝑖𝑖(𝑑𝑑) − 𝜑𝜑𝑖𝑖(𝐷𝐷) ≥ 𝐾𝐾, (2) proof: we will prove the statement by induction. for 𝑖𝑖 = 0 the statement obviously is true, let it is true from 1 to i-1 step inclusive. consider the step i. denote 𝑀𝑀1 = 𝑋𝑋𝜑𝜑𝑖𝑖−1[1, 𝜑𝜑𝑖𝑖−1(𝑥𝑥) − 1] and 𝑀𝑀2 = 𝑋𝑋𝜑𝜑𝑖𝑖−1[𝜑𝜑𝑖𝑖−1(𝑧𝑧) + 1, 𝑛𝑛]. we will show, that if for some vertices 𝑎𝑎 and 𝑏𝑏 conditions (1) are fulfilled, then there should exist vertices d’ and d’ which satisfy the conditions (2). the only possible case, when 𝜑𝜑𝑖𝑖−1(𝑎𝑎) > 𝜑𝜑𝑖𝑖−1(𝑏𝑏) can hold when a = z and bϵm. in this case vertices x and y can be taken instead of 𝐷𝐷 and 𝑑𝑑. really, any vertex from 𝑀𝑀 is adjacent to 𝑑𝑑 = 𝑦𝑦 and 𝜑𝜑𝑖𝑖(𝑑𝑑) − 𝜑𝜑𝑖𝑖(𝐷𝐷) = 𝜑𝜑𝑖𝑖(𝑦𝑦) − 𝜑𝜑𝑖𝑖(𝑥𝑥) ≥ 𝐾𝐾. in other cases 𝜑𝜑𝑖𝑖−1(𝑎𝑎) < 𝜑𝜑𝑖𝑖−1(𝑏𝑏). let’s analyze possible cases. case 1. 𝑏𝑏𝐷𝐷𝑀𝑀1. then 𝑎𝑎𝐷𝐷𝑀𝑀1. since at the step 𝑖𝑖 only the numbers of vertices from 𝑀𝑀 ∪ {𝑧𝑧} can be changed, then at d’𝐷𝐷𝑀𝑀1 ∪ 𝑀𝑀2 taking 𝐷𝐷 = 𝐷𝐷′ and 𝑑𝑑 = 𝑑𝑑′, it is easy to see that for 𝐷𝐷 and 𝑑𝑑 at 𝜑𝜑𝑖𝑖 conditions (2) are fulfilled. if 𝑑𝑑′ = 𝑧𝑧, then from the fact that 𝑧𝑧 ⊂̇ 𝑞𝑞 for any 𝑞𝑞ϵm, it follows that each vertex from 𝑋𝑋𝜑𝜑𝑖𝑖−1[𝜑𝜑𝑖𝑖−1(𝐷𝐷 ′), 𝜑𝜑𝑖𝑖−1(𝑏𝑏)] is adjacent to each vertex from m. note that some vertex from m receives the number 𝜑𝜑𝑖𝑖−1(𝑧𝑧) at 𝜑𝜑𝑖𝑖. then taking 𝐷𝐷 = 𝐷𝐷′ and 𝑑𝑑 = 𝜑𝜑𝑖𝑖 −1�𝜑𝜑𝑖𝑖−1(𝑧𝑧)�, one can observe, that for 𝐷𝐷 and 𝑑𝑑 conditions (2) will be fulfilled. if d’ϵm, then as 𝜑𝜑𝑖𝑖(𝑞𝑞) ≥ 𝜑𝜑𝑖𝑖−1(𝑞𝑞) for each 𝑞𝑞ϵm, we will have 𝜑𝜑𝑖𝑖(d’) − 𝜑𝜑𝑖𝑖(𝐷𝐷′) ≥ 𝜑𝜑𝑖𝑖−1(d’) − 𝜑𝜑𝑖𝑖−1(𝐷𝐷′) and we can take 𝐷𝐷 = 𝐷𝐷′ and 𝑑𝑑 = 𝑑𝑑′ case 2. 𝑏𝑏 = 𝑧𝑧. then 𝑎𝑎𝐷𝐷𝑀𝑀1. if 𝐷𝐷′ ∈ 𝑀𝑀1, then we will put 𝐷𝐷 = 𝐷𝐷′ and 𝑑𝑑 = 𝑑𝑑′, and if 𝐷𝐷′ ∈ 𝑀𝑀, then 𝐷𝐷 = 𝑏𝑏 = 𝑧𝑧 and 𝑑𝑑 = 𝑑𝑑′. then from the fact that 𝑧𝑧 ⊂̇ 𝑞𝑞 for each 𝑞𝑞 ∈ 𝑀𝑀, it follows that for 𝐷𝐷 and 𝑑𝑑 at 𝜑𝜑𝑖𝑖 conditions (2) are fulfilled. case 3. 𝑏𝑏 ∈ 𝑀𝑀 ∪ 𝑀𝑀1. at first let’s assume that for any vertex from 𝑋𝑋𝜑𝜑𝑖𝑖−1[𝜑𝜑𝑖𝑖−1(x), 𝜑𝜑𝑖𝑖−1(𝑏𝑏) − 1] occurs 𝜑𝜑0(𝑞𝑞) < 𝜑𝜑0(𝑏𝑏). then 𝑎𝑎𝐷𝐷𝑀𝑀1. if 𝑏𝑏 ∈ 𝑀𝑀, then in place of 𝐷𝐷 one can take the vertex 𝑥𝑥 and in place of 𝑑𝑑 – vertex 𝑦𝑦, i.e., all vertices from 𝑀𝑀 are adjacent to 𝑦𝑦. if 𝑏𝑏 ∈ 𝑀𝑀2 and 𝐷𝐷′ ∈ 𝑀𝑀2, then during the transition from 𝜑𝜑𝑖𝑖−1 to 𝜑𝜑𝑖𝑖, nothing is changed for a and b, therefore we can take 𝐷𝐷 = 𝐷𝐷′ and 𝑑𝑑 = 𝑑𝑑′. let 𝑏𝑏 ∈ 𝑀𝑀2 and 𝐷𝐷′ ∈ 𝑀𝑀. since 𝑧𝑧 is adjacent to 𝑑𝑑′, then all vertices from 𝑀𝑀 will be adjacent to 𝑑𝑑′, and one can take 𝐷𝐷 = 𝑧𝑧 (certainly only after receiving the number 𝜑𝜑𝑖𝑖−1(x) by z) and 𝐷𝐷 = 𝐷𝐷′. now let q be a vertex from 𝑋𝑋𝜑𝜑𝑖𝑖−1[𝜑𝜑𝑖𝑖−1(x), 𝜑𝜑𝑖𝑖−1(𝑏𝑏) − 1], the index of which is greater than the index of b, and let c be the vertex with the greatest number among them at 𝜑𝜑𝑖𝑖−1. let 𝑏𝑏 ∈ 𝑀𝑀2. then ∈ 𝑀𝑀2 ∪ {𝑧𝑧} . really, if 𝑐𝑐 ∈ 𝑀𝑀, then from 𝑧𝑧 ⊂̇ 𝑐𝑐 we will have 𝜑𝜑0(𝑧𝑧) > 𝜑𝜑0(𝑐𝑐) and therefore – 𝜑𝜑0(𝑧𝑧) > 𝜑𝜑0(𝑏𝑏), which will be at odds with the selection of c. but if cϵm2∪{z}, then at 𝜑𝜑𝑖𝑖−1 there are 𝐷𝐷′ and 𝑑𝑑′, satisfying (2), at that 𝐷𝐷′ ∈ 𝑀𝑀2, and as their d. muradian 77 numbers are not changed during the transition from 𝜑𝜑𝑖𝑖−1 to 𝜑𝜑𝑖𝑖, then one can take 𝐷𝐷 = 𝐷𝐷′ and 𝑑𝑑 = 𝑑𝑑′. let 𝑏𝑏 ∈ 𝑀𝑀. if 𝑎𝑎 ∈ 𝑀𝑀1, then one can take 𝐷𝐷 = 𝑥𝑥 and 𝑑𝑑 = 𝑦𝑦. let 𝑎𝑎 ∈ 𝑀𝑀. let’s analyze the step 𝑖𝑖. the inequality 𝜑𝜑𝑖𝑖(𝑎𝑎) < 𝜑𝜑𝑖𝑖(𝑏𝑏) means that at 𝜑𝜑𝑖𝑖−1 there was a vertex c’, such that 𝜑𝜑𝑖𝑖−1(𝑎𝑎) < 𝜑𝜑𝑖𝑖−1(c’) < 𝜑𝜑𝑖𝑖−1(𝑏𝑏), whereas at 𝜑𝜑𝑖𝑖 : 𝜑𝜑𝑖𝑖(c’) > 𝜑𝜑𝑖𝑖(𝑏𝑏) , i.e., the vertex was belonging to some nonempty set sr, and c’ = ar. therefore, considering the pair of vertices (c’, b) at 𝜑𝜑𝑖𝑖−1, by the inductive conjecture there are vertices 𝐷𝐷′ and 𝑑𝑑′, satisfying (2), the numbers of which are not changed during the transition from 𝜑𝜑𝑖𝑖−1 to 𝜑𝜑𝑖𝑖, i.e., one can take 𝐷𝐷 = 𝐷𝐷′ and 𝑑𝑑 = 𝑑𝑑′. this proves the lemma. before proving that the algorithm stops without creating a layout with the bandwidth 𝐾𝐾 only on graphs having the bandwidth above 𝐾𝐾, we will define a graph called a generalized 1caterpillar. definition: let 𝐴𝐴𝑖𝑖,𝑉𝑉𝑖𝑖 (𝑖𝑖 ∈ 1, 𝑚𝑚��) be disjoint sets and ai ≠ ∅ for all (𝑖𝑖 ∈ 1, 𝑚𝑚)��. a graph h = (f,e) with the set of vertices 𝐹𝐹 = ⋃ 𝐴𝐴𝑖𝑖 ∪ ⋃ 𝑉𝑉𝑖𝑖 𝑚𝑚 𝑖𝑖=1 𝑚𝑚 𝑖𝑖=1 is called a generalized 1-caterpillar if (x,y)ϵe if and only if 𝑥𝑥 ∈ 𝑉𝑉𝑖𝑖, 𝑦𝑦 ∈ 𝐴𝐴𝑖𝑖 (𝑖𝑖 ∈ 1, 𝑚𝑚��), or 𝑥𝑥 ∈ 𝐴𝐴𝑖𝑖 , 𝑥𝑥 ∈ 𝐴𝐴𝑖𝑖+1 (𝑖𝑖 ∈ 1, 𝑚𝑚 − 1��), or 𝑥𝑥, 𝑦𝑦 ∈ 𝐴𝐴𝑖𝑖 (𝑖𝑖 ∈ 1, 𝑚𝑚��). lemma 4: if the number of vertices of the graph h is more than (𝑚𝑚 + 1)(𝐾𝐾 + 1) − ∑ 𝐴𝐴𝑖𝑖 𝑚𝑚 𝑖𝑖=1 , then b(h) > k. proof: let 𝑝𝑝 be the number of vertices of 𝐻𝐻 and 𝑝𝑝 > (𝑚𝑚 + 1)(𝐾𝐾 + 1) − ∑ 𝐴𝐴𝑖𝑖 𝑚𝑚 𝑖𝑖=1 . let’s assume that b(h) ≤ k. let 𝜑𝜑 be a layout with the smallest bandwidth for h, and without losing generality, let’s assume that 𝜑𝜑−1(1) ∈ 𝐴𝐴𝑖𝑖 ∪ 𝑉𝑉𝑖𝑖 and 𝜑𝜑−1(𝑝𝑝) ∈ 𝐴𝐴𝑗𝑗 ∪ 𝑉𝑉𝑗𝑗 at some 𝑖𝑖, 𝑗𝑗 (1 ≤ i ≤ j ≤ m). using the assumption b(h) ≤ k, it is easy to prove the following statement: if 𝑧𝑧1𝐷𝐷𝐴𝐴𝑡𝑡 for some t, and 𝑧𝑧2 – vertex from 𝐴𝐴𝑡𝑡+1 with the smallest number, then 𝜑𝜑(𝑧𝑧2) ≤ 𝜑𝜑(𝑧𝑧1) + 𝐾𝐾 − |𝐴𝐴𝑡𝑡+1| + 1. solving this recurrent inequalities we will obtain that if z is a vertex from 𝐴𝐴𝑗𝑗 with the smallest number, then 𝜑𝜑(𝑧𝑧) ≤ (𝑗𝑗 − 𝑖𝑖 + 1)(𝐾𝐾 + 1) − ∑ |𝐴𝐴𝑡𝑡| 𝑗𝑗 𝑡𝑡=𝑖𝑖 + 1. but z is adjacent to 𝜑𝜑 −1(𝑝𝑝), therefore 𝐾𝐾 ≥ 𝑝𝑝 − 𝜑𝜑(𝑧𝑧) > (𝑚𝑚 + 1)(𝐾𝐾 + 1) − �|𝐴𝐴𝑡𝑡| 𝑚𝑚 𝑡𝑡=1 − (𝑗𝑗 − 𝑖𝑖 + 1)(𝐾𝐾 + 1) + �|𝐴𝐴𝑡𝑡| = 𝑗𝑗 𝑡𝑡=𝑖𝑖 (𝑚𝑚 − 𝑗𝑗 + 𝑖𝑖)(𝐾𝐾 + 1) − �|𝐴𝐴𝑡𝑡| 𝑖𝑖−1 𝑡𝑡=1 − � |𝐴𝐴𝑡𝑡| 𝑚𝑚 𝑡𝑡=𝑗𝑗+1 − 1. besides |𝐴𝐴𝑡𝑡| ≤ 𝐾𝐾 + 1 for all t ϵ1, 𝑚𝑚��, therefore: 𝐾𝐾 ≥ 𝑝𝑝 − 𝜑𝜑(𝑧𝑧) > (𝑚𝑚 − 𝑗𝑗 + 𝑖𝑖)(𝐾𝐾 + 1) − (𝑚𝑚 − 𝑗𝑗 + 𝑖𝑖 − 1)(𝐾𝐾 + 1) − 1 = 𝐾𝐾, which leads to a contradiction. this proves the lemma. now we will state the last necessary property of layouts from φ0. lemma 5: let 𝜑𝜑𝐷𝐷𝛷𝛷0 and let 𝑢𝑢1, 𝑢𝑢2, 𝑢𝑢3 be vertices with the following properties: 𝜑𝜑(𝑢𝑢1) < 𝜑𝜑(𝑢𝑢2) < 𝜑𝜑(𝑢𝑢3), (𝑢𝑢1, 𝑢𝑢3)𝐷𝐷𝑈𝑈 and 𝜑𝜑0(𝑢𝑢2) < 𝜑𝜑0(𝑢𝑢3). then (𝑢𝑢1, 𝑢𝑢2)𝐷𝐷𝑈𝑈. proof: let’s assume that 𝑢𝑢1, 𝑢𝑢2 are not adjacent. then from 𝜑𝜑(𝑢𝑢1) < 𝜑𝜑(𝑢𝑢2) we will have 𝜑𝜑0(𝑢𝑢1) < 𝜑𝜑0(𝑢𝑢2). but (𝑢𝑢1, 𝑢𝑢3)𝐷𝐷𝑈𝑈 and therefore 𝛤𝛤−(𝑢𝑢3�) ⊂ 𝛤𝛤−(𝑢𝑢3�) and 𝜑𝜑0(𝑢𝑢3) < 𝜑𝜑0(𝑢𝑢2), which contradicts the conditions of the lemma. this proves the lemma. theorem: if the algorithm at some step stops without creating for the graph g a layout with bandwidth k, then b(g) > k. a polynomial algorithm for the minimum bandwidth of interval graphs 78 proof: let’s assume that the situation described in the formulation of the theorem occurs at step i+1: each vertex from 𝑋𝑋𝜑𝜑𝑖𝑖[𝜑𝜑𝑖𝑖(𝑥𝑥), 𝜑𝜑𝑖𝑖(𝑦𝑦)] is adjacent to y. if there is no any vertex among them, the index of which is greater than 𝜑𝜑0(𝑦𝑦), then by lemma 5, the subgraph induced by the set 𝑋𝑋𝜑𝜑𝑖𝑖[𝜑𝜑𝑖𝑖(𝑥𝑥), 𝜑𝜑𝑖𝑖(𝑦𝑦)] is a clique with the vertex number over k+1 and therefore b(g) > k. let’s assume that there is a vertex in 𝑋𝑋𝜑𝜑𝑖𝑖[𝜑𝜑𝑖𝑖(𝑥𝑥), 𝜑𝜑𝑖𝑖(𝑦𝑦) − 1], the index of which is greater than 𝜑𝜑0(𝑦𝑦) and let а1 have the greatest number among them. denote 𝑏𝑏1 = 𝑦𝑦. then for а1 and 𝑏𝑏1conditions (1) are fulfilled and therefore there exist vertices d1 from 𝑋𝑋𝜑𝜑𝑖𝑖[𝜑𝜑𝑖𝑖(а1) + 1, 𝜑𝜑𝑖𝑖(𝑏𝑏1)] and 𝑏𝑏2 , such that 𝜑𝜑𝑖𝑖(𝑏𝑏2) − 𝜑𝜑𝑖𝑖(𝐷𝐷1) ≥ 𝐾𝐾 and all vertices from 𝑋𝑋𝜑𝜑𝑖𝑖[𝜑𝜑𝑖𝑖(𝐷𝐷1), 𝜑𝜑𝑖𝑖(𝑏𝑏1)] are adjacent to 𝑏𝑏2. denote 𝑅𝑅0 = 𝑋𝑋𝜑𝜑𝑖𝑖[𝜑𝜑𝑖𝑖(𝑥𝑥), 𝜑𝜑𝑖𝑖(𝐷𝐷1) − 1] and 𝐴𝐴1 = 𝑋𝑋𝜑𝜑𝑖𝑖[𝜑𝜑𝑖𝑖(𝐷𝐷1), 𝜑𝜑𝑖𝑖(𝑏𝑏1)]. then let 𝑎𝑎2 be the vertex with the greatest number from 𝑋𝑋𝜑𝜑𝑖𝑖[𝜑𝜑𝑖𝑖(𝑏𝑏1) + 1, 𝜑𝜑𝑖𝑖(𝑏𝑏2)], the index of which is equal or greater than 𝜑𝜑0(𝑏𝑏2) (equality of indices is understood as a simple coincidence: 𝑎𝑎2 = 𝑏𝑏2). let 𝑎𝑎2 ≠ 𝑏𝑏2. then 𝑎𝑎2, 𝑏𝑏2 satisfy the conditions (1) and therefore there exist vertices d2 from 𝑋𝑋𝜑𝜑𝑖𝑖[𝜑𝜑𝑖𝑖(а2) + 1, 𝜑𝜑𝑖𝑖(𝑏𝑏2)] and 𝑏𝑏3, for which the conditions (2) are fulfilled (after taking 𝐷𝐷 = 𝐷𝐷1, 𝑑𝑑 = 𝑏𝑏3 in (2)). denote 𝑅𝑅1 = 𝑋𝑋𝜑𝜑𝑖𝑖[𝜑𝜑𝑖𝑖(𝑏𝑏1) + 1, 𝜑𝜑𝑖𝑖(𝐷𝐷2) − 1] and 𝐴𝐴2 = 𝑋𝑋𝜑𝜑𝑖𝑖[𝜑𝜑𝑖𝑖(𝐷𝐷2), 𝜑𝜑𝑖𝑖(𝑏𝑏2)]. let’s continue this procedure. as the graph is finite, then the sets 𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚 and 𝑅𝑅0, 𝑅𝑅1, … , 𝑅𝑅𝑚𝑚 will be obtained, such that 𝑅𝑅𝑗𝑗 = 𝑋𝑋𝜑𝜑𝑖𝑖�𝜑𝜑𝑖𝑖�𝑏𝑏𝑗𝑗� + 1, 𝜑𝜑𝑖𝑖�𝐷𝐷𝑗𝑗+1� − 1� 𝑎𝑎𝑎𝑎 𝑚𝑚 ≥ 1, 𝐴𝐴𝑗𝑗 = 𝑋𝑋𝜑𝜑𝑖𝑖�𝜑𝜑𝑖𝑖�𝐷𝐷𝑗𝑗�, 𝜑𝜑𝑖𝑖�𝑏𝑏𝑗𝑗�� 𝑎𝑎𝑎𝑎 j ϵ1, 𝑚𝑚��, at that 𝜑𝜑𝑖𝑖�𝑏𝑏𝑗𝑗� − 𝜑𝜑𝑖𝑖�𝐷𝐷𝑗𝑗−1� ≥ 𝐾𝐾 (j ϵ1, 𝑚𝑚 + 1��, 𝐷𝐷0 = 𝑥𝑥), every vertex from 𝐴𝐴𝑗𝑗 is adjacent to 𝑏𝑏𝑗𝑗+1 (j ϵ1, 𝑚𝑚��), every vertex from 𝑅𝑅0 is adjacent to 𝑏𝑏1 and there are no vertices in 𝑅𝑅0 , the indices of which are above 𝜑𝜑0(𝑏𝑏𝑚𝑚+1). from lemma 5 we know that every vertex from 𝑅𝑅0 is adjacent to every vertex from 𝐴𝐴1, every vertex from 𝐴𝐴𝑗𝑗 is adjacent to every vertex from 𝐴𝐴𝑗𝑗+1 (j ϵ1, 𝑚𝑚 − 1��) and every vertex from 𝐴𝐴𝑚𝑚 is adjacent to every vertex from 𝑅𝑅𝑚𝑚. let u be an arbitrary vertex from 𝑅𝑅𝑗𝑗 (j ϵ1, 𝑚𝑚 − 1��). we will show, that if u is not adjacent to any vertex from 𝐴𝐴𝑗𝑗 (𝐴𝐴𝑗𝑗+1), then it is adjacent to all vertices from 𝐴𝐴𝑗𝑗+1 (correspondingly: 𝐴𝐴𝑗𝑗), where j ϵ1, 𝑚𝑚 − 1��). really, assume the contrary: 𝑢𝑢1𝐷𝐷а𝑗𝑗, 𝑢𝑢2𝐷𝐷а𝑗𝑗+1 and both are adjacent to u. from 𝜑𝜑𝑖𝑖(𝑢𝑢) < 𝜑𝜑𝑖𝑖(𝑢𝑢2) we have 𝜑𝜑0(𝑢𝑢) < 𝜑𝜑0(𝑢𝑢2). then applying lemma 5 to the triple 𝑢𝑢1, 𝑢𝑢2, 𝑢𝑢3, we will get that 𝑢𝑢1 is adjacent to 𝑢𝑢, which leads to a contradiction. therefore every vertex 𝑢𝑢ϵ𝑅𝑅𝑗𝑗 (j ϵ1, 𝑚𝑚 − 1��) is adjacent to all vertices of at least one of the sets 𝐴𝐴𝑗𝑗, 𝐴𝐴𝑗𝑗+1. let 𝑉𝑉1 be a set consisting of all vertices of 𝑅𝑅0 and those vertices from 𝑅𝑅1, which are adjacent to all vertices of 𝐴𝐴1. let 𝑉𝑉𝑗𝑗 be a set consisting of all vertices of 𝑅𝑅𝑗𝑗−1\𝑉𝑉𝑗𝑗−1 and those vertices from 𝑅𝑅𝑗𝑗, which are adjacent to all vertices of 𝐴𝐴𝑗𝑗 (j ϵ1, 𝑚𝑚 − 1��), and 𝑉𝑉𝑚𝑚 = (𝑅𝑅𝑚𝑚−1\𝑉𝑉𝑚𝑚−1) ∪ 𝑅𝑅𝑚𝑚. it is easy to see that the graph g contains as a subgraph a generalized 1-caterpillar satisfying the conditions of lemma 4. this proves the theorem. 5. estimation of algorithm complexity from the definition of 𝜑𝜑0 and its second property we have 𝐵𝐵𝜑𝜑0(𝐺𝐺) ≤ ∆ − 1, where ∆ is the maximal degree of vertices. on the other hand we have a trivial lower bound: 𝐵𝐵𝜑𝜑0(𝐺𝐺) ≥ 𝐵𝐵(𝐺𝐺) ≥ � ∆ 2 � . therefore �∆ 2 � ≤ 𝐵𝐵(𝐺𝐺) ≤ ∆ − 1. first we will estimate the running time of the step i. the vertex z can be found checking |𝑀𝑀| vertices and |𝑀𝑀| elementary operations are sufficient for the swap of sets (𝑀𝑀, {𝑧𝑧}) (for the assignment of numbers). the rest part of the step i, i.e., the problem of finding vertices 𝑎𝑎𝑗𝑗 as well as realization of swaps ({𝑎𝑎𝑗𝑗}, 𝑆𝑆𝑗𝑗) is equivalent to one pass of known bubble algorithm, and d. muradian 79 therefore will require no more than |𝑀𝑀| comparisons of indices and |𝑀𝑀| operations for the reassignments of numbers. therefore the step i will require o(|𝑀𝑀|) elementary operations. as the vertex set 𝑀𝑀 forms a clique (the corresponding intervals contain the interval �̂�𝑧), then |𝑀𝑀| ≤ 𝐾𝐾 + 1. besides the length of the edge (x, y) no more than 2𝐾𝐾, because �∆ 2 � ≤ 𝐾𝐾 ≤ ∆ − 1. so, in order to decrease the length of the edge (𝑥𝑥, 𝑦𝑦) to 𝐾𝐾 no more than 𝐾𝐾 steps are needed. therefore, to achieve a situation where 𝑦𝑦 is not adjacent to an edge with length over 𝐾𝐾, 𝑂𝑂(𝐾𝐾2) elementary operations are sufficient. finally, due to the fact that �∆ 2 � ≤ 𝐾𝐾 ≤ ∆ − 1, the bandwidth minimization problem for an interval graph with number of vertices n and with maximal vertex degree ∆, can be solved using 𝑂𝑂(∆2𝑛𝑛 log2 ∆) elementary operations. references [1] l. h. harper, “optimal numberings and isoperimetric problems on graphs”, journal of combinatorial theory, vol.1, no. 3, pp. 385–393, 1966. [2] c. papadimitriou, “the np-completeness of the bandwidth minimization problem”, computing, vol. 16, pp. 263-270, 1976. [3] d. muradian, “the bandwidth minimization problem for cyclic caterpillars with hair length 1 is np-complete”, theoretical computer science, “selected papers in honour of lawrence harper”, vol. 307, no. 3, pp. 567-572, 2003. [4] y.-l. lai, bandwidth, edgesum and profile of graphs. ph.d. thesis, dept. of computer science, western michigan univ, 1997. [5] t. kloks, d. kratsch and h. muller, “bandwidth of chain graphs”, information processing letters, vol. 68, no. 6, pp. 313-315, 1998. [6] s. f. assman, peck et al., “the bandwidth of caterpillars with hair of length 1 and 2”, siam journal on algebraic and discrete methods, vol. 2, pp. 387-393, 1981. [7] д. о. мурадян, “полиномиальный алгоритм для нахождения минимаксных нумераций графов интервалов”, академия наук арм. сср, в. 82, pp. 64-66, 1986. [8] d. kleitman and r. vohra, “computing the bandwidth of interval graphs”, siam j. discrete math., vol. 3, no. 3, pp. 373-375, 1990. [9] r. mahesh, c. p. rangan and a. srinivasan, “on finding the minimum bandwidth of interval graphs”, information and computation, vol. 95, no. 2, pp. 218-224, 1991. [10] a. p sprague, “an o(n log n) algoritm for bandwidth of interval graphs”, siam journal on discrete mathematics, vol. 7, no. 2, pp. 213-220, 1994. submitted 21.07.2016, accepted 24.10.2016. http://dblp.uni-trier.de/db/journals/siamdm/siamdm3.html%23kleitmanv90 http://dblp.uni-trier.de/db/journals/siamdm/siamdm3.html%23kleitmanv90 a polynomial algorithm for the minimum bandwidth of interval graphs 80 պոլինոմիալ բարդությամբ ալգորիթմ ինտերվալ գրաֆների բարձրությունը գտնելու համար դ. մուրադյան ամփոփում դիցուք g-ն կապակցված գրաֆ է x գագաթների և u կողերի բարձրությամբ: յուրաքանչյուր փոխմիարժեք 𝜑𝜑 համապատասխանություն, որ գագաթների x բազմությունը արտապատկերում է {1, 2, . . . , |x|} բազմության վրա, կոչվում է g գրաֆի համարակալում: 𝐵𝐵𝜑𝜑(𝐺𝐺) = max | 𝜑𝜑(𝑢𝑢) − 𝜑𝜑(𝑣𝑣)| թիվը, որտեղ մաքսիմումը վերցվում է ըստ գրաֆի բոլոր կողերի, սահմանվում է որպես 𝜑𝜑 համարակալման բարձրություն: g գրաֆի բարձրությունը սահմանվում է որպես b(g) = min𝐵𝐵𝜑𝜑(𝐺𝐺), որտեղ մինիմումը վերցվում է ըստ գրաֆի բոլոր համարակալումների: ինտերվալ գրաֆը սահմանվում է որպես թվային առանցքի վրա տրված ինտերվալների ինչ-որ ընտանիքի հատումների գրաֆ: աշխատանքում բերվում է ինտերվալ գրաֆների բարձրությունը գտնող պոլինոմիալ բարդությամբ ալգորիթմ: ալգորիթմն ունի o�n∆2log(∆)� բարդություն, որտեղ n-ը գրաֆի գագաթների քանակն է, իսկ δ-ն՝ գագաթների մեծագույն աստիճանը: алгоритм полиномиальной сложности для нахождения высоты графов интервалов д. мурадян аннотация пусть g=(x, u) – граф со множеством вершин x и ребер u. каждое взаимнооднозначное отображение { }|x| , ,2 ,1: →xϕ назовем его нумерацией. при этом число |𝜑𝜑(𝑥𝑥) − 𝜑𝜑(𝑦𝑦)| назовем длиной ребра (x,y), а числа 𝐵𝐵𝜑𝜑(𝐺𝐺) = max(𝑥𝑥,𝑦𝑦)∈𝑈𝑈 |𝜑𝜑(𝑥𝑥) − 𝜑𝜑(𝑦𝑦)| и 𝐵𝐵(𝐺𝐺) = min 𝜑𝜑 𝐵𝐵𝜑𝜑(𝐺𝐺), где минимум берется по всевозможным нумерациям графа g, соответственно высотой нумерации 𝜑𝜑 и графа g. граф интервалов определяется как граф пересечений семейства интервалов данных на числовой прямой. в настоящей работе приводится алгоритм полиномиальной сложности для нахождения высоты произвольного графа интервалов. алгоритм имеет сложность o�n∆2log(∆)�, где n – количество вершин, а δ – максимальная степень вершин графа. 1. introduction 2. preliminaries 3. algorithm 4. proof of the algorithm’s correctness 5. estimation of algorithm complexity d:\sbornik\...\nas.dvi mathematical problems of computer science 36, 7{12, 2012. t he degr ee of unsolvability of the completion semantics for gener al logic p r ogr ams l e vo n a . h a yka z ya n yerevan state university e-mail levon@rock.com abstract the completion semantics considers interpretations that satisfy a special ¯rst-order theory was ¯rst introduced in [1]. these interpretations include but are not limited to herbrand interpretations. nevertheless, in logic programming the restriction to herbrand interpretations is very desirable. as [2] remarks, however, this results in a non-recursively enumerable semantics. in this paper we show the ¦11-completeness of the completion semantics with restriction to herbrand interpretations. refer ences [1 ] k . l . cla r k, \ n e g a t io n a s fa ilu r e " , in h. gallaire, j . m inker, l ogic and d atabeses, p p 2 9 3 -3 2 3 , p le n u m , 1 9 7 8 . [2 ] j. c. s h e p h e r d s o n , \ n e g a t io n a s fa ilu r e , c o m p le t io n a n d s t r a t ī c a t io n " , in d . m . gabbay, c. j . hogger, j . a. r obinson, l ogic p rogramming, p p 3 5 5 -4 1 9 , cla r e n d o n p r e s s , 1 9 9 8 . [3 ] j. w . l lo yd , r . w . to p o r , \ ma kin g p r o lo g m o r e e xp r e s s ive " , j ournal of l ogic p rogramming, vo l. 1 , p p . 2 2 5 -2 4 0 , 1 9 8 4 . [4 ] j. w . l lo yd , f oundations of l ogic p rogramming, s p r in g e r -v e r la g , 1 9 9 4 . [5 ] h . r o g e r s , the theory of r ecursive f unctions and e ®ective computability, mc gr a wh ill, 1 9 6 7 . àý¹ñ³ýñ³óí³í ïñ³ù³µ³ý³ï³ý íñ³·ñ»ñç ÷³ïù³ý ë»ù³ýïçï³ûç ³ýéáõí»éçáõãû³ý ³ëïç×³ýá è. ð³ûï³½û³ý ²ù÷á÷áõù ö³ïù³ý ë»ù³ýïçï³ý ý»ñùáõíí»é ¿ [1] ³ßë³ïáõãûáõýáõù, áõñ ¹çï³ñïíáõù »ý çýï»ñåñ»ï³óç³ý»ñ, áñáýù µ³í³ñ³ñáõù »ý ñ³ïáõï ³é³ççý ï³ñ·ç ï»ëáõãû³ý: ²ûë çýï»ñåñ»ï³óç³ý»ñá ý»ñ³éáõù »ý, µ³ûó ã»ý ë³ñù³ý³÷³ïíáõù ð»ñµñ³ýç çýï»ñåñ»ï³óç³ý»ñáí: ²ûýáõ³ù»ý³ûýçí, ïñ³ù³µ³ý³ï³ý íñ³·ñ³íáñù³ý ù»ç 7 8 the degree of unsolvability of the completion semantics for general logic programs ð»ñµñ³ýç çýï»ñåñ»ï³óç³ý»ñáí ë³ñù³ý³÷³ïáõùá ëçëï ó³ýï³éç ¿: ê³ï³ûý çýãå»ë ýßíáõù ¿ [2] ³ßë³ïáõãûáõýáõù, ¹³ µ»ñáõù ¿ áã é»ïáõñëçíáñ»ý ãí³ñï»éç ë»ù³ýïçï³ûç: êáõûý ³ßë³ïáõãûáõýáõù óáõûó ¿ ïñíáõù ÷³ïù³ý ë»ù³ýïçï³ûç ¦ 11éñçíáõãûáõýá ð»ñµñ³ýç çýï»ñåñ»ï³óç³ý»ñáí ë³ñù³ý³÷³ïù³ý ¹»åùáõù: ñòåïåíü íåðàçðåøèìîñòè ñåìàíòèêè çàìûêàíèÿ îáîáùåííûõ ëîãè÷åñêèõ ïðîãðàìì ë. àéêàçÿí àííîòàöèÿ ñåìàíòèêà çàìûêàíèÿ áûëà ââåäåíà â ðàáîòå [1], â êîòîðîé ðàññìàòðèâàþòñÿ èíòåðïðåòàöèè, óäîâëåòâîðÿþùèå ñïåöèàëüíîé òåîðèè ïåðâîãî ïîðÿäêà. ýòè èíòåðïðåòàöèè âêëþ÷àþò â ñåáÿ ýðáðàíîâñêèå èíòåðïðåòàöèè, íî íå îãðàíè÷èâàþòñÿ èìè. òåì íå ìåíåå â ëîãè÷åñêîì ïðîãðàììèðîâàíèè îãðàíè÷åíèå ýðáðàíîâñêèìè èíòåðïðåòàöèÿìè âåñüìà æåëàòåëüíî. îäíàêî, êàê îòìå÷åíî â ðàáîòå [2], ýòî ïðèâîäèò ê íå ðåêóðñèâíî ïåðå÷èñëèìîé ñåìàíòèêå. â äàííîé ðàáîòå ïîêàçûâàåòñÿ ¦ 11 -ïîëíîòà ñåìàíòèêè çàìûêàíèÿ ïðè îãðàíè÷åíèè ýðáðàíîâñêèìè èíòåðïðåòàöèÿìè. mathematical problems of computer science 48, 42–49, 2017. image caption generation and object detection via a single model aghasi s. poghosyan institute for informatics and automation problems of nas ra e-mail: agasy18@gmail.com abstract automated semantic information extraction from the image is a difficult task. there are works which can extract image caption or object names and their coordinates. this work presents a merged single model of object detection and automated caption generation systems. the final model extracts from image caption and object coordinates with their names without losing accuracy according to initial models. keywords: neural networks, image caption, object detection, deep learning, rnn, lstm 1. introduction automatically describing the content of an image is a fundamental problem in artificial intelligence that connects computer vision and natural language processing. the content can be partially described via image caption and objects names and their locations. this is significantly harder than the well-studied image classification [1] or object recognition. these studies can help visually impaired people better understand the content of images on the web, also it can have a great impact on search engines and in robotics, for example, self driving cars. automatically generated image caption should contain the main object names, their properties, relations, and actions. moreover, the generated caption should be expressed through a natural language like english. there are a number of works approaching this problem. some of them [2, 3, 4] offer combining the existing image object detection and sentence generation systems. but there is a more efficient solution [5] that offers a joint model. it takes an image and generates the caption, which describes the image adequately. the lastest achievements in statistical machine translation were actively used in image caption generation tasks. the reason for this is mainly the proven achievement of greater results when using a powerful sequential model trained by maximizing the probability of the correct translation for the input sentence. these models [6, 5, 7] are based on recurrent neural networks (rnns). the model encodes the variable length input into the fixed length vector representation. this representation enables conversion of the input sentence into the target sentence or the input image into the target image caption. 42 ag. poghosyan 43 neural nets have become a leading method for high quality object detection in recent years. modern object detectors based on convolutional neural network (cnn) [8] networks, such as faster region-based convolutional neural network (faster r-cnn) [9], region-based fully convolutional network (r-fcn) [10], multibox [11], single shot detector (ssd) [12] and yolo: real-time object detection [13], are now good enough to be deployed in consumer products (e.g., google photos, pinterest visual search) and some of them have been shown to be fast enough to run on mobile devices. there is work [14] which present a multi-model neural network method closely related to the human visual system that automatically learns to describe the content of images. the model consists of two sub-models: an object detection and localization model, which extract the information of objects and their spatial relationship in images respectively; besides, a deep recurrent neural network (rnn)-based on long short-term memory (lstm) units with an attention mechanism for sentences generation. each word of the description is automatically aligned to different objects of the input image when it is generated. it is similar to the attention mechanism of the human visual system. this work present a merged model of object detection and automated caption generation systems. for object detection we will choose faster r-cnn [9] based on inception [15] and for caption generation show and tell [5]. these two models are based on inception image classification model. this will allow as save all quality characteristics of the initial models. 2. object detection the r-cnn paper by girshick et al. [16] was among the first modern incarnations of convolutional network-based detection. inspired by recent successes in image classification [17], the r-cnn method took a straightforward approach of cropping externally computed box proposals out of an input image and running a neural net classifier on these crops. this approach can be expensive, however, because many crops are necessary, leading to significant duplicated computation from overlapping crops. fast r-cnn [9] alleviated this problem by pushing the entire image once through a feature extractor then cropping from an intermediate layer so that crops share the computation load of feature extraction. in the faster r-cnn detection happens in two stages. at the first stage, called the region proposal network (rpn), images are processed by a feature extractor, and features at some selected intermediate level are used to predict class-agnostic box proposals. l(a, i; θ) = α · 1[a is positive] · `loc(φ(ba; a) − floc(i; a, θ)) + β · `cls(ya, fcls(i; a, θ)), (1) the loss function for this first stage takes the form of equation 1 using a grid of anchors tiled in space, scale and aspect ratio. at the second stage, these (typically 300) box proposals are used to crop features from the same intermediate feature map which are subsequently fed to the remainder of the feature extractor in order to predict a class and class-specific box refinement for each proposal. the loss function for this second stage box classifier takes the form of equation 1 using the proposals generated from the rpn as anchors. notably, one does not crop proposals directly from the image and re-run crops through the feature extractor, which would be a duplicated computation. however, there is a part of computation that must be performed once per region, and, thus, the run time depends on the number of regions proposed by the rpn. determining classification and regression targets for each anchor requires matching anchors to groundtruth instances. common approaches include greedy bipartite matching (e.g., 44 image caption generation and object detection via a single model based on jaccard overlap) or many-to-one matching strategies in which bipartiteness is not required, but matchings are discarded if jaccard overlap between an anchor and groundtruth is too low. paper [18] refers to these strategies as bipartite or argmax, respectively. the model [18] uses argmax matching throughout with thresholds set as suggested in the original paper [9]. after matching, there is typically a sampling procedure designed to bring the number of positive anchors and negative anchors to some desired ratio. to encode a groundtruth box with respect to its matching anchor, the model uses the box encoding function φ(ba; a) = [10 · xcwa , 10 · yc ha , 5 · log w, 5 · log h] (also used by [16, 9]). the scalar multipliers 10 and 5 are typically used in all of these prior works [18, 9, 16]. for our work we will use pretrained faster-rcnn based on inception classifier [18]. we will extract high level features before object detector. extracted features will be used to create an image embedding vector. 3. caption generaton the model encodes the variable length input into the fixed length vector representation. this representation enables conversion of the input sentence into the target sentence or the input image into the target image caption. the last model was being trained to maximize p (s|i) likelihood to generate the target sequence of words s = {s1, s2, ...} for an input image i, where each word st comes from a given dictionary, that describes the image adequately. show and tell [5] model can generate image descriptions with recurrent neural network. it maximizes the probability of the correct caption for the given image, log p(s|i; θ) = n∑ t=0 log p(s|i, s0, ..., st−1; θ), (2) where (s|i) is a training example pair. while training, we optimize the sum of the log probabilities for the whole training set using adagrad [19]. p(s|i, s0, ..., st−1; θ) probability will correspond to the t step (iteration) of recurrent neural network (rnn) based model. the variable number of words that are conditioned upon, up to t − 1 is expressed by a fixed length hidden state or memory ht. after every iteration for the new input, memory will be updated by using a non-linear function f . ft+1 = f (ht, xt). (3) in this work, we will select mixed 7c layer from google inception [15] (we will use object detector’s inception [18]) and append average pooling layer which will have 2048-dimensional output for image description. also, we will append fully connected neural layer with ne neurons, which will convert 2048-dimensional vector into ne dimensional vector. ne is an image-words embedding vectors dimensionality [20]. the output vector x1 of fully connected layer will be the first feed vector for rnn, x−1 = m ixed7c ∗ wi + bi, (4) where wi ∈ r2048xne and bi ∈ rne are trainable parameters for image embedding. we also have lookup embedding matrix wl ∈ rdxne , where d is the dictionarys words count. each row of the matrix represents a word embedding in image-word embedding space. each xi (where i ≥ 0) is the corresponding row at index (si)(equation 5). xi = w si e . (5) ag. poghosyan 45 for f from equation 3 we use a long short-term memory (lstm) [21], which has shown state-of-the-art performance on sequence generation tasks, such as translation or image caption generation. long short-term memory (lstm) is an rnn cell. it helps in solving rnn training time problems like vanishing and exploding gradients [21], which is a significant problem for rnns. lstm is commonly used in machine translation, sequence generation and image description generation tasks. paper [5] uses recurrent neural network with an lstm cell to generate image caption. from a construction perspective, lstm is a memory cell c encoding knowledge at every iteration of what inputs have been seen up to this iteration. later this knowledge is used for subsequent word generation (10, 11). behavior of the cell is controlled by three gates: an input gate, an output gate and a forget gate. each gate is a vector of real number elements ranging from 0 to 1. in particular, the forget gate is responsible for controlling whether to forget the cells old value, the input gate controls the permission for reading a new input value and finally the output gate controls the permission to output the new value from the cell. this is done by multiplying the given gate with the corresponding value (9, 10). the definition of the lstm is as follows: it = σ(wixxt + wimmt−1), (6) ft = σ(wf xxt + wf mmt−1), (7) ot = σ(woxxt + wommt−1), (8) ct = ft ¯ ct−1 + it ¯ h(wcxxt + wcmmt−1), (9) mt = ot ¯ ct, (10) pt+1 = sof tmax(wpm ∗ mt), (11) in (6)-(11) equations it, ot, ft are input, output and forget gates, correspondingly, ct is a cell memory in step t and mt is an output of the lstm for step (iteration) t. wix, wim, wf x, wf m, wox, wom, wcx, wcm are trainable parameters (variables) of the lstm.¯ represents the product with a gate value. sigmoid σ(·) and h(·) hyperbolic tangent are nonlinearities of the lstm. equation 11 will produce a probability distribution pt+1 over all words in the dictionary, where wpm is a trainable parameter. the lstm model is trained to predict the probability for the next word of an image caption after it has observed all the previous words in the captions and image features. for easier training lstm is represented in unrolled form, which is a copy of the lstm memory for the image and each word of the sentence. also all lstms share the same parameters. thus, x−1 is the first input for the first lstm. initial state of the lstm is c−1 zero-filled memory. for the next lstms, inputs correspond to the word embedded vectors. also, all recurrent connections are converted into feed-forward connections. loss function will be sum of the negative log likelihood of the correct word at each step: l(i, s) = − n∑ t=1 log p(st). for training, we have used adagrad instead of multi-batch stochastic gradient descent. we have trained on microsoft common objects in context (mscoco) [22] image dataset and keep the same metrics from the original work in caption generation task. [5]. inference has been made via using beam search which gives us variants for the best scored sentence after many predictions in table 1. 46 image caption generation and object detection via a single model table 1: image caption generation and object detection via a single model. 1) a group of elephants standing next to each other . 2) a group of elephants standing in a pen . 3) a group of elephants standing next to each other in a zoo. 1) a group of people standing on top of a sandy beach . 2) a group of people standing on top of a beach . 3) a group of people standing on a beach next to the ocean . 1) a baseball player holding a bat on a field . 2) a baseball player swinging a bat on a field . 3) a baseball player swinging a bat at a ball 1) a man sitting in a chair holding a banana . 2) a man sitting in a chair holding a hot dog . 3) a man sitting in a chair holding a banana 4. conclusion we have built an image caption generation model on top of object detection model. as our chosen object detection model has same object classifier (feature extractor) as our chosen captions generation model, we have the kept all accuracy metrics from both initial works. thus, we have created a single model that can generate an image caption and detect objects names and their locations. ag. poghosyan 47 references [1] o. russakovsky, j. deng, h. su, j. krause, s. satheesh, s. ma, z. huang, a. karpathy, a. khosla, m. bernstein, et al., “imagenet large scale visual recognition challenge,” international journal of computer vision, vol. 115, no. 3, pp. 211–252, 2015. 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[11] c. szegedy, s. reed, d. erhan, d. anguelov, and s. ioffe, “scalable, high-quality object detection,” arxiv preprint arxiv:1412.1441, 2014. [12] w. liu, d. anguelov, d. erhan, c. szegedy, s. reed, c.-y. fu, and a. c. berg, “ssd: single shot multibox detector,” in european conference on computer vision. springer, 2016, pp. 21–37. [13] j. redmon, s. divvala, r. girshick, and a. farhadi, “you only look once: unified, realtime object detection,” in proceedings of the ieee conference on computer vision and pattern recognition, 2016, pp. 779–788. 4 8 image caption generation and object detection via a single model [1 4 ] z. y a n g , y .-j. zh a n g , y . h u a n g e t a l., \ im a g e c a p t io n in g wit h o b je c t d e t e c t io n a n d lo c a liz a t io n ," a r x iv p r e p r in t a r x iv:1 7 0 6 .0 2 4 3 0 , 2 0 1 7 . [1 5 ] c. s z e g e d y, v . v a n h " in p roceedings of the ie e e conference on computer vision and p attern r ecognition, p p . 2 8 1 8 { 2 8 2 6 , 2 0 1 6 . [1 6 ] r . gir s h ic k, j. d o n a h u e , t. d a r r e ll a n d j. ma lik, \ r ic h fe a t u r e h ie r a r c h ie s fo r a c c u r a t e o b je c t d e t e c t io n a n d s e m a n t ic s e g m e n t a t io n " , in p roceedings of the ie e e conference on computer vision and pattern recognition, p p . 5 8 0 { 5 8 7 , 2 0 1 4 . [1 7 ] a . k r iz h e vs ky, i. s u t s ke ve r a n d g. e . h in t o n , \ im a g e n e t c la s s ī c a t io n wit h d e e p c o n vo lu t io n a l n e u r a l n e t wo r ks " , in advances in neural information processing systems, p p . 1 0 9 7 1 1 0 5 , 2 0 1 2 . [1 8 ] j. h u a n g , v . r a t h o d , c. s u n , m. zh u , a . k o r a t t ika r a , a . fa t h i, i. fis c h e r , z. w o jn a , y . s o n g , s . gu a d a r r a m a e t a l., \ s p e e d / a c c u r a c y t r a d e -o ®s fo r m o d e r n c o n vo lu t io n a l o b je c t d e t e c t o r s " , a r x iv p r e p r in t a r x iv:1 6 1 1 .1 0 0 1 2 , 2 0 1 6 . [1 9 ] m. d . ze ile r , \ a d a d e lt a : a n a d a p t ive le a r n in g r a t e m e t h o d " , a r x iv p r e p r in t a r x iv:1 2 1 2 .5 7 0 1 , 2 0 1 2 . [2 0 ] t. miko lo v, k . ch e n , g. co r r a d o a n d j. d e a n , \ e ± c ie n t e s t im a t io n o f wo r d r e p r e s e n t a t io n s in ve c t o r s p a c e " , a r x iv p r e p r in t a r x iv:1 3 0 1 .3 7 8 1 , 2 0 1 3 . [2 1 ] s . h o c h r e it e r a n d j. s c h m id h u b e r , \ l o n g s h o r t -t e r m m e m o r y" , neural computation, vo l. 9 , n o . 8 , p p . 1 7 3 5 { 1 7 8 0 , 1 9 9 7 . [2 2 ] t.-y . l in , m. ma ir e , s . b e lo n g ie , j. h a ys , p . p e r o n a , d . r a m a n a n , p . d o lla r a n d c. l . zit n ic k, \ mic r o s o ft c o c o : co m m o n o b je c t s in c o n t e xt " , in e uropean conference on computer vision, s p r in g e r , p p . 7 4 0 { 7 5 5 , 2 0 1 4 . submitted 30.08.2017, accepted 05.12.2017. ä³ïï»ñç í»ñý³·ñç ·»ý»ñ³óáõùá ¨ ûµû»ïïý»ñç ñ³ûïý³µ»ñáõùá ù»ï ùá¹»éç ùççáóáí ². äáõáëû³ý ²ù÷á÷áõù ä³ïï»ñç ù³ëçý çù³ëï³µ³ý³ï³ý çýýáñù³óç³ûç ³íïáù³ï³óí³í ëï³óáõùá µ³ñ¹ ëý¹çñ ¿: î³ý ³ßë³ï³ýùý»ñ, áñáýù ·»ý»ñ³óýáõù »ý å³ïï»ñç í»ñý³·çñá ï³ù ·ïýáõù ûµû»ïïý»ñç ïááñ¹çý³ïý»á í»ñççýý»ñçë ³ýí³ýáõùý»ñç ñ»ï ù»ïï»õ: ²ûë ³ßë³ï³ýùá ý»ñï³û³óýáõù ¿ ù»ï ³ùµáõç³ï³ý ùá¹»é, áñá ï³ñáõ³ýáõù ¿ ·»ý»ñ³óý»é å³ïï»ñç í»ñý³·çñá ¨ ûµû»ïïý»ñç ³ýí³ýáõùý»ñá çñ»ýó ïááñ¹çý³ïý»ñáí: ag. poghosyan 4 9 ãåíåðàöèÿ çàãîëîâêà èçîáðàæåíèÿ è îáíàðóæåíèå îáúåêòà ñ ïîìîùüþ åäèíîé ìîäåëè à. ïîãîñÿí àííîòàöèÿ àâòîìàòè÷åñêîå èçâëå÷åíèå ñåìàíòè÷åñêîé èíôîðìàöèè èç èçîáðàæåíèÿ ñëîæíàÿ çàäà÷à. ñóùåñòâóþò ðàáîòû, êîòîðûå ìîãóò èçâëåêàòü çàãîëîâêè èçîáðàæåíèé èëè èìåíà îáúåêòîâ è èõ êîîðäèíàòû. ýòà ðàáîòà ïðåäñòàâëÿåò ñîáîé îáúåäèíåííóþ åäèíóþ ìîäåëü îáíàðóæåíèÿ îáúåêòîâ è àâòîìàòè÷åñêîãî ôîðìèðîâàíèÿ çàãîëîâêîâ. aghasi_42--48.pdf (p.1-6) abstract_42--48.pdf (p.7-8) mathematical problems of computer science 41, 131---134, 2014. 131 a method of constructing and using the tree of possible transformations from unl to the natural language levon r. hakobyan institute for informatics and automation problems of nas ra e-mail: levon.r.hakobyan@gmail.com abstract an approach to improving the transformation process from unl to natural language is described. we introduce the tree of possible transformations, which lets us generate all the possible transformation paths for the current sentence and rule set. the implementation of the algorithm in undl platform is described. keywords: unl, machine translations, transformation rules, grammar. 1. introduction some new methods for improving the transformation rules from unl to natural language are described. the sentences in universal networking language (unl) contain some semantical information which is usually represented in different natural languages. such semantical information is formalized in unl by a structure of an oriented linked graph [1]. the information usually contained in a sentence on a natural language is transformed in unl using oriented binary labeled links (which are treated as “relations’ between nodes or nodes). nodes are represented by “universal words”, or simply “uw”; they express the corresponding concepts. the semantics of the sentence in unl is expressed also by the so-called “attributes” which contain some information about modality of the used concepts. we will consider one of the main systems for the unl processing, namely, the undl platform. it has been developed by undl foundation in recent years [2]. the terms “analysis” and “generation” we will use as the names of the transformation process, respectively, from a natural language to unl, and from unl to the natural language. these processes are implemented in undl platform respectively by ian and eugen projects [2]. the current transformation algorithm applies a sequence of the rules to the current sentence. this sequence is created by using some kind of priorities. but other rules could be used for the current sequence. so, we need some tools which let us find the most appropriate sequence of the rules to be applied to the sentence. in the existing transformation algorithms it is not easy to understand which rules sequence should be used. a method of constructing and using the tree of possible transformations from unl132 to solve this problem we suggest extending the existing algorithms by adding functionality for the construction of the tree of possible transformations, also known as the tree of possible paths of transformation. in this article we describe the tree of possible paths of transformation and its implementation in undl platform system. in the first part we give a short description of the transformation process done by the generator, and the description of the tree of possible paths of the transformation. in the second part we give algorithms for the construction of the tree and information about classes and methods, used in the implementation in the undl platform environment. our approach is developed for transformations from unl to the natural language. but it could be used also for transformations from natural language to unl, with some modifications. 2. the tree of possible transformations transformations in the undl platform system are implemented by consequently applying transformation rules on the unl sentence. in case of the analyzer, in the result of this process the linear structure is transformed into a semantic graph. in case of the generator, the semantic graph is transformed into a linear structure. in the process of the generation when a given unl sentence is transformed to a sentence in the natural language the given dictionary and the set of transformation rules are used. any transformation rule has the form (α, β), where α is a predicate (i.e. condition which can be true or false for a given current sentence), and β is an algorithm defining the transformation of the considered current sentence. we assume that some number p such that 0 ≤ p ≤ 255 is attached to any transformation rule; this number is called “priority coefficient”. if there are several rules which can be applied to a considered sentence then the rule having the greatest priority coefficient is chosen. the process of transformation includes a finite number of steps. on any step only one transformation rule can be applied. the process is assumed to be completed when there is no rule which can be applied to the considered sentence; this sentence is admitted as the result of the process. by changing the priorities, we can obtain more than one sequence of applicable rules. using this we can construct the tree of possible paths of transformation. node in this tree is a structure which contains, besides other additional data, a unl sentence and a rule, by applying which the sentence was obtained. in the tree every sub node contains, besides other additional data, a corresponding rule and a unl sentence, which was obtained by applying the rule to the unl sentence of the parent node. in root node the rule should be null, and the sentence should be original to be transformed. the leaves obtained after the construction of the tree, will store all possible transformation results. 3. the construction of the tree as we know, undl platform is a web application. computational time is very important for us, because it is bounded by server response time. vertices of the tree are implemented by the objects of the treenode class. the objects of this class contain, besides some basic and additional data about the vertex, links to a parent and the children of the vertex. every object of the treenode class has its unique id which lets us operate with such kind of objects; even then they are identical by other parameters. l. hakobyan 133 the tree is stored in the tree class. static object of that class is created for every web request. the tree class contains, besides some other data and methods, a layers object which is implemented by the hash-table. the key in that hash-table is an integer which shows the number of the layer in the tree. and its value is a linked list which stores links to vertices of the current layer, which are implemented by the objects of the treenode class. this solution lets us have an access to vertices in the tree in two ways: first by the original way by recursive searching from root vertex, and second by accessing it by using the index and other parameters. the second method lets us access to vertices faster. in this article we describe two methods for the implementation of the tree in the undl platform environment. the first method is a depth-first search. in the undl platform a functionality was implemented, which lets us apply rules with the corresponding indices by passing a sequence of that indices. in the transformation process on each step, other rules which could be applied on this step, besides the applied rule, are registered by activating the corresponding feature. this lets us get the first transformation path, and some free vertices after the first call of the transformation function. a free vertex is a vertex from which the construction of the tree could be continued. after that, a loop is used to get the next free vertex. for that vertex the corresponding sequence of indices are constructed. this sequence is a path from the root to the current free vertex. after that, a function of transformation with this sequence is called. the loop continues as long as there are free vertices in the tree. the second method for the construction of the tree of possible transformation is based on the breadth-first search. there is a great probability of occurring of equal sentences in different vertices of the same layer, because on every transformation path the same subset of the rules could be applied in another sequence. this lets us group equal vertices and continue the construction of the tree only for one vertex from a group. in this way, we can avoid analyzing some paths in the tree which leads to the same results and reduces the computational time. in the current implementation, the breadth-first search is used up to some predefined layer in the tree. after that, the depth-first search is used for the corresponding vertices. 4. conclusion in this article we introduce the tree of possible transformations and describe its implementation in the undl platform system. the tree of possible transformations lets us obtain all possible transformation results for the current sentence and rule set as the leaves of the tree. this lets the user compare these results with reference translation. algorithms for the evaluation of translations like bleu and meteor can be used to compare transformation results. the most appropriate transformation path can be found by using these tools. this feature can be used as a tool for automated editing. the experiments showed that the tree of possible transformations of the comparatively simple sentence (containing no more than 5 uws) can be constructed by the corresponding programs using test drive corpus in reasonable time and corresponding results set will contain no more than 130 possible transformations. a method of constructing and using the tree of possible transformations from unl134 the main problem of using the methods described above is a computational complexity because of a huge amount of vertices on testing on real examples. this also leads to the problem of great amount of information, which makes the working process uncomfortable for the user. therefore, further researches should be done to make the algorithm work faster and to create an effective method for the users to deal with data which are generated in the result of the algorithm. references [1] h. uchida, m. zhu, and t. della senta, a gift for a millennium, ias/unu, 1999. [2] undl foundation web site, [online]. available: http://www.undlfoundation.org/. [3] deconverter specification version 2.6, unl center /undl foundation 2002 unl center, enconverter specifications, version 3.3 tokyo, 2001 submitted 24.12.2013, accepted 02. 03. 2014. unl լեզվից դեպի բնական լեզու տրանսֆորմացման հնարավոր ուղիների ծառի կառուցման մեթոդը լ. հակոբյան ամփոփում հոդվածում նկարագրված է unl լեզվից դեպի բնական լեզու տրանսֆորմացման գործընթացի բարելավման մեթոդը: ներկայացված է տրանսֆորմացման հնարավոր ուղիների ծառը, որը թույլ է տալիս ստանալ բոլոր հնարավոր տրանսֆորմացման ուղիները տվյալ նախադասության և կանոնների խմբի համար: հոդվածում նկարագրված է նաև համապատասխան ալգորիթմների իրականցումը undl platform համակարգում: метод построения и использования дерева возможных направлений трансформации с языка unl на естественный язык л. акопян аннотация описывается подход к улучшению процесса трансформации с языка unl на естественный язык. дается метод построения дерева возможных путей трансформации, который позволяет генерировать всевозможные пути трансформации для данного предложения и набора правил. описана также реализация соответствующих алгоритмов в системе undl platform. d:\sbornik\...\article.dvi mathematical problems of computer science 35, 104{108, 2011. on the p ower s of dead-end recognizing systems in the class of t wo-e lement sets concer ning oper ations of i nter section and complement s e yr a n m. v a r d a n ya n institute for informatics and automation problems of nas of ra e-mail: seyranv@ipia.sci.am abstract the notion of n-recognizing system in the class of two-element sets concerning the operations of intersection and complement is de¯ned. the set of powers of dead-end n-recognizing systems of such kind is estimated. refer ences [1 ] ñ. ì. âàðäàíÿí, “î äëèíàõ òóïèêîâûõ ðàñïîçíàþùèõ ñèñòåì â êëàññå äâóõýëåìåíòíûõ ïîäìíîæåñòâ”, äàí íàí ðà, òîì 107, n 1, ñ ñ. 37 43, 2007. 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[5] f. harary, graph theory, addison wesley. reading ma 1969. ð³ïù³ý ¨ éñ³óù³ý ·áñíáõáõãûáõýý»ñç ýï³ïù³ù³µ »ñïï³ññ³ýç µ³½ùáõãûáõýý»ñç ¹³ëáõù ×³ý³ãáõ ÷³ïáõõ³ûçý ñ³ù³ï³ñ·»ñç ñ½áñáõãûáõýý»ñç ù³ëçý ê. ì³ñ¹³ýû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ë³ñù³ýíáõù ¿ n-ï³ññ³ýç µ³½ùáõãû³ý n-×³ý³ãáõ ñ³ù³ï³ñ· ñ³ëï³óáõãûáõýá áëï µ³½ùáãûáõýý»ñç ñ³ïù³ý áõ éñ³óù³ý ·áñíáõáõãûáõýý»ñç ¨ »ñïï³ññ³ýç »ýã³µ³½ùáõãûáõýý»ñç ¹³ëáõù ·ý³ñ³ïíáõù ¿ ÷³ïáõõ³ûçý n-×³ý³ãáõ ñ³ù³ï³ñ·»ñç ñ½áñáõãûáõýý»ñç µáéáñ ñý³ñ³íáñ ³ñå»ùý»ñá: 1 0 4 microsoft word tpel.doc mathematical problems of computer science 35, 14--18, 2011. 14 implementation of a search parallelization method on high performance computing structures grigor h. geoletsyan and hrant g. geoletsyan institute for informatics and automation problems of nas ra russian-armenian (slavonic) university abstract we describe the implementation of modified initialization and load balancing methods during parallelization of search algorithms for solution of optimization problems on high performance computing structures. most optimization problems in discrete mathematics related to applications in new research areas are np-hard. parallelization of search algorithms using static and dynamic load-balancing (distribution of computational resources) and their implementation on high performance computing structures, in particular, on computer clusters is an essential tool for obtaining effective solutions to such problems. references [1] г. геолецян, “эвристический алгоритм решения задачи ограниченного вершинного покрытия двудольного графа”, доклады нан ра, т. 107, н. 1, стр. 44–48, 2007. 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[10] f. yu, c-h. tsai, y-w.hang, h-y. lin, d. t. lee, s-y. kuo, “efficient еxact spare allocation via boolean satisfiability”, proceedings of the 20th ieee international symposium on defect and fault tolerance in vlsi systems (dft’05), oct., monterey, california, 9p, 2005. g. geoletsyan, h. geoletsyan 15 ð³ï³ñïù³ý ½áõգ³ñ»é³óù³ý ù»ãá¹ç çñ³ï³ý³óáõùá µ³ñóñ ³ñï³¹ñõáõ³ï³ýáõãû³ùµ ñ³ßíáõ³ï³ý ï³éáõóí³íùý»ñáõù ¶. ¶»áé»óû³ý ¨ ð. ¶»áé»óû³ý ²ù÷á÷áõù üï³ñ³գñí³í ¿ ñ³ù³ï³ñգí³í ñ³ï³ñïù³ý ³éգáñçãùý»ñç ½áõգ³ñ»é³óù³ý ëï½µý³µ»ñù³ý ¨ µ³é³ýë³íáñù³ý ùá¹çýçï³óí³í ù»ãá¹ç çñ³ï³ý³óáõùá µ³ñóñ ³ñï³¹ñáõ³ï³ýáõãû³ùµ ñ³ßíáõ³ï³ý ï³éáõóí³íùý»ñç íñ³ ûåïçù³é³óù³ý ëý¹çñý»ñç éáõíù³ý å³ù³ý³ï: d:\sbornik\...\tpel\tpel.dvi mathematical problems of computer science 36, 51{56, 2012. a n ote on m aximum weight i ndependent set in outer -r ectangle gr aph e d u a r d t. p ilip o s ya n russian-armenian state university e-mail: eduard.piliposyan@gmail.com abstract an outer-rectangle graph is the intersection graph of rectangles lying inside a rectangular box and having exactly one edge on the boundary of the box. we present a polynomial-time algorithm for the problem of computing a maximum weight independent set in 2-side outer-rectangle graphs where any two rectangles lying on same edge of box do not intersect. keywords: weighted independent set, rectangle graphs, dynamic programming. refer ences [1 ] p . ch a le r m s o o k a n d j. ch u z h o y.\ ma xim u m in d e p e n d e n t s e t o f r e c t a n g le s " . in 20th annual acm -siam sod a, p p . 8 9 2 { 9 0 1 , 2 0 0 9 . [2 ] p . k . a g a r wa l, m. j. va n k r e ve ld , a n d s . s u r i. \ l a b e l p la c e m e n t b y m a xim u m in d e p e n d e n t s e t in r e c t a n g le s " . comput. geom., 1 1 ( 3 { 4 ) : p p .2 0 9 { 2 1 8 , 1 9 9 8 . [3 ] s . d o e r s c h le r a n d h . fr e e m a n . \ a r u le -b a s e d s ys t e m fo r d e n s e -m a p n a m e p la c e m e n t " . communications of the acm , 3 5 ( 1 ) , p p . 6 8 { 7 9 , 1 9 9 2 . [4 ] r . j. fo wle r , m. s . p a t e r s o n , a n d s . l . ta n im o t o . \ op t im a l p a c kin g a n d c o ve r in g in t h e p la n e a r e n p -c o m p le t e " . inform. p rocess. l ett., 1 2 ( 3 ) , p p .1 3 3 { 1 3 7 , 1 9 8 1 . [5 ] t. a s a n o . d i± culty of the maximum independent set problem on intersection graphs of geometric objects. in 6 t h icta g, 1 9 9 1 . [6 ] p . b e r m a n , b . d a s gu p t a , s . mu t h u kr is h n a n , a n d s . r a m a s wa m i. \ e ± c ie n t a p p r o xim a t io n a lg o r it h m s fo r t ilin g a n d p a c kin g p r o b le m s wit h r e c t a n g le s " . j . algorithm, 4 1 , 2 0 0 1 . [7 ] l . l e win -e yt a n , j. n a o r , a n d a . or d a . \ a d m is s io n c o n t r o l in n e t wo r ks wit h a d va n c e r e s e r va t io n s " . algorithmica, 4 0 ( 4 ) : p p .2 9 3 { 3 0 4 , 2 0 0 4 . [8 ] t. m. ch a n a n d s . h a r -p e le d . \ a p p r o xim a t io n a lg o r it h m s fo r m a xim u m in d e p e n d e n t s e t o f p s e u d o -d is ks " . in 25th annual acm socg, p p . 3 3 3 { 3 4 0 , 2 0 0 9 . [9 ] g. h . ch e n , m. t. k u o , a n d j. p . s h e u . \ a n o p t im a l t im e a lg o r it h m fo r ¯ n d in g a m a xim u m we ig h t in d e p e n d e n t s e t in a t r e e " . b it, vo l. 2 3 , p p . 3 5 3 | 3 5 6 , 1 9 8 8 . [1 0 ] g. fr a h lin g , u . fa ig le .a combinatorial algorithm for weighted stable sets in bipartite graphs. d is c r e t e a p p lie d m a t h e m a t ic s , 2 0 0 6 . 5 1 5 2 a note on maximum weight independent set in outer-rectangle graph [1 1 ] m. p a l a n d g. p . b h a t t a c h a r je e , a sequential algorithm for ¯nding a maximum weight k-independent set on interval graphs, intern. j . computer m athematics, vo l. 6 0 , p p . 2 0 5 -2 1 4 , 1 9 9 6 . ²é³í»é³·áõûý ïßéáí ³ýï³ë µ³½ùáõãûáõýý»ñ ³ñï³ùçý áõõõ³ýïûáõýý»ñç ·ñ³ýý»ñáõù ¾. öçéçåáëû³ý ²ù÷á÷áõù ²ñï³ùçý áõõõ³ýïûáõýý»ñç ·ñ³ýý ³ûýåçëç áõõõ³ýïûáõýý»ñç ñ³ïáõùý»ñç ·ñ³ý ¿, áñáýù µáéáñá ý»ñ³éí³í »ý ùç¨ýáõûý áõõõ³ýïûáõý ßñç³ý³ïç ù»ç ¨ áñáýó ×çßï ù»ï ïáõùá ñ»ýí³í ¿ ³û¹ ßñç³ý³ïç áñ¨¿ ïáõùç íñ³: ²ßë³ï³ýùáõù ý»ñï³û³óí³í ¿ ³ñï³ùçý áõõõ³ýïûáõýý»ñç ·ñ³ýáõù ³é³í»é³·áõûý ïßéáí ³ýï³ë µ³½ùáõãûáõý ï³éáõóáõ µ³½ù³ý¹³ù³ûçý µ³ñ¹áõãû³ùµ ³é·áñçãù ³ûý ¹»ù»ñç ñ³ù³ñ, »ñµ áõõõ³ýïûáõýý»ñá ñ»ýí³í »ý ßñç³ý³ïç ïáõù»ñçó ùç³ûý »ñïáõëç íñ³, ¨ ßñç³ý³ïç ùç¨ýáõûý ïáõùçý ñ»ýí³í áõõõ³ýïûáõýý»ñá ã»ý ñ³ïíáõù: íåçàâèñèìûå ìíîæåñòâà ìàêñèìàëüíîãî âåñà â âíåøíå ïðÿìîóãîëüíûõ ãðàôàõ ý. ïèëèïîñÿí àííîòàöèÿ âíåøíåïðÿìîóãîëüíûé ãðàô ýòî ãðàô ïåðåñå÷åíèé ïðÿìîóãîëüíèêîâ, êîòîðûå âñå ðàñïîëîæåíû âíóòðè íåêîòîðîãî ïðÿìîóãîëüíèêà ðàìêè, è òî÷íî îäíà ñòîðîíà êîòîðûõ íàõîäèòñÿ (îïèðàåòñÿ) íà êàêîé-òî ñòîðîíå ðàìêè. â ðàáîòå ïðåäñòàâëåí ïîëèíîìèàëüíûé àëãîðèòì ïîñòðîåíèÿ íåçàâèñèìîãî ìíîæåñòâà ìàêñèìàëüíîãî âåñà â âíåøíåïðÿìîóãîëüíîì ãðàôå â ñëó÷àå, êîãäà ïðÿìîóãîëüíèêè îïèðàþòñÿ òîëüêî íà äâå ñòîðîíû ðàìêè, è îïèðàþùèåñÿ íà îäíó è òó æå ñòîðîíó ðàìêè ïðÿìîóãîëüíèêè íå ïåðåñåêàþòñÿ. microsoft word a. gevorgyan_7.doc mathematical problems of computer science 34, 2010. 23 mathematical problems of spin-glass systems in onedimension ashot s gevorkyan, institute for informatics and automation problems, nas of armenia spin glass models are very useful for investigations of a number of important and difficult applied problems of physics, chemistry, material science, biology, evolution, organization dynamics, hard-optimization, environmental and social structures, human logic systems, financial mathematics etc (see for example [1-10]). it is possible to divide the considered meanfield models of spin-glasses into two types. the first consists of the true random-bond models, where the coupling between interacting spins are taken to be independent random variables [1113]. the solution of these models was obtained by n-replica trick [11-13] and required the invention of sophisticated schemes of replica-symmetry breaking [13-14]. in the models of second type, the bond-randomness is expressed in terms of some underlining hidden siterandomness and is thus of a superficial nature. it has been pointed out in the works, however, that this feature retains an important physical aspect of true spin-glasses, viz. that they are random with respect to the positions of magnetic impurities. the main way of investigation of spin-glasses systems is a method of numerical simulation. but, numerical studies of spin-glass problem is difficult to accomplish and, in general, only for a small number of spins, the numerical simulation is effectively solvable. for development of new ideas and creation of program packages for numerical simulation of spin-glass systems the investigation of 1d model is very important. the modified 1d model can be useful also for analysis of series of real problems. 24 fig. 1. the red line shows visualization of numerical data while the green line shows analytic approximation by cauchy function. the statistical properties of ensemble of disordered 1d steric spin-chains (ssc) of various lengths are investigated. using 1d spin-glass type classical hamiltonian, the recurrent transcendental equations for stationary points and corresponding conditions for the construction of stable 1d sscs are found. the ideal ensemble of spin-chains (non-interacting spin-chains) is analyzed and the latent interconnections between random angles and interaction constants for each set of three nearest-neighboring spins are found. it is analytically proved and by numerical calculation is shown that the distribution of interaction constant satisfy levy's alpha-stable distribution law [15] (see fig. 1 and reference [16]). energy distribution in ensemble is calculated depending on different conditions of possible polarization of spin-chains. it is specifically shown that the dimensional effects in the form of set of local maximums in the energy distribution arise when the number of spin-chains m< 0 r ig h t ly p ( » ¸ t) · m»=t: 2 ) th e s e c o n d ch e b ys h e v in e qu a lit y ( [4 ]) . l e t t h e a b o ve r a n d o m va r ia b le » h a ve a d is p e r s io n d». th e n fo r a n y t > 0 r ig h t ly p ( j» ¡ m »j ¸ t) · d»=t2: 1 6 asymptotic estimates of the number of solutions of systems of equations with partial boolean functions 3 ) fo r a n y x > 1 µ 1 ¡ 1 x ¶x < e¡1 : 4 ) fo r a n y n a t u r a l n a n d m2 = o( n) c m n » nm m! : 5 ) fo r a n y n a t u r a l n a n d 1 · m · n c m n < ³ en m ´m : 6 ) l e t b ( k; n; p ) = c k n p kqn¡k, wh e r e 0 < p; q < 1 ; p + q = 1 . th e n fo r r > np n¡rx j=0 b( r + j; n; p) < b( r; n; p) ( r + 1 ) q= ( r + 1 ¡ ( n + 1 ) p) ( t h e e s t im a t e o f t h e \ t a il" o f t h e b in o m ia l d is t r ib u t io n ( [4 ]) ) . l e t s b e a s ys t e m in r ( n; l ) . a r r a n g in g ( t r a n s p o s it io n s b y a ll t h e va r ia t io n s ) t h e e qu a t io n s in s , we o b t a in ! n e w s ys t e m s , d i®e r in g fr o m e a c h o t h e r b y t r a n s p o s it io n o f t h e e qu a t io n s . th u s , fr o m t h e s e t r ( n; l ) we o b t a in a n e w s e t r0 ( n; l ) o f o r d e r e d a n d n o n r e p e t it ive ( n o t c o n t a in in g t h e e qu iva le n t e qu a t io n s ) s ys t e m s . it 's e vid e n t t h a t jr0 ( n; l ) j = jr ( n; l ) jl!: ( 2 ) l e t a lm o s t a ll t h e s ys t e m s o f t h e s e t r0 ( n; l ) h a ve t h e p r o p e r t y e , wh ic h is in va r ia n t fo r a n y t r a n s p o s it io n o f t h e e qu a t io n s o f t h e s ys t e m . it 's e a s y t o s e e , t h a t a lm o s t a ll t h e s ys t e m s o f t h e s e t r( n; l ) will a ls o h a ve t h e p r o p e r t y e . th u s , fo r t h e p r o o f o f t h e th e o r e m it will b e e n o u g h t o c o n s id e r t h e s e t r0 ( n; l ) in s t e a d o f r ( n; l ) . n e xt , we d e n o t e b y r 00 ( n; l ) e xp a n s io n o f t h e s e t r0 ( n; l ) in t h e s ys t e m s fr o m r0 ( n; l ) a llo we d a s a m e e qu a t io n s . it is e a s y t o s e e , t h a t jr00 ( n; l ) j = 3 l2n : ( 3 ) fr o m ( 2 ) , ( 3 ) a n d 4 ) we o b t a in , t h a t jr0 ( n; l ) j jr00 ( n; l ) j = l!cl 32 n 3 l2 n ! 1 ; wh e n l2 = o ¡ 3 2 n ¢ ( n ! 1 ) . th u s , if l2 = o ¡ 3 2 n ¢ , a n y a s s e r t io n fo r a lm o s t a ll s ys t e m s o f t h e s e t r00 ( n; l ) is t r u e fo r a lm o s t a ll t h e s ys t e m s o f t h e s e t r0 ( n; l ) . w e c o n s id e r r00 ( n; l ) a s a s p a c e o f e ve n t s , wh e r e e ve r y e ve n t s 2 r00 ( n; l ) t a ke s p la c e wit h t h e p r o b a b ilit y 1 =jr00 ( n; l ) j = 3 ¡l2n . co n s id e r t h e r a n d o m va lu e »s ( ~®) , wh ic h is c o n n e c t e d wit h s 2 r0 ( n; l ) a s fo llo ws : »s ( ~®) = 1 ; if ~® is t h e s o lu t io n o f t h e s ys t e m s , a n d »s ( ~® ) = 0 in a n o t h e r c a s e . fr o m t h e d e ¯ n it io n it fo llo ws , t h a t t h e n u m b e r o f t h e s ys t e m s 2 r0 ( n; l ) , fo r wh ic h ~® is a s o lu t io n , e qu a l t o 3 l(2 n¡1). fr o m t h is a n d ( 3 ) it fo llo ws , t h a t p ( »s ( ~®) = 1 ) = 3 ¡l. e. yeghiazaryan 1 7 co n s id e r a n o t h e r r a n d o m va lu e v = p ~®2bn »s ( ~® ) . r a n d o m va lu e v h a s a b in o m ia l d is t r ib u t io n , b e c a u s e p ( v = j ) = c j 2n 3 ¡lj ( 1 ¡ 3 ¡l ) 2n¡j : h e n c e , mv = 2 n 3 ¡l a n d dv = 2 n 3 ¡l ¡ 1 ¡ 3 ¡l ¢ , wh e r e mv a n d dv a r e t h e m a t h e m a t ic a l e xp e c t a t io n a n d d is p e r s io n o f t h e r a n d o m va lu e v, a c c o r d in g ly. l e t n ¡ ` lo g 3 ! 1 wh e n n ! 1. it m e a n s , t h a t m v = 2 n 3 ¡l = 2 n¡l log 3 ! 1 wh e n n ! 1. u s in g t h e ch e b is h e v's in e qu a t io n 2 ) wh e n t = mv= p n ¡ l lo g 3 , we o b t a in , p ¡ jv ¡ m vj ¸ mv= p n ¡ l lo g 3 ¢ · ( n ¡ l lo g 3 ) ( 1 ¡ 3 ¡l ) = 2 n¡l log 3 ! 0 wh e n n ! 1. h e n c e a n d fr o m t h e d e ¯ n it io n o f r a n d o m va lu e v it fo llo ws , t h a t a lm o s t a ll t h e s ys t e m s o f t h e s e t r00 ( n; l ) h a ve t h e n u m b e r o f s o lu t io n s , wh ic h a s ym p t o t ic a lly e qu a ls mv. s in c e u n d e r n ¡ ` lo g 3 ! 1 is p e r fo r m e d l2 = o ¡ 3 2 n ¢ , a lm o s t a ll t h e s ys t e m s o f t h e s e t r( n; l ) h a ve a ls o n u m b e r o f s o lu t io n s , a s ym p t o t ic a lly e qu a l t o m v = 2 n 3 ¡l. th e ¯ r s t s t a t e m e n t o f t h e th e o r e m is p r o ve d . l e t n ¡ ` lo g 3 ! ¡1 wh e n n ! 1. th e n mv = 2 n 3 ¡l = 2 n¡l log 3 ! 0 ( n ! 1) . u s in g ch e b is h e v's ¯ r s t in e qu a t io n wh e n t =l, we o b t a in p ( v ¸ 1 ) ! 0 wh e n n ! 1 a n d t h e r e fo r e , p ( v = 0 ) ! 1 wh e n n ! 1. h e n c e it fo llo ws , t h a t a lm o s t a ll t h e s ys t e m s s o f t h e s e t r00 ( n; l ) h a ve n o s o lu t io n . th e r e fo r e , wh e n l2 = o ¡ 3 2 n ¢ t h e s e c o n d s t a t e m e n t o f t h e th e o r e m is p r o ve d . it is e a s y t o s e e , t h a t fo r t h e g r e a t e r va lu e s o f t h e p a r a m e t e r l t h e s t a t e m e n t o f t h e th e o r e m a ls o h o ld s ( t h e n u m b e r o f s o lu t io n s o f t h e s ys t e m d o e s n o t in c r e a s e wit h t h e n u m b e r o f e qu a t io n s ) . n o w le t n ¡ ` lo g 3 b e r e s t r ic t e d , wh e n n ! 1. th e n mv = 2 n 3 ¡l = 2 n¡l log 3 is a ls o r e s t r ic t e d wh e n n ! 1. u s in g t h e in e qu a t io n s 6 ) , 5 ) a n d 3 ) , we o b t a in p ( v > r ) = 2n¡rx i=0 cr+i2n 3 ¡l(r+i) ( 1 ¡ 3 ¡l ) 2n¡r¡i · cr2n 3 ¡lr ( 1 ¡ 3 ¡l ) 2 n¡r£ £ ( r + 1 ) ¡ 1 ¡ 3 ¡l ) ± ¡ r + 1 ¡ ( 2 n + 1 ) 3 ¡l ¢ · ( e 2 n 3 ¡lr¡1 ) r · µ em v r ¶ ! 0 ; wh e n r ! 1 , b e c a u s e mv is r e s t r ic t e d . th e r e fo r e , fo r a lm o s t a ll t h e s ys t e m s o f t h e s e t r00 ( n; l ) t h e t h ir d s t a t e m e n t o f t h e th e o r e m h o ld s . s in c e n ¡ ` lo g 3 is r e s t r ic t e d , t h e n l2 = o ¡ 3 2 n ¢ is p e r fo r m e d , a n d , t h e r e fo r e , fo r a lm o s t a ll t h e s ys t e m s o f t h e s e t r ( n; l ) t h e t h ir d s t a t e m e n t o f t h e th e o r e m h o ld s . th e o r e m is c o m p le t e ly p r o ve d . refer ences [1 ] m. ge r i a n d d . jo h n s o n , computers and intractability, ( in r u s s ia n ) , mo s c o w, mir , 1 9 8 2 . [2 ] e . v . y e g h ia z a r ya n , \ me t r ic p r o p e r t ie s o f s ys t e m s o f b o o le a n e qu a t io n s " , d an armenian ssr ,( in r u s s ia n ) , vo l. 7 2 , n o .2 , p p . 6 7 { 7 2 , 1 9 8 1 . [3 ] e . v . y e g h ia z a r ya n , \ e s t im a t e s r e la t e d t o t h e n u m b e r o f s o lu t io n s o f b o o le a n e qu a t io n s " , coll. tasks of cybernetics. combinatorial analysis and graph theory, ( in r u s s ia n ) mo s c o w, p p . 1 2 4 { 1 3 0 , 1 9 8 0 . [4 ] w . fe lle r , an introduction to p robability theory and its applications, ( in r u s s ia n ) , vo l. 1 , mo s c o w, mir , 1 9 7 6 . 1 8 asymptotic estimates of the number of solutions of systems of equations with partial boolean functions submitted 15.11.2015, accepted 22.01.2016 ø³ëý³ïç µáõéû³ý ýáõýïóç³ý»ñáí ñ³í³ë³ñáõùý»ñç ñ³ù³ï³ñ·»ñç éáõíáõùý»ñç ù³ý³ïç ³ëçùåïáïçï ·ý³ñ³ï³ï³ýý»ñ ¾. ºõç³½³ñû³ý ²ù÷á÷áõù îñíáõù »ý ù³ëý³ïç (áã ³ù»ñáõñ»ù áñáßí³í) µáõéû³ý ýáõýïóç³ý»ñçó ï³½ùí³í ñ³í³ë³ñáõùý»ñç ñ³ù³ï³ñ·»ñç éáõíáõùý»ñç ù³ý³ïç ³ëçùåïáïçï ·ý³ñ³ï³ï³ýý»ñ §ïçåçï¦ ¹»åùç ñ³ù³ñ: àñèìïòîòè÷åñêèå îöåíêè ÷èñëà ðåøåíèé ñèñòåì óðàâíåíèé ñ ÷àñòè÷íûìè áóëåâûìè ôóíêöèÿìè ý. åãèàçàðÿí àííîòàöèÿ îïðåäåëÿþòñÿ àñèìïòîòè÷åñêèå îöåíêè ÷èñëà ðåøåíèé ñèñòåì óðàâíåíèé ñ ÷àñòè÷íûìè (íå âñþäó îïðåäåëåííûìè) áóëåâûìè ôóíêöèÿìè äëÿ ”òèïè÷íîãî” ñëó÷àÿ. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 45, 59--66, 2016. secure multiparty computations for collaboration between competing service providers davit h. danoyan yerevan state university e-mail: danoyan@gmail.com abstract we introduce a platform for secure computations and describe the platform workflow. we detail the golreich-micalli-widgerson protocol employed for secure multiparty computation in our platform with optimization techniques applied to it. being based on white-box cryptography, the underlying oblivious transfer protocol avoids the use of expensive public key operations and provides good overhead compared to similar systems. also we point out some useful application scenarios and provide a real world application design based on our platform. keywords: secure multiparty computation, oblivious transfer, white-box cryptography 1. introduction secure computations were introduced by yao in [19] searching a solution to the millionaires problem, where two millionaires want to find out who is the richest of them without revealing to each other or any other party of their actual property. years later several approaches were developed for the solution of this problem and its generalized version with 𝑛𝑛 millionaires as yao’s garbled circuit based protocol [12], gmw [18] and others. although those protocols were valid solutions for the problem, the computational resources available at that time were insufficient for using them in practice. the first practical implementation of a secure computation protocol was presented in [1] nearly a decade ago. later, several enhancements and new implementations have been developed [2]. many applications were proposed for online auctions [5], e-commerce [7], data mining [6], secure computations outsourcing [4], etc. in this paper we present a new application scenario, where several service providers possessing specific private information wish to make secure computations on their collective data. in section 2 we will present some background on the gmw multiparty computation protocol and our optimization techniques. section 3 will be the brief description of our platform used for the secure computations in the specific application. the applications’ motivation and description can be found in section 4 and section 5 is the conclusion of this paper. 59 60 secure multiparty computations for collaboration between competing service providers 2. gmw protocol the gmw protocol is intended for computation of any function that is possible to represent as a single pass boolean circuit, where the permitted operations (circuit gates) are 𝑋𝑋𝑋𝑋𝑋𝑋 and 𝐴𝐴𝐴𝐴𝐴𝐴. as the protocol avoids the need of trusted third parties, communication is required between all pairs of participants. unlike the yao’s garbled circuits protocol, where one party is the circuit generator and the other party is the circuit evaluator, here all the parties possess the circuit locally and each of them plays an evaluator with input shares of himself and the other participants. 2.1 secret sharing each party generates 𝑛𝑛 shares for each input wire 𝑤𝑤 (𝑠𝑠𝑤𝑤1, … , 𝑠𝑠𝑤𝑤𝑤𝑤) randomly so, that ⊕𝑖𝑖=1 𝑤𝑤 𝑠𝑠𝑤𝑤𝑖𝑖 = 𝑠𝑠𝑤𝑤. share generation doesn’t really require much effort to satisfy the above equation. party 𝑃𝑃𝑗𝑗 generates n-1 shares randomly for all other parties 𝑃𝑃𝑖𝑖, 𝑖𝑖 ≠ 𝑗𝑗. to comply with the equation, it just has to adjust its own share correspondingly 𝑠𝑠𝑤𝑤𝑗𝑗 = (⊕i≠j 𝑠𝑠𝑤𝑤𝑖𝑖) ⊕ 𝑠𝑠𝑤𝑤. after the share generation, each share 𝑠𝑠𝑤𝑤𝑖𝑖 is sent to the corresponding participant 𝑃𝑃𝑖𝑖. now, having shares for all input wires, each participant proceeds to circuit evaluation. after the evaluation, each participant 𝑃𝑃𝑖𝑖 would have its own value 𝑠𝑠𝑤𝑤𝑖𝑖, so for each output wire 𝑤𝑤 of the circuit we again will have an n-tuple (𝑠𝑠𝑤𝑤1, … , 𝑠𝑠𝑤𝑤𝑤𝑤). to combine those values into a single value for each wire the protocol suggests a specific way of gate evaluation, according to which the ntuples for each output wire will play as shares of parties, and the final value will be computed with 𝑠𝑠𝑤𝑤 = ⊕𝑖𝑖=1 𝑤𝑤 𝑠𝑠𝑤𝑤𝑖𝑖 formula. as the circuit consists of xor and and gates only, below we provide the methods for both gates’ evaluation that should be used for all input, intermediate and output gates of the circuit and show that they operate as expected. 2.2 gate evaluation evaluation of 𝑋𝑋𝑋𝑋𝑋𝑋 gates. suppose we have an 𝑋𝑋𝑋𝑋𝑋𝑋 gate with input wires 𝑢𝑢 and 𝑣𝑣 and an output wire 𝑤𝑤. all parties already have their shares for the input wires (𝑠𝑠𝑢𝑢1, … , 𝑠𝑠𝑢𝑢𝑤𝑤) and (𝑠𝑠𝑣𝑣1, … , 𝑠𝑠𝑣𝑣𝑤𝑤). each party 𝑃𝑃𝑖𝑖 computes the value of the output wire in the following way: 𝑠𝑠𝑤𝑤𝑖𝑖 = 𝑠𝑠𝑢𝑢𝑖𝑖 ⊕ 𝑠𝑠𝑣𝑣𝑖𝑖 . the 𝑛𝑛-tuple (𝑠𝑠𝑤𝑤1, … , 𝑠𝑠𝑤𝑤𝑤𝑤) would be a valid sharing for 𝑤𝑤’s value, because 𝑠𝑠𝑤𝑤 = ⊕𝑖𝑖=1 𝑤𝑤 𝑠𝑠𝑤𝑤𝑖𝑖 = ⊕𝑖𝑖=1 𝑤𝑤 (𝑠𝑠𝑢𝑢𝑖𝑖 ⊕ 𝑠𝑠𝑣𝑣𝑖𝑖) = (⊕𝑖𝑖=1 𝑤𝑤 𝑠𝑠𝑢𝑢𝑖𝑖) ⊕ ( ⊕𝑖𝑖=1 𝑤𝑤 𝑠𝑠𝑣𝑣𝑖𝑖) = 𝑠𝑠𝑢𝑢 ⊕ 𝑠𝑠𝑣𝑣. evaluation of 𝐴𝐴𝐴𝐴𝐴𝐴 gates suppose we have an 𝐴𝐴𝐴𝐴𝐴𝐴 gate with input wires 𝑢𝑢 and 𝑣𝑣 and an output wire 𝑤𝑤. the shares of input values of 𝑢𝑢 and 𝑣𝑣 are (𝑠𝑠𝑢𝑢1, … , 𝑠𝑠𝑢𝑢𝑤𝑤) and (𝑠𝑠𝑣𝑣1, … , 𝑠𝑠𝑣𝑣𝑤𝑤). the output value 𝑠𝑠𝑤𝑤 should be computed so, that 𝑠𝑠𝑤𝑤 = 𝑠𝑠𝑢𝑢 ∧ 𝑠𝑠𝑣𝑣 = (⊕𝑖𝑖=1 𝑤𝑤 𝑠𝑠𝑢𝑢𝑖𝑖) ∧ ( ⊕𝑖𝑖=1 𝑤𝑤 𝑠𝑠𝑣𝑣𝑖𝑖) = = (⊕𝑖𝑖=1 𝑤𝑤 (𝑠𝑠𝑢𝑢𝑖𝑖 ∧ 𝑠𝑠𝑣𝑣𝑖𝑖)) ⊕ (⊕𝑖𝑖<𝑗𝑗 ((𝑠𝑠𝑢𝑢𝑖𝑖 ∧ 𝑠𝑠𝑣𝑣𝑗𝑗) ⊕ (𝑠𝑠𝑢𝑢𝑗𝑗 ∧ 𝑠𝑠𝑣𝑣𝑖𝑖))). the first sum ⊕𝑖𝑖=1 𝑤𝑤 (𝑠𝑠𝑢𝑢𝑖𝑖 ∧ 𝑠𝑠𝑣𝑣𝑖𝑖) can be computed locally for every participant, but for evaluation of the second sum one should communicate with other participants. to avoid the collection of individual shares of other parties, oblivious transfer protocols are used for 𝐴𝐴𝐴𝐴𝐴𝐴 gate evaluation in the following way. d. danoyan 61 as 𝑃𝑃𝑖𝑖 has to interact with several 𝑃𝑃𝑗𝑗-s to compute each element of ⊕𝑖𝑖<𝑗𝑗 ��𝑠𝑠𝑢𝑢𝑖𝑖 ∧ 𝑠𝑠𝑣𝑣𝑗𝑗� ⊕ �𝑠𝑠𝑢𝑢𝑗𝑗 ∧ 𝑠𝑠𝑣𝑣𝑖𝑖��, 𝑃𝑃𝑗𝑗 generates a random value 𝑟𝑟𝑗𝑗 {𝑖𝑖,𝑗𝑗} and composes four values with 𝑟𝑟𝑗𝑗 {𝑖𝑖,𝑗𝑗} and its shares for input wires of the gate being computed 𝑠𝑠𝑢𝑢𝑗𝑗 and 𝑠𝑠𝑣𝑣𝑗𝑗. 𝑟𝑟𝑗𝑗 {𝑖𝑖,𝑗𝑗}, 𝑟𝑟𝑗𝑗 {𝑖𝑖,𝑗𝑗} ⊕ 𝑠𝑠𝑢𝑢𝑗𝑗 , 𝑟𝑟𝑗𝑗 {𝑖𝑖,𝑗𝑗} ⊕ 𝑠𝑠𝑣𝑣𝑗𝑗, 𝑟𝑟𝑗𝑗 {𝑖𝑖,𝑗𝑗} ⊕ 𝑠𝑠𝑢𝑢𝑗𝑗 ⊕ 𝑠𝑠𝑣𝑣𝑗𝑗. then the parties invoke a 1 − 𝑜𝑜𝑢𝑢𝑜𝑜 − 𝑜𝑜𝑜𝑜 − 4 oblivious transfer with 𝑃𝑃𝑖𝑖 as receiver and 𝑃𝑃𝑗𝑗 as sender. 𝑃𝑃𝑖𝑖 requests one of the abovementioned values with its index composed of its shares for the input wires 𝑠𝑠𝑢𝑢𝑖𝑖, 𝑠𝑠𝑣𝑣𝑖𝑖. 𝑃𝑃𝑖𝑖 computes the required sum with the received value 𝑟𝑟𝑖𝑖 {𝑖𝑖,𝑗𝑗} 𝑟𝑟𝑖𝑖 {𝑖𝑖,𝑗𝑗} ⊕ 𝑟𝑟𝑗𝑗 {𝑖𝑖,𝑗𝑗} = (𝑠𝑠𝑢𝑢𝑖𝑖 ∧ 𝑠𝑠𝑣𝑣𝑗𝑗) ⊕ (𝑠𝑠𝑢𝑢𝑗𝑗 ∧ 𝑠𝑠𝑣𝑣𝑖𝑖). the proof of correctness of the 𝐴𝐴𝐴𝐴𝐴𝐴 gates’ computation method described above can be found in section 7.5.2.2 of [3]. so 𝑃𝑃𝑖𝑖, after getting all 𝑟𝑟𝑗𝑗 {𝑖𝑖,𝑗𝑗}-s from all parties with 𝑗𝑗 > 𝑖𝑖, can compute its share for the output wire 𝑤𝑤. 𝑠𝑠𝑤𝑤𝑖𝑖 = (𝑠𝑠𝑢𝑢𝑖𝑖 ∧ 𝑠𝑠𝑣𝑣𝑖𝑖) ⊕ (⊕𝑖𝑖<𝑗𝑗 ��𝑠𝑠𝑢𝑢𝑖𝑖 ∧ 𝑠𝑠𝑣𝑣𝑗𝑗� ⊕ �𝑠𝑠𝑢𝑢𝑗𝑗 ∧ 𝑠𝑠𝑣𝑣𝑖𝑖��. given the methods of share based evaluation of both type gates, each party computes the circuits’ output value shares. the computation of the full value of an output wire can be done after each participant sends the result of its computations to others. note, that computation of an 𝑋𝑋𝑋𝑋𝑋𝑋 gate share values is free in terms of network communication and almost free, in terms of computation (𝑛𝑛 𝑋𝑋𝑋𝑋𝑋𝑋 operation). however, the computation of 𝐴𝐴𝐴𝐴𝐴𝐴 gates takes � 𝑛𝑛 2� ot invocations. 2.3 optimizations white-box oblivious transfer oblivious transfer is an essential cryptographic primitive heavily used in secure computations. it was introduced by rabin in [13] as a protocol where the sender sends a message to the receiver with 1 2 probability and does not reveal if the receiver got the message or not. a modified version of this protocol was introduced later by even et al. in [14], currently known as the basic 1-out-of2 ot protocol. this protocol assumes that the receiver has a selection bit 𝑠𝑠𝑠𝑠{0,1} and the sender has two bits 𝑚𝑚0, 𝑚𝑚1𝑠𝑠{0,1} . after protocol execution the receiver gets 𝑚𝑚𝑠𝑠 and nothing about 𝑚𝑚1−𝑠𝑠 and the sender does not reveal 𝑠𝑠. unfortunately this protocol depends on public-key operations, which require exponentiation operations which are pretty expensive, considering the huge amount of ot operations required for secure computations (in yao’s protocol ots are used for all input bits of one of the participants, in gmw ots are used for all 𝐴𝐴𝐴𝐴𝐴𝐴 gates). in [11] a new approach to ot was introduced using white-box cryptography techniques instead of public-key operations, allowing to reduce cost of an ot in several orders of magnitude, in case of many invocations, as reported. wbot extension 62 secure multiparty computations for collaboration between competing service providers another optimization technique for reducing the ot invocation count is ot extension introduced by beaver in [17] and later enhanced in [16]. we have constructed similar optimization for white-box ot, allowing to reduce the required ot invocation count to a predetermined fixed number. the details of the extension can be found in [8]. 2.4 gmw protocol extension here we introduce an extended version of the gmw protocol, that enables involvement of so called passive parties that don’t participate in the computation process, while having input and expecting output. suppose 𝑛𝑛 + 𝑚𝑚 parties wish to compute a common function based on private inputs, only 𝑚𝑚 of which are passive. in this scenario we suggest the active (participating in computation) parties to behave as they do in the original protocol and share their inputs among 𝑛𝑛 active parties with the method described in section 2.2. each passive participant also generates 𝑛𝑛 shares for each of its input, but doesn’t keep a share for himself and distributes all shares among the active parties. the latter compute the function based on the secret shares as in the original protocol and distribute output shares among all parties. the security requirement for this protocol are also different – here we require at least one non-corrupted active party. 3. platform description the platform processes a file containing a program described in an imperative language for to be computed by the participants. beside the function description the file contains specification on the intended inputs and expected outputs. the program defines computations in a manner they would be carried out by a virtual trusted party possessing all input parameters. our platform consist of three main modules – a compiler, and two modules responsible for two-party and multiparty secure computations respectively. below we briefly describe the main modules. compiler we implemented a compiler generating an equivalent boolean circuit on basis { 𝐴𝐴𝐴𝐴𝐴𝐴, 𝑋𝑋𝑋𝑋𝑋𝑋, 1 } implementing the described function. the compiler generates an initial boolean circuit and passes it through various stages of optimizations trying to minimize the number of gates in generated circuit. another major goal of the compiler is the minimization of 𝐴𝐴𝐴𝐴𝐴𝐴 gates and circuit depth. for this purpose we use low depth circuit constructions developed mainly for vlsi applications. to deal with large circuits efficiently, the compiler generates a file containing the usage count for each gate. when a gate is processed, by a participant, the initial usage count is read from this file, and each time the output of the gate is accessed, the associated counter is decremented. data associated with the gate is removed from memory once the counter hits zero. in this way we minimize the number of gates simultaneously kept in memory during evaluation of the circuit. two-party module although it is possible to perform secure computations with two participants using gmw protocol, in our platform we have a specialized module for this task. for this purpose we have implemented yao’s garbled circuits protocol [12]. the module works on the output of the compiler module. detailed description can be found in [9]. d. danoyan 63 multiparty module the multiparty module is intended for secure computations with more than 2 participants. the module implements the goldreich-micalli-widgerson protocol[18] with several enhancements described in [15] and in the previous section. it also operates on a boolean circuit outputted by the compiler module. 4. applications here we will describe the process of combining multiple delivery services into a unified platform that involves several members. in fact, the application described is suitable for using in different contexts like combining any delivery or transportation services where the members are service providers, who do not want to disclose their current locations or the locations they are available to cover within the fixed time to their competitors, but eager to cooperate with them to have their share in unified ordering system, thus increasing their order counts and service efficiency. concerning the motivation of client to use the unified system, let’s compare the actions needed for getting the fastest service. in case of having an access to the unified application, the client just has to give the application her location and confirm the order. all the computation and decision processes are completed without the clients’ involvement. in absence of such a system one should contact several service providers (possibly with different interfaces), give them its location, compare the offers of different providers, find the best of them and finally put an order. in the latter case the client also has to trust the information it got from service providers. suppose the service providers are taxi service companies with one or more cabs and the client has no priorities for choosing particular service other than fast pick-up. taxi services do not want to make their locations visible to rival companies to avoid them getting advantage by better positioning of their cabs (for example, this can be done by a company with significantly more cabs). all cab locations are also hidden from the client, because a competitor can possibly act as a client to find out the others’ cab locations and get advantage. having the application scenario described we show several methods as candidates for the secure computation itself. this methods are basically computing the minimum value of distances between a client as one point, and provider instances as candidates for the second point. different distance measurement algorithms are presented below. the main purpose of this application is to find the nearest item in the unified database of locations to a submitted location. with the computation securing method already appointed, we still need to have accurate location detecting and network connection hardware for the service providers and individual cabs. in this work we consider the mentioned hardware implemented in a black-box manner – we do not consider their technical and efficiency details. we also assume that the computing parties have identical maps as well and their locations follow common coordinate system, to avoid misinterpretation of function arguments. considering the distance measurement we have several methods as a candidate. euclid distance the most general version of the distance measurement function is computation of the euclid distance between the customer and all the provider instances (e.g. cabs). with the coordinates given in two dimensional format, the customer’s location denoted as (𝑥𝑥𝑐𝑐, 𝑦𝑦𝑐𝑐) and for 𝑖𝑖-th provider instance (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖) considering m provider instances, the following function should be computed to determine the “winning” bid: min i chess board. fig 7. chess move module. chess move module has 2 parts: 1. chess move (represents chess moves) 2. chess move validator (provides mechanism for validating chess moves) chess move module api for part 1:  ucimovetosymbolicmove(theucimove) converts the move received using uci to symbolic representation.  symbolicmovetoabstractmove(thesymbolicmove) converts the symbolic representation of chess move to abstract one. chess move module api for part 2:  isvalid(themove) checks if the move is valid.  checkendofgame(thechessgamestate) checks theend of game for specified chess game state. 3.6. chess game module this module allows to handle chess games. chess game is related to chess board, chess piece, chess move, chess player, chess communication. chess game is a sequence of game states. fig. 8. chess game module. chess game can have different properties:  simple attack  move with no effect  castling possibility  etc. chess game module api: development of a modular tool for regulating and analyzing activities in chess 114  create() creates new chess game  initialize() initializes new chess game.  run() runs chess game with specified options.  checkeffect() checks the effect on game like castling effect, etc.  registermove() registers move for particular chess game. 3.7. chess game logging module this module allows to handle chess games logging. logging can be done for game state or for the whole game. fig 9. chess game logging module. there are different approaches for description of chess board state. currently software supports fen [7]. the logging of the whole game can be done in pgn format [8]. chess game logger module api:  converttologgingformat() converts chess data to logging format.  convertfromloggingformat() converts logging data to chess data. 4. usage scenarios the developed tool usage scenarios are based on calling modules api functions related to particular components. calling modules api functions can be organized in any sequence of basic commands that provides flexible use cases mechanism. basic commands for ui:  modules shows available modules.  module [module name] shows api for module.  module [module name] -component [component name] -function [module api function name] -options [list of options values separated by coma]: calls the function from particular module api by component name that implements it with specified options values.  quit quits the application. example (run chess engine as process from file with location "./engine" and make a new session for it): module chessenginesessionmodule -component chessengineasprocess function createnewenginesession -options "./engine" 5. conclusion 1. software is developed for analyzing and regulating activities in chess. the software and its design provide users with the following advantages: a. possibility to add modules to software without violating the existing ones. b. possibility to add components to modules without violating the module structure. l. berberyan 115 c. possibility to make custom scenarios for defining new software functionality call paths. 2. the software is validated by running under popular desktop pc platforms (windows 7, ubuntu, mac os). 3. developed approach and software can be used for.  chess engines comparison through organizing games between them.  chess engines communicating tasks.  chess engines assessment. in future specified software modules and their components can be improved, also new ones can be added to provide more developed tool for particular usage, for example, tutoring. developed software can be used particularly for new chess engine assessment, for example, the solver [9] designed under the guidance of edward pogossian. acknowledgements author expresses gratitude to edward pogossian and sedrak grigoryan for counseling this research, also to emma danielyan for helping to improve the article. references [1] arena chess gui website. [online]. available: https://www.playwitharena.com [2] e. pogossian, “on assessment of performance of systems by combining on-the-job and expert attributes scales”, proceedings of international conference csit 2015, yerevan, armenia, pp. 331—334, 2015. [3] s. v. grigoryan and l. s. berberyan, “developing interactive personalized tutors in chess", transactions of the iiap nas of ra, mathematical problems of computer science, vol. 44, pp. 116-132, 2015. [4] e. gamma, r. helm, r. johnson and j. vlissides, design patterns, software engineering, object-oriented programming, addison-wesley, 1994. [5] r. huber and s. meyer-kahlen, uci (universal chess interface), ccc, november 28, 2000. [6] xboard website. [online]. available: https://www.gnu.org/software/xboard/ [7] s. j. edwards, “forsyth-edwards notation”, portable game notation specification and implementation guide, 03.12.1994. [8] s. j. edwards, “portable game notation”, portable game notation specification and implementation guide, 03.12.1994. [9] k. khachatryan and s. grigoryan, “java programs for matching situations to the meanings of ssrgt games”, proceedings of seua annual conference, yerevan, armenia, pp. 135-141, 2013. submitted 01.08.2016, accepted 15.11.2016. development of a modular tool for regulating and analyzing activities in chess 116 շախմատային գործունեության վերահսկման և վերլուծման մոդուլային գործիքի կառուցում լ. բերբերյան ամփոփում տվյալ աշխատանքում առաջարկվում են ծրագրային ապահովում շախմատային տարբեր տեսակի գործունեության վերահսկման և վերլուծման ապահովման համար և իր կառուցվածքի մոտեցում: աշխատանքը սկսվում է տարածված շախմատային գործունեության վերահասկման և վերլուծման ծրագրային ապահովման քննարկմամբ՝ նշելով դրանց առավելություններն ու թերությունները, հատկապես շախմատային շարժիչների համեմատության վերաբերյալ: ապա ներկայացվում է բարելավված ծրագրային ապահովման կառուցման մոտեցում շախմատային տարբեր տեսակի գործունեության վերահսկման և վերլուծման ապահովման համար հիմնված ծրագրային մոդուլների առանձնացման և նրանցից յուրաքանչյուրի համար api սահմանելու վրա: այնուհետև նկարագրվում է առաջարկվող մոտեցման իրականացումը, մասնավորապես բաղադրիչները իրականացնող մոդուլների api-ն, մասնավորապես մոդուլների api-ն իրականացնող բաղադրիչները նույնպես անկախ են իրարից, որն ապահովում է ճկուն ձևախման մեխանիզմներ: տրամադրվում են նաև մշակված ծրագրային ապահովման օգտագործման սկզբունքներ` հիմնված բազային հրամանների կանչերի մանիպուլյացիաների վրա: разработка модульного инструмента регулирования и анализа активностей в шахматах л. берберян аннотация в данной работе предлагаются программное обеспечение для регулирования и анализа различных видов активности в шахматах и подход к его конструированию. статья начинается с анализа распространенных программ для регулирования и анализа различных видов активности в шахматах, перечисляя некоторые их достоинства и недостатки, в особенности, касающиеся сравнения шахматных движков. далее определяется улучшенный подход к конструированию программного обеспечения для регулирования и анализа различных видов активности в шахматах, основанный на разделении модулей программного обеспечения и определении api для каждого из них. затем описывается реализация предложенного подхода, в частности компоненты, реализующие api модулей, независимы друг от друга, что обеспечивает гибкие механизмы для различных видов манипуляций. также приводятся сведения для использования разработанного программного обеспечения, основанные на манипуляции вызовов базовых команд. mathematical problems of computer science 41, 103--113, 2014. 103 on an algebraic classification of multidimensional recursively enumerable sets expressible in formal arithmetical systems1 seda n. manukian institute for informatics and automation problems of nas ra e-mail: zaslav@ipia.sci.am abstract algebraic representations of multidimensional recursively enumerable sets which are expressible in formal arithmetical systems based on the signatures ),,,0(  s , ),,,0( s , ),,0( s , where 1)(  xxs , are introduced and investigated. the equivalence is established between the algebraic and logical representations of multidimensional recursively enumerable sets expressible in the mentioned systems. keywords: predicate formula, universal algebra, recursively enumerable set, mathematical structure, deductive system, formal arithmetic. 1. introduction in this paper algebraic representations of multidimensional recursively enumerable sets (res) described in some subsystems of peano’s formal arithmetic ([1], [2], [3]) are introduced and investigated. similar problems concerning two-dimensional reses are considered in [4], [5], [6]. but the structure of algebraic representations of multidimensional reses differs from the structure of algebraic representations of two-dimensional ones. it was necessary to introduce essential changes in the notions used in [4], [5], [6] for the description of such algebraic representations. however, as it will be proved below, the relations between algebraic and logical representations of multidimensional reses are similar to those described in [5]. theorems 2.1, 2.2, 2.3 (see below) about such relations will be formulated in sec.2 and proved in sec.3. 2. main definitions and results let us give the definitions of notions used below (cf. [7], [8]). an n-dimensional arithmetical set, where 1n , is defined in a natural way as a set of n-tuples ),...,,( 21 nxxx , where nxxx ,...,, 21 are 1 this work was supported by state committee of science, mes ra, in frame of the research project № scs 13b321. on an algebraic classification of multidimensional recursively enumerable sets expressible104 nonnegative integers 0, 1, 2, .... an n-dimensional arithmetical predicate is defined as a predicate which is true on some n-dimensional arithmetical set and false out of it. the notion of recursively enumerable set (res) is defined as in [1]. an algebra is defined as a “universal algebra” ([9], [10]) with a fixed set of basic elements. thus, any algebra is described by a main set m , by a set of operations ,..., 21 ff on m (in general not everywhere defined), and a set of basic elements ,..., 21 aa in m ([5]). we say that an element ma is inductively representable in a given algebra ,...),,...;,;( 2121 aaffm if it can be obtained by the operations ,..., 21 ff from the basic elements ,..., 21 aa . the notions of a subalgebra and a proper subalgebra of a given algebra are defined in a natural way (for example, as in [5]). we will consider the following operations on multidimensional reses (cf [14]). 1) the operations of union  and intersection  of reses are defined in a usual way (note that these operations are applied only to reses having equal dimensions). 2) the operation i of projection for n-dimensional res a concerning i-th co-ordinate, where ni 1 , is defined by the following generating rule (g.r.): if axxx n ),...,,( 21 , then )(),...,,,...,,( 1121 axxxxx inii  . 3) the operation  of cartesian product for n-dimensional res a and m-dimensional res b is defined by the following g.r.: if axxx n ),...,,( 21 , and byyy m ),...,,( 21 , then bayyyxxx mn ),...,,,,...,,( 2121 . 4) the operation ijt of transposition of i-th and j-th co-ordinates in n-dimensional res a , where nji  ,1 , is defined by the following g.r.: if axxx n ),...,,( 21 , then )(),...,,,...,,,,...,,( 111121 atxxxxxxxxx ijnjijiji  . 5) the operation  of transitive closure for a res a having an even dimension 2n is defined by the following generating rules: (a) if axxx n ),...,,( 221 , then axxx n ),...,,( 221 ; (b) if ayyyxxx nn ),...,,,,...,,( 2121 and azzzyyy nn ),...,,,,...,,( 2121 , then azzzxxx nn ),...,,,,...,,( 2121 . the following reses are used as basic elements for the considered algebras (cf. [7]): }0|{0  xxz ; }1|),{(  xyyxr ; }|),,{( yxzzyxadd  ; }|),{( yxyxq  ; }|),{( yxyxj  . examples: rq  ; )())(( 11 addadd  . let us define the algebras 0 , 1 , 2 , 3 . the main set for these algebras is the set of all multidimensional reses having the dimensions 1n . the list of operations for all these algebras is ),,,,( ijt . the lists of basic elements are as follows (cf. [7]): ),,( 0 addrz for 0 , ),,( 0 qrz for 1 , ),,( 0 jrz for 2 , ),( 0 rz for 3 . note: the introduced algebras are different from the algebras denoted by 0 , 1 , 2 , 3 in [5]. the algebras having these notations in [5] we will denote below by 0 ~  , 1 ~  , 2 ~  , 3 ~  . the relations between the algebras 0 , 1 , 2 , 3 and 0 ~  , 1 ~  , 2 ~  , 3 ~  will be considered in sec.3. s. manukian 105 the notion of a predicate formula based on the logical operations  ,,,,&, (as other notions connected with it, for example, the notion of a term) is defined in a usual way ([1], [3], [11], see also [5]). a signature is defined in a usual way as any set of constants symbols, functional symbols, predicate symbols. we say that a formula f (respectively, a term t ) belongs to a given signature  (or is a formula (respectively, a term) in the signature  ) if all the constants symbols, functional symbols, predicate symbols contained in f (respectively, all the constants symbols and functional symbols contained in t ) belong to  . we will consider the signatures ),,,0( s , ),,,0( s , ),,0( s , where s is an one-dimensional functional symbol; these signatures will be denoted below respectively by hn , ln , sn (cf. [7]). note that similar notations are used in [7] as the notations of the corresponding mathematical structures (however, the structure corresponding to the signature ),,,0( s is denoted in [7] (and in [3]) by an ). the arithmetical interpretation of a predicate formula belonging to one of these signatures and containing no other free variables except nxxx ,...,, 21 is defined in a natural way as an ndimensional arithmetical predicate; the functional symbol s is interpreted as the function 1)(  xxs , and other symbols in the mentioned signatures are interpreted in a natural way. the deductive systems of formal arithmetic in the signatures hn , ln , sn are defined as in ([1], [3], [11]-[13]; see also [6]); we will denote these deductive arithmetical systems respectively by hded , lded , sded (cf. [6]). for example, the system hded is equivalent to m. presburger’s system described in [11]-[13]. we say that formulas f and g (respectively terms t and s ) are equivalent in the framework of the corresponding deductive system if the formula )(&)( fggf  (respectively, the formula st  ) is deducible in this system. if the formulas f and g or the terms t and s are equivalent in hded (respectively, lded or sded ), we will say that they are hded -equivalent (respectively, lded -equivalent or sded -equivalent). all mentioned systems of formal arithmetic are complete ([3], [11]-[13]). we say that a set  of predicate formulas belonging to one of the mentioned signatures admits the elimination of quantifiers (in the framework of the corresponding deductive system) if for any predicate formula f belonging to  a formula g belonging to г can be constructed so that g does not contain quantifiers and is equivalent to f in the framework of the corresponding deductive system. the sets of all predicate formulas belonging to the signatures hn , ln , sn admit the elimination of quantifiers in the framework of the corresponding deductive systems hded , lded , sded ([3], [11]-[13]). by )(ts n , where 0n , and t is a term, we denote the term )...))((...( tsss , where the symbol s is repeated n times ( )(0 ts is t ). by n we denote the term )0(ns . we say that a k-dimensional arithmetical set a is represented (or representable) by a formula f belonging to one of the mentioned signatures and containing free variables kxxx ,...,, 21 , if the following condition holds: the arithmetical interpretation of the formula obtained by the substitution of the terms knnn ,...,, 21 for the variables kxxx ,...,, 21 in f is true if and only if annn k ),...,,( 21 . we say that a k-dimensional arithmetical set a is represented (or representable) in hded (respectively, lded , sded ) by a formula f in hn (respectively ln , sn ) if it is represented by some formula f  equivalent to f in hded (respectively, lded , sded ). for example, the (n+1)-dimensional res }...|),,...,,{( 2121 yxxxyxxx nn  is represented in hded by the formula ))0(...( 21 yszxxxz n  in hn . on an algebraic classification of multidimensional recursively enumerable sets expressible106 a formula f in the signature sn is said to be positive if it contains no other logical symbols except  ,,&, and all the symbols  of negation contained in it relate to elementary subformulas containing no more than one variable (cf. [5], [7]). theorem 2.1: a multidimensional res is inductively representable in the algebra 0 (respectively 1 , 2 ) if and only if it is represented in hded (respectively, lded , sded ) by a formula in hn (respectively, ln , sn ). theorem 2.2: a multidimensional res is inductively representable in the algebra 3 if and only if it is represented in sded by a positive formula in sn . theorem 2.3: every next algebra in the sequence 0 , 1 , 2 , 3 is a proper subalgebra of the preceding one. theorems 2.1 and 2.2 are formulated (without proofs and in some other terms) in [7]. 3. proofs of theorems in this section the proofs of theorems 2.1, 2.2, 2.3 will be given. we will consider the following sets. by v we denote the set of all non-negative integers 0,1,2,..., by kv we denote the set of all k-tuples ),...,,( 21 kxxx where 1k , and all ix are nonnegative integers. by o we denote the 1-dimensional empty set, by ko we denote the kdimensional empty set. by e and 1q we denote the sets }|),{( yxyxe  and }|),{(1 yxyxq  . obviously, all these sets are represented in the following deductive systems: v in deds by the formula xx  , kv in deds by the formula )&...&&( 2211 kk xxxxxx  , o in deds by the formula )(xsx  , ko in deds by the formula ))(&...&)(&)(( 2211 kk xsxxsxxsx  , e in deds by the formula yx  , 1q in dedl by the formula )()( yxyx  . lemma 3.1: the sets v , kv , o , ko , e , 1q , q , j are inductively representable in the following algebras: the sets v , kv , o , ko , e in all algebras 0 , 1 , 2 , 3 , the sets 1q , q in the algebras 0 and 1 , the set j in the algebras 0 , 1 , 2 . the proof is given by the following equalities: )(2 rv  ; vvvv k  ... , where the symbol v is repeated k times; ))(( 121 rtro  ; oooo k  ... , where the symbol o is repeated k times; )))(()(( 122 rtvvre  ; eqq 1 ; )(11 addq  ; ))()(( 12 rvvqq  ; )(12 qtqj  . corollary: every next algebra in the sequence 0 , 1 , 2 , 3 is a subalgebra of the preceding one. s. manukian 107 lemma 3.2: any res inductively representable in 0 (respectively, 1 , 2 ) can be represented in hded (respectively, lded , sded ) by a formula in hn (respectively, ln , sn ). proof: the basic sets for 0 are represented by the formulas 0x , )(xsy  , yxz  ; similarly, the basic sets for 1 are represented by the formulas 0x , )(xsy  , yx  ; for 2 by the formulas 0x , )( xsy  , )( yx  . if arithmetical sets a and b having equal dimensions are represented by formulas f and g , then the sets ba and ba are represented by the formulas )( gf  and )&( gf . if an n-dimensional arithmetical set a is represented by a formula f containing free variables nxxx ,...,, 21 , then the set )( ai , where ni 1 , is represented by the formula )(fxi . if an n-dimensional arithmetical set a is represented by a formula f containing only free variables nxxx ,...,, 21 , and an m-dimensional arithmetical set b is represented by a formula g containing only free variables myyy ,...,, 21 , then the set ba is represented by the formula gf & , where the formula g is obtained from g by the substitution of variables mnnn xxx  ,...,, 21 for myyy ,...,, 21 in g . if an n-dimensional arithmetical set a is represented by a formula f containing free variables nxxx ,...,, 21 , then the formula )( atij , where nji  ,1 , is represented by a formula f obtained from f by a corresponding replacement of free variables. this completes the proof. now we will give the proof of the statement opposite to the statement of lemma 3.2. in what follows any term in hn having the form )...( xxx  , where the variable x is repeated k times, will be shortly denoted by kx . the notation kx will denote the term 0 when 0k ; it will denote the term x when 1k . we will consider below the following sets. (1) the set kz , where k is a constant, 0k ; it is a one-dimensional set containing only the number k . (2) the set kw , where k is a constant, 0k ; it is a one-dimensional set containing all the numbers x such that kx  . (3) the set kr , where k is a constant, 1k ; it is a two-dimensional set )}(|),{( xsyyx k . (4) the set keadd , where k is a constant, 0k ; it is a two-dimensional set }|),{( kxyyx  . (5) the set ),,...,,( 21 qkkklin nexp , where 1n , and qkkk n ,,...,, 21 are constants, 01 k , 02 k , ..., 0nk , 0q ; it is an (n+1)-dimensional set }...|),,...,,{( 221121 yqxkxkxkyxxx nnn  . (6) the set ),( yxcongrk , where k is a constant, 2k ; it is a two-dimensional set )})(mod(|),{( kyxyx  . clearly, all these sets are represented by formulas in the following deductive systems: kz is represented by the formula )( kx  in dedh, dedl, deds; kw is represented by the formula ))(( 1 zsxz k in dedh, dedl, deds; kr is represented by the formula )(xsy k in dedh, on an algebraic classification of multidimensional recursively enumerable sets expressible108 dedl, deds; keadd is represented by the formula kxy  in dedh; q),k,...,k,linexp(k n21 is represented in dedh by the formula yqxkxkxk nn  ...2211 ; ),( yxcongrk , where 2k is represented in dedh by the formula ))()(( xkzyykzxz  . lemma 3.3: the sets kz , where 0k , the sets kw , where 0k , the sets kr , where 1k , the sets keadd , where 0k , the sets q),k,...,k,linexp(k n21 , where 1n , 0ik for ni 1 , 0q , the sets ),( yxcongrk , where 2k are inductively representable in the following algebras: kz , kw and kr in 0 , 1 , 2 , 3 ; keadd , q),k,...,k,linexp(k n21 and ),( yxcongrk in 0 . the proof is given by the following equalities (note that the sets 0z and r are included as basic elements in all the algebras 0 , 1 , 2 , 3 ): ))((11 rvzz kk  ; )(10 rw  ; ))((11 rvww kk  ; rr 1 ; ))()((21 rvvrr kk  ; 00 zveadd  ; )))()()((( 223 2 111 vetaddvveaddeadd kk  ; )))()()(((),( 2222 addvvzvveaddqkexplin qk  ; )))()()),,...,((((),,,...,,( 123213,12221 addvveaddvvqkkklintqlkkklin n l n nnnnnn    expexp ; ))))()(((()))()((( 21112 2 11 addvveaddtaddvveaddcongr kkk  . lemma 3.4: any term in hn is hded -equivalent to a term having the form qxkxkxk nn  ...2211 , where q is a nonnegative integer constant. any formula in hn is hded -equivalent to a formula which can be obtained by & and  from subformulas having the form st  or ))(mod( kst  , where t and s are terms, and k is an integer constant, 2k . this lemma is proved (in other terms) in [3], [4], [11] (cf. [6], lemma 4.1). lemma 3.5: any res represented in hded by a formula in hn is inductively representable in the algebra 0 . proof: let f be a formula in hn . let us denote by nxxx ,...,, 21 the list of all free variables contained in f . using lemma 3.4 we conclude that there exists a formula f which is hded equivalent to f and can be obtained by & and  from subformulas having the form st  or ))(mod( kst  , where t and s have the form described in lemma 3.4. without loss of generality we can suppose that the list of variables in all mentioned terms t and s coincides with the list nxxx ,...,, 21 (indeed, if some variable ix is missing in a corresponding sum, then we can add to this sum the summand ix0 ; the order of summands in all considered sums can be unified using the operation ijt ). we see that the formula f is hded -equivalent to some formula which can be obtained by & and  from the formulas having the form st  or ))(mod( kst  in which all the terms t and s have the form qxkxkxk nn  ...2211 , where 0ik for ni 1 , and 0q . n-dimensional sets represented by the formulas of such kind can be described in 0 by the following expressions: the set represented by the formula having the s. manukian 109 form st  by the expression of the form )))()),,...,,(()),,...,,(((( ""2 " 12,1 '' 2 ' 121 qvvqkkkexplintvqkkkexplin n nnnnnn   ; the set represented by the formula having the form ))(mod( kst  by the expression of the form )))()),,...,,(()),,...,,(((( 212,12121 k n nnnnnn congrvvqkkkexplintvqkkkexplin    . thus, any n-dimensional set represented by the formula f can be described in 0 applying the operations  and  to the expressions having the forms mentioned above. this completes the proof. lemma 3.6: any term in ln has the form )(xs k or )0(ks , where x is a variable. any formula in ln is lded -equivalent to a formula which can be obtained by & and  from subformulas having the form )( st  , where t and s are terms. this lemma is proved (in other terms) in [3], [5], [11] (cf. [6], lemma 4.2). lemma 3.7: any res represented in lded by some formula in ln is inductively representable in the algebra 1 . proof: let f be a formula in ln . let us denote by nxxx ,...,, 21 the list of all free variables in f . we suppose that 2n (the case 2n is considered in a similar way). using lemma 3.6 we conclude that f is lded -equivalent to some formula f  which can be obtained by & and  from subformulas of the form )( st  , where t and s are terms. let us consider the case when t and s contain only the variables 1x and 2x (the general case is reduced to the mentioned one using the operation ijt ). we will denote the variables 1x and 2x by x and y . using lemma 3.6 we see that in the subformula )( st  the term t has one of the forms )(xs k , )( ys k , )0(ks , where 0k ; the term s has one of the forms )(xs l , )( ys l , )0(ls , where 0l . thus, there are 9 possible forms of the subformula )( st  . if )( st  has the form )()( ysxs lk  , then the ndimensional set represented by this formula is 21 ))()((    n lk vqvvr when lk  ; it is 2 233 ))()((    n kl vrvtvq , when lk  , and 2 nvq when lk  . if )( st  has the form )()( xsxs lk  , then the n-dimensional set represented by this formula is no when lk  , and is nv when lk  . if )( st  has the form )0()( lk sxs  , then the n-dimensional set represented by this formula is no when lk  , and is 1110 )...(    n kl vzzz when lk  . the remaining forms of the formula )( st  are considered in a similar way. thus, the ndimensional res represented by the formula f in lded is obtained by  and  from sets inductively representable in 1 . this completes the proof. lemma 3.8: any term in sn has the form )(xs k or )0(ks , where x is a variable. any formula in sn is sded -equivalent to a formula which can be obtained by & and  from subformulas having the form )( st  or )( st  , where t and s are terms. this lemma is actually proved (in other terms) in [3], [11] (cf. [5], lemma 3.8). on an algebraic classification of multidimensional recursively enumerable sets expressible110 lemma 3.9: any res represented in sded by formula in sn is inductively representable in the algebra 2 . proof: the proof is similar to that of lemma 3.7. let f be a formula in sn . let us denote by nxxx ,...,, 21 the list of all free variables contained in f . we suppose (as in the proof of lemma 3.7) that 2n . using lemma 3.8 we conclude that f is sded -equivalent to some formula f which can be obtained by & and  from subformulas having the forms )( st  and )( st  . as in the proof of lemma 3.7 we consider the case when t and s contain only variables 1x and 2x ; we will denote these variables by x and y . using lemma 3.8 we see that in the subformulas )( st  and )( st  the term t has one of the forms )(xs k , )( ys k , )0(ks , where 0k ; the term s has one of the forms )(xs l , )( ys l , )0(ls , where 0l . thus, there are 9 possible forms of the subformula )( st  and 9 possible forms of the subformula )( st  . if )( st  has the form )()( ysxs lk  , then the n-dimensional set represented by this formula is 2   n lk vr when lk  ; it is 2 12 )(    n kl vrt when lk  , and 2 nve when lk  . the n-dimensional set represented by the formula ))()(( ysxs lk  is ))()((2 jvvr lk   when lk  , it is )))()((( 212 jvvrt kl   when lk  , and 2 nvj when lk  . the n-dimensional set represented by the formula )()( xsxs lk  is no when lk  ; it is nv when lk  . the n-dimensional set represented by the formula ))()(( xsxs lk  is nv when lk  ; it is no when lk  . the n-dimensional set represented by the formula )0()( lk sxs  is no when lk  ; it is 1   n kl vz when lk  . the n-dimensional set represented by the formula ))0()(( lk sxs  is nv when lk  ; it is 1 110 )...(    n klkl vwzzz when lk  ; and 1 0  nvw when lk  . the remaining forms of the formulas )( st  and )( st  are considered in a similar way. thus, the n-dimensional res represented by the formula f is obtained by  and  from sets inductively representable in 2 . this completes the proof. the proof of theorem 2.1 is obtained now using lemmas 3.2, 3.5, 3.7, 3.9. lemma 3.10: the set of positive formulas in sn admits the elimination of quantifiers in the framework of sded . the proof follows from the considerations in [3], because it is easily seen that the method of elimination of quantifiers in sn described in [3] gives for any positive formula f in sn some positive formula g such that g does not contain quantifiers and is sded -equivalent to f . lemma 3.11: any positive formula in sn is sded -equivalent to a formula which can be obtained by & and  from subformulas having the form )( st  or )( st  , where t and s s. manukian 111 are terms of the form )(xs k or )0(ks , and any subformula of the form )( st  contains no more than one variable. the proof is easily obtained using lemmas 3.8 and 3.10. lemma 3.12: any res inductively representable in 3 can be represented in sded by a positive formula in sn . proof: the basic sets in 3 are represented in sn by the positive formulas 0x and )(xsy  . it is easily seen that the transformation of formulas generated in the proof of lemma 3.2 by the operations iji t,,,,  gives positive formula being applied to positive formulas. this completes the proof. lemma 3.13: any res represented in sded by a positive formula in sn is inductively representable in the algebra 3 . proof: let f be a positive formula in sn . let us denote by nxxx ,...,, 21 the list of all free variables in f . similarly to the proof of lemmas 3.7 and 3.9 we suppose that 1n . using lemma 3.11 we conclude that f is sded -equivalent to a positive formula which can be obtained by & and  from positive subformulas of the form )( st  or )( st  , where t and s are terms in sn . it is easily seen that any n-dimensional set represented by a formula of the form )( st  is described by the expressions considered in the proof of lemma 3.9. now let us consider the sets represented by subformulas of the form )( st  . let us recall that any positive formula of the form )( st  contains no more than one variable. the single variable contained in )( st  we denote by x and suppose that it coincides with the variable 1x in the list nxxx ,...,, 21 (the general case is considered similarly). then the formula )( st  has the form ))0()(( lk sxs  , where 0k , 0l . but the inductive representations of the set represented by this formula in 2 are described in the proof of lemma 3.9; it is easily seen that these representations are also representations in 3 . thus, the n-dimensional res represented by the formula f is obtained by  and  from sets inductively representable in 3 . this completes the proof. the proof of theorem 2.2 is obtained now using lemmas 3.12 and 3.13. lemma 3.14: any multidimensional res is inductively representable in 0 ~  (respectively, 1 ~  , 2 ~  , 3 ~  ) if and only if it is two-dimensional and is inductively representable in 0 (respectively, 1 , 2 , 3 ). proof: let us recall that we denote by 0 ~  , 1 ~  , 2 ~  , 3 ~  the algebras denoted in [5] by 0 , 1 , 2 , 3 . as it is proved in [5], (in other terms) a two-dimensional res is inductively representable in 0 ~  (respectively, 1 ~  , 2 ~  ) if and only if it is represented in hded on an algebraic classification of multidimensional recursively enumerable sets expressible112 (respectively, lded , sded ) by a formula in hn (respectively, ln , sn ); a two-dimensional res is inductively representable in 3 ~  if and only if it is represented in sded by a positive formula in sn . now the statement of lemma 3.14 is obtained using theorems 2.1 and 2.2 proved above. proof of theorem 2.3: as it is proved in [5], there exists a two-dimensional res 1a (respectively, 2a , 3a ) which is inductively representable in 1 ~  (respectively, 2 ~  , 3 ~  ) but not in 0 ~  (respectively, 1 ~  , 2 ~  ). the statement of theorem 2.3 is now obtained using lemma 3.14. let us note that the list of operations in the algebras 0 , 1 , 2 , 3 is similar to that considered in [14]. let us note that if we add the operation of transitive closure to this list then any multidimensional res will be inductively representable in the extended algebra 3 . such statement is proved in [15] (see [15], lemma 1). acknowledgement the author is grateful to professor patrick cegielski for setting the problem considered in this paper, for his attention to this work, for valuable advice and notes. references [1] s. c. kleene, introduction to metamathematics, d. van nostrand comp., inc., new york-toronto, 1952. [2] e. mendelson, introduction to mathematical logic, d. van nostrand comp., inc., princeton-toronto-new york-london, 1964. [3] h. enderton, a mathematical introduction to logic, 2nd ed., san diego, harcourt, academic press, 2001. [4] s. n. manukian, “algebras of recursively enumerable sets and their applications to fuzzy logic”, journal of mathematical sciences, vol. 130, no. 2, pp. 4598-4606, 2005. [5] s. n. manukian. “on the representation of recursively enumerable sets in weak arithmetics”, transactions of the iiap of nas ra, mathematical probelms of computer science, vol. 27, pp. 90--110, 2006. [6] s. n. manukian, “some algebraical and logical properties of two-dimensional arithmetical sets representable in presburger’s system”, transactions of the iiap of nas ra, mathematical probelms of computer science, vol. 37, pp. 64--74, 2012. [7] s. manukian, “on the inductive representation of many-dimensional recursively enumerable sets definable in some arithmetical structures”, proceedings of the international conference “computer science and information technologies”, csit-09, yerevan, armenia, pp. 51--53, 2009. [8] s. n. manukian. “classification of many-dimensional arithmetical sets represented in m. presburger’s system”, reports of the national academy of science of armenia, (in russian), vol. 111, no. 2, pp. 114--120, 2011. [9] g. graetzer, universal algebra, 2nd edition, new-york-heidelberg-berlin, 1979. [10] a. i. malcev, algebraic systems, springer verlag, 1973. [11] d. hilbert and p. bernays, grundlagen der mathematik, band i.zweite auflage, berlinheidelberg-new york, springer verlag, 1968. s. manukian 113 [12] m. presburger, “über die vollständigkeit eines gewissen system der arithmetik ganzer zahlen, in welchem die addition als einzige operation hervortritt”, comptes rendu de i congres der mathematiciens des pays slaves, warszawa, pp. 92--101, 1930. [13] r. stansifer, presburger’s article on integer arithmetic: remarks and translation, department of computer science, cornell university, ithaca, new york, 1984. [14] g. s. tseytin, “one method of representation for the theory of algorithms and enumerable sets”, transactions of steklov institute of the russian academy of sciences, (in russian), vol. 72, pp. 69--98, 1964. [15] s. n. manukian. “on the structure of recursively enumerable fuzzy sets”, transactions of the iiap of nas ra and ysu, mathematical probelms of computer science, (in russian), vol. 17, pp. 86--91, 1997. submitted 14.12.2013, accepted 24. 02. 2014. ձևայնացված թվաբանական համակարգերում արտահայտելի բազմաչափ անդրադարձ թվարկելի բազմությունների որոշ հանրահաշվական դասակարգման մասին ս. մանուկյան ամփոփում սահմանվում և հետազոտվում են ),,,0(  s , ),,,0( s , ),,0( s (որտեղ 1)(  xxs ) սիգնատուրաների վրա հիմնված ձևայնացված թվաբանության համակարգերի մեջ արտահայտելի բազմաչափ անդրադարձ թվարկելի բազմությունների հանրահաշվական ներկայացումները: հաստատվում է համարժեքություն նշված տիպի անդրադարձ թվարկելի բազմությունների հանրահաշվական և տրամաբանական ներկայացումների միջև: об алгебраической классификации многомерных рекурсивно перечислимых множеств, выразимых в формальных арифметических системах с. манукян аннотация вводятся и исследуются алгебраические представления многомерных рекурсивно перечислимых множеств, которые выразимы в системах формальной арифметики, основанных на сигнатурах ),,,0(  s , ),,,0( s , ),,0( s , где 1)(  xxs . устанавливается эквивалентность между алгебраическими и логическими представлениями многомерных рекурсивно перечислимых множеств, выразимых в указанных системах. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 45, 99--105, 2016. discretization of geometric models arthur j. keyan institute for informatics and automation problems of nas ra e-mail: artur.keyan@gmail.com abstract we are interested in 2d cutting and packing problem with irregularly shaped objects. in previous work the author has presented the description of the two dimensional modeling program of irregularly shaped objects. the next step of achieving the optimal arrangement of any irregular shape details on the surface is the discretization of geometric models. in this paper a solution to this problem is given. the solution is described in steps. keywords: optimal cutting of materials, two dimensional modeling, packing problem 1. introduction the cutting and bin packing problem is one of the classic and challenging problems in the field of computer science. this problem has been studied for over 30 years and to this date work is going on to find new improvised approaches and better results. cutting and packing to this date has ever new applications and especially the variants of bin packing are important to information technology. this problem has many potential applications in different real world industries (like transportation, logistics, information technology, etc.). some of the applications are scheduling television programming, cutting stock problem, cloud computing, truck loading problem and many more. there are different variants of the problem and similarly several approaches for tackling them, a lot of papers have been published relating to this problem. since bin packing belongs to the class of np hard problems, it is really difficult to come up with a polynomial time algorithm which solves the problem to give an optimal solution. so as a result, approximation algorithms are presented to find the closest possible solution to the optimal. 99 discretization of geometric models 100 2d bin packing problem is a generalization of 1d bin packing problem and is an np-hard problem, too. the potential use of 2d bin packing in many real time and industrial applications is a motivating factor in studying this problem. this 2d bin packing is used in applications like packing items in warehouses and trucks, cutting stock problems (cutting rectangles from sheets of a given size) and many more. these algorithms can be broadly classified into two categories  online bin packing algorithms  offline bin packing algorithms online bin packing algorithms pack items in the bin as per the input sequence. these algorithms do not have knowledge of the next items in the input sequence whereas the offline algorithm has knowledge of the next item in the input sequence required for bin packing and can possibly arrange them in a particular order before packing the items in the bins. we are interested in the offline version of the 2d bin packing problem. some results are known in this field for rectangles of fixed size. particularly, in 1982 chung [1] gave an approximation algorithm, a better one was given in 2002 by caprara [2]. in 2009 bansal [3] presented a randomized algorithm which improved the previous ones. he also showed that the two-dimensional bin packing problem does not admit an asymptotic polynomial-time approximation scheme. a. m. chwatal and s. pirkwieser [4] in 2011 were solving the two-dimensional bin packing problem with variable bin sizes by greedy randomized adaptive search procedures and variable neighborhood search. in 2011, f. eisenbrand, d. palvolgyi and t. rothvo [5] presented a paper called bin packing via discrepancy of permutations. a. layeb and s. chenche [6] came up with a paper titled a novel grasp algorithm for solving the bin packing problem which was published in 2012. in the same year g. perboli, r. tadei, m. m. baldi [7] published a paper on the stochastic generalized bin packing problem and along with crainic they presented another paper on branch-and-price and beam search algorithms for the generalized bin packing problem [8]. this is not the full list of papers on cutting and packing problem, but these papers indicate the amount of work and research performed in this field and emphasize the importance of the considered problem. we are interested in 2d cutting and packing problem with irregularly shaped objects. in [9] the author presented the description of the two dimensional modeling program of irregularly shaped objects. the next step of achieving the optimal arrangement of any irregular shape details on the surface is the discretization of geometric models. in this paper a solution to this problem is given. the solution is described in steps. in the second section the circumscribing rectangle with the minimal area for geometric models is found. this brings the problem to the optimal distribution of the rectangles on rectangle. the process of discretization is given in section three. 2. finding the circumscribing rectangle with minimal area to solve this task we construct circumscribed rectangles with the sides parallel to the axes of coordinates by rotating the element around its center and determining which of them is the one with the minimal area. theoretically we have to rotate the element in all possible angles and build circumscribed rectangles, then we have to find the rectangle with minimum area. but one can notice that it is enough to rotate only 0 90 degrees, as after every 90 degrees the area is repeated. the finding of the minimal area is realized by the method of comparing. the first rectangle is considered as the one with minimal area, after we rotate the element in 1 degree and compare the new area of circumscribed rectangle with the previous position and if it is smaller it is accepted as a. keyan 101 the rectangle with minimal area, otherwise just passing forward up to 90 degrees. when we find the rectangle with minimal area we rotate the detail back to the corresponding angle. 3. discretization at this step the circumscribed rectangle must be divided into small squares. the size of this squares depends on the user’s decision who must solve the trade-off problem: accuracy of the measurements against speed of the work. on default the factor of discretization equals 30 pixels. the number of squares is given by the following formulas: 𝑁𝑁ℎ = 𝐿𝐿 + (𝑙𝑙 − 1) 𝑙𝑙 , (1) 𝑁𝑁𝑣𝑣 = 𝐻𝐻 + (𝑙𝑙 − 1) 𝑙𝑙 , where 𝑁𝑁ℎ and 𝑁𝑁𝑣𝑣 are, correspondingly, the quantities of horizontal and vertical squares, 𝐿𝐿 and 𝐻𝐻 are the width and height of the circumscribed rectangle with the minimal area and 𝑙𝑙 is the square size or discretization factor. as the factor of the discretization must be a whole number, the obtained values are round up to the nearest integer. the result of the division is given in figure 1. fig. 1. circumscribed rectangle is divided into small squares. the squares that are empty must be removed. the result is given in figure 2. discretization of geometric models 102 fig. 2. empty squares are removed. the same result for smaller discretization factor (5 pixels) is depicted in figure 3. fig. 3. small discretization factor. in the final stage to each rectangle we correspond a matrix with 1 for each square and 0 otherwise (fig. 4). 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 1 1 fig. 4. corresponding matrix. this will help to simplify the solution of the next problem of optimal distribution. for example, to rotate the image on 90 degrees the transpose matrix will correspond (fig. 5). a. keyan 103 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 1 1 1 1 1 fig. 5. rotation on 90 degrees. to restore the image corresponding to each matrix, the number of the detail is stored with the corresponding matrix. 4. conclusion in the paper the process of discretization of geometric models is described. first the circumscribing rectangle with the minimal area for geometric models is found to bring the problem to the optimal distribution of the rectangles on rectangle. in the next step the circumscribed rectangle is divided into small squares and the squares that are empty are removed. to simplify the solution of the next problem of optimal distribution each rectangle is associated with a binary matrix. references [1] f. chung, m. garey and d. johnson, “on packing two-dimensional bins”, siam j. alg. disc. meth., vol. 3, no. 1, pp. 66–76, 1982. [2] a. caprara, “packing 2-dimensional bins in harmony”, proceedings of the symposium on foundations of computer science (focs), pp. 490–499, 2002. [3] n. bansal, a. caprara and m. sviridenko, “a new approximation method for set covering problems with applications to multidimensional bin packing”, siam j. comput. vol. 39, no. 4, pp. 1256–1278, 2009. [4] a. m. chwatal and s. pirkwieser, “solving the two-dimensional bin-packing problem with variable bin sizes by greedy randomized adaptive search procedures and variable neighborhood search”, eurocast, vol. 1, pp. 456-463, 2011. [5] f. eisenbrand, d. palvoelgyi and t. rothvoss, “bin packing via discrepancy of permutations”, symposium on discrete algorithms (soda 2011), san francisco, usa, pp. 22-25, 2011. discretization of geometric models 104 [6] a. layeb and s. chenche, “a novel grasp algorithm for solving the bin packing problem”, international journal of information engineering and electronic business (ijieeb), vol. 4, no. 2, pp. 8-14, 2012. [7] g. perboli, r. tadei and m. m. baldi, “the stochastic generalized bin packing problem”, discrete applied mathematics, vol. 160, no. 7-8, pp. 1291—1297, 2012. [8] m. m. baldi, t. g. crainic, g. perboli and r. tadei, “branch-and-price and beam search algorithms for the generalized bin packing problem”, cirrelt, montreal, canada, pp. 1--18, 2012. [9] a. keyan, “modeling of irregularly shaped two – dimensional objects”, transactions of the ipia of the nas ra, mathematical problems of computer science, vol.44, pp. 154--160, 2015. submitted 20.10.2015, accepted 18.02.2016 երկրաչափական մոդելների դիսկրետացում ա. քեյան ամփոփում մեզ հետաքրքրում է կամայական ուրվագծով երկչափ օբյեկտների ձևման և փաթեթավորման խնդիրը: նախորդ աշխատանքում հեղինակը ներկայացրել է կամայական ուրվագծով օբյեկտների երկչափ մոդելավորման ծրագրի նկարագրությունը: կամայական ուրվագծով օբյեկտների մակերևույթի վրա օպտիմալ բաշխման հաջորդ քայլը երկրաչափական մոդելների դիսկրետացումն է: այս աշխատանքում լուծված է այդ խնդիրը: նախ գտնում ենք երկրաչափական մոդելին արտագծած ուղղանկյուն նվազագույն մակերեսով՝ խնդիրը հանգեցնելու ուղղանկյուններն ուղղանկյան վրա օպտիմալ բաշխմանը: հաջորդ քայլում արտագծած ուղղանկյունը բաժանում ենք փոքր քառակուսիների և դատարկ քառակուսիները հեռացնում: օպտիմալ բաշխման խնդրի լուծումը հեշտացնելու նպատակով յուրաքանչյուր ուղղանկյանը համապատասխանեցվում է երկուական մատրից: a. keyan 105 дискретизация геометрических моделей а. кеян аннотация нас интересует задача 2d раскроя и упаковки объектов неправильной формы. в предыдущей работе автор представил описание программы двухмерного моделирования объектов неправильной формы. следующим шагом оптимального распределения деталей неправильной формы на поверхности является дискретизация геометрических моделей. в данной работе решена эта задача. сначала находится описанный прямоугольник с минимальной площадью для геометрической модели, чтобы привести к проблеме оптимального распределения прямоугольников на прямоугольнике. на следующем шаге описанный прямоугольник делится на небольшие квадраты и пустые квадраты удаляются. чтобы упростить решение следующей задачи оптимального распределения каждому прямоугольнику ставится в соответствие бинарная матрица. 2. finding the circumscribing rectangle with minimal area 3. discretization microsoft word w11_2_.doc mathematical problems of computer science 37, 88--95, 2012. 88 optimal permissible placement by the height of the transitive directed tree with one root armen khachaturyan yerevan state university e-mail: khachaturyanarmen@gmail.com abstract in paper [1] we have introduced a new concept – the transitive directed tree with one root and have formulated its minimum permissible placement problem by the height. in the papers [1] and [2] we have introduced a couple of new concepts and obtained necessary conditions for the solution of that problem. in the present paper using the results and the introduced concepts from the papers [1] and [2] we obtain an optimal polynomial algorithm for the problem solution and present the proof of its optimality. keywords: a transitive directed graph, an optimal placement. references 1. a. h. khachaturyan, “necessary conditions for optimal permissible placement by the height of the transitive directed tree with one root”, mathematical problems of computer science, vol. 36, pp. 104-114, 2012. 2. a. h. khachaturyan, “necessary conditions for optimal permissible placement by the height of the transitive directed tree with one root (part second)”, mathematical problems of computer science, vol. 37, pp. 102-107, 2012. 3. a.h. khachaturyan, “the optimal permissible placement by the height of the transitive oriented tree containing one vertex of branching”, mathematical problems of computer science, vol. 30, pp. 71-75, 2008. 4. m.r. garey, d.s. johnson, computers and intractability: a guide to the theory of npcompleteness. san francisco, ca: w.h. freeman, 1979. 5. f. gavril, “some np-complete problems on graphs,” proc.11th conf. on information sciences and systems, johns hopkins university, baltimore, md, pp. 91-95, 1977. 6. m.r. garey, r.l. graham, d.s. johnson and d.e. knuth, “complexity results for bandwidth minimization”, siam j. appl. math., vol. 34, pp. 477–495. 1978. 7. m.r. garey, d.s. johnson and l. stockmeyer, “some simplified np-complete graph problems”, theor. comput. sci., vol. 1, pp. 237–267. 1976. 8. ch.h. papadimitriou, “the np-copleteness of the bandwidth minimization problem”, computing, v. 16, pp. 263–270. 1976. 9. a.v. petrosyan, s.e. markosyan, yu.g. shukuryan, mathematical problems of automation and projection of calculating-machine. yer., (in russian). 1977. 10. g.g. geoletsyan, “flat placement of the vertices of tree with minimization of width”, dan arm. ssr, issue 56, no. 4, pp. 202–207 (in russian). 1973. a. khachaturyan 89 11. l. m. goldberg and i. a. klipker, “minimum placement of trees on a line,” technical report, physico-technical institute of low termeperatures, academy of sciences of ukraina ssr, ussr. 1976. 12. y. shiloach, “a minimum linear arrangement algorithm for undirected trees” report, dept. of applied mathematics, weizmann institute, rehovot, israel. 1976. 13. d. adolphson and t.c. hu, “optimal linear ordering”, siam j. appl. math., vol. 25, no. 3, pp. 403–423. 1973. 14. c. berge, the theory of graphs and its applications. new york: wiley, 1962. ø»ï ³ñù³ïáí ïñ³ý½çïçí ûñç»ýï³óí³í í³éç ûåïçù³é ãáõûé³ïñ»éç ï»õ³¹ñáõùý áëï µ³ñóñáõãû³ý ². ê³ã³ïáõñû³ý ²ù÷á÷áõù [1] ñá¹í³íáõù ù»ýù ý»ñùáõí»é »ýù ýáñ ñ³ëï³óáõãûáõý` ù»ï ³ñù³ïáí ïñ³ý½çïçí ûñç»ýï³óí³í í³é ¨ ó¨³ï»ñå»é ýñ³ áëï µ³ñóñáõãû³ý ûåïçù³é ãáõûé³ïñ»éç ï»õ³¹ñù³ý ëý¹çñá: [1] ¨ [2] ñá¹í³íý»ñáõù ù»ýù ý»ñùáõí»é »ýù ùç ß³ñù ýáñ ñ³ëï³óáõãûáõýý»ñ ¨ ëï³ó»é ³û¹ ëý¹ñç éáõíù³ý ³ýññ³å»ßï å³ûù³ýý»ñ: ú·ï³·áñí»éáí [1] ¨ [2] ñá¹í³íý»ñáõù ëï³óí³í ³ñ¹ûáõýùý»ñá ¨ ý»ñùáõíí³í ñ³ëï³óáõãûáõýý»ñá, ëáõûý ñá¹í³íáõù ù»ýù ëï³ó»é »ýù ýßí³í ëý¹ñç éáõíù³ý ûåïçù³é µ³½ù³ý¹³ù³ûçý ³é·áñçãùá ¨ ³é³ç³ñï»é ýñ³ ûåïçù³éáõãû³ý ³å³óáõûóá: îïòèìàëüíàÿ äîïóñòèìàÿ ðàññòàíîâêà ïî âûñîòå òðàíçèòèâíî îðèåíòèðîâàííîãî äåðåâà ñ îäíèì êîðíåì à. õà÷àòóðÿí аннотация â ñòàòüå [1] ìû ââåëè íîâóþ êîíöåïöèþ – òðàíçèòèâíî îðèåíòèðîâàííîå äåðåâî ñ îäíèì êîðíåì è ñôîðìóëèðîâàëè çàäà÷ó åãî îïòèìàëüíî äîïóñòèìîé ðàññòàíîâêè ïî âûñîòå. â ñòàòüÿõ [1] è [2] ìû ââåëè íåñêîëüêî íîâûõ êîíöåïöèé è ïîëó÷èëè íåîáõîäèìûå óñëîâèÿ äëÿ ðåøåíèÿ ýòîé çàäà÷è. èñïîëüçóÿ ðåçóëüòàòû è ââåäåííûå êîíöåïöèè èç ñòàòüåé [1] è [2], ìû â íàñòîÿùåé ñòàòüå ïîëó÷èëè îïòèìàëüíûé ïîëèíîìèàëüíûé àëãîðèòì äëÿ ðåøåíèÿ çàäà÷è è ïðåäñòàâèëè äîêàçàòåëüñòâî åãî îïòèìàëüíîñòè. microsoft word tpel.doc mathematical problems of computer science 35, 137--139, 2011. 137 compact high performance algorithm for convolutional decoder arsen hakhoumian, tigran zakaryan, aramazd muzhikyan and vahan nikoghosyan institute of radiophysics and electronics, armenian national ac.sci. alikhanian 1, ashtarak, 0203, armenia abstract an efficient algorithm for convolutional decoder running in the mode of soft decision (1/2, k=7, 3-bit decision) in a fpga virtex-ii is proposed. the results show that although increasing the length of the cutoff buffer improves the signal to noise ratio by 2db, but it can significantly increase calculation time, which exponentially depends on the length of the cutoff buffer. at the optimal length of the cutoff buffer equal to 7, the obtained throughput is estimated up to 14mbps. references [1] m hosemann. r.habendorf and g.fettweis, “hardware-software codesign of a 14.4mbit 64 state viterbi decoder for an application-apecific digital signal processor”, ieee workshop on signal processing systems, pp. 45-50, 2003. [2] m.röder and r.hamzaoui, “fast tree-trellis list viterbi decoding”, ieee transactions on communications, vol. 54, no. 3, pp. 453-461, 2006 [3] j.campos and r.cumplido, “a runtime reconfigurable architecture for viterbi decoding”, 3rd international conference on electrical and electronics engineering , pp. 1-4, 2006. ´³ñóñ ï³ï³ñáõ³ï³ýáõãû³ùµ ÷³ãáõûã³ûçý ³å³ïá¹³íáñù³ý ë»õù ³é·áñçãù ². ð³ëáõùû³ý, î. ¼³ù³ñû³ý, ². øáõåçïû³ý ¨ ì. üçïáõáëû³ý ²ù÷á÷áõù ²é³ç³ñïíáõù ¿ ³ñ¹ûáõý³í»ï ³å³ïá¹³íáñçãç ÷³ãáõûã³ûçý ³é·áñçãù, áñý ³ßë³ïáõù ¿ áñáßáõù ï³û³óý»éáõ ÷³÷áõï é»åçùáõù (1/2, k=7, áñáßáõù ï³û³óý»éáõ 3 µçã) fpga virtex-ii ñ³ù³ï³ñ·áõù: êï³óí³í ³ñ¹ûáõýùý»ñá óáõûó »ý ï³éçë, áñ ïïñù³ý µáõý»ñç »ñï³ñáõãû³ý ïñïý³ïç ù»í³óáõùá ãý³û³í ¨ ñ³ý·»óýáõù ¿ ³½¹³ýß³ý/³õùáõï ñ³ñ³µ»ñáõãû³ý 2 ¹´-áí é³í³óù³ýá, ï³ñáõ ¿ ¿³ï³ýáñ»ý ù»í³óý»é ñ³ßí³ñïç å³ù³ý³ïá, áñá óáõóã³ûçý ï³ëí³íáõãûáõý áõýç ïïñù³ý µáõý»ñç »ñï³ñáõãûáõýçó: k=7 ûåïçù³é »ñï³ñáõãû³ý ñ³ù³ñ ëï³óíáõù ¿ 14 øµ/í ³é³í»é³·áõûý áõý³ïáõãûáõýá: microsoft word markosyan_17.doc математические вопросы кибернетики и вычислительной техники 34, 2010. 45 инновационные перспективы инфокоммуникаций в армении м.в.маркосян д.т.н., доцент ернии сс, ра славянский университет уровень развития каждой страны определяется уровнем внедрения ик в повседневную жизнь граждан. особенно это социальная сфера медицина, городское хозяйство (жкх), образование, финансовое обслуживание, магазины, транспорт туризм, правительство: э-медицина, э-город, э-образование, э-банк, э-магазины, э-транспорт, эправительство. притом, необходимо отметить, что навсегда решать ту или иную задачу невозможно, т.к. ,в зависимости от уровня развития общества, задачи меняются и меняются требования к их решению. следовательно, «решенные» задачи подлежат к пересмотру. естественно, решения каких -то задач изменяются исходя из новых технических возможностей, появляющихся в следствие развития новой техники и технологий, представления, передачи и обработки данных. естественно и то, что какие то задачи функционально не будут меняться. широкое внедркние ик приводит к возникновению новых рабочих мест по обслуживанию техники и повышает требования к знаниям работников. в армении работают более 200 компаний в области it, в которых работают более 5200 специалистов. количество компаний 200, из них больше половины с иностранным капиталом количество специалистов > 5200 российский/снг 18% европейский 22% сша/канада 56% компании в основном работают в областях: чип дизайн; разработка сапр; web дизайн; асу; асу тп; прикладное программирование в области асни; разработка и производство коммуникационных средств; построение сетей; обслуживание сетей; разработка коммуникационных проектов; предоставление коммуникационных услуг; организация радио и теле вещания. анализ мировых тенденций развития общества дает нам основание утверждать, что глобальными и основными задачами 21-го века в области ик будут: развитие мобильных инфокоммуникаций развитие ик среды и средств для борьбы с террором развитие ик среды и средств экологического мониторинга развитие ик среды и средств общего языкового пространства разработка средств борьбы с киберпреступностью разработка ик средств развития телемедицины microsoft word w6_new.doc mathematical problems of computer science 37, 45--52, 2012. 45 an algorithm of digital image interpolation lilit g. minasyan state engineering university of armenia abstract the effective algorithms for reducing and increasing the size of images using the orthogonal wavelet-like transformation are presented. the experimental results show an improvement in terms of psnr in comparison to the well-known image interpolation algorithms. keywords: interpolation, orthogonal transform, image resizing algorithm. references [1] j. mukherjee, s. k. mitra, “image resizing in the compressed domain using subband dct,” ieee trans. circuits and systems for video technology, vol. 12, no. 7, pp. 620-627, 2002. [2] s. a. martucci, “image resizing in the discrete cosine transform domain,” ieee proc. int. conf. image processing, vol. 2, pp. 244-247, 1995. [3] j. scott, r. tutwiler and m. pusateri, “hyper-spectral content aware resizing,” ieee conf. applied imagery pattern recognition workshop, pp. 1-7, 2008. [4] s. khachatryan, l. minasyan, “on synthesis and application of an orthogonal discrete transform for image compression,” journal of engineering academy of armenia, t4, № 1. yerevan, pp.124-129, 2007 (in russian). [5] s. khachatryan and l. minasyan, “design of orthogonal transform on partion of unity method and their application to image compression, “ int. workshop on local and non local approximation in image processing, lnla’2008, tuusula, finland, pp. 145-152, 2008. [6] s. khachatryan and l. minasyan, “on algorithms of digital image interpolation,” mathema tics in high school, t. 6, № 2, pp. 32-41, 2010. [7] r. gonzalez and r. woods, digital image processing, 2nd ed. upper saddle river, new jersey: prentice-hall, inc., 2002. [8] t. w. parks and c. s. burrus. digital filter design. new york: john wiley & sons, pp. 209-213, 1987. [9] a. v. oppenheim and r. w. schafer. “discrete-time signal processing”. upper saddle river, nj: prentice-hall, pp. 450-454, 1999. [10] r. keys, “cubic convolution interpolation for digital image processing,” ieee trans. on signal processing, acoustics, speech, and signal processing 29 (6), pp.1153-1160, 1981. [11] k. turkowski and s. gabriel, “filters for common resampling tasks”. in andrew s. glassner. graphics gems i. academic press. pp. 147–165. isbn 9780122861659, 1990. an algorithm of digital image interpolation 46 [12] w. burger, m. j. burge, principles of digital image processing: core algorithms. springer. pp. 231–232. isbn 9781848001947, 2009. [13] g. schaub, “genuine fractals 4.1; resampling with gf might make the megapixel race moot,” march, shutterbug, 2006. [14] j. canny, “a computational approach to edge detection,” ieee transactions on pami, 8(6), pp. 679–698, 1986. թվային պատկերների միջարկման մի ալգորիթմի մասին լ. մինասյան ամփոփում առաջարկված են պատկերների չափերի փոքրացման և մեծացման արդյունավետ ալգորիթմներ'վեյվլեթ տիպի օրթոգոնալ ձևափոխության կիրառմամբ: կատարված են համեմատություններ պատկերների միջարկման հայտնի ալգորիթմների արդյունքների հետ: об одном алгоритме интерполяции цифрового изображения л. минасян аннотация предложены эффективные алгоритмы уменьшения и увеличения размеров изображений с применением ортогонального вейвлетообразного преобразования. приведены сравнения с результатами известных алгоритмов интерполяции изображений. parandzem_aram_harourounian.dvi mathematical problems of computer science 35, 77{85, 2011. m any h ypotheses lao t esting with rejection of decision for ar bitr ar ily var ying object e vg u e n i h a r o u t u n ia n , p a r a n d z e m h a ko b ya n a n d a r a m y e s s a ya n institute for informatics and automation problems of nas of ra e-mail: evhar@ipia.sci.am, par h@ipia.sci.am abstract the model of multiple statistical hypotheses testing with possibility rejecting to make choices between hypotheses concerning discrete arbitrarily varying object is investigated. the optimal procedure of decisions is shown. the matrix of optimal asymptotical interdependencies (reliability{reliability functions) of all possible pairs of the error probability exponents (reliabilities) is studied for arbitrarily varying object with the current states sequence known to the statistician. refer ences [1 ] w . h o e ®d in g , \ a s ym p t o t ic a lly o p t im a l t e s t s fo r m u lt in o m ia l d is t r ib u t io n s ," the annals of m athematical statistics, vo l. 3 6 , p p . 3 6 9 { 4 0 1 , 1 9 6 5 . [2 ] i. cs is z ¶a r a n d g. l o n g o , \ on t h e e r r o r e xp o n e n t fo r s o u r c e c o d in g a n d fo r t e s t in g s im p le s t a t is t ic a l h yp o t h e s e s " , studia sc. m ath. hungarica, vo l. 6 , p p . 1 8 1 { 1 9 1 , 1 9 7 1 . [3 ] g. tu s n ¶a d y, \ on a s ym p t o t ic a lly o p t im a l t e s t s ," annals of statatistics, vo l. 5 , n o . 2 , p p . 3 8 5 -3 9 3 , 1 9 7 7 . [4 ] g. l o n g o a n d a . s g a r r o , \ th e e r r o r e xp o n e n t fo r t h e t e s t in g o f s im p le s t a t is t ic a l h yp o t h e s e s : a c o m b in a t o r ia l a p p r o a c h " , j ournal of combinatorics and informational system science, vo l. 5 , n o . 1 , p p . 5 8 -6 7 , 1 9 8 0 . [5 ] l . b ir g ¶ e , \ v it e s s e s m a xim a ls d e d ¶ c r o is s e n c e d e s e r r e u r s e t t e s t s o p t im a u x a s s o c ie ¶ s " . z. w a h r s c h . ve r w. ge b ie t e , vo l. 5 5 , p p . 2 6 1 { 2 7 3 , 1 9 8 1 . [6 ] s . n a t a r a ja n , \ l a r g e d e r iva t io n s h yp o t h e s is t e s t in g a n d s o u r c e c o d in g fo r ¯ n it e ma r ko v c h a in s " , ie e e trans. inform. theory, vo l. 3 1 , n o . 3 , p p . 3 6 0 { 3 6 5 , 1 9 8 5 . [7 ] m. gu t m a n , \ a s ym p t o t ic a lly o p t im a l c la s s i¯ c a t io n fo r m u lt ip le t e s t s wit h e m p ir ic a lly o b s e r ve d s t a t is t ic s " , ie e e transactions on information theory, vo l. 3 5 , n o . 2 , p p . 4 0 1 { 4 0 8 , 1 9 8 9 . [8 ] v . a n a n t h a r a m , \ a la r g e d e r ia t io n s a p p r o a c h t o e r r o r e xp o n e n t in s o u r c e c o d in g a n d h yp o t h e s e s t e s t in g " , ie e e trans. inform. theory, vo l. 3 6 , n o . 4 , p p . 9 3 8 { 9 4 3 , 1 9 9 0 . [9 ] t. s . h a n , \ h yp o t h e s is t e s t in g wit h t h e g e n e r a l s o u r c e " , ie e e transactions on information theory, vo l. 4 6 , n o . 7 , p p . 2 4 1 5 -2 4 2 7 , 2 0 0 0 . [1 0 ] e . a . h a r o u t u n ia n , \ ma n y s t a t is t ic a l h yp o t h e s e s : in t e r d e p e n d e n c e o f o p t im a l t e s t 's e r r o r p r o b a b ilit ie s e xp o n e n t s " , ( in r u s s ia n ) , a b s t r a c t o f t h e r e p o r t o n t h e 3 r d a llu n io n s c h o o l-s e m in a r , "p rogram-algorithmical software for applied multi-variate statistical analysis", ts a kh ka d z o r , p a r t 2 , p p . 1 7 7 { 1 7 8 , 1 9 8 8 . 7 7 7 8 many hypotheses lao testing with rejection of decision for arbitrarily varying object [1 1 ] e . a . h a r o u t u n ia n , \ l o g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f m u lt ip le s t a t is t ic a l h yp o t h e s e s " , p roblems of control and information theory, vo l. 1 9 ( 5 -6 ) , p p . 4 1 3 { 4 2 1 , 1 9 9 0 . [1 2 ] e . h a r o u t u n ia n a n d p . h a ko b ya n , \ on m u lt ip le h yp o t h e s e s t e s t in g b y in fo r m e d s t a t is t ic ia n fo r a r b it r a r ly va r yin g o b je c t a n d a p p lic a t io n t o s o u r c e c o d in g " , transactions of iiap of nas of r a and of ysu, m athematical p roblems of computer science, vo l. 2 3 , p p . 3 6 -4 6 , 2 0 0 4 . [1 3 ] f. w . fu a n d s . y . s h e n , \ h yp o t h e s is t e s t in g fo r a r b it r a r ily va r yin g s o u r c e wit h e xp o n e n t ia l-t yp e c o n s t r a in t ," ie e e transactions on information theory, vo l. 4 4 , n o . 2 , p p . 8 9 2 -8 9 5 , 1 9 9 8 . [1 4 ] r . f. a h ls we d e , e . a lo ya n a n d e . h a r o u t u n ia n , \ on lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l h yp o t h e s is t e s t in g fo r a r b it r a r ily va r yin g s o u r c e wit h s id e in fo r m a t io n ," l ecture notes in computer science, volume 4123, "general theory of information transfer and combinatorics", springer, p p . 4 5 7 -4 6 1 , 2 0 0 6 . [1 5 ] m. s . n iku lin , \ on o n e r e s u lt o f l . n . b o ls h e v fr o m t h e o r y o f h yp o t h e s e s s t a t is t ic a l t e s t in g " , ( in r u s s ia n ) , studies on m athematical statistics. notes of scienti¯c seminars of saint-p etersburg b ranch of the m athematical institute, vo l. 7 , p p . 1 2 9 { 1 3 7 , 1 9 8 6 . [1 6 ] i. cs is z ¶a r a n d j. k äo r n e r , \ in fo r m a t io n t h e o r y: c o d in g t h e o r e m s fo r d is c r e t e m e m o r yle s s s ys t e m s " , academic press., n e w y o r k, 1 9 8 1 . [1 7 ] r . e . b la h u t , \ h yp o t h e s is t e s t in g a n d in fo r m a t io n t h e o r y," ie e e transactions on information theory, vo l. 2 0 , n o . 4 , p p . 4 0 5 { 4 1 7 , 1 9 7 4 . [1 8 ] t. m. co ve r a n d j. a . th o m a s , e lements of information theory. s e c o n d e d it io n , n e w y o r k, w ile y, 2 0 0 6 . [1 9 ] i. cs is z ¶a r a n d p . s h ie ld s , \ in fo r m a t io n t h e o r y a n d s t a t is t ic s : a t u t o r ia l" , f oundations and trends in communications and information theory, 2 0 0 4 . [2 0 ] r . f. a h ls we d e a n d e . a . h a r o u t u n ia n , \ on lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f h yp o t h e s e s a n d id e n t i¯ c a t io n " , l ecture notes in computer science, volume 4123, "general theory of information transfer and combinatorics", springer, p p . 4 6 2 { 4 7 8 , 2 0 0 6 . [2 1 ] e . h a r o u t u n ia n , m. h a r o u t u n ia n a n d a . h a r u t yu n ya n , \ r e lia b ilit y c r it e r ia in in fo r m a t io n t h e o r y a n d in s t a t is t ic a l h yp o t h e s is t e s t in g " , f oundations and trends in communications and information theory, vo l. 4 , n o . 2 -3 , 2 0 0 8 , 1 7 1 p . [2 2 ] e . h a r o u t u n ia n a n d p . h a ko b ya n , \ mu lt ip le h yp o t h e s e s l a o t e s t in g fo r m a n y in d e p e n d e n t o b je c t s " , in t e r n a t io n a l jo u r n a l \ s c h o la r ly r e s e a r c h e xc h a n g e " , p p . 1 -6 , 2 0 0 9 . î³ù³û³ï³ýáñ»ý ÷á÷áëíáõ ûµû»ïïç ýï³ïù³ùµ áñáßáõùçó ññ³å³ñù³ùµ µ³½ù³ïç í³ñï³íý»ñç è²ú ëïáõ·áõùá º. ð³ñáõãûáõýû³ý, ö. ð³ïáµû³ý ¨ ². ºë³û³ý ²ù÷á÷áõù àõëáõùý³ëçñí»é ¿ ï³ù³û³ï³ýáñ»ý ÷á÷áëíáõ ûµû»ïïç í»ñ³µ»ñû³é áñáßáõùçó ññ³å³ñáõùáí µ³½ù³ïç í³ñï³íý»ñç íç×³ï³·ñ³ï³ýáñ»ý ëïáõ·ù³ý ·áñíáýã³óá, »ñµ í³ñï³íý»ñç ùçç¨ áýïñáõãûáõýá ï³ï³ñíáõù ¿ áëï ³ýï³ë ¹çï³ñïáõùý»ñç ³ñ¹ûáõýùý»ñç: òáõûó ¿ ïñí»é áñáßáõùý»ñç áý¹áõýù³ý ûåïçù³é áýã³ó³ï³ñ·á: î³ù³û³ï³ýáñ»ý ÷á÷áëíáõ ûµû»ïïç í»ñ³µ»ñû³é, áñç íç×³ïý»ñá ñ³ûïýç »ý e. haroutunian, p. hakobyan and a. yessayan 7 9 íç×³ï³·ñçý ñ»ï³½áïí»é ¿ ñý³ñ³íáñ ëë³éý»ñç ½áõû·»ñç óáõóçãý»ñç (ñáõë³éçáõãûáõýý»ñç) ÷áëï³åí³íáõãûáõýý»ñç ù³ïñçóá: àñå»ë ñ»¨³ýù, ³ñï³íí»é »ý ³ý÷á÷áë ûµû»ïïç í»ñ³µ»ñû³é ññ³å³ñù³ý áñáßáõùáí µ³½ù³ïç í³ñï³íý»ñç è²ú ëïáõ·ù³ý ëë³éý»ñç ñáõë³éçáõãûáõýý»ñç ½áõû·»ñç ÷áëï³åí³íáõãûáõýý»ñá ³ñï³ñ³ûïáõ ýáõýïóç³ý»ñá: начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 47, 76--78, 2017. mechanisms for administrating tasks and editing data bases in hybrid email/sms info-communicational systems andranik e. mkhitaryan institute for informatics and automation problems of nas ra e-mail: and.mkhitaryan@gmail.com abstract info-communicational system ‘mail2sms’ [1] is a flexible mechanism to notify the receivers of email about the delivered e-mail by sms messages. in such systems every user should be able to control the list of allowed/disallowed notifiers. in this article a possible solution to this problem is considered by special commands. keywords: mail2sms, e-mail, sms, notification electronic mail (email), along with indisputable advantages, has some qualities characteristic to those of traditional mail services delivering the letter to the addressee’s “poste restante” mailbox. another system for delivering letters is sms, which, not being a postal system, also performs such functions. one of the differences of sms from email is the delivery of a mail right to the addressee’s telephone with this excluding the uncertainty over time of when the user will check his/her mailbox. the problem of uncertainty over time of reading an email in addressee’s mailbox can be solved by using post informers (sms servers) connected with the mail server and regional cellular network sending sms to notify the addressee about receipting an email [2-5]. other technologies performing the same tasks may also be used. technologies use sms-servers as independent, substantive info-communicational resources. sms server wherein operates with mail’s copy which is automatically sent to the email address of smsserver (function “cc”). a similar technology is used in mail2sms.asnet.am (www. asnet.am/sms applications) post informer for sending an sms as a notification about an email sent to him/her. as a targeting mechanism to sms-server a domain outlined with markers is used in the email address part “subject” (markers are used for identifying the email, targeting the sms-server) which is a more optional variant for operating applications of sms-server. functions in the domain “subject” (in mail2sms limited only to telephone to which the smsnotification is to be sent) can be significantly expanded if consider the marked domain in “subject” as a command line, where commands can be written connecting other applications to sms-server. it should be noted that the instrument of “command line” is an open resource 76 a. mkhitaryan 77 allowing the other developers to insert their own operation into the system with email/sms services. similar combinations, “hybrid” technologies can be used not only as systems for notifications, but also for other applications using email/sms technologies. these technologies are attractive with that they allow different sms applications to be combined into a single infocommunicational resource by the "one-window" principle. this, in addition to the direct assignment, allows the user, if necessary, to use the address part of the mail while formatting it as a targeting mechanism to the sms-server for connecting to the desired service and managing its functions from the list of options available in this system. as an example here, we cite some available basic list of options implemented by commands from “subject”:  *sms to… (#phone)* send an sms to the specified phone number by ‘#phone’  *notice to…(#phone)* along with email send sms notification to recipient  *save…(username/#phone)* record attributes of recipients in the database  *add to black list* add someone to the recipient's “black list” to disallow to send sms notifications.  *del from black list* delete blocked users from recipient’s “black list” references [1] a.nanassian and k. khachatryan, “mail2sms.asnet.am – the alert system of incoming”, proceedings of international conference computer science and information technologies, csit, yerevan, armenia, pp. 459--461, 2013. [2] [online]. available: https://help.mail.ru/mail-help/settings/notifications [3] [online]. available: http://iglous.ru/besplatnyj-sposob-poluchat-sms-uvedomleniya-opochte/ [4] d. gevorkyan, k. khachatryan, a. nanassian, a. petrosyan, g. petrosyan, v. sahakyan and e. vardanyan, “mail informerselective incoming instand phone notification system”, proceedings of international conference computer science and information technologies, csit, yerevan, armenia, pp. 466--467, 2009. [5] д. геворкян, а.нанасян and к. хачатрян, “новые web ресурсы asnet.am”, proceedings of international conference computer science and information technologies, csit, yerevan, armenia, pp. 311--312, 2011. submitted 14.12.2016, 20.01.2017 accepted. https://help.mail.ru/mail-help/settings/notifications http://iglous.ru/besplatnyj-sposob-poluchat-sms-uvedomleniya-o-pochte/ http://iglous.ru/besplatnyj-sposob-poluchat-sms-uvedomleniya-o-pochte/ mechanisms for administrating tasks and editing data bases in hybrid email/sms systems 78 հիբրիդային email/sms ինֆո-կոմունիկացիոն համակարգերի առաջադրանքներով ղեկավարման մեխանիզմները և տվյալների բազաների խմբագրումը ա. մխիթարյան ամփոփում աշխատանքում առաջարկվում է ինֆո-կոմունիկացիոն համակարգերում օգտատերերի կողմից թույլատրելի/անթույլատրելի ծանուցիչների ցուցակը կառավարող մեխանիզմ: նկարագրված համակարգում յուրաքանչյուր օգտատեր հնարավորություն ունի էլ. փոստի “subject”-ում օգտագործել համապատասխան հրամաններ սև ցուցակում որևէ մեկին ավելացնելու կամ հանելու նպատակով: механизмы управления заданиями и редактированием базы данных в гибридных email/sms инфокоммуникационных системах а. мхитарян аннотация в работе обсуждается метод управления и редактирования «черными списками» входящих электронных писем. система позволяет пользователю электронной почты использовать один из двух возможных команд в поле «subject» для редактирования черного списка. d:\sbornik\...\article.dvi mathematical problems of computer science 35, 46{52, 2011. realization of 1d classic h eisenber g spin-glass system on gp u h a ko b g. a b a jya n institute for informatics and automation problems of nas of ra e-mail habajyan@ipia.sci.am abstract new high performance parallel algorithm for simulation of 1d classic heisenberg spin-glass system is developed. the realization is done on graphics processing units (gpu). numerical simulations of the classic heisenberg spin glass model show that this is a good example of applications that may bene¯t of the gpu computing capabilities. gpu performance time and memory statistics are presented. refer ences [1 ] a . p . y o u n g ( e d .) , spin glasses and r andom f ields, w o r ld s c ie n t i¯ c , s in g a p o r e , 1 9 9 8 . [2 ] r . fis c h a n d a . b . h a r r is , \ s p in -g la s s m o d e l in c o n t in u o u s d im e n s io n a lit y" , p hys. r ev. l et., 47, p . 6 2 0 , 1 9 8 1 . [3 ] a . s . ge vo r kya n e t a l., \ n e w m a t h e m a t ic a l c o n c e p t io n a n d c o m p u t a t io n a lg o r it h m fo r s t u d y o f qu a n t u m 3 d d is o r d e r e d s p in s ys t e m u n d e r t h e in ° u e n c e o f e xt e r n a l ¯ e ld " , trans. on comput. sci., vii, l ncs, p p . 1 3 2 -1 5 3 , s p in g e r -v e r la g e , 1 0 .1 0 0 7 / 9 7 8 -3 -6 4 2 1 1 3 8 9 -5 8 . [4 ] a . s . ge vo r kya n , h . g. a b a jya n a n d h . s . s u kia s ya n . a r x iv: c o n d -m a t .d is -n n 1 0 1 0 .1 6 2 3 v1 . [5 ] n v id ia cu d a , c p r o g r a m m in g gu id e , ve r s io n 3 .2 , 2 0 1 0 . [6 ] n v id ia co m p u t e v is u a l p r o ¯ le r u s e r gu id e , 2 0 1 0 . [7 ] l . n yla n d , m. h a r r is , j. p r in s , \ fa s t n -b o d y s im u la t io n wit h cu d a , in : gp u ge m s 3 " , addison-w esley p rofessional, ch . 3 1 , p p . 6 7 7 6 9 5 , 2 0 0 7 . [8 ] j. d . owe n s , d . l u e b ke , n . go vin d a r a ju , m. h a r r is , j. k r u äg e r , a . l e fo h n , t. j. p u r c e ll, \ a s u r ve y o f g e n e r a l-p u r p o s e c o m p u t a t io n o n g r a p h ic s h a r d wa r e " , computer graphics f orum, vo l. 2 6 , is s u e 1 , p p . 8 0 1 1 3 , 2 0 0 7 . [9 ] d . y u e n , j. w a n g , l . jo h n s s o n , c. h . ch i, y . s h i( e d s .) , gp u solutions to multi-scale problems in science and engineering, s p r in g e r , 1 s t e d it io n , 2 0 1 1 . 1d ¸³ë³ï³ý ñ»û½»ýµ»ñ· ëåçý-³å³ïç h³ù³ï³ñ·ç çñ³ï³ý³óáõùá ·ñ³ýçï³ï³ý åñáó»ëáñý»ñç íñ³ 4 6 h. abajyan 4 7 ð. ²µ³çû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù µ»ñí³í ¿ 1d ¹³ë³ï³ý ð»û½»ýµ»ñ· ëåçý-³å³ïç ñ³ù³ï³ñ·ç ùá¹»é³íáñù³ý ýáñ µ³ñóñ ³ñï³¹ñáõ³ï³ýáõãû³ý ½áõ·³ñ»é ³é·áñçãù: ²é·áñçãùý çñ³ï³ý³óí³í ¿ ñ³ýçï³ï³ý äñáó»ëáñý»ñç (ä) íñ³: âí³ûçý ëçùáõéû³óç³ý óáõûó ¿ ï³éçë, áñ ¹³ë³ï³ý ð»û½»ýµ»ñ· ëåçý-³å³ïç ùá¹»éá ïçñ³éáõãû³ý é³í ûñçý³ï ¿ ä ñ³ßíáõ³ï³ý ñý³ñ³íáñáõãûáõýý»ñç û·ï³·áñíù³ý ï»ë³ýïûáõýçó: ²ßë³ï³ýùáõù ý³¨ µ»ñí³í »ý ³ñ³·³·áñíáõãû³ý å³ù³ý³ï³ûçý ¨ ñçßáõáõãû³ý íç×³ï³·ñ³ï³ý ïíû³éý»ñá: microsoft word asatryan_16.doc mathematical problems of computer science 36, 128-132, 2012. 128 a method for quality assessment of image resizing algorithms david asatryan, vardan kurkchiyan and marine bagramyan institute for informatics and automation problems of nas of ra e-mail: dasat@ipia.sci.am russian-armenian (slavonic) university e-mail: vakur87@gmail.com, bagmarish@gmail.com abstract in this paper, a novel technique for quality assessment of image resizing algorithms is proposed. the technique is based on consecutive application of a resizing algorithm with different resizing coefficients so that the size of the last image coincide with the size of the original image. the mean square deviation between the last and the original image characterizes the quality of applied resizing algorithm, which can be called the “backlash” of that algorithm. numerical results of experiments are given to show the effectiveness of proposed technique. references 1. daniel vaquero, matthew turk, kari pulli et al. “a survey of image retargeting techniques”. applications of digital image processing, xxxiii, andrew g. tescher, editor: proc. spie, 7798, 779814, 2010. 2. young-jin liu, xi luo, yu-ming xuan, wen-feng chen, xiao-lan fu. “image retargeting quality assessment”. eurographics 2011, volume 30, number 2, 2011. 3. b. girod, "what's wrong with mean squared error?" in a. b. watson (ed.), visual factors of electronic image communications, mit press, pp. 207-220, 1993. 4. wang z., bovik a.c. “a universal image quality index”, ieee signal processing letters, vol. 9, no. 3, pp. 81-84, 2002. 5. wang z., bovik a.c. modern image quality assessment, san rafael, ca: morgan & claypool. 2006. 6. asatryan d., egiazarian k. “quality assessment measure based on image structural properties”, proc. of the international workshop on local and non-local approximation in image processing. inland, helsinki, pp. 70-73, 2009. 7. асатрян д.г., куркчиян в.в., баграмян м.м., “сравнительный анализ качества алгоритмов масштабирования изображения”. вестник государственного d. asatryan, v. kurkchiyan, m. bagramyan инженерного университета армении: моделирование, оптимизация, управление. вып. 14, т. 2, сс. 70-76, 2011. 8. asatryan d., egiazarian k., kurkchiyan v. “orientation estimation with applications to image analysis and registration”. international journal on information theories and applications, vol. 17, no 4, pp. 303-311, 2010. 9. www.benvista.com/products պատկերի մասշտաբավորման ալգորիթմների որակի հետազոտման մեթոդիկա դ. ասատրյան, վ. քուրքչիյան, մ. բաղրամյան ամփոփում պատկերի մասշտաբավորման տարբեր ալգորիթմների որակի գնահատման առաջարկված մեթոդիկան հիմնված է այդ ալգորիթմների` տարբեր գործակիցներով բազմակի և հաջորդաբար կիրառման վրա այնպես, որ վերջում ստացված պատկերի չափսերը համընկնեն բնօրինակի չափսերի հետ: այս եղանակով ստացված պատկերների միջին քառակուսիական շեղումը բնորոշում է կիրառված մասշտաբավորման ալգորիթմի որակը և կոչվում է ալգորիթմի “խաղացք”: բերվել են առաջարկված մեթոդիկայի արդյունավետությունը ցուցադրող փորձերի արդյունքներ: методика исследования качества алгоритмов масштабирования изображения д. асатрян, в. куркчиян, м. баграмян предложена методика оценивания качества различных алгоритмов масштабирования изображения, основанная на многократном и последовательном масштабировании этих алгоритмов с различными коэффициентами масштабирования таким образом, чтобы размеры полученного в конце изображения совпали с размерами оргинала. среднеквадратическое отклонение полученных таким способом изображений характеризует качество алгоритма масштабирования и называется «люфтом» алгоритма. приведены результаты численных экспериментов, иллюстрирующих эффективность предложенного подхода. a method for quality assessment of image resizing algorithms 130 d:\sbornik\...\tpel1.dvi mathematical problems of computer science 33, 54{58, 2010. on i nter val t otal color ings of doubly convex b ipar tite gr aphs p e t r o s a . p e t r o s ya n yz a n d a n i s . s h a s h ikya n z yinstitute for informatics and automation problems of nas of ra, zdepartment of informatics and applied mathematics, ysu e-mail: pet petros@ipia.sci.am, anishashikyan@gmail.com abstract a bipartite graph g = (u; v; e) is doubly convex if all its vertices from the u can be numbered 1; 2; : : : ; juj and all vertices from v can be numbered 1; 2; : : : ; jv j in such a way that for any vertex of g the set of numbers assigned to neighbors is an interval of integers. an interval total t-coloring of a graph g is a total coloring of g with colors 1; 2; : : : ; t such that at least one vertex or edge of g is colored by i; i = 1; 2; : : : ; t, and the edges incident to each vertex v together with v are colored by dg(v)+1 consecutive colors, where dg(v) is the degree of a vertex v in g. in this paper we prove that all doubly convex bipartite graphs have an interval total coloring. furthermore, we give some bounds for the minimum and the maximum span in interval total colorings of these graphs. refer ences [1 ] a .s . a s r a t ia n , t.m.j. d e n le y, r . h a g g kvis t , b ip a r t it e gr a p h s a n d t h e ir a p p lic a t io n s , ca m b r id g e u n ive r s it y p r e s s , ca m b r id g e , 1 9 9 8 . [2 ] p .a . p e t r o s ya n ,\ in t e r va l t o t a l c o lo r in g s o f c o m p le t e b ip a r t it e g r a p h s " , p roceedings of the csit conference, p p . 8 4 -8 5 , 2 0 0 7 . [3 ] p .a . p e t r o s ya n ,\ in t e r va l t o t a l c o lo r in g s o f c e r t a in g r a p h s " , m athematical p roblems of computer science, vol. 31, p p . 1 2 2 -1 2 9 , 2 0 0 8 . [4 ] p .a . p e t r o s ya n , a .s . s h a s h ikya n ,\ on in t e r va l t o t a l c o lo r in g s o f t r e e s " , m athematical p roblems of computer science, vol. 32, p p . 7 0 -7 3 , 2 0 0 9 . [5 ] p .a . p e t r o s ya n , n .a . k h a c h a t r ya n ,\ in t e r va l t o t a l c o lo r in g s o f g r a p h s wit h a s p a n n in g s t a r " , m athematical p roblems of computer science, vol. 32, p p . 7 8 -8 5 , 2 0 0 9 . [6 ] p .a . p e t r o s ya n , a .s . s h a s h ikya n ,\ on in t e r va l t o t a l c o lo r in g s o f b ip a r t it e g r a p h s " , p roceedings of the csit conference, p p . 9 5 -9 8 , 2 0 0 9 . [7 ] p .a . p e t r o s ya n , a .y u . to r o s ya n ,\ in t e r va l t o t a l c o lo r in g s o f c o m p le t e g r a p h s " , p roceedings of the csit conference, p p . 9 9 -1 0 2 , 2 0 0 9 . [8 ] a .s . s h a s h ikya n ,\ on in t e r va l t o t a l c o lo r in g s o f b ip a r t it e g r a p h s " , ma s t e r 's th e s is , y e r e va n s t a t e u n ive r s it y, 2 0 0 9 , 2 9 p . [9 ] d .b . w e s t , in t r o d u c t io n t o gr a p h th e o r y, p r e n t ic e -h a ll, n e w je r s e y, 1 9 9 6 . 5 4 p. petrosyan, a. shashikyan 5 5 [1 0 ] h .p . y a p , to t a l co lo r in g s o f gr a p h s , l e c t u r e n o t e s in ma t h e m a t ic s 1 6 2 3 , s p r in g e r v e r la g , 1 9 9 6 . ºñï³ïç áõéáõóçï »ñïïáõù³ýç ·ñ³ýý»ñç ùçç³ï³ûù³ûçý éç³ï³ï³ñ ý»ñïáõùý»ñç ù³ëçý ä. ä»ïñáëû³ý ¨ ². þ³ßçïû³ý ²ù÷á÷áõù g »ñïïáõù³ýç ·ñ³ýá ïáãíáõù ¿ »ñï³ïç áõéáõóçï, »ã» g ·ñ³ýç ·³·³ãý»ñá ï³ñ»éç ¿ ³ûýå»ë ñ³ù³ñ³ï³é»é, áñ ³ù»ý ùç ïáõùç ó³ýï³ó³í ·³·³ã ñ³ñ¨³ý éçýç ùûáõë ïáõùç ñ³çáñ¹³ï³ý ç¹»ùëý»ñáí ·³·³ãý»ñç ñ»ï: g ·ñ³ýç éç³ï³ï³ñ ý»ñïáõùá 1 ; 2 ; :::t ·áõûý»ñáí ï³ýí³ý»ýù ùçç³ï³ûù³ûçý éç³ï³ï³ñ t-ý»ñïáõù, »ã» ³ù»ý ùç i ·áõûýáí, i = 1 ; 2 ; :::; t ý»ñïí³í ¿ ³éýí³½ý ù»ï ·³·³ã ï³ù ïáõ ¨ ûáõñ³ù³ýãûáõñ v ·³·³ãçý ïçó ïáõ»ñá ¨ ³û¹ ·³·³ãá ý»ñïí³í »ý dg ( v ) + 1 ñ³çáñ¹³ï³ý ·áõûý»ñáí, áñï»õ -áí ýß³ý³ïí³í ¿ v ·³·³ãç ³ëïç×³ýá g ·ñ³ýáõù: ²ûë ³ßë³ï³ýùáõù ³å³óáõóíáõù ¿, áñ µáéáñ »ñï³ïç áõéáõóçï »ñïïáõù³ýç ·ñ³ýý»ñá áõý»ý ùçç³ï³ûù³ûçý éç³ï³ï³ñ ý»ñïáõù: ²í»éçý, ³ßë³ï³ýùáõù ïñíáõù »ý áñáß ·ý³ñ³ï³ï³ýý»ñ ³ûë ·ñ³ýý»ñç ùçç³ï³ûù³ûçý éç³ï³ï³ñ ý»ñïáõùý»ñáõù ù³ëý³ïóáõ ·áõûý»ñç ýí³½³·áõûý ¨ ³é³í»é³·áõûý ñý³ñ³íáñ ãíç ñ³ù³ñ: mathematical problems of computer science 54, 53–68, 2020. udc 004.8 improving uav object detection through image augmentation karen m. gishyan university of bath, bath, united kingdom e-mail: karen.gishyan@bath.edu abstract ground-image based object detection algorithms have had great improvements over the years and provided good results for challenging image datasets such as coco and pascal voc. these models, however, are not as successful when it comes to unmanned aerial vehicle (uav)-based object detection and commonly performance deterioration is observed. it is due to the reason that it is a much harder task for the models to detect and classify smaller objects rather than medium-size or large-size objects, and drone imagery is prone to variances caused by different flying altitudes, weather conditions, camera angles and quality. this work explores the performance of two state-of-art-object detection algorithms on the drone object detection task and proposes image augmentation 1 procedures to improve model performance. we compose three image augmentation sequences and propose two new image augmentation techniques and further explore their different combinations on the performances of the models. the augmenters are evaluated for two deep learning models, which include model-training with high-resolution images (1056×1056 pixels) to observe their overall effectiveness. we provide a comparison of augmentation techniques across each model. we identify two augmentation procedures that increase object detection accuracy more effectively than others and obtain our best model using a transfer learning 2 approach, where the weights for the transfer are obtained from training the model with our proposed augmentation technique. at the end of the experiments, we achieve a robust model performance and accuracy, and identify the aspects of improvement as part of our future work. keywords: computer vision, deep learning, image processing 1. introduction deep learning-based computer vision algorithms for object detection in images and videos have had much success in the last decade. object identification and detection from unmanned aerial vehicles (uavs) have widely gained the attention of researchers, as uav 1image augmentation is a technique to expand the training dataset by creating modified versions of images in the dataset through different methods of processing. 2transfer learning is a research problem in machine learning that focuses on storing knowledge gained while solving one problem and applying it to a different but related problem. 53 54 improving uav object detection through image augmentation and computer vision-based applications can be successfully deployed in search-and-rescue, surveillance, agricultural crop identification and counting [1, 2]. however, compared to the object detection models that are trained on ground-based images, there is a significant decline in the performance of detecting objects of such models when applied to uav-based images. as drones fly at various altitudes and there is a constant change in the viewing angles, uav-based models have to deal with more visual appearances of the same object to work successfully. as an example, uavs can look at one object from front-view to side view in a very short period of time, which can result in arbitrary aspect ratios of the objects [3]. this work proposes image processing and augmentation techniques and evaluates their performance on yolov3-spp and yolov5-large object detection models. we compose augmentation methods such as (3 × 4) grid augmentation and geometric sequence that successfully improve performance across multiple models, and their combination through transfer learning helps to achieve the highest overall mean average precision (map) 0.5 accuracy with yolov3-spp. we also identify procedures where giving significantly more annotations 3 to the model during the training process does not improve and sometimes even impairs performance, and we verify the previous findings that augmentation techniques are effective when there are certain amount of images in the training dataset, which come from the same distribution as the validation set, otherwise in most of the cases we do not observe performance improvement. the augmentation procedures are evaluated across images with (1056 × 1056) input dimensions, which we regard as high resolution images for our experiments. high resolution images, however, translate to more computational resources and an incremental increase in the image size is also not guaranteed to improve performance. in spite of the significant amount of annotations in the images in the dataset that we use, there still exists a class imbalance, meaning the model may see an instance of a car class much more frequently than an instance of a bicycle class, resulting in training overfitting and decreasing the overall model accuracy during validation, which we aim to improve with our proposed augmentation methods. our results show that (3×4) grid augmentation and (3×5) grid augmentation with transfer learning with random resampling methodology can successfully improve accuracy. as part of future work, instead of random sampling, we will explore grid augmentation variations with an oversampling methodology and specifically target the images containing imbalanced classes. 2. methodology 2.1. models in this work, for conducting our experiments, we use two model variants, which use onestage object detection paradigm. the models are yolov3 with spatial pyramid pooling (yolov3-spp) from the works of [4, 5, 6] and yolov5-large [7]. to solve the problem of yolov2 backbone’s low ability to do feature extraction and inability to make a full use of multi-scale local region features, [5] propose dc-spp-yolo (dense connection and spatial pyramid pooling based yolo) to improve the accuracy of yolov2. they design a new spatial pyramid pooling introduced to collect and concatenate the multi-scale local region 3annotation is the process of labeling images, which provides information to the computer vision model about the objects in the image. coco bounding box annotations store (x-top left, y-top left, width, height) values of the given object in the image in a json format, while pascal voc annotations store (x-top left, y-top left, x-bottom right, y-bottom right) values of the object in an xml format. k. gishyan 55 features for more comprehensive learning. they also show that dc-spp-yolo is more accurate than yolov2. we use a combination of spatial pyramid pooling and yolov3 by [6] and a recently developed yolov5-large model by [7] for these experiments. [7] mentions that yolov5-large achieves an average precision (ap) 0.5 score of 66.5% on a coco testdev dataset. in spite of being the official yolov5 repository, the model is fairly new, and the official paper is still to be released. 2.2. data this work uses a subset of visdrone-det2019 4 dataset as a starting point for the experiments, and later scales upon it. the original dataset includes 10 categories, which are pedestrian, person, car, van, bus, truck, motor, bicycle, awning-tricycle and tricycle. for these 10 categories, there are about 540,000 bounding boxes, 6471 training, 548 validation and 1580 testing images [8]. we select 1000 images having a height of less than 768 pixels from the original training dataset for the experiment, which are later readjusted to 1056 × 1056 input image dimensions and use it for conducting our image augmentation experiments. these images are later divided into 700 training, 200 validation and 100 test sets. the original 10 categories are reduced to 7, and our class list contains all default categories except motor, awning-tricycle and tricycle. by using image augmentation sequences and techniques, we later generate 700 new images equal to the number of images in our training dataset, and our final dataset includes 1400 training, 200 validation and 100 test images. we keep 11.7% of our total images for validation, compared to the original dataset, where the number of validation images is 6.3% of the total. 2.3. image transformations and augmentations we present three image augmentation sequences and name them blending sequence, blurring sequence, geometric sequence, and 2 new image augmentation techniques and name them (3 × 4) grid augmentation, and (3 × 5) grid augmentation, which are applied to the training dataset to generate 700 new images, and for the main experiment, we compare the performance of the augmentation techniques against the base results5, or simply no augmentation results. to test whether the obtained results can be further approved, we later experiment with a combination of sequences and grid augmentation for the model providing the best result across the experiments. the first three sequences are explored as part of the data-warping augmentations, while the grid augmentations are explored as part of random-resampling augmentation to help decrease class imbalance. each augmentation sequence includes from five to seven augmenters 6 and is grouped according to its relevant augmentation type. using such approach allows each image in our dataset to undergo a unique image transformation procedure obtained from each augmentation sequence. after the augmentation procedures, we mostly end up with training images, which may be a little complex in form and not always observed in real life, however, as it will be seen from the results, most of them do positively affect the performance, and some of them do it much 4visdrone dataset is a large drone-based dataset containing images and videos particularly made for object-detection and object-tracking tasks. 5base results are obtained from evaluating the models on the original 700 images without any augmentation, and the respective model is called a base model. 6augmenter is a distinctive image transformation technique. 56 improving uav object detection through image augmentation more effectively than the rest. each augmenter from a sequence is selected with a certain probability, and in the end one new augmented image is generated per image in the original dataset. with a defined probability, depending on the augmentation sequence, the original image undergoes from zero to two image processing procedures for each sequence, the probabilities varying for each one. we leave a small chance that no augmenter is applied for a given image. the grid augmentations are stochastic, as they select images randomly, while blending, blurring and geometric sequences are deterministic, so for a set of images they always produce the same results making the experiments reproducible. 2.3.1. blending sequence blending sequence includes five different blending augmenters. for each image, either one or two of the five available augmenters is selected. each augmenter itself, has a 90 % probability of happening when selected. the reason that the augmenters are not assigned full probabilities of happening is to prevent them from altering the default dynamics and over damaging the image. this can be better conceptualized for the case when two augmenters are selected, this method giving a small chance that not both of them are applied all the time. if for a given image two augmenters are selected, there is a 1 % probability that none of them is applied, and 10% probability that at least one is not applied. the blending sequence includes alpha frequency noise blending with a nearest upscale method, alpha mask blending, alpha checkerboard blending with a hue upscale method, alpha linear gradient blending and alpha simplex noise blending with a linear upscale method. alpha frequency noise blending uses frequency noise masks for blending two images. the alpha masks are sampled from varying frequency noises, and as we use the nearest neighbour upsampling, it mostly results in the creation of blobs with sharp edges. alpha mask blending augmenter generates an (h, w) or (h, w, c) channel-wise mask for each image, where h is the height, w is the width and c is the number of channels of the image. the mask is later used to alpha blend pixels between the foreground augmenter branch and the background branch. clouds are added as a part of this augmenter, as can be observed at the bottom right hand side of fig. 1b. alpha checkerboard blending uses a checkerboard pattern for blending the images. (r × c) grid, where r is the number of rows and c is the number of columns, is placed on each image. for this experiment, r is 1, and c is a number uniformly drawn from the [1,4] interval. the value for the hue of images is randomly drawn from the [-80,80] interval. alpha linear gradient blending blends two images along a vertical gradient. here the gradient is applied to a pooled-image, and the starting and ending coordinates of y-coordinate is chosen at random. this randomness causes the gradient to increase either at the top or at the bottom [9]. alpha simplex noise blending detects edges inside the image, which are marked black and white, then uses simplex noise to alpha blend with the original image. here we use a linear upscaling method, which creates rectangles having smooth edges. a sample blending transformation example can be seen in fig. 1. 2.3.2. blurring sequence this sequence includes four blur-related augmenters and three contrast-related augmenters, and together they comprise the blurring sequence. for each image, one or two of the available augmenters are selected. as blurring sequence includes more augmenters, each augmenter has a 95% probability of being executed when selected (see fig. 2). the augmenters are: k. gishyan 57 (a) original image. (b) transformed image. fig. 1. blending sequence transformation. additive gaussian noise, gaussian blur, motion blur, mean shift blur, histogram equalization, sigmoid contrast and linear contrast. additive gaussian noise adds gaussian noise to the image. for each pixel, the noise is sampled from f(0, s) where f follows a normal distribution, and s is sampled once per image. we choose s as 70. gaussian blur uses a gaussian kernel for blurring the images. the standard deviation for the kernel is selected randomly from the range of [1,3] for each image. the motion blur augmenter applies a motion blur with a kernel size of 10 × 10 and a blur angle from the range of [-30,30], selected randomly per image. the mean shift blur augmenter applies a pyramidic mean shift filter to each image. histogram equalization augmenter transforms the image into rgb color space, applies histogram equalization and the input channels are transformed back to the original color space. for the sigmoid contrast output, we use a (3,10) tuple. the multiplier for the sigmoid function is selected randomly from the interval of [3,10] per image, and the cutoff value, which defines the shift of the sigmoid function in the horizontal direction, is selected randomly from the range of [0.4,0.6] per image. higher values result in darker pixels [9], and we select the range accordingly. linear contrast scales each pixel to 127+α ∗ (a − 127), where a is selected randomly from the range of [0.6,1.4] per image. the default range is preserved. (a) original image. (b) transformed image. fig. 2. blurring sequence transformation. 2.3.3. geometric sequence for each image as before, either one or two augmenters get selected. for this sequence, each augmenter has a 93 % probability of being applied to the image, when selected. the 58 improving uav object detection through image augmentation geometric sequence has the following augmenters: elastic transformation, 90 degree rotation, rotation from -30 to 30, polar warping, shear transformation in y direction and shear transformation in x direction. elastic transformation uses displacement fields and (a) original image. (b) transformed image. fig. 3. geometric sequence transformation. moves pixels around the figure. the augmenter has an alpha parameter, and we pass a tuple of (20, 30), and each image is transformed with a random value from the range of the tuple. we specifically choose low values for the tuple to introduce pixel distortion. 90 degree rotation rotates an image clockwise with a multiple of 90 degrees, resulting in various flipping transformations. rotation from -30 to 30 rotates an image with a randomly chosen value from the interval of [-30,30]. polar warping transforms an image into polar representation, which results in circular-warped patterns in the image. this transformation can be partially observed in subfigure 3b. shear transformation in y direction applies a shear transformation across the y axis, and shear transformation in x direction applies shear transformation across the x axis (fig. 3b). the transformations intervals, from which the values are selected randomly for the transformation are [30,50] and [30,70], respectively. the values are chosen to replicate realistic viewing scenarios that a drone may be exposed to during a real-time flight. during a geometric transformation, however, some of the objects start to lie outside of the image and thus some of the bounding-box coordinates get dropped. 2.3.4. grid augmentations 2.3.5. (3 × 4) grid and (3 × 5) grid [10], in their recent work on yolov4, use a new data augmentation technique first proposed by [6], which combines randomly chosen four images together during the model training process to increase image variability. they name this technique mosaic augmentation and demonstrate that it helps to increase the map accuracy score on coco dataset. we extend the idea of mosaic augmentation and propose two new data augmentation techniques and name them (3 × 4) grid augmentation and (3 × 5) grid augmentation, the latter being used with transfer learning during the model training, which combine 3 images horizontally and 4 and 5 images vertically, respectively. the combined images with their bounding boxes for (3 × 4) grid augmentation can be seen in fig. 4 7. it is a common practice and sometimes a requirement to choose the image size of the deep learning network as a multiple of 16 or 32 pixels. for this reason, for the (3 × 4) grid augmentation, 7a visual example of (3 × 5) grid augmentation is not provided as it follows a transformation pattern similar to (3 × 4) grid augmentation. k. gishyan 59 each image is resized to (352 × 352), which is thought to be an optimal size considering the increase in width and height that we get after grid augmentation. each image, thus, has a square shape. the images are then combined horizontally and vertically, resulting in (1056 × 1408) size images, including annotations of 12 randomly selected images, which are resized accordingly. for the (3 × 5) grid augmentation, a random batch of 3 images is selected once a time. the 3 images are resized to the minimum image height in the given batch, and the widths are re-adjusted according to the new image height. the results in the creation of horizontal images. a random batch of 5 horizontally combined images are then selected to be combined vertically. the images are now resized based on the minimum image width of the given batch, and the heights are adjusted accordingly. this method allows us to preserve the relative sizes of the images, as well as different widths and heights. these differences, however, are sometimes not visible, as most of the images have relatively similar shapes, and this may create grid augmentations, where each image in the grid may appear to be square in shape. for the (3 × 5) grid augmentation, we join images, which have undergone a blur sequence transformation, unlike (3 × 4) grid augmentation, where the images are selected from the subset of original visdrone dataset. grid augmentation methods significantly increase the number of small object in a given image, and most of the grid-image combinations may be realistically analogous to the real world, as the drone sees objects from various settings during a real flight, especially from higher altitudes, an effect obtained from grid augmenting transformation. though technically the proposed technique allows us to combine more images both horizontally and vertically, we should be careful about the computational costs, because if the given image has a number of annotations on average, the image after grid transformation contains a × w × h annotations on average, where w and h are the horizontal and vertical numbers of combined images. 2.3.6. blurring sequence and grid augmentation for the model that provides the best map 0.5 score, we test an augmentation sequence, where instead of blurring the images then combining them into a grid-like image as done for (3 × 5) grid augmentation, we apply a blurring sequence augmentation to the whole grid-augmented image. 2.3.7. all augmentation sequences and grid augmentation as a final experiment, we select 700 images that have been transformed using (3 × 4) grid augmentation and divide them into 3 equal proportions. we transform the first, second, and third proportions using blending sequence, blurring sequence, geometric sequence, respectively. the results for blending sequence and geometric sequence can be seen by looking at fig. 5. for training the model, we use only the transformed 700 images, unlike the rest of the experiments, which use 1400 images. combining images into a grid-like form also combines all separate annotations into one. we provide a summary of augmentation sequences including the total number of annotations for each technique in table 1. we observe that grid augmentation techniques significantly increase the amount of annotations in the dataset, most of them being generated from (3×5) grid augmentation, which, over the original dataset with no augmentation, contains 15.3 times more labels. 60 improving uav object detection through image augmentation (a) (3 × 4) grid-augmented image. (b) (3 × 4) grid-augmented image with bounding boxes. fig. 4. (3 × 4) grid-augmented image. (a) blending sequence applied to a (3 × 4) gridaugmented image. (b) geometric sequence applied to a (3 × 4) gridaugmented image. fig. 5. augmentation sequences applied to (3 × 4) grid-augmented images. k. gishyan 61 table 1: augmentation sequences applied to the original 700 training images. augmentation sequence no bounding box annotations total no of images no augmentation 40309 700 blending sequence 40309 1400 blurring sequence 40309 1400 geometric sequence 37967 1400 (3 × 4) grid 471233 1400 (3 × 5) grid 618256 1400 blurring sequence and (3 × 4) grid 471233 1400 all sequences and (3 × 4) grid 458044 700 a blending, blurring, geometric sequences, (3 × 4) grid, (3 × 5) grid are applied to the original 700 images. b blurring sequence and (3 × 4) grid, all sequences and (3 × 4) grid are applied to 700 images that have been (3 × 4) grid-augmented. 2.4. transfer learning the trainings begin from a pretrained checkpoint for both models. yolov3-spp uses yolov3-spp-ultralytics weights, and yolov5-large uses yolov5l weights. using pretrained backbones helps to speed up learning and gives more improved performance than training the models from randomly-initialized weights, however, we believe the performance can be further improved if we use weights that have been trained specifically on the drone-object detection footage, as the pretrained weights available for initialization work well with larger objects, but there is a significant performance reduction when evaluated on a small object detection task. for this reason, we conduct transfer learning based on the weights that we have obtained from our experiments and as the results will later show it helps to further improve performance and obtain a better model. we use the model’s weights, which have been trained on the dataset without augmentation (700 images) as an initialization point for yolov3-spp and yolov5-large and the weights that have been trained on the dataset with (3×4) grid-augmented images as an initialization point for yolov3-spp trained with geometric augmentation as part of the performance-improvement experiment with the best model. 3. experiments and results 3.1. model specifications 3.2. yolov3-spp yolov3-spp trained on higher resolution images provides the better results for both experiments. the base model provides map 0.5 score of 39.7 %. we further improve this performance up to 46.2 % with (3 × 4) grid augmentation, which becomes the procedure providing the highest increase in map score amongst all the models without using transfer learning8. geometric sequence and blending sequence provide almost equal results in terms 8without transfer learning means not using custom weights obtained from our base model. 62 improving uav object detection through image augmentation table 2: parameters of yolov3-spp and yolov5-large. parameters yolov3-spp yolov5-large batch size 2 2 epoch size 15 15 channels 3 3 validation interval 1 1 initial learning rate 0.001 0.01 image size (1056 × 1056) (1056 × 1056) of map 0.5; 45.3% and 45.2%, respectively. we will later see that the geometric sequence and the pretrained weights of (3 × 4) grid sequence together provide the highest results obtained for this work. the other augmenter, the blurring sequence provides a result of 45.1%, however, the results of (3 × 5) augmentation with transfer learning is almost just as good as the base model, while containing about 620,000 more annotations than the base model and using pretrained weights as initialization weights for training. this shows that extensive augmentation procedure is not guaranteed to improve performance. we observe very smooth learning and increase in precision by looking at fig. 6. summary results can be found in table 3. fig. 6. map 0.5: yolov3-spp results with (1056 × 1056) image size. table 3: best results summary for yolov3-spp: (1056 × 1056) size. rank model map 0.5 f1 score 1 (3 × 4) grid augmentation 46.2% 48.9% 2 geometric sequence 45.3% 47.1% 3 blending sequence 45.2% 47.7% no augmentation 39.7% 41.2% k. gishyan 63 3.3. yolov5-large with yolov5-large, we obtain an map 0.5 of 29.2%, achieved by (3×5) grid augmentation with transfer learning. further results can be found in table 4, showing that blending sequence and blurring sequence are important augmentation procedures for yolov5-large with map 0.5 scores of 27.0 % and 26.3 %, respectively. we also observe an increase in accuracy provided by the augmentation sequences over the base model. fig. 7. map 0.5: yolov5-large results with (1056 × 1056) image size. table 4: best results summary for yolov5-large: (1056 × 1056) size. rank model map 0.5 map 0.5:0.95 1 (3 × 5) grid augmentation with transfer learning 29.2% 14.0% 2 blending seqence 27.0% 14.6 % 3 blurring sequence 26.3 % 14.2 % no augmentation 22.5% 12.9% 4. further improvements we conduct a few more experiments to see if the model performance can be further improved. we make one more experiment with yolov5-large and 3 more experiments with yolov3spp. by training a model with blending sequence augmentation using pretrained weights from the base model training we are able to obtain the best model for yolov5-large, with an map 0.5 score of 36.5%, which is a 14% improvement from the base model. we choose blending sequence as it is the second-best augmenter for yolov5-large, in terms of map 0.5 score. for the yolov3-spp model, we choose the weights obtained from (3 × 4) grid augmentation for yolov3-spp and initialize it for our second-best geometric sequencetransformed dataset for training. we obtain map 0.5 score of 48.6%, which is a 2.4% improvement over our previous best score, and 8.9% improvement over our base model. we 64 improving uav object detection through image augmentation also provide two examples where we train the models with the augmentation methods defined in 2.3.6., where we perform blurring sequence transformation on the dataset containing 700 original and 700 (3 × 4) grid-augmented images, and another experiment, where we take 700 training images, which have undergone a (3 × 4) grid-augmentation and apply blurring, blending and geometric sequence transformations, each augmenter transforming 1/3 of the images, as described in section 2.3.7.. moreover, for this augmenter, which was our most complex transformation, the map decreases by 5.9% compared to the base model. we observe that more annotations do not strictly increase accuracy. as a note, these final two augmentation experiments where we did not observe improvements contained no images from the original dataset, unlike the rest of the experiments. the experiments with the best overall results for yolov3-spp and yolov5-large can be observed from fig. 8 and fig. 9. fig. 8. map 0.5: yolov5-large (1056 × 1056) size with improved performance. fig. 9. map 0.5: yolov3-spp (1056 × 1056) size with improved performance. k. gishyan 65 (a) inference 1. (b) inference 2. fig. 10. inference results of yolov3-spp. 5. inference results from the inference results of the best yolov3-spp model from fig. 10a, we can observe that the model performs well on identifying almost all of the objects in the defined categories, however, we also see in fig. 10b that there are pedestrians in the left hand part of the image that are not detected, so we identify room for improvement. 6. conclusion this work explored multiple-object detection task for drones using a component of visdronedet2019 dataset. we constructed three-image-processing sequences each containing from 5 to 7 augmenters, and named them blending sequence, blurring sequence, geometric sequence. in an effort to reduce data overfitting with random resampling, we also developed grid augmentation techniques where we combined images in a custom-shape (3 × 4) square form and (3 × 5) form where the relative dimensions of the images were preserved after combination. we named these techniques (3 × 4) grid augmentation and (3 × 5) grid augmentation. we conducted multiple experiments where we tested how our augmentation techniques affected the performance of the models with an aim to identify the techniques that work best for the drone object detection task. we identified three augmentation procedures that help to improve the model performance the most, which are using geometric sequence, (3 × 4) grid augmentation transformations and (3 × 5) grid augmentation with transfer learning. we considered (3 × 4) grid augmentation as the overall best and geometric sequence as the second best augmentation method and the best one among all three sequences. we further conducted a few more experiments by using transfer learning from custom weights, and combining our sequences with grid augmenters in different forms for two of our best models; yolov5-large and yolov3-spp with (1056 × 1056) image size, achieving further increase in accuracy. we also showed that some augmentation techniques did not work well and resulted in performance deterioration, despite containing significant amount of annotations, as we saw when applying blurring sequence to the (3 × 4) gridaugmented dataset, and when applying all of our sequences to 700 (3 × 4) grid-transformed images. these datasets for training contained no images from the original dataset, and we can state that when training and validation datasets are not from the same setting, the augmentation procedures will most certainly not be successful. on the other hand, as a result of our further experiments, we obtained our overall best model by using the weights of the model trained with (3×4) grid augmentation for training a model with geometric sequence 66 improving uav object detection through image augmentation augmentation, and improved the performance of the previous best result by 2.4% to map 0.5 score of 48.6% achieved with a yolov3-spp model. the dataset sizes differ and the original dataset contains 3 more classes, so it is challenging to compare this setting to the original, however, we note that the winner of the visdrone-det 2019 challenge achieved an ap 0.5 score of 54.0%, and 47.98% was the tenth best score [8]. we take these results as benchmark results for future improvement. references [1] l jangwon, j wang, d crandall, s šabanovic and g fox, “real-time, cloud-based object detection for unmanned aerial vehicles” first ieee international conference on robotic computing (irc), taichung, taiwan, doi: 10.1109/irc.2017.77, pp. 36 43, 2017. [2] a. carrio, c. sampedro, a. rodriguez-ramos and p. campoy, “a review of deep learning methods and applications for unmanned aerial vehicles” journal of sensors, https://doi.org/10.1155/2017/3296874, 2017. [3] zh. wu, k. suresh, p. narayanan, h. xu, h. kwon and zh. wang, “delving into robust object detection from unmanned aerial vehicles: a deep nuisance disentanglement approach”, proceedings of the ieee international conference on computer vision, pp. 1201–1210, 2019. [4] j. redmon and a. farhadi, “yolov3: an incremental improvement”, arxiv:1804.02767v1, 2018. [5] zh. huang and j. wang, “dc-spp-yolo: dense connection and spatial pyramid pooling based yolo for object detection”, arxiv:1903.08589, 2019. [6] glenn jocher. ultralytics yolov3. (2019). https://github.com/ultralytics/yolov3 [7] glenn jocher. ultralytics yolov5. (2020). https://github.com/ultralytics/yolov5 [8] dheeraj reddy pailla et al., “visdrone-det2019: the vision meets drone object detection in image challenge results”, ieee/cvf international conference on computer vision workshop (iccvw), doi: 10.1109/iccvw.2019.00030, 2019. [9] a. jung. imgaug. (2020). online.available: https://github.com/aleju/imgaug [10] a. bochkovskiy, c.-yao wang and hong-yuan mark lia, “yolov4: optimal speed and accuracy of object detection”, arxiv preprint arxiv:2004.10934, 2020. submitted 22.07.2020, accepted 19.11.2020. k. gishyan 6 7 ²âê-çó ëï³óí³í ³é³ñï³ý»ñç ×³ý³ãù³ý µ³ñ»é³íáõù å³ïï»ñç ³áõ·ù»ýï³óç³ûç ùççáóáí î³ñ»ý ø. ¶çßû³ý ´³ãç ð³ù³éë³ñ³ý e-mail: karen.gishyan@bath.edu ²ù÷á÷áõù ²é³ñï³ý»ñç ñ³ûïý³µ»ñù³ý ³é·áñçãùý»ñá, áñáýù ñçùýí³í »ý ó³ù³ù³ûçý å³ûù³ýý»ñáõùëï³óí³íéáõë³ýï³ñý»ñçíñ³, í»ñççýï³ñçý»ñçáýã³óùáõùµ³ñ»é³íí»é ¨ ù»í ñ³çáõáõãûáõýý»ñ »ý ·ñ³ýó»é ³ûýåçëç µ³ñ¹ ïíû³éý»ñç ñ³í³ù³íáõý»ñç íñ³, çýãåçëçù »ý coco-ý ¨ pascal voc-á: ²ûë ùá¹»éý»ñá, ³ûýáõ³ù»ý³ûýçí, ³ûýù³ý ¿é é³í ³ñ¹ûáõýù»ñ ã»ý ·ñ³ýóáõù ²âê-ç íñ³ ñçùýí³í ³é³ñï³ý»ñç ñ³ûïý³µ»ñù³ý å³ù³ý³ï ¨ ñ³×³ë ýï³ïíáõù ¿ ï³ï³ñáõ³ï³ýáõãû³ý í³ïã³ñ³óáõù: ä³ï×³éý ³ûý ¿, áñ ùá¹»éý»ñç ñ³ù³ñ ß³ï ³í»éç µ³ñ¹ ¿ ñ³ûïý³µ»ñ»é ¨ ¹³ë³ï³ñ·»é ³é³í»é ÷áùñ ûµû»ïïý»ñá, ç ï³ñµ»ñáõãûáõý ùçççý ¨ ëáßáñ ã³÷ç ûµû»ïïý»ñç, ¨ ¹ñáýý»ñçó ³ñí³í 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ï»ñåáí, ù³ýª ùûáõëý»ñá ¨ ëï³ýáõù, áý¹ñ³ýáõñ ³éù³ùµ, ù»ñ é³í³·áõý ùá¹»éáª û·ï³·áñí»éáí ÷áë³ýóù³ý áõëáõóù³ý ùáï»óáõù, áñï»õ ÷áë³ýóù³ý ïßçéý»ñá ó»éù »ý µ»ñíáõù áõëáõó³ý»éáí ùá¹»éá ù»ñ ïáõùçó ³é³ç³ñïí³í ³áõ·ù»ýï³óçáý ù»ãá¹áí: öáñó³ñïáõùý»ñç ³ñ¹ûáõýùáõù ù»ýù ëï³ýáõù »ýù ùá¹»éç é³í ï³ï³ñáõ³ï³ýáõãûáõý ¨ ×ß·ñïáõãûáõý ¨ ù³ïý³ýßáõù µ³ñ»é³íù³ý ³ëå»ïïý»ñ áñå»ë ñ»ï³·³ ³ßë³ï³ýùç ù³ë: ´³ý³éç µ³é»ñ` ñ³ù³ï³ñ·ã³ûçý ï»ëáõáõãûáõý, ëáñá áõëáõóáõù, å³ïï»ñç ùß³ïáõù: 6 8 improving uav object detection through image augmentation óëó÷øåíèå îáíàðóæåíèÿ îáúåêòîâ áïëà ñ óâåëè÷åíèåì èçîáðàæåíèÿ êàðåí m. ãèøÿí áàòñêèé óíèâåðñèòåò e-mail: karen.gishyan@bath.edu àííîòàöèÿ àëãîðèòìû îáíàðóæåíèÿ îáúåêòîâ íà îñíîâå íàçåìíûõ èçîáðàæåíèé çíà÷èòåëüíî óëó÷øèëèñü çà ïîñëåäíèå ãîäû è äàëè õîðîøèå ðåçóëüòàòû äëÿ ñëîæíûõ íàáîðîâ äàííûõ èçîáðàæåíèé, òàêèõ êàê coco è pascal voc. îäíàêî ýòè ìîäåëè íå òàê óñïåøíû, êîãäà äåëî äîõîäèò äî îáíàðóæåíèÿ îáúåêòîâ ñ ïîìîùüþ áïëà, è îáû÷íî íàáëþäàåòñÿ óõóäøåíèå õàðàêòåðèñòèê. ýòî ñâÿçàíî ñ òåì, ÷òî äëÿ ìîäåëåé ãîðàçäî ñëîæíåå îáíàðóæèâàòü è êëàññèôèöèðîâàòü áîëåå ìåëêèå îáúåêòû, ÷åì îáúåêòû ñðåäíåãî èëè áîëüøîãî ðàçìåðà, à èçîáðàæåíèÿ ñ äðîíîâ ïîäâåðæåíû òðàíñëÿöèîííûì îòêëîíåíèÿì, âûçâàííûìè ðàçíîé âûñîòîé ïîëåòà, ïîãîäíûìè óñëîâèÿìè, ðàêóðñàìè è êà÷åñòâîì êàìåðû. â ýòîé ðàáîòå èññëåäóåòñÿ ýôôåêòèâíîñòü äâóõ ñîâðåìåííûõ àëãîðèòìîâ îáíàðóæåíèÿ îáúåêòîâ â çàäà÷å îáíàðóæåíèÿîáúåêòîâ ñïîìîùüþäðîíàèïðåäëàãàþòñÿïðîöåäóðûóâåëè÷åíèÿ èçîáðàæåíèÿ äëÿ ïîâûøåíèÿ ïðîèçâîäèòåëüíîñòè ìîäåëè. ìû ñîñòàâëÿåì òðè ïîñëåäîâàòåëüíîñòè óâåëè÷åíèÿ èçîáðàæåíèÿ è ïðåäëàãàåì äâà íîâûõ ìåòîäà óâåëè÷åíèÿ èçîáðàæåíèÿ, à òàêæå äîïîëíèòåëüíî èññëåäóåì èõ ðàçëè÷íûå êîìáèíàöèè íà õàðàêòåðèñòèêàõ ìîäåëåé. àóãìåíòåðû îöåíèâàþòñÿ äëÿ äâóõ ìîäåëåé ãëóáîêîãî îáó÷åíèÿ, êîòîðûå âêëþ÷àþò îáó÷åíèå ìîäåëè ñ èçîáðàæåíèÿìè ñ âûñîêèì ðàçðåøåíèåì (1056 x 1056 ïèêñåëåé), ÷òîáû îöåíèòü èõ îáùóþ ýôôåêòèâíîñòü. ìû ïðåäîñòàâëÿåì ñðàâíåíèå ìåòîäîâ óâåëè÷åíèÿ äëÿ êàæäîé ìîäåëè. ìû îïðåäåëÿåì äâå ïðîöåäóðû äîïîëíåíèÿ, êîòîðûå ïîâûøàþò òî÷íîñòü îáíàðóæåíèÿ îáúåêòîâ áîëåå ýôôåêòèâíî, ÷åì äðóãèå, è ïîëó÷àåì íàøó ëó÷øóþ îáùóþ ìîäåëü ñ èñïîëüçîâàíèåì ïîäõîäà ñ ïåðåíîñîì îáó÷åíèÿ, ïðè êîòîðîì âåñà äëÿ ïåðåíîñà ïîëó÷àþòñÿ â ðåçóëüòàòå îáó÷åíèÿ ìîäåëè ñ ïîìîùüþ ïðåäëàãàåìîé íàìè òåõíèêè óâåëè÷åíèÿ. â êîíöå ýêñïåðèìåíòîâ ìû äîñòèãàåì íàäåæíûõ õàðàêòåðèñòèê è òî÷íîñòè ìîäåëè è îïðåäåëÿåì àñïåêòû óëó÷øåíèÿ êàê ÷àñòü íàøåé áóäóùåé ðàáîòû. êëþ÷åâûå ñëîâà: êîìïüþòåðíîå çðåíèå, ãëóáîêîå îáó÷åíèå, îáðàáîòêà èçîáðàæåíèÿ. 05_karen_gishyan_54_53_68 (1) karen+ mathematical problems of computer science 51, 7--20, 2019. udc 519.712.6 complexity of error-correcting code based on nearest neighbor search algorithm1 levon h. aslanyan and hayk e. danoyan institute for informatics and automation problems of nas ra e-mail: lasl@sci.am, hed@ipia.sci.am abstract the nearest neighbor search algorithm considered in this paper is well known (elias algorithm). it uses error-correcting codes and constructs appropriate hash-coding schemas. these schemas preprocess the data in the form of lists. each list is contained in some sphere, centered at a code-word. the algorithm is considered for the cases of perfect codes, so the spheres and, consequently, the lists do not intersect. as such codes exist for the limited set of parameters, the algorithm is considered for some other generalizations of perfect codes, and then the same data point may be contained in different lists. a formula of time complexity of the algorithm is obtained for these cases, using coset weight structures of the mentioned codes. keywords: nn search, best match, hash-coding schema, perfect codes, uniformly packed codes, quasi-perfect codes. 1. introduction let 𝐸𝐸 = {0,1}. consider cartesian degree 𝐸𝐸𝑛𝑛, which is known as the set of vertexes of ndimensional unit cube. for any 𝑥𝑥, 𝑦𝑦 ∈ 𝐸𝐸𝑛𝑛 denote by 𝑑𝑑(𝑥𝑥, 𝑦𝑦) the hamming distance between vectors x and y. for an arbitrary 𝑥𝑥 ∈ 𝐸𝐸𝑛𝑛 denote by 𝑆𝑆𝑟𝑟𝑛𝑛(𝑥𝑥) the sphere of radius r, centred at x, i.e., 𝑆𝑆𝑟𝑟𝑛𝑛(𝑥𝑥) = {𝑦𝑦 𝑦𝑦 ∈ 𝐸𝐸𝑛𝑛, 𝑑𝑑(𝑥𝑥, 𝑦𝑦) ≤ 𝑟𝑟⁄ } and by 𝑂𝑂𝑟𝑟𝑛𝑛(𝑥𝑥) denote the shell of radius r, centred at x, i.e., 𝑂𝑂𝑟𝑟𝑛𝑛(𝑥𝑥) = {𝑦𝑦 𝑦𝑦 ∈ 𝐸𝐸𝑛𝑛, 𝑑𝑑(𝑥𝑥, 𝑦𝑦) = 𝑟𝑟⁄ }. we will denote by 𝑐𝑐𝑐𝑐𝑟𝑟(𝑥𝑥) the carrier of vector 𝑥𝑥 = 1 partially supported by grants № 18rf-144, and № 18t-1b407 of the science committee of the ministry of education and science of armenia 7 complexity of error-correcting code based on nearest neighbor search algorithm 8 (𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) then 𝑐𝑐𝑐𝑐𝑟𝑟(𝑥𝑥) = {𝑖𝑖 𝑥𝑥𝑖𝑖 = 1, 𝑖𝑖 = 1, … , 𝑛𝑛⁄ }. denote by 𝑤𝑤(𝑥𝑥) the weight of vector 𝑥𝑥, i.e., 𝑤𝑤(𝑥𝑥) = ∑ 𝑥𝑥𝑖𝑖 𝑛𝑛 𝑖𝑖=1 . let us have a subset 𝐹𝐹 ⊆ 𝐸𝐸𝑛𝑛 and a vector 𝑥𝑥 ∈ 𝐸𝐸𝑛𝑛. let us consider the problem of finding the set of nearest neighbors of 𝑥𝑥 from 𝐹𝐹. more precisely it is required to find the set 𝐹𝐹𝑥𝑥 = {𝑦𝑦 ∈ 𝐹𝐹 𝑑𝑑(𝑥𝑥, 𝑦𝑦) = 𝑑𝑑(𝑥𝑥, 𝐹𝐹)}⁄ . hash coding schemaes are considered [1,2] for preprocessing the data. a brief description of such schemes is brought below. hash function is defined as a function ℎ: 𝐸𝐸𝑛𝑛 → 𝑉𝑉, where 𝑉𝑉 = {𝑣𝑣1, … , 𝑣𝑣𝑁𝑁} is a finite set of 𝑁𝑁 elements [1]. cases are usually considered, when 𝑉𝑉 = 𝐸𝐸𝑘𝑘, 𝑘𝑘 ≤ 𝑛𝑛. the subset f is represented as a union of 𝑁𝑁 disjoint sets (lists). denote by 𝐵𝐵𝑖𝑖 the set {𝑥𝑥 ∈ 𝐸𝐸𝑛𝑛 ℎ(𝑥𝑥) = 𝑣𝑣𝑖𝑖⁄ }. the 𝑖𝑖-th list 𝐿𝐿𝑖𝑖 stores those vectors belonging to 𝐹𝐹, which have the same hash value, i.e., 𝐿𝐿𝑖𝑖 = {𝑥𝑥 ∈ 𝐹𝐹 ℎ(𝑥𝑥) = 𝑣𝑣𝑖𝑖⁄ } or 𝐿𝐿𝑖𝑖 = 𝐵𝐵𝑖𝑖⋂𝐹𝐹, 𝑖𝑖 = 1, … , 𝑁𝑁. hash coding schemae is called balanced if |𝐵𝐵𝑖𝑖| = 2𝑛𝑛 𝑁𝑁⁄ . the elias algorithm [2] considers blocks 𝐵𝐵𝑖𝑖 ordering them by their distances at vector 𝑥𝑥. mention that we must have an efficient method to find all blocks 𝐵𝐵𝑗𝑗1, 𝐵𝐵𝑗𝑗2, … , 𝐵𝐵𝑗𝑗𝑠𝑠(𝑗𝑗) located at distance 𝑗𝑗 from 𝑥𝑥 if such blocks exist. after the step of ordering, the algorithm examines the lists 𝐿𝐿𝑗𝑗1, 𝐿𝐿𝑗𝑗2, … , 𝐿𝐿𝑗𝑗𝑠𝑠(𝑗𝑗) one after another by increasing 𝑗𝑗𝑡𝑡. let the best match distance be denoted by 𝛿𝛿 (also the current value of the best match distance in the algorithm). due to 𝐹𝐹 ≠ ∅ initialization of 𝛿𝛿 will happen in some step. now, if the current values obey 𝛿𝛿 < 𝑗𝑗 algorithm stopes the work. all blocks with higher distances than 𝛿𝛿 at 𝑥𝑥 do not need to be examined. in the reminder case 𝛿𝛿 ≥ 𝑗𝑗, examining the nonempty list 𝐿𝐿𝑗𝑗𝑡𝑡 the algorithm can change the best match distance 𝛿𝛿, also refreshing the current best match set, or the 𝛿𝛿 will remain unchanged and the current best match set will be updated. the pseudocode of the algorithm is brought below: elias algorithm // 𝑛𝑛 is the word length, 𝑁𝑁 is the number of blocks input 𝑥𝑥, 𝐹𝐹, // 𝐹𝐹 ≠ ∅ integer 𝛿𝛿 = ∞, // the current best match distance set 𝑆𝑆 = ∅, // 𝑆𝑆-is the current set of vectors of 𝐹𝐹 located at distance 𝛿𝛿 from 𝑥𝑥 integer 𝑗𝑗 = −1, // current distance of blocks under consideration from 𝑥𝑥 while(𝑗𝑗 < 𝛿𝛿) { j++, if(𝑠𝑠(𝑗𝑗) ≠ 0) // 𝑠𝑠(𝑗𝑗) is the number of blocks located at distance 𝑗𝑗 from 𝑥𝑥 for(integer i=0, i 𝑛𝑛. taking into account formulas (4) and (5) and the coset weight structures for considered codes, we may formulate the following: proposition 1: the complexity of the algorithm for the hash function defined by [2𝑚𝑚 − 1, 2𝑚𝑚 − 𝑚𝑚 − 1,3]1 hamming code ℋ𝑚𝑚 is: α�hℋm� = 1 2m � v(j) ��ai ℋm + (2m − 1)ai e1+ℋm� j+1 i=0 � 0≤j≤2m−1 , (6) where by 𝑐𝑐𝑖𝑖 is denoted any fixed vector of weight 𝑖𝑖. proposition 2: for the golay code 𝒢𝒢 the complexity of the algorithm is: 𝛼𝛼�ℎ𝒢𝒢� = � 𝑉𝑉(𝑗𝑗) 0≤𝑗𝑗≤23 �� 1 211 𝐴𝐴𝑖𝑖 𝒢𝒢 + 23 211 𝐴𝐴𝑖𝑖 e1+𝒢𝒢 + 253 211 𝐴𝐴𝑖𝑖 e2+𝒢𝒢 + 5819 211 𝐴𝐴𝑖𝑖 e3+𝒢𝒢� 𝑗𝑗+3 𝑖𝑖=0 . (7) complexity of error-correcting code based on nearest neighbor search algorithm 14 proposition 3: for the code ℋ�𝑚𝑚 the complexity of the algorithm is: 𝛼𝛼�ℎℋ�𝑚𝑚� = � 𝑉𝑉(𝑗𝑗) �� 1 2𝑚𝑚+1 𝐴𝐴𝑖𝑖 ℋ�𝑚𝑚 + 1 2 𝐴𝐴𝑖𝑖 e1+ℋ�𝑚𝑚 + 2𝑚𝑚 − 1 2𝑚𝑚+1 𝐴𝐴𝑖𝑖 𝑢𝑢+ℋ�𝑚𝑚� 𝑗𝑗+2 𝑖𝑖=0 � 0≤𝑗𝑗≤2𝑚𝑚 , (8) where by 𝑢𝑢𝑖𝑖 is denoted any fixed vector of weight 2, the first coordinate of which is equal to 1. proposition 4: for two error correcting bch code 𝔅𝔅𝑚𝑚 of length 2𝑚𝑚 − 1 for odd 𝑚𝑚 the complexity of the algorithm is: 𝛼𝛼�ℎ𝔅𝔅𝑚𝑚� = � 𝑉𝑉(𝑗𝑗) 0≤𝑗𝑗≤2𝑚𝑚−1 �� 1 22𝑚𝑚 𝐴𝐴𝑖𝑖 𝔅𝔅𝑚𝑚 + 2𝑚𝑚 − 1 22𝑚𝑚 𝐴𝐴𝑖𝑖 e1+𝔅𝔅𝑚𝑚 + 𝑗𝑗+3 𝑖𝑖=0 + (2𝑚𝑚 − 1)(2𝑚𝑚−1 − 1) 22𝑚𝑚 𝐴𝐴𝑖𝑖 e2+𝔅𝔅𝑚𝑚 + 22𝑚𝑚−1 + 2𝑚𝑚−1 − 1 22𝑚𝑚 𝐴𝐴𝑖𝑖 e3+𝔅𝔅𝑚𝑚�. (9) 5. numeric results even having formulas, it is hard to imagine the practical complexities of things. to compare the results with those in [2], we provide numeric results for propositions 1 to 4 to demonstrate the complexity of the algorithm for each case of hash-coding schema. table 1 demonstrates the formula (6) in cases of hamming code of length 2𝑚𝑚 − 1 for 𝑚𝑚 = 4. table 2 demonstrates the formula (7) in case of golay code. tables 3 and 4 demonstrate the formula (8) in cases of extended hamming code of length 2𝑚𝑚 − 1 for 𝑚𝑚 = 4 and 𝑚𝑚 = 5. table 5 demonstrates the formula (9) in case of bch codes of length 2𝑚𝑚 − 1 for 𝑚𝑚 = 5. table 1: complexity of the algorithm in case of hamming code of length 15. subset generation probability 𝑝𝑝 average number of considered blocks the percentage relation of considered elements and cardinality of subset 1 1 0.05 2-1 4.28 0.21 2-2 6.2 0.3 2-3 10.1 0.49 2-4 17.31 0.85 2-5 26.3 1.28 2-6 42.27 2.06 2-7 67.67 3.3 2-8 107.23 5.24 2-9 170.38 8.32 2-10 268.45 13.11 2-11 418.26 20.42 2-12 638.09 31.16 2-13 901.17 44.0 2-14 1001.82 48.92 2-15 823.24 40.2 l. aslanyan and h. danoyan 15 table 2: complexity of the algorithm in case of golay code of length 23 subset generation probability 𝑝𝑝 average number of considered blocks the percentage relation of considered elements and cardinality of subset 1 1 0.02 2-1 3.16 0.08 2-2 4.26 0.10 2-3 5.45 0.13 2-4 8.54 0.21 2-5 12.86 0.31 2-6 17.14 0.42 2-7 24.51 0.6 2-8 36.97 0.90 2-9 51.85 1.27 2-10 75.16 1.83 2-11 109.80 2.68 2-12 157.04 3.83 2-13 227.57 5.55 2-14 325.28 7.94 2-15 463.76 11.32 2-16 654.12 15.97 2-17 911.53 22.25 2-18 1249.29 30.50 2-19 1674.63 40.88 2-20 2176.78 53.14 table 3: the case of extended hamming code of length 𝑛𝑛 = 16. subset generation probability p average number of considered blocks the percentage relation of considered elements and cardinality of subset 1 4.28 0.89 2-1 13.03 2.72 2-2 17.83 3.72 2-3 25.47 5.32 2-4 39.69 8.29 2-5 56.14 11.73 2-6 80.80 16.89 2-7 119.08 24.89 2-8 171.14 35.77 2-9 247.85 51.81 2-10 354.80 74.17 2-11 502.63 105.07 complexity of error-correcting code based on nearest neighbor search algorithm 16 2-12 699.58 146.24 2-13 947.74 198.12 2-14 1196.9 250.20 2-15 1235.31 258.23 2-16 977.51 204.34 table 4: the case of extended hamming code of length n=32. subset generation probability p average number of considered blocks the percentage relation of considered elements and cardinality of subset 1 8.26 0.0001 2-1 47.01 0.0005 2-2 66.43 0.0008 2-3 82.93 0.001 2-4 147.70 0.001 2-5 280.41 0.003 2-6 419.46 0.005 2-7 568.55 0.007 2-8 976.12 0.012 2-9 1731.29 0.02 2-10 2572.65 0.03 2-11 4038.7 0.04 2-12 7117.07 0.08 2-13 11173.6 0.13 2-14 18011.1 0.22 2-15 30662.2 0.37 2-16 48773 0.6 2-17 81535.8 1.00 2-18 133110 1.63 2-19 218976 2.69 2-20 358907 4.42 2-21 587880 7.24 2-22 957978 11.79 2-23 1.55722∙106 19.17 2-24 2.511422∙106 30.16 2-25 4.0295∙106 49.63 2-26 6.3936∙106 78.74 2-27 1.0009∙107 123.27 2-28 1.53811∙107 189.44 2-29 2.29847∙107 283.09 2-30 3.16976∙107 390.41 2-31 3.4639∙107 462.64 2-32 2.82105∙107 347.46 l. aslanyan and h. danoyan 17 table 5: the case of two error correcting bch code of length n=31. subset generation probability 𝑝𝑝 average number of considered blocks the percentage relation of considered elements and cardinality of subset 1 4.87 0.001 2-1 20.23 0.004 2-2 27.93 0.006 2-3 34.07 0.007 2-4 54.72 0.01 2-5 94.71 0.02 2-6 135.65 0.03 2-7 179.04 0.04 2-8 284.66 0.06 2-9 463.67 0.10 2-10 658.39 0.15 2-11 987.25 0.22 2-12 1597.11 0.37 2-13 2363.61 0.54 2-14 3624.62 0.84 2-15 5683.15 1.32 2-16 8556.05 1.98 2-17 13333.3 3.09 2-18 20290.5 4.71 2-19 31196.3 7.25 2-20 47458 11.03 2-21 72165.5 16.77 2-22 108815 25.29 2-23 162915 37.87 2-24 241404 56.11 2-25 353024 82.06 2-26 507518 117.977 2-27 713305 165.813 complexity of error-correcting code based on nearest neighbor search algorithm 18 2-28 971293 225.785 2-29 1.22855∙106 285.587 2-30 1.26797∙106 294.749 2-31 1.00312∙106 233.184 6. conclusion the nearest neighbor search algorithm considered in this paper is well known (elias algorithm). it uses hash-coding schemas for data preprocessing. these schemas partition the space into non intersecting blocks of the same cardinality. it is known that the algorithm is optimal when these blocks are spheres. such partitions may be obtained by error-correcting codes. the algorithm is considered for the cases of perfect codes, so the spheres and, consequently, the lists do not intersect. as such codes exist for a limited set of parameters, the algorithm is considered for some other generalizations of perfect codes, and then the same data point may be contained in different lists. a formula of time complexity of the algorithm is obtained for these cases. these formulas show the area of practical use of the algorithm: the algorithm encounters “the course of dimensionality”, i.e., as the word length grows, the algorithm turns into an exhaustive search in a file. references [1] d. e. knuth, the art of computer programming, vol. 3, sorting and searching, second edition, eddison-wesley, 1998. [2] r. rivest, “on the optimality of elias’s algorithm for performing best-match searches”, information processing, pp. 678–681, 1974. [3] l. h. aslanyan and h. e. danoyan, “on the optimality of the hash-coding type nearest neighbour search algorithm”, selected works of 9th csit conference, pp. 1-6, 2013. [4] f. j. mac-williams and n. j. a. sloane, the theory of error-correcting codes, amsterdam, the netherlands: north-holland, 1986. [5] v. a. zinoviev and v. k. leont’ev, “the nonexistence of perfect codes over galois fields”, problems of control and information theory, vol. 2, no. 2, pp. 123-132, 1973. [6] l. a. bassalygo, g. v. zaitsevand v. a. zinoviev, “uniformly packed codes”, problemy peredachi informatsii, vol. 10, no. 1, pp. 9-14, 1974. [7] t. baicheva, i. bouyukliev and s. dodunekov, “binary and ternary linear quasiperfect codes with small dimensions”, ieee transactions of information theory, vol. 54, no. 9, pp. 4335-4339, 2008. [8] e. m. gabidulin, a. a. davydov and l. m. tombak, “linear codes with covering radius 2 and other new covering codes,” ieee transactionsof information theory, vol. 37, no. 1, pp. 219–224, 1991. l. aslanyan and h. danoyan 19 [9] a. a. davydov and a. yu. drozhzhina-labinskaya, “constructions, families, and tables of binary linear covering codes,” ieee transactions of information theory, vol. 40, no. 4, pp. 1270–1279, jul. 1994. [10] t. etzion and b. mounits, “quasi-perfect codes with small distance”, ieee trans. inform. theory, vol. 51, no. 11, pp. 3938-3946, 2005. [11] t. etzion and g. greenberg, “constructions for perfect mixed codes and other covering codes,” ieee transactions of information.theory, vol. 39, no. 1, pp. 209– 214, 1993. [12] d. gorenstein, w. peterson and n. zierler, “two error-correcting bose-chaudhuri codes are quasi perfect”, information and control, no. 3, pp. 291-294, 1960. [13] t. kasami, s. lin and w. peterson, “some results on the weight distributions of bch codes”, ieee trans. information theory, vol. 12, no. 2, pp. 274-277, 1966. [14] p. charpin “weight distributions of cosets of two-error-correcting bch codes, extended or not”, ieee transactions of information.theory, vol. 40, no. 5, pp. 1425– 1442, 1991. submitted 18.02.2019, accepted 15.04.2019. սխալ ուղղող կոդերի վրա հիմնված մոտակա հարևանների փնտրման ալգորիթմի բարդությունը լևոն հ. ասլանյան և հայկ է. դանոյան հհ գաա ինֆորմատիկայի և ավտոմատացման պրոբլեմների ինստիտուտ e-mail: lasl@sci.am, hed@ipia.sci.am ամփոփում հայտնի է մոտակա հարևանների փնտրման հաշ-կոդավորման տիպի էլիասի ալգորիթմը: ալգորիթմում կարող են կիրառվել սխալ ուղղող կոդերը՝ հաշկոդավորման սխեմա կառուցելու համար: այդ սխեմաները տվյալները ներկայացնում են ցուցակների տեսքով: կամայական ցուցակ իրենից ներկայացնում է որևէ գնդի ենթաբազմություն (հեմինգի տարածությունում), որի կենտրոնը նշված սխալ ուղղող կոդի կոդային բառ է: ալգորիթմը դիտարկվում է կատարյալ կոդերի համար, հետևաբար կոդային բառերի շուրջ համապատասխան շառավղով գնդերը չեն հատվում, հետևաբար նշված ցուցակները նույնպես չեն հատվի: քանի որ կատարյալ կոդեր գոյություն ունեն պարամետրերի սպեցիֆիկ արժեքների համար, ալգորիթմը դիտարկվել է կատարյալ կոդերի այլ ընդհանրացումներով ստացվող հաշկոդավորման սխեմաների համար, որտեղ, սակայն, միևնույն էլեմենտը կարող է պատկանել մի քանի ցուցակների: նշված դեպքերի համար ստացվել են ալգորիթմի complexity of error-correcting code based on nearest neighbor search algorithm 20 բարդության բանաձևերը՝ կախված կոդի հարակից դասերի կշռային կառուցվածքներից: բանալի բառեր` մոտակա հարևանների փնտրում, լավագույն համընկնումների փնտրում, հաշ-կոդավորման սխեմա, կատարյալ կոդեր, հավասարաչափ փաթեթավորված կոդեր, քվազիկատարյալ կոդեր сложность алгоритма поиска ближайших соседей с кодами исправляющими ошибки левон а. асланян и айк э. даноян институт проблем информатики и автоматизации нан ра e-mail: lasl@sci.am, hed@ipia.sci.am аннотация известен алгоритм поиска ближайших соседей (алгоритм элиаса). алгоритм использует коды, исправляющие ошибки для построения схем хеш-кодирования. эти схемы представляют данные в форме списков. каждый список содержится в некоторой сфере, центром которого является некоторое кодовое слово. алгоритм рассматривается для случаев совершенных кодов, поэтому указанные сферы и, следовательно, списки не пересекаются. поскольку совершенные коды существуют для очень специфичного набора параметров, алгоритм рассматривается для некоторых обобщений совершенных кодов, когда одна и та же точка данных может содержаться в разных списках. для указанных случаев получены формулы временной сложности алгоритма с использованием весовой структуры смежных классов кодов. ключевые слова: поиск ближайших соседей, поиск наилучших совпaдений, схемы хеш-кодирования, совершенные коды, равномерно упакованные коды, квазисовершенные коды 3_mossine_19--22.dvi mathematical problems of computer science 48, 19{22, 2017. on long cycles in gr aphs in t er ms of degr ee sequences mo s s in e s . k o u la kz ia n institute for informatics and automation problems of nas ra e-mail: mossine@hotmail.com abstract let g be a graph on n vertices with degree sequence ± = d1 · d2 · ::: · dn. let m be the number of connected components of g, c the circumference the order of a longest cycle, p the order of a longest path in g and ¾s the minimum degree sum of an independent set of s vertices. in this paper it is shown that in every graph g, c ¸ d¾m+m + 1. this bound is best possible and generalizes the earliest lower bound for the circumference due to dirac (1952): c ¸ ± + 1 = d1 + 1. as corollaries, we have: (i) c ¸ d±+1 + 1; (ii) if d¾m+m ¸ p ¡ 1, then c = p; (iii) if g is a connected graph with d±+1 ¸ p ¡ 1, then g is hamiltonian; (iv) if d¾m+m ¸ n ¡ 1, then g is hamiltonian. keywords: hamilton cycle, longest cycle, circumference, minimum degree, degree sequence. 1 . in t r o d u c t io n w e c o n s id e r o n ly ¯ n it e u n d ir e c t e d g r a p h s wit h n e it h e r lo o p s n o r m u lt ip le e d g e s . a g o o d r e fe r e n c e fo r a n y u n d e ¯ n e d t e r m s is [1 ]. th e s e t o f ve r t ic e s o f a g r a p h g is d e n o t e d b y v ( g) a n d t h e s e t o f e d g e s b y e ( g) . l e t n b e t h e o r d e r ( t h e n u m b e r o f ve r t ic e s ) o f g, c t h e o r d e r o f a lo n g e s t c yc le ( c a lle d c ir c u m fe r e n c e ) in g a n d p t h e o r d e r o f a lo n g e s t p a t h . th e m in im u m d e g r e e in g is d e n o t e d b y ± a n d t h e m in im u m d e g r e e s u m o f a n in d e p e n d e n t s e t o f s ve r t ic e s will b e d e n o t e d b y ¾s. l e t d1; d2; :::; dn b e t h e d e g r e e s e qu e n c e in g wit h ± = d1 · d2 · ::: · dn. w e u s e n ( v ) t o d e n o t e t h e s e t o f a ll n e ig h b o r s o f ve r t e x v a n d d( v ) = jn ( v ) j t o d e n o t e t h e d e g r e e o f ve r t e x v. a p a t h ( s im p le p a t h ) o f o r d e r m is a s e qu e n c e o f d is t in c t ve r t ic e s v1; :::; vm, d e n o t e d b y v1v2:::vm, s u c h t h a t vi¡1vi is a n e d g e fo r a ll 2 · i · m. s im ila r ly. a c yc le o f o r d e r m is a s e qu e n c e o f d is t in c t ve r t ic e s v1; :::; vm, d e n o t e d b y v1v2:::vmv1, s u c h t h a t vi¡1vi a n d vmv1 a r e e d g e s fo r a ll 2 · i · m. in p a r t ic u la r , fo r m = 2 , v1v2v1 is a c yc le o f o r d e r 2 ; a n d fo r m = 1 , v1v1 is a c yc le o f o r d e r 1 . s o , b y t h e d e ¯ n it io n , e ve r y ve r t e x ( e d g e ) c a n b e c o n s id e r e d a s a c yc le o f o r d e r 1 ( 2 , r e s p e c t ive ly) .a g r a p h g is h a m ilt o n ia n if g c o n t a in s a h a m ilt o n c yc le , t h a t is a s im p le s p a n n in g c yc le . w e wr it e a c yc le ( a p a t h ) q wit h a g ive n o r ie n t a t io n b y ¡! q . th e r e ve r s e s e qu e n c e o f ve r t ic e s o f ¡! q is d e n o t e d b y ã¡ q . fo r x 2 v ( q ) , we d e n o t e t h e p r e d e c e s s o r o f x o n ¡!q ( if s u c h 1 9 2 0 on long cycles in graphs in terms of degree sequences ve r t ic e s e xis t ) b y x¡. w e u s e p = x ¡! p y t o d e n o t e a p a t h wit h e n d ve r t ic e s x a n d y in t h e d ir e c t io n fr o m x t o y. th e e a r lie s t a n d s im p le s t lo we r b o u n d fo r t h e c ir c u m fe r e n c e wa s o b t a in e d in 1 9 5 2 d u e t o d ir a c [2 ] in t e r m s o f m in im u m d e g r e e ±. t heor em a: [2 ]. in every graph, c ¸ ± + 1 . th e o r e m a is b e s t p o s s ib le . h o we ve r , t h e b o u n d c ¸ ± + 1 in th e o r e m a is e qu iva le n t t o c ¸ d1 + 1 , wh ic h is fa r fr o m b e in g b e s t p o s s ib le . in t h is p a p e r we p r e s e n t a n im p r o ve m e n t o f th e o r e m a in t e r m s o f d e g r e e s e qu e n c e s wit h o u t a n y a d d it io n a l c o n d it io n s . t heor em 1: l et g be a graph with m connected components and degree sequence ± = d1 · d2 · ::: · dn. then c ¸ d¾m+m + 1 . if g = mk±+1, t h e n c = ± + 1 = dm±+m + 1 = d¾m+m + 1 . th is g r a p h e xa m p le s h o ws t h a t t h e b o u n d d¾m+m + 1 in th e o r e m 1 c a n n o t b e r e p la c e d b y d¾m+m + 2 . n e xt , le t g = m ( k± + k±+1 ) . th e n d¾m+m+1 = dm±+m+1 = 2 ± = c. th is g r a p h e xa m p le s h o ws t h a t t h e lo we r b o u n d d¾m+m +1 in th e o r e m 1 c a n n o t b e r e p la c e d b y d¾m+m+1 +1 . th u s , th e o r e m 1 is b e s t p o s s ib le in a ll r e s p e c t s . cor ollar y 1: l et g be a graph with degree sequence ± = d1 · d2 · ::: · dn. then c ¸ d±+1 + 1 . th e g r a p h k± + ( ± + 1 ) k1 is ±-c o n n e c t e d wit h d1 = d2 = ::: = d±+1 = ± a n d d±+2 = 2 ±. s in c e c = 2 ± a n d d±+2 + 1 = 2 ± + 1 , t h e b o u n d c ¸ d±+1 + 1 in co r o lla r y 1 c a n n o t b e r e p la c e d b y c ¸ d±+2 + 1 . th u s , co r o lla r y 1 is b e s t p o s s ib le e ve n fo r h ig h c o n n e c t e d g r a p h s . th e n e xt t h r e e s t a t e m e n t s c a n b e o b t a in e d fr o m th e o r e m 1 e a s ily. t heor em 2: l et g be a graph with m connected components and degree sequence ± = d1 · d2 · ::: · dn. if d¾m+m ¸ p ¡ 1 , then c = p. cor ollar y 2: l et g be a graph with degree sequence ± = d1 · d2 · ::: · dn. if d±+1 ¸ p ¡ 1 , then c = p. t heor em 3: l et g be a graph with degree sequence ± = d1 · d2 · ::: · dn. if d±+1 ¸ p ¡ 1 , then g is hamiltonian. t heor em 4: l et g be a graph with degree sequence ± = d1 · d2 · ::: · dn. if d±+1 ¸ n ¡ 1 , then g is hamiltonian. 2 . p r o o fs p r oof of t heor em 1. l e t h1; h2; :::; hm b e t h e c o n n e c t e d c o m p o n e n t s o f g a n d le t¡! p = u ¡! p v b e a lo n g e s t p a t h in h1. cle a r ly, n ( u ) µ v ( p ) . a s s u m e t h a t ( a ) p is c h o s e n in h1 s o t h a t d( u ) is m a xim u m . l e t x1; x2; :::; xt b e t h e e le m e n t s o f n ( u ) o c c u r r in g o n ¡! p in a c o n s e c u t ive o r d e r , wh e r e t = d( u ) ¸ ±. ob s e r ve t h a t fo r e a c h i 2 f 1 ; 2 ; :::; tg, x¡i ã¡ p uxi ¡! p v is a lo n g e s t p a t h in h1, im p lyin g t h a t n ( x¡i ) µ v ( p ) ( i = 1 ; 2 ; :::; t ) : b y ( a ) , d ( u ) ¸ d( x¡i ) ( i = 1 ; 2 ; ::; t) ; d( u ) ¸ d ( v ) : ( 1 ) m. koulakzian 2 1 p u t c = u ¡! p xtu. cle a r ly, c ¸ jv ( c ) j ¸ t + 1 = d ( u) + 1 : b y ( 1 ) , d( u ) ¸ maxfd ( x¡1 ) ; d( x¡2 ) ; :::; d( x¡t ) ; d( v ) g; im p lyin g t h a t c ¸ d( u ) + 1 ¸ maxfd( x¡1 ) ; d( x¡2 ) ; :::; d( x¡d(u) ) ; d( v ) g + 1 : a n a lo g o u s ly, fo r e a c h i 2 f 1 ; 2 ; :::; mg, we c a n ¯ n d p a ir wis e d is t in c t ve r t ic e s yi1; yi2; :::; yid(ui)+1 in hi s u c h t h a t c ¸ m a xfd( yi1 ) ; d( yi2 ) ; :::; d( yid(ui)+1 ) g + 1 ; wh e r e ui 2 v ( hi ) . s in c e t h e ve r t ic e s y11; y 1 2 ; :::; y 1 d(u1)+1 ; y21; y 2 2 ; :::; y 2 d(u2)+1 ; :::::::::::::::::::::::::: ym1 ; y m 2 ; :::; y m d(um)+1 a r e a ll p a ir wis e d is t in c t , fo r s o m e p a ir wis e d is t in c t ve r t ic e s z1; z2; :::; zd(u1)+d(u2)+:::+d(um)+m we h a ve c ¸ m a xfd ( z1 ) ; d( z2 ) ; :::; d( zd(u1)+d(u2)+:::+d(um)+m ) g + 1 ¸ m a xfd1; d2; :::; dd(u1)+d(u2)+:::+d(um)+mg + 1 = dd(u1)+d(u2)+:::+d(um)+m + 1 : ob s e r vin g a ls o t h a t fu1; u2; :::; umg is a n in d e p e n d e n t s e t o f ve r t ic e s , we o b t a in t h e d e s ir e d b o u n d c ¸ d¾m+m + 1 . p r oof of t heor em 2. b y th e o r e m 1 , c ¸ d¾m+m + 1 ¸ p. if c ¸ p + 1 , t h e n c le a r ly t h e c yc le o f o r d e r a t le a s t p + 1 c o n t a in s a p a t h wit h a t le a s t p + 1 ve r t ic e s , c o n t r a d ic t in g t h e fa c t t h a t t h e lo n g e s t p a t h in g h a s e xa c t ly p ve r t ic e s . h e n c e , c = p. p r oof of t heor em 3. l e t g b e a c o n n e c t e d g r a p h wit h d±+1 ¸ p ¡ 1 . s in c e m = 1 a n d d¾m+m ¸ p ¡ 1 , b y th e o r e m 2 , c = p. l e t ¡! c b e a lo n g e s t c yc le in g o f le n g t h p. if p = n, t h e n g is h a m ilt o n ia n . l e t p < n. s in c e g is c o n n e c t e d , we h a ve vu 2 e ( g ) fo r s o m e v 2 v ( c ) a n d u 2 g ¡ c. th e n uv ¡! c v¡ is a p a t h o n p + 1 ve r t ic e s , a c o n t r a d ic t io n . th e o r e m 4 fo llo ws fr o m th e o r e m 2 im m e d ia t e ly. 2 2 on long cycles in graphs in terms of degree sequences refer ences [1 ] j. a . b o n d y a n d u .s .r . mu r t y, graph theory with applications, ma c m illa n , l o n d o n a n d e ls e vie r , n e w y o r k, 1 9 7 6 . [2 ] g. a . d ir a c , \ s o m e t h e o r e m s o n a b s t r a c t g r a p h s " , p roc. l ondon, m ath. soc., vo l. 2 , p p . 6 9 -8 1 , 1 9 5 2 . submitted 22.08.2017, accepted 06.12.2017. ¶ñ³ýý»ñáõù »ñï³ñ óçïé»ñç ù³ëçý ³ëïç×³ý³ûçý ñ³çáñ¹³ï³ýáõãûáõýý»ñç é»½íáí ø. øáõé³ù½û³ý ²ù÷á÷áõù ¸çóáõù g-ý ± ýí³½³·áõûý ³ëïç×³ý ¨ ± = d1 · d2 · ::: · dn ³ëïç×³ý³ûçý ñ³çáñ¹³ï³ýáõãûáõý áõý»óáõ n ·³·³ã³ýç ·ñ³ý ¿: ¶ñ³ýç ï³å³ïóí³íáõãû³ý µ³õ³¹ñçãý»ñç ù³ý³ïá ïýß³ý³ï»ýù m-áí, ³ù»ý³»ñï³ñ óçïéç »ñï³ñáõãûáõýá` cáí, çëï s ³ýï³ë ·³·³ãý»ñç ³ëïç×³ýý»ñç ýí³½³·áõûý ·áõù³ñá` ¾s -áí: ü»ñï³ ³ßë³ï³ýùáõù ³å³óáõóíáõù ¿, áñ ï³ù³û³ï³ý ·ñ³ýáõù c ¸ d¾m+m + 1 : êï³óí³í ·ý³ñ³ï³ï³ýá »ýã³ï³ ã¿ µ³ñ»é³íù³ý ¨ áý¹ñ³ýñ³óýáõù ¿ 1952-çý ¸çñ³ïç ïáõùçó ëï³óí³í c ¸ ± + 1 = d1 + 1 ·ý³ñ³ï³ï³ýá: î äëèííåéøèõ öèêëàõ ãðàôà â òåðìèíàõ ïîñëåäîâàòåëüíîñòè ñòåïåíåé âåðøèí ì. êóëàêçÿí àííîòàöèÿ ïóñòü ±=d1·d2·::: ·dn ïîñëåäîâàòåëüíîñòü ñòåïåíåé âåðøèí n-âåðøèííîãî ãðàôà g ñ ìèíèìàëüíîé ñòåïåíüþ ±. ×èñëî êîìïîíåíò ñâÿçíîñòè ãðàôà g îáîçíà÷àåòñÿ ÷åðåç m, äëèíà äëèííåéøåãî öèêëà ÷åðåç c, à ìèíèìàëüíîå ÷èñëî ñóìì ñòåïåíåé s íåçàâèñèìûõ âåðøèí -÷åðåç ¾s. â íàñòîÿùåé ðàáîòå äîêàçûâàåòñÿ, ÷òî â ëþáîì ãðàôå g, c ¸ d¾m+m + 1 . ïîëó÷åííàÿ îöåíêà íåóëó÷øàåìà è îáîáùàåò îöåíêó c ¸ ± + 1 = d1 + 1 äèðàêà (1952). microsoft word 15.doc mathematical problems of computer science 35, 109--115, 2011. 109 segmentation of mammography images enhanced by histogram equalization armen sahakyan institute for informatics and automation problems of nas of ra e-mail: armensahakyan@gmail.com abstract the mammography is the most effective procedure for an early diagnosis of the breast cancer. in this paper, a technique for detecting masses in mammographic images will be presented. image enhancement techniques, based on histogram equalization, are shown and used before image segmentation. threshold technique is proposed for an image segmentation which is a very critical task in any image processing. enhancement methods are implemented on a mammogram and accordingly, a comparison between the methods for better threshold is carried out. keywords: mammography, image enhancement, histogram equalization, image segmentation, threshold references: [1] l.-m. wun, r. m. merrill, and e. j. feuer, "estimating lifetime and age-conditional probabilities of developing cancer," lifetime data analysis, vol. 4, pp. 169-186, 1998. [2] "who cancer facts," http://www.who.int/cancer/en/, 2009. [3] swanson, g. m. (1992). breast cancer in the 1990's. journal of american medical women’s association, v. 47, p. 140-148. [4] d. e. stewart, et al., "attributions of cause and recurrence in long-term breast cancer survivors," psycho-oncology, vol. 10, pp. 179-183, 2001. [5] h. d. cheng and h. xu, "a novel fuzzy logic approach to mammogram contrast enhancement," information sciences, vol. 148, pp. 167-184, 2002. [6] shapiro, linda g. & stockman, george c. (2002). "computer vision". prentice hall. isbn 0-13-030796-3 [7] i. larrabide, a. a. novotny, r. a. feij´oo, and e. taroco, "a medical image enhancement algorithm based on topological derivative and anisotropic diffusion," in proceedings of the xxvi iberian latin-american congress on computational methods in engineering, guarapari, esp´ırito santo, brazil, 2005. [8] r. c. gonzalez and r. e. woods, digital image processing, 3 ed.: pearson prentice hall, 2007. [9] w. k. pratt, digital image processing. john wiley & sons, new york, 2001. [10] j. alex stark, “adaptive image contrast enhancement using generalizations of histogram equalization” ieee trans. on image processing, vol. 9, no. 5, pp. 889-896, may 2000. 110 [11] jin, yinpeng; fayad, laura m.; laine, andrew f., “contrast enhancement by multiscale adaptive histogram equalization”, proc. of wavelets: applications in signal and image processing ix, vol. no. 4478, pp. 206-213, dec. 2001. [12] s. m. pizer, et al., "adaptive histogram equalization and its variations," computer vision, graphics, and image processing, vol. 39, pp. 355-368, 1987. մամոգրաֆիկ պատկերների սեգմենտացիայի բարելավվումը հիստոգրամային հավասարեցման եղանակով ա. սահակյան ամփոփում մամոգրաֆիան կրծքի քաղցկեղի վաղ ախտորոշման համար արդյունավետ եղանակներից է: աշխատանքում ներկայացվում է մամոգրաֆիկ պատկերներում կուտակումների հայտնաբերման մեթոդը: ուսումնասիրվում են հիստոգրամային հավասարեցման վրա հիմնված պատկերների բարելավման մեթոդներ: այս մեթոդները օգտագործվում են մինչ պատկերի սեգմենտավորումը, որի համար առաջարկվում է շեմային մեթոդը: բերված են իրական մամոգրամների բարելավվման փորձնական արդյունքները և շեմային արդյունքների համեմատությունը: d:\sbornik\...\article.dvi mathematical problems of computer science 40, 34{38, 2013. on n eyman-p ear son p r inciple in m ultiple h ypotheses t esting e vg u e n i a . h a r o u t u n ia n a n d p a r a n d z e m m. h a ko b ya n institute for informatics and automation problems of nas ra e-mail: eghishe@ipia.sci.am, par h@ipia.sci.am abstract the aim of this paper is to newly generalize the classical neyman-pearson lemma to the case of more than two simple hypotheses. keywords: multiple hypotheses, optimal statistical test, error probability, neyman-pearson lemma. 1 . in t r o d u c t io n th e n e ym a n -p e a r s o n le m m a is t h e fo u n d a t io n o f t h e m a t h e m a t ic a l t h e o r y o f s t a t is t ic a l h yp o t h e s is t e s t in g . w e c a ll t h e s t a t is t ic a l h yp o t h e s is e a c h s u p p o s it io n s t a t e m e n t wh ic h m u s t b e ve r ī e d c o n c e r n in g t h e p r o b a b ilit y d is t r ib u t io n o f a n o b s e r va b le r a n d o m o b je c t . th e t a s k o f s t a t is t ic ia n is t o c o n s t r u c t a n a lg o r it h m ( t e s t ) fo r e ®e c t ive d e t e c t io n o f t h e h yp o t h e s is wh ic h is r e a liz e d . th e d e c is io n m u s t b e m a d e o n t h e b a s e o f ve c t o r o f r e s u lt s o f n in d e p e n d e n t id e n t ic a lly d is t r ib u t e d e xp e r im e n t s , c a lle d a s a m p le a n d d e n o t e d b y x 4 = ( x1; :::; xn; :::; xn ) , t h e e le m e n t s o f x n , wh e r e x is t h e s p a c e o f p o s s ib le r e s u lt s o f e a c h e xp e r im e n t . th e p r in c ip le o f n e ym a n -p e a r s o n p la ys a c e n t r a l r o le in b o t h t h e t h e o r y a n d p r a c t ic e o f s t a t is t ic s . th e r e e xis t s a va s t lit e r a t u r e wh e r e t h e t h e o r y o f t h e h yp o t h e s is t e s t in g a n d t h e n e ym a n p e a r s o n le m m a a r e e xp o u n d e d in d e t a il [1 ]{ [1 0 ]. th e p a r a d ig m o f n e ym a n -p e a r s o n is fr e qu e n t ly u s e d in d i®e r e n t a p p lic a t io n s [1 1 ]{ [1 3 ]. b u t t h e m o s t p a r t o f t h e s e t e xt s is d e d ic a t e d t o t h e c a s e o f t wo h yp o t h e s e s . th e p o s s ib ilit y o f e xt e n s io n o f fu n d a m e n t a l l e m m a t o t h e c a s e o f m u lt ip le h yp o t h e s e s is m e n t io n e d in [3 ]. s in c e t h e t e s t in g o f h yp o t h e s is is a c t u a l in a p p lic a t io n s we p r e s e n t o u r ve r s io n o f t h e l e m m a fo r t h e c a s e o f t h r e e , o r m o r e h yp o t h e s e s . th e id e a o f t h is s t u d y wa s fo r m u la t e d in [4 ]. 2 . p r o b le m s t a t e m e n t a n d r e s u lt fo r m u la t io n l e t p ( x ) b e t h e s p a c e o f a ll p r o b a b ilit y d is t r ib u t io n s ( p d s ) o n x . l e t x b e r v t a kin g va lu e s in x wit h o n e o f m c o n t in u o u s p d s gm 2 p ( x ) , m = 1 ; m. l e t t h e s a m p le x = 3 4 e. haroutunian, p. hakobyan 3 5 ( x1; :::; xn; :::; xn ) , xn 2 x , n = 1 ; n , b e a ve c t o r o f r e s u lt s o f n in d e p e n d e n t o b s e r va t io n s o f x. b a s e d o n d a t a s a m p le a s t a t is t ic ia n m a ke s a d e c is io n wh ic h o f t h e p r o p o s e d h yp o t h e s e s hm : g = gm, m = 1 ; m , is c o r r e c t . th e p r o c e d u r e o f d e c is io n m a kin g is a n o n -r a n d o m iz e d t e s t 'n ( x ) , it c a n b e d e ¯ n e d b y d ivis io n o f t h e s a m p le s p a c e x n o n m d is jo in t s u b s e t s am, m = 1 ; m. th e s e t am, m = 1 ; m, c o n s is t s o f ve c t o r s x fo r wh ic h t h e h yp o t h e s is hm is a d o p t e d . w e s t u d y t h e p r o b a b ilit ie s o f t h e e r r o n e o u s a c c e p t a n c e o f h yp o t h e s is hl p r o vid e d t h a t hm is t r u e ®ljm ( 'n ) 4 = gnm ( al ) = x x: x2a l gnmx ) ; m; l = 1 ; m; m 6= l: if t h e h yp o t h e s is hm is t r u e , b u t it is n o t a c c e p t e d t h e n t h e p r o b a b ilit y o f e r r o r is t h e fo llo win g : ®mjm ( 'n ) 4 = x l:l 6=m ®ljm ( 'n ) = 1 ¡ gnm ( am ) ; m = 1 ; m: fo r t h e g ive n p r e a s s ig n e d va lu e s 0 < ®¤1j1; ® ¤ 2j2; ::::; ® ¤ m¡1jm ¡1 < 1 we c h o o s e n u m b e r s t1, t2, ..., tm ¡1 a n d s e t s am, m = 1 ; m, s u c h t h a t a¤1 = ( x : m in ã g1 ( x ) g2 ( x ) ; :::; g1 ( x ) gm ( x) ! > t1 ) ; 1 ¡ gn1 ( a¤1 ) = ®¤1j1; a¤2 = a¤1 \ ( x : m in ã g2 ( x ) g3 ( x ) ; :::; g2 ( x ) gm ( x ) ! > t2 ) ; 1 ¡ gn2 ( a¤2 ) = ®¤2j2; :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: a¤m¡1 = a¤1 \ a¤2 \ ::: \ a¤m ¡2 \ ( x : gm¡1 ( x) gm ( x ) > tm ¡1 ) ; 1 ¡ gnm¡1 ( a¤m ¡1 ) = ®¤m¡1jm¡1; a n d a¤m = x n ¡ ( a¤1 [ a¤2 [ ::: [ a¤m ¡1 ) = a¤1 \ a¤2 \ ::: \ a¤m¡1: th e c o r r e s p o n d in g e r r o r p r o b a b ilit ie s a r e d e n o t e d b y ®¤ljm ( 'n ) ; m; l = 1 ; m ¡ 1 : t heor em: the test determined by the sets a¤1, a¤2, ....., a¤m is optimal in the sense that, for each other test de¯ned by the set b1, b2, ...., bm with the corresponding error probabilities ¯ljm; m; l = 1 ; m, if ¯1j1 · ®¤1j1; then m a x( ¯1j2; ¯1j3; :::; ¯1jm ) ¸ m a x( ®¤1j2; ®¤1j3; ::::; ®¤1jm ) ; if ¯2j2 · ®¤2j2; then m a x( ¯2j3; ¯2j4; :::; ¯2jm ) ¸ m a x( ®¤2j3; ®¤2j4; ::::; ®¤2jm ) ; ::::::::::::::::::::::::::::::::::::::::::::::: and if ¯m ¡1jm¡1 · ®¤m¡1jm¡1; then ¯m ¡1jm ¸ ®¤m¡1jm : fo r s im p lic it y o f fo r m u la t io n s we p r e s e n t t h e p r o o f o f t h e th e o r e m fo r m = 3 . 3 6 on neyman-pearson principle in multiple hypotheses testing in t h a t c a s e fo r t h e g ive n va lu e s 0 < ®¤1j1; ® ¤ 2j2 < 1 a n d c h o s e n n u m b e r s t1 a n d t2 s e t s a¤m, m = 1 ; 3 , a r e t h e fo llo win g : a¤1 = ( x : m in ã g1 ( x) g2 ( x) ; g1 ( x ) g3 ( x ) ! > t1 ) ; 1 ¡ gn1 ( a¤1 ) = ®¤1j1; a¤2 = a¤1 \ ( x : g2 ( x ) g3 ( x ) > t2 ) ; 1 ¡ gn2 ( a¤2 ) = ®¤2j2; a n d a¤3 = x n ¡ ( a¤1 [ a¤2 ) : th e c o r r e s p o n d in g e r r o r p r o b a b ilit ie s a r e d e n o t e d b y ®¤ljm ( 'n ) ; m; l = 1 ; 3 : w e m u s t p r o ve t h a t t h e t e s t d e t e r m in e d b y t h e s e t s a¤1, a¤2 a n d a¤3 is o p t im a l in t h e s e n s e t h a t , fo r e a c h o t h e r t e s t d e ¯ n e d b y t h e s e t s b1, b2 a n d b3 wit h t h e c o r r e s p o n d in g e r r o r p r o b a b ilit ie s ¯ljm, m; l = 1 ; 3 , if ¯1j1 · ®¤1j1, t h e n m a x( ¯1j2; ¯1j3 ) ¸ m a x( ®¤1j2; ®¤1j3 ) , a n d if ¯2j2 · ®¤2j2 t h e n ¯2j3 ¸ ®¤2j3. p r oof: l e t ©a ¤m a n d © b m b e t h e in d ic a t o r fu n c t io n s o f t h e d e c is io n r e g io n s a¤m a n d bm. fo r a ll x = ( x1; x2; :::; xn ) 2 x n , t h e fo llo win g in e qu a lit y is c o r r e c t ( ©a ¤1 ( x ) ¡ ©b 1 ( x ) ) ( g1 ( x ) ¡ m a x( t1g2 ( x ) ; t1g3 ( x) ) ) ¸ 0 : mu lt ip lyin g a n d t h e n s u m m in g o ve r x n we o b t a in ©a ¤1 ( x ) g1 ( x) ¡ ©a ¤1 ( x) m a x( t1g2 ( x) ; t1g3 ( x) ) ¡©b 1 ( x) g1 ( x ) + ©b 1 ( x) m a x( t1g2 ( x) ; t1g3 ( x ) ) ¸ 0 ; x x: x2x n h © a ¤1 ( x) g1 ( x) ¡ ©a ¤1 ( x) m a x( t1g2 ( x ) ; t1g3 ( x ) ) ¡© b 1 ( x ) g1 ( x) + © b 1 ( x) m a x( t g2 ( x) ; t1g3 ( x) ) ] ¸ 0 ; x x: x2a ¤1 [g1 ( x ) ¡ t1 m a x( g2 ( x ) ; g3 ( x) ) ] ¡ x x: x2b 1 [g1 ( x) ¡ t1 m a x( g2 ( x) ; g3 ( x ) ) ] ¸ 0 ; 1 ¡ ®¤1j1 ¡ t1 m a x( ®¤1j2; ®¤1j3 ) ¡ ( 1 ¡ ¯1j1 ) + t1 m a x( ¯1j2; ¯1j3 ) ¸ 0 ; ¡¯1j1 + ®¤1j1 · t1[¡ m a x( ®¤1j2; ®¤1j3 ) + m a x( ¯1j2; ¯1j3 ) ]: w e s e e n o w t h a t fr o m ¯1j1 · ®¤1j1, it fo llo ws t h a t m a x( ¯1j2; ¯1j3 ) ¸ m a x( ®¤1j2; ®¤1j3 ) . th e p r o o f o f t h e o t h e r c a s e is s im ila r . th e fo llo win g in e qu a lit y t a ke s p la c e fo r a ll x 2 x n ( ©a ¤2 ( x) ¡ ©b 2 ( x) ) ( g2 ( x) ¡ t2g3 ( x) ) ¸ 0 : mu lt ip lyin g a n d a ft e r t h a t s u m m in g o ve r x n we g e t © a ¤2 ( x) g2 ( x ) ¡ ©a ¤2 ( x ) t2g3 ( x ) ¡ ©b 2 ( x ) g2 ( x ) + ©b 2 ( x ) t2g3 ( x ) ¸ 0 ; x x: x2x n h ©a ¤ 2 ( x) g2 ( x ) ¡ ©a ¤ 2 ( x ) t2g3 ( x ) ) ¡ ©b 2 ( x ) g2 ( x ) + ©b 2 ( x ) t2g3 ( x ) i ¸ 0 ; e. haroutunian, p. hakobyan 3 7 x x: x2a ¤ 2 [g2 ( x) ¡ t2g3 ( x) ] ¡ x x: x2b 2 [g2 ( x ) ¡ t2g3 ( x ) ] ¸ 0 ; 1 ¡ ®¤2j2 ¡ t2®¤2j3 ¡ ( 1 ¡ ¯2j2 ) + t2¯2j3 ¸ 0 ; ¡¯2j2 + ®¤2j2 · t2 ( ¯¤2j3 ¡ ®¤2j3 ) : it is c le a r t h a t if ¯2j2 · ®¤2j2, t h e n ¯2j3 ¸ ®¤2j3. th e t h e o r e m is p r o ve d . 3 . co n c lu s io n in t h is p a p e r we g e n e r a liz e d n e ym a n -p e a r s o n c r it e r io n o f o p t im a lit y fo r m a n y c o n t in u o u s h yp o t h e s e s . w h e n d is t r ib u t io n s o f x a r e d is c r e t e t h e l e m m a c a n b e r e fo r m u la t e d wit h u s e o f r a n d o m iz a t io n a s it is n o t e d in [3 ], [4 ] a n d [7 ]. b a ye s ia n t e s t in g wa s c o n s id e r e d fo r t h e c a s e o f t wo a n d m o r e h yp o t h e s e s in [4 ], [5 ]. it is d e s ir a b le h a ve in t e n t io n t o c o n s id e r m u lt yh yp o t h e s e s b a ye s ia n t e s t in g fo r t h e m o d e l c o n s is t in g o f m a n y o b je c t s . a c kn o wle d g e m e n t th is wo r k wa s s u p p o r t e d in p a r t b y s cs o f me s o f r a u n d e r th e m a t ic p r o g r a m n o s cs 1 3 { 1 a 2 9 5 . refer ences [1 ] j. n e ym a n a n d e . s . p e a r s o n , \ on t h e p r o b le m o f t h e m o s t e ± c ie n t t e s t s o f s t a t is t ic a l h yp o t h e s e s " , p h il. tr a n s . r o y. s o c . l o n d o n , s e r . a , 2 3 1 , p p . 2 8 9 -3 3 7 , 1 9 3 3 . [2 ] j. n e ym a n , f irst course in p robability and statistics, h o lt , r in e h a r t a n d w in s t o n , n e w y o r k, 1 9 5 0 . [3 ] e . l . l e h m a n a n d j.p . r o m a n o , testing statistical hypotheses, th ir d e d it io n . s p r in g e r , n e w y o r k, 2 0 0 5 . [4 ] a . a . b o r o vko v, m athematical statistics, in r u s s ia n , n a u ka , mo s c o w, 1 9 9 7 . [5 ] h . l . v a n tr e e s , d etection, e stimation and m odulation theory, p a r t 1 . n e w y o r k: w ile y, 1 9 6 8 . [6 ] m. h . d e gr o o t , p robability and statistics, 2 n d e d ., r e a d in g , ma , a d d is o n -w e s le y, 1 9 8 6 . [7 ] m. g. k e n d a ll a n d a . s t u a r t , the advanced theory of statistics, 2 , in fe r e n c e a n d r e la t io n s h ip , th ir d e d it io n . h a fn e r p u b lis h in g c o m p a n y, l o n d o n , 1 9 6 1 . [8 ] a . k . b e r a , \ h yp o t h e s is t e s t in g in t h e 2 0 -t h c e n t u r y wit h a s p e c ia l r e fe r e n c e t o t e s t in g wit h m is s p e c i¯ e d m o d e ls " , in : \ s t a t is t ic s fo r t h e 2 1 -s t c e n t u r y. me t h o d o lo g ie s fo r a p p lic a t io n s o f t h e fit ir e " , ma r c e l d e kke r , in c ., n e w y o r k, b a s e l, p p . 3 3 -9 2 , 2 0 0 0 . [9 ] i. cs is z ¶a r a n d j. k äo r n e r , information theory: coding theorems for d iscrete m emoryless systems, a c a d e m ic p r e s s , n e w y o r k, 1 9 8 1 , ( r u s s ia n t r a n s la t io n , mir , mu s c o w, 1 9 8 5 ) . 3 8 on neyman-pearson principle in multiple hypotheses testing [1 0 ] t. m. co ve r a n d j. a . th o m a s , e lements of information theory, s e c o n d e d it io n . w ile y, n e w y o r k, 2 0 0 6 . [1 1 ] e . l e vit a n a n d n . me r h a v, \ a c o m p e t it ive n e ym a n -p e a r s o n a p p r o a c h t o u n ive r s a l h yp o t h e s is t e s t in g wit h a p p lic a t io n s " , ie e e transactions on information theory, vo l. 4 8 , n o . 8 , p p . 2 2 1 5 { 2 2 2 9 , 2 0 0 2 . [1 2 ] p . mo u lin , \ a n e ym a n -p e a r s o n a p p r o a c h t o u n ive r s a l e r a s u r e a n d lis t d e c o d in g " , isit, to r o n t o , ca n a d a , ju ly 6 -1 1 , 2 0 0 8 . [1 3 ] p .-n . ch e n , \ ge n e r a l fo r m u la s fo r t h e n e ym a n -p e a r s o n t yp e -ii e r r o r e xp o n e n t s u b je c t t o ¯ xe d a n d e xp o n e n t ia l t yp e -i e r r o r b o u n d s " , ie e e transactions on information theory, vo l. 4 2 , n o . 1 , p p . 3 1 6 { 3 2 3 , 1 9 9 6 . [1 4 ] c. c. l e a n g a n d d . h . jo h n s o n , \ on t h e a s ym p t o t ic s o f m-h yp o t h e s is b a ye s ia n d e t e c t io n " , ie e e transactions on information theory, vo l. 4 3 , n o . 1 , p p . 2 8 0 { 2 8 2 , 1 9 9 7 . [1 5 ] e . a . h a r o u t u n ia n , \ n e ym a n -p e a r s o n p r in c ip le fo r m o r e t h a n t wo h yp o t h e s e s " , abstracts of armenian m athematical union annual session d edicated to 90 anniversary of r afael alexandrian, y e r e va n , p p .4 9 { 5 0 , 2 0 1 3 . submitted 22.08.2013, accepted 10.10.2013. ü»ûù³ý-äçñëáýç ëï½µáõýùá µ³½ù³ïç í³ñï³íý»ñç ï»ëï³íáñù³ý í»ñ³µ»ñû³é º. ð³ñáõãûáõýû³ý ¨ ö. ð³ïáµû³ý ²ù÷á÷áõù ²ûë ³ßë³ï³ýùáõù ý»ñï³û³óí³í ¿ ü»ûù³ý-äçñëáýç ¹³ë³ï³ý é»ùù³ûç áý¹ñ³ýñ³óáõùá »ñïáõëçó ³í»éç í³ñï³íý»ñç í»ñ³µ»ñû³é: î ïðèíöèïå íåéìàíà-ïèðñîíà ïðè ïðîâåðêå ìíîãèõ ãèïîòåç å. àðóòþíÿí è ï. àêîïÿí àííîòàöèÿ öåëü íàñòîÿùåé ñòàòüè ïðåäñòàâèòü íîâîå îáîáùåíèå êëàññè÷åñêîé ëåììû íåéìàíà-ïèðñîíà äëÿ áîëåå ÷åì äâóõ ãèïîòåç. d:\sbornik\...\aram_13\aram.dvi mathematical problems of computer science 33, 95{101, 2010. on reliability appr oach to m ultiple h ypotheses t esting and i denti¯cation of p r obability distr ibutions of t wo stochastically coupled objects e vg u e n i h a r o u t u n ia n a n d a r a m y e s s a ya n institute for informatics and automation problems of nas of ra evhar@ipia.sci.am abstract this paper is devoted to logarithmically asymptotically optimal hypotheses testing and identi¯cation for a model consisting of two stochastically related objects. it is supposed that l1 possible probability distributions are known for the ¯rst object and the second object is distributed according to one of l1 £ l2 given conditional distributions depending on the distribution index and the current observed state of the ¯rst object. the matrix of interdependencies of all possible pairs of the error probability exponents in asymptotically optimal tests of distributions of both objects is studied. the identi¯cation of the distributions of two objects gives an answer to the question whether r1-th and r2-th distributions occurred, or not on the ¯rst and the second objects, correspondingly. refer ences [1 ] e . a . h a r o u t u n ia n , \ l o g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f m u lt ip le s t a t is t ic a l h yp o t h e s e s " , p roblems of control and information theory, vo l. 1 9 ( 5 -6 ) , p p . 4 1 3 { 4 2 1 , 1 9 9 0 . [2 ] r . f. a h ls we d e a n d e . a . h a r o u t u n ia n , \ on lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f h yp o t h e s e s a n d id e n t ī c a t io n " . l e c t u r e n o t e s in co m p u t e r s c ie n c e , vo l. 4 1 2 3 , \ ge n e r a l th e o r y o f in fo r m a t io n tr a n s fe r a n d co m b in a t o r ic s " , s p r in g e r , p p . 4 6 2 { 4 7 8 , 2 0 0 6 . [3 ] e . a . h a r o u t u n ia n , \ r e lia b ilit y in m u lt ip le h yp o t h e s e s t e s t in g a n d id e n t ī c a t io n " . p r o c e e d in g s o f t h e n a to a s i, y e r e va n 2 0 0 3 , n a to s c ie n c e s e r ie s , iii: co m p u t e r a n d s ys t e m s s c ie n c e s , vo l. 1 9 8 , ios p r e s s , p p . 1 8 9 { 2 0 1 , 2 0 0 5 . [4 ] e . a . h a r o u t u n ia n a n d p . m. h a ko b ya n , \ on l a o t e s t in g o f m u lt ip le h yp o t h e s e s fo r p a ir o f o b je c t s " , m athematical p roblems of computer science, vo l. x x v , p p . 9 2 { 1 0 0 , 2 0 0 6 . [5 ] e . a . h a r o u t u n ia n a n d p . m. h a ko b ya n , \ on id e n t ī c a t io n o f d is t r ib u t io n s o f t wo in d e p e n d e n t o b je c t s " , m athematical p roblems of computer science, vo l. x x v iii, p p . 1 1 4 { 1 1 9 , 2 0 0 7 . [6 ] e . a . h a r o u t u n ia n a n d p . m. h a ko b ya n , \ mu lt ip le h yp o t h e s e s l a o t e s t in g fo r m a n y in d e p e n d e n t o b je c t " , international journal \scholarly r esearch e xchange", 1 5 p p , 2 0 0 9 . 9 5 9 6 on reliability approach to hypotheses testing and idententi¯cation of distributions of two coupled objects [7 ] e . a . h a r o u t u n ia n a n d a . o. y e s s a ya n , \ on h yp o t h e s e s t e s t in g fo r t wo d i®e r e n t ly d is t r ib u t e d o b je c t s " . m athematical p roblems of computer science, vo l. x x v i, p p . 9 1 { 9 6 , 2 0 0 6 . [8 ] e . a . h a r o u t u n ia n a n d a . o. y e s s a ya n , \ on lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l h yp o t h e s is t e s t in g fo r p a ir o f s t a t is t ic a lly d e p e n d e n t o b je c t s " , m athematical p roblems of computer science, vo l. x x ix , p p . 9 7 { 1 0 3 , 2 0 0 7 . [9 ] e . a . h a r o u t u n ia n a n d n . a . gr ig o r ya n , \ on r e lia b ilit y a p p r o a c h fo r t e s t in g o f m a n y d is t r ib u t io n s fo r p a ir o f ma r ko v c h a in s " , m athematical p roblems of computer science, vo l. x x ix , p p . 8 9 { 9 6 , 2 0 0 7 . [1 0 ] a . o. y e s s a ya n , \ on r e lia b ilit y a p p r o a c h t o id e n t i¯ c a t io n o f p r o b a b ilit y d is t r ib u t io n s o f t wo s t a t is t ic a lly d e p e n d e n t o b je c t s " , m athematical p roblems of computer science, vo l. x x x ii, p p . 6 5 { 6 9 , 2 0 0 9 . [1 1 ] e . a . h a r o u t u n ia n a n d l . n a va e i, \ on o p t im a l id e n t ī c a t io n o f ma r ko v c h a in d is t r ib u t io n s u b je c t t o t h e r e lia b ilit y c r it e r ia " , m athematical p roblems of computer science, vo l. x x x ii, p p . 5 6 { 6 4 , 2 0 0 9 . [1 2 ] i. cs is z ¶a r a n d j. k äo r n e r , information theory: coding theorems for d iscrete m emoryless systems. n e w y o r k: a c a d e m ic p r e s s , 1 9 8 1 ( r u s s ia n t r a n s la t io n , mir , mo s c o w, 1 9 8 5 ) . [1 3 ] e . a . h a r o u t u n ia n , m. e . h a r o u t u n ia n , a n d a . n . h a r u t yu n ya n , \ r e lia b ilit y c r it e r ia in in fo r m a t io n t h e o r y a n d in s t a t is t ic a l h yp o t h e s e s t e s t in g " , f oundations and trends in communications and information theory, vo l. 4 , n o . 2 -3 , 2 0 0 8 . êïáë³ëïçïáñ»ý ï³ëû³é ûµû»ïïý»ñç ñ³í³ý³ï³ý³ûçý µ³ßëáõùý»ñç ýï³ïù³ùµ í³ñï³íý»ñç ëïáõ·ù³ý ¨ ýáõûý³ï³ý³óù³ý ñáõë³éçáõãû³ý ùáï»óù³ý ù³ëçý º. ². ð³ñáõãûáõýû³ý ¨ ². ú. ºë³û³ý ²ù÷á÷áõù ¸çï³ñïí³í »ý ëïáë³ëïçïáñ»ý ï³ëû³é »ñïáõ ûµû»ïïý»ñç ýï³ïù³ùµ í³ñï³íý»ñç ëïáõ·ù³ý ¨ ýáõûý³ï³ý³óù³ý ëý¹çñý»ñá: ²é³ççý ûµû»ïïá ï³ñáõ ¿ µ³ßëí³í éçý»é ïñí³í ñ³í³ý³ï³ý³ûçý µ³ßëáõùý»ñçó ù»ïáí, çëï »ñïñáñ¹áª ï³ëí³í ³é³ççýçó, ïñí³í å³ûù³ý³ï³ý ñ³í³ý³ï³ý³ûçý µ³ßëáõùý»ñçó ù»ïáí: àõëáõùý³ëçñí»é ¿ í³ñï³íý»ñç ûåïçù³é ï»ëï³íáñù³ý ¹»åùáõù »ñïáõ ûµû»ïïý»ñç ýï³ïù³ùµ ëë³éý»ñç ñ³í³ý³ï³ýáõãûáõýý»ñç óáõóçãý»ñç (ñáõë³éçáõãûáõýý»ñç) ÷áëï³ëí³íáõãûáõýá ¨ ëï³óí»é ¿ »ñïáõ ëïáë³ëïçïáñ»ý ï³ëû³é ûµû»ïïý»ñç ñ³í³ý³ï³ý³ûçý µ³ßëáõùý»ñç ³ëçùåïáïáñ»ý ûåïçù³é ýáõûý³ï³ý³óù³ý ëý¹ñç éáõíáõùá: d:\sbornik\...\tpel.dvi mathematical problems of computer science 36, 13-16, 2012. a n ote on i nter val e dge-color ings of gr aphs r a fa ye l r . k a m a lia n yz a n d p e t r o s a . p e t r o s ya n yx yinstitute for informatics and automation problems of nas of ra, zdepartment of applied mathematics and informatics, rau xdepartment of informatics and applied mathematics, ysu e-mail: rrkamalian@yahoo.com, pet petros@ipia.sci.am abstract an edge-coloring of a graph g with colors 1; : : : ; t is called an interval t-coloring if for all colors are used, and the colors of edges incident to each vertex of g are distinct and form an interval of integers. in this note we prove that if a connected graph g with n vertices admits an interval t-coloring, then t · 2n ¡ 3. we also show that if g is a connected r-regular graph with n vertices that has an interval t-coloring and n ¸ 2r + 2, then this upper bound can be improved to 2n ¡ 5. refer ences [1 ] a .s . a s r a t ia n , c.j. ca s s e lg r e n , \ on in t e r va l e d g e c o lo r in g s o f ( ®; ¯ ) -b ir e g u la r b ip a r t it e g r a p h s " , d iscrete m ath. 307, p p . 1 9 5 1 -1 9 5 6 , 2 0 0 6 . [2 ] a .s . a s r a t ia n , r .r . k a m a lia n , \ in t e r va l c o lo r in g s o f e d g e s o f a m u lt ig r a p h " , appl. m ath. 5, p p . 2 5 -3 4 , 1 9 8 7 . [3 ] a .s . a s r a t ia n , r .r . k a m a lia n , \ in ve s t ig a t io n o n in t e r va l e d g e -c o lo r in g s o f g r a p h s " , j . combin. theory ser. b 62, p p . 3 4 -4 3 , 1 9 9 4 . [4 ] m.a . a xe n o vic h , \ on in t e r va l c o lo r in g s o f p la n a r g r a p h s " , congr. numer. 159, p p . 7 7 -9 4 , 2 0 0 2 . [5 ] k . gia r o , m. k u b a le , m. ma la ¯ e js ki, \ co n s e c u t ive c o lo r in g s o f t h e e d g e s o f g e n e r a l g r a p h s " , d iscrete m ath. 236, p p . 1 3 1 -1 4 3 , 2 0 0 1 . [6 ] r .r . k a m a lia n , in t e r va l e d g e -c o lo r in g s o f g r a p h s , d o c t o r a l th e s is , n o vo s ib ir s k, 1 9 9 0 . [7 ] r .r . k a m a lia n , p .a . p e t r o s ya n , \ a n o t e o n u p p e r b o u n d s fo r t h e m a xim u m s p a n in in t e r va l e d g e -c o lo r in g s o f g r a p h s " , d iscrete m ath. 312, p p . 1 3 9 3 -1 3 9 9 , 2 0 1 2 . [8 ] p .a . p e t r o s ya n , \ in t e r va l e d g e -c o lo r in g s o f c o m p le t e g r a p h s a n d n-d im e n s io n a l c u b e s " , d iscrete m ath. 310, p p . 1 5 8 0 -1 5 8 7 , 2 0 1 0 . 1 3 1 4 a note on interval edge-colorings of graphs ¶ñ³éáõù ·ñ³ýý»ñç ùçç³ï³ûù³ûçý ïáõ³ûçý ý»ñïáõùý»ñç ù³ëçý è. ø³ù³éû³ý ¨ ä. ä»ïñáëû³ý ²ù÷á÷áõù g ·ñ³ýç ïáõ³ûçý ýñïáõùá 1 ; : : : ; t ·áõûýñáí ï³ýí³ý»ýù ùçç³ï³ûù³ûçý t–ý»ñïáõù, »ã» µáéáñ ·áõûý»ñá û·ï³·áñíí»é »ý ¨ ûáõñ³ù³ýãûáõñ ·³·³ãçý ïçó ïáõ»ñá ý»ñïí³í »ý ½áõû· ³é ½áõû· ï³ñµ»ñ ¨ ñ³çáñ¹³ï³ý ·áõûý»ñáí: ²ûë ³ßë³ï³ýùáõù ³å³óáõóíáõù ¿, áñ »ã» ·³·³ã³ýç g ï³å³ïóí³í ·ñ³ýá áõýç ùçç³ï³ûù³ûçý t–ý»ñïáõù, ³å³ t · 2 n¡ 3 : ü³¨ ³ßë³ï³ýùáõù óáõûó ¿ ïñíáõù, áñ »ã» n ·³·³ã³ýç g ï³å³ïóí³í r–ñ³ù³ë»é ·ñ³ýá áõýç ùçç³ï³ûù³ûçý t-ý»ñïáõù ¨ n ¸ 2 r + 2 , ³å³ t · 2 n ¡ 5 : çàìåòêà î èíòåðâàëüíûx ðåáåðíûx ðàñêðàñêàx ãðàôîâ ð. êàìàëÿí è ï. ïåòðîñÿí àííîòàöèÿ èíòåðâàëüíîé t-ðàñêðàñêîé ãðàôà g íàçîâåì ïðàâèëüíóþ ðàñêðàñêó ðåáåð g â öâåòà 1 ; :::; t ïðè êîòîðîé â êàæäûé öâåò i, 1 · i · t îêðàøåíî xîòÿ áû îäíî ðåáðî ãðàôà g, è ðåáðà, èíöèäåíòíûå êàæäîé âåðøèíå g, îêðàøåíû â ïîñëåäîâàòåëüíûå öâåòà. â íàñòîÿùåé ðàáîòå äîêàçàíî, ÷òî åñëè ñâÿçíûé ãðàô g ñ n âåðøèíàìè èìååò èíòåðâàëüíóþ t–ðàñêðàñêó, òî t · 2 n ¡ 3 . òàêæå â äàííîé ðàáîòå ïîêàçàíî, ÷òî åñëè ñâÿçíûé r-ðåãóëÿðíûé ãðàô g ñ n âåðøèíàìè èìååò èíòåðâàëüíóþ t-ðàñêðàñêó è n ¸ 2 r + 2 , òî t · 2 n ¡ 5 d:\sbornik\...\article.dvi mathematical problems of computer science 37, 7{16, 2012. upper and lower b ounds of b iometr ic i denti¯cation e-capacity ma r ia m e . h a r o u t u n ia n , l ilit a . te r -v a r d a n ya n a n d a r t h u r r . mu r a d ya n institute for informatics and automation problems of nas ra armar@ipia.sci.am, lilit@sci.am, mur¡art@yahoo.com abstract in this paper we introduce a new concept of e-capacity for biometric indenti¯cation system, which is the generalization of the capacity studied by willems et al [1]. we investigate this function by constructing upper and lower bounds. when e ! 0 we derive the lower and upper bounds of the channel capacity which coincides with the capacity obtained in [1]. keywords: biometric identi¯cation system, identi¯cation capacity, e-capacity bounds, error exponents. refer ences [1 ] f. w ille m s , t. k a lke r , j. go s e lin g , a n d j.-p . l in n a r t z , \ on t h e c a p a c it y o f a b io m e t r ic a l id e n t i¯ c a t io n s ys t e m " , international symposium on information theory, y o ko h a m a , ja p a n , p . 8 2 , 2 0 0 3 . [2 ] s . p a n ka n t i, r . m. b o lle a n d a . ja in , \ b io m e t r ic s -t h e fu t u r e o f id e n t ī c a t io n " , ie e e computer, vo l. 3 3 , n o . 2 , p p . 4 6 4 9 , fe b r u a r y, 2 0 0 2 . [3 ] t. ig n a t e n ko a n d f. w ille m s , " b io m e t r ic s e c u r it y fr o m a n in fo r m a t io n -t h e o r e t ic a l p e r s p e c t ive " , f oundations and trends in communications and information theory, vo l. 7 , n o 2 -3 , p p . 1 3 5 -3 1 6 , 2 0 1 2 . [4 ] e . a . h a r o u t u n ia n , \ on b o u n d s fo r e-c a p a c it y o f d mc" , ie e e transactions on information theory, vo l. 5 3 , n o . 1 1 , p p . 4 2 1 0 -4 2 2 0 , 2 0 0 7 . [5 ] e . a . h a r o u t u n ia n , m. e . h a r o u t u n ia n a n d a . n . h a r u t yu n ya n , " r e lia b ilit y c r it e r ia in in fo r m a t io n t h e o r y a n d in s t a t is t ic a l h yp o t h e s is t e s t in g " , f oundations and trends in communications and information theory, vo l. 4 , n o 2 -3 , p p . 9 7 -2 6 3 , 2 0 0 8 . [6 ] m. e . h a r o u t u n ia n , \ e s t im a t e s o f e-c a p a c it y a n d c a p a c it y r e g io n s fo r m u lt ip le -a c c e s s c h a n n e l wit h r a n d o m p a r a m e t e r " , l ecture notes in computer science, vo l. 4 1 2 3 , s p r in g e r v e r la g , p p . 1 9 6 -2 1 7 , 2 0 0 6 . [7 ] m. e . h a r o u t u n ia n , s . a . to n o ya n , \ r a n d o m c o d in g b o u n d o f in fo r m a t io n h id in g ec a p a c it y" , p roc. of ie e e international symposium on information theory, p . 5 3 6 , u s a , ch ic a g o , 2 0 0 4 . [8 ] t. m. co ve r a n d j. a . th o m a s , e lements of information theory, w ile y, n e w y o r k, 1 9 9 1 . 7 8 upper and lower bounds of biometric identi¯cation e-capacity [9 ] i. cs is z ¶a r a n d j. k äo r n e r , information theory: coding theorems for d iscrete m emoryless systems, a c a d e m ic p r e s s , n e w y o r k, 1 9 8 1 . [1 0 ] i. cs is z ¶a r , \ th e m e t h o d o f t yp e s " , ie e e transactions on information theory, vo l. 4 4 , n o . 6 , p p . 2 5 0 5 -2 5 2 3 , 1 9 9 8 . î»ýë³ã³÷³ï³ý ýáõûý³ï³ý³óù³ý ñ³ù³ï³ñ·ç e-áõý³ïáõãû³ý í»ñçý ¨ ëïáñçý ·ý³ñ³ï³ï³ýý»ñá ø. ð³ñáõãûáõýû³ý, è. î»ñ-ì³ñ¹³ýû³ý ¨ ². øáõñ³¹û³ý ²ù÷á÷áõù ðá¹í³íáõù ý»ñùáõííáõù ¿ ï»ýë³ã³÷³ï³ý ýáõûý³ï³ý³óù³ý ñ³ù³ï³ñ·ç eáõý³ïáõãû³ý ýáñ ñ³ëï³óáõãûáõýá, áñý áý¹ñ³ýñ³óýáõù ¿ ìçé»ùëç ¨ áõñçßý»ñç [1] áõëáõùý³ëçñ³í áõý³ïáõãû³ý ·³õ³÷³ñá: ð»ï³½áïíáõù ¿ ³û¹ ýáõýïóç³ý‘ í»ñçý ¨ ëïáñçý ·ý³ñ³ï³ï³ýý»ñç ï³éáõóù³ý ùççáóáí: ºñµ e ! 0 , ù»ýù ëï³ýáõù »ýù ï³åáõõáõ áõý³ïáõãû³ý í»ñçý ¨ ëïáñçý ·ý³ñ³ï³ï³ýý»ñá, áñáýù ñ³ùáýïýáõù »ý [1]áõù ëï³óí³í áõý³ïáõãû³ý ñ»ï: âåðõíÿÿ è íèæíÿÿ ãðàíèöû e-ïðîïóñêíîé ñïîñîáíîñòè áèîìåòðè÷åñêîé ñèñòåìû èäåíòèôèêàöèè ì. àðóòþíÿí, ë. òåð-âàðäàíÿí è à. ìóðàäÿí àííîòàöèÿ â ñòàòüå ââîäèòñÿ íîâîå ïîíÿòèå å-ïðîïóñêíîé ñïîñîáíîñòè äëÿ áèîìåòðè÷åñêîé ñèñòåìû èäåíòèôèêàöèè, êîòîðàÿ ÿâëÿåòñÿ îáîáùåíèåì ïðîïóñêíîé ñïîñîáíîñòè, èçó÷åííîé âèëåìñîì è äð. â [1]. ôóíêöèÿ èññëåäóåòñÿ ïóòåì ïîñòðîåíèÿ âåðõíåé è íèæíåé ãðàíèö. êîãäà e ! 0 , ìû ïîëó÷àåì âåðõíþþ è íèæíþþ ãðàíèöû ïðîïóñêíîé ñïîñîáíîñòè êàíàëà, êîòîðûå ñîâïàäàþò ñ ïðîïóñêíîé ñïîñîáíîñòüþ, ïîëó÷åííîé â [1]. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 44, 51--58, 2015. structure-based technique for object detecting in uav imagery david g. asatryan institute for informatics and automation problems of nas ra e-mail: dasat@ipia.sci.am abstract in this paper, the problem of automated detection of anomalies in the video, filmed in the controlled area on board of the uav is considered. the complexity of the problem associated with the effect of multiple factors on the quality of the image is pointed out. expediency of processing methods using the structural properties of the image, stable with respect to a number of noise and distortion is justified. we propose two approaches to use the structural properties of an image. the first approach is based on full hierarchical segmentation and simplification of the image and usage of the appropriate parameters of segments. the efficiency of the approach applied to the problem of detection of smoke and fire is shown. the second approach is based on modeling and analysis of the image gradient field. then a method of detection of anomalies in the image is proposed. a few examples illustrating the operation of the proposed procedures are considered and effectiveness of approaches is shown. keywords: uav, anomaly search, image structural properties, total segmentation, the gradient field, proximity measure. 1. introduction the problem of detecting anomalies arises in many areas of science and technology. for example, anomalies, tracked from space, could be oil spills on the sea surface, major forest fires, man-made pollution of the atmosphere, hidden plantations of narcotic plants among other crops, and others. in order to detect such objects unmanned aerial vehicles (uav) are also widely used, which are often almost the only means used for search and detection of objects of interest. 51 structure-based technique for object detecting in uav imagery 52 in this paper, we consider the problem of automated detection of anomalies in the video, filmed on uav in the controlled areas. the anomaly is understood in a broad sense, including relative to objects of both known and unknown images. as it is known, the video captured on board uavs simultaneously undergoes the variety of effects and distortions hampering efficient processing of information. the main distortion of an image is occurred due to the rapid changes of uav flight trajectory, resulting in an image of the same object to be different in the different frames. in addition, images of the same objects in the adjacent frames of a video sequence differ both in size and perspective, and the images themselves are fluctuating due to the vibration of uav board and other factors. therefore, for the efficient processing of information received uav, you must either create special approaches and methods or improve the existing ones. at the same time it should be noted that the effectiveness of digital image analysis depends largely on the available prior information about the properties of the image characterizing features of a scene and numerical values of its major parameters. on the other hand, since the analysis results ultimately are estimated by a person, the used methods should take into account the properties and capabilities of human visual system (hvs). in recent years, particular attention is paid to methods of analysis using structural properties and image features that are consistent to hvs decisions. under the structure of the image is commonly understood the set of existing standard elements in it and the links between them that allow making a judgment on the content of the considered scene. herewith, as structural (morphological) elements often the image homogeneous area, as well as the edges of homogeneous regions are considered. to find the structural elements they use a number of morphological operations to perform various image transformations [1]. there is also a huge variety of methods based on the use of different combinations of morphological operations, allowing analyzing the scene specifically for a particular purpose. it follows that, for the efficient processing of video, filmed on board the uav, it is advisable to apply the methods of image analysis which provide results that are independent or weakly dependent on the above type of interference. therefore, we restrict ourselves to those situations in which when exposed to this type of interference the structure of the image remains fairly stable. in this paper, we describe two methods of analysis based on using structural properties of an image, allowing detecting anomalies on images obtained from video filming by uav. results of application of both methods to real data are given. 2. two approaches to analyzing the structure of an image in this paper, we follow the statement that image structure which is perceived by hvs is basically identified via mutual dislocations of available segments and edges. recently, guided by this statement, we have proposed two approaches to the analysis of the image structure. a. total segmentation and simplification of image. the algorithm of total segmentation and description of the corresponding program system is done in [2]. let’s describe briefly the algorithm. d. asatryan 53 let i i( m,n )= be the pixels intensity matrix of a grayscale image, m , ,..., n ;= −0 1 1 n , ,...,m= −0 1 1, where { }i( m,n ) , ,...,∈ 0 1 255 . let the numbers { }j , ,...,θ ∈ 1 2 254 , j , ,...,l= +1 2 1 , { }l , ,...,∈ 1 2 254 satisfy the inequalities l...θ θ θ< < <1 2 . then the interval of intensities is divided into l +1 subintervals lt , t ,..., t +1 2 1 . call a set of pixels cs of the image i connected, if any pixel of that set has at least a “neighbor” from the same set. call a connected set of pixels cs the segment, if all pixels of that set have intensity belonging to the same subinterval from lt , t ,..., t +1 2 1 . let a segmentation algorithm ξ be chosen, and we have all segments ksss ,...,, 21 of an image i as a segmentation result. this means that а) k i i s s = = 1  , i , ,...,k ,= 1 2 б) i js s∩ = 0 , if i j≠ , i, j , ,...,k= 1 2 . then we say that the total segmentation is performed. total hierarchical segmentation means the sequential application of segmentation procedure to the given image at maxl , ,..., l= 1 2 , where maxl ≤ 254 . the procedure of substitution of intensities of all pixels of each segment for its average value is called an image simplification. the described procedure of total segmentation can easily be generalized to the case of color image. for that it is necessary to decompose the color image into components of the used color space, perform total segmentation of each color component and then the derived components convert to color image. the values of parameter l can differ for different components. thus, the result of total segmentation characterizes the image structure which is varying weakly at different values of parameter l . in other words, the number, dislocation and relative sizes of segments remain almost the same. this fact explains the hvs property to recognize the scene content at different sizes and orientations of an image. described technique was applied to the problems of recovery of damaged images [3] as well as in smoke or fire detection procedures using uav imagery [4]. 2.2. modeling and analyzing the gradient field of an image. the basis for usage of gradient field is the well known fact that hvs confidently recognizes the content of an image by the aggregate of available edges. methods of edge detection in an image are basically based on gradient field analysis using different mathematical methods [1]. one of the approaches to gradient field processing is its presentation as a two-dimensional random variable and creating certain statistical models for image structure investigation [5]. let h hg g (m,n)= and v vg g (m,n)= be the horizontal and vertical gradients of an image i correspondingly, and m m(m,n)= be the gradient magnitude, where n),(mg+n),(mg=n),m(m, 2v 2 h . (1) a. model for dominant orientation of an image is based on the usage of the term of orthogonal regression [6], the equation of which is as follows: ( ) ( )( ) ( )h h hv h h v v v v hv h h v v g g g g c µ ρ µ µ µ ρ σ σ σ σ  − − − − − + =  −    2 2 2 2 2 2 21 1 , (2) structure-based technique for object detecting in uav imagery 54 where hµ , vµ , hσ , vσ mathematical expectation and mse of random variables hg and vg , hvρ is the coefficient of correlation between them, c – is a constant. dominant orientation α of an image is determined as follows: ( ) hv h v h v hv * tg ρ α σ σ σ σ ρ = − + − + 22 2 2 2 2 2 4 . (3) models (1)-(3) were applied in different tasks for analyzing of images with different sizes and orientations [6, 7]. b. model for distribution of magnitude gradient. let’s assume that the gradient magnitude (1) is a random variable with weibull distribution density ,0,exp),;( 1 ≥               −      = − x xx xf ηη σσσ η ση (4) where 0>η shape parameter, 0>σ scale parameter. it must be noted that the fact of using only two parameters in this model has a significant advantage in certain tasks of image processing, connected with searching in big data bases [8, 9]. particularly, images similarity assessment measure is proposed in [5], which shows the proximity of images having gradient magnitudes with weibull distribution densities f ( x; , )η σ1 1 1 and ),;( 222 σηxf . the measure is determined by the formula as follows: ),max(),max( ),min(),min( 2121 21212 σσηη σσηη =w , 10 2 ≤< w , (5) where the gradients are calculated using sobel operator and the parameters in (5) are estimated by the method of moments. 3. technique for anomaly detection on an image. let us have an image of a scene on which the images of certain objects of unknown form, sizes and orientation can be there. a feature by which the object can be characterized is its difference from the surrounding space by structural properties. it is required to detect an area which contains the object and select it as clearly as possible. such statement of this problem has many uncertainties, but very often even an approximate solution of the specified task has a practical significance because the reliability of solution can be increased by multiple observations from different positions and trajectories of uav. the goal of the present investigation is to show that the information about structural properties of the image helps to solve the problem of anomaly detection by uav imagery. the algorithm of object searching includes the following steps. step 1. select on a scene a part of image which wittingly doesn’t contain any anomaly. the size of the part must exceed the size of the searching object a little. step 2. using sliding window consequently estimate the proximity measure w 2 between the chosen part (at step 1) and the current slide. step 3. select the current slide if w 2 exceeds the threshold given beforehand. the result of performance of these steps will differ from the surrounding space by structure. it can be noted that probabilities of true detecting and false alarm errors at the described procedure application significantly depends on signal/noise ratio and can exceed the admissible values. therefore, the detected anomalies must be investigated additionally by another method. d. asatryan 55 3. results of experiments some experiments were performed to detect anomalies of different types. results of two experiments are given below. experiment 1. let's suppose that a tank or another big machine is hidden in a forest, and we want to detect an anomalous part by uav imagery. in fig. 1a an image is given of size 307x214 from video frame which presumably contains an object. estimating the situation visually we can note that the image of tank and its nearest surroundings differ from the remaining part of the whole image. in fig. 1b (slightly increased for demonstrability) an image is shown of size 47x45 from other frame of video which doesn’t wittingly contain an anomaly. performing the above described procedure we have detected an anomaly, i.e., the part of the whole image with the lowest similarity with the chosen image of fig. 1b. in fig. 1c the binarized image of similarity map is shown, calculated using similarity measure w 2 , which allows to illustrate the corresponding part of the whole image with an anomaly object (i.e., hidden tank). fig. 1. results of searching for detecting an anomaly. avideo frame with hidden object, b – image part without anomaly, c – binarized map of proximity measure, d – the part of whole image with minimal values of proximity measure. experiment 2. searching of object of interest using google map web mapping service. here we solve the following problem. suppose that formerly someone took a picture of an urban quarter using uav (fig. 2a). later he wanted to find that territory using google map web mapping service for determination of more detailed information on that territory (more precise coordinates, dislocation of objects, etc.). suppose that the chosen part of google map (fig. 2b) contains the object of interest, but distance, orientation, perspective and other information regarding the picture of fig. 2a are unknown. the goal of this experiment is to find the object of interest in whole image of fig. 2b (i.e., a part which corresponds to the image of fig. 2a). a b c d https://en.wikipedia.org/wiki/web_mapping https://en.wikipedia.org/wiki/web_mapping https://en.wikipedia.org/wiki/web_mapping structure-based technique for object detecting in uav imagery 56 using the above described searching procedure with sliding window (by analogy with experiment 1) the part of map can be found which is maximally similar to the image of fig. 2a (see rectangular frame in fig. 2c). in fig. 2d the increased image of that frame is shown. it can be seen that the frame contains the necessary information though the scale and orientation differ from the object of interest. for further specification of scene the described procedure can be repeated for another part of map, which contains the object of interest more precisely. fig. 2. results of searching the object of interest on a map. a – chosen area of interest. b – part of google map presumably containing the object of interest. c – found part of map with maximal values of proximity measure. d – increased image of found part. a b c d d. asatryan 57 4. conclusions in this paper, a problem of automated detection of anomalies in the video, filmed in the controlled area on board of the uav is considered. due to complexity of the problem associated with the influence of multiple factors on the quality of uav imagery it is reasonable to use the structural properties of images which are stable with respect to a number of noise and distortion. the proposed approaches and procedures allow solving effectively the problem of detection of anomalies in images filmed by uav. the considered examples of applications show the effectiveness of the proposed approaches and procedures. references [1] r. c.gonzales and r.e. woods, digital image processing. 2-nd edition, prentice hall, 2002. [2] d. g. asatryan, g.s. sazhumyan and h. s. shahverdyan, “technique for coherent segmentation of image and applications”, mathematical problems of computer science, iiap, yerevan, armenia, vol. 28, pp. 88-93, 2007. [3] [online]. available: http://www.rau.am/cct/ [4] d. asatryan and s. hovsepyan, “vision based technique for smoke and fire detection in monitored forest terrain”, in proceedings of csit 2015, yerevan, pp. 129-132, 2015. [5] d. asatryan and k. egiazarian. “quality assessment measure based on image structural properties”, proc. of the international workshop on local and non-local approximation in image processing, finland, helsinki, pp. 70-73, 2009. [6] d. asatryan, k. egiazarian and v. kurkchiyan, “orientation estimation with applications to image analysis and registration”, international journal “information theories and applications”, vol. 17, no. 4, pp. 303-311, 2010. [7] d. asatryan, v. kurkchiyan and m. bagramyan. “a method for quality assessment of image resizing algorithms”, mathematical problems of computer science, vol. 36, pp. 128-132, 2012. [8] д. г. асатрян, в. в. куркчиян и л. р. харатян, “метод классификации текстур с использованием структурных характеристик изображения”, компьютерная оптика, n3, с. 574-579, 2014. [9] d. asatryan and m. zakaryan, “novel approach to content-based video indexing and retrieval by using a measure of structural similarity of frames”, international journal "information content and processing", vol. 2, no. 1, pp. 71-81, 2015. submitted 04.08.2015, accepted 25.11.2015 structure-based technique for object detecting in uav imagery 58 աթս տեսանկարահանված պատկերի կառուցվածքի հիման վրա օբյեկտի հայտնաբերման մեթոդ դ. ասատրյան ամփոփում անօդաչու թռչող սարքերը (աթս) լայնորեն կիրառվում են տնտեսական, բնապահպանական, ռազմական և այլ բնագավառներում: սույն աշխատանքը նվիրված է հսկվող տարածքում նկարահանման միջոցով օբյեկտներ հայտնաբերելու խնդրին, հենվելով պատկերի կառուցվածքային հատկությունների օգտագործման մեթոդաբանության վրա: առաջարկվել է պատկերի կառուցվածքը բնորոշող երկու մոդել, որոնցից մեկը կապված է պատկերի լրիվ հատվածավորման և պարզեցման արդյունքների օգտագործման, իսկ մյուսը՝ պատկերում առկա եզրագծերի և սահմանների բազմության վիճակագրական վերլուծության հետ: առաջարկվել են պատկերում ընդհանուր ֆոնից օբյեկտի՝ որպես անոմալ տիրույթի, առանձնացման ընթացակարգեր: մասնավորապես, դիտարկվել են ծխի և կրակի, ինչպես նաև google map համակարգից ստացված քարտեզների վրա նախօրոք գրանցված օբյեկտի հայտնաբերման օրինակներ: իրական տվյալների վրա ցույց է տրվել առաջարկված մոդելների և համապատասխան ավտոմատացված համակարգի արդյունավետությունը: метод обнаружения объекта по видеосъемке бла, основанный на использовании структуры изображения д. асатрян аннотация беспилотные летающие аппараты (бла) широко используются в народном хозяйстве, в задачах защиты природы, в военном деле и других областях. статья посвящена задаче обнаружения объектов по видеосъемкам бла на контролируемой местности, основываясь на методологии использования структурных свойств изображения. предложены две модели, характеризующие структуру изображения, одна из которых связана с использованием результатов полной сегментации и упрощения изображения, а другая основана на статистическом анализе множества имеющихся в изображении краев и границ. предложена процедура обнаружения объекта на общем фоне – как аномалии в изображении. в частности, рассмотрены примеры обнаружения дыма и огня, а также обнаружения предварительно регистрированного объекта по карте, полученной при помощи системы google map. на реальных данных показана эффективность предложенных моделей и автоматизированной системы обнаружения объектов. microsoft word tpel.doc mathematical problems of computer science 33, 121--126, 2010. 121 linear cryptanalysis of block ciphers in the cluster computational environment melsik kyuregyan, ofelya manukyan and edita harutyunyan institute for informatics and automation problems of nas of ra e-mail: melsik@ipia.sci.am, manofa81@yahoo.com abstract this paper presents some results concerning synthesis of new cryptosystems equivalent to safer+ and safer++ to perform their linear cryptanalysis in the cluster computational environment. a parallel software package “linearcryptanalyser” is developed to find such "armenian shuffles" which were chosen as secure against differential cryptanalysis and now will be checked if they are also secure against linear cryptanalysis. the research is focused on both theoretical and practical aspects of existence of linked i/o sums. the software package “linearcryptanalyser” analyzes the existence of linked i/o sums and the absence of such sums will indicate cryptоresistance of block ciphers against last-round attack. . references 1. j. l. massey, g. h. khachatrian and m. k. kuregian, ``nomination of safer+ as candidate algorithm for the advanced encryption standard (aes)”, submission document from cylink corporation to nist, june 1998. 2. j. l. massey, g. h. khachatrian and m. k. kuregian, ``nomination of safer++ as candidate algorithm for the new european schemes for signatures, integrity, and encryption (nessie)”, submission document from cylink corporation, 2000. 3. c. harpes, ``cryptanalysis of iterated block ciphers”, eth series in information processing, editor: james l. massey. v. 7, hartung-gorre verlang konstanz, 1996. 4. c. harpes, g. g. kramer and j. l. massey, ``a generalization of linear cryptanalysis and the applicability of matsui’s piling-up lemma”, presented at eurocrypt ’95. 5. c. harpes, ``a generalization of linear cryptanalysis applied to safer”, signal and info. proc. lab., ch-8092 zurich, march 9, 1995. 122 linear cryptanalysis of block ciphers in the cluster computational environment ´éáï³ûçý í³íï³·ñù³ý ñ³ù³ï³ñ·»ñç ·í³ûçý í»ñí³ýáõùá ïé³ëï»ñ³ûçý ñ³ßíáõ³ï³ý ñ³ù³ï³ñ·áõù ø. îûáõñ»õû³ý, ú. ø³ýáõïû³ý ¨ ¾. ð³ñáõãûáõýû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ýï³ñ³·ñí³í »ý ³ñ¹ûáõýùý»ñ safer+ ¨ safer++ µéáï³ûçý í³íï³·ñù³ý ñ³ù³ï³ñ·»ñç ýáñ ï³ñµ»ñ³ïç ï³éáõóù³ý í»ñ³µ»ñû³é: êï»õíí»é ¿ ½áõ·³ñ»é ñ³ßí³ñïý»ñç “linearcryptanalyser” íñ³·ñ³ß³ñ, áñç û·ýáõãû³ùµ ÷ýïñíáõù »ý ³ûýåçëç "armenian shuffle" ïááñ¹çý³ï³ûçý ï»õ³÷áëáõãûáõýý»ñ, áñáýó ñ³ù³å³ï³ëë³ý µéáï³ûçý í³íï³·ñù³ý ñ³ù³ï³ñ·»ñá ïéçý»ý ï³ûáõý ¹çý»ñ»ýóç³é ¨ ·í³ûçý í»ñéáõíáõãûáõýý»ñç ýï³ïù³ùµ: ð»ï³½áïáõãûáõýý»ñá ï³ï³ñí»é »ý ï³å³ïóí³í ùáõïù/»éù ·áõù³ñý»ñç ·áûáõãû³ý çýãå»ë ï»ë³ï³ý, ³ûýå»ë ¿é ïçñ³é³ï³ý ï»ë³ýïûáõýý»ñçó: “linearcryptanalyser” ½áõ·³ñ»é ñ³ßí³ñïý»ñç íñ³·ñ³ß³ñá ãáõûé ¿ ï³éçë áõëáõùý³ëçñ»é ï³å³ïóí³í ùáõïù/»éù ·áõù³ñý»ñç ·áûáõãû³ý ñ³ñóá, çëï ýù³ý ï³å³ïóí³í ·áõù³ñý»ñç µ³ó³ï³ûáõãûáõýá íï³ûáõù ¿ áõëáõùý³ëçñíáõ µéáï³ûçý í³íï³·ñù³ý ñ³ù³ï³ñ·»ñç ï³ûáõýáõãû³ý ù³ëçý ·í³ûçý ïñçåïá³ý³éç½ç ýï³ïù³ùµ: microsoft word tpel.doc математические вопросы кибернетики и вычислительной техники 33, 201--208, 2010. 201 информационные особенности геохимии хрома в присеванском офиолитовом поясе армении артак э. антонян и марина р. геворкян ереванский государственный университет аннотация в статье с приминением статистических приемов расмотрено поведение хрома в породах офиолитовой ассоцации присеванского пояса. распределение cr в гипербазитах и базитах пояса сравнимо с признанными кларками. хром в виде cr3+ главным образом скоцентрирован в гипербазитах– перидотитах, дунитах и серпентинитах. средние содержания расчитаны методом вариационной статистики и показаны в виде гистограмм. с помощью модуля 3d – analyst arcview построена геохимическая карта распределения cr на территории северо-восточного берега озера севан. литература [1] с. б. абовян, геология и полезные ископаемые северо-восточного побережья озера севан. изд. ан армсср. ер. 1961. [2] а. п. виноградов , “среднее содержание химических элементов в главных типах изверженных горных пород земной коры.” ж. геохимия, № 7, 1962. [3] р. г геворкян и м. р. геворкян , “офиолитовая палеокеаническая кора армении (южный кавказ)”, ер.: геоид, 2003. [4] k. k. turekian and k.h. wedepohl , “distibution of the elements in some majorunits of the earth’s crust” geological society of armenia. № 72, 1961. քրոմի երկրաքիմիûç çý áñù³óçáý ³é³ýóý³ñ³ïïáõãûáõýý»ñá մերձսևանյան օֆիոլիտային գոտու ապարներում ².²ýïáýû³ý ¨ ø. ¶¨áñ·û³ý ²ù÷á÷áõù ìç×³ï³·ñ³ï³ý »õ³ý³ïý»ñáí áõëáõùý³ëçñí³í ¿ ùñáùç í³ñùá ø»ñó-ê¨³ýû³ý ûýçáéçï³ûçý ·áïáõ ³å³ñý»ñáõù: øñáùç µ³ßëáõùá ·áïáõ ñçå»ñµ³½çïý»ñáõù ¨ µ³½çïý»ñáõù ñ³ù³ýù³ý ¿ ñ³ûïýç ïé³ñïý»ñçý: øñáùá cr3+ çáýç ï»ëùáí ñçùý³ï³ýáõù ï»ýïñáý³óí³í ¿ ñçå»ñµ³½çïý»ñáõù` å»ñç¹áïçïý»ñáõù, ¹áõýçïý»ñáõù ¨ ë»ñå»ýïçýçïý»ñáõù: øñáùç ùçççý å³ñáõý³ïáõãûáõýý»ñá ñ³ßí³ñïí³í »ý í³ñç³óçáý ëï³ïçëïçï³ûç »õ³ý³ïáí ¨ ý»ñï³û³óí³í »ý ñçëïá·ñ³ùý»ñç ï»ëùáí: 3d – analyst arcview ùá¹áõéç û·ýáõãû³ùµ ï³½ùí³í ¿ ê¨³ý³ é×ç ñûáõëçë-³ñ¨»éû³ý ³÷áõù ùñáùç µ³ëßù³ý »ñïñ³ùçùç³ï³ý ù³ñï»½á: 13_rafayel_barseghyan.dvi mathematical problems of computer science 39, 101{105, 2013. sear ching i mages by using color str uctur e r a fa ye l v . b a r s e g h ya n institute for informatics and automation problems of nas ra e-mail: barseghyan@gmail.com abstract in this paper a color-structured image search method is proposed which allows to classify large image collections by color and quickly ¯nd the interested images. according to the proposed method a software system is developed that demonstrates the e±ciency and accurancy of the proposed method. keywords: image search, color structure, visual similarity. 1 . in t r o d u c t io n in r e c e n t ye a r s im a g e s e a r c h e n g in e s , s u c h a s go o g le im a g e s [1 ], y a n d e x im a g e s [2 ], b in g im a g e s [3 ],tin e ye [4 ], p ic s e a r c h [5 ] a n d m a n y o t h e r s , a r e b e in g r a p id ly d e ve lo p e d . th e s e s e a r c h e n g in e s a llo w t o ¯ n d t h e d e s ir e d im a g e s b y u s in g va r io u s c r it e r ia s u c h a s c r e a t io n d a t e s , o r ie n t a t io n , s iz e , c o lo r a n d vis u a l s im ilia r it y ( d u p lic a t e s ) . me t h o d s fo r c o n t e n t b a s e d im a g e s e a r c h in g c a n b e d ivid e d in t o t wo m a jo r t yp e s : vis u a l r e r a n kin g [6 ], [7 ] a n d p r e d ic t io n -b a s e d r e ¯ n e m e n t [8 ]. in t e n t s e a r c h [8 ] p r o vid e s a m e c h a n is m t o a llo w t h e u s e r s t o s e le c t a fe w im a g e s o f in t e r e s t , a n d a u t o m a t ic a lly m in e t h e u s e r 's in t e n t t o r e o r d e r im a g e s e a r c h r e s u lt s . in t h is p a p e r , a s im p le m e t h o d fo r ¯ n d in g im a g e s b a s e d o n e xp lic it in t e n t is p r e s e n t e d . th e d e ve lo p e d s o ft wa r e a llo ws t o ¯ n d t h e d e s ir e d im a g e s b y u s in g t h e p e r c e n t a g e o f t h e d e s ir e d c o lo r s . in [9 ] a n d [1 0 ] yo u c a n s e e a s ys t e m wh ic h is s im ilia r t o t h e s u g g e s t e d m e t h o d . [9 ] wa s d e ve lo p e d a n d s u p p o r t e d b y id e e l a b s [1 1 ] ca n a d ia n -b a s e d c o m p a n y wh ic h o wn e d tin e ye [4 ]. mu lt ic o lo r s e a r c h l a b [9 ] m a ke s c o lo r -b a s e d im a g e s e a r c h wit h in m o r e t h e n 1 0 m illio n im a g e s fr o m flic kr [1 2 ] wh ic h h a s a lic e n s e o f cr e a t ive co m m o n s . [9 ] h a s a c o n t r o l o f d e s ir e d c o lo r s c o n c e n t r a t io n in im a g e s . th e u s e r s c a n s e le c t u p t o 5 c o lo r s a n d c o n t r o l c o lo r c o n c e n t r a t io n b a la n c e . a s a c o n s e qu e n c e o f in c r e a s e in c o n c e n t r a t io n p e r c e n t o f o n e c o lo r t h e o t h e r c o lo r s c o n c e n t r a t io n p e r c e n t d e c r e a s e s . in [9 ] ( s e e fig u r e 1 ) b a s e c o lo r s p a le t t e u s e d c o n t a in s 2 5 6 c o lo r s . in [1 3 ] u s e d p a le t t e c o n t a in s 2 8 c o lo r s , go o g le 's ( s e e fig u r e 2 ) , b in g 's a n d p ic s e a r c h 's c o lo r s p a le t t e c o n t a in s 1 2 c o lo r a n d y a n d e x's ( s e e fig u r e 3 ) p a le t t e c o n t a in s 9 c o lo r s . 2 . me t h o d in t h e ¯ r s t s t e p o f o u r a lg o r it h m we n e e d t o c o r r e s p o n d e a c h p ixe l c o lo r o f e ve r y im a g e t o o u r c o lo r fr o m o u r b a s e c o lo r s p a le t t e . a s b a s e c o lo r s t h e go o g le 's c o lo r p a le t t e is u s e d . fo r 1 0 1 1 0 2 searching images by using color structure t h is we n e e d t o g e t r gb va lu e s o f e a c h p ixe l in o u r c o lo r p a le t t e . a ft e r t h a t we n e e d t o ¯ n d t h e c lo s e s t va lu e o f c o lo r in r gb s ys t e m fo r e a c h p ixe l. a s a d is t a n c e m e t r ic e m e a n s qu a r e e r r o r is u s e d . th e m a t h e m a t ic a l fo r m u la is p r e s e n t e d b e lo w: d = ( pixelpal:rgb:r ¡ pixelimg:rgb:r ) 2 + ( pixelpal:rgb:g ¡ pixelimg:rgb:g ) 2 + + ( pixelpal:rgb:b ¡ pixelimg:rgb:b ) 2 ( 1 ) d¡ > min wh e r e pixelpal:rgb is a r g b va lu e o f p a le t t e c o lo r a n d pixelpal:rgb:r, pixelpal:rgb:g a n d pixelpal:rgb:b a r e , r e s p e c t ive ly, t h e r e d , g r e e n a n d b lu e va lu e s o f r g b . 3 . r e a liz a t io n th is s e c t io n is d ivid e d in t o fo u r lo g ic a l p a r t s : ge t t in g t h e im a g e s fr o m flic kr , in d e xin g ( a n a lyz in g ) t h e im a g e s , s t o r in g t h e in d e xe d r e s u lt s a n d s e a r c h in g wit h in t h e s t o r e d d a t a . ² th e ¯ r s t s t e p in c o r p o r a t e s m in in g flic kr fo r im a g e s . fo r t u n a t e ly, flic kr p r o vid e s a wa y o f d o in g t h is : it a llo ws t h e s o ft wa r e o f o u t s id e u s e r s t o in t e r a c t wit h t h e ir a p i [1 4 ]. w e h a ve c h o s e n a ll im a g e s wh ic h a r e t a g g e d a s " a r m e n ia " a n d lic e n s e d a s " cr e a t ive co m m o n s " . flic kr s e a r c h r e t u r n e d 1 6 2 5 0 im a g e s m e e t in g t h e s e c r it e r ia . fo r d o in g t h is wit h flic kr a p i we n e e d t o s ig n u p a s a flic kr d e ve lo p e r in o r d e r t o r e t r ie ve t h e " a p i k e y " . on e o f t h e lim it a t io n s o f t h e flic kr a p i is t h a t o n e c a n o n ly r e t r ie ve a m a xim u m o f 5 0 0 im a g e s p e r p a g e fo r a s p e c ī e d s e a r c h t e r m . it is in t e r e s t in g fa c t t h a t t h e d o wn lo a d e d im a g e s c o n t a in o n ly 3 9 im a g e s in gif fo r m a t a n d 4 3 im a g e s in p n g fo r m a t , t h e o t h e r s a r e c o d e d b y jp e g fo r m a t . ² a ft e r d o wn lo a d in g t h e r e qu ir e d im a g e s we s t a r t t o a n a lys e t h e m . fo r d o in g t h a t we h a ve wr it t e n a n in d e xe r . th e in d e xe r wa s wr it t e n b y u s in g im a g e ma g ic k [1 5 ]. im a g e ma g ic k is a wid e ly kn o wn s o ft wa r e lib r a r y t o c r e a t e , e d it , c o m p o s e , o r c o n ve r t b it m a p im a g e s . it c a n r e a d a n d wr it e im a g e s in a va r ie t y o f fo r m a t s ( o ve r 1 0 0 ) ( t h e fu ll lis t o f t h e s u p p o r t e d fo r m a t s o n e c a n s e e in [1 6 ]) . w e c a lc u la t e t h e p e r c e n t s o f p a le t t e 's c o lo r s in g ive n im a g e . w e t a ke o n ly t h a t c o lo r s wh ic h h a ve m o r e t h a n o n e p e r c e n t e xis t a n c e in t h e g ive n im a g e . ² th e m o s t c r it ic a l p a r t o f o u r s ys t e m is s t o r in g in fo r m a t io n o f t h e a n a lys e d im a g e s . w e c a n u s e c la s s ic s ql d a t a s t o r e s u c h a s s ql it e , mys ql o r p o s t g r e s ql , b u t fo r h a vin g m o r e s p e e d wh e n d o in g s e a r c h we c h o o s e n e w t yp e s o f d a t a s t o r e r e d is [1 7 ]. r e d is is a s c h e m e le s s , fa s t a n d r e lia b le d a t a b a s e e n g in e . r e d is s t a n d s o u t fr o m m a n y o t h e r n o s ql d a t a b a s e s wit h it s p o we r fu ll d a t a t yp e s . fo r o u r s ys t e m t h e b e s t c h o ic e o f d a t a t yp e is t h e " s o r t e d s e t s " . w e s t o r e t h e a n a lys e d d a t a a s color id percent img id wh e r e img id -is t h e id e n t ifa c t io n n u m b e r o f im a g e , color id is t h e id e n t ifa c t io n n u m b e r o f c o lo r fr o m p a le t t e a n d percent is t h e p e r c e n t o f color id in t h e g ive n im a g e . if t h e g ive n c o lo r d o e s n 't e xis t in t h e g ive n im a g e we s t o r e 0 . r. barseghyan 1 0 3 ² fo r s e a r c h in g im a g e s wit h t h e r e qu ir e d c o n c e n t r a t io n o f c o lo r s we wr it e qu e r ie s t o o u r d a t a b a s e . fo r g e t t in g im a g e s wh ic h c o n t a in t h e r e qu ir e d p e r c e n t s o f t h e d e s ir e d c o lo r we n e e d t o o p e r a t e o n s c o r e s . w e c a n d o t h is b y u s in g t h e " z r a n g e s c o r e " o p e r a t o r . th e o p e r a t o r " z r a n g e s c o r e " r e t u r n s a s e t . zrangebyscore color id percent < input percent a ft e r g e t t in g t h e r e qu ir e d s e t s fo r a ll d e s ir e d c o lo r s we d o s e t s in t e r s e c t io n . to e xc lu d e c o lo r s we n e e d t o d o a d d it io n a l qu e r y like t h is zrangebyscore color id percent ¡ inf 0 4 . co n c lu s io n th e d e ve lo p e d s o ft wa r e s ys t e m c a n h e lp t h e o wn e r s o f la r g e im a g e a r c h ive s t o e a s ily ¯ n d t h e d e s ir e d im a g e s . to g e t m o r e r e le va n t im a g e s t h e s ys t e m s u p p o r t s m u lt ic o lo r s e a r c h a n d a llo ws t h e u s e r t o in c lu d e / e xc lu d e c o lo r s a n d c o n t r o l p e r c e n t a g e c o lo r o f t h e d e s ir e d im a g e . th e d e ve lo p e d s o ft wa r e s ys t e m wo r ks fa s t , fo r e xa m p le , t h e s e a r c h p r o c e s s in o u r c o lle c t io n wit h 1 6 2 5 0 im a g e s t a ke s o n ly 0 ,0 7 s e c o n d s . refer ences [1 ] h t t p :/ / im a g e s .g o o g le .c o m [2 ] h t t p :/ / im a g e s .ya n d e x.r u [3 ] h t t p :/ / b in g .c o m / im a g e s [4 ] h t t p :/ / t in e ye .c o m [5 ] h t t p :/ / p ic s e a r c h .c o m [6 ] w . h . h s u , l . s . k e n n e d y, a n d s .-f. ch a n g , \ r e r a n kin g m e t h o d s fo r vis u a l s e a r c h ," ie e e m ultim edia, vo l. 1 4 , n o . 3 , p p .1 4 -2 2 , 2 0 0 7 . [7 ] y . jin g a n d s . b a lu ja , " p a g e r a n k fo r p r o d u c t im a g e s e a r c h . in w w w ," w w w 2 0 0 8 , b e ijin g , ch in a , p p . 3 0 7 3 1 6 , 2 0 0 8 . [8 ] j. cu i, f. w e n , a n d x . ta n g , \ in t e n t s e a r c h : in t e r a c t ive o n -lin e im a g e s e a r c h r e r a n kin g ," p roc. 16th acm int'l conf. m ultimedia, p p . 9 9 7 -9 9 8 , 2 0 0 8 . [9 ] h t t p :/ / la b s .t in e ye .c o m / m u lt ic o lr [1 0 ] h t t p :/ / la b s .t in e ye .c o m / m u lt ic o lo u r [1 1 ] h t t p :/ / id e e in c .c o m [1 2 ] h t t p :/ / ° ic kr .c o m [1 3 ] h t t p :/ / d r ib b b le .c o m [1 4 ] h t t p :/ / ° ic kr .c o m / s e r vic e s / a p i/ [1 5 ] h t t p :/ / im a g e m a g ic k.o r g [1 6 ] h t t p :/ / im a g e m a g ic k.o r g / s c r ip t / fo r m a t s .p h p [1 7 ] h t t p :/ / r e d is .io 1 0 4 searching images by using color structure figure 1: tineye multicolr search lab interface. figure 2: google's image search interface. figure 3: yandex's image search interface. submitted 16.01.2013, accepted 25.02.2013. r. barseghyan 1 0 5 ä³ïï»ñý»ñç áñáýáõù áëï ·áõý³ûçý ñ³ïï³ýçßý»ñç è. ´³ñë»õû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ³é³ç³ñïí³í ¿ ù»ãá¹, áñá ãáõûé ¿ ï³éçë áñáý»é å³ïï»ñý»ñª û·ï³·áñí»éáí ¹ñ³ýó ·áõý³ûçý ñ³ïï³ýçßý»ñá: øß³ïí³í ù»ãá¹á ïû·ýç å³ïï»ñý»ñç ù»í ñ³í³ù³íáõ áõý»óáõý»ñçý ñ»ßïáõãû³ùµ ·ïý»é å³ñ³ýçíáõ å³ïï»ñý»ñá: øß³ïí»é ¿ íñ³·ñ³ûçý ñ³ù³ï³ñ·, áñáª áñå»ë ÷ýïñù³ý å³ñ³ù»ïñ, û·ï³·áñíáõù ¿ ó³ýï³éç ·áõûýç (·áõûý»ñç) ³éï³ûáõãûáõýá/ µ³ó³ï³ûáõãûáõýá: ð³ù³ï³ñ·á ãáõûé ¿ ï³éçë ï³é³í³ñ»é å³ñ³ýçíáõ ·áõûýç (·áõûý»ñç) ³éï³ûáõãû³ý ùçç³ï³ûùá: ð³ù³ï³ñ·á ÷áñó³ñïí»é ¿ û·ï³ï»ñ»ñç ïáõùçó flickr ¿é»ïïñáý³ûçý é»ëáõñë í»ñµ»éýí³í “armenia” åçï³ïáí ýß³·ñí³í ¨ “creative commons” éçó»ý½ç³ áõý»óáõ ýï³ñý»ñç µ³½ùáõãû³ý íñ³: ïîèñê èçîáðàæåíèÿ ïî öâåòîâûì õàðàêòåðèñòèêàì ð. áàðñåãÿí àííîòàöèÿ â ðàáîòå ïðåäëîæåí ìåòîä äëÿ ïîèñêà èçîáðàæåíèé ïî öâåòîâûì õàðàêòåðèñòèêàì. ðàçðàáîòàííûé ìåòîä ïðèçâàí ïîìî÷ü âëàäåëüöàì áîëüøèõ àðõèâîâ èçîáðàæåíèé áûñòðî è âèçóàëüíî ïðîñòî íàéòè íóæíûå èçîáðàæåíèÿ. â ñîîòâåòñòâèè ñ ïðåäëîæåííûì ìåòîäîì áûëî ðàçðàáîòàíî ïðîãðàììíîå îáåñïå÷åíèå, êîòîðîå ïîçâîëÿåò èñêàòü íóæíûå èçîáðàæåíèÿ êàê ïî âõîæäåíèþ îïðåäåëåííîãî öâåòà (îïðåäåëåííûõ öâåòîâ), òàê è èñêëþ÷åíèþ îïðåäåëåííîãî öâåòà (îïðåäåëåííûõ öâåòîâ), à òàêæå ïîçâîëÿåò çàäàòü ïðîöåíò âõîæäåíèÿ öâåòà (öâåòîâ). ðàçðàáîòàííîå ïðîãðàììíîå îáåñïå÷åíèå áûëî òåñòèðîâàíî íà áàçå èçîáðàæåíèé, êîòîðûå áûëè çàãðóæåíû ïîëüçîâàòåëÿìè â èíòåðíåò ñåðâèñ flickr è áûëè ïîìå÷åíû ìåòêîé ”armenia” è èìåëè ëèöåíçèþ ”creative commons”. d:\sbornik\...\tpel_new.dvi mathematical problems of computer science 35, 19{25, 2011. upper b ounds for the m aximum span in i nter val t otal color ings of gr aphs p e t r o s a . p e t r o s ya n y a n d n e r s e s a . k h a c h a t r ya n z yinstitute for informatics and automation problems of nas of ra, zdepartment of informatics and applied mathematics, ysu e-mail: pet petros@ipia.sci.am, xachnerses@gmail.com abstract an interval total t-coloring of a graph g is a total coloring of g with colors 1; 2; : : : ; t such that at least one vertex or edge of g is colored by i, i = 1; 2; : : : ; t, and the edges incident to each vertex v together with v are colored by dg(v) + 1 consecutive colors, where dg(v) is the degree of a vertex v in g. in this paper we prove that if a connected graph g with n vertices admits an interval total t-coloring, then t · 2n ¡ 1. furthermore, we show that if g is a connected r-regular graph with n vertices which has an interval total t-coloring and n ¸ 2r + 2, then this upper bound can be improved to 2n ¡ 3. we also give some other upper bounds for the maximum span in interval total colorings of graphs. refer ences [1 ] a .s . a s r a t ia n , c. j. ca s s e lg r e n ,\ on in t e r va l e d g e c o lo r in g s o f ( ®; ¯ ) -b ir e g u la r b ip a r t it e g r a p h s " , d iscrete m athematics 307, p p . 1 9 5 1 -1 9 5 6 , 2 0 0 6 . [2 ] a .s . a s r a t ia n , r .r . k a m a lia n ,\ in t e r va l c o lo r in g s o f e d g e s o f a m u lt ig r a p h " , appl. m ath. 5, p p . 2 5 -3 4 , 1 9 8 7 ( in r u s s ia n ) . [3 ] a .s . a s r a t ia n , r .r . k a m a lia n ,\ in ve s t ig a t io n o n in t e r va l e d g e -c o lo r in g s o f g r a p h s " , j . combin. theory ser. b 62, p p . 3 4 -4 3 , 1 9 9 4 . [4 ] m.a . a xe n o vic h ,\ on in t e r va l c o lo r in g s o f p la n a r g r a p h s " , congr. numer. 159, p p . 7 7 9 4 , 2 0 0 2 . [5 ] m. b e h z a d ,\ gr a p h s a n d t h e ir c h r o m a t ic n u m b e r s " , p h .d . t h e s is , mic h ig a n s t a t e u n ive r s it y, 1 9 6 5 . [6 ] k . gia r o , m. k u b a le a n d m. ma la ¯ e js ki,\ co n s e c u t ive c o lo r in g s o f t h e e d g e s o f g e n e r a l g r a p h s " , d iscrete m ath. 236, p p . 1 3 1 -1 4 3 , 2 0 0 1 . [7 ] r .r . k a m a lia n ,\ in t e r va l e d g e -c o lo r in g s o f g r a p h s " , d o c t o r a l th e s is , n o vo s ib ir s k, 1 9 9 0 ( in r u s s ia n ) . [8 ] p .a . p e t r o s ya n ,\ in t e r va l t o t a l c o lo r in g s o f c o m p le t e b ip a r t it e g r a p h s " , p roceedings of the csit conference, p p . 8 4 -8 5 , 2 0 0 7 . [9 ] p .a . p e t r o s ya n ,\ in t e r va l t o t a l c o lo r in g s o f c e r t a in g r a p h s " , m athematical p roblems of computer science, vol. 31, p p . 1 2 2 -1 2 9 , 2 0 0 8 . 1 9 2 0 upper bounds for the maximum span in interval total colorings of graphs [1 0 ] p .a . p e t r o s ya n , a .s . s h a s h ikya n ,\ on in t e r va l t o t a l c o lo r in g s o f t r e e s " , m athematical p roblems of computer science, vol. 32, p p . 7 0 -7 3 , 2 0 0 9 . [1 1 ] p .a . p e t r o s ya n , a .y u . to r o s ya n ,\ in t e r va l t o t a l c o lo r in g s o f c o m p le t e g r a p h s " , p roceedings of the csit conference, p p . 9 9 -1 0 2 , 2 0 0 9 . [1 2 ] p .a . p e t r o s ya n , \ in t e r va l e d g e -c o lo r in g s o f c o m p le t e g r a p h s a n d n-d im e n s io n a l c u b e s " , d iscrete m athematics 3 1 0 , p p . 1 5 8 0 -1 5 8 7 , 2 0 1 0 . [1 3 ] v .g. v iz in g , \ ch r o m a t ic in d e x o f m u lt ig r a p h s " , d o c t o r a l th e s is , n o vo s ib ir s k, 1 9 6 5 ( in r u s s ia n ) . [1 4 ] d .b . w e s t , in t r o d u c t io n t o gr a p h th e o r y, p r e n t ic e -h a ll, n e w je r s e y, 1 9 9 6 . [1 5 ] h .p . y a p , to t a l co lo r in g s o f gr a p h s , l e c t u r e n o t e s in ma t h e m a t ic s 1 6 2 3 , s p r in g e r v e r la g , 1 9 9 6 . ¶ñ³ýý»ñç ùçç³ï³ûù³ûçý éç³ï³ï³ñ ý»ñïáõùý»ñáõù ù³ëý³ïóáõ ·áõûý»ñç ³é³í»é³·áõûý ñý³ñ³íáñ ãíç í»ñçý ·ý³ñ³ï³ï³ýý»ñ ä. ². ä»ïñáëû³ý, ü. ². ê³ã³ïñû³ý g ·ñ³ýç éç³ï³ï³ñ ý»ñïáõùá i = 1 ; 2 ; : : : ; t ·áõûý»ñáí ï³ýí³ý»ýù ùçç³ï³ûù³ûçý éç³ï³ï³ñ t–ý»ñïáõù, »ã» ³ù»ý ùç i ·áõûýáí, i = 1 ; 2 ; : : : ; t ý»ñïí³í ¿ ³éýí³½ý ù»ï ·³·³ã ï³ù ïáõ ¨ ûáõñ³ù³ýãûáõñ v ·³·³ãçý ïçó ïáõ»ñá ¨ ³û¹ ·³·³ãá ý»ñïí³í »ý dg ( v ) + 1 ñ³çáñ¹³ï³ý ·áõûý»ñáí, áñï»õ dg ( v ) -áí ýß³ý³ïí³í ¿ v ·³·³ãç ³ëïç×³ýá g ·ñ³ýáõù: ²ûë ³ßë³ï³ýùáõù ³å³óáõóíáõù ¿, áñ »ã» n ·³·³ã³ýç g ï³å³ïóí³í ·ñ³ýá áõýç ùçç³ï³ûù³ûçý éç³ï³ï³ñ t–ý»ñïáõù, ³å³ t · 2 n ¡ 1 : ²í»éçý, óáõûó ¿ ïñíáõù, áñ »ã» n ·³·³ã³ýç g ï³å³ïóí³í r–ñ³ù³ë»é ·ñ³ýá áõýç ùçç³ï³ûù³ûçý éç³ï³ï³ñ t–ý»ñïáõù ¨ n ¸ 2 r + 2 , ³å³ t · 2 n ¡ 3 : ü³¨ ³ßë³ï³ýùáõù ïñíáõù »ý ·ñ³ýý»ñç ùçç³ï³ûù³ûçý éç³ï³ï³ñ ý»ñïáõùý»ñáõù ù³ëý³ïóáõ ·áõûý»ñç ³é³í»é³·áõûý ñý³ñ³íáñ ãíç ³ûé ·ý³ñ³ï³ï³ýý»ñ: mathematical problems of computer science 46, 26–36, 2016. review of white-box implementations of aes block cipher and known attacks martun m. karapetyan institute for informatics and automation problems of nas ra e-mail: martun.karapetyan@gmail.com abstract conventional encryption algorithms are designed to be secure in the “black-box” context, i.e. the attacker has access to the input and output of the algorithm, but cannot observe the intermediate values generated during the software execution. yet in some cases, the encryption algorithm runs in a hostile environment, where the attacker can see not only the input and output values but also has full access to all the internal values and can change the execution at will. white-box cryptography algorithms are designed to be executed in such untrusted environments and are said to operate in the white-box attack context. a white-box implementation of aes cipher was first presented by chow, eisen, johnson and van oorschot in 2002 [1], which was shown to be insecure against the bge attack presented by billet, gilbert and ech-chatbi in 2004 [2]. in 2010, another white-box aes implementation was presented by karroumi, which was supposed to withstand the bge attack [3]. in 2013, de mulder, roelse, and preneel showed, that karroumis and chows implementations are equivalent, i.e. the bge attack can be successfully applied to both [4]. they also presented several optimizations, which reduce the work factor of the attack to 222 work steps. in this paper we will review both aes implementations and the bge attacks. keywords: cryptography, white-box, aes, bge attack, review. 1. introduction in the “black-box” encryption model a cryptographic operation is executed in a trusted environment. the attacker, whose main goal is to extract the cryptographic key, observes the input and output of encryption/decryption operations, but has no access to the internal values generated by the algorithm. conventional encryption algorithms were designed to be secure in this context. in some cases, cryptographic software runs on a device controlled by a hostile user. in this case the attacker sees any intermediate value generated by the execution of a cryptographic operation by observing the memory of the device and can change the execution routine at will. examples of such software are digital rights management (drm) systems or any client software running on a cloud. white-box implementations of encryption algorithms are designed to run on these devices, and are said to operate in the white-box attack context. white-box algorithms are implemented as series of look-up from tables, which contain the cryptographic key in such a way to prevent its extraction. there 26 m. karapetyan 27 were numerous attempts to design a white-box implementation of the advanced encryption standard (aes), all of which were later broken. a white-box implementation of aes cipher was first presented by chow, eisen, johnson and van oorschot in 2002 [1]. an attack presented by billet, gilbert and ech-chatbi in 2004 (bge attack) showed that the secure key can be extracted from chows implementation in 230 work steps [2]. in 2010, karroumi presented a modified version of chows algorithm based on dual ciphers, which was designed to withstand the bge attack and increase the work factory of it to 293 [3]. in 2013, de mulder, roelse , and preneel proved that karroumis and chows implementations are identical and the bge attack can be applied to karroumis implementation with minor modifications [4]. also several speed optimizations were presented, after which just 222 work steps are required to extract the key. in this paper we will review both implementations of aes and the bge attack applied on them. the paper is organized as follows: in the sections 2 black-box encryption of aes is briefly represented. section 3 is devoted to the chow’s implementation of aes white-box encryption. section 4 contains the bge attack. section 5 briefly describes karroumi’s implementation. section 6 comments on the bge attack for karoumi’s implementation. the paper ends with the conclusion. 2. aes black-box encryption aes is a substitution-permutation network cipher for symmetric encryption also known as the rijndael cipher [5]. it supports key lengths of 128, 192 or 256 bits and has 10, 12 or 14 rounds, respectively. aes-128 will be considered as the primary setting in the rest of this paper. each round updates a 16-byte state and consists of four operations: subbytes, shiftrows, mixcolumns and addroundkey, except the final round, where the mixcolumns operation is omitted. the 128-bit state is interpreted as a 4x4 matrix of 8-bit values. subbytes operation substitutes each value of the matrix ai,j with s(ai,j), where s(ai,j) values are built using inversion in gf(28) modulo an irreducible polynomial m(x) = x8 + x4 + x3 + x + 1. shiftrows transformation shifts 2nd, 3rd and 4th rows of the table 1, 2 and 3 times to the left, respectively. mixcolumns is a transformation applied on the columns of the state matrix. each column is multiplied with a fixed matrix mc =   2 3 1 1 1 2 3 1 1 1 2 3 3 1 1 2  . addroundkey operation simply xors the state with the next key, generated by the key schedule. for purpose of creating a white-box implementation for aes, we can view the algorithm in a different manner. one can notice, that subbytes and shiftrows operations can be safely switched without any change in the output. also because the shiftrows is a linear transformation, addroundkey(ki) followed by shiftrows is identical to shiftrows followed by addroundkey(ki), where ki is the result of applying shiftrows operations on ki. these allow us to change the aes structure to the one in figure 2. 28 review of white-box implementations of aes block cipher and known attacks fig. 1. structure of aes black-box encryption. fig. 2. aes subbytes operation. 3. chows aes white-box implementation in 2002, chow, eisen, johnson and van oorschot proposed the first white-box implementation of aes-128 [1]. instead of calculating the function ek on the plaintext, another function ek = g · ek · f −1 is computed, where g and f are input and output encodings, m. karapetyan 29 fig. 3. aes shiftrows operation. fig. 4. aes mixcolumns operation. which are randomly generated independent of k. since in some cases it’s infeasible to have input and output encodings of the desired bit length, some encodings can be represented as a concatenation of smaller bijections. a bijection f of size n = n1 + n2 + + nk can be built from a list of smaller bijections fi, where fi has size ni, and for any n-bit vector b = (b1, b2, ..., bn) f(b) = f1(b1, ..., bn1)||f2(bn1+1, ..., bn1 + n2)...fk(bn1+...+nk1+1, ..., bn). in this case we say f = f1||f2||...||fk, and call f a concatenated encoding. the encryption algorithm is implemented as a sequence of look-ups from different look-up tables. the output 30 review of white-box implementations of aes block cipher and known attacks fig. 5. aes algorithm with modified structure. encoding of any table matches the input encoding of the table following it, so the encodings can cancel each other. after the encryption routine is over, the value of ek = g · ek · f −1 is properly computed, as any intermediate encodings are cancelled out. we will present an unprotected implementation first, to describe the tables we’ll need, and then describe the modified protected version of the tables. for each round, addroundkey and subbytes transformations can be made using 16 look-up tables that map 1 byte to 1 byte for each round. these look-up tables are called t-boxes, and are defined as follows: t ri (x) = s(x ⊕ k̃r−1[i]), for i = 0..15, and r = 1..9, (1) t 10i (x) = s(x ⊕ k̃9[i]) ⊕ k10[i], for i = 0..15. (2) it’s obvious, that t-boxes have no security and the attacker can easily extract the keys from the t-boxes, if those were provided. so additional encoding is applied on the t-boxes, before they can be used. after the state vector passes through the t-boxes, mixcolumns operations must be applied on it. mixcolumns operation multiplies each 4 bytes of the state vector [x1, x2, x3, x4] with the mc =   2 3 1 1 1 2 3 1 1 1 2 3 3 1 1 2   matrix. this can be accomplished by the so called tyi 3.1 unprotected implementation m. karapetyan 31 tables, where ty0(x) = x · [02 01 01 03] t , (3) ty1(x) = x · [03 02 01 01] t , (4) ty2(x) = x · [01 03 02 01] t , (5) ty3(x) = x · [01 01 03 02] t . (6) so tyi-boxes are 1 byte to 4 byte tables. the outputs of each 4 consecutive tyi tables must be xor-ed to get the output of the round, i.e. ty0(x)+ty1(x)+ty2(x)+ty3(x). xor tables are used for this purpose: xor(x, y) = x ⊕ y. (7) xor tables operate on 2 nibbles (4-bit values), so they are 1 byte to 4 bits mapping tables. the xor of two 32 bit values can be computed using 8 copies of these xor tables. one can notice, that t-boxes and tyi boxes can be combined into a single table. these tables will look as follows: ty0(t r i (x)) = s(x + k r−1[i]) ∗ [03 02 01 01]t . (8) ty1(t r i (x)) = s(x + k r−1[i]) ∗ [03 02 01 01]t . (9) ty2(t r i (x)) = s(x + k r−1[i]) ∗ [03 02 01 01]t . (10) ty3(t r i (x)) = s(x + k r−1[i]) ∗ [03 02 01 01]t . (11) then the outputs of these tables must be passed through xor boxes to get the round’s output. so there will be 144 composed t − box/tyi tables, 864 xor tables and 16 t-boxes total for all the rounds of aes together. these boxes are not secure against key extraction. the next section will change these tables to apply some defense mechanisms against different attacks. as there are just 256 possible values for kr−1[i], one can build a t − box/tyi table for each of these values, and check if the table we provide matches any of those. this will allow an attacker to extract the key from t − box/tyi tables. in order to prevent this, input and output encodings are applied to all the tables. the encodings are applied in a networked fashion, so all the encodings except the input encoding of the first round f and the output encoding of the last round g cancel out each other, and the function ek = g · ek · f −1 is calculated by using these tables. here g and f are concatenated encodings of 128 bits, made of 16 bijections of 8 bits each. these encodings dramatically increase the total number of possible tables. there are (16!)2 possible input encodings and (16!)8 output encodings per table. it’s obvious that for a fixed input key, the tables constructed for different output encodings are all different. this means that there are at least (16!)8 possible tables, which makes it impossible to the attacker to enumerate. as the input/output encodings provide the confusion step for the tables, linear transformations are applied to produce the diffusion step. the table input/outputs are multiplied with randomly generated matrices over gf(2) which are called mixing bijections. 16 8-bit to 8-bit mixing bijections are randomly generated and applied at the inputs of each round except the first. lets denote mixing bijection for 3.2 protected implementation 32 review of white-box implementations of aes block cipher and known attacks byte i in round r as lir. another 4 32-bit to 32-bit mixing bijections mbri , i = 1..4, r = 1..9 are applied to all the rounds outputs except the last one. so after these changes, the results that we get after applying xor tables will need to be multiplied with the inverse of the mixing bijection (mbri ) − 1 and (l r+1 i ) − 1, so the effect of mb is cancelled out, and the inverse of lr+1i is applied, so it can get cancelled in the next round. this is done with the same technique as the mixcolumns step, and additional xor tables are created for this. the total size of the lookup tables of this implementation is 770,048 bytes, and there are 3104 lookups during each execution [1]. for a more detailed tutorial on chows whitebox aes refer to [6]. 4. bge attack bge attack, introduced by billet, gilbert and ech-chatbi in 2004 [2], targets not a single table from chows implementation, but a group of tables, which form the aes round. as each table has good confusion and diffusion properties, it’s hard to extract the key from a single table, but attacking a group of tables is more reasonable. the aes round is viewed as four 32-bit to 32-bit mappings rjr, where the structure of rrj is shown in figure 6. p r i are a combination of input encodings and mixing bijections, qri are a combination of mixing bijections and output encodings. all the intermediate encodings between different tables have been cancelled out. the bge attack consists of 3 phases: 1) the values of p ri s and q r i s are recovered up to an unknown affine transformation. this allows us to change transformations p ri s and q r i s with affine transformations p̃ r i and q̃ri . 2) the values of p ri and q r i are recovered completely. 3) having p ri and q r i , the aes-128 key is extracted. let’s denote the inputs of rr0 as x0, x1, x2, x3, and the outputs as y0, y1, y2, y3. y0 = q0(02 · t r0 · p r 0 (x0) ⊕ 03 · t r 1 · p r 1 (x1) ⊕ 01 · t r 2 · p r 2 (x2) ⊕ 01 · t r 3 · p r 3 (x3)), (12) y1 = q1(01 · t r0 · p r 0 (x0) ⊕ 02 · t r 1 · p r 1 (x1) ⊕ 03 · t r 2 · p r 2 (x2) ⊕ 01 · t r 3 · p r 3 (x3)), (13) y2 = q2(01 · t r0 · p r 0 (x0) ⊕ 01 · t r 1 · p r 1 (x1) ⊕ 02 · t r 2 · p r 2 (x2) ⊕ 03 · t r 3 · p r 3 (x3)), (14) y3 = q3(03 · t r0 · p r 0 (x0) ⊕ 01 · t r 1 · p r 1 (x1) ⊕ 01 · t r 2 · p r 2 (x2) ⊕ 02 · t r 3 · p r 3 (x3)); (15) now, having these tables, we must find a transformation q̃ri , such that q̃ r i = q r i ∗ ari , i.e. differs from qri by an unknown affine transformation a r i . so y0 is a function of 4 parameters x0, x1, x2, x3. let’s fix the values of x1 and x2 to c1 and c2, respectively, and give 2 different values to x3, namely c3 and c3. we will get 2 functions: y0(x0, c1, c2, c3) = q0(02 · tr · pr(x0)) ⊕ bc1,c2,c3, (16) y0(x0, c1, c2, c3) = q0(02 ∗ tr ∗ pr(x0)) ⊕ bc1,c2,c3. (17) from these 2 equations we get: y0(x0, a1, c2, c3) · y0(x0, a2, c2, c3)−1 = q0(q−10 (x) + b)whereb = bc1,c2,c3 ⊕ bc1,c2,c3. (18) m. karapetyan 33 fig. 6. rr0 mapping, 1 of the 4 mappings that make the aes round. if we vary the value of c3 so it can take all the possible 256 values in gf(2 8), the value of b will also take all the 256 different values. this gives us all the 256 functions q0(q −1 0 (x) + b) for all values of b. this set of bijections forms a commutative group. bilet et al provide a technique to recover the value of q0 up to an unknown affine transformation, given these group of bijections. once this is done, we can change all the tables to use q̃i instead of qi, which significantly weakens the confusion properties of the tables. this allows key extraction, which is described in detail in [2]. 5. karroumis aes white-box implementation in 2010 mohamed karroumi presented a modification of chows aes white-box algorithm, which was supposed to withstand the bge attack [3]. the algorithm is based on usage of aes dual ciphers. aes dual ciphers were first presented in [7] with a list of 240 ciphers. this list was further expanded to 61,200 ciphers that are dual to aes in [8]. for each of these dual ciphers, there exists an affine transformation ∆ that maps aes plaintext p, ciphertext c and key k into plaintext pdual, ciphertext cdual and key kdual of a dual aes, i.e. pdual = ∆(p), cdual = ∆(c) and kdual = ∆(k). in karroumis white-box implementation, for each of 10 rounds of aes a dual-aes is selected randomly. subbytes constants, mixcolumns matrix and the key of the corresponding dual-aes cipher are used for building the t − box/tyi 34 review of white-box implementations of aes block cipher and known attacks tables. in order to get the same output values for the same input values as chow’s aes algorithm, affine transformation ∆r must be applied at the input of each round, and ∆r−1 at the output of the round. the outputs of each round of this implementation will be different from chows outputs as different subbytes and mixcolumns values are used, but the final round outputs will match. fig. 7. aes white-box round structure of karroumi’s modification. one can notice, that as each 4 output bytes of an aes round depend on only 4 input bytes, 4 different dual ciphers may be used in each round, so employing 40 different randomly chosen dual ciphers in total. these changes will not affect the general structure of the tables, so the speed and memory requirement will be identical to chows implementation. karroumi argues that the attacker will need to brute-force these randomly chosen dual ciphers, which will increase the security to 291. the detailed description of this implementation can be found in [3]. 6. bge attack on karoumis implementation it is shown in [4] that an encoded dual aes subround can be represented as an encoded aes subround with the same key. this lets the attacker convert karroumi’s tables into chow’s tables, and apply the bge attack the exact same way. a detailed explanation on how to do the conversion can be found in section 4.1 of [4]. m. karapetyan 3 5 in t h is p a p e r we r e vie we d ch o w's a n d k a r r o u m i's wh it e -b o x im p le m e n t a t io n s o f a e s a lg o r it h m a n d b r ie ° y d e s c r ib e d t h e b ge a t t a c k wh ic h wa s s u c c e s s fu lly a p p lie d o n b o t h im p le m e n t a t io n s . s o fa r n o kn o wn s e c u r e a e s wh it e -b o xe s we r e c r e a t e d , a ll t h e kn o wn im p le m e n t a t io n s we r e b r o ke n wit h a wo r k fa c t o r le s s t h a n 2 30. refer ences [1 ] s . ch o w, p . e is e n , h . jo h n s o n a n d p . c. va n oo r s c h o t , \ w h it e -b o x c r yp t o g r a p h y a n d a n a e s im p le m e n t a t io n " , in 9th annual w orkshop on selected areas in cryptography (sac 2002), a u g .1 5 -1 6 , p p . 1 { 1 8 , 2 0 0 2 . [2 ] o. b ille t , h . gilb e r t a n d c. e c h -ch a t b i, \ cr yp t a n a lys is o f a wh it e -b o x a e s im p le m e n t a t io n " , in selected areas in cryptography(sac), p p . 2 2 7 -2 4 0 , 2 0 0 4 . [3 ] m. k a r r o u m i, \ p r o t e c t in g wh it e -b o x a e s wit h d u a l c ip h e r s " , in k yung-hyune r hee and d aehun nyang, editors, information security and cryptology icisc 2010, of l ecture notes in computer science, springer b erlin heidelberg, vo l.6 8 2 9 , p p . 2 7 8 { 2 9 1 , 2 0 1 1 . [4 ] y . d e mu ld e r , p . r o e ls e a n d b . p r e n e e l, \ r e vis it in g t h e b ge a t t a c k o n a wh it e -b o x a e s im p le m e n t a t io n " , [on lin e ]. a va ila b le : h t t p :/ / e p r in t .ia c r .o r g / 2 0 1 3 / 4 5 0 .p d f. [5 ] n a t io n a l in s t it u t e o f s t a n d a r d s a n d te c h n o lo g y ( n is t) , \ a d va n c e d e n c r yp t io n s t a n d a r d ( a e s ) " , ¯ p s p u b lic a t io n 1 9 7 , 2 6 n o v. 2 0 0 1 . [6 ] ( 2 0 1 2 ) j. a . mu ir , \ a tu t o r ia l o n w h it e -b o x a e s " , m athematics in industry [on lin e ]. a va ila b le : h t t p :/ / www.c c s l.c a r le t o n .c a / ja m u ir / p a p e r s / wb -a e s -t u t o r ia l.p d f. [7 ] ( 2 0 0 2 ) e . b a r ka n a n d e . b ih a m , " th e b o o k o f r ijn d a e ls " , cr yp t o lo g y e p r in t a r c h ive , r e p o r t 2 0 0 2 / 1 5 8 , [on lin e ]. a va ila b le : h t t p :/ / e p r in t .ia c r .o r g / 2 0 0 2 / 1 5 8 . [8 ] a .b ir yu ko v, c. d e ca n n i` e r e , a . b r a e ke n , b . p r e n e e l, \ a t o o lb o x fo r c r yp t a n a lys is : l in e a r a n d a ± n e e qu iva le n c e a lg o r it h m s " , e ur ocr yp t 2003. l ncs, , vo l. 2 6 5 6 , p p . 3 3 { 5 0 , 2 0 0 3 . ´³ó ïá¹áí (whitebox) í³íï³·ñù³ý ñ³ù³ï³ñ·»ñç í»ñéáõíáõãûáõý ¨ ñ³ûïýç ñ³ñó³ïù³ý ù»ãá¹ý»ñç ñ»ï³½áïáõãûáõý ø. î³ñ³å»ïû³ý ²ù÷á÷áõù êï³ý¹³ñï í³íï³·ñù³ý ³é·áñçãùý»ñá ý³ë³ï»ëí³í »ý “ë¨ ïáõ÷ç” ñ³ù³ï»ùëïáõù ³ýíï³ý· ³ßë³ï»éáõ ñ³ù³ñ, ³ûëçýùýñ̀³ñó³ïíáõá ï³ñáõ ¿ áõëáõùý³ëçñ»é ³é·áñçãùç ùáõïùç ¨ »éùç ³ñå»ùý»ñá, µ³ûó ãç ï³ñáõ ï»ëý»é ³é·áñçãùç ³ßë³ï³ýùç áýã³óùáõù íñ³·ñç ·»ý»ñ³óñ³í ùçç³ýïû³é ³ñå»ùý»ñá: ê³ï³ûý, áñáß ¹»åù»ñáõù, í³íï³·ñù³ý ³é·áñçãùý ³ßë³ïáõù ¿ ûï³ñ ùçç³í³ûñáõù, áñï»õ ñ³ñó³ïíáõá ï»ëýáõù ¿ áã ùç³ûý ³é·áñçãùç ùáõïùý áõ »éùá, ³ûé ý³¨ ó³ýï³ó³í ùçç³ýïû³é ³ñå»ù` ·»ý»ñ³óí³í ³é·áñçãùç ïáõùçó ¨ ó³ýïáõãû³ý ¹»åùáõù ï³ñáõ ¿ ÷á÷áë»é submitted 08.07.2016, accepted 04.11.2016. 7 7. co n c lu s io n 3 6 review of white-box implementations of aes block cipher and known attacks êðèïòîãðàôè÷åñêèå àëãîðèòìû ðàáîòàþùèå ïî ïðèíöèïó áåëîãî ÿùèêà è èçâåñòíûå àòàêè ì. êàðàïåòÿí àííîòàöèÿ îáû÷íûå ñèììåòðè÷íûå êðèïòîãðàôè÷åñêèå àëãîðèòìû ðàñ÷èòàíû íà áåçîïàñíîñòü â òàê íàçûâàåìîé ñðåäå “÷åðíîãî ÿùèêà”, ò.å. àòàêóþùèé èìååò äîñòóï ê âõîäíûì è âûõîäíûì äàííûì àëãîðèòìà, íî íå ìîæåò âèäåòü ïðîìåæóòî÷íûå çíà÷åíèÿ, ãåíåðèðóåìûå ïðè èñïîëíåíèè. èíîãäà êðèïòîãðàôè÷åñêèå ïðîãðàììû ðàáîòàþò â íåçàùèùåííîé ñðåäå, ãäå àòàêóþùèé èìååò äîñòóï íå òîëüêî ê âõîäíûì è âûõîäíûì äàííûì àëãîðèòìà, íî òàêæå ê ëþáîìó ïðîìåæóòî÷íîìó çíà÷åíèþ ãåíåðèðóåìîìó àëãîðèòìîì. àòàêóþùèé òàêæå ìîæåò èçìåíèòü ïðîìåæóòî÷íûå çíà÷åíèÿ èëè ñàì àëãîðèòì ïî ñîáñòâåííîìó æåëàíèþ. àëãîðèòìû, ðàáîòàþùèå ïî ïðèíöèïó áåëîãî ÿùèêà, ðàñ÷èòàíû äëÿ áåçîïàñíîé ðàáîòû â òàêîé ñðåäå. ïåðâîå èñïîëíåíèå àëãîðèòìà aes, ðàáîòàþùåå ïî ïðèíöèïó áåëîãî ÿùèêà, áûëî ñîçäàíî ×î, åíñåíîì, äæîíñîíîì è âàí îðøîòîì â 2002 ãîäó, êîòîðûé áûë óäà÷íî àòàêîâàí ìåòîäîì àòàêè “bge”, ïðåäëîæåííîì â 2004 ãîäó. â 2010 ãîäó äðóãîé ìåòîä ðåàëèçàöèè àëãîðèòìà aes ïî ïðèíöèïó áåëîãî ÿùèêà áûë ïðåäëîæåí êàðóìè, íî â 2013 ãîäó äå ìþëäåð, ðîëñå è ïðåíèë ïîêàçàëè, ÷òî ìåòîäû êàðóìè è ×î èäåíòè÷íû, ò.å. àòàêà ”bge” ìîæåò áûòü óñïåøíî ïðèìåíåíà ê îáîèì ìåòîäàì. â äàííîé ñòàòüå ïðåäñòàâëåíû ìåòîäû èñïîëíåíèÿ àëãîðèòìà aes ïî ïðèíöèïó áåëîãî ÿùèêà è èçâåñòíûå àòàêè íà íèõ. ³é·áñçãùç ³ßë³ï³ýùá: ´³ó ïá¹áí í³íï³·ñù³ý ³é·áñçãùý»ñá ý³ë³ï»ëí³í »ý ³ßë³ï»éáõ ³ûëåçëç ûï³ñ ùçç³í³ûñ»ñáõù: aes í³íï³·ñù³ý ³é·áñçãùç µ³ó ïá¹áí çñ³ï³ý³óù³ý ëë»ù³ ³é³ççýá ³é³ç³ñïí»é ¿ âáç, ²ûë»ýç, æáýëáýç ¨ ì³ý úñßáïç ïáõùçó 2002ã.-çý, áñç íñ³ ñ³çáõ ñ³ñó³ïù³ý ù»ãáõ ³é³ç³ñïí»ó 2004ã.-çý, çëï ñ³ñó³ïù³ý ù»ãá¹á ïáãí»ó “bge ñ³ñó³ïáõù”: 2010-çý aes-ç ù»ï ³ûé µ³ó ïá¹áí çñ³ï³ý³óáõù ³é³ç³ñïí»ó î³éáõùçç ïáõùçó, ë³ï³ûý 2013-çý ¸» øáõé¹»ñá, èáéëá ¨ äñ»ýçéá óáõûó ïí»óçý, áñ î³éáõùçç ¨ âáç çñ³ï³ý³óáõùý»ñá ñ³ù³ñå»ù »ý, ³ûëçýùý` “bge ñ³ñó³ïáõùá” ñ³çáõáõãû³ùµ ³ßë³ïáõù ¿ ý³¨ ³û¹ ëë»ù³ûç íñ³: ²ûë ñá¹í³íáõù ù»ýù ïí»ñéáõí»ýù aes-ç ³é³ç³ñï³í µ³ó ïá¹áí í³íï³·ñù³ý ëë»ù³ý»ñá ¨ ¹ñ³ýó ýï³ïù³ùµ ñ³çáõáõãû³ùµ ïçñ³éí³í ñ³ñó³ïù³ý ù»ãá¹ý»ñá: whitebox_aes_review_final.pdf (p.1) 03.pdf (p.2-11) m_abstract.pdf (p.10-11) microsoft word tpel.doc mathematical problems of computer science 33, 162--171, 2010. 162 approximation algorithm for wireless network interference minimization hakob l. aslanyan department of informatics, university of geneva, switzerland e-mail: hakob.aslanyan@unige.ch abstract interference minimization problem in wireless sensor and ad-hoc networks are considered. that is to assign a transmission radius to each node of network, to make it connected and at the same time to minimize the maximum number of overlapping transmission ranges on each node of network. additional means of topology control besides the connectivity is blocking the long line connections at the receiver level. we propose a polynomial time approximation algorithm which finds connected network with at most ))log(( 2nopto  interference where opt is the minimal interference of the given network, and n is the number of network nodes. the lower bound for this problem, where a general distance function is considered, has been proven to be )(log no . the algorithm is known which finds a network where maximum interference is bounded by )( no . references [1] p. von rickenbach, s. schmid, r. wattenhofer and a. zollinger, “a robust interference model for wireless ad-hoc networks”, proceedings of parallel and distributed processing symposium, vol. 13, pp. 239a, 2005. [2] m. m. halldorsson and t. tokuyama, “minimizing interference of a wireless ad-hoc network in a plane”, theoretical computer science, vol. 402, issue 1, pp. 29-42, 2006. [3] d. bil`o and g. proietti, “on the complexity of minimizing interference in ad-hoc and sensor networks”, proc. 2nd international workshop on algorithmic aspects of wireless sensor networks, algosensors 2006, in: lncs, vol. 4240, pp. 13–24, 2006. [4] f. kuhn, p. von rickenbach, r. wattenhofer, e. welzl, and a. zollinger, “interference in cellular networks: the minimum membership set cover problem”, proc. 11th cocoon, volume 3595 of lncs, pp. 188–198. springer, 2005. [5] k. moaveni-nejad and x.y. li, “low-interference topology control for wireless adhoc networks”, ad hoc & sensor wireless networks: an international journal 1(1-2) (2005). [6] t. johansson and l. carr-motyckova, “reducing interference in ad hoc networks through topology control”, proc of the 3 rd acm joint workshop on foundations of mobile computing (dialm-pomc), acm press, pp. 17-23, 2005. [7] m. benkert, j. gudmundsson, h. haverkort and a. wolff, “constructing minimuminterference networks”, computational geometry: vol. 40, issue 3, pp. 179-194, 2008. h. aslanyan 163 [8] m. fussen, r. wattenhofer and a. zollinger, “interference arises at the receiver”, proc. of the int. conf. on wireless networks, communications, and mobile computing (wirelesscom), pp. 1-6, 2005. [9] d. johnson, “approximation algorithms for combinatorial problems”, journal of computer and system sciences, pp. 256–278, 1974. [10] u. feige, “a threshold of ln n for approximating set cover”, journal of the acm (jacm), 45(4), pp. 634–652, 1998. [11] e. d. demaine, m. t. hajiaghayi, u. feige, and m. r. salavatipour, “combination can be hard: approximability of the unique coverage problem”, proc. 17th soda, pp. 162–171, acm press, 2006. [12] n. garg, v. v. vazirani, and m. yannakakis, “primal-dual approximation algorithms for integral flow and multicut in trees”, algorithmica, 18(1), pp. 3–20, 1997. [13] j. guo and r. niedermeier, “exact algorithms and applications for tree-like weighted set cover”, journal of discrete algorithms, vol. 4, issue 4, pp. 608-622, 2006. [14] s. mecke and d. wagner, “solving geometric covering problems by data reduction”, proc. 12th esa, vol. 3221 of lncs, pp. 760–771. springer, 2004. [15] n. ruf and a. schöbel, “set covering with almost consecutive ones property”, discrete optimization, 1(2), pp. 215–228, 2004. [16] h. aslanyan, “greedy set cover estimations”, int. conference “computer science and information technologies”, yerevan, pp. 143-144, 2003. [17] s. meguerdichian, f. koushanfar, m. potkonjak, and m. srivastava, “coverage problem in wirelss ad-hoc sensor networks”, ieee infocom, pp. 1380–1387, 2001. [18] h. breu and d. g. kirkpatrick, “unit disk graph recognition is np-hard”, computational geometry: theory and applications, 9(1-2), pp. 3–24, 1998. [19] j.n. al-karaki and a.e. kamal, “routing techniques in wireless sensor networks: a survey”, ieee wireless communications, 11(6), pp. 6-28, 2004. [20] k. pahlavan and a. h. levesque, “wireless information networks”, john wiley & sons, new york, ny, usa, 1995. ²ýé³ñ ó³ýó»ñáõù çýï»ñý»ñ»ýëç ùçýçùç½³óù³ý ùáï³íáñ ³é·áñçãù ð. ²ëé³ýû³ý ²ù÷á÷áõù ¸çï³ñïí»é ¿ ³ýé³ñ ó³ýó»ñáõù çýï»ñý»ñ»ýëç ùçýçùç½³óù³ý ëý¹çñá, áñá ñ»ï¨û³éý ¿. ó³ýóç ûáõñ³ù³ýãûáõñ ñ³ý·áõûóçý í»ñ³·ñ»é ñ³õáñ¹³ïóù³ý ß³é³íçõ ³ûýå»ë, áñ ó³ýóá éçýç ï³å³ïóí³í ¨ ùç¨ýáõûý å³ù³ý³ï ó³ýóáõù ù³ùëçù³é çýï»ñý»ñ»ýë áõý»óáõ ñ³ý·áõûóç çýï»ñý»ñ»ýëá (ñ³ý·áõûóý áý¹·ñïáõ ñ³õáñ¹³ïóù³ý ßñç³ýý»ñç ù³ý³ïá) éçýç ùçýçù³é: ²ûë ëý¹ñç ñ³ù³ñ ³é³ç³ñïíáõù ¿ µ³½ù³ý¹³ù³ûçý å³ù³ý³ïáõù ³ßë³ïáõ ùáï³íáñ ³é·áñçãù, áñá ·ïýáõù ¿ ï³å³ïóí³í ó³ýó, áñáõù ûáõñ³ù³ýãûáõñ ñ³ý·áõûóç çýï»ñý»ñ»ýëá ãç ·»ñ³½³ýóáõù ))log(( 2nopto  -ý: ²ûëï»õ opt -á ïñí³í n ñ³ý·áõûóý»ñáí ó³ýóç ùçýçù³é çýï»ñý»ñ»ýëý ¿: ð³ûïýç ¿, áñ ëý¹çñá ù»ïñçï³ï³ý ï³ñ³íáõãûáõýáõù ¹çï³ñï»éáõ ¹»åùáõù (áñá ³ßë³ï³ýùáõù ¹çï³ñïí³í ¹»åùý ¿), )(log no -á ùáï³ñïù³ý ëïáñçý ë³ñù³ýý ¿: ð³ûïýç ¿ ý³¨ µ³½ù³ý¹³ù³ûçý ³é·áñçãù, áñá ·ïýáõù ¿ ï³å³ïóí³í ó³ýó ³é³í»é³·áõûýá )( no çýï»ñý»ñ»ýëáí: microsoft word geolecyan_18.doc îçµ»éý»ïçï³ûç ¨ ñ³ßíáõ³ï³ý ï»ëýçï³ûç ù³ã»ù³ïçï³ï³ý ñ³ñó»ñ 34, 2010. 46 բարձրագույն որակավորման խնդիրները եվ հայաստանի գիտահետազոտական է-ինֆրակառուցվածքը հրանտ գեոլեցյան բարձրագույն գիտական որակավորումը` խթանում է գիտական և գիտամանկավարժական կադրերի որակի բարձրացումը, նպաստում է գիտության մեջ նոր գիտելիքի ստեղծման գործում անձնական ավանդի ճանաչմանը` գիտական աստիճանների և գիտական կոչումների շնորհման միջոցով: 1934թ. խսհմ-ում և արդեն մեկ տարի անց խորհրդային հայաստանում սկսվեց գիտական աստիճանաշնորհման գործընթացը: հհ բարձրագույն որակավորման հանձնաժողովի ստեղծումից մինչ այսօր մոտ վեց հազար գիտնական ստացել են գիտությունների դոկտորի և թեկնածուի գիտական աստիճանների հայաստանյան եռալեզու վկայագրեր: այսօր բոհ-ի առջև դրված են երկու կարևոր խնդիր. առաջինը` գիտական կադրերի որակավորման գնահատման ապահովումը, երկրորդը` հնարավորություն ապահովել գիտական հանրությանը պաշտպանած ատենախոսությունների նյութերի օգտագործման համար: է-ինֆրակառուցվածքը` բոհ-ի հիմնական խնդիրների լուծմանն ուղղված արդյունավետ գործիք է: այն ապահովում է ինֆորմացիայի պահպանումը, փնտրումը, վերլուծությունը, ներկայացումը և հաղորդումը: նախատեսվում է ատենախոսությունների սեղմագրերի տեղադրումը մասնագիտական խորհուրդների կայքերում, հետագայում բոհ-ի համապատասխան կայքերում տեղադրելով ամբողջական ատենախոսությունները: ռդ-ում ներդրված “антиплагиат” համակարգի նմանակի ստեղծումը` ժամանակակից ինֆորմացիոն տեխնոլոգիաների օգտագործմանն ուղղված բոհ-ի երկրորդ կարևոր խնդիրն է, ինչը թույլ կտա բացահայտել անբարեխիղճ հայցորդներին: բոհ-ի ինֆորմացիոն ապահովման համակարգում գործում են գիտնականների և աստիճանաշնորհման տվյալների կայքեր: այդպիսի կայքերը պետք է գտնվեն աշխատանքային վիճակում, պարբերաբար նորացվեն, լուծվեն ինֆորմացիոն անվտանգության, մյուս համակարգերի հետ համագործակցության և այլ կարևոր խնդիրներ: բոհ-ի աշխատանքում ինֆորմացիոն տեխնոլոգիաները սկզբունքորեն անհրաժեշտ են աշխատանքի արդյունավետության և որակի բարձրացման գործում, միաժամանակ չդառնալով ինքնանպատակ, չստեղծելով նոր խնդիրներ գիտնականների և գիտական որակավորման համակարգի համար: microsoft word h.sarukhanyan_2.doc mathematical problems of computer science 34, 2010. 10 orthogonal transforms for digital signal and image processing hakob sarukhanyan institute for informatics and automation problems of nas ra abstract in this report there are presented some primary results obtained in digital signal and image processing laboratory of the institute for informatics and automation problems of nas ra. fast discrete orthogonal transforms: the computation of unitary transforms is a complicated and time-consuming task. however, it would not be possible to use the orthogonal transforms in signal and image processing applications without effective algorithms to calculate them. note that both complexity issues–efficient software and circuit implementations are the heart of the most applications. an important question in many applications is how to achieve the highest computation efficiency of the discrete orthogonal transforms (dot) [1]. the suitability of unitary transforms in each of the above applications depends on the properties of their basis functions as well as on the existence of fast algorithms, including parallel ones. a fast dot is an efficient algorithm to compute the dot and its inverse with an essentially smaller number of operations than direct matrix multiplication. recall that the dot of the sequence )(nf is given by     1 0 1 ,1,0],[][][ n n nn nkknfky  where ]}[{ kn is an orthogonal system. it follows that the determination of each y[k] requires n multiplication and n – 1 addition. because we have to evaluate y[k] for ,1,0  nk it follows that the direct determination of dft requires n(n – 1) operations, which means that the number of multiplication and additions/subtractions is proportional to 2n , i.e., the complexity of dft is ).( 2no general concept in design of the fast dot algorithms: a fast transform ftn may be achieved by factoring the transform matrix nt by the multiplication of k sparse matrices. typically, nn 2 and ,1212 fffft nnn  where if are very sparse matrices so that the complexity of multiplying by if is ),( no .,1 ni  fast fourier transform: particularly, the fourier matrix nf of order nn 2 can be represented as ,112211 babababf nnnn  where ra and ia are sparse matrices of the following form 11     .,1, ,1,1, 222 12 2 1 2 0 222 1 1111 niihib nrwwwiia ini rn rnrnrnrnr i r       ,2         h  is a sign of kronecker product, and        k m jw km 2 exp .therefore the fast fourier transform has the complexity )log( 2 nno . the nn 2 -point fourier transform of a },,,{ 110  nxxxx  with assumption ]1[]1[  nxx can be represented as (see also [2,3]) .1,0,]14[]14[]4[][ 1 0 1 0 1 0 4 4 4 4 2 2          nnwkxwwkxwwkxnx n n n n n n k nkn n k nkn n k nk by introducing the notations ;1,0,]4[][ 2 1 0 0 2 2     n k nk nwkxny n n ,]14[][ 1 0 1 4 4    n n k nkwkxny ,1,0,]14[][ 4 1 0 2 4 4     n k nk nwkxny n n fft can be realized as follows         .1,0,][][][][ ,][][][][ ,][][][][ ,][][][][ 421404 3 2104 2 21404 210         nn n n n nn n n n n n n n n n nn n n n n nnywnywjnynx nywnywnynx nywnywjnynx nywnywnynx the number of operations for a realization of fft given below .6)1( 9 2 9 38 log 3 4 ,2)1( 9 2 9 16 log 3 8 2 2 log 2 log 2     n n n n nnnc nnnc fast hadamard transform: the hadamard matrix nh of order nn 2 can be factorized as follows: ,121 ffffh nnn  where   .,,2,1,222 1 niihif inii   it is not difficult to show that nn 2 -point fast hadamard transform (fht) has the complexity )log( 2 nno (note that fht requires only the additions and subtraction operations). later, in [4] were developed more general fht which have the complexity  1log 2  knnknd with assumption that hadamard matrix nh of order n has the following representation kkn avavavh  2211 where iv are k-size (-1,+1) mutualy orthogonal vectors, and kja j ,1,  are nk n  size (0,-1,+1) matrices with the following conditions .,,2,1, ,,,2,1,,,0 , ,)1,1( ,,,2,1,,, 1 1 kiiaa kjijiaa iaa matrixaisa kjijiaa k nk n i t i j t i nk n k i t ii k i i ji             12 a-s theorem and its generalization: in 1981 agaian and sarukhanyan [5] have offered a construction method of hadamard matrix (h-matrix) of order mn/2 proceeding from existence h-matrices of order m and n (see figure below). later, this result has been named by multiplicative theorem a-s (the multiplicative theorem of agaian-sarukhanyan) [6]. in the same place, the existence of h-matrix of order mnpq/16 is proved based on a-s theorem. further, by using of the multiplicative method there have been developed methods of construction hybrid orthogonal matrices and corresponding fast transform algorithms. last years a-s methodwas successfully applied at construction of lapped transforms and filter banks, and also at construction antipodal paraunitary matrices and their application in orthogonal frequency division multiplexing (ofdm) systems [7, 8]. references 1. n. ahmed and k. r. rao, orthogonal transforms for digital signal processing. springer-verlag, new york (1975). 2. r. yavne, “an economical method for calculating the discrete fourier transform,” proc. afips fall joint computer conf., vol. 33, 1968, pp. 115–125. 3. m. frigo and s. g. johnson, “a modified split-radix fft with fewer arithmetic operations,” ieee transactions on signal processing, vol. 55, pp. 111-119, 2007. 4. h. sarukhanyan. decomposition of the hadamard matrices and fast hadamard transform. computer analysis of images and patterns, lecture notes in computer science, vol. 1296, 1997, pp. 575-581 5. s. agaian, h. sarukhanyan. recurrence formulae construction of williamson’s type matrices. math. notes, vol. 30, no.34, 1981, pp. 796-804 6. j. seberry and m. yamada, “hadamard matrices, sequences, and block designs,” in contemporary design theory: a collection of surveys, j. h. dinitz and d. r. stinson, eds. new york: wiley, 1992. 7. see-may phoong, yuan-pei lin. lapped hadamard transforms and filter banks, ieee, icassp 2003, vol. 6, 2003, pp. 509-512. 8. see-may phoong, kai-yen chang. antipodal paraunitary matrices and their application to ofdm systems, ieee trans. on signal processing, vol. 53, no. 4, 2005, pp. 1374-1386. d:\sbornik\...\article_eng.dvi mathematical problems of computer science 34, 2010. an appr oach to gener al for m recur sive e quation solutions h r a n t b . ma r a n d jia n institute for informatics and automation problems of nas ra e-mail: hrant marandjian@sci.am r e s e a r c h in ¯ n d in g s o lu t io n s o f fu n c t io n a l e qu a t io n s h a ve a lo n g h is t o r y. in 1 7 9 1 a . l e g e n d r e [4 ] p o s e d t h e p r o b le m o f ¯ n d in g t h e c o n t in u o u s s o lu t io n s o f t h e e qu a t io n f ( x + y ) = f ( x ) + f ( y ) ( 1 ) it wa s d e a lt wit h a s l e g e n d r e a n d ga u s s , b u t it s s o lu t io n , a s we ll a s o t h e r s im ila r e qu a t io n s , b u t m a n a g e d t o g e t b y a . ca u c h y [1 ]. l a t e r t h e fu n c t io n a l e qu a t io n s we r e in ve s t ig a t e d b y m a n y a u t h o r s . th e p r e s e n t a u t h o r b e g a n r e s e a r c h in s e a r c h o f c o m p u t a b le s o lu t io n s t o g e n e r a l fo r m r e c u r s ive e qu a t io n s ( gfr e ) ( s e e [5 ], [6 ]) : f1[f1; f2; : : : ; fn]( x1; x2; : : : ; xm ) ' g1[f1; f2; : : : ; fn]( x1; x2; : : : ; xm ) ; f2[f1; f2; : : : ; fn]( x1; x2; : : : ; xm ) ' g2[f1; f2; : : : ; fn]( x1; x2; : : : ; xm ) ; ; ( 2 ) fk[f1; f2; : : : ; fn]( x1; x2; : : : ; xm ) ' gk[f1; f2; : : : ; fn]( x1; x2; : : : ; xm ) ; wh e r e xi a r e n u m e r ic va r ia b le s ,fia r e va r ia b le s d e n o t in g t h e d e s ir e d fu n c t io n s , fi, gi r e c u r s ive ( c o m p u t a b le ) fu n c t io n a ls . in [5 ] it wa s s h o wn t h a t t h e p r o b le m o f t h e e xis t e n c e o f c e r t a in s o lu t io n s t o t h e s ys t e m s o f t h e t yp e ( 2 ) is §03¡ c o m p le t e if lo o kin g fo r p a r t ia l s o lu t io n s a n d is §11¡ c o m p le t e if lo o kin g fo r t o t a l s o lu t io n s . it is e vid e n t , t h a t a n y s ys t e m o f d i®e r e n t ia l, in t e g r a l o r e ve n m ixe d t yp e o f im p lic it e qu a t io n s , r e wr it t e n in a fo r m s u it a b le fo r n u m e r ic a l c o m p u t a t io n , b r in g s t o fo r m o f a gfr e . h e n c e it is n a t u r a l t o fa c e t o p r o b le m s c o n c e r n in g t h e s e a r c h fo r s o lu t io n s t o t h e s e e qu a t io n s . p . co llin s a n d d . b a u m a n [2 ] e xa m in e d s e a r c h fo r c o m p u t a b le s o lu t io n s t o o r d in a r y d i®e r e n t ia l e qu a t io n s . th e a u t h o r s h u m o r o u s ly c a lle d t h e m e t h o d , t h e y h a ve d e ve lo p e d , a m e t h o d " b y a t h o u s a n d m o n ke ys ." n e ve r t h e le s s , t o s o lve im p o r t a n t s c ie n t i¯ c o r t e c h n ic a l p r o b le m , we n e e d s o lve s u c h s ys t e m s o f e qu a t io n s . in t h is p a p e r , we wo u ld like t o illu s t r a t e a n a p p r o a c h t h a t s o m e t im e s c a n h e lp t o ¯ n d s o lu t io n ( s ) . a t it s h e a r t a r e t h e d e t a ils o f t h e p r o o f o f t h e t h e o r e m o n n e c e s s a r y a n d s u ± c ie n t c o n d it io n s fo r c o m p u t a b le s o lu t io n e xis t e n c e t o e qu a t io n s o f t h is t yp e ( [5 ], [6 ]) . th e p r o p o s e d m e t h o d c o n s is t s o f t h e s e le c t io n o f r e p la c e m e n t s o f o c c u r r e n c e s o f t h e u n kn o wn fu n c t io n s in e xp r e s s io n s wit h p r o p e r ly m a t c h e d fu n c t io n a ls a n d t h e n a p p lyin g t h e t h e o r e m o n t h e jo in t ¯ xe d p o in t . fo r s im p lic it y we will r e s t r ic t o u r s e lve s wit h a s im p le c a s e t h a t s h o ws t h e n e e d e d 2 6 2 7 s t e p s . e xample. s u p p o s e we a r e g ive n t h e fo llo win g t r a n s c e n d e n t a l e qu a t io n : [f ( n) =2 ] + ( f ( n ) ) 2f (n¡1) ' ( 2 f ( n ¡ 2 ) ) f (n+1) + f ( n ¡ 2 ) : ( 3 ) e a s y t o s e e t h a t t h e m a in o b s t a c le s a r e t h e p r e s e n c e o f t h e " in t e g e r p a r t " a n d s u m m a t io n fu n c t io n s . to e lim in a t e t h e s e p a r t s , w e d e ¯ n e a n a u xilia r y fu n c t io n a l h a s fo llo ws : h[g]( n) ) ' 2 g ( n ¡ 2 ) : ( 4 ) r e p la c in g in ( 3 ) t h e o c c u r r e n c e o f f ( n ) wit h t h e d e ¯ n it io n o f h[f]( n ) a n d m a kin g c a n c e lla t io n , we g e t : ( 2 f ( n ¡ 2 ) 2f (n¡1) ' ( 2 f ( n ¡ 2 ) ) f (n+1): ( 5 ) a t t h e n e xt s t e p we r e p la c e in ( 5 ) f ( n + 1 ) wit h h[f]( n + 1 ) a n d o b t a in ( 2 f ( n ¡ 2 ) 2f (n¡1) ' ( 2 f ( n ¡ 2 ) ) 2f (n¡1): ( 6 ) t h a t is it is a n id e n t it y. e a s y t o s e e t h a t t h e s .c. k le e n e 's [3 ] ¯ xe d p o in t s o f t h e e qu a t io n ( 4 ) a r e t h e fu n c t io n s s a t is fyin g t h e c o n d it io n f ( n) ' 2 f ( n ¡ 2 ) : ( 7 ) co n s e qu e n t ly, a s a s o lu t io n t o ( 3 ) we c a n t a ke a n y fu n c t io n s a t is fyin g t h e d e ¯ n it io n ( 7 ) ( a d d in g in it ia l va lu e s ) . refer ences [1 ] ca u c h y a .-l .,cours d'analyse de l' ¶e cole r oyale p olytechnique, p r e m iµ e r e p a r t ie , a n a lys e a lg ¶ e b r iqu e , p a r is , 1 8 2 1 [oe u vr e s ( 2 ) 3 , p a r is , 1 8 9 7 ]. [2 ] co llin s p ., gr a »c a d . s .: e ®ective computability of solutions of ordinary d i®erential e quations. the thousand m onkeys approach,e le c t r o n ic n o t e s in th e o r e t ic a l co m p u t e r s c ie n c e 2 2 1 , 2 0 0 8 , p p .1 0 3 -1 1 4 . [3 ] k le e n e s . c., introduction to m etamathematics. d . v a n n o s t r a n d co ., in c ., n e w y o r k, to r o n t o , 1 9 5 2 . [4 ] l e g e n d r e a . m., e l¶ements de g¶eometrie. p a r is , 1 7 9 1 . n o t e ii. [5 ] ma r a n d jia n h .b ., selected topics in r ecursive f unction theory in computer science, d th , l yn g b y, 1 9 9 0 . p p 3 1 -4 7 . [6 ] ma r a n d jia n , h .b ., general f orm r ecursive e quations i, in : co m p u t e r s c ie n c e l o g ic . s e le c t e d p a p e r s , l e c t u r e n o t e s in co m p u t e r s c ie n c e , vo l. 9 3 3 , s p r in g e r , b e r lin , h e id e lb e r g , 1 9 9 5 , p p . 5 0 1 -5 1 1 . mathematical problems of computer science 53, 14–20, 2020. udc 510.6 investigation of monotonous properties for frege systems arsen a. hambardzumyan yerevan state university e-mail: arsen.hambardzumyan2@ysumail.am abstract in this paper, we investigate the relations between the frege proof lines of minimal tautologies and the results of substitutions in them. we show that there is a sequence of tautologies ψn, each of which has a unique minimal tautology ϕn, such that for every n the frege proof lines of ϕn are an order more than the frege proof lines for ψn. keywords: minimal tautology, frege systems, proof lines, monotonous and strong monotonous of proof systems. 1. introduction the minimal tautologies play a main role in the proof complexity area. really, all propositional formulas, the proof complexities of which are investigated in many well-known papers, are minimal tautologies. there is a traditional assumption that a minimal tautology shouldn’t be more complicated than any substitution in it. this idea was first revised by anikeev in [1]. he introduced the notion of a monotonous proof system and gave examples of monotonous and non-monotonous systems, but both of them are not complete systems. in [2]-[5], the notion of strongly monotonous systems for propositional proof systems is additionally introduced and the properties of monotonous and strongly monotonous for many propositional proof systems of classical and non-classical logics are investigated. some of the investigated systems (resolution systems, cut-free sequent systems) are monotonous systems, in each of which the proof lines of all minimal tautologies are not more than the proof lines for the results of substitutions in them. some others are not monotonous (systems based on the splitting method, elimination systems), the proof lines of some substituted formulas in each of which can be less than the proof lines of some corresponding minimal tautologies. it was proved that frege systems are not strongly monotonous, but the question of monotonicity of frege systems still remained open. in this paper we prove that frege systems are not monotonous: it is shown that there is a sequence of tautologies ψn, each of which has a unique minimal tautology ϕn, such that for every n the frege proof lines of ϕn are an order more than the frege proof lines of ψn. 14 a. hambardzumyan 15 2. preliminaries we will use the current concepts of a propositional formula, subformula, elementary subformula, a classical tautology, frege proof systems for classical propositional logic, proof and proof complexity [1]-[6]. let us recall some of them. definition 1: a frege system f uses a denumerable set of propositional variables, a finite, complete set of propositional connectives; f has a finite set of inference rules defined by a figure of the form a1a2...an b (the rules of inference with zero hypotheses are the axioms schemes); f must be sound and complete, i.e. for each rule of inference a1a2...an b every truth-value assignment, satisfying a1a2 . . .an, also satisfies b, and f must prove every tautology. in the theory of proof complexity, one of the main characteristics of the proof is tcomplexity, defined as the number of proof steps. let φ be a proof system and ϕ be a tautology. we denote by tφϕ the minimal possible value of t-complexity for all φ-proofs of the tautology ϕ. definition 2: two proof systems are called t-linearly equivalent if any proof in one system can be modified to a proof of the same tautology in another system, so that the t-complexity of the proof is increased not more than linearly. in [6], it is proved that all frege systems are t-linearly equivalent. the particular choice of a language for the presented propositional formulas is immaterial in this consideration. however, for some technical reasons we assume that the language contains only logical connectives ¬,⊃ and assume that ∨ and & are presented in traditional ways through ¬,⊃. we assume that f has the well-known inference rule modus ponens. by |ϕ| we denote the size of the formula ϕ, defined as the number of all logical signs entries in it. it is obvious that the full size of the formula, which is understood to be the number of all symbols, is bounded by some linear function in |ϕ|. definition 3: a tautology is called minimal if the replacement result of all occurrences for each of its non-elementary subformulas by some new variable is not a tautology. we denote by s(ϕ) the set of all formulas, every of which is the result of some substitution in a minimal tautology ϕ. definition 4: the proof system φ is called t-monotonous if for every tautology ψ there is a minimal tautology ϕ, such that ψ ∈ s(ϕ) and tφϕ = t φ ψ. definition 5: the proof system φ is called t-strongly monotonous if for every tautology ψ there is no minimal tautology ϕ, such that ψ ∈ s(ϕ) and tφϕ > t φ ψ. 3. main results here we give the main theorem, but first we should give the following auxiliary statements. lemma 1: let p(a) be a tautology, a be some of its subformulas of size n, and p(a) not contain q. there is a modified image of the subformula a to a′(q), and there are not more than 2n ti(q) tautologies that can be proved in a constant number of steps, so that no minimal tautology can be obtained from p ′(a′,q) = (q ⊃ p(a′(q)))&t1(q)& . . . &ti(q)& . . . tautology by replacing with new variables neither any of the formulas a′(q),t1(q), . . . ,ti(q), . . ., nor their non-elementary subformulas. 16 investigation of monotonous properties for frege systems proof: let’s describe the modification of each subformula u of a, while we will add new conjuncts, i.e., ti(q)s. 1. if u is a variable, then its modification is itself u ′ = u. no conjunct will be added. 2. if u = (u1 ⊃ u2), and q is true(1), then u is equivalent to (u1 ⊃ (q ⊃ u2)). the modified image of u will be u ′ = (u ′1 ⊃ (q ⊃ u ′2)), where u ′1 and u ′2 are modified images of u1 and u2, correspondingly. we will add new conjuncts ((¬q) ⊃ (u ′1 ⊃ (q ⊃ u ′2))) and ((¬q) ⊃ (q ⊃ u ′2)). if we replace (u ′1 ⊃ (q ⊃ u ′2)) in p ′, then ((¬q) ⊃ (u ′1 ⊃ (q ⊃ u ′2))) will not remain a tautology. if we replace (q ⊃ u ′2), then ((¬q) ⊃ (q ⊃ u ′2)) will not remain a tautology. 3. if u = (¬u1), then we should modify u1 to u ′1. the modified image of u will be u ′ = (¬u ′1). then we add (u ′1 ⊃ (¬(¬u ′1))) in case u ′1 is a tautology, and ((¬(¬u ′1)) ⊃ u ′1) otherwise as a conjunct. if we replace (¬u ′1) or (¬(¬u ′1)) in p ′, then the added conjunction won’t remain a tautology. we can replace u ′1 in case of ¬, and u ′1 and u ′2 in the case of ⊃ in added conjuncts, but p ′(a′,q) won’t remain a tautology, because the corresponding conjuncts were added to p ′, which won’t allow replacing these formulas. the necessity of the given modifications is presented in appendix a. we will add q ⊃ to p(a′(q)) because we assumed q is true(1) when modifying ⊃. so, no minimal can be obtained by replacing any subformula of a or ti(q) in (q ⊃ p(a′(q)))& . . . &ti(q) . . .. lemma 1 is proved. lemma 2: for any formula a of size n, there exists a proof of q ` a′(q) ⊃ a with o(n) steps. proof: for any (u1 ⊃ (q ⊃ u2)) 1. ` (q ⊃ ((u1 ⊃ (q ⊃ u2))) ⊃ (u1 ⊃ u2)) 2. ` ((u1 ⊃ (q ⊃ u2)) ⊃ (u1 ⊃ u2)) modus ponens of premise q and 1. 3. ` (q ⊃ ((u1 ⊃ u2) ⊃ (u1 ⊃ (q ⊃ u2)))) 4. ` ((u1 ⊃ u2) ⊃ (u1 ⊃ (q ⊃ u2))) modus ponens of premise q and 3. ... x1 ` (a′ ⊃ a) from the resulting equivalences using the replacement theorem (replacement theorem 6 [7]) and taking into account that the following formulas are tautologies: ((d ∼ d′) ⊃ ((c ∼ c′) ⊃ ((d ⊃ c) ⊃ (d′ ⊃ c′)))), ((d ∼ d′) ⊃ ((¬d) ∼ (¬d′))). using constant proofs of each of these tautologies and constructing a step by step connecting to two equivalent formulas, not the old ones, but equivalent formulas, x1 = o(n). lemma 2 is proved. consider the formula p = a∨ (p ⊃ p), where p,q,t,r are not present in a. obviously, t∨ (p ⊃ p) is minimal of p. if a is a minimal tautology, then a ∨ r is also minimal of p . we can construct p ′ applying lemma 1 to p : p ′(a′(q)) = (q ⊃ (a′(q) ∨ (p ⊃ p)))& . . . &ti(q)& . . . a. hambardzumyan 17 it’s obvious that p ′ can be proved in o(n) steps and its only minimal is as follows: m(q,r,a) = (q ⊃ (a′(q) ∨ r))& . . . &ti(q)& . . . suppose it can be proved in f(n) steps. the following three tautologies can be proved in at most f(n) steps. (q ⊃ (a′(q) ∨ (¬r)))& . . . &ti(q)& . . . ((¬q) ⊃ (a′(¬q) ∨ r))& . . . &ti((¬q))& . . . ((¬q) ⊃ (a′(¬q) ∨ (¬r)))& . . . &ti((¬q))& . . . now we can prove that t φ a = o(t φ m + |a|) for any minimal tautology a. to do this, we will use the deduction theorem, which will linearly increase the t-complexity of proof. now we will show o(t φ m + |a|) step proof of (¬r),q ` a. (¬r),q ` ... step t φ m . ` m. using the shortest proof of m from the empty set of premises. step t φ m + 1. ` (q ⊃ (a′ ∨ r)). by removing & in m. step t φ m + 2. ` q. it’s a premise. step t φ m + 3. ` (a′ ∨ r). using the modus ponens rule for t φ m + 2 and t φ m + 1. step t φ m + 4. ` ((¬r) ⊃ (a′ ∨ r) ⊃ a′). it’s a tautology of form ((¬r) ⊃ (a∨ r) ⊃ a) step t φ m + 5. ` (¬r). it’s a premise. step t φ m + 6. ` ((a′ ∨ r) ⊃ a′). using modus ponens for t φ m + 5 and t φ m + 4. step t φ m + 7. ` a′. using modus ponens for t φ m + 3 and t φ m + 6. step t φ m + ca. (a ′ ⊃ a). from lemma 2. step t φ m + ca + 1. a. using modus ponens for t φ m + 7 and t φ m + ca, where ca = o(|a|) by lemma 2. we can prove the following with the same number of steps: (¬r), (¬q) `((¬q) ⊃ (a′(¬q) ∨ r))& . . . &ti(¬q)& . . . (¬(¬r)),q ` (q ⊃ (a′(q) ∨ (¬r)))& . . . &ti(q)& . . . (¬(¬r)), (¬q) ` ((¬q) ⊃ (a′(¬q) ∨ (¬r)))& . . . &ti(¬q)& . . . now, using the deduction rule, we can state that there is an o(|a| + f(n)) step proof of the following 4 formulas (q ⊃ ((¬r) ⊃ a)) ((¬q) ⊃ ((¬r) ⊃ a)) (q ⊃ ((¬(¬r)) ⊃ a)) ((¬q) ⊃ ((¬(¬r)) ⊃ a)) from the first two, we will get ((¬r) ⊃ a) in a constant number of steps. from the third and forth, we will get ((¬(¬r)) ⊃ a) in a constant number of steps. from these two we get a with a constant number of steps. so, we got a proof of a with o(|a| + tφm ) steps. note that the proof of (q ⊃ ((¬r) ⊃ a)) has been used 4 times. there exists a sequence of formulas ai, such that t φ ai = ω( |ai|3/2 log3 2 (|ai|) ) by [8]. by the proof above, the following holds t φ ai = o(|ai| + t φ mi ), where mi is the only minimal of p ′(a′i). 18 investigation of monotonous properties for frege systems from the above two estimates we can state that t φ mi = ω( |ai|3/2 log3 2 (|ai|) ). using the lemma 1, the derivation of p ′(a′i) is o(|ai|). of these two, we can say that the only minimal of p ′(a′i) is proved in more steps than the tautology itself for some i. thus, the frege system φ is not monotonous. so, the following was proved: theorem 1: frege systems are not t-monotonous, and consequently, are not t-strongly monotonous. 4. conclusion we prove, that frege systems are not t-monotonous and, as consequence neither t-strongly monotonous. as we state above that all the investigated proof systems are not t-strongly monotonous, then the question of the existence of t-strongly monotonous systems still remains open. 5. acknowledgement this work was supported by the ra mes state committee of science within the framework of the research project 18t-1b034. i am grateful to my supervisor, professor of ysu anahit chubaryan for her encouragement and fruitful discussions, and also for helpful suggestions, which improved and expanded the results. 6. appendix a let’s show that the modifications in lemma 1 need to be made in each subformula of a, but not only in a, in the example of p(a) = a∨ (p ⊃ p). suppose a = ((a ⊃ b) ⊃ ((¬b) ⊃ (¬a))) after the modifications only in the outer implication, the resulting formula will be as follows: p ′ = (q ⊃ ((a ⊃ b) ⊃ (q ⊃ ((¬b) ⊃ (¬a)))) ∨ (p ⊃ p)) &((¬q) ⊃ ((a ⊃ b) ⊃ (q ⊃ ((¬b) ⊃ (¬a))))) &((¬q) ⊃ (q ⊃ ((¬b) ⊃ (¬a)))) let’s replace the formulas (a ⊃ b) and ((¬b) ⊃ (¬a)) in p ′ with t,r correspondingly. the resulting formula will be: m ′ = (q ⊃ (t ⊃ (q ⊃ r)) ∨ (p ⊃ p)) &((¬q) ⊃ (t ⊃ (q ⊃ r))) &((¬q) ⊃ (q ⊃ r)) the resulting tautology m ′ is minimal. thus, p ′ hasn’t unique minimal, as stated in theorem. so, the modification in every subformula in formula a is necessary. a. hambardzumyan 1 9 references [1 ] à. ñ. àíèêååâ, î íåêîòîðîé êëàññèôèêàöèè âûâîäèìûõ ïðîïîçèöèîíàëüíûõ ôîðìóë, ìàòåìàòè÷åñêèå çàìåòêè, ò. 11, í. 2, ññ. 165-174, 1 9 7 2 . 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[8 ] a . a . ch u b a r ya n , h . a . ta m a z ya n a n d a . s . ts h it o ya n , \ s o m e im p r o ve m e n t o f lo we r b o u n d s fo r s t e p s a n d s iz e s o f p r o o fs in fr e g e s ys t e m s " , sciences of e urope, vo l. 1 , n o . 3 7 , p p . 3 9 -4 4 , 2 0 1 9 . submitted 18.03.2020, accepted 26.06.2020. øáýáïáýáõãû³ý ñ³ïïáõãû³ý ñ»ï³½áïáõù üñ»·»ç ñ³ù³ï³ñ·»ñáõù ²ñë»ý ². ð³ùµ³ñóáõùû³ý ºñ¨³ýç å»ï³ï³ý ñ³ù³éë³ñ³ý e-mail: hambardzumyanarsen99@gmail.com ²ù÷á÷áõù êáõûý ³ßë³ï³ýùáõù ñ»ï³½áïí»é »ý ùçýçù³é ýáõûý³µ³ýáõãûáõýý»ñç ¨ ýñ³ýó ù»ç ï³ï³ñí³í ï»õ³¹ñáõãûáõýý»ñç ³ñ¹ûáõýùý»ñç ýí³½³·áõûý ³ñï³íù³ý ù³û»ñç ñ³ñ³µ»ñáõãûáõýá üñ»·»ç ñ³ù³ï³ñ·»ñáõù: ²å³óáõóí³í ¿, áñ ·áûáõãûáõý áõýç ³ûýåçëç ãn áã ùçýçù³é ýáõûý³µ³ýáõãûáõýý»ñç ñ³çáñ¹³ï³ýáõãûáõý, áñáýóçó ûáõñ³ù³ýãûáõñý áõýç ùç³ï ³ûýåçëç ùçýçù³é 'n ýáõûý³µ³ýáõãûáõý, áñ üñ»·»ç ñ³ù³ï³ñ·»ñáõù í»ñççýý»ñç ³ñï³íáõùý»ñç ýí³½³·áõûý ù³ûé»ñç ù³ý³ïý áëï ï³ñ·ç ³í»éç ù»í ¿, ù³ý ãn µ³ý³ó¨»ñç ³ñï³íáõùý»ñç ýí³½³·áõûý ù³ûé»ñç ù³ý³ïá: ´³ý³éç µ³é»ñ` ùçýçù³é ýáõûý³µ³ýáõãûáõý, üñ»·»ç ñ³ù³ï³ñ·»ñ, ³ñï³íù³ý ù³ûé»ñç ù³ý³ï, ³ñï³íù³ý ñ³ù³ï³ñ·»ñç ùáýáïáýáõãûáõý ¨ ëçëï ùáýáïáýáõãûáõý: 2 0 investigation of monotonous properties for frege systems èññëåäîâàíèå ñâîéñòâà ìîíîòîííîñòè ñèñòåì ôðåãå àðñåí à. àìáàðöóìÿí åðåâàíñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò e-mail: hambardzumyanarsen99@gmail.com àííîòàöèÿ â íàñòîÿùåé ðàáîòå äëÿ ñèñòåì ôðåãå èññëåäîâàíî ñîîòíîøåíèå êîëè÷åñòâà øàãîâ âûâîäîâ ìèíèìàëüíûõ òàâòîëîãèé è ðåçóëüòàòîâ ïîäñòàíîâêè â íèõ. äîêàçàíî, ÷òî ñóùåñòâóåò òàêàÿ ïîñëåäîâàòåëüíîñòü íåìèíèìàëüíûõ òàâòîëîãèé ãn, êàæäàÿ èç êîòîðûõ èìååò åäèíñòâåííóþ ìèíèìàëüíóþ 'n è äëÿ êàæäîãî n êîëè÷åñòâî ìèíèìàëüíûõ øàãîâ âûâîäîâ â ñèñòåìàõ ôðåãå ôîðìóë 'n íà ïîðÿäîê áîëüøå êîëè÷åñòâà ìèíèìàëüíûõ øàãîâ âûâîäîâ â ñèñòåìàõ ôðåãå ôîðìóë ãn. êëþ÷åâûå ñëîâà: ìèíèìàëüíàÿ òàâòîëîãèÿ, ñèñòåìû ôðåãå, êîëè÷åñòâî øàãîâ âûâîäîâ, ìîíîòîííîñòü è ñòðîãàÿ ìîíîòîííîñòü ñèñòåì âûâîäîâ. 02_arsen_hambardzumyan_53 02 d:\sbornik\...\tpel.dvi mathematical problems of computer science 35, 53{62, 2011. random coding b ound for e-capacity region of asymmetr ic b r oadcast channel with stochastic e ncoding n a s r in a fs h a r institute for informatics and automation problems of nas of ra abstract e-capacity (rate-reliability) region of the asymmetric broadcast channel with stochastic encoding is studied. the asymmetric broadcast channel involves two discrete memoryless channels with a common input. a common message is transmitted to both receivers and one private message to the intended receiver. we derive an inner bound for rate-reliability region. key words: asymmetric broadcast channel, e-capacity, error exponent, random coding bound, rate-reliability region, stochastic encoding. refer ences [1 ] t. m. co ve r , \ b r o a d c a s t c h a n n e ls " , ie e e trans. inform. theory, vo l. it-1 8 , n o 1 , p p . 2 -1 4 , 1 9 7 2 . 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[1 2 ] c. e . s h a n n o n , \ p r o b a b ilit y o f e r r o r fo r o p t im a l c o d e s in ga u s s ia n c h a n n e ls " , b ell syst. tech. j ., vo l. 3 8 , n o 5 , p p . 6 1 1 -6 5 9 , 1 9 5 9 . è³ûý³ë÷ûáõé ³ýñ³ù³ã³÷ ëïáë³ëïçï ïá¹³íáñù³ùµ ï³åáõõáõ e-áõý³ïáõãû³ý ïçñáõûãç å³ï³ñ³ñ³ý ïá¹³íáñù³ý ·ý³ñ³ï³ï³ýá ü. ²íß³ñ ²ù÷á÷áõù ðá¹í³íáõù áõëáõùý³ëñíáõù ¿ é³ûý³ë÷ûáõé ³ýñ³ù³ã³÷ ëïáë³ëïçï ïá¹³íáñù³ùµ ï³åáõõçý: î³éáõóí³í ¿ e-áõý³ïáõãû³ý (³ñ³·áõãûáõý-ñáõë³éçáõãûáõý) ïçñáõûãç å³ï³ñ³ï³ý ïá¹³íáñù³ý ·ý³ñ³ï³ï³ýá: 1dspin-glass_tpel.dvi mathematical problems of computer science 35, 86{98, 2011. statistical p r oper ties of i deal e nsemble of disor der ed 1d ster ic spin-chains a s h o t s . ge vo r kya n , h a ko b g. a b a jya n a n d h a yk s . s u kia s ya n institute for informatics and automation problems of nas of ra e-mail g ashot@sci.am, habajyan@ipia.sci.am, haikarin@netsys.am abstract the statistical properties of ensemble of disordered 1d steric spin-chains (ssc) of various length are investigated. using 1d spin-glass type classical hamiltonian, the recurrent trigonometrical equations for stationary points and corresponding conditions for the construction of stable 1d sscs are found. the ideal ensemble of spin-chains is analyzed and the latent interconnections between random angles and interaction constants for each set of three nearest-neighboring spins are found. it is analytically proved and by numerical calculation is shown that the interaction constant satis¯es le¶vy's alpha-stable distribution law. energy distribution in ensemble is calculated depending on di®erent conditions of possible polarization of spin-chains. it is speci¯cally shown that the dimensional e®ects in the form of set of local maximums in the energy distribution arise when the number of spin-chains m << n 2x (where nx is the number of spins in a chain) while in the case when m / n 2x energy distribution has one global maximum and the ensemble of spin-chains satis¯es birkho®'s ergodic theorem. e®ective algorithm for parallel simulation of problem which includes calculation of di®erent statistic parameters of 1d sscs ensemble is elaborated. refer ences [1 ] k . b in d e r a n d a . p . y o u n g , \ s p in g la s s e s : e xp e r im e n t a l fa c t s , t h e o r e t ic a l c o n c e p t s , a n d o p e n qu e s t io n s " , r ev. m od. p hysics, vo l. 5 8 , n o . 4 , p p . 8 0 1 -9 7 6 , 1 9 8 6 . 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[7 ] k . v . r . ch a r y, g. go vil, \ n mr in b io lo g ic a l s ys t e m s : fr o m mo le c u le s t o h u m a n " , f ocus on structural b iology 6, s p r in g e r , p . 5 1 1 , 2 0 0 8 . 8 6 a. gevorkyan, h. abajyan and h. sukiasyan 8 7 [8 ] e . b a a ke , m. b a a ke a n d h . w a g n e r , \ is in g qu a n t u m ch a in is a e qu iva le n t t o a mo d e l o f b io lo g ic a l e vo lu t io n " , p hys. r ev. l et., 78( 3 ) , p p . 5 5 9 -5 6 2 , 1 9 9 7 . [9 ] a . s . ge vo r kya n e t a l., \ n e w ma t h e m a t ic a l co n c e p t io n a n d co m p u t a t io n a lg o r it h m fo r s t u d y o f qu a n t u m 3 d d is o r d e r e d s p in s ys t e m u n d e r t h e in ° u e n c e o f e xt e r n a l fie ld " , trans. on comput. sci., v ii, l n cs 1 3 2 -1 5 3 , s p in g e r -v e r la g e , 1 0 .1 0 0 7 / 9 7 8 -3 6 4 2 -1 1 3 8 9 -5 8 . [1 0 ] s . f. e d wa r d s a n d p . w . a n d e r s o n , th e o r y o f s p in g la s s e s , j. p h ys . f 9, p . 9 6 5 , 1 9 7 5 . [1 1 ] j. vo n n e u m a n , \ p h ys ic a l a p p lic a t io n s o f t h e e r g o d ic h yp o t h e s is " , p roc. nat. acad. sci. usa, 18( 3 ) : p p . 2 6 3 -2 6 6 ( 1 9 3 2 ) . [1 2 ] g. d . b ir kh o ®, \ w h a t is e r g o d ic t h e o r e m ?" , american m athematical m onthly, vo l. 4 9 , n o . 4 , p p . 2 2 2 -2 2 6 , 1 9 3 1 . [1 3 ] s . fläu g g e , p ractical quantum mechanics i, s p r in g e r -v e r la g , b e r lin -h e id e lb e r g n e w y o r k, 1 9 7 1 . [1 4 ] m. r . s p ie g le , \ th e o r y a n d p r o b le m s o f p r o b a b ilit y a n d s t o c h a s t ic s " , new-york, m cgraw-hill, p p . 1 1 4 -1 1 5 , 1 9 9 2 . [1 5 ] i. ib r a g im o v a n d y u . l in n ik, independent and stationary sequences of r andom variables, w o lt e r s -n o o r d h o ® p u b lis h in g gr o n in g e n , th e n e t h e r la n d s , 1 9 7 1 . [1 6 ] j. p . n o la n , \ s t a b le d is t r ib u t io n s : m o d e ls fo r h e a vy t a ile d d a t a ( 2 0 0 9 -0 2 -2 1 ) . en:wikipedia:org=stable=distribution. [1 7 ] h . g. k a t z g r a b e r , a . k . h a r t m a n n a n d a . p . y o u n g , \ n e w in s ig h t s fr o m o n e d im e n s io n a l s p in g la s s e s " , a r x iv:0 8 0 3 .3 4 1 7 v1 [c o n d -m a t .d is -n n ], 2 0 0 8 . æ¹»³é³ï³ý 1d ï³ñ³í³ï³ý ãï³ñ·³íáñí³í ëåçý-ßõã³ý»ñç ñ³ùáõûãç íç×³ï³·ñ³ï³ý ñ³ïïáõãûáõýý»ñá ². ¶¨áñ·û³ý, ð. ²µ³çû³ý ¨ ð. êáõùç³ëû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù áõëáõùý³ëçñí³í »ý ï³ñµ»ñ »ñï³ñáõãû³ùµ ç¹»³é³ï³ý 1d ï³ñ³í³ï³ý ãï³ñ·³íáñí³í ëåçý-ßõã³ý»ñç ñ³ùáõûãç (î.â.¸.ê.þ.ð) íç×³ï³·ñ³ï³ý ñ³ïïáõãûáõýý»ñá: ú·ï³·áñí»éáí 1d ëåçý-³å³ïç ïçåç ¹³ë³ï³ý ñ³ùçéïáýç³ýá, ·ïýí»é »ý ëï³óçáý³ñ ï»ïç é»ïáõñ»ýï »é³ýïûáõý³ã³÷³ï³ý ñ³í³ë³ñáõùý»ñá ¨ ñ³ù³å³ï³ëë³ý å³ûù³ýý»ñ‘ 1d ï³ûáõý î.â.¸.ê.þ.ð. ï³éáõó»éáõ ñ³ù³ñ: ì»ñéáõíí»é ¿ ëåçý-ßõã³ý»ñç ç¹»³é³ï³ý ñ³ùáõûãá ¨ ûáõñ³ù³ýãûáõñ 3 ³ù»ý³ùáï ëåçý»ñç µ³½ùáõãû³ý ñ³ù³ñ ·ïýí»é »ý ³ýïûáõý³ûçý ¨ ÷áë³½¹»óáõãû³ý ñ³ëï³ïáõýç ùçç¨ ã³ùýí³í ï³å»ñá: ²ý³éçïçïáñ»ý ³å³óáõóí»é ¿ ¨ ãí³ûçý ñ³ßí³ñïý»ñç ùççáóáí óáõûó ¿ ïñí»é, áñ ÷áë³½¹»óáõãû³ý ñ³ëï³ïáõýá µ³í³ñ³ñáõù ¿ è»íçç ³éý³-ï³ûáõý µ³ßëù³ý ûñ»ýùçý: ð³ùáõûãç ¿ý»ñ·ç³ûç µ³ßëáõùá ñ³ßíí³í ¿ ï³ëí³í ëåçý-ßõã³ý»ñç ï³ñµ»ñ ñý³ñ³íáñ µ¨»é³óí³íáõãû³ý å³ûù³ýý»ñçó: ø³ëý³íáñ³å»ë óáõûó ¿ ïñí³í, áñ >ý»ñ·ç³ûç µ³ßëù³ý ù»ç éáï³é ù³ùëçùáõùý»ñç µ³½ùáõãû³ý ï»ëùáí ³é³ç³ýáõù »ý ã³÷³ûçý >ý»ïïý»ñ, »ñµ ëåçý-ßõã³ý»ñç ù³ý³ïá m << ( nx ) 2 (áñï»õ nx ßõã³ûáõù ëåçýý»ñç ù³ý³ïý ¿): ²ûý ¹»åùáõù, »ñµ m ( nx ) 2, ¿ý»ñ·ç³ûç µ³ßëáõùý áõýç 1 ·éáµ³é ù³ùëçùáõù ¨ ëåçý ßõã³ý»ñç ñ³ùáõûãá µ³í³ñ³ñáõù ¿ ´çñïñáýç >ñ·á¹çï ã»áñ»ùçý: êï»õíí³í ¿ ³ñ¹ûáõý³í»ï ³é·áñçãù ëý¹ñç ½áõ·³ñ»é ùá¹»é³íáñù³ý ñ³ù³ñ, áñá ý»ñ³éáõù ¿ 1d î.â.¸.ê.þ. ñ³ùáõûãç ï³ñµ»ñ íç×³ï³·ñ³ï³ý å³ñ³ù»ïñ»ñç ñ³ßí³ñïá: начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 39, 125--128, 2013. 125 moving objects detection on colored video images suren b. alaverdyan institute for informatics and automation problems of nas ra e-mail: souren@ipia.sci.am abstract the background covering formulas of images are obtained. with their help the algorithm of detection of moving objects on colored video images is given. keywords: image, detection. 1. introduction when solving various problems of colored image processing, the image is represented in a gray scale ( ) ⁄ or to accelerate processing and reduce the used memory excluding the possible damage of the original image (loss of contour points or appearance of false contour points). in fig. 2 the difference between the original and the gray scale images is given, where the values of ( )-th point are as follows: | | | | | | fig 1. original fig. 2. difference it is clear that the components artificially added to the original image, influence the result of the problem being solved. . while obtaining the image in image processing theory, the image device accuracy, image quantization and, certainly, the environment which influences the formation of the image noise, are principally important. , moreover, since for the problems the detection of objects on images or their segmentation the problem of general filters' synthesis is not solved, then in practice to mailto:souren@ipia.sci.am moving objects detection on colored video images 126 solve these problems different approaches are used -threshold restriction, brightness interval transformation, cluster analysis etc. – [1, 2]. in this article a new approach of moving objects detection on video images, obtained by a stationary camera, is given. 2. image background change by standard image a mathematical definition of the background does not exist, it is interpreted as the background of image [3, 4]. let { ̅̅ ̅̅ ̅ ̅̅̅̅̅} ( ), (red, green and blue colour channels of a point), and the standard image with the size . to reduce the background of the image background of the image we find the following statistical characteristics of both images: mean values ∑ ∑ , ∑ ∑ , mean square deviations √ ∑ ( ) , √ ∑ ( ) . the general formula for the new values of the image background of the image represented as ( ) , ̅̅̅̅ ̅̅ ̅̅̅̅ ̅ , (1) let if and let if below we give an example of the background change by formula (1). fig. 3. original image fig. 4. standard image fig. 5. resulting image thus, the parameters ( , ) can determine the approximate value of the image background. considering statistical moments of higher orders one can obtain a more exact value of background. 3. object detecting to avoid artificially covering the image with new data (fig. 2) we do not use in this work a gray scale image to detect objects. there are two principal methods used to detect moving video objects on video streams: determination of image stationary part and statistical methods with threshold restriction. in this work to detect the objects the original image is transformed into a negative image background space by formula (1) to reduce the probability of loss of dark colored objects. since the negative mean value is equal to the formula (1) takes the following form: s. alaverdyan 127 ( ) , ̅̅̅̅ ̅̅ ̅̅̅̅ ̅ . to obtain a binary (black-white) image, calculate the inter frame gradient using the following formula: {| ( ) ( ) | ( ) } , ̅̅ ̅̅ ̅̅ ̅̅̅̅ ̅ where ( ) √∑ ∑ ( ( ) ( ) ) , is the frame number, ( ) interval of ( ) point is used. the threshold is determined as follows: ∑ ∑ or . both of threshold values give good results. binary image { ̅̅̅̅ ̅̅ ̅̅̅̅ ̅} is obtained as follows: { below the results of object detecting algorithms are given. fig. 6. original image with negative background fig. 7. objects fig. 8. original image with negative background fig. 9. objects moving objects detection on colored video images 128 references [1] у. прэтт, цифровая обработка изображений, м. мир, т.2, 1982. [2] t. ko, s. soatto and d. estrin, “background subtraction on distributions”, proceedings of the 10th european conference on computer vision: part iii, pp. 276-289, 2008. [3] v. konushin and a. konushin, “improvement of background subtraction by mask constraints”, proc. graphicon, pp. 96-99, 2010. [4] p viola, m jones, “rapid object detection using a boosted cascade of simple features”, ieee computer society conference on computer vision and pattern recognition, pp. 511-518, 2001. submitted 30.11.2012, accepted 15.02.2013. տեսաշարում գունավոր պատկերների վրա շարժվող օբյեկտների հայտնաբերում ս. ալավերդյան ամփոփում աշխատանքում ստացված են պատկերների ֆոնային տեղափոխության բանաձևեր, բերված է նրանց կիրառության միջոցով տեսաշարում գունավոր պատկերների վրա շարժվող օբյեկտների հայտնաբերման ալգորիթմը: обнаружение движущихся объектов на видеоизображениях с. алавердян аннотация приведен новый алгоритм нахождения движущихся объектов на цвeтных видеоизображениях. приведены также результаты работы алгоритма. d:\sbornik\...\article1.dvi mathematical problems of computer science 33, 59{65, 2010. statistical p ostpr ocessing of the output fr om n umer ical weather p r ediction system tig r a n v . k h o t s a n ya n , v la d im ir g. s a h a kya n a n d ir in a a . s a fa r ya n institute for informatics and automation problems of nas ra abstract this article is devoted to the investigation of relation between really observed meteorological parameters and the forecast of the same parameters produced by a numerical weather prediction system. a statistical correction method of the output from the model of the system is suggested which improves the forecast by coe±cient of overlapping, root mean square error and pearson's correlation. refer ences [1 ] t.n . k r is h n a m u r t i, a n d l . b o u n o u a ,\ a n in t r o d u c t io n t o n u m e r ic a l w e a t h e r p r e d ic t io n te c h n iqu e s " , b oca r aton f l us:, cr c p ress, 1 9 9 6 . 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[7 ] n oa a 's n a t io n a l w e a t h e r s e r vic e . cu r r e n t mos fo r e c a s t p r o d u c t io n , h t t p :/ / www.n ws .n o a a / m d l/ s yn o p / p r o d u c t s .p h p [8 ] j. p . d a lla va lle , m. c. e r ic ks o n , a n d j. c. ma lo n e y, \ mo d e l o u t p u t s t a t is t ic s ( mos ) g u id a n c e fo r s h o r t -r a n g e p r o je c t io n s " , 20th conference of w eather analysis and f orecasting, seattle, amer. m eteor.soc., p r e p r in t s , p p .1 -1 1 , 2 0 0 4 . [9 ] c. ma z b r a n , s . s a n d g a t h e , e . k a ln a y, \ mos , p e r fe c t p r o g a n d r e a n a lys is " , m on. w ea. r ev.,1 3 4 , p p . 6 5 8 -6 6 3 , 2 0 0 6 . [1 0 ] f. s c h m id a n d a . s c h m id t , \ n o n p a r a m e t r ic e s t im a t io n o f c o e ± c ie n t o f o ve r la p p in g t h e o r y a n d e m p ir ic a l a p p lic a t io n " , computational statistics and d ata analysis, vo l. 5 0 , p p . 1 5 8 3 -1 5 9 6 , 2 0 0 6 . 5 9 6 0 statistical postprocessing of the output from numerical weather prediction system [1 1 ] t.v . k h o t s a n ya n , \ n u m e r ic a l we a t h e r p r e d ic t io n e n vir o n m e n t fo r t h e t e r r it o r y o f a r m e n ia " ,m athematical p roblems of computer science vo l. 3 3 , 8 3 { 9 0 , 2 0 1 0 . [1 2 ] t. gn e it in g , f. b a la b d a o u i a n d a . e . r a ft e r y, \ p r o b a b ilis t ic fo r e c a s t s , c a lib r a t io n a n d s h a r p n e s s " , technical r eport, n 4 8 3 , p p . 1 -3 1 , d e p a r t m e n t o f s t a t is t ic s , u n ive r s it y o f w a s h in g t o n , 2 0 0 5 . [1 3 ] h t t p :/ / www.c lu s t e r .a m / wr f/ s t a t [1 4 ] d . s . w ilks , t. m. h a m ill, \ co m p a r is o n o f e n s e m b le -mos u s in g gfs r e fo r e c a s t " , mo n . w e a . r e v.1 3 5 , p p . 2 3 8 0 -2 3 8 9 , 2 0 0 7 . ºõ³ý³ïç ãí³ûçý ï³ýë³ï»ëù³ý ñ³ù³ï³ñ·ç ³ñ¹ûáõýùý»ñç íç×³ï³·ñ³ï³ý í»ñ³ùß³ïáõù î. êáó³ýû³ý, ì. ê³ñ³ïû³ý ¨ æ. ê³ý³ñû³ý ²ù÷á÷áõù ²ßë³ï³ýùá ýíçñí³í ¿ çñ³ï³ýáõù ¹çï³ñïí³í ¨ ï³ýë³ï»ëí³í ùç¨ýáõûý û¹»ñ¨áõã³µ³ý³ï³ý å³ñ³ù»ïñ»ñç ñ³ñ³µ»ñáõãû³ý ñ»ï³½áïù³ýá: ²é³ç³ñïí³í ¿ ï³ýë³ï»ëí³í ïíû³éý»ñç áõõõù³ý íç×³ï³·ñ³ï³ý »õ³ý³ï, áñá µ³ñ»é³íáõù ¿ ï³ýë³ï»ëáõùá áëï í³íïí³íáõãû³ý ·áñí³ïóç, ùçççý ù³é³ïáõë³ûçý ëë³éç ³ñù³ïç ¨ äçñëáýç ñ³ñ³µ»ñ³ïóáõãû³ý ·áñí³ïçóá: d:\sbornik\...\idenm2.dvi mathematical problems of computer science 33, 91{94, 2010 remar ks about reliable i denti¯cation of p r obability distr ibutions of t wo i ndependent objects e vg u e n i a . h a r o u t u n ia n a n d p a r a n d z e m m. h a ko b ya n institue for informatics and automation problems of nas of ra e-mail: evhar@ipia.sci.am abstract in this paper we present some additions to results on logarithmically asymptotically optimal identi¯cation of probability distributions of two independent objects, published by authors in 2007. refer ences [1 ] r . f. a h ls we d e a n d e . a . h a r o u t u n ia n , \ on lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f h yp o t h e s e s a n d id e n t ī c a t io n " . l e c t u r e n o t e s in co m p u t e r s c ie n c e , vo l. 4 1 2 3 ,\ ge n e r a l th e o r y o f in fo r m a t io n tr a n s fe r a n d co m b in a t o r ic s " s p r in g e r , p p . 4 6 2 { 4 7 8 , 2 0 0 6 . [2 ] e . a . h a r o u t u n ia n a n d p . m. h a ko b ya n , \ on id e n t i¯ c a t io n o f d is t r ib u t io n s o f two in d e p e n d e n t ob je c t s " , m athematical p roblems of computer science, vo l. 2 8 , p p . 1 1 4 { 1 1 9 , 2 0 0 7 , [3 ] l . n a va e i , \ on r e lia b le id e n t ī c a t io n o f t wo in d e p e n d e n t ma r ko v c h a in " , m athematical p roblems of computer sciences , vo l. 3 2 , p p . 7 5 -7 9 , 2 0 0 9 . [4 ] e . a . h a r o u t u n ia n , \ l o g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f m u lt ip le s t a t is t ic a l h yp o t h e s e s " , p roblems of control and information theory, vo l. 1 9 ( 5 -6 ) , p p . 4 1 3 { 4 2 1 , 1 9 9 0 . ¸çïáõáõãûáõýý»ñ »ñïáõ ³ýï³ë ûµû»ïïý»ñç ñ³í³ý³ï³ý³ûçý µ³ßëáõùý»ñç ñáõë³éç ýáõûý³ï³ý³óù³ý í»ñ³µ»ñû³é º. ð³ñáõãûáõýû³ý ¨ ö. ð³ïáµû³ý ²ù÷á÷áõù ðá¹í³íáõù áõëáõùý³ëçñíáõù ¿ »ñïáõ ³ýï³ë ûµû»ïïý»ñç µ³ßëáõùý»ñç éá·³ñãùáñ»ý ³ëçùåïáïáñ»ý ûåïçù³é ýáõûý³ï³ý³óáõùá: þïïí»é »ý ý³ëáñ¹ 2007ã. ñá¹í³íáõù ýï³ïí³í ã»ñáõãûáõýý»ñá: 9 1 d:\sbornik\...\tpel.dvi mathematical problems of computer science 36, 63{69, 2012. designing a n ew fr amewor k for the p atient a®ected with h ear t p r oblem i-s h ya n h wa n g y a n d d . b e yb u t ya n z yyuan ze university chung-li, taiwan zinstitute for informatics and automation problems of nas of ra e-mail: ishwang@saturn.yzu.edu.tw, beybutyandavid@gmail.com abstract according to recent research healthcare is becoming more expensive and less accessible for general public. cellular phones and gsm networks are considered as cheapest and most accessible means of communications, so they can be used as one of the best devices in creating healthcare systems. during recent years several commercial tools are introduced to monitor and improve health. each of the present pieces of software in the ¯eld of heart diseases accounts for a separate task. in present work we try to de¯ne a new frame work and software that can cover all kinds of heart disorders is an ideal for patients and that they can use it with the least know-how and e®ort. we designed the software so, that it should be easy to use, independent from any operating system and hardware of cell phone, independent from place as well as it can predict heart attack and in case of emergency, immediately send the alarm. keywords: mobile healthcare, gsm, ecg signal, heart attack, visual data, predict, software, java. refer ences [1 ] th e u n ive r s it y o f ma in e the u.s. health care system: best in the world, or just the most expensive? h t t p :/ / d ll.u m a in e .e d u / b le / u .s . 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[5 ] mo b ile h e a lt h : th e p o t e n t ia l o f m o b ile t e le p h o n y t o b r in g h e a lt h c a r e t o t h e m a jo r it y. 2 0 0 9 . a va ila b le a t h t t p :/ / id b d o c s .ia d b .o r g / ws d o c s / g e t d o c u m e n t .a s p x?d o c n u m 1 7 2 8 0 6 1 , a u g u s t 1 , 2 0 1 0 . 6 3 6 4 designing a new framework for the patient a®ected with heart problem [6 ] r . a . cla r k, s .c. in g lis , f.a . mc a lis t e r , a .l . s m it h , " te le m o n it o r in g o r s t r u c t u r e d t e le p h o n e s u p p o r t p r o g r a m m e s fo r p a t ie n t s wit h c h r o n ic h e a r t fa ilu r e : s ys t e m a t ic r e vie w a n d m e t a -a n a lys is " .b r m ed j ; 3 3 4 , p p . 9 4 2 -9 5 0 , 2 0 0 7 . [7 ] moca . ca r e a n ywh e r e -w ir e le s s h e a lt h c a r e , mit moca p r o je c t . a va ila b le a t www.m o c a m o b ile .o r g = la s t a c c e s s e d oc t o b e r 2 0 , 2 0 0 9 . 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[2 7 ] s .b . p a t il, y .s . k u m a r a s wa m y, \ in t e llig e n t a n d e ®e c t ive h e a r t a t t a c k p r e d ic t io n s ys t e m u s in g d a t a m in in g a n d a r t i¯ c ia l n e u r a l n e t wo r k" , e uropean j ournal of scienti¯c r esearch, vo l. 3 1 , n o . 4 , p p . 6 4 2 -6 5 6 , 2 0 0 9 . [2 8 ] h .g. l e e , k .y . n o h , h .k . p a r k, k .h . r yu , \ p r e d ic t in g c o r o n a r y a r t e r y d is e a s e fr o m h e a r t r a t e va r ia b ilit y u s in g c la s s ī c a t io n a n d s t a t is t ic a l a n a lys is " . in: 7th ie e e international conference on computer and information technology, p p . 5 9 -6 4 , 2 0 0 7 . [2 9 ] s u n d e ve lo p e r n e t wo r k-mid p gu i p r o g r a m m in g : p r o g r a m m in g t h e p h o n e in t e r fa c e h t t p :/ / d e ve lo p e r s . s u n .c o m / t e c h t o p ic s / m o b ilit y/ m id p / a r t ic le s / u i/ s . [3 0 ] l . h e s lo p , s . w e e d in g , l . d a ws o n a n d o t h e r . \ im p le m e n t a t io n is s u e s fo r m o b ile -wir e le s s in fr a s t r u c t u r e a n d m o b ile h e a lt h c a r e c o m p u t in g d e vic e s fo r a h o s p it a l wa r d s e t t in g " , j . m ed. syst, vo l. 3 4 , p p . 5 0 9 -5 1 8 , 2 0 1 0 . [3 1 ] e . jo va n vo c , a . mile n ko vic , c. ot t o , \ a wir e le s s b o d y a r e a n e t wo r k o f in t e llig e n t m o t io n s e n s o r s fo r c o m p u t e r a s s is t e d p h ys ic a l r e h a b ilit a t io n " , j ournal of neuro-engineering and r ehabilitation, vo l. 2 , n o . 6 , ma r c h 2 0 0 5 . üáñ íñ³·ñ³ûçý ùáï»óáõùý»ñç ý³ë³·íáõù ëñï³ýáã³ûçý ñçí³ý¹áõãûáõý áõý»óáõý»ñç ñ³ù³ñ æ-þû³ý ðí³ý· ¨ ¸. ´»ûµáõãû³ý ²ù÷á÷áõù àëï í»ñççý ñ»ï³½áïáõãûáõýý»ñç ³ñ¹ûáõýùý»ñç ³éáõç³å³ñáõãûáõýá ¹³éýáõù ¿ ³í»éç ã³ýï ¨ áã ñ³ë³ý»éç ñ³ýñáõãû³ýá: ´çç³ûçý ñ»é³ëáëý»ñá ¨ gsm ó³ýó»ñá ñ³ù³ñíáõù »ý ³ù»ý³¿å³ý ¨ ³é³í»é ù³ïã»éç ï³åç ùççáóý»ñ: ð»ï¨³µ³ñ, ³éáõç³å³ñáõãû³ý ñ³ù³ï³ñ·áõù ýñ³ýù ï³ñáõ »ý í³é³û»é áñå»ë é³í³·áõûý ë³ñù³íáñáõùý»ñ: ì»ñççý ï³ñçý»ñç áýã³óùáõù ³é³ç³ñïí»é »ý áñáß ïáù»ñóçáý ë³ñù³íáñáõùý»ñ, áñáýù ãáõûé³ïñáõù »ý í»ñ³ñëï»é ¨ µ³ñ»é³í»é ñçí³ý¹ç ³éáõç³ï³ý íç×³ïá: êñïç ñçí³ý¹áõãûáõýý»ñç áéáñïáõù ³ù»ý ùç íñ³·çñ û·ï³·áñííáõù ¿ »½³ïç ëý¹ñç éáõíù³ý ñ³ù³ñ: ü»ñï³ ³ßë³ï³ýùáõù ÷áñó»é »ýù áñáß³ïç³óý»é ³ßë³ï³ýùç ³ûýåçëç ßñç³ý³ï³ý»ñ áõ íñ³·çñ, áñáýù ïï³ñáõ³ý³ý í³íï»é ëñïç ³ßë³ï³ýùç µáéáñ ïçåç ³ýï³ñ·³íáñí³íáõãûáõýý»ñá: ðçí³ý¹ý»ñç ñ³ù³ñ ¹³ ïéçýç ç¹»³é³ï³ý ³ûýù³ýáí, áñ ýñ³ýù ïï³ñáõ³ý³ý ³ûý û·ï³·áñí»é ùçýçù³é áõëáõóù³ý ¨ ç³ýù»ñç ·ýáí: ìñ³·çñá ·ñí³í ¿ ³ûýå»ë, áñ ³ûý éçýç ñ»ßï ïçñ³é»éç, ³ýï³ë ó³ýï³ó³í ûå»ñ³óçáý ñ³ù³ï³ñ·çó 6 6 designing a new framework for the patient a®ected with heart problem ¨ µçç³ûçý ñ»é³ëáëç ï³éáõóí³íùçó: ²ûý å»ïù ¿ ³ýï³ë éçýç ï»õçó, çýãå»ë ý³¨ ï³ñáõ³ý³ ï³ýë³·áõß³ï»é ëñïç ï³ãí³íá, çëï ³ñï³ï³ñ· å³ï³ñ³ñç ¹»åùáõù ùç³ý·³ùçó ³½¹³ýß³ý ï³é: ðàçðàáîòêà íîâûõ ñòðóêòóðíûõ ðåøåíèé äëÿ áîëüíûõ ñ ñåðäå÷íî-ñîñóäèñòûìè çàáîëåâàíèÿìè è-øèàí àâàíã è ä. áåèáóòÿí àííîòàöèÿ ñîãëàñíî ïîñëåäíèì èññëåäîâàíèÿì ìåäèöèíñêîå îáñëóæèâàíèå ñòàíîâèòñÿ áîëåå äîðîãîñòîÿùåé è ìåíåå äîñòóïíîé äëÿ øèðîêîé îáùåñòâåííîñòè. ñîòîâûå òåëåôîíû è gsm-ñåòè ñ÷èòàþòñÿ ñàìûìè äåøåâûìè è íàèáîëåå äîñòóïíûìè ñðåäñòâàìè êîììóíèêàöèè, ïîýòîìó îíè ìîãóò áûòü èñïîëüçîâàíû â êà÷åñòâå îäíîé èç íàèáîëåå ýôôåêòèâíûõ óñòðîéñò ïðè ñîçäàíèè ñèñòå ìçäðàâîîõðàíåíèÿ. â ïîñëåäíèå ãîäû áûëè ïðåäëîæåíû ðÿä êîììåð÷åñêèõ èíñòðóìåíòîâ ïî êîíòðîëþ è óëó÷øåíèþ çäîðîâüÿ. êàæäàÿ èç ñóùåñòâóþùèõ ïðîãðàìì â îáëàñòè çàáîëåâàíèé ñåðäöà ðàñ÷èòàíà íà ðåøåíèå îòäåëüíîé êîíêðåòíîé çàäà÷è. â íàñòîÿùåé ðàáîòå ìû ïîïûòàëèñü îïðåäåëèòü íîâûå ðàìêè ðàáîòû è ïðîãðàììíîãî îáåñïå÷åíèÿ, êîòîðîå ìîæåò îõâàòèòü âñå âèäû ñåðäå÷íûõ çàáîëåâàíèé è ÿâëÿåòñÿ èäåàëüíûì äëÿ ïàöèåíòîâ, òàê ÷òî îíè ìîãóò èñïîëüçîâàòü åãî ëåãêî è áåç ñïåöèàëüíîé ïîäãîòîâêè. ìû ñîñòàâèëè ïðîãðàììíîå îáåñïå÷åíèå òàêèì îáðàçîì, ÷òîáû îíî áûëî ïðîñòûì â èñïîëüçîâàíèè, íåçàâèñÿùèì îò îïåðàöèîííîé ñèñòåìû è óñòðîéñòâà ñîòîâîãî òåëåôîíà, à òàêæå îò ãåîãðàôè÷åñêîãî ìåñòîïîëîæåíèÿ. îíî òàêæå ìîæåò ïðåäñêàçàòü ñåðäå÷íûé ïðèñòóï, à â ñëó÷àå ÷ðåçâû÷àéíîé ñèòóàöèè íåçàìåäëèòåëüíî ïîäàòü ñèãíàë òðåâîãè. microsoft word 16.doc mathematical problems of computer science 35, 116--127, 2011. 116 effective, secure and robust cdma digital image watermarking in ycbcr channel using dwt2 mehdi khalili 1 and david g. asatryan 2 1 institute for informatics and automation problems of nas ra 2 russian-armenian (slavonic) university e-mail: khalili@ipia.sci.am 1 , dasat@ipia.sci.am 2 abstract digital watermarking has been widely used in digital rights management and copyright protection. in this paper, a new cryptographic cdma watermark scheme is proposed. compared to the existing watermarking techniques, this proposed watermarking scheme combine transparency, security and robustness that none of the existing schemes can do. in this paper, a wavelet-based watermarking approach for hiding watermark image in color host images is proposed. the experimental results show that the proposed approach provides extra imperceptibility, security and robustness against jpeg compression and different noise attacks such as gaussian and “salt & pepper” compared to the similar proposed methods such as [7]. moreover, the proposed approach has no need of the original image to extract watermarks. references [1] jian. ren, “a cryptographic watermarking technique for multimedia signals”, springer science, business media, august 2008. [2] s.-h. wang and y.-p lin, “wavelet tree quantization for copyright protection watermarking”, ieee trans, image process , vol.13, no. 2, pp. 154–165, 2004. [3] h.s. malvar, d.a.f florêncio, “improved spread spectrum: a new modulation technique for robust watermarking”, ieee trans. signal proc., vol. 51, no. 4, pp. 898–905, 2003. [4] x.g. xia, c. g. boncelet and g. r. arce, “wavelet transform based watermark for digital images”, optics express, vol. m, no. 12, pp.497-511, 1998. [5] m. khalili, “a comparison between digital images watermarking in two different color spaces using dwt2”, proceedings of the 7th international conference on computer science and information technologies, pp. 158-162, yerevan, armenia, 2009. [6] m. khalili and d. asatryan, “effective digital image watermarking in ycbcr color space accompanied by presenting a novel technique using dwt”, mathematical problems of computer science, vol. 33, pp. 150—161, 2010. m. khalili, d. asatryan 117 [7] y. fang, j. huang and yun q. shi, “image watermarking algorithm applying cdma”, ieee, 2003. [8] qian-chuan zhong, qing-xin, pinglizhang, “a satial domain color watermarking scheme based on chaos”, ieee, 2008. [9] x.g. xia, c. g. boncelet and g. r. arce, “wavelet transform based watermark for digital images”, optics express, vol. m, no. 12, pp.497-511, 1998. [10] c. lin, “face detection in complicated backgrounds and different illumination conditions by using ycbcr color space and neural network”, elsevier, pattern recognition letters 28 (2007), pp. 2190–2200, 2007. [11] m.-s. hsieh, “wavelet-based image watermarking and compression”, ph-d thesis, institute of computer science and information engineering national central university, taiwan, 2001. [12] a. william and irizarry-cruz, “fpga implementation of a video watermarking algorithm”,, m.s. thesis, university of puerto rico mayaguez campus, 2006. [13] f. a. p. petitcolas, “watermarking schemes evaluation” ieee signal processing, vol. 17, no. 5, pp. 58–64, 2000. [14] arvind kumar parthasarathy, “improved content based watermarking for images”, m.s. thesis, submitted to the graduate faculty of the louisiana state university and agricultural and mechanical college in partial fulfillment of the requirements, 2006. dwt2 ïçñ³éù³ùµ ycbcr ï³åáõõáõù cdma å³ïï»ñç ãí³ûçý ³ñ¹ûáõý³í»ï, íëï³ñ»éç ¨ ï³ûáõý çñ³ýßáõù ø. ê³éçéç ¨ ¸. ²ë³ïñû³ý ամփոփում âí³ûçý çñ³ýßáõùá é³ûýáñ»ý ïçñ³éíáõù ¿ ñ»õçý³ï³ûçý çñ³íáõýùç å³ßïå³ýáõãû³ý µý³·³í³éáõù: ðá¹í³íáõù ³é³ç³ñïí»é ¿ çñ³ýßù³ý cdma í³íï³·ñ³ï³ý ýáñ ëë»ù³: ²é³ç³ñïí³í çñ³ýßù³ý ëë»ù³ý ñ³ù³ïóáõù ¿ ã³÷³ýóçïáõãû³ý, íëï³ñ»éçáõãû³ý ¨ ï³ûáõýáõãû³ý ³ûýåçëç ñ³ïïáõãûáõýý»ñ, áñáýù ç ½áñáõ ã¿ ³å³ñáí»éáõ ·áûáõãûáõý áõý»óáõ çñ³ýßù³ý ëë»ù³ý»ñçó áã ù»ïá: êáõûý ñá¹í³íáõù ·áõý³íáñ å³ïï»ñý»ñáõù çñ³ýçßç ï»õ³ï³ûù³ý ñ³ù³ñ ³é³ç³ñïí»é ¿ í»ûíé»ïý»ñç ïçñ³éù³ý íñ³ ñçùýí³í ùáï»óáõù,: öáñóç ïíû³éý»ñá óáõûó »ý ï³éçë, áñ ³é³ç³ñïí³í ùáï»óáõùá ýù³ý³ïçå ³ûé ù»ãá¹ý»ñç (çýãå»ë ûñçý³ï` [7]) ýï³ïù³ùµ ³å³ñáíáõù ¿ éñ³óáõóçã ã³÷³ýóçïáõãûáõý, íëï³ñ»éçáõãûáõý ¨ ï³ûáõýáõãûáõý jpeg ë»õùù³ý ¨ ï³ñµ»ñ å³ï³ñ³ï³ý ³õùáõïý»ñç, çýãå»ë ûñçý³ï` ·³áõëû³ý, “³õ ¨ åõå»õ” ï»ë³ïç, ýï³ïù³ùµ: ²í»éçý, ³é³ç³ñïí³í ùáï»óáõùá çñ³ýçßç ³ñï³íù³ý ñ³ù³ñ ãç å³ñ³ýçáõù çñ³ýçßç µýûñçý³ïç ³éï³ûáõãûáõýá: microsoft word v. sahakyan_5.doc mathematical problems of computer science 34, 2010. 18 about the queue organization in the multiprocessor computing systems vladimir sahakyan (vladimir.sahakyan@sci.am) the development of distributed and parallel computing systems is caused by the tasks of information search, multi-agent intelligent systems, and storage of large amounts of data and research on various scientific directions. many tasks of modeling in science and technology require more and more accuracy and timeliness of their calculation, i.e. fast computational resources. before these similar tasks could be solved within the common powerful computers, today there are already tasks requiring decisions within large supercomputing facilities. the development of scientific research in e-infrastructure can be implemented on a combination of computational resources within the grid-system. the implementation of grid technologies allows building management systems of distributed computing resources in which it is not important for the user at what particular node of network his task is executed. grid is different from services of internet ([1]) which supports not only work with the distributed information, but also uses the services of distributed computing capacities for performance of users’ applications. in the base of grid-system there are powerful computing clusters consisting of many multiprocessor nodes. as a rule, clusters solve the problems in which the interaction between separate computing nodes is organized by means of transferring messages. grid-systems are not intended for decision of parallel tasks, but are intended for the solution of batch tasks, when each separate parallel task is executed entirely on one node. the queue system of tasks manages different tasks. the operational performance of task in grid-environments requires organizing the dynamic distribution and the management of resources. for that new models and methods are required. one of the most important aspects of computing systems is the availability of a flexible system of queuing. it should be noted that the parameters of running tasks contain information about the number of required processors, the quantity of required memory and disk memory with input and output data, as well as may contain temporary parameters such as duration and the terms of running tasks. in this case, there are some tasks of optimum use of computing resources, by means of redistributing tasks between queues inside the grid-system. these tasks also include the process of stopping the running tasks, transferring them to a new turn and restart to continuation. the optimal use of processor time in multiprocessor systems of cluster type depends from many factors, such as the method of receptions and productions tasks in queue, the definition of order performance, the opportunities of computing resources dynamic distributions ([2, 3]), the opportunities of transferring task on various performance phases into minimally necessary environment or performance stop with opportunity of continuation, etc. reception tasks in the system for service plays important role in organization of its process. the opportunity of distributed processes interactions in determined intervals of time requires the synchronization and simultaneous performance as in one so in different computing systems. therefore, 19 the acceptance of tasks from queue for the service imposes the responsibility on scheduler for supplying its duly performance. let a computing system consisting from n processors (n≥1) act with flow tasks. every task is characterized by three parameters (ν, τ, ω), where ν -the number of processors, necessary for performance tasks, τ maximal time, required for performance, ω admitted time for holding tasks in queue before performance beginning. in received moment, the task can be accepted for service only on condition of its performance opportunities in term. in opposite case it receives failure in service. the algorithm of checking conditions for opportunities is described for various restrictions on order performance. references 1. ian foster, carl kesselman, steven tuecke, the anatomy of the grid: enabling scalable virtual organizations, journal international journal of high performance computing applications, volume 15 issue 3, august 2001 2. t.grigoryan ,v.sahakyan dynamic resource manager for clusters, proceedings of conference. computer science and information technolgies, 2005 3. astsatryan h., yu.shoukouryan, v.sahakyan, m. daydé, a.hurault, m.pantel, e.caron, a gridaware web interface with advanced service trading for linear algebra calculations, high performance computing for computational science, vecpar2008, lecture notes in computer science 5336 springer 2008, pp. 150-159, toulouse, france, june 24-28, 2008. mathematical problems of computer science 41, 114---121, 2014. 114 an automated system for correction of rules of transformation from universal networking language into natural language1 aram a. avetisyan and levon r. hakobyan institute for informatics and automation problems of nas ra e-mail: a.avetisyan@undlfoundation.org, levon.r.hakobyan@gmail.com abstract an approach to automated correction of unl grammar rules is implemented. the development process of the rule sets could be made in a more effective way if some tools are used which can find the rules responsible for possible errors and make suggestions about its correction. several definitions for the formalization of the transformation process and the algorithm for such kind of functionality are created. the implementation of the algorithm in the undl platform and some examples are described. keywords: unl, machine translations, transformation rules, grammar. 1. introduction unl (universal networking language) is a meta-language representing semantic information. its main purpose is to store “the meaning” of natural language texts in a language-independent format. each sentence in unl is a oriented linked graph [1,2,3]. any sentence in unl is a semantic network believed to be logically precise, humanly readable and computationally tractable. in the unl approach, information conveyed by natural language is represented sentence by sentence, as a graph composed of a set of oriented binary labeled links (referred to as “relations”) between nodes (the “universal words”, or simply “uw”), which stand for concepts. uws can also be annotated with “attributes" representing modality [1,2]. today one of the main systems for the unl processing is undl platform. undl platform is a web-based application which was developed by undl foundation [3]. transformation process from natural language to unl is called analysis and from unl to natural language generation. in undl platform the analyzer (i.e. algorithm which implements the analysis) was developed under the ian project, and the generator (i.e. algorithm which implements the generation) under the eugene project [3]. 1 this work was supported by state committee of science, mes ra, in frame of the research project scs 13-b321 a. avetisyan and l. hakobyan 115 transformation applications use dictionaries and grammar rule sets for the corresponding natural language. therefore, one of the main goals of the unl project is the development of grammar rule sets. but the process of creating and editing such rules can be a big challenge for users. in this article we introduce an approach to the automation of the rules correction process using real examples. we introduce some new definitions, describe the corresponding algorithm for responsible rules searching and give basic information about its implementation in the undl platform system. 2. the problem setting transformations in the undl platform system are implemented by consequently applying transformation rules on the unl sentence. in case of analyzer, the result of this process is transformation from the linear structure to the semantic graph. in case of generator, it is transformation from the semantic graph to the linear structure. transformation from unl to natural language has the following input parameters: dictionary, unl sentence and grammar rule set. there are two types of grammar rules: transformation rules (t-rules) and disambiguation rules (d-rules) [4]. transformation rule is a pair (α, β), where the left side α is a condition statement, and the right side β is an action to be performed over α. disambiguation rule is a pair (α, p), where the left side α is a statement and the right side p is an integer from 0 to 255 that indicates the priority coefficient which is taken into account for the implementation of α. without loss of generality, let us assume that there are no d-rules, and attach priorities to the corresponding t-rules. the transformation process consists of several iterations (steps). on each step one t-rule is applied. rule applying process continues until there are no rules which can be applied to the sentence. on each step, transformation algorithm loops through all t-rules. if a t-rule with a satisfying condition is found, that rule will be applied to the current sentence, i.e. an action of current t-rule will be applied to the current sentence. when the algorithm finishes, the user gets a transformation result and trace-data. trace-data contains information about all steps of the transformation process applied to the sentence. in fact, this is the description of the transformation process. but because of a huge amount of information, this description is not easy for the users to deal with. if the user finds mistakes in the result of transformation, he has to look for a t-rule that caused the current mistake to appear in the result. this process is not really effective for the user. this approach can become even more ineffective if the user also wants to find dependencies between rules and to make a decision on how applying some rules affect on choosing of the rules on the next steps. therefore, one of the main goals of the unl project is creating additional tools for users to help them for improving the grammar rule sets. as we mentioned above, the result of the transformation from unl to the natural language is a linear structure with nodes. therefore, possible mistakes are contained in one or more nodes. and the user can mark the corresponding nodes whose transformation should be corrected. proceeding from the above, we can define our goal as the development of the corresponding functionality, to help us find the rules that caused the specified node’s state. to find an appropriate solution we should, at first, introduce some definitions about the rule’s responsibility for the corresponding structure of the transformation result. an automated system for correction of rules of transformation116 in this paper, we consider the method for a development of the corresponding functionality. we also introduce an algorithm for responsible rules searching and give basic information about its implementation in the undl platform system. our approach is developed for transformations from unl to natural language. but with some modifications it could be used also for transformations from natural language to unl. 3. one possible solution of the problem as we mentioned above, during the transformation process from unl to natural language, a transformation from the semantic graph to a linear graph takes place. as a result the user gets a linear graph which contains one node for each word in the sentence in the natural language. so every mistake is contained in nodes from the unl sentence obtained during the transformation. this makes possible to mark nodes which contain mistakes. we assume that the source unl sentence and dictionaries are correct. considering this we claim that the correctness of the transformation result totally depends on transformation rules. 3.1. basic definitions at first, we should give some basic definitions of the roles of rules in the transformation process. as we have mentioned, the unl sentence is a graph, so sub-graphs may be considered. rule is called affected on current node or relation if the current node or relation has been created or modified by applying the rule, or this rule’s application has triggered the creation or modification of the mentioned node or relation in future steps. rule is called responsible for some sub-graph if its application has resulted from the corresponding sub-graph appearance in the transformation process. rule is called indirectly responsible for some sub-graph if its application has triggered a rule responsible for this sub-graph in future steps. of course, several rules can be responsible for a single sub-graph. taking these definitions into consideration, we can rephrase our goal as: functionality for searching all rules responsible for current nodes. 3.2. approach our approach consists of two parts: basic part and modules. basic part is an algorithm that reviews the steps of the transformation process and marks affected steps. the modules part consists of semi-independent logical modules. in the current implementation, the module is a function. each of these modules should be used for a specific type of errors. for example, there is a module for syntactic errors. basic algorithm reviews all transformation steps, marks some of them which have affected on current nodes, decides which module should be called and prepares data to be passed to that module depending on user input. on the second stage the module starts processing. this module analyzes all rules marked as affected on current node by basic algorithm and marks the rules responsible for current nodes. a. avetisyan and l. hakobyan 117 3.3. implementation in our implementation, during the transformation process the transformation function already saves some critical information about transformation steps. this feature lets us start analyzing steps immediately after the transformation process. in undl platform, the steps of the implemented transformation process are as follows: function match is used to check, if the sentence satisfies the rule condition. it returns the object of the matchset class. this object is empty if there is no sub-graph which satisfies the current condition. if match function finds a sub-graph which satisfies the condition showing that this rule is applicable to the current sentence, it returns the object of the matchset class which contains a copy of the current sub-graph. if there is a corresponding sub-graph, the rule applying takes place. rule applying is implemented by the execute function. this function returns the object of the outputset class, which contains a copy of the sub-graph, obtained by rule applying. in fact, the object of the matchset class contains data about nodes and relations of the subgraph which satisfies the condition. and the object of the outputset class contains data about nodes and relations, which were obtained by rule applying. we have developed the tracer class, which stores all critical information for searching responsible rules. on every step our algorithm saves matchset and outputset objects, the applied rule, and the current sentence obtained by rule applying, in the object of the tracenode class. the tracer class contains a path object which is implemented by the hash-table. in this hashtable the key is a number of applied rule, and its corresponding value is an object of the tracenode. step number and tracenode object created on that step are added to the path object on every step of the transformation process. so, after transformation the tracer object stores all information about transformation steps we need. in order to find the affected rules, unique identifiers of the corresponding nodes are allocated. the analyzepath function is called with these identifiers as parameters. the analyzepath function looks through all tracenode objects that are stored in path hashtable from the last to the first. outputset and matchset of the corresponding tracenode are analyzed on every step. in fact, a rule can be claimed as affected on node, if the identifier of that node is contained in outputset or matchset. for every step with an affected rule, an object of markeditem class is being created. the markeditem stores the link to the current tracenode and some additional information. the markeditems set is then passed to the corresponding module. so, the basic algorithm finds all rules affected on current node. on the second stage, a module for deep error processing starts. this module loops through all the steps marked by the basic algorithm and analyzes them. it also analyses outputset and matchset, but the processing logic depends on the error type. in rules sets usually rules, which are used for some kind of sentence post processing, occur. they remove unused brackets and add whitespaces to sentences. these rules applying usually does not have a semantic meaning and these rules applying does not have a significant impact on the nl sentence obtained by the transformation process. so, in our algorithm we have the ability to ignore such kind of post processing rules. an automated system for correction of rules of transformation118 3.4. computational complexity computational complexity is very important for us, because undl platform is a webapplication and its computational time is bounded with the server response time. so we should show that the upper bound for computational time of our algorithm is appropriate for us. if we take maximal count of all nodes and relations for all steps as m, and count of steps in transformation process as n, we claim that the estimated algorithm time is о(2(n * 2m)). indeed, the maximal count of steps analyzed on the current stage, for the first and second stages, is equal to n. and the maximal possible count of reviewed nodes and relations (taking into consideration that both outputset and matchset are reviewed) is equal to 2m. from here we get an upper bound for the current stage of the algorithm which equals to n * 2m. therefore, the whole algorithm upper bound is equal to 2(n * 2m). 4. examples 4.1 the basic algorithm during our development and testing process we used the test drive corpora developed by undl foundation [5]. we used the following sentence with the “pre#1” identifier: [s: pre#1] {org} the book on the table {/org} {unl} plc(book:01.@def, table:02.@def.@on) {/unl} [/s] after transformation we get the “the book on the table” result (it is equal to the original). and it has the following linear representation: #l("the":06, " ":09) #l(" ":09, "book":01) #l("book":01, " ":0a) #l(" ":0a, "on":04) #l("on":04, " ":0b) #l(" ":0b, "the":08) #l("the":08, " ":0c) #l(" ":0c, "table":02.@on) we mark the “:08” node to get all rules responsible for that node, and run our basic algorithm. as a result, we get this set of rules which are marked as responsible for the “:08” node. when we get this information, the corresponding module can be called to deeply analyze the marked rules. during this process two rules were ignored as post processing rules. they are shown in table 2. a. avetisyan and l. hakobyan 119 table 1 rul e id rule depth matchset outputset 781 np(%x;%y):=(%y,+>blk)( %x,+>blk); 29 np:03(table:02.@ on, the:08) #l:03(the:08, table:02.@on) 711 ns(%x;%y):=np(%x;%y); 25 ns:03(table:02.@ on, the:08) np:03(table:02.@on, the:08) 685 xp(%x,n;%y):=ns(%x;%y); 22 xp:03(table:02.@ on, the:08) ns:03(table:02.@on, the:08) 650 /[acdijnpv]s/(%x;%y,^spe c):=xp(%x;%y,+spec); 16 ns:03(table:02.@ on, the:08) xp:03(table:02.@on, the:08) 625 (%x,n,@def):=(ns(%x,@def;%y,[the],+lex=d,+po s=art),+lex=n); book.@def > ns(book,the) 8 [table:02.@def.@ on] [sc:03(ns:03(table:02.@ on, the:08))] table 2 rule id rule 816 ((%x)(%y)):=(%x)(%y); 817 (%x,>blk)(%y,^blk,^put,^stail):=(%x,->blk)(" ",+blk)(%y); 4.2. an error specific module here we consider an example of one simple module which could be a prototype for more complex modules creation. we used the following sentence with the “ver#1” identifier from test drive corpora: “he arrived”. let’s assume we get the following transformation “he arrives”. here is an error in the last letter. user shows that error and inputs the right letter “d”. so the basic algorithm returns the following subset of steps to be analyzed by the module: table 3 rule id rule 419 (@past,^att=@past):=(-@past,+att=@past,+when); 45 (%x,v,@past):=(%x,-@past,-vbl,+ate=pas); 166 (%x,^inflected,flx):=(%x,!flx,+inflected); on the last step analyzing (the algorithm goes from the last to the first), the algorithm assumes that “s” was added on that step by comparing the matchset and the outputset. after that, the algorithm analyzes the rule and assumes that on the last step the “s” letter was added by applying “flx” of the current node because of the “ate=pas” attribute. therefore, this is the reason of current error. so this is the responsible rule. an automated system for correction of rules of transformation120 after that, the algorithm starts to analyze the other rules from the subset in order to find indirectly responsible rules. now the algorithm looks for the steps which are responsible for the “ate=pas” attribute of the current node and/or the matchset of the last overviewed step. the algorithm assumes that the second step is indirectly responsible, because the “ate=pas” attribute was added on that step and the matchset of the last overviewed step is equal to the ouputset of the current step. the last step is analyzed by the same way. on the previous overviewed step the “ate=pas” attribute was added because of the “att=@past” attribute. so, on this step the algorithm looks for the reason of the “@past” attribute and the corresponding equality between the outputset of the current overviewed step and the matchset of the previous overviewed step. after that, because there are no other steps, the algorithm assumes that there are two indirectly and one directly responsible rule as it is shown on table 4. table 4 rule id rule type of responsibility 419 (@past,^att=@past):=(@past,+att=@past,+when); indirectly 45 (%x,v,@past):=(%x,-@past,-vbl,+ate=pas); indirectly 166 (%x,^inflected,flx):=(%x,!flx,+inflected); directly and finally we get recommendation to replace “pas:=0>"s";” part of the corresponding dictionary entry attribute by “pas:=0>"d";” string. as we see, the modules for error processing entirely depend on the type of the error. in general, on every step a module should analyze the matchset and the outputset of the current step, the matchset of the previous step, the condition and the action of the rule and the attributes of the corresponding nodes. the modules should be intelligent enough to be able to algorithmically analyze the structures and the goals of the rules. 5. conclusion today, a development of the methods for generating of transformation rules is one of the important parts of the unl project. the main goal of such methods is a creation of tools, which would be able to increase efficiency of the rules development process. several definitions for the formalization of some aspects of the transformation process have been introduced above. the algorithm for responsible rules searching and the structures, used in the current implementation under the undl platform project, are described. also the examples of an error processing module and the basic algorithm were given. we propose here the first implementation of the mentioned method. it should be improved during testing and the analysis of new cases of errors processing. researches will be done in order to create new algorithms and improve the existing ones for more precise automation of the error analyzing process. references [1] h. uchida, m. zhu, and t. della senta, a gift for a millennium, ias/unu, 1999. [2] r. martins, unl wordnet, mackenzie university, 2008. a. avetisyan and l. hakobyan 121 [3] undl foundation web site, [online]. available: http://www.undlfoundation.org/ [4] deconverter specification version 2.6, unl center /undl foundation 2002. unl center, enconverter specifications, version 3.3 tokyo, 2001. [5] test drive corpus, undl foundation, [online]. available: http://www.unlweb.net/wiki/eugene#test_drive submitted 14.12.2013, accepted 22. 02. 2014. unl լեզվից դեպի բնական լեզու տրանսֆորմացման կանոնների ուղղման ավտոմատացված համակարգ ա. ավետիսյան և լ. հակոբյան ամփոփում հոդվածում նկարագրված է unl լեզվի քերականության կանոնների ուղղման ավտոմատացված մեթոդը: կանոնների մշակման գործընթացը կարող է ավելի արդյունավետ լինել, եթե օգտագործվեն համապատասխան գործիքներ, որոնք թույլ կտան գտնել կանոններ՝ պատասխանատու հնարավոր սխալների համար և ստանալ հանձնարարական դրանց շտկման հնարավոր մեթոդի վերաբերյալ: հոդվածում բերված են որոշ սահմանումներ տրանսֆորմացման ընթացքը ֆորմալիզացնելու համար և նկարագրված է դրան համապատասխան ալգորիթմը: հոդվածում նկարագրված է նաև համապատասխան ալգորիթմի իրականցումը undl platform համակարգում և բերված են դրա կիրառման առանձին օրինակներ: автоматизированная система для корректирования правил трансформации с языка unl на естественный язык а. аветисян и л. акопян аннотация в статье описан метод автоматизированного корректирования правил unl грамматики. процесс разработки правил может быть более эффективным с использованием инструментов позволяющих найти правила, ответственных за возможные ошибки, и получить рекомендации о методе их исправления. в статье даны некоторые определения с целью формализации процесса трансформации и приведен соответствующий алгоритм. в статье также описаны реализация алгоритма в системе undl platform и некоторые примеры его использования. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 44, 77--84, 2015. video shot detection method based on histogram comparison procedure manuk k. zakaryan russian armenian (slavonic) university e-mail: zakaryanmanuk@yahoo.com abstract video segmentation and shot detection are the main components in automatic video indexing, archiving, editing and retrieval. for the last fifteen years a lot of techniques have been proposed some of which have reliable performances in the video processing, especially for the video cut detection, key frame extraction and browsing technique. they have a big range of usage mainly in medical diagnosis, military, retail catalog, traffic control, security, etc. in this paper we provide a comparative analysis of histogram and certain structural properties based methods for video cut detection problems. advantages and disadvantages of the mentioned methods are shown. keywords: video segmentation, shot detection, image similarity measure, histogram. 1. introduction saying content-based image retrieval we understand an extraction of a range of images which is relevant with the given image from a large database of images. here we first need to get the feature of each image data and store it in a table. the next step is to perform some similarity measurement algorithm to find similar image data. the main question firstly comes to mind “what is the best feature to choose?” there are many image features like color, texture, shape. the first step is to extract these features from image. there are several video segmentation algorithms currently proposed in the literature which are based on various statistical properties of the shots. we briefly consider some of them.  pixel values direct comparison method one of the well-known and widely used methods based on mean square error method (psnr).  color distribution method mostly used method is color histogram comparison.  optical flow method mainly used when there is a panning or zooming of video camera.  object edges method  macroblock type method 77 video shot detection method based on histogram comparison procedure 78 nevertheless, as it is mentioned in [1] they are developed for the specific range of problems and may sometimes be not useful for a wide variety of video sequences. more precise analysis of digital video segmentation is given in [2]. in this paper we will mainly discuss the image histogram comparison and the proposed method named w2. it is known that human is more sensitive to the color features of an image than to shape or texture. existing retrieval systems based on color features are more popular. here are some methods which extract color features histogram, color moments, color correlogram, color coherence vector [3], etc. sometimes it is not effective to use only color-based features for similarity measure, because it may produce a problem in similar color distribution in different images. so this kind of methods are mainly combined with some other similarity measures to exclude this disadvantage. 2. color histogram analysis based method histograms are collected counts of data organized into a set of predefined bins. saying data we are not restricting it to be a color. the data collected can be whatever feature we find useful to describe our image. in our case it will be intensity values. let's imagine a matrix which contains information about an image pixels intensities in the range from 0 to 255. to compare two histograms to express how well both histograms match we choose a metrics χ2 mostly known as chi-square metrics [4]. let us have two histograms h1 and h2 that we want to compare. n is the number of pixels in the first image, and m is the number of pixels in the second image. so the difference d(h1, h2) between these two histograms will be expressed by this formula: ,),( 1 2 21 ∑ = +       − = k i ii ii m m n n m m n n hhd (1) where k is the number of bin segments, ni and mi are the numbers of pixels in i-th bin for h1 and h2 histograms, correspondingly. for segmentation problems we are splitting the video to the same size of frames, as in the experiments m = n. for shot boundary detection we use this method and compare consecutive frames histograms with (1) equation and where it gets its maximum value there we consider that cut has taken place. another metrics that is used for comparison of histograms is correlation method [5]. to compare h1 and h2 histograms we are using the following equation: ( )( ) ( ) ( ) , )()( )()( ),( 2 22 2 11 2211 21 ∑∑ ∑ −− −− = ii i hihhih hihhih hhd (2) where ,)( 1 ∑= j kk jhn h and n is the total number of bins. histogram intersection (hi), proposed by swain and ballard, is a straightforward method to calculate the matching rate between two histograms for this purpose [6]. ( ).)(),(min),( 2121 ∑= i ihihhhd (3) m. zakaryan 79 the resultant fractional matching value between 0 and 1 is actually the proportion of pixels from the model image that has corresponding pixels of the same color in the target image. a higher histogram matching rate indicates higher similarity between the model image and the target image. bhattacharyya distance. bhattacharyya coefficient is one of the criteria, which gives a measure of similarity between the probability density functions (spectral information) of two images [7, 8]. it is a divergence-type measure which has a straightforward geometric interpretation: .)(),( 1 1),( 21 2 21 21 ∑−= i ihih nhh hhd (4) 3. experimental results we will apply the usage of the above mentioned methods for video anni008.wmv taken from famous trecvid library of video sequences. segmentation results are shown in figures below. the video contains 15 shots. fig. 1. comparison of histogram values with chi-square method. in fig.1 we provide experimental results comparing frames with chi-square method. we got 13 cuts that have been accurately detected and 2 cuts that have not been detected. the threshold value was 0.3. video shot detection method based on histogram comparison procedure 80 fig. 2. comparison of histogram values with correlation method. in fig.2 results with correlation method are provided. it accurately detected 13 cuts, but 2 cuts were not detected and there was also one false cut. threshold value was 0.65. fig.3. comparison of histogram values with bhattacharyya method. fig.3 illustrates comparison results with bhattacharyya method. 14 cuts have been detected, and there is one missed cut and one false cut. threshold was selected equal to 0.39. real cuts that have been detected real cuts that have not been detected false cuts m. zakaryan 81 fig. 4. comparison of histogram values with intersection method. the results are given in figures from 1 to 4. as seen from the data, all 4 methods did not give absolute results. as it was mentioned previously the main two disadvantage of this method is that different images may have the same color distribution (cuts that have not been detected) and the change of brightness may totally affect the histogram values (false cuts). in figure 5, 2 structurally different images are illustrated that have almost the same histograms. fig. 5. consecutive frame of the cut and corresponding histograms. in experimental results we also provide a comparison of these methods with the proposed method which is based on structural properties of the image and called w2 )c,(c)b,(b )c,(c)b,(b =w 2121 21212 maxmax minmin , ,10 2 ≤< w (5) where the corresponding parameters b and c are represented as statistical estimations got from the corresponding samples of gradient magnitude [9, 10]. video shot detection method based on histogram comparison procedure 82 frame 1 frame 2 frame 3 fig. 6. frames to be compared. in table 1 we provide comparison results of four histogram methods and also provide the results got with w2 method. it was done for consecutive frames given in fig.6. as we can see correlation and w2 methods give results in range [0, 1] and if the consecutive frames are similar they give values near 1, for chi-square and bhattacharyya methods, the more similar are images the less are results near 0. intersection method gives higher values if images are similar and decreased when the cut appears. table 1. metric results of comparison of 3 consecutive frames with 4 histogram-based and w2 methods. 4. conclusion in this paper we have performed comparative analysis of histogram-based methods. the experimental results have shown that though they work fine in most cases of video segmentation problems, but they have two main disadvantages: the histogram comparison may give a wrong result when the brightness of similar frames has been changed, it explains false cuts that we saw in test results. it very often is happening when there are light effects in the scene or it has been taken outside. second problem is that two different images may have the same intensity and that is why we got cuts that had not been detected. in some cases it can occur that one of the considered methods provides himself better than the others. so to achieve more accurate results it is suggested to combine the mentioned methods. references [1] r. lienhart, “reliable transition detection in videos: a survey and practitioners guide”, international journal of image and graphics, vol. 1, no. 3, pp. 469--486, 2001. [2] i. koprinska and s. carrato , “temporal video segmentation: a survey”, signal process.: image communication, vol. 16, no. 5, pp. 477 – 500, 2001. method fr. 1 fr. 1 fr. 1 fr. 2 fr.2 fr. 3 chi-square correlation bhattacharyya intersection w2 0.0000 1.0000 0.0000 24.4915 1.0000 0.0104 0.9466 0.2213 24.3512 0.9513 0.0923 0.5676 0.5106 13.0642 0.6087 m. zakaryan 83 [3] r. brunelli and o. mich, “histograms analysis for image retrieval”, pattern recognition, vol. 34, no.8, pp. 1625-1637, 2001 [4] x. jiang, t. sun, j. li and xi chen, “a novel shot edge detection algorithm based on chi-square histogram and macro-block statistics”, international symposium on information science and engieering, vol. 2, pp. 604 607, 2008. [5] m. sizintsev, o. n toronto, k. g. derpanis and a. hogue, “histogram-based search: a comparative study”, ieee conference on computer vision and pattern recognition, cvpr 2008, pp. 1 8, june 2008. [6] w. jia, h. zhang, x. he and q. wu, “a comparison on histogram based image matching methods”, ieee international conference on video and signal based surveillance, 2006. avss '06, p.97, nov 2006. [7] m. s. khalid and m. bilal malik, “biased nature of bhattacharyya coefficient in correlation of gray-scale objects”, proceedings of the 4th international symposium on image and signal processing and analysis, pp. 209 214, sept. 2005. [8] s. dubuisson, “the computation of the bhattacharyya distance between histograms without histograms”, 2nd international conference on image processing theory tools and applications (ipta), pp. 373 378, july 2010. [9] d. asatryan and k. egiazarian, “quality assessment measure based on imagestructural properties”, proc. of the international workshop on local and non-local approximation in image processing, finland, helsinki, pp. 70-73, 2009. [10] d. asatryan and m. zakaryan, “method for video shot detection and separation”, international journal "information models and analysis", vol. 3, no. 3, p. 247-251, 2014. submitted 04.08.2015, accepted 20.11.2015 սյունապատկերների համեմատման եղանակով տեսահաջորդականության հատվածավորման մեթոդ մ. զաքարյան ամփոփում տեսահաջորդականության հատվածավորումը և անցումների հայտնաբերումը տեսանյութի ինդեքսավորման, արխիվացման, խմբագրման և որոնման մեթոդների հիմնական բաղադրիչներն են։ վերջին 15 տարիներին առաջարկվել են բազմաթիվ եղանակներ, որոնց նպատակն է հատվածավորումը հուսալիորեն իրականացնել հատկապես տեսահատվածների կտրուկ անցումների համար։ սույն հոդվածում ուսումնասիրվել են տեսահաջորդականության սյունապատկերների, ինչպես նաև պատկերի կառուցվածքային որոշակի հատկությունների վրա հիմնված եղանակներ: բերվել են նշված եղանակների համեմատական վերլուծության արդյունքներ: video shot detection method based on histogram comparison procedure 84 метод сегментации видеопоследовательности, основанный на сравнении гистограмм м. закарян аннотация сегментация видеопоследовательности является основным компонентом при индексации, архивации, редактировании и поиске видеоматериалов. за последние 15 лет были предложены многочисленные методы, предназначенные для надёжной сегментации, особенно при резких переходах видеосегментов. в настоящей работе исследованы методы, основанные на гистограммы видеопоследовательности, а также на определённые структурные свойства изображения. приведены результаты сравнительного анализа указанных методов. mathematical problems of computer science 44, 133-137, 2015. 133 on optimality of safer-256 diffusion knarik m. kyuregyan abstract in this paper it is shown that the new block cipher safer-256 provides an optimal diffusion in the sense that the cipher is resistant against differential cryptanalysis attack after minimum possible number of rounds. keywords: diffusion, shuffle, byte permutation, differential cryptanalysis. 1. introduction a new 256-bit size block cipher of safer family named safer-256 consists of 6 rounds followed by an output transformation. to develop a new 256-bit size block cipher we have investigated a construction of regular cipher safer+ [1] and modified safer+ [2]. as a result we have found, that we can provide the cipher resistance against differential cryptanalysis after minimum possible number of rounds (𝑟 = 5) in the case of using four iterations of byte shuffling (in case of 3 times shuffle it will be secure against differential cryptanalysis after 6 rounds at the time) (fig. 1). the classical criteria for the resistance against differential cryptanalysis [3] allow to compute the maximum expected probability that the given input difference leads to a given sequence of output differences after successive rounds in the cipher. diffusion that ensures the small changes in each round input and results in large changes in the round output, allows to reduce the number of rounds, improving speed of implementation, while ensuring a security against differential cryptanalysis. the purpose of this paper is to show an optimality of the diffusion of a new cipher safer-256 with 256-bit block length and key size [4]. this fact also insures higher processing speed for relevant ciphers while keeping their security level intact against differential cryptanalysis attack. all research has been done by the prior developed c++ code. 2. optimal diffusion of safer-256 a detailed construction and a differential cryptanalysis of the new cipher safer-256 is presented in [4]. the round function is built from four layers: institute for informatics and automation problems of nas ra e-mail: knarikyuregyan@gmail.com mailto:gurgenkh@aua.am,%20melsik@ipia.sci.am,%20knarikyuregyan@gmail.com on the optimality of safer-256 diffusion 134 1. xor/addition layer: bytes 1,2,3,4,9,10,11,12,17,18,19,20,25,26,27,28 of the round input are xor-ed with bytes 1,2,3,4,9,10,11,12,17,18,19,20,25,26,27,28 of round first subkey. bytes 5,6,7,8,13,14,15,16,21,22,23,24,29,30,31,32 of the round input are added modulo 256 with bytes 5,6,7,8,13,14,15,16,21,22,23,24,29,30,31,32 of round first subkey. 2. nonlinear layer: the 45(x) transformation is applied to bytes 1,2,3,4,9,10,11,12,17,18, 19,20,25,26,27,28 of the output xor/addition layer (with the convention 45(128)mod257 = 0) and log45(x) transformation is applied to bytes 5,6,7,8,13,14,15,16, 21,22,23,24,29,30,31,32, where x ∈ ℤ256. 3. addition/ xor layer: bytes 1,2,3,4,9,10,11,12,17,18,19,20,25,26,27,28 of the output of the nonlinear layer added modulo 256 with bytes 1,2,3,4,9,10,11,12,17,18,19,20,25,26, 27,28 of round second subkey. bytes 5,6,7,8,13,14,15,16,21,22,23,24,29,30,31,32 of the output of the nonlinear layer are xor-ed with bytes 5,6,7,8,13,14,15,16,21,22,23,24,29, 30,31,32 of round second subkey. 4. invertible linear transformation layer. at first “armenian shuffle” is applied on the input of this layer, which is the coordinate permutation [25, 28, 29, 32, 17, 20, 21, 24, 13, 16, 9, 12, 5, 8, 1, 4, 3, 2, 7, 6, 11, 10, 15, 14, 27, 26, 31, 30, 19, 18, 23, 22] with the meaning that the first output byte is the 25th input byte, the second output byte is the 28th input byte, etc. then 2-pht linear transformation is applied to the output, that maps a byte pair  2,1 xx to the byte pair  21,212 xxxx  where addition is modulo 256. the effect of the four layers of “armenian shuffle”+2-pht transform on the 32 byte output of the addition/ xor layer is to postmultiply 32 byte output with m matrix in fig. 1. one sees from fig. 1 that every row of the matrix m contains at least eight 1’s (all odd-numbered rows contain exactly eight 1’s and the remaining rows contain exactly thirteen 1’s), which means that every input block with a single non-zero byte will produce an output block with at least eight output non-zero bytes. moreover, there are changes of an input bytes that will cause only this minimum number of output bytes to change, namely a change by 128 in any odd-numbered symbol position. the diffusion provided by the matrix m is highly resistant to differential cryptanalysis. if odd-coordinate bytes and even-coordinate bytes in 32-byte block are permuted separately, then in the case of certain permutations out of (16!)2 possible permutations we can obtain matrices with the above mentioned property. the following permutations are some of these, which are obtained from software researches: 17 20 21 24 25 28 29 32 13 16 9 12 5 8 1 4 3 2 7 6 11 10 15 14 19 18 23 22 27 26 31 30 17 20 21 24 25 28 29 32 13 16 9 12 5 8 1 4 3 2 7 6 11 10 15 14 19 18 23 22 31 30 27 26 17 20 21 24 25 28 29 32 13 16 9 12 5 8 1 4 3 2 7 6 11 10 15 14 23 22 19 18 31 30 27 26 17 20 21 24 25 28 29 32 13 16 9 12 5 8 1 4 3 2 7 6 15 14 11 10 23 22 19 18 31 30 27 26 17 20 21 24 25 28 29 32 13 16 9 12 1 4 5 8 7 6 3 2 15 14 11 10 23 22 19 18 31 30 27 26 17 20 21 24 25 28 29 32 9 12 13 16 1 4 5 8 7 6 3 2 15 14 11 10 23 22 19 18 31 30 27 26 17 20 21 24 29 32 25 28 9 12 13 16 1 4 5 8 7 6 3 2 15 14 11 10 23 22 19 18 31 30 27 26 21 24 17 20 29 32 25 28 9 12 13 16 1 4 5 8 7 6 3 2 15 14 11 10 23 22 19 18 31 30 27 26 21 24 17 20 25 28 29 32 13 16 9 12 5 8 1 4 3 2 7 6 11 10 15 14 19 18 23 22 27 26 31 30 21 24 17 20 29 32 25 28 13 16 9 12 5 8 1 4 3 2 7 6 11 10 15 14 19 18 23 22 27 26 31 30 25 28 29 32 17 20 21 24 13 16 9 12 5 8 1 4 3 2 7 6 11 10 15 14 19 18 23 22 27 26 31 30 25 28 29 32 17 20 21 24 13 16 9 12 5 8 1 4 3 2 7 6 11 10 15 14 27 26 31 30 19 18 23 22 25 28 29 32 17 20 21 24 5 8 1 4 13 16 9 12 11 10 15 14 3 2 7 6 27 26 31 30 19 18 23 22 k. kyuregyan 135 f ig 1 . s a f e r -2 5 6 i n v e rt ib le l in e a r tr a n sf o rm a ti o n m a tr ix on the optimality of safer-256 diffusion 136 13 16 9 12 5 8 1 4 17 20 21 24 25 28 29 32 19 18 23 22 27 26 31 30 3 2 7 6 11 10 15 14 3 2 7 6 11 10 15 14 19 18 23 22 27 26 31 30 17 20 21 24 25 28 29 32 13 16 9 12 5 8 1 4 7 6 3 2 11 10 15 14 19 18 23 22 27 26 31 30 17 20 21 24 25 28 29 32 13 16 9 12 5 8 1 4 7 6 3 2 11 10 15 14 19 18 23 22 27 26 31 30 17 20 21 24 25 28 29 32 13 16 9 12 1 4 5 8 11 10 15 14 3 2 7 6 19 18 23 22 27 26 31 30 17 20 21 24 25 28 29 32 13 16 9 12 5 8 1 4 11 10 15 14 3 2 7 6 19 18 23 22 27 26 31 30 17 20 21 24 25 28 29 32 5 8 1 4 13 16 9 12 11 10 15 14 3 2 7 6 19 18 23 22 27 26 31 30 17 20 21 24 25 28 29 32 5 8 1 4 13 16 9 12 11 10 15 14 3 2 7 6 23 22 19 18 27 26 31 30 17 20 21 24 25 28 29 32 5 8 1 4 13 16 9 12 11 10 15 14 3 2 7 6 23 22 19 18 31 30 27 26 21 24 17 20 29 32 25 28 5 8 1 4 13 16 9 12 21 24 17 20 29 32 25 28 5 8 1 4 13 16 9 12 11 10 15 14 3 2 7 6 23 22 19 18 31 30 27 26 17 20 21 24 29 32 25 28 5 8 1 4 13 16 9 12 11 10 15 14 3 2 7 6 23 22 19 18 31 30 27 26 17 20 21 24 29 32 25 28 5 8 1 4 13 16 9 12 11 10 15 14 3 2 7 6 23 22 19 18 27 26 31 30 21 20 17 24 25 28 29 32 13 16 9 12 5 8 1 4 3 2 7 6 11 10 15 14 19 18 23 22 27 26 31 30 17 24 21 20 25 28 29 32 13 16 9 12 5 8 1 4 3 2 7 6 11 10 15 14 19 18 23 22 27 26 31 30 17 20 21 24 25 28 29 32 13 16 9 12 5 8 1 4 3 2 7 6 11 14 15 10 19 18 23 22 27 26 31 30 some of these permutations provide the best possible diffusion. 3. conclusion in this paper it is shown that diffusion provided by safer-256 algorithm is optimal from the point of differential analysis. a complete characterization of the corresponding invertible linear transformation matrix that insures an optimum transformation diffusion i.e., provides a resistance against differential cryptanalysis after minimum number of rounds is also given. acknowledgement the author is thankful to prof. gurgen khachatrian and dr. melsik kureghyan for very useful discussions and comments. references [1] j. l. massey, g. khachatrian and m. kyuregian, “nomination of safer+ as candidate algorithm for the advanced encryption standard (aes)”, nist aes proposl, 1998. [2] k. kyuregyan, “some modifications of safer+”, in reports of nas ra, vol. 115, no 1, pp. 33-39, yerevan, armenia, 2015. [3] e. biham and a. shamir, “differential cryptanalysis of des-like cryptosystem”, advances in cryptology-crypto’90, lecture notes in computer science, heidelberg and new york, springer, no. 537, pp. 212-241, 1990. [4] g. h. khachatrian, m. k. kyureghyan and k. m. kyuregyan, “design and cryptanalysis of a new encryption algorithm safer-256”, transactions of iiap nas ra, mathematical problems of computer science, vol. 42, pp. 97-106, 2014. submitted 20.07.2015, accepted 27.11.2015 k. kyuregyan 137 safer-256 համակարգի դիֆուզիայի օպտիմալությունը ք.կյուրեղյան ամփոփում այս հոդվածում ցույց է տրված, որ safer-256 բլոկային ծածկագրական համակարգը ապահովում է օպտիմալ դիֆուզիա այն իմաստով, որ այն կրիպտոկայուն է դիֆերենցիալ վերլուծության նկատմամբ մինիմալ թվով ռաունդների դեպքում: об оптимальности диффузии safer-256 к. кюрегян аннотация в данной статье показано, что блоковый шифр safer-256 обеспечивает оптимальную диффузию, в том смысле, что шифр устойчив по отнoшению к дифференциальному анализу после минимально возможного количества раундов. mathematical problems of computer science 45, 127--137, 2016. linear cryptanalysis of safer-256 knarik m. kyuregyan institute for informatics and automation problems of nas ra e-mail: knarikyuregyan@gmail.com abstract in this paper linear cryptanalysis of 256-bit block cipher safer256 is presented. safer-256 is secure against differential cryptanalysis after 5 rounds. keywords: linear cryptanalysis. 1. introduction in paper [1] the detailed construction and security analysis against differential analysis of block cipher safer-256 is introduced. in this paper the result of linear cryptanalysis [2] of safer-256 is presented. we follow the terminology and notation in [2]. the round function is applied to 256 bit plaintext x = x1x2 … x32 with 6 times. the round function consists of the following four layers: 1. xor/add, the first round key is “added” to the round input. bytes 1,2,3,4,9,10,11,12,17,18,19,20,25,26,27,28 are added together bit-by-bit modulo 2 ( xor) bytes 5,6,7,8,13,14,15,16,21,22,23,24,29,30,31,32 are added together modulo 256 (add): u = xor/add(x, k2i−1). 2. non-linear (nl), where the values of bytes 1,2,3,4,9,10,11,12,17,18,19,20,25,26,27,28 are converted to 45xmodulo257 (with the convention that 45128 is represented as 0) and the values of bytes 5,6,7,8,13,14,15,16,21,22,23,24,29,30,31,32 are converted to log45x (with the convention that the output log450 is represented by 128): v = nl(u). 3. add/xor, by this layer the second round key is inserted. the round key bytes 1,2,3,4,9,10,11,12,17,18,19,20,25,26,27,28 are added to the corresponding output bytes of linear layer modulo 256 (add) while the round key bytes 5,6,7,8,13, 14,15,16,21, 22,23,24, 29,30,31,32 are added to the corresponding output bytes of linear layer modulo 2 ( xor): w = add/xor(v, k2i). 4. invertible linear transform or pseudo-hadamard transform (pht) consisting of four times applied 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑠𝑠ℎ𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝐴𝐴 and four times applied eight 2-pht boxes: y = pht(w), is equivalent to y = wm, where m is called an invertible linear transformation matrix of safer256 introduced in fig 1. 127 mailto:knarikyuregyan@gmail.com linear cryptanalysis of safer-256 128 fi g. 1 . i nv er tib le li ne ar tr an sf or m at io n m at ri x of s a fe r -2 56 .. k. kyuregyan 129 2. linear cryptanalysis of safer-256 we will find effective homomorphic i/o sums to a cascade of half-rounds of safer-256. at first we find all binary-valued homomorphisms for add/xor and for xor/add. there are 28 − 1 binary-valued homomorphisms for 8 bit xor, namely the functions defined as 𝑢𝑢𝑎𝑎2(𝑉𝑉2) ∶= 𝐴𝐴2 ∘ 𝑉𝑉2, where 𝐴𝐴2 is a non-zero binary 8-tuple " ∘ " operation denotes the modulo two “dot product”. there is only one binary-valued homomorphism for modulo 256 addition, namely the function 𝑢𝑢𝑎𝑎1 where 𝐴𝐴1 = 0000 0001 = 01 (hex notation of byte). hence, there exist 2144 − 1 balanced homomorphisms for add/xor, namely the functions 𝑢𝑢𝑎𝑎 defined as 𝑢𝑢𝑎𝑎(𝑉𝑉) = 𝐴𝐴 ∘ 𝑉𝑉 where 𝐴𝐴 lies in the set of 256-tuples. 𝒜𝒜 = {𝐴𝐴: 𝐴𝐴 ∈ {0,1}256\{00}; 𝐴𝐴1, 𝐴𝐴2, 𝐴𝐴3, 𝐴𝐴4, 𝐴𝐴9, 𝐴𝐴10, 𝐴𝐴11, 𝐴𝐴12, 𝐴𝐴17, 𝐴𝐴18, 𝐴𝐴19, 𝐴𝐴20, 𝐴𝐴25, 𝐴𝐴26, 𝐴𝐴27, 𝐴𝐴28} ∈ {00, 01}}. similarly, there are 2144 − 1 balanced homomorphism for xor/add, namely the functions 𝑢𝑢𝑏𝑏(𝑉𝑉) = 𝑏𝑏 ∗ 𝑉𝑉, where 𝑏𝑏 lies in the set ℬ = {𝑏𝑏: 𝑏𝑏 ∈ {0,1}256\{00}; 𝑏𝑏5, 𝑏𝑏6, 𝑏𝑏7, 𝑏𝑏8, 𝑏𝑏13, 𝑏𝑏14, 𝑏𝑏15, 𝑏𝑏16, 𝑏𝑏21, 𝑏𝑏22, 𝑏𝑏23, 𝑏𝑏24, 𝑏𝑏29, 𝑏𝑏30, 𝑏𝑏31, 𝑏𝑏32} ∈ {00, 01}}. the set of all homomorphic functions for xor/add and the set of all homomorphic functions for add/xor are subsets of the set of all linear binary-valued functions. at first we consider the part of round (half-round) containing the pht function. the following lemma specifies all homomorphic i/o sums that have non-zero imbalance. the input function must be balanced and homomorphic for add/xor; the output function must be balanced and homomorphic for xor/ add. there are (2144 − 1)2 such i/o sums, namely s𝑎𝑎,𝑏𝑏 pht−hr ∶= 𝑢𝑢𝑎𝑎(v)⨁𝑢𝑢𝑏𝑏(y) 𝐴𝐴 ∈ 𝒜𝒜 , 𝑏𝑏 ∈ ℬ. lemma 1: for the 𝑃𝑃𝑃𝑃𝑃𝑃-half-round, the only homomorphic i/o sums that have non-zero imbalance are the 232 − 1 guaranteed i/o sums obtained by 𝑋𝑋𝑋𝑋𝑋𝑋-ing together any positive number of the 32 guaranteed i/o sums listed in table 1. since 𝐼𝐼�𝑆𝑆𝑎𝑎,𝑏𝑏 pht−hr|𝑘𝑘2𝑖𝑖� = 𝐼𝐼�𝑆𝑆𝑎𝑎,𝑏𝑏 pht� for any 𝑘𝑘2𝑖𝑖 ∈ {0,1}256, where 𝑆𝑆𝑎𝑎,𝑏𝑏 pht ∶= 𝑢𝑢𝑎𝑎(𝑊𝑊)⨁ 𝑢𝑢𝑏𝑏(𝑌𝑌), (𝐴𝐴 ∈ 𝒜𝒜 and 𝑏𝑏 ∈ ℬ) is an i/o sum for the pht function alone, we will look for i/o sums 𝑆𝑆𝑎𝑎,𝑏𝑏 pht with non-zero imbalance instead of looking for 𝑆𝑆𝑎𝑎,𝑏𝑏 pht−hr with non-zero imbalance. so, our main purpose is to find 𝐴𝐴 ∈ 𝒜𝒜 and 𝑏𝑏 ∈ ℬ such that 𝑆𝑆𝑎𝑎,𝑏𝑏 pht sum has the form 𝑊𝑊𝛼𝛼⨁𝜙𝜙(𝑊𝑊256, … , 𝑊𝑊𝛼𝛼+1, 𝑊𝑊𝛼𝛼−1, … , 𝑊𝑊0) for some input bit 𝑊𝑊𝛼𝛼, since this implies that the i/o sum imbalance is 0. first we consider pht function. table 2 shows three kind of dependences for some input and output bits. by using table 2 we can iteratively show that i/o sums 𝑆𝑆𝑎𝑎,𝑏𝑏 pht with none-zero imbalance cannot contain any of the output bits 𝑌𝑌𝐴𝐴𝑗𝑗, where 𝐴𝐴 = 1,2,3,4,9,10,11,12,17,18,19,20,25,26,27,28 and 𝑗𝑗 = 1,2,3,4,5,6,7. finally, we consider the 232 − 1 balanced output functions obtained as linear combinations of the remaining 32 output bits that might occur if 𝑏𝑏 ∈ ℬ. for each of these output functions, we found all input functions such that 𝑆𝑆𝑎𝑎,𝑏𝑏 pht doesn’t depend linearly on any input bit. it is easy to show that �(𝐴𝐴, 𝑏𝑏); 𝐴𝐴, 𝑏𝑏 ∈ {0,1}256, 𝐼𝐼�s𝑎𝑎,𝑏𝑏 pht� = 1� is a subgroup of (𝒜𝒜 × ℬ, ⨁256bits). (in fact if 𝐼𝐼�s𝑎𝑎,𝑏𝑏 pht� = 1 and 𝐼𝐼�s𝑎𝑎′,𝑏𝑏′ pht � = 1 for some 256 tuples 𝐴𝐴′ and 𝑏𝑏′, then 𝐼𝐼�s𝑎𝑎⨁𝑎𝑎′,𝑏𝑏⨁𝑏𝑏′ pht � = 𝐼𝐼�s𝑎𝑎,𝑏𝑏 pht⨁s𝑎𝑎′,𝑏𝑏′ pht � = 𝐼𝐼�s𝑎𝑎,𝑏𝑏 pht� ∙ 𝐼𝐼�s𝑎𝑎′,𝑏𝑏′ pht � = 1 as well, whch shows that 𝒜𝒜 × ℬ is closed under "⨁".) thus, we obtain all i/o sums with non-zero imbalance, as is done in the statement of the lemma. linear cryptanalysis of safer-256 130 table 1. effective i/o sums for the pht-function. (𝐴𝐴, 𝑏𝑏) 𝑢𝑢𝑏𝑏(𝑌𝑌) 𝑢𝑢𝑎𝑎(𝑊𝑊) 𝐼𝐼�𝑆𝑆𝑎𝑎,𝑏𝑏 pht� (00000000000110011001000010010001, 10000000000000000000000000000000) 𝑌𝑌10 𝑊𝑊120⨁𝑊𝑊130⨁𝑊𝑊160⨁𝑊𝑊170⨁𝑊𝑊200⨁𝑊𝑊250⨁𝑊𝑊280⨁𝑊𝑊320 1 (00000000000110011111000011110101, 01000000000000000000000000000000) 𝑌𝑌20 𝑊𝑊120⨁𝑊𝑊130⨁𝑊𝑊160⨁𝑊𝑊170⨁𝑊𝑊180⨁𝑊𝑊190⨁𝑊𝑊200⨁ 𝑊𝑊250⨁𝑊𝑊260⨁𝑊𝑊270⨁𝑊𝑊280⨁𝑊𝑊300⨁𝑊𝑊320 1 (10010001100100000001100100000000, 00100000000000000000000000000000) 𝑌𝑌30 𝑊𝑊10⨁𝑊𝑊40⨁𝑊𝑊80⨁𝑊𝑊90⨁𝑊𝑊120⨁𝑊𝑊200⨁𝑊𝑊210⨁𝑊𝑊240 1 (10010001100100000001111101000110, 00010000000000000000000000000000) 𝑌𝑌40 𝑊𝑊10⨁𝑊𝑊40⨁𝑊𝑊80⨁𝑊𝑊90⨁𝑊𝑊120⨁𝑊𝑊200⨁𝑊𝑊210⨁ 𝑊𝑊220⨁𝑊𝑊230⨁𝑊𝑊240⨁𝑊𝑊260⨁𝑊𝑊300⨁𝑊𝑊310 1 (00011001000000001001000110010000, 00001000000000000000000000000000) 𝑌𝑌50 𝑊𝑊40⨁𝑊𝑊50⨁𝑊𝑊80⨁𝑊𝑊170⨁𝑊𝑊200⨁𝑊𝑊240⨁𝑊𝑊250⨁𝑊𝑊280 1 (00011001000000001111010111110000, 00000100000000000000000000000000) 𝑌𝑌60 𝑊𝑊40⨁𝑊𝑊50⨁𝑊𝑊80⨁𝑊𝑊170⨁𝑊𝑊180⨁𝑊𝑊190⨁𝑊𝑊200⨁ 𝑊𝑊220⨁𝑊𝑊240⨁𝑊𝑊250⨁𝑊𝑊260⨁𝑊𝑊270⨁𝑊𝑊280 1 (00000110010001100100011000000000, 00000010000000000000000000000000) 𝑌𝑌70 𝑊𝑊60⨁𝑊𝑊70⨁𝑊𝑊100⨁𝑊𝑊140⨁𝑊𝑊150⨁𝑊𝑊180⨁𝑊𝑊220⨁𝑊𝑊230 1 (00000110010001100100111100011001, 00000001000000000000000000000000) 𝑌𝑌80 𝑊𝑊60⨁𝑊𝑊70⨁𝑊𝑊100⨁𝑊𝑊140⨁𝑊𝑊150⨁𝑊𝑊180⨁𝑊𝑊210⨁ 𝑊𝑊220⨁𝑊𝑊230⨁𝑊𝑊240⨁𝑊𝑊280⨁𝑊𝑊290⨁𝑊𝑊320 1 (01100000011001000110010000000000, 00000000100000000000000000000000) 𝑌𝑌90 𝑊𝑊20⨁𝑊𝑊30⨁𝑊𝑊100⨁𝑊𝑊110⨁𝑊𝑊140⨁𝑊𝑊180⨁𝑊𝑊190⨁𝑊𝑊220 1 (01111001011011010110010000000000, 00000000010000000000000000000000) 𝑌𝑌100 𝑊𝑊20⨁𝑊𝑊30⨁𝑊𝑊40⨁𝑊𝑊50⨁𝑊𝑊80⨁𝑊𝑊100⨁𝑊𝑊110⨁ 𝑊𝑊130⨁𝑊𝑊140⨁𝑊𝑊160⨁𝑊𝑊180⨁𝑊𝑊190⨁𝑊𝑊220 1 (10010000100100010000000000011001, 00000000001000000000000000000000) 𝑌𝑌110 𝑊𝑊10⨁𝑊𝑊40⨁𝑊𝑊90⨁𝑊𝑊120⨁𝑊𝑊160⨁𝑊𝑊280⨁𝑊𝑊290⨁𝑊𝑊320 1 (10010000100100010100011000011111, 00000000000100000000000000000000) 𝑌𝑌120 𝑊𝑊10⨁𝑊𝑊40⨁𝑊𝑊90⨁𝑊𝑊120⨁𝑊𝑊160⨁𝑊𝑊180⨁𝑊𝑊220⨁ 𝑊𝑊230⨁𝑊𝑊280⨁𝑊𝑊290⨁𝑊𝑊300⨁𝑊𝑊310⨁𝑊𝑊320 1 (10010001000000000001100100001001, 00000000000010000000000000000000) 𝑌𝑌130 𝑊𝑊10⨁𝑊𝑊40⨁𝑊𝑊80⨁𝑊𝑊200⨁𝑊𝑊210⨁𝑊𝑊240⨁𝑊𝑊290⨁𝑊𝑊320 1 (11010111000001100001100100001001, 00000000000001000000000000000000) 𝑌𝑌140 𝑊𝑊10⨁𝑊𝑊20⨁𝑊𝑊40⨁𝑊𝑊60⨁𝑊𝑊70⨁𝑊𝑊80⨁𝑊𝑊140⨁𝑊𝑊150⨁ 𝑊𝑊200⨁𝑊𝑊210⨁𝑊𝑊240⨁𝑊𝑊290⨁𝑊𝑊320 1 (00001001000110010000000010010001, 00000000000000100000000000000000) 𝑌𝑌150 𝑊𝑊50⨁𝑊𝑊80⨁𝑊𝑊120⨁𝑊𝑊130⨁𝑊𝑊160⨁𝑊𝑊250⨁𝑊𝑊280⨁𝑊𝑊320 1 (01101101011110010000000010010001, 00000000000000010000000000000000) 𝑌𝑌160 𝑊𝑊20⨁𝑊𝑊30⨁𝑊𝑊50⨁𝑊𝑊60⨁𝑊𝑊80⨁𝑊𝑊100⨁𝑊𝑊110⨁𝑊𝑊120⨁ 𝑊𝑊130⨁𝑊𝑊160⨁𝑊𝑊250⨁𝑊𝑊280⨁𝑊𝑊320 1 (01100100011000000000000001100100, 00000000000000001000000000000000) 𝑌𝑌170 𝑊𝑊20⨁𝑊𝑊30⨁𝑊𝑊60⨁𝑊𝑊100⨁𝑊𝑊110⨁𝑊𝑊260⨁𝑊𝑊270⨁𝑊𝑊300 1 (01101101011110010000000001100100, 00000000000000000100000000000000) 𝑌𝑌180 𝑊𝑊20⨁𝑊𝑊30⨁𝑊𝑊50⨁𝑊𝑊60⨁𝑊𝑊80⨁𝑊𝑊100⨁𝑊𝑊110⨁𝑊𝑊120⨁ 𝑊𝑊130⨁𝑊𝑊160⨁𝑊𝑊260⨁𝑊𝑊270⨁𝑊𝑊300 1 (01000110000001100000000001000110, 00000000000000000010000000000000) 𝑌𝑌190 𝑊𝑊20⨁𝑊𝑊60⨁𝑊𝑊70⨁𝑊𝑊140⨁𝑊𝑊150⨁𝑊𝑊260⨁𝑊𝑊300⨁𝑊𝑊310 1 (01000110000001100001100101001111, 00000000000000000001000000000000) 𝑌𝑌200 𝑊𝑊20⨁𝑊𝑊60⨁𝑊𝑊70⨁𝑊𝑊140⨁𝑊𝑊150⨁𝑊𝑊200⨁𝑊𝑊210⨁ 𝑊𝑊240⨁𝑊𝑊260⨁𝑊𝑊290⨁𝑊𝑊300⨁𝑊𝑊310⨁𝑊𝑊320 1 (00000000010001100100011000000110, 00000000000000000000100000000000) 𝑌𝑌210 𝑊𝑊100⨁𝑊𝑊140⨁𝑊𝑊150⨁𝑊𝑊180⨁𝑊𝑊220⨁𝑊𝑊230⨁𝑊𝑊300⨁𝑊𝑊310 1 (10010000110101110100011000000110, 00000000000000000000010000000000) 𝑌𝑌220 𝑊𝑊10⨁𝑊𝑊40⨁𝑊𝑊90⨁𝑊𝑊100⨁𝑊𝑊120⨁𝑊𝑊140⨁𝑊𝑊150⨁ 𝑊𝑊160⨁𝑊𝑊180⨁𝑊𝑊220⨁𝑊𝑊230⨁𝑊𝑊300⨁𝑊𝑊310 1 (00011001000010011001000100000000, 00000000000000000000001000000000) 𝑌𝑌230 𝑊𝑊40⨁𝑊𝑊50⨁𝑊𝑊80⨁𝑊𝑊130⨁𝑊𝑊160⨁𝑊𝑊170⨁𝑊𝑊200⨁𝑊𝑊240 1 (01111001011011011001000100000000, 00000000000000000000000100000000) 𝑌𝑌240 𝑊𝑊20⨁𝑊𝑊30⨁𝑊𝑊40⨁𝑊𝑊50⨁𝑊𝑊80⨁𝑊𝑊100⨁𝑊𝑊110⨁ 𝑊𝑊130⨁𝑊𝑊140⨁𝑊𝑊160⨁𝑊𝑊170⨁𝑊𝑊200⨁𝑊𝑊240 1 (01100100000000000110000001100100, 00000000000000000000000010000000) 𝑌𝑌250 𝑊𝑊20⨁𝑊𝑊30⨁𝑊𝑊60⨁𝑊𝑊180⨁𝑊𝑊190⨁𝑊𝑊260⨁𝑊𝑊270⨁𝑊𝑊300 1 (01100100000000001111000011110101, 00000000000000000000000001000000) 𝑌𝑌260 𝑊𝑊20⨁𝑊𝑊30⨁𝑊𝑊60⨁𝑊𝑊170⨁𝑊𝑊180⨁𝑊𝑊190⨁𝑊𝑊200⨁ 𝑊𝑊250⨁𝑊𝑊260⨁𝑊𝑊270⨁𝑊𝑊280⨁𝑊𝑊300⨁𝑊𝑊320 1 (01000110000000000000011001000110, 00000000000000000000000000100000) 𝑌𝑌270 𝑊𝑊20⨁𝑊𝑊60⨁𝑊𝑊70⨁𝑊𝑊220⨁𝑊𝑊230⨁𝑊𝑊260⨁𝑊𝑊300⨁𝑊𝑊310 1 (11010111100100000000011001000110, 00000000000000000000000000010000) 𝑌𝑌280 𝑊𝑊10⨁𝑊𝑊20⨁𝑊𝑊40⨁𝑊𝑊60⨁𝑊𝑊70⨁𝑊𝑊80⨁𝑊𝑊90⨁𝑊𝑊120⨁ 𝑊𝑊220⨁𝑊𝑊230⨁𝑊𝑊260⨁𝑊𝑊300⨁𝑊𝑊310 1 (00000000011001000110010001100000, 00000000000000000000000000001000) 𝑌𝑌290 𝑊𝑊100⨁𝑊𝑊110⨁𝑊𝑊140⨁𝑊𝑊180⨁𝑊𝑊190⨁𝑊𝑊220⨁𝑊𝑊260⨁𝑊𝑊270 1 (00000000011001001111010111110000, 00000000000000000000000000000100) 𝑌𝑌300 𝑊𝑊100⨁𝑊𝑊110⨁𝑊𝑊140⨁𝑊𝑊170⨁𝑊𝑊180⨁𝑊𝑊190⨁𝑊𝑊200⨁ 𝑊𝑊22_0⨁𝑊𝑊24_0⨁𝑊𝑊250⨁𝑊𝑊260⨁𝑊𝑊270⨁𝑊𝑊280 1 (00000000100100010000100100011001, 00000000000000000000000000000010) 𝑌𝑌310 𝑊𝑊90⨁𝑊𝑊120⨁𝑊𝑊160⨁𝑊𝑊210⨁𝑊𝑊240⨁𝑊𝑊280⨁𝑊𝑊290⨁𝑊𝑊320 1 (00000110110101110000100100011001, 00000000000000000000000000000001) 𝑌𝑌320 𝑊𝑊60⨁𝑊𝑊70⨁𝑊𝑊90⨁𝑊𝑊100⨁𝑊𝑊120⨁𝑊𝑊140⨁𝑊𝑊150⨁ 𝑊𝑊160⨁𝑊𝑊210⨁𝑊𝑊240⨁𝑊𝑊280⨁𝑊𝑊290⨁𝑊𝑊320 1 k. kyuregyan 131 𝑊𝑊 28 7 6 5 4 3 2 1 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑊𝑊 27 7 6 5 4 3 2 1 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑊𝑊 26 7 6 5 4 3 2 1 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑊𝑊 25 7 6 5 4 3 2 1 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑊𝑊 20 7 6 5 4 3 2 1 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑊𝑊 19 7 6 5 4 3 2 1 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑊𝑊 18 7 6 5 4 3 2 1 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑊𝑊 17 7 6 5 4 3 2 1 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑊𝑊 12 7 6 5 4 3 2 1 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑊𝑊 11 7 6 5 4 3 2 1 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑊𝑊 10 7 6 5 4 3 2 1 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑌𝑌2 5 7 6 5 4 3 2 1 0 𝑌𝑌2 6 7 6 5 4 3 2 1 0 𝑌𝑌2 7 7 6 5 4 3 2 1 0 𝑌𝑌2 8 7 6 5 4 3 2 1 0 𝑌𝑌2 9 0 𝑌𝑌3 0 0 𝑌𝑌3 1 0 𝑌𝑌3 2 0 k. kyuregyan 135 the half round containing the nonlinear function nl and find homomorphic i/o sums for nl that have non-zero imbalance. here the input function is homomorphic for xor/add and the output function is homomorphic for add/xor. such i/o sums can be obtained by summing i/o sums for its exp and log blocks. for the function exp with the input byte u1 and output byte v1, the only homomorphic i/o sums are 𝑆𝑆𝑎𝑎1,𝑏𝑏1 exp = 𝑢𝑢𝑎𝑎1(u1)⨁ 𝑢𝑢𝑏𝑏1(v1), for 𝐴𝐴1 ∈ 𝐵𝐵8\{00}; 𝑏𝑏1 = 01. the most effective ones are obtained when (𝐴𝐴1, 𝑏𝑏1) is equal to (𝑐𝑐𝑐𝑐, 01) or (𝑢𝑢𝑢𝑢, 01) (the imbalance being 28 128 ) or to (86,01), (𝑏𝑏𝑢𝑢, 01), (𝑐𝑐0, 01) or (𝑢𝑢7,01) (the imbalance being 24 128 ). computing all these i/o sums for exp, establishes the following. remark 1: 𝐼𝐼�𝑆𝑆01,01𝐸𝐸𝐸𝐸𝐸𝐸 � = 𝐼𝐼�𝑆𝑆02,01𝐸𝐸𝐸𝐸𝐸𝐸 � = 𝐼𝐼�𝑆𝑆03,01𝐸𝐸𝐸𝐸𝐸𝐸 � = 0. furthermore, for all 𝐴𝐴1 and 𝑏𝑏1 in 𝐵𝐵8, if 𝐴𝐴17 = 0, then 𝐼𝐼�𝑆𝑆𝑎𝑎1,𝑏𝑏1 𝐸𝐸𝐸𝐸𝐸𝐸 � = 0. for the function log with the input u2 and output v2, the only homomorphic i/o sums are 𝑆𝑆𝑎𝑎2,𝑏𝑏2 log = 𝑢𝑢𝑎𝑎2(u2)⨁ 𝑢𝑢𝑏𝑏2(v2), for 𝐴𝐴2 = 01; 𝑏𝑏2 ∈ 𝐵𝐵8\{00}. their imbalance is easily deduced since 𝐼𝐼�𝑆𝑆𝑎𝑎1,𝑏𝑏1 exp � = 𝐼𝐼�𝑆𝑆𝑏𝑏1,𝑎𝑎1 log �. remark 2: for all 𝐴𝐴1 and 𝑏𝑏1 in 𝐵𝐵8, if 𝑏𝑏17 = 0, then 𝐼𝐼�𝑆𝑆𝑎𝑎1,𝑏𝑏1 𝐿𝐿𝐿𝐿𝐿𝐿 � = 0. finally we have link i/o sums for successive half rounds. theorem 1: the procedure for finding effective homomorphic i/o sums doesn’t find an i/o sum with non-zero imbalance for cascade of half rounds taken in the same order as they are used in safer-256 and containing at least two 𝑃𝑃𝑃𝑃𝑃𝑃-layers. proof: let 𝑃𝑃𝑎𝑎,𝑏𝑏 pht−hr, 𝑃𝑃𝑏𝑏,𝑐𝑐 nl and 𝑃𝑃𝑐𝑐,𝑑𝑑 pht−hr be linked homomorphic half-round threefold sums with maximizing key function. if 𝑃𝑃𝑎𝑎,𝑏𝑏 pht−hr and 𝑃𝑃𝑐𝑐,𝑑𝑑 pht−hr have none zero imbalance, the 256 bit none zero masks a, b, c, d, can have a 1 only in the two least significant bits of each byte (bits of byte are numbered from 7 for the most significant bit to 0 for the least significant bit) (lemma 1). then 𝐼𝐼�𝑃𝑃𝑏𝑏,𝑐𝑐 nl� = 0 since the i/o sum average key imbalance is also 0 (remark 1) therefore, the sum of the three half-round threefold sums has imbalance 0. one of the most effective i/o sums is 𝑆𝑆𝑎𝑎,𝑏𝑏 nl−pht−nl, where 𝐴𝐴2, 𝐴𝐴10 and 𝑏𝑏13, 𝑏𝑏14, 𝑏𝑏31, 𝑏𝑏32 are either 𝑐𝑐𝑐𝑐 or 𝑢𝑢𝑢𝑢and other bytes of 𝐴𝐴 and 𝑏𝑏 are zero. their imbalance is � 28 128 � 6 , because 𝐼𝐼�𝑆𝑆𝑓𝑓𝑓𝑓,01 exp � = 𝐼𝐼�𝑆𝑆𝑐𝑐𝑑𝑑,01 exp � = 𝐼𝐼�𝑆𝑆01,𝑓𝑓𝑓𝑓 log � = 𝐼𝐼�𝑆𝑆01,𝑐𝑐𝑑𝑑 log � = 28 128 and because 𝐼𝐼�𝑆𝑆𝑎𝑎,𝑏𝑏 pht� = 𝐼𝐼�𝑆𝑆𝑎𝑎1⨁𝑎𝑎2⨁𝑎𝑎3⨁𝑎𝑎4,𝑏𝑏1⨁𝑏𝑏2⨁𝑏𝑏3⨁𝑏𝑏4 pht � = 𝐼𝐼�𝑆𝑆𝑎𝑎1,𝑏𝑏1 pht � ∙ 𝐼𝐼�𝑆𝑆𝑎𝑎2,𝑏𝑏2 pht � = 𝐼𝐼�𝑆𝑆𝑎𝑎3,𝑏𝑏3 pht � = 𝐼𝐼�𝑆𝑆𝑎𝑎4,𝑏𝑏4 pht � = 1, if 𝐴𝐴1⨁𝐴𝐴2⨁𝐴𝐴3⨁𝐴𝐴4 = (01 00 00 01 00 00 00 01 00 00 00 00 00 00 00 00 00 00 00 01 01 00 00 01 00 00 00 00 01 00 00 01 )⨁ (01 01 00 01 00 01 01 01 00 00 00 00 00 01 01 00 00 00 00 01 01 00 00 01 00 00 00 00 01 00 00 01) ⨁ (00 00 00 00 00 00 00 00 01 00 00 01 00 00 00 01 00 00 00 00 01 00 00 01 00 00 00 01 01 00 00 01) ⨁ (00 00 00 00 00 01 01 00 01 01 00 01 00 01 01 01 00 00 00 00 01 00 00 01 00 00 00 01 linear cryptanalysis of safer-256 136 01 00 00 01) = (00 01 00 00 00 00 00 00 00 01 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00) = 𝐴𝐴 (hex notation), 𝑏𝑏1⨁𝑏𝑏2⨁𝑏𝑏3⨁𝑏𝑏4 = (00 00 00 00 00 00 00 00 00 00 00 00 01 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00)⨁ (00 00 00 00 00 00 00 00 00 00 00 00 00 01 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00)⨁ (00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 01 00) ⨁ (00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 01 (00 00 00 00 00 00 00 00 00 00 00 00 01 01 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 01 01) = 𝑏𝑏 (hex notation). thus, we have proved that safer-256 is secure against the linear cryptanalysis after only three of its suggested six rounds. acknowledgement the author is thankful to prof. gurgen khachatrian and dr. melsik kureghyan for very useful discussions and comments. references [1] g. h. khachatrian, m. k. kyureghyan, k. m. kyuregyan, design and cryptanalysis of a new encryption algorithm safer-256. transactions of iiap nas ra, mathematical problems of computer science vol. 42, pp. 97-106, 2014. [2] h. carlo, cryptanalysis of iterated block ciphers, eth series in information processing (ed massey), v. 7, konstanz: hartung-gorre verlag, 1996. submitted 05.11.2016, accepted 03.02.2016 k. kyuregyan 137 safer-256 համակարգի գծային վերլուծությունը ք.կյուրեղյան ամփոփում այս հոդվածում ներկայացված է 256 բիթ բլոկի երկարության safer-256 բլոկային ծածկագրական համակարգի գծային վերլուծությունը, որը դիֆերենցիալ վերլուծության նկատմամբ կայուն է 5 ռաունդից հետո: линейнный криптоанализ алгоритма safer-256 к. кюрегян аннотация в данной статье представлен линейный криптоанализ блочного шифра safer-256, которая устойчива по отношению к дифференциальному анализу после 5 раындов. [1] g. h. khachatrian, m. k. kyureghyan, k. m. kyuregyan, design and cryptanalysis of a new encryption algorithm safer-256. transactions of iiap nas ra, mathematical problems of computer science vol. 42, pp. 97-106, 2014. [2] h. carlo, cryptanalysis of iterated block ciphers, eth series in information processing (ed massey), v. 7, konstanz: hartung-gorre verlag, 1996. d:\tex_522\01_mok.dvi mathematical problems of computer science 52, 7{13, 2019. on i nitial segments of t ur ing degr ees containing simple t -m itotic but not wtt-m itotic sets a r s e n h . mo ka t s ia n institute for informatics and automation problems of nas ra e-mail: arsenmokatsian@gmail.com abstract we consider the properties of computably enumerable (c.e.) turing degrees containing sets, which possess the property of a t -mitotic splitting but don't have a wtt-mitotic splitting. it is proved that for any noncomputable c.e. degree b there exists a degree a, such that a · b and a contains a simple t -mitotic set, which is not wtt-mitotic. keywords: mitotic set, t -reducibility, wtt-reducibility, simple set, contiguous degree. 1 . in t r o d u c t io n w e s h a ll u s e t h e n o t io n s a n d t e r m in o lo g y in t r o d u c e d in s o a r e [1 ], r o g e r s [2 ]. n o t a t io n s . w e d e a l wit h s e t s a n d fu n c t io n s o ve r t h e n o n n e g a t ive in t e g e r s ! = f 0 ; 1 ; 2 ; : : :g. l e t !ev = fx : ( 9k ) ( x = 2 k ) g; !od = fx : ( 9k ) ( x = 2 k + 1 ) g. l e t 'e b e t h e e th p a r t ia l c o m p u t a b le fu n c t io n in t h e s t a n d a r d lis t in g ( s o a r e [1 , p . 1 5 , p . 2 5 ]) . if a µ ! a n d e 2 !, le t ©ae ( x ) = ©e ( a : x) = fega ( x) ( s e e s o a r e [1 , p p . 4 8 -5 0 ]) . âa d e n o t e s t h e c h a r a c t e r is t ic fu n c t io n o f a, wh ic h is o ft e n id e n t i¯ e d wit h a a n d wr it t e n s im p ly a s a ( x) . l e t '( x) # d e n o t e s t h a t '( x ) is d e ¯ n e d , '( x) " d e n o t e s t h a t '( x ) is u n d e ¯ n e d . we = dom 'e = fx : 'x ( x ) #g. 'e; at s+1 ( x ) # d e n o t e s 'e; s+1 ( x ) # & 'e; s ( x ) ". x 2 we;at s+1 d e n o t e s x 2 we;s+1 ¡ we;s. in t h e lit e r a t u r e , tu r in g r e d u c ib ilit y is u s u a lly t a ke n a s t h e m a in r e d u c ib ilit y. if t h e wo r d \ r e d u c ib ilit y" is u s e d wit h o u t a fu r t h e r e xp la n a t io n , it m e a n s , a s a r u le , tu r in g r e d u c ib ilit y. if t h e t e r m \ d e g r e e o f u n s o lva b ilit y" is u s e d wit h o u t a fu r t h e r e xp la n a t io n , t h e t -d e g r e e is u s u a lly m e a n t . de¯nition 1: th e use function u ( a; e; x; s) is 1 + ( t h e m a xim u m n u m b e r u s e d in c o m p u t a t io n if ©aes ( x ) # ) , a n d = 0 , o t h e r wis e . th e u s e fu n c t io n u ( a; e; x) is u ( a; e; x; s) if ©aes ( x ) # fo r s o m e s, a n d is u n d e ¯ n e d if ©aes ( x ) ". 7 8 on initial segments of turing degrees containing simple t -mitotic but not wtt-mitotic sets de¯nition 2: a is computable in (turing reducible to) b, wr it t e n a·t b, if a = ©be fo r s o m e e ( s o a r e [1 , p . 5 0 ]) . de¯nition 3: a is weak truth-table reducible to b, wr it t e n a ·wtt b, if ( 9e) [a = ©be & ( 9 c o m p u t a b le f ) ( f ( x ) ¸ u( b; e; x ) ) ] ( wh e r e u ( b; e; x ) is t h e u s e fu n c t io n fr o m d e ¯ n it io n 1 ) ( r o g e r s [2 , p . 1 5 8 ]) . de¯nition 4: if a is a n o n c o m p u t a b le c o m p u t a b ly e n u m e r a b le ( c .e .) s e t , t h e n a splitting of a is a p a ir a1, a2 o f d is jo in t c .e . s e t s s u c h t h a t a1 s a2 = a ( d o wn e y , s t o b [3 , p . 4 ]) . de¯nition 5: c.e . s e t a is t -mitotic (wtt-mitotic), if t h e r e is a s p lit t in g a1, a2 o f a s u c h t h a t a1 ´t a2 ´t a ( a1 ´wtt a2 ´wtt a) ( d o wn e y , s t o b [3 , p . 8 3 ]) . de¯nition 6: ( i) a s e t is immune, if it is in ¯ n it e b u t c o n t a in s n o in ¯ n it e c .e . s e t . ( ii) a s e t is simple, if a is c .e . a n d ¹a is im m u n e ( s o a r e [1 , p . 7 8 ]) . de¯nition 7: a c .e . d e g r e e a is contiguous if fo r e ve r y p a ir a; b o f c .e . s e t s in a, a ´wtt b ( d o wn e y, s t o b [3 , p . 4 5 ]) . n o t e t h a t e a c h c o n t ig u o u s d e g r e e , b y d e ¯ n it io n , d o e s n 't c o n t a in t -m it o t ic s e t s , wh ic h a r e n o t wtt-m it o t ic . l a c h la n p r o ve d t h e e xis t e n c e o f n o n m it o t ic c .e . s e t ( l a c h la n [4 ]) . l a d n e r p r o ve d t h e e xis t e n c e o f c o m p le t e ly m it o t ic c .e . d e g r e e ( l a d n e r [5 ]) . l a d n e r a n d s a s s o [6 ] p r o ve d , t h a t fo r e ve r y n o n z e r o c .e . d e g r e e b t h e r e is a n o n z e r o c .e . d e g r e e a · b s u c h t h a t a is c o n t ig u o u s ( s e e a ls o d o wn e y, s t o b [3 ]) . th u s , t h e r e is a n in ¯ n it e c la s s o f c o n t ig u o u s d e g r e e s , a n d t h e s e d e g r e e s , a s we h a ve m e n t io n e d , d o n 't c o n t a in t -m it o t ic s e t s , wh ic h a r e n o t wtt-m it o t ic .] in g r a s s ia ( [7 ]) p r o ve d t h e d e n s it y o f n o n m it o t ic c .e . s e t s ( in r ) ( s e e a ls o d o wn e y, s la m a n [8 ]) . e . j. gr i± t h s ( [9 ]) p r o ve d t h e fo llo win g th e o r e m : th e r e e xis t s a lo w c .e . d e g r e e u s u c h t h a t if v is a c .e . d e g r e e a n d u · v, t h e n v is n o t c o m p le t e ly m it o t ic . 2 . p r e lim in a r ie s , b a s ic mo d u le s t heor em 1: f or any noncomputable c.e. degree b there is a degree a such that a · b and a contains a simple t -mitotic, but not wtt-mitotic set. proof. ( s ke t c h ) l e t h b e a g e n e r a l c o m p u t a b le fu n c t io n t h a t m a p s ! t o !2. l e t ( ªi; ãi ) d e n o t e s t h e p a ir ( ©i0; 'i0 ) fo r a ll i, wh e r e h( i ) = ( i0; i1 ) ( n o t e t h a t ªi is wtt-r e d u c t io n wit h ãi, d e n o t in g t h e c o r r e s p o n d in g u s e fu n c t io n ) . it is kn o wn ( l a d n e r [1 0 ]) t h a t t h e c o m p u t a b ly e n u m e r a b le s e t a is t -m it o t ic , , a is t -a u t o r e d u c ib le , a n d s im ila r ly, t h e c o m p u t a b ly e n u m e r a b le s e t a is wtt-m it o t ic , , a is wtt-a u t o r e d u c ib le ( d o wn e y, s t o b [3 ], s e e a ls o tr a kh t e n b r o t [1 1 ]) . th e r e fo r e , in o r d e r t o a c h ie ve n o n -wtt-m it o t ic it y, it is e n o u g h fo r u s t o a c h ie ve n o n -wtta u t o r e d u c ib ilit y. th u s , t o p r o ve o u r t h e o r e m , it is n e c e s s a r y t o c o n s t r u c t s u c h a c .e . s e t a, s o t h a t t h e fo llo win g r e qu ir e m e n t s a r e m e t . a. mokatsian 9 re : ( 9 x ) : ( ªe ( a sfxg; x ) = a ( x ) ) , if ( 8z · y ) ( ãe ( z ) #) . pe : ( we is in ¯ n it e ) ) ( 9 x ) ( x 2 we & x 2 a ) . n o t e t h a t s a t is fyin g t h e re r e qu ir e m e n t ( fo r a ll e ) p r o vid e s t h e in ¯ n it y o f t h e s e t ¹a. or d e r t h e r e qu ir e m e n t s in t h e fo llo win g p r io r it y r a n kin g : r0; p0; r1; p1; : : : ; rn; pn; : : : l e t l ( e; s) = m a xfx : ( 8y < x) ( ªe;s ( as sfyg : y ) = a ( y ) & ( 8z · x) ( ªe ( z ) #) ) g. th e m a in s t r a t e g y fo r s a t is fyin g re is t h e fo llo win g : we s e le c t a n u m b e r ( t h e s o -c a lle d fo llo we r ) x ( wh ic h s h o u ld b e a c c o m p a n ie d b y t wo m o r e e le m e n t s x¡ 2 a n d x¡ 1 , a n d p o s s ib ly, t h e t h ir d x̂; a n e xa c t d e ¯ n it io n o f t h e a t t e n d a n t n u m b e r s o f t h e fo llo we r x, n a m e ly ( x ¡ 2 ) , ( x ¡ 1 ) , x̂, will b e g ive n h e r e in a ft e r ) , we wa it u n t il l ( e; s) > x a n d e n u m e r a t e x in as+1, if ( 8z < x ) ( ãe;s ( z ) # ) , s e t t in g r ( e; s + 1 ) = u ( x; e; s) , wh e r e u ( x; e; s) = u ( ªe;s ( as sfxg; x) ) . a n b-p e r m it t in g p r o c e d u r e is in t r o d u c e d in o r d e r t o p r o vid e a·t b ( wh e r e b is a c .e . s e t fr o m d e g r e e b) . to s a t is fy t h e r e qu ir e m e n t o f re a t e a c h s t a g e , we h a ve a ¯ n it e s e t o f fo llo we r s x1;s; < : : : < xn;s. in t h is c o n s t r u c t io n , a m o d ī c a t io n o f t h e b-p e r m it t in g m e t h o d is u s e d . w e t r e a t t h e in t e r va l [0 , . . . , i] a s a llo win g fo r xi;s. to s a t is fy t h e r e qu ir e m e n t o f pe a t e a c h s t a g e , we h a ve a ¯ n it e s e t o f fo llo we r s y1;s; < : : : < yn;s. fo r r e qu ir e m e n t pe, t h e u s u a l b-p e r m it t in g m e t h o d is u s e d . th e c o n s t r u c t io n will b e s u c h t h a t if e ve n t u a lly we h a ve ªe ( a s fxg; x) = a( x ) & ãe is a t o t a l fu n c t io n , t h e n it will b e p o s s ib le t o c o m p u t e b. th e g r o u n d o f s a t is fa c t io n s fo r r e qu ir e m e n t s o f re a n d pe will b e g ive n b e lo w. 2 .1 b a s ic mo d u le fo r re th e fo llo we r s x1;s; : : : ; xn;s s a t is fy t h e fo llo win g r u le s b e lo w. appointment. if xi;s is c u r r e n t ly d e ¯ n e d a n d xi+1;s is n o t , t h e n if l ( e; s) > xi;s, d e c la r e xi;s a s active, a n d s e t xi+1;s+1 = ¹z ( z ¸ s + 2 & ( 9 k ) ( z = 2 k ) ) . s e t ~r ( e; s + 1 ) = m a x( u ( xk;s; e; s) : k · i) . to g e t a n id e a o f t h e r e s t r ic t io n fu n c t io n ~r ( e; s) , we g ive it s d e ¯ n it io n , a lt h o u g h it is n o t u s e d in t h e b a s ic m o d u le . w e s a y t h a t xi;s is superactive, if xi;s ¡ 2 a n d xi;s ¡ 1 b e lo n g t o as. p ermission. if xi;s is a c t ive a n d i 2 bat s, t h e n if ( i) ( 9 j > i) [xj;s is s u p e r a c t ive & xj;s =2 as], le t j0 = ¹z [xz;s is s u p e r a c t ive & xj;s =2 as]. th e n we e n u m e r a t e t h e n u m b e r s xj0; s, x̂j0; s in t o as+1 ( wh e r e x̂j0;s = ãe ( xj0;s ) ) . ca n c e l xk;s , fo r a ll ( k > j0 ) . w e will d o t h e s a m e wit h t h e a c c o m p a n yin g e le m e n t s o f t h e c o r r e s p o n d in g fo llo we r s . ( ii) if ( i) a n d ( :9 j ) [xj;s 2 as] d o e s n o t h o ld , t h e n we e n u m e r a t e t h e n u m b e r s xi;s¡ 2 ; xi;s¡ 1 in t o as+1. ca n c e l xk;s , fo r a ll ( k > i ) . w e will d o t h e s a m e wit h t h e a c c o m p a n yin g e le m e n t s o f t h e c o r r e s p o n d in g fo llo we r s . fo r a n y i s u c h t h a t t h e fo llo we r xi;s is n o t c a n c e le d a t t h e e n d o f t h e p a r t \ p e r m is s io n " o f t h e b a s ic m o d u le a n d is a c t ive , le t 's s e t xi;s+1 = xi;s. w e will d o t h e s a m e wit h t h e a c c o m p a n yin g e le m e n t s o f t h e c o r r e s p o n d in g fo llo we r s . th e \ c a n c e lla t io n " r u le , wh ic h is p r e s e n t in t h e p r o o f o f th e o r e m 4 .8 ( d o wn e y, s la m a n [8 ]) , in t h is c a s e it will b e n e c e s s a r y t o n o t e t h e e ®e c t o f t h e r e qu ir e m e n t s o f rj a n d pj ( wh e r e j < e) o n s a t is fyin g t h e r e qu ir e m e n t re, b u t n o t t o d e s c r ib e t h e b a s ic m o d u le it s e lf fo r re. 1 0 on initial segments of turing degrees containing simple t -mitotic but not wtt-mitotic sets 2 .2 b a s ic mo d u le fo r pe th e fo llo we r s y1;s; : : : ; yn;s s a t is fy t h e fo llo win g r u le s b e lo w. appointment. if yi;s is c u r r e n t ly d e ¯ n e d a n d yi+1;s is n o t , t h e n if ( 9z ) ( z 2 we z ¸ yi;s ) , d e c la r e yi;s a s a c t ive , a n d s e t yi+1;s+1 = ¹z ( z ¸ s & ( 9 k ) ( z = 2 k ) ) . p ermission. if yi;s is a c t ive a n d i 2 bat s, t h e n e n u m e r a t e t h e n u m b e r s yi;s; yi;s + 1 ; z a n d z + 1 in t o as+1. th e \ c a n c e lla t io n " r u le , wh ic h is p r e s e n t in t h e p r o o f o f th e o r e m 4 .8 ( d o wn e y, s la m a n [8 ]) , in t h is c a s e it will b e n e c e s s a r y t o n o t e t h e e ®e c t o f t h e r e qu ir e m e n t s o f rj ( wh e r e j · e ) o n s a t is fyin g t h e r e qu ir e m e n t pe, b u t n o t t o d e s c r ib e t h e b a s ic m o d u le it s e lf fo r pe. 3 . v e r ī c a t io n o f l e m m a s lemma 1: suppose that ãe is total and ( 8x ) ( ªe ( a s fxg; x ) #) . then ( 9 y ) : ( ( ªe ( a s fyg; y ) = a ( y ) ) . thus, the requirement re is satis¯ed. proof. s u p p o s e o t h e r wis e . w e s h o w t h a t b is c o m p u t a b le . n o t e t h a t s in c e we o n ly c o n s id e r t h e s a t is fa c t io n o f t h e b a s is m o d u le fo r re ( t h a t is , we d o n o t t a ke in t o a c c o u n t t h e e ®e c t o f t h e r e qu ir e m e n t s rj a n d pj ( wh e r e j < e) o n t h e s a t is fa c t io n o f t h e r e qu ir e m e n t re ) , it is o b vio u s t h a t c o n d it io n s ( i) , ..., ( iv) a r e m e t . ( i) a ll t h e xi;s e ve n t u a lly b e c o m e p e r m a n e n t ly d e ¯ n e d , t h a t is lim s xi;s = xi e xis t s wit h xi =2 a. ( ii) on c e xk is d e ¯ n e d a t s t a g e t, ( 8s > t ) ( u ( xk; e; t ) = u ( xk; e; s) = u( e; xk ) ) . ( iii) ( 8i ) ( xi+1 > m a xfu( e; xk ) : k · i g) . ( iv) it c a n b e e ®e c t ive ly r e c o g n iz e d , wh e n ( i) o c c u r s . two c a s e s a r e p o s s ib le : ( a ) ( 9 m) ( 8 k > m ) [xk ¡ 2 =2 a]; ( b ) ( 8 m) ( 9 k > m ) [xk ¡ 2 2 a]. fo r b o t h c a s e s ( ( a ) a n d ( b ) ) , it will b e p r o ve d t h a t b is c o m p u t a b le ( a n d t h u s , t h e a s s u m p t io n t h a t l e m m a 1 is fa ls e will le a d t o a c o n t r a d ic t io n wit h t h e s u p p o s it io n o f n o n c o m p u t a b ilit y o f b ) . n o w, if ( a ) h o ld s , we p r o ve t h a t b is c o m p u t a b le . if c o n d it io n s ( i) , ..., ( iv) a r e s a t is ¯ e d , we s h o w h o w t o c o m p u t e b ( t h a t is , t h e c h a r a c t e r is t ic fu n c t io n o f t h e s e t b; r e m in d t h a t we o ft e n id e n t ify t h e s e t b wit h it s c h a r a c t e r is t ic fu n c t io n ) . l e t f k x d e n o t e s t h e r e s t r ic t io n o f f t o a r g u m e n t s y < x, a n d a k x d e n o t e s âa k x. l e t s0 b e s u c h a s t a g e t h a t b k m + 1 = bs0 k m + 1 a n d a k xm+1 = as0 k xm+1. l e t q 2 ! a n d q > m. e ®e c t ive ly c o m p u t e a s t a g e s s o t h a t xq+1 is d e ¯ n e d , t h a t is xq+1 = xq+1;s ( in t h a t c a s e , in fa c t , s > s0 ) . th e n xq is a c t ive , xq 2 a a n d s in c e xq+1 is t h e ¯ n a l va lu e o f t h e q + 1 -t h fo llo we r , t h e c o m p u t a t io n s o f u ( xj; e; s) a r e t r u e fo r a ll j · q. in t h is c a s e q 2 b , q 2 bs, b e c a u s e o t h e r wis e it wo u ld le a d t o t h e fa c t t h a t xq ¡ 2 a. mokatsian 1 1 wo u ld h a ve e n t e r e d t h e s e t , c o n t r a r y t o o u r a s s u m p t io n t h a t c a s e ( a ) h o ld s . n o w s u p p o s e t h a t c a s e ( b ) h o ld s . l e t u s p r o ve t h a t in t h is c a s e a ls o b is c o m p u t a b le . if c o n d it io n s ( i) , ..., ( iv) a r e fu l̄ lle d , we s h o w h o w t o c o m p u t e b. l e t q 2 !. e ®e c t ive ly c o m p u t e s u c h a s t a g e s a n d a n u m b e r p s o t h a t p = ¹z ( z ¸ q & xz¡2 2 a & xz+1 = xz+1;s ) . th e n xp is a c t ive , xp =2 as a n d s in c e xp+1 is t h e ¯ n a l va lu e o f t h e p + 1 -t h fo llo we r , t h e n u ( xj; e; s) c o m p u t a t io n s a r e t r u e fo r a ll j · p. in t h is c a s e q 2 b , q 2 bs, s in c e o t h e r wis e ( t h a t is , if q e n t e r s b a ft e r t h e s t a g e s) t h is will le a d t o t h e e n t r y p in t o a a n d s a t is fa c t io n o f t h e r e qu ir e m e n t re, wh ic h will c o n t r a d ic t t h e in it ia l a s s u m p t io n t h a t l e m m a 1 is fa ls e . l e m m a 1 is p r o o ve d . lemma 2: suppose that we is an in¯nite set. then ( 9 z ) ( z 2 we & z 2 a ) . thus, the requirement pe is satis¯ed. proof. s u p p o s e o t h e r wis e . w e s h o w t h a t b is c o m p u t a b le . l e t ~r ( e) = lim s ~r ( e; s) . a lt h o u g h t h e u s e o f t h is fu n c t io n in t h e d e s c r ip t io n o f t h e b a s is m o d u le fo r pe is n o t n e c e s s a r y, a n in d ic a t io n o f t h is fu n c t io n c le a r ly s h o ws t h e e ®e c t o f t h e r e qu ir e m e n t s rj ( wh e r e j · e ) o n t h e s a t is fa c t io n o f t h e r e qu ir e m e n t s pe wh e n c o n s t r u c t in g t h e s e t a. l e t t0 b e s u c h t h a t ( 8s ¸ t0 ) ~r ( e; s) = ~r ( e ) . th e n it is o b vio u s , t h a t a ll t h e yi;s b e c o m e p e r m a n e n t ly d e ¯ n e d ( i.e ., 8i 9 ( t ¸ t0 ) ( 8s) ( yi;t = yi;s = yi ) ) wit h yi =2 a. in fa c t , if t h e r e e xis t e d k s u c h t h a t yk 2 a, t h e n , b y c o n s t r u c t io n , t h e r e wo u ld e xis t z s u c h t h a t z 2 we t a. a s s u m in g t h e o p p o s it e o f t h e s t a t e m e n t o f t h e p r o p o s it io n , we s h o w h o w b c a n b e c o m p u t e d . l e t q 2 !. fin d t ¸ t0 s u c h t h a t yq is p e r m a n e n t ly d e ¯ n e d . th e n q 2 b , q 2 bt, s in c e o t h e r wis e q0 s e n t r y in t o b wo u ld m e e t pe. l e m m a 2 is p r o o ve d . 4 . co n c lu s io n n o t e t h a t t h e c o h e r e n c e o f c o n s t r u c t io n s t o s a t is fy t h e r e qu ir e m e n t s re a n d pe ( fo r a ll e) is n o t d i± c u lt , s in c e s a t is fyin g t h e r e qu ir e m e n t s re a n d pe ( fo r a ll e ) r e qu ir e s a ¯ n it e n u m b e r o f s t e p s . w e a ls o n o t e t h a t t h e in d ic a t e d m e t h o d o f c o n s t r u c t in g t h e s e t a ( b a s e d o n t h e c o n s t r u c t io n s fo r t h e b a s ic m o d u le s ) will r e s u lt in t h e s e t a t !ev b e in g t -e qu iva le n t t o t h e s e t a t !od. th e s e r e m a r ks a llo w u s t o c o m p le t e t h e p r o o f o f t h e t h e o r e m . n o t e t h a t it fo llo ws fr o m t h e a b o ve t h e o r e m t h a t b e lo w a n y n o n c o m p u t a b le c .e . d e g r e e t h e r e is a n in ¯ n it e n u m b e r o f n o n c o m p u t a b le c .e . d e g r e e s wit h t h e a b o ve m e n t io n e d p r o p e r t y ( s in c e t h e d e g r e e a ( m e n t io n e d in t h e t h e o r e m ) , c o n t a in in g a s im p le s e t , is a n o n c o m p u t a b le c .e . d e g r e e ) . 1 2 on initial segments of turing degrees containing simple t -mitotic but not wtt-mitotic sets references [1 ] r . i. s o a r e , r ecursively e numerable sets and d egrees, s p r in g e r -v e r la g , 1 9 8 7 . [2 ] h . r o g e r s , theory of r ecursive f unctions and e ®ective computability, mc gr a w-h ill b o o k co m p a n y, 1 9 6 7 . [3 ] r . g. d o wn e y a n d m. s t o b , \ s p lit t in g th e o r e m s in r e c u r s io n th e o r y" , ann. p ure appl. l ogic, vo l. 6 5 , p p . 1 -1 0 6 , 1 9 9 3 . [4 ] a . h . l a c h la n , th e p r io r it y me t h o d . i, ze it s c h r ift fr m a t h e m a t is c h e l o g ik u n d gr u n d la g e n d e r ma t h e m a t ik, vo l. 1 3 , p p . 1 -1 0 , 1 9 6 7 . [5 ] r . l a d n e r , \ a c o m p le t e m it o t ic r e c u r s ive ly e n u m e r a b le d e g r e e " , trans. amer. m ath. soc., 1 8 4 , p p . 4 9 7 -5 0 7 , 1 9 7 3 . [6 ] r . l a d n e r a n d l . s a s s o , \ th e we a k t r u t h -t a b le d e g r e e s o f r e c u r s ive ly e n u m e r a b le s e t s " , arch. m ath. l ogik grundlang grundlagen der m athematik, vo l. 8 , p p . 4 2 9 -4 4 8 , 1 9 7 5 . [7 ] m. in g r a s s ia , p -generecity for recursive enumerable degrees, p h d . th e s is , u n ive r s it y o f illin o is , 1 9 8 1 . [8 ] r . g. d o wn e y a n d t. a . s la m a n , \ co m p le t e ly mit o t ic r . e . d e g r e e s " , ann. p ure appl. l ogic, vo l. 4 1 , p p . 1 1 9 -1 5 2 , 1 9 8 9 . [9 ] e . j. gr i± t h s , completely m itotic turing d egrees, j ump classes and e numeration d egrees, p h .d . th e s is , u n ive r s it y o f w is c o n s in -ma d is o n , 1 9 9 8 . [1 0 ] r . l a d n e r , \ mit o t ic r e c u r s ive ly e n u m e r a b le s e t s " , the j ournal of symbolic l ogic, vo l. 3 8 , n o . 2 , p p . 1 9 9 -2 1 1 , 1 9 7 3 . [1 1 ] b . a . tr a kh t e n b r o t , \ on a u t o r e d u c ib ilit y" ( in r u s s ia n ) , d okl. akad. nauk sssr , vo l. 1 9 2 , p p . 1 2 2 4 -1 2 2 7 , 1 9 7 0 . submitted 04.07.2019, accepted 22.11.2019. ä³ñ½ t -ùçãáïçï, µ³ûó áã wtt-ùçãáïçï µ³½ùáõãûáõýý»ñ å³ñáõý³ïáõ ãûáõñçý·û³ý ³ëïç×³ýý»ñç áñáß ñ³ïïáõãûáõýý»ñç í»ñ³µ»ñû³é ²ñë»ý ð. øáï³óû³ý ðð ¶²² æýýáñù³ïçï³ûç ¨ ³íïáù³ï³óù³ý åñáµé»ùý»ñç çýëïçïáõï e-mail: arsenmokatsian@gmail.com ²ù÷á÷áõù ð»ï³½áïíáõù»ý t -ùçãáïçï ïñáñù³ý ñ³ïïáõãû³ùµ ûåïí³í, µ³ûó wtt-ùçãáïçï ïñáñáõù ãáõý»óáõ µ³½ùáõãûáõýý»ñ å³ñáõý³ïáõ é»ïáõñëçíáñ»ý ãí³ñï»éç (é.ã.) ãûáõñçý·û³ý ³ëïç×³ýý»ñç ñ³ïïáõãûáõýý»ñá: ²å³óáõóí³í ¿, áñ ï³ù³û³ï³ý áã é»ïáõñëçí (é.ã.) a ³ëïç×³ýç ñ³ù³ñ ·áûáõãûáõý áõýç áã é»ëáõñëçí (é.ã) ³ûýåçëç b ³ëïç×³ý, áñ b · a ¨ b-ý å³ñáõý³ïáõù ¿ å³ñ½ t ùçãáïçï, µ³ûó áã wttùçãáïçï µ³½ùáõãûáõý: a. mokatsian 1 3 ´³ý³éç µ³é»ñ` ùçãáïçï µ³½ùáõãûáõý, t -ñ³ý·»óáõù, wtt-ñ³ý·»óáõù, å³ñ½ µ³½ùáõãûáõý, ½áõ·³ïóí³í ³ëïç×³ý: î íåêîòîðûõ ñâîéñòâàõ òüþðèíãîâûõ ñòåïåíåé, ñîäåðæàùèõ ïðîñòûå t -ìèòîòè÷åñêèå ìíîæåñòâà, íå ÿâëÿþùèåñÿ wtt-ìèòîòè÷åñêèìè àðñåí à. ìîêàöÿí èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà e-mail: arsenmokatsian@gmail.com àííîòàöèÿ èññëåäóþòñÿ ñâîéñòâà ðåêóðñèâíî ïåðå÷èñëèìûõ (ð.ï.) òüþðèíãîâûõ ñòåïåíåé, ñîäåðæàùèõ ìíîæåñòâà, êîòîðûå îáëàäàþò ñâîéñòâîì -ìèòîòè÷åñêîãî ðàçáèåíèÿ, íî íå èìåþò wtt-ìèòîòè÷åñêîãî ðàçáèåíèÿ. äîêàçàíî, ÷òî äëÿ ëþáîé íåðåêóðñèâíîé (ð.ï.) ñòåïåíè a ñóùåñòâóåò íåðåêóðñèâíàÿ (ð.ï.) ñòåïåíü b, òàêàÿ ÷òî b · a è b ñîäåðæèò ïðîñòîå t -ìèòîòè÷åñêîå ìíîæåñòâî, êîòîðîå íå ÿâëÿåòñÿ wtt-ìèòîòè÷åñêèì. êëþ÷åâûå ñëîâà: ìèòîòè÷åñêîå ìíîæåñòâî, t -ñâîäèìîñòü, wtt-ñâîäèìîñòü, ïðîñòîå ìíîæåñòâî, ñöåïëåííàÿ ñòåïåíü. microsoft word amirranjabar_22.doc mathematical problems of computer science 33, 172--182, 2010. 172 estimate for the arithmetical cost of an algebraic multigrid preconditioner amir ranjbar yerevan state university, islamic azad university of minoodasht (iran) ranjbar@minoodasht.ac.ir abstract a multigrid preconditioner for the matrix arising in finite element approximation of model elliptic boundary value problem is proposed. hierarchical triangular grids with bisection form the basis of multigrid construction. the main purpose of the paper is to evaluate the arithmetical cost of a preconditioner step. references 1. yu.a. kuznetsov, “algebraic multigrid domain decomposition methods”, sov. j. numer. anal. math. modelling, vol.4, no.5, pp. 351-379, 1989. 2. yu.r. hakopian and yu.a. kuznetsov, “algebraic multigrid/substructuring preconditioners on triangular grids”, sov. j. numer. anal. math. modelling, vol.6, no.6, pp. 453-483, 1991. 3. o. axelsson, yu.r. hakopian and yu.a. kuznetsov, “multilevel preconditioning for perturbed finite element matrices”, ima j. numer. anal., vol.17, pp. 125-149, 1997. 4. yu.r. hakopian, “algebraic multilevel/substructuring preconditioner in finite element method with piecewise quadratic approximation”, “mathematical problems of computer science”, vol.21, pp. 164-180, 2000. 5. yu.r. hakopian, “algebraic multilevel preconditioners for third order finite element approximation”, algebra, geometry & their applications (seminar proceedings), yerevan state university, armenia, vol.1, pp. 20-39, 2001. 6. yu.r. hakopian and a.s. harutyunyan, “two-level preconditioners for serendipity finite element matrices”, linear algebra appl., vol.13, no.10, pp. 847-864, 2006. 7. g. strang and g. fix, “an analysis of the finite element method”, prentice-hall englewood cliffs, n.j., 1973. 8. o.c. zenkiewich and k. morgan, “finite elements and approximation”, wiley, ny, 1983. 9. o. axelsson and p.s.vassilevski, “algebraic multilevel preconditioning methods”,i, numer. math., vol. 56, pp. 157-177, 1989. 10. o. axelsson. iterative solution methods, cambridge university press, 1994. a.ranjbar 173 ð³ýñ³ñ³ßí³ï³ý µ³½ù³ó³ýó³ûçý í»ñ³å³ûù³ý³íáñçãç ãí³µ³ý³ï³ý ·ýç ·ý³ñ³ï³ï³ýá ². è³ýçµ³ñ ²ù÷á÷áõù ²ßë³ï³ýùáõù ï³éáõóíáõù ¿ µ³½ù³ó³ýó³ûçý í»ñ³å³ûù³ý³íáñçã ùá¹»é³ûçý ¿éçåï³ï³ý »½ñ³ûçý ëý¹ñç í»ñç³íáñ ï³ññ³ûçý ùáï³ñïù³ý ³ñ¹ûáõýùáõù ëï³óíáõ ù³ïñçóç ñ³ù³ñ: ´³½ù³ó³ýó³ûçý ï³éáõóí³íùç ñçùùáõù áýï³í »ý ñç»ñ³ñëç³ï³ý »é³ýïûáõý ó³ýó»ñ: ðá¹í³íç ñçùý³ï³ý ýå³ï³ïý ¿ ·ý³ñ³ï»é í»ñ³å³ûù³ý³íáñù³ý ù³ûéç ãí³µ³ý³ï³ý ·çýá: начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 43, 52--56, 2015. reconstruction of distorted images souren b. alaverdyan institute for informatics and automation problems of nas ra e-mail: souren@ipia.sci.am abstract the non-focused images quality arising algorithm based on winer filtration is presented in the paper. filtration is realized in spectral domain of image. keywords: filter, image, spectrum, wiener. 1. introduction during the image registration there often appear distortions of different types depending on registering devices characteristics (permission), technical situation of location and also peculiarities of the area being registered. on images obtained by optical devices there can be violations of focal distance; there can appear diffusions when moving objects are being registered, etc. we consider the image as a function of two variables 𝑓𝑓(𝑥𝑥, 𝑦𝑦) which is the projection of two or three dimensional fields of view, where (𝑥𝑥, 𝑦𝑦) is a coordinate of any point of plane and 𝑓𝑓(𝑥𝑥, 𝑦𝑦) is the light intensity in the point (𝑥𝑥, 𝑦𝑦). we’ll consider a problem of optically [1] registered distorted images reconstructon in spectral area, because optical systems focus the falling light and that can be expressed by fourie transform, so the image reconstruction problem reduces the solving of integral equations of second order. 2. image reconstruction let 𝑔𝑔(𝑥𝑥, 𝑦𝑦) be the given image and 𝑓𝑓(𝑥𝑥, 𝑦𝑦) be the reconstructing image. then the following equation [2] takes place: 𝑔𝑔(𝑥𝑥, 𝑦𝑦) = �𝑓𝑓(𝑢𝑢, 𝑣𝑣)ℎ(𝑥𝑥, 𝑦𝑦, 𝑢𝑢, 𝑣𝑣)𝑑𝑑𝑢𝑢𝑑𝑑𝑣𝑣, (1) where the function ℎ(𝑥𝑥, 𝑦𝑦, 𝑢𝑢, 𝑣𝑣) is called an image registering system’s impulse response (output value corresponding to unit impulse). to solve this equation we’ll give some assumptions. 52 mailto:souren@ipia.sci.am s. alaverdyan 53 definition 1: the system is called space-invariant, if its impulse function response depends on the difference between the input(𝑥𝑥, 𝑦𝑦) and output(𝑥𝑥, 𝑦𝑦) planes coordinates: ℎ(𝑥𝑥, 𝑦𝑦, 𝑢𝑢, 𝑣𝑣) = ℎ(𝑥𝑥 − 𝑢𝑢, 𝑦𝑦 − 𝑣𝑣). for such system the equation (1) will be represented as 𝑔𝑔(𝑥𝑥, 𝑦𝑦) = �𝑓𝑓(𝑢𝑢, 𝑣𝑣)ℎ(𝑥𝑥 − 𝑢𝑢, 𝑦𝑦 − 𝑣𝑣)𝑑𝑑𝑢𝑢𝑑𝑑𝑣𝑣, (2) which is usually called a convolution. equation (2) can also be represented as 𝑔𝑔(𝑥𝑥, 𝑦𝑦) = 𝑓𝑓(𝑥𝑥, 𝑦𝑦) ∗ ℎ(𝑥𝑥, 𝑦𝑦). (3) since 𝑓𝑓(𝑥𝑥, 𝑦𝑦) is a function of image describing the range of vision, and 𝑔𝑔(𝑥𝑥, 𝑦𝑦) is a function of registered image, we can see that ℎ(𝑥𝑥, 𝑦𝑦) is a noise describing function. in general case linear filtration algorithms are realized by transforms of type (2) having the following discrete representation 𝑔𝑔𝑖𝑖,𝑗𝑗 = � � 𝑓𝑓𝑘𝑘,𝑙𝑙ℎ𝑘𝑘−𝑖𝑖+𝑟𝑟 2� ,𝑙𝑙−𝑗𝑗+𝑟𝑟 2� , 𝑖𝑖 ∈ � 𝑟𝑟 2� , 𝑀𝑀 + 𝑟𝑟 2� �, 𝑗𝑗 ∈ � 𝑟𝑟 2� , 𝑁𝑁 + 𝑟𝑟 2� �. 𝑗𝑗+𝑟𝑟/2 𝑙𝑙=𝑗𝑗−𝑟𝑟/2 𝑖𝑖+𝑟𝑟/2 𝑘𝑘=𝑖𝑖−𝑟𝑟/2 (4) 𝑀𝑀 is the number of image rows, 𝑁𝑁 is the number of image columns, the sum includes the points of rectangular with centre(𝑖𝑖, 𝑗𝑗) and 2𝑟𝑟 + 1 sides. before calculation of transform (4) all sides of image should be already widened by rectangular layers of width 𝑟𝑟/2. in spectral domain the linear filtration algorithm is also based on convolution theorem, so instead of calculating by formula (4) it can be realized by the following formula: 𝐺𝐺(𝑢𝑢, 𝑣𝑣) = 𝐹𝐹(𝑢𝑢, 𝑣𝑣)𝐻𝐻(𝑢𝑢, 𝑣𝑣), (5) where 𝐺𝐺, 𝐹𝐹, 𝐻𝐻 are fourier transforms of functions 𝑔𝑔, 𝑓𝑓, ℎ. note, that complex multiplication is realized by all 𝑢𝑢, 𝑣𝑣 frequencies. now we’ll represent the mathematical model of the system: 𝑓𝑓(𝑥𝑥, 𝑦𝑦) -input image function ( undistorted), ℎ(𝑥𝑥, 𝑦𝑦) noise causing function, 𝑛𝑛(𝑥𝑥, 𝑦𝑦) total noise, 𝑔𝑔(𝑥𝑥, 𝑦𝑦) distorted image (fuzzified, unfocused). so we have the following representation of the process: 𝑔𝑔(𝑥𝑥, 𝑦𝑦) = 𝑓𝑓(𝑥𝑥, 𝑦𝑦) ∗ ℎ(𝑥𝑥, 𝑦𝑦) + 𝑛𝑛(𝑥𝑥, 𝑦𝑦). (6) it is required to find the impulse characteristic function which will be for the system the best reconstruction function by mean square deviation reconstruction of distorted images 54 𝜎𝜎 = � 1 𝑚𝑚𝑛𝑛 � ��𝑓𝑓𝑖𝑖,𝑗𝑗−𝑓𝑓𝑖𝑖,𝑗𝑗� 2 𝑛𝑛−1 𝑗𝑗=0 𝑚𝑚−1 𝑖𝑖=0 → 𝑚𝑚𝑖𝑖𝑛𝑛. the problem solution for linear stationary processes was given by wiener, the detailed proof is given in [3]. the best approximating filter’s spectral representation of function 𝑓𝑓(𝑥𝑥, 𝑦𝑦) is represented as [3] 𝐹𝐹�(𝑢𝑢, 𝑣𝑣) = 1 𝐻𝐻(𝑢𝑢, 𝑣𝑣) ∙ |𝐻𝐻(𝑢𝑢, 𝑣𝑣)|2 |𝐻𝐻(𝑢𝑢, 𝑣𝑣)|2 + 𝑆𝑆𝑛𝑛(𝑢𝑢. 𝑣𝑣) 𝑆𝑆𝑓𝑓(𝑢𝑢. 𝑣𝑣) � 𝐺𝐺(𝑢𝑢, 𝑣𝑣), (7) where 𝑆𝑆𝑛𝑛(𝑢𝑢. 𝑣𝑣) is the spectral density of additive noise and 𝑆𝑆𝑓𝑓(𝑢𝑢. 𝑣𝑣)𝑓𝑓(𝑥𝑥, 𝑦𝑦) is the spectral density of the function. generally these values are unknown. the ratio 𝑆𝑆𝑛𝑛(𝑢𝑢. 𝑣𝑣)/𝑆𝑆𝑓𝑓(𝑢𝑢. 𝑣𝑣) is the inverse value of signal-noise value. its value in time domain is considered acceptable if it is in the interval of 30-40 decibels. the noises induced by focal distance violations on the images registered by the optical devices mainly depend on the light dispersion problem described by the following two functions: ℎ(𝑥𝑥, 𝑦𝑦) = 1 2𝜋𝜋𝜎𝜎2 𝑒𝑒− 𝑥𝑥2+𝑦𝑦2 2𝜎𝜎2 , ℎ(𝑥𝑥, 𝑦𝑦) = � 1 𝜋𝜋𝑟𝑟2 , 𝑖𝑖𝑓𝑓 𝑥𝑥2 + 𝑦𝑦2 < 𝑟𝑟2 0, 𝑖𝑖𝑓𝑓 𝑥𝑥2 + 𝑦𝑦2 ≥ 𝑟𝑟2. if the image doesn’t include an additive noise then 𝑛𝑛(𝑥𝑥, 𝑦𝑦) = 0 and the formula (7) is represented as 𝐹𝐹�(𝑢𝑢, 𝑣𝑣) = 𝐺𝐺(𝑢𝑢, 𝑣𝑣)/𝐻𝐻(𝑢𝑢, 𝑣𝑣) , (8) and is called an inverse filter. 3. inverse filters indeterminacy appears when because of some device errors during image registering under some frequencies the value of denominator 𝐻𝐻(𝑢𝑢, 𝑣𝑣) of equation (8) is equal to 0. in such cases the value of spectrum corresponding to this value of image is set equal to zero. as a result, on the filtered image there appear obvious horizontal or vertical (sometimes curved) phenomena. to reduce such occurrences we offer to realize the lowfrequency interpolation in spectral domain: 𝐹𝐹� (𝑢𝑢, 𝑣𝑣) = � � 𝑠𝑠𝑖𝑖,𝑗𝑗, 𝑢𝑢+𝑤𝑤 𝑗𝑗=𝑣𝑣−𝑤𝑤 𝑢𝑢+𝑤𝑤 𝑖𝑖=𝑢𝑢−𝑤𝑤 (9) where s. alaverdyan 55 𝑠𝑠𝑖𝑖,𝑗𝑗 = ⎩ ⎨ ⎧𝐹𝐹(𝑖𝑖, 𝑗𝑗) sin(2𝜋𝜋𝑓𝑓𝑖𝑖) 𝑖𝑖 sin(2𝜋𝜋𝑓𝑓𝑗𝑗) 𝑗𝑗 , 𝑖𝑖𝑓𝑓 𝑖𝑖 > 0, 𝑗𝑗 > 0, 2𝜋𝜋𝑓𝑓𝐹𝐹(𝑖𝑖, 𝑗𝑗), 𝑖𝑖𝑓𝑓 𝑖𝑖 = 0 𝑜𝑜𝑟𝑟 𝑗𝑗 = 0, 0, 𝑖𝑖𝑓𝑓 𝑖𝑖 < 0 𝑜𝑜𝑟𝑟 𝑗𝑗 < 0. in case of 𝑖𝑖 = 𝑤𝑤, 𝑗𝑗 = 𝑤𝑤 , 𝐹𝐹�(𝑢𝑢, 𝑣𝑣) = 2𝜋𝜋𝑓𝑓, 𝑓𝑓 ∈ (0; 0,5). there are many internet investigations and program realizations of this problem. i think, the system smartdeblur-1.27-win is one of the best program realizations, but its mathematical apparatus is not presented in the work. the program realization of the method(5)-(9) presented in this paper has been fulfilled. the result of the system work and comparison with system smartdeblur-1.27-win [4] are given below. a b c fig. 1. a) input image including gauss noise with domain of dispersion 𝜎𝜎 = 3 and radius 𝑟𝑟 = 5, 𝑓𝑓 = 0.45; b) the result of program realization of developed system; c) the result of smartdeblur-1.27-win system work. references [1] x. chen, j. yong,“image deblur in gradient domain”,optical enginering, vol. 49, no. 11,117003, 2010. [2] у. пратт, цифровая обработка изображений, мир, москва 1982. [3] н. ахмед, к.р. рао,ортогональные преобразования при обработке цифровых сигналов, москва, “связь”, стр. 164-181, 1980. [4] [online]. available: http://smartdeblur.net/ submitted 10.12.2014, accepted 12. 02. 2015. http://smartdeblur.net/ reconstruction of distorted images 56 չֆոկուսացված պատկերների որակի բարձրացում ս. ալավերդյան ամփոփում աշխատանքում ներկայացվում է չֆոկուսացված պատկերների որակի բարձրացման նոր ալգորիթմ, որը հիմնված է վիների ֆիլտրացիայի վրա: ֆիլտրացիան իրականացվում է պատկերի սպեկտրալ տիրույթում: улучшение качества расфокусированных изображений с. алавердян аннотация в работе представляется новый алгоритм улучшения качества расфокусированных изображений, который основан на фильтре винера. фильтрация выполняется в спектральной области изображения. d:\sbornik\...\efficiency.dvi mathematical problems of computer science 33, 5{10, 2010. e ±ciency of depth-restr icted substitution rules h a ko b n a lb a n d ya n institute for informatics and automation problems of nas of ra e-mail: hakob nalbandyan@yahoo.com abstract we compare the proof complexities in frege systems with a substitution rule without any restrictions and with depth-restricted substitution rule. we prove that frege system with well-known substitution rule and frege system with depth-restricted substitution rule are polynomially equivalent by size, but the ¯rst system has exponential speed-up over the second system by steps. refer ences [1 ] s . a . co o k, a . r . r e c kh o w,\ th e r e la t ive e ± c ie n c y o f p r o p o s it io n a l p r o o f s ys t e m s " , j ournal of symbolic l ogic, vo l. 4 4 , p p . 3 6 { 5 0 , 1 9 7 9 . [2 ] g. ce jt in , a . ch u b a r ya n , \ on s o m e b o u n d s t o t h e le n g t h s o f lo g ic a l p r o o fs in c la s s ic a l p r o p o s it io n a l c a lc u lu s " , ( in r u s s ia n ) , trudy vycisl.centra an arm ssr i yerevan univ. 8, p p . 5 7 { 6 4 , 1 9 7 5 . [3 ] a . a . ch u b a r ya n , \ th e c o m p le xit y in fr e g e p r o o fs wit h s u b s t it u t io n " , m athem. p roblems of computer science, vo l. 2 2 , p p . 7 { 1 1 , 2 0 0 1 . [4 ] s . r . b u s s , \ s o m e r e m a r ks o n le n g t h s o f p r o p o s it io n a l p r o o fs " , arch. m ath. l ogic, vo l. 3 4 ,p p . 3 7 7 -3 9 4 , 1 9 9 5 . [5 ] a . a . ch u b a r ya n , a r m . ch u b a r ya n , h . n a lb a n d ya n , \ co m p a r is o n o f t h e e ± c ie n c y o f fr e g e s ys t e m s wit h r e s t r ic t e d s u b s t it u t io n r u le s " , cs it, y e r e va n , p p . 3 1 -3 2 , 2 0 0 9 . [6 ] a . a . ch u b a r ya n , a r m . ch u b a r ya n , h . n a lb a n d ya n , \ e ± c ie n c y o f we a k s u b s t it u t io n r u le s " , e s m o f a s l , l c-0 9 , s o ¯ a , p . 3 7 . 5 6 e±ciency of depth-restricted substitution rules êáñáõãû³ùµ ë³ñù³ý³÷³ï ï»ë³¹ñù³ý ï³ýáýý»ñç ³ñ¹ûáõý³í»ïáõãûáõýá ð. ü³éµ³ý¹û³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ñ³ù»ù³ïíáõù »ý áëï ³ñï³íáõùý»ñç »ñïáõ µ³ñ¹áõãû³ý µýáõã³·áñçãý»ñç (»ñï³ñáõãûáõý ¨ ù³ûé»ñç ù³ý³ï) üñ»·»ç ñ³ù³ï³ñ·ç µ³½ù³ïç ï»õ³¹ñù³ý ¨ ëáñáõãû³ùµ ë³ñù³ý³÷³ï ï»õ³¹ñù³ý ï³ýáýáí »ñïáõ áý¹é³ûýáõùý»ñ: ²å³óáõóí³í ¿, áñ áëï ³ñï³íù³ý »ñï³ñáõãû³ý µ³½ù³ïç ¨ ëáñáõãû³ùµ ë³ñù³ý³÷³ï ï»õ³¹ñù³ý ï³ýáýý»ñáí üñ»·»ç ñ³ù³ï³ñ·»ñá µ³½ù³ý¹³ùáñ»ý ñ³ù³ñå»ù »ý, ë³ï³ûý áëï ù³ûé»ñç ù³ý³ïç µ³½ù³ïç ï»õ³¹ñù³ý ï³ýáýáí üñ»·»ç ñ³ù³ï³ñ·ý áõýç óáõóã³ûçý ³ñ³·³óáõù ëáñáõãû³ùµ ë³ñù³ý³÷³ï ï»õ³¹ñù³ý üñ»·»ç ñ³ù³ï³ñ·»ñç ýï³ïù³ùµ: identification.dvi mathematical problems of computer science 33, 127{134, 2010. data e ncr yption, authentication and key p r edistr ibution schemes for wir eless sensor n etwor ks a r a a le xa n ia n , h a ko b a s la n ya n a n d a r m e n s o g h o ya n yerevan state university abstract this paper investigates some security aspects of wireless sensor networks. particularly symmetric encryption, authentication and key predistribution schemes are proposed. encryption and authentication schemes are based on a secret key which is a table of strong generators of a permutation group (as per sims algorithm). the ciphertext of a plaintext of size n is a single permutation over n = 2n + 1 elements. encryption/decryption processes are simple and easy to perform for a wireless node with limited abilities. key predistribution scheme deals with all the node and network limitations and requirements existing in wireless sensor networks and gives a secure key predistribution scheme under the assumption that nodes are not physically captured for t0 time after being deployed, where t0 is time necessary for invoking the key establishment phase of proposed scheme, which is in practice should be some seconds. refer ences [1 ] y e e w e i l a w, je r o e n d o u m e n a n d p ie t e r h a r t e l, \ s u r ve y a n d b e n c h m a r k o f b lo c k c ip h e r s fo r wir e le s s s e n s o r n e t wo r ks " , acm transactions on sensor networks tosn, vo l. 2 , n u m b e r 1 , p p . 6 5 -9 3 , 2 0 0 6 . 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[1 5 ] s . a . ca m t e p e a n d b . y e n e r , co m b in a t o r ia l d e s ig n o f ke y d is t r ib u t io n m e c h a n is m s fo r wir e le s s s e n s o r n e t wo r ks , ie e e /acm transactions on networking (ton), vo l. 1 5 , n u m b e r 2 , p p . 3 4 6 -3 5 8 , 2 0 0 7 . [1 6 ] y . x ia o , v . k r is h n a r a yi, b o s u n , x . d u , fe i h u a n d m. ga llo wa y, \ a s u r ve y o f ke y m a n a g e m e n t s c h e m e s in wir e le s s s e n s o r n e t wo r ks " , computer communications, vo l. 3 0 , n u m b e r 1 1 -1 2 , p p . 2 3 1 4 -2 3 4 1 , 2 0 0 7 . [1 7 ] h . ch a n , a . p e r r ig , a n d d . s o n g , \ k e y d is t r ib u t io n te c h n iqu e s fo r s e n s o r n e t wo r ks ', w ireless sensor networks, t. zn a t i e t a l., e d s , 2 0 0 4 . [1 8 ] a . a le xa n ya n , a lg e b r a ( gr o u p s , r in g s , fie ld s ) , y e r e va n u n ive r s it y p u b lis h e r , in a r m e n ia n , 2 0 0 6 ,. [1 9 ] m. b r o wn , d . ch e u n g , d . h a n ke r s o n , j. l . h e r n a n d e z , m. k ir ku p a n d a . me n e z e s , \ p gp in c o n s t r a in e d wir e le s s d e vic e s " , 9th use nix security symposium, 2 0 0 0 . [2 0 ] d . w . ca r m a n , p .s . k r u u s a n d b . j. ma t t ., \ co n s t r a in t s a n d a p p r o a c h e s fo r d is t r ib u t e d s e n s o r n e t wo r ks s e c u r it y" , n a i l a b , s e p t e m b e r , n u m b e r 0 0 -0 1 0 , 2 0 0 0 . [2 1 ] s e r e g e l a n g , a lg e b r a , s p r in g e r s c ie n c e +b u s in e s s me d ia , in c ., 2 0 0 2 . [2 2 ] b .l v a n d e r v a r d e n , a lg e b r a , s p r in g e r v e r la g , 1 9 6 8 . a. alexanian, h. aslanyan and a. soghoyan 1 2 9 ²ýé³ñ ë»ýëáñ³ûçý ó³ýó»ñç ïíû³éý»ñç í³íï³·ñù³ý, çýùýáõãû³ý ñ³ëï³ïù³ý ¨ µ³ý³éçý»ñç µ³ßëù³ý ëë»ù³ý»ñ ñ³ù³ñ ². ²é»ùë³ýû³ý, ð. ²ëé³ýû³ý ¨ ². êáõáû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ¹çï³ñïí³í »ý ³ýé³ñ ë»ýëáñ³ûçý ó³ýó»ñç ³å³ñáíáõãû³ýá í»ñ³µ»ñáõ áñáß ëý¹çñý»ñ: ø³ëý³íáñ³å»ë ³é³ç³ñïíáõù »ý ïíû³éý»ñç ëçù»ïñçï í³íï³·ñù³ý, çýùýáõãû³ý ñ³ëï³ïù³ý ¨ µ³ý³éçý»ñç µ³å³ýù³ý ëë»ù³ý»ñ: ì³íï³·ñù³ý ¨ çýùýáõãû³ý ñ³ëï³ïù³ý ëë»ù³ý»ñç µ³ý³éç ¿ ñ³ý¹çë³ýáõù ï»õ³¹ñáõãûáõýý»ñç ëùµç áõå»õ íýçãý»ñç µ³½ùáõãû³ý ³õûáõë³ïá (çýãå»ë êçùëç ³é·áñçãùáõù): øáõïù³ûçý ïíû³éý»ñç n ã³÷³ýç ( n µ³ûã å³ñáõý³ïáõ) µéáïá í³íï³·ñ»éáõó ëï³óíáõù ¿ n = 2 n + 1 ï³ññ»ñçó ï³½ùí³í ù»ï ï»õ³¹ñáõãûáõý: ì³íï³·ñù³ý ¨ í»ñí³ýù³ý ·áñíáõáõãûáõýý»ñá å³ñ½ »ý ¨ ï³ñáõ »ý ï³ï³ñí»é ë³ñù³ý³÷³ï ñý³ñ³íáñáõãûáõýý»ñ áõý»óáõ ³ýé³ñ ë»ýëáñ³ûçý ñ³ý·áõûóç ïáõùçó: ´³ý³éçý»ñç µ³å³ýù³ý ëë»ù³ý ñ³ßíç ¿ ³éýáõù ³ýé³ñ ë»ýëáñ³ûçý ó³ýó»ñáõù ³éï³ µáéáñ ë³ñù³ý³÷³ïáõùý»ñý áõ å³ñ³ýçý»ñá ¨ ï³éçë ¿ µ³ý³éçý»ñ µ³å³ýù³ý ³å³ñáí ëë»ù³ ³ý»éáí ùç³ûý ù»ï »ýã³¹ñáõãûáõý, ³ûý ¿‘ ñ³ý·áõûóý»ñá ï»õ³¹ñí»éáõó ñ»ïá t0 å³ù³ý³ïç áýã³óùáõù ã»ý »ýã³ñïíáõù ýç½çï³ï³ý ñ³ñó³ïù³ý: ²ûëï»õ t0 -ý ó³ýóç ñ³ý·áõûóý»ñç ùçç¨ ñ³õáñ¹³ïóù³ý µ³ý³éçý»ñç ñ³ëï³ïù³ý ñ³ù³ñ ³ýññ³å»ßï å³ù³ý³ïý ¿, áñá çñ³ï³ýáõù ï¨áõù ¿ í³ûñïû³ýý»ñ: mathematical problems of computer science 42, 81--84, 2014. 81 research and deployment of improved web server protection methods arthur s. petrosyan and gurgen s. petrosyan institute for informatics and automation problems of nas ra e-mail: arthur@sci.am, gurgen@sci.am abstract since the world wide web service remains the most widely used internet service, the protection of web servers becomes more and more important, especially in view of vulnerability, being found from day to day. this article describes the research work done in academic scientific research computer network of armenia (asnet-am) managed by the institute for informatics and automation problems (iiap) of the national academy of sciences of the republic of armenia (nas ra), targeted to the research and deployment of the improved methods for web server protection. special attention was given to obtaining the best solution for web server protection methods in apache/php-based shared web hosting environment. keywords: www, web hosting environment, web server, apache, php. 1. introduction in a shared web hosting environment, different hosted websites share the whole server but each client has its own set of resources. a number of websites share a single server. it is an economic solution for those websites which do not have high traffic and high storage requirements. in such shared hosting environment everything that interacts with a server is a threat to any of the hosted websites. the following analogy helps understanding the issues of shared hosting environment. a shared server is like an apartment building, where all the resources like water supply, power supply, parking lot, etc. are shared with other people in the apartment building. in case of water or power supply failure, every apartment faces the impact. similarly in shared web hosting, the web server software (apache http server project [1]), requires a control over the files to be served to the client which immediately poses a security concern. if the domains have an ability to run scripts or if the domains have an access to the shell, then in shared hosting environment, one client can modify the files of another client. though in a multiuser operating system like linux read/write/execute, privileges can be provided to different user groups (user/group/other) yet through a simple php script, files outside the own home directory can be accessed. this is research and deployment of improved web server protection methods82 because in default web server configuration all hosted websites are served from the same username/uid and, thus, have the same privileges at the operating system level. even when using pre-packaged software solutions, you need to allow the hosting server to have read, write and execute access to your files and, thus, expose vulnerability to other clients. moreover, though the php functions like exec(), shell_exec() provide flexibility to the developers, yet they pose adverse security problems. most of the websites require some image uploads from the web and if the client on shared hosting does not have a server permission then these uploads will not move to the destination directory. the common solution is to give all the users 777 (read/write/execute) access to the destination directory. this is a common solution but what it has provided is an easy way to hack the files of other users sharing the same server. 2. restriction on php level an important php feature, which can be helpful in shared hosting environment, is open_basedir. it can be used to limit the files that can be accessed by php to the specified directory-tree, including the file itself. when some php script tries to access the filesystem, for example using include, or fopen(), the location of the file is checked. when the file is outside the directory-tree, specified within the ‘open_basedir’, php will refuse to access it. the typical ‘open_basedir’ setting should be: open_basedir = /tmp/:/var/www/ this will protect the other system directories from accessing by any php script. but the problem is that the ‘open_basedir’, which is generally set in the php global configuration file ‘php.ini’, can be overridden in the local .htaccess file at the directory level, in case the apache web server ‘allowoverride’ setting is set to ‘all’ or at least ‘options’, and even within the php script (using the function ‘ini_set()’). thus, the use of ‘open_basedir’ in this configuration does not produce the desired effect. also setting ‘open_basedir’ in the ‘php.ini’ file globally for the entire web server, will only help to protect external system directories. but the neighboring websites using the same shared web server will still be able to access each other’s directories at php script level. it means that in case the website is broken, other website data will be potentially available to the intruder too. the solution to the above issue is to define ‘open_basedir’ as so called php_admin_value. php documentation shows the settings defined as ‘php_admin_value’ can’t be overridden in ‘.htaccess’or ‘ini_set()’ [2]. even more, ‘open_basedir’ can be defined as ‘php_admin_value’ for each virtual host separately, so the issue mentioned above will be solved and we will get a separate ‘open_basedir’ setting, determined for each website (virtual hosting), which can’t be changed from ‘.htaccess’ or ‘ini_set()’. in this way it is possible to determine a separate php scripts execution area for each hosted website, so that it does not have access to the whole system, and even to files of neighboring websites using the same shared web server. although php developer team is mentioning that the use of ‘open_basedir’ feature is “a convenience to system administrators and should in no way be thought of as a complete security framework” [3], using the ‘open_basedir’ helps to limit the potential threat area. a. petrosyan and g. petrosyan 83 3. filesystem permissions level protection php level restrictions described above are good enough in case of proper configuration. but to obtain the best solution for web server protection in shared hosting environment it would be important to have additional protection on the filesystem permissions level. this task can be implemented in different ways and as a result of research work done in asnet-am to deploy the improved methods for web server protection the package apache 2 itk mpm was chosen to be most effective [4]. apache 2 itk mpm (just mpm-itk for short) is a multi-processing module (mpm) for the apache web server. mpm-itk allows to run each of the virtual websites under a separate userid (uid and groupid (gid). this means, that a desired additional protection on the filesystem permissions level can be obtained. the typical ‘mpm_itk_module’ setting for each virtualhost in apache configuration should include: assignuserid user1 group1 this will force apache wer server to fork a process with 'user1' and 'group1' for serving requests to this website. as a result the neighboring websites using the same shared web server will not be able to access either each other’s directories, or the system directories, based on the filesystem permissions. this solution can be counted much more effective, since it provides protection on a system level and, thus, not only php, but any scripts and configuration files for one virtual website no longer is accessible for all the others. it means that in case any website is broken, only this website data will be available to the intruder. an important note on mpm-itk usage is that it is based on prefork method and not threads, (i.e., extra fork is done per request). on one hand, it means that mpm-itk can support running a non-thread-aware code (like many php extensions) without problems. on the other hand, mpmitk performance is lower when compared with threads. but our decision was to have less performance benefits and gain a more powerful method of protection on the filesystem permissions level. 4. conclusion deployment of the improved methods for web server protection includes implementation of multilevel security means. thus, the best solution for web server protection in apache/phpbased shared web hosting environment can be described as a combination of properly configured open_basedir php level restrictions and additional protection on the filesystem permissions level with modules like mpm-itk. references [1] apache http server project, [online]. available: http://httpd.apache.org [2] description of core php.ini directives, [online]. available: http://php.net/manual/en/ini.core.php research and deployment of improved web server protection methods84 [3] a note on security in php, [online]. available: http://php.net/security-note.php [4] apache 2 itk mpm, [online]. available: http://mpm-itk.sesse.net/ submitted 29.07.2014, accepted 27.11.2014. վեբ սերվերների պաշտպանության բարելավված մեթոդների հետազոտություն և մշակում ա. պետրոսյան և գ. պետրոսյան ամփոփում քանի որ world wide web (համաշխարհային սարդոստայնի) ծառայությունը առանցքային դեր ունի ներկայիս ցանցային տեխնոլոգիաների ոլորտում, վեբ սերվերների պաշտպանությունը կենսական նշանակություն ունի: հաշվի առնելով վերջին տարիներին լայն տարածում ստացած վեբ սերվերներին ուղղված հարձակումները, շատ կարևոր են դառնում world wide web ծառայության պաշտպանության արդյունավետ մեթոդների մշակումն ու ներդրումը: հոդվածում նկարագրված են asnet-am հայաստանի ակադեմիական գիտահետազոտական կոմպյուտերային ցանցում կատարված ուսումնասիրության արդյունքները, որոնց նպատակն է մշակել վեբ սերվերների պաշտպանության արդյունավետ մեթոդներ: исследование и реализация улучшенных методов защиты веб серверов а. петросян и г. петросян аннотация услуга world wide web (всемирной паутины) является ключевой в современном мире сетевых коммуникаций, поэтому защита веб серверов имеет жизненно важное значение. в связи с различными формами атак на веб сайты, внедрение эффективных методов защиты веб серверов становится очень важным. в статье описаны исследования, направленные на определение эффективных способов защиты от атак на веб сайты. представлены результаты научно-исследовательской работы, проделанной в академической научно-исследовательской компьютерной сети армении (asnet-am), направленные на применение улучшенных методов защиты веб серверов. особое внимание уделено определению наилучшего решения для защиты веб серверов в среде виртуального разделяемого хостинга, на основе программных пакетов apache/php. mathematical problems of computer science 40, 39---43, 2013. 39 a parallel algorithm for geoprocessing of vegetation indices albert g. saribekyan institute for informatics and automation problems of nas ra e-mail: albert_saribekyan@ipia.sci.am abstract geographic information systems (gis) play a vital role in environment-related issues, from which well-known is the calculation of vegetation indices (vi) using the satellite images. in this paper a parallel algorithm for geoprocessing of vi’s is introduced with appropriate benchmarkings, which were performed using highperformance computing (hpc) resources. keywords: gis, grass, vegetation indices, chunk, region, ndvi, gari, gvi. 1. introduction earth remote sensing plays an important role in our reality which is immediately associated with satellite images. satellite images contain large amount of useful information. the process of extracting this information and understanding it may require huge computing power and processing time. distributed computing can reduce the processing time by providing more computational power. geographic resources analysis support system (grass) gis [1] is an open source software/tool, which has been used to process satellite images. inside grass, different modules have been developed for processing satellite images. there are some operations and modules that are widely used in the grass gis, for instance, the calculation of vi [2] is a very important and frequently used operation, so it was used as a test example for this study. in order to make the calculation of vi more convenient with grass-gis a special module i.vi [3] has been developed to calculate 13 different vi from raw spatial data. grass gis module i.vi works with raster images (rows x columns). different band raster [4] images are required for calculation of different indices:  ndvi (normalized difference vi), rvi (ratio vi) red and nir (near infrared)  arvi (atmospherically resistant vi) red, nir and blue  gvi (green vi) red, nir infrared, blue, green, band5 and band7 a parallel algorithm for geoprocessing of vegetation indices40  gari (green atmospherically resistant vi) red, nir, blue, green  rvi (ratio vi) red and nir  ipvi (infrared percentage vi) red and nir  dvi (difference vi) red and nir  pvi (perpendicular vi) band5 and band7  wdvi (weighted difference vi) red and nir  savi (soil adjusted vi) red and nir  msavi (modified soil adjusted vi ) red and nir  msavi2 (second modified soil adjusted vi ) red and nir  gemi (global environmental monitoring vi ) red and nir grass functions are used to extract row wise data from the specific band images and store them in buffers, and then each column value is extracted sequentially from the buffers and sent them for generating the specific vi values. thus, after completing the vi from row buffers, the row wise vi values are put back into an output image and this process is applying recursively for each row. 2. parallelization algorithm a parallelization algorithm that splits the geographic region of the target location into specified number of chunks has been implemented after series of experiments using several parallelization approaches. the algorithm allows to calculate each chunk separately in a parallel way and then it combines all the parts together when it finishes the calculations. the parallelization approach consists of three main stages:  splitting  calculation  merging it is known that in grass gis, a region refers to a geographic area with some defined boundaries, based on a specific map coordinate system (wgs 84[5]) and map projection. one of the features of grass gis is to perform calculations only on that part of raster data which is defined by the current region. so by changing the coordinates of the current region it is possible to change the part on the raster image on which the calculation will take place. this feature is used to perform the first stage of parallelization – splitting. a shell script has been developed, which in the first step reads the northern and southern boundaries of the image and after dividing the area between those boundaries into specified number of pieces, defines new northern and southern boundaries for each so called chunk, these new defined boundaries are then saved into the named regions. the region's boundaries are given as the northernmost, southernmost, easternmost, and westernmost points that define its extent (cell edges). the north and south boundaries are commonly called northings, while the east and west boundaries are called eastings. each mapset has a current geographic region, which defines the geographic area for raster analyses. raster data, in case of necessity is resembled, in order to meet the cell resolutions of the current geographic region setting. each grass gis location has a fixed geographic region, called the default geographic region that defines the extent of the database. the current region can easily be reset to the default region. each mapset may contain any number of pre-defined and named geographic regions, called saved regions. any of these pre-defined geographic regions is possible to select by name in order to define it as a current geographic region. in the next stage, the script runs a specified number of parallel processes; each of this process sets current region of the default mapset permanent to one of the saved regions and does the a.saribekyan 41 calculation on its part of image. after the parallel calculation is completed, there comes a time for merging and exporting the processed chunks of data. in the third stage of parallelization approach the merging process is being started and begins to combine the split regions on which the calculation has already been done. after that the complete image will be obtained. this operation has been done using gdal_merge.py [6] utility of geospatial data abstraction library (gdal) [7]. the main important aspect is that all input images have to be in the same coordinate system. the gdal_merge.py operates outside grass so the chunks of the image need to be exported and merged outside grass. so, when all the data is processed all the chunks saved as a raster pictures exported from grass and they will be combined with gdal_merge.py utility (see fig. 1). fig. 1. the structure of parallelization algorithm 3. benchmarkings in order to find out a general parallelization approach for operations with different complexity, a set of benchmarkings have been carried out. as a target data for the experiments the landsat thematic mapper (tm) image is used [8], which consists of 7 bands and 30 m (98 ft) spatial resolution for bands used in vi calculations (bands 1-5, 7). every band consists of 7130 rows and 7844 columns, so the total number of cells is 55927720. the experiments have been carried out using grass gis 7.0.svn and some computational resources (1 node with the following parameters 8 cores, intel e5420, 8 gb ram) of the armenian national grid infrastructure [9]. as there are at least two ways to split the image into chunks (horizontal and vertical), the both ways have been implemented to find out the optimal option. since horizontal splitting of chunks is faster (speedup of vertical efficiency is 26.5% instead of horizontal – 35.2%) than those with vertical one, then for the future experiments the horizontal splitting is used. from technical point of view, the calculation of ndvi index is the simplest one among 13 vis of i.vi. it uses only 2 bands, redchan and nirchan (3-rd and 4-th bands of landsat tm/etm+ images). the algorithm of calculation consists of 3 operations: ndvi = (nirchan redchan) / (nirchan + redchan). in case of gari, it uses 4 bands, redchan, nirchan, bluechan and greenchan (3rd, 4-th, 1-st and 2-nd bands of landsat tm/etm+ images). the algorithm of calculation of gari is more complex: gari = nirchan (greenchan (bluechan redchan)) / (nirchan(greenchan(bluechan redchan))). the gvi is the last studied and the most studied complex vi. it uses 6 bands of spatial image bluechan, greenchan, redchan, nirchan, chan5chan and chan7chan. the calculation algorithm is also more complex than the previous examples: gvi = (-0.2848 * bluechan 0.2435 * a parallel algorithm for geoprocessing of vegetation indices42 greenchan 0.5436 * redchan + 0.7243 * nirchan + 0.0840 * chan5chan-0.1800 * chan7chan). fig. 2 presents the parallel calculation speedup of each of these indices. fig. 2. parallelization of vi calculations 4. conclusion the series of experiments show that the suggested parallel algorithm significantly decreases the calculation time using the standards vi modules of grass gis. the i.vi module of grass gis is used for the experiments mainly concentrated on 3 from 13 vis, which differ from each other from technical point of view. the results show that the eficiency of the parallelization depends on the complexity of the algorithm of the calculation of indices. it is planned to continue the experiments with other grass gis modules, too. acknowledgements the author expresses his gratitude to the head of hpc laboratory dr. hrachya astsatryan for supervising the work, as well as the same laboratory engineers andranik hayrapetyan and wahi narsisian for very valuable discussions. references [1] m. neteler and h. mitasova, open source gis: a grass gis approach, second edition, kluwer academic publishers, 2003. [2] c. ünsalan and k. l. boyer, “linearized vegetation indices, multispectral satellite image understanding”, advances in computer vision and pattern recognition, pp 19-39, 2011. [3] i.vi calculates different types of vegetation indices, [online]. available: a.saribekyan 43 http://grass.osgeo.org/grass70/manuals/i.vi.html [4] raster bands, [online]. available: http://edndoc.esri.com/arcsde/9.2/concepts/rasters/entities/rasterbands.htm [5] defense mapping agency, "geodesy for the layman", chapter 8. [6] description of gdal_merge.py utility. [online]. available: http://www.gdal.org/gdal_merge.html [7] g. brent hall, michael g. leahy, open source approaches in spatial data handling, springer berlin heidelberg, 2008. [8] national aeronautics and space administration, "landsat 7 science data users handbook", [online]. available: http://landsathandbook.gsfc.nasa.gov/pdfs/landsat7_handbook.pdf [9] h. astsatryan, v. sahakyan and yu. shoukourian, “brief introduction of armenian national grid initiative”, book of abstracts of 4th international conference "distributed computing and grid-technologies in science and education" (grid’2010), pp. 30-31, june 28 july 3, dubna, russia, 2010. submitted 19.08.2013, accepted 08.10.2013. զուգահեռալգորիթմբուսականությանցուցիչների աշխարհագրականհաշվարկներիհամար ա.սարիբեկյան ամփոփում երկրատեղեկատվական համակարգերը մեծ նշանակություն ունեն շրջակա միջավայրի հետազոտման խնդիրներում, որոնցից առավել հայտնի է բուսականության ցուցիչների ստացումը տիեզերական պատկերների միջոցով: տվյալ աշխատանքում ներկայացված է բուսականության ցուցիչների հաշվարկման համար զուգահեռ ալգորիթմ` համապատասխան վերլուծություններով, որոնք իրականացվել են բարձր արտադրողականությանհաշվողականհամակարգերիվրա: параллельный алгоритм для геообработки вегетационных индексов а. сарибекян аннотация геоинформационные системы имеют большое значение в исследованиях задач окружающей среды, из которых больше известно получение вегетационных индексов от спутниковых изображений. в данной статье представлен параллельный алгоритм для вычисления вегетационных индексов с соответствующими анализами, которые были выполнены на высокопроизводительных вычислительных ресурсах. 02_darbinyan_18_05_2017.dvi mathematical problems of computer science 47, 15{29, 2017. on cyclability of digr aphs with a m anoussakis-type condition s a m ve l k h . d a r b in ya n institute for informatics and automation problems of nas ra e-mail: samdarbin@ipia.sci.am abstract let d be a digraph of order n ¸ 4 and y be a non-empty subset of vertices of d. let for any pair u, v of distinct vertices of y the digraph d contain a path from u to v and a path from v to u. suppose d satis¯es the following conditions for every triple x; y; z 2 y such that x and y are nonadjacent: if there is no arc from x to z, then d(x) + d(y) + d+(x) + d¡(z) ¸ 3n ¡ 2. if there is no arc from z to x, then d(x) + d(y) + d+(z) + d¡(x) ¸ 3n ¡ 2. we prove that there is a directed cycle in d which contains all the vertices of y , except possibly one. this result is best possible in some situations and gives an answer to a question of li, flandrin and shu (discrete mathematics, 307 (2007) 1291-1297). keywords: digraphs, cycles, hamiltonian cycles, cyclability. 1 . in t r o d u c t io n fo r c o n ve n ie n c e o f t h e r e a d e r , t e r m in o lo g y a n d n o t a t io n s will b e g ive n in d e t a ils in s e c t io n 2 . a s e t s o f ve r t ic e s in a d ir e c t e d g r a p h d ( a n u n d ir e c t e d g r a p h g) is s a id t o b e c yc la b le in d ( in g ) if d ( if g) c o n t a in s a d ir e c t e d c yc le ( u n d ir e c t e d c yc le ) t h r o u g h a ll t h e ve r t ic e s o f s. th e r e a r e m a n y we ll-kn o wn c o n d it io n s wh ic h g u a r a n t e e t h e c yc la b ilit y o f a s e t o f ve r t ic e s in a n u n d ir e c t e d g r a p h . mo s t o f t h e m c a n b e s e e n a s r e s t r ic t io n s o f h a m ilt o n ia n c o n d it io n s t o t h e c o n s id e r e d s e t o f ve r t ic e s ( s e e [1 , 2 , 3 , 4 ]) . l e t 's p r o vid e s o m e e xa m p le s b e lo w: t heor em a: ( r . s h i [3 ]) . l et g be a 2-connected undirected graph of order n. if s is a subset of the vertices of g and d( x ) ¸ n= 2 for all vertices x 2 s, then s is cyclable in g. t heor em b : ( r . s h i [3 ]) . l et g be a 2-connected undirected graph of order n. if s is a subset of the vertices of g and d ( x) + d( y ) ¸ n for any two nonadjacent vertices x 2 s and y 2 s, then s is cyclable in g. n o t ic e t h a t th e o r e m s a a n d b g e n e r a liz e t h e c la s s ic a l t h e o r e m s o n h a m ilt o n ic it y o f d ir a c a n d or e , r e s p e c t ive ly. in vie w o f t h e n e xt t h e o r e m s we n e e d t h e fo llo win g d e ¯ n it io n s . l e t d b e a d ig r a p h o f o r d e r n ¸ 3 a n d s b e a n o n -e m p t y s u b s e t o f ve r t ic e s o f d. fo llo win g [5 ], we s a y t h a t a d ig r a p h d is s-s t r o n g ly c o n n e c t e d if fo r a n y p a ir x; y o f d is t in c t ve r t ic e s o f s t h e d ig r a p h d c o n t a in s a p a t h fr o m x t o y a n d a p a t h fr o m y t o x. a me yn ie l s e t m is a s u b s e t o f ve r t ic e s o f a d ig r a p h d s u c h t h a t d ( x) + d ( y ) ¸ 2 n ¡ 1 fo r e ve r y p a ir o f d is t in c t ve r t ic e s x, y in m wh ic h a r e n o n a d ja c e n t in d. 1 5 1 6 on cyclability of digraphs with a manoussakis-type condition fo r g e n e r a l d ir e c t e d g r a p h s ( d ig r a p h s ) t h e r e a r e n o t a s m a n y c o n d it io n s in lit e r a t u r e a s fo r u n d ir e c t e d g r a p h s t h a t g u a r a n t e e t h e e xis t e n c e o f a d ir e c t e d c yc le wit h t h e g ive n p r o p e r t ie s ( in p a r t ic u la r , s u ± c ie n t c o n d it io n s fo r t h e e xis t e n c e o f a h a m ilt o n ia n c yc le s in d ig r a p h s ) . th e m o r e g e n e r a l a n d c la s s ic a l o n e s a r e t h e fo llo win g t h e o r e m o f m. me yn ie l: t heor em c: ( m. me yn ie l [6 ]) . l et d be a strongly connected digraph of order n ¸ 2 . if the vertex set of d is a m eyniel set, then d is hamiltonian. n o t ic e t h a t me yn ie l's t h e o r e m is a g e n e r a liz a t io n o f we ll-kn o wn c la s s ic a l t h e o r e m s o f gh o u ila -h o u r i [7 ] a n d w o o d a ll [8 ]. a b e a u t ifu l s h o r t p r o o f o f me yn ie l's t h e o r e m c a n b e fo u n d in [9 ] ( s e e a ls o [1 0 ], p p . 3 9 9 -4 0 0 ) . in [1 1 ], t h e fo llo win g wa s p r o ve d : t heor em d: ( s . d a r b in ya n [1 1 ]) . l et d be a strongly connected digraph of order n ¸ 3 and y be a subset of vertices of d. if jy j = n ¡ 1 and y is a m eyniel set, then d is hamiltonian or contains a cycle of length n ¡ 1 . fr o m th e o r e m d we o b t a in t h e fo llo win g c o r o lla r ie s . cor ollar y 1: l et d be a strongly connected digraph of order n ¸ 3 . if d has n ¡ 1 vertices of degree at least n, then d is hamiltonian or contains a cycle of length n ¡ 1 . cor ollar y 2: l et d be a strongly connected digraph of order n ¸ 3 and y be a subset of vertices of d. if jy j = n ¡ 1 and y is a m eyniel set, then d has a cycle that contains all the vertices of y . a s u ± c ie n t c o n d it io n fo r c yc la b ilit y in d ig r a p h s wit h t h e c o n d it io n o f me yn ie l's t h e o r e m wa s g ive n b y k . a . b e r m a n a n d x . l iu [1 2 ]. th e y im p r o ve d th e o r e m f b y p r o vin g t h e fo llo win g g e n e r a liz a t io n o f t h e we ll-kn o wn t h e o r e m o f me yn ie l. t heor em e : ( k . b e r m a n a n d x . l iu [1 2 ]) . l et d be a strongly connected digraph of order n. then every m eyniel set m of d lies in a directed cycle. l a t e r h . l i, e . fla n d r in a n d j. s h u [5 ] p r o ve d t h e fo llo win g g e n e r a liz a t io n o f th e o r e m e . t heor em f: ( h . l i, e . fla n d r in a n d j. s h u [5 ]) . l et d be a digraph of order n and m be a m eyniel set in d. if d is m -strongly connected, then d contains a cycle through all the vertices of m . l e t d b e a d ig r a p h o f o r d e r n. w e s a y t h a t a n o n -e m p t y s u b s e t y o f t h e ve r t ic e s o f d s a t is ¯ e s c o n d it io n a0 if fo r e ve r y t r ip le o f t h e ve r t ic e s x; y; z in y s u c h t h a t x a n d y a r e n o n a d ja c e n t : if t h e r e is n o a r c fr o m x t o z, t h e n d ( x) + d( y ) + d+ ( x ) + d¡ ( z ) ¸ 3 n ¡ 2 . if t h e r e is n o a r c fr o m z t o x, t h e n d( x ) + d ( y ) + d¡ ( x ) + d+ ( z ) ¸ 3 n ¡ 2 . y . ma n o u s s a kis [1 3 ] p r o ve d a s u ± c ie n t c o n d it io n fo r h a m ilt o n ic it y o f d ig r a p h s t h a t in vo lve s t r ip le s r a t h e r t h a n p a ir s o f ve r t ic e s . t heor em g: ( y . ma n o u s s a kis [1 3 ]) . l et d be a strongly connected digraph of order n ¸ 4 . if v ( d ) satis¯es condition a0, then d is hamiltonian. h . l i, fla n d r in a n d s h u [5 ] p u t a qu e s t io n t o kn o w if t h is t h e o r e m o f ma n o u s s a kis ( o r t h e s u ± c ie n t c o n d it io n s o f h a m ilt o n ic it y o f d ig r a p h s o f b a n g -je n s e n , gu t in a n d l i [1 4 ] o r o f b a n g -je n s e n , gu o a n d y e o [1 5 ]) h a s a c yc la b le ve r s io n . in t h is p a p e r we p r o ve t h e fo llo win g t h e o r e m wh ic h g ive s s o m e a n s we r s t o t h e a b o ve qu e s t io n wh e n a s u b s e t y 6= ; o f t h e ve r t ic e s o f a d ig r a p h d s a t is ¯ e s c o n d it io n a0 a n d t h e d ig r a p h d is y -s t r o n g ly c o n n e c t e d . t heor em 1: l et d be a digraph of order n ¸ 4 and let y be a non-empty subset of the vertices of d. suppose that d is y -strongly connected and the subset y satis¯es condition a0. then d contains a cycle through all the vertices of y maybe except one. remar k 1: the following example shows that there is a digraph d which contains a s. darbinyan 1 7 nonempty subset y of v ( d ) such that d is y -strongly connected and the subset y satis¯es condition a0 but d has no cycle that contains all the vertices of y . to s e e t h is , le t g a n d h b e t wo a r b it r a r y d is jo in t d ig r a p h s wit h jv ( g ) j = m ¸ 2 a n d jv ( h ) j = n ¡ m ¸ 4 . l e t y 2 v ( h ) a n d x; z 2 v ( g ) , x 6= z. a s s u m e t h a t d( y; h ) = 2 ( n ¡ m ¡ 1 ) , g c o n t a in s a h a m ilt o n ia n c yc le , d+ ( x; g) = m ¡ 1 a n d d( z; g ) = 2 ( m ¡ 1 ) . fr o m g a n d h we fo r m a n e w d ig r a p h d wit h v ( d ) = v ( g ) [ v ( h ) a s fo llo ws : a d d a ll t h e p o s s ib le a r c s ux; xu, wh e r e u 2 v ( h ) n fyg, a n d t h e a r c yx. a n e a s y c o m p u t a t io n s h o ws t h a t d( y ) + d( z ) + d¡ ( y ) + d+ ( x) = 4 n ¡ m ¡ 6 ¸ 3 n ¡ 2 ; s in c e m · n ¡ 4 . th u s , we h a ve t h a t t h e s e t y = fx; y; zg s a t is ¯ e s c o n d it io n a0. d is y -s t r o n g ly c o n n e c t e d a n d h a s n o c yc le t h a t c o n t a in s a ll t h e ve r t ic e s o f y . ou r p r o o fs a r e b a s e d o n t h e a r g u m e n t s o f [5 , 1 3 ]. 2 . te r m in o lo g y a n d n o t a t io n in t h is p a p e r we c o n s id e r ¯ n it e d ig r a p h s wit h o u t lo o p s a n d m u lt ip le a r c s . te r m in o lo g y a n d n o t a t io n s n o t d e s c r ib e d b e lo w fo llo w [1 6 ]. th e ve r t e x s e t a n d t h e a r c s e t o f a d ig r a p h d a r e d e n o t e d b y v ( d ) a n d a ( d ) , r e s p e c t ive ly. th e o r d e r o f d is t h e n u m b e r o f it s ve r t ic e s . fo r a n y x; y 2 v ( d ) , we a ls o wr it e x ! y if xy 2 a ( d ) . if xy 2 a ( d ) , t h e n we s a y t h a t x d o m in a t e s y o r y is a n o u t -n e ig h b o u r o f x a n d x is a n in -n e ig h b o u r o f y. if x ! y a n d y ! z we wr it e x ! y ! z. two d is t in c t ve r t ic e s x a n d y a r e a d ja c e n t if xy 2 a( d ) o r yx 2 a( d ) ( o r b o t h ) . if t h e r e is n o a r c fr o m x t o y we s h a ll u s e t h e n o t a t io n xy =2 a ( d ) . w e le t n + ( x ) , n¡ ( x ) d e n o t e t h e s e t o f o u t -n e ig h b o u r s , r e s p e c t ive ly t h e s e t o f in n e ig h b o u r s o f a ve r t e x x in a d ig r a p h d. if a µ v ( d ) , t h e n n + ( x; a) = a \ n + ( x ) a n d n¡ ( x; a ) = a \ n¡ ( x) . th e o u t -d e g r e e o f x is d+ ( x ) = jn + ( x) j a n d d¡ ( x) = jn¡ ( x) j is t h e in -d e g r e e o f x. s im ila r ly, d+ ( x; a ) = jn + ( x; a) j a n d d¡ ( x; a ) = jn ¡ ( x; a) j. if x 2 v ( d ) a n d a = fxg we s o m e t im e s wr it e x in s t e a d o f fxg. th e d e g r e e o f t h e ve r t e x x in d is d e ¯ n e d a s d( x ) = d+ ( x) + d¡ ( x ) ( s im ila r ly, d( x; a ) = d+ ( x; a) + d¡ ( x; a ) ) . th e s u b d ig r a p h o f d in d u c e d b y a s u b s e t a o f v ( d ) is d e n o t e d b y dhai o r hai fo r b r e vit y. th e p a t h ( r e s p e c t ive ly, t h e c yc le ) c o n s is t in g o f t h e d is t in c t ve r t ic e s x1; x2; : : : ; xm ( m ¸ 2 ) a n d t h e a r c s xixi+1, i 2 [1 ; m ¡ 1 ] ( r e s p e c t ive ly, xixi+1, i 2 [1 ; m ¡ 1 ], a n d xmx1 ) , is d e n o t e d b y x1x2 ¢ ¢ ¢ xm ( r e s p e c t ive ly, x1x2 ¢ ¢ ¢ xmx1 ) . th e le n g t h o f a c yc le o r p a t h is t h e n u m b e r o f it s a r c s . w e s a y t h a t x1x2 ¢ ¢ ¢ xm is a p a t h fr o m x1 t o xm o r is a n ( x1; xm ) -p a t h . a n ( x; y ) -p a t h p is a n ( x; y ) -p a t h if x 2 x, y 2 y a n d v ( p ) \ ( x [ y ) = fx; yg, wh e r e x a n d y a r e s o m e s u b s e t s o f t h e ve r t ic e s o f a d ig r a p h d. give n a ve r t e x x o f a d ir e c t e d p a t h p o r a d ir e c t e d c yc le c, we u s e t h e n o t a t io n s x+ a n d x¡ fo r t h e s u c c e s s o r a n d t h e p r e d e c e s s o r o f x ( o n p o r o n c ) a c c o r d in g t o t h e o r ie n t a t io n a n d in c a s e o f a m b ig u it y, we p r e c is e p o r c a s a s u b s c r ip t ( t h a t is x+p ...) . a c yc le ( r e s p e c t ive ly, a p a t h ) t h a t c o n t a in s a ll t h e ve r t ic e s o f d is a h a m ilt o n ia n c yc le ( r e s p e c t ive ly, is a h a m ilt o n ia n p a t h ) . a d ig r a p h is h a m ilt o n ia n if it c o n t a in s a h a m ilt o n ia n c yc le . fo r a c yc le c := x1x2 ¢ ¢ ¢ xkx1 o f le n g t h k, t h e s u b s c r ip t s c o n s id e r e d m o d u lo k, i.e ., xi = xs fo r e ve r y s a n d i s u c h t h a t i ´ s ( m o d k ) . if p is a p a t h c o n t a in in g a s u b p a t h fr o m x t o y we le t p [x; y] d e n o t e t h a t s u b p a t h . s im ila r ly, if c is a c yc le c o n t a in in g ve r t ic e s x a n d y, c[x; y] d e n o t e s t h e s u b p a t h o f c fr o m x t o y. if c is a c yc le a n d p is a p a t h in a d ig r a p h d, o ft e n we will wr it e c in s t e a d o f v ( c ) a n d p in s t e a d o f v ( p ) . a d ig r a p h d is s t r o n g ly 1 8 on cyclability of digraphs with a manoussakis-type condition c o n n e c t e d ( o r , ju s t , s t r o n g ) if t h e r e e xis t s a p a t h fr o m x t o y a n d a p a t h fr o m y t o x fo r e ve r y p a ir o f d is t in c t ve r t ic e s x; y. l e t c b e a n o n -h a m ilt o n ia n c yc le in a d ig r a p h d. fo r t h e c yc le c, a c-b yp a s s is a p a t h o f le n g t h a t le a s t t wo wit h b o t h e n d -ve r t ic e s o n c a n d n o o t h e r ve r t ic e s o n c. if ( x; y ) -p a t h p is a c-b yp a s s wit h v ( p ) \ v ( c ) = fx; yg, t h e n we c a ll t h e le n g t h o f t h e p a t h c[x; y] t h e g a p o f p wit h r e s p e c t t o c. if we c o n s id e r a s u b s e t o f ve r t ic e s s µ v ( d ) , we d e n o t e t h e ve r t ic e s o f s b y s-ve r t ic e s a n d t h e n u m b e r o f s-ve r t ic e s in a c yc le is c a lle d it s s-le n g t h . th e s u b d ig r a p h o f a d ig r a p h d in d u c e d b y a s u b s e t a o f v ( d ) is d e n o t e d b y dhai, o r hai fo r b r e vit y. th e c o n ve r s e d ig r a p h o f a d ig r a p h d is t h e d ig r a p h o b t a in e d fr o m d b y r e ve r s in g a ll a r c s o f d. fo r in t e g e r s a a n d b, a · b, le t [a; b] d e n o t e t h e s e t o f a ll in t e g e r s wh ic h a r e n o t le s s t h a n a a n d a r e n o t g r e a t e r t h a n b. 3 . p r e lim in a r ie s w e n o w c o lle c t t h e t o o ls wh ic h we n e e d in p r o o f o f o u r t h e o r e m . in t h e fo llo win g , we o ft e n u s e t h e fo llo win g d e ¯ n it io n : de¯nition 1: l et p = x1x2 : : : xm (m ¸ 2 ) be a path in a digraph d and q = y1y2 : : : yk be a path in hv ( d ) n v ( p ) i (possibly, k = 1 ). assume that there is an i 2 [1 ; m ¡ 1 ] such that xiy1 and ykxi+1 2 a( d ) . in this case d contains the path x1x2 : : : xiy1y2 : : : ykxi+1 : : : xm and we say that q can be inserted into p . th e fo llo win g l e m m a s 1 a n d 2 a r e s lig h t ly m o d i¯ e d ve r s io n s o f le m m a b y h äa g g kvis t a n d th o m a s [1 7 ] a n d o f le m m a b y b o n d y a n d th o m a s s e n [9 ], r e s p e c t ive ly ( t h e ir p r o o fs a r e n o t t o o d i± c u lt ) . th e y will b e u s e d e xt e n s ive ly in t h e p r o o f o f o u r r e s u lt . lemma 1: l et ck := x1x2 : : : xkx1, k ¸ 2 , be a non-hamiltonian cycle in a digraph d. m oreover, assume that there exists a path q := y1y2 : : : yr, r ¸ 1 , in hv ( d ) n v ( ck ) i. if d¡ ( y1; ck ) + d + ( yr; ck ) ¸ k + 1 , then for all m 2 [r + 1 ; k + r] the digraph d contains a cycle cm of length m with vertex set v ( cm ) µ v ( ck ) [ v ( q) . lemma 2: l et p := x1x2 : : : xk, k ¸ 2 , be a non-hamiltonian path in a digraph d. m oreover, assume that there exists a path q := y1y2 : : : yr, r ¸ 1 , in hv ( d ) n v ( p ) i. if d¡ ( y1; p ) + d + ( yr; p ) ¸ k + d¡ ( y1; fxkg) + d+ ( yr; fx1g ) ; then there is an i 2 [1 ; k ¡ 1 ] such that xiy1 and yrxi+1 2 a ( d ) , i.e., d contains a path from x1 to xk with vertex set v ( p ) [ v ( q ) , i.e., q can be inserted into p . th e fo llo win g le m m a fr o m [5 ] is a s lig h t ly m o d ī e d ve r s io n o f mu lt i-in s e r t io n l e m m a d u e t o b a n g -je n s e n , gu t in a n d h . l i ( s e e [1 6 ], l e m m a 5 .6 .2 0 ) . lemma 3: ( h . l i, e . fla n d r in . j. s h u [5 ]) . l et d be a digraph and let p be an ( a; b ) -path in d. l et q be a path in hv ( d ) n v ( p ) i and s be a subset of v ( q ) . if every vertex of s can be inserted into p , then there exists an ( a; b ) -path r such that v ( p ) [s µ v ( r) µ v ( p ) [v ( q ) . th e fo llo win g le m m a a ls o wa s p r o ve d in [5 ]. lemma 4: ( h . l i, e . fla n d r in . j. s h u [5 ]) . l et d be a digraph of order n and s ½ v ( d ) , s 6= ; be a m eyniel set. assume that d is s-strongly connected and c is a cycle in d of maximum s-length. if s is an s-vertex of v ( d ) n v ( c ) , then d contains a c-bypass through s. b y t h e in s p e c t io n o f t h e p r o o f in [1 2 ] o n e c a n s t a t e l e m m a 4 in t h e fo llo win g fo r m ( it s p r o o f is t h e s a m e a s t h e p r o o f o f l e m m a 4 ) . s. darbinyan 1 9 lemma 5: l et d be a digraph of order n and c be a non-hamiltonian cycle in d. l et x be an arbitrary vertex not on this cycle c and d contain no c-bypass through x. if in d there are ( c; x ) and ( x; c ) -paths, then the following holds: (i). if x is adjacent to some vertex y of c, then d is not 2-strong, d ( x; v ( c ) n fyg ) = 0 and d( x ) + d( z ) · 2 n ¡ 2 for all vertices z 2 v ( c ) n fyg. (ii). assume that x and any vertex of c are nonadjacent, i.e., d ( x; v ( c ) ) = 0 . l et p be a shortest ( c; x ) -path with fug = v ( p ) \ v ( c ) and q be a shortest ( x; c ) -path with fvg = v ( q ) \ v ( c ) (possibly, u = v). then d( x ) + d( z ) · 2 n ¡ 2 for all vertices z 2 v ( c ) , maybe except one from fu; vg. in [1 3 ], t h e fo llo win g wa s p r o ve d : lemma 6: ( y . ma n o u s s a kis [1 3 ]) . l et d be a digraph of order n and v ( d ) satisfy condition a0. assume that there are two distinct pairs of nonadjacent vertices x; y and x; z in d. then either d( x ) + d( y ) ¸ 2 n ¡ 1 or d ( x) + d ( z ) ¸ 2 n ¡ 1 . it is n o t d i± c u lt t o s h o w t h a t we c a n s t a t e l e m m a 6 in t h e fo llo win g m u c h s t r o n g e r fo r m : lemma 7: l et d be a digraph of order n and y be a subset of v ( d ) . assume that y satis¯es condition a0 and contains two distinct pairs of nonadjacent vertices x; y and x; z. then either d( x ) + d( y ) ¸ 2 n ¡ 1 or d ( x) + d ( z ) ¸ 2 n ¡ 1 . fo r t h e p r o o f o f o u r r e s u lt we a ls o n e e d t h e fo llo win g s im p le le m m a . lemma 8: l et d be a digraph of order n. assume that xy =2 a( d ) and the vertices x, y in d satisfy the degree condition d+ ( x ) + d¡ ( y ) ¸ n ¡ 2 + k, where k ¸ 1 . then d contains at least k internally disjoint ( x; y ) -paths of length two. th e fo llo win g le m m a a ls o wa s p r o ve d in [5 ]. lemma 9: ( h . l i, e . fla n d r in . j. s h u [5 ]) . l et d be a digraph of order n and s µ v ( d ) , s 6= ; be a m eyniel set. if d is s-strongly connected, then any two s-vertices s and s0 are contained in a cycle of d such that they are at distance at most two on this cycle. w e c a n s t a t e l e m m a 9 in t h e fo llo win g fo r m . lemma 10: l et d be a digraph of order n. assume that a pair of distinct vertices x; y in d satis¯es the degree condition d( x ) + d( y ) ¸ 2 n ¡ 1 . if d is fx; yg-strongly connected, then the vertices x and y are contained in a cycle of d such that they are at distance at most two on this cycle. n o w we will p r o ve t h e fo llo win g le m m a . lemma 11: l et d be a digraph of order n and y be a subset of vertices of d with jy j ¸ 4 . assume that d is y -strongly connected and the subset y satis¯es condition a0. if c is a non-hamiltonian cycle in d which contains at least two y -vertices and y 2 v ( d ) n v ( c ) is an arbitrary y -vertex, then d contains a c-bypass through y. p r oof of lemma 11. if t h e c yc le c c o n t a in s a t le a s t t h r e e y -ve r t ic e s , t h e n t h e le m m a im m e d ia t e ly fo llo ws fr o m l e m m a s 5 a n d 7 . w e m a y t h e r e fo r e a s s u m e t h a t c c o n t a in s e xa c t ly t wo y -ve r t ic e s , s a y x a n d u, a n d t h e r e e xis t s a y -ve r t e x, s a y y, in b := v ( d ) n v ( c ) s u c h t h a t in d t h e r e is n o c-b yp a s s t h r o u g h y. fr o m jy j ¸ 4 it fo llo ws t h a t b c o n t a in s a t le a s t t wo y -ve r t ic e s . l e t z b e a n a r b it r a r y y -ve r t e x in b o t h e r t h a n y. w e will c o n s id e r t wo c a s e s d e p e n d in g u p o n t h e va lu e o f d( y; c ) . case 1: d( y; c ) ¸ 1 . w it h o u t lo s s o f g e n e r a lit y, a s s u m e t h a t t h e ve r t e x y is a d ja c e n t t o a ve r t e x w o f v ( c ) . if w =2 fu; xg, t h e n fr o m l e m m a 5 ( i) it fo llo ws t h a t y; u a n d y; x a r e d is t in c t p a ir s o f n o n a d ja c e n t ve r t ic e s o f y , d( y ) + d ( u) · 2 n ¡ 2 a n d d( y ) + d ( x) · 2 n ¡ 2 , wh ic h c o n t r a d ic t s 2 0 on cyclability of digraphs with a manoussakis-type condition l e m m a 7 . w e m a y t h e r e fo r e a s s u m e t h a t w 2 fx; ug, fo r e xa m p le , le t w = u a n d yu 2 a( d ) . s in c e we a s s u m e d t h a t d h a d n o c-b yp a s s t h r o u g h y, b y l e m m a 5 ( i) we h a ve d ( y ) + d ( x) · 2 n ¡ 2 a n d d( y; v ( c ) n fug ) = 0 : ( 1 ) n o w we c o n s id e r t wo s u b c a s e s xz 2 a( d ) , xz =2 a ( d ) . subcase 1.1. xz 2 a( d ) . th e n zy =2 a ( d ) ( fo r o t h e r wis e , xzyu is a c-b yp a s s t h r o u g h y, wh ic h c o n t r a d ic t s o u r a s s u m p t io n t h a t d h a s n o c-b yp a s s t h r o u g h y ) . th e r e fo r e , t h e t r ip le o f y -ve r t ic e s x; y; z s a t is ¯ e s c o n d it io n a0, i.e ., d( y ) + d( x ) + d¡ ( y ) + d+ ( z ) ¸ 3 n ¡ 2 : th is t o g e t h e r wit h d( y ) + d ( x) · 2 n ¡ 2 ( b y ( 1 ) ) im p ly t h a t d+ ( z ) + d¡ ( y ) ¸ n. h e n c e , b y l e m m a 8 , z ! v ! y fo r s o m e ve r t e x v o t h e r t h a n u. fr o m d( y; v ( c ) n fug) = 0 ( b y ( 1 ) ) it fo llo ws t h a t v 2 b. th u s , xzvyu is a c-b yp a s s t h r o u g h y, a c o n t r a d ic t io n . subcase 1.2. xz =2 a( d ) . th e n b y c o n d it io n a0 we h a ve d( y ) + d( x ) + d+ ( x) + d¡ ( z ) ¸ 3 n ¡ 2 : th e r fo r e , b y ( 1 ) , d+ ( x ) + d¡ ( z ) ¸ n, a n d h e n c e b y l e m m a 8 a n d xy =2 a ( d ) , t h e r e e xis t s a ve r t e x v o t h e r t h a n u a n d y s u c h t h a t x ! v ! z. it is e a s y t o s e e t h a t vy =2 a ( d ) a n d zy =2 a ( d ) . in t h is s u b c a s e , a g a in we h a ve t h a t d+ ( z ) + d¡ ( y ) ¸ n. h e n c e , b y l e m m a 8 , z ! a ! y fo r s o m e ve r t e x a o t h e r t h a n u a n d v. b y ( 1 ) , a =2 v ( c ) . th e r e fo r e , a 2 b n fy; z; vg a n d vzayu o r xvzayu is a c-b yp a s s t h r o u g h y wh e n v 2 c o r n o t , r e s p e c t ive ly, a c o n t r a d ic t io n . th e d is c u s s io n o f ca s e 1 is c o m p le t e d . case 2: d ( y; c ) = 0 . b y l e m m a 5 ( ii) , we h a ve e it h e r d( y ) + d( x ) · 2 n ¡ 2 o r d ( y ) + d ( u ) · 2 n ¡ 2 . w it h o u t lo s s o f g e n e r a lit y, a s s u m e t h a t d( y ) + d( x ) · 2 n ¡ 2 : ( 2 ) th is t o g e t h e r wit h c o n d it io n a0 im p ly t h a t d+ ( y ) + d¡ ( u ) ¸ n a n d d+ ( u ) + d¡ ( y ) ¸ n: ( 3 ) th is t o g e t h e r wit h l e m m a 8 im p ly t h a t t h e r e a r e ve r t ic e s a a n d v ( p o s s ib ly, a = v ) o t h e r t h a n z s u c h t h a t u ! v ! y a n d y ! a ! u. ob s e r ve t h a t v a n d a a r e n o t o n c s in c e d( y; c ) = 0 . fir s t c o n s id e r t h e c a s e d( z; v ( c ) n fug ) 6= 0 . w it h o u t lo s s o f g e n e r a lit y, a s s u m e t h a t w 2 v ( c ) n fug a n d zw 2 a ( d ) ( fo r t h e c a s e wz 2 a ( d ) we will c o n s id e r t h e c o n ve r s e d ig r a p h o f d ) . if yz 2 a ( d ) , t h e n uvyzw is a c-b yp a s s t h r o u g h y, a c o n t r a d ic t io n . w e m a y t h e r e fo r e a s s u m e t h a t yz =2 a ( d ) . th e n fr o m ( 2 ) a n d c o n d it io n a0 it fo llo ws t h a t d+ ( y ) + d¡ ( z ) ¸ n. th e r e fo r e , b y l e m m a 8 , fo r s o m e ve r t e x b 2 b n fvg, y ! b ! z, a n d h e n c e , uvybzw is a c-b yp a s s t h r o u g h y, wh ic h is a c o n t r a d ic t io n . n o w c o n s id e r t h e c a s e d ( z; v ( c ) n fug) = 0 . th e n t h e ve r t ic e s z a n d x a r e n o n a d ja c e n t . fr o m ( 2 ) a n d c o n d it io n a0 it fo llo ws t h a t d+ ( x ) + d¡ ( z ) ¸ n a n d d+ ( z ) + d¡ ( x ) ¸ n: s. darbinyan 2 1 fr o m d+ ( z ) + d¡ ( x ) ¸ n a n d l e m m a 8 it fo llo ws t h a t t h e r e a r e a t le a s t t wo ( z; x ) -p a t h s o ft h e le n g t h t wo . subcase 2.1. there is a ( z; x ) -paths of the length two, say z ! b ! x, such that b =2 fu; vg. in t h is s u b c a s e , yz =2 a ( d ) a n d yb =2 a( d ) ( fo r o t h e r wis e , uvyzbx o r uvybx is a c-b yp a s s t h r o u g h y, fo r yz 2 a( d ) a n d fo r yb 2 a ( d ) , r e s p e c t ive ly) . s in c e x; y; z a r e y -ve r t ic e s , fr o m c o n d it io n a0 a n d ( 2 ) it fo llo ws t h a t d + ( y ) + d¡ ( z ) ¸ n. n o w u s in g l e m m a 8 a n d t h e fa c t s t h a t yz =2 a ( d ) a n d yb =2 a( d ) , we o b t a in t h a t t h e r e e xis t s a ve r t e x q 2 b n fv; b; zg s u c h t h a t y ! q ! z. th u s , uvyqzbx is a c-b yp a s s t h r o u g h y, a c o n t r a d ic t io n . subcase 2.2. there is no w 2 b n fvg such that z ! w ! x. th e n fr o m l e m m a 8 a n d d+ ( z ) + d¡ ( x ) ¸ n it fo llo ws t h a t d+ ( z ) + d¡ ( x) = n a n d zv; vx; zu 2 a( d ) ( i.e ., t h e r e a r e e xa c t ly t wo ( z; x ) -p a t h s o f t h e le n g t h t wo ) . n o w u s in g t h e in e qu a lit y d+ ( u ) +d¡ ( y ) ¸ n ( b y ( 3 ) ) a n d l e m m a 8 we c o n c lu d e t h a t t h e r e e xis t a t le a s t t wo ( u; y ) -p a t h s o f t h e le n g t h t wo . if t h e r e is a p a t h u ! c ! y s u c h t h a t c is o t h e r t h a n v a n d z, t h e n we m a y c o n s id e r t h e p a t h s u ! c ! y a n d z ! v ! x. fo r t h e s e p a t h s we h a ve t h e a b o ve c o n s id e r e d c a s e ( b =2 fu; vg) . w e m a y t h e r e fo r e a s s u m e t h a t t h e r e is n o c 2 b n fv; zg s u c h t h a t u ! c ! y. fr o m t h is a n d l e m m a 8 it fo llo ws t h a t d+ ( u ) + d¡ ( y ) = n a n d u ! z ! y s in c e d+ ( u ) + d¡ ( y ) ¸ n ( b y ( 3 ) ) . th is t o g e t h e r wit h vx 2 a ( d ) im p ly t h a t yv =2 a( d ) ( if yv 2 a ( d ) , t h e n uzyvx is a c-b yp a s s t h r o u g h y ) . n o w b y c o n d it io n a0 a n d ( 2 ) we h a ve d( y ) + d ( x) = 2 n ¡ 2 ; s in c e x; y; u a r e y -ve r t ic e s , x; y a r e n o t a d ja c e n t a n d uy =2 a( d ) . th e la s t e qu a lit y im p lie s t h a t d+ ( y ) + d¡ ( x ) ¸ n ¡ 1 o r d¡ ( y ) + d+ ( x) ¸ n ¡ 1 . if d+ ( y ) +d¡ ( x ) ¸ n¡ 1 , t h e n , s in c e yv =2 a ( d ) a n d d ( y; c ) = 0 , fr o m l e m m a 8 it fo llo ws t h a t t h e r e e xis t s a ve r t e x z1 2 bnfvg s u c h t h a t y ! z1 ! x. th e r e fo r e , uvyz1x is a c-b yp a s s t h r o u g h y, a c o n t r a d ic t io n . w e m a y t h e r e fo r e a s s u m e t h a t d+ ( y ) + d¡ ( x ) · n ¡ 2 . th e n d¡ ( y ) +d+ ( x) ¸ n s in c e d ( y ) +d( x ) = 2 n¡ 2 . th e r e fo r e , s in c e d( y; c ) = d( z; v ( c ) nfug ) = 0 , b y l e m m a 8 t h e r e e xis t s a ve r t e x z2 2 b n fag s u c h t h a t x ! z2 ! y. h e n c e , xz2yau is a c-b yp a s s t h r o u g h y, wh ic h is a c o n t r a d ic t io n . th is c o n t r a d ic t io n c o m p le t e s t h e p r o o f o f l e m m a 1 1 . 4 . p r o o f o f t h e ma in r e s u lt fo r r e a d e r s c o n ve n ie n c e , a g a in we will fo r m u la t e t h e m a in r e s u lt . t heor em 1: l et d be a digraph of order n and let y be a nonempty subset of the vertices of d, where jy j ¸ 2 . suppose that d is y -strongly connected and the subset y satis¯es condition a0. then d contains a cycle through all the vertices of y maybe except one. p r oof: if jy j = 2 , t h e n t h e r e is n o t h in g t o p r o ve . if jy j = 3 , t h e n fr o m y -s t r o n g ly c o n n e c t e d n e s s o f d it fo llo ws t h a t y is a n in d e p e n d e n t s e t . b y l e m m a 7 , fo r s o m e t wo ve r t ic e s o f y , s a y x a n d y, we h a ve d( x ) + d ( y ) ¸ 2 n ¡ 1 . th e r e fo r e , b y l e m m a 1 0 , t h e r e is a c yc le in d t h a t c o n t a in s t h e ve r t ic e s x a n d y. in t h e s e qu e l, a s s u m e t h a t jy j ¸ 4 . n o w s u p p o s e t h a t t h e s u b s e t y o f t h e ve r t ic e s o f d s a t is ¯ e s t h e s u p p o s it io n o f t h e t h e o r e m b u t a n y c yc le in d d o e s n o t c o n t a in a t le a s t t wo y -ve r t ic e s . b y ma n o u s s a kis ' t h e o r e m , we m a y a s s u m e t h a t y 6= v ( d ) . s in c e d is y -s t r o n g ly c o n n e c t e d , u s in g l e m m a s 7 a n d 1 0 we o b t a in t h a t in d t h e r e e xis t s a c yc le wh ic h c o n t a in s a t le a s t t wo y -ve r t ic e s . if c is a n o n -h a m ilt o n ia n c yc le in d wh ic h c o n t a in s a t le a s t t wo y -ve r t ic e s a n d y 2 v ( d ) n v ( c ) is a n a r b it r a r y y -ve r t e x, t h e n fr o m l e m m a 1 1 it fo llo ws t h a t d c o n t a in s a c-b yp a s s t h r o u g h 2 2 on cyclability of digraphs with a manoussakis-type condition y. in d we c h o o s e a c yc le c a n d a c-b yp a s s p0 t h r o u g h a y -ve r t e x o f v ( d ) n v ( c ) s u c h t h a t ( a ) c c o n t a in s a s m a n y ve r t ic e s o f y a s p o s s ib le ( c c o n t a in s a t le a s t t wo y -ve r t ic e s ) , ( b ) t h e g a p o f c-b yp a s s p0 is m in im u m , s u b je c t t o ( a ) ( b y l e m m a 1 1 , fo r t h e c yc le c t h e r e e xis t s a c-b yp a s s t h r o u g h a n y y -ve r t e x n o t o n c ) , a n d ( c ) t h e le n g t h o f c-b yp a s s p0 is m in im u m , s u b je c t t o ( a ) a n d ( b ) . in t h e s e qu e l we a s s u m e t h a t t h e c yc le c := x1x2 : : : xmx1 a n d t h e c-b yp a s s p0 := x1z1 : : : zkyzk+1 : : : ztxa+1 s a t is fy t h e c o n d it io n s ( a ) -( c ) , wh e r e 1 · a · m ¡ 1 a n d y 2 v ( d ) n v ( c ) is a y -ve r t e x ( p o s s ib ly, z1 = y o r y = zt ) . s in c e t h e c yc le c h a s t h e m a xim u m y -le n g t h , it fo llo ws t h a t a ¸ 2 a n d c[x2; xa] c o n t a in s a y -ve r t e x. n o t e t h a t t h e g a p o f c-b yp a s s p0 is e qu a l t o a. d e n o t e p := z1 : : : zkyzk+1 : : : zt, e := p [z1; zk] a n d l := p [zk+1; zt]. s in c e t h e g a p a is m in im a l, t h e ve r t e x y is n o t a d ja c e n t t o a n y ve r t e x o f c[x2; xa], i.e , d( y; c[x2; xa]) = 0 . th e r e fo r e , b y l e m m a 2 , d( y; c ) = d( y; c[xa+1; x1]) · m ¡ a + d¡ ( y; fx1g ) + d+ ( y; fxa+1g ) ; ( 4 ) s in c e a n y y -ve r t e x o f b := v ( d ) n v ( c ) c a n n o t b e in s e r t e d in t o c. fr o m t h e m in im a lit y o f t h e p a t h p it fo llo ws t h a t d( y; v ( p ) ) · jv ( p ) j + 1 ¡ d¡ ( y; fx1g ) ¡ d+ ( y; fxa+1g ) : ( 5 ) n o t ic e t h a t jc[x2; xa]j = a ¡ 1 a n d jc[xa+1; x1]j = m ¡ a + 1 . fir s t ly fo r t h e c yc le c a n d c-b yp a s s p0 := x1z1 : : : zkyzk+1 : : : ztxa+1 we p r o ve t h e fo llo win g t wo c la im s . claim 1: there is a y -vertex, say y1, in c[x2; xa] such that y; y1 are nonadjacent, d( y ) + d( y1 ) · 2 n ¡ 2 and y1 cannot be inserted into c[xa+1; x1]. m oreover, (i). the path p contains exactly one y -vertex, namely only y. (ii). any y -vertex of c[x2; xa] other than y1 can be inserted into c[xa+1; x1]. (iii). there are three ( xa+1; x1 ) -paths, say p1; p2 and p3, with vertex set c[xa+1; x1] [ f1, c[xa+1; x1] [ f2 and c[xa+1; x1] [ f3, respectively, where f1 µ c[x2; xa], f2 µ c[x2; y¡1 ], f3 µ c[y+1 ; xa] ( if y1 = x2, then c[x2; y¡1 ] = ;, if y1 = xa, then c[y+1 ; xa] = ; ) and f1 ( respectively, f2; f3 ) contains all the y -vertices of c[x2; xa] n fy1g ( respectively, all the y -vertices of c[x2; y ¡ 1 ], all the y -vertices of c[y + 1 ; xa]) . p r oof: s in c e we a s s u m e d t h a t t h e c yc le c h a s m a xim u m y -le n g t h , fr o m l e m m a 3 it fo llo ws t h a t s o m e y -ve r t e x, s a y y1, o f c[x2; xa] c a n n o t b e in s e r t e d in t o c[xa+1; x1]. h e n c e , u s in g l e m m a 2 , we o b t a in d( y1; c ) = d( y1; c[x2; xa]) + d( y1; c[xa+1; x1]) · 2 a ¡ 4 + m ¡ a + 2 = m + a ¡ 2 : ( 6 ) fr o m t h e m in im a lit y o f c[x2; xa] it fo llo ws t h a t t h e ve r t ic e s y a n d y1 a r e n o n a d ja c e n t . p u t r := v ( d ) n ( v ( c ) [ v ( p ) ) . n o w we wa n t t o c o m p u t e t h e s u m o f d e g r e e y a n d y1. b y m in im a lit y o f c[x2; xa] we h a ve d ( y1; r) + d ( y; r ) · 2 jrj a n d d( y; c[x2; xa]) = 0 : ( 7 ) fr o m t h e m in im a lit y o f c[x2; xa] a ls o it fo llo ws t h a t d+ ( y1; fz1; : : : ; zkg ) = d¡ ( y1; fzk+1; : : : ; ztg ) = 0 : ( 8 ) s. darbinyan 2 3 th e r e fo r e , d( y1; p ) · jp j ¡ 1 s in c e y a n d y1 a r e n o n a d ja c e n t . th is t o g e t h e r wit h t h e a b o ve in e qu a lit ie s ( 4 ) -( 7 ) g ive d ( y ) + d ( y1 ) = d( y; r ) + d( y1; r) + d ( y; p ) + d ( y; c ) + d( y1; p ) + d ( y1; c ) · 2 jrj + 2 jp j + 2 m ¡ 2 = 2 n ¡ 2 : th u s , d( y ) +d( y1 ) · 2 n¡ 2 fo r a n y y -ve r t e x y o f p a n d fo r a n y y -ve r t e x y1 o f c[x2; xa] wh ic h c a n n o t b e in s e r t e d in t o c[xa+1; x1]. th is t o g e t h e r wit h l e m m a 7 im p ly t h a t p c o n t a in s o n ly o n e y -ve r t e x, n a m e ly y, a n d a n y y -ve r t e x o f c[x2; xa] d i®e r e n t fr o m y1 c a n b e in s e r t e d in t o c[xa+1; x1]. fr o m t h is a n d l e m m a 3 it im m e d ia t e ly fo llo ws t h e t h ir d a s s e r t io n o f t h e c la im . cla im 1 is p r o ve d . b y cla im 1 , we h a ve d ( y1 ) + d( y ) · 2 n ¡ 2 : ( 9 ) claim 2: l et y1 be a y -vertex of c[x2; xa] which cannot be inserted into c[xa+1; x1]. then d( y1; p ) = 0 . p r oof: s u p p o s e , o n t h e c o n t r a r y, t h a t d( y1; p ) ¸ 1 . th e n fr o m ( 8 ) it fo llo ws t h a t e it h e r ziy1 2 a ( d ) o r y1zj 2 a ( d ) , fo r s o m e i 2 [1 ; k] o r j 2 [k+1 ; t], r e s p e c t ive ly. l e t y1zj 2 a ( d ) . w e c o n s id e r t h e c yc le c1 := p3c[x1; y1]p [zj; zt]xa+1. th is c yc le c o n t a in s a ll t h e y -ve r t ic e s o f c, a n d h e n c e , h a s m a xim u m y -le n g t h . it is e a s y t o s e e t h a t q := x1z1 : : : zkyzk+1 : : : zj is a c1-b yp a s s t h r o u g h y. b y t h e c h o ic e o f t h e c yc le c a n d c-b yp a s s p0 we h a ve t h a t y1 = xa, i.e ., t h e c-g a p o f p0 a n d c1-g a p o f q a r e e qu a l b u t t h e p a t h z1 : : : zkyzk+1 : : : zj¡1 is s h o r t e r t h a n t h e p a t h z1 : : : zkyzk+1 : : : zt, wh ic h c o n t r a d ic t s ( c ) . th e r e fo r e , y1zj =2 a ( d ) fo r a ll j 2 [k + 1 ; t]. s im ila r ly, we c a n p r o ve t h a t ziy1 =2 a ( d ) fo r a ll i 2 [1 ; k]. th u s , d¡ ( y1; fz1; : : : ; zkg ) = d+ ( y1; fzk+1; : : : ; ztg ) = 0 wh ic h t o g e t h e r wit h ( 8 ) im p ly t h a t d ( y1; p ) = 0 . cla im 2 is p r o ve d . l e t x b e a n a r b it r a r y y -ve r t e x in b = v ( d ) n v ( c ) o t h e r t h a n y. cla im 1 ( i) im p lie s t h a t x is n o t o n p . w e d is t in g u is h t wo c a s e s a c c o r d in g a s in hb n ( p n fyg ) i t h e r e e xis t s a p a t h wit h e n d -ve r t ic e s x a n d y o r n o t . case 1: in hb n ( p n fyg ) i there exists a path from x to y or there exists a path from y to x. w it h o u t lo s s o f g e n e r a lit y, we m a y a s s u m e t h a t in hb n ( p n fyg ) i t h e r e is a n ( x; y ) -p a t h ( fo r o t h e r wis e , we c o n s id e r t h e c o n ve r s e d ig r a p h o f d ) . l e t h b e a s h o r t e s t ( x; y ) -p a t h in hb n ( p n fyg ) i. ob s e r ve t h a t x1x =2 a ( d ) , s in c e o t h e r wis e , if x1x 2 a ( d ) , t h e n t h e p a t h p1 ( cla im 1 ( iii) ) t o g e t h e r wit h t h e a r c x1x a n d t h e p a t h s h a n d p0[y; xa+1] fo r m s a c yc le , s a y c1, wh ic h c o n t a in s m o r e y -ve r t ic e s t h a n c, wh ic h c o n t r a d ic t s o u r a s s u m p t io n t h a t c h a s m a xim u m y -le n g t h ( c1 c o n t a in s a ll y -ve r t ic e s o f c, e xc e p t y1, b u t c o n t a in s y -ve r t ic e s x a n d y ) . p u t r0 := b n ( v ( p ) [ v ( h ) ) a n d h0 := h[x+h ; y¡h ] ( if x+h = y, t h e n h0 = ; ) . fr o m t h e m in im a lit y o f t h e g a p a ( o r o f t h e e xis t e n c e o f t h e p a t h p3 ) it fo llo ws t h a t y1x =2 a ( d ) ( t h e r e fo r e e it h e r xy1 2 a( d ) o r x a n d y1 a r e n o n a d ja c e n t ) a n d d+ ( y1; r0 ) + d ¡ ( x; r0 ) · jr0j: ( 1 0 ) subcase 1.1. xy1 2 a ( d ) . fr o m l e m m a 2 it fo llo ws t h a t d¡ ( x; p1 ) + d + ( y1; p1 ) · jp1j: ( 1 1 ) 2 4 on cyclability of digraphs with a manoussakis-type condition s in c e x1x =2 a ( d ) a n d t h e a r c xy1 c a n n o t b e in s e r t e d in t o p1 ( fo r o t h e r wis e , d c o n t a in s a c yc le wh ic h c o n t a in s a ll y -ve r t ic e s o f c a n d y -ve r t ic e s x; y, wh ic h is a c o n t r a d ic t io n ) . fr o m t h e m in im a lit y o f t h e g a p a, t h e e xis t e n c e o f t h e p a t h s p2; p3 ( b y cla im 1 ) a n d cla im 2 it fo llo ws t h a t d+ ( y1; p [ h ) = d¡ ( x; c[x2; xa]) = d¡ ( x; p ) = 0 : ( 1 2 ) cle a r ly, d+ ( y1; s ) · jsj ¡ 1 a n d d¡ ( x; h0 ) · jh0j; ( 1 3 ) wh e r e s := c[x2; xa] ¡ p1. b y a d d in g t h e a b o ve r e la t io n s ( 1 0 ) -( 1 3 ) , we o b t a in d+ ( y1 ) + d ¡ ( x) = d+ ( y1; r0 ) + d ¡ ( x; r0 ) + d ¡ ( x; p1 ) + d + ( y1; p1 ) + d + ( y1; s ) + d + ( y1; p [ h ) +d¡ ( x; s ) + d¡ ( x; h0 ) + d¡ ( x; p ) · jr0j + jp1j + jsj + jh0j ¡ 1 · n ¡ 2 : th is t o g e t h e r wit h ( 9 ) g ive d( y ) + d( y1 ) + d ¡ ( x ) + d+ ( y1 ) · 3 n ¡ 4 ; wh ic h c o n t r a d ic t s c o n d it io n a0, s in c e y; y1 a n d x a r e y -ve r t ic e s , y; y1 a r e n o n a d ja c e n t a n d y1x =2 a ( d ) . subcase 1.2. the vertices x and y1 are nonadjacent. w e will d is t in g u is h t wo s u b c a s e s , a c c o r d in g a s t h e r e e xis t s a ( y; x ) -p a t h in hb n ( p n fyg ) i o r n o t . subcase 1.2.1. in hb n ( p n fyg ) i there is no ( y; x ) -path, in particular yx =2 a( d ) . th e n , c le a r ly d+ ( y; r0 [ h0 ) + d¡ ( x; r0 [ h0 ) · jr0 [ h0j = jr0j + jh0j: ( 1 4 ) it is n o t d i± c u lt t o s e e t h a t t h e p a t h h c a n n o t b e in s e r t e d in t o c. h e n c e , fr o m l e m m a 1 it fo llo ws t h a t d¡ ( x; c ) + d+ ( y; c ) · jcj: ( 1 5 ) mo r e o ve r , d¡ ( x; p ) = d¡ ( x; e ) + d¡ ( x; l ) · jlj; s in c e d¡ ( x; e ) = 0 ; r e c a ll t h a t e := p [z1; zk] a n d l := p [zk+1; zt], a n d b y t h e m in im a lit y o f p , we h a ve d+ ( y; p ) = d+ ( y; e ) + d+ ( y; l) · jej + 1 ; s in c e d+ ( y; l) · 1 : h e n c e , d¡ ( x; p ) + d+ ( y; p ) · jlj + jej + 1 = jp j: th e la s t in e qu a lit y t o g e t h e r wit h ( 1 4 ) , ( 1 5 ) a n d ( 9 ) im p ly t h a t d( y ) + d( y1 ) + d ¡ ( x ) + d+ ( y ) · 2 n ¡ 2 + jrj + jh0j + jcj + jp j = 3 n ¡ 3 ; wh ic h c o n t r a d ic t s c o n d it io n a0, s in c e y; y1 a n d x a r e y -ve r t ic e s , y; y1 a r e n o n a d ja c e n t a n d yx =2 a ( d ) . subcase 1.2.2. in hb n ( p n fyg ) i there is a ( y; x) -path. fir s t c o n s id e r t h e c a s e wh e n in hb n ( p [ h n fx; yg ) i t h e r e is a ( y; x ) -p a t h . l e t q b e a s h o r t e s t ( y; x ) -p a t h in hb n ( p [ h n fx; yg ) i. s. darbinyan 2 5 l e t r1 := b n ( p [ h [ q) . w e wa n t t o c o m p u t e t h e d e g r e e s u m o f t h e ve r t ic e s x a n d y1. fr o m t h e m in im a lit y o f t h e c[x2; xa] a n d t h e e xis t e n c e o f t h e p a t h s h a n d q it fo llo ws t h a t d ( x; r1 ) + d( y1; r1 ) · 2 jr1j a n d d( y1; h0 [ q0 ) = 0 ; ( 1 6 ) wh e r e q0 := q[y+q; x ¡ q] ( h e r e if y + q = x, t h e n q 0 = ; ) . b y cla im 2 , d( y1; p ) = 0 . th is a n d ( 6 ) im p ly t h a t d ( y1; c [ p ) · m + a ¡ 2 : ( 1 7 ) n o w we c o n s id e r t h e ve r t e x x. it is n o t d i± c u lt t o s e e t h a t x c a n n o t b e in s e r t e d in t o p ( fo r o t h e r wis e t h e r e e xis t s a ( z1; zt ) -p a t h wit h ve r t e x s e t v ( p ) [fxg wh ic h t o g e t h e r wit h t h e a r c s x1z1; ztxa+1 a n d t h e p a t h p1 ( cla im 1 ) fo r m s a c yc le wh ic h c o n t a in s a ll t h e y -ve r t ic e s o f c e xc e p t y1 b u t c o n t a in s y -ve r t ic e s x a n d y, t h is c o n t r a d ic t s t h e a s s u m p t io n t h a t c h a s t h e m a xim u m y -le n g t h ) . th e r e fo r e , b y l e m m a 2 , d( x; p ) · jp j + 1 : ( 1 8 ) fr o m t h e m in im a lit y o f t h e p a t h s h a n d q it fo llo ws t h a t d( x; q0 ) · jq0j + 1 a n d d( x; h0 ) · jh0j + 1 : ( 1 9 ) s in c e t h e g a p a is m in im a l, we o b t a in t h a t d ( x; c[x2; xa]) = 0 . u s in g t h e p a t h p1 ( cla im 1 ) , it is n o t d i± c u lt t o s e e t h a t x1x =2 a ( d ) a n d xxa+1 =2 a( d ) . th e r e fo r e , b y l e m m a 2 , d( x; c ) = d( x; c[xa+1; x1]) · m ¡ a, s in c e x c a n n o t b e in s e r t e d in t o c. s u m m in g t h e a b o ve in e qu a lit ie s ( 1 6 ) -( 1 9 ) a n d t h e la s t in e qu a lit y, a n e a s y c o m p u t a t io n s h o ws t h a t d( y1 ) + d( x ) · 2 jr1j + 2 m + jp j + jq0j + jh0j + 1 · 2 n ¡ 2 : th is t o g e t h e r wit h d( y1 ) + d( y ) · 2 n ¡ 2 ( b y ( 9 ) ) c o n t r a d ic t l e m m a 7 , s in c e y1; x a n d y1; y a r e t wo d is t in c t p a ir s o f n o n a d ja c e n t ve r t ic e s in y . n o w c o n s id e r t h e c a s e wh e n a n y ( y; x ) -p a t h in hb n ( p n fyg) i h a s a c o m m o n in t e r n a l ve r t e x wit h ( x; y ) -p a t h h. th e n , in p a r t ic u la r , t h e ve r t ic e s y a n d x a r e n o n a d ja c e n t . l e t t b e a s h o r t e s t ( y; x ) -p a t h in hb n ( p n fyg ) i. d e n o t e t 0 := t [y+t ; x ¡ t ] a n d r2 := b n ( p [ h0 [ t 0 [ fxg ) . ob s e r ve t h a t jh0j ¸ 1 a n d jt 0j ¸ 1 s in c e y a n d x a r e n o n a d ja c e n t . n o w we wa n t t o c o m p u t e t h e s u m d+ ( x ) + d¡ ( y ) . it is e a s y t o s e e t h a t d+ ( x; r2 ) + d ¡ ( y; r2 ) · jr2j; ( 2 0 ) s in c e fo r o t h e r wis e in hb n ( p nfyg ) i t h e r e e xis t s a m in im a l ( y; x ) -p a t h wh ic h h a s n o c o m m o n in t e r n a l ve r t e x wit h t h e t . ob s e r ve t h a t ( y; x ) -p a t h t c a n n o t b e in s e r t e d in t o c[xa+1; x1] ( fo r o t h e r wis e in d t h e r e is a c yc le wh ic h c o n t a in s m o r e y -ve r t ic e s t h a n t h e c yc le c ) . n o t ic e t h a t xxa+1 =2 a( d ) ( fo r o t h e r wis e , if xxa+1 2 a ( d ) , t h e n t h e p a t h s p [z1; y], t a n d p1 ( cla im 1 ) a n d t h e a r c s xxa+1, x1z1 fo r m a c yc le wh ic h h a s m o r e y -le n g t h t h a n t h e c yc le c ) . th e r e fo r e , b y l e m m a 2 we h a ve d+ ( x; c[xa+1; x1]) +d ¡ ( y; c[xa+1; x1]) · m¡a+d¡ ( y; fx1g ) +d+ ( x; fxa+1g ) · m¡a+1 : ( 2 1 ) fr o m t h e m in im a lit y o f c[x2; xa] ( i.e ., o f t h e g a p a) a n d t h e e xis t e n c e o f t h e p a t h t it fo llo ws t h a t d¡ ( y; c[x2; xa]) = d + ( x; c[x2; xa]) = 0 : ( 2 2 ) 2 6 on cyclability of digraphs with a manoussakis-type condition b y t h e m in im a lit y o f p we h a ve d¡ ( y; p ) = d¡ ( y; e ) + d¡ ( y; l) · jlj + 1 : it is n o t d i± c u lt t o s e e t h a t d+ ( x; p ) = d+ ( x; e ) + d+ ( x; l ) · jej; s in c e if xzj 2 a( d ) fo r s o m e j 2 [k + 1 ; t], t h e n u s in g t h e p a t h s t , p1 a n d t h e s u b p a t h s p [z1; y], p [zj ; zt] we c a n o b t a in a c yc le wh ic h c o n t a in s m o r e y -ve r t ic e s t h a n c. th e la s t t wo in e qu a lit ie s im p ly t h a t d¡ ( y; p ) + d+ ( x; p ) · jlj + jej + 1 = jp j: ( 2 3 ) it r e m a in s t o c o m p u t e d+ ( x; h0 [ t 0 ) a n d d¡ ( y; h0 [ t 0 ) . d e n o t e t 00 := t 0 n h0. fr o m t h e m in im a lit y o f t h e p a t h h it fo llo ws t h a t d+ ( x; h0 ) = d¡ ( y; h0 ) = 1 . th is t o g e t h e r wit h t h e a b o ve e xp r e s s io n s ( 2 0 ) -( 2 3 ) im p ly t h a t d+ ( x) +d¡ ( y ) = d+ ( x; r2 ) +d ¡ ( y; r2 ) +d + ( x; c[xa+1; x1]) +d ¡ ( y; c[xa+1; x1]) +d + ( x; c[x2; xa]) +d¡ ( y; c[x2; xa]) + d + ( x; p ) + d¡ ( y; p ) + d+ ( x; h0 ) + d¡ ( y; h0 ) + d+ ( x; t 00 ) + d¡ ( y; t 00 ) · jr2j + m ¡ a + 1 + jp j + 2 + d+ ( x; t 00 ) + d¡ ( y; t 00 ) = jr2j + jcj + jp j + 3 + d+ ( x; t 00 ) + d¡ ( y; t 00 ) ¡ a: ( 2 4 ) a s s u m e t h a t jh0j ¸ 2 , t h e n d+ ( x; t 00 ) +d¡ ( y; t 00 ) · jt 00j, s in c e o t h e r wis e in hbn ( p nfyg ) i t h e r e is a n ( x; y ) -p a t h s h o r t e r t h a n h. th e la s t in e qu a lit y t o g e t h e r wit h ( 2 4 ) g ive d+ ( x ) + d¡ ( y ) · jr2j + jcj + jp j + 3 + jt 00j ¡ a + jh0j ¡ jh0j = n + 2 ¡ a ¡ jh0j · n ¡ 2 ; s in c e a ¸ 2 a n d jh0j ¸ 2 . th is t o g e t h e r wit h ( 9 ) im p ly t h a t d( y ) + d ( y1 ) + d ¡ ( y ) + d+ ( x) · 3 n ¡ 4 ; ( 2 5 ) wh ic h c o n t r a d ic t s c o n d it io n a0 s in c e x; y; y1 a r e y -ve r t ic e s , y; y1 a r e n o n a d ja c e n t a n d xy =2 a( d ) . n o w a s s u m e t h a t jh0j = 1 . w e m a y a s s u m e t h a t jt 0j = 1 ( fo r o t h e r wis e , we c o n s id e r t h e c o n ve r s e d ig r a p h o f d ) . it fo llo ws t h a t t 00 = ;. n o w u s in g ( 9 ) , ( 2 4 ) , a ¸ 2 a n d jh0j ¸ 1 , we s e e t h a t a g a in ( 2 5 ) is t r u e , wh ic h is a c o n t r a d ic t io n . th e d is c u s s io n o f ca s e 1 is c o m p le t e d . case 2: in hb n ( p n fyg ) i there is no path between the vertices x and y. in particular, x and y are nonadjacent. l e t r3 := b n ( p [ fxg ) . th e n it is e a s y t o s e e t h a t d( x; r3 ) + d( y; r3 ) · 2 jrj; s in c e in hb n ( p n fyg ) i t h e r e is n o p a t h b e t we e n x a n d y. u s in g l e m m a s 1 a n d 2 we o b t a in t h a t d ( x; p ) · jp j + 1 a n d d ( x; c ) · m; s in c e t h e ve r t e x x c a n b e in s e r t e d n e it h e r in t o p n o r in c. th e la s t t h r e e in e qu a lit ie s t o g e t h e r wit h ( 4 ) a n d ( 5 ) im p ly t h a t d( y ) + d( x ) · 2 jr3j + jp j + m + m ¡ a + jp j + 2 · 2 n ¡ a · 2 n ¡ 2 : th is t o g e t h e r wit h ( 9 ) c o n t r a d ic t l e m m a 7 , s in c e fx; yg a n d fy; y1g a r e t wo d is t in c t p a ir s o f n o n a d ja c e n t ve r t ic e s o f y . th e d is c u s s io n o f ca s e 2 is c o m p le t e d a n d wit h it t h e p r o o f o f t h e t h e o r e m is a ls o c o m p le t e d . s. darbinyan 2 7 5 . co n c lu d in g r e m a r ks ob s e r ve t h a t t h e e xa m p le o f t h e d ig r a p h in r e m a r k 1 is n o t 2 -s t r o n g ly c o n n e c t e d a n d jy j = 3 . w e b e lie ve t h a t t h e fo llo win g m a y b e t r u e . conjectur e 1: l et d be a digraph of order n ¸ 4 and let y be a nonempty subset of vertices of d which satis¯es condition a0. then d has a cycle that contains all the vertices of y if either (i) or (ii) or (iii) below is satis¯ed: (i) d is 2-strongly connected. (ii) d is y -strongly connected and jy j ¸ 4 . (iii) for any ordered pair of distinct vertices x; y of y there are two internally disjoint paths from x to y in d. c. th o m a s s e n [1 8 ] ( fo r n = 2 k + 1 ) a n d t h e a u t h o r [1 9 ] ( fo r n = 2 k ) p r o ve d t h e fo llo win g t h e o r e m . t heor em h : ( c.th o m a s s e n [1 8 ], s . d a r b in ya n [1 9 ]) . l et d be a digraph of order n ¸ 5 with minimum degree at least n ¡ 1 and with minimum semi-degree at least n=2 ¡ 1 . then d is hamiltonian or belongs to a non-empty ¯nite family of non-hamiltonian digraphs, which are characterized. a qu e s t io n wa s p u t in [2 0 ]: l e t d b e a d ig r a p h o f o r d e r n ¸ 5 a n d le t t 6= ; b e a s u b s e t o f v ( d ) . a s s u m e t h a t d is s t r o n g ly c o n n e c t e d ( o r d is t -s t r o n g ly c o n n e c t e d ) a n d e ve r y ve r t e x o f t h a s a d e g r e e a t le a s t n ¡ 1 a n d h a s a n o u t d e g r e e a n d a n in d e g r e e a t le a s t n=2 ¡ 1 . d o e s d h a ve a c yc le t h a t c o n t a in s a ll t h e ve r t ic e s o f t ?. fo r n = 2 m + 1 in [2 0 ] it wa s p r o ve d : if d is strongly connected and contains a cycle of length n ¡ 1 , then d has a cycle containing all the vertices of t unless some extremal cases. a c kn o wle d g m e n t s w e wo u ld like t o t h a n k t h e r e fe r e e s fo r m a n y u s e fu l s u g g e s t io n s , wh ic h h a ve c o n s id e r a b ly im p r o ve d t h e p r e s e n t a t io n o f t h e p a p e r . refer ences [1 ] b . b o llo b a s , g. b r ig h t we ll, \ cyc le s t h r o u g h s p e c ī e d ve r t ic e s " , combinatorica, vo l. 1 3 , n o . 2 , p p . 1 1 7 -1 5 5 , 1 9 9 3 . 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[1 9 ] s . k h . d a r b in ya n , \ a s u ± c ie n t c o n d it io n fo r t h e h a m ilt o n ia n p r o p e r t y o f d ig r a p h s wit h la r g e s e m id e g r e e s " , akademy nauk armyan. ssr d okl., vo l. 8 2 , n o . 1 , p p . 6 -8 , 1 9 8 6 ( s e e a ls o a r x iv: 1 1 1 1 .1 8 4 3 v1 [m a t h .co] 8 n o v 2 0 1 1 ) . [2 0 ] s . k h . d a r b in ya n a n d i.a . k a r a p e t ya n , \ on c yc le s t h r o u g h ve r t ic e s o f la r g e s e m id e g r e e in d ig r a p h s " , m athematical p roblems of computer science, vo l. 3 9 , p p .1 0 6 -1 1 8 , 2 0 1 3 . submitted 02.11.2016, accepted 26.01.2017 ø³ýááõë³ïçëç ïçåç å³ûù³ýçý µ³í³ñ³ñáõ ïáõùýáñáßí³í ·ñ³ýý»ñç óçïéçïáõãû³ý ù³ëçý ê. ¸³ñµçýû³ý ²ù÷á÷áõù ü»ñï³ ³ßë³ï³ýùáõù óáõûó ¿ ïñíáõù áñ »ã» ïáõùýáñáßí³í ·ñ³ýç ·³·³ãý»ñç y »ýã³µ³½ùáõãûáõýá µ³í³ñ³ñáõù ¿ d ïáõùýáñáßí³í ·ñ³ýý»ñç ñ³ù³ñ ø³ýááõë³ïçëç s. darbinyan 2 9 ñ³ùçéïáýû³ýáõãû³ý µ³í³ñ³ñ å³ûù³ýçý (j. graph theory, 16, 1992), ³å³ ³û¹ ·ñ³ýá å³ñáõý³ïáõù ¿ óçïé, áñý ³ýóýáõù ¿ y »ýã³µ³½ùáõãû³ýá å³ïï³ýáõ µáéáñ ·³·³ãý»ñáí µ³óç ·áõó» ù»ïçó: êï³óí³í ³ñ¹ûáõýùá éáõíáõù ¿ èçç, üé³ý¹ñçýç ¨ þáõç (discrete mathematics, 307, 2007) ïáõùçó ³é³ç³ñï³í ëý¹çñá: î öèêëè÷íîñòè îðãðàôîâ ïðè óñëîâèé òèïà ìàíîóñàêèñà ñ. äàðáèíÿí àííîòàöèÿ â ðàáîòå äîêàçàíî, ÷òî åñëè ïîäìíîæåñòâî y âåðøèí îðãðàôà d óäîâëåòâîðÿåò äîñòàòî÷íîìó óñëîâèþ ãàìèëüòîíîâñòè ìàíîóñàêèñà (j. graph theory, 16, 1992 ), òî â d ñóùåñòâóåò êîíòóð, êîòîðûé ñîäåðæèò ïî êðàéíåé ìåðå jy j¡ 1 âåðøèí ïîäìíîæåñòâà y . ïîëó÷åííûé ðåçóëüòàò ðåøàåò çàäà÷ó ëè, ôëàíäðèí è øó (discrete mathematics, 307, 2007). microsoft word l. aslanyan_9.doc mathematical problems of computer science 34, 2010. 28 pattern recognition with mathematical logic and discrete analysis levon aslanyan institute for informatics and automation problems information society technologies center joint laboratory of computational biology institute for information theories and applications 1, p.sevak, 0014 yerevan, lasl@sci.am pattern recognition undergoes an important development for many years. as a research area this is not uni-modular like classic mathematical sciences, there are a number of sub disciplines inside such as – feature selection, object and feature ranking, analogy measuring, supervised and unsupervised classification, etc. the same time pattern recognition is indeed an integrated theory studying object descriptions and their diverse classification models. this is a collection of mathematical, statistical, heuristic and inductive techniques of fundamental role in executing the intellectual tasks, typical for a human being, – but on computers. many applied problems with multidimensional experimental data use the object description that is often non-classical, that is, not exclusively in terms of only numerical or only categorical features, but simultaneously by both kinds of values. sometimes, the missing or unreliable value is introduced so that finally we deal with mixed and incomplete descriptions of objects as elements of cartesian product of feature values, without any algebraic, logical or topological properties assumed in applied area. how then, do we select in these cases the most informative features, classify a new object given a (training) sample or find the relationships between all objects based on a certain measure of similarity? logic combinatorial pattern recognition (lcpr) is a research area formed since 70’s that uses a mix of discrete descriptors with similarities, separation, frequencies, integration, corrections, and optimization, and solves the whole spectrum of pattern recognition tasks. this approach is originated by the work [dmitriev et al, 1966] that transfers the engineering domain technique of tests for electrical schemes [chegis, yablonskii, 1958] to the feature selection and object classification area. the applied task of [dmitriev et al, 1966] address prognosis of mineral resources. testor theory [dmitriev et al, 1966] is based on a restriction of feature sets with preservation of the basic learning set property. a feature subset },,{ kiii xxxt  21  is called a testor (or test) of l , if prjection of l on t keeps the classes nonintersecting (learning set property). irreducible is the testor where no one ji x may be eliminated with conservation of the learning set property. further a feature ranking is made taking into consideration the frequency of belonging ji x to the testors (irreducible testors). testor based supervised classification algorithms are constructed by use of frequency similarity measures. which is the structural property used from the learning 29 set? this is the pair wise difference of learning set elements from different classes (learning set property). we may suppose that this is a consequence of the well accepted compactness hypothesis. on formal basis testor technology is well visible on binary tables. now it is known that constructing all irreducible testors is an np hard problem. approximations are studied as well. there is a high similarity with association rule mining models, especially in learning the monotone set structures – be it with frequent itemsets in associative rule mining or irreducible tests and testors in this theory. kora and logic separation. [vaintsvaig, 1973] introduced one of the basic concept in lcpr – the kora algorithm. kora is constructing elementary conjunctions c that intersect with only one class ic of learning set l . it is easy to imagine the corresponding interval c of n dimensional binary cube ne . c does not contain contradictory knowledge of learning the set l . contrary, [aslanyan, 1975] considers all irreducible conjunction forms that intersect with only one class ic of the learning set l . this is not the generating idea of this work but is the consequence of the logic separation (ls) principle. the work, factually for the first time, is considering learning set elements in a non-disjoint manner. the potential function concept is used that an element spreads its similarity, reducing by the distance measure, which interrupts facing the different class object. an extension may consider not only pairs of learning set elements – one spreading the similarity measure and the second interrupting that but also arbitrary subsets/fragments of learning set. several comments: logically, it is evident that the best learning algorithm is also best suited to the learning set (at least the learning set is reconstructable by the information used by the algorithm). this also may use the recognition hypothesis when available. the same learning set fragments play a crucial role in determining the best suited recognition algorithm to the given learning data set. after the methodological discussion it is worth to mention further developments with the voting algorithm, algorithmic correction procedure and other approaches of advanced logiccombinatorial pattern recognition [zhuravlev, 1998] . references [chegisyablonskii, 1958] chegis i. a. and yablonskii s. v., logical methods for controlling electrical systems, proceedings of mathematical institute after v. a. steklov, 51: 270-360, moscow (in russian), 1958. [dmitriev et al, 1966] dmitriev a. n., zhuravlev yu. i. and krendelev f.p. on mathematical principles of object and phenomena classification, discrete analysis, 7: 3-15, novosibirsk (in russian), 1966. [bongard, 1967] bongard m.m. the problem of cognition, moscow, 1967. [vaintsvaig, 1973] vaintsvaig m.n. pattern recognition learning algorithm kora, pattern recognition learning algorithms, moscow, sovetskoe radio, 1973, pp.110-116. [yablonskii, 1958] yablonskii s. v., functional constructions in k-valued logic, proceedings of mathematical institute after v. a. steklov, 51: 5-142, moscow (in russian), 1958. [zhuravlev, 1958] zhuravlev yu. i., on separating subsets of vertices of n-dimensional unit cube, proceedings of mathematical institute after v. a. steklov, 51: 143-157, 1958. [baskakova, zhuravlev, 1981] baskakova l.v., zhuravlev yu.i. model of recognition algorithms with representative and support sets, journal of comp. math. and math. phys., 1981, vol. 21, no.5, pp. 1264-1275. [zhuravlev, 1978] zhuravlev yu.i. on an algebraic approach to the problems of pattern recognition and classification, problemy kybernetiki, vol. 33, 1978, pp.5-68. 30 [zhuravlev, nikiforov, 1971] zhuravlev yu.i., nikiforov v.v. recognition algorithms based on calculation of estimates. cybernetics, vol. 3, 1971, pp.1-11. [aslanyan, 1975] aslanyan l. h. on a pattern recognition method based on the separation by the disjunctive normal forms. kibernetika, 5, (1975), 103 -110. [zhuravlev, 1977] zhuravlev yu.i. correct algebras over the sets of incorrect (heuristical) algorithms: i, cybernetics, no.4, 1977, pp.5-17., ii, cybernetics, no.6, 1977., iii, cybernetics, no.2, 1978. [aslanyan, zhuravlev, 2001] aslanyan l., zhuravlev yu,.logic separation principle, computer science & information technologies conference, yerevan, september 17-20, 2001, 151-156. [zhuravlev, 1998] zhuravlev yu. i., selected scientific works (in russian), magistr, moscow, 1998, 417 p. [aslanyan, castellanos, 2006] aslanyan l. and castellanos j., logic based pattern recognition – ontology content (1), itech-06, 20-25 june 2006, varna, bulgaria, proceedings, pp. 61-66. [aslanyan, ryazanov, 2008] aslanyan l. and ryazanov v., logic based pattern recognition ontology content (2), information theories and applications, issn 1310-0513, 2008, volume 15, number 4, pp. 314-318. [aslanyan, 1975] aslanyan l. h. on a pattern recognition method based on the separation by the disjunctive normal forms. kibernetika, 5: 103-110, 1975. [aslanyan, 1979] aslanyan l. h. the discrete isoperimetry problem and related extremal problems for discrete spaces, problemy kibernetiki, 36: 85-128, 1979. [aslanyan, sahakyan, 2009] l. aslanyan, h. sahakyan, chain split and computations in practical rule mining, intenational book series, information science and computing, book 8, classification, forecasting, data mining, pp. 132-135, 2009. [aslanyan, sahakyan, 2010] l. aslanyan, h. sahakyan, composite block optimized classification data structures, classification, forecasting, data mining conference, varna, 2224 june, 2010. d:\sbornik\...\chubaryan4.dvi mathematical problems of computer science 33, 66{72, 2010. on the n umber of i r r educible linear ised cover ings for subsets in finite fields h o vh a n n e s k . n u r ija n ya n department of applied mathematics and informatics, yerevan state university hovikn@gmail.com abstract we present lower and upper bounds for the number of irreducible linearised coverings of subsets in a ¯nite ¯eld with characteristic greater than 2. in case of ¯nite ¯eld with characteristic equal to 2, these bounds are obtained by alexanian. refer ences [1 ] â. ãàáðèåëÿí, “î ìåòðè÷åñêèõ õàðàêòåðèñòèêàõ, ñâÿçàííûõ ñ ïîêðûòèÿìè ïîäìíîæåñòâ êîíå÷íûõ ïîëåé ñìåæíûìè êëàññàìè ëèíåéíûõ ïîäïðîñòðàíñòâ”, èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè, åðåâàí, 2004. [2] à. àëåêñàíÿí, äèçúþíêòèâíûå íîðìàëüíûå ôîðìû íàä ëèíåéíûìè ôóíêöèÿìè (òåîðèÿ è ïðèëîæåíèÿ), åðåâàíñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò 1990. ì»ñç³íáñ ¹³ßï»ñç »ýã³µ³½ùáõãûáõýý»ñç ·í³ûý³óíáõ ÷³ïáõõ³ûçý í³íïáõûãý»ñç ù³ý³ïç ù³ëçý ð. üáõñçç³ýû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ý»ñï³û³óí³í ¿ 2-çó ù»í µýáõã³·ñçãáí í»ñç³íáñ ¹³ßïç »ýã³µ³½ùáõãûáõýý»ñç ÷³ïáõõ³ûçý ·í³ûý³óíáõ í³íïáõûãý»ñç ù³ý³ïç í»ñçý ¨ ëïáñçý ë³ñù³ýý»ñá: 2 µýáõã³·ñçã áõý»óáõ í»ñç³íáñ ¹³ßïç ¹»åùáõù, ³û¹ í»ñçý ¨ ëïáñçý ë³ñù³ýý»ñá ëï³óí³í »ý ²é»ùë³ýû³ýç ïáõùçó: 6 6 начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 44, 59--66, 2015. video-based automated system for pavement surface quality monitoring gurgen h. hakobyan national polytechnic university of armenia e-mail: hakobyan.gurgen@yahoo.com abstract nowadays, an important task is to evaluate the pavement condition, in order to perform necessary works for keeping the road safe. performing such kind of surveys and calculations on the roads only by humans is ineffective not only concerning time but it also requires more costs and may be unsafe for pavement condition inspectors. in this paper we introduce an automated monitoring system, which can classify cracks and artifacts from pavement video images and illustrate the processed images results during image acquisition. firstly, a threshold value is counted using the image histogram. then, image processing operations are applied, namely binarization and segmentation on each frame. afterwards, the image is filtered from small segments which don't contain crack information. finally, the defect percent of each frame is counted and displayed on the graph real-time, where areas with cracks are distinguishable for pavement management operators. this information is helpful for pavement condition rating. experimental data were from real road images in armenia. keywords: pavement, crack, automated quality monitoring, binarization, segmentation. 1. introduction in pavement management there is always a need of surface quality assessment. it requires image acquisition, image processing, distress evaluation. before 1980, all these inspections were accomplished manually [1]. doing these kinds of surveys and calculations on the roads only by humans is ineffective not only concerning time but it also requires more costs and may be unsafe for pavement condition inspectors. also this method of field inspection poses several drawbacks, such as: slow, labour intensive and expensive, subjective approach generating inconsistencies and inaccuracies in the determination of pavement condition; inflexible and not providing an absolute measure of the surface, it has poor repeatability since the assessment of the given pavement section may differ from one survey to another, could expose a serious safety hazard to 59 video based automated system for pavement surface quality monitoring 60 the surveyors due to high speed and high volume traffic [2]. thus, over the last decades, a demand of automated digital imaging systems came out for pavement quality monitoring. nowadays, this task can be automated with noninvasive techniques to be more comfortable, less dangerous for employees and users of the road but also more efficient and less expensive than manual methods [1]. also with an automated image processing system, processing of survey images may be done more accurately and objectively. pavement quality is characterized by many factors, including the surface deflection from regulated standards as well as different types of defects, caused by regular pavement exploitation. crack is the most common distress type of asphalt pavement, which is classified in 4 main types: longitudinal (linear), block, transverse, alligator. types of cracks are shown in figure 1. crack has three characteristics: darker than its surrounding (because of crack form, many light rays cannot be reflected from crack to camera); continuous (crack can be thinner than aggregates but it is always longer than them); dominant orientation (on all length of crack or on each segment) [3]. longitudinal transverse block alligator fig. 1. main types of pavement cracks. crack detection is a difficult task, because most of the time it is represented weakly on the image and depending on pavement texture, may be low contrasted. in case of low contrast, preprocessing operations are needed, such as image normalization and equalization. there are various automated systems for crack detection, assessment, classification [1-5]. a typical pavement management system consists of image data collection, verification, processing and analysis. in [3] an automatic crack detection and classification are introduced, which can detect small cracks as 1mm in width, joint and bridged types of cracks. first, pre-processing is applied to images to remove lane-marking. then an anisotropy measure is calculated to detect road defects. finally, a back-propagation neural network is used to classify the images into four classes: defect-free, crack, joint and bridged. although, in that work, the road images are labelled by human operators. in [4] a method for automated detection and assessment of potholes, cracks and patches from video clips of highways is proposed. in that method, potholes, cracks and patches are detected and quantified automatically using cddmc algorithm. several devices are also developed, which monitor condition of road surface speedily and with better quality of data. known examples are csiro’s road crack detection vehicle, and roadware’s wisecrax, crack detection system. unfortunately, by the commercial nature of these systems, information on their algorithms of image processing for the crack detection is limited [6]. as described above, the main difficulty of effective detection of abnormal pavement areas (in particular cracks) is the variability of image characteristics (such as brightness, contrast, etc.). so, any method of automated pavement abnormal area detection should have enough flexible algorithms, which will be adaptable for concrete conditions under which the monitoring is executed. previously, we developed an image morphological analysis technique [7], which allows to classify cracks and artifacts on each frame by using special procedures. g. hakobyan 61 in this paper we propose an automated pavement quality monitoring system, by using statistical analysis methods on pavement image segments. with this system pavement management operators will be able to easily detect distressed areas real-time and rate the pavement quality. 2. image processing procedure for pavement quality assessment the main ideology for pavement quality assessment, by means of photo or video processing, relies on researches of morphological properties of processed image. this kind of processing includes known algorithms of binarization, segmentation, edge detection, etc. fig. 2. work concept of automated system and image acquisition vehicle. figure 2 illustrates a short scheme which includes the main steps of proposed procedure for pavement quality assessment. the proposed monitoring method is based on the following operations: image acquisition, threshold selection, binarization, segmentation, filtration, crack detection, pavement quality assessment. short descriptions of the mentioned procedures are given below. image acquisition. images are collected from the roads in armenia with size 320x240. the camera is mounted on the special car (as it is shown in fig. 2) which contains a mounted camera on it and a powerful light illumination which will make image pre-processing operations, such as normalization, equalization, etc., easier. depending on driving speed the frequency of shots is counted the way that all images together cover the whole canvas of pavement. in order to apply our image processing methods each frame is converted into 8 bit grayscale image, processed live and also stored in the database. the captured data is sent to image processing software, which is described in the next sections. threshold selection. usually, cracks and potholes in pavement images are characterized by dark intensity values, which are also noticeable from histograms. in order to detect them we use image processing methods such as binarization and segmentation. for image binarization we image acquisition threshold selection binarization and segmentation crack detection pavement quality assessment segments filtration video based automated system for pavement surface quality monitoring 62 need to find an optimal threshold value, for extraction of foreground in the image. an example of pavement video images with and without a crack and their histograms are shown in figure 3. as it is visible on histograms (c, d), pavement images contain only one mode, which makes it almost impossible to separate cracks from background using the known threshold selection methods (such as otsu method [8]). a b c d fig. 3. images with and without crack and their histogram. there is a small difference between histograms with and without cracks, but it's not possible to definitely detect cracks just by histogram difference, especially during automated image processing. in this paper, we count threshold using bα generalized parameter, which defines the portion of "black" pixels inside the image pixels array after binarization. by using the bα parameter we preliminarily fixate the array of pixels, which are connected with each other and form all the "black" segments of the image. depending on pavement structure those segments will be classified as cracks or artifacts. binarization, segmentation and filtration. following [7], on each frame using its bt threshold, we apply binarization. the bt threshold is counted according to bα value from pixels intensity histogram of the relevant image. then full segmentation is applied to binary image. a segment is considered an array of "black" pixels each of which has at least one neighbor pixel from the same array. notice that in the end the problem of segments classification boils down into search of threshold value of segments sizes st , which will split the segments into cracks and artifacts. segments which have less pixels than st in total characterize the pavement surface which doesn't contain cracks, the rest characterize cracks on the pavement. threshold value st may be counted in a couple of ways depending on various conditions. in the paper the simple method of "three-sigma limits" is used, i.e., 𝑇𝑇𝑠𝑠 = �̂�𝜇𝑏𝑏 + 3𝜎𝜎�𝑏𝑏 formula is applied, where �̂�𝜇𝑏𝑏 and 𝜎𝜎�𝑏𝑏 are values of mean and standard deviation of segment sizes. after counting st , the segments smaller than that value are removed from the image. crack detection and pavement quality assessment. after these operations we have all the necessary information for counting crack area inside each frame (d%) ,100*% n segkd = g. hakobyan 63 where segk is the total count of "black" pixels of all segments after filtration, n count of all pixels of the image. after applying this processing on all images of video sequence and adding d% to the chart, we can monitor and distinguish areas which contain cracks. a. developed software an image processing software is developed based on binarization and segmentation algorithms [9]. it can be used for crack detection and assessment from static images [7] as well as for processing video stream images of pavement and its quality monitoring. currently, the following operations are available: extract pixels matrix from images, build image from pixels matrix, automated threshold selection based on selected α value, binarization, segmentation, segments filtration based on minimum and maximum input values of segments, results exportation(.xls(x), .csv) as well as video stream images processing and building live chart which shows defect percent of each frame. all processing operations require 8 bit grayscale images. interface of the software is shown on figure 4. video stream processing example results are shown on the next section. during video images processing, the software is able to send each result data chunk to remote server. it can be useful if there is a need for a remote data examination or it is required to see the results from more than one location. the image processing may be done in the following ways:  static image processing. if bα is defined, binarization threshold and segments min-length for filtration (described above) are counted automatically. it's also possible to input all those values manually. software will output detected crack, total count of segments and defect percent.  video stream processing. takes input parameter bα , outputs each frame detected crack image, defect percent and builds live chart (figure 5) and stores results in the database. for more accuracy it's also possible to change the factor value 𝜎𝜎�𝑏𝑏. the software was developed using java programming language. fig. 4. software state during processing. video based automated system for pavement surface quality monitoring 64 3. experimental results for illustration of achieved results a chart is shown in figure 5, which represents d% values of 128 images from the same video sequence. the chart is being built on-time automatically during image acquisition and processing. as you can see from the chart, d% takes smaller values on the first images and high values on the last images. those values represent small and big cracks, respectively. depending on intensity variations on some of the shots there can be several jumps on the charts, which can be ignored. fig. 5. pavement monitoring chart. the corresponding image examples are shown in table 1 (row 3). on the first row of the table frame numbers are shown. on the second row the distances from the beginning of image acquisition are shown. images after all steps of processing described above (binarization, segmentation, small segments filtration) and defect percent values of each frame are shown on the fourth and fifth rows, respectively. table 1. processed data during monitoring. frame n 1 20 40 60 82 90 100 110 120 distance (m) 0 2.5 5 7.5 10.25 11.25 12.5 13.5 15 image binarization, segmentation , filtration d(%) 0.03 0.2 0.02 0.04 0.2 0.9 1 0.7 0.3 g. hakobyan 65 4. conclusion proposed automated system is based on image processing algorithms such as binarization and segmentation. it can be used in pavement management for surface condition monitoring. on real experimental data it is shown that by using this system it will be easy for pavement management operators to distinguish areas which contain cracks or other type of pavement distresses. references [1] s. chambon and j. m. molinard, “automatic road pavement assessment with image processing: review and comparison", international journal of geophysics, vol. 2011, article id 989354 [2] m. mustaffar, t. c. ling and o. c. puan, “automated pavement imaging 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[4] l. huidrom, l. k. das and s. k. sud, “method for automated assessment of potholes, cracks and patches from road surface video clips”, 2nd conference of transportation research group of india, pp. 312-321, 2013. [5] l. li, l. sun, g. ning and s. tan, “automatic pavement crack recognition based on bp neural network, swarm intelligence”, promet – traffic&transportation, vol. 26, pp. 1122, 2014 [6] t. sy nguyen, m.avila, s. begot and j.-c. bardet, “detection of defects in road surface by a vision system”, electrotechnical conference, the 14th ieee mediterranean, ajaccio, france, pp.847 – 851, 2008. [7] д. г. асатрян и г. o. акопян, “методика морфологического анализа изображения и оценивания качества дорожного покрытия”, в е с т н и к российско-армянского (славянского) университета, №1, pp. 36-44, 2015. [8] n. otsu, “a threshold selection method from gray-level histograms”, ieee trans. syst. mancybern, vol.9, pp. 62–66, 1979. [9] d. g asatryan, g. s. sazhumyan and h. s. shahverdyan, “technique for coherent segmentation of image and applications”, transactions of iiap of nas ra, mathematical problems of computer science, vol. 28, pp. 88–93, 2007. submitted 04.08.2015, accepted 24.11.2015 video based automated system for pavement surface quality monitoring 66 տեսանկարահանման վրա հիմնված ավտոմատացված համակարգ՝ ճանապարհային ծածկույթի որակի հսկում իրականացնելու համար գ. հակոբյան ամփոփում մեր օրերում կարևոր խնդիր է ճանապարհային ծածկույթի որակի գնահատումը՝ ճանապարհների անվտանգությունը և արդյունավետությունը ապահովելու նպատակով: նման աշխատանքների իրականացումը առանց համապատասխան տեխնիկական միջոցների ժամանակատար է, պահանջում է մեծ ծախսեր և վտանգավոր է չափումներ իրականացնող անձնակազմի համար: հոդվածում առաջարկվում է ավտոմատացված համակարգ, որը հնարավորություն է տալիս տեսանկարահանման օգնությամբ ճանապարհային ծածկույթի պատկերներից հայտնաբերել ծածկույթի արատները և համապատասխան մշակման միջոցով գնահատել ծածկույթի որակը: արդյունքները արտապատկերվում են համակարգում տեղակայված համակարգչի էկրանի վրա: автоматизированная система контроля качества дорожного покрытия на основе видеообработки г. акопян аннотация в наши дни, для обеспечения качества и эффективности дорожного покрытия, важным вопросом является оценка состояния поверхности покрытия. такого рода исследования без соответствующих технических средств требуют не только больше времени, но и больших затрат и могут быть опасными для специалистов, которые проводят исследования. в статье предлагается автоматизированная система контроля качества, которая дает возможность с помощью соответствующей обработки выявлять дефекты дорожного покрытия и оценить качество покрытия. результаты отображаются на экране компьютера, который вмонтирован в предлагаемую автоматизированную систему. microsoft word title.doc mathematical problems of computer science 35, 128--136, 2011. 128 tools development of ring computing network structure improvement hayk b. avsharyan and ruben g. kirakossian yerevan state university abstract the current state of computer networks is discussed. the need to create the tools which will improve the structure of the network is justified. a mathematical model proposed for the improvement of the network structure. on the basis of the proposed model an algorithm which improves the structure of the ring computer network was developed. references [1] д. абрагин, “телекоммуникационные сети нового поколения,” первая миля, №2, сс. 28 – 31, 2009. [2] и. кривошеев, “к развитию глобального информационного общества,” телецентр, №3 (11), сс. 55 – 63, 2005. [3] а. а чернов, “становление глобального информационного общества: проблемы и перспективы,” дашков и ко, 232 p., 2004. [4] h. ab. m. hatamleh, “structural optimization of computer networks with the technology of mpls with restrictions on the quality of service,” european journal of scientific research, pp. 597-601, 2009. [5] a. a. назаров, а. н. туенбаева, “исследование марковской модели компьютерной сети с нестационарным входящим потоком,” вестник томского государственного университета, № 16, сс. 87-93, 2006. [6] հ.բ. ավշարյան, “հաշվողական համակարգերի կառուցվածքային լավարկումը նմանակային մոդելավորմամբ,” հայաստանի ճարտարագիտական ակադեմիայի լրաբեր, հատոր 7, համար 1, էջ 153-157, 2010. [7] г. б. авшарян, р. г. киракосян “метод проектирования компьютерной сети,” известия нан ра и гиуа, сер. th. 2010. t. lxiii, №4, сс 296 – 301, 2010. [8] р.м. ларин, а.в. плясунов, а.в. пяткин, методы оптимизации. примеры и задачи, нгу, 120 с, 2003. h. avsharyan, r.kirakossian 129 úõ³ï³ó¨ ñ³ßíáõ³ï³ý ó³ýóç ï³éáõóí³íùç µ³ñ»é³íù³ý ·áñíçù³ûçý ùççáóý»ñç ùß³ïáõù ð. ²íß³ñû³ý ¨ è. îçñ³ïáëû³ý ²ù÷á÷áõù øýý³ñïíáõù ¿ ñ³ßíáõ³ï³ý ó³ýó»ñç ý»ñï³ íç×³ïá: ðçùý³íáñíáõù ¿ ñ³ßíáõ³ï³ý ó³ýó»ñç ï³éáõóí³íùá µ³ñ»é³íáõ ·áñíçù³ûçý ùççáóý»ñ ëï»õí»éáõ ³ýññ³å»ßïáõãûáõýá: ²é³ç³ñïíáõù ¿ ûõ³ï³ó¨ ñ³ßíáõ³ï³ý ó³ýóç ï³éáõóí³íùç µ³ñ»é³íù³ý ù³ã»ù³ïçï³ï³ý ùá¹»é: ²é³ç³ñïíáõ ù³ã»ù³ïçï³ï³ý ùá¹»éç ñçù³ý íñ³ ùß³ïí»é ¿ ûõ³ï³ó¨ ñ³ßíáõ³ï³ý ó³ýóç ï³éáõóí³íùç µ³ñ»é³íù³ý ³é·áñçãù: d:\sbornik_45_tex\...\nik.dvi mathematical problems of computer science 45, 19{26, 2016. on some ver sions of conjectur es of b ondy and jung zh o r a g. n iko g h o s ya n ¤ institute for informatics and automation problems of nas ra e-mail: zhora@ipia.sci.am abstract most known fundamental theorems in hamiltonian graph theory (due to dirac, ore, nash-williams, bondy, jung and so on) are related to the length of a longest cycle c in a graph g in terms of connectivity · and the length p of a longest path in g ¡ c, for the special cases when · · 3 and p · 1 (if p = ¡1 then v (g ¡ c) = ; and c is called hamiltonian; and if p = 0 then v (g ¡ c) is an independent set of vertices and c is called dominating). bondy (1980) and jung (2001) conjectured a common generalization of these results in terms of degree sums including p and · as parameters. these conjectures still are open in the original form. in 2009, the minimum degree c¡versions (c the length of a longest cycle in v (g ¡ c)) of conjectures of bondy and jung are shown to be true by the author (discrete math, v.309, 2009, 19251930). in this paper, using another result of the author (graphs and combinatorics, v.29, 2013, 1531-1541), a number of analogous sharp results are presented including both p and c¡minimum degree versions of conjectures of bondy and jung without connectivity conditions, inspiring a number of new strengthened and extended versions of conjectures of bondy and jung. keywords: hamilton cycle, dominating cycle, long cycles, bondy's conjecture, jung's conjecture. 1 . in t r o d u c t io n th e g e n e r a liz e d c o n je c t u r e s o f b o n d y [1 ] ( 1 9 8 0 ) a n d ju n g [2 ] ( 2 0 0 1 ) in c lu d e a n u m b e r o f m o s t kn o wn fu n d a m e n t a l r e s u lt s ( t h e m in im u m d e g r e e a n d t h e d e g r e e s u m ve r s io n s ) in h a m ilt o n ia n g r a p h t h e o r y c o n c e r n in g h a m ilt o n a n d d o m in a t in g c yc le s a s s p e c ia l c a s e s d u e t o d ir a c [3 ], or e [4 ], n a s h -w illia m s [5 ], b o n d y [6 ],[1 ] ju n g [7 ],[8 ],[9 ] a n d s o o n . th e s e c o n je c t u r e s s t ill a r e o p e n in t h e o r ig in a l fo r m . u s in g s o m e e a r lie r r e s u lt s o f t h e a u t h o r [1 0 ], [1 1 ] ( 2 0 0 9 , 2 0 1 3 ) , in t h is p a p e r s o m e ve r s io n s o f c o n je c t u r e s o f b o n d y a n d ju n g a r e s h o wn t o b e t r u e , in s p ir in g a n u m b e r o f n e w s t r e n g t h e n e d a n d e xt e n d e d ve r s io n s o f t h e s e c o n je c t u r e s . a ll r e s u lt s a r e s h a r p . th r o u g h o u t t h is a r t ic le we c o n s id e r o n ly ¯ n it e u n d ir e c t e d g r a p h s wit h o u t lo o p s o r m u lt ip le e d g e s . a g o o d r e fe r e n c e fo r a n y u n d e ¯ n e d t e r m s is [1 2 ]. th e s e t o f e d g e s o f a g r a p h g is d e n o t e d b y e ( g ) . if q is a p a t h o r a c yc le , t h e n t h e le n g t h o f q, d e n o t e d b y jqj, is je ( q) j. e a c h ve r t e x a n d e d g e in g c a n b e in t e r p r e t e d a s s im p le c yc le s o f le n g t h s 1 a n d 2 , r e s p e c t ive ly. fo r a lo n g e s t c yc le c in g, le t p a n d c d e n o t e t h e le n g t h s o f a lo n g e s t p a t h a n d a lo n g e s t c yc le in gnc, r e s p e c t ive ly. p u t p = ¡ 1 wh e n 1 9 2 0 on some versions of conjectures of bondy and jung v ( gnc ) = ;. if e it h e r p = ¡ 1 o r c = 0 , t h e n c is c a lle d a h a m ilt o n c yc le . n e xt , if e it h e r p = 0 o r c = 1 , t h e n c is c a lle d a d o m in a t in g c yc le . fu r t h e r , if c · ¸ ¡ 1 fo r a n in t e g e r ¸, t h e n c is c a lle d a cd¸ ( c yc le d o m in a t in g ) -c yc le . in p a r t ic u la r , cd1-c yc le s a r e h a m ilt o n c yc le s a n d cd2-c yc le s a r e d o m in a t in g c yc le s . l e t g b e a g r a p h o f o r d e r n a n d m in im u m d e g r e e ±. th e d e g r e e s u m o f s s m a lle s t d e g r e e s a m o n g s p a ir wis e n o n a d ja c e n t ve r t ic e s will b e d e n o t e d b y ¾s. in 1 9 8 0 , b o n d y [1 ] c o n je c t u r e d a c o m m o n g e n e r a liz a t io n o f we ll-kn o wn t h e o r e m s o f or e [4 ] ( 1 9 6 0 , ¸ = 1 ) a n d b o n d y [1 ] ( 1 9 8 0 , ¸ = 2 ) in t e r m s o f p. conjectur e a: [1 ]. l et g be a ¸-connected ( ¸ ¸ 1 ) graph and c a longest cycle in g. if 1 ¸ + 1 ¾¸+1 ¸ n + 2 ¸ + 1 + ¸ ¡ 2 ; then p · ¸ ¡ 2 . fo r ¸ = 3 , co n je c t u r e a h a s b e e n ve r ī e d in 1 9 8 7 b y zo u [1 3 ]. th e m in im u m d e g r e e ve r s io n o f co n je c t u r e a c o n t a in s t wo fu n d a m e n t a l t h e o r e m s o n t h is s u b je c t d u e t o d ir a c [3 ] ( 1 9 5 2 , ¸ = 1 ) a n d n a s h -w illia m s [5 ] ( 1 9 7 1 , ¸ = 2 ) a s s p e c ia l c a s e s . conjectur e b : [1 ]. l et g be a ¸-connected ( ¸ ¸ 1 ) graph and c a longest cycle in g. if ± ¸ n + 2 ¸ + 1 + ¸ ¡ 2 ; then p · ¸ ¡ 2 . fo r ¸ = 3 , co n je c t u r e b h a s b e e n ve r ī e d in 1 9 8 1 b y ju n g [9 ]. in 2 0 0 9 , t h e a u t h o r p r o ve d [1 0 ] t h a t e a c h lo n g e s t c yc le in g u n d e r t h e c o n d it io n o f co n je c t u r e b , is a cdm in f¸;±¡¸+1g-c yc le . t heor em a: [1 0 ]. l et g be a ¸-connected ( ¸ ¸ 1 ) graph and c a longest cycle in g. if ± ¸ n + 2 ¸ + 1 + ¸ ¡ 2 ; then c is a cdm in f¸;±¡¸+1g-cycle. th e o r e m a is s h o wn in [1 0 ] t o b e b e s t p o s s ib le . ob s e r vin g t h a t b e in g a cd¸-c yc le im p lie s c · ¸ ¡ 1 ( b y t h e d e ¯ n it io n ) , th e o r e m a im p lie s t h e fo llo win g r e s u lt . cor ollar y a: l et g be a ¸-connected ( ¸ ¸ 1 ) graph and c a longest cycle in g. if ± ¸ n + 2 ¸ + 1 + ¸ ¡ 2 ; then c · m in f¸ ¡ 1 ; ± ¡ ¸g. th u s , t h e m in im u m d e g r e e c-ve r s io n o f b o n d y's c o n je c t u r e is t r u e wit h s o m e s t r e n g t h e n in g . th e t r a n s it io n fr o m m in im u m d e g r e e c-ve r s io n o f b o n d y's c o n je c t u r e t o d e g r e e s u m zh. nikoghosyan 2 1 p-ve r s io n ( t h a t is t h e s o lu t io n o f b o n d y's c o n je c t u r e ) n o w c a n b e c o n s id e r e d a s a t e c h n ic a l p r o b le m . s in c e th e o r e m a a n d co r o lla r y a a r e e qu iva le n t , we c a n s t a t e t h a t co r o lla r y a is s h a r p a s we ll. in t h is p a p e r we p r e s e n t t wo a n a lo g o u s s h a r p r e s u lt s wit h o u t c o n n e c t ivit y c o n d it io n s c o n c e r n in g b o t h p a n d c-ve r s io n s . t heor em 1: l et g be a graph and c a longest cycle in g. if ± ¸ n + 2 ¸ + 1 + ¸ ¡ 2 ; for some positive integer ¸, then either p · m in f¸ ¡ 2 ; ± ¡ ¸ ¡ 1 g or p ¸ m a xf¸; ± ¡ ¸ + 1 g. t heor em 2: l et g be a graph and c a longest cycle in g. if ± ¸ n + 2 ¸ + 1 + ¸ ¡ 2 ; for some positive integer ¸, then either c · m in f¸ ¡ 1 ; ± ¡ ¸g or c ¸ m a xf¸ + 1 ; ± ¡ ¸ + 2 g. w e s h a ll s h o w t h a t th e o r e m 1 a n d th e o r e m 2 a r e s h a r p . p u t g1 = ( ¸ + 2 ) k¸¡1 + k¸+1. s in c e ± = 2 ¸ ¡ 1 , p = ¸ ¡ 2 = ± ¡ ¸ ¡ 1 a n d c = ¸ ¡ 1 = ± ¡ ¸, t h e g r a p h g1 s h o ws t h a t t h e b o u n d s ¸ ¡ 2 a n d ± ¡ ¸ ¡ 1 in th e o r e m 1 , a n d t h e b o u n d s ¸ ¡ 1 , ± ¡ ¸ in th e o r e m 2 , a r e s h a r p . n o w p u t g2 = ¸k¸+1 + k¸¡1. s in c e ± = 2 ¸ ¡ 1 , p = ¸ = ± ¡ ¸ + 1 a n d c = ¸ + 1 = ± ¡ ¸ + 2 , t h e n t h e g r a p h g2 s h o ws t h a t t h e b o u n d s ¸ a n d ± ¡ ¸ + 1 in th e o r e m 1 , a n d t h e b o u n d s ¸ + 1 , ± ¡ ¸ + 2 in th e o r e m 2 a r e s h a r p a s we ll. fin a lly, le t g3 = ( ± ¡ ¸ + 2 ) k¸ + k±¡¸+1. th e n ± = ( n + 1 ) = ( ¸ + 1 ) + ¸ ¡ 2 , p = ¸ ¡ 1 a n d c = ¸, im p lyin g t h a t t h e b o u n d ( n + 2 ) =( ¸ + 1 ) + ¸ ¡ 2 in th e o r e m s 1 a n d 2 c a n n o t b e r e la xe d . in vie w o f co r o lla r y a a n d th e o r e m s 1 -2 , co n je c t u r e s a a n d b c a n b e c o n s id e r a b ly s t r e n g t h e n e d . mo r e o ve r , t h e y c a n b e n a t u r a lly e xt e n d e d o n a c c o u n t o f t h e c-ve r s io n . conjectur e 1: l et g be a ¸-connected ( ¸ ¸ 1 ) graph and c a longest cycle in g. if 1 ¸ + 1 ¾¸+1 ¸ n + 2 ¸ + 1 + ¸ ¡ 2 ; then p · m in n ¸ ¡ 2 ; 1 ¸ + 1 ¾¸+1 ¡ ¸ ¡ 1 o ; c · m in n ¸ ¡ 1 ; 1 ¸ + 1 ¾¸+1 ¡ ¸ o : conjectur e 2: l et g be a ¸-connected ( ¸ ¸ 1 ) graph and c a longest cycle in g. if ± ¸ n + 2 ¸ + 1 + ¸ ¡ 2 ; then p · m in f¸ ¡ 2 ; ± ¡ ¸ ¡ 1 g. n o w we t u r n t o t h e lo n g c yc le ve r s io n s o f co r o lla r y a a n d th e o r e m s 1 -2 . 2 2 on some versions of conjectures of bondy and jung in 2 0 0 1 , ju n g [2 ] c o n je c t u r e d a c o m m o n g e n e r a liz a t io n o f t wo fu n d a m e n t a l t h e o r e m s in h a m ilt o n ia n g r a p h t h e o r y d u e t o d ir a c [3 ] ( 1 9 5 2 , ¸ = 2 ) a n d ju n g [8 ] ( 1 9 7 8 , ¸ = 3 ) . conjectur e c: [2 ] l et g be a ¸-connected ( ¸ ¸ 1 ) graph and c a longest cycle in g. if p ¸ ¸ ¡ 2 , then jcj ¸ (̧ ± ¡ ¸ + 2 ) . th e d e g r e e s u m ve r s io n o f co n je c t u r e c c o n t a in in g t h e t h e o r e m s o f b o n d y [6 ] ( 1 9 7 1 , ¸ = 2 ) , b e r m o n d [1 4 ] ( 1 9 7 6 , ¸ = 2 ) , l in ia l [1 5 ] ( 1 9 7 6 , ¸ = 2 ) , fr a is s e a n d ju n g [7 ] ( 1 9 8 9 , ¸ = 3 ) a s s p e c ia l c a s e s c a n b e fo r m u la t e d a s fo llo ws . conjectur e 3: l et g be a ¸-connected ( ¸ ¸ 1 ) graph and c a longest cycle in g. if p ¸ ¸ ¡ 2 , then jcj ¸ ¸ ³ 1 ¸ ¾¸ ¡ ¸ + 2 ´ : in 2 0 0 9 , t h e a u t h o r p r o ve d [1 0 ] t h e fo llo win g . t heor em b : [1 0 ]. l et g be a ( ¸ + 1 ) -connected ( ¸ ¸ 0 ) graph and c a longest cycle in g. then either jcj ¸ ( ¸ + 1 ) ( ± ¡ ¸ + 1 ) or c is a cdm in f¸;±¡¸g-cycle. th e o r e m b is s h o wn in [1 0 ] t o b e b e s t p o s s ib le a n d c le a r ly is e qu iva le n t t o t h e fo llo win g . t heor em c: [1 0 ]. l et g be a ¸-connected ( ¸ ¸ 1 ) graph and c a longest cycle in g. then either jcj ¸ ¸ ( ± ¡ ¸ + 2 ) or c is a cdm in f¸¡1;±¡¸+1g-cycle. u s in g t h e d e ¯ n it io n o f cd¸-c yc le s , we g e t t h e fo llo win g r e s u lt . cor ollar y b : l et g be a ¸-connected ( ¸ ¸ 1 ) graph and c a longest cycle in g. if c ¸ m in f¸ ¡ 1 ; ± ¡ ¸ + 1 g then jcj ¸ (̧ ± ¡ ¸ + 2 ) . th u s , c-ve r s io n o f ju n g 's c o n je c t u r e is t r u e wit h s o m e s t r e n g t h e n in g . th e t r a n s it io n fr o m m in im u m d e g r e e c-ve r s io n o f ju n g 's c o n je c t u r e t o d e g r e e s u m p-ve r s io n ( t h a t is t h e s o lu t io n o f ju n g 's c o n je c t u r e ) is a t e c h n ic a l p r o b le m . in t h is p a p e r we p r e s e n t t wo a n a lo g o u s s h a r p r e s u lt s wit h o u t c o n n e c t ivit y c o n d it io n s . t heor em 3: l et g be a graph and c a longest cycle in g. if m in f¸ ¡ 2 ; ± ¡ ¸g · p · m a xf¸ ¡ 2 ; ± ¡ ¸g; for some positive integer ¸, then jcj ¸ (̧ ± ¡ ¸ + 2 ) . t heor em 4: l et g be a graph and c a longest cycle in g. if m in f¸ ¡ 1 ; ± ¡ ¸ + 1 g · c · m a xf¸ ¡ 1 ; ± ¡ ¸ + 1 g; for some positive integer ¸, then jcj ¸ (̧ ± ¡ ¸ + 2 ) . zh. nikoghosyan 2 3 to s h o w t h a t t h e c o n d it io n s in th e o r e m s 3 a n d 4 c a n n o t b e r e la xe d , a s s u m e ¯ r s t t h a t m in f¸ ¡ 2 ; ± ¡ ¸g = ¸ ¡ 2 , t h a t is ¸ ¡ 2 · ± ¡ ¸. p u t h1 = ( ± ¡ ¸ + 4 ) k¸¡2 + k±¡¸+3. cle a r ly p = ¸ ¡ 3 , c = ¸ ¡ 2 a n d jcj = ( ¸ ¡ 1 ) ( ± ¡ ¸ + 3 ) . r e c a llin g t h a t ¸ ¡ 2 · ± ¡ ¸, we g e t ( ¸ ¡ 1 ) ( ± ¡ ¸ + 3 ) = (̧ ± ¡ ¸ + 2 ) + ( 2 ¸ ¡ ± ¡ 2 ) ¡ 1 < (̧ ± ¡ ¸ + 2 ) ; im p lyin g t h a t t h e b o u n d s ¸ ¡ 2 a n d ¸ ¡ 1 in th e o r e m s 3 a n d 4 c a n n o t b e r e la xe d . n o w a s s u m e t h a t m in f¸ ¡ 2 ; ± ¡ ¸g = ± ¡ ¸, t h a t is ¸ ¡ 2 ¸ ± ¡ ¸. p u t h2 = ( ¸ + 2 ) k±¡¸ + k¸+1. cle a r ly p = ± ¡ ¸ ¡ 1 , c = ± ¡ ¸ a n d jcj = ( ¸ + 1 ) ( ± ¡ ¸ + 1 ) . s in c e ¸ ¡ 2 ¸ ± ¡ ¸, we h a ve ( ¸ + 1 ) ( ± ¡ ¸ + 1 ) < (̧ ± ¡ ¸ + 2 ) , im p lyin g t h a t t h e b o u n d s ± ¡ ¸ a n d ± ¡ ¸ + 1 in th e o r e m s 3 a n d 4 c a n n o t b e r e la xe d a s we ll. in vie w o f co r o lla r y b a n d th e o r e m s 3 -4 , co n je c t u r e c a n d co n je c t u r e 3 c a n b e c o n s id e r a b ly s t r e n g t h e n e d . mo r e o ve r , t h e y c a n b e n a t u r a lly e xt e n d e d o n a c c o u n t o f t h e c-ve r s io n . conjectur e 4: l et g be a ¸-connected ( ¸ ¸ 1 ) graph and c a longest cycle in g. if either p ¸ m in n ¸ ¡ 2 ; 1 ¸ ¾¸ ¡ ¸ o or c ¸ m in n ¸ ¡ 1 ; 1 ¸ ¾¸ ¡ ¸ + 1 o ; then jcj ¸ ¸ ³ 1 ¸ ¾¸ ¡ ¸ + 2 ´ : conjectur e 5: l et g be a ¸-connected ( ¸ ¸ 1 ) graph and c a longest cycle in g. if p ¸ m in f¸ ¡ 2 ; ± ¡ ¸g, then jcj ¸ (̧ ± ¡ ¸ + 2 ) . to p r o ve th e o r e m s 1 -4 , we n e e d t h e fo llo win g t wo t h e o r e m s [1 1 ] b y t h e a u t h o r . t heor em d: [1 1 ] ( 2 0 1 3 ) . l et g be a graph and c a longest cycle in g. then jcj ¸ ( p + 2 ) ( ± ¡ p ) . t heor em e : [1 1 ] ( 2 0 1 3 ) . l et g be a graph and c a longest cycle in g. then jcj ¸ ( c + 1 ) ( ± ¡ c + 1 ) . 2 . p r o o fs p r oof of t heor em 1. b y t h e h yp o t h e s is , n · ( ¸ + 1 ) ( ± ¡ ¸ + 2 ) ¡ 2 . on t h e o t h e r h a n d , we h a ve n ¸ jcj + p + 1 . s in c e jcj ¸ ( p + 2 ) ( ± ¡ p ) ( b y th e o r e m d ) , we h a ve n ¸ ( p + 2 ) ( ± ¡ p) + p + 1 = ( p + 2 ) ( ± ¡ p + 1 ) ¡ 1 : th u s ( ¸ + 1 ) ( ± ¡ ¸ + 2 ) ¸ ( p + 2 ) ( ± ¡ p + 1 ) + 1 ; wh ic h is e qu iva le n t t o ( ¸ ¡ p ¡ 1 ) ( ± ¡ ¸ ¡ p ) ¸ 1 : th e n we h a ve e it h e r ¸ ¡ p ¡ 1 ¸ 1 a n d ± ¡ ¸ ¡ p ¸ 1 ; 2 4 on some versions of conjectures of bondy and jung wh ic h is e qu iva le n t t o p · m in f¸ ¡ 2 ; ± ¡ ¸ ¡ 1 g, o r ¸ ¡ p ¡ 1 · ¡ 1 a n d ± ¡ ¸ ¡ p · ¡ 1 ; wh ic h is e qu iva le n t t o p ¸ m a xf¸; ± ¡ ¸ + 1 g. p r oof of t heor em 2. b y t h e h yp o t h e s is , n · ( ¸ + 1 ) ( ± ¡ ¸ + 2 ) ¡ 2 . on t h e o t h e r h a n d , we h a ve n ¸ jcj + c. s in c e jcj ¸ ( c + 1 ) ( ± ¡ c + 1 ) ( b y th e o r e m e ) , we h a ve n ¸ ( c + 1 ) ( ± ¡ c + 1 ) + c = ( c + 1 ) ( ± ¡ c + 2 ) ¡ 1 ; im p lyin g t h a t ( ¸ + 1 ) ( ± ¡ ¸ + 2 ) ¸ ( c + 1 ) ( ± ¡ c + 2 ) + 1 ; th is is e qu iva le n t t o ( ¸ ¡ c) ( ± ¡ ¸ ¡ c + 1 ) ¸ 1 : th e n we h a ve e it h e r ¸ ¡ c ¸ 1 a n d ± ¡ ¸ ¡ c + 1 ¸ 1 ; wh ic h is e qu iva le n t t o c · m in f¸ ¡ 1 ; ± ¡ ¸g, o r ¸ ¡ c · ¡ 1 a n d ± ¡ ¸ ¡ c + 1 · ¡ 1 ; wh ic h is e qu iva le n t t o c ¸ m a xf¸ + 1 ; ± ¡ ¸ + 2 g. p r oof of t heor em 3. w e d is t in g u is h t wo c a s e s . case 1. m in f¸ ¡ 2 ; ± ¡ ¸g = ¸ ¡ 2 . b y t h e h yp o t h e s is , ¸ ¡ 2 · p · ± ¡ ¸. th e n ( p ¡ ¸ + 2 ) ( ± ¡ p ¡ )̧ ¸ 0 ; wh ic h is e qu iva le n t t o ( p + 2 ) ( ± ¡ p) ¸ (̧ ± ¡ ¸ + 2 ) : s in c e jcj ¸ ( p + 2 ) ( ± ¡ p) ( b y th e o r e m d ) , we h a ve jcj ¸ ¸ ( ± ¡ ¸ + 2 ) . case 2. m in f¸ ¡ 2 ; ± ¡ ¸g = ± ¡ ¸. b y t h e h yp o t h e s is , ± ¡ ¸ · p · ¸ ¡ 2 , im p lyin g t h a t ( p ¡ ¸ + 2 ) ( ± ¡ p ¡ )̧ ¸ 0 a n d we c a n a r g u e a s in ca s e 1 . p r oof of t heor em 4. w e d is t in g u is h t wo c a s e s . case 1. m in f¸ ¡ 1 ; ± ¡ ¸ + 1 g = ¸ ¡ 1 . b y t h e h yp o t h e s is , ¸ ¡ 1 · c · ± ¡ ¸ + 1 . th e n ( c ¡ ¸ + 1 ) ( ± ¡ c ¡ ¸ + 1 ) ¸ 0 ; wh ic h is e qu iva le n t t o ( c + 1 ) ( ± ¡ c + 1 ) ¸ ¸ ( ± ¡ ¸ + 2 ) : s in c e jcj ¸ ( c + 1 ) ( ± ¡ c + 1 ) ( b y th e o r e m e ) , we h a ve jcj ¸ (̧ ± ¡ ¸ + 2 ) . case 2. m in f¸ ¡ 1 ; ± ¡ ¸ + 1 g = ± ¡ ¸ + 1 . b y t h e h yp o t h e s is , ± ¡ ¸ + 1 · c · ¸ ¡ 1 , im p lyin g t h a t ( c ¡ ¸ + 1 ) ( ± ¡ c ¡ ¸ + 1 ) ¸ 0 a n d we c a n a r g u e a s in ca s e 1 . zh. nikoghosyan 2 5 refer ences [1 ] j. a . b o n d y, \ l o n g e s t p a t h s a n d c yc le s in g r a p h s o f h ig h d e g r e e " , r esearch r eport cor r 80-16. university of w aterloo, w a t e r lo o , on t a r io , 1 9 8 0 . 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[1 0 ] zh . g. n iko g h o s ya n , \ d ir a c -t yp e g e n e r a liz a t io n s c o n c e r n in g la r g e c yc le s in g r a p h s " , d iscrete m athematics, vo l. 3 0 9 , n o . 8 , p p . 1 9 2 5 -1 9 3 0 , 2 0 0 9 . [1 1 ] zh . g. n iko g h o s ya n , \ a d va n c e d l o we r b o u n d s fo r t h e cir c u m fe r e n c e " , graphs and combinatorics, vo l. 2 9 , n o . 5 , p p . 1 5 3 1 -1 5 4 1 , 2 0 1 3 . [1 2 ] j. a . b o n d y a n d u .s .r . mu r t y, graph theory with applications, ma c m illa n , l o n d o n a n d e ls e vie r , n e w y o r k, 1 9 7 6 . [1 3 ] y . zo u , \ a g e n e r a liz a t io n o f a t h e o r e m o f ju n g " , j . nanjing normal university natural science e dition, vo l. 2 , p p . 8 -1 1 , 1 9 8 7 . [1 4 ] j. c. b e r m o n d , \ on h a m ilt o n ia n wa lks " , congressus numerantium, vo l. 1 5 , p p . 4 1 -5 1 , 1 9 7 6 . [1 5 ] n . l in ia l, \ a lo we r b o u n d o n t h e c ir c u m fe r e n c e o f a g r a p h " , d iscrete m athematics, vo l. 1 5 , n o . 3 , p p . 2 9 7 -3 0 0 , 1 9 7 6 . submitted 12.10.2015, accepted 20.01.2016 ´áý¹çç ¨ úáõý·ç í³ñï³íý»ñç áñáß ï³ñµ»ñ³ïý»ñç ù³ëçý ä. üçïáõáëû³ý ²ù÷á÷áõù ¸çóáõù c-ý g ·ñ³ýç ³ù»ý³»ñï³ñ óçïéý ¿, ·-ý` ï³å³ïóí³íáõãû³ý µýáõã³·ñçãá, çëï p-ý` g-c-ç ³ù»ý³»ñï³ñ ßõã³ûç »ñï³ñáõãûáõýá: ð³ùçéïáýû³ý ·ñ³ýý»ñç ï»ëáõãû³ý ³é³í»é ñ³ûïýç ñçùý³ñ³ñ ³ñ¹ûáõýùý»ñá` ëï³óí³í ¸çñ³ïç, úñ»ç, 2 6 on some versions of conjectures of bondy and jung ü»ß-ìçéû³ùëç, ´áý¹çç, úáõý·ç ¨ ³ûéáó ïáõùçó, çñ»ýóçó ý»ñï³û³óýáõù »ý c óçïéç »ñï³ñáõãû³ý ·ý³ñ³ï³ï³ýý»ñ` ³ñï³ñ³ûïí³í ·ñ³ýç ·³·³ãý»ñç ýí³½³·áõûý ³ëïç×³ýáí ï³ù ³ëïç×³ý³ûçý ·áõù³ñý»ñáí ¨ ·, p µýáõã³·ñçãý»ñç ù³ëý³íáñ ³ñå»ùý»ñáí, »ñµ · · 3 ¨ p · 1 (p = ¡1 ¹»åùá ñ³ù³ñå»ù ¿ v ( g ¡ c ) = ; å³ûù³ýçý, ¨ c-ý ïáãíáõù ¿ ñ³ùçéïáýû³ý óçïé; çëï p = 0 ¹»åùáõù v ( g¡c ) -ý ·³·³ãý»ñç ³ýï³ë µ³½ùáõãûáõý ¿, ¨ c-ý ïáãíáõù ¿ ¹áùçý³ýï óçïé): ´áý¹çç (1980) ¨ úáõý·ç (2001) áý¹ñ³ýñ³óí³í í³ñï³íý»ñá ñçùýíáõù »ý ³ëïç×³ý³ûçý ·áõù³ñý»ñç íñ³, áñï»õ p -ý ¨ ·-ý ñ³ý¹»ë »ý ·³éçë áñå»ë áý¹ñ³ýñ³ï³ý å³ñ³ù»ïñ»ñ` áý¹·ñï»éáí í»ñáñçßû³é ³ñ¹ûáõýùý»ñá áñå»ë ù³ëý³íáñ ¹»åù»ñ: ²ûë í³ñï³íý»ñá áý¹ñ³ýáõñ ¹»åùáõù ùýáõù »ý ãéáõíí³í: 2009-çý ñ»õçý³ïç ïáõùçó éáõíí»é »ý ´áý¹çç ¨ úáõý·ç í³ñï³íý»ñç ýí³½³·áõûý ³ëïç×³ý³ûçý c¡ ï³ñµ»ñ³ïý»ñá, áñï»õ cý ý»ñï³û³óýáõù ¿ v ( g ¡ c ) -ç ³ù»ý³»ñï³ñ óçïéç »ñï³ñáõãûáõýá (discrete math, v.309, 2009, 1925-1930): ðçùýí»éáí ñ»õçý³ïç ù»ï ³ûé ³ßë³ï³ýùç íñ³ (graphs and combinatorics, v.29, 2013, 15311541), ý»ñï³ ³ßë³ï³ýùáõù ³å³óáõóíáõù »ý ùç ù³ýç ñ³ù³ýù³ý ³ñ¹ûáõýùý»ñ` áý¹·ñï»éáí ´áý¹çç ¨ úáõý·ç ýí³½³·áõûý ³ëïç×³ý³ûçý p ¨ c ï³ñµ»ñ³ïý»ñá ³é³ýó ï³å³ïóí³íáõãû³ý å³ûù³ýç, áñáýù çñ»ýó ñ»ñãçý íýáõù »ý ³ûë í³ñï³íý»ñç ùç ù³ýç ýáñ áõå»õ³óí³í ¨ áý¹é³ûýí³í ï³ñµ»ñ³ïý»ñ: êï³óí³í µáéáñ ³ñ¹ûáõýùý»ñá é³í³óý»éç ã»ý: î íåêîòîðûõ âåðñèÿõ ãèïîòåç áîíäè è þíãà æ. íèêîãîñÿí àííîòàöèÿ íàèáîëåå èçâåñòíûå ôóíäàìåíòàëüíûå òåîðåìû â òåîðèè ãàìèëüòîíîâîñòè ãðàôîâ (àâòîðû: äèðàê, îðå, íåø-âèëüÿìñ, áîíäè, þíã è ò.ä.) ïðåäñòàâëÿþò ðàçëè÷íûå îöåíêè äëèíû äëèííåéøåãî öèêëà c ãðàôà g â òåðìèíàõ ìèíèìàëüíîé ñòåïåíè âåðøèí èëè ñóìì ñòåïåíåé, ñâÿçíîñòè · è äëèíû p äëèííåéøåé öåïè â g ¡ c äëÿ ÷àñòíûõ ñëó÷àåâ êîãäà · · 3 è p · 1 (â ñëó÷àå p = ¡ 1 èìååò ìåñòî v ( g ¡ c ) = è c íàçûâàåòñÿ ãàìèëüòîíîâûì öèêëîì; åñëè p = 0 òî v ( g ¡ c ) ÿâëÿåòñÿ íåçàâèñèìûì ìíîæåñòâîì âåðøèí è c íàçûâàåòñÿ äîìèíàíòíûì öèêëîì). îáîáùåííûå ãèïîòåçû áîíäè (1980) è þíãà (2001) îñíîâàíû íà ñóììàõ ñòåïåíåé, ãäå p è · ÿâëÿþòñÿ ïàðàìåòðàìè, âêëþ÷àþùèå âûøåóïàìÿíóòûå ðåçóëüòàòû êàê ÷àñòíûå ñëó÷àè. ýòè ãèïîòåçû îñòàþòñÿ íåðåøåííûìè. â 2009 ã àâòîð ðåøèë c-âåðñèè ãèïîòåç áîíäè è þíãà íà îñíîâå ìèíèìàëüíîé ñòåïåíè, ãäå c îáîçíà÷àåò äëèíó äëèííåéøåãî öèêëà â v ( g¡c ) (discrete math, v.309, 2009, 1925-1930). íà îñíîâå äðóãîãî ðåçóëüòàòà àâòîðà (graphs and combinatorics, v.29, 2013, 1531-1541), â ðàáîòå äîêàçûâàåòñÿ ñïðàâåäëèâîñòü íåêåòîðûõ âåðñèé ãèïîòåç áîíäè è þíãà, âêëþ÷àþùèå p è c-âåðñèè áåç óñëîâèÿ ñâÿçíîñòè, êîòîðûå â ñâîþ î÷åðåäü ïîðîæäàþò íîâûå óñèëåííûå è ðàñøèðåííûå âåðñèè ýòèõ ãèïîòåç. âñå ïîëó÷åííûå ðåçóëüòàòû íåóëó÷øàåìû. mathematical problems of computer science 40, 55--67, 2013. 55 hierarchical cluster analysis for partially synthetic data generation levon h. aslanyan and vardan h. topchyan institute for informatics and automation problems of nas ra e-mail: lasl@sci.am, vardan.topchyan@gmail.com abstract limiting the risk of information disclosure is now common for statistical agencies. one of the widespread approaches is to release the synthetic, public use of microdata sets. to put it another way, thanks to the multiple imputations the sensitive variables of original data are replaced by new/synthetic values. this paper introduces the method for partially synthetic data generation based on hierarchical cluster analysis. keywords: confidentiality, multiple imputation, synthetic data, hierarchical clustering. 1. introduction submission of sociological and/or economic data to the public structures is an integral part of procedures in statistical organizations. however, this task should not assume the risk of disclosure of sensitive or personal information. analysis of the published data in this area [1] indicates the presence of diverse approaches/methods for solving such problems, including variable recoding, swapping data, and adding noise values. although these methods replace the original data, protecting the information in this way may lead to distortion of relationships between the different segments of the data set, which in turn can lead to erroneous conclusions/inferences on the stage of data analysis, such as methods of standard statistical processing. an alternative approach to solve this problem, which also tries to maintain functional relationships between the segments of the data set simultaneously, is the approach of fully synthetic data generation [2]. in this case, the statistical organization should: (i) randomly and independently record the general format and content of critical information units, as well as integrate them into the corresponding set of expected synthetic data; (ii) establish new/synthetic values in the information units by the selected strategy; (iii) provide a number of generated synthetic data sets to the public. there are various methods [3]-[5] for generating fully synthetic data providing the receipt of meaningful results using standard statistical methods. in spite of advantages of the fully synthetic data, the process of generating these data is quite time-consuming. in this regard, statistical organizations often use partially synthetic data hierarchical cluster analysis for partially synthetic data generation56 which is a mix of original and synthesized data [6]. for example, the statistical agency may seek to protect the confidentiality of certain records, or may not allow the identification of several records. in connection with that, the synthetic values generated only for certain variables and the values of the other variables remain changed. as in the case of fully synthetic data, partially synthetic data are also providing the restriction of the disclosure risk, allowing to obtain meaningful results by using standard statistical analysis. note that, due to its nature, the use of partially synthetic data provide more accurate statistical calculations. for the same reason, the risk of disclosure is higher than in case of the fully synthetic data. however, there are several algorithms [7]-[9] for generating partially synthetic data which are used by many statistical agencies (us federal reserve board, us bureau of the census, etc.) that indicate the perspectives of this method. the above mentioned situation was considered as a basis for the analysis of non-parametric methods for generating partially synthetic data sets used for calculation of simple estimands (average, standard deviation, etc.) and for construction of data driven linear regression models [3]. published literature [10] shows that one of the most known non-parametric approaches is the method of sequential imputations of variables [11]. [11] uses cart (classification and regression trees) model [12] for this reason. in this case, in non-parametric method [11] the hierarchical clustering model can be assumed to substitute cart as an alternative approach. the paper is organized as follows. section 2 reviews the description of partially synthetic data, the principle of their generation and analysis. section 3 illustrates how the hierarchical clustering model can be used in a similar to cart way, for generating partially synthetic data. section 4 concludes the discussion about using hierarchical clustering in partially synthetic data generation. 2. partially synthetic data 2.1 creation of partially synthetic data for partially synthetic data definition, we use notation [13]. the process of generating partially synthetic data consists of two parts: (i) pretreatment/preprocessing of data; (ii) replacement of the corresponding/tagged values with the synthetic one. formally, this process can be described as follows. let u be the set of records/information units, },...,2,1{ nuuuu  , where each information unit )1( niiu  is characterized by the p attributes/ variables, },...,2,1{ pyyyy  . during preprocessing information units and confidential variables (rows and columns of matrix u) are selected, and the threshold conditions for these variables are set. 1y 2y  py u 1u 11y 12y  py1 2u 21y 22y  py2  nu 1ny 2ny  npy l. aslanyan ,v. topchyan 57 let )( nnn  be the number of randomly selected information units that will be considered in the current observation, denote those units by },..., 2 , 1 { ni uiuiu . similarly, )( pdd  is the number of confidential variables, },..., 2 , 1 { d jyjyjy . in addition, n-block ),...,2,1( niiii  and p-block ),...,2,1( pjjjj  of numbers are defined as follows: , 1, { , ,..., } 1 2 0, { , ,..., } 1 2 u u u ur i i in ir u u u ur i i in       1 ,r n  , 1, { , ,..., } 1 2 0, { , ,..., } 1 2 y y y yj j jk d j k y y y yj j jk d       1 .k p  the process of determining the information units and confidential variables is presented in the following scheme. as a result, ),( nrepurepuobsu  matrix is defined. here repu is a ][ dn  matrix with the values of confidential variables },...,,{ 21 djjj yyy and nrepu )]([ dpn  is a matrix of other variables values (replaced vs. not replaced). 1j    pj 1y    py u 1i 1u 11y py1         ni nu 1ny npy hierarchical cluster analysis for partially synthetic data generation58 next, the d jcjcjc ,..., 2 , 1 threshold conditions are set for the d jyjyjy ,..., 2 , 1 variables. based on these conditions, the indicator matrix ][ dnz  is defined. repu nrepu 1j y  djy  pjy obsu 1i u 11 ji y  d jiy 1  p jiy 1 2i u 12 ji y  d jiy 2  p jiy 2  ri u 1jiry  dr jiy  pr jiy  ni u 1jiny  dn jiy  pn jiy 1j c  djc 1j y  djy repu 1i u 11 jiy  d jiy 1 2i u 12 ji y  d jiy 2  ri u 1jir y  dr jiy  ni u 1jiny  dn jiy z 11 j z  d jz1 12 j z  d jz2  1rj z  d rjz  1nj z  d njz l. aslanyan ,v. topchyan 59 this matrix describes the need to replace the corresponding values of the confidential variables. thus, during preprocessing ),( nrepurepuobs u  and z matrices are defined. the resulting, observed data set are denoted as ),,( znrepurepud  . the second part of the partially synthetic data generation is the process of replacement. namely, based on ),,( znrepurepud  and the selected method/algorithm, the corresponding values of nrepu matrix are substituted with the synthetic one. replacements are made independently m times to generate m different partially synthetic data sets: ),( nrepu i synuisd  , 1 ,i m  where i synu a matrix of imputed (replaced) values of i-th synthetic data set. the values in u rep are the same in all synthetic data sets, isd , mi 1 . thus, the generated partially synthetic data sets },...,2,1{ msdsdsdsynd  are the information that are provided to the corresponding organizations and the public. 2.2 analysis of synthetic data based on the released synthetic data sets },...,2,1{ msdsdsdsynd  , the corresponding organization, in other words the analyst, makes inferences about some population quantity )(yqq  ( for example, q can be the average of interest or the coefficients in a regression model). in each synthetic data set isd ),...,2,1( mi  , the analyst estimates q , by some value iq , and estimates the variance of iq with some estimator iv . it is assumed that the analyst determines the iq and iv as if isd was in fact collected data from a random sample of u . such technique is usual in area of missing value statistics which needs to develop approaches of unbiased data generation [3]. the approach used in this article is to consider mivq ii ,...,2,1),,(  in similar to [3] as a sufficient characterization of the synthetic databases synd , and construct an approximate posterior distribution of q given synd , )|pr( syndq , in analogy with the theory of multiple imputation for missing data [5]. in that case the analyst can obtain valid inferences for q by combining the results of iq and iv ),...,2,1( mi  . the following quantities are needed for inferences (see [3]): 1 1 , m m i im q q      2 1 1 , 1 m m i i mm q qb     hierarchical cluster analysis for partially synthetic data generation60 1 1 , m m i im v v    the analyst then can use q m to estimate the scalar q and vbt mmm m  ) 1 1( to estimate the variance of q m . 3. hierarchical clustering model for generation of partially synthetic data the 2 base approaches we touch in data approximation are as the probabilistic distribution approximation on one hand and the heuristic data extensions on the other. even when the first one seems more fundamental the heuristic approaches are more interpretable and applicable especially when used by not professionals. such methods are also fast and can replace successfully the theoretical counterparts in many cases. having the example of using cart model in synthetic data generation we will try to understand the role and power of hierarchical cluster analysis in the same role. it is more important to understand the inferences of these 2 approaches that complement each other. cart when generated is pruned like the cauterization, and cauterization is evaluated where to stop by the use of additional quality estimates. 3.1 algorithm for imputations our studies are based on hierarchical agglomerative clustering [14] for generating partially data sets. the principle of synthetic data generation is the same as in the algorithm that considered in [11]. the only difference is that this model is used as a tool for estimating a conditional distribution for sensitive variables in the space where data need to be joined/replaced. that means that the replacement of sensitive variables occurs sequentially (in descending order of number of replacement values). moreover, for generating new/synthetic values the bayesian bootstrapping [15] is used. formally, this process can be described as follows. assume that the variables },...,2,1{ pyyyy  are continuous. without loss of generality, suppose that from the set of variables },...,2,1{ pyyyy  the first d is confidential. in addition, the matrix ),( nrepurepuobsu  consists of the first n records of the information units set u . first, for each variable )1( dk k y  , based on the indicator matrix z , the number of its replacements,   n i z ik 1 is computed. as a result, d yyy ,...,2,1 variables are sorted in descending order of the computed values and denoted as: )( ,..., )2( , )1( d yyy . after that, sequentially confidential variable imputations are processed. let, )(k y be the current variable. firstly, for )(k y we estimate the conditional distribution in the space where data need to be replaced by using hierarchical agglomerative clustering. obviously, clustering will be produced based on information units satisfying the following condition: l. aslanyan ,v. topchyan 61 niz ki ,...,1,1)(  . consider that the quantity of the mentioned units equals to k n . at the beginning of clustering each information unit is considered as a single cluster: },...,2,1{ k ncccinit c  . after that the sequential process of clusters merging begins: each time it brings together a pair of closest clusters, where the distance between them is taken as a distance of their centers, and as an integrated distance measure the euclidean distance is used:    p j c rjc ijrcicd 1 )( 2 ),( , where ),...,1( cipcici  and ),...,1( crpcrcr  are centers of clusters ri cc , , respectively. to limit the disclosure risk this process continues until there is a possibility of clusters merging in accordance with one of the homogeneity validity measures [16]: spr (semi-partial rsquared),rmsstd (root-mean-square standard deviation), etc. as a result we get the set of current/final clusters }',...,' 2 ,' 1 { t ccc fin c  . further, in each cluster ' l c )1( tl  new values for )(k y are generated by using bayesian bootstrap procedure. consider ly as a set of values of )(k y in the corresponding cluster ' l c , },..., 2 , 1 { l l n ylylyly  . bayesian bootstrap draws values based on some donor pool. in this algorithm ly is taken for ' l c as a donor pool. bayesian bootstrap method proceeds as follows: 1. draw )1(  l n uniform random numbers in the range )1,0( and sort these numbers in ascending order: 1, )1( ,..., 2 , 1 ,0 0    l n a l n aaaa . 0 = 0 1 = 12 3 −1 −2 −1 2. draw l n uniform random numbers in range ]1,0( : l n u l n uuu , )1( ,..., 2 , 1  0 = 0 1 = 12 3 −1 −2 −1 1 2 3 4 hierarchical cluster analysis for partially synthetic data generation62 3. for each i u )1( l ni  determined interval in which it is contained: ], 1 ( j a j a i u   and replace l i y value by l j y . note that some information units involved in clustering, may include new / synthetic values of )1( ,..., )2( , )1( kyyy variables. in order to maintain imputation consistency for all possible combinations of )(k y value and )1( ,..., )2( , )1( kyyy new values corresponding information units are searched, after what )(k y imputations are done among them. thus, as a result of sequential imputations of confidential variables values generated set of partially synthetic data. the whole process is repeated independently m times and generated synthetic databases synd are provided to the public. 3.2 simulation studies for the above mentioned method our simulation studies are based on public release data from 2011 r.a. household’s integrated living conditions survey which consist of 7872n records. of the entire set of attributes/variables that characterize these units, we are interested in only six of them. the interest variables descriptions are presented in table 1. table 1: description of variables used in empirical studies variable type range notes monitory income numeric(18,10) 0 – 3512850 food purchase numeric(18,10) 0 – 555600 food consume numeric(18,10) 0 193191.8083296074 nonfood purchase numeric(18,10) 0 – 965100 expenditures numeric(18,10) 6256.3126710940 4672865.1342223603 13.643% of house holders have total income more than 225000 amd total income numeric(18,10) 0 3529697.3182837632 13.795% of house holders have expenditures more than 175000 amd next, we assume that total income and expenditures are sensitive variables and set threshold conditions for total income and expenditures are as follows; total income 225000 and expenditures 175000 . in empirical studies each observed data set d consists of 315n randomly sampled households from the 7872 households. as a result of simulation 5m partially synthetic data sets msdsdsd ,...,, 21 are constructed for each d . each ),...,2,1( misdi  is generated using the algorithm presented earlier. clustering for each sensitive variable is performed on the basis of the units that satisfy the threshold conditions for that variable. in turn, as a validity measure we use minimal count of units in cluster and minimal count of distinct values of sensitive variable in cluster. in this simulation we require minimum ten units with at least three distinct values in each cluster. l. aslanyan ,v. topchyan 63 table2: simple estimands for sensitive variables. table 3: regression model. dependent variable: total income. independent variables: monitory income, food purchase, expenditures. table 4: regression model. dependent variable: total income. independent variables: monitory income, expenditures. estimand q avg. q 5 coefficients constant 8868.7488 11893.24448 monitory income 0.9638 0.89952 expenditures 0.072 0.09412 r 0.9818 0.93268 estimand q avg. q 5 % of h.h. with total income > 300000 5.206344 5.3860208 % of h.h with expenditures > 230000 6.031744 6.2291976 average of total income 132357.290182 132611.3233328 average of expenditures 109301.86175 109548.668466 standard deviation of total income 94569.735448 95539.39169016 standard deviation of expenditures 77458.3434 77376.82509436 estimand q avg. q 5 coefficients constant 11279.3796 14181.77272 monitory income 0.9662 0.90748 food purchase -0.2676 -0.20408 expenditures 0.1652 0.16512 r 0.9844 0.93436 hierarchical cluster analysis for partially synthetic data generation64 table 5: regression model. dependent variable: expenditures. independent variables: total income, food purchased, food consumed, nonfood purchased. table 6: regression model. dependent variable: expenditures. independent variables: food purchased, food consumed, food nonpurchased. estimand q avg. q 5 coefficients constant -168.794 3725.97836 food purchased 1.0334 1.05348 food consumed 1.0162 1.204032 food nonpurchased 1.1082 0.95308 r 0.9838 0.90272 tables 2 – 6 summarize the results of simulation for a variety of estimands. inferences are made using the methods presented in section 2.2. for simple estimands (table 2) the averages of synthetic point estimates are close to their corresponding q. the average of parameter r1 for each regression models for synthetic data sets (tables 3-6) are greater 0.900 which indicates that these models are worth considering. in addition, the averages of regression coefficients are close to original values. so, the analyst will make the same inferences as in the case of actual data. in case for disclosure risk of each sensitive variable jl y (l=1, 2,…, d), we assume that the analyst would estimate jl y value of ir ’s unit by averaging the replaced values , , , . r l r l m rep k k y ji ji y   1 r shows how much the independent variables explain the dependent variable estimand q avg. q 5 coefficients constant -1664.4878 -825.07984 total income 0.026 0.06612 food purchased 1.017 1.00768 food consumed 0.9924 0.90364 food nonpurchased 1.0886 0.906 r 0.9842 0.90692 l. aslanyan ,v. topchyan 65 to assess that risk we calculate the root mean squared error (rmse) and relative root mean squared error (relrmse) of this estimator for each information unit: , ,2 2( ) ( ) / (( 1) ), ,, ,, 1 m rep k rmse y y y m myj ji ir rj jl lir irjl lir l k       ,re /,, , lrmse rmse yjirj jli ir rl l  for any data set, the distributions of the jlir rmse , and jlir relrmse , across all units with replaced values can be examined to ensure sufficient variability in imputations. table 7 displays averages of these quantities across all simulations. median of relmses is typically around 13.5%, which indicates that imputations for most records have a wide range of uncertainty. in case when data owner requires larger errors in terms of decreasing sensitive variables disclosure risk, stricter validity measure criteria can be used in clustering. table 7: sensitive variables limitation in simulation studies. 4. conclusion the simulation results show that the proposed clustering model can be used as an alternative approach in partially synthetic data generation in the similar to the cart way. the only limitation is that the attributes/variables characterizing the information units must be continuous. the foregoing can 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[14] c.d. manning, p. raghavan and h. schutze, introduction to information retrieval, cambridge university press, 2008. [15] d.b. rubin, “the bayesian bootstrap”, the annals of statistics, vol. 9, pp. 130–134, 1981. [16] m. halkidi, y. batistakis and m. vazirgiannis, “clustering validity checking methods: part ii”, acm new york, ny, usa, vol. 31, pp. 19-27, 2002. submitted 05.09.2013, accepted 18.10.2013. l. aslanyan ,v. topchyan 67 մասնակի սինթետիկ տվյալների գեներացիայի հիերարխիկ կլաստերային վերլուծություն լ. ասլանյան և վ. թոփչյան ամփոփում կոնֆիդենցիալ տեղեկությունների բացահայտման ռիսկի նվազեցումը այսօրվա դրությամբ հանդիսանում է վիճակագրական ընկերությունների հիմնական խնդիրներից մեկը։ այդ խնդրի լուծման համար կիրառվող ամենահայտնի մեթոդներից մեկն է այսպես կոչված սինթետիկ տվյալների բազմությունների մշակման և տրամադրման մեթոդը։ այլ կերպ ասած, կոնֆիդենցիալ փոփոխականների արժեքները փոխարինվում են նոր սինթետիկ արժեքներով։ տվյալ հոդվածում ներկայացված է մասնակի սինթետիկ տվյալների ստեղծման/գեներացիայի մի մեթոդ, որի հիմքում ընկած է հիերարխիկ կլաստերային վերլուծությունը: иерархический кластерный анализ для генерации частично синтетических данных л. асланян и в. топчян аннотация ограничение риска раскрытия конфиденциальной информации на сегодняшний день является одной из основных задач статистических агентств. одним из широко применяемых подходов является предоставление синтетических множеств данных. иными словами, благодаря множественным замещениям значения конфиденциальных переменных исходных данных заменяются новыми синтетическими значениями. в данной статье представлен метод генерации частично синтетических данных на основе иерархического кластерного анализа. d:\sbornik\...\article.dvi mathematical problems of computer science 35, 99{103, 2011. i r r educibility of some composite p olynomials over finite fields ¤ s e r g e y a b r a h a m ya n a n d e d it a h a r u t yu n ya n institue for informatics and automation problems of nas of ra e-mail serj.abrahamyan@gmail.com, edita@ipia.sci.am abstract given the ¯eld f with q elements and of characteristics p and an irreducible polynomial p (x) = cnx n +cn¡1x n¡1+c1xn¢¢¢c0 over f. we prove that the composite polynomial (dxp ¡ dx)n p ³ axp¡ax+c dxp¡dx ´ is irreducible over f under certain conditions. also a recursive construction of sequences of irreducible polynomials of degree n2k(k = 1; 2; 3; ¢ ¢ ¢) over f2s is given. refer ences [1 ] s . e . a b r a h a m ya n , \ co n s t r u c t io n o f ir r e d u c ib le p o lyn o m ia ls o ve r fin it e fie ld s " , p roceedings of 12th international w orkshop, casc 2010, in l ecture notes in computer science, vo l. 6 2 4 4 , ge r d t , p p . 3 -4 , 2 0 1 0 . [2 ] s . d . co h e n , \ on ir r e d u c ib le p o lyn o m ia ls o f c e r t a in t yp e s in ¯ n it e ¯ e ld s " , p roc. cambridge p hilos. soc., vo l. 6 6 , p p . 3 3 5 { 3 4 4 , 1 9 6 9 . [3 ] r . l id l a n d h . n ie d e r r e it e r . fin it e fie ld s . ca m b r id g e u n ive r s it y p r e s s , 1 9 8 7 . [4 ] m. k . k yu r e g ya n , \ r e c u r r e n t m e t h o d s fo r c o n s t r u c t in g ir r e d u c ib le p o lyn o m ia ls o ve r fq o f o d d c h a r a c t e r is t ic s " , f inite f ields appl., vo l. 9 , p p . 3 9 { 5 8 , 2 0 0 3 . [5 ] m. k . k yu r e g ya n , \ it e r a t e d c o n s t r u c t io n s o f ir r e d u c ib le p o lyn o m ia ls o ve r ¯ n it e ¯ e ld s wit h lin e a r ly in d e p e n d e n t r o o t s " , f inite f ields , appl., vo l. 1 0 , p p . 3 2 3 { 4 3 1 , 2 0 0 4 . [6 ] m. k . k yu r e g ya n , \ r e c u r r e n t m e t h o d s fo r c o n s t r u c t in g ir r e d u c ib le p o lyn o m ia ls o ve r fq o f o d d c h a r a c t e r is t ic s ii" , f inite f ields, appl., vo l. 1 2 , p p . 3 5 7 { 3 7 8 , 2 0 0 6 . [7 ] m. k . k yu r e g ya n a n d g. m. k yu r e g ya n , \ ir r e d u c ib le co m p o s it io n s o f p o lyn o m ia ls o ve r fin it e fie ld s " , d esign, codes and cryptography, a va ila b le o n lin e : is s n :0 9 2 5 -1 0 2 2 2 0 1 0 . [8 ] a . me n e z e s , i.f. b la ke , x . ga o , r . c. mu llin , s . a . v a n s t o n e , t. y a g h o o b ia n . applications of f inite f ields, k lu we r a c a d e m ic p u b lis h e r s , b o s t o n d o r d r e c h t l a n c a s t e r , 1 9 9 3 . [9 ] r . r . v a r s h a m o v, \ a g e n e r a l m e t h o d o f s yn t h e s iz in g ir r e d u c ib le p o lyn o m ia ls o ve r ga lo is ¯ e ld s " , soviet m ath. d okl., vo l. 2 9 , 3 3 4 { 3 3 6 , 1 9 8 4 . ¤the work was supported by armenian target programm 04.10.31. 9 9 1 0 0 irreducibility of some composite polynomials over finite fields àñáß µ³õ³¹ñû³é µ³½ù³ý¹³ù»ýñç ãµ»ñí»éçáõãûáõýá í»ñç³íáñ ¹³ßï»ñç íñ³ ê. ²µñ³ñ³ùû³ý ¨ ¾. ð³ñáõãûáõýû³ý ²ù÷á÷áõù ²ßë³ï³ýùá ýíçñí³í ¿ í»ñç³íáñ ¹³ßï»ñç íñ³ ãµ»ñíáõ µ³½ù³ý¹³ùý»ñç µ³õ³¹ñáõãûáõýý»ñç ï³éáõóù³ýá: îñí³í ¿ q ñ½áñáõãû³ùµ, p µýáõã³·ñçãáí fq ¹³ßïá ¨ fq ¹³ßïç íñ³ p ( x ) = cnx n + cn¡1x n¡1+c1xn¢¢¢c0 ãµ»ñíáõ µ³½ù³ý¹³ùá: òáõûó ¿ ïñí³í, áñ ( dxp ¡ dx ) n p ³ axp¡ax+c dxp¡dx ´ µ³õ³¹ñáõãûáõýá ãµ»ñíáõ ¿ fq ¹³ßïç íñ³ áñáß³ïç å³ûù³ýý»ñç ¹»åùáõù: ²í»éçý, ïñí³í ¿ f2s ¹³ßïç íñ³ n2 k( k = 1 ; 2 ; 3 ; ¢ ¢ ¢ ) ³ëïç×³ýç ãµ»ñíáõ µ³½ù³ý¹³ùý»ñç ï³éáõóù³ý ñ³çáñ¹³ï³ý »õ³ý³ï: d:\sbornik\...\article.dvi mathematical problems of computer science 29, 2007, 26{32. on a gener alization of i nter val e dge color ings of gr aphs p . a . p e t r o s ya n y a n d h . z. a r a ke lya n z yinstitue for informatics and automation problems of nas of ra e-mail: pet petros@ipia.sci.am zdepartment of informatics and applied mathematics, ysu e-mail: arak hakob@yahoo.com abstract an interval edge (t; h)¡coloring (h 2 z+) of a graph g is a proper coloring ® of edges of g with colors 1; 2; : : : ; t such that at least one edge of g is colored by i; i = 1; 2; : : : ; t and the colors of edges incident with each vertex v satisfy the condition dg(v) ¡ 1 · max s (v; ®) ¡ min s (v; ®) · dg(v) + h ¡ 1; where dg(v) is the degree of a vertex v and s (v; ®) is the set of colors of edges incident with v. in this paper we investigate some properties of interval edge (t; h)¡colorings. refer ences [1 ] a .s . a s r a t ia n , r .r . k a m a lia n , in t e r va l c o lo r in g s o f e d g e s o f a m u lt ig r a p h , appl. m ath. 5 ( 1 9 8 7 ) , y e r e va n s t a t e u n ive r s it y, p p . 2 5 -3 4 . [2 ] r .r . k a m a lia n , in t e r va l e d g e co lo r in g s o f gr a p h s , d o c t o r a l d is s e r t a t io n , th e in s t it u t e o f ma t h e m a t ic s o f t h e s ib e r ia n b r a n c h o f t h e a c a d e m y o f s c ie n c e s o f u s s r , n o vo s ib ir s k, 1 9 9 0 . [3 ] f. h a r a r y, gr a p h th e o r y, a d d is o n -w e s le y, r e a d in g , ma ,1 9 6 9 . [4 ] v .g. v iz in g , th e c h r o m a t ic in d e x o f a m u lt ig r a p h , k ibernetika 3 ( 1 9 6 5 ) , p p . 2 9 -3 9 . [5 ] d .b . w e s t , in t r o d u c t io n t o gr a p h th e o r y, p r e n t ic e -h a ll, n e w je r s e y, 2 0 0 1 . [6 ] r .r . k a m a lia n , in t e r va l c o lo r in g s o f c o m p le t e b ip a r t it e g r a p h s a n d t r e e s , p r e p r in t o f t h e co m p u t in g ce n t r e o f t h e a c a d e m y o f s c ie n c e s o f a r m e n ia , 1 9 8 9 , 1 1 p . 2 6 p. a. petrosyan, h. z. arakelyan 2 7 ¶ñ³ýý»ñç ùçç³ï³ûù³ûçý ïáõ³ûçý ý»ñïáõùý»ñç áý¹ñ³ýñ³óù³ý ù³ëçý ä. ². ä»ïñáëû³ý ¨ ð. ¼. ²é³ù»éû³ý ²ù÷á÷áõù g ·ñ³ýç ïáõ»ñç ×çßï ® ý»ñïáõùá 1 ; 2 ; :::; t ·áõûý»ñáí ï³ýí³ý»ýù ùçç³ï³ûù³ûçý ïáõ³ûçý ( t; h) -ý»ñïáõù (h 2 z+), »ã» ³ù»ý ùç i ·áõûýáí, i = 1 ; 2 ; :::; t ý»ñïí³í ¿ ³éýí³½ý ù»ï ïáõ ¨ ûáõñ³ù³ýãûáõñ ·³·³ãçý ïçó ïáõ»ñç ·áõûý»ñá µ³í³ñ³ñáõù »ý ñ»ï¨û³é å³ûù³ýçý‘ dg ( v ) ¡ 1 · m a x s ( v; ®) ¡ m in s ( v; ®) · dg ( v ) + h¡ 1 , áñï»õ dg ( v ) -ý v ·³·³ãç ³ëïç×³ýý ¿ g-áõù, çëï s ( v; ® ) v ·³·³ãçý ïçó ïáõ»ñç ·áõûý»ñç µ³½ùáõãûáõýý ¿: ²ûë ³ßë³ï³ýùáõù ñ»ï³½áïíáõù »ý ùçç³ï³ûù³ûçý ïáõ³ûçý ( t; h) ý»ñïáõùý»ñç áñáß ñ³ïïáõãûáõýý»ñ: microsoft word armen khachaturyan_23.doc mathematical problems of computer science 33, 183--186, 2010. 183 the optimal permissible placement by the height of the transitive oriented treecontaining one vertex of branching armen khachaturyan yerevan state university, faculty of informatics and applied mathemathics e-mail armenarmenian@mail.ru abstract in this paper we consider the optimal permissible placement by the height of the following type of transitive oriented graphs (the height of the vertex is the number of the arcs passing through the vertex): the graph consists of chains of quantity 2s branching after the end of the chain. the problem was solved by the algorithm of ss log complexity. references 1. в. а. евстигнеев, “применение теории графов в программировании”,1985. 2. а. в. петросян, с. е. маркосян, ю. г. шукурян, “математические вопросы автоматизации и проектирования эвм”.ереван, 1977. 3. г. г. геолецян, “плоское размещение вершин дерева на линии с минимизацией ширины.” дан апм сср, вып. 56, но-4, стр.202-207, 1973 4. m.r. garey, r.l. graham, d.s. johnson, and d.e. knuth “complexity results for bandwidth minimization”, siam j. appl. math. 34, pp. 477-495, 1978. 5. m.r. garey, d.s. johnson, and l. stockmeyer “some simplified np-complete graph problems”, theor. comput. sci.1, pp. 237-267, 1976. 6. ch. h. papadimitriou, “the np-copleteness of the bandwidth minimization problem”, computing 16, pp.263-270, 1976. 7. d. adolphson, and t. c. hu, “optimal linear ordering”, siam j. appl. math., vol.25,no.3, pp. 403423, 1973. 184 the optimal permissible placement ø»ï ×ûáõõ³íáñù³ý ·³·³ã å³ñáõý³ïáõ ïñ³ý½çïçí ûñç»ýï³óí³í í³éç ûåïçù³é ãáõûé³ïñ»éç ï»õ³¹ñáõùý áëï µ³ñóñáõãû³ý ². ê³ã³ïáõñû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ß³ñ³¹ñí³í »ý ñ»ï¨û³é ïçåç ïñ³ý½çïçí ûñç»ýï³óí³í ·ñ³ýý»ñç ûåïçù³é ãáõûé³ïñ»éç ñ³ù³ñ³ï³éáõùý áëï µ³ñóñáõãû³ý (·³·³ãç µ³ñóñáõãûáõý ³ë»éáí ñ³ëï³ýáõù »ýù ·³·³ãç íñ³ûáí ³ýóýáõ ³õ»õý»ñç ù³ý³ïá)` ·ñ³ýá ï³½ùí³í ¿ ßõã³ûçó ¨ ßõã³ûç í»ñççó ×ûáõõ³íáñíáõ 2s ù³ý³ïáõãû³ùµ ßõã³ý»ñçó: êý¹çñá éáõíí»é ¿ ss log µ³ñ¹áõãû³ý ³é·áñçãùáí áñï»õ s -á ¹çï³ñïí³í ·ñ³ýç ×ûáõõ³íáñíáõ ßõã³ý»ñç ù³ý³ïý ¿ : microsoft word 12_sona_hakobyan_new mathematical problems of computer science 42, 113--120, 2014. 113 fast slant-hadamard transform algorithm sonik r. hakobyan yerevan state university e-mail: sonahakobyan@ysu.am abstract in this paper, we investigate an efficient algorithm for computation of parametric slant-hadamard transforms. we present the slant-hadamard matrix of order n2 as a product of sparse matrices, develop the appropriate fast slant-hadamard transform and its complexity. in the end we present the detailed example of factoring of slanthadamard transform matrix of order 8 and matlab code for implementation slanthadamard transform. keywords: slant-hadamard transform algorithm, sparse matrix. 1. introduction a phenomenon characteristic of digital images is the presence of approximately constant or uniformly changing gray levels over a considerable distance or area. the slant transform is specifically defined for efficient representation of such images. it is a piecewise linear basis that follows the spirit of walsh transform, possessing a discrete saw tooth-like basis vector which efficiently represents linear brightness variations along an image line. the slant transform has been used for signal compression and pattern recognition as well as for intel’s “indo” video compression algorithm [1-2]. the slant transform has the best compaction performance among the non-sinusoidal fast orthogonal transforms. historically, enomoto and shibata conceived the first, eight-point slant transform in 1971 andused it in tv image encoding. its major innovation is given by the slant vector, which can properly follow the gradual changes in brightness of natural images. pratt, welch, and chen have generalized it to any order n 2 and compared its performance with other transforms [2]. the slantlet transform has been successfully applied in compression and denoising. currently, slant transforms are usually constructed via hadamard or haar transforms [5-7]. the most common fast slant transform methods [5] are based on the factorization of the slant transform matrix into a set of largely sparse matrices. fast slant-hadamard transform algorithm 114 2. the slant-hadamard transform the slant-hadamard transform is defined as x = sx [8], where s the slant-hadamard transform matrix of order n is generated recursively by the following formula: where 2 2 n p = − , 2 n n = , m i denotes an identity matrix of order m, and the parameters n a and n b are given by 1/ 2 2 2 1/ 2 1 12 2 3 1 , , (2) 4 1 4 1 n n n n a b n n + +    − = =    − −    and ( )       − = 11 11 2 1 1s is just a hadamard matrix of order 2 [8]. example 2.1: below we present the slant-hadamard transform matrices of order 4 and 8 obtained from (1) and (2) 1 1 1 1 1 1 1 1 7 5 3 1 1 3 5 7 1 1 1 1 1 / 21 3 1 1 3 3 1 1 3 3 31 1 1 / 5 7 1 9 17 17 9 1 71 15 5 5 5 1 / 105(2) , (3) 1 1 1 1 1 1 1 1 1 1 1 14 8 3 31 1 1 1 1 1 1 1 1 1 1 / 5 5 5 5 5 1 3 3 1 1 3 3 1 1 / 5 1 3 3 1 1 3 3 1 s s ⋅ ⋅ ⋅ ⋅ ⋅ − − − −   − − − −  − −  − − − −  = = − − − − − −    − − − −− −    − − − − − − − − 3. construction of parametric slant-hadamard transform the forward and inverse parametric slant-hadamard transforms of order 2 n (n =1, 2, 3,...), are defined as xsx n 2 = , xsx t n 2 = [9-10], respectively, where x is an input data vector and ns 2 is generated recursively as 2 1 )( =ns nn ba 01 p o ×2 nn ba− 01 p o ×2       − − × )1( )1( nso ons , (1) 2×po pi 2×po pi nn ab− 10 p o ×2 nn ab 10 − p o ×2 2×po pi 2×po pi− s. hakobyan 115      ⊗=         = − −− −− 1 11 11 2222 1 22 22 22 1 2 nn nn nn nn siq so os qs , 1>n , 1 , 2 2 2 s h + +  = =   + −  where m o denotes the zero matrix of order m, “+” corresponds to 1 and “-” to -1, ⊗ denotes the kronecker product, and n q 2 is the recursion kernel matrix defined as as in [9-10], we introduce the expressions for n a 2 and n b 2 to construct the parametric slant-hadamard transforms matrices of order 2 n n n n n a 2 22 22 2 )2(4 )2(3 β− = − − , n n n n n b 2 22 2 22 2 )2(4 2 β β − − = − − , (3) where 22 2 22 22 −− ≤≤− n n n β , 1 2 =a . it can be shown that for 22 2 2 − > n nβ the slant-hadamard transforms matrices lose their orthogonality. the parametric slant transform matrices fulfill the requirements of the classical slant transform matrix. however, the parametric slant transform matrix is a parametric matrix with parameters n 2 84 ...,,, βββ . � when ,1 2 84 ===== ββββ n l we obtain a classical slant transform � when 22 2 2 − = n n β for all n 2 β , 2≥n , we obtain a walsh-hadamard transform. � when , 2 84 ββββ ==== n l for 44 ≤≤− β , we obtain a constant beta slant transform. � when n 2 84 ... βββ ≠≠≠ , 22 2 22 22 −− ≤≤− n n n β , ...,4,3,2=n , we refer to this case as a multiple betas slant transform [9-10]. in this case some of n 2 β can be equal but not all of them. for n=2 and 0.4 4 −=β we have the following multiple betas slant transform matrix of order 4. =nq 2 nn ba 22 01 p o ×2 nn ba 22 01 − p o ×2 2×po pi 2×po pi nn ab 22 10 − p o ×2 nn ab 22 10 − p o ×2 2×po pi 2×po pi fast slant-hadamard transform algorithm 116 1 1 1 1 3 2 3 2 3 2 3 2 1 5 5 5 5 (2) . 1 1 1 14 3 2 3 2 3 2 3 2 5 5 5 5 s     + − − + − −   =   − −    − + + − − −    remark 3.1: parametric slant transform matrices fulfill the following requirements of classical slant transform: � its first row vector is of constant value. � its second row vector represents the parametric slant vector. � its basis vectors are orthonormal. � it has a fast algorithm. remark 3.2: it is easy to verify that the parametric slant transform matrix can be represented as follows: , 2, 4 41 , 2,2 2 2 2 m h for n s n c h for nn n n      = = > where         ⊗= − − 1 1 2 0 0 2 22 n n nn c c mc , (4) where 22 1 ,22 imp n =−= − . kmo × denotes a zero matrix of size ,km × mi denotes an identity matrix of order m, ⊗ is the operator of kronecker product, nh 2 is the walsh-hadamard matrix of order n 2 , and the parameters na 2 and nb 2 are given in (3). =nm 2 nb 2 0 01 p o ×2 0 00 2 na po ×2 2×po pi 2×po ppo × na 2 0 00 p o ×2 0 10 2 nb− po ×2 2×po ppo × 2×po pi s. hakobyan 117 4. fast slant-hadamard transform algorithm the parametric slant-hadamard transform matrix can be represented [11-12] as ( )( ) ,2, 2 1 11 222222 ≥⊗⊗= −− nsiihms nnnn n where 22 im = , nm 2 is given in (4). the slant transform matrix of order 2 n can be factored [11-12] as ∏ = −− == n i innn ssssss n 1 1212 ... , where ))(( 1 22222 −−− ⊗⊗⊗= iiniin ihimisi . below we represent in detail sparse matrices decomposition of slant-hadamard matrix of order 8. consider the slant-hadamard matrix 1238 8 1 ssss = for 1 84 === βββ . we have 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 , 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 s − − = − − 5/1 5/1 5/1 5/1 21210000 10100000 12120000 01010000 00002121 00001010 00001212 00000101 2 ⋅ ⋅ ⋅ ⋅ − − − − − − =s 21/1 21/1 10001000 01000100 00450045 00100010 10001000 01000100 00540054 00010001 3 ⋅ ⋅ − − − − − =s below the 8-point transform steps are given in detailed ( xssss 1238 = , scaling coefficient 8 1 is omitted). step 1: , 1 xsy = (the number of additions is 8). 76 76 54 54 32 32 10 10 7 6 5 4 3 2 1 0 1 7 6 5 4 3 2 1 0 11000000 11000000 00110000 00110000 00001100 00001100 00000011 00000011 xx xx xx xx xx xx xx xx x x x x x x x x xs y y y y y y y y y − + − + − + − + =⋅ − − − − === step 2: , 2 ysz = (the number of additions is 12 and of multiplications is 8). fast slant-hadamard transform algorithm 118 0 0 2 1 0 2 1 3 2 1 3 3 0 2 1 3 2 4 64 4 6 5 75 5 76 7 1/ 5 1/ 5 1/ 5 1 0 1 0 0 0 0 0 (2( ) ( ))2 1 2 1 0 0 0 0 1/ 5 0 1 0 1 0 0 0 0 ( ( ) 2( ))1 2 1 2 0 0 0 0 1/ 5 0 0 0 0 1 0 1 0 (2( ) ( ))0 0 0 0 2 1 2 1 1/ 5 0 0 0 0 0 1 0 1 ( (0 0 0 0 1 2 1 2 1/ 5 y y y y y y y y y y y y y y y y z s y y yy y y y yy y yy y + − + + ⋅− ⋅ −− − − + + ⋅− ⋅ = = ⋅ = + − + + ⋅− ⋅ −− −− ⋅ 4 6 5 7 1/ 5 . ) 2( ))y y y y− + + ⋅ step 3: , 3 zsx = (the number of additions is 10 and of multiplications is 4). 0 40 0 4 1 51 2 62 3 73 3 1 54 5 0 4 1 5 6 2 6 7 3 7 4 / 21 5 / 21 / 21 4 / 21 1 0 0 0 1 0 0 0 ( ) ( )4 5 0 0 4 5 0 0 1/ 21 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 . 0 1 0 0 0 1 0 0 5 ( ) ( )5 4 0 0 5 4 0 0 1/ 21 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 z zz z z z zz z zz z zz x s z z zz z z z z z z z z z z z + − + +− ⋅ + + = = ⋅ = −− − − + +− ⋅ −− −− the total number of additions is 30 and the total number of multiplication is 12. now calculate the number of operations for ns 2 transform. 1 1 1 1 1 2 2 22 2 2 2 2 2 2 2 2 ( )( ) ( ) . i i n i i n i i n i i n i i i i i s i m i h i i m i i i i − − − − − − − − −    = ⊗ ⊗ ⊗ = ⊗ ⊗   −   the x ii ii i ii ii in                 − ⊗ −− −− − 11 11 22 22 2 transform requires only n 2 addition operations, and the ymi iin )( 22 ⊗− transform requires 6 2n i−⋅ operations ( 4 2 n i− ⋅ multiplications and 2 2 n i− ⋅ additions). therefore, the s i requires 12 2n i n+ − + addition and 4 2 n i− ⋅ multiplication operations. so, the fast slant-hadamard transform of order n2 requires 22)1(2 −+= + n nc n addition and 42 1 2 −= +× n nc multiplication operations( 2>n ), with 84 = + c and 4 4 = × c , where + m c and x m c are, respectively, the number of additions and multiplications required for the m – point fast algorithm [8]. % 1d fsht matlab code function m=m2n(n) if n==1 m=eye(2); else p=2^(n-1)-2; m=2^(2*n-2); a=sqrt(3*m/(4*m-1)) ; b=sqrt((m-1)/(4*m-1)); s. hakobyan 119 m=[[1 0 ;0 b] zeros(2,p) [0 0;a 0] zeros(2,p); zeros(p,2) eye(p) zeros(p,2) zeros(p,p); [0 0 ; 0 a] zeros(2,p) [0 1 ;-b 0] zeros(2,p); zeros(p,2) zeros(p,p) zeros(p,2) eye(p)]; end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function y=fsht(n,x) y=x; for i=1:n y=kron(eye(2^(n-i)),m2n(i))* kron(eye(2^(n-i)),kron(hadamard(2),eye(2^(i-1))))*y; end end the 2d slant-hadamard transform can be defined as sht(f) = sh * f * hs ′ , where f is the image and hsandsh ′ are slant-hadamard matrix and its transpose [9]. 5. cunclusion in this paper, an efficient algorithm for computation of parametric slant-hadamard transforms is investigated. the slant-hadamard matrix of order n 2 is presented as a product of sparse matrices and the appropriate fast slant-hadamard transform and its complexity are developed. in the end the detailed example of factoring of slant-hadamard transform matrix of order 8 and matlab code for implementation slant-hadamard transform on 1d case are presented. later the slant-hadamard transformation will be discussed which will be used in image processing. references [1] a. k. jain, fundamentals of digital image processing. prentice hall inc, englewood cliffs, nj 1989. [2] m. m. anguh and r. r. martin, “a truncation method for computing slant transforms with applications to image processing”, ieee transactions on communications, vol. 43, no. 6, pp. 21032110, 1995. [3] i. w. selesnick, “the slantlet transform”, ieee transactions on signal processing, vol. 47, no. 5, pp. 1304-1313, 1999. [4] n. ahmed and k. r. rao, orthogonal transforms for digital signal processing, new york, springer-verlag, 1975. [5] j. f. yang and c. p. fang, “centralized fast slant transform algorithms”, ieice transactions on fundamentals of electronics,communications and computer sciences, vol. e80a, no. 4, pp. 705711, 1997. [6] h. sarukhanyan, s. agaian, k. egiazarian and j. astola, “reversible hadamard transforms”, facta universities (nis), series: electronics and energetics, vol. 20, no. 3, pp. 309-330, 2007. [7] s. agaian, h. sarukhanyan, k. egiazarian and j. astola, hadamard transforms, spie press, bellingham, washington, usa, 2011. [8] s. agaian, k. tourshan and j. p. noonan, “parametric slant-hadamard transforms with applications”, ieee signal processing letters, vol. 9, no. 11, pp. 375-377, 2002. fast slant-hadamard transform algorithm 120 [9] s. agaian, k. tourshan and j. p. noonan, “parametric slant-hadamard transforms”, is&t/spie’s 15 th annual symposium electronic imaging science and technology, image processing: algorithms and systems ii, pp. 20-24 , santan clara, california, 2003. [10] h. sarukhanyan, “hadamard matrices: construction methods and applications”, in proc. of the workshop on transforms and filter banks, 35p., tampere, finland, 1998. [11] h. sarukhanyan, a. anoyan, s. agaian, k. egiazarian and j. astola, “fast hadamard transforms”, proc. of international ticsp workshop on spectral methods and multirate signal processing, smmsp'2001, pula, croatia, pp. 33-40, 2001. [12] s. agaian, h .sarukhanyan and k. tourshan, “new classes of sequential slant hadamard transform”, proc. of international ticsp workshop on spectral methods and multirate signal processing, smmsp’02, toulouse, france, 4p., 2002. [13] s. minasyan, d. guevorkian, s. agaian and h. sarukhanyan, ”on “slant-like” fast orthogonal transforms of arbitrary order”, vipromcom-2002, 4th eurasip – ieee region 8 international symposium on video/image processing and multimedia communications, zadar, croatia, p. 6, 2002. submitted 02.08.2014, accepted 28.11.2014. սլանթ-հադամարի ձևափոխության արագ ալգորիթմ ս. հակոբյան ամփոփում ուսումնասիրվում է պարամետրով սլանթ–հադամարի ձևափոխության արդյունավետ ալգորիթմը: n 2 կարգի սլանթ-հադամարի մատրիցը ներկայացվում է նոսր մատրիցների արտադրյալի տեսքով, ինչպես նաև մշակվում են սլանթհադամարի արագ ձևափոխությունն ու նրա բարդությունը: վերջում մանրամասն ներկայացվում են 8-րդ կարգի սլանթ-հադամարի ձևափոխության մատրիցի ֆակտորիզացիան և սլանթ-հադամարի ձևափոխությունը իրականացնող matlab կոդը: алгоритм быстрого преобразования сланта-адамара с. акопян аннотация в данной работе мы исследуем эффективный алгоритм параметрического преобразования сланта-адамара. матрица сланта-адамара порядка n2 представлена в виде произведения разреженных матриц. разработаны соответствующее быстрое преобразование слантаадамара и его сложность. представлены подробный пример факторизации матрицы преобразования и matlab код для реализации преобразования сланта-адамара . microsoft word tpel.doc mathematical problems of computer science 36, 91--98, 2012. 91 on some kinds of constructive fuzzy logics hovhannes r. bolibekyan and igor d. zaslavsky yerevan state university institute for informatics and automation problems of nas of ra e-mail: bolibekhov@ysu.am, zaslav@ipia.sci.am abstract some modification of the extended fuzzy constructive logic [14] is considered. it is proved that this modification is actually equivalent to the intuitionistic fuzzy logic introduced in [9]. references [1] h. bolibekyan and i. zaslavsky. “on the relations between two systems of fuzzy intuitionistic logic”. logic colloquium 2010, paris, july 25-31, université paris diderot, la halle aux farines, abstracts of contributed talks, p. 4, 2010. [2] h. b. enderton. a mathematical introduction to logic, 2nd edition, san diego, harcourt, academic press, 2001. [3] a. heyting. intuitionism (an introduction). north-hall publ. comp., amsterdam, 1956. [4] s. c. kleene. introduction to metamathematics. d. van nostrand comp., inc., new-yorktoronto, 1952. [5] s. n. manukian. “on some properties of recursively enumerable fuzzy sets”. in proceedings of the conference “computer science and information technologies”, csit-99, yerevan, armenia, pp. 5-6, 1999. [6] v. novak. fuzzy sets and their applications. adam hilger, bristol, 1989. [7] v. novák, i. perfilieva, j. močkoř. mathematical principles of fuzzy logic. kluwer academic publishers, 1999. [8] e. specker. “nicht konstructiv beweisbare sätze der analysis”. journ. of symb. logic, vol. 14, №3, pp. 145-158, 1949. [9] g. takeuti and s. titani. “intuitionistic fuzzy logic and intuitionistic fuzzy set theory”. journ. of symb. logic, vol. 49, №3, pp. 851-866, 1984. [10] j. yen, r. langari. fuzzy logic, intelligence, control, and information. prentice-hall, upper saddle river, new jersey, 1999. [11] l. zadeh. fuzzy sets. information and control, vol. 8, pp. 338-353, 1965. [12] и. д. заславский. “о конструктивной истинности суждений и некоторых нетрадиционных системах конструктивной логики”. труды вц ан арм. сср и егу, “математические вопросы кибернетики и вычислительной техники”, т.8, с. 99-153, 1975.  this research was supported by grants #11-1b023 and #11-1b189 of the government of the republic of armenia. on some kinds of constructive fuzzy logics 92 [13] и. д. заславский. “нечеткая конструктивная логика”. записки научных семинаров поми, “исследования по конструктивной математике и математической логике xi”, т. 358, с. 130-152, 2008. [14] и. д. заславский. “расширенная нечеткая конструктивная логика”. (to appear in “записки научных семинаров поми”). [15] б. а. кушнер. лекции по конструктивному математическому анализу. м., “наука”, 1973. [16] с. н. манукян. “о структуре нечетких рекурсивно перечислимых множеств”. труды института проблем информатики и автоматизации, “математические вопросы кибернетики и вычислительной техники”, т. 17, с. 86-91, 1997. [17] с. н. манукян. “некоторые алгебры рекурсивно перечислимых множеств и их приложения к нечеткой логике”. записки научных семинаров поми, “теория сложности вычислений viii”, т. 304, с. 75-98, 2003. [18] а. а. марков. “о конструктивной математике”. труды миан ссср, т. 67, с. 8-14, 1962. [19] г. с. цейтин, “один способ изложения теории алгорифмов и перечислимых множеств”. труды миан ссср, т. 72, с. 69-98, 1964. [20] н. а. шанин. “о конструктивном понимании математических суждений”. труды миан ссср, т. 52, с. 266-311, 1958. [21] н. а. шанин. “конструктивные вещественные числа и конструктивные функциональные пространства”. труды миан ссср, т. 67, с. 15-294, 1962 îáýëïñáõïïçí áã å³ñ½áñáß ïñ³ù³µ³ýáõãûáõýý»ñç áñáß ï³ñ³ï»ë³ïý»ñç ù³ëçý ð. ´áéçµ»ïû³ý ¨ æ. ¼³ëé³íëïç ²ù÷á÷áõù ¸çï³ñïíáõù ¿ áý¹é³ûýí³í áã å³ñ½áñáß ïáýëïñáõïïçí ïñ³ù³µ³ýáõãû³ý [14] áñáß í»ñ³÷áëáõãûáõý: ²å³óáõóíáõù ¿, áñ ³û¹ í»ñ³÷áëáõãûáõýá ÷³ëïáñ»ý ñ³ù³ñå»ù ¿ [9] ³ßë³ï³ýùáõù ë³ñù³ýí³í æýïáõçóçáýçëï³ï³ý áã å³ñ½áñáß ïñ³ù³µ³ýáõãû³ýá: î íåêîòîðûõ ðàçíîâèäíîñòÿõ íå÷åòêèõ êîíñòðóêòèâíûõ ëîãèê î. áîëèáåêÿí è è. çàñëàâñêèé àííîòàöèÿ ðàññìàòðèâàåòñÿ ìîäèôèêàöèÿ ðàñøèðåííîé íå÷åòêîé êîíñòðóêòèâíîé ëîãèêè [14]. äîêàçûâàåòñÿ, ÷òî äàííàÿ ìîäèôèêàöèÿ ôàêòè÷åñêè ýêâèâàëåíòíà è íòóèöèîíèñòñêîé íå÷åòêîé ëîãèêå, îïðåäåëåííîé â [9]. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 48, 50--56, 2017. multiplatform use-after-free and double-free detection in binaries * grigor s. keropyan, vahagn g. vardanyan, hayk k. aslanyan, shamil f. kurmangaleev and sergey. s. gaissaryan ivannikov institute for system programming of the ras e-mail: {goqor, vaag, hayk, kursh, ssg}@ispras.ru abstract use-after-free (uaf) defects are a class of memory corruption bugs, which occur when a program continues to use a pointer after it has been freed. double-free (df) defects arise when the same memory is freed more than once. the developed platform is capable to analyze binaries of several architectures (x86, x86-64, mips, power-pc, arm) and is based on program static analysis approach. for program analysis sdg (system dependence graph) machine-independent representation is used. sdg combines call graph, control and data flow graphs of the program. the tool consists of two main components: sdg generation and analysis of the obtained sdg. sdg generation is implemented using ida pro [1] disassembler and binnavi [2] static analysis platform. experimental results prove the scalability and effectiveness of the developed framework. the tool is tested on several test suits such as juliet [3]. it also has detected a number of well-known bugs in real-world projects ․ keywords: binary static analysis, use-after-free, dangling pointer detection. 1. introduction detecting bugs in software is an important task to help to improve security and reliability. it can be done at any time during the software lifecycle. ideally, all bugs are found during the testing phase before the software is deployed. however, software testing does not find all possible bugs at any given time. moreover, when a programmer uses memory unsafe languages, such as c/c++, these bugs can lead to security violations. despite this, these programming languages are still popular because of high performance. a pointer pointing to a memory location that has been deleted (or freed) is called a dangling pointer. uaf defect arises in case of the usage of dangling pointer and df defect arises in case of deleting (or freeing) it. there is a line of research specifically focusing on detecting uaf defects. in general, they can be detected through both static and dynamic analyses. static analysis methods are often 50 * the paper is supported by rfbr grant 18-07-01154 mailto:goqor@ispras.ru mailto:vaag@ispras.ru mailto:hayk@ispras.ru mailto:kursh@ispras.ru mailto:ssg@ispras.ru g. keropyan, v.vardanyan, h. aslanyan, s. kurmangaleev and s. gaissaryan 51 based on the disassembly process. dynamic analysis methods try to trace the program execution and search for memory state to carry out the analysis. in this paper, we introduce a new approach for detecting uaf defects which supports multiple target architectures and is based on graph representation of program. at first, a system dependence graph (sdg) is generated, which combines the call graph, interprocedural data dependences, the control flow graph and the data flow graph. the background for sdg generation is binnavi static analysis platform and reil (the reverse engineering intermediate language) representation [14]. finally, the main algorithm analyzes sdg for detecting uaf and df defects. fig. 1 represents the main architecture of the proposed tool. 2. related work several works are dedicated to source code analysis. n. nethercote develops an instrument named memcheck using valgrind [4] which can check all of the memory read / write operations and the interception of calls like malloc, new, free, delete. it is good for detecting memory management problems, but cannot deal with aliasing problems. cppcheck [5] can detect possible memory errors, and provides a simple analysis such as mismatching allocation and deallocation. more detailed analysis is done by polyspace [6] and frama-c [7]. the platforms gather several analysis techniques into a single collaborative extensible framework. these tools are mainly focused on safety verification and have several plugins for detecting security violations. usually the source code is not available and a binary analysis is required. additionally, the binary analysis can detect bugs, which do not exist in source code and are generated by compiler during various transformations. dynamic analysis solutions ( [8], [9], [10]) for detecting uaf defects generally introduce a high performance overhead, and may miss lots of bugs because they are only able to explore a small portion of the program execution space. in [11] the authors represent the tool gueb for static finding of uaf defects in binary files. at first, they track heap operations and address transfers, taking into account aliases, using a value analysis. then they use these results to statically identify the uaf defects. finally, they extract subgraphs, for each uaf, describing sequentially where the dangling pointer is created, freed and used. however, the whole process of detecting uaf is not automatic. for instance, the user should manually identify all wrappers to free/malloc functions, which, mainly are not applicable for large binaries. s. cesare represents an approach [12] for detecting uaf defects using decompilation and data flow analysis. as decompilation is not sound, program behavior may be underapproximated leading to false negatives. another tool, which is based on static analysis, is represented by d. dewey et al. [13]. they use gen-kill analysis for detecting available expressions, then they use data flow analysis for determining uaf defects. dataflow analysis is implemented on small number of x86 assembler instructions, hence their tool works only on x86 binaries. the suggested approach in the article makes possible to detect uaf and df bugs in binaries of several architectures. unlike gueb it does not need user interaction. the tool does not require high performance overhead and detects defects with 30% false positive rate in average. multiplatform use -after-free and double-free detection in binaries 52 fig.1 tool architecture. 3.1 sdg generation system dependence graph is one of the most detailed structures for representing semantics of the program. it combines call graph, interprocedural data dependences, control and data flow graphs. at the first stage of sdg generation target program is disassembled using ida pro. this allows to import disassembled binary into binnavi platform. binnavi platform is used to recover the structure of the target program, generate data, control and interprocedual dependencies between instructions and translate program to reil representation. reil representation is independent of target architecture and allows to analyze binaries from different architectures such as x86, x86_64, arm, mips and powerpc. furthermore, reil representation has only 17 atomic instructions and only two instructions for memory access (stm: store to memory, ldm: load from memory) which allows to simplify the analysis process. at first stage of sdg generation algorithm tries to find all free functions of the program. then the algorithm tries to slice program into independent pieces. slicing of the program is performed on the call graph. for each free function call, algorithm generates all the paths that are leading to that function. slicing allows to drop away the significant part of program and perform analysis only on the parts which contain memory free operations. the length of the paths can be configured by the user. finally, the algorithm generates sdgs for each slice. vertices of these sdgs correspond to reil instruction and edges correspond to control dependences, data and interprocedural data dependences between instructions. for each reil instruction an sdg vertex is constructed. two vertices are connected if there are data (interprocedural or intraprocedural) or control dependencies between the corresponding reil instructions (fig.2). generated slices may contain many common parts (functions). so, sdg generation algorithm uses dynamic programming approach: information about dependencies for all functions is cached in database and reused during sdg generation for each slice. another big advantage of the program slicing approach is the possibility to organize g. keropyan, v.vardanyan, h. aslanyan, s. kurmangaleev and s. gaissaryan 53 whole analysis parallel on each independent slice. this is especially important while analyzing big binaries (over 30 mb) where the size of the recovered graphs can grow enormously huge. fig.2 slice example. 3.2 detecting uaf and df the main analysis for detecting uaf and df bugs is performed on generated sdg slices. root of these slices is the call to the system free function. we start analysis from the system free function and try to find freed pointer (argument of free function). it can be easily done by using the existing data dependences. the difficulty arises when the freed pointer may have several aliases. performing alias analysis on the whole program can bring huge performance overhead. in our approach, we have implemented a simple analysis on sdg which tries to find all aliases for the given pointer within a function using available data and control dependencies. there are several possible cases for handling found freed variable: a. freed pointer is defined in the function, which contains call to free function. b. freed pointer is in global scope. c. freed pointer is passed as an argument to the observing function. first and second cases algorithm handles by traversing sdg by control dependencies starting from the free function call. traversing by control dependencies allows to find paths (if any) from free to use of the pointer (or one of its aliases). then, using data dependencies, algorithm checks redefinitions of pointer on each found path. if there are no redefinition on one of the paths, algorithm reports uaf or df error. third case, when the freed pointer is passed to the observing function as an argument, algorithm marks that function as free and adds it to the free functions list. then the whole analysis of finding uaf is repeated until no new free function is found. using sdg as an intermediate representation makes analysis path sensitive and interprocedual. however, there are some limitations in the current stage of the development. the main limitation is that during dependencies construction we assume that all arguments are passed to the functions by stack. support of different calling conversation is in active state of development. multiplatform use -after-free and double-free detection in binaries 54 3. experimental results suggested tool is tested against several well-known test suits such as juliet [3]. results show that we can find all the uaf and df errors presented in this suit with 5% false positive rate in average. the tool is also tested on dozens of real word programs that contain real uaf and df bugs. in the table 1 the list of projects with uaf and df defects are presented. these defects were admitted and fixed by the projects maintainers in the newer versions. last three columns of the table contain analysis results of suggested tool. all tests are performed on the server with 20 physical intel xeon 2.3 ghz processors. analysis time column in the table proves effectiveness of sdg splitting technique for parallel analysis. table 1: analysis result on several well-known projects containing uaf and df defects. project name version project size analysis time uaf and df count false positive 5 cores 10 cores 20 cores jasper 1.900.1 1 mb 231s 177s 166s 3 0% giflib 5.1.2 50 kb 13s 10s 10s 1 0% libtiff 4.0.3 1 mb 278s 160s 100s 12 33% gnomenettool 3.8.1 336 kb 74 52 48 1 0% openslp 1.2.1 700 kb 60s 56s 40s 4 50% libssh 0.5.2 700 kb 190 168 99 19 21% the presented tool is also capable to find uaf and df bugs in gnome-nettool and giflib packages of debian distributive. false positive rate for examined programs is competitive with other well-known techniques [9], [11], [12]. moreover, the presented tool is multiplatform and can be used for analyzing binaries for different target architectures. 4. conclusion and future work in the paper a binary static analysis tool is presented that detects uaf and dg defects on several modern architectures. the analysis is performed on system dependence graph (sdg) of the target program. the proposed algorithm includes two steps: generation of sdgs for program slices where the root of each slice is a “free” function and analysis of generated slices. alias analysis is performed on each graph to avoid performance overhead of finding aliases in the whole program. the approach makes it possible to perform the analysis in parallel for each independent slice. development and improvement of the tool is in active state. we plan to add support for different calling conventions and continue testing on different real-world programs and libraries to reduce false positive rate and improve quality of analysis. g. keropyan, v.vardanyan, h. aslanyan, s. kurmangaleev and s. gaissaryan 55 references [1] [online]. available: www.hex-rays.com/products/ida [2] [online]. available: www.zynamics.com/binnavi.html [3] [online]. available: www.samate.nist.gov/srd/testsuite.php [4] n. nethercote, "dynamic binary analysis and instrumentation", phd dissertation, university of cambridge, 2004. [5] [online]. available: www.cppcheck.sourceforge.net [6] [online]. available: www.mathworks.com/training-schedule/polyspace-code-prover-for-cccode-verification.html [7] p. cuoq, f. kirchner, n.kosmatov, v. prevosto, j. signoles and b. yakobowski, "framac—a software analysis perspective," sefm, pp. 233-247, 2012. [8] w. xu, d. c. duvarney and r. sekar, "an efficient and backwards-compatible transformation to ensure memory safety of c programs," acm sigsoft software engineering notes, vol. 29, pp. 117-126, 2004. [9] j. caballero, g. grieco, m. marron and a. nappa, "undangle: early detection of dangling pointers in use-after-free and double-free vulnerabilities," proceedings of the 2012 international symposium on software testing and analysis, pp. 133-143, 2012. [10] b. lee, c. song, y. jang and t. wang, "preventing use-after-free with dangling pointers nullification," in ndss symposium 2015, pp. 17-32, 2015. [11] j. feist, l. mounier and ml. potet, "statically detecting use after free on binary code," journal of computer virology and hacking techniques, vol. 10, no. 3, pp. 211-217, 2014. [12] s. cesare, "detecting bugs using decompilation and data flow analysis," in black hat usa, 2013. [13] d. dewey, b. reaves and p. traynor, "uncovering use-after-free conditions in compiled code," in 10th international conference on availability, reliability and security, 2015. [14] www.zynamics.com/binnavi/manual/html/reil_language.htm submitted 02.08.2017, accepted 20.11.2017. երկուական կոդում ազատված հիշողության օգտագործման և հիշողության կրկնակի ազատման սխալների հայտնաբերման բազմապլատֆորմ համակարգ գ. քերոբյան, վ․վարդանյան, հ․ասլանյան, շ․ կուրմանգալեև և ս․ գայսարյան ամփոփում ազատված հիշողության օգտագործման սխալները հիշողության աշխատանքի հետ կապված սխալների դաս է, որն առաջանում է, երբ ծրագիրը շարունակում է օգտագործել ազատված հիշողությունը։ մշակված համակարգը հնարավորություն է multiplatform use -after-free and double-free detection in binaries 56 տալիս վերլուծել տարբեր ճարտարապետությունների (x86, x86-64, mips, power-pc, arm) երկուական կոդ և հիմնված է կոդի ստատիկ անալիզի վրա։ ծրագրի անալիզի համար օգտագործվում է sdg (system dependence graph) մեքենայից անկախ ներկայացում: sdg-ն ներառում է ծրագրի կանչերի, ղեկավարման և կախվածությունների գրաֆները։ sdg-ն կառուցվում է՝ հիմնվելով ida pro [1] դիսասսեմբլերի և binnavi [2] ստատիկ անալիզի համակարգի վրա։ փորձարարական արդյունքները ապացուցում են ներկայացված համակարգի ընդլայնելիությունն ու արդյունավետությունը։ գործիքը թեստավորվել է բազմաթիվ թեստավորման համակարգերի վրա, որոնցից է juliet-ը [3]: այն նաև հայտնաբերել է բազմաթիվ հայտնի սխալներ կիրառական ծրագրերում։ мультиплатформное нахождение ошибок использования памяти после освобождения и повторного освобождения памяти в бинарном коде г. керопян, в. вартанян, а. асланян, ш. курмангалеев и с. гайсарян аннотация ошибки использования памяти после освобождения — это класс ошибок памяти, который возникает, когда программа продолжает использовать память после его освобождения. разработанная платформа способна анализировать бинарные коды нескольких архитектур (x86, x86-64, mips, power-pc, arm) и основан на статическом подходе. для анализа программ используется sdg (system dependence graph), независимое от машины представление. sdg объединяет графы вызовов, управления и потоков данных программы. инструмент состоит из двух основных компонентов: генерации sdg и анализа полученных sdg. генерация sdg реализована с использованием дисассемблера ida pro [1] и платформы статического анализа binnavi [2]. экспериментальные результаты доказывают масштабируемость и эффективность разработанного платформа. инструмент тестировался на разных тестовых наборах, таких как juliet [3]. он также обнаружил ряд известных дефектов в реальных проектах. references mathematical problems of computer science 53, 57--62, 2020. udc 004 development of management system for eduroam database updated specification arthur s. petrosyan and gurgen s. petrosyan institute for informatics and automation problems of nasra e-mail: arthur@sci.am, gurgen@sci.am abstract this paper presents a system developed to simplify the eduroam database management. the goal of the eduroam database is to provide the necessary information needed for the operation of the eduroam service and related supporting services like eduroam monitoring, service location maps and usage statistics, as well as the eduroam cat tool. eduroam database specifications have been updated to the second version and the update has recently become mandatory, requiring many changes for databases of all national roaming operators (nro). the solution presented here can be useful for nro administrators to simplify the eduroam database management. keywords: eduroam, nro, wi-fi, wireless, monitoring, statistics, database. 1. introduction eduroam (education roaming) [1] has become a well-known secure wireless network roaming access service used by the international research and education community. it enables scientists, researchers, teachers, students from different institutions to securely gain wi-fi internet access at any eduroam-enabled institution worldwide. the eduroam service organization model assumes that in each country there is a national roaming operator (nro), mostly operated by the national research and education network (nren) of that country. the eduroam database has been built as a central database with a mechanism that automatically collects data from nros to enable the operation of the eduroam service and related supporting services like eduroam monitoring, service location maps and usage statistics, as well as the eduroam cat tool. eduroam database specifications have been updated to the second version and the update have recently become mandatory [2], requiring many changes for all national roaming operators (nro). this paper presents a system developed for managing the eduroam database with specifications updated to the second version. 57 mailto:arthur@sci.am mailto:gurgen@sci.am development of management system for eduroam database updated specification 58 2. developed management system according to the new second version (v.2.0) of eduroam database specification [3], each nro should provide general data in the defined xml or json format. the data should be available at the specific, predefined urls: http://www.eduroam./general/ and must be accessible from the eduroam database server site monitor.eduroam.org. data for the eduroam database v.2.0 can be validated by xml [4] or json validators [5]. the developed management system is fully compliant with the eduroam database v.2.0 specification and provides a flexible web interface to manage data for a particular nro. we have chosen to implement only the json version in our system and omit the xml format, since one of them is required, not both, and json seems to be preferred by the eduroam database server-site monitoring. mysql/maria db was chosen as a backend for the implementation of eduroam database specification. appropriate mysql tables were designed and created for datasets (i.e., database tables) and respective fields (attributes) containing general data as described in [3]. the web interface created for database management provides the possibility to add/edit both the institutions and service locations. the main page of the web interface looks like the one presented in fig. 1. fig. 1. main page. the created web interface is english language-based. text fields in the database are bilingual (english and armenian), but only english fields are required. as a result, the system can be used by any nro worldwide. a. petrosyan and g. petrosyan 59 a separate contact table is created for each institution, and contact persons can be added from the web interface and then selected as responsible for multiple institutions (fig. 2). it is allowed to add up to five contact person per institution. fig. 2. contacts page. next, the institution can be added by filling in the required fields in the form presented in fig. 3. fig. 3. institution page. the locations interface then allows specifying the location data for each registered institution (fig. 4). required fields include location gps coordinates. the web interface of our system allows us to easily select a location on the map and put the appropriate coordinates into the form. this makes filling in the data much easier. development of management system for eduroam database updated specification 60 fig. 4. locations page. after having the data of institutions and service locations ready in the database, we can generate a json file, as well as validate it with the validators of central eduroam database server site. after generation, the system places the files in a special folder on the web server, the access to which is open only for the ip address of the central monitoring server eduroam, which downloads them from nros on a daily basis, checks them for compliance with the requirements, and updates them. to prevent data loss, all old previously generated json files are stored in a separate backup folder with the date and time of their generation. all the data can be edited and verified at any time via the web interface. after adding an organization, it is possible to add/edit/delete/sort the eduroam service locations for the organization in accordance with the database structure. in particular, the following fields are available: name, address, city, coordinates, connection status, contact persons, type of service location. in addition, there are specific fields such as: type of wi-fi coverage (single/area/mobile), name of the wi-fi network (ssid) that in most cases is “eduroam”, encryption (wpa, wpa2), number of access points and availability (physical access restrictions or no restrictions). 3. service location map along with the backend management system described above, a frontend service location map was also created. it takes data from the same institutions and service locations database and presents it on the nros website. the current version of it is part of armenian nro website [6] and is based on openstreetmap [7] and leaflet [8]. a. petrosyan and g. petrosyan 61 4. advantages similar eduroam database management systems were developed by greek nren grnet and called djnro [9], as well as by czech nren cesnet [10]. but the main issue of using this djnro is that it has service locations coordinates support based on the google maps api, which has become a paid service since 2018. instead, in our work we used the free openstreetmap/leaflet [7] [8] alternatives. the cesnet solution turned out to be hard to implement on our site because of many separated scripts and insufficiently documented solution. 5. conclusion this solution may be interesting for any nros worldwide. it is currently implemented in asnet-am for managing the armenian part of eduroam database. the solution we presented is all-in-one for nros functions regarding the management of eduroam database, as well as for presenting this data on the service location map. references [1] online]. available: https://www.eduroam.org/ [2] [online]. available: https://monitor.eduroam.org/fact_eduroam_db.php [3] [online]. available: https://monitor.eduroam.org/eduroam-database/v2/docs/eduroamdatabase-ver17102017.pdf [4] [online]. available: https://monitor.eduroam.org/eduroamdatabase/v2/scripts/xml_validation_test.php [5] [online]. available: https://monitor.eduroam.org/eduroamdatabase/v2/scripts/json_validation_test.php [6] [online]. available: https://www.eduroam.am/ [7] [online]. available: https://www.openstreetmap.org [8] [online]. available: https://leafletjs.com/ [9] [online]. available: https://github.com/grnet/djnro [10] [online]. available: https://github.com/cesnet/eduroam-db submitted 10.12.2019, accepted 04.05.2020. https://github.com/grnet/djnro development of management system for eduroam database updated specification 62 նորացված ձևաչափի eduroam տվյալների շտեմարանի կառավարման համակարգի մշակում արթուր ս. պետրոսյան և գուրգեն ս. պետրոսյան հհ գաա ինֆորմատիկայի և ավտոմատացման պրոբլեմների ինստիտուտ e-mail: arthur@sci.am, gurgen@sci.am ամփոփում այս հոդվածում ներկայացված է eduroam տվյալների շտեմարանի կառավարումը պարզեցնելու նպատակով մշակված համակարգը: eduroam տվյալների շտեմարանի նպատակը eduroam ծառայության և դրա հետ կապված աջակցության ծառայությունների համար անհրաժեշտ տեղեկատվություն տրամադրելն է, ինչպիսիք են` eduroam-ի մոնիտորինգը, ծառայության գտնվելու վայրի քարտեզները և օգտագործման վիճակագրությունը, ինչպես նաև` eduroam cat գործիքը: eduroam տվյալների շտեմարանը վերջերս թարմացվել է նոր ձևաչափին համապատասխան, ինչը մեծ փոփոխություններ է պահանջում բոլոր ազգային ռոումինգի օպերատորների համար (nro): այստեղ ներկայացված լուծումը կարող է օգտակար լինել nro-ների ադմինիստրատորների համար: բանալի բառեր` eduroam, nro, wi-fi, անլար, մոնիտորինգ, վիճակագրություն, տվյալների շտեմարան: разработка системы управления обновленной базой данных eduroam артур с. петросян и гурген с. петросян институт проблем информатики и автоматизации нан ра e-mail: arthur@sci.am, gurgen@sci.am аннотация в статье представлена система упрощающая управление базой данных eduroam. цель базы данных eduroam предоставить необходимую информацию для работы службы eduroam и связанных с ней вспомогательных услуг, таких как мониторинг eduroam, карты расположения служб и статистика использования, а также инструмент cat eduroam. спецификации базы данных eduroam были обновлены до второй версии, что требует значительных изменений баз сервиса для всех национальных операторов роуминга (nro). представленное здесь решение может быть полезным для администраторов управления базами данных eduroam nro. ключевые слова: eduroam, nro, wi-fi, беспроводная связь, мониторинг, статистика, база данных. mailto:arthur@sci.am mailto:gurgen@sci.am mailto:arthur@sci.am mailto:gurgen@sci.am d:\tex_522\main.dvi mathematical problems of computer science 52, 14{20, 2019. u d c 5 1 9 .1 relative lengths of p aths and cycles in 2-connected gr aphs zh o r a g. n iko g h o s ya n institute for informatics and automation problems of nas ra e-mail: zhora@ipia.sci.am abstract let l be the length of a longest path in a 2-connected graph g and c the circumference the length of a longest cycle in g. in 1952, dirac proved that c > p 2l, by noting that "actually c ¸ 2 p l, but the proof of this result, which is best possible, is rather complicated". let l1; l2; :::; lm be a vine on a longest path of g. in this paper, using the parameter m, we present a more general sharp bound for the circumference c including the bound c ¸ 2 p l as an immediate corollary, based on elementary arguments. keywords: longest cycle, longest path, circumference, vine. 1 . in t r o d u c t io n w e c o n s id e r o n ly u n d ir e c t e d g r a p h s wit h n o lo o p s o r m u lt ip le e d g e s . l e t g b e a 2 -c o n n e c t e d g r a p h . w e u s e c a n d l t o d e n o t e t h e c ir c u m fe r e n c e ( t h e le n g t h o f a lo n g e s t c yc le ) a n d t h e le n g t h ( t h e n u m b e r o f e d g e s ) o f a lo n g e s t p a t h o f g. a g o o d r e fe r e n c e fo r a n y u n d e ¯ n e d t e r m s is [1 ]. in 1 9 5 2 , d ir a c [2 ] p r o ve d t h e fo llo win g . t heor em a: [2 ]. in every 2-connected graph, c > p 2 l. in t h e s a m e p a p e r [2 ], d ir a c c o n s id e r e d a s h a r p ve r s io n o f th e o r e m a b y n o t in g t h a t " a c t u a lly c ¸ 2 p l, b u t t h e p r o o f o f t h is r e s u lt , wh ic h is b e s t p o s s ib le , is r a t h e r c o m p lic a t e d " . a n a lo g o u s qu e s t io n s we r e s t u d ie d fo r k-c o n n e c t e d g r a p h s wh e n k ¸ 3 b y b o n d y a n d l o c ke ( [4 ],[5 ]) . in t h is p a p e r , u s in g a n e w p a r a m e t e r , we p r e s e n t a m o r e g e n e r a l s h a r p b o u n d fo r t h e c ir c u m fe r e n c e c in 2 -c o n n e c t e d g r a p h s in t e r m s o f l a n d t h e le n g t h o f a vin e o n a lo n g e s t p a t h o f g, in c lu d in g t h e b o u n d c ¸ 2 p l a s a c o r o lla r y, b a s e d o n e le m e n t a r y a r g u m e n t s . in o r d e r t o fo r m u la t e t h is r e s u lt , we n e e d s o m e a d d it io n a l d e ¯ n it io n s a n d n o t a t io n s . th e s e t o f ve r t ic e s o f a g r a p h g is d e n o t e d b y v ( g) a n d t h e s e t o f e d g e s b y e ( g) . if q is a p a t h o r a c yc le , t h e n t h e le n g t h o f q, d e n o t e d b y l ( q ) , is je ( q ) j t h e n u m b e r o f e d g e s in q. w e wr it e a c yc le q wit h a g ive n o r ie n t a t io n b y ¡! q . fo r x; y 2 v ( q ) , we d e n o t e b y 1 4 zh. nikoghosyan 1 5 x ¡! q y t h e s u b p a t h o f q in t h e c h o s e n d ir e c t io n fr o m x t o y. w e u s e p = x ¡! p y t o d e n o t e a p a t h wit h e n d ve r t ic e s x a n d y in t h e d ir e c t io n fr o m x t o y. w e s a y t h a t ve r t e x z1 p r e c e d e s ve r t e x z2 o n ¡! q if z1, z2 o c c u r o n ¡! q in t h is o r d e r , a n d in d ic a t e t h is r e la t io n s h ip b y z1 á z2. w e will wr it e z1 ¹ z2 wh e n e it h e r z1 = z2 o r z1 á z2. l e t p = x ¡! p y b e a p a t h . a vin e o f le n g t h m o n p is a s e t fli = xi ¡! l iyi : 1 · i · mg o f in t e r n a lly-d is jo in t p a t h s s u c h t h a t ( a ) v ( li ) \ v ( p ) = fxi; yig ( i = 1 ; :::; m) , ( b ) x = x1 á x2 á y1 ¹ x3 á y2 ¹ x4 á ::: ¹ xm á ym¡1 á ym = y o n p . th e m a in r e s u lt is t h e fo llo win g . t heor em 1: l et g be a 2-connected graph. if fl1; l2; :::; lmg is a vine on a longest path of g, then c ¸ 8 >>< >>: 2l m+1 + m+1 2 ; when m is odd; 2l¡ 1 2 m+1 + m+1 2 ; when m is even: e quivalently, theorem 1 can be formulated as follows, implying d irac's conjecture as an immediate corollary. t heor em 2: l et g be a 2-connected graph. if fl1; l2; :::; lmg is a vine on a longest path of g, then c ¸ 8 >>< >>: q 4 l + ( c ¡ m ¡ 1 ) 2; when m is odd; q 4 l + ( c ¡ m ¡ 1 ) 2 ¡ 1 ; when m is even: cor ollar y 1: in every 2-connected graph, c ¸ 2 p l. note that if m is odd, then c > 2 p l. the following lemma guarantees the existence of at least one vine on a longest path in a 2-connected graph. t he vine lemma: [3 ]. l et g be a k-connected graph and p a path in g. then there are k ¡ 1 pairwise-disjoint vines on p . 2 . p r o o fs p r oof of t heor em 1. l e t p = x ¡! p y b e a lo n g e s t p a t h in g a n d le t fli = xi ¡! l iyi : 1 · i · mg b e a vin e o f le n g t h m o n p . p u t 1 6 relative lengths of paths and cycles in 2-connected graphs li = xi ¡! l iyi ( i = 1 ; :::; m) ; a1 = x1 ¡! p x2; am = ym¡1 ¡! p ym; ai = yi¡1 ¡! p xi+1 ( i = 2 ; 3 ; :::; m ¡ 1 ) ; bi = xi+1 ¡! p yi ( i = 1 ; :::; m ¡ 1 ) ; l ( ai ) = ai ( i = 1 ; :::; m ) ; l ( bi ) = bi ( i = 1 ; :::; m ¡ 1 ) : u s in g t h e g ive n vin e l1; l2; :::; lm, we c o n s t r u c t a n u m b e r o f a p p r o p r ia t e c yc le s a n d o b t a in a lo we r b o u n d fo r t h e c ir c u m fe r e n c e a s a m e a n o f t h e ir le n g t h s . fir s t , we p u t q0 = m[ i=1 ( ai [ li ) ; qi = m¡i[ j=i+1 ( aj [ lj ) [ bi [ bm¡i; wh e r e i 2 f 1 ; 2 ; :::; ( m ¡ 1 ) = 2 g wh e n m is o d d , a n d i 2 f 1 ; 2 ; :::; ( m ¡ 2 ) =2 g wh e n m is e ve n . s in c e l ( li ) ¸ 1 ( i = 1 ; 2 ; :::; m) a n d a1 ¸ 1 , am ¸ 1 , we h a ve c ¸ l ( q0 ) = mx i=1 l ( li ) + a1 + am + m¡1x i=2 ai ¸ m + a1 + am: ( 1 ) case 1. m is o d d . fo r e a c h i 2 f 1 ; 2 ; :::; ( m ¡ 1 ) = 2 g, we h a ve c ¸ l ( qi ) = bi + bm¡i + m¡ix j=i+1 ( aj + l ( lj ) ) ¸ bi + bm¡i + m¡ix j=i+1 aj + m ¡ 2 i: ( 2 ) b y s u m m in g ( 1 ) a n d ( 2 ) , we g e t m + 1 2 c ¸ m¡1 2x i=0 l ( qi ) ¸ m¡1x i=1 bi + mx i=1 ai + m + 1 2 m ¡ 2 m¡1 2x i=1 i: s in c e l = pm¡1 i=1 bi + pm i=1 ai, we h a ve m + 1 2 c ¸ l + m + 1 2 m ¡ m 2 ¡ 1 4 = l + ( m + 1 ) 2 4 ; im p lyin g t h a t c ¸ 2 l m + 1 + m + 1 2 : case 2. m is e ve n . a s in ca s e 1 , fo r e a c h i 2 f 1 ; 2 ; :::; ( m ¡ 2 ) =2 g, c ¸ l ( qi ) ¸ bi + bm¡i + m¡ix j=i+1 aj + m ¡ 2 i: ( 3 ) zh. nikoghosyan 1 7 case 2.1. 1 2 c ¸ b m 2 . b y s u m m in g ( 1 ) , ( 3 ) a n d 1 2 c ¸ b m 2 , we g e t m 2 c + 1 2 c ¸ ã m¡1x i=1 bi + mx i=1 ai ! + m¡2 2x i=0 ( m ¡ 2 i) = l + m 2 m ¡ 2 m¡2 2x i=0 i = l + m ( m + 2 ) 4 ; im p lyin g t h a t c ¸ 2 l ¡ 1 2 m + 1 + m + 1 2 : case 2.2. 1 2 ( c + 1 ) · b m 2 . p u t r0 = b m 2 [ m 2[ i=1 ( ai [ li ) ; rm = b m 2 [ m[ i= m+2 2 ( ai [ li ) : fu r t h e r , fo r e a c h i 2 f 1 ; 2 ; :::; m¡2 2 g, we p u t ri = b m 2 [ bi [ m 2[ j=i+1 ( aj [ lj ) ; rm¡i = b m 2 [ bm¡i [ m¡i[ j= m+2 2 ( aj [ lj ) : th e n c le a r ly, c ¸ l ( r0 ) = b m 2 + m 2x i=1 ( ai [ l ( li ) ) ¸ b m 2 + m 2x i=1 ai + m 2 ; ( 4 ) c ¸ l ( rm ) = b m 2 + mx i= m+2 2 ( ai [ l ( li ) ) ¸ b m 2 + mx i= m+2 2 ai + m 2 : ( 5 ) fu r t h e r m o r e , fo r e a c h i 2 f 1 ; 2 ; :::; m¡2 2 g, c ¸ l ( ri ) = b m 2 + bi + m 2x j=i+1 ( aj + l ( lj ) ) ¸ b m 2 + bi + m 2 ¡ i; ( 6 ) 1 8 relative lengths of paths and cycles in 2-connected graphs c ¸ l ( rm¡i ) = b m 2 + bm¡i + m¡ix j= m+2 2 ( aj + l ( lj ) ) ¸ b m 2 + bm¡i + m 2 ¡ i: ( 7 ) b y s u m m in g ( 4 ) , ( 5 ) , ( 6 ) a n d ( 7 ) , we g e t mc ¸ mb m 2 + mx i=1 ai + ã m¡1x i=1 bi ¡ b m 2 ! + m m 2 ¡ 2 m¡2 2x i=1 i = ( m ¡ 1 ) b m 2 + ã mx i=1 ai + m¡1x i=1 bi ! + m2 + 2 m 4 ¸ ( m ¡ 1 ) ( c + 1 ) 2 + l + m2 + 2 m 4 ; im p lyin g t h a t c ¸ 2 l m + 1 + ( m + 2 ) 2 ¡ 6 2 ( m + 1 ) ¸ 2 l ¡ 1 2 m + 1 + m + 1 2 : th e o r e m 1 is p r o ve d . p r oof of t heor em 2. b y ( 1 ) , c ¸ m + a1 + a2 ¸ m + 2 . l e t c = m + y + 2 fo r s o m e in t e g e r y ¸ 0 . b y s u b s t it u t in g m = c ¡ y ¡ 2 in th e o r e m 1 , we g e t c ¸ q 4 l + ( y + 1 ) 2 = q 4 l + ( c ¡ m ¡ 1 ) 2 wh e n m is o d d ; a n d c ¸ q 4 l + y2 = q 4 l + ( c ¡ m ¡ 2 ) 2 wh e n m is e ve n . th e o r e m 2 is p r o ve d . to s h o w t h e s h a r p n e s s o f t h e b o u n d s in th e o r e m s 1 a n d 2 , le t p = x ¡! p y b e a p a t h a n d le t fli = xi ¡! l iyi : 1 · i · mg b e a vin e o n p . p u t li = xi ¡! l iyi ( i = 1 ; :::; m) ; a1 = x1 ¡! p x2; am = ym¡1 ¡! p ym; ai = yi¡1 ¡! p xi+1 ( i = 2 ; 3 ; :::; m ¡ 1 ) ; bi = xi+1 ¡! p yi ( i = 1 ; :::; m ¡ 1 ) ; l ( ai ) = ai ( i = 1 ; :::; m ) ; l ( bi ) = bi ( i = 1 ; :::; m ¡ 1 ) : l e t y ¸ 0 b y a n in t e g e r a n d a1 = am = y 2 + 1 ; a2 = a3 = ::: = am¡1 = 0 ; bi = bm¡i = y 2 + i + 1 ( i = 1 ; 2 ; :::; b( m ¡ 1 ) =2 c) : zh. nikoghosyan 1 9 if m is o d d , t h e n it is e a s y t o s e e t h a t c = m + y + 2 = 2 l m + 1 + m + 1 2 = q 4 l + ( c ¡ m ¡ 1 ) 2: if m is e ve n , we p u t bm=2 = y 2 + m+2 2 , im p lyin g t h a t c = m + y + 2 = 2 l ¡ 1 2 m + 1 + m + 1 2 = q 4 l + ( c ¡ m ¡ 1 ) 2 ¡ 1 : th u s , t h e b o u n d s in th e o r e m s 1 a n d 2 a r e b e s t p o s s ib le . references [1 ] j.a . b o n d y a n d u .s .r . mu r t y, graph theory with applications, ma c m illa n , l o n d o n a n d e ls e vie r , n e w y o r k, 1 9 7 6 . [2 ] g.a . d ir a c , \ s o m e t h e o r e m s o n a b s t r a c t g r a p h s " , p roc. l ondon, m ath. soc., vo l. 2 , p p . 6 9 -8 1 , 1 9 5 2 . [3 ] j. a . b o n d y, b asic graph theory: p aths and circuits, h a n d b o o k o f c o m b in a t o r ic s , vo l. 1 ,2 , e ls e vie r , a m s t e r d a m , 1 9 9 0 . [4 ] j. a . b o n d y a n d s . c. l o c ke , \ r e la t ive le n g t h s o f p a t h s a n d c yc le s in k-c o n n e c t e d g r a p h s " , annals of d iscrete m athematics, vo l. 8 , p p . 2 5 3 -2 5 9 , 1 9 8 0 . [5 ] s .c. l o c ke , \ r e la t ive le n g t h s o f p a t h s a n d c yc le s in k-c o n n e c t e d g r a p h s " , j . of comb. theory, s e r ie s b 3 2 , p p . 2 0 6 -2 2 2 , 1 9 8 2 . submitted 25.05.2019, accepted 10.10.2019. þõã³ý»ñç ¨ óçïé»ñç ñ³ñ³µ»ñ³ï³ý »ñï³ñáõãûáõýý»ñç ù³ëçý 2-ï³å³ïóí³í ·ñ³ýý»ñáõù äáñ³ ¶. üçïáõáëû³ý ðð ¶²² æýýáñù³ïçï³ûç ¨ ³íïáù³ï³óù³ý åñáµé»ùý»ñç çýëïçïáõï e-mail: zhora@ipia.sci.am ²ù÷á÷áõù ¸çóáõù l-á 2-ï³å³ïóí³í g ·ñ³ýç ³ù»ý³»ñï³ñ ßõã³ûç »ñï³ñáõãûáõýý ¿, çëï c-ý` ³ù»ý³»ñï³ñ óçïéç »ñï³ñáõãûáõýá: ¸çñ³ïá 1952-çý óáõûó ïí»ó, áñ c > p 2 l, ùç³å³ù³ý³ï ýß»éáí, áñ çñ³ï³ýáõù c ¸ 2 p l, áñá ñý³ñ³íáñ é³í³·áõûýý ¿, µ³ûó ³ûë ·ý³ñ³ï³ï³ýç ³å³óáõûóá µ³í³ï³ý³ã³÷ µ³ñ¹ ¿: ¸çóáõù l1; l2; :::; lmá g ·ñ³ýç ³ù»ý³»ñï³ñ ßõã³ûç íñ³ ùç µ³õ»õ ¿: ü»ñï³ ³ßë³ï³ýùáõù m å³ñ³ù»ïñç û·ýáõãû³ùµ µvñíáõù ¿ ùç ³í»éç áý¹ñ³ýáõñ ·ý³ñ³ï³ï³ý, áñï»õçó 2 0 relative lengths of paths and cycles in 2-connected graphs c ¸ 2 p l ·ý³ñ³ï³ï³ýá µëáõù ¿ áñå»ë ³ýùçç³ï³ý ñ»ï¨³ýù` ñçùýí³í áã µ³ñ¹ ¹³ïáõáõãûáõýý»ñç íñ³: ´³ý³éç µ³é»ñ` ³ù»ý³»ñï³ñ óçïé, ³ù»ý³»ñï³ñ ßõã³, ³ù»ý³»ñï³ñ óçïéç »ñï³ñáõãûáõý, µ³õ»õ: îòíîñèòåëüíûå äëèíû öåïåé è öèêëîâ â 2-ñâÿçíûõ ãðàôàõ æîðà ã. íèêîãîñÿí èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà e-mail: zhora@ipia.sci.am àííîòàöèÿ ïóñòü l îáîçíà÷àåò äëèíó äëèííåéøåé öåïè ãðàôà g, à c îáîçíà÷àåò äëèíó äëèííåéøåãî öèêëà. â 1952ã. äèðàê äîêàçàë, ÷òî c > p 2 l, îòìåòèâ, ÷òî ”â ñàìîì äåëå èìååò ìåñòî c ¸ 2 p l, ÷òî óëó÷øèòü íåâîçìîæíî, íî äîêàçàòåëüñòâî ýòîé îöåíêè äîñòàòî÷íî ñëîæíî”. ïóñòü l1; l2; :::; lm ïëþù íà äëèííåéøåé öåïè ãðàôà g. â íàñòîÿùåé ðàáîòå ïðèâîäèòñÿ íîâàÿ áîëåå îáùàÿ îöåíêà, îòêóäà âûòåêàåò ñïðàâåäëèâîñòü îöåíêè c ¸ 2 p l êàê íåïîñðåäñòâåííîå ñëåäñòâèå, îñíîâàííîå íà ýëýìåíòàðíûõ ñîîáðàæåíèé. êëþ÷åâûå ñëîâà: äëèííåéøèé öèêë, äëèíííåéøàÿ öåïü, äëèíà äëèííåéøåãî öèêëà, ïëþù. 18 mathematical problems of computer science 56, 18--34, 2021. udc 004.932 single image joint motion deblurring and super-resolution using the multi-scale channel attention modules misak t. shoyan national polytechnic university of armenia e-mail: misakshoyan@gmail.com abstract during the last decade, deep convolutional neural networks have significantly advanced the single image super-resolution techniques reconstructing realistic textural and spatial details. in classical image super-resolution problems, it is assumed that the low-resolution image has a certain downsampling degradation. however, complicated image degradations are inevitable in real-world scenarios, and motion blur is a common type of image degradation due to camera or scene motion during the image capturing process. this work proposes a fully convolutional neural network to reconstruct high-resolution sharp images from the given motion blurry low-resolution images. the deblurring subnetwork is based on multi-stage progressive architecture, while the super-resolution subnetwork is designed using the multi-scale channel attention modules. a simple and effective training strategy is employed where a pretrained frozen deblurring module is used to train the super-resolution module. the deblurring module is unfrozen in the last training phase. experiments show that, unlike the other methods, the proposed method reconstructs relatively small structures and textural details while successfully removing the complex motion blur. the implementation code and the pre-trained model are publicly available at https://github.com/misakshoyan/joint-motion-deblur-and-sr. keywords: motion deblurring, super-resolution, channel attention. 1. introduction single image super-resolution (sisr) addresses the problem of recovering a sharp, highresolution (hr) image from a given low-resolution (lr) image. the super-resolution (sr) problem has become very popular during the last decade. its solution is beneficial for a wide range of applications such as object detection, object recognition, surveillance, etc. image sr is an ill-posed problem since multiple possible hr images correspond to a single lr image. mailto:misakshoyan@gmail.com https://github.com/misakshoyan/joint-motion-deblur-and-sr m. shoyan 19 in classical sr problems, the lr image is assumed to be the bicubically downsampled version of the hr image with known or small degradations. however, complicated image degradations are inevitable in real-world scenarios, and image blur is a common type of image degradation. during the image capturing process, the lr image may be degraded by various blur effects, such as motion blur, resulting from camera or scene motion during the exposure. therefore, upscaling lr images with the sr technique will cause distorted hr results. so, it is essential to effectively combine deblurring and sr techniques to address the motion blurry image super-resolution problem. the motion deblurring problem is also highly ill-posed as multiple possible deblurred images correspond to a single motion blurry image. in this work, the problem of restoring the hr sharp image from a given motion blurry lr image is addressed. the image degradation process consists of two parts for the motion blurry image superresolution problem: blurring and downsampling. so, to tackle this joint problem, both deblurring and sr problems should be solved. during the last decade, deep convolutional neural networks and transformer-based [1] architectures have significantly advanced the image deblurring [2-7] and sr [8-12] techniques. although both the image deblurring [2-7] and sr [8-12] methods generate state-of-the-art results, naively cascading them sequentially does not reconstruct blurry images well, as it is shown in [13]. there are several reasons: first, the error estimated from the first module will be magnified by the second module leading to error accumulation. second, these two tasks are correlated, and it is sub-optimal to employ the feature extraction and image reconstruction phases twice. the features extracted from the deblurring module can be reused in the sr module to reconstruct the spatial details of the image. several recent methods jointly solve the motion blurry image super-resolution problem. zhang et al. [14] proposed a dual-branch architecture to extract deblurring and sr features parallel and fuse them by the recursive gate module. however, the proposed deblurring and sr modules extract independent features, leading to sub-optimal results when blur is significant. shoyan et al. [13] propose a single branch architecture by reusing the features extracted by the hierarchical layers of mprnet [3] for image super-resolution. they refine the features extracted by mprnet [3] and propagate them to the reconstruction module. however, their reconstruction module is not large enough to fully reuse the features extracted from the deblurring module. recently, the ntire 2021 image deblurring challenge [15] was held where the image deblurring track 1 (low resolution) addresses the joint image deblurring and super-resolution problem. several methods were proposed under the competition track 1 in the challenge. bai et al. [16] developed a cascaded non-local residual network (cnlrn). the deblurring subnetwork is based on encoder-decoder architecture like srn [2]. as a super-resolution subnetwork, they develop a non-local residual network to increase the resolution of the reconstructed features progressively. they propose a non-local block based on the self-attention mechanism [1] and use it in the sr subnetwork, thus achieving the 3rd psnr [17] / ssim [18] / lpips [19] scores in the ntire 2021 challenge [15]. however, as shown in fig. 1, their method fails to remove the complex motion blur and recover enough sharpness for relatively small structures present in the lr blurry image. also, the proposed network has a large number of parameters (~81m) and, therefore, has high computational complexity. xi et al. [20] proposed a pixel-guided dual-branch attention network (pdan). they design a residual spatial and channel attention module (rsca) for feature extraction. they propose a hard pixel example mining (hpem) loss to pay more attention to pixels with complicated degradation. their method achieves the 2nd psnr/ssim scores in the ntire 2021 challenge [15]. however, the network performance mainly depends on the deep architecture and synthesized dataset. the network has about 61m parameters and considerable computational complexity as [16]. also, the source code, pre-trained model, and network implementation details, such as the number of blocks used in feature extraction, reconstruction, and deblurring modules, are not publicly available, making their results nonreproducible. xu et al. [21] proposed an enhanced deep pyramid network (edpn). they adjust a single image joint motion deblurring and super-resolution using the multi-scale channel attention modules 20 video restoration architecture to the blurry image super-resolution problem. five replicated images are generated from the input image and fed into the network to extract the degraded lr image’s self-scale and cross-scale similarity information. their proposed pyramid progressive transfer (ppt) module performs feature alignment on the replicated images while the pyramid self-attention (psa) module fuses the aligned features. the proposed network achieves the 1st psnr/ssim/lpips scores in the ntire 2021 challenge [15]. however, no pre-trained model is available for reproducing the results. also, the publicly available source code is one of the ablation studies and has inconsistencies with some implementation details described in the paper. blurry lr image ground truth hr blurry lr input shoyan et al. cnlrn proposed method fig. 1. qualitative comparison between shoyan et al. [13], cnlrn [16], and the proposed method. please zoom in for the best view. to tackle the problems mentioned above, a single branch architecture is proposed to effectively estimate the hr sharp image from the given motion blurry lr image. as shown in fig. 1 (as well as in the results section), in contrast to the other methods, the proposed method removes the complex motion blur from the small structures full of edges and generates an hr sharp image closer to the ground truth. compared to the architectures suggested in [16] and [20], the proposed method has fewer parameters and, therefore, less computational complexity (see the results section). since mprnet [3] generates contextually and spatially enriched features in its 3rd stage, the proposed architecture reuses these deblurring features in the reconstruction module to recover the hr sharp image. the reconstruction module is designed based on multiscale channel attention modules (ms-cam) [22] to process the small structures better. to train the whole network, a simple and effective training strategy is employed where 1) firstly, the deblurring module is trained, then 2) the sr module is trained using the pre-trained frozen deblurring module, and finally, 3) the deblurring module is unfrozen for joint end-to-end training. the contributions of this paper are as follows: • an end-to-end single branch architecture is proposed to effectively reconstruct the hr sharp image from the given motion blurry lr image. • a reconstruction module is designed based on ms-cam blocks [22] to better process the small structures in the lr blurry image. • a simple and effective training strategy is exploited where the pre-trained frozen deblurring module is used as a feature extractor to train the sr module. then the whole network is jointly trained with the unfrozen deblurring module. the source code and the pre-trained model are publicly available at https://github.com/misakshoyan/joint-motion-deblur-and-sr. https://github.com/misakshoyan/joint-motion-deblur-and-sr m. shoyan 21 2. related work this section briefly reviews the motion deblurring, super-resolution, motion blurry superresolution techniques and the related attention mechanism. motion deblurring. the motion deblurring problem for dynamic scenes is highly illposed since the blur kernel is spatially varying and unknown (non-uniform blind deblurring problem). conventional blind motion deblurring methods jointly estimate the blur kernel and sharp image by solving computationally expensive optimization problems [23, 24]. simplified priors are employed to regularize the solution space, such as dark channel prior [24], totalvariation prior [23], etc. however, the simplified assumptions on the blur model make these techniques non-generalizable for real-world applications. recently, deep convolutional neural networks (cnn) have achieved significant success in image deblurring [2-7]. since the blur kernel estimation is not practical in real-world applications, recent methods directly map the blurry image to the corresponding sharp image by designing an end-to-end image deblurring architecture. nah et al. [6] proposed a deep multi-scale cnn called deepdeblur to mimic the conventional coarse-to-fine optimization approaches. tao et al. [2] proposed a scale recurrent network (srn) and improved the deepdeblur [6] network by sharing the network weights at different scales where each scale employs an encoder-decoder architecture. zamir et al. [3] proposed a multi-stage progressive image restoration architecture called mprnet. they employ an encoder-decoder subnetwork in the first and second stages to learn contextually enriched features due to large receptive fields. in contrast to the first and second stages, the last stage does not contain a downsampling operation. it employs a singlescale pipeline to generate spatially enriched features via channel attention blocks (cab) [3]. a supervised attention module (sam) is introduced to control the information flow between the stages, which refines the features before propagating them to the next stage. the mprnet achieves competitive results in motion deblurring, image denoising, and deraining problems. chu et al. [7] further improved the mprnet results by exploring the statistics distribution inconsistency between training (with image patches) and testing (with full-size image) phases. they argue that the operations, which aggregate the global spatial statistics along the entire spatial dimension, may lead to a statistics distribution shift between training and testing phases since different size images are used in these phases. this statement mainly refers to the global average pooling operation used in cab. to address this issue, they propose a test-time local statistics converter (tlsc) mechanism to calculate the mean value in a local window for each pixel rather than calculating a single mean value for the entire spatial dimension. they replace the global average pooling layer of cab with tlsc-based local average pooling layers for each pixel at the test-time without re-training or fine-tuning the network. zamir et al. [4] proposed a transformer-based [1] encoder-decoder architecture called restormer. they calculate the selfattention [1] across channels rather than the spatial dimension, thus implicitly encoding the global context with linear complexity. super-resolution. like the motion deblurring problem, the sr problem is also highly ill-posed. early approaches employ interpolation techniques such as bicubic and bilinear interpolations [25, pp. 77-78]. some methods rely on other techniques such as neighbor embedding [26], sparse coding [27], etc. cnn-based methods [8-10] have achieved unprecedented success in single image sr during the last decade. zhang et al. [8] designed a very deep residual channel attention network (rcan). they focus on learning high-frequency information by employing residual in residual structure with long and short skip connections to bypass the abundant low-frequency information. the channel attention (ca) mechanism is proposed to model the interdependencies among feature channels inspired by the squeeze and excitation mechanism [28]. they design residual channel attention block (rcab) using conv single image joint motion deblurring and super-resolution using the multi-scale channel attention modules 22 relu-conv structure followed by ca block and construct the network based on rcabs. dai et al. [9] designed a second-order attention network (san) by introducing second-order channel attention (soca) block to learn the channel-wise feature interdependencies better and to focus on more informative features using second-order channel statistics. they exploit share-source skip connections to bypass more abundant low-frequency information present in the lr image. niu et al. [10] proposed a holistic attention network (han) based on rcan [8]. the proposed layer attention module (lam) allows the network to learn the interrelationships between features of different layers and focus on more informative layers. a channel-spatial attention module (csam) is introduced to learn the inter-channel and intra-channel dependencies for the last layer of the network. however, the discussed sr methods assume known or small degradations and amplify the blur present in the lr image, as shown in [13]. therefore, the sr network needs also to incorporate motion deblurring techniques. joint motion deblurring and super-resolution. several recent methods solve the motion blurry image super-resolution problem by incorporating motion deblurring and sr techniques in a unified network. zhang et al. [14] proposed a gated fusion network (gfn) to extract deblurring and sr features separately by designing a dual-branch architecture. the gate module fuses these features, and the reconstruction module generates an hr sharp image using the fused features. however, in their design, the sr branch operates on the blurry lr image, which limits the reconstruction ability of the network to generate sharp hr results, as shown in [13]. in contrast to gfn, shoyan et al. [13] proposed a single branch architecture. they suggest reusing the features extracted by the hierarchical layers of mprnet [3] since it generates contextually and spatially enriched features in its deep layers. these features are then refined and fed to the reconstruction module to generate the four times upscaled sharp hr image. however, it seems that their reconstruction module could be a bit larger to reuse the features extracted from the deblurring module fully. several methods were proposed in the scope of ntire 2021 image deblurring challenge [15] under the image deblurring track 1 (low resolution) to solve the joint motion deblurring and sr problem. bai et al. [16] cascaded the deblurring and super-resolution modules in a unified network. the deblurring module employs an encoder-decoder architecture like srn [2], without a recurrent mechanism. as a super-resolution subnetwork, they develop a non-local residual network that contains rcabs [8] and self-attention-based [1] non-local blocks. the non-local block aims to model the global information for residual blur removal. a gradient loss function is developed to preserve the edges of the reconstructed hr image. they also propose a progressive upsampling mechanism to increase the resolution of the reconstructed features progressively and achieve the top-3 psnr [17] / ssim [18] / lpips [19] scores in the lowresolution track 1 of the ntire 2021 challenge [15]. however, as shown in fig.1 (see also the results section), this method fails to recover enough sharpness in its hr output when the relatively small structures of the lr blurry image are full of edges. also, the non-local blocks and progressive upsampling mechanism have considerable computational complexity due to the self-attention mechanism [1]. in addition, the deblurring module has about 70m parameters because of the large convolutional kernel size. xi et al. [20] proposed a pixel-guided dual-branch attention network. the proposed residual spatial and channel attention module aims to better extract informative features by fusing cross-channel and spatial information. the deblurring module is based on an encoderdecoder architecture that employs residual blocks. unlike the other architectures, the deblurring module is used only in the training stage for computational efficiency and helps the network extract and learn more useful deblurring information. the sr module is mainly composed of convolutional layers and pixel shuffling layers [29]. they argue that some pixels of the lr m. shoyan 23 degraded image may contain a large amount of blur and downsampling degradation. in contrast, the other pixels may be only affected by downsampling, so they propose the hard pixel example mining loss to pay more attention to pixels with complicated degradation. their network achieves top-2 psnr/ssim scores on track 1 of the ntire 2021 challenge [15]. however, the network has considerable computational complexity as [16] since it has about 61m parameters. unlike other methods, the network’s performance depends on the additionally synthesized dataset (~72k images) used to fine-tune the network. in addition, the source code and the implementation details of the network (such as the number of blocks used in feature extraction, reconstruction, and deblurring modules) are not publicly available for researchers to reproduce the results. xu et al. [21] adjusted a video restoration architecture to the blurry image superresolution problem by feeding five replicated images into the network to fully exploit the degraded lr image's self-scale and cross-scale similarity information. their proposed pyramid progressive transfer (ppt) module employs a pyramid structure and aims to generate attention masks to progressively transfer the self-similarity information. the pyramid self-attention (psa) module aggregates and reweights the transferred features with a pyramid structure. their proposed network achieves the best psnr/ssim/lpips scores in the ntire 2021 image deblurring challenge [15]. however, the pre-trained model is not provided for researchers. the publicly available source code is one of the ablation studies of edpn that has some inconsistencies with the paper. therefore, the method results are not reproducible, like [20]. attention mechanism. the attention mechanism aims to mimic the human visual perception to pay attention to the informative part of the input. the self-attention [1] mechanism allows modeling the global dependencies between each word in a sentence or pixel in the image. hu et al. [28] proposed the squeeze and excitation block (se) to model the interdependencies between the feature channels. dai et al. [9] designed the second-order channel attention block by replacing the global average pooling operation of the se block with the global covariance pooling operation [9] to capture higher-order feature statistics for more discriminative representations. to solve the problems of scale variation and small objects, dai et al. [22] designed the multi-scale channel attention module (ms-cam) for aggregating both the local and global feature contexts within the channel attention. the global average pooling operation employed in the se block emphasizes the globally distributed large objects while potentially ignoring the locally distributed small objects. therefore, in addition to the global branch, a local branch is added into se to simultaneously aggregate the global and local contexts within a multiscale channel attention mechanism. 3. proposed method this work proposes a single-branch network architecture to solve the joint motion deblurring and sr problem. unlike the existing methods, the proposed method successfully reconstructs the relatively small degraded structures (see the results section). the network reuses the contextually and spatially enriched features generated by mprnet [3] for image superresolution. the sr module is designed by employing residual in residual structure based on multi-scale channel attention modules (ms-cam [22]), which simultaneously aggregate the local and global feature contexts within the channel attention to better process the small objects. network architecture. to solve the motion deblurring and sr problems jointly, the network should extract both contextually and spatially informative features. the encoderdecoder architectures [2, 4] are effective in encoding the contextually informative features, which are helpful for the motion deblurring problem. the single-scale pipelines [8-10] extract single image joint motion deblurring and super-resolution using the multi-scale channel attention modules 24 spatially-enriched features that are informative for the sr task. the mprnet [3] naturally offers a twofold functionality since it employs both encoder-decoder and single-scale pipelines. the proposed network consists of three main parts: deblurring, feature transformation, and super-resolution (see fig. 2). deblurring module. the deblurring module is based on mprnet [3] and employs multistage progressive architecture with three stages (see fig. 2). an encoder-decoder architecture is employed in the first and second stages of mprnet based on u-net architecture [30] to generate contextually informative features due to the large receptive field. each scale of the encoder and decoder networks uses 2 channel attention blocks (cab) [3]. also, each skip connection between encoder and decoder uses 2 cabs. the cab block incorporates conv-relu-conv structure and squeeze-and-excitation block (se) [28] with the residual connection. it reweights the input feature map by emphasizing more informative channels. the 3rd stage employs a single-scale original-resolution subnetwork (orsnet) [3] and aims to generate spatially informative features without downsampling and upsampling operations. it contains 3 originalresolution blocks (orb) [3]. each orb is a residual group with 8 cabs followed by a convolution layer. the supervised attention module (sam) [3] and the cross-stage feature fusion (csff) [3] mechanism are employed to control the information flow between two consecutive stages. the sam module uses the ground-truth supervisory image. it generates attention maps to suppress the less informative features of the current stage and to propagate only the more informative ones to the next stage. the csff mechanism propagates the intermediate contextualized features of the previous stage to the next stage. it aims to compensate for the information loss due to repeated upsampling and downsampling operations performed in the first and second stages. thus, the sam module and csff mechanism allow the orsnet to combine the generated spatially informative features with the contextually informative features of the previous stage and allow the mprnet to produce both contextually and spatially enriched features in its 3rd stage. fig. 2. the architecture of the proposed method. please, zoom in for the best view. feature transformation module. the feature transformation module refines the contextually and spatially enriched features generated by mprnet [3] before propagating them to the super-resolution module. it contains a cab block followed by a convolutional layer. a cab block is applied on mprnet generated features to emphasize the more informative features for sr and suppress the less informative ones. since the third stage of mprnet operates on 128 m. shoyan 25 channels, a convolutional layer transforms the cab-generated features by decreasing the number of channels from 128 to 64. it aims to achieve a trade-off between computational complexity and accuracy for the sr. super-resolution module. the super-resolution module takes the transformed features as input and employs a single-scale pipeline to reconstruct the hr sharp image. the image sr problem can be treated as a process to recover the high-frequency information (like regions full of edges) as much as possible since the low-frequency information is less informative (like uniform regions). thus, the abundant low-frequency information can be forwarded to the final hr output with relatively less processing. inspired by rcan [8], a residual-in-residual (rir) structure is exploited to bypass the abundant low-frequency information via long and short skip connections and make the network focus on learning the high-frequency information. the rir structure stacks 17 residual groups (rg), where each residual group contains 10 residual multiscale channel attention blocks (rms-cab) followed by a convolution layer (see fig. 3). after the rir structure, the pixel-shuffling layer [29] follows to upscale the spatial resolution of the features four times. then a convolutional layer is used to generate a colored image from 64 channels. fig. 3. the rms-cab block and ms-cam [22] block. please, zoom in for the best view. the employed rms-cab block is inspired by cab [3]. it consists of a conv-reluconv structure and ms-cam [22] block with the residual connection. different from cab, it employs ms-cam [22] block instead of se block [28] to aggregate both the local and global feature contexts. the se and ms-cam blocks aim to capture the interdependencies among feature channels and reweight them by amplifying the more informative channels and suppressing the less informative ones. in contrast to the se block, the ms-cam block captures the information from the globally distributed large and locally distributed small objects, while the se block ignores the signals present in small objects. the ms-cam block employs global and local branches (see fig. 3) to implement the channel attention in two scales. the global branch applies a global average pooling operation and calculates the mean value for each channel to capture the channel-wise feature statistics globally. then two point-wise convolutions follow with relu activation between them. unlike the global branch, the local branch operates on the original spatial resolution and does not exploit the global average pooling operation. the features generated by two branches are aggregated via broadcasting addition. then a sigmoid activation follows, and the final features are element-wise multiplicated with the input. training strategy and loss function. the whole network can be trained jointly from scratch, but this approach causes sub-optimal results. the reason is that for the most part of the joint training, the deblurring module will generate blurry results, which will hamper the training process of the sr module. as a result, the network will not learn the exact mapping between blurry lr and sharp hr images. since it would be beneficial for the sr module to operate on deblurred features, a simple and effective training strategy is exploited, consisting of three phases. single image joint motion deblurring and super-resolution using the multi-scale channel attention modules 26 in the first training phase, only the deblurring module (mprnet) is trained. the mprnet generates lr deblurred image at each stage, denoted as (�̂�1, �̂�2 , �̂�3 ). since, in the proposed architecture, the third stage of the mprnet does not produce a deblurred image, a convolutional layer is added to generate the �̂�3 image for the training phase only. the deblurring loss is defined as ℒdb = ∑ ℒchar(l̂s , l) + λ𝑒𝑑𝑔𝑒 ∗ ℒedge(l̂s, l) 3 s=1 , ℒchar = 1 n ∑ √||l̂𝑠 𝑖 − l𝑖 ||2 + ε2 𝑁 𝑖=1 , ℒedge = 1 n ∑ √||δl̂𝑠 𝑖 − δl𝑖 ||2 + ε2 𝑁 𝑖=1 , where ℒdb denotes the deblurring loss, n is the number of training images, and l is the bicubic downsampled version of the ground-truth hr image. as in [3], ℒchar and ℒedge denote the charbonnier loss [31] and the edge loss [3], δ denotes the laplacian operator. ε and λ𝑒𝑑𝑔𝑒 were empirically set to 10−3 and 0.05, respectively. in the second training phase, only the sr module is trained (including the transformation module) by employing the pre-trained frozen deblurring module. it is beneficial for the sr module since, in contrast to joint training from scratch, it starts to operate on fully processed deblurred features and does not have to deal with blurry features now. to preserve the structural details, like edges, the combination of pixel-wise and gradient losses (ℒpixel and ℒgrad) is employed as in [16]: ℒpixel = 1 n ∑‖ĥ𝑖 − h𝑖 ‖ 1 𝑁 𝑖=1 , ℒgrad = 1 n ∑‖∇ĥ𝑖 − ∇h𝑖 ‖ 1 𝑁 𝑖=1 , ℒsr = ℒpixel + λ𝑔𝑟𝑎𝑑ℒgrad , where ĥ and h are the reconstructed and ground-truth high-resolution images, respectively. ℒsr denotes the sr loss, ∇ is the image gradient operator and λ𝑔𝑟𝑎𝑑 is set to 0.1 as in [16]. in the last training phase, the deblurring module is unfrozen, and the whole network is trained jointly. the deblurring loss is employed only in the first and second stages of the mprnet. this trick aims to force the third stage of mprnet to pay more attention to generating the sr features during the last phase of the training. the following combination of loss functions is exploited ℒ = ℒsr + αℒdb, where α was empirically set to 0.5 as in [13]. implementation details. the reds dataset [32] is used to train the proposed network. it contains 24.000 training images, 3000 validation images, and 3000 testing images. the validation dataset is used for quantitative evaluations since the ground-truth images of the testing m. shoyan 27 set are not available. the size of the lr blurry and ground-truth hr images is equal to 320x180 and 1280x720, respectively. the proposed method is trained with the three-phase training strategy as discussed above. the training patch size for the blurry lr and ground-truth hr images is 64x64 and 256x256, respectively, which are randomly cropped from the training set. randomly horizontal and vertical flips are applied combined with rotation for data augmentation. the adam optimizer [33] is used with parameters 𝛽1 = 0.9, 𝛽2 = 0.999 and 𝜖 = 10 −8. the initial learning rate and the batch size for the three phases of training are set to (10−4, 10), (10−4, 3), and (10−5, 3), respectively. during the training of each phase, the learning rate was gradually modified to obtain the best results. the network is trained and evaluated on a single ‘nvidia geforce gtx 1660 ti with max-q design’ gpu. at the test time, the tlsc mechanism [7] is applied to the whole network to address the issue of statistics distribution shift between the training and testing phases. the local window size of tlsc was empirically set to 96x96. the proposed method takes about 1.1 seconds to recover a 1280x720 size hr image from the 320x180 size lr blurry input. 4. results the proposed network is evaluated on the reds validation dataset [32] both quantitatively and qualitatively. the psnr [17] / ssim [18] scores are calculated using the computation codes released by [16] for a fair comparison. table 1 summarizes the quantitative comparison results between the existing methods on the val300 [16] dataset derived from the reds validation set by sampling each tenth image from the validation set. table 1: quantitative comparison results on the val300 dataset [16]. * denotes the results cited from [16], # denotes the self-ensemble strategy [12]. methods bicubic* gfn* rcan* [mprnet + rcan] * [srn + rcan] * shoyan et al cnlrn proposed method psnr 23.848 26.635 27.338 27.550 27.610 27.164 27.770 / 27.922# 27.620 / 27.740# ssim 0.6481 0.7447 0.7661 0.7740 0.7745 0.7610 0.7784 / 0.7813# 0.7735 / 0.7760# as shown in table 1, the psnr score of the sr method rcan [8] is at least 0.29db less than the proposed method since the blur degradation hampers the performance of sr techniques. the existing sr methods [8-10] employ only a single-scale pipeline, which fails to extract the contextual information to remove the blur degradation. the joint method gfn [14] does not perform well since the sr branch operates on the blurry lr image. the joint method proposed by shoyan et al. [13] generates promising results (see fig. 1, fig. 4), but their reconstruction module is not large enough to fully reuse the contextually and spatially enriched features extracted from mprnet [3]. the cascaded approaches of deblurring and sr methods (mprnet [3] + rcan [8], srn [2] + rcan [8]) still obtain less psnr/ssim scores than the proposed method since they employ the squeeze-and-excitation mechanism [28]. in contrast, the proposed method exploits the benefits of ms-cam [22] blocks to better process the relatively small structures present in the lr blurry image. the joint method cnlrn [16] obtains higher single image joint motion deblurring and super-resolution using the multi-scale channel attention modules 28 psnr/ssim results than the proposed method. however, the cnlrn [16] fails to reconstruct enough sharpness for relatively small structures in the lr blurry image and generates smoothed texture details, as shown in fig. 4. the self-ensemble strategy [12] is also employed to increase the performance of the proposed method by running the model on 8 augmented lr images followed by inverse transformation and averaging, resulting in 27.740/0.7760 db psnr/ssim scores, respectively. blurry lr image ground truth hr blurry lr input shoyan et al. cnlrn proposed method blurry lr image ground truth hr blurry lr input shoyan et al. cnlrn proposed method blurry lr image ground truth hr blurry lr input shoyan et al. cnlrn proposed method fig. 4. qualitative comparison results on the validation dataset. please, zoom in for the best view. m. shoyan 29 table 2: quantitative comparison results on the reds [32] validation dataset. methods cnlrn pdan edpn proposed method psnr 27.828 27.890 28.010 27.660 ssim 0.7794 0.7798 0.8203 0.7745 #params 83․06m 61.00m 13.34m 34.12m table 2 summarizes the quantitative comparison results for the proposed method and three topranked methods in ntire 2021 [15] challenge (cnlrn [16], pdan [20], edpn [21]). the methods were evaluated on the reds [32] validation set. the results of pdan [20] and edpn [21] were cited from their papers. as shown in table 2, the proposed method obtains less psnr/ssim scores than the other methods, but it has fewer parameters (~34m) and, therefore, less computational complexity compared to cnlrn (~83m) and pdan (~61m). it makes the model more applicable in scenarios when computational resources are limited. it should be mentioned that the results of pdan are not reproducible since the source code, pre-trained model, and network implementation details are not publicly available. it also refers to edpn since no pre-trained model is provided, while the publicly available source code is one of the authors’ ablation studies. the qualitative comparison results on the reds validation set are demonstrated in fig. 4. as it is shown, the cnlrn [16] and the method proposed by shoyan et al. [13] fail to recover the small structures present in the lr blurry image. instead, the proposed method successfully removes the complex motion blur from the relatively small structures and reconstructs fine texture details. as shown, unlike the other methods, the hr image generated by this method is closer to the ground-truth hr image. 5. ablation study the ablation experiments were performed on the val300 dataset. table 3 summarizes the effect of different modifications in terms of psnr [17] / ssim [18] metrics on the rgb channel for the proposed method. the baseline model employs only the deblurring and transformation modules followed by upsampling layer [29]. as a model-1, the sr module is added to the baseline model with 12 rgs, each containing 10 rms-cab blocks followed by a convolution layer. table 3: ablation study on the val300 dataset [16]. modifications models baseline model-1 model-2 model-3 baseline ✓ ✓ ✓ ✓ 120 rms-cab ✓ 170 rms-cab ✓ ✓ tlsc ✓ psnr 27.098 27.453 27.501 27.620 ssim 0.7530 0.7696 0.7713 0.7735 single image joint motion deblurring and super-resolution using the multi-scale channel attention modules 30 this change brings ~0.36db improvement in terms of psnr. when the number of rgs is increased to 17, the psnr score is improved from 27.453db to 27.501db (model-2). finally, the tlsc mechanism [7] is employed on the whole network (model-3) at the test time, achieving a 27.620db score in terms of psnr. the models were trained with the proposed three-phase training strategy by gradually modifying the learning rate to obtain the best results. to show the effect of the proposed training strategy, the model-1 is trained jointly from scratch. it achieves 27.371db in terms of psnr, about 0.08db less than the same model trained with the proposed three-phase training strategy. 6. conclusion this paper proposes an end-to-end single branch architecture to reconstruct a sharp hr image from the given motion blurry lr image. the proposed method reuses the contextually and spatially enriched features extracted from the mprnet [3] in the super-resolution subnetwork. the super-resolution subnetwork is designed based on ms-cam [22] blocks. a three-phase training strategy is exploited where a pre-trained frozen deblurring module is used as a feature extractor to train the sr module. the deblurring module is unfrozen in the last phase of the training. experiments show that, unlike the other methods, the proposed method successfully removes the complex motion blur from the relatively small structures and reconstructs fine texture details. the source code and the pre-trained model are available at https://github.com/misakshoyan/joint-motion-deblur-and-sr. references [1] a. vaswani et al. “attention is all you need”, arxiv preprint arxiv:1706.03762, 2017. 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[33] d. kingma and j. ba, “adam: a method for stochastic optimization”, arxiv preprint arxiv:1412.6980, 2014. submitted 18.05.2021, accepted 14.09.2021. պատկերից շարժման հետևանքով առաջացած շաղվածության հեռացում և պատկերի կետայնության բարձրացում՝ օգտագործելով շերտային ուշադրության բազմամասշտաբ մոդուլները միսակ տ․ սհոյան հայաստանի ազգային պոլիտեխնիկական համալսարան e-mail: misakshoyan@gmail.com ամփոփում վերջին տասնամյակում խորը փաթույթային նեյրոնային ցանցերը զգալիորեն զարգացրել են պատկերի կետայնության բարձրացման մեթոդները՝ վերակառուցելով պատկերի կառուցվածքային և տարածական իրատեսական մանրամասներ։ պատկերի կետայնության բարձրացման դասական խնդիրներում ենթադրվում է, որ ցածր կետայնության պատկերն ունի կետայնության նվազման որոշակի դեգրադացիա։ սակայն, իրական սցենարներում, պատկերի բարդ դեգրադացիաներն անխուսափելի են և շարժման հետևանքով առաջացած շաղվածությունը պատկերի դեգրադացիայի տարածված տեսակ է, որն առաջանում է պատկերի նկարահանման գործընթացում՝ տեսախցիկի կամ տեսարանի շարժման mailto:misakshoyan@gmail.com m. shoyan 33 հետևանքով։ այս աշխատանքում առաջարկվում է լրիվ փաթույթային նեյրոնային ցանց՝ տրված ցածր կետայնությամբ և շարժման հետևանքով առաջացած շաղվածություն պարունակող պատկերներից բարձր կետայնության հստակ պատկերներ վերակառուցելու համար։ շաղվածության հեռացման ենթացանցը հիմնված է բազմափուլային պրոգրեսիվ ճարտարապետության վրա, մինչդեռ կետայնության բարձրացման ենթացանցը նախագծելու համար օգտագործվում են շերտային ուշադրության բազմամասշտաբ մոդուլները։ կիրառվում է ուսուցանման պարզ և արդյունավետ եղանակ, որտեղ նախապես ուսուցանված և սառեցված շաղվածության հեռացման մոդուլն օգտագործվում է կետայնության բարձրացման մոդուլն ուսուցանելու համար։ ուսուցման վերջին փուլում շաղվածության հեռացման մոդուլը նույնպես ուսուցանվում է։ կատարված փորձերը ցույց են տալիս, որ ի տարբերություն մյուս մեթոդների, առաջարկվող մեթոդը վերակառուցում է համեմատաբար փոքր դետալները և կառուցվածքային մանրամասները՝ միաժամանակ հաջողությամբ հեռացնելով շարժման հետևանքով առաջացած բարդ շաղվածությունը։ իրականացման կոդը և նախապես ուսուցանված մոդելը հասանելի են https://github.com/misakshoyan/joint-motion-deblurand-sr կայքում։ բանալի բառեր՝ շարժման հետևանքով առաջացած շաղվածության հեռացում, կետայնության բարձրացում, շերտային ուշադրություն։ удаление размытости вызванной движением и увеличение разрешения одного изображения с использованием многомасштабных модулей внимания канала мисак т. сгоян национальный политехнический университет армении e-mail: misakshoyan@gmail.com аннотация за последнее десятилетие глубокие сверточные нейронные сети значительно продвинули методы увеличения разрешения одного изображения, реконструируя реалистичные текстурные и пространственные детали. в классических задачах увеличении разрешения изображения предполагается, что изображение с низким разрешением имеет определенную деградацию понижения разрешении. однако, в реальных сценариях, сложные деградации изображения неизбежны, и размытие изображения вызванное движением является распространенным типом деградации, которое происходит в процессе съемки изображения из-за движения камеры или сцены. в этой работе предлагается полностью сверточная нейронная сеть для реконструкции четких изображений с высоким разрешением из заданных размытых изображений https://github.com/misakshoyan/joint-motion-deblur-and-sr https://github.com/misakshoyan/joint-motion-deblur-and-sr mailto:misakshoyan@gmail.com single image joint motion deblurring and super-resolution using the multi-scale channel attention modules 34 вызванных движением с низким разрешением. подсеть удаления размытости основана на многоступенчатой прогрессивной архитектуре, в то время как подсеть увеличения разрешения разработана с использованием многомасштабных модулей внимания каналов. используется простая и эффективная стратегия обучения, в которой предварительно обученный и замороженный модуль удаления размытости используется для обучения модуля увеличения разрешения. модуль удаления размытости размораживается на последнем этапе обучения. эксперименты показывают, что, в отличие от других методов, предлагаемый метод реконструирует относительно небольшие структуры и текстурные детали, успешно удаляя сложное размытие вызванное движением․ код реализации и предварительно обученная модель общедоступны по адресу https://github.com/misakshoyan/joint-motion-deblur-and-sr. ключевые слова: удаление размытости вызванной движением, увеличениe разрешения, внимание канала. https://github.com/misakshoyan/joint-motion-deblur-and-sr mathematical problems of computer science 45, 138–142, 2016. image visual similarity based on high level features of convolutional neural networks aghasi s. poghosyan, hakob g. sarukhanyan institute for informatics and automation problems of nas ra e-mail: agasy18@gmail.com, hakop@ipia.sci.am abstract nowadays, the task of similar content retrieval is one of the central topics of interest in academic and industrial worlds. there are numerous techniques that are both dealing good with structured data and unstructured such as texts, respectively. however, in this paper we present a technique for retrieval of similar image content. we embed images to n dimensional feature space using convolutional neural networks and perform the nearest neighbor search afterwards. at the end, several distance metrics and their influence on the outcome are discussed. we are rather interested in the proportion of related content than in the additional ranking. thus, the evaluation of results is based on precision and recall. we have selected 6 major categories from imagenet dataset to assess the performance. keywords: image retrieval, convolutional neural networks, distance metrics. 1. introduction the content-based image retrieval(cbir) [1] has a broad application from professional database searching to search engines in the internet. in these applications one uses a sample image as a query item and the cbir system responds with the list of relevant items. usually, the resulting list uses underlying several factors. first, it uses a method to derive a knowledge about the contents of the image, such as interest point-based detection methods (sift [2] surf [3]). secondly, it uses extracted features and a distance metric to measure similarity between the query image and the images in the database. at last, one can derive a ranking method in order to show the query result. we perform the steps desrcibed above using the following procedure. in order to embed the image into the n dimensional feature space we use the feature vector extracted from pool5/7x7 s1 layer of googlenet [5] convolutional neural network (cnn). googlenet is trained on the imagenet [4] dataset and highly effective in object classification and localization task. the pool5/7x7 s1 layer is the last layer in the cnn on which the classification task is performed. in other words, it consists of high level features of the image. later we use and compare the performance of twelve distance measures to assess the similarity. we do not use any additional ranking methods for sorting the result. in the current case, results are sorted according to the similarity between the query image and the images in the database. 138 ag. poghosyan, h. sarukhanyan 139 the rest of the paper is organized as follows. in the second section we discuss the feature layer of googlenet we used. later, we give an overview to the twelve distance metrics we mentioned above. afterwards, we show our evaluation results based on precision/recall metrics. at the end we discuss advantages and disadvantages of this method and further improvements. 2. high level feature extraction the googlenet is a cnn that aims to localize and classify objects in images. it is trained on the imagenet dataset and shows 6.6% top-5 accuracy error on image classification task. in googlenet the last layer before the classification layer is 1024-dimensional vector and called pool5/7x7 s1. this layer extracts the highest level features of the image and gives the most descriptive information about the objects within it. thus, the features extracted by pool5/7x7 s1 cover the majority information contained within the image. we structure our content by propagating each image through googlenet and storing pool5/7x7 s1 layer information to feature vector database (fvd). from now on by database we mean fvd. for query images we follow the same procedure, except the saving part. specifically, we propagate the query image through goolgenet to embed the extracted feature vector to the same 1024-dimensional space. 3. distance metrics the next important step for our task is to choose a distance metric that will maximize the performance. now we briefly discuss each metric under our consideration below. for a pair of n-dimensional real-valued vectors u and v the distances are defined as follows:∑ i |ui − vi|, (manhattan) (1) ||u − v||2, (euclidean) (2) 1 − u · v ||u||2||v||2 , (cosine) (3) where u · v is the dot product of u and v.∑ |ui − vi|/ ∑ |ui + vi|, (bray − curtis) (4) d(u, v) = ∑ i |ui − vi| |ui| + |vi| , (canberra) (5) where ui and vi are 0 for given i, then the fraction 0/0 = 0 is used in the calculation. max i |ui − vi|, (chebyshev) (6) 1 − (u − ū) · (v − v̄) ||(u − ū)||2||v − v̄||2 , (correlation) (7) where ū is the mean of the elements of u and x. 140 image visual similarity based on high level features of convolutional neural networks we use a subset image-net [4] dataset to assess the retrieval performance. we extract 6 major categories in total size of 5500 images. in fig. 1 the frequency distribution per category is depicted. in order to increase the precision of our measurements we limit each category to have 440 images. in this way we can ensure that recall metric will not depend on the sample size of the category and will provide more reliable results. fig. 1. image-net subset per categories image count bar. 4. evaluation precision and recall are common measures on evaluating performance of information retrieval tasks. precision is defined as the percentage of relevant items in retrieved items. and recall as the percentage of relevant items that were retrieved in total relevant items. specifically precision = |{relevant items} ∩ {retrieved items}| |{retrieved items}| (8) recall = |{relevant items} ∩ {retrieved items}| |{relevant items}| (9) in our experiment we fixed then the number of retrieved items (neighbors) to 100. per image category average precision has been computed by: pq ′ = ∑ kϵaq p(ik) |aq| , q = 1, 2, ...n, (10) ag. poghosyan, h. sarukhanyan 141 where n is the number of categories, aq is qth category, p(ik) is the precision for k th image. per image category average recall has been computed by: rq ′ = ∑ kϵaq r(ik) |aq| , q = 1, 2, ...n, (11) where n is the number of categories, aq is qth category, r(ik) is recall for k th image. the average precision and average recall are given by: p ′ = n∑ q=1 pq ′ n , (12) and r′ = n∑ q=1 rq ′ n . (13) in order to evaluate our method we constructed a similarity matrix for the image database. for each image within the image database we retrieved 100 neighbors and calculated precision and recall for each of them. first we have evaluated results per distance metric and averaged them across image categories: tables 1, 2. we can see that on average correlation metric performs better than the other metrics. however, there are image categories where correlation shows lower results than the other metrics. table 1. metrics comparison precisions. manhattan cosine euclidean braycurtis canberra chebyshev correlation construction 0.862 0.928 0.852 0.924 0.912 0.635 0.932 vertebrate 0.603 0.781 0.561 0.782 0.813 0.440 0.800 device 0.814 0.835 0.768 0.836 0.856 0.429 0.845 food 0.974 0.949 0.963 0.945 0.948 0.717 0.956 invertebrate 0.699 0.851 0.654 0.839 0.838 0.364 0.865 tree 0.953 0.944 0.960 0.944 0.917 0.795 0.940 mean 0.818 0.881 0.793 0.878 0.881 0.563 0.890 table 2. metrics comparison recall. manhattan cosine euclidean braycurtis canberra chebyshev correlation construction 0.196 0.211 0.193 0.210 0.207 0.144 0.212 vertebrate 0.137 0.177 0.127 0.177 0.184 0.100 0.181 device 0.185 0.189 0.174 0.190 0.194 0.097 0.192 food 0.221 0.215 0.219 0.214 0.215 0.163 0.217 invertebrate 0.159 0.193 0.148 0.190 0.190 0.082 0.196 tree 0.216 0.214 0.218 0.214 0.208 0.180 0.213 mean 0.185 0.200 0.180 0.199 0.200 0.128 0.202 5. conclusion this paper investigated another technique for cbir and evaluated the proposed method against different similarity measures. we have shown that using high level features from googlenet in combination with correlation distance metric can lead to promising results. 1 4 2 image visual similarity based on high level features of convolutional neural networks refer ences [1 ] r . d a t t a , j. l i a n d j. z. w a n g , \ co n t e n t -b a s e d im a g e r e t r ie va l a p p r o a c h e s a n d t r e n d s o f t h e n e w a g e " , the p ennsylvania state university, university p ark, p a 16802, 2 0 0 5 . 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[5 ] c. s z e g e d y, w . l iu , y . jia , p . s e r m a n e t , s . r e e d , d . a n g u e lo v, d . e r h a n , v . v a n h o u c ke a n d a . r a b in o vic h , \ go in g d e e p e r wit h co n vo lu t io n s " , arxiv, n o . 1 4 0 9 .4 8 4 2 , 2 0 1 4 . submitted 15.09.2015, accepted 25.01.2016 ä³ïï»ñý»ñç ï»ëáõ³ï³ý ýù³ýáõãûáõýá ñçùýí³í ÷³ãáõûã³ûçý ý»ûñáý³ûçý ó³ýó»ñç µ³ñóñ ï³ñ·ç ñ³ïïáõãûáõýý»ñç íñ³ ². äáõáëû³ý, ð. ê³ñáõë³ýû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ý»ñï³û³óí³í ¿ ÷³ãáõûã³ûçý ý»ûñáý³ûçý ó³ýó»ñç` µ³ñóñ ï³ñ·ç ñ³ïïáõãûáõýý»ñç íñ³ ñçùýí³í ï»ëáõ³ï³ý ýù³ý å³ïï»ñý»ñç áñáýù³ý ñ³ù³ï³ñ·ç í»ñéáõíáõãûáõý: òáõûó ¿ ïñíáõù, áñ googlenet-ç µ³ñóñ ï³ñ·ç ñ³ïïáõãûáõýý»ñç í»ïïáñý»ñç ñ³ù³ñ ñ»é³íáñáõãû³ý ýáõýïóç³ûç áýïñáõãûáõýá ù»í ³½¹»óáõãûáõý áõýç áñáýáõ ñ³ù³ï³ñ·ç ×ß·ñïáõãû³ý íñ³, ¨ í»ñççýçë ñ³ù³ñ é³í³·áõûý ³ñ¹ûáõýùý»ñá ëï³óíáõù »ý, »ñµ ïáé»éû³óç³ý ¹çï³ñïíáõù ¿ áñå»ë ñ»é³íáñáõãû³ý ýáõýïóç³: âèçóàëüíàÿ ñõîæåñòü èçîáðàæåíèé, îñíîâàííàÿ íà ñâîéñòâàõ âûñøåãî óðîâíÿ êîíâîëþöèîííûõ íåéðîííûõ ñåòåé à. ïîãîñÿí, à. ñàðóõàíÿí àííîòàöèÿ â äàííîé ðàáîòå ïðåäñòàâëåíî èññëåäîâàíèå ñèñòåìû ïîèñêà âèçóàëüíî ïîõîæèõ èçîáðàæåíèé, îñíîâàííîé íà ñâîéñòâàõ âûñøåãî óðîâíÿ êîíâîëþöèîííûõ íåéðîííûõ ñåòåé. ïîêàçàíî, ÷òî âûáîð ôóíêöèè ðàññòîÿíèÿ äëÿ ñâîéñòâ âûñøåãî óðîâíÿ googlenet èìååò áîëüøîå âëèÿíèå íà òî÷íîñòü ïîèñêîâîé ñèñòåìû. äëÿ ïîëó÷åíèÿ ëó÷øèõ ðåçóëüòàòîâ èç ïðîñìîòðåííûõ ôóíêöèé âûäåëÿåòñÿ ôóíêöèÿ êîððåëÿöèè. 17.pdf (p.1-4) aghasi_abstract.pdf (p.5) mathematical problems of computer science 45, 111—121, 2016. 111 linear cryptanalysis of modified safer + algorithm melsik k kyureghyan, knarik m. kyuregyan institute for informatics and automation problems of nas ra e-mail: melsik@ipia.sci.am, knarikyuregyan@gmail.com abstract in paper [1] a modified safer+ [2] algorithm was presented. it was shown that those modifications made it possible to speed up a modified algorithm implementation about 1,7 times and obtain the same security level as compared with safer+ in terms of differential analysis. in this paper a linear cryptanalysis of modified safer+ algorithm is presented and it is shown that it has the same resistance against linear cryptanalysis as compared with an original safer+. keywords: block cipher, linear cryptanalysis, balanced function. 1. introduction linear cryptanalysis is one of the powerful cryptanalyses against iterated block ciphers. it was introduced by matsui as a theoretical attack on des and it is in fact a known-plaintext attack: that is a cryptanalyst has an access to the set of some plaintexts and their corresponding ciphertexts. matsui exploits a cipher’s weakness that he quantifies in terms of “unbalanced linear expression” that involves plaintext bits, ciphertext bits (actually bits from the second last round output) and subkey bits. a linear expression is unbalanced if the equation is satisfied with the probability different from 1 2⁄ when the plaintext and the keys are uniformly random and independent. the cryptanalyst applies the last round attack using this linear expression for the entire cipher excluding the last round. he or she guesses wrong keys and by processing a large number of plaintext/ciphertext pairs and tries to find the last round key. the success of linear cryptanalysis depends on the original linear expression being more imbalanced than the linear expression obtained with wrong keys. in this paper we will consider a linear cryptanalysis of modified safer+ [1], whose round function is depicted in fig. 1. let x = x1x2 … x16 be the 16 bytes input of the 𝑖-th round. the round function consist of the following four layers: mailto:melsik@ipia.sci.am,%20%20knarikyuregyan@gmail.com linear cryptanalysis of modified safer+ algorithm 112 1. xor/add, first round key is “added” to the round input either modulo 2 ( xor) or modulo 256 (add): u = xor/add(x, k2i−1). 2. non-linear (nl), where each byte is subjected to either the non-linear function exp: x → 45xmodulo257 (except that 45128 is taken as 0 rather than −1 = 257) or its inverse function log: v = nl(u). 3. add/xor, by this layer the second round key is inserted: w = add/xor(v, k2i). invertible linear transform or pseudo-hadamard transform (pht) consisting of four times applied 𝐴𝑟𝑚𝑒𝑛𝑖𝑎𝑛 𝑠ℎ𝑢𝑓𝑓𝑙𝑒 and four times applied eight 2-pht boxes: y = pht(w), is equivalent to  .256mod 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1814122212112141 11624142422124181 1418114212211221 24116218214411422 1221111221441812 22412114228411614 1211181111224422 14121162121248442 2122421418111112 41248224116211214 2212112141181214 24141241811162224 1142221824121111 12822411644142121 4111241112122218 81214412142224116 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1                                                                                                                                                         w w w w w w w w w w w w w w w w y y y y y y y y y y y y y y y y 2. preliminaries in this paper we follow the terminology and notation in [3]. a binary-valued function is said to be 𝑏𝑎𝑙𝑎𝑛𝑐𝑒𝑑 if it takes on the value 0 for exactly half of its possible arguments and the value 1, otherwise. an 𝐼/𝑂 𝑠𝑢𝑚 for the 𝑖𝑡ℎ round, denoted by 𝑆(𝑖), is a modulo-two sum of balanced binary-valued function 𝑓𝑖 of the round input 𝑋 (𝑖) = 𝑌(𝑖−1) and a balanced binary-valued function 𝑔𝑖 of the round output 𝑌 (𝑖), that is 𝑆(𝑖) ∶= 𝑓𝑖 (𝑌 (𝑖−1)) ⊕ 𝑔𝑖 (𝑌 (𝑖 )), where ⊕ denotes modulo two addition, i.e., the xor operation. the functions 𝑓𝑖 and 𝑔𝑖 are called the 𝑖𝑛𝑝𝑢𝑡 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 and the 𝑜𝑢𝑡𝑝𝑢𝑡 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛, respectively, of the i/o sum 𝑆 (𝑖). i/o sums for successive rounds are said to be 𝑙𝑖𝑛𝑘𝑒𝑑 if the output function of each of these i/o sums, except the last, coincides with the input function of the following i/o sum (i.e., 𝑔𝑖 = 𝑓𝑖+1). when 𝑆 (1), 𝑆(2), … , 𝑆(𝜌)are linked, then their modulo-two sum is also i/o sum, namely 𝑆(1…𝜌) ∶= ⨁ 𝑆(𝑖) 𝜌 𝑖=1 = 𝑓1(𝑌 (1)) ⊕ 𝑔𝜌(𝑌 (𝜌)), where 𝑌(1) = 𝑋(1) = 𝑋 is the input to the first round, and is called a 𝜌-𝑟𝑜𝑢𝑛𝑑 𝐼/𝑂 𝑠𝑢𝑚. m. kyureghyan, k. kyuregyan 113 fig. 1. design of the 𝑖-th round of modified safer+. linear cryptanalysis depends critically on the notation of “𝑖𝑚𝑏𝑎𝑙𝑎𝑛𝑐𝑒” for binary-valued random variables. the 𝑖𝑚𝑏𝑎𝑙𝑎𝑛𝑐𝑒 𝐼(𝑉) of a binary-valued random variable 𝑉 is the real number |2𝑃[𝑉 = 0] − 1|, where 𝑃[𝑉 = 0] is the probability that 𝑉 takes on the value 0. note that 0 ≤ 𝐼(𝑉) ≤ 1 𝑅𝑜𝑢𝑛𝑑 𝐼𝑛𝑝𝑢𝑡 (16 𝐵𝑦𝑡𝑒𝑠) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 xor xor xor xor add add add add xor xor xor xor add add add add exp exp exp exp log log log log exp exp exp exp log log log log add add add add xor xor xor xor add add add add xor xor xor xor nl half round 𝐾2𝑖−1 𝐾2𝑖 2-pht 2-pht 2-pht 2-pht 2-pht 2-pht 2-pht 2-pht modified safer+ specific 𝐴𝑟𝑚𝑒𝑛𝑖𝑎𝑛 𝑆ℎ𝑢𝑓𝑓𝑙𝑒 [7 12 9 14 5 8 13 10 11 4 3 6 15 2 1 16] pht half round modified safer+ specific 𝐴𝑟𝑚𝑒𝑛𝑖𝑎𝑛 𝑆ℎ𝑢𝑓𝑓𝑙𝑒 [7 12 9 14 5 8 13 10 11 4 3 6 15 2 1 16] 2-pht 2-pht 2-pht 2-pht 2-pht 2-pht 2-pht 2-pht modified safer+ specific 𝐴𝑟𝑚𝑒𝑛𝑖𝑎𝑛 𝑆ℎ𝑢𝑓𝑓𝑙𝑒 [7 12 9 14 5 8 13 10 11 4 3 6 15 2 1 16] 2-pht 2-pht 2-pht 2-pht 2-pht 2-pht 2-pht 2-pht modified safer+ specific 𝐴𝑟𝑚𝑒𝑛𝑖𝑎𝑛 𝑆ℎ𝑢𝑓𝑓𝑙𝑒 [7 12 9 14 5 8 13 10 11 4 3 6 15 2 1 16] 2-pht 2-pht 2-pht 2-pht 2-pht 2-pht 2-pht 2-pht linear cryptanalysis of modified safer+ algorithm 114 with equality on the left if and only if 𝑉 is constant random variable and with equality on the right if and only if 2𝑃[𝑉 = 0] = 1 2⁄ . in linear cryptanalysis we always assume that the plaintext and all round keys are independent and uniformly random over the appropriate sets. in practice, however, the full key is usually produced from a user selected secret key by key scheduling algorithm. sometimes we explicitly fix keys by specifying e. g., 𝐾(1…𝜌) = 𝑘(1…𝜌), which implies that the statistics of the random experiment are the conditional statistics given that 𝐾 (1…𝜌) = 𝑘(1…𝜌). a round iterated block cipher of block-size 𝑛 encrypts a plaintext x by 𝜌 successive application of a keyed round function with different key in each round. the full key will be denoted by 𝐾(1…𝜌) = (𝐾(1), 𝐾(2), … , 𝐾(𝜌)), where 𝐾(𝑖) is the round key applied in the 𝑖-th round for 𝑖 = 1,2, … , 𝜌. note that, in the ciphers of safer family, 𝐾(𝑖) = (𝐾2𝑖−1, 𝐾2𝑖 ). the 𝑘𝑒𝑦-𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑖𝑚𝑏𝑎𝑙𝑎𝑛𝑐𝑒 𝐼(𝑆(1…𝜌)|𝐾(1…𝜌)) of the i/o sum 𝑆 (1…𝜌) is the random variable whose value when 𝐾(1…𝜌)=𝑘(1…𝜌) is the key-dependent imbalance of 𝑆(1…𝜌), i.e., 𝐼(𝑆(1…𝜌)|𝑘(1…𝜌)). the 𝑎𝑣𝑒𝑟𝑎𝑔𝑒-𝑘𝑒𝑦 𝑖𝑚𝑏𝑎𝑙𝑎𝑛𝑐𝑒 𝐼(̅𝑆(1…𝜌)) of the i/o sum 𝑆(1…𝜌) is the expectation of this key-dependent imbalance computed under the assumption that the round keys are chosen independently and uniformly at random, i.e., 𝐼(̅𝑆(1…𝜌)) ∶= 𝐸[𝐼(𝑆(1…𝜌)|𝐾(1…𝜌))] = 1 |𝒦 𝜌| ∑ 𝐼(𝑆(1…𝜌)|𝑘(1…𝜌))𝑘(1…𝜌)∈𝒦 𝜌 , where |𝒦𝜌| denotes the cardinality of the set of full keys. an i/o sum is said to be 𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 if it has a “large” average-key imbalance, and is said to be 𝑔𝑢𝑎𝑟𝑎𝑛𝑡𝑒𝑒𝑑 if its average-key imbalance is 1, the maximum possible. a 𝑡ℎ𝑟𝑒𝑒𝑓𝑜𝑙𝑑 𝑠𝑢𝑚 𝑇(𝑖) for the 𝑖-th round is a modulo-two sum of three terms: the first, a balanced binary-valued function 𝑓𝑖 of the round input 𝑋 (𝑖) = 𝑌(𝑖−1); the second, a balanced binary-valued function 𝑔𝑖 of the round output 𝑌 (𝑖); the third, some binary-valued function ℎ𝑖 of the round key 𝐾(𝑖), i.e., 𝑇 (𝑖) = 𝑓𝑖 (𝑌 (𝑖−1)) ⊕ 𝑔𝑖 (𝑌 (𝑖 )) ⊕ ℎ𝑖 (𝐾 (𝑖 )). the function ℎ𝑖 is the 𝑘𝑒𝑦 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 of the threefold sum 𝑇 (𝑖) and 𝑆(𝑖) = 𝑓𝑖 (𝑌 (𝑖−1)) ⊕ 𝑔𝑖 (𝑌 (𝑖 )) is the 𝑝𝑎𝑟𝑒𝑛𝑡 𝐼/𝑂 𝑠𝑢𝑚 for 𝑇 (𝑖). theorem 1.1: (threefold sum imbalance and i/o sum average-key imbalance) let 𝑇(1…𝜌) = 𝑆(1…𝜌) ⊕ ℎ𝑖 (𝐾 (1…𝜌)) be a threefold sum. then the average-key imbalance of the parent i/o sum 𝑆(1…𝜌) is lower bounded by the threefold sum imbalance, i.e., 𝐼(̅𝑆(1…𝜌)) ≥ 𝐼(𝑇 (1…𝜌)). moreover, equality holds if and only if ℎ is equal to { 0 𝑖𝑓 𝑃[𝑆 = 0/𝐾 = 0] > 1 2⁄ 1 𝑖𝑓 𝑃[𝑆 = 0/𝐾 = 0] < 1 2⁄ 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦 𝑖𝑓 𝑃[𝑆 = 0/𝐾 = 0] = 1 2⁄ (in this case the function ℎ is called a maximizing key function for 𝑇(1…𝜌)) or is the complement of such a function. a threefold sum is said to be 𝑔𝑢𝑎𝑟𝑎𝑛𝑡𝑒𝑒𝑑 if it has imbalance 1. it follows from theorem 1.1 that 𝑇(1…𝜌) = 𝑆(1…𝜌) ⊕ ℎ(𝐾(1…𝜌)) is guaranteed if only if the parent i/o sum 𝑆 (1…𝜌) is guaranteed and ℎ is a maximizing key function. lemma 1.1: (matsui’s piling-up lemma) the imbalance of a modulo-two sum of independent binary-valued variables 𝑉(1), 𝑉(2), … , 𝑉 (𝜌) is the product of their imbalance, i.e., m. kyureghyan, k. kyuregyan 115 𝐼 (⨁ 𝑉(𝑖) 𝜌 𝑖=1 ) = ⨅(𝐼(𝑉)(𝑖)) 𝜌 𝑖=1 . remark 1.1: if 𝑇(1), 𝑇(2), … , 𝑇(𝜌) are independent threefold sums, then it follows from lemma 1 that 𝐼 (⨁ 𝑇(𝑖) 𝜌 𝑖=1 ) = ⨅(𝐼(𝑇)(𝑖)), 𝜌 𝑖=1 where ⨁ 𝑇(𝑖) 𝜌 𝑖=1 is 𝜌-round threefold sum provided that 𝑇 (1), 𝑇(2), … , 𝑇(𝜌) are linked. the round functions of our cipher can be written as 𝑌(𝑖 ) = 𝑅 𝐾(𝑖) (𝑖) (𝑌(𝑖−1 )) = 𝜙(𝑌(𝑖−1 ) ⊗𝑖 𝐾2𝑖−1, 𝐾2𝑖 ), where ⊗𝑖 is the group operation by which one of the round keys is inserted in the 𝑖-th round (fig. 1). we will show that this structure guarantees the independence of any “homomorphic” one-round threefold sum and the input to that round. definition 1.1: consider an iterated block cipher whose 𝑖-th round function inserts the key 𝐾2𝑖−1 with a group operation ⊗𝑖 at the input to each round (fig. 1). let 𝐵 𝑛 denote the set of binary 𝑛tuples, very often considered as 𝐵𝑛 = {0, 1, 2, … , 2𝑛 − 1}. a homomorphism from a group (𝐵𝑛, ⨂) onto the group (𝐵, ⨁) is called 𝑏𝑖𝑛𝑎𝑟𝑦-𝑣𝑎𝑙𝑢𝑒𝑑 ℎ𝑜𝑚𝑜𝑚𝑜𝑟𝑝ℎ𝑖𝑠𝑚 𝑓𝑜𝑟 ⨂1. an i/o sum for the rounds 𝑖 to 𝑗 (𝑖 ≤ 𝑗) is homomorphic if the input function is a binary-valued homomorphism for ⊗𝑖 and the output function is a binary-valued homomorphism for ⊗𝑗+1. a threefold sum is ℎ𝑜𝑚𝑜𝑚𝑜𝑟𝑝ℎ𝑖𝑐 if the parent i/o sum is homomorphic. if for all rounds 𝑖, the operation ⊗𝑖 is the bitwise xor operation in 𝐵 𝑛, then the only binaryvalued homomorphisms are the linear function 𝑙𝑎(𝑥) = 𝑎 ∘ 𝑥 for 𝑥 in 𝐵 𝑛, where 𝑎 is a non zero 𝑛-tuple and 𝑎 ∘ 𝑥 denotes the scalar product with operations of 𝐺𝐹(2). an i/o sum (or a threefold sum) whose input and output functions are 𝑙𝑎 and 𝑙𝑏, respectively, will be called 𝑙𝑖𝑛𝑒𝑎𝑟 with 𝑙𝑖𝑛𝑒𝑎𝑟 𝑚𝑎𝑠𝑘 (𝑎, 𝑏). theorem 1.2: consider the cascade of 𝜌 rounds with round functions 𝑅(1), 𝑅(2), … , 𝑅(𝜌) for which 𝑌(𝑖 ) = 𝑅 𝐾(𝑖) (𝑖) (𝑌(𝑖−1 )) = 𝜙𝑖 (𝑌 (𝑖−1 ) ⊗𝑖 𝐾2𝑖−1, 𝐾2𝑖 ), where ⊗𝑖 denotes a group operation in 𝐵 𝑛 and where 𝜙𝑖 (. , 𝑘2𝑖 ) is a bijection on 𝐵 𝑛 for all 𝑘2𝑖 (fig. 1). let 𝑇 (𝑖) = 𝑓𝑖 (𝑌 (𝑖−1)) ⊕ 𝑓𝑖+1(𝑌 (𝑖)) ⊕ (𝑓𝑖 (𝐾2𝑖−1) ⊕ ℎ𝑖 (𝐾2𝑖 )) (1) be a homomorphic threefold sum for the 𝑖-th round, so that 𝑇(1), 𝑇(2), … , 𝑇(𝜌) are linked. then, the average-key imbalance for the parent i/o sum 𝑆(1…𝜌) of the threefold sum 𝑇(1…𝜌) ≔ ⨁ 𝑇(𝑖) 𝜌 𝑖=1 is lower bounded by the product of the one-round threefold sum imbalance, i.e., 𝐼(̅𝑆(1…𝜌)) ≥ ∏ 𝐼(𝑇(𝑖)). 𝜌 𝑖=1 (2) note that, as 𝜙𝑖 (. , 𝑘2𝑖 ) is a bijection and 𝑋 is uniformly distributed, 𝑌 (0), 𝑌(1), … , 𝑌(𝜌−1) are uniformly distributed. thus, 𝐼(𝑇(𝑖)) for 𝑖 = 2, 3, … , 𝜌 can be computed quite easily. for a given 𝑆(1…𝜌) we can evaluate the right side of (2) for many choices of linked homomorphic one-round threefold sums whose sum has 𝑆(1…𝜌) as its parent i/o sum. we can then use the maximum such threefold-sum imbalance as an approximation 𝐼(̅𝑆(1…𝜌)), i.e., 1 𝑓 is such homomorphism if, for all 𝑈, 𝑉 ∈ 𝐵𝑛 𝑓(𝑈⨂𝑉) = 𝑓(𝑈)⨁𝑓(𝑉) and if 𝑓 is not identically zero. linear cryptanalysis of modified safer+ algorithm 116 𝐼(̅𝑆(1…𝜌)) ≈ max 𝑓2,…,𝑓𝜌 ℎ1,…,ℎ𝜌 ∏ 𝐼(𝑇(𝑖)), 𝜌 𝑖=1 (3) where 𝑇(𝑖) is given by (1). the following is an effective procedure for finding this maximum. 𝑃𝑟𝑜𝑐𝑒𝑑𝑢𝑟𝑒 𝑓𝑜𝑟 𝑓𝑖𝑛𝑑𝑖𝑛𝑔 𝑎𝑛 𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝜌-𝑟𝑜𝑢𝑛𝑑 ℎ𝑜𝑚𝑜𝑚𝑜𝑟𝑝ℎ𝑖𝑐 𝐼/𝑂 𝑠𝑢𝑚 𝑓𝑜𝑟 𝑎 𝑐𝑖𝑝ℎ𝑒𝑟 𝑤ℎ𝑜𝑠𝑒 𝑟𝑜𝑢𝑛𝑑 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠 𝑖𝑛𝑠𝑒𝑟𝑡 𝑎 𝑘𝑒𝑦 𝑏𝑦 𝑢𝑠𝑖𝑛𝑔 𝑎 𝑔𝑟𝑜𝑢𝑝 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑎𝑡 𝑡ℎ𝑒 𝑖𝑛𝑝𝑢𝑡. 1. for 𝑖 = 1,2, … , 𝜌 + 1 find the set ℋ𝑖 of all binary-valued functions on 𝐵 𝑛 that are binary valued homomorphisms for ⊗𝑖. 2. for 𝑖 = 1,2, … , 𝜌 find the imbalance of all 𝑖-th round homomorphic threefold sums with input function 𝑓𝑖 in ℋ𝑖 , output function 𝑓𝑖+1 in ℋ𝑖+1, and maximizing key function. discard the threefold sums with small imbalance. 3. consider each possible choice of 𝜌 linked threefold sums containing one of the threefold sums found in step 2 in each round. use the right side of (3) as an estimate of the imbalance of the 𝜌-round i/o sums. output the 𝜌-round i/o sums having the largest estimated imbalance. 3. linear cryptanalysis of modified safer+ now we will find effective homomorphic i/o sums to a cascade of half-rounds of modified safer+ algorithm. we first find all binary-valued homomorphisms for add/xor and for xor/add. there are 28 − 1 binary-valued homomorphisms for 8 bit xor, namely the functions defined as 𝑙𝑎2(𝑉2) ∶= 𝑎2 ∘ 𝑉2, where 𝑎2 is a non-zero binary 8-tuple " ∘ " operation denotes the modulo two “dot product”. there is only one binary-valued homomorphism for modulo 256 addition, namely the function 𝑙𝑎1 where 𝑎1 = 0000 0001 = 01 (hex notation of byte). thus, there exist 272 − 1 balanced homomorphisms for add/xor, namely the functions 𝑙𝑎 defined as 𝑙𝑎(𝑉) = 𝑎 ∘ 𝑉 where 𝑎 lies in the set of 128-tuples. 𝒜 = {𝑎: 𝑎 ∈ {0,1}128\{00}; 𝑎1, 𝑎2, 𝑎3, 𝑎4, 𝑎9, 𝑎10, 𝑎11, 𝑎12} ∈ {00, 01}}. similarly, there are 272 − 1 balanced homomorphism for xor/add, namely the functions 𝑙𝑏 (𝑉) = 𝑏 ∗ 𝑉, where 𝑏 lies in the set ℬ = {𝑏: 𝑏 ∈ {0,1}128\{00}; 𝑏5, 𝑏6, 𝑏7, 𝑏8, 𝑏13, 𝑏14, 𝑏15, 𝑏16} ∈ {00, 01}}. the set of all homomorphic functions for xor/add and the set of all homomorphic functions for add/xor are subsets of the set of all linear binary-valued functions. we now consider the part of round (half-round) containing the pht function. the following lemma specifies all homomorphic i/o sums that have non-zero imbalance. the input function must be balanced and homomorphic for add/xor; the output function must be balanced and homomorphic for xor/ add. there are (272 − 1)2 such i/o sums, namely s𝑎,𝑏 pht−hr ∶= 𝑙𝑎(v)⨁𝑙𝑏 (y) 𝑎 ∈ 𝒜 , 𝑏 ∈ ℬ. lemma 2.1: for the 𝑃𝐻𝑇-half-round, the only homomorphic i/o sums that have non-zero imbalance are the 216 − 1 guaranteed i/o sums obtained by 𝑋𝑂𝑅-ing together any positive number of the 16 guaranteed i/o sums listed in table 1. m. kyureghyan, k. kyuregyan 117 table 1. effective i/o sums for the pht-function. since 𝐼(𝑆𝑎,𝑏 pht−hr|𝑘2𝑖 ) = 𝐼(𝑆𝑎,𝑏 pht) for any 𝑘2𝑖 ∈ {0,1} 128, where 𝑆𝑎,𝑏 pht ∶= 𝑙𝑎(𝑊)⨁ 𝑙𝑏(𝑌), (𝑎 ∈ 𝒜 and 𝑏 ∈ ℬ) is an i/o sum for the pht function alone, we will look for i/o sums 𝑆𝑎,𝑏 pht with non-zero imbalance instead of looking for 𝑆𝑎,𝑏 pht−hr with non-zero imbalance. so our main purpose is to find 𝑎 ∈ 𝒜 and 𝑏 ∈ ℬ such that 𝑆𝑎,𝑏 pht sum has the form 𝑊𝛼 ⨁𝜙(𝑊127, … , 𝑊𝛼−1, 𝑊𝛼+1, …, 𝑊0) for some input bit 𝑊𝛼, since this implies that the i/o sum imbalance is 0. first we consider pht function. table 2 shows three kind of dependences for some input and output bits. for example, for 𝑌102 we conclude that 𝑌102 = 𝑊11⨁𝑊22⨁𝑊92⨁𝑊102⨁𝑊111⨁𝑊121⨁ 𝜙(𝑊21, 𝑊91, 𝑊101; 𝑊10, 𝑊20, 𝑊30, 𝑊40, 𝑊90, 𝑊100, 𝑊110, 𝑊120; 𝑊5, 𝑊6, 𝑊7, 𝑊8, 𝑊13, 𝑊14, 𝑊15, 𝑊16) for some function 𝜙. (𝑎, 𝑏) 𝑙𝑏 (𝑌) 𝑙𝑎(𝑊) 𝐼(𝑆𝑎,𝑏 pht) (0100000101001010, 1000000000000000) 𝑌10 𝑊20⨁𝑊80⨁𝑊100⨁𝑊130⨁𝑊150 1 (0100010111001110, 0100000000000000) 𝑌20 𝑊20⨁𝑊60⨁𝑊80⨁𝑊90⨁𝑊100⨁𝑊110⨁ 𝑊130⨁𝑊140⨁𝑊150 1 (1010010001000001, 0010000000000000) 𝑌30 𝑊10⨁𝑊30⨁𝑊60⨁𝑊100⨁𝑊160 1 (1111010001000011, 0001000000000000) 𝑌40 𝑊10⨁𝑊20⨁𝑊30⨁𝑊40⨁𝑊60⨁𝑊100⨁ 𝑊150⨁𝑊160 1 (0000011010010100, 0000100000000000) 𝑌50 𝑊60⨁𝑊70⨁𝑊90⨁𝑊120⨁𝑊140 1 (0101011010110100, 0000010000000000) 𝑌60 𝑊20⨁𝑊40⨁𝑊60⨁𝑊70⨁𝑊90⨁𝑊110⨁ 𝑊120⨁𝑊140 1 (0101100100000010, 0000001000000000) 𝑌70 𝑊20⨁𝑊40⨁𝑊50⨁𝑊80⨁𝑊150 1 (0111110101000010, 0000000100000000) 𝑌80 𝑊20⨁𝑊30⨁𝑊40⨁𝑊50⨁𝑊60⨁𝑊80⨁ 𝑊100⨁𝑊150 1 (0000001010010101, 0000000010000000) 𝑌90 𝑊70⨁𝑊90⨁𝑊120⨁𝑊140⨁𝑊160 1 (0000001111011101, 0000000001000000) 𝑌100 𝑊70⨁𝑊80⨁𝑊90⨁𝑊100⨁𝑊120⨁ 𝑊130⨁𝑊140⨁𝑊160 1 (0101000001101000, 0000000000100000) 𝑌110 𝑊20⨁𝑊40⨁𝑊100⨁𝑊110⨁𝑊130 1 (0101001001111001, 0000000000010000) 𝑌120 𝑊20⨁𝑊40⨁𝑊70⨁𝑊100⨁𝑊110⨁𝑊120⨁ 𝑊130⨁𝑊160 1 (0001100100100100, 0000000000001000) 𝑌130 𝑊40⨁𝑊50⨁𝑊80⨁𝑊110⨁𝑊140 1 (1001100100110101, 0000000000000100) 𝑌140 𝑊10⨁𝑊40⨁𝑊50⨁𝑊80⨁𝑊110⨁𝑊120⨁ 𝑊140⨁𝑊160 1 (1010010000010001, 0000000000000010) 𝑌150 𝑊10⨁𝑊30⨁𝑊60⨁𝑊60⨁𝑊120⨁𝑊160 1 (1010110100010101, 0000000000000001) 𝑌160 𝑊10⨁𝑊30⨁𝑊50⨁𝑊60⨁𝑊80⨁𝑊120⨁ 𝑊140⨁𝑊160 1 linear cryptanalysis of modified safer+ algorithm 118 table 2. dependencies for certain bits of the pht output 𝑌 on certain bits of the pht input 𝑊 "0"-no dependence, "1"-binary linear dependence (i.e., complementing this input bit complements the corresponding output bit), "n"non-linear binary dependence. input bit output bit 𝑊1 7 6 5 4 3 2 1 𝑊2 7 6 5 4 3 2 1 𝑊3 7 6 5 4 3 2 1 𝑊4 7 6 5 4 3 2 1 𝑊9 7 6 5 4 3 2 1 𝑊10 7 6 5 4 3 2 1 𝑊11 7 6 5 4 3 2 1 𝑊12 7 6 5 4 3 2 1 𝑌1 7 6 5 4 3 2 1 0 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑌2 7 6 5 4 3 2 1 0 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑌3 7 6 5 4 3 2 1 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑌4 7 6 5 4 3 2 1 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑌5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑌6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑌7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑌8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑌9 7 6 5 4 3 2 1 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 𝑌10 7 6 5 4 3 2 1 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 𝑌11 7 6 5 4 3 2 1 0 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑌12 7 6 5 4 3 2 1 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 n n n n n n 0 1 n n n n n 0 0 1 n n n n 0 0 0 1 n n n 0 0 0 0 1 n n 0 0 0 0 0 1 n 0 0 0 0 0 0 1 0 0 0 0 0 0 0 𝑌13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑌14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑌15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑌16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 m. kyureghyan, k. kyuregyan 119 secondly, 𝑆𝑎,𝑏 pht depends linearly on some 𝑊𝛼 if 𝑙𝑏 (𝑌) depends linearly on 𝑊𝛼 and 𝑎𝛼 = 0. since 𝑎 ∈ 𝒜 , then the rows of table 2 contain input bits that cannot appear in 𝑙𝑎(𝑊). whenever 𝑙𝑏 (𝑌) contains an output bit that depends linearly on such a 𝑊𝛼 and contain no other output bit that depends on 𝑊𝛼 𝐼(𝑆𝑎,𝑏 pht) = 0. by using table 2 we can iteratively show that i/o sums 𝑆𝑎,𝑏 pht with none-zero imbalance cannot contain any of the output bit 𝑌𝑖𝑗 , where 𝑖 = 1,2,3,4,9,10,11,12 and 𝑗 = 1,2,3,4,5,6,7. finally, we consider the 216 − 1 balanced output functions obtained as linear combinations of the remaining 16 output bits that might occur if 𝑏 ∈ ℬ. for each of these output functions, we found all input functions such that 𝑆𝑎,𝑏 pht doesn’t depend linearly on any input bit. it is easy to show that ((𝑎, 𝑏); 𝑎, 𝑏 ∈ {0,1}128, 𝐼(s𝑎,𝑏 pht) = 1) is a subgroup of (𝒜 × ℬ, ⨁256bits). (in fact if 𝐼(s𝑎,𝑏 pht) = 1 and 𝐼(s 𝑎′,𝑏′ pht ) = 1 for some 128 tuples 𝑎′ and 𝑏′, then 𝐼(s 𝑎⨁𝑎′,𝑏⨁𝑏′ pht ) = 𝐼(s𝑎,𝑏 pht⨁s 𝑎′,𝑏′ pht ) = 𝐼(s𝑎,𝑏 pht) ∙ 𝐼(s 𝑎′,𝑏′ pht ) = 1 as well, whch shows that 𝒜 × ℬ is closed under "⨁".) thus, we obtain all i/o sums with non-zero imbalance, as is done in the statement of the lemma. we next consider the half round containing the nonlinear function nl and find homomorphic i/o sums for nl that have non-zero imbalance. here the input function is homomorphic for xor/add and the output function is homomorphic for add/xor. such i/o sums can be obtained by summing i/o sums for its exp and log blocks. for the function exp with the input byte u1 and output byte v1, the only homomorphic i/o sums are 𝑆𝑎1,𝑏1 exp = 𝑙𝑎1(𝑈1)⨁ 𝑙𝑏1(𝑉1), for 𝑎1 ∈ 𝐵 8\{00}; 𝑏1 = 01. the most effective ones are obtained when (𝑎1, 𝑏1) is equal to (𝑐𝑑, 01) or (𝑓𝑓, 01) (the imbalance being 28 128 ) or to (86,01), (𝑏𝑓, 01), (𝑐0, 01) or (𝑓7,01) (the imbalance being 24 128 ). computing all these i/o sums for exp , establishes the following. remark 2.1: 𝐼(𝑆01,01 exp ) = 𝐼(𝑆02,01 exp ) = 𝐼(𝑆03,01 exp ) = 0. furthermore, for all 𝑎1 and 𝑏1 in 𝐵8, if 𝑎17 = 0, then 𝐼(𝑆𝑎1,𝑏1 𝐸𝑋𝑃 ) = 0. for the function log with the input u2 and output v2, the only homomorphic i/o sums are 𝑆𝑎2,𝑏2 log = 𝑙𝑎2(u2)⨁ 𝑙𝑏2(v2), for 𝑎2 = 01; 𝑏2 ∈ 𝐵 8\{00}. their imbalance is easily deduced since 𝐼(𝑆𝑎1,𝑏1 exp ) = 𝐼(𝑆𝑏1,𝑎1 log ). remark 2.2: for all 𝑎1 and 𝑏1 in 𝐵8, if 𝑏17 = 0, then 𝐼(𝑆𝑎1,𝑏1 log ) = 0. finally we have link i/o sums for successive half rounds. theorem 2.1: the procedure for finding effective homomorphic i/o sums doesn’t find an i/o sum with non-zero imbalance for cascade of half rounds taken in the same order as they are used in modified safer+ and containing at least two pht-layers. proof: let 𝑇𝑎,𝑏 pht−hr, 𝑇𝑏,𝑐 nl and 𝑇𝑐,𝑑 pht−hr be linked homomorphic half-round threefold sums with maximizing key function. if 𝑇𝑎,𝑏 pht−hr and 𝑇𝑐,𝑑 pht−hr have none zero imbalance, the 128 bit none zero masks a, b, c, d, can have a 1 only in the two least significant bits of each byte (bits of byte are numbered from 7 for the most significant bit to 0 for the least significant bit) (lemma 2.1). then 𝐼(𝑇𝑏,𝑐 nl) = 0 since the i/o sum average key imbalance is also 0 (remark 2.1) therefore, the sum of the three half-round threefold sums has imbalance 0. one of the most effective i/o sums is 𝑆𝑎,𝑏 nl−pht−nl, where 𝑎2, 𝑎4, 𝑎11 and 𝑏5, 𝑏6 are either 𝑐𝑑 or 𝑓𝑓and other bytes of 𝑎 and 𝑏 are zero. their imbalance is ( 28 128 ) 5 , because linear cryptanalysis of modified safer+ algorithm 120 𝐼(𝑆𝑓𝑓,01 exp ) = 𝐼(𝑆𝑐𝑑,01 exp ) = 𝐼(𝑆01,𝑓𝑓 log ) = 𝐼(𝑆01,𝑐𝑑 log ) = 28 128 and because 𝐼(𝑆𝑎,𝑏 pht) = 𝐼(𝑆𝑎1⨁𝑎2,𝑏1⨁𝑏2 pht ) = 𝐼(𝑆𝑎1,𝑏1 pht ) ∙ 𝐼(𝑆𝑎2,𝑏2 pht ) = 1, if 𝑎1⨁𝑎2 = (00 00 00 00 00 01 01 00 01 00 00 01 00 01 00 00)⨁ (00 01 00 01 00 01 01 00 01 00 01 01 00 01 00 00) = (00 01 00 01 00 00 00 00 00 00 01 00 00 00 00 00) = 𝑎 (hex notation), 𝑏1⨁𝑏2 = ( 00 00 00 00 01 00 00 00 00 00 00 00 00 00 00 00)⨁ (00 00 00 00 00 01 00 00 00 00 00 00 00 00 00 00) = (00 00 00 00 01 01 00 00 00 00 00 00 00 00 00 00) = 𝑏 (hex notation). 4. conclusion we have proved that the procedure for finding effective homomorphic i/o sums, cannot find an i/o sum with none-zero imbalance for two rounds of modified safer+. thus, as in the case with regular safer+, a modified safer+ is also secure against linear cryptanalysis after only three of its suggested six rounds. acknowledgement the authors are thankful to prof. gurgen khachatrian for very useful discussions and comments. references [1] k. kyuregyan, “some modifications of safer+”, in reports of nas ra, vol. 115, no 1, pp. 33--39, yerevan, armenia, 2015. [2] j. l. massey, g. khachatrian and m kyuregian, “nomination of safer+ as candidate algorithm for the advanced encryption standard”, (aes). nist aes proposal, 1998. [3] carlo h.: cryptanalysis of iterated block ciphers, eth series in information processing (ed massey), v. 7, konstanz: hartung-gorre verlag, 1996. submitted 05.11.2016, accepted 03.02.2016 m. kyureghyan, k. kyuregyan 121 ձևափոխած safer+ ալգորիթմի գծային վերլուծությունը մ կյուրեղյան, ք.կյուրեղյան ամփոփում [2] հոդվածում ներկայացված են safer+ համակարգի ձևափոխությունները, որոնք ալգորիթմի արագագործությունը բարելավում են մոտավորապես 1,7 անգամ, դիֆերենցիալ վերլուծության նկատմամբ ապահովելով նույն կրիպտոկայունությունը, ինչ՝ safer+ համակարգի ալգորիթմը: այս հոդվածում ներկայացված է ձևափոխված safer+ համակարգի կրիպտոկայունությունը բլոկային ծածկագրական համակարգերի մյուս ամենաարդյունավետ կրիպտովերլուծության, գծային վերլուծության նկատմամբ: ցույց է տրվել, որ safer+ համակարգը ձևափոխություններից հետո (ձևափոխած safer+) գծային վերլուծության նկատմամբ ևս ունի միևնույն կրիպտոկայունությունը, ինչ՝ safer+ համակարգը: линейнный криптоанализ модифицированного алгоритма safer+ м. кюрегян, к. кюрегян аннотация в статье [2] представлены некоторые модификации системы safer+, которые примерно в 1,7 раза увеличивают скорость алгоритма, обеспечивая такую же криптостойкость по отношению к дифференциальному анализу, как алгоритм системы safer+. в этой статье представлена криптостойкость модифицированной системы safer+ по отношению к линейному анализу, который являетса одним из эффективных аттак против итеративных блочных шифров. было доказано, что после модификации система safer+ (модифицированный safer+) имеет ту же криптостойкость по отношению к линейному анализу, что и система safer+. microsoft word 17_vahe_ arakelyan.doc mathematical problems of computer science 39, 129--134, 2013. 129 vulnerable security problems in learning management system (lms) moodle vahe a. arakelyan institute for informatics and automation problems of nas of ra e-mail: a_vahe@sci.am abstract in this paper the security mechanisms of lms moodle are investigated. some vulnerable parts have been discovered and some solutions and methods are proposed to make the system more secure and safe. keywords: moodle, security, attack. 1. introdaction moodle is a content management system (lms) specially designed for creating high-quality online courses. “moodle” is an abbreviation, which is obtained from the expression “modular object-oriented dynamic learning environment’’. moodle has been translated into many languages and is used in 197 countries. the leader of the above mentioned system is martin dougiamas from australia. it is an open source system and a lot of programmers can take part in the developing of it. moodle is written in php language, using sql-database. having a number of functions and modules moodle lms competes with famous commercial learning systems such as blackboard, desire2learn, sharepoint lms and so on. the advantage of the distributed open-source software is that it allows to modify the system specific study program characteristics, and, if necessary, add new modules. the rapid progress in information technology allows the use of computers as a teaching effective means. automation of the learning process is realized by means of computer training programs and the use of electronic textbooks, which are used not only for magnetic storage media (laser disc) to the application, but also for local and global computer networks. by means of the use of global computer network it creates a specialized information-educational field which allows to realize and carry out training, using modern technologies. to implement it in informational educational sphere as well as in local and global computer networks, the effective use of e-learning materials, high-operative development are essential and necessary. the goal of the creation of electronic training materials is to improve the quality of efficiency of the learning process and mastering of specialists. in the field of higher education e-learning, materials can be used as an additional educational resource, which allows to organize the teacher’s control on student’s independent vulnerable security problems in learning management system 130 work correctly. thus, higher education will involve a step by step contribution of virtual learning environment technologies, particularly in distance learning. so very soon higher education will be conducted with the use of virtual learning environment. the yerevan state pedagogical university and the magistracy of the national academy of sciences of ra plan to introduce this system, due to which training will be available to students who live in different regions of armenia and even beyond the republic, for those who wish to get highly qualified and modern education. at the same time in virtual learning environment (vle) electronic educational materials for students are the basic source of information [8]. 2. vulnerabilites in moodle such important disadvantages which can make the system weak in case of attacks are presented in this paper. they are classified into four groups: authentication, availability, confidentiality and integrity. distance learning systems or vles are client/server web programs, which develop requests coming from the user internet browser [5]. polls processing system often uses strict security rules that require data, such as, e.g., databases and files. 2.1. authentication attacks authentication and session management include all aspects, which are necessary to identify the user and manage the active session. identification in this process is considered to be a very dangerous process, even in case of complicated mechanisms it can be broken because of the insufficient management functions of identification data, such as an opportunity of password change or the password may be forgotten and can be restored, a password remembering feature, or transferring plain date during authentication. the attackers can use these data and hack the system. 2.2. dos attacks the main purpose of attacks is to make the education system unavailable for users. the most popular one is denial of service attack. under dos attack can be any of the web services, including banking systems. there are two types of dos attacks, logical and flooding. logical attacks use the vle’s existing errors and carry out such requests to respond them, that makes the system an endless cycle of actions. during the flooding attack a large number of requests are made simultaneously and the system is not able to manage and respond to all of them. in both cases the system productivity is reduced, or it stops to serve the endusers. 2.3. confidentiality attacks confidentiality attacks are submissive kind of attacks which allow prohibited access to confidential resources and data. the main purpose of an attacker is not the data alteration but data access and distribution. the most frequent confidentiality flaws are insecure cryptographic storage, insecure direct objects reference, improper error handling and information leakage. 2.4. integrity attacks the target of this type of attacks is to create, modify or even destroy the existing e-learning system data. there are different types of integrity attacks: buffer overflow, cross site request forgery (xsrf/csrf)), cross site scripting (xss), injection flaws, upload and run malicious files, failure to restrict url access. harmful files can be uploaded as homework or just music, v. arakelyan 131 which are not controlled by the learning system. in the field of student data (e.g. form field) the attacker can write such a code, which is a command for incorrect actions. with the help of xsrf the attacker can create a dynamic web site, which concludes the harmful script and it can steal the web browser from the user. xsrf attack method can be applied by the learning system users. cross site request forgery (xsrf/csrf) is a client side attack which exploits faith that an lms has for the user. when the user is logged into lms, the attacker can deceive his browser by making a request to one of lms tasks, which will cause a change on the server. buffer overflow attack [5] occurs when a lms module (e.g. libraries, drivers, server components) tries to store data into an available buffer without validating its size by inserting larger values than expected. in case of failure to restrict url access, some lms resources are limited to a small subset of advantaged users (e.g. administrators). 3. effective attacks against moodle security studies have found that these types of attacks can hack moodle security. • username prediction • password prediction • session hijacking 3.1. password prediction a brute-force attack can be applied because of a structural defect of the moodle vle identification methods. the attacker sends requests with blank cookie fields, so the login failure count becomes zero when the cookie field is blank in the request. as a result of this, the attacker may try an infinite number of passwords. 3.2. username prediction the user name prediction is also possible to implement via a brute force method. multiple requests are sent to the system with different usernames and the same password. in case of the existing username the system responds later than the other non-existent usernames. 3.3. session attack two session attacks are effective against moodle: session hijacking and session fixation. session hijacking is a part of the eavesdropping attacks, where an attacker listens to the communication between the client and the server trying to find the payload inside: in this case the http request, information that can be used to impersonate the user and take control of his or her session. moodle manages its sessions trough two values to identify an active session: moodlesession and moodlesessiontest. these values are stored in the cookie that is sent to each http request inside the header of the message. in order to impersonate a target user, an attacker must obtain such values. session fixation attack also targets the session data of a user. however, this attack is classified as an active attack [3] or an interception attack. instead of eavesdroping the communication between a target user and the server, the attacker intercepts the http request of the target user. vulnerable security problems in learning management system 132 4. solutions to moodle vulnerabilities as it was shown in the previous section, moodle is vulnerable to the following attacks: session fixation, session hijacking, prediction of usernames and prediction of passwords by brute force. such vulnerabilities can be avoided by modifying certain portions of the code and adding new functions. these modifications will be described later [2]. 4.1. to use ssl over the entire site moodle already has an option to use ssl over certain critical actions. however, such method cannot prevent session fixation, session hijacking and username prediction. in order to avoid such attacks the entire site must create ssl connections with its clients. this can be done by adding a php scripts that change the content of the object that holds the environment configuration named cfg. inside cfg there are the following four variables that are ssl related. • themewww. this variable holds the location of resources for building the graphical interface as a full url string. the script has to change the http protocol for https (ssl request). • wwwroot. moodle uses this variable to know the url assigned to it for a quick navigation. the script has to change the http protocol for https. • loginhttps. the value of this flag is retrieved from the database and when it is on the login page, it is encrypted trough ssl. the script turns it on, even if the main configurations say otherwise. • httpstheme. when the loginthttps is turned on, the original source code changes the url protocol from http to https. this script also changes this value to https, overriding the original loginhttps value. • the script also has to change the value of the global flag httpspagerequired to true. this flag is a part of the moodle default configuration. such fixes were implemented in a script called buap_security that is invoked when the main configuration script config.php is called on every user request. the security server configuration page was also changed to reflect the new option of ciphering the entire site by modifying the source code of the security.php located inside the admin package. 4.2. id session regeneration the ssl protocol cannot protect the site from session fixation when the user requests a connection for the first time if that petition is done by http protocol without ssl. this was fixed by generating a new id session when the user is authenticated by login/password matching. php allows to change the id by placing the instruction session_regenerate_id after switching privileges, at the line 2684, from the source code moodlelib.php stored inside the lib package. 4.3. login with captcha the authentication service, implemented as a login page, can be a subject of automatic brute force attack. the login page can be protected by using captcha [1]. the login page was added with the official captcha implementation known as recaptcha. this makes the authentication service stronger against the brute force attacks automated with software tools. moodleib.php has to be modified in order to check the new data associated with the recaptcha library. v. arakelyan 133 4.4. correct permissions loading as the author suggested in [6], usernames stored inside moodle can be predicted because the permissions are loaded before username and password checking, which make responses to take longer when a request with a valid username is sent. this was fixed by just changing the order of the actions taken by the authenticate_user_login method coded in the moodlelib.php file. 4.5. username obfuscation as a part of its authentication implementation, moodle stores the username inside the cookie of the http protocol. this field is obfuscated with the rc4 algorithm using a private key defined in the source code. this is highly insecure because even with ssl an attacker can capture the user cookie of older sessions and can decrypt it with the information of the source code. in order to avoid such decryption, a new configuration file was implemented allowing the administrator of moodle to change the obfuscation private key and even to choose the algorithm used; instead of using the moodle implementation of rc4 the new configuration file implements the mcrypt library [4] of php. it is possible to change also the algorithm used for storing passwords. currently, moodle uses md5 for storing passwords inside its database, but there are works in [7] that show how an md5 hash can be broken. the new configuration file also offers a possibility for selecting a new hash algorithm available in the hash library [4] of php. however, when changing this parameter, the administrator of moodle must be aware that every password hash previously stored with the old algorithm, even the password of the administrator, will become invalid. references [1] a. l. von, m. blum, n. j. hopper, and j. langford, “captcha: using hard ai problems for security”, conference on the theory and applications of cryptographic techniques, eurocrypt, pp. 294-311, 2003. [2] j. carlos, g. hernández and c. m. a. león, “moodle security vulnerabilities”, 5th international conference on electrical engineering, computing science and automatic control (cce), isbn: 978-1-4244-2499-3, 2008. [3] m. eric, “network security a beginner’s guide”, 2 ed., mcgraw-hill/osborne, pp.19-24, 2003. [4] s. chris, m. southwell, “pro php security”, apress, pp. 66-68, 80-85, 2005. [5] stapic’ z., t. orehovački, m. ðanic', “determination of optimal security settings for lms moodle”, proceedings of 31st mipro international convention on information systems security, opatija, vol. 5, pp. 84-89, 2008. [6] d. stuttard and m. pinto, the web application hacker’s handbook: discovering and expl oiting security flaws, wiley publishing inc., 2007. [7] x. wang, d. feng, x. lai and h. yu, “collisions for hash functions md4, md5, haval128 and ripenmd”, in cryptology eprint archive, http://eprint.iacr.org, (accessed 25 january 2012), report 2004/199, 2004. [8] m. zenha-rela and r. carvalho, “work in progress: self evaluation through monitored peer review using the moodle platform”, in frontiers in education conference, 36th annual., san diego, ca: ieee, pp. 230-241, 2006. submitted 14.11.2012, accepted 06.02.2013. vulnerable security problems in learning management system 134 անվտանգության խոցելի խնդիրները ուսուցման կառավարման moodle համակարգում վ. առաքելյան ամփոփում հոդվածում հետազոտված են ուսուցումը կառավարող moodle համակարգի անվտանգությունը ապահովող մեխանիզմները։ հայտնաբերվել են որոշ խոցելի հատվածներ, առաջարկվել են լուծումներ և մեթոդներ, որոնց կիրառման դեպքում համակարգը կդառնա ավելի պաշտպանված և անվտանգ։ уязвимые проблемы безопасности в системе управления обучением moodle в. аракелян аннотация в этой статье исследованы механизмы безопасности системы управления обучением moodle. обнаружены некоторые уязвимые части и предложены некоторые решения и методы, с использованием которых можно сделать систему более надежной и безопасной. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 43, 57--61, 2015. encoding and decoding procedures for double ±1 error correcting linear code over ring 𝑍𝑍5 hamlet k. khachatryan institute for informatics and automation problems of nas ra e-mail: hamletxachatryan.08@gmail.com abstract from practical point of view the codes over 𝑍𝑍2𝑚𝑚 or 𝑍𝑍2𝑚𝑚+1 are interesting, because they can be used in 22𝑚𝑚 –qam (quadrature amplitude modulation) schemes. in this paper a construction of encoding and decoding procedures for double ±1 error correcting optimal(12,8) linear code over ring 𝑍𝑍5 is presented. keywords: error correcting codes, codes over the ring 𝑍𝑍5, encoding and decoding procedures. 1. introduction codes over finite rings, particularly over integer residue rings and their applications in coding theory have been studied for a long time. errors happening in the channel are basically asymmetrical; they also have a limited magnitude and this effect is particularly applicable to flash memories.there are many constructed codes capable to correct up to two errors of value ±1 .the earliest paper discussing the codes over the ring 𝑍𝑍𝐴𝐴 of integers modulo 𝐴𝐴 are due to blake [1], [2]. the optimality criteria for the linear code over fixed ring 𝑍𝑍𝑚𝑚 was considered in 2 ways in [3]. first of all, recall that the code of the length n is optimal-1 if it has a minimum possible number of parity check symbols. secondly, optimality-2 criteria for the code is that for a given number of parity check symbols it has a maximum possible length. here, we propose to construct encoding and decoding algorithms for the optimal codes. the code presented in this paper satisfies the optimality criteria-1([3]). at this point we do not know any codes that satisfy the optimality criteria-2. there have been encoding and decoding procedures for the (4, 2) code over ring 𝑍𝑍9 in [4]. implementation of codes over large alphabets is much more difficult compared with small alphabets. in this paper a construction of encoding and decoding procedures of (12, 8) linear code over ring 𝑍𝑍5 correcting double ±1 errors is presented. 57 encoding and decoding procedures for double ±1 error correcting linear code over ring 𝒁𝒁𝟓𝟓 58 2. presentation of the code let a linear (12, 4) code over ring z5 be given by the following parity check matrix 𝐻𝐻: 𝐻𝐻 = � 1 1 1 1 1 0 1 2 3 4 1 1 0 1 2 3 4 2 2 2 2 2 1 1 3 2 4 4 2 3 2 4 4 2 1 1 1 1 1 1 1 3 2 4 4 2 0 4 �. a linear code over z5 given by the parity check matrix 𝐻𝐻 can correct up to two errors of the type ±1, because 𝐻𝐻 has a property according to which all the syndromes resulting from adding and subtracting operations between any two columns of the matrix 𝐻𝐻 are different (±hi± hj and hi≠hj)(proof of this you can see in [3]). in this case the number of combinations for each code word that can be corrected is (1 + 12 ∗ 2 + (12 𝑐𝑐ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 2) ∗ 4) = 289. for encoding every vector in 𝑍𝑍5 we should have the generator matrix 𝐺𝐺. for this we should construct a combinatorial equivalent matrix 𝐻𝐻′ from matrix 𝐻𝐻. 𝐻𝐻′ = � 1 0 0 0 3 1 0 3 4 3 4 0 0 1 0 0 2 2 4 4 0 2 4 3 0 0 1 0 0 0 2 4 2 1 3 1 0 0 0 1 1 3 2 0 4 4 3 3 �. in this matrix all 289 possible syndromes will be different, too. from [5] we know, that 𝐺𝐺𝐻𝐻′𝑇𝑇 = 0. (1) if h′ = [−pt|in−k], then g = [ik|p] (the reverse statement is also true), where ik is a 𝑘𝑘 ∗ 𝑘𝑘 identity matrix and p is a 𝑘𝑘 ∗ (𝑛𝑛 − 𝑘𝑘) matrix[5]. thus, we can construct the generator matrix 𝐺𝐺: 𝐺𝐺 = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 2 3 0 4 1 0 0 0 0 0 0 0 4 3 0 2 0 1 0 0 0 0 0 0 0 1 3 3 0 0 1 0 0 0 0 0 2 1 1 0 0 0 0 1 0 0 0 0 1 0 3 1 0 0 0 0 1 0 0 0 2 3 4 1 0 0 0 0 0 1 0 0 1 1 2 2 0 0 0 0 0 0 1 0 0 2 4 2 0 0 0 0 0 0 0 1⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ . h. khachatryan 59 3. encoding and decoding procedures 3.1 encoding procedure in our scheme the message was presented by 8-tuples in z5. let 𝐺𝐺 be a generator matrix for (12,8) linear code. 𝑣𝑣 = (𝑎𝑎1, 𝑎𝑎2, 𝑎𝑎3, … , 𝑎𝑎8) is an arbitrary 8-tuple, and consider the vector 𝑢𝑢 that is the linear combination of columns 𝐺𝐺 with 𝑎𝑎𝑖𝑖 is the 𝑖𝑖𝑡𝑡ℎ coefficient. 𝑢𝑢 = 𝑣𝑣𝐺𝐺 = (𝑐𝑐1, 𝑐𝑐2, 𝑐𝑐3, 𝑐𝑐4, 𝑎𝑎1, 𝑎𝑎2, 𝑎𝑎3, … , 𝑎𝑎8), where the first 4 components of the code vector are the check symbols, the next 8 components are information symbols and 𝑐𝑐𝑗𝑗 = �∑ 𝑎𝑎𝑖𝑖𝑝𝑝𝑖𝑖𝑗𝑗 𝑘𝑘 𝑖𝑖=1 �𝑚𝑚𝑜𝑜𝑚𝑚5. (2) example. let (3 4 0 0 2 1 1 4) be the message vector in 𝑍𝑍5. from (2) we can obtain check symbols. for example, the first check symbol is c1: 𝑐𝑐1 = (3 ∗ 2) + (4 ∗ 4) + (0 ∗ 0) + (0 ∗ 2) + (2 ∗ 1) + (1 ∗ 2) + (1 ∗ 1) + (4 ∗ 0) = = 6 + 16 + 0 + 0 + 2 + 2 + 1 + 0 = 27𝑚𝑚𝑜𝑜𝑚𝑚5 = 2 . similarly, we can find other 3 check symbols: 𝑐𝑐2 = 3, 𝑐𝑐3 = 3, 𝑐𝑐4 = 3. after performing other multiple operations with matrix 𝐺𝐺 we obtain this encoded vector: (2 3 3 3 3 4 0 0 2 1 1 4). 3.2 decoding procedure in this section we describe the decoding procedure: 1. receiver multiplies the vector with every column of matrix h′ and gets the syndrome s = vh′. if s = (0,0,0,0) then there were not any errors in the received vector. 2. if the calculated syndrome s is a nonzero vector, then there are some errors. this (12,8) code can correct only up to two errors with magnitude ±1. we know that all possible syndromes of matrix h′ are different (±hi, ±hj and hi ≠ hj) (the number of them is 288 and syndrome (0,0,0,0)). after calculating the syndrome the receiver knows from which two columns of the matrix h′ the syndrome was resulted, consequently, he can find the two corresponding components of the vector, where the error was occurred and the direction of the error (if +hi, then upward direction or if −hi downward direction). on the other hand, if in the table of syndromes we do not have the resulted syndrome, then we cannot correct this kind of errors. 3. after finding the error components the receiver adds or subtracts 1 from them (he adds if downward, else subtracts) and obtains the sent code vector (𝒄𝒄𝟏𝟏, 𝒄𝒄𝟐𝟐, 𝒄𝒄𝟑𝟑, 𝒄𝒄𝟒𝟒, 𝒂𝒂𝟏𝟏, 𝒂𝒂𝟐𝟐, 𝒂𝒂𝟑𝟑, … , 𝒂𝒂𝟖𝟖). so (𝒂𝒂𝟏𝟏, 𝒂𝒂𝟐𝟐, 𝒂𝒂𝟑𝟑, … , 𝒂𝒂𝟖𝟖) is our message vector. encoding and decoding procedures for double ±1 error correcting linear code over ring 𝒁𝒁𝟓𝟓 60 example. (2 3 3 3 3 4 0 0 2 1 1 4) is an encoded vector from the previous example. let there occur 2 errors in the channel, and the receiver get the vector (2 3 3 3 3 4 0 4 2 2 1 4). after performing multiple operations with rows of matrix 𝐻𝐻′ 𝑡𝑡ℎ𝑜𝑜 receiver obtains the syndrome (0 3 2 4). next from the table of syndromes he finds the corresponding columns, now they are -8 and 10. hence, the syndrome (0 3 2 4) was resulted from adding a negated column 8 of matrix 𝐻𝐻 to column −3 +3 −4 +2 −4 +1 0 +4 = 0 −2 −3 4 (𝑚𝑚𝑜𝑜𝑚𝑚5) = (0 3 2 4), (because in 𝑍𝑍5, 0 = 5, −1 = 4, −2 = 3 , −3 = 2, −4 = 1). hence, the error positions of encoded vector are 8 and 10 (in 8𝑡𝑡ℎ downward direction and in 10𝑡𝑡ℎ upward). so, he adds 1 to 8𝑡𝑡ℎ component and subtracts 1 from 10𝑡𝑡ℎ of vector (2 3 3 3 3 4 0 4 2 2 1 4) and obtains the sent encoded vector (2 3 3 3 3 4 0 0 2 1 1 4). consequently, the message vector 𝑖𝑖𝑜𝑜 (3 4 0 0 2 1 1 4). 4. conclusion in this paper a construction of encoding and decoding procedures of optimal-1 (12,8) linear code over ring 𝑍𝑍5 correcting double ±1 errors is presented. we propose that this approach can be extended for constructing similar procedures for the optimal codes over other rings 𝑍𝑍𝑛𝑛 and the research in this direction will follow. references [1] i. blake, “codes over certain rings”, information and control, vol. 20, pp. 396-404, 1972. [2] i. blake, “codes over integer residue rings”, information and control, vol. 29, 1975, 295-300. [3] g. khachatrian and h. morita, “construction of optimal 1 double error correcting linear codes over ring z5 ”, 3th international workshop on advances in communications, boppard, germany, pp. 10-12, may 2014. [4] h. kostadinov, n. manev and h. morita, “double ±1-error correctable codes and their applications to modulation schemes”, proc. elev. intern. workshop acct, pamporovo,june 16-22, pp 155-160,2008. [5] w. w. peterson and e.j. weldon, error-correcting codes, m.i.t. press, cambridge, mass.,1972. submitted 15.10.2014, accepted 12. 02. 2015. h. khachatryan 61 կոդավորման և ապակոդավորման ալգորիթմը z5 օղակում ±1 մեծությամբ կրկնակի սխալ ուղղող կոդերի համար հ. խաչատրյան ամփոփում պրակտիկ տեսանկյունից մեծ հետաքրքրություն են առաջացնում 𝒁𝒁𝟐𝟐𝟐𝟐 կամ 𝒁𝒁𝟐𝟐𝟐𝟐+𝟏𝟏 օղակների վրա դիտարկված կոդերը, քանի որ նրանք ունեն լայն կիրառություն 𝟐𝟐𝟐𝟐𝟐𝟐 –qam մոդուլյացիոն սխեմաներում: այս հոդվածի շրջանակներում ներկայացված է կոդավորման և ապակոդավորման ալգորիթմը z5 օղակում ±1 մեծության մինչև 2 սխալ ուղղող օպտիմալ (12, 8) գծային կոդի համար: алгоритм кодирования и декодирования в кольце z5 для кодов исправляющих двойные ошибки размера ±1 г. хачатрян аннотация с практической точки зрения большой интерес вызывают коды на кольцах 𝑍𝑍2𝑚𝑚 и 𝑍𝑍2𝑚𝑚+1, так как они имеют широкое применение в 22𝑚𝑚 –к а м модуляционных схемах. в данной статье представлен алгоритм кодирования и декодирования в кольце z5 для оптимального (12,8) линейного кода, исправляющего до двух ошибок размера ±1. encoding and decoding procedures for double ±1 error correcting linear code over ring ,𝑍-5. hamlet k. khachatryan from practical point of view the codes over ,𝑍-2𝑚. or ,𝑍-2𝑚+1. are interesting, because they can be used in ,2-2𝑚. –qam (quadrature amplitude modulation) schemes. in this paper a construction of encoding and decoding procedures for double ±1 erro... 1. introduction 2. presentation of the code d:\sbornik\...\tpel.dvi mathematical problems of computer science 32, 23{26, 2009. t he m inimum linear ar r angement p r oblem on b ipar tite ¡-or iented gr aphs l e vo n r . r a fa ye lya n yerevan state university levonr@synopsys.com abstract in general case the minimum linear arrangement (minla) problem is np-complete. it is np-complete also for bipartite graphs. in this paper it is proved that a minimum linear arrangement for bipartite ¡-oriented graphs can be found in polynomial time. the formula for the cost of optimal arrangement is given as well. refer ences [1 ] m. r . ga r e y, d . s . jo h n s o n ,computers and intractability. a g u id e t o t h e t h e o r y o f n p -c o m p le t e n e s s . fr e e m a n a n d co m p a n y, 1 9 7 9 . [2 ] s . e ve n , y . s h ilo a h , n p -c o m p le t e n e s s o f s e ve r a l a r r a n g e m e n t p r o b le m s , r e p o r t n o . 4 3 , d e p t . o f co m p u t e r s c ie n c e , te c h n io n , h a ifa , is r a e l, 1 9 7 5 . øçýçù³é ·í³ûçý ñ³ù³ñ³ï³éáõù ëý¹ñç éáõíáõùá »ñïïáõù³ýç ¡ -ïáõùýáñáßí³í ·ñ³ýý»ñç ñ³ù³ñ è. è³ý³û»éû³ý ²ù÷á÷áõù ¶ñ³ýç ùçýçù³é ·í³ûçý ñ³ù³ñ³ï³éáõù ëý¹çñá áý¹ñ³ýáõñ ¹»åùáõù npéñçí ¿: ²ûý npéñçí ¿ ý³¨ »ñïïáõù³ýç ·ñ³ýý»ñç ñ³ù³ñ: ²ûëï»õ ³å³óáõóí³í ¿, áñ »ñïïáõù³ýç ¡ -ïáõùýáñáßí³í ·ñ³ýý»ñç ñ³ù³ñ ùçýçù³é ·í³ûçý ñ³ù³ñ³ï³éáõù ï³ñ»éç ¿ ·ïý»é µ³½ù³ý¹³ù³ûçý å³ù³ý³ïáõù: îñí³í ¿ ý³¨ ûåïçù³é éáõíù³ý ³ñå»ùý»ñç ñ³ßí³ñïáõ µ³ý³ó¨á: 2 3 mathematical problems of computer science 57, 7–17, 2022. doi:10.51408/1963-0082 udc 004.934 emotion classification of voice recordings using deep learning narek t. tumanyan weizmann institute of science, rehovot, israel e-mail: narek.tumanyan@weizmann.ac.il abstract in this work, we present methods for voice emotion classification using deep learning techniques. to processing audio signals, our method leverages spectral features of voice recordings, which are known to serve as powerful representations of temporal signals. to tackling the classification task, we consider two approaches to processing spectral features: as temporal signals and as spatial/2d signals. for each processing method, we use different neural network architectures that fit the approach. classification results are analyzed and insights are presented. keywords: voice sentiment detection, mood recognition, speech emotion recognition, cepstral features. 1. introduction the problem that is addressed in this work is the emotion classification from voice recording. formally, given some representation x of voice recording data and a set of n emotion labels/classes {y1, y2, ..., yn}, the aim is to come up with a classifier f(x) = yi that maps x to a label yi ∈ {y1, ..., yn}. practically, having such a classifier f can have a wide range of applications, such as recommendation systems of movies or music driven by users’ mood, systems for tracking the emotional state and satisfaction of clients through time, security systems for preventing harmful actions based on emotion, and so on. previous attempts to tackle the voice emotion classification problem include svm-based algorithms of classifying voice into 5 categories angry, happy, neutral, sad, or excited [1], which also considers the facial expression of the speaker during speech as an additional signal. glüge et al. [2] propose a deep neural network extreme learning method with efficient performance on small datasets. eskimez et al. [3] tackle the speech emotion recognition problem through an unsupervised approach, by which they come up with meaningful speech representations by learning the underlying structure of the data, which aids in solving the main task. bertero et al. [4] introduce a convolutional neural network (cnn)-based approach of 3-label (“angry”, “happy”, “sad”) emotion recognition of speech, where they use the standard pulse-code modulation (pcm) temporal representation of the audio signal as 7 8 emotion classification of voice recordings using deep learning input. mirsamadi et al. [5] propose a 4-label (“angry”, “happy”, “sad”, “neutral”) speech emotion recognition model based on long short term memory network (lstm) architecture and local attention, and base their model on mel-frequency cepstral coefficients (mfcc), fast fourier transform (fft), fundamental frequency and zero-crossing rate features of the audio. in our setups, we experiment with both cnn-based and lstm-based architectures and consider 8 emotional labels for classification, which are described in section 2. in this paper, we use cepstral features as representations of voice data, particularly, we utilize mel-frequency cepstral coefficients (mfcc) for representing the audio signal. we experiment with two views for processing mfccs: processing them as sequential data in the time domain, and processing them as spatial data. for each of the approaches, we use the appropriate neural network architecture. specifically, for processing mfccs as temporal data, we utilize long short term memory networks (lstm), and for processing mfcc as spatial/2d data, we make use of convolutional neural networks (cnn). 2. datasets in our setup, we consider 8 emotion labels for classification. the databases used in the paper are as follows: ryerson audio-visual database of emotional speech and song (ravdess) [6], surrey audio-visual expressed emotion (savee) [7] and toronto emotional speech set (tess) [8]. each item in the datasets is a recording of an actor that pronounces some statement with a certain expressed emotion. voice recordings in the databases come in a .wav format, which describes the amplitude of air pressure oscillations in the temporal domain. each voice recording has an emotion label attached to it. the ravdess database has 24 actors that pronounce 2 phrases: “kids are talking by the door“ and “dogs are sitting by the door” with 2 intensities: normal and high each repeated twice. neutral emotion has no high intensity so it is only repeated twice. the emotion labels are: “neutral”, “calm”, “happy”, “sad”, “angry”, “fearful”, “disgust”, “surprised”. tess dataset has 2 actors, young and old, and both of them are female. there are 2800 voices in total with each phrase being of the form “say the word x, where x stands for some word. recordings in the tess dataset have the same labeled emotions as in ravdess, except for the calm label, which is absent in this dataset. savee dataset has 4 english male actors with 480 voice recordings. 7 emotions are present, with the calm emotion missing. in total, there are 4720 samples. the distribution of samples and classes is summarised in table 1 and in table 2. table 1: summary of datasets used. database num of recordings num of actors emotion labels ravdess 1440 24 8 savee 480 4 7 tess 2880 2 7 n. tumanyan 9 table 2: number of voice recordings per emotion label across all databases. neutral calm sad fear anger surprises happiness disgust 616 192 652 652 652 652 652 652 3. method 3.1 feature extraction to extract audio features from voice recordings, we use librosa library for python [9]. it handles most of the transformations done to voice recordings to get final features used for classification. the first step before extracting features is to resample voice recording files to obtain their time domain and amplitude representation. voice recordings from our databases have different original sampling rates, which range from 22khz to 48khz. however, the content that we are trying to analyze from those recordings are the human voices themselves. normally, the human voice ranges from low range frequencies 300hz to higher ranges of 4 10khz. this means that we can use lower sampling rates to resample our voice recording. we chose 22.05khz sampling rate, which preserves all human voices in original audio recordings and also preserves some possible frequency deviations from the normal range, which can be caused by pronouncing high-frequency tones, e.g. fricatives. the result is a floating-point time series describing the amplitude of air pressure oscillations from a mean frequency of 0 at each time point. thus, we obtain a time-domain representation of the signal. an example is illustrated below in fig. 1. fig. 1. sample waveform representation of a voice recording signal. having the temporal signal representation of the voice signal, we then process it to obtain its spectral features, which serves as the main data representation for our models. 10 emotion classification of voice recordings using deep learning 3.2 spectral features extraction conceptually, given a temporal signal x(t), we can represent it as a combination of periodic functions of varying frequencies [10]: x(t) = ∞∫ −∞ x(w)ejwtdw, where w is the frequency of the corresponding periodic function. thus, having the coefficients x(w) is equivalent to having the original signal x(t), and we can use these coefficients as a representation of the temporal signal in the frequency space. to achieving such a representation, the fourier transform operation is used [10]. since we are dealing with discrete data, the equivalent operation used is discrete fourier transform (dft), which converts discrete temporal signal x[n] of length k to a representation of this signal in frequency space by obtaining the coefficients / intensities x[k] for each frequency k [11]: x[k] = k∑ n=1 x[n]e−i2πkn/n; 1 ≤ k ≤ k. in signal processing, frequency decomposition is often performed by dividing the signals into time intervals of specified window size and performing dft on each windowed signal, thus coming up with frequency components in multiple time intervals. such representation of a signal is called the short-time fourier transform (stft) of a signal [10]. for audio signals, in some cases, more sophisticated representations of the signal based on stft are necessary for higher efficiency. mel-frequency cepstral coefficients, a.k.a. mfccs, are features, which represent a given signal by cepstral energy coefficients at specific short intervals of time. the advantage of mfcc features is that they represent the signal in a way that is close to the signal perception by the human ear, which, is intuitively achieved by applying smaller window-sized cepstral filters on low frequencies on a signal and increasing the window size of the filters as the considered frequency increases. the reason behind such intuition is that the human ear perceives frequencies in lower ranges much better than in higher ones. hence, higher resolution at lower ranged frequencies is used while computing mfccs [12]. in its final form, the mfcc of a signal can be represented simply as a function pi(k), where the outputted value is the intensity of k-th cepstral coefficient in i-th temporal frame index. an example of an extracted mfcc feature is demonstrated in fig. 2 fig. 2. sample mfcc representation of a voice recording signal. n. tumanyan 11 3.3 architectures and results 3.3.1 long short term memory networks (lstms) considering the temporal nature of the data in hand, i.e., the voice recordings that are represented as magnitudes of air pressure (amplitude) across time, and the computed mfcc’s that are a time series of energy coefficient values, it is sensible to use architectures that are by design intended for processing sequential data and have the appropriate inductive bias. one example of such architectures are long short term memory networks (lstm) [13], which are a variant of recursive neural networks (rnn). the main idea behind lstm is the usage of feedback connections for preventing the vanishing gradient problem. the architecture of lstm used is summarized in fig. 3. fig. 3. the architecture of the trained lstm model. fig. 4. roc curves of the trained lstm model. each curve corresponds to an emotion label. 12 emotion classification of voice recordings using deep learning mfcc sequences are fed into an lstm recurrent layer with a hidden dimension of size 1024. there are 2 lstm layers stacked on top of each other, meaning that the outputs of the first layer are processed by the second one. this increases the perceptiveness of the network towards the features present in the sequence. due to the dataset being small, we used dropout with high probability (p = 0.5) on the outputs of the first lstm unit to prevent overfitting. the output of the last lstm layer is then passed to a multilayer perceptron (mlp), which outputs an 8-dimensional vector representing the logits of each emotion label. the network was trained using only the ravdess dataset. the recordings of the 1st and 2nd actors (one male and one female) were used as a testing set, the rest of the recordings were used for training the network. adam optimizer with learning rate of 0.0005 was used and the loss function to minimize was cross entropy loss given by: l (ŷi) = log ( exp (ŷi)∑ j exp (ŷj) ) , l (ŷi) = − ∑ i yil (ŷi) , where {ŷi} are the estimated class labels, and {yi} are the ground-truth labels. the classification results and comparison to the existing relevant method are demonstrated in table 3. the receiver operating characteristic curves (roc curves) of the results are shown in fig. 4. 3.3.2 convolutional neural networks (cnns) as stated in subsection 3.2, the mfcc of a recording can be observed as a 2d feature map of a signal, with one dimension being the temporal dimension and the other being the cepstral coefficient dimension. thus, a possible approach to working with mfcc’s is processing them as spatial signals. convolutional neural networks (cnn) are one of the most prominent architectures used for processing spatial data due to their shift equivariance, their inductive bias in searching for local patterns, and many other inherent benefits. thus, we consider solving the voice emotion classification task by training a cnn on extracted mfcc data. for mfcc calculation, the window size of 4096 and the overlap of between subsequent windows were chosen. decreasing the window size by half degrades the performance of the network. on average, these settings produced better results. 4096 for a window size is good because it allows computing the fft of length 4096 on that window to capture frequency spectrum of up to 4khz. this means that the majority of human speech in those recordings is captured in each window. after calculating mfccs for every recording and padding sequences with less length than the longest sequence, we obtain input matrices to our network of size (40 x 160) where at each sequence point we have 40 mfccs. the architecture of the cnn used is depicted in fig. 5, and the method is summarized as follows: there are 3 convolutional layers in the network followed by average pooling layers of size (2x2). the last layer is a fully connected layer that maps output of convolutional layers to an 8 length vector. log softmax activation is applied to use cross entropy loss. each layer has 32 kernels of parameters. the first layer has kernels of size (10x3), and it is deliberately chosen to be narrow and heighty to capture features from change of mfccs through the sequence. between layers, leaky rectified linear unit (relu) activation function given as h(x)=max(x, 0)+0.01*min(0, x)is used both to enable fast training and to prevent neurons n. tumanyan 13 from dying. leaky relu adds a small slope to non activated neurons thus preventing them from becoming 0 and not contributing to backpropagation in later epochs [14]. since our dataset is very small, we used dropout with high probability (p = 0.5) as well as l2 regularization to prevent overfitting, which penalizes the sum of squares of the weights of the model. fig. 5. the architecture of the trained cnn model. all recordings of the 1st and 2nd actors, one male and one female from the ravdess database were used for testing, which the neural network was not trained on. all remaining recordings were used for training. we used adam optimizer with a learning rate of 0.00005 and l2 regularization with decay of 10−4. the final loss function becomes: l(ŷi) = log ( exp (ŷi)∑ j exp (ŷj) ) , l (ŷi) = − ∑ i yil (ŷi) + λ ∑ w∈w w2, where w is the set of all trainable weights of the cnn. the classification results and comparison to the existing related method are summarized in table 3. average roc area under curve (auc) for all classes was 0.927. roc curves for all classes are demonstrated in fig. 6. table 3: classification results. architecture train test mirsamadi et al. [5] bertero et al. [4] accuracy accuracy test accuracy test accuracy lstm 93.58% 65% 63.5% cnn 96% 67.5% 66.1% as it can be observed, the network captures some emotions more easily than others. for instance, neutral, calm, angry and surprise were captured better than the rest. roc-auc metric also suggests that the model learned meaningful representations for the task. 14 emotion classification of voice recordings using deep learning fig. 6. roc curves of the trained cnn model. each curve corresponds to an emotion label. overall, the results show that the models managed to learn meaningful representations from the training procedure. in table 3, we compare our results to the lstm-based method of mirsamadi et al. [5], which was trained and tested on the iemocap benchmark [16] with a 4-label (“angry”, “happy”, “sad”, “neutral”) classification setting, as well as to the cnn-based method of bertero et al. [4], which was trained and tested on the ted-lium benchmark [15] with a 3-label (“angry”, “happy”, “sad”) classification setting. as it can be observed, our method gains superior results on our 8-label classification setting. in contrast to the 2 methods, we leverage only the mfcc representation of the signal, which highlights the efficiency of the mfcc representation and its usage with deep learning methods for the task. 4. discussion and conclusion this paper proposes deep learning approaches for the voice emotion classification problem. particularly, cnn and lstm architectures were trained on mfcc features of voice recordings, depending on processing mfccs either as a spatial signal or as a sequential signal. the results indicate that the networks have learned meaningful representations from the training data. a possible future direction for improving the classification performance of the pron. tumanyan 15 posed models could be adding augmentations to the audio data. the recent advancements in using transformers [17] for multi-modal representation learning [18] and the expressiveness of the resulting feature space can also be a promising direction for solving the speech emotion recognition task. references [1] e. mower, m. j. mataric and s. narayanan,“a framework for automatic human emotion classification using emotion profiles”, ieee transactions on audio, speech, and language processing, vol. 19, no. 5, pp. 1057–1070, 2010. 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[12] m. sahidullah and g. saha, “design, analysis and experimental evaluation of block based transformation in mfcc computation for speaker recognition”, speech communication, vol. 54, no. 4, pp. 543–565, 2012. 1 6 emotion classi¯cation of voice recordings using deep learning [1 3 ] s . h o c h r e it e r a n d j. s c h m id h u b e r , \ l o n g s h o r t -t e r m m e m o r y" , neural computation, vo l. 9 , n o . 8 , p p . 1 7 3 5 { 1 7 8 0 , 1 9 9 7 . [1 4 ] b . x u , n . w a n g , t. ch e n a n d m. l i, \ e m p ir ic a l e va lu a t io n o f r e c t ī e d a c t iva t io n s in c o n vo lu t io n a l n e t wo r k" , cor r , vo l. a b s / 1 5 0 5 .0 0 8 5 3 , 2 0 1 5 . [1 5 ] a . r o u s s e a u a n d p . d e le g lis e , \ e n h a n c in g t h e te d -l iu m c o r p u s wit h s e le c t e d d a t a fo r la n g u a g e m o d e lin g a n d m o r e te d t a lks " , international conference on l anguage r esources and e valuation, r e ykja vik, ic e la n d , p p . 3 9 3 5 { 3 9 3 9 , 2 0 1 4 . [1 6 ] c. b u s s o , m. b u lu t , ch i-ch u n l e e , a . k a z e m z a d e h , e . mo we r , s . k im , j. n . ch a n g , s . l e e , a n d s . s . n a r a ya n a n , \ ie m o c a p : in t e r a c t ive e m o t io n a l d ya d ic m o t io n c a p t u r e d a t a b a s e " , l anguage r esources and e valuation , vo l. 4 2 , n o . 4 , p p . 3 3 5 { 3 5 9 , 2 0 0 8 . [1 7 ] a . v a s wa n i, n . s h a z e e r , n . p a r m a r , j. u s z ko r e it , l . jo n e s , a . n . go m e z , l . k a is e r a n d i. p o lo s u kh in , \ a t t e n t io n is a ll yo u n e e d " , cor r , vo l. a b s / 1 5 0 5 .0 0 8 5 3 , 2 0 1 7 . [1 8 ] h . a kb a r i, l . y u a n , r . qia n , w .h . ch u a n g , s .-fu ch a n g , y . cu i a n d b . go n g , \ v a tt: tr a n s fo r m e r s fo r m u lt im o d a l s e lf-s u p e r vis e d le a r n in g fr o m r a w vid e o , a u d io a n d t e xt ," advances in neural information p rocessing systems, 2 0 2 1 . submitted 18.02.2022, accepted 27.05.2022. êáñá áõëáõóù³ý íñ³ ñçùýí³í ó³ûý³·ñáõãûáõýý»ñç ¿ùáóç³ý»ñç ¹³ë³ï³ñ·ù³ý ù»ãá¹ý»ñ ü³ñ»ï î. âáõù³ýû³ý ì»ûóù³ýç ¶çïáõãûáõýý»ñç ð³ù³éë³ñ³ý, è»ëáíáï, æëñ³û»é e-mail: narek.tumanyan@weizmann.ac.il ²ù÷á÷áõù ´³ý³éç µ³é»ñ` ó³ûýç ïñ³ù³¹ñáõãû³ý ×³ý³ãáõù, ëáëùç ¿ùáóç³ûç ¹³ë³ï³ñ·áõù, ñ³×³ë³ï³ý ñ³ïï³ýçßý»ñ: îíû³é ñá¹í³íáõù ý»ñï³û³óíáõù »ý ëáñá áõëáõóù³ý íñ³ ñçùýí³í ó³ûý³·ñáõãûáõýý»ñç ¹³ë³ï³ñ·ù³ý ù»ãá¹ý»ñ: ²áõ¹çá ³½¹³ýß³ýý»ñá ùß³ï»éáõ ñ³ù³ñ û·ï³·áñííáõù »ý ó³ûý³·ñáõãûáõýý»ñç ñ³×³ë³ï³ý ïíû³éý»ñ, áñáýù ñ³ûïýç »ý å³ù³ý³ï³ûçý ³½¹³ýß³ýý»ñç ³ñ¹ûáõý³í»ï ý»ñï³û³óù³ùµ: ¸³ë³ï³ñ·ù³ý ëý¹çñá éáõí»éáõ ñ³ù³ñ ñá¹í³íáõù ñ³ßíç »ý ³éýíáõù ñ³×³ë³ï³ý ñ³ïï³ýçßý»ñç ùß³ïù³ý »ñïáõ ùáï»óáõù` áñå»ë å³ù³ý³ï³ûçý ³½¹³ýß³ýý»ñç ùß³ïù³ý ùáï»óáõù ¨ áñå»ë ï³ñ³í³ï³ý ³½¹³ýß³ýý»ñç ùß³ïù³ý ùáï»óáõù: úáõñ³ù³ýãûáõñ ùáï»óù³ý ñ³ù³ñ ïçñ³éíáõù »ý ñ³ù³å³ï³ëë³ý ³ññ»ëï³ï³ý ó³ýó»ñç ùá¹»éý»ñ: ü»ñï³û³óíáõù ¿ ¹³ë³ï³ñ·ù³ý ³ñ¹ûáõýùý»ñç í»ñéáõíáõãûáõý, ï³ï³ñíáõù »ý »½ñ³ï³óáõãûáõýý»ñ: n. tumanyan 1 7 êëàññèôèêàöèÿ ýìîöèé â ãîëîñå ñ èñïîëüçîâàíèåì ãëóáîêîãî îáó÷åíèÿ íàðåê ò. òóìàíÿí èíñòèòóò âåéöìàíà, ðåõîâîò, èçðàèëü e-mail: narek.tumanyan@weizmann.ac.il àííîòàöèÿ â ýòîé ñòàòüå ìû ïðåäñòàâëÿåì ìåòîäû êëàññèôèêàöèè ýìîöèé â ãîëîñå ñ èñïîëüçîâàíèåì ìåòîäîâ ãëóáîêîãî îáó÷åíèÿ. äëÿ îáðàáîòêè àóäèîñèãíàëîâ, äàííûé ìåòîä èñïîëüçóåò ÷àñòîòíûå ïðèçíàêè èçâëå÷åííûå èç ãîëîñîâûõ çàïèñåé, êîòîðûå, êàê èçâåñòíî, ñëóæàò ìîùíûì ïðåäñòàâëåíèåì âðåìåííûõ ñèãíàëîâ. äëÿ ðåøåíèÿ çàäà÷è êëàññèôèêàöèè, â äàííîé ðàáîòå ðàññìàòðèâàþòñÿ äâà ïîäõîäà îáðàáîòêè ÷àñòîòíûõ ïðèçíàêîâ: êàê âðåìåííûå ñèãíàëû è êàê ïðîñòðàíñòâåííûå/2d-ñèãíàëû. äëÿ êàæäîãî èç ïîäõîäîâ ìû èñïîëüçóåì ïîäõîäÿùèå àðõèòåêòóðû íåéðîííûõ ñåòåé. áûëè ïðîàíàëèçèðîâàíû ðåçóëüòàòû êëàññèôèêàöèè è ïðåäñòàâëåíû âûâîäû. êëþ÷åâûå ñëîâà: îïðåäåëåíèå íàñòðîåíèÿ ïî ãîëîñó, ðàñïîçíàâàíèå íàñòðîåíèÿ, êëàññèôèêàöèè ýìîöèé â ãîëîñå, ÷àñòîòíûå ïðèçíàêè. 01_tumanyan_57_7_17 (1) 01a d:\sbornik\...\333.dvi mathematical problems of computer science 33, 135{143, 2010. studies of the m atr ix gener ator statistical featur es s a r g is h o va s a p ya n yerevan physics institute, yerevan, armenia e-mail: ske13@yerphi.am abstract the statistical features for the optimized matrix generator of pseudorandom numbers are studied. the â2 and kolmogorov-smirnov tests are applied for very long sequences of the tested pseudorandom numbers. refer ences [1 ] n . z. a ko p o v, g. k . s a vvid y a n d n .g te r -a r u t yu n ia n , ma t r ix g e n e r a t o r o f p s e u d o r a n d o m n u m b e r , p . 5 7 3 -5 7 9 , 1 9 9 1 [2 ] d .e k n u t h , th e a r t o f t h e c o m p u t e r p r o g r a m m in g , v o l. 2 s e m i n u m e r ic a l a lg o r it h m s , a d d is o n -w e s le y, 1 9 8 1 [3 ] n . z. a ko p o v, g.g. a t h a n a s iu , e .g. flo r a t o s , g.k . s a vvid y, p e r io d o f k s ys t e m g e n e r a t o r o f p s e u d o r a n d o m n u m b e r s , cr e a t e u n ive r s it y p r e p r in t cr e te .th / 1 2 / 9 5 [4 ] b . ja n s s o n , r a n d o m n u m b e r ge n e r a t o r s , s t o c kh o lm , 1 9 6 6 . [5 ] g.a . ga lp e r in , n .i. ch e r n o w, b illia r d i i k h a o s , 1 9 6 7 . [6 ] f. gu t b r o d , n e w t r e n d s in p s e u d o -r a n d o m n u m b e r g e n e r a t io n , d e s y t-9 8 -0 1 , 1 9 9 8 . [7 ] f. ja m e s , mo n t e ca r lo t h e o r y a n d p r a c t ic e , ce r n ge n e va , 1 9 8 0 . [8 ] j.h a h r e n s a n d u . d e it e r , \ e xt e n s io n o f fo r s yt h e 's m e h o d fo r r a n d o m s a m p lin g fr o m t h e n o r m a l d is t r iu t io n " , m ath.comp27, p p 9 2 7 -9 3 7 , 1 9 7 3 . [9 ] u . d ie t e r , \ p s e u d o -r a n d o m n u m b e r s . th e e xa c t d is r ib u t io n o f p a ir s " , m ath comp, 25, p p . 8 5 5 -8 8 3 , 1 9 7 1 . [1 0 ] d . ca r e y a n d d . d r ija r d , \ mo n t e ca r lo p h a s e s p a c e wit h lim it e d t r a n s ve r s e m o m e n t u m " , j .comp. p hys, vo l. 2 8 ,p p . 3 2 7 -3 5 6 , 1 9 7 8 . [1 1 ] p .l `e c u ye r . e ± c ie n t a n d p o r t a b le r a n d o m n u m b e r g e n e r a t o r s . co m . a cm3 1 : 7 4 2 , 1 9 8 8 . [1 2 ] i.m. co b o l, ch is le n n ie m e t o d i mo n t e -ca r lo , n a u ka , 1 9 7 3 . [1 3 ] f.ja m e s , a r e vie w o r p r a c t ic a l r a n d o m n u m b e r g e n e r a t o r s , oc t o b e r 1 9 8 8 . 1 3 5 1 3 6 studies of the matrix generator statistical features ø³ïçó³ûçý ·»ý»ñ³ïáñç ëï³ïçëïçï ñ³ïï³ýçßý»ñç áõëáõùý³ëçñáõãûáõýý»ñ ê. ðáí³ë³÷û³ý ²ù÷á÷áõù àõëáõùý³ëçñí³í »ý åë»íá¹áå³ï³ñ³ï³ý ãí»ñç ûåïçùç½³óí³í ù³ïñçó³ûçý ·»ý»ñ³ïáñç ëï³ïçëïçï ñ³ïï³ýçßý»ñá: îçñ³éíáõù »ý â2 ¨ îáéùá·áñáí-êùçñýáí ï»ëï»ñá ß³ï »ñï³ñ ñ³çáñ¹³ï³ýáõãûáõýý»ñ áõý»óáõ åë»áõ¹á å³ï³ñ³ï³ý ãí»ñç ñ³ù³ñ: 06_d_asatryan's article (accepted) 54 mathematical problems of computer science 52, 54--60, 2019. udc 004.93 no-reference quality assessment of an image resizing algorithms david g. asatryan institute for informatics and automation problems of nas ra russian-armenian university e-mail: dasat@ipia.sci.am abstract in this paper, a no-reference method is proposed for quality assessment of an image resizing methods based on interpolation algorithms. quality refers to the image blur. to this end, it is proposed to use a previously developed measure based on a statistical assessment of the weibull distribution shape parameter, adopted as a model for gradient magnitude of an image. the results of numerical experiments are presented that allow us to evaluate and compare the quality of various image resizing algorithms. keywords: image resizing, interpolation, blur, quality assessment, gradient magnitude, weibull distribution. 1. introduction image resizing (scaling) methods are widely used in various problems of theory and technique of image processing. an image, once obtained in some way, can undergo multiple changes to meet the needs of the practice, in particular, resizing in one direction or another. a variety of needs and a huge market have led to the need to develop effective and convenient algorithms and resizing tools suitable for use in a wide range of technical applications, from space research to mobile communications. however, the use of any resizing algorithms changes the content of the image, thereby affecting its quality. moreover, each mathematical method that underlies the resizing algorithm works well for a certain class of images and may not be suitable for other classes. this explains the wide variety of mathematical methods developed to date and implemented in the form of software modules and systems, including the widely known image processing systems. in parallel, image processing specialists have long performed comparative studies of the quality of resizing methods and proposed appropriate recommendations for their use (see, for example, [1]). d. asatryan 55 to select the appropriate method for resizing images of a given class, it is necessary to apply a certain measure to assess the quality of the resulting image. at the same time, depending on the image information available to the user and the problem to be solved, its quality can be characterized in different ways. for example, if there is an original, by resizing which a test image is obtained, then the quality of the latter can be estimated by the degree of the observed distortion, which inevitably appears when resizing. however, in the absence of the original (standard) or any other additional information about the original, one has to use only the information that can be extracted from the image by analyzing its available properties. the literature suggests many types of image quality measures, known as standardless or noreference. this work is devoted to the application of the previously developed no-reference measure for assessing image quality using its structural properties. 2. a brief overview of the problem status quality assessment algorithms are divided into three categories, depending on the completeness of the information used for this: 1. full use of standards (full reference), 2. partial use of standards (reduced reference), 3. lack of standards (no-reference). with full reference, it is assumed that the latter is of an acceptable high quality, and the test image is compared with it by assessing their discrepancy. many such measures have been proposed in the literature, and we restrict ourselves to a reference to the papers [2-4], which are based on the use of various structural properties of an image. we also point out the survey papers [5-6], in which other measures for assessing an image quality are considered. an important feature of the applicability of the comparison method with the standard, related to the need to assess the similarity of images with different sizes, should be noted. in [7], for this problem, we used the method [3] based on the weibullian model of gradient magnitude, which is briefly described below. with the partial use of standards, they are based on certain properties or features that allow you to make decisions regarding the quality of the test image. the most difficult category is represented by no-reference algorithms that do not require specific meaningful information about the test image. the main difficulty in this case is due to the fact that there are a huge number of types of distortion to which the image can be subjected, and often nothing is known about the specific type of distortion of the test image. therefore, creating algorithms that work equally well for all types (or at least for a large number of types) of distortion is an important, but rather difficult task. in this paper, we restrict ourselves to a reference to the review paper [6], which is devoted to the comparative analysis of various known no-reference methods for assessing an image quality. it should be noted that in the literature the quality of a resized image is often evaluated visually, based only on a subjective analysis of the observed distortions. for example, figure 1 shows the original image from the database [8] and its fivefold reduced sample (images with equal sizes are shown in the text to save space). distortions are clearly visible on the resized image; however, it is impossible to visually assess the degree of these distortions, which emphasizes the urgency of developing and studying formal methods for this task. no-reference quality assessment of an image resizing algorithms 56 original resized fig. 1. illustration of distortion when resizing. the present paper is devoted to a comparative analysis of the quality of some of the most commonly used resizing algorithms based on well-known methods of numerical interpolation, using the recently proposed and thoroughly studied no-reference measure based on the use of image structural properties [9-10]. as shown in [9], the weibull distribution shape parameter can be used as a measure of image blur, and in [10] the ability of this measure to evaluate the image quality for other types of distortion is shown. we used the results of [8], which describes the tid2013 image database containing 3000 images obtained from 25 originals, distorted by 24 different types with five levels each. in this work, the results of an extensive experiment on visual assessment of quality according to a point system by a large number of people in different countries are presented. as a result of processing these data, each of the 3000 images is assigned a numerical rating of the quality estimated visually (mean opinion score mos). using the tid2013 database allows you to evaluate the quality of any procedure used both in the task of assessing image quality with the full reference procedures, and in the noreference situation. an example of applying the approach using the no-reference measure is [10]. in this article, this procedure is used to assess the quality of the image obtained as a result of resizing the original, which allows us to make certain judgments about the applicability of some resizing methods. 3. quality assessment method and experimental results mathematical model. following [3], we will accept for the magnitude of the image gradient a model based on the weibull distribution, the density of which is given by the formula ,0,exp),;( 1 ≥           −     = − x xx xf ηη λλλ η ηλ (1) where 0η > is the shape parameter, 0λ > is the scale parameter. the magnitudes of the gradients are estimated using the sobel operator, and the distribution parameters are estimated from the totality of all the magnitudes (1). in [9], it was proposed to take the parameter value as a measure of image blur (the larger the measure value, the greater the blur). details of the procedure can be found in [8–9]. description of experimental material. for the experiments, the originals of 25 images of the base [8] were selected, some of which are shown in fig. 2. d. asatryan 57 1,219 1,086 1,043 1,042 0,684 0,624 0,601 0,556 fig. 2. original images of the tid2013 base with high measure values (upper row) and with low measure values (lower row). resizing was performed using the well-known interpolation algorithms implemented in many popular graphic editors: lanczos, nearest neighbor, bilinear, bspline, gaussian. the following values were selected for the resizing factor (rf): 0.25, 0.5, 0.66, 0.75, 1, 1.25, 1.5, 1.75 and 2.0. before analyzing the results, we make one important point. according to the methodology [9] of the no-reference estimation of blurriness, the more blurry the image, the greater the measure value. it was noted that for sufficiently blurry images, this value approximates to “2” and only slightly exceeds this value in certain cases, and for images with a fairly clearly defined structure, this value is much less than 2. however, it should also be noted that the value of the measure for an arbitrary image depends on its structure and specific content, and it can hardly be interpreted as an absolute estimate of blur. therefore, when considering the value of the measure obtained for a fixed image, the problem arises of studying the dependence of the quality of the images obtained, perhaps, by various transformations of this sample, including when resizing it. the results of numerical experiments are as follows:. 1. figure 2 shows the original images of the tid2013 database with the lowest and highest measure values (indicated under the corresponding images). examining these images and visually assessing the degree of their blurring and comparing the corresponding values of the measure, we can note their correspondence with a certain degree of reliability. 2. let’s consider the first image in the list of the tid2013 database (it is placed second in figure 2 in the upper row) and perform the resizing operation according to the selected algorithms. figure 3 shows graphs of the dependence of the blur measure on the rf. here, the abscissa axis shows the resizing levels from 1 to 9, corresponding to the above values of the resizing factor from 0.25 to 2.0. the dependency curves show that all five resizing methods behave approximately the same way, i.e., as the rf increases, the blur generally decreases, and when the rf approaches maximum values, the degree of blur more or less stabilizes (with the exception of the bilinear method, in which the decrease in blur is significant, which apparently appears due to the features of this image). it should be noted that the trend noted above is a formal confirmation of the correspondence between the visual perceptions of the effect of reducing blur due to the smoothing of the picture of intensity transitions in the image. it is also important to note that for rf <1 and rf> 1, the branches of the curves are located differently relative to each other. in other words, for some rf values, one resizing method may have an advantage, while for other rf values, another. no-reference quality assessment of an image resizing algorithms 58 fig. 3. dependence of a fixed image blur on resizing method. а b c d fig. 4. dependence of blur on the resizing algorithm for the images shown in fig. 2, a lanszos algorithm, b nearest neighbor, c – bilinear, d – bspline. 3. it is of interest to compare the nature of the change in blur from rf for images with different initial blur values. figure 4 shows the graphs of the dependence of the blur when resizing eight images from the tid2013 database, shown in fig. 2. we note that, firstly, the same properties hold as above. in addition, despite the fact that the results are presented only for four resizing methods (similar results are also obtained for other methods); the nature of the dependence is preserved. however, there is a difference in the values and stabilization rate of the degree of d. asatryan 59 blurring depending on its value in the absence of resizing (i.e., at rf = 1). in other words, the higher the degree of blur of the original, the less this image is subject to blur when resizing. 4. conclusions the paper proposes a method of no-reference estimation of the quality of image resizing algorithms based on the application of the previously developed image blur estimation technique. the technique is based on modeling the image gradient magnitude data using a two-parameter weibull distribution. in this case, a statistical estimation of the shape parameter of this distribution is used as a blur measure. it is shown that this measure allows you to compare the quality of various resizing methods, which are based on numerical interpolation methods. it can also be noted that the application of the obtained results in practice can be more useful in combination with the results of assessing the similarity of the resized image to the original, carried out by adequate methods. references [1] a. parker, r. v. kenyon and d. troxel, “comparison of interpolating methods for image sampling”, ieee transactions on medical imaging, vol. m1-2, no. 1, p. 31-39, 1983. [2] z.wang, a.c.bovik, “modern image quality assessment”,synthesis lectures on image, video, and multimedia processing,vol. 2, no. 1, pp. 1–156, 2006. [3] d.asatryan, k. egiazarian,‘quality assessment measure based on image structural properties”,proc. of the international workshop on local and non-local approximation in image processing, finland, helsinki, 19-21 august, pp. 70-73, 2009. [4] w.xue,l.zhang, x.mou and a.c. bovik, “gradient magnitude similarity deviation: a highly efficient perceptual image quality index”,ieee transactions on image processing, vol. 23, no. 2, pp.684-695, 2014. [5] o.sebastian and s.arkadiusz,”the survey of subjective and objective methods for quality assessment of 2d and 3d images”, theoretical and applied informatics,vol. 26,no. 1, 2, pp. 39 – 67, 2014. [6] в.в.старовойтов и ф.в. старовойтов, “сравнительный анализ безэталонных мер оценки качества цифровых изображений”, системный анализ и прикладная информатика, № 1, сс. 24-32, 2017. [7] d.g.asatryan, “gradient-based technique for image structural analysis and applications”, computer optics, vol. 43, no. 2, pp. 245-250, 2019. [8] n.ponomarenko, l. jin, o. ieremeiev, v. lukin, k. egiazarian, j. astola, b.vozel, k. chehdi, carli, f. battisti, et al. “image database tid2013: peculiarities, results and perspectives”, signal processing: image communication,vol.30, pp. 57–77, 2015. [9] д. г.асатрян, оценивание степени размытости изображения путём анализа градиентного поля,компьютерная оптика,вып. 41, №6, сс. 957-962,2017. [10] d. a. asatryan, “novel technique for no-reference image quality assessment”,proceedings of international conference on computer science and information technologies, yerevan, armenia,pp. 201-203, 2019. submitted 10.05.2019, accepted 15.11.2019. no-reference quality assessment of an image resizing algorithms 60 պատկերի մասշտաբավորման ալգորիթմների որակի առանց էտալոնի գնահատումը դավիթ գ. ասատրյան հհ գաա ինֆորմատիկայի և ավտոմատացման պրոբլեմների ինստիտուտ հայ-ռուսական համալսարան e-mail: dasat@ipia.sci.am ամփոփում հոդվածում դիտարկվում է պատկերի մասշտաբավորման ալգորիթմների որակի՝ առանց էտալոնի գնահատման եղանակ, որը հենված է միջարկման մեթոդների վրա: որակ ասելով հասկանում ենք պատկերի ճապաղվածության աստիճանը, որն առաջանում է մասշտաբավորման հետևանքով: դրա համար առաջարկվում է օգտագործել նախկինում մշակված չափանիշը, որը հենված է որպես պատկերի գրադիենտի մագնիտուդի մոդել ընդունված վեյբուլի բաշխման ձևի պարամետրի վիճակագրական գնահատականի վրա: բերվում են թվային փորձարկումների արդյունքներ, որոնք թույլ են տալիս գնահատել և համեմատել պատկերի մասշտաբավորման տարբեր ալգորիթմների որակը: բանալի բառեր` պատկերի մասշտաբավորում, միջարկում, ճապաղվածություն, որակի գնահատում, գրադիենտի մագնիտուդ, վեյբուլի բաշխում: безэталонное оценивание качества алгоритмов масштабирования изображения давид г. асатрян институт проблем информатики и автоматизации нан ра российско-армянский университет e-mail:dasat@ipia.sci.am аннотация в статье рассматривается метод безэталонного оценивания качества алгоритмов масштабирования изображения, основанный на методах интерполирования. под качеством понимается степень размытия изображения, вызванного масштабированием. для этого предлагается использовать разработанную ранее меру, основанную на статистическую оценку параметра формы распределения вейбулла, принятого в качестве модели для магнитуды градиента изображения. приводятся результаты численных экспериментов, которые позволяют оценивать и сравнивать качество различных алгоритмов масштабирования изображения. ключевые слова. масштабирование изображения, интерполирование, размытость, оценивание качества, магнитуда градиента, распределение вейбулла начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 44, 138--144, 2015. the optimal approach for kolmogorov-smirnov test calculation in high dimensional space norayr z. akopov and narek h. martirosyan yerevan physics institute e-mail: -narek@mail.yerphi.am abstract numerical estimation of the kolmogorov-smirnov discrepancy dn in high dimensional space is an extremely time and memory consuming problem. new approach with the minimal bin number, which essentially reduces the time and memory requirements, to perform the dn tests in two and more dimensional space is discussed. keywords: statistics, kolmogorov–smirnov test, goodness-of-fit tests, prng. 1. introduction one of the most important tasks nowadays is the numerical experiments on super computers with the modeling of the sophisticated physics system. the monte carlo method is used for solving the problems where the high dimensional integration is involved. to use the monte carlo method for analysis of the high energy physics experimental and theoretical problems we have to solve the problem of the quality of the pseudo-random generators, which should have a strong statistical feature, also a large period of sequences and a high speed of generating of the pseudorandom number. the kolmogorov-smirnov (ks) test is used to compare how well the empirical cumulative distribution function (ecdf) of a sample fits the cumulative distribution function (cdf) of the reference distribution by computing the maximum distance 𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 between these two functions. the ks test is applied to exactly continuous data, and for dimensions 𝑑𝑑 ≥ 2 we have to compute the distance on infinite points then take a maximum, because when 𝑑𝑑 ≥ 2 for cdf and ecdf we have d-dimensional surfaces. the ks test is widely used as a powerful statistical test to check the quality of different pseudo-random number generators (prngs). 138 https://en.wikipedia.org/wiki/empirical_distribution_function https://en.wikipedia.org/wiki/empirical_distribution_function https://en.wikipedia.org/wiki/cumulative_distribution_function n. akopov and n. martirosyan 139 several algorithms for computing the two-dimensional ks test were proposed [1-3]. we propose another approach for 2 and higher dimensions. our method is based on discretization of the space introducing a binning technique applied to continuous multidimensional data. this technique does not correspond to the usual ks test principle, but we show that it is possible to compute 𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 very precisely with the quite computationally efficient algorithm taking minimal bin number. here our ks test results are obtained using the well-tested uniform mersenne twister prng [4], which is included in cern library [5]. in the proposed paper we describe a new technique, which allows to reduce essentially the number of the bins and correspondingly decrease the needed time and memory used. 2. formalism in one-dimensional (1d) case with n random numbers to check if they come from uniform distribution in the (0, 1) range, for which the cdf is equal to: 𝐹𝐹(𝑥𝑥) = 𝑥𝑥, we need to compute 𝐷𝐷𝑁𝑁. usual (unbinned) approach [68] in 1d case looks as follows:  n numbers should be sorted in increasing order {𝑥𝑥1, 𝑥𝑥2, … , 𝑥𝑥𝑁𝑁}, 𝑥𝑥𝑖𝑖 ≤ 𝑥𝑥𝑖𝑖+1  then 𝐷𝐷𝑁𝑁 is computed in the following way defining ecdf as: 𝐹𝐹𝑁𝑁(𝑥𝑥) = 𝑘𝑘 𝑁𝑁 , 𝑥𝑥 = 𝑥𝑥1, 𝑥𝑥2, … , 𝑥𝑥𝑁𝑁, where k is the number of random points out of n with 𝑥𝑥1, 𝑥𝑥2, … , 𝑥𝑥𝑘𝑘 ≤ 𝑥𝑥 𝐷𝐷𝑁𝑁 = max0≤𝑚𝑚≤1 |(𝐹𝐹𝑁𝑁(𝑥𝑥) − 𝑥𝑥)| = max1≤𝑖𝑖≤𝑁𝑁 �� 𝑖𝑖 𝑁𝑁 − 𝑥𝑥𝑖𝑖� , � 𝑖𝑖 − 1 𝑁𝑁 − 𝑥𝑥𝑖𝑖��. the unbinned approach can be also applied for 2d case following the algorithm described in [1], which has been used for the performed studies. in the 1d binned approach the (0, 1) interval is divided into 𝑛𝑛 bins, then n random numbers are distributed in 𝑛𝑛 bins according to their values. the ecdf is again defined as: 𝐹𝐹𝑁𝑁(𝑥𝑥) = 𝑘𝑘 𝑁𝑁 . the difference is that here 𝑥𝑥 is already the bin edge: 𝑥𝑥 = 𝑖𝑖 𝑛𝑛 , 𝑖𝑖 = 1,2, … , 𝑛𝑛. for example, if 𝑛𝑛 = 10, we have the bins edges like: {0.1, 0.2, … , 0.9,1} . therefore, the advantage of the method is that if n is quite large then ecdf can be defined with a relatively small number of points (𝑛𝑛 < 𝑁𝑁). 𝐷𝐷𝑁𝑁 = max0≤𝑚𝑚≤1 |(𝐹𝐹𝑁𝑁(𝑥𝑥) − 𝑥𝑥)| = max1≤𝑖𝑖≤𝑛𝑛 |𝐹𝐹𝑁𝑁(𝑥𝑥𝑖𝑖) − 𝑥𝑥𝑖𝑖| = max1≤𝑖𝑖≤𝑛𝑛 � ∑ 𝑌𝑌𝑘𝑘𝑖𝑖𝑘𝑘=1 𝑁𝑁 − 𝑖𝑖 𝑛𝑛 �. the optimal approach for kolmogorov-smirnov test calculation in high dimensional space 140 in 1d case there exists the theoretical distribution function for the value of 𝐾𝐾𝑁𝑁 = √𝑁𝑁 𝐷𝐷𝑁𝑁 (also the mean value and dispersion for 𝐾𝐾𝑁𝑁), which is used to make a decision to accept or reject the hypothesis concerning the uniform distribution of the tested sample of random numbers. this binned method can be generalized for dimensions 2, 3 and higher. in two-dimensional case for 𝑥𝑥1 ≤ 𝑋𝑋1, 𝑥𝑥2 ≤ 𝑋𝑋2 the corresponding expression for 𝐷𝐷𝑁𝑁 is: 𝐷𝐷𝑁𝑁 = max0≤𝑚𝑚1≤1 0≤𝑚𝑚2≤1 |(𝐹𝐹𝑁𝑁(𝑥𝑥1, 𝑥𝑥2) − 𝑥𝑥1𝑥𝑥2)| = max1≤𝑖𝑖≤𝑛𝑛 1≤𝑗𝑗≤𝑛𝑛 � ∑ ∑ 𝑌𝑌𝑘𝑘𝑚𝑚 𝑗𝑗 𝑚𝑚=1 𝑖𝑖 𝑘𝑘=1 𝑁𝑁 − 𝑖𝑖𝑖𝑖 𝑛𝑛2 �. where 𝑌𝑌𝑘𝑘𝑚𝑚 is the number of observations that actually do fall into category {𝑘𝑘, 𝑚𝑚}, 𝑁𝑁 is the count of all observations, n is the number of bins used along the x and y. in d-dimension case of 𝑥𝑥1 ≤ 𝑋𝑋1, 𝑥𝑥2 ≤ 𝑋𝑋2, … … . , 𝑥𝑥𝑑𝑑 ≤ 𝑋𝑋𝑑𝑑 the expression for ks test is as follows: 𝐷𝐷𝑁𝑁 = max0≤𝑚𝑚1≤1 0≤𝑚𝑚2≤1… 0≤𝑚𝑚𝑑𝑑≤1 |(𝐹𝐹𝑁𝑁(𝑥𝑥1, 𝑥𝑥2, … , 𝑥𝑥𝑑𝑑) − 𝑥𝑥1𝑥𝑥2…𝑥𝑥𝑑𝑑)| = max1≤𝑖𝑖1≤𝑛𝑛 1≤𝑖𝑖2≤𝑛𝑛… 1≤𝑖𝑖𝑑𝑑≤𝑛𝑛 � ∑ ∑ … ∑ 𝑌𝑌𝑘𝑘1𝑘𝑘2…𝑘𝑘𝑑𝑑 𝑖𝑖𝑑𝑑 𝑘𝑘𝑑𝑑 𝑖𝑖2 𝑘𝑘2=1 𝑖𝑖1 𝑘𝑘1=1 𝑁𝑁 − 𝑖𝑖1𝑖𝑖2 … 𝑖𝑖𝑑𝑑 𝑛𝑛𝑑𝑑 �. with any of schemes to calculate the average 𝐾𝐾𝑁𝑁 described above (binned and unbinned) we can estimate also the statistical uncertainties δ𝐾𝐾𝑁𝑁 making sampling for calculated values of 𝐾𝐾𝑁𝑁 and 𝐾𝐾𝑁𝑁2 with the certain number m of the used samples. the mean value for 𝐾𝐾𝑁𝑁 averaging over m samples is given by: < 𝐾𝐾𝑁𝑁 >= 1 𝑀𝑀 �𝐾𝐾𝑁𝑁 𝑖𝑖 , 𝑀𝑀 𝑖𝑖=1 and mean squared 𝐾𝐾𝑁𝑁 is defined as: < 𝐾𝐾𝑁𝑁2 >= 1 𝑀𝑀 �(𝐾𝐾𝑁𝑁 𝑖𝑖 𝑀𝑀 𝑖𝑖=1 )2, then we can calculate 𝜎𝜎𝐾𝐾𝑁𝑁 2 as: 𝜎𝜎𝐾𝐾𝑁𝑁 2 =< 𝐾𝐾𝑁𝑁2 > −< 𝐾𝐾𝑁𝑁 >2. and finally, based on the central limiting theorem we can calculate the statistical uncertainty δ𝐾𝐾𝑁𝑁 as: δ𝐾𝐾𝑁𝑁 = 1 √𝑀𝑀 �𝜎𝜎𝐾𝐾𝑁𝑁 2 . 3. results and discussion the first set of the obtained results is related to the unbinned approach, where in 1d case we can compare not only the obtained average < 𝐾𝐾𝑁𝑁 > with the predicted theoretical value of 0.8687311605.. [7], but also the observed and theoretical (th) distributions for 𝐾𝐾𝑁𝑁 (see fig. 1). n. akopov and n. martirosyan 141 fig 1. th (curve) and observed (red points) for kn distributions. on fig. 2 one can see the observed distribution obtained for 2d case, which is systematically shifted in respect to 1d th distribution. fig 2. th distribution (curve) for 1d and observed distribution (red points) for 2d cases. then the next set of results, which is related to the binned method, is shown in figs. 3-4. in fig.3 one can see that for 𝑁𝑁 = 105 the number of 𝑛𝑛 = 104 bins is enough to get the exact distribution which was the goal of this paper. the optimal approach for kolmogorov-smirnov test calculation in high dimensional space 142 fig 3. 1d case: th distribution (curve) and binned distributions (colored points). in fig. 4 the < 𝐾𝐾𝑁𝑁 > dependence on the number of bins is presented for 1, 2 and 3d cases. here one can recognize the clear dependence of the obtained values for < 𝐾𝐾𝑁𝑁 > on the number of the bins (n) used. taking into account the obtained saturating shape on the n for such dependences we performed the fit with the following fitting function: 𝑓𝑓𝑓𝑓𝑖𝑖𝑓𝑓 𝑝𝑝 → (𝑥𝑥) = 𝑝𝑝1 + 𝑝𝑝2 exp(−𝑝𝑝3𝑥𝑥). the meaning of the parameter 𝑝𝑝1 corresponds to the saturation level, which can be achieved with quite large value of n. although, using very high values for the bins number it is impossible in practice to calculate the 𝐾𝐾𝑁𝑁 in the sense of the needed cpu time and memory, especially in case of higher (>4) dimensional space. that is why the idea to use a limited number of points over the n, then estimate the saturation parameter (𝑝𝑝1), and then make a correction for the average value of𝐾𝐾𝑁𝑁, estimated with e.g., n=10, introducing the correction factor: 𝐶𝐶10 = 𝑝𝑝1 − < 𝐾𝐾𝑁𝑁(𝑛𝑛 = 10) >, seems to be very interesting and effective in order to realize the procedure of the < 𝐾𝐾𝑁𝑁 > estimation in case of high dimensional space. also the exponential coefficient (𝑝𝑝3), which regulates the speed of convergence to the saturation level, can be used to check the quality of different prngs. it should be noted, that the used fitting function is quite stable in respect to the variation of the fitting points used. in 3d case with the total number of points (ntotal=11), this variation was 11, 9, 7, 5. in 2d case with the ntotal=19: 19, 15, 11, 7, and in 1d case with ntotal=75: 75, 55, 35, 15. this is an important feature of the functional form used for a fit, because in case of essentially higher dimensions one can compute with the binned method a limited number of points, perform the fit and estimate the saturation level. in further studies the authors are going to provide the dependence as a function of the space dimension, which is very interesting due to the lack of knowledge on th distribution for high dimension space. n. akopov and n. martirosyan 143 fig. 4. average kn dependence on the number of bins (n). the 1, 2 and 3d cases are presented with the fit parameters for each case. different colors along the curves correspond to the variation of the number of fitting points used (see explanation in the text). 4. conclusion the optimization of the kolmogorov-smirnov discrepancy dn calculation with the binned method for high dimensional spaces by means of reducing the used bin number is performed. developed approach allows to extend the estimation of dn to very high dimensional spaces with reasonable requirements to the cpu time and memory, and, thus, to provide very important tests of the prngs, which are used to apply the monte carlo method for solving of essentially high dimensional physics problems. references [1] a. justel, d. pena and r. zamar, “a multivariate kolmogorov-smirnov test of goodness of fit”, statistics & probability letters 35, pp. 251-259, 1997. [2] j. a. peacock, “two-dimensional goodness-of-fit testing in astronomy”, monthly notices royal astronomy society, vol. 202, pp. 615-627, 1983. [3] g. fasano and a. franceschini, “a multidimensional version of the kolmogorov-smirnov test”, monthly notices royal astronomy society, vol. 225, pp. 155-170, 1987. the optimal approach for kolmogorov-smirnov test calculation in high dimensional space 144 [4] m. matsumoto and t. nishimura, “mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator”, acm transactions on modeling and computer simulation, vol. 8, pp. 3-30, 1998. [5] [online]. available: https://cern.ch/cernlib [6] m. a. stephens, “edf statistics for goodness of fit and some comparisons”, journal of the american statistical association (american statistical association), vol. 69, pp. 730–737, 1974. [7] g. marsaglia, w. w. tsang and j. wang, “evaluating kolmogorov’s distribution”, journal of statistical software, vol. 8, pp. 1-4, 2003. [8] d. knuth, the art of computer programming, v.2, addison-wesley publishing company, 1981. submitted 30.07.2015, accepted 19.11.2015 օպտիմալ մոտեցում կոլմոգորով-սմիրնով թեստի հաշվարկի համար բարձր չափողականությամբ տարածությունում ն. ակոպով և ն. մարտիրոսյան ամփոփում կոլմոգորով-սմիրնով թեստի 𝐷𝐷𝑁𝑁 թվային գնահատականը բարձր չափողականությամբ տարածությունում բավականին ժամանակատար և հիշողություն պահանջող խնդիր է: առաջարկվող հոդվածում քննարկվում է նվազագույն բինների քանակով նոր մոտեցումը, որը էապես նվազեցնում է ժամանակի և հիշողության պահանջները 𝐷𝐷𝑁𝑁 մեծությունը 2 և ավելի բարձր չափողականությամբ տարածությունում հաշվելու համար: оптимальный подход для численных реализаций теста колмогоровасмирнова в пространствах высокой размерности н. акопов и н. мартиросян аннотация численные оценки дискрепанса колмогорова-смирнова 𝐷𝐷𝑁𝑁 в пространствах высокой размерности являются существенной проблемой в плане необходимых ресурсов пaмяти и времени. в статье обсуждается новый подход для вычисления величины 𝐷𝐷𝑁𝑁 в пространствах двух и более высоких размерностей с минимальным числом разбиений, который существенно снижает требования к памяти и времени. http://www.jstatsoft.org/v08/i18/paper microsoft word article.doc ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 29, 2007, 33–35. 33 о моделировании процесса принятия решений при форсмажорных ситуациях ваге с. саратикян институт проблем информатики и автоматизации нан ра e-mail: saratikyan@yahoo.com аннотация предлагается один из возможных подходов по разработке имитационной модели процесса принятия решений при форсмажорных ситуациях. 1. форд л.р. “потоки в сетях” 519.8/ф79 1996г 2. галабурда в.г. “оптимальное планирование грузопотоков” 658/г15 1985г 3. типовые требования к регистрации, отображению, прогнозированию, учету и анализу движения поездов в автоматизированных системах диспетчерского контроля и управления (дк, дц) на диспетчерских участках и в железнодорожных узлах, руководящий документ мпс рф, санкт-петербург, 1999 г. 4. саратикян в.с. автоматизация процесса принятия решений при поездообразовании на межгосударственных железнодорожных стыках // соискатель. -2005. – №1(2). – с.81-84 5. кутыркин а.в., разработка моделей и алгоритмов решения функциональных задач управления транспортными системами и производством, автореферат диссертации на соискание ученой степени доктора технических наук, москва, 2004г ²ñï³ï³ñ· çñ³íç×³ïý»ñáõù áñáßáõùý»ñç áý¹áõýù³ý ·áñíáýã³óç ùá¹»é³íáñù³ý ù³ëçý ì. ê. ê³ñ³ïçïû³ý ²ù÷á÷áõù ²é³ç³ñïíáõù ¿ ·ý³ñ³ï³ï³ý ï³é ³ñï³ï³ñ· çñ³íç×³ïý»ñáõù áñáßáõùý»ñç áý¹áõýù³ý ·áñíáýã³óáõù ï³ï³ñíáõ í×çéý»ñçý, áñáýó ý³ëáñáù ñý³ñ³íáñ ã¿ ³é·áñçãù³íáñ»é, ëï»õí»é ·ý³ñ³ï³ï³ýý»ñç µ³½³ ¨ ñ»ï³·³ûáõù ýù³ý³ïçå çñ³íç×³ïç ùá¹»é³íáñù³ý ýå³ï³ïáí û·ïí»é ³û¹ µ³½³ûçó: haroutunian_37_43.dvi mathematical problems of computer science 46, 37{43, 2016. on optimality of t ests of m -ar y h ypotheses for fixed n umber of i ndependent obser vations e vg u e n i a . h a r o u t u n ia n institute for informatics and automation problems of nas ra e-mail: eghishe@sci.am abstract the paper is devoted to the design of optimal approaches of testing of multiple simple hypotheses with samples of a ¯xed number of independent observations. keywords: statistical hypothesis testing, error probability, compairing of tests, optimal test, neyman pearson approach. 1 . in t r o d u c t io n w e a d d r e s s t h e c la s s ic a l d e t e c t io n p r o b le m . l e t m ¸ 2 s im p le h yp o t h e s e s , wh ic h we n o t e h1, ..., hm , c o r r e s p o n d t o d is t r ib u t io n s g1; :::; gm , g ive n o n t h e ¯ n it e s p a c e x . it is n o t kn o wn wh ic h h yp o t h e s is is t r u e . th e o b s e r ve d s a m p le is a ve c t o r x = ( x1; x2; ; :::; xn ) in n-d im e n t io n a l s p a c e x n o f n in d e p e n d e n t o b s e r va t io n s id e n t ic a lly d is t r ib u t e d a c c o r d in g t o o n e o f h yp o t h e s e s . th e s t a t is t ic ia n u s in g t h e o b t a in e d s a m p le m u s t g u e s s wh ic h h yp o t h e s is is t r u e . it is n e c e s s a r y t o c h o o s e a n o p t im a l c r it e r io n fo r m a kin g s u c h a d e c is io n . th e p r o c e d u r e o f d e c is io n is c a lle d a t e s t . d e ¯ n in g a t e s t c a n b e fo r m u la t e d a s p a r t it io n in g o f t h e s p a c e x n in t o m d is jo in t s e t s x n1 ,..., x nm o n e a c h o f wh ic h o n e o f t h e h yp o t h e s e s h1, ..., hm will b e , r e s p e c t ive ly, a c c e p t e d . th e qu a lit y o f a t e s t ' m a y b e c h a r a c t e r iz e d b y t h e m a t r ix o f e r r o r p r o b a b ilit ie s ®ljm = ®ljm ( ') = gm ( x 2 x nl ) ; l; m = 1 ; m ; l 6= m; ®m = ®mjm ( ') = gm ( x =2 x nm ) ; m = 1 ; m; wh e r e ®ljm is t h e p r o b a b ilit y t o a c c e p t hl, wh e n hm is t r u e , a n d ®mjm ( ') is t h e p r o b a b ilit y o f r e je c t io n o f h yp o t h e s is hm wh e n it is t r u e . e vid e n t ly ®m = x l 6=m ®ljm; m = 1 ; m: r ic h lit e r a t u r e is d e d ic a t e d t o t h e p r o b le m o f h yp o t h e s is t e s t in g , we qu o t e [1 -3 ]. th e c a s e o f m u lt ip le h yp o t h e s e s t e s t in g is c o n s id e r e d in d e t a il in [4 ] a n d [5 ]. to c o m p a r e t e s t s t h e r e e xis t d i®e r e n t a p p r o a c h e s . a b a ye s a p p r o a c h is b a s e d o n a s s u m p t io n t h a t t h e o b s e r va t io n s a r e g o ve r n e d b y g ive n p r o b a b ilit ie s c a lle d t h e a p r io r i p r o b a b ilit ie s . in t h is c a s e t h e s e t o f t e s t s is o r d e r e d b y t h e va lu e s o f t h e a ve r a g e p r o b a b ilit ie s o f e r r o r s o f t e s t s [4 ]. 3 7 3 8 on optimality of tests of m -ary hypotheses for fixed number of independent observations n e ym a n p e a r s o n a p p r o a c h [6 ] p r im a r ily wa s fo r m u la t e d fo r t wo h yp o t h e s e s . in c a s e o f m h yp o t h e s e s t h e e r r o r p r o b a b ilit ie s a r e ¯ xe d fo r m = 1 ; m ¡ 1 . th e t e s t is fo u n d b y m a xim iz in g t h e p r o b a b ilit y 1 ¡ ®m o f c o r r e c t d e t e c t io n o f t h e h yp o t h e s is hm . a n o t h e r d ir e c t io n o f in ve s t ig a t io n s o f qu a lit y a n d o p t im a lit y o f t e s t s is t h e a s ym p t o t ic a p p r o a c h wh e n t h e le n g t h o f s a m p le is n o t lim it e d a n d wh e n t h e vo lu m e n o f t h e s a m p le s g o e s t o t h e in ¯ n it y, e r r o r p r o b a b ilit ie s ®ljm d e c r e a s e e xp o n e n t ia lly a s 2 ¡neljm a n d t h e in t e r d e p e n d e n c e o f t h e s e e xp o n e n t ia l c o e ± c ie n t s eljm, c a lle d r e lia b ilit ie s , c h a r a c t e r iz e t h e t e s t . th is d ir e c t io n is fo u n d e d b y h o e ®d in g [7 ] a n d t h e c o r r e s p o n d in g s e qu e n c e o f t e s t s is c a lle d lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l ( l a o) , o r e xp o n e n t io n a l r a t e o p t im a l ( e r o) [7 ,8 ]. th e p r o b le m s o f l a o t e s t s fo r m u lt ip le h yp o t h e s e s fo r d ive r s e m o d e ls we r e s t u d ie d in [9 -1 4 ]. in t h e m in im a x a p p r o a c h t h e c o m p a r is io n o f m a xim a l va lu e s o f e r r o r p r o b a b ilit ie s o f a t e s t a ls o a llo ws t o o r d e r t h e s e t o f a ll t e s t s . th e in t e r c h a n g e o f id e a s a n d m e t h o d s o f s t a t is t ic s a n d in fo r m a t io n t h e o r y is p r e s e n t e d in [1 5 ,1 6 ]. ou r p u r p o s e in t h is p a p e r is c o m p a r in g d i®e r e n t t e s t s a n d in t r o d u c in g a n o r d e r o n t h e s e t o f a ll t e s t s fo r m ¸ 2 h yp o t h e s e s wh e n b o t h , lis t o f h yp o t h e t ic a l d is t r ib u t io n s a n d t h e vo lu m e n o f s a m p le s , a r e ¯ xe d . w e p r o c e e d a n a lo g ic a lly a s in [4 ], wh e r e t h e b a ye s ia n t e s t s a r e s t u d ie d . 2 . fo r m u la t io n o f r e s u lt s w e b e g in b y c o n s id e in g t h e c a s e o f m h yp o t h e t ic a l d is t r ib u t io n s wh e n a p r io r i p r o b a b ilit ie s a r e a b s e n t . it is c le a r t h a t t h e t e s t is b e t t e r if it s e r r o r p r o b a b ilit e s a r e s m a lle r . de¯nition 1: an average error probability ®( ') for test ' is ®( ') = 1 2 m x l;m ®ljm = 1 m mx 1 ®m: de¯nition 2: the test '0 we call optimal if ®( '0 ) = m in ' ®( ') : n o w it is p o s s ib le t o a r r a n g e t h e s e t o f t e s t s ' a c c o r d in g t o t h e va lu e s o f t h e a va r a g e e r r o r p r o b a b ilit ie s ®( ') . in t h e fo llo win g t h e o r e m we s t a t e t h e c o n s t r u c t io n o f o p t im a l t e s t . l e t p r o b a b ilit y d is t r ib u t io n s gm, m = 1 ; m , h a ve d e n s it ie s fm ( x ) wit h r e s p e c t t o s o m e m e a s u r e ¹, t h e n t h e like lih o o d fu n c t io n is fm ( x) = nq n=1 fm ( xn ) ; m = 1 ; m: t heor em 1: 1. the average probability of error ®( ') of any test ' satis¯es the inequality ®( ') ¸ 1 ¡ 1 m z m a x m fm ( x) ¹ n ( dx ) : ( 1 ) 2. in order the test '0 be optimal it is necessary and su± cient that for almost all with respect to measure g ( g( x ) = 1 m mp m=1 gm ( x ) ) values of x the test satis¯es the relations '0 ( x ) = hm; if fm ( x) = m a x l fl ( x ) : ( 2 ) f or such '0 relation (1) becomes an equality. e. haroutunian 3 9 it is n e c e s s a r y t o n o t e t h a t wh e n t wo o r m o r e va lu e s fm ( x ) a r e m a xim a l it d o e s n o t m a t t e r wh ic h o f t h e s e h yp o t h e s e s t o c h o o s e a n d we c a n m a ke a r a n d o m d e c is io n , t h is will n o t c h a n g e t h e va lu e o f ®( '0 ) . it is n o t d i± c u lt t o a p p ly t h is c r it e r io n t o t wo p a r t ic u la r e xt r e m a l c a s e s . if a ll d e n s it ie s a r e e qu a l f1 ( x) = :::: = fm ( x) it is n o t p o s s ib le t o c o n s t r u c t a t e s t wit h ® ( ' 0 ) le s s t h a n m¡1 m . in t h e o t h e r c a s e , wh e n x n = ms m=1 x nm a n d gm ( x nm ) = 1 , gm ( x nl ) = 0 , m = 1 ; m, l 6= m, t h e b e s t t e s t h a s ® ( '0 ) = 0 . p r oof: 1 . b y d e ¯ n it io n 1 fo r e a c h t e s t ' fo r m h yp o t h e s e s t h e a va r a g e e r r o r p r o b a b ilit y o f t h e t e s t ' is ® ( ') = 1 m mx m=1 ®m; wh e r e ®m = gm ( '( x ) 6= m) = z x:'(x) 6=m fm ( x ) ¹ n ( dx ) : th e n ® ( ') = 1 m mx m=1 z x:'(x)6=m fm ( x) ¹ n ( dx ) = = 1 ¡ 1 m mx m=1 z x:'(x)=m fm ( x ) ¹ n ( dx ) ¸ ¸ 1 ¡ 1 m z m a x m fm ( x ) ¹ n ( dx ) : 2 . op t im a l t e s t '0 d e ¯ n e d in ( 2 ) r e a c h e s t h e lo we r b o u n d in ( 1 ) , t h a t is c o n d it io n ( 2 ) is s u ± c ie n t . to p r o ve t h e n e c e s s it y o f ( 2 ) s u p p o s e t h a t t h e o p t im a l t e s t ' is s u c h t h a t '( x) = hm wit h fm ( x ) < m a xl fl ( x ) fo r x 2 a wit h a s e t a o f p o s it ive p r o b a b ilit y g ( a) > 0 . s u c h t e s t ' c a n b e im p r o ve d o n a b y g ivin g fo r x 2 a, '0 ( x) = hm1 , wit h fm1 ( x ) = m a xl fl ( x ) . r e a lly ® ( '0 ) < ®( ') b e c a u s e ®( ') ¡ ®( '0 ) = 1 m z a [fm1 ( x ) ¡ fm ( x) ]¹n ( dx) > 0 : n o w we s h a ll c o n s id e r t h e m in im a x a p p r o a c h . w e will c o m p a r e t e s t s wit h m a xim a l va lu e s o f e r r o r p r o b a b ilit ie s . de¯nition 3: d enote maximal error probability in the matrix for each test ' by ®( ') = m a x m ®mjm ( ') : it is c le a r t h a t it is p o s s ib le t o p u t in o r d e r t e s t s b y va lu e s o f ® ( ') . w e n a m e m in im a x t h e t e s t ' wh ic h h a s t h e m in im a l va lu e o f ®( ') ®( ') = m in ' ® ( ') : t heor em 2: the optimal test ' such that ®1 ( ') = ®2 ( ') = ::: = ®m ( ') ( 3 ) 4 0 on optimality of tests of m -ary hypotheses for fixed number of independent observations will be a minimax test. p r oof: fo r e ve r y t e s t ' u s in g ( 3 ) we h a ve ®( ') ¸ mx m=1 1 m ®m ( ') ¸ mx m=1 1 m ®m ( ') = ®( ') : n o w we t a ke in t o c o n s id e r a t io n t h e n o t io n o f t h e m o s t p o we r fu l t e s t in a c la s s o f t e s t s . in o r d e r t o c o m p a r e t e s t s it is p o s s ib le t o in t r o d u c e t h e c la s s e s h a vin g g ive n va lu e s o f e r r o r p r o b a b ilit ie s ®1, ..., ®m ¡1 k®1;:::;®m¡1 = f' : ®m ( ') = ®m; m = 1 ; m ¡ 1 g a n d t h e n o r d e r t e s t s b y t h e va lu e s o f ®m ( ') , n a t u r a lly t h e s m a lle r ®m ( ') c o r r e s p o n d s t o t h e b e t t e r t e s t . de¯nition 4: a test e' 2 k®1;:::;®m¡1 is the most powerful test in the class k®1;:::;®m¡1 if for any ' from the same class ®m ( e') · ®m ( ') : if m = 2 t h e o p t im a l t e s t e' is g ive n b y t h e fu n d a m e n t a l l e m m a o f n e ym a n -p e a r s o n [2 ,3 ,6 ,1 6 ]. fo r m o r e t h a n t wo h yp o t h e s e s m ¸ 2 t h e s o lu t io n c a n b e fo u n d b y g e n e r a liz a t io n o f t h e n e ym a n -p e a r s o n l e m m a [1 7 ]. th is im p o r t a n t r e s u lt d e s e r ve s t o b e r e c u r e d . a s it wa s n o t e d in [1 5 ] t h e c a s e n = 1 c o n t a in s t h e g e n e r a l o n e a n d t h e r e is n o n e e d t o r e s t r ic t a t t e n t io n t o m u lt ip le in d e p e n d e n t d r a win g s . fo r g ive n p r e a s s in g n e d va lu e s 0 < ®¤1j1; ® ¤ 2j2; :::; ® ¤ m¡1jm ¡1 < 1 we c h o o s e n u m b e r s t1; t2; :::; tm ¡1 a n d s e t s a¤m, m = 1 ; m, s u c h t h a t a¤1 = ( x : m in ã g1 ( x) g2 ( x) ; :::; g1 ( x) gm ( x) ! > t1 ) ; 1 ¡ g1 ( a¤1 ) = ®¤1j1; a¤2 = a¤1 \ ( x : m in ã g2 ( x ) g3 ( x ) ; :::; g2 ( x ) gm ( x ) ! > t2 ) ; 1 ¡ g2 ( a¤2 ) = ®¤2j2; ........... a¤m¡1 = a¤1 \a 2 ¤ \ :::: \ a¤m ¡2 \ ( x : gm¡1 ( x ) gm ( x) ) > tm¡1 ) ; 1 ¡gm¡1 ( a¤m ¡1 ) = ®¤m ¡1jm¡1; a n d a¤m = x n ¡ ( a¤1 [ a¤2 [ ::: [ a¤m ¡1 ) = a¤1 \ a 2 ¤ \ :::: \ a¤m¡2 \ a¤m¡1: th e c o r r e s p o n d in g e r r o r p r o b a b ilit ie s a r e d e n o t e d b y ®¤ljm ( 'n ) ; l; m = 1 ; m : t heor em 3: ( ge n e r a liz a t io n o f n e ym a n -p e a r s o n l e m m a ) the test determined by sets a¤1, a¤2, ..., a¤m is optimal in the sense that, for each other test de¯ned by the sets b1, b2, ..., bm with the corresponding error probabilities ¯ljm, l; m = 1 ; m , if ¯mjm · ®¤mjm, for some m 2 [1 ; m ¡ 1 ], then there exists at least one index j, j 2 [m+1 ; m] such that ¯mjj ¸ ®¤mjj: e. haroutunian 4 1 p r oof: l e t ©a ¤m a n d © b m b e t h e in d ic a t o r fu n c t io n s o f t h e d e c is io n r e g io n s a¤m a n d bm; m = 1 ; m . fo r a ll x = ( x1; x2; :::; xn ) 2 x n , t h e fo llo win g in e qu a lit y is c o r r e c t ( ©a ¤m ( x) ¡ ©b m ( x) ) ( gm ( x) ¡ m a x( tmgm+1 ( x) ; :::; tmgm ( x ) ) ) ¸ 0 : mu lt ip lyin g a n d t h e n s u m m in g o ve r x n we o b t a in x x:x2x n h © a ¤m ( x ) gm ( x ) ¡ ©a ¤m ( x) m a x( tmgm+1 ( x) ; :::; tmgm ( x ) ) ¡ ¡©b m ( x) gm ( x) + ©b m ( x ) m a x( tmgm+1 ( x ) ; :::; tmgm ( x ) ) ¸ 0 ] ; x x:x2a m¤ [gm ( x ) ¡ tm m a x( gm+1 ( x ) ) ; :::; gm ( x ) ] ¡ ¡ x x:x2b m [gm ( x) ¡ tm m a x( gm+1 ( x ) ) ; :::; gm ( x) ] ¸ 0 : a c c o r d in g t o t h e d e ¯ n it io n o f e r r o r p r o b a b ilit y we o b t a in t h e fo llo win g in e qu a lit y 1 ¡ ®¤mjm ¡ tm m a x( ®¤mjm+1; :::; ®¤mjm ) ¡ ( 1 ¡ ¯mjm ) + tm m a x( ¯mjm+1; :::; ¯mjm ) ¸ 0 ; ¡¯mjm + ®¤mjm · tm h ¡ m a x( ®¤mjm+1; :::; ®¤mjm ) + m a x( ¯mjm+1; :::; ¯mjm ) i : w e s e e n o w t h a t fr o m ¯mjm · ®¤mjm it fo llo ws t h a t m a x( ¯mjm+1; :::; ¯mjm ) ¸ m a x( ®¤mjm+1; :::; ®¤mjm ) : fr o m t h is it fo llo ws t h a t if t h e m a xim a l is ¯mjj , j 2 [m + 1 ; m], t h e n ¯mjj ¸ ®¤mjj. disscussion: th is p a p e r d e a ls wit h s o m e c e n t r a l b a s ic r e s u lt s o f t h e th e o r y o f t e s t in g s t a t is t ic a l h yp o t h e s e s o u g h t t o b e in c lu d e d in t e xt b o o ks . d i®e r e n t a p p r o a c h e s o f c o n s t r u c t in g o p t im a l t e s t s fo r ¯ n it e ly m a n y s im p le h yp o t h e s e s b y a p p lic a t io n o f s a m p le s o f ¯ xe d le n g t h a r e p r e s e n t e d . a c kn o wle d g m e n t th e a u t h o r t h a n ks t h e r e vie we r fo r s u g g e s t io n s o n im p r o vin g t h e e xp o s it io n o f t h e m a t e r ia l. refer ences [1 ] r . r . b a h a d u r , \ s t o c h a s t ic c o m p a r is o n o f t e s t s " , a n n . ma t h . s t a t is t . vo l. 3 1 , n o 3 , p p . 2 7 6 -2 9 5 , 1 9 6 0 . 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[1 7 ] e .a . h a r o u t u n ia n , \ n e ym a n -p e a r s o n p r in c ip le fo r m o r e t h a n t wo h yp o t h e s e s " , a b s t r a c t s o f a n n u a l s e s s io n o f t h e a r m e n ia n ma t h e m a t ic a l u n io n , y e r e va n , p p . 4 7 -4 8 , 2 0 1 3 . submitted 12.08.2016, accepted 10.11.2016. üçùëí³í ãíáí ³ýï³ë ¹çï³ñïáõùý»ñç m -³ï³ý í³ñï³íý»ñç ï»ëï»ñç ûåïçù³éáõãû³ý ù³ëçý º. ð³ñáõãûáõýû³ý ²ù÷á÷áõù ðá¹í³íá ýíçñí³í ¿ ýçùëí³í ãíáí ³ýï³ë ¹çï³ñïáõùý»ñç ýùáõßý»ñç ñçù³ý íñ³ µ³½ù³ãçí å³ñ½ í³ñï³íý»ñç ï»ëï³íáñù³ý ûåïçù³é ùáï»óáõùý»ñç ï³éáõóù³ýá: e. haroutunian 4 3 îá îïòèìàëüíîñòè òåñòîâ m-àðíûõ ãèïîòåç ïðè ôèêñèðîâàííîì ÷èñëå íåçàâèñèìûõ íàáëþäåíèé å. àðóòþíÿí àííîòàöèÿ ñòàòüÿ ïîñâÿùåíà ïîñòðîåíèþ îïòèìàëüíûõ ïîäõîäîâ ê òåñòèðîâàíèþ ìíîãèõ ïðîñòûõ ãèïîòåç ïðè ôèêñèðîâàííîì ÷èñëå íåçàâèñèìûõ íàáëþäåíèé. article_barseghyan.dvi mathematical problems of computer science 33, 73{82, 2010. automated p ap algor ithm for i nter fer ometr ic p hase reconstr uction y u r i a . b a r s e g h ya n yerevan state university e-mail: yuri.barseghyan@gmail.com abstract in this work the problem of interferometric phase reconstruction is considered. an automatic windows size selection algorithm is proposed which in combination with earlier developed pap (pointwise approximation of phase) algorithm provides stable results even for bad quality of interferogramms. experimental results show that combined algorithm demonstrates better performance in comparison with its original version. refer ences [1 ] gr a h a m l . c. \ s yn t h e t ic in t e r fe r o m e t e r r a d a r fo r t o p o g r a p h ic m a p p in g " . in p roceedings of the ie e e , 1 9 7 4 , v. 6 2 , p . 7 6 3 . 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[1 2 ] k . h o a n d j. k a h n , \ e xa c t p r o b a b ilit y d e n s it y fu n c t io n fo r p h a s e m e a s u r e m e n t in t e r fe r o m e t r y" . j . opt. soc. amer. a, vo l. 1 2 , p p . 1 9 8 4 -1 9 8 9 , 1 9 9 5 . [1 3 ] j. l e e , k . h o p p e l, s . ma n g o , a n d a . mille r , \ in t e n s it y a n d p h a s e s t a t is t ic s o f m u lt ilo o k p o la r im e t r ic a n d in t e r fe r o m e t r ic s a r im a g e r y" . ie e e trans. geosci. r emote sensing, vo l. 3 2 , p p . 1 0 1 7 -1 0 2 8 , 1 9 9 4 . [1 4 ] s . ma d s e n , \ s p e c t r a l p r o p e r t ie s o f h o m o g e n e o u s a n d n o n h o m o g e n e o u s r a d a r im a g e s " , ie e e trans. aerosp. e lectron. syst., vo l. a e s -2 3 , p p . 5 8 3 -5 8 8 , 1 9 8 7 . [1 5 ] c. r a t h je n , \ s t a t is t ic a l p r o p e r t ie s o f p h a s e -s h ift a lg o r it h m s " , j . opt. soc. amer. a, vo l. 1 2 , p p . 1 9 9 7 -2 0 0 8 , 1 9 9 5 . [1 6 ] a . go ld e n s h lu g e r a n d a . n e m ir o vs ki, \ on s p a t ia l a d a p t ive e s t im a t io n o f n o n p a r a m e t r ic r e g r e s s io n " , m athematical m ethods of statistic, vo l. 6 , p p . 1 3 5 -1 7 0 , 1 9 9 7 . 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gpu accelerators hrachya v. astsatryan and edita e. gichunts institute for informatics and automation problems of nas ra e-mail: hrach@sci.am, editagich@ipia.sci.am abstract while finding the solutions of hermitian matrix eigenproblem it is a key issue to find an efficient version of algorithms of symmetric tridiagonal solutions. in this paper these algorithms are compared for complex hermitian matrices in hybrid systems. the methods were carried out on the tesla c1060 and tesla k40 gpu accelerators and the performances are presented as between the methods, as well as between the accelerators. keywords: gpu accelerator; hybrid architectures, tridiagonal matrices, mrrr, d&c, bi, algorithm. 1. introduction finding the matrix eigenvalues and eigenvectors is one of the central issues in linear algebra. in particular, the solutions to the eigenproblem are in lapack, scalapack and plapack packages, in the systems with general and distributed memory, respectively. matrix eigensolutions can achieve a higher performance through magma library in hybrid architecture on gpu accelerators. the aim of magma [1,2] library is the realization of lapack library sub-programs in the architecture of hybrid systems. due to the development of productivity of ranking algorithms, this problem is overcome in hybrid architecture. finding of eigenproblem solutions of hermitian matrix, as a rule, takes place through the following three stages: 1. through the householder transformation the matrix is reduced to a tridiagonal form. 2. the tridiagonal matrix solutions are found through one of the following algorithms:  qr iteration [3,4] ,  bisection for the eigenvalues and inverse iteration for the eigenvectors(bi) [5,6],  divide & conquer method (d&c) [7,8], 1. back transformation to find the eigenvectors for the full problem from the eigenvectors of the tridiagonal problem. 93 performance comparisons of eigensolutions algorithms of a symmetric tridiagonal matrix on gpu 94 in fact, the algorithm performance may depend on the matrix, platform, and on the underlying linear algebra libraries blas, lapack, atlas and magma. magma library is used to realize the projects on gpu accelerator. the reduction of matrix to tridiagonal form in all cases is carried out through the householder transformation, as well as through magma_chetrd function of magma library. it should also be noted that in all releases of magma library for complex hermitian matrices the eigenproblem solving function with qr iteration is missing. therefore, only the remaining (d & c), (mrrr), (bi) three cases will be considered. in this work the solutions of hermitian problem are presented by means of three methods on gpu accelerators. moreover, the performance comparisons for three algorithms are presented as between the methods, as well as between the accelerators. section 2 briefly presents the mentioned algorithms. section 3 states about the resources required for the implementation of programs. sections 4 and 5 give the results of the mentioned three algorithms on gpu accelerators for both standard and generalized forms of eigensolutions of complex hermitian matrices, respectively. section 6 covers the conclusion of the obtained results. 2. description of algorithms 2.1.divide and conquer. the divide-and-conquer method can be described in terms of a binary tree where each node corresponds to a submatrix and its eigenpairs, obtained through recursively dividing the matrix in halves; see the exposition in [12]. the tree is processed bottom up, starting with submatrices of size 25 or smaller. dc uses qr to solve the small eigenproblems and then computes the eigenpairs of a parent using the already computed eigenpairs of the children. a parent’s eigenvalues can be computed as solutions of a secular equation. the eigenvector computation consists of two steps. the first one is a relatively inexpensive scaling step. the second one, which is most of the work, multiplies the eigenvectors of the current matrix by the eigenvector matrix accumulated so far. this step uses the level 3 blas (blas 3) routine gemm (dense matrix-matrix multiply). in the worst case, dc is an o(n3) algorithm. 2.2.multiple relatively robust representations. since its introduction in the 1990s, much has been written about the mrrr algorithm, its theoretical foundation is discussed in several publications [ 13, 14, 15, 16] and practical aspects of efficient and robust implementations are discussed in [17, 18, 19, 20, 21, 22]. one could say, with an implementation of the algorithm in the widely used lapack library and the description of (parts of) the algorithm in textbooks such as [23], mrrr has become mainstream. mrrr is a sophisticated variant of inverse iteration that avoids gram–schmidt orthogonalization and thus becomes an o(n2) algorithm. the algorithm can be described in terms of a (generally irregular) representation tree. the root node describes the entire spectrum of t, and the children define gradually refined eigenvalue approximations. the overall complexity of the algorithm depends on the clustering of the eigenvalues. if some eigenvalues of t agree to d digits on average, then the algorithm has to do work proportional to dn2. the algorithm uses a random perturbation to ensure with high probability that eigenvalues cannot be too strongly clustered; see [24] for details. mrrr cannot make use of higher-level blas. 2.3.bisection and inverse iteration. bisection based on sturm sequences requires o(nk) operations to compute k eigenvalues of t. if the distance between the eigenvalues is large enough (relative to ||t||), then computing the corresponding eigenvector by inverse iteration also is an o(nk) process. if, however, the eigenvalues are not well separated, gram–schmidt orthogonalization is employed to try to achieve numerically orthogonal eigenvectors. in this case h. astsatryan and e. gichunts 95 the complexity of the algorithm increases to o(nk2). in the worst case where almost all eigenvalues of t are “clustered,” the complexity can increase to o(n3). furthermore, from the accuracy point of view this procedure is not guaranteed to be reliable; see [11, 25]. neither bisection nor inverse iteration make use of higher-level blas. 3. software development the gpu equipment in hybrid systems, due to its unique architecture, is used as an accelerator to process the limited computational applications. the popularity of gpu-based hybrid systems started with the release of the nvidia compute unified device architecture (cuda) and the extensions of industry-standard programming languages, such as c, c++ and fortran, which made the gpus easier to program, allowing the developers to exploit the computational power of modern gpu devices. to realize the programs, the required software is presented on tesla c1060 and tesla k40 gpu accelerators. the architecture of tesla c1060 consists of 240 processor cores, using the maximum capacity of parallelization. it is endowed with a high bandwidth transmission of messages between cpu and gpu, and also has 4 gb of global memory, 512-bit gddr3 memory interface and cuda c programming environment. the operation system on tesla c1060 is ubuntu 12.04.5 lts, and cuda4 programming environment was used for realization of programs. magma 1.3.0 package was installed. lapack-3.4.2, clapack-3.2.1 and atlas-3.8.0 packages were installed for this library compilation. gcc-4.4, gfortran-4.4 and nvcc compilers were used. during the compilation a number of references of static and dynamic libraries were made in make.incfile, such as libf77blas.a, libcblas.a, libf2c.a, libcublas.so, libcudart.so, libstdc ++.so, libpthread.so, libdl.so. magma 1.3.0 package contains libmagma.a and libmagmablas.a libraries. the architecture of tesla k40 consists of 2880 cuda processor cores. it is endowed with much higher bandwidth 288 gb/s of message transfer between cpu and gpu, having 12 gb of global memory, gddr5 memory interface, and cuda c programming environment. the operation system of tesla k40is ubuntu 14.04.2 lts.cuda7 programming environment was used for the realization of programs. magma 1.6.1 package was installed in accordance with cuda7 environment. for the compilation of magma library the lapack-3.4.2, clapack-3.2.1 and atlas-3.10.0 packages were installed. gcc-4.8, gfortran-4.8, g ++ 4.8 and nvcc compilers were used. such references were made in make.inc file on libf77blas.a, libcblas.a, libf2c.a, libcublas.so, libcudart.so, libstdc ++. so, libpthread.so, libdl.so static and dynamic libraries. magma 1.6.1 package contains libmagma.a and libmagma_sparse.a libraries. 4. performance results of eigenproblem solutions for a standard form finding of eigenproblem solutions of complex hermitian matrices for az = λz standard form is carried out with the help of the following functions of magma library:  magma_xheevdx (d&c),  magma_xheevr (mrrr),  magma_xheevx (bi). moreover, x can be c or z in complex and double complex cases, respectively. figures 1 and 2 show the time-dependent graphs for three methods in the case of standard form on tesla c1060 and tesla k40 gpu accelerators, respectively. performance comparisons of eigensolutions algorithms of a symmetric tridiagonal matrix on gpu 96 fig. 1. tesla c1060, standard eingensolvers. fig. 2. tesla k40, standard eingensolvers. since the global memory of tesla c1060 is 4 gb, and the matrix a should be wholly moved to the global memory of gpu, hence, the maximum dimensionality of the input complex matrix can be 12288*12288. in the case of tesla k40 gpu accelerator it can be 31744 * 31744 as its global memory is 12 gb. the results show that mrrr algorithm, in a standard form, for 2-2.5 times concedes the dc algorithm. for example, on tesla c1060 accelerator in case of 12288*12288 maximum dimensional input matrix, the dc algorithm is carried out at gpu_time = 258sec., and in the case of mrrr algorithm it is fulfilled atgpu_time = 462sec. on tesla k40 gpu accelerator in case of 31744*31744 maximum input-dimensional complex matrix, the dc algorithm is carried out atgpu_time = 603sec., and in case of mrrr algorithm at gpu_time = 1680sec. . figures 3 and 4 show in a standard form case the comparisons of gpu accelerators of timedependent dc and mrrr algorithms with equal amounts of input matrices. the maximum dimensionality of the input matrix will be equal to the possible maximum dimensionality of the matrix used on tesla c1060. fig. 3. dc algorithm. fig. 4. mrrr algorithm. obviously, tesla c1060 much concedes the tesla k40 by its architecture, but let’s present the results in the case of these two algorithms. for example, in case of the input matrix with 12288*12288 maximum dimensionality on tesla c1060 accelerator, the dc algorithm is implemented atgpu_time = 258sec., whereas on tesla k40 it is implemented atgpu_time = 50sec.. mrrr algorithm is implemented on tesla c1060 accelerator atgpu_time = 462sec., whereas on tesla k40 at gpu_time = 150sec.. 0 200 400 600 800 1000 1200 1400 0 5000 10000 15000 evdx evr evx ti m e (s ec on ds ) n 0 500 1000 1500 2000 0 20000 40000 evdx evr evx n ti m e (s ec on ds ) 0 50 100 150 200 250 300 0 10000 20000 dc c1060 dc k40 ti m e (s ec on ds ) n 0 100 200 300 400 500 0 10000 20000 mrrr c1060 mrrr k40 ti m e (s ec on d) n h. astsatryan and e. gichunts 97 5. performance results of eigenproblem solutions for generalized form finding of eigensolutions of complex hermitian matrices for az= λbz generalized form is carried out with the help of the following functions of magma library:  magma_xhegvdx ( d&c),  magma_xhegvr (mrrr),  magma_xhegvx (bi). x can be c or z in complex and double complex cases, respectively. figures 5 and 6 show the time-dependent graphs for three methods in the case of generalized form on tesla c1060 and tesla k40 gpu accelerators, respectively. fig. 5.tesla c1060, generalized eingensolvers fig. 6. tesla k40, generalized eingensolvers since for a generalized form a and b matrices should be wholly moved to the global memory of gpu accelerators, therefore in case of tesla c1060 the maximum dimensionalities of input matrices will be 9216*9216, and in case of tesla k40 they will be 21504*21504. the results show that for a generalized form together with the increase in dimensionalities of input matrices the mrrr algorithm concedes the dc algorithm for 1.5-4times. for example, on tesla c1060 accelerator in case of 9216*9216 maximum dimensional input matrices, the dc algorithm is carried out at gpu_time=150sec., and in the case of mrrr algorithm it is fulfilled at gpu_time = 265sec. on tesla k40 gpu accelerator in case of 21504*21504 maximum inputdimensional complex matrices, the dc algorithm is carried out at gpu_time = 603sec., and in case of mrrr algorithm at gpu_time = 1005sec.. figures 7 and 8 show in a generalized form case the comparisons of gpu accelerators of time-dependent dc and mrrr algorithms with equal amounts of input matrices. the maximum dimensionality of the input matrix will be equal to the possible maximum dimensionality of the matrix used on tesla c1060. 0 200 400 600 800 1000 1200 1400 0 5000 10000 gvdx gvr gvx n ti m e (s ec on ds ) 0 200 400 600 800 1000 1200 0 20000 40000 gvdx gvr gvx n ti m e( se co nd s) performance comparisons of eigensolutions algorithms of a symmetric tridiagonal matrix on gpu 98 fig. 7. dc algorithm fig. 8. mrrr algorithm for the generalized form the results of these two algorithms will be presented. for example, in the case of input matrices with 9216*9216 maximum dimensionality on tesla c1060 accelerator, the dc algorithm is implemented at gpu_time = 150sec., whereas on tesla k40 it is implemented at gpu_time = 29sec. mrrr algorithm is implemented on tesla c1060 accelerator at gpu_time = 266sec., whereas on tesla k40 at gpu_time = 106sec.. 6. conclusion the performance studies of solutions of a symmetric tridiagonal matrix were carried out on both accelerators tesla c1060 and tesla k40. our assessment on this issue considers the speed of the bisection and inverse iteration (bi), the divide-and-conquer method (dc), and the method of multiple relatively robust representations of (mrrr) algorithms in complex hermitian matrices. the conclusions are as follows:  dc and mrrr algorithms, in case of large matrices, are rather quickly than in case of bi. test results show that the programs using bi algorithm, in both cases of standard and generalized forms, are 20 times slower than the programs using dc and mrrr algorithms.  the results showed that the mrrr algorithm, as in both cases of standard and generalized forms, as well as on both gpu accelerators is inferior to the dc algorithm. but for the implementation of dc algorithm much more workspace is required, than for the implementation of mrrr algorithm. therefore, the choice of the algorithms dc and mrrr depends on the user, which algorithm is more efficient for the intended issue.  comparing the work of two gpu accelerators, we see that on tesla k40 in case of an input matrix with the same dimensionality, dc algorithm is implemented 5 times faster than on tesla c1060, and the mrrr algorithm is implemented 3 times faster. references [1] “magma matrix algebra on gpu and multicore architectures”, [online]. available:http://icl.cs.utk.edu/magma/, 2014. 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[25] i. s. dhillon, “current inverse iteration software can fail”, bit numerical mathematics, vol. 38 pp. 685–704, 1998. submitted 31.08.2015, accepted 25.11.2015 սիմետրիկ երեք անկյունագծային մատրիցի սեփական լուծումների ալգորիթմների արտադրողականությունների համեմատությունները gpu արագագործիչների վրա հ. ասցատրյան և է. գիչունց ամփոփում հերմիտյան մատրիցի սեփական լուծումները գտնելիս շատ կարևոր խնդիր է հանդիսանում սիմետրիկ երեքանկյունագծային մատրիցի լուծումների ալգորիթմներից արդյունավետ տարբերակի որոշումը: աշխատանքում համեմատվում են այս ալգորիթմները կոմպլեքս հերմիտյան մատրիցների դեպքում հիբրիդային համակարգերում: մեթոդները կիրառվել են tesla c1060 և tesla k40 gpu արագագործիչների վրա և ներկայացված են արտադրողականությունները ինչպես մեթոդների միջև, այնպես էլ արագագործիչների միջև: сравнение производительности алгоритмов вычисления собственных решений симметричных трехдиагональных матриц на графических процессорах gpu г. асцатрян и э. гичунц аннотация при нахождении решений собственных значений и векторов эрмитовых матриц большую важность представляет проблема определения эффективного варианта из алгоритмов нахождения решений симметричных трехдиагональных матриц. в данной работе эти алгоритмы сравниваются для случая комплексных эрмитовых матриц в гибридных системах. методы были применены на графических процессорах teslac1060 и teslak40 и представлены прозводительности как для методов, так и между графическими процессорами. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 41, 93--102, 2014. 93 a probabilistic model of user navigation in a digital interactive publication vahram h. darbinyan state engineering university of armenia (polytechnic) e-mail: dvahram@gmail.com abstract a probabilistic model of user navigation in a digital interactive publication is considered. digital interactive publication is a paged media where user can navigate either by flipping pages back and forth or using a link for navigation to another page. user’s navigation by links is analyzed in detail. for that case a special weight is assigned to any navigation from the current page to another page. a method of calculating these weights is proposed basing on a publication type and structure. considerations include both server-side and client-side analysis. a mapping of the introduced weights to probabilities of navigation to each page is proposed also. these probabilities allow predicting user behavior during the paged media navigation. experiments with a representative sample set of digital publications have been performed to illustrate the validity of the suggested model. keywords: digital interactive publication, user behavior, user navigation, navigation model, navigation prediction, server-side analysis, client-side analysis. 1. introduction user navigation is widely used in modern web based interactive technologies. many websites and web based services have issues that users become disoriented: they lose sense of location and direction, so service providers try to come up with best practices of website structure. user’s behavior and action prediction is also practiced in attempts of enhancing user experience [1, 2]. prediction of the nextaction can be very beneficial for many web-based services and it can allow preparing and caching the expected next requested page, starting transfer of a video ortaking other actions that can improve the user performance overall. when visiting a website the user’s behavior is unpredictable in general. the visitor can follow any link from the menu. there are special tools that allow analyzing the user’s behavior in website [3]. they provide statistical and analytical data that allow building websites which are more suited for a particular goal.but still generalizing the visitor’s behavior in websites is a complicated problem and it’s hard to create a general applicable model to benefit from. digital interactive publications (dip) are unique media [4] where users’ behavior is more predictable in comparison with a regular website. while people browsing the internet usuallyare in pursuit of some specific information,the objectivewhen reading a digital magazine usually is a probabilistic model of user navigation in a digital interactive publication 94 leisure and entertainment. in internet pages a visitor usually navigates from page to another by links seeking the information he needs or is interested in. unless there is a certain “wizard” interface with “next” and “previous” buttons, the user behavior is random and unique for each website [1, 5, 6]. taking into account the mentioned predictability of a user behavior when reading a digital publication a probabilistic model of user’s navigation can be created. such a model would allow estimating probability of user’s next action via the built model. using statistical data for the user’s action prediction is generally easier and more accurate. at the same time a diversity of behaviors for different users when reading the same publication is not big and, therefore, the decisions made on statistical data are usually accurate if some other aspects such as used device are considered too. the main drawback of statistics-based decision is the availability of data. a publication that has been published for a while and has been read multiple times leads to corresponding data which are usually available, while new publications lack such statistics. in that case we can’t benefit from being able to predict users’ next action for enhancing the user performance as well as for making other decisions. the above described unavailability of data is the case when such a model can be extremely helpful as it would allow predicting the user’s behavior without using statistical data. there exist user behavior models based on hyperlink analysis that allow predicting user’s navigation via navigation history and link structures [1, 7, 8]. in [8] a navigation behavior pattern is proposed, which can be used to predict visitor’s navigation through the web. however, the existing models are designed for websites and do not incorporate important aspects of the investigated object such as a publication structure or type, which are the key for predicting user behavior in the case of reading a dip. a navigation model defines the content, structure, and metadata of the navigation. it specifies the items to include and the hierarchy of those items. navigation models typically include the following resources [9]:  pages (individual pages and page hierarchies)  external links  content (individual content items or the results of a content query)  other navigation models modeling of user navigation via page flipping and links will be concentrated on further. section 2 considers how the publication type affects user navigation. an influence of user devices on the navigation will be discussed too. based on these factors weights will be calculated for each navigational event: flipping a page and navigating via a link. in section 3 the way of calculation of the proposed weights is described. weights are calculated for each page depending on page’s displacement relative to the current page (e.g., if the page is next or previous page for the current one) and surface of links from current to the considered page. mapping of weights to probabilities and building of a probabilistic model is described too. section 4 incorporates client-side data into the built probabilistic model. using the clientside information will enhance the accuracy of calculations in cases when navigation prediction is performed on a user device. section 5 describes experiments performed with a sample set of digital publications. data illustrating the validity of the suggested model are adduced there. the conclusion summarizes the obtained results and outlines further investigations. v. darbinyan 95 2. server-side factors that affect user navigation many factors affect users’ behavior, e.g.,a structure of the publication, its content, interactive elements like videos and links and many others. these factors vary in their effect and those that have most significant effect are considered below. these data are available at the server-side and are intended for prediction of users’ navigation actions via server. 2.1. types of publications types of digital publications differ in waysusers read them. behavior of a userreading a fiction book differs from behavior when a scientific/research article or a magazine is read [10]. thereforethe model to be built should take into account this difference. to make the consideration observable we suggest presenting the existing variety of publication types: magazines, catalogues, photo albums, articles, journals, manuals, newspapers, brochures, etc. distributed into the following threebig groups. sequential content: the group will contain publication types that have mostly monolithic and sequential content, like fiction books where the content is a single story and is supposed to be read sequentially. here the navigation is sequential and links are rarely used. bookmarks are used to continue reading, but from the moment reading is restarted from a given page the navigation once again becomes sequential. discrete content: this group is comprised of catalogues, manuals and publications of other similar types. the navigation here is mostly non sequential, the content consists of separate articles, sections and titles that are rarely read altogether. few users will read a manual completely at once, while many users will be referencing it periodically exploiting the links in table of content for navigation. mixed content: publications like newspapers, magazines, and journals are forming the group, where depending on a use case, the content can be read either with a sequential navigation, e.g. flipping through the content or using the links. 2.2. device type depending on a user device and the size of the screen the behavior of the user might be totally different [11, 12]. here too user navigation in a dip can be either sequential or link-oriented depending on the device and screen size. in case of sequential behavior there is a greater chance the user will be scrolling through pages and there are not less cases when the navigation is linkoriented. the following way of describing this difference in the model is suggested. chance of a link to be clicked is described by a pair of values: l and c. l stands for the coefficient of clicking a link on any page other than the cover page. the chance of clicking a link on the cover page is bigger, so c will be used to denote that case. heuristically calculated values are used for these coefficients. the values are calculated basing on data gathered from more than 250000 digital publications and more than 100000 publishers. the gathered data is analyzed from the navigation point of view in terms of page flipping and link usage. initial statistical processing a probabilistic model of user navigation in a digital interactive publication 96 was performed using live digital interactive publication service. over 80 million records of user navigation were analyzed; more than 30 million among them were in publications considered as having sequential navigation and more than 50 million navigations in publications considered as having link-oriented navigation. in more detail this distribution is adduced below. sequential navigation: here over 4.7 million navigations were from cover page and almost 25.3 million navigations were from non-cover pages. among over 4.7 million navigations from cover page 1.7 million were using links, which brings to ~36% of the total number of navigations. among 25.3 million navigations from pages other than cover page, over 5.2 million were using links, which is ~21% of the total number of navigations. link-oriented navigation: here over 43,3 million were navigations from pages that are not cover page and 6.8 million were navigations from cover pages. among them, ~46% (19.9 million) of navigations from non-cover pages use links. ~52% (3.5 million) of navigations from cover page use links. thus in the case of sequential navigation the values of coefficients can be taken as l=0.21 and c=0.36 [12]. when user navigation is link-oriented the chance of user clicking a link is higher compared to sequential navigation. the link-oriented navigation is common for devices with small displays or slow internet connection where users choose to use links instead of flipping from page to page [11]. in this case coefficients of link navigation are higher: l=0.46 and c=0.52. to express this formally a notation l(x,nm) is introduced for link navigation factor, where x is a number of the current page and nm is the navigation mode: 1 – corresponds to sequential, 2 – to link-oriented. ( ) 2.2. structure of publication other important aspect that affects users’ behavior when reading a digital publication is a structure of the publication itself, more specifically the presence of links from one page to another. if there is a link on page x to page y the chance of page y being visited when the user is on page x is bigger. if the page x is a cover page, the chances are even bigger. another factor to be taken into account while modeling the navigation from page x to page y is not only the mere presence of a link from page x to page y, but the size and location of the link on the page. links that are larger are more likely to be used compared to smaller ones. links can also be placed over the whole page which makes the probability of them being used even higher (sometimes by mistake). assuming there is a number of links on page x, let’s name that set of link objects can be expressed as: { } where is number of links on the page x. let y be a page lead by some links from the set . a subset of containing all links that lead to y is denoted as : v. darbinyan 97 { } where m is number of links on the page x leading to the page y. the chance of navigating from page x to page y using links is correlated to surface of links on the page x that lead to the page y. fs(x,y)notation is introduced to denote that chance. for calculation of fs(x,y) the sum of surfaces s(x,y) for all links to the page y from the page x has to be calculated. ( ) ∑ ( ) where is the number of the linkobject from the set and ( ) is the surface of that linkobject. also a total sum of link object surfaces is required for fs(x,y) calculation. s(x) stands for that sum. ( ) ∑ ( ) where ( ) is surface of link object from set . finally, fs(x,y)is calculated using the following formula: ( ) { ( ) ( ) ( ) ( ) 3. creating the model of user navigation it implies from the considerations above that in our model the chance of user navigation from page x to page y depends on the following factors:  publication type,  number of the current page,  does the current page coincide with the previous page?  set of links from the current page to the target page (number of links and their surface) the way of calculating this chance is adduced below. 3.1. calculating weights for pages when the user is on page x, three following actions can be taken in terms of navigation: 1. navigate to next page 2. navigate to previous page 3. use a link to navigate to some other page a probabilistic model of user navigation in a digital interactive publication 98 for predicting which page is going to be navigated by the user while being on the page x, it is suggested to use special weights for all pages to be navigated. these weights should be calculated beforehand. the weight of a given page is correlated to the chance of being navigated. weights are calculated basing on page’s dislocation relative to the current page and also links from the current page to the considered one. three measures are introduced below: is introduced to indicate if the number y of the next page immediately succeeds the number x of the current page. will have value 1 if the condition of being the immediately succeeding numberis true and 0 in cases when it is not true. to indicate the fact that the number y of a givenpage is the immediately preceding the number xof the current page, is introduced. similarly, when and in all other cases.for the value 1, and are mutually exclusiveas any page can’t be simultaneously previous and next page for a given page.thus and can’t have value 1 at the same time.presence of links on the page x to the page yshould increasethe weight of the page y. the measure is to reflect that: ( ) ( ) the total weight for the page yis calculated by summing the values of considered measures: where and are coefficients of influence for the corresponding measures. they are calculated heuristically. initial calculation of and .analysis of gathered statistical data on navigation actions is performed. from the same ~80 million navigation records mentioned above, almost 50 million records were navigations via page flipping. among those over 42 million records were navigations to the next page, which is ~53% of total 80 million navigations. navigations to previous page are counted in more than 7 million records: ~9% of the total. thus for and values like 53 and 9 can be used. as already includes the chance ( ) of a link to be used, a value 100 is used for . tuning of and in the process of use.initially calculated values should be adjusted during their use. if according to the built model abehavior of a user is described asusing a link for navigation, but instead a page was just flipped to the next page, the coefficient of page flipping ( ) should be increased and the coefficient of using a link should be reduced. in the casethe prediction was correct, no action should be taken against the coefficients. per our experiments after a sufficient number of publication readings the coefficients are reaching some stable values. so it is assumed further that each page has a calculated weight for navigation. 3.2. mapping weights to probabilities sum of weights for each page of a given publication is needed to calculate probabilities for pages. will be used to denote that sum of page weights. it is calculated by the formula below: v. darbinyan 99 ∑ where p is the number of pages of the publication that is being analyzed. stands for weight of a page with the number i. ( ) notation is introduced to express the probability of page y being navigated while the user is on page x. ( ) is calculated using the following formula: ( ) ⁄ where is the weight of the page y and is sum of weights of all pages. based on the calculated probabilities, pages are sorted by their probability values in descending order. the page with highest probability is predicted to be the next in navigation sequence. 4. using client-side information the model proposed can be complemented with the client-side data to enhance the accuracy. though using the client-side information limits the usage of the model to only client side, at the same time it provides higher accuracy. some devices cannot fit the whole page of the publication on the screen, or they require zooming in to provide acceptable reading performance [11]. in the case of zoomed page or scrolling, not all links on current page are visible to the user, thus the chance of usage for some of them is reduced. considering that on many devices where page zooming or scrolling is required, the user’s navigation is link-oriented the accurate calculation of probability of navigating via a link is paramount. in case of zoomed pages or scrolling, when only a part of the page is visible at a moment, instead of calculating the surface of all links to a certain page from the current one, surface of visible links should be calculated. so a set of visible links should be considered instead of and set of visible links leading to page y should substitute : { }, { }, accordingly ( ) will be calculated by the following formula: ( ) ∑ ( ) where ( ) is surface of visible link object from set .this case also implies comparing ( )to visible part of the screen instead of the surface of the whole page. ( )denotes the visible surface of the page and is calculated similarly to the above. thus ( ) is calculated according to thebelow: a probabilistic model of user navigation in a digital interactive publication 100 ( ) { ( ) ( ) ( ) ( ) 5. evaluating the validity of the suggested model experiments have been conducted on a sample set of digital interactive publications. the set consists of publications of all groups: publications with sequential content, discrete and mixed contents. the suggested model is implemented in digital interactive publication client-side reader software. the client software is provided with the necessary publication metadata from the server side: number of publication pages, type of the publication, weight factors. the software is also “aware” of the user device it is being run on. base on the available data and user’s current page user’s navigation is predicted. according the prediction resources of the predicted page are being pre-loaded, to reduce wait time for the resources and enhance user experience while reading the digital interactive publication. after the user navigates to any other page, both the actual destination page number and predicted page number are sent to server. these values are stored in server database. based on the stored values accuracy of the suggested model is evaluated. the experimental set of publications consisted of 27000 publications, published between dec 1, 2013 and mar 1, 2014. this set includes:  23324 magazines  531 catalogues  120 photo albums  380 e-books  17 e-cards  146 articles  27 essays  189 journals  364 manuals  553 newspapers  239 portfolios  897 reports  665 brochures  37 comics during the period of model evaluation 24 million predictions were made. the accuracy of predictions made using the suggested model ranged between 67 to 79% depending on publication type. 6. conclusion a probabilistic model for user navigation was proposed that allows predicting users’ next action while reading a digital interactive publication. the model is used for action prediction in digital interactive publication services to predict user’s navigation and to pre-load the page user v. darbinyan 101 is most probably about to visit, thus enhancing the user experience. the model is also used to substitute statistical data when such is not available. the further development of the suggested model is connectedwith adopting the model to behavior of each user instead of having a common model for all readers of the publication. also elements of machine learning methods will substitute gradually the heuristics based calculations. references [1] e. herder, “modeling user navigation”, lecture notes in computer science, vol. 2702, pp. 417--419, 2003. [2] n. wunderlich, f. wangenheim and m. jo bitner, “high tech and high touch: a framework for understanding user attitudes and behaviors related to smart interactive services”, journal of service research, february 2014. [3] m. eirinaki, m. vazirgiannis, “web mining for web personalization”, acm transactions on internet technology, vol. 3, no. 1, pp. 1--27, feb 2003. [4] w. rosenblatt, s. mooney and w. trippe, digital interactive publications (dip) are unique media, john wiley & sons, inc., new york, ny, usa, 2001. [5] v. h. darbinyan and r. h. vardanyan, “detecting ddos attacks on digital interactive publication hosting services”, proceedings of seua series "information technologies, electronics, radio engineering", yerevan, armenia, vol. 15, pp. 20-25, 2012. [6] r. h. vardanyan, “an adaptive method for visualization of the interactive multimedia publications”, proceedings of the conference computer science and information technologies, yerevan, armenia, pp. 332--334, 2011. [7] w. t. fu and p. pirolli, “snif-act: a cognitive model of user navigation on the world wide web”, human-computer interaction, vol. 22, no. 4, pp. 355--412, 2007. [8] j. borges and m. levene, “data mining of user navigation patterns”, lecture notes in computer science, vol. 1836, pp. 92-112, 2000. [9] d. cohn and t. hofmann, “the missing link – a probabilistic model of document content and hypermedia connectivity”, advances in neural information processing systems13 proceedings of the 2000 conference, pp. 430--436, 2000. [10] r. chen, a. rose, b.b. bederson, “how people read books online: mining and visualizing web logs for use information”, lecture notes in computer science, vol. 5714, pp. 364--369, 2009. [11] r.h. vardanyan, “adapting the interactive multimedia publication content to mobile devices”, proceedings of the yerevan state university. series, yerevan, armenia, vol. 1, pp. 51--54, 2013. [12] v. h. darbinyan, “analysis of user behavior when reading digital interactive magazine online on various devices”, proceedings of the conference computer science and information technologies, sep. 23-27, yerevan, armenia, pp. 445--446, 2013. submitted 25.12.2013, accepted 26.02.2014. a probabilistic model of user navigation in a digital interactive publication 102 թվային ինտերակտիվ հրապարակման մեջ օգտագործողի ուղորդման հավանականային մոդել վ. դարբինյան ամփոփում հոդվածում նկարագրված է թվային ինտերակտիվ հրապարակումներում օգտագործողի ուղորդման հավանականային մոդել: թվային ինտերակտիվ հրապարակումը էջային կառուցվածքով միջավայր է, որտեղ օգտագործողի ուղորդումը հնարավոր է կամ էջից էջ առաջ կամ ետ թերթելով, կամ հղումներ օգտագործելով: հոդվածում վերլուծված է օգտագործողի վարքագիծը: ամեն էջ ուղորդման տեղի ունենալուն վերագրված է որոշակի կշիռ, որը հաշվարկվում է հրապարակման տիպից և կառուցվածքից ելնելով: կշիռներից կախված, ամեն էջի համար հաշվարկված է այդ էջ ուղորդման հավանականությունը: вероятностная модель навигации пользователя в цифровых интерактивных публикациях в. дарбинян аннотация в данной статье предложена вероятностная модель навигации пользователя в цифровых интерактивных публикациях. цифровые интерактивные публикации являются средой со страничной организацией, где навигация пользователя происходит или по средствам листания страниц, или же использованием ссылок. приведен анализ поведения пользователя. навигации с текущей страницы публикации на каждую другую страницу присваевается вес. вес расчитывается исходя из типа публикации и ее структуры. вероятности навигации на каждую страницу расчитываются исходя из весов. microsoft word naghashyanpogosyan_24.doc mathematical problems of computer science 33, 187--195, 2010. 187 developing java software for representation, acquisition and management of strategy knowledge zaven naghashyan1 and edward pogossian1,2 1state engineering university of armenia 1,2institute for informatics and automation problems of nas of ra e-mail: lasl@sci.am, david@dm-lab.sci.am url: http://dm-lab.sci.am abstract technical solutions for representing and managing concepts of strategy knowledge for java based software package and the framework of personalized and communalized knowledge acquisition is described. the following six units are considered game tree organization, instances matching oriented classes, dynamic class hierarchy management, web based graphical interface and relations builder procedure as well as ideology of knowledge learning and evaluation. the solutions develop and enhance classical object oriented approaches to be used for representation and management of the concepts of strategy knowledge, their learning and evaluation. . references 1. e. pogossian, v. vahradyan and a. grigoryan, “on competing agents consistent with expert knowledge”, lecture notes in computer science, ais-adm-07: the international workshop on autonomous intelligent systems agents and data mining, st. petersburg, russia, pp. 229-241, june 6-7, 2007. 2. t. bagdasaryan, a. grigoryan and z. nagashyan, “developing of a scripting language interpreter for regular acquisition of expert knowledge”, international conference in computer sciences and information technologies, pp. 187-191, yerevan, 2009. 3. su lixin, h. mikko, lipasti. dynamic class hierarchy mutation http://www.ece.wisc.edu/~pharm/papers/cgo4.pdf 4. r. joel brandt; goldman, j. kenneth, run-time modification of the class hierarchy in a live java development environment http://cse.wustl.edu/research/lists/technical%20reports/attachments/642/403_hierarchy.pdf 5. java language specification http://java.sun.com/docs/books/jls/ 6. jude official site http://jude.change-vision.com/jude-web/index.html 7. ibm rational rose http://www-01.ibm.com/software/rational/ 8. linked lists basics http://cslibrary.stanford.edu/103/ 9. e. pogossian, “combinatorial game models for security systems”, nato arw on "security and embedded systems", porto rio, patras, greece, aug., pp. 8-18, 2005. 188 developing java software for representation, acquisition and management of strategy knowledge 10. э. погосян, в.ваградян, а.григорян, “эксперименты согласования знаний экспертов с принятием решений в этюдах рети и нoдареишвили”. proceedings of ipia in mathematics and computer sciences, p. 20, 2007. 11. e. pogossian, adaptation of combinatorial algorithms, (in russian), pp. 293,yerevan, 1983. 12. e. pogossian, specifying personalized expertise. international association for development of the information society (iadis): international conference cognition and exploratory learning in digital age (celda 2006), barcelona, spain, pp.151-159, 2006. java íñ³·ñ³ß³ñç ùß³ïáõù é³½ù³í³ñ³ï³ý ·çï»éçùý»ñç ý»ñï³û³óù³ý, áýï³éù³ý ¨ ï³½ù³ï»ñåù³ý ýå³ï³ïáí ¼. ü³õ³ßû³ý ¨ ¾. äáõáëû³ý ²ù÷á÷áõù ðá¹í³íáõù ýï³ñ³·ñí³í »ý java íñ³·ñ³íáñù³ý é»½íç ñçù³ý íñ³ ëï»õíí³í íñ³·ñ³ß³ñçý ïçñ³éí³í ï»ëýçï³ï³ý éáõíáõùý»ñá ¨ ³ýóý³íáñí³í áõ ñ³ë³ñ³ï³ûý³óí³í ·çï»éçùý»ñç áýï³éù³ý ñçùý³ïù³ëùá: ¸çï³ñïí³í »ý ñ»ï¨û³é í»ó ã»ù³ý»ñá` ë³õ³ûçý í³éç ï³½ù³ï»ñåáõùá, ýùáõßý»ñç áýïñù³ý ñ³ù³ñ ¹³ë»ñç ùß³ïáõùá, ¹çý³ùçï ¹³ë»ñç ëïáñ³¹³ë³ûçý ï³½ù³ï»ñåáõùá, í»µ çýï»ñý»ûëç ùß³ïáõùá, ¹³ë»ñç ùçç¨ ñ³ñ³µ»ñáõãûáõýý»ñç ï³éáõóù³ý áýã³ó³ï³ñ·á çýãå»ë ý³¨ ·çï»éçùçý»ñç áõëáõóù³ý ¨ ·ý³ñ³ïù³ý ·³õ³÷³ñý»ñá: îçñ³éí³í ùáï»óáõùý»ñá ½³ñ·³óýáõù ¨ áý¹é³ûýáõù »ý ¹³ë³ï³ý ûµû»ïïáñáßí³í íñ³·ñ³íáñù³ý ù»ãá¹ý»ñá, ãáõûé ï³éáí û·ï³·áñí»é ¹ñ³ýù ëïñ³ï»·ç³ý»ñáõù ·çï»éçùý»ñç ñ³ëï³óáõãûáõýý»ñá ñ³ù³ï³ñ·ãç ùççáóáí ý»ñï³û³óý»éáõ, ï³½ù³ï»ñå»éáõ, áõëáõó³ý»éáõ ¨ ·ý³ñ³ï»éáõ ñ³ù³ñ: microsoft word pertosian.doc ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 23, 2004, 175–177. 175 особенности построения диалоговых систем в комплексах компьютерной телефонии давид петросян èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà àííîòàöèÿ обсуждаются возможные принципы и механизмы оптимального построения диалоговых систем компьютерной телефонии. литература [1] http://www.comptek/ru/telephony/board/recognition.html [2] http://www.comptek.ru/telephony îáùåûáõï»ñ³ûçý ñ»é³ëáëï³åç ñ³ù³éçñý»ñáõù »ñïëáë³ïó³ï³ý ñ³ù³ï³ñ·»ñç ï³éáõóù³ý ³é³ýóý³ñ³ïïáõãûáõýý»ñá ¸. ä»ïñáëû³ý ²ù÷á÷áõù øýý³ñïíáõù »ý ïáùåûáõý»ñ³ûçý ñ»é³ëáë³ï³åç »ñïëáë³ïó³ï³ý ñ³ù³ï³ñ·»ñç ûåïçù³é ï³éáõóù³ý ñý³ñ³íáñ ëï½µáõýùý»ñá ¨ ï³éáõóí³íùý»ñá: microsoft word yu. shoukourian_1.doc mathematical problems of computer science 34, 2010. 7 on creation of computational infrastructure for development of science in armenia. yuri shoukourian the vital necessity of new research methods based on the usage of perspective computing resources, data collections and scientific tools (i. e. e-science) promises to assist to scientific inventions. numerous advanced countries have accepted the direction of e-science, which gives high importance to the construction and development of e-infrastructures assisting it. an important part of e-science is the computational science [1, 2, 3]. the computational science is a rapidly growing multidisciplinary field that uses advanced computing capabilities to understand and solve complex problems. cs fuses three distinct elements [2]: numerical and non-numerical algorithms and modeling and simulation software developed to solve science (e.g., biological, physical, and social), engineering, and humanities problems; advanced system hardware, software, networking and data management components developed through computer and information science to solve computationally demanding problems; the computational infrastructure that supports both science and engineering problem solving and development of computer and information science. the application of computers and mathematical methods for the decision of scientific and technical tasks was one of the basic tasks put in front of science in armenia still in the initial period of creation of computer facilities and cybernetics in our country. an essential progress is certainly made in development of connection of the armenian scientific and educational community with two modern and important directions of computational infrastructure: the european geant network and grid-structure for e-science within the framework of the european seventh frame program. now it is beginning to be formed the socalled virtual research environments in armenia, which allow considerably to expand opportunities of scientific cooperation and uses of the distributed databases. simultaneously it has begun to be formed scientific databases in the field of astrophysics, archeology, seismology, etc. [4, 5, 9, 10, 11]. currently e-infrastructures are formed of five interconnected components [2] : network – a network with the large bandwidth for information interchange uniting affiliated scientific-educational networks, computer centers, forming a global network; grids – service allowing by means of the internet to supply distribution of computing capacities and memory of the data; the scientific data – the purpose is the maintenance of regulation of their accelerated growth, development of new methods of their storage and use, 8 maintenance of uniformity at distribution; global virtual communities – uniting multidisciplinary cooperation of the international cooperation. there are problems on all above-stated components in armenia. they can be grouped in the following way: effective use of e-infrastructures for development of computer and information science; choice of scientific research directions for the creation and development of the corresponding computational solvers; creation of the general program of development of a computational infrastructure for e-science in armenia in view of the above-stated components and formation of some conceptual documents determining the basic ways of its realization. effective use of e-infrastructure for the development of computer and information science there exists a dynamic link between the development of e-infrastructures and that of informatics. let us consider the levels of e-infrastructure which are network (switches, routers, cables, etc.), resources (supercomputers, servers, memories and sensors), middleware (computing access, communication service, grid service, operation system, etc.), applications and service (virtual organizations, users’ interfaces). taking into consideration that hardware performance has exponential growth and scientific researches require processing of very large scale data, it is possible to predict numerous problems coherent to computer and information science. so a necessity arises to reconsider the modification and development of mathematical, engineering research priorities of the mentioned sphere in armenia in assistance to the creation of an advanced e-infrastructure in the country. choice of scientific research directions for the creation and development of the computational solvers for the scientific problems here the following problems are important.  to collect the research areas requiring multidisciplinary approaches and high performance computing means in armenia.  to move the possibilities of e-infrastructures for science into the over scientific spheres (e-government, e-education, e-healthcare, etc.). there are some achievements on the solutions to the both problems. there has been a report on the first problem in csit and other conferences, thus corresponding virtual organizations have been formed [4-8]. for the second problem the following results are attained: the weather research and forecast model for the territory of armenia is in elaboration by the armenian state hydrometeorology for an operative use, as well as for the protection and elaboration of seismic data, elf (earthquake location finding) program package installation for the early announcement of an operative decision of earthquake basic parameters (close to real time) [9]. national program of development of e-infrastructure in armenia the importance of e-science has grown up for the elaboration of the program of development of e-infrastructure in armenia. its main aim must be in creation and formation of such an infrastructure based on perspective approach which will provide accomplishment of high-level scientific researches during the next 5-10 years. it must assist to the integration into european scientific area as well as to the formation of the generation of young scientists. one of the most important portals will be the collaboration with international and national centers via geant network and armenian national grid initiative (armngi) [12]. new public projects must be put forward by means of which the above stated program will be 9 performed. the mentioned program is appropriate to be elaborated by the participation of all the interested organizations of the academic and educational system. references 1. j. dongarra. overview of high-performance computing. eecs 594, lecture1 pp. 1-56, spring 2008. 2. ict infrastructures for e-science. communication from the european commission, com 108 final, 5.3.2009, brussels, belgium. 3. h. marandjian, yu. shoukourian. computational science in armenia (invited talk). casc 2010:203, lncs, 6244, 12th international workshop, september 2010, tsakhkadzor, armenia. 4. h. astsatryan, v. sahakyan, yu. shoukourian. the armcluster project: creation of highperformance computation cluster and databases in armenia. ii international conference on parallel computations and control problems (paco ‘2004), october 4-6, 2004, moscow, russia. 5. yu. shoukourian. scientific computing environment in armenia (experience and prospects). csit-2005, presentation, yerevan, armenia. 6. e. davtyan, h. karapetyan, a. kocharyan, k. shahbazyan, yu. shoukourian. implementation of one dimensional systolic arrays on a beowulf cluster computer. proceedings of int. conf. on computer science and information technologies, pp. 54-60, 2005, yerevan, armenia. 7. h. astsatryan, v. sahakyan, yu. shoukourian. a grid-aware web portal with advanced service trading for linear algebra calculations. 8th international conference of high performance computing for computational science–vecpar’2008, june 24-28, 2008, toulouse, france. 8. h. astsatryan, v. sahakyan, yu. shoukourian. brief introduction of armenian national grid initiative. book of abstracts of 4th international conference "distributed computing and grid-technologies in science and education" (grid'2010), june 28 july 3, 2010, dubna, russia. 9. h. astsatryan, v. sahakyan, yu. shoukourian. implementing and evaluating the weather research and forecast model for the territory of armenia. proceedings of csit conference 2009, pp. 490-493, 2009, yerevan, armenia. 10. yu. shoukourian. development modern computational and information environment for research and education community in armenia. 5-th international conference on computer science and information technologies, 24 september 28 september, 2007, yerevan, armenia (presentation). 11. yu. shoukourian. the vital needs of armenian national research and education networks. nato advanced networking workshop “armenian national research and education networks: achievements, problems and solutions”, 12-14 november, 2007, yerevan, armenia (presentation). 12. http://www.grid.am d:\sbornik\...\article.dvi mathematical problems of computer science 40, 13{22, 2013. n on-hamiltonian gr aphs with given t oughness zh o r a g. n iko g h o s ya n 1 institute for informatics and automation problems of nas ra e-mail: zhora@ipia.sci.am abstract in 1973, chv¶atal introduced the concept of toughness ¿ of a graph and conjectured that there exists a ¯nite constant t0 such that every t0-tough graph (that is ¿ ¸ t0) is hamiltonian. to solve this challenging problem, all e®orts are directed towards constructing non-hamiltonian graphs with toughness as large as possible. the last result in this direction is due to bauer, broersma and veldman, which states that for each positive ², there exists a non-hamiltonian graph with 9 4 ¡ ² · ¿ < 9 4 . the following related broad-scale problem, reminding the well-known pancyclicity or hypohamiltonicity, arises naturally: whedher there exists a non-hamiltonian graph with a given toughness. we conjecture that if there exist a non-hamiltonian t-tough graph then for each rational number a with 0 < a · t there exists a non-hamiltonian graph whose toughness is exactly a. in this paper we prove this conjecture for t = 9 4 ¡ ² by using a number of additional modi¯ed building blocks to construct the required graphs. keywords: hamilton cycle, toughness of graph. 1 . in t r o d u c t io n on ly ¯ n it e u n d ir e c t e d g r a p h s wit h o u t lo o p s o r m u lt ip le e d g e s a r e c o n s id e r e d . th e s e t o f ve r t ic e s o f a g r a p h g is d e n o t e d b y v ( g) a n d t h e s e t o f e d g e s b y e ( g ) . th e o r d e r a n d t h e in d e p e n d e n c e n u m b e r o f g a r e d e n o t e d b y n a n d ®, r e s p e c t ive ly. fo r s a s u b s e t o f v ( g) , we d e n o t e b y gns t h e s u b g r a p h o f g in d u c e d b y v ( g ) ns. th e n e ig h b o r h o o d o f a ve r t e x x 2 v ( g) is d e n o t e d b y n ( x) . a g r a p h g is h a m ilt o n ia n if g c o n t a in s a h a m ilt o n c yc le , i.e . a c yc le o f le n g t h n. a g o o d r e fe r e n c e fo r a n y u n d e ¯ n e d t e r m s is [5 ]. th e c o n c e p t o f t o u g h n e s s o f a g r a p h wa s in t r o d u c e d in 1 9 7 3 b y ch v¶a t a l [6 ]. l e t ! ( g ) d e n o t e t h e n u m b e r o f c o m p o n e n t s o f a g r a p h g. a g r a p h g is t-t o u g h if jsj ¸ t! ( gns ) fo r e ve r y s u b s e t s o f t h e ve r t e x s e t v ( g) wit h ! ( gns ) > 1 . th e t o u g h n e s s o f g, d e n o t e d ¿ ( g) , is t h e m a xim u m va lu e o f t fo r wh ic h g is t-t o u g h ( t a kin g ¿ ( kn ) = 1 fo r a ll n ¸ 1 ) . mu c h o f t h e r e s e a r c h o n t h is s u b je c t h a ve b e e n in s p ir e d b y t h e fo llo win g c o n je c t u r e d u e t o ch v¶a t a l [6 ]. conjectur e 1: there exists a ¯nite constant t0 such that every t0-tough graph is hamiltonian. to s o lve t h is c h a lle n g in g p r o b le m , a ll e ®o r t s a r e d ir e c t e d t o wa r d s c o n s t r u c t in g n o n h a m ilt o n ia n g r a p h s wit h t o u g h n e s s a s la r g e a s p o s s ib le . in [6 ], ch v¶a t a l c o n s t r u c t e d a n in ¯ n it e fa m ily o f n o n -h a m ilt o n ia n g r a p h s wit h ¿ = 3 2 , a n d t h e n th o m a s s e n [[4 ]; p: 1 3 2 ] fo u n d 1 g. g. nicoghossian (up to 1997). 1 3 1 4 non-hamiltonian graphs with given toughness n o n -h a m ilt o n ia n g r a p h s wit h ¿ > 3 2 . l a t e r e n o m o t o e t a l. [7 ] h a ve fo u n d n o n -h a m ilt o n ia n g r a p h s wit h ¿ ¸ 2 ¡ ² fo r e a c h p o s it ive ². th e la s t r e s u lt in t h is d ir e c t io n is d u e t o b a u e r , b r o e r s m a a n d v e ld m a n [2 ] in s p ir e d b y s p e c ia l c o n s t r u c t io n s in t r o d u c e d in [1 ] a n d [3 ]. t heor em a: f or each positive ² > 0 , there exists a non-hamiltonian graph with 9 4 ¡ ² < ¿ < 9 4 . th e fo llo win g r e la t e d b r o a d -s c a le p r o b le m , r e m in d in g t h e we ll-kn o wn p a n c yc lic it y o r h yp o h a m ilt o n ic it y, a r is e s n a t u r a lly. p r oblem: d o e s t h e r e e xis t a n o n -h a m ilt o n ia n g r a p h wit h t h e g ive n t o u g h n e s s ? th e fo llo win g m e t a c o n je c t u r e s e e m s r e a s o n a b le . conjectur e 2: if there exists a non-hamiltonian t-tough graph then for each rational number a with 0 < a · t there exists a non-hamiltonian graph whose toughness is exactly a. in t h is p a p e r we p r o ve t h is c o n je c t u r e fo r t = 9 4 ¡ ². t heor em 1: f or each rational number t with 0 < t < 9 4 , there exists a non-hamiltonian graph g with ¿ ( g) = t. th e o r e m 1 p r o vid e s a ls o a c o m p le t e b a c kg r o u n d fo r fu r t h e r in ve s t ig a t io n t o wa r d s ¯ n d in g n o n -h a m ilt o n ia n g r a p h s wit h t o u g h n e s s a t le a s t 9 4 . 2 . p r e lim in a r ie s to p r o ve th e o r e m 1 , we n e e d b o t h n e w a n d o ld g r a p h c o n s t r u c t io n s . de¯nition 1. l e t l(1) b e a g r a p h o b t a in e d fr o m c8 = w1w2:::w8w1 b y a d d in g t h e e d g e s w2w4; w4w6; w6w8 a n d w2w8. p u t x = w1 a n d y = w5. th is is t h e we ll-kn o wn b u ild in g b lo c k l u s e d t o o b t a in ( 9 4 ¡ ²) -t o u g h n o n -h a m ilt o n ia n g r a p h s ( s e e fig u r e 1 in [2 ]) . in t h is p a p e r we will u s e a n u m b e r o f a d d it io n a l m o d ī e d b u ild in g b lo c ks . de¯nition 2: l et l(2) be the graph obtained from l(1) by deleting the edges w1w2, w2w8 and identifying w2 with w8. de¯nition 3: l et l(3) be the graph obtained from l(1) by adding a new vertex w9 and the edges w4w9, w6w9. de¯nition 4: l et l(4) be the graph obtained from the triangle w1w2w3w1 by adding the vertices w4; w5 and the edges w1w4, w3w5. p ut x = w4 and y = w5. de¯nition 5: f or each l 2 fl(1); l(2)g, de¯ne the graph g( l; x; y; l; m ) ( l; m 2 n ) as follows. take m disjoint copies l1; l2; :::; lm of l, with xi; yi the vertices in li corresponding to the vertices x and y in l ( i = 1 ; 2 ; :::; m) . l et fm be the graph obtained from l1 [ ::: [ lm by adding all possible edges between the pairs of vertices in x1; :::; xm; y1; :::; ym. l et t = kl and let g ( l; x; y; l; m ) be the join t _ fm of t and fm. th e fo llo win g c a n b e c h e c ke d e a s ily. claim 1: the vertices x and y are not connected by a hamilton path of l(i) ( i = 1 ; 2 ; 3 ) . th e p r o o f o f t h e fo llo win g r e s u lt o c c u r s in [1 ]. claim 2: l et h be a graph and x; y two vertices of h which are not connected by a hamilton path of h. if m ¸ 2 l + 1 then g ( h; x; y; l; m) is non-hamiltonian. 3 . p r o o f o f th e o r e m 1 b y t h e d e ¯ n it io n , t h e t o u g h n e s s ¿ ( g) is a r a t io n a l n u m b e r . l e t t b e a n y r a t io n a l n u m b e r wit h 0 < t < 9 4 a n d le t t = a b fo r s o m e in t e g e r s a; b. zh. nikoghosyan 1 5 case 1: 0 < a b < 1 . l e t ka;b b e t h e c o m p le t e b ip a r t it e g r a p h g = ( v1; v2; e ) wit h ve r t e x c la s s e s v1 a n d v2 s u c h t h a t jv1j = a a n d jv2j = b. s in c e ab < 1 , we h a ve ®( g ) = b > ( a + b ) = 2 a n d t h e r e fo r e , ka;b is a n o n -h a m ilt o n ia n g r a p h . cle a r ly, ¿ · jv1j=! ( gnv1 ) = a=b. ch o o s e s ½ v ( g) s u c h t h a t ¿ ( g) = jsj=! ( gns ) . p u t s \ vi = si a n d jsij = si ( i = 1 ; 2 ) . if vins 6= ; ( i = 1 ; 2 ) t h e n c le a r ly ! ( gns ) = 1 , wh ic h is im p o s s ib le b y t h e d e ¯ n it io n . h e n c e , vins = ; fo r s o m e i 2 f1 ; 2 g. case 1.1: i = 2 . it fo llo ws t h a t ¿ = b + s1 a ¡ s1 ¸ b a : r e c a llin g t h a t ¿ · a b , we h a ve b2 · a2, c o n t r a d ic t in g t h e h yp o t h e s is a b < 1 . case 1.2: i = 1 . it fo llo ws t h a t ¿ = s2 + a b ¡ s2 ¸ a b ; im p lyin g im m e d ia t e ly t h a t ¿ = a b . case 2: a b = 1 . l e t g b e a g r a p h o b t a in e d fr o m c6 = v1v2:::v6v1 b y a d d in g a n e w ve r t e x v7 a n d t h e e d g e s v1v7; v4v7; v2v6. cle a r ly, g is n o t h a m ilt o n ia n a n d ¿ ( g) = 1 . case 3: 1 < a b < 3 2 . case 3.1: a b < 3 2 ¡ 1 b . l e t v1; v2; v3 b e p a ir wis e d is jo in t s e t s o f ve r t ic e s : v1 = fx1; x2; :::; xa¡b+1g; v2 = fy1; y2; :::; ybg; v3 = fz1; z2; :::; zbg: jo in e a c h xi t o a ll t h e o t h e r ve r t ic e s a n d e a c h zi t o e ve r y o t h e r zj a s we ll a s t o t h e ve r t e x yi wit h t h e s a m e s u b s c r ip t i. ca ll t h e r e s u lt in g g r a p h g. ch o o s e w ½ v ( g) s u c h t h a t ¿ ( g) = jw j=! ( gnw ) . p u t m = jw \ v3j. cle a r ly, w is a m in im a l s e t wh o s e r e m o va l fr o m g r e s u lt s in a g r a p h wit h ! ( gnw ) c o m p o n e n t s . a s w is a c u t s e t , we h a ve v1 ½ w a n d m ¸ 1 . fr o m t h e m in im a lit y o f w we e a s ily c o n c lu d e t h a t v2 \ w = ; a n d m · b ¡ 1 . th e n we h a ve jw j = m + a ¡ b + 1 a n d ! ( gnw ) = m + 1 . h e n c e , ¿ ( g ) = jw j ! ( gnw ) = m in 1·m·b¡1 m + a ¡ b + 1 m + 1 = a b : to s e e t h a t g is n o n -h a m ilt o n ia n , le t u s a s s u m e t h e c o n t r a r y, i.e . le t c b e a h a m ilt o n c yc le in g. d e n o t e b y f t h e s e t o f e d g e s o f c h a vin g a t le a s t o n e e n d ve r t e x in v2. s in c e v2 is in d e p e n d e n t , we h a ve jf j = 2 jv2j. on t h e o t h e r h a n d , t h e r e a r e a t m o s t 2 jv1j e d g e s in f h a vin g o n e e n d ve r t e x in v1 a n d a t m o s t jv3j e d g e s in f h a vin g o n e e n d ve r t e x in v3. th u s , 2 b = 2 jv2j = jf j · 2 jv1j + jv3j = 2 ( a ¡ b + 1 ) + b = 2 a ¡ b + 2 : b u t t h is is e qu iva le n t t o a=b ¸ 3 =2 ¡ 1 =b, c o n t r a d ic t in g t h e h yp o t h e s is . case 3.2: a b ¸ 3 2 ¡ 1 b . b y c h o o s in g a s u ± c ie n t ly la r g e q 2 n wit h a b = aq bq < 3 2 ¡ 1 bq ; 1 6 non-hamiltonian graphs with given toughness we c a n a r g u e a s in ca s e 3 .1 . case 4: a b = 3 2 . a n e xa m p le o f a n o n -h a m ilt o n ia n g r a p h wit h ¿ = 3 =2 is o b t a in e d wh e n in t h e p e t e r s e n g r a p h , e a c h ve r t e x is r e p la c e d b y a t r ia n g le . case 5: 3 2 < a b < 7 4 . claim 3: f or l ¸ 2 and m ¸ 1 , ¿ ³ g ³ l(2); x; y; l; m ´´ = l + 3 m 1 + 2 m : p r oof. l e t g = g ( l(2); x; y; l; m) fo r s o m e l ¸ 2 a n d m ¸ 1 . ch o o s e s µ v ( g) s u c h t h a t ! ( gns ) > 1 ; ¿ ( g ) = jsj ! ( gns ) : ob vio u s ly, v ( t ) µ s. d e ¯ n e si = s \ v ( li ) , si = jsij, a n d le t !i b e t h e n u m b e r o f c o m p o n e n t s o f linsi t h a t c o n t a in n e it h e r xi n o r yi ( i = 1 ; :::; m ) . th e n ¿ ( g ) = l + pm i=1 si c + pm i=1 !i ¸ l + pm i=1 si 1 + pm i=1 !i ; wh e r e c = ( 0 if xi; yi 2 s fo r a ll i 2 f 1 ; :::; mg 1 o t h r wis e . it is e a s y t o s e e t h a t !i · 2 ; si ¸ 3 2 !i ( i = 1 ; :::; m ) : th e n ¿ ¸ l + 3 2 pm i=1 !i 1 + pm i=1 !i = l ¡ 3 2 1 + pm i=1 !i + 3 2 ¸ l ¡ 3 2 1 + 2 m + 3 2 = l + 3 m 1 + 2 m : s e t u = v ( t ) [ u1 [ ::: [ um, wh e r e ui is t h e s e t o f ve r t ic e s o f li wit h t h e d e g r e e a t le a s t 4 in li ( i = 1 ; :::; m ) . th e p r o o f o f cla im 3 is c o m p le t e d b y o b s e r vin g t h a t ¿ ( g) · juj ! ( gnu ) = l + 3 m 2 m + 1 : case 5.1:b = 2 k + 1 for some integer k. co n s id e r t h e g r a p h g( l(2); x; y; a ¡ 3 2 ( b ¡ 1 ) ; b¡1 2 ) . case 5.1.1: a b · 7 4 ¡ 9 4b . b y t h e h yp o t h e s is , m = b ¡ 1 2 ¸ 2 µ a ¡ 3 2 ( b ¡ 1 ) ¶ + 1 = 2 l + 1 : b y cla im 2 , g is n o t h a m ilt o n ia n . cle a r ly b ¸ 3 , im p lyin g t h a t m = ( b ¡ 1 ) =2 ¸ 1 . if a b ¸ 3 2 + 1 2b t h e n l = a ¡ 3 2 ( b ¡ 1 ) ¸ 2 a n d b y cla im 3 , ¿ ( g ) = a b . n o w le t a b < 3 2 + 1 2b . b y c h o o s in g a s u ± c ie n t ly la r g e in t e g e r q wit h a b = aq bq ¸ 3 2 + 1 2 bq ; zh. nikoghosyan 1 7 we c a n a r g u e a s in t h e p r e vio u s c a s e . case 5.1.2: a b > 7 4 ¡ 9 4b . b y c h o o s in g a s u ± c ie n t ly la r g e in t e g e r q wit h a b = aq bq · 7 4 ¡ 9 4 bq ; we c a n a r g u e a s in ca s e 5 .1 .1 . case 5.2: b = 2 k for some integer k. co n s id e r t h e g r a p h g0 o b t a in e d fr o m g( l(2); x; y; l; m ) b y r e p la c in g lm wit h l (3). claim 4: f or l ¸ 2 and m ¸ 1 , ¿ ( g0 ) = l + 3 m + 1 2 ( m + 1 ) : p r oof: ch o o s e s µ v ( g0 ) s u c h t h a t ! ( g0ns ) > 1 a n d ¿ ( g0 ) = jsj=! ( g0ns ) . ob vio u s ly, v ( t ) µ s. d e ¯ n e si = s \ v ( li ) , si = jsij, a n d le t !i b e t h e n u m b e r o f c o m p o n e n t s o f linsi t h a t c o n t a in n e it h e r xi n o r yi ( i = 1 ; :::; m ) . s in c e si ¸ 32 !i ( i = 1 ; :::; m ¡ 1 ) a n d sm ¸ 43 !m, we h a ve ¿ ( g0 ) ¸ l + pm i=1 si c + pm i=1 !i ¸ l + 3 2 pm¡1 i=1 !i + 4 3 !m 1 + pm i=1 !i = l ¡ 1 6 !m 1 + pm i=1 !i + 3 2 ; wh e r e c = 0 if xi; yi 2 s fo r a ll i 2 f 1 ; :::; mg a n d c = 1 , o t h e r wis e . ob s e r vin g a ls o t h a t !i · 2 ( i = 1 ; :::; m ¡ 1 ) a n d !m · 3 , we o b t a in ( l ¡ 2 ) mx i=1 !i + 1 3 ( m + 1 ) !m · ( l ¡ 2 ) ( 2 m + 1 ) + ( m + 1 ) · 2 l ( m + 1 ) : b u t t h is is e qu iva le n t t o l ¡ 1 6 !m 1 + pm i=1 !i + 3 2 ¸ l ¡ 2 2 ( m + 1 ) + 3 2 ; im p lyin g t h a t ¿ ( g0 ) ¸ l ¡ 2 2 ( m + 1 ) + 3 2 = l + 3 m + 1 2 ( m + 1 ) : s e t u = v ( t ) [ u1 [ ::: [ um, wh e r e ui is t h e s e t o f ve r t ic e s o f li wit h t h e d e g r e e a t le a s t 4 in li ( i = 1 ; :::; m ) . th e p r o o f o f cla im 4 is c o m p le t e d b y o b s e r vin g t h a t ¿ ( g0 ) · juj ! ( gnu ) = l + 3 m + 1 2 ( m + 1 ) : co n s id e r t h e g r a p h g0 wit h m = b 2 ¡ 1 a n d l = a ¡ 3 2 b + 2 . cle a r ly m = b 2 ¡ 1 ¸ 1 a n d l = a ¡ 3 2 b + 2 ¸ 2 . b y cla im 4 , ¿ ( g0 ) = a b . if a b · 7 4 ¡ 3 b t h e n m ¸ 2 l + 1 , a n d b y cla im 2 , g0 is n o t h a m ilt o n ia n . ot h e r wis e , b y c h o o s in g a s u ± c ie n t ly la r g e q wit h a b = aq bq · 7 4 ¡ 3 b ; we c a n a r g u e a s in t h e p r e vio u s c a s e . case 6: 7 4 ¡ ² < a b · 2 . 1 8 non-hamiltonian graphs with given toughness l e t m = m1 + m2 ¸ 2 l + 1 a n d le t g00 b e t h e g r a p h o b t a in e d fr o m g ( l(1); x; y; l; m ) b y r e p la c in g li wit h l (2) ( i = m1 + 1 ; m1 + 2 ; :::; m ) . b y cla im 2 , g 00 is n o t h a m ilt o n ia n . claim 5: f or l ¸ 2 , m ¸ 1 and m2 ¸ l ¡ 2 , ¿ ( g00 ) = l + 3 m2 2 m2 + 1 : p r oof: ch o o s e s µ v ( g00 ) s u c h t h a t ¿ ( g00 ) = jsj=! ( g00ns ) . ob vio u s ly, v ( t ) µ s. d e ¯ n e si = s\v ( li ) , si = jsij, a n d le t !i b e t h e n u m b e r o f c o m p o n e n t s o f linsi t h a t c o n t a in n e it h e r xi n o r yi ( i = 1 ; :::; m ) . s in c e si ¸ 2 !i ( i = 1 ; :::; m1 ) a n d si ¸ 32 !i ( i = m1 + 1 ; :::; m) , we h a ve ¿ ( g00 ) ¸ l + pm1 i=1 si + pm i=m1+1 si c + pm i=1 !i ¸ l + 2 pm1 i=1 !i + 3 2 pm i=m1+1 !i 1 + pm i=1 !i l + 1 2 pm1 i=1 !i ¡ 32 + 3 2 ( 1 + pm i=1 !i ) 1 + pm i=1 !i = 2 l + pm1 i=1 !i ¡ 3 2 ( 1 + pm i=1 !i ) + 3 2 ; wh e r e c = 0 if xi; yi 2 s fo r a ll i 2 f 1 ; :::; mg a n d c = 1 , o t h e r wis e . ob s e r vin g t h a t !i · 2 ( i = 1 ; :::; m ) , we o b t a in ( 2 l ¡ 3 ) mx i=m1+1 !i ¡ ( 2 m2 ¡ 2 l + 4 ) m1x i=1 !i · 4 lm2 ¡ 6 m2: b u t t h is is e qu iva le n t t o 2 l + pm1 i=1 !i ¡ 3 2 ( 1 + pm i=1 !i ) + 3 2 ¸ 2 l ¡ 3 2 ( 2 m2 + 1 ) + 3 2 ; im p lyin g t h a t ¿ ( g00 ) ¸ 2 l ¡ 3 2 ( 2 m2 + 1 ) + 3 2 = l + 3 m2 2 m2 + 1 : s e t u = v ( t ) [ u1 [ ::: [ um, wh e r e ui is t h e s e t o f ve r t ic e s o f li wit h t h e d e g r e e a t le a s t 4 in li ( i = 1 ; :::; m ) . th e p r o o f o f cla im 5 is c o m p le t e d b y o b s e r vin g t h a t ¿ ( g00 ) · juj ! ( gnu ) = l + 3 m2 2 m2 + 1 : case 6.1: b = 2 k + 1 for some integer k. co n s id e r t h e g r a p h g00 wit h m2 = b¡1 2 a n d l = a ¡ 3 2 ( b ¡ 1 ) . case 6.1.1: a b ¸ 3 2 + 1 2b . s in c e a b · 2 , we h a ve m2 = b ¡ 1 2 ¸ a ¡ 3 2 ( b ¡ 1 ) ¡ 2 = l ¡ 2 : n e xt , s in c e a b ¸ 3 2 + 1 2b , we h a ve l = a ¡ 3 2 ( b ¡ 1 ) ¸ 2 . b y cla im 5 , ¿ ( g00 ) = a b . case 6.1.2: a b < 3 2 + 1 2b . b y c h o o s in g a s u ± c ie n t ly la r g e in t e g e r q wit h a b = aq bq ¸ 3 2 + 1 2 bq ; zh. nikoghosyan 1 9 we c a n a r g u e a s in ca s e 6 .1 .1 . case 6.2: b = 2 k for some integer k. co n s id e r t h e g r a p h g000 o b t a in e d fr o m g00 b y r e p la c in g lm wit h l (3). claim 6: fo r l ¸ 2 , m ¸ 1 a n d m2 ¸ l ¡ 2 , ¿ ( g000 ) = l + 3 m2 + 1 2 ( m2 + 1 ) : p r oof: ch o o s e s µ v ( g000 ) s u c h t h a t ¿ ( g000 ) = jsj=! ( g000ns ) . ob vio u s ly, v ( t ) µ s. d e ¯ n e si = s\v ( li ) , si = jsij, a n d le t !i b e t h e n u m b e r o f c o m p o n e n t s o f linsi t h a t c o n t a in n e it h e r xi n o r yi ( i = 1 ; :::; m) . s in c e si ¸ 2 !i ( i = 1 ; :::; m1 ) , si ¸ 32 !i ( i = m1 + 1 ; :::; m ¡ 1 ) a n d sm ¸ 43 !m, we h a ve ¿ ( g000 ) ¸ l + pm1 i=1 si + pm¡1 i=m1+1 si + sm c + pm i=1 !i ¸ l + 2 pm1 i=1 !i + 3 2 pm¡1 i=m1+1 !i + 4 3 !m 1 + pm i=1 !i = l + 1 2 pm1 i=1 !i ¡ 16 !m + ( 3 2 pm1 i=1 !i + 3 2 pm i=m1+1 ) 1 + pm i=1 !i = l + 1 2 pm1 i=1 !i ¡ 16 !m 1 + pm i=1 !i + 3 2 ; wh e r e c = 0 if xi; yi 2 s fo r a ll i 2 f 1 ; :::; mg a n d c = 1 , o t h e r wis e . ob s e r vin g t h a t !i · 2 ( i = 1 ; :::; m ¡ 1 ) a n d !m · 3 , we o b t a in ( l ¡ 2 ) mx i=m1+1 !i + 1 3 ( m2 + 1 ) !m ¡ ( m2 ¡ l + 3 ) m1x i=1 !i · l + 2 lm2 + 2 : b u t t h is is e qu iva le n t t o l + 1 2 pm1 i=1 !i ¡ 16 !m 1 + pm i=1 !i + 3 2 ¸ l ¡ 2 2 ( m2 + 1 ) + 3 2 ; im p lyin g t h a t ¿ ( g000 ) ¸ l ¡ 2 2 ( m2 + 1 ) + 3 2 = l + 3 m2 + 1 2 ( m2 + 1 ) : s e t u = v ( t ) [ u1 [ ::: [ um, wh e r e ui is t h e s e t o f ve r t ic e s o f li wit h t h e d e g r e e a t le a s t 4 in li ( i = 1 ; :::; m ) . th e p r o o f o f cla im 6 is c o m p le t e d b y o b s e r vin g t h a t ¿ ( g000 ) · juj ! ( gnu ) = l + 3 m2 + 1 2 ( m2 + 1 ) : co n s id e r t h e g r a p h g000 wit h m2 = b 2 ¡ 1 a n d l = a ¡ 3 2 b + 2 . case 6.2.1: a b · 2 ¡ 1 b . b y t h e h yp o t h e s is , m2 = b 2 ¡ 1 ¸ ³ a ¡ 3 2 b + 2 ´ ¡ 2 = l ¡ 2 . n e xt , s in c e a b > 7 4 ¡ ² > 3 2 , we h a ve l = 3 2 b + 2 ¸ 2 . b y cla im 6 , ¿ ( g000 ) = a b . 2 0 non-hamiltonian graphs with given toughness case 6.2.2: a b > 2 ¡ 1 b . b y c h o o s in g a s u ± c ie n t ly la r g e in t e g e r q wit h a b = aq bq · 2 ¡ 1 bq , we c a n a r g u e a s in ca s e 6 .2 .1 . case 7: 2 < a b < 9 4 . case 7.1: b = 2 k + 1 for some integer k. case 7.1.1: a b · 9 4 ¡ 11 4b . ta ke t h e g r a p h g ³ l(1); x; y; a ¡ 2 b + 2 ; b¡1 2 ´ . s in c e a b > 2 , we h a ve l = a ¡ 2 b + 2 ¸ 2 . n e xt , t h e h yp o t h e s is a b · 9 4 ¡ 11 4b is e qu iva le n t t o m = b ¡ 1 2 ¸ 2 ( a ¡ 2 b + 2 ) + 1 = 2 l + 1 : b y cla im 1 , g ³ l(1); x; y; a ¡ 2 b + 2 ; b¡1 2 ´ is n o t h a m ilt o n ia n . th e t o u g h n e s s ¿ ³ g ³ l(1); x; y; a ¡ 2 b + 2 ; b¡1 2 ´´ c a n b e d e t e r m in e d e xa c t ly a s in t h e p r o o f o f th e o r e m a [2 ], ¿ ã g ã l(1); x; y; a ¡ 2 b + 2 ; b ¡ 1 2 !! ¸ l + 4 m 2 m + 1 = a b : case 7.1.2: a b > 9 4 ¡ 11 4b . b y c h o o s in g a s u ± c ie n t ly la r g e in t e g e r q wit h aq bq = a b · 9 4 ¡ 1 1 4 bq ; we c a n a r g u e a s in ca s e 7 .1 .1 . case 7.2: b = 2 k for some positive integer k. ta ke t h e g r a p h g0000 o b t a in e d fr o m g ³ l(1); x; y; a ¡ 2 b + 2 ; b 2 ´ b y r e p la c in g lm wit h l (4). s in c e a b > 2 , we h a ve l = a ¡ 2 b + 2 > 2 . w e h a ve a ls o m = b 2 > 1 , s in c e b ¸ 3 . claim 7: f or l ¸ 2 and m ¸ 1 , ¿ ( g0000 ) = l + 4 m ¡ 2 2 m : p r oof: ch o o s e s µ v ( g000 ) s u c h t h a t ¿ ( g000 ) = jsj=! ( g000ns ) . ob vio u s ly, v ( t ) µ s. d e ¯ n e si = s \ v ( li ) , si = jsij, a n d le t !i b e t h e n u m b e r o f c o m p o n e n t s o f linsi t h a t c o n t a in n e it h e r xi n o r yi ( i = 1 ; :::; m) . s in c e si ¸ 2 !i ( i = 1 ; :::; m) , !i · 2 ( i = 1 ; :::; m ¡ 1 ) a n d !m · 1 , we h a ve ¿ ( g0000 ) = l + pm i=1 si c + pm i=1 !i ¸ l + 2 pm i=1 !i 1 + pm i=1 !i = l ¡ 2 1 + pm i=1 !i + 2 ¸ l ¡ 2 2 m + 2 = l + 4 m ¡ 2 2 m ; wh e r e c = 0 if xi; yi 2 s fo r a ll i 2 f 1 ; :::; mg a n d c = 1 , o t h e r wis e . s e t u = v ( t ) [ u1 [ ::: [ um, wh e r e ui is t h e s e t o f ve r t ic e s o f li wit h t h e d e g r e e a t le a s t 4 in li ( i = 1 ; :::; m) . th e p r o o f o f cla im 7 is c o m p le t e d b y o b s e r vin g t h a t ¿ ( g0000 ) · juj ! ( gnu ) = l + 4 m ¡ 2 2 m : case 7.2.1: a b · 9 4 ¡ 3 b . zh. nikoghosyan 2 1 b y t h e h yp o t h e s is , m ¡ 1 = b 2 ¡ 1 ¸ 2 ( a ¡ 2 b + 2 ) + 1 = 2 l + 1 : b y cla im 2 , g0000 is n o t h a m ilt o n ia n a n d b y cla im 7 , ¿ ( g0000 ) = a b . case 7.2.2: a b > 9 4 ¡ 3 b . b y c h o o s in g a s u ± c ie n t ly la r g e in t e g e r q wit h aq bq = a b · 9 4 ¡ 3 3 bq ; we c a n a r g u e a s in ca s e 7 .2 .1 . th e o r e m 1 is p r o ve d . refer ences [1 ] d . b a u e r , h .j. b r o e r s m a , j. va n d e n h e u ve l a n d h .j. v e ld m a n , \ on h a m ilt o n ia n p r o p e r t ie s o f 2 -t o u g h g r a p h s " , j . graph theory, vo l. 1 8 , p p . 5 3 9 -5 4 3 , 1 9 9 4 . [2 ] d . b a u e r , h .j. b r o e r s m a a n d h .j. v e ld m a n , \ n o t e ve r y 2 -t o u g h g r a p h is h a m ilt o n ia n " , d iscrete appl. m ath., vo l. 9 9 , p p . 3 1 7 { 3 2 1 , 2 0 0 0 . [3 ] d . b a u e r a n d e . s c h m e ic h e l, \ to u g h n e s s , m in im u m d e g r e e a n d t h e e xis t e n c e o f 2 fa c t o r s " , j . graph theory, vo l. 1 8 , p p . 2 4 1 -2 5 6 , 1 9 9 4 . [4 ] j. c. b e r m o n d , selected topics in graph theory, in : l . b e in e ke , r .j. w ils o n ( e d s ) , a c a d e m ic p r e s s , l o n d o n a n d n e w y o r k, 1 9 7 8 . [5 ] j.a . b o n d y a n d u .s .r . mu r t y, graph theory with applications, ma c m illa n , l o n d o n a n d e ls e vie r , n e w y o r k, 1 9 7 6 . [6 ] v . ch v¶a t a l, \ to u g h g r a p h s a n d h a m ilt o n ia n c ir c u it s " , d iscrete m ath., vo l. 5 , p p . 2 1 5 2 2 8 , 1 9 7 3 . [7 ] h . e n o m o t o , b . ja c ks o n , p . k a t e r in is a n d a . s a it o , \ to u g h n e s s a n d t h e e xis t e n c e o f k-fa c t o r s " , j . graph theory, vo l. 9 , p p . 8 7 -9 5 , 1 9 8 5 . submitted 05.09.2013, accepted 18.10.2013. îñí³í ïáßïáõãû³ùµ áã ñ³ùçéïáýû³ý ·ñ³ýý»ñ ä. üçïáõáëû³ý ²ù÷á÷áõù ¶ñ³ýç ïáßïáõãû³ý ¿ µýáõã³·ñçãá ý»ñùáõí»é ¿ êí³ï³éá 1973-çý: àëï êí³ï³éç í³ñï³íç, ·áûáõãûáõý áõýç ³ûýåçëç t0 í»ñç³íáñ ãçí, áñ ï³ù³û³ï³ý t0-ïáßï ·ñ³ý (çýãá ýß³ý³ïáõù ¿, áñ ¿ ¸ t0 ) ñ³ùçéïáýû³ý ¿: ²ûë ù³ñï³ññ³í»ñ³ûçý ëý¹ñç éáõíù³ý µáéáñ ç³ýù»ñá ¨ ï»ëýçï³ý áõõõí³í »ý ³é³í»é³·áõûý ïáßïáõãûáõý áõý»óáõ áã ñ³ùçéïáýû³ý ·ñ³ýý»ñç ï³éáõóù³ýá, ³ýï»ë»éáí ³í»éç ó³íñ ïáßïáõãû³ý ·ñ³ýý»ñá: ²ûë áõõõáõãû³ùµ ëï³óí³í í»ñççý ³ñ¹ûáõýùá, áñá ëï³ó»é »ý ´³áõ»ñá, ´ñá»ñëù³ý ¨ ì»é¹ù³ýá, åý¹áõù ¿, áñ ï³ù³û³ï³ý ¹ñ³ï³ý " ãíç ñ³ù³ñ ·áûáõãûáõý áõýç áã ñ³ùçéïáýû³ý ·ñ³ý, áñç ïáßïáõãûáõýá ·ïýíáõù ¿ 9 4 ¡" · ¿ < 9 4 ë³ñù³ýý»ñáõù: ð»ï¨û³é ³í»éç áý¹·ñïáõý ëý¹çñá çñ ¹ñí³íùáí ñçß»óýáõù ¿ å³ýóçïéçïáõãû³ý ¨ ñçåáñ³ùçéïáýû³ý ·ñ³ýý»ñç ·áûáõãû³ý 2 2 non-hamiltonian graphs with given toughness ëý¹çñý»ñá. ·áûáõãûáõý áõýç, ³ñ¹ûáù áã ñ³ùçéïáýû³ý ·ñ³ý ïñí³í ïáßïáõãû³ùµ: àëï ù»ñ í³ñï³íç, »ã» ·áûáõãûáõý áõýç áã ñ³ùçéïáýû³ý t-ïáßï ·ñ³ý, ³å³ ( 0 ; t] çýï»ñí³éçý å³ïï³ýáõ ï³ù³û³ï³ý a é³óçáý³é ãíç ñ³ù³ñ ·áûáõãûáõý áõýç áã ñ³ùçéïáýû³ý ·ñ³ý, áñç ïáßïáõãûáõýá áõõçõ ¿: ü»ñï³ ³ßë³ï³ýùáõù ³ûë í³ñï³íá ³å³óáõóíáõù ¿ ( 0 ; 9 4 ) ùçç³ï³ûùçý å³ïï³ýáõ ï³ù³û³ï³ý é³óçáý³é ãíç ñ³ù³ñ: ä³ñ³ýçí³í ·ñ³ýý»ñá ï³éáõó»éáõ ñ³ù³ñ û·ï³·áñíí»é »ý ùç ß³ñù ýáñ ï³éáõóí³íù³ûçý ùç³íáñý»ñ, áñáýù ï³ñµ»ñíáõù »ý ·ñ³ï³ýáõãû³ý ù»ç ñ³ûïýç ùç³íáñý»ñçó: íåãàìèëüòîíîâûå ãðàôû ñ çàäàííîé æåñòêîñòüþ æ. íèêîãîñÿí àííîòàöèÿ ïîíÿòèå æåñòêîñòè ¿ ( g ) ãðàôà g áûëî ââåäåíî õâàòàëîì â 1973 ãîäó. ïî èçâåñòíîé ãèïîòåçå õâàòàëà, ñóùåñòâóåò êîíå÷íîå ÷èñëî t0 òàêîå, ÷òî êàæäûé t0-æåñòêèé ãðàô (ýòî îçíà÷àåò, ÷òî ¿ ¸ t0) ãàìèëüòîíîâ. äëÿ ðåøåíèÿ ýòîé ñòèìóëèðóþùåé ïðîáëåìû âñå óñèëèÿ êîíöåíòðèðîâàëèñü íà ïîñòðîåíèå íåãàìèëüòîíîâûõ ãðàôîâ ñ ìàêñèìàëüíîé æåñòêîñòüþ, íå óäåëÿÿ âíèìàíèÿ íà ãðàôû ñ íèçêîé æåñòêîñòüþ. ïîñëåäíèé ðåçóëüòàò â ýòîì íàïðàâëåíèè, êîòîðûé ïîëó÷èëè áàóåð, áðîåðñìà è âåëüäìàí, óòâåðæäàåò, ÷òî äëÿ êàæäîãî ïîëîæèòåëüíîãî ÷èñëà ", ñóùåñòâóåò íåãàìèëüòîíîâ ãðàô ñ æåñòêîñòüþ 9 4 ¡ " · ¿ < 9 4 . ïîäîáíî çàäà÷àì ïàíöèêëè÷íîñòè è ñóùåñòâîâàíèÿ ãèïîãàìèëüòîíîâûõ ãðàôîâ, ìû ðàññìàòðèâàåì áîëåå åìêóþ çàäà÷ó: ñóùåñòâóåò ëè íåãàìèëüòîíîâ ãðàô ñ çàäàííîé æåñòêîñòüþ. ïî íàøåé ãèïîòåçå, åñëè ñóùåñòâóåò íåãàìèëüòîíîâ t-æåñòêèé ãðàô, òî äëÿ ëþáîãî ðàöèîíàëüíîãî ÷èñëà a èç èíòåðâàëà ( 0 ; t], ñóùåñòâóåò íåãàìèëüòîíîâ ãðàô, æåñòêîñòü êîòîðîãî ðàâíà a. â äàííîé ðàáîòå ìû äîêàçûâàåì ýòó ãèïîòåçó äëÿ ëþáîãî ðàöèîíàëüíîãî ÷èñëà èç èíòåðâàëà ( 0 ; 9 4 ) . äëÿ ïîñòðîåíèÿ òðåáóåìûõ ãðàôîâ áûëè èñïîëüçîâàíû íåêîòîðûå íîâûå äîïîëíèòåëüíûå êîíñòðóêòèâíûå áëîêè. untitled an improved algorithm for generation of truncated normal distributed random numbers vahe v. sahakyan institute for informatics and automation problems of nas ra e-mail: vahe@spir.me abstract in this paper we discuss the computational problems of random numbers generation distributed by truncated normal distribution. it is shown that the standard methods and libraries have a limit for truncation point caused by the limit on the smallest number representable by double precision format. theoretically the problems arise starting from the truncation point ≈ 40, but in practical calculations the limit is lower, starting from ≈ 8.5. an improved method is represented, based on the combination of two approximation algorithms, which with the represented coefficients has 4.5 times more coverage interval than the standard methods. keywords: random numbers generation, truncated normal distribution, error function, inverse error function. 1. introduction the need of generation of numbers distributed by truncated normal distribution arises across many statistical calculations and modelling problems. the family of normal and truncated normal distributions is defined using the error function, and their calculation is strictly related to the calculation of the error function and the inverse error function. unfortunately, this functions are hard to compute numerically. particularly, in case of the error function difficulties are visible in tail regions: it convergences to 0 or 1 values very fast and the floating point representation of the resulting values lose their accuracy. obviously, the inverse case will convergence to +∞ or −∞ values very fast for arguments near 1 or 0. due to these issues the generation of numbers distributed by truncated normal distribution requires a different approach. in this paper we are going to address possible ways to overcome those difficulties and generate numbers distributed by such distribution. 2. definitions the probability density function (pdf) of standard normal distribution is defined over the interval (−∞, +∞). that function is usually denoted as φ(0, 1, x) and is represented by the following formula: 73 mathematical problems of computer science 42, 73–80, 2014. 74 an improved algorithm for generation of truncated normal distributied random numbers f0hxl f1hxl 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 f0hxl f1hxl 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 φ(0, 1, x) = 1√ 2π e− x 2 2 . in this article we will observe a distribution defined only on the positive range [0,∞), often called a half-normal standard distribution (n+). the normal distribution is symmetric about the origin, hence, it is enough to examine the half-normal case and reconstruct the normal distribution from it. the same is true for the truncated case. pdf of half-normal standard distribution on interval [0, +∞) is defined as follows: f(x) = { 2 φ(0, 1, x) x ≥ 0 0 x < 0 =    √ 2 π e− x 2 2 x ≥ 0 0 x < 0 . we will say, that a is distributed by half-normal standard distribution truncated by z (n+ z ), if a follows n+ and satisfies a ≥ z, where z ≥ 0. the pdf of this distribution will be: fz(x) =        √ 2 π e − x 2 2 erfc ( z √ 2 ) x > z 0 x ≤ z , (1) where erfc is a complementary error function, which is defined as erfc(x) = 2√ π ∫ ∞ x e−t 2 dt. the cumulative density function (cdf) of n+ z can be easily found by integrating (1) on the interval (−∞, x], and it will be fz(x) = ∫ x −∞ fz(t) dt =        1− erfc ( x √ 2 ) erfc ( z √ 2 ) x > z 0 x ≤ z . (2) fig. 1. pdf and cdf of half-normal standard distribution and truncated by 1 half-normal standard distributions. v. sahakyan 75 calculated expected 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 1 erfc−1(erfc(x)) using [5] for erfc−1 and [6] for erfc. 3. numbers generation there are two common approaches to generate numbers distributed by an arbitrary distribution. the first one is the von neumann’s rejection technique [10]. on truncated normal distribution this technique has very a low efficiency, because the probability that the generated number satisfies x > z, where x ∼ n+ is equal to f(z). in case, if z = 5 the f(5) ≈ 3 ∗ 10−6, so approximately 99.9997% of probes will be rejected. obviously, in applications this method is not acceptable. the second approach is the inverse transform sampling method [2]. it claims that if u is uniformly distributed on the interval (0, 1), then f −1 z (u) follows the distribution in which cdf is fz. we will take u = 1−u to make the equation simple, and in this case the inverse of (2) will be f −1 z (u) = √ 2 erfc−1 ( erfc ( z√ 2 ) u ) . (3) note, that during the inverse function calculation we are not considering the case when x < z, because in that case fz(x) = 0, therefore the probability of that is 0, so it goes beyond our interests. as we can see, computation of f −1 z requires computation of erfc and erfc−1. the argument range of erfc is [0,∞), therefore, the values are ranged in [0, 1]. the u is a random number from the range (0, 1), hence, the argument range for erfc−1 is [0, 1]. from a theoretical point of view the expression f −1 z is correct, however, from computational perspective it could imply numerical errors. for example, if z = 40, then erfc ( 40√ 2 ) ≈ 7.3 × 10−350, which is not representable in double precision format [3] and resulting in 0, therefore also erfc−1(0) = ∞. figure 2 shows the actual numerical calculation results. it is clearly visible, that starting from values near 6 the result is ∞, even though theoretically it is becoming not computable starting from values higher than 28.28. the break point corresponds to the value z = 6 √ 2 ≈ 8.49. conclusion from here is that independently from precision and accuracy of calculation, when erfc and erfc−1 are computed separately, starting from z ≥ 40 the final result will always be ∞ in double precision representation. obviously, their calculation should be done combined. fig. 2. this graphic shows calculated and expected values of 76 an improved algorithm for generation of truncated normal distributied random numbers 4. numerical approximations the usual approach to the computation of erfc, is to divide the axis into segments and use different polynomials and asymptotic approximations for those segments ([1], [6], [7], [8] and [9]). let us have a look at approach discussed in [1]. at first erfc can be approximated in the following form: erfc(x) = t exp ( −x2 + p(t) ) x > 0, (4) where t = 2 2 + x , and p(t) is polynomial (0 ≤ t ≤ 1) which is found by chebyshev polynomial approximation. the sample coefficients for 28 degree polynomial are shown in table 1. now let us have a look at the calculation of erfc−1. as suggested in [1], we apply the halley’s method on g(x) = erfc(x) − y and solve it for g(x) = 0. in that case the iteration variable will be xn+1 = xn − 2g(xn)g ′(xn) 2[g′(xn)] 2 −g(xn)g′′(xn) = xn + erfc(xn)−y 2√ π exp(−xn2)−xn(erfc(xn)−y) . here, the following form of approximation is used for the initial value of xn: erfc−1(x) ≈ r(v), 0 < x < 1, (5) where r(v) is a rational polynomial and v = √ −2 log ( x 2 ) . (6) r has the following form and the coefficients are found by rational polynomial approximation [1] r(v) = 0.70711v3 + 15.6585v2 + 11.5099v −36.4132 v2 + 22.1444v + 22.3164 . table 1. coefficients of polynomial p(t). degree column corresponds to degree of variable t in polynomial p . the represented form of both of these approximations are not the best ones from accuracy and performance perspective, but they have very simple and convenient form for further transformations. accuracy can also be improved by using higher degree polynomials. 5. improved algorithm as was shown in section 3, with standard approach and using standard libraries the computation of (3) is impossible for values more than 6 √ 2. here we will describe a combined method from two algorithms described in the previous section. taking into consideration the argument of erfc−1 in (3) and (4), (6) will get v. sahakyan 77 degree value degree value 0 −1.265512123484646 14 8.81583262585187∗101 1 9.99999999999995∗10−1 15 −2.449786109546412∗102 2 3.750000000013982∗10−1 16 5.592802242564082∗102 3 8.33333331763132∗10−2 17 −1.037357191852551∗103 4 −8.59374904138571∗10−2 18 1.541879449019999∗103 5 −1.437503619675783∗10−1 19 −1.824712058124052∗103 6 −9.17876834732462∗10−2 20 1.714187964988546∗103 7 2.940960653857621∗10−2 21 −1.272415091307403∗103 8 1.351072101958271∗10−1 22 7.388485218111491∗102 9 9.90519562132564∗10−2 23 −3.293364188005826∗102 10 1.566341639700399∗10−1 24 1.090336950910116∗102 11 −1.389503257698621 25 −2.530009680666059∗101 12 5.910981818651237 26 3.677189095748008 13 −2.552672826806604∗101 27 −2.522015791327478∗10−1 v = √ √ √ √ √−2 log   t u exp ( −z2 2 + p(t) ) 2   = √ z2 −2p(t)−2 log(t u) + log(4), where t = 4 4 + √ 2z . by this simple insertion we combine both computation algorithms and as a result exempt from exponential part in (4). based on this computation the algorithm will be defined as follows: function inversetruncatedcdf(z, u) y ← erfc(z)∗u t ← 4/(4 + sqrt(2)∗z) v ← sqrt(z ∗z −2∗p(t)−2∗ log(t∗u) + log(4)) x ← r(v) for i ← 1, 4 do a ← erfc(x)−y b ← 2/sqrt(π)∗ exp(−x∗x)−x∗a if b = 0 then break end if x ← x + a/b end for return x end function table 1: coefficients of polynomial p(t). degree column corresponds to degree of variable t in polynomial p . 78 an improved algorithm for generation of truncated normal distributied random numbers pdf histogram 9.8 10.0 10.2 10.4 10.6 10.8 0 2 4 6 8 10 pdf histogram 19.9 20.0 20.1 20.2 20.3 20.4 0 5 10 15 20 6 + 10 and n+20, respectively. the algorithm consists of maximum 4 loops of halley’s method. this amount was taken by applying a series of numerical experiments and the results show that the increase of loops doesn’t improve the result. drawback of this algorithm is that with the increase of z the initial x is also increasing. with coefficients represented in paper for z = 39 we will have x ≈ 28.2833 and the value of exp(−x ∗ x) ≈ 3.87 ∗ 10−348, which is already not representable in double precision format. because of that the halley’s method improvement steps are not possible, so accuracy is lost. nevertheless, with current coefficients it is covering range [0, 38] with maximum error smaller than 10−14. the result can be improved by taking better approximation for initial x, which means using a higher degree polynomial for r. inversetruncatedcdf can be used for random number generation following distribution n+ z . the following algorithm can be used for that purpose: function randtruncatednormal(z) u ← rand() ⊲ uniformly distributed random number from interval (0, 1) r ← inversetruncatedcdf(z, u) return √ 2∗ r end function in figure 3 the result of numbers generation are shown using the above algorithm. 6. conclusion and remarks the problems related to limits of double precision number representation are making impossible the calculation of such formulas in tail regions. the suggested way is one of the approaches that is improving the coverage range. another way would be the use of chebyshev multi-variate approximation, but as a rule a better coverage means higher degree of polynomial and consequently a slow computational time. the choice of algorithm is strictly dependent on the initial problem and here we represented a relatively light-weight method, also with a possibility to be improved by increasing degrees of approximation polynomials p and r. samples calculated by randtruncatednormal function for n fig. 3. probability density function (pdf) and histogram of 10 v. sahakyan 7 9 refer ences [1 ] w . h . p r e s s , s . te u ko ls ky, w . t. v e t t e r lin g , a n d b . p . fla n n e r y, numerical r ecipes: the art of scienti¯c computing, th ir d e d it io n , n e w y o r k, 2 0 0 7 . 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[on lin e ]. a va ila b le : h t t p :/ / www.n e t lib .o r g / c e p h e s [7 ] j. f. h a r t , e t a l., computer approximations, th e s ia m s e r ie s in a p p lie d ma t h e m a t ic s , n e w y o r k, 1 9 6 8 [8 ] w . j. co d y, \ r a t io n a l ch e b ys h e v a p p r o xim a t io n s fo r t h e e r r o r fu n c t io n " m ath. comp., vo l. 2 3 , p p . 6 3 1 -6 3 7 , 1 9 6 9 [9 ] gn u p r o je c t , gn u s c ie n t i¯ c l ib r a r y. [on lin e ]. a va ila b le : h t t p s :/ / www.g n u .o r g / s o ft wa r e / g s l/ [1 0 ] j. vo n n e u m a n n , \ v a r io u s t e c h n iqu e s u s e d in c o n n e c t io n wit h r a n d o m d ig it s . mo n t e ca r lo m e t h o d s " , nat. b ureau standards, vo l. 1 2 , p p . 3 6 -3 8 , 1 9 5 1 . submitted 05.09.2014, accepted 28.11.2014. îïñí³í ëï³ý¹³ñï ýáñù³é µ³ßëí³í ãí»ñç ·»ý»ñ³óù³ý µ³ñ»é³íí³í ³é·áñçãù ì. ê³ñ³ïû³ý ²ù÷á÷áõù ðá¹í³íáõùùýý³ñïí³í»ýïïñí³íëï³ý¹³ñïýáñù³éµ³ßëí³íãí»ñç·»ý»ñ³óù³ý ·áñíáýã³óáõù ³é³ç³óáõ ñ³ßíáõ³ï³ý ëý¹çñý»ñá: ²å³óáõûóí»é ¿, áñ ëï³ý¹³ñï »õ³ý³ïý»ñá ¨ ·ñ³¹³ñ³ýý»ñá áõý»ý ïïñù³ý ï»ïç áýïñáõãû³ý ë³ñù³ý³÷³ïáõù, áñá å³ûù³ý³íáñí³í ¿ ïñïý³ïç ×ßïáõãû³ùµ ãí»ñç ý»ñï³û³óù³ý ë³ñù³ý³÷³ïáõùý»ñáí: î»ë³ï³ýáñ»ý ëý¹çñý»ñý ³é³ç³ýáõù »ý ¼ 4 0 -çó ù»í ïïñù³ý ï»ïç ñ³ù³ñ, ë³ï³ûý ·áñíý³ï³ýáõù ³û¹ ë³ñù³ýá ß³ï ³í»éç ÷áùñ ¿` ¼ 8 :5 -çó ëïë³í: ²é³ç³ñïí³í ¿ ýáñ »õ³ý³ï, áñç ñçùùáõù »ñïáõ ùáï³ñïù³ý ³é·áñçãùý»ñç ùç³íáñáõùý ¿: îçñ³é»éáí ñá¹í³íáõùý»ñï³û³óí³í·áñí³ïçóý»ñá` ïïñù³ýï»ïçñ³ù³ñëï³óíáõù ¿ 4.5³ý·³ù ³í»éç ù»í ñ³ßí³ñï»éç ïçñáõûã, ù³ý ëï³ý¹³ñï »õ³ý³ïý»ñç ¹»åùáõù: 8 0 an improved algorithm for generation of truncated normal distributied random numbers óëó÷øåííûé àëãîðèòì ãåíåðàöèè ñëó÷àéíûõ ÷èñåë ñ óñå÷åííûì íîðìàëüíûì ðàñïðåäåëåíèåì â. ñààêÿí àííîòàöèÿ â ýòîé ðàáîòå ðàññìîòðåíû âû÷èñëèòåëüíûå çàäà÷è ãåíåðàöèè ñëó÷àéíûõ ÷èñåë ñ óñå÷åííûì íîðìàëüíûì ðàñïðåäåëåíèåì. â ðàáîòå ïîêàçàíî, ÷òî ñòàíäàðòíûå ìåòîäû è áèáëèîòåêè èìåþò îãðàíè÷åíèÿ òî÷êè óñå÷åíèÿ. ýòè îãðàíè÷åíèÿ îáóñëîâëåíû ëèìèòîì íà íàèìåíüøåå ÷èñëî, ïðåäñòàâèìîå ñ äâîéíîé òî÷íîñòüþ.òåîðåòè÷åñêè, ñëîæíîñòè âîçíèêàþò íà÷èíàÿ ñ óñå÷åíèÿ â òî÷êå ¼ 4 0 , íî â ïðàêòè÷åñêèõ ðàñ÷åòàõ ïðåäåë íàìíîãî íèæå, íà÷èíàÿ ñ ¼ 8 :5 . ïðåäñòàâëåí óñîâåðøåíñòâîâàííûé ìåòîä, ñîçäàííûé íà îñíîâå êîìáèíàöèè äâóõ àïïðîêñèìàöèîííûõ àëãîðèòìîâ. â ñðàâíåíèè ñî ñòàíäàðòíûìè ìåòîäàìè, ïðè ïîêàçàííûõ êîýôôèöèåíòàõ, äàííûé ìåòîä îáåñïå÷èâàåò â 4.5 ðàç áîëüøèé èíòåðâàë ïîêðûòèÿ. 7_vahe_sahakyan.pdf (p.1-6) 1_vahe.pdf (p.1-6) microsoft word s. shokurian_3.doc mathematical problems of computer science 34, 2010. 13 multidimensional tapes and multitape automata: recent results and beyond samvel shoukourian it educational and research center yerevan state university possible implications of three recent results in theory of automata and automata languages [1] are discussed. the first result is the theorem [2], which states that the equivalence problem of multidimensional multitape automata can be reduced to the equivalence problem of onedimensional multitape automata and, thus, is also solvable. as a corollary to this, we also get, that the functional equivalence problem in the class of program schemata on a nondegenerate basis of rank unity is solvable. although the equivalence problem of program schemata on a nondegenerate basis was investigated actively by many researchers the latter problem was open since early 60-ies [3]. introduction of a special binary coding for elements of a free partially commutative semigroup and its justification [4] leads to consideration of a multidimensional tape cells instead of semigroup elements and, therefore, to an opportunity of their comparison as integer vectors when analyzing behaviors of two automata on a given semigroup. specifically, it becomes possible to measure the distance between two automata during their movement on different representations of the same element of the semigroup. for the introduced coding the movement of automata on words containing more non-commutative symbols is “faster” than on words containing more commutative symbols: the distance between the initial cell and the reached cell is less in the latter case. coding brings to the second result a new combinatorial proof of the equivalence problem of deterministic finite multitape automata [5]. notions of an essential diagonal of a length i for a multidimensional tape and a set of states reached by an automaton at a given essential diagonal are introduced. it is shown that for if there are essential diagonals d1 and d2 with repeating sets of reached states and for each state and each cell achieved in the considered state in d1 then there is a path from that cell to a cell in d2 where the same state is reached. moreover, if automata are equivalent and representations of a given semigroup element that automata move on are different, then the distance between cells achieved in d1 by these automata and the distance between corresponding cells achieved in d2 coincide. in 1973 m. bird proved that the equivalence problem for two-tape automata is solvable [5]. in 1991 t. harju and j. karhumäki proved the solvability of the equivalence problem for multitape automata without any restriction on the number of tapes [6]. it is based on a purely algebraic technique. per t. harju “the main argument of the proof uses an embedding result for ordered groups into division rings”. the suggested purely combinatorial proof of solvability [7] is similar to the solution suggested by m. bird [5], but instead of a transformation of source automata to a commutative 14 diagram, the commutativity assumptions are taken into account via a special multidimensional tape [3][4] used for coding execution traces. the third result is the application of the developed technique for solving another automata related problem [8] open since early 70-ies [9, 10]: when is the problem of equivalence for regular expressions over a partially commutative alphabet solvable? the obtained results directly lead to consideration of the following open problems: 1. the introduced trace coding can be used for representation of sets of word tuples accepted by multitape automata. semantics of operations used in the obtained description, which we call an extended regular expression, should be adjusted to operations introduced within coding. basing on this new notation for regular expressions problems of analysis and synthesis for multitape automata can be considered similarly to corresponding theorems for ordinary automata. 2. minimization problem for multitape automata can be considered as well. instead of equivalence classes based on the length of words in the case of one-tape automata the equivalence classes here should be considered in connection with the length of essential diagonals of the multidimensional tape. references 1. m.o. rabin, d. scott, finite automata and their decision problems, ibm j. res. dev. 3 (2) (1959) 114–125. 2. h.a. grigorian, s.k. shoukourian, the equivalence problem of multidimensional multitape automata, j. comput. system sci. 74 (7) (208) 1131–1138. 3. m.s. paterson, equivalence problems in a model of computation, ph.d. thesis, cambridge university, 1967. 4. а.б.годлевский, а.а.летичевский, с.к.шукурян, о сводимости проблемы функциональной эквивалентности схем программ над невырожденным базисом ранга единица к эквивалентности автоматов с многомерными лентами, кибернетика и системный анализ, № 6, с. 1-7, 1980. 5. bird, m.: the equivalence problem for deterministic two-tape automata j. comput. system sci. 4 (3) (1973) 220–249. 6. harju t., karhumäki j., the equivalence problem of multitape finite automata, theoret. comput. sci. 78 (2) (1991) 347–355. 7. a.a.letichevsky, a.s.shoukourian, and s. k. shoukourian, the equivalence problem of deterministic multitape finite automata: a new proof of solvability using a multidimensional tape, proceedings of international conference “languages and automata: theory and application lata’2010, trier”, lecture notes in computer science 6031, pp. 392–402, 2010. 8. a. с. шукурян, эквивалентность регулярных выражений над частично коммутативным алфавитом, кибернетика и системный анализ, №3, стр. 3-11, 2009. 9. в. н. редько, об алгебре коммутативных событий, украинский математический журнал, 16, № 2, стр. 185-195, 1964. 10. в. а. тузов,проблемы разрешения для граф-схем с перестановочными операторами. ii, кибернетика, №5, стр. 28-32, 1971. microsoft word tpel.doc mathematical problems of computer science 33, 144--149, 2010. 144 construction of explicit irreducible polynomials over 2f in cluster computational environment ofelya manukyan and melsik kyuregyan institute for informatics and automation problems of nas of ra e-mail: manofa81@yahoo.com, melsik@ipia.sci.am abstract this paper describes a method for constructing families of explicit irreducible polynomials over 2f . the proposed method allows construction of explicit polynomials of higher degree over 2f from a given sequence of primitive polynomials. a computational algorithm has been developed and implemented on base of this method. the program is realized in the most effective way possible in cluster computational environment. allocation and distribution of memory resources have been implemented in a careful manner, since data size increases drastically with increasing of the amount of computations required. program paralleling is performed using data paralleling, i.e. data is distributing among all processors, which ran the same program and each of which builds the subsequent irreducible polynomial, and finally a sequence of all the irreducible polynomials in explicit form is obtained. moreover, the program also searches for the polynomial with the lowest possible weight among of all the polynomials of the same degree. references [1] m. kyuregyan, “recurrent methods for constructing irreducible polynomials over )2( sgf ”, finite fields and their applications 8, pp. 52-68, 2002. [2] r. lidl and h. niederreiter, finite fields , cambridge university press 1987. [3] a. j. menezes, i. f. blake, x. gao, r. c. mullin, s. a. vanstone and t. yaghoobian, “applications of finite fields”, kluwer academic publishers, boston, dordrecht, lancaster, 1993. o. manukyan, m. kyuregyan 145 2f í»ñç³íáñ ¹³ßïç íñ³ ³ýí»ñ³í»éç µ³½ù³ý¹³ùý»ñç µ³ó³ñ³ûï ï»ëùáí ï³éáõóù³ý »õ³ý³ï ïé³ëï»ñ³ûçý ñ³ßíáõ³ï³ý ñ³ù³ï³ñ·áõù ú. ø³ýáõïû³ý ¨ ø. îûáõñ»õû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ýï³ñ³·ñí³í ¿ 2f í»ñç³íáñ ¹³ßïç íñ³ ³ýí»ñ³í»éç µ³½ù³ý¹³ùý»ñç µ³ó³ñ³ûï ï»ëùáí ï³éáõóù³ý ùç »õ³ý³ï, áñá ñý³ñ³íáñçýë ¿ý»ïïçíáñ»ý ùß³ïí»é ¨ çñ³ï³ý³óí»é ¿ ïé³ëï»ñ³ûçý ñ³ßíáõ³ï³ý ñ³ù³ï³ñ·áõù: ðçßáõáõãû³ý é»ëáõñëý»ñç µ³ßëáõùá` ñçßáõáõãû³ý ñ³ïï³óáõùý áõ ³½³ïáõùá ï³ï³ñí»é »ý ëý³ûáõ³µ³ñ, ù³ýç áñ ñ³ßí³ñïý»ñç áýã³óùáõù ïíû³éý»ñç »ñï³ñáõãûáõýý»ñá ß³ï ³ñ³· ³×áõù »ý: îé³ëï»ñ³ûçý ñ³ßíáõ³ï³ý ñ³ù³ï³ñ·áõù íñ³·ñç ½áõ·³ñ»é³óáõùá ï³ï³ñí»é ¿ áëï ïíû³éý»ñç, ³ûëçýùý ïíû³éý»ñá µ³ßëíáõù »ý åñáó»ëáñý»ñç ùçç¨, µáéáñ åñáó»ëáñý»ñá ³ßë³ïáõù »ý ùç¨ýáõûý íñ³·ñáí ¨ ûáõñ³ù³ýãûáõñá ï³éáõóáõù ¿ ñ»ñã³ï³ý ³ýí»ñ³í»éç µ³½ù³ý¹³ùá: ²ñ¹ûáõýùáõù ëï³ýáõù »ýù ³ýí»ñ³í»éç µ³½ù³ý¹³ùý»ñç ñ³çáñ¹³ï³ýáõãûáõý ¨ µ³óç ³û¹ ÷ýïñáõù »ýù ùç¨ýáõûý ³ëïç×³ýç ñý³ñ³íáñçýë ÷áùñ ïßçé áõý»óáõ ³ýí»ñ³í»éç µ³½ù³ý¹³ùá: начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 42, 121—126, 2014. 121 benchmarking of gpu nvidia cuda, cublas and magma libraries based on matrix multiplication problem edita e. gichunts institute for informatics and automation problems of nas ra e-mail: editagich@ipia.sci.am abstract solving linear systems of equations is a fundamental problem in scientific computing. many scientific computer applications need a high-performance matrix algebra. the major hardware developments always influenced the new developments in linear algebra libraries. nowadays major chip manufacturers are developing next-generation microprocessor designs that integrate multicore cpu and gpu components [1]. the main aim is to benchmark cublas and magma libraries on matrix multiplication problem using the tesla c1060 graphical processing unit. keywords: parallel computing, matrix algebra, graphical processor. 1. introduction today multi-core processor computers have become wide-spread. the software-hardware architecture cuda which emerged with the advent of nvidia g80 chip allows performing of calculations using nvidia graphics processors. nvidia tesla is nvidia's brand name for their products targeting stream processing and/or general purpose gpu. products utilize gpus from the g80 series onward. with 240 processor cores and a standard c compiler that simplifies application development, tesla scales to solve the world’s most important computing challenges-more quickly and accurately. the technology cuda which presents the nvidia software–hardware computational architecture is based on the extension of the c programming language. it provides an opportunity of using its memory for parallel computations. cuda assists functioning of algorithms on gpu graphics processors. cuda provides an effective transformation of data from the system to video-memory [2]. the project magma is directed to the development of a library for the linear algebra tasks. it is similar to the lapack library, but it is targeted to the hybrid architecture, starting from multicore processors and gpu systems. the investigation of magma is based on the idea that the benchmarking of gpu nvidia cuda, cublas and magma libraries 122 solution of complex tasks in the hybrid environment has a potential to derive optimal programming solutions. proceeding from this idea we apply the library to implementing the linear algebra algorithms on multi-core and graphics processor systems [3]. these days there are numerous works devoted to the investigation of parallel systems performance either on the clustering system or multi-core processor base. the problem elaborated in this article is significantly important for producing an effective performance of optimal parallelization for the linear algebra tasks for the case of contemporary gpu nvidia capacities, envisioned for simple methods of programming as well as cublas(cuda basic linear algebra subroutines) and magma software packages and multiplication of matrices with one point numbers . 2. the problem statement and main results the efficiency of the matrix multiplication is elaborated. the calculations were implemented for the three cases. only the graphics processor gpu (tesla c1060, 1296.0 mhz clock, 4095.8 mb memory, capability 1.3) with involving the cuda technology was used in the first case. the cublaslibrary of cuda was used in the second case and the magmalibrary in the third one. the computational system is controlled with the ubuntu 12.04.1 lts operational system. the nvidia cuda 4 with its libraries is loaded, the atlas library is installed, gcc-4.4, nvcc, gfortran-4.4 compilers are used. the magma 1.3 library is installed. lapack library is required to install before compilation of magma. 2.1 the first case every program written on the nvidia cuda irrespective its significance can be divided into a number of general steps. the gpu does not possess an access to the operative memory, hence, the programmer has to care about all the sources necessary for the kernel work which should be placed in the memory of the video-card. for this task three main functions, namely cudamalloc, cudamemcpy and cudafree from cuda sdk , are applied. it is worth to mention also that the function cudamemcpy has an additional parameter, which denotes the direction in which the information loading (from cpu to gpu or the contrary) takes place. the process of configuration is easy and consists in indicating the size of the net and blocks. the main problem on the next step is the selection of the optimal balance between the number of blocks and their size. the nvidia suggests using blocks with 128 or 256 flows. the kernel is called like a conventional c function in the programming language. the only real difference is the requirement to indicate the predetermined sizes for the net and blocks during the call of the kernel. after completing the kernel implementation with applying the cudamemcpy function it is necessary to copy back the results to operative memory with indicating the reverse direction of copying. in the way similar to the case of the memory flow interruption for any other c program it is necessary to release all the provided resources. the functions provided by the kernel are written in a file with extension cu, which is compiled during the application of the program nvcc. in the result of compilation the program code is divided into two parts: the first one supposed for performing on cpu, the second containing codes ptx objects for the gpu. to define the data patters necessary for treatment the variables blockdim ¨ blockidx which contain the size of the block and an index for the corresponding kernel in action are used. e. gichunts 123 the general way of a task solution on cuda consists in the following: 1. obtaining data for calculations. 2. duplicate the data in gpu memory. 3. perform calculations by means of gpu kernel function. 4. duplicate the calculated data in the memory of cpu from the gpu. 5. observe the results. 6. release the resources. 2.2 the second and third cases the cublas library is an implementation of the blas (a basic linear algebra subprograms) on top of the nvidia cuda. it provides an input opportunity to computational resources of the graphics processor nvidia. the library is of the corresponding api level i.e., there is no direct interaction with the driver cuda. cublas is connected to one gpu and it is not connected in parallel with several gpu. the package magma r is a new collection of matrix class objects that implements procedures with matrix and linear operations. operations are performed using the library magma with the c sub-programs, which are optimized for parallel processing on the graphic card and multikernel processors. (http://icl.cs.utk.edu/magma/). it presents a gpu for the matrix algebra and multi-kernel architecture. the general form of the task solution through magma and cublas libraries consists in the following:  formation of matrices and vectors in the memory of gpu,  allocating the data in the memory domain,  sequential call of the cublas functions,  loading the results from the domain of gpu memory to the cpu. cublas and magma contain auxiliary functions designed for the creation and deletion of objects, as well as for data registration and information extraction in the memory of gpu. the plot on fig. 1 shows the dynamics of change of time of implementation in dependence of the volume of input data n. examining the results we came to the conclusion that depending on data volume either cublas or magma library could have higher productivity. it is worth to mention that magma is more effective in case of large data, such as, e.g., 1 gigabyte, but cublas more frequently wins while working with data of 100 megabyte order. however, the productivity of gputesla c1060 which uses global memory is many times less in comparison with magma and cublas. two time scales were considered in the process of the performance estimating, namely the absolute time of computation and the data input/output time. for both magma and cublas libraries no increase in speed of the input and output was observed, resulting in that the increase in speed was observed only during operation performing, and since the computational time depends on the cube of the matrix sizes while the volume of input/output is a function of the matrix size squares. benchmarking of gpu nvidia cuda, cublas and magma libraries 124 transferred matrices should be placed in gpu memory completely, which leads to the restriction of the total size of the two matrices with 4 gb. fig. 1. the dynamics of change of time of implementation in dependence of the volume of input data n. 16000*16000*8 bytes make 2 gb, since two matrices are entered the bound for the data size is 4 gb. thus, gpu tesla c1060 works with two matrices of maximum size 16000*16000. 3. related work at present there are numerous studies on the productivity of parallel systems, both on the basis of cluster systems, and multi-core cpu and gpu nvidia. but there hasn’t been carried out any detailed study of the capabilities of modern gpu nvidia applying a standard way of writing parallel programs, using the global memory gpu and also program libraries cublas and magma for solving matrix multiplication with single-precision numbers. 4. conclusion several libraries have been installed during this study. three cases are investigated. the productivity of graphics processor for the task of non-symmetric matrices is investigated. to attain the maximal efficiency with the gpu nvidia cuda it is necessary to use the libraries magma and cublas which in cases under investigation provide acceleration to 1618 times more than in case of global memory using. -50 0 50 100 150 200 250 300 350 400 450 0 2000 4000 6000 8000 10000 12000 14000 16000 t im e ( se c o n d s) n gpu cublas magma e. gichunts 125 references [1] l. seiler, d. carmean, e. sprangle, t. forsyth, m. abrash, p. dubey, s. junkins, a. lake, j. sugerman, r. cavin, r. espasa, e. grochowski, t. juan and p. hanrahan, “larrabee: a many-core 86 architecture for visual computing”, acm trans. graph., vol. 27, no. 3, pp. 1–15, 2008. [2] j. nickolls, i. buck, m. garland and k. skadron,“scalable parallel programming with cuda”, presentation by christian hansen article published in acm queue, march, page 2, 2008. [3] s.tomov, r.nath, h. ltaief and j. dongarra,“dense linear algebra solvers for multicore with gpu accelerators”, proc. of ipdps'10, atlanta, ga, january 15, pp. 1—2. 2010. submitted 08.09.2011, accepted 26.11.2014. gpu nvidia cuda–ի, cublas և magma գրադարանների արտադրողականության գնահատականը մատրիցի բազմապատկման խնդրի լուծման ժամանակ է. գիչունց ամփոփում գծային հավասարումների համակարգերի լուծումը հիմնարար խնդիր է գիտական հաշվարկներում: բարձր արտադրողականության մատրիցային հանրահաշիվը պահանջվում է շատ գիտական կոմպյուտերային կիրառություններում: հիմնական տեխնիկական զարգացումները միշտ ազդել են գծային հանրահաշվի գրադարանների նոր զարգացումների ստեղծման վրա: ներկայումս խոշոր միկրո-չիպային արտադրողներ մշակում են նոր սերնդի չիպեր, որոնք ընդգրկում են բազմամիջուկային cpu և gpu բաղադրիչները [1]: աշխատանքի գլխավոր նպատակն է cublas և magma գրադարանների թեստավորումը մատրիցի բազմապատկման խնդրի համար` tesla c1060 գրաֆիակական պրոցեսորի օգտագործմամբ: benchmarking of gpu nvidia cuda, cublas and magma libraries 126 оценка производительности gpunvidiacuda, библиотек cublas и magma при решении задач матричного умножения э. гичунц аннотация решение систем линейных уравнений является фундаментальной проблемой в научных вычислениях. высоко-производительная матричная алгебра требуется во многих научных компьютерных приложениях. основные аппаратные разработки всегда влияли на создание новых разработок библиотек линейной алгебры. в настоящее время главные производители микросхем разрабатывают схемы нового поколения, которые интегрируют компоненты с много-ядерными cpu и gpu [1]. главная цель работы-это тестирование библиотек cublas и magma для задачи умножения матриц с использованием графического процессора teslac1060. 404 not found d:\sbornik\...\tpel\tpel.dvi mathematical problems of computer science 32, 27{34, 2009. t he stable set n umber for the str ong p r oduct of gener alized cycles s e va k h . b a d a lya n a n d s t e p a n e . ma r ko s ya n yerevan state university sevak badalyan@yahoo.com abstract the strong product of an odd cycle and a generalized cycle and the strong product of two generalized cycles are investigated. for both cases a method is given to construct a stable set of vertices in product graph to achieve the known upper bound ®(g£h) · ½(g)£®(h) in case some conditions hold. for the stable set number of strong product of generalized cycles a lower bound is found. refer ences [1 ] c. e . s h a n n o n , \ th e z e r o -e r r o r c a p a c it y o f a n o is y c h a n n e l" , tr a n s . 1 9 5 6 s ym p . in fo r m a t io n th e o r y, in s t . r a d io e n g . it-2 , 8 -1 9 . 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[8 ] c. b e r g e , graphs and hypergraphs, n o r t h -h o lla n d , a m s t e r d a m , 1 9 7 3 . 2 7 2 8 the stable set number for the strong product of generalized cycles àôå»õ ³ñï³¹ñû³éç ³ýï³ëáõãû³ý ãçíá áý¹ñ³ýñ³óí³í óçïé»ñç ñ³ù³ñ ê. ´³¹³éû³ý, ê. ø³ñïáëû³ý ²ù÷á÷áõù êáõûý ³ßë³ï³ýùáõù áõëáõùý³ëçñíáõù ¿ ·ñ³ýý»ñç áõå»õ ³ñï³¹ñû³éç ³ýï³ëáõãû³ý ãíç ï³åá ³ñï³¹ñçãý»ñç ³ýï³ëáõãû³ý ãí»ñç ñ»ï, »ñµ ³ñï³¹ñçãý»ñçó ù»ïá ï³ù »ñïáõëý ¿é áý¹ñ³ýñ³óí³í óçïé »ý: ¶ïýí³í ¿ ýßí³í ³ýï³ëáõãû³ý ãçíá áñáß å³ûù³ýý»ñç ¹»åùáõù ¨ ïñí³í ¿ ù»ãá¹ ñ³ù³å³ï³ëë³ý ³ýï³ë µ³½ùáõãûáõýá ï³éáõó»éáõ ñ³ù³ñ: àý¹ñ³ýñ³óí³í óçïé»ñç áõå»õ ³ñï³¹ñû³éç ³ýï³ëáõãû³ý ãíç ñ³ù³ñ ·ïýí³í ¿ ëïáñçý ·ý³ñ³ï³ï³ý: 404 not found d:\sbornik\...\naira.dvi mathematical problems of computer science 33, 102{110, 2010. e r r or p r obability e xponents of m ultiple h ypotheses t esting i llustr ations n a ir a m. gr ig o r ya n state engineering university of armenia nar.gri@gmail.com abstract the paper presents an application of multiple hypothesis testing for two markov sources with virtual example in terms of text categorization problem. some numerical experiments concerning markov sources are considered. our goal is to present numerical illustrations of interdependencies of error probability exponents as a supplementary to our previous theoretical paper [9]. refer ences [1 ] l . b ir g ¶ e , \ v it e s s m a xim a ls d e d ¶ c r o is s a n c e d e s e r r e u r s e t t e s t s o p t im a u x a s s o c ie s " , z. w ahrsch. verw. gebiete, vo l. 5 5 , p p . 2 6 1 ¡ 1 7 3 , 1 9 8 1 . [2 ] r . f. a h ls we d e a n d e . a . h a r o u t u n ia n , \ on lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f h yp o t h e s e s a n d id e n t ī c a t io n " , l ecture notes in computer science, vo l. 4 1 2 3 ,\ ge n e r a lth e o r y o f in fo r m a t io n tr a n s fe r a n d co m b in a t o r ic s " , s p r in g e r , p p . 4 6 2 { 4 7 8 , 2 0 0 6 . [3 ] s . n a t a r a ja n , \ l a r g e d e via t io n s , h yp o t h e s e s t e s t in g , a n d s o u r c e c o d in g fo r ¯ n it e ma r ko v c h a in s " , ie e e trans. inform. theory, vo l 3 1 , n o . 3 , p p . 3 6 0 -3 6 5 , 1 9 8 5 . [4 ] e . a . h a r o u t u n ia n , \ on a s ym p t o t ic a lly o p t im a l c r it e r ia fo r ma r ko v c h a in s " , ( in r u s s ia n ) , f irst w orld congress of b ernoulli society, s e c t io n 2 , vo l. 2 , n o . 3 , p p . 1 5 3 -1 5 6 , 1 9 8 9 . [5 ] e . a . h a r o u t u n ia n , \ a s ym p t o t ic a lly o p t im a l t e s t in g o f m a n y s t a t is t ic a l h yp o t h e s e s c o n c e r n in g ma r ko v c h a in " , ( in r u s s ia n ) , 5-th intern. vilnius conferance on p robability theory and m athem. statistics, vo l. 1 ( a -l ) , p p . 2 0 2 -2 0 3 , 1 9 8 9 . [6 ] m. gu t m a n , \ a s ym p t o t ic a lly o p t im a l c la s s ī c a t io n fo r m u lt ip le t e s t s wit h e m p ir ic a lly o b s e r ve d s t a t is t ic s " , ie e e tr a n s . in fo r m . th e o r y, vo l 3 5 , n o 2 , ma r c h , 4 0 1 -4 0 8 , 1 9 8 9 . [7 ] e . a . h a r o u t u n ia n , \ l o g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f m u lt ip le s t a t is t ic a l h yp o t h e s e s " , p roblems of control and information theory, vo l. 1 9 , n o . 5 -6 , p p . 4 1 3 ¡ 4 2 1 , 1 9 9 0 . [8 ] e . a . h a r o u t u n ia n a n d p . m. h a ko b ya n , \ on lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l h yp o t h e s e s t e s t in g o f t h r e e d is t r ib u t io n s fo r p a ir o f o b je c t s " , m athematical p roblems of computer science, vo l. 2 4 , p p . 7 6 { 8 1 , 2 0 0 5 . [9 ] e . a . h a r o u t u n ia n a n d n . m. gr ig o r ya n , \ r e lia b ilit y a p p r o a c h fo r t e s t in g o f m a n y d is t r ib u t io n s fo r p a ir o f ma r ko v c h a in s " , m athematical p roblems of computer science, vo l. 2 8 , p p . 1 0 9 ¡1 1 6 , 2 0 0 7 . 1 0 2 n. grigoryan 1 0 3 [1 0 ] e . a . h a r o u t u n ia n , m. e . h a r o u t u n ia n , a n d a . n . h a r u t yu n ya n , \ r e lia b ilit y cr it e r ia in in fo r m a t io n th e o r y a n d in s t a t is t ic a l h yp o t h e s e s te s t in g " , f oundation and trends in comunications and information theory, vo l. 4 , n o . 2 ¡ 3 , p p . 9 7 ¡ 2 6 3 , 2 0 0 8 . [1 1 ] i. cs is z ¶a r a n d j. k äo r n e r , information theory: coding theorems for discrete memoryless systems, a c a d e m ic p r e s s , n e w y o r k, 1 9 8 1 . [1 2 ] c. e . s h a n n o n , \ a ma t h e m a t ic a l t h e o r y o f c o m m u n ic a t io n " , b e ll s ys t e m te c h n ic a l jo u r n a l , vo l. 2 7 , p p . 3 7 9 ¡ 4 2 3 , 6 2 3 ¡ 6 5 6 , ju ly, oc t o b e r , 1 9 4 8 [1 3 ] p . f. b r o wn , v . j. d e lla p ie t r a , p . v . d e s o u z a , j. c. l a i, r . l . me r c e r , \ cla s s -b a s e d n -g r a m m o d e ls o f n a t u r a l la n g u a g e " , computational l inguistics, vo l. 1 8 ( 4 ) , p p . 4 6 7 ¡ 4 7 9 , 1 9 9 2 . [1 4 ] l . d . b a ke r a n d a . k . mc ca llu m , \ d is t r ib u t io n a l c lu s t e r in g o f wo r d s fo r t e xt c la s s ī c a t io n " , p roceedings of the 21st annual international acm sigir conference on r esearch and development in information retrieval , p p . 9 6 ¡1 0 3 , 1 9 9 8 . [1 5 ] j. l . e lm a n , \ fin d in g s t r u c t u r e in t im e " , cognitive science, vo l. 1 4 , 1 7 9 ¡2 1 1 , 1 9 9 0 . ´³½ù³ïç í³ñï³íý»ñç ëïáõ·ù³ý ëë³éý»ñç ñ³í³ý³ï³ýáõãûáõýý»ñç óáõóçãý»ñç éáõë³µ³ýáõùá ûñçý³ïý»ñáí ü.¶ñç·áñû³ý ²ù÷á÷áõù àõëáõùý³ëçñí»é ¿ µ³½ù³ïç í³ñï³íý»ñç ëïáõ·ù³ý ëý¹çñá »ñïáõ ù³ñïáíû³ý ³õµûáõñý»ñçó µ³õï³ó³í ñ³õáñ¹³ïóáõãû³ý ñ³ù³ï³ñ·ç ¹»åùáõù ¨ ýßí»é »ý áñáß ïçñ³éáõãûáõýý»ñ: ø³ëý³ïç ûñçý³ïç ¹»åùáõù ï³ï³ñí»é »ý ãí³ûçý ñ³ßí³ñïý»ñ ¨ ý»ñï³û³óí»é ñáõë³éçáõãû³ý ù³ïñçóç ï³ññ»ñç ·ñ³ýçï³ï³ý å³ïï»ñý»ñ: 404 not found 54 mathematical problems of computer science 55, 54--61, 2021 udc 004.8 designing and implementing a method of data augmentation using machine learning aren k. mayilyan national polytechnic university of armenia e-mail: mayilyan96@gmail.com abstract efficiency of neural network (nn) models depend on the parameters given and the input data. due to the complexity of environmental conditions and limitations the data for nn models, especially for the case of images, can be insufficient. to overcome this problem data augmentation has been used to enlarge the dataset. the task is to generate diverse set of images from a small set of images for nn training. due to data augmentation transformation, 3105 new images out of 345 input data were created for classification, detection and image segmentation. keywords: machine learning, convolutional neural networks, data augmentation, burn degrees. 1. introduction nowadays image classification problems are mainly solved via convolutional neural networks (cnn). deep learning cnn needs a huge number of images for the model to be trained effectively. if the issue of data scarcity is faced, the simple, yet effective techniques such as transformations may pose a limited solution [1, 2]. data augmentation techniques in data analysis are used to generate slightly modified copies or create synthetic data from a real dataset and hence artificially increase its size. cnn is known to be invariant, meaning that it can robustly classify objects with different transformations. hence, the increase of relevant data in the dataset will result in a better accuracy for the cnn model. besides adding more images to our dataset, the data augmentation tools are beneficial for having images in a limitless set of conditions. in real-world scenarios pictures can be taken in different orientation, location, scale, brightness, etc. therefore, various transformations of the same image such as rotation and cropping, can enhance the utility of the training set and increase the performance of ml algorithms. there are two options for data augmentation: mailto:mayilyan96@gmail.com a. mayilyan 55  online augmentation or augmentation on the fly is applied in real time. this method performs transformations on the mini-batches and then fits into the model. this means that the online augmentation will see different images at each epoch. this kind of expansion is preferred for larger datasets, otherwise there would be an explosive increase in size.  offline version transforms each image in the training set by rotating, cropping, etc. as a result, the size of the training dataset increases by a factor equal to the number of transformations performed on the image. offline data augmentation is preferred for relatively smaller datasets as the goal is having more images to train for the model [3, 4]. 2. description of the dataset the collected dataset consists of around 400 images of human skin burns. the pictures of the burns are taken from different angles mainly in a monotone background. the dataset is labeled into three classes according to their burn degrees, examples of which are presented in fig. 1. fig.1. some examples of images from the dataset before designing the model, the dataset is divided into two sets: training and test. on the first one the model is trained, while the second one is used to determine the accuracy of the designed model. in our case 345 out of 400 images form the training dataset. as the dataset was initially labeled into 3 different classes, the training and test sets should consist of items of three classes with each class containing nearly equal number of images. in our case we have approximately 120 images in each class for the training set and 30 images in each class for the test set (fig. 2). designing and implementing a method of data augmentation using machine learning 56 fig.2. original train and test sets according to three types of burn degrees 3. description of the method images are represented as arrays in our data. colored images are a mixture of red, blue and green (rgb). the pixels of the images are tiny blocks of information arranged in the form of a 2d grid, and the depth of a pixel is the color information. data augmentation basically shifts or transforms these pixels to get a new image. in our data we have resized all the images to be of the shape of (150, 150, 3), where the first two numbers show the size of rows and columns of the matrix that form the 2d grid, and the third number indicating the rgb coloring. if we separate the images into three (150, 150, 1)-shaped matrices, we will get three separate red, blue and green colored pictures (fig.3.) fig.3. rgb representation of a picture. here are five basic and powerful augmentation techniques that we used to increase our dataset [5]. a. mayilyan 57 table 1: 5 data augmentation techniques used in the program, their application and results data augmentation technique methods used in the program result of the technique flip np.fliplr(image) np.flipud(image) flipping images horizontally and vertically rotate rotate(image, angle = 45) rotate(image, angle = -45) rotating images by a random angle chosen by the program crop function defined by the user to crop the main part of the image. randomly sampling a section from the original image, then resizing the original image size. this process is known as random cropping brightness adjustment adjust_gamma(image, gamma = 0.5, gain = 1) adjust_gamma(image, gamma = 2, gain = 1) changing the original image’s darkness/brightness noise random_noise(image) adding noise to the image, hence having blurred image with slightly different pixel values the flip and rotate transformations are alike. particularly, flipping the image horizontally or vertically is the same thing as rotating the image by 90 or 180 degrees. in order to rotate our images around its center by specified number of degrees, we take the rgb values at every 2d location, rotate it as needed, and then write these values in the new location. thus, having the location coordinates x and y, we apply the transformation matrix and get the new location for the same rgb value. the calculation is done with multiplication of the old coordinates with the transformation matrix, as shown in formula 1. [ 𝑥∗ 𝑦 ∗ ]=[ 𝑐𝑜𝑠θ 𝑠𝑖𝑛θ −𝑠𝑖𝑛θ 𝑐𝑜𝑠θ ] [ 𝑥 𝑦 ], where θ is the angle by which we want to rotate, x* and y* are the new coordinates, and x and y are the old ones. in our case we have used 90, 180, 45 and -45 degrees. after having the images flipped horizontally and vertically, rotation of the images by 45 and -45 degrees is the most effective, because two other lesser degrees would result in the same original image, e. g. 10 degrees, and, rotating by a large degree will be very close, as to flipping horizontally or vertically. hence, rotating by 45 degrees, which is the mean value of 0 and 90 degrees, is the best choice for our images. after these 2 techniques we generate four rotated images, which we append to our training set. cropping could be done by randomly choosing a smaller rectangle on our image and cutting out that part. however, for the crop technique the essential condition is having the burn image fully displayed on our image even after cropping it. in order to reach this goal, in our code we first divide the y axis of our image into three equal parts, having the low, middle and high parts. then we choose a random number that would belong to the low part, and another random number from the high part. as a result, we have two points on y axis. same goes for the x axis. by having 2 designing and implementing a method of data augmentation using machine learning 58 points on the x axis and 2 on the y axis, we then form the rectangle that includes the middle part of the image and which will be cropped out. by doing so we generate one image from each original one and add to the training set. brightness adjustment means adjustment of the rgb value. in our technique we have multiplied each of r, g, b values by two constants. in the first case by 0.5, and as 0.5 is less than 1, then it made our images darker. in the second case we multiplied by 2, which made our images brighter. as a result of this technique, we generated two more images. the last thing we have applied to our images is adding some blur. most common are box blur and gaussian noise. for the box blur we took a kernel, which is a 5x5 matrix, rolled on the image, and took the average color of pixels inside that matrix. as for the gaussian noise, we add random variables from standard normal distribution to our rgb values. based on these two techniques we get two new images, therefore, overall nine images from each image [10][11]. with application of our data augmentation techniques on each image (fig. 4) we generated 3105 new images and, hence, together with the original data a new training set consisting of 3450 images [6][7] was produced. in order to be able to fit into the cnn model, each image should preserve the size of the original dataset, which is 3-dimensional array of (150, 150, 3)-shape. however, in comparison with the original dataset, not all the newly created ones have the necessary shape. it occurs as a result of transforming the arrays. cnn models should have inputs of the same size, hence, it is crucial to bring all the data into the same shape before fitting to the model. the reshaping was done individually for each image after their transformation. after changing the sizes of the images, the information isn’t lost: it is just demonstrated via different shaped arrays [9]. for the transformations such as flip, crop, brightness adjustment, gaussian noise, etc., the images will retain all the information. for some cases, such as for the rotations, the image does not have any information about things outside its boundary. as we see in fig.4, the picture in the 2nd row and 1st column has black fillings outside its boundary. in such cases we need to make some assumptions. there are different ways of doing so: constant filling the unknown region with some constant value edge the edge values are extended after the boundary reflect the image pixel values are reflected along the image boundary symmetric at the boundary of reflection, a copy of the edge pixels is made wrap the image is repeated beyond its boundary. as our images are mainly taken in a monochromatic background, the space beyond the image’s boundary is assumed to be the constant 0 at every point and is displayed with black color [10]. after the application of all the above-mentioned techniques and bringing all the images to the same size, for each image we get nine copies with a slight difference. the augmented images are demonstrated in fig.4. there are other types of data augmentation such as generative adversarial networks (gan). it is questionable why nn data augmentation is not preferred while using cnn model. the answer lies in the data itself. the images that we have are not easily distinguishable even by the human eye. for example, if we were to classify buildings from forests, then with gan we could generate the same picture of the building in summer, autumn, spring, winter and, hence, have four more pictures in this case. in our case by generating new images using gan, we can turn an image with 2nd degree burn into a 3rd degree burn image and decrease the accuracy of the cnn model. a. mayilyan 59 fig. 4. application of the data augmentation techniques on an image. after the transformations we have two training datasets. the first one is the original one containing 345 unique images, and the second one is the augmented, containing overall 3450 images. in order to understand whether data augmentation is beneficial for the application of the cnn model, we need to plug the original training set of 345 to the model, get the accuracy and then plug in the second augmented training set of 3450 and get the accuracy. the result will be seen by comparing the two accuracies. in our case we applied convolutional neural networks and got 59% accuracy out of original dataset and 75% accuracy out of the augmented dataset. 4. conclusion data augmentation gives the opportunity to add more training data into the model, prevent data scarcity for better models, reduce data overfitting, create variability in data and resolve class imbalance issues in classification. for classification of human skin burns, the newly generated images should not lose much information from the original image. in comparison with our techniques, the other data augmentation tool generative adversarial network, generates synthetic copies of an image, basically mixtures of images. in our case it is not expedient, as the difference between classes is very slight, and even a small mixture of images will result in bad input data. besides all these advantages the transformation of the images reduces costs of collecting and labeling data. to fully take advantage of the data augmentation, the below mentioned steps were taken: observe data to understand which augmentation tools are better to use; implement and apply those tools on each image to produce 3105 new items (from 345 original images) adding to our training set; designing and implementing a method of data augmentation using machine learning 60 create two training datasets. the first one is the original dataset consisting of 345 images and the second is the augmented dataset, containing 3450 images (from which 345 were the original images and 3105 newly generated); fit both training sets into a cnn and get the corresponding models; test the two models on the test set and get the accuracies for each. thus, as a result of data augmentation the training set increased from 345 items to 3450 and the cnn accuracy from 59% to 75%. references [1] a. kwasigroch, a. mikolajczyk and m. grochowski, “deep convolutional neural as a decision support tool in medical problems-malignant melanoma case study”, trends in advanced intelligent control, optimization and automation, advances in intelligent systems and computing, springer, cham, vol 577, pp. 848-856, 2017. [2] z. chen, z. gaochen, r. gao, k. mao, p. wang, r. yan and zhao, deep learning and its applications to machine health monitoring: a survey. corr, abs/1612.07640, 2016. [3] m. frid-adar, e. klang, m. amitai, j. goldberger and h. greenspan, “synthetic data augmentation using gan for improved liver lesion classification”, arxivprepr. arxiv180102385, 2018. [4] t. a. rutkowski and f. prokopiuk, “identification of the contamination source location in the drinking water distribution system based on the neural network classifier”, the 10th ifac symposium on fault detection, supervision and safety of technical processes august 29-31, warsaw, poland, 2018. [5] a. haydarornek and m. ceylan, “comparison of traditional transformations for data augmentation in deep learning of medical thermography”, telecommunications and signal processing (tsp) 2019 42nd international conference, pp. 191-194, 2019. [6] s.-l. wannipa, w. wettayaprasit and p. aiyarak, “convolutional neural networks using mobilenet for skin lesion classification", computer science and software engineering (jcsse) 2019 16th international joint conference on, pp. 242-247, 2019. [7] b. jahić, nicolas and g. benoîtries, “software engineering for dataset augmentation using generative adversarial networks”, software engineering and service science (icsess) 2019 ieee 10th international conference, pp. 59-66, 2019. [8] i. laina, ch. rupprecht, v. belagiannis, f. tombari and n. navab, “deeper depth prediction with fully convolutional residual networks”, corr, abs/1606.00373, 2016. [9] i. goodfellow, y. bengio and a. courville, deep learning, mit press, 2016. [10] (04 oct 2021) [online]. available: https://en.wikipedia.org/wiki/box_blur [11] (04 oct 2021) [online]. available: https://en.wikipedia.org/wiki/gaussian_blur submitted 10.02.2021, accepted 20.04.2021 https://en.wikipedia.org/wiki/box_blur https://en.wikipedia.org/wiki/gaussian_blur a. mayilyan 61 մեքենայական ուսուցմամբ տվյալների բազայի ընդլայնման մեթոդի մշակումը և կիրառումը արեն կ․ մայիլյան հայաստանի ազգային պոլիտեխնիկական համալսարան e-mail: mayilyan96@gmail.com ամփոփում նեյրոնային ցանցի մոդելների էֆֆեկտիվությունը պայմանավորված է տրված պարամետրերով և մուտքային տվյալներով: արտաքին գործոններով պայմանավորված դժվարություններից և սահմափակումներից ելնելով՝ նեյրոնային ցանցի մոդելների համար հավաքագրված տվյալները սովորաբար բավարար չեն հատկապես պատկերների դեպքում: այս խոչընդոտը հաղթահարելու նպատակով օգտագործվում են մեքենայական ուսուցման մեխանիզմներ, որոնք թույլ են տալիս ընդլայնելու տվյալների նախնական բազան: վերջինս ներկայացնող մուտքային պատկերի վերափոխումամբ ստեղծվում է պատկերների բազմազան հավաքածու, հնարավորություն տալով արդյունավետորեն լուծելու դասակարգման, նույնականացման և սեգմենտավորման/մասնատման խնդիրներ: բանալի բառեր` մեքենայական ուսուցում, կոնվոլյուցիոն նեյրոնային ցանցեր, տվյալների ընդլայնում, այրվածքի աստիճաններ: разработка и применение метода расширения базы данных с помощью машинного обучения арен к. маилян национальный политехнический университет армении e-mail: mayilyan96@gmail.com аннотация эффективность моделей нейронных сетей (nn) зависит от заданных параметров и входных данных. из-за сложности и ограничений, обусловленных внешними факторами, данных для моделей nn, довольно часто бывает недостаточно, особенно в случае изображений. чтобы решить эту проблему, обычно используется расширение данных для увеличения их набора. преобразование входных данных создает разнообразный набор изображений из небольшого набора изображений для классификации, обнаружения или сегментации изображения. ключевые слова: машинное обучение, сверточная нейронная сеть, нейронные сети, расширение данных, степени ожога. mailto:mayilyan96@gmail.com mailto:mayilyan96@gmail.com d:\final_47\...\nikoghosyan.dvi mathematical problems of computer science 47, 30{36, 2017. on longest cycles in 2-connected gr aphs mo s s in e s . k o u la kz ia n a n d zh o r a g. n iko g h o s ya n institute for informatics and automation problems of nas ra e-mail: mossine@hotmail.com, zhora@ipia.sci.am abstract for a graph g, n denotes the order (the number of vertices) of g, c the order of a longest cycle in g (called circumference), p the order of a longest path and ± the minimum degree. in 1952, dirac proved: (i) if g is a 2-connected graph, then c ¸ minfn; 2±g. the bound 2± in (i) was enlarged independently by bondy (1971), bermond (1976) and linial (1976) in terms of ¾2 the minimum degree sum of two nonadjacent vertices: (ii) if g is a 2-connected graph, then c ¸ minfn; ¾2g. in this paper two further extensions of (i) and (ii) are presented by incorporating p and the length of a vine on a longest path of g as new parameters along with n, ± and ¾2. keywords: hamilton cycle, dominating cycle, longest cycle, longest path, minimum degree, degree sums. 1 . in t r o d u c t io n w e c o n s id e r o n ly ¯ n it e u n d ir e c t e d g r a p h s wit h n e it h e r lo o p s n o r m u lt ip le e d g e s . a g o o d r e fe r e n c e fo r a n y u n d e ¯ n e d t e r m s is [2 ]. th e s e t o f ve r t ic e s o f a g r a p h g is d e n o t e d b y v ( g) a n d t h e s e t o f e d g e s b y e ( g) . l e t n b e t h e o r d e r ( t h e n u m b e r o f ve r t ic e s ) o f g, c t h e o r d e r o f a lo n g e s t c yc le ( c a lle d c ir c u m fe r e n c e ) in g a n d p t h e o r d e r o f a lo n g e s t p a t h . th e m in im u m d e g r e e s u m o f t wo n o n a d ja c e n t ve r t ic e s in g is d e n o t e d b y ¾2. in p a r t ic u la r , t h e m in im u m d e g r e e ¾1 is d e n o t e d b y ±. w e u s e n ( v ) t o d e n o t e t h e s e t o f a ll n e ig h b o r s o f ve r t e x v a n d d( v ) = jn ( v ) j t o d e n o t e t h e d e g r e e o f ve r t e x v. a g r a p h g is h a m ilt o n ia n if g c o n t a in s a h a m ilt o n c yc le , t h a t is a s im p le s p a n n in g c yc le . a c yc le c o f g is c a lle d a d o m in a t in g c yc le if e ve r y e d g e o f g h a s a t le a s t o n e o f it s e n d ve r t ic e s o n c, o r , e qu iva le n t ly, if g ¡ v ( c ) c o n t a in s n o e d g e s . th e e a r lie s t n o n t r ivia l lo we r b o u n d fo r t h e c ir c u m fe r e n c e wa s o b t a in e d in 1 9 5 2 d u e t o d ir a c [4 ] in t e r m s o f ± a n d n. t heor em a: [4 ]. in every 2-connected graph, c ¸ m in fn; 2 ±g. th e b o u n d 2 ± in th e o r e m a wa s e n la r g e d in d e p e n d e n t ly b y b o n d y [1 ], b e r m o n d [3 ] a n d l in ia l [5 ] in t e r m s o f ¾2. t heor em b : [1 ],[3 ],[5 ]. in every 2-connected graph, c ¸ m in fn; ¾2g. in t h is p a p e r t wo fu r t h e r e xt e n s io n s o f t h e s e r e s u lt s a r e p r e s e n t e d b y in c o r p o r a t in g p a n d t h e le n g t h o f a vin e o n a lo n g e s t p a t h o f g in c o r r e s p o n d in g b o u n d s a s n e w p a r a m e t e r s a lo n g wit h n, ± a n d ¾2. th e vin e 's d e ¯ n it io n n e e d s s o m e a d d it io n a l n o t a t io n . if q is a p a t h o r a c yc le , t h e n t h e le n g t h o f q, d e n o t e d b y l ( q) , is je ( q ) j t h e n u m b e r o f e d g e s in q. w e wr it e a c yc le q wit h a g ive n o r ie n t a t io n b y ¡! q . fo r x; y 2 v ( q ) , we d e n o t e 3 0 m. koulakzian and zh. nikoghosyan 3 1 b y x ¡! q y t h e s u b p a t h o f q in t h e c h o s e n d ir e c t io n fr o m x t o y. fo r x 2 v ( q) , we d e n o t e t h e s u c c e s s o r a n d t h e p r e d e c e s s o r o f x o n ¡! q ( if s u c h ve r t ic e s e xis t ) b y x+ a n d x¡, r e s p e c t ive ly. w e u s e p = x ¡! p y t o d e n o t e a p a t h wit h e n d ve r t ic e s x a n d y in t h e d ir e c t io n fr o m x t o y. w e s a y t h a t ve r t e x z1 p r e c e d e s ve r t e x z2 o n a p a t h ¡! q if z1, z2 o c c u r o n ¡! q in t h is o r d e r , a n d in d ic a t e t h is r e la t io n s h ip b y z1 á z2. w e will wr it e z1 ¹ z2 wh e n e it h e r z1 = z2 o r z1 á z2. l e t p = x ¡! p y b e a p a t h . a vin e o f le n g t h m o n p is a s e t fli = xi ¡! l iyi : 1 · i · mg o f in t e r n a lly-d is jo in t p a t h s s u c h t h a t ( a ) v ( li ) \ v ( p ) = fxi; yig ( i = 1 ; :::; m) , ( b ) x = x1 á x2 á y1 ¹ x3 á y2 ¹ x4 á ::: ¹ xm á ym¡1 á ym = y o n p . th e fo llo win g r e s u lt g u a r a n t e e s t h e e xis t e n c e o f a t le a s t o n e vin e in a 2 -c o n n e c t e d g r a p h . lemma: (t he vine lemma) [4 ]. l et g be a k-connected graph and p a path in g. then there are k ¡ 1 pairwise-disjoint vines on p . in t h e p a p e r , we o b t a in a lo we r b o u n d fo r t h e c ir c u m fe r e n c e in t e r m s o f n, ¾2 a n d t h e le n g t h m o f a vin e o n a lo n g e s t p a t h o f g. t heor em 1: l et g be a 2-connected graph and fl1; l2; :::; lmg be a vine on a longest path of g. then c ¸ m in fn; ¾2 + m ¡ 2 g: th e m in im u m d e g r e e ve r s io n o f th e o r e m 1 fo llo ws im m e d ia t e ly. cor ollar y 1: l et g be a 2-connected graph and fl1; l2; :::; lmg be a vine on a longest path of g. then c ¸ m in fn; 2 ± + m ¡ 2 g: if m = 1 in th e o r e m 1 , t h e n c le a r ly g is h a m ilt o n ia n . th e r e fo r e , th e o r e m 1 is a n e xt e n s io n o f th e o r e m s a a n d b b y in c o r p o r a t in g p a r a m e t e r m a lo n g wit h n a n d ¾2. n e xt , we o b t a in a lo we r b o u n d fo r t h e c ir c u m fe r e n c e c in t e r m s o f ¾2 a n d p. t heor em 2: l et g be a 2-connected graph. then c ¸ 8 >>>>>>< >>>>>>: p wh e n p · ¾2; p ¡ 1 wh e n ¾2 + 1 · p · ¾3 ¡ 2 ; q 2 p ¡ 1 0 + 1 4 ( ¾2 ¡ 7 ) 2 + 12 ( ¾2 + 1 ) wh e n p ¸ ¾3 ¡ 1 : th e o r e m 2 c a n b e c o n s id e r e d a s a n o t h e r e xt e n s io n o f th e o r e m s a a n d b . in d e e d , if p · ¾2, t h e n b y th e o r e m 2 , c ¸ p, im p lyin g t h a t c = p = n ¸ m in fn; ¾2g. n e xt , if ¾2 + 1 · p · ¾3 ¡ 2 , t h e n b y th e o r e m 2 , c ¸ p ¡ 1 ¸ ¾2 ¸ m in fn; ¾2g. fin a lly, if p ¸ ¾3 ¡ 1 , t h e n o b s e r vin g t h a t 2 ¾3 ¸ 3 ¾2, we g e t s 2 p ¡ 1 0 + 1 4 ( ¾2 ¡ 7 ) 2 ¸ s 2 ( ¾3 ¡ 1 ) ¡ 1 0 + 1 4 ( ¾2 ¡ 7 ) 2 = 1 2 ( ¾2 ¡ 1 ) ; a n d b y th e o r e m 2 , c ¸ ¾2 ¸ m in fn; ¾2g. 3 2 on longest cycles in 2-connected graphs th e m in im u m d e g r e e ve r s io n o f th e o r e m 2 fo llo ws im m e d ia t e ly. cor ollar y 2: l et g be a 2-connected graph. then c ¸ 8 >>>>>>< >>>>>>: p wh e n p · 2 ±; p ¡ 1 wh e n 2 ± + 1 · p · 3 ± ¡ 2 ; q 2 p ¡ 1 0 + ( ± ¡ 7 2 ) 2 + ± + 1 2 wh e n p ¸ 3 ± ¡ 1 : th e s p e c ia l c a s e s c ¸ p a n d c ¸ p ¡ 1 in th e o r e m 2 c a n b e in t e r p r e t e d in t e r m s o f h a m ilt o n a n d d o m in a t in g c yc le s b y t h e fo llo win g t wo p r o p o s it io n s . p r oposition 1: [6 ]. a connected graph is hamiltonian if and only if c = p. p r oposition 2: [6 ]. l et g be a connected graph with c ¸ p ¡ 1 . then every longest cycle in g is a dominating cycle. to s h o w t h a t t h e b o u n d s in co r o lla r y 2 ( a s we ll a s in th e o r e m 2 ) a r e s h a r p , o b s e r ve ¯ r s t t h a t in g e n e r a l, p ¸ c, t h a t is c = p wh e n p · 2 ±, im p lyin g t h a t t h e b o u n d c ¸ p in co r o lla r y 2 c a n n o t b e r e p la c e d b y c ¸ p + 1 . on t h e o t h e r h a n d , t h e g r a p h k±;±+1 wit h p = 2 ± + 1 a n d c = 2 ± = p ¡ 1 s h o ws t h a t t h e c o n d it io n p · 2 ± c a n n o t b e r e la xe d t o p · 2 ± + 1 . in a d d it io n , t h e g r a p h k±;±+1 wit h c = p s h o ws t h a t t h e b o u n d c ¸ p ¡ 1 ( wh e n 2 ± + 1 · p · 3 ± ¡ 2 ) c a n n o t b e r e p la c e d b y c ¸ p. fu r t h e r , t h e g r a p h k2 + 3 k±¡1 wit h n = p = 3 ± ¡ 1 a n d c = 2 ± · p ¡ 2 s h o ws t h a t t h e c o n d it io n p · 3 ± ¡ 2 c a n n o t b e r e la xe d t o p · 3 ± ¡ 1 . fin a lly, t h e s a m e g r a p h k2 + 3 k±¡1 wit h p = 3 ± ¡ 1 a n d c = 2 ± = s 2 p ¡ 1 0 + µ ± ¡ 7 2 ¶ + ± + 1 ; 2 s h o ws t h a t t h e b o u n d q 2 p ¡ 1 0 + ( ± ¡ 7 2 ) + ± + 1 2 in co r o lla r y 2 c a n n o t b e im p r o ve d t o q 2 p ¡ 1 0 + ( ± ¡ 7 2 ) + ± + 1 . th e fo llo win g t h e o r e m will b e u s e fu l. t heor em c: [6 ]. l et g be a 2-connected graph. then either ( i ) c ¸ p ¡ 1 or ( ii) c ¸ ¾3 ¡ 3 or ( iii ) · = 2 and p ¸ ¾3 ¡ 1 . 2 . p r e lim in a r ie s th e fo llo win g le m m a c a n b e p r o ve d b y s t a n d a r d a r g u m e n t s ( c a lle d d ir a c a n d or e a r g u m e n t s ) . lemma 1: l et g be a connected graph and p = x ¡! p y a longest path in g. ( i ) if xz; yz¡ 2 e ( g ) for some z 2 v ( x+¡!p y ) , then c = p = n, that is g is hamiltonian. ( ii) if d( x ) + d( y ) ¸ p, then c = p = n. ( iii ) l et z1; z2 2 v ( p ) and z1 á z2. if xz; yz 62 e ( g ) for each z 2 v ( z+1 ¡! p z¡2 ) , then either c = p or p ¸ d( x ) + d( y ) ¡ 2 + jz1 ¡! p z2j. th e n e xt le m m a is c r u c ia l fo r t h e p r o o f o f th e o r e m s 1 a n d 2 . lemma 2: l et g be a 2-connected graph and fl1; l2; :::; lmg be a vine on a longest path of g. then c ¸ 2 p ¡ 1 0 m + 1 + 4 : m. koulakzian and zh. nikoghosyan 3 3 3 . p r o o fs p r oof of lemma 2: l et p = x ¡! p y be a longest path in g. p ut li = xi ¡! l iyi ( i = 1 ; :::; m) ; a1 = x1 ¡! p x2; am = ym¡1 ¡! p ym; ai = yi¡1 ¡! p xi+1 ( i = 2 ; 3 ; :::; m ¡ 1 ) ; bi = xi+1 ¡! p yi ( i = 1 ; :::; m ¡ 1 ) ; jaij ¡ 1 = ai ( i = 1 ; :::; m ) ; jbij ¡ 1 = bi ( i = 1 ; :::; m ¡ 1 ) : b y c o m b in in g a p p r o p r ia t e li; ai; bi, we fo r m m + 1 d i®e r e n t c yc le s t o o b t a in a lo we r b o u n d fo r t h e c ir c u m fe r e n c e a s t h e m e a n o f t h e ir o r d e r s . q1 = m[ i=1 ai [ m[ i=1 li; q2 = m¡1[ i=1 ai [ bm¡1 [ m¡1[ i=1 li; q3 = m[ i=2 ai [ b1 [ m[ i=2 li; ri = bi [ ai+1 [ bi+1 [ li+1 ( i = 1 ; :::; m ¡ 2 ) : s in c e jlij ¸ 2 ( i = 1 ; :::; m ) , we h a ve c ¸ jq1j = mx i=1 ai + mx i=1 ( jlij ¡ 1 ) ¸ mx i=1 ai + m; c ¸ jq2j = bm¡1 + m¡1x i=1 ai + m¡1x i=1 ( jlij ¡ 1 ) ¸ bm¡1 + m¡1x i=1 ai + m ¡ 1 ; c ¸ jq3j = b1 + mx i=2 ai + mx i=2 ( jlij ¡ 1 ) ¸ b1 + mx i=2 ai + m ¡ 1 ; c ¸ jrij = bi + ai+1 + bi+1 + jli+1j ¡ 1 ¸ bi + ai+1 + bi+1 + 1 ( i = 1 ; :::; m ¡ 2 ) : b y s u m m in g , we g e t ( m + 1 ) c ¸ ( 2 mx i=1 ai + 2 m¡1x i=1 bi ) + 2 m¡1x i=2 ai + 4 m ¡ 4 ¸ 2 ( mx i=1 ai + m¡1x i=1 bi + 1 ) + 4 m ¡ 6 = 2 p + 4 m ¡ 6 ; im p lyin g t h a t c ¸ 2 p ¡ 1 0 m + 1 + 4 : l e m m a 2 is p r o ve d . 3 4 on longest cycles in 2-connected graphs p r oof of t heor em 1: if m = 1 , t h e n xy 2 e ( g) a n d b y l e m m a 1 ( i ) , c = p. l e t m ¸ 2 . p u t li = xi ¡! l iyi ( i = 1 ; :::; m ) a n d le t ai; bi; ai; bi; qi b e a s d e ¯ n e d in t h e p r o o f o f l e m m a 2 . w e c h o o s e l1; l2; :::; lm s o a s t o m in im iz e m a s we ll a s b1 a n d bm¡1. case 1: m = 2 . it fo llo ws t h a t n ( x ) [ n ( y ) µ v ( a1 [ a2 ) . b y l e m m a 1 ( iii ) , e it h e r c = p o r p = a1 + a2 + b1 + 1 ¸ d( x ) + d( y ) ¡ 1 + b1, im p lyin g t h a t c ¸ jq1j = a1 + a2 + 2 ¸ d( x ) + d ( y ) = d( x ) + d( y ) + m ¡ 2 : case 2: m = 3 . l e t xz1; yz2 2 e ( g) fo r s o m e z1; z2 2 v ( p ) . if z2 á z1, t h e n fxz1; yz2g is a vin e c o n s is t in g o f t wo p a t h s ( e d g e s ) a n d we c a n a r g u e a s in ca s e 1 . b y t h e c h o ic e o f l1; l2; l3, n ( x ) µ v ( a1 [ a2 ) ; n ( y ) µ v ( a2 [ a3 ) a n d z1 ¹ z2 fo r e a c h z1 2 n ( x ) a n d z2 2 n ( y ) . th e r e fo r e , a1 + a2 + a3 ¸ d ( x) + d ( x) ¡ 2 a n d c ¸ jq1j = a1 + a2 + a3 + 3 ¸ d( x ) + d( x ) + 1 = d( x ) + d( x ) + m ¡ 2 : case 3: m ¸ 4 . b y t h e c h o ic e o f l1; l2; :::; lm, n ( x ) µ v ( a1 [ a2 ) ; n ( y ) µ v ( am¡1 [ am ) a n d z1 á z2 fo r e a c h z1 2 n ( x ) a n d z2 2 n ( y ) . ob s e r vin g a ls o t h a t a1 + a2 ¸ d( x ) ¡ 1 ; am¡1 + am ¸ d( y ) ¡ 1 ; we g e t c ¸ jq1j = mx i=1 ai + m = ( a1 + a2 + am¡1 + am ) + m¡2x i=3 ai + m ¸ d ( x ) + d( y ) ¡ 2 + m¡2x i=3 ai + m ¸ d ( x) + d ( y ) + m ¡ 2 : th e o r e m 1 is p r o ve d . p r oof of t heor em 2: l e t p = x ¡! p y b e a lo n g e s t p a t h in g. case 1: p · ¾2. if xy 2 e ( g) , t h e n b y l e m m a 1 ( i) , c = p. l e t xy 62 e ( g) . th e n d ( x) + d( y ) ¸ ¾2 ¸ p a n d b y l e m m a 1 ( ii) , a g a in c = p. case 2: ¾2 + 1 · p · ¾3 ¡ 2 . if c ¸ ¾3 ¡ 3 , t h e n b y t h e h yp o t h e s is , c ¸ p ¡ 1 . n e xt , if · = 2 a n d p ¸ ¾3 ¡ 1 , t h e n p ¸ ¾3 ¡ 1 ¸ p + 1 , a c o n t r a d ic t io n . h e n c e , b y th e o r e m c, c ¸ p ¡ 1 . m. koulakzian and zh. nikoghosyan 3 5 case 3: p ¸ ¾3 ¡ 1 . s in c e g is 2 -c o n n e c t e d , t h e n b y t h e v in e l e m m a , t h e r e is a vin e fl1; :::; lmg o n p . b y th e o r e m 1 , m · c ¡ d( x ) ¡ d( y ) + 2 · c ¡ ¾2 + 2 . u s in g l e m m a 2 , we g e t c ¸ 2 p ¡ 1 0 m + 1 + 4 ¸ 2 p ¡ 1 0 c ¡ ¾2 + 3 + 4 ; im p lyin g t h a t c ¸ s 2 p ¡ 1 0 + 1 4 ( ¾2 ¡ 7 ) 2 + 1 2 ( ¾2 + 1 ) : th e o r e m 2 is p r o ve d . refer ences [1 ] j.a . b o n d y, \ l a r g e c yc le s in g r a p h s " , d iscrete m ath., vo l. 1 , p p . 1 2 1 -1 3 1 , 1 9 7 1 . [2 ] j. a . b o n d y a n d u .s .r . mu r t y, graph theory with applications, ma c m illa n , l o n d o n a n d e ls e vie r , n e w y o r k, 1 9 7 6 . [3 ] j. c. b e r m o n d , \ on h a m ilt o n ia n wa lks " , congr numer, vo l. 1 5 , p p . 4 1 -5 1 , 1 9 7 6 . [4 ] g. a . d ir a c , \ s o m e t h e o r e m s o n a b s t r a c t g r a p h s " , p roc. l ondon, m ath. soc., vo l. 2 , p p . 6 9 -8 1 , 1 9 5 2 . [5 ] n . l in ia l, \ a lo we r b o u n d o n t h e c ir c u m fe r e n c e o f a g r a p h " , d iscrete m ath., vo l. 1 5 , p p . 2 9 7 -3 0 0 , 1 9 7 6 . [6 ] k . oz e ki a n d t. y a m a s h it a , \ l e n g t h o f lo n g e s t c yc le s in a g r a p h wh o s e r e la t ive le n g t h is a t le a s t t wo " , graphs and combin., vo l. 2 8 , p p . 8 5 9 -8 6 8 , 2 0 1 2 . submitted 22.10.2016, accepted 17.01.2017. 2-ï³å³ïóí³í ·ñ³ýý»ñç ³ù»ý³»ñï³ñ óçïé»ñç ù³ëçý ø. øáõé³ù½û³ý ¨ ä. üçïáõáëû³ý ²ù÷á÷áõù ¸çóáõù n-á ýß³ý³ïáõù ¿ g ·ñ³ýç ·³·³ãý»ñç ù³ý³ïá, c-ý g-ç` ³ù»ý³»ñï³ñ óçïéç »ñï³ñáõãûáõýá, p-ý` ³ù»ý³»ñï³ñ ßõã³ûç ·³·³ãý»ñç ù³ý³ïá ¨ ±-ý` ·ñ³ýç ýí³½³·áõûý ³ëïç×³ýá: 1952-çý ¸çñ³ïá ³å³óáõó»ó, áñ (i) »ã» g-ý 2-ï³å³ïóí³í ·ñ³ý ¿, ³å³ c ¸ m in fn; 2 ±g: ¸çñ³ïç 2 ± ý»ñùçý ·ý³ñ³ï³ï³ýá çñ³ñçó ³ýï³ë áý¹é³ûý»óçý ´áý¹çý (1971), ´»ñùáý¹á ¨ èçýç³éá (1976), û·ï³·áñí»éáí ¾2 (áã ñ³ñ¨³ý »ñïáõ ·³·³ãý»ñç ³ëïç×³ýý»ñç ýí³½³·áõûý ·áõù³ñá) å³ñ³ù»ïñá, (ii) »ã» g-ý 2-ï³å³ïóí³í ·ñ³ý ¿, ³å³ c ¸ m in fn; ¾2g: ü»ñï³ ³ßë³ï³ýùáõù (i) ¨ (ii) áý¹é³ûýáõùý»ñá ³í»éç »ý áý¹é³ûýíáõù` ·ý³ñ³ï³ï³ýý»ñç ù»ç ý»ñùáõí»éáí p-ý ¨ g ·ñ³ýç ³ù»ý³»ñï³ñ ßõã³ûç µ³õ»õç »ñï³ñáõãûáõýá áñå»ë ýáñ å³ñ³ù»ïñ»ñ n, ±, ¾2 å³ñ³ù»ïñ»ñç ïáõùçý: 3 6 on longest cycles in 2-connected graphs î äëèííåéøèõ öèêëàõ 2-ñâÿçíûõ ãðàôîâ ì. êóëàêçÿí è æ. íèêîãîñÿàí àííîòàöèÿ ïóñòü n, c, p è ± îáîçíà÷àþò ÷èñëî âåðøèí ãðàôà g, äëèíà äëèííåéøåãî öèêëà, ÷èñëî âåðøèí äëèííåéøåé öåïè è ìèíèìàëüíàÿ ñòåïåíü ãðàôà. â 1952 ãîäó äèðàê äîêàçàë, ÷òî (i) åñëè g ÿâëÿåòñÿ 2-ñâÿçíûì ãðàôîì, òî c ¸ m in fn; 2 ±g: ýòó îöåíêó íåçàâèñèìî ðàñøèðèëè áîíäè (1971), áåðìîíä è ëèíèàë (1976) ñ ïîìîùüþ ïàðàìåòðà ¾2 (ìèíèìàëüíàÿ ñóììà ñòåïåíåé äâóõ íå ñîñåäíèõ âåðøèí): (ii) åñëè g ÿâëÿåòñÿ 2-ñâÿçíûì ãðàôîì, òî c ¸ m in fn; ¾2g. â íàñòîÿùåé ðàáîòå ïðåäñòàâëåíû äâå íîâûå ðàñøèðåíèÿ îöåíîê (i) è (ii) ïîìîùüþ ïàðàìåòðîâ p è äëèíû ïëþùà äëèííåéøåé öåïè ãðàôà g íà ðÿäó ñ ïàðàìåòðàìè n, ±, ¾2. microsoft word tigransargsyanpogosyan_25.doc mathematical problems of computer science 33, 196--200, 2010. 200 developing game model for migration process of static timing analysis tools tigran sargsyan state engerineering university of armenia e-mail: sartig17@yahoo.com abstract the aim of the work is to find an adequate game model for static timing analysis (sta) migration process and develop method for evaluating the efficiency of actions on each step of migration. for comparing timing models generated by sta tool an automated comparison utility was created which should be used for finding the best strategy in each cycle of migration process references 1. j. bhasker and r. chadha «static timing analysis for nanometer designs» janet shapiro «strategic planning toolkit» http://www.civicus.org 2. e. pogossian, adaptation of combinatorial algorithms, (in russian), yerevan., pp. 293, 1983. 3. e. pogossian: “combinatorial game models for security systems”, nato arw on "security and embedded systems", porto rio, patras, greece, aug., pp. 8-18, 2005. 4. e. pogossian, “on measures of performance of functions of human mind”, 6th international conference in computer science and information technologies, csit2007, pp. 149-154, yerevan, 2007. 5. manual page for diff : http://ss64.com/bash/diff.html 6. example of sta tool: http://www.synopsys.com/tools/implementation/signoff/pages/nanotime.aspx 7. information on liberty format: http://www.opensourceliberty.org/ http://www.synopsys.com/community/interoperability/pages/libertylibmodel.aspx 8. lectures on sta: http://www.csee.umbc.edu/~cpatel2/links/641/slides/lect05_lib.pdf êï³ïçï å³ù³ý³ï³ûçý վ»ñéáõíáõãû³ý ·áñíçùý»ñç ùç·ñ³óç³ûç åñáó»ëç ë³õ³ûçý ùá¹»éç ùß³ïáõù տ. սարգսյան ²ù÷á÷áõù ²ûë ³ßë³ï³ýùç ýå³ï³ïý»ñý »ý`êï³ïçï ä³ù³ý³ï³ûçý ì»ñéáõíáõãû³ý ·áñíçùý»ñç ùç·ñ³óç³ûç åñáó»ëç ³¹»ïí³ï ùá¹»éç ëï»õíáõùá ¨ ùç·ñ³óç³ûç ûáõñ³ù³ýãûáõñ ù³ûéáõù ó»éý³ñïí³í ·áñíáõáõãûáõýý»ñç ³ñ¹ûáõý³í»ïáõãû³ý ·ý³ñ³ïù³ý ù»ãá¹ç ùß³ïáõùá: êï»õíí»é ¿ å³ù³ý³ï³ûçý ùá¹»éý»ñç(liberty ýáñù³ïáí) ñ³ù»ù³ïáõãû³ý íñ³·ñ³ûçý ·áñíçù, áñý û·ï³·áñíí»éáõ ¿ ùç·ñ³óç³ûç ûáõñ³ù³ýãûáõñ ÷áõéáõù ññ³ù³ýý»ñç ×ßï·ñïù³ý é³í³·áõûý ëïñ³ï»·ç³ûç ÷ýïñù³ý ñ³ù³ñ: microsoft word 5.doc математические вопросы кибернетики и вычислительной техники 38, 13--14, 2012. 13 об асимптотических оценках числа решений систем булевых уравнений с зависимыми переменными э. в. егиазарян кафедра дискретной математики и теоретической информатики егу известно, что многие вопросы функционирования дискретных управляющих систем сводятся к решению систем булевых уравнений, либо к определению числа их решений. таковы, например, некоторые задачи синтеза цифровых автоматов, нахождения внутренне и внешне устойчивых множеств в графе, вопросы теории тестов и др. естественно предполагать, что множество решений системы будет сужаться с ростом числа уравнений и при достаточно большом количестве последних будет пустым. в работах [1,2 ] обосновывается это предположение для “типичного” случая, а также приводятся оценки числа решений “почти всех” систем из l неэквивалентных уравнений как с n независимыми, так и с m зависимыми ( и n независимыми ) переменными. в настоящей работе исследуется класс уравнений с “ неявными” m зависимыми переменными. приведем необходимые определения. пусть  1)( rrm -такое семейство множеств, что | )(rm при r ( m обозначает число элементов множества m ), а )(rm e -подмножество тех элементов из )(rm , которые обладают заданным свойством e . говорят, что почти все элементы множества )(rm обладают свойством e , если 1)(/)( rmrm e при n (“типичный” случай). всюду под log понимается логарифм по основанию 2. обозначим через lns , множество всех систем из l уравнений вида   ,...,1 1,,1      l xxf i ni  где    lixxf ni ,...,1,,,1  попарно отличающиеся булевые функции от переменных nxxx ,...,, 21 пусть  1,0b ,  nibb inn  1,),,...,,(~/~ 21  . набор n ni b ),.....,,( ~ 21  называется решением системы (1) , если ,...,1 1),.....,,( 21      li nif  сопоставим каждой системе s множества s ln , целочисленный параметр )(s , равный числу ее решений. справедлива следующая теорема 1 (см [1,2]). 1 . е с л и   nn  , то для почти всех систем s множества s ln , имеет место lns 2~)( . об асимптотических оценках числа решений систем булевых уравнений с зависимыми переменными 14 2. е с л и   nn  , то почти все системы s множества s ln , не имеют решений. 3. е с л и n ограничено при ln, ,то для почти всех систем s множества s ln , число решений ограничено сверху произвольной функцией )(n , удовлетворяющей условию )(n при n . обозначим через mlns ,, множество всех систем из l уравнений вида , где  xxy njj ,,1  ( mj ,1 ) –неизвестные булевые функции ,зависящие от переменных nxxx ,...,, 21 ; ff ji  , при ji  . набор булевых функций     xxxx nmn ,,,,,, 111   называется решением системы (4), если при подстановке в (4) вместо переменных y j функций  xx nj ,,1   mj ,1 все уравнения обращаются в тождества. сопоставим каждой системе s множества mlns ,, целочисленный параметр )(st , равный числу ее решений. справедлива следующая теорема 2 (см [2]). 1 . е с л и   mnnlm , , то для почти всех систем s множества mlns ,, имеет место nlmst 2)(2~)(  . 2. е с л и 1]2ln)21([log  lnlm для достаточно больших m и n , то почти все системы s множества mlns ,, имеют хотя бы одно решение. 3. е с л и 0]2ln)21([log  lnlm для достаточно больших m и n , то почти все системы s множества mlns ,, не имеют решений. рассмотрим теперь множество mlns ,,/ всех систем уравнений вида   ,...,1 1,,1      li yyf mi  , ff ji  при ji  . относительно числа решений )(st систем из mlns ,,/ справедлива следующая теорема 3. 1 . е с л и   nmnm ,)log(  ,то для почти всех систем множества mlns ,,' имеет место n lmst 2)(2~)(  . 2. е с л и   lmlm , , то почти все системы s множества mlns ,,' не имеют решений. список литературы 1.егиазарян э.в. оценки, связанные с числом решений булевых уравнений. сб. вопросы кибернетики, комбинаторный анализ и теория графов, москва, 1981. 2.егиазарян э.в. метрические свойства систем булевых уравнений. дан арм.сср, т. 72, n2, 1981.        li x yynxf mi ,...,1 1,,,,... 11  microsoft word tigran.doc mathematical problems of computer science 33, 83--90, 2010. 83 numerical weather prediction environment for the territory of armenia tigran v. khotsanyan institute for informatics and automation problems of nas of ra e-mail: tkhotsanyan@ipia.sci.am abstract this article is devoted to the creation and development of a numerical weather prediction environment for the territory of armenia. one of the main features of this environment is an easy and clear web interface supporting to make numerical weather forecasts and analyze the results. references [1] armstatehidromet, http://www.meteo.am [2] weather research & forecasting model website, http://www.wrf-model.org [3] i. foster, c. kesselman, s. tuecke, “the anatomy of the gridenabling scalable virtual organizations”, international journal supercomputer applications, 15(3), pp. 200-222, 2001 [4] i. foster, c. kesselman, “computational grids”, in “the grid: blueprint for a new computing infrastructure" chpter, morgan-kaufman, pp. 15-51, 1999. [5] see-grid-sci, http://www.see-grid-sci.eu/ [6] egi design study web pages, http://web.eu-egi.eu/ [7] official web-site of the armenian national grid initiative foundation, http://www.grid.am [8] wrf model graphic tools : rip4, http://www.mmm.ucar.edu/wrf/users/graphics/rip4/rip4.htm [9] t. khotsanyan, h. astsatryan, a. mirzoyan, v. sahakyan, yu. shoukourian, h. melkonyan, a. hovsepyan, z. petrosyan, v. kotroni, “implementing and evaluating the weather research and forecast model for the territory of armenia”, seventh international conference on computer science and information technologies (csit’2009), yerevan, armenia, 2009, pp. 490-493. [10] national centers for enviromental prediction (ncep), http://www.ncep.noaa.gov [11] t. khotsanyan, http://www.cluster.am/wrf/, wrf [12] t. khotsanyan, http://www.cluster.am/wrf/stat, statistical analysis 84 numerical weather prediction environment for the territory of armenia ºõ³ý³ïç ãí³ûçý ï³ýë³ï»ëù³ý ùçç³í³ûñ ð³û³ëï³ýç ï³ñ³íùç ñ³ù³ñ î. êáó³ýû³ý ²ù÷á÷áõù ²ßë³ï³ýùá ýíçñí³í ¿ ð³û³ëï³ýç ï³ñ³íùç ñ³ù³ñ »õ³ý³ïç ãí³ûçý ï³ýë³ï»ëù³ý ùçç³í³ûñç ëï»õíù³ýý áõ ùß³ïù³ýá: ²ûë ùçç³í³ûñç ï³ñ¨áñ³·áõûý ýå³ï³ïý»ñçó ù»ïý ¿ ³å³ñáí»é û·ï³·áñíáõçý å³ñ½ í»µ ùçç³í³ûñ ï³ï³ñ»éáõ »õ³ý³ïç ãí³ûçý ï³ýë³ï»ëáõù ¨ ùß³ï»éáõ ³û¹ ï³ýë³ï»ëù³ý ³ñ¹ûáõýùý»ñá: mariam_haroutunian.dvi mathematical problems of computer science 42, 17{27, 2014. i nfor mation t heor etical analysis of b iometr ic gener ated secr et key shar ing m odel ma r ia m e . h a r o u t u n ia n 1 a n d n a r e k s . p a h le va n ya n 2 1institute for informatics and automation problems of nas ra 2gyumri state pedagogical institute e-mail: armar@ipia.sci.am, narek@ravcap.com abstract we investigate the biometric generated secret key sharing system. we consider the generalization of the secret key rate, studied by ignatenko and willems [1]: the notion of e-achievable secret key rate is introduced. the lower and upper bounds for the largest e-achievable secret key rate are obtained. when e tends to zero, the limits of our bounds coincide and are equal to the largest achievable secret key rate stated in [1]. keywords: biometric systems, generated secret key, e-achievable rate, achievable secret rate. 1 . in t r o d u c t io n in r e c e n t ye a r s t h e b io m e t r ic s e c r e c y s ys t e m s a r e b e in g wid e ly u s e d , r a n g in g fr o m b o r d e r c o n t r o l a t a ir p o r t s t o a c c e s s c o n t r o l in va r io u s s ys t e m s s u c h a s c o m p u t e r lo g -in , id c a r d s , e -p a s s p o r t s . th e p u r p o s e o f u s in g b io m e t r ic s e c r e c y s ys t e m s is t o a vo id a lo t o f r e la t e d p r o b le m s wit h t h e u s a g e o f t r a d it io n a l p a s s wo r d s . fo r in s t a n c e , s im p le p a s s wo r d s c a n b e e a s ily g u e s s e d o r b r o ke n b y b r u t e -fo r c e a t t a c ks , wh ile m o r e c o m p le x p a s s wo r d s a r e d i± c u lt t o r e m e m b e r . fu r t h e r m o r e , wh e n a s in g le p a s s wo r d is c o m p r o m is e d , it m a y o p e n m a n y o t h e r \ g a t e s " , b e c a u s e m o s t p e o p le u s e t h e s a m e p a s s wo r d fo r a u t h e n t ic a t io n in d i®e r e n t lo c a t io n s . fin a lly, a p a s s wo r d c a n b e s h a r e d wit h a n o t h e r p e r s o n a n d t h e r e will b e n o wa y t o kn o w wh o t h e a c t u a l u s e r is . th e a b o ve m e n t io n e d lim it a t io n s c a n b e s o lve d , if t h e b io m e t r ic s e c r e c y s ys t e m s wo u ld b e u s e d t o a u t h e n t ic a t e a n in d ivid u a l. b e s id e s , t h e a u t h e n t ic a t io n b io m e t r ic s e c r e c y s ys t e m s c a n b e u s e d in id e n t i¯ c a t io n , e xa m in a t io n s , p a ym e n t p r o c e s s in g s a n d in o t h e r va r io u s a p p lic a t io n s . s u c h s ys t e m s a r e m o r e s e c u r e t h a n t h e t r a d it io n a l p a s s wo r d -b a s e d a u t h e n t ic a t io n s ys t e m s , b e c a u s e t h e b io m e t r ic p r o p e r t ie s c a n n o t b e lo s t o r fo r g o t t e n , t h e y a r e d i± c u lt t o fa ls ify o r d u p lic a t e , s h a r e , a n d d is t r ib u t e . in m a n y a p p lic a t io n s , s u c h a s , e xa m in a t io n s , t h e p e r s o n is r e qu ir e d t o b e p r e s e n t a t t h e t im e a n d p o in t o f a u t h e n t ic a t io n . mo r e o ve r , t h e r e a r e a c c e s s s c e n a r io s , wh ic h r e qu ir e a p a r t ic ip a t io n o f m u lt ip le p r e vio u s ly r e g is t e r e d u s e r s fo r a s u c c e s s fu l a u t h e n t ic a t io n o r t o g e t a n a c c e s s g r a n t fo r a c e r t a in e n t it y. fo r in s t a n c e , t h e r e a r e c r yp t o g r a p h ic c o n s t r u c t io n s kn o wn a s s e c r e t s h a r in g s c h e m e s , wh e r e a s e c r e t ke y is s p lit in t o s h a r e s a n d d is t r ib u t e d a m o n g s t t h e u s e r s in s u c h a wa y t h a t it c a n b e r e c o n s t r u c t e d 1 7 1 8 information theoretical analysis of biometric generated secret key sharing model o n ly wh e n t h e n e c e s s a r y n u m b e r o f s e c r e t ke y h o ld e r s c o m e s t o g e t h e r . th e r e ve a le d s e c r e t c a n t h e n b e u s e d fo r e n c r yp t io n o r a u t h e n t ic a t io n . on e o f s u c h a p p lic a t io n s c o u ld b e s h a r in g o f a b a n k a c c o u n t b y fa m ily m e m b e r s . h o we ve r , t h e u s a g e o f b io m e t r ic s e c r e c y s ys t e m s h a s it s o wn d is a d va n t a g e s . a s t h e b io m e t r ic d a t a a r e g a t h e r e d fr o m in d ivid u a ls u n d e r e n vir o n m e n t a l c o n d it io n s a n d t h e c h a n n e ls a r e e xp o s e d t o n o is e t h e b io m e t r ic s e c r e c y s ys t e m m a y a c c e p t a n im p o s t o r o r r e je c t a n a u t h o r iz e d in d ivid u a l. b a s ic a lly, it 's n o t p o s s ib le t o b u ild a n id e a l b io m e t r ic s e c r e c y s ys t e m , it c a n b e in fo r m a t io n -t h e o r e t ic a l s e c u r e u p t o a c e r t a in le ve l. ig n a t e n ko a n d w ille m s [1 ] h a ve m e n t io n e d t h a t a p e r fe c t s ys t e m fo r a s e c u r e b io m e t r ic a u t h e n t ic a t io n h a s t o s a t is fy t h r e e r e qu ir e m e n t s . fir s t ly, b io m e t r ic d a t a h a ve t o b e p r iva t e , t h a t m e a n s t h e r e fe r e n c e in fo r m a t io n s t o r e d in d a t a b a s e s h o u ld n o t r e ve a l t h e a c t u a l b io m e t r ic d a t a . s e c o n d ly, r e fe r e n c e d a t a t h a t a r e c o m m u n ic a t e d fr o m a d a t a b a s e t o t h e p la c e wh e r e a n a c c e s s c a n b e g r a n t e d o r d e n ie d h a ve t o b e fa u lt -t o le r a n t t o e a ve s d r o p p in g . a n d ¯ n a lly r e fe r e n c e d a t a s t o r e d in d a t a b a s e h a ve t o b e r e s ilie n t t o b r u t e -fo r c e a t t a c ks . n o n e t h e le s s , a s p r a c t ic e s h o ws p e o p le d o n o t fe e l c o m fo r t a b le in p r o vid in g t h e ir b io m e t r ic in fo r m a t io n t o a la r g e a m o u n t o f o u t wa r d ly s e c u r e d a t a b a s e s , b e c a u s e o n e c a n n o t fu lly t r u s t t h e s e c u r it y im p le m e n t a t io n s o f t h ir d p a r t ie s , a n o t h e r r e a s o n is t h a t t h e d a t a b a s e m ig h t b e c o m p r o m is e d fr o m in s id e , wh ic h will a llo w a n o wn e r o f a d a t a b a s e t o m is u s e b io m e t r ic in fo r m a t io n . a n d ¯ n a lly p e o p le h a ve lim it e d b io m e t r ic r e s o u r c e s , s o \ id e n t it y t h e ft " h a s m u c h m o r e s e r io u s im p a c t s t h a n a \ s im p le " t h e ft o f a c r e d it c a r d . in t h is p a p e r we r e vis it t h e p r o b le m o f g e n e r a t in g s e c r e t ke ys fr o m b io m e t r ic d a t a p r o vid e d in [1 ]. w e in t r o d u c e t h e n o t io n o f e-a c h ie va b le s e c r e t ke y r a t e a n d o b t a in t h e lo we r a n d u p p e r b o u n d s fo r t h e la r g e s t e-a c h ie va b le s e c r e t ke y r a t e . w h e n e t e n d s t o z e r o , t h e lim it s o f o u r r e s u lt s c o in c id e wit h t h e la r g e s t a c h ie va b le s e c r e t ke y r a t e d e ¯ n e d in [1 ]. 2 . r e la t e d w o r k s e c u r it y c o n c e r n s r e la t e d t o t h e u s e o f b io m e t r ic d a t a in d i®e r e n t s e c r e c y s ys t e m s we r e r a is e d a lo n g t im e a g o . fr o m t h e in fo r m a t io n -t h e o r e t ic a l p e r s p e c t ive t h e b io m e t r ic s e c r e c y s ys t e m s we r e s t u d ie d b y o's u lliva n a n d s c h m id [2 ] a n d w ille m s e t a l [3 ]. w ille m s [3 ] in ve s t ig a t e d t h e fu n d a m e n t a l p r o p e r t ie s o f t h e b io m e t r ic id e n t ī c a t io n s ys t e m . it h a s b e e n s h o wn t h a t it is im p o s s ib le t o r e lia b ly id e n t ify m o r e p e r s o n s t h a n c a p a c it y wh ic h is a n in h e r e n t c h a r a c t e r is t ic o f a n y id e n t ī c a t io n s ys t e m . b y a n a lo g y wit h n o t io n o f e-c a p a c it y o r r a t e -r e lia b ilit y fu n c t io n in t r o d u c e d fo r d is c r e t e m e m o r yle s s c h a n n e ls b y e . h a r o u t u n ia n [4 ] in [5 ] t h e n e w c o n c e p t o f id e n t ī c a t io n e-c a p a c it y fo r b io m e t r ic a l id e n t ī c a t io n s ys t e m wa s in t r o d u c e d . th e a u t h o r s d e r ive d t h e u p p e r a n d lo we r b o u n d s o f b io m e t r ic id e n t ī c a t io n s ys t e m . l a t e r in [6 ] t h e a u t h o r s in ve s t ig a t e d t h e r a t e -r e lia b ilit y fu n c t io n fo r b io m e t r ic id e n t i¯ c a t io n p r o t o c o l wit h a r a n d o m p a r a m e t e r . th e p r o b le m o f g e n e r a t in g s e c r e t ke ys fr o m b io m e t r ic d a t a is c lo s e ly r e la t e d t o t h e c o n c e p t o f s e c r e t s h a r in g , wh ic h wa s in t r o d u c e d b y ma u r e r [7 ] a n d b y a h ls we d e a n d cs is z a r [8 ]. th is p r o b le m in b io m e t r ic s e t t in g wa s c o n s id e r e d b y ig n a t e n ko a n d w ille m s [1 ]. u n like in t r a d it io n a l s e c r e t ke y s h a r in g , wh e r e t h e s e c r e t ke y is b e in g g e n e r a t e d a n d s h a r e d b e t we e n t e r m in a ls , in b io m e t r ic s e c r e c y s ys t e m s a s e c r e t ke y is g e n e r a t e d d u r in g a n e n r o llm e n t p r o c e d u r e in wh ic h t h e b io m e t r ic d a t a a r e o b s e r ve d fo r t h e ¯ r s t t im e . th e s e c r e t ke y is t o b e r e c o n s t r u c t e d a ft e r t h e s e b io m e t r ic d a t a a r e o b s e r ve d a g a in , d u r in g a n a t t e m p t t o g e t a n a c c e s s . r e lia b le b io m e t r ic s e c r e c y s ys t e m s e xt r a c t h e lp e r d a t a fr o m t h e b io m e t r ic in fo r m a t io n a t t h e t im e o f e n r o llm e n t , a s b io m e t r ic m e a s u r e m e n t s a r e t yp ic a lly n o is y. th e s e h e lp e r d a t a m. haroutunian and n. pahlevanyan 1 9 c o n t r ib u t e t o r e lia b le r e c o n s t r u c t io n o f t h e s e c r e t ke y. th e d e t a ile d d e s c r ip t io n o f t h is m o d e l is g ive n in t h e n e xt s e c t io n . mo r e d e t a ile d r e vie w o n in fo r m a t io n -t h e o r e t ic a p p r o a c h e s o f b io m e t r ic s e c r e c y s ys t e m s c a n b e fo u n d in [9 ]. 3 . b io m e t r ic ge n e r a t e d s e c r e t k e y s h a r in g mo d e l b e fo r e we s t a r t in t r o d u c in g t h e m o d e l le t 's d e ¯ n e s o m e c o n ve n t io n s t h a t a r e a p p lie d wit h in t h is p a p e r . ca p it a l le t t e r s a r e u s e d fo r r a n d o m va r ia b le s ( r v ) x; y t a kin g va lu e s in t h e ¯ n it e a lp h a b e t s x ; y, c o r r e s p o n d in g ly. th e c a r d in a lit y o f t h e a lp h a b e t x is d e n o t e d b y jx j. b io m e t r ic g e n e r a t e d s e c r e t ke y s h a r in g is o n e o f t h e a va ila b le m o d e ls o f b io m e t r ic s e c r e t ke y s h a r in g s ys t e m . th e m o d e l is r e p r e s e n t e d in fig u r e 1 . fig. 1. biometric generated secret key sharing model. th e m o d e l is b a s e d o n a b io m e t r ic s o u r c e wit h d is t r ib u t io n fq ( x; y ) ; x 2 x ; y 2 yg. th is s o u r c e p r o d u c e s x ´ xn = ( x1; x2; :::; xn ) o f n s ym b o ls fr o m t h e ¯ n it e a lp h a b e t x a n d a s e c o n d s e qu e n c e y ´ yn = ( y1; y2; :::; yn ) o f n s ym b o ls fr o m t h e ¯ n it e a lp h a b e t y. th e ¯ r s t s e qu e n c e is c a lle d a n e n r o llm e n t s e qu e n c e , a n d t h e s e c o n d s e qu e n c e a n a u t h e n t ic a t io n s e qu e n c e . fu r t h e r m o r e , t h e s e c o n d s e qu e n c e y n is a n o is y ve r s io n o f t h e ¯ r s t s e qu e n c e x n . l e t u s d e n o t e q( x; y ) = q1 ( x ) q2 ( yjx ) ; x 2 x ; y 2 y: w e a s s u m e t h a t qn ( x; y ) = ny n=1 q ( xn; yn ) : th e n c o n s id e r a n e n c o d e r t h a t e xp lo r e s t h e e n r o lm e n t s e qu e n c e xn . fr o m t h is s e qu e n c e in b io m e t r ic g e n e r a t e d s e c r e t ke y s h a r in g m o d e l t h e e n c o d e r g e n e r a t e s a s e c r e t s 2 f1 ; 2 ; :::; jsjg a n d t h e n a p u b lic h e lp e r -m e s s a g e ( h e lp e r d a t a ) m 2 f 1 ; 2 ; :::; jmjg. th a t m e a n s t h a t f ( x n ) = ( s; m ) ; wh e r e b y f ( ¢ ) we d e n o t e t h e e n c o d e r fu n c t io n . th e h e lp e r -m e s s a g e is s e n t t o t h e d e c o d e r . th e d e c o d e r e xp lo r e s t h e a u t h e n t ic a t io n s e qu e n c e y n a n d p r o d u c e s a n e s t im a t e ŝ o f t h e s e c r e t s u s in g t h e r e c e ive d h e lp e r d a t a m, h e n c e g ( y n ; m ) = ŝ; wh e r e b y g ( ¢; ¢ ) we d e n o t e t h e d e c o d e r fu n c t io n . th e c h a n n e l b e t we e n t h e e n c o d e r a n d t h e d e c o d e r is e xp e c t e d t o b e p u b lic . w e a s s u m e t h a t a n a t t a c ke r h a s a n a c c e s s t o t h a t c h a n n e l, s o h e c a n s e e a ll t h e p u b lic in fo r m a t io n b u t c a n n o t m o d ify. th e in fo r m a t io n o u t ° o w is d e s c r ib e d in t e r m s o f m u t u a l in fo r m a t io n , a n d t h e s iz e o f t h e s e c r e t ke y{ in t e r m s o f e n t r o p y. fin g e r p r in t s a n d ir is e s c a n b e m o d e le d a s s u c h b io m e t r ic s o u r c e s . 2 0 information theoretical analysis of biometric generated secret key sharing model th e im p o r t a n t p a r a m e t e r s o f a b io m e t r ic s e c r e c y s ys t e m in c lu d e t h e s iz e o f s e c r e t ke y a n d t h e in fo r m a t io n t h a t t h e h e lp e r d a t a le a k o n t h e b io m e t r ic o b s e r va t io n . th a t le a k o f b io m e t r ic in fo r m a t io n is c a lle d a p r iva c y le a ka g e . th e p r iva c y le a ka g e s h o u ld b e s m a ll, t o a vo id t h e b io m e t r ic d a t a o f a n in d ivid u a l t o b e c o m e c o m p r o m is e d . mo r e o ve r , t h e s e c r e t ke y le n g t h s h o u ld b e la r g e t o m in im iz e t h e p r o b a b ilit y t h a t t h e s e c r e t ke y is g u e s s e d . it is t h e g o a l o f b o t h s id e s ( e n c o d e r a n d d e c o d e r ) t o p r o d u c e a s e c r e t ke y a s la r g e a s p o s s ib le , t h a t s a t is i¯ e s t h is c o n d it io n p r fs 6= ŝg ¼ 0 , t h is m e a n s t h a t p r o b a b ilit y t h a t t h e e s t im a t e d s e c r e t ŝ is n o t e qu a l t o t h e g e n e r a t e d s e c r e t s is c lo s e t o z e r o . th e b io m e t r ic g e n e r a t e d s e c r e t ke y s h a r in g m o d e l m u s t s a t is fy t h e fo llo win g r e qu ir e m e n t s [1 ] p r fs 6= ŝg ¼ 0 ( r e lia b ilit y) , 1 n h ( s ) ¼ 1 n lo g 2 jsj ( s e c r e t u n ifo r m it y) , 1 n h ( s ) is a s la r g e a s p o s s ib le ( s e c r e t ke y r a t e ) , 1 n i ( s ^ m ) ¼ 0 ( s e c r e c y le a ka g e ) , 1 n i ( x n ^ m ) is a s s m a ll a s p o s s ib le ( p r iva c y le a ka g e ) . h e r e is a d e ¯ n it io n fo r t h e a c h ie va b le s e c r e t ke y r a t e s t a t e d in [1 ]. a s e c r e t ke y r a t e r, fo r r ¸ 0 , is c a lle d a c h ie va b le if fo r a ll ± > 0 a n d a ll n la r g e e n o u g h , t h e r e e xis t e n c o d e r s a n d d e c o d e r s s u c h t h a t p r fs 6= ŝg · ±; 1 n h ( s ) + ± ¸ 1 n lo g 2 jsj ¸ r ¡ ±; 1 n i ( s ^ m ) · ±: a h ls we d e a n d cs is z ¶a r [8 ] p r o p o s e d a n d p r o ve d a t h e o r e m , wh ic h s t a t e d t h a t fo r a s o u r c e t yp e m o d e l t h e la r g e s t a c h ie va b le s e c r e t ke y r a t e r is e qu a l t o m u t u a l in fo r m a t io n i ( x ^y ) . u n like t h e o r ig in a l p r o o f, wh ic h is b a s e d o n s t r o n g t yp ic a lit y, a n o t h e r p r o o f o f t h e s a m e t h e o r e m , b u t fo r b io m e t r ic g e n e r a t e d s e c r e t m o d e l, u s in g we a k t yp ic a lit y is g ive n in [1 ]. in t h e n e xt s e c t io n we will in t r o d u c e n e w c o n c e p t o f e-a c h ie va b le s e c r e t ke y r a t e , wh ic h d i®e r s fr o m t r a d it io n a l s e c r e t ke y r a t e b y h a vin g a m o r e s o lid r e qu ir e m e n t a n d e xp o n e n t ia lly d e c r e a s e s t h e e r r o r p r o b a b ilit y in p r a c t ic e . 4 . e-a c h ie va b le s e c r e t k e y r a t e de¯nition: a secret key rate r ( e ) , for r ( e ) ¸ 0 , is called e-achievable if for all ± > 0 , e > 0 and n large enough, there exists a code such that p r fs 6= ŝg · 2 ¡n(e¡±); 1 n h ( s ) + ± ¸ 1 n lo g 2 jsj ¸ r ( e ) ¡ ±; 1 n i ( s ^ m ) · ±: m. haroutunian and n. pahlevanyan 2 1 w e s h a ll u s e t h e fo llo win g p d in t h e fo r m u la t io n o f r e s u lt : q1 = fq1 ( x ) ; x 2 x g ; q2 = fq2 ( yjx ) ; y 2 y; x 2 x g; p1 = fp1 ( x ) ; x 2 x g ; p2 = fp2 ( yjx ) ; y 2 y; x 2 x g; q = fq ( x; y ) ; x 2 x ; y 2 yg ; p = fp ( x; y ) ; x 2 x ; y 2 yg: w e r e fe r t o [4 ] , [1 0 ] , [1 1 ] a n d [1 2 ] fo r n o t io n s o f d ive r g e n c e d ( p jjq ) , m u t u a l in fo r m a t io n ip ( x ^ y ) , in fo r m a t io n -t h e o r e t ic qu a n t it ie s . th e p r o o fs a r e b a s e d o n t h e m e t h o d o f t yp e s . w e d e n o t e b y t np1 ( x ) t h e s e t o f ve c t o r s x o f t yp e p1, b y t n p ( x; y ) t h e s e t o f ve c t o r s ( x; y ) o f t yp e p . w e u s e s o m e kn o wn p r o p e r t ie s [4 ], [1 0 ], [1 1 ], [1 2 ]. fo r ( x; y ) 2 t np ( x; y ) qn ( x; y ) = e xp f¡n ( hp ( x; y ) + d ( p jjq ) ) g; ( 1 ) jt np ( x; y ) j · e xp fnhp ( x; y ) g; ( 2 ) qn ( t np ( x; y ) ) · e xp f¡n d ( p jjq ) g; ( 3 ) d ( p jjq ) = d ( p1jjq1 ) + d ( p2jjq2jp1 ) : ( 4 ) ou r m a in r e s u lt is s t a t e d in t h e fo llo win g t h e o r e m . t heor em: f or biometric generated-secret model the largest e-achievable secret key rate r ( e ) is lower bounded by rr ( e ) = m in p :d(p jjq)·e jip ( x ^ y ) + d ( p jjq ) ¡ ej+ ( 5 ) a n d u p p e r b o u n d e d b y rsp ( e ) = m in p :d(p jjq)·e ip ( x ^ y ) : ( 6 ) h e r e t h e n o t a t io n jaj+ is u s e d fo r m a x( a, 0 ) . cor ollar y: w hen e ! 0 , the limits of our bounds coincide and are equal to the largest achievable secret key rate de¯ned in [1]: lim e!0 rr ( e ) = lim e!0 rsp ( e ) = iq ( x ^ y ) : 5 . p r o o f o f th e o r e m p r oof of upper b ound. l e t e > 0 a n d t h e e r r o r p r o b a b ilit y s a t is fy t h e c o n d it io n p r fs 6= ŝg · e xp f¡n ( e ¡± ) g. it m e a n s t h a t x x2x n qn1 ( x ) 1 jsj x s2s qn2 n y n ¡ g¡1m ( s) ¯̄ ¯xm ( s) o · e xp f¡n ( e ¡ ± ) g; 2 2 information theoretical analysis of biometric generated secret key sharing model wh e r e g¡1m ( s) = fy : gm ( y ) = sg: fo r a n y p we c a n wr it e x s2s x xm(s)2t np1 (x) qn1 ( xm ( s) ) £ qn2 n t np ( y jxm ( s) ) ¡ g¡1m ( s) ¯̄ ¯xm ( s) o · jsj e xp f¡n ( e ¡ ± ) g: qn1 ( xm ( s) ) a n d q n 2 ( yjxm ( s) ) a r e c o n s t a n t fo r va r io u s x a n d y o f ¯ xe d t yp e p , h e n c e , we c a n wr it e x s2s x xm(s)2t np1 (x) n¯̄ ¯t np ( y jxm ( s) ) ¯̄ ¯ ¡ ¯̄ ¯t np ( y jxm ( s) ) \ g¡1m ( s) ¯̄ ¯ o £ qn1 ( xm ( s) ) £ qn2 ( yjxm ( s) ) · jsj e xp f¡n ( e ¡ ± ) g: a c c o r d in g t o ( 3 ) x s2s x xm(s)2t np1 (x) n¯̄ ¯t np ( y jxm ( s) ) ¯̄ ¯ ¡ ¯̄ ¯t np ( y jxm ( s) ) \ g¡1m ( s) ¯̄ ¯ o £ e xp f¡n ( d ( p jjq ) + hp2 ( y jx ) ) g · jsj e xp f¡n ( e ¡ ± ) g: ( 7 ) it fo llo ws fr o m t h e d e ¯ n it io n o f d e c o d in g fu n c t io n g t h a t fo r t h e g ive n m t h e s e t s g¡1m ( s) a r e d is jo in t , t h e r e fo r e x s2s ¯̄ ¯t np ( y jxm ( s) ) \ g¡1m ( s) ¯̄ ¯ · jt np ( y ) j: th e n fr o m ( 7 ) we h a ve x s2s ¯̄ ¯t np ( y jxm ( s) ) ¯̄ ¯ ¡ jsj e xp f¡n ( e ¡ ± ) g e xp f¡n ( d ( p jjq ) + hp ( y jx ) ) g · jt np ( y ) j: h e n c e , jsj · e xp fn ip ( x ^ y ) g ( n + 1 ) ¡jxjjy j ¡ e xp fn ( d ( p jjq ) ¡ e ¡ ± ) g: th e r ig h t -h a n d s id e o f t h is in e qu a lit y c a n b e m in im iz e d b y t h e c h o ic e o f p ke e p in g t h e d e n o m in a t o r p o s it ive , wh ic h t a ke s p la c e fo r la r g e n, wh e n d ( p jjq) · e ¡ ±: achievability par t of theor em. code constr uction. w e c o n s id e r t h o s e t yp e s p t h a t d ( p jjq ) · e, fr o m ( 4 ) it fo llo ws t h a t d ( p1jjq1 ) · e a n d d ( p2jjq2jp1 ) · e: l e t u s d e n o t e t nq1 ( e ) = [ p1:d(p1jjq1)·e t np1 ( x ) ; t nq ( e ) = [ p :d(p jjq)·e t np ( x; y ) : m. haroutunian and n. pahlevanyan 2 3 w e d e ¯ n e a r a n d o m p a r t it io n o f t nq1 ( e ) in t o jmj b in s . th e e n c o d e r in d e p e n d e n t ly a s s ig n s a h e lp e r la b e l ( in d e x o f t h e b in ) m 2 f 1 ; 2 ; :::jmjg t o e a c h s e qu e n c e x wit h t h e p r o b a b ilit y p r fm ( x ) = mg = 1 =jmj: th e n we d e ¯ n e a s e c o n d r a n d o m p a r t it io n o ve r t nq1 ( e ) wit h jsj b in s , a n d t h e e n c o d e r a s s ig n s a r a n d o m la b e l ( b in -in d e x o f t h is s e c o n d p a r t it io n ) s 2 f 1 ; 2 ; :::jsjg t o e a c h s e qu e n c e x wit h t h e p r o b a b ilit y p r fs ( x ) = sg = 1 =jsj: th e e n c o d e r e xp lo r e s t h e s e qu e n c e x a n d d e t e r m in e s t h e s e c r e t la b e l s a n d h e lp e r la b e l m. th e e n c o d e r s e n d s t h e h e lp e r la b e l m t o t h e d e c o d e r . th e d e c o d e r a ft e r h a vin g o b s e r ve d y s e qu e n c e lo o ks fo r a u n iqu e s e qu e n c e x wit h t h e h e lp e r la b e l m s u c h t h a t ( x; y ) 2 tp ( x; y ) a n d d ( p jjq ) is m in im a l. e r r or pr obability. fr o m ( 3 ) we o b t a in t h a t fo r a n y p s u c h t h a t d ( p jjq ) > e t h e p r o b a b ilit y o f t h e e ve n t is s m a ll e n o u g h qn ( t np ( x; y ) ) · e xp f¡neg: ( 8 ) th e d e c o d e r c a n m a ke a n e r r o r if fo r t h e g ive n m t h e s e c r e t s wa s d e t e r m in e d , b u t t h e r e e xis t s ŝ 6= s, s u c h t h a t fo r s o m e p̂ ( xm ( s) ; ym ) 2 tp ( x; y ) ; ( x̂m ( ŝ) ; ym ) 2 tp̂ ( x; y ) a n d d ( p̂ jjq ) · d ( p jjq) : th e m a t h e m a t ic a l e xp e c t a t io n o f t h is e ve n t c a n b e u p p e r b o u n d e d b y t h e fo llo win g e xp r e s s io n : x p;p̂ :d(p̂ jjq)·d(p jjq) x ŝ6=s x x2x n x y2yn q n ( x; y ) £ p r f ( xm ( s) ; ym ) 2 t np ( x; y ) g £ p r f ( x̂m ( ŝ) ; ym ) 2 t np̂ ( x; y ) g: th e ¯ r s t p r o b a b ilit y is d i®e r e n t fr o m z e r o o n ly if x 2 t np1 ( x ) a n d y 2 t n p ( y ) : h e n c e , t h e e xp r e s s io n will h a ve t h e fo llo win g fo r m : x p;p̂ :d(p̂ jjq)·d(p jjq) x ŝ6=s x x2t n p1 (x) x y2t n p (y ) q n ( x; y ) £ p r f ( xm ( s) ; ym ) 2 t np ( x; y ) g £ p r f ( x̂m ( ŝ) ; ym ) 2 t np̂ ( x; y ) g: ta kin g in t o a c c o u n t t h a t fo r a n y p̂ jsj ¡ 1 · e xp fn ( ip̂ ( x ^ y ) + d ( p̂ jjq ) ¡ e ¡ ±1 ) g a n d ( 1 ) , ( 2 ) t h e la s t e xp r e s s io n will n o t b e g r e a t e r t h a n x p;p̂ :d(p̂ jjq)·d(p jjq) e xp fn ( ip̂ ( x ^ y ) + d ( p̂ jjq) ¡ e ¡ ±1 ) g£ e xp f¡n ( hp ( x; y ) + d ( p jjq ) ) g £ e xp fn ( hp ( x ) + hp ( y ) ) g£ 2 4 information theoretical analysis of biometric generated secret key sharing model e xp f¡n ( ip ( x ^ y ) ¡ ±2 ) g £ e xp f¡n ( ip̂ ( x ^ y ) ¡ ±3 ) g = x p;p̂ :d(p̂ jjq)·d(p jjq) e xp fn ( d ( p̂ jjq ) ¡ e ¡ ( ±1 + ±2 + ±3 ) ) g£ e xp f¡nd ( p jjq ) g · e xp f¡n ( e ¡ ±0 ) g: ( 9 ) th e n fr o m ( 8 ) a n d ( 9 ) we s t a t e t h a t fo r n la r g e e n o u g h t h e r e e xis t s a c o d e wit h la b e lin g s m a n d s s u c h t h a t p r fs 6= ŝg · e xp f¡n ( e ¡ ±0 ) g + e xp f¡ne ) g · e xp f¡n ( e ¡ ± ) g: fo r t h e r e s t o f t h e p r o o f we s t a t e t h a t s in c e p r fs 6= ŝg · e xp f¡n ( e ¡ ± ) g fo r n la r g e e n o u g h , t h e r e e xis t s a t le a s t o n e p a ir o f la b e lin g s m; s; s u c h t h a t h ( m ) · lo g jmj = n m a x p :d(p jjq)·e ( hp ( xjy ) ¡ d ( p jjq ) + e + ²1 ) : ( 1 0 ) h ( s ) · lo g jsj = n m in p :d(p jjq)·e ( ip ( x ^ y ) + d ( p jjq) ¡ e ¡ ²2 ) : ( 1 1 ) unifor mity. l e t x̂n b e t h e e s t im a t e o f x n b a s e d o n s a n d m; t h e n we ¯ n d t h a t h ( x n ) = h ( x n ; s; m ) = h ( s ) + h ( mjs ) + h ( xn js; m ) · h ( s ) + h ( m ) + h ( xn js; m; x̂ n ) · h ( s ) + h ( m ) + npe lo g jx j + 1 ; t h e la s t s t e p fo llo ws fr o m fa n o 's in e qu a lit y. h e n c e , fr o m ( 1 0 ) a n d ( 1 1 ) we h a ve h ( s ) ¸ h ( xn ) ¡ h ( m ) ¡ n pe lo g jx j ¡ 1 ¸ nh ( x ) ¡ n m a x p :d(p jjq)·e ( hp ( xjy ) ¡ d ( p jjq ) + e + ² ) ¡ n e xp f¡n ( e ¡ ± ) g lo g jx j ¡ 1 ¸ lo g jsj ¡ n ( ²1 ¡ ²2 ) ¡ n e xp f¡n ( e ¡ ± ) g lo g jx j ¡ 1 : secr ecy. n o w we s t u d y t h e s e c r e c y. i ( s ^ m ) = h ( s ) + h ( m ) + h ( s; m ) = h ( s ) + h ( m ) ¡ h ( s; m; x n ) + h ( xn js; m ) = h ( s ) + h ( m ) ¡ h ( xn ) + h ( xn js; m; x̂ n ) · h ( s ) + h ( m ) ¡ n h ( x ) + n p1 lo g jx j + 1 : fr o m ( 1 0 ) a n d ( 1 1 ) we o b t a in h ( s ) + h ( m ) ¡ nh ( x ) + np1 lo g jx j + 1 · n ( m in p :d(p jjq)·e ip ( x ^ y ) + m a x p :d(p jjq)·e hp ( xjy ) ¡ h ( x ) ) + n ( ²1 ¡ ²2 ) + n e xp f¡n ( e ¡ ± ) g lo g jx j + 1 : fin a lly, we s e e t h a t t h e s e c r e c y le a ka g e is s m a ll fo r n la r g e e n o u g h 1 n i ( s ^ m ) · e xp f¡n ( e ¡ ± ) g lo g jx j + ( ²1 ¡ ²2 ) + 1 n : th e t h e o r e m is p r o ve d . m. haroutunian and n. pahlevanyan 2 5 6 . p r iva c y l e a ka g e th e fo llo win g p r o p o s it io n g ive s t h e p r iva c y le a ka g e c o r r e s p o n d in g t o t h e m a xim u m ea c h ie va b le s e c r e t ke y r a t e in t h e b io m e t r ic g e n e r a t e d s e c r e t ke y s h a r in g m o d e l. p r oposition: in a biometric generated secret key sharing model for e-achievable secret key rate the privacy leakage is 1 n iq1 ( m ^ x n ) = m a x p :d(p jjq)·e ( hp ( xjy ) + d ( p jjq ) ¡ e ) : ( 1 2 ) p r oof: a s m is a fu n c t io n o f x n we h a ve fr o m ( 1 1 ) i ( x n ^ m ) = h ( m ) ¸ h ( xn ) ¡ h ( s ) ¡ npe lo g jx j ¡ 1 ¸ nhq1 ( x ) ¡ n m in p :d(p jjq)·e ( ip ( x ^ y ) + d ( p jjq ) ¡ e ¡ ²2 ) ¡ n 2 ¡n (e¡±) lo g jx j ¡ 1 ¸ n m a x p :d(p jjq)·e ( hp ( xjy ) + d ( p jjq) ¡ e ¡ ²2 ) ¡ n 2 ¡n(e¡±) lo g jx j ¡ 1 : on t h e o t h e r h a n d fr o m ( 1 0 ) h ( m ) · n m a x p :d(p jjq)·e ( hp ( xjy ) ¡ d ( p jjq) + e + ²1 ) : d ivid in g b o t h s id e s b y n , a n d fo r n ! 1 we o b t a in ( 1 2 ) . remar k: the above proposition gives the privacy leakage if we apply the coding scheme outlined in the achievability proof. however, it may be possible to achieve a smaller privacy leakage depending on the secret-key rate. this problem will be considered in the future work. 7 . co n c lu s io n w e s t u d ie d t h e b io m e t r ic g e n e r a t e d -s e c r e t m o d e l fo r d is c r e t e i.i.d b io m e t r ic s o u r c e s . th e n e w c o n c e p t o f e-a c h ie va b le s e c r e t ke y r a t e is in t r o d u c e d a n d t h e e xp r e s s io n s fo r t h e lo we r a n d u p p e r b o u n d s o f la r g e s t r a t e a r e o b t a in e d . th is n o t io n is t h e g e n e r a liz a t io n o f t h e a c h ie va b le s e c r e t ke y r a t e a s it t e n d s t o t h e la s t wh e n e t e n d s t o z e r o . th e p r o o fs a r e b a s e d o n t h e s t r o n g t yp ic a lit y. a ls o a n e xp r e s s io n fo r p r iva c y le a ka g e , wh ic h c o r r e s p o n d s t o t h e la r g e s t e-a c h ie va b le s e c r e t ke y r a t e is o b t a in e d . refer ences [1 ] t. ig n a t e n ko a n d f. m. w ille m s , \ b io m e t r ic s e c u r it y fr o m a n in fo r m a t io n -t h e o r e t ic a l p e r s p e c t ive ," f oundations and trends in communications and information theory, vo l. 7 , n o . 2 -3 , p p . 1 3 5 { 3 1 6 , 2 0 1 2 . [2 ] j. a . o's u lliva n a n d n . a . s c h m id , \ l a r g e d e via t io n s p e r fo r m a n c e a n a lys is fo r b io m e t r ic s r e c o g n it io n " , p roc. 40th annual allerton conf. on communication, control, and computing, p p . 1 { 1 0 , oc t . 2 0 0 2 . 2 6 information theoretical analysis of biometric generated secret key sharing model [3 ] f. w ille m s , t. k a lke r , j. go s e lin g a n d j.-p . l in n a r t z , \ on t h e c a p a c it y o f a b io m e t r ic a l id e n t i¯ c a t io n s ys t e m ," in information theory, 2003. p roceedings. ie e e international symposium on information theory, y o ko h a m a , ja p a n , 2 0 0 3 , p . 8 2 . [4 ] e . h a r o u t u n ia n , \ on b o u n d s fo r e-c a p a c it y o f d mc," ie e e transactions on information theory, vo l. 5 3 , n o . 1 1 , p p . 4 2 1 0 { 4 2 2 0 , 2 0 0 7 . [5 ] m. h a r o u t u n ia n , a . mu r a d ya n a n d l . te r -v a r d a n ya n , \ u p p e r a n d lo we r b o u n d s o f b io m e t r ic id e n t ī c a t io n e-c a p a c it y, " transactions of iiap of nas of r a, m athematical p roblems of computer science, vo l. 3 6 , p p . 1 { 1 0 , 2 0 1 2 . [6 ] m. h a r o u t u n ia n a n d l . te r -v a r d a n ya n , \ in ve s t ig a t io n o f e-c a p a c it y fo r b io m e t r ic id e n t i¯ c a t io n p r o t o c o l wit h r a n d o m p a r a m e t e r ," transactions of iiap of nas of r a, m athematical p roblems of computer science, vo l. 3 9 , p p . 8 8 { 9 3 , 2 0 1 3 . [7 ] u . ma u r e r , \ s e c r e t ke y a g r e e m e n t b y p u b lic d is c u s s io n fr o m c o m m o n in fo r m a t io n ," ie e e transactions on information theory, vo l. 3 9 , n o . 3 , p p . 7 3 3 { 7 4 2 , ma y 1 9 9 3 . [8 ] r . a h ls we d e a n d i. cs is z a r , \ co m m o n r a n d o m n e s s in in fo r m a t io n t h e o r y a n d c r yp t o g r a p h y. i s e c r e t s h a r in g ," ie e e transactions on information theory, vo l. 3 9 , n o . 4 , p p . 1 1 2 1 { 1 1 3 2 , 1 9 9 3 . [9 ] y . ch e n a n d a . h . v in c k, \ fr o m p a s s wo r d t o b io m e t r ic s : h o w fa r c a n we g o ," p roceidings 7th asia-e urope w orkshop on concepts in information theory, b o p p a r d , ge r m a n y, p p . 1 { 8 , 2 0 1 1 . [1 0 ] e . a . h a r o u t u n ia n , m. e . h a r o u t u n ia n a n d a . n . h a r u t yu n ya n , \ r e lia b ilit y c r it e r ia in in fo r m a t io n t h e o r y a n d in s t a t is t ic a l h yp o t h e s is t e s t in g ," f oundations and trends in communications and information theory, vo l. 4 , n o . 2 -3 , p p . 9 7 { 2 6 3 , 2 0 0 8 . [1 1 ] i. cs is z a r , \ th e m e t h o d o f t yp e s ," ie e e transactions on information theory, vo l. 4 4 , n o . 6 , p p . 2 5 0 5 { 2 5 2 3 , 1 9 9 8 . [1 2 ] t. m. co ve r a n d j. a . th o m a s , e lements of information theory, 2nd e dition, n e w y o r k, w ile y-in t e r s c ie n c e , 2 0 0 6 . submitted 15.08.2014, accepted 22.11.2014. ¶»ý»ñ³óí³í ·³õïýç µ³ý³éç ÷áë³ý³ïáõ ï»ýë³ã³÷³ï³ý ùá¹»éç çýýáñù³óçáý-ï»ë³ï³ý ñ»ï³½áïáõãûáõý ø. ð³ñáõãûáõýû³ý ¨ ü. ö³ñé¨³ýû³ý ²ù÷á÷áõù ð»ï³½áïíáõù ¿ ·»ý»ñ³óí³í ·³õïýç µ³ý³éç ÷áë³ý³ïáõ ï»ýë³ã³÷³ï³ý ñ³ù³ï³ñ·á: ü»ñùáõíí»é ¿ ·³õïýçùç e ñ³ë³ý»éç ³ñ³·áõãûáõý ýáñ ñ³ëï³óáõãûáõýá, áñý áý¹ñ³ýñ³óýáõù ¿ æ·ý³ï»ýïáûç ¨ ìçé»ùëç [1] áõëáõùý³ëçñ³í ·³õïýçùç ñ³ë³ý»éç ³ñ³·áõãû³ý ·³õ³÷³ñá: î³éáõóí»é »ý ·³õïýçùç e ñ³ë³ý»éç ³é³í»é³·áõûý ³ñ³·áõãû³ý í»ñçý ¨ ëïáñçý ·ý³ñ³ï³ï³ýý»ñá: ºñµ e ! 0 , ëï³óí³í ·ý³ñ³ï³ï³ýý»ñç ë³ñù³ýý»ñá ñ³ùáýïýáõù »ý ¨ ñ³í³ë³ñ »ý [1]-áõù ëï³ó³í ·³õïýç µ³ý³éáõñ³ë³ý»éç ³ñ³·áõãû³ý ù»í³·áõûý ³ñå»ùçý: m. haroutunian and n. pahlevanyan 2 7 èíôîðìàöèîííî-òåîðåòè÷åñêèé àíàëèç áèîìåòðè÷åñêîé ìîäåëè ðàñïðåäåëåíèÿ ñãåíåðèðîâàííîãî ñåêðåòíîãî êëþ÷à ì. àðóòþíÿí è í. ïàéëåâàíÿí àííîòàöèÿ ðàññìàòðèâàåòñÿ áèîìåòðè÷åñêàÿ ñèñòåìà ðàñïðåäåëåíèÿ ñãåíåðèðîâàííîãî ñåêðåòíîãî êëþ÷à. ââîäèòñÿ íîâîå ïîíÿòèå e-äîñòèæèìîé ñêîðîñòè ñåêðåòà, êîòîðîå ÿâëÿåòñÿ îáîáùåíèåì äîñòèæèìîé ñêîðîñòè ñåêðåòà, èçó÷åííîé èãíàòåíêî è âèëåìñîì â [1]. ïîñòðîåíû âåðõíÿÿ è íèæíÿÿãðàíèöûíàèáîëüøåé e-äîñòèæèìîé ñêîðîñòè ñåêðåòà. êîãäà e ! 0 , ïðåäåëûïîëó÷åííûõ ãðàíèö ñîâïàäàþò ñ íàèáîëüøåé äîñòèæèìîé ñêîðîñòüþ ñåêðåòíîãî êëþ÷à, ïîëó÷åííîé â [1]. 404 not found начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 44, 145--153, 2015. application of multivariate statistical analysis in process control tigran z. khachikyan and sahak m. narimanyan yerevan state university department of probability theory and mathematical statistics e-mail: tkhach@inbox.ru, sahakn@yandex.ru abstract due to significant increase of information systems and its intensive usage in our everyday life, several problems like automatic identification of system faults, finding times of drastic change in stochastic characteristics as well as locating those characteristics, which “went out of control” need to be addressed. to solve these problems, we propose an algorithm based on multivariate statistical analysis. the algorithm is implemented with the r software environment and tested on custom metrics for vesta server and other groups of random metrics. keywords: principal component analysis, level of significance, t2 statistics, q statistics, loading matrix, score matrix, eigenvalues, covariance matrix, upper critical value. 1. introduction due to significant increase of information systems and its intensive usage in our everyday life, several problems like automatic identification of system faults, finding times of drastic change in stochastic characteristics as well as locating those characteristics, which “went out of control” need to be addressed. the normal process is usually conditioned by some characteristics, which may correlate with each other. in that case, analysis of individual characteristics may lead to significant errors due to different confidence intervals as well as impossibility of identification of joint level of significance. we used principal component analysis (pca), hotelling’s criteria based on 2t statistics, which is known to be uniformly the most powerful test (the null hypothesis for a vector of average values) in the class of all randomized tests invariant to transformations of similarity, to solve this problem. we also used q -statistics for the residual matrix of dataset after pca prediction. 2. method let ( , ,..., )1 2x x x x m= be a multivariate random variable, individual components of which characterize the state of a system having joint normal probability density function 145 mailto:tkhach@inbox.ru mailto:sahakn@yandex.ru 146 application of multivariate statistical analysis in process control { } 1 12 2( ) (2 ) | | exp ( ) ( ) , m tf x x xπ µ µ − − −= σ − − σ − 𝑥𝑥 = ( 𝑥𝑥1, 𝑥𝑥2, … , 𝑥𝑥𝑚𝑚), where 𝜇𝜇 a vector of average values, σ is a covariance matrix. to test the hypothesis 0 0:h µ µ= in multivariate case, let us use the hotelling’s 2t statistics ([1]) ( ) ( )2 10 0 t t n x s xµ µ−= − − , where n , n m> , is a sample size, s is a sample estimate of covariance matrix σ , is a mean vector of x . we will find values of 2t statistics for each instant time 𝑡𝑡 (𝑡𝑡 = 1, 2, … ). for this, we take historical data of fixed sample size n before time t , and shifting time t forward, while keeping the sample size unchanged. we will denote these values of 2t statistics as 2tt . if the covariance matrix σ is known, then the 2t statistics has a distribution 2χ and 2 2 (1 )t qcr mχ α= − , where 2 (1 )q mχ α− is the quantile of 2 mχ distribution with significance level α . and when the covariance matrix σ is unknown, the upper critical value of 2t statistics is calculated as ( 1)2 (1 , , ) m n t qf m n mcr n m α − = − − − , where qf is the quantile of fisher’s distribution with parameters ( , )m n m− , α is a significance level. the 2t statistics has a probability density function 1 1 21 2 1 2 ( ) , 0. 1 2 2 2 nm n x x n f x xm n m m n +−−+ γ + = > − + γ γ                         we will consider that the studied system operates normally, when 22 crt tt < . the method of pc ([2]) is one of the ways to reduce the size of datasets, while losing minimum amount of information. for our system and in case of multivariate random variable, we need to construct such an orthogonal coordinate system to transform correlated variables into new uncorrelated variables. sample divergence in relation to principal components is organized in decreasing order. we take so many principal components that the summarized sample divergence of pc is comprising 95% of the total divergence. using this method, we get a loading matrix and a score matrix. the values of these new variables form the factor scores, and these scores can be interpreted geometrically as the projections of the observations onto the principal components. the loadings are simply the coordinates of original variables in the principal components space. the loading matrix p has dimensions ( )m k× , where m is a space dimension and k is the quantity of principal components. the scoring matrix t has dimensions ( )n k× , where n is the sample size, k the number of principal components. the residual matrix is tr x tp= − , where is the dataset matrix, ( , ,..., )1 2x x x x m= . using the method of principal components we decrease dataset space dimension and hence, lose some t. khachikyan and s. narimanyan 147 information. in order to estimate the influence of other parameters on our system, let us consider -statistics for the residual matrix introduced by jackson ([2]). the –statistics is the following , where r is the vector-column of the residual matrix r . the upper critical value of q statistics is = , where 1 , 1, 2,3, m i i j j k iθ λ = + = =∑ 2 1 31 .0 23 2 h θ θ θ = − here сα is a quantile of standard normal distribution with significance level 1 α− and jλ are eigenvalues of sample covariance matrix s . 3. implementation 1) let , , ,...0 1 2t t t be the arrival times of dataset, and 1t t consti i− =− . let ∆ be the length of the interval for historical data. we will investigate the data matrix in the time interval , , 0,1,...t t kk k− ∆ =   2) at time 0t , we remove those variables from the data matrix, which are constant (do not change over that time interval). the resulting matrix represents a multivariate data sample at time 0t . we can normalize this matrix then. 3) we can employ pc method and as a result can find those components, which provide 95% of total divergence, score matrix, loading matrix, and residual matrix. 4) we can calculate the statistics at time 0t and at time 0t . then we calculate both statistics and at time 0t .if 22 crtt < and crqq < , then our system functions normally. otherwise, if 22 crtt ≥ or crqq ≥ the null-hypothesis is declined and it is assumed that the system “went out of control”, i.e., malfunctioning occurred. an automatic alert messaging to system administrators can be organized to take measures. 5) calculation of weights for individual parameters in 2t and q statistics takes place. those parameters, which have significant weight, can be considered as cause for both the 2t and q statistics to go out of control. for the next time 1t , we go back to point 1) and start over and again. 4. computerized and real-life example in the real-life example below, the system monitors 16 different parameters of vesta system working on real multivariate data. implementation of the suggested algorithm and using pc method, the following results are obtained. 148 application of multivariate statistical analysis in process control f ig . 1 . 𝑄𝑄 st at is tic s tr an sf or m ed to u ni ty 𝑄𝑄 /𝑄𝑄 𝑐𝑐𝑐𝑐 . a t t im e 14 :1 2 th e va lu e of 𝑄𝑄 s ta tis tic s is g re at er th an 1 , s o th e sy st em “ w en t o ut o f c on tr ol ” . t. khachikyan and s. narimanyan 149 f ig . 2 . a t t im e 14 :1 2 th e gr ap h de pi ct s th e m et ri cs , w ei gh t o f w hi ch is th e hi gh es t, re su lti ng th e va lu e of q s ta tis tic s is g re at er th an 1 . 150 application of multivariate statistical analysis in process control f ig . 3 . t he g ra ph o f 𝑇𝑇 2 tr an sf or m ed to u ni ty 𝑇𝑇 2 /𝑇𝑇 𝑐𝑐𝑐𝑐2 . a t t im e 16 :3 2 th e va lu e of 𝑇𝑇 2 st at is tic s is g re at er th an 1 , s o th e sy st em “ w en t o ut o f c on tr ol ”. t. khachikyan and s. narimanyan 151 fi g. 4 . a t t im e 16 :3 2 th e gr ap h de pi ct s th e m et ri cs , w ei gh t o f w hi ch is th e hi gh es t, re su lti ng th e va lu e of 𝑇𝑇 2 s ta tis tic s to b e gr ea te r t ha n 1. 152 application of multivariate statistical analysis in process control 5. conclusion principal component analysis (pca) is the most popular multivariate statistical technique and it is used by almost all scientific disciplines. pca method can be successfully applied to provide solutions for many it-related problems like automatic identification of system faults, finding times of drastic change in stochastic characteristics as well as locating those characteristics, which “went out of control”. a particular problem for system administrators is to monitor performance of custom metrics due to both absence of relevant thresholds and quantity of such metrics, which in some situations can be significant. the suggested algorithm used principal component analysis (pca), hotelling’s criteria based on 2t statistics, which is known to be uniformly the most powerful test (the null hypothesis for a vector of average values) in the class of all randomized tests invariant to transformations of similarity, to solve this problem. q statistics is also used for the residual matrix of dataset after pca prediction. the algorithm is applied to real-life situation with vesta system monitoring 16 different parameters on real multivariate data. the algorithm enables system administrators to identify event times when the system “went out of control” as well as to locate the “problematic” parameters causing such problems. automatic alert messaging and control mechanism can be organized to warn system administrators to take measures. references [1] t. w. anderson, an inroduction to multivariate statistical analysis, 3rd ed., wiley series in probability and mathematical statistics, 2003. [2] j. e. jackson, a user’s guide to principal components, wiley series in probability and mathematical statistics, 1991. submitted 23.07.2015, accepted 27.11.2015 բազմաչափ վիճակագրական վերլուծության կիրառություն գործընթացների կառավարման ոլորտում տ. խաչիկյան և ս. նարիմանյան ամփոփում մեր առօրյա կյանքում ինֆորմացիոն համակարգերի զգալի աճի և նրանց ինտենսիվ օգտագործման պատճառով, կարիք է առաջանում բազմաթիվ խնդիրների հետ առնչվել, ինչպիսիք են` համակարգի անսարքությունների ավտոմատ հայտնաբերումը, ստոխաստիկ բնութագրիչների կտրուկ փոփոխությունների պահերի որոշումը, ինչպես նաև` այն բնութագրիչների վերհանումը, որոնք «դուրս են եկել կառավարումից»: այդպիսի խնդիրների t. khachikyan and s. narimanyan 153 լուծման նպատակով առաջարկում ենք ալգորիթմ՝ հիմնված բազմաչափ վիճակագրական վերլուծության վրա։ ալգորիթմը իրականացված է r ծրագրավորման լեզվով և փորձարկված է վեստա սերվերի պատահական բնութագրիչների համար, ինչպես նաև այլ պատահական մետրիկների խմբերի համար։ применение многомерного статистического анализа для контроля процессов т. хачикян и с. нариманян аннотация в связи со значительным увеличением информационных систем и их интенсивным использованим в нашей повседневной жизни, возникают проблемы, такие как автоматическая идентификация неисправностей системы, нахождение времен резкого изменения стохастических характеристик, а также определение тех характеристик, которые "вышли из-под контроля". для решения таких задач, мы предлагаем алгоритм основанный на многомерном статистическом анализе. алгоритм реализован в среде программирования r и тестирован на случайных метриках сервера веста и в других группах случайных метрик. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 41, 122---130, 2014. 122 image denoising using wavelet transform and cuda hovhannes m. bantikyan state engineering university of armenia e-mail: bantikyan@gmail.com abstract the discrete wavelet transform has a huge number of applications in science, engineering, mathematics and computer science. most notably, it is used for signal coding to represent a discrete signal in a more redundant form, often as a preconditioning for data compression. beginning in the 1990s, wavelets have been found to be a powerful tool for removing noise from a variety of signals (denoising). in this paper the implementation of dwt (discrete wavelet transform)-based denoising algorithm in parallel manner on graphics processing unit is presented, using the cuda technology. keywords: image denoising, discrete wavelet transform, haar wavelet, daubechies wavelet, parallel computing, gpgpu, cuda programming. 1. introduction the wavelet transform, originally developed as a tool for the analysis of seismic data, has been applied in areas as diverse as signal processing, video and image coding, compression, data mining and seismic analysis. the theory of wavelets bears a large similarity to fourier analysis, where a signal is approximated by superposition of sinusoidal functions. a problem, however, is that the sinusoids have an infinite support, which makes fourier analysis less suitable to approximate sharp transitions in a function or signal. wavelet analysis overcomes this problem by using small waves, called wavelets, which have a compact support. one starts with a wavelet prototype function, called a basic wavelet or mother wavelet. then a wavelet basis is constructed by translated and dilated (i.e., rescaled) versions of the basic wavelet. the fundamental idea is to decompose a signal into components with respect to this wavelet basis, and to reconstruct the original signal as a superposition of wavelet basis functions; therefore we speak of a multiresolution analysis. if the shape of the wavelets resembles that of the data, the wavelet analysis results in a sparse representation of the signal, making wavelets an interesting tool for data compression and noise removal [1]. there is a wide range of applications in which denoising is important. examples are medical image analysis, data mining, radio astronomy and many more. each application has its special requirements. for example, noise removal in medical signals requires specific care, since h. bantikyan 123 denoising which involves smoothing of the noisy signal (e.g., using a low-pass filter) may cause the loss of fine details [2]. 2. discrete wavelet transform the main idea of (the first generation) wavelet decomposition for finite 1-d signals is to start from a signal , with n samples (we assume that n is a power of 2). then we apply convolution filtering of by a low pass analysis filter h and downsample the result by a factor of 2 to get an “approximation” signal (or “band”) of length n/2, i.e., half the initial length. similarly, we apply convolution filtering of by a high pass analysis filter g, followed by downsampling, to get a detail signal (or “band”) . then we continue with and repeat the same steps to get further approximation and detail signals and of length n/4. this process is continued a number of times, say j. here j is called the number of levels or stages of the decomposition. the explicit decomposition equations for the individual signal coefficients are: ∑ ∑ , where and are the coefficients of the filters h and g. note that only the approximation bands are successively filtered, the detail bands are left “as is”. this process is presented graphically in fig. 1, where the symbol 2 (enclosed by a circle) indicates downsampling by a factor of 2. this means that after the decomposition the initial data vector is represented by one approximation band and j detail bands , , . . . , . the total length of these approximation and detail bands is equal to the length of the input signal [1]. fig. 1. structure of the forward wavelet transform with j stages: recursively split a signal into approximation bands and detail bands . signal reconstruction is performed by the inverse wavelet transform: first upsample the approximation and detail bands at the coarsest level j, then apply synthesis filters ̃ and ̃ to them, and add the resulting bands. (in the case of orthonormal filters, such as the haar basis, the synthesis filters are essentially equal to the analysis filters.) again this is done recursively. this process is presented graphically in fig. 2, where the symbol 2 indicates upsampling by a factor of 2. fig. 2. structure of the inverse wavelet transform with j stages: recursively upsample, filter and add approximation signals and detail signals . image denoising using wavelet transform and cuda 124 in case of images we first apply dwt for all rows and then for all columns. in fig. 3 dwt of lena image is presented. fig. 3. dwt of lena image with levels j = 1 (left) and j = 2 (right). in this paper are implemented haar and daubechies 2 (db2) discrete wavelet transforms. table 1 shows filter coefficients for discrete haar wavelet transform [9]. table 1. filter coefficients of discrete haar wavelet decomposition low-pass filter h0 = 0.7071067812 h1 = 0.7071067812 high-pass filter g0 = -0.7071067812 g1 = 0.7071067812 reconstruction low-pass filter h0 = 0.7071067812 h1 = 0.7071067812 high-pass filter g0 = 0.7071067812 g1 = -0.7071067812 table 2 shows filter coefficients for discrete daubechies 2 wavelet transform [9]. table 2. filter coefficients of discrete daubechies 2 (db2) wavelet decomposition low-pass filter h0 = -0.1294095226 h1 = 0.2241438680 h2 = 0.8365163037 h3 = 0.4829629131 high-pass filter g0 = -0.4829629131 g1 = 0.8365163037 g2 = -0.2241438680 g3 = -0.1294095226 reconstruction low-pass filter h0 = 0.4829629131 h1 = 0.8365163037 h2 = 0.2241438680 h3 = -0.1294095226 high-pass filter g0 = -0.1294095226 g1 = -0.2241438680 g2 = 0.8365163037 g3 = -0.4829629131 3. wavelet thresholding consider an image of size n×n. assume that it is corrupted by independent and identically distributed (i.i.d) zero mean, white gaussian noise with standard deviation σ. the corrupted image, denoted by , is given as follows: h. bantikyan 125 (1) the orthogonality of dwt (assuming that orthogonal wavelets are used with periodic boundary conditions) leads to the feature that white noise is transformed into white noise. after applying the two-dimensional orthogonal discrete wavelet transform (dwt) to (1), we have (2) where , , and , i, j = 1, 2, ···, n. since dwt is an orthogonal transform, is also an i.i.d gaussian random variable, i.e., is n (0, σ 2 ) [3]. the coefficients of the wavelet transform are usually sparse. that is, most of the coefficients in a noiseless wavelet transform are effectively zero. therefore, we may reformulate the problem of recovering f as one of recovering the coefficients of f which are relatively "stronger" than the gaussian white noise background. that is, the coefficients with small magnitude can be considered as pure noise and should be set to zero. the approach, in which each coefficient is compared with a threshold in order to decide whether it constitutes a desirable part of the original signal or not, is called wavelet thresholding. the thresholding of the wavelet coefficients is usually applied only to the detail coefficients of y rather than to the approximation coefficients , since the latter ones represent 'low-frequency' terms that usually contain important components of the signal, and are less affected by the noise. the thresholding extracts the significant coefficients by setting to zero the coefficients the absolute value of which is below a certain threshold level, which is to be denoted by λ. the thresholded wavelet coefficients are obtained using either a hard or a soft thresholding rule given respectively by: ( ) { | | | | ( ) { | | the hard thresholding rule is usually referred to simply as wavelet thresholding, whereas the soft thresholding rule is usually referred to as a wavelet shrinkage, since it "shrinks" the coefficients with high amplitude towards zero. the thresholding rules are depicted in fig. 4. fig. 4. thresholding, λ=3. image denoising using wavelet transform and cuda 126 most of the algorithms using the thresholding approach try to estimate the optimal value of λ. however, the first step in these algorithms usually involves the estimation of the noise level σ. assuming simply that σ is proportional to the standard deviation of the coefficients is clearly not a good estimator, unless f is reasonably flat. a popular estimate of the noise level σ was proposed by donoho and johnstone [5]. this is based on the last level of the detail coefficients, according to the median absolute deviation: ̂ {| |} (3) where n is the number of coefficients in subband and the factor in the denominator is the scale factor which depends on the distribution of , and is equal to 0.6745 for a normally distributed data. setting for all levels simply to the universal bound √ , where s is the size of the image equal to , provides good results. this method was proposed by donoho and johnstone in [5] and is known as visushrink. 4. parallel implementation and results gpu can process large volume data in parallel when working in single instruction multiple data (simd) mode. in november 2006, the compute unified device architecture (cuda) which is specialized in compute intensive highly parallel computation is unveiled by nvidia. a cudacapable gpu is referred to as a device and the cpu as a host. thread is the finest grain unit of parallelism in cuda [8]. thousands of threads are able to run concurrently on the device. threads are grouped into the warps. size of a warp is usually 32 threads. warp is the smallest unit that can be processed by multiprocessors. warps scheduled across processors of one multiprocessor are coupled into the thread blocks. block is a unit of the resource assignment. typical size of a thread block depends on the particular application what the optimal size of a thread block is to ensure the best utilization of the device. thread blocks form a grid. grid can be viewed as a 1-dimensional, 2dimensional or 3-dimensional array. fig. 5 is depicting the described hierarchy. fig.5. nvidia cuda programming model [8] h. bantikyan 127 first we briefly describe the process to obtain denoised image using 2d-dwt thresholding. each image is processed as follows: 1. perform multiscale decomposition on the image corrupted by gaussian noise. the 2-d orthogonal wavelet transform dwt on a noisy image y is performed up to j th level to generate several subbands. 2. compute threshold λ at subband (using equation 3). 3. for each subband (except the low pass residual), apply λ and then use the soft or hard thresholding method to the noisy coefficients to get the noiseless wavelet coefficients. 4. finally take the inverse wavelet transform of the resultant image obtained in step 3 in order to obtain a denoised image. fig.6. 2d dwt parallel implementation. fig. 6 shows how we launch a kernel for dwt. let’s consider an image with size mxn. number of blocks will be m and number on threads in each block will be n/2. we will have mxn/2 threads working parallel. each thread will perform k th g and h filter operation. threads of each block calculate dwt of one row. this single kernel launch will result dwt of all rows of matrix. then we transpose the matrix, and now by launching the kernel we will get dwt of all columns of matrix. at last we again transpose the matrix. if we compare the proposed cuda version of the dwt with a sequential cpu implementation, we can see a large speedup for large images. table 3 shows the execution time of dwt on cpu and gpu. experiments are performed on cpu: intel(r) core(tm) i3-2100 3,10ghz, and gpu: geforce gt 630, max threads per block: 1024, max blocks in kernel launch: 65,535. we take into account the time needed to copy data and results to and from the gpu. table 3. haar dwt j=1 execution time on cpu and gpu image size 512x512 1024x1024 2048x2048 sequential on cpu 47ms 218ms 1020ms parallel on gpu 16ms 46ms 210ms to compute the threshold λ we need to estimate σ in hhj subband. σ estimation is required to calculate median on hhj subband, and for that task we implement the parallel merge sort algorithm (fig. 7). image denoising using wavelet transform and cuda 128 fig.7. merge sort parallel implementation this algorithm performs in steps. in each step cuda kernel runs n/2 j-1 threads to perform merge operation. to go to the next step we have to wait until the calculation in current step is finished (we call cuda function __syncthreads). after computing threshold we perform thresholding operation to all high pass subbands. we choose either hard or soft thresholding method. at last we perform inverse dwt to thresholded. fig. 8 shows an example of a noisy and denoised image. fig.8. lena original image (left), noisy with variance 0.01 (middle), denoised with haar dwt j=2 (right). in table 4 some experimental results are presented for dwt thresholding for different levels. here we show psnr of original and denoised images. haar and daubechies 2 (db2) wavelets are used, and tested on noisy images with variance 0.01 (psnr = 18) and 0.04 (psnr = 14). second row in this table corresponds to fig. 8. table 4. psnr of original and denoised images wavelet level variance = 0.01 variance = 0.04 haar 1 23.1161 20.2064 haar 2 22.9204 23.7507 haar 3 21.8677 22.8585 daubechies 2 1 22.5815 19.7278 daubechies 2 2 23.3683 23.8697 daubechies 2 3 21.5698 23.4152 h. bantikyan 129 5. conclusion in this paper we have presented and evaluated an implementation of the 2d discrete wavelet transform and wavelet thresholding for cuda-enabled devices. a brief introduction to the wavelet transform and thresholding has been provided prior to explaining our parallelization strategy. we have compared the proposed cuda version of the dwt with a sequential implementation. we have computed psnrs of denoised images for different transformation levels based on visushrink algorithm. as we can see, we gain in performance using parallel gpu algorithm. parallelizing the sequential and simple algorithms will be beneficial to control code complexity and minimize execution time of the process. references [1] w. j. van der laan, c. j. andrei and j. b. t. m. roerdink, “accelerating wavelet lifting on graphics hardware using cuda”, parallel and distributed systems, ieee transactions, vol. 22, pp. 132--146, 2011. [2] (2012) r. cohen, “signal denoising using wavelets”, [online]. available: http://tx.technion.ac.il/~rc/ [3] h. om and m. biswas, “a new image denoising scheme using soft-thresholding”, journal of signal and information processing, vol. 3, pp. 360--363, 2012. [4] p. porwik and a. lisowska, “the haar–wavelet transform in digital image processing: its status and achievements”, machine graphics and vision, vol. 13, pp.79--98, 2004. [5] d. l. donoho and j. m. johnstone, “ideal spatial adaptation by wavelet shrinkage”, biometrika, vol. 81, pp. 425--455, 1994. [6] i. w. selesnick, wavelet transforms a quick study, physics today magazine, september 27, 2007. [7] (2012) nvidia cuda c programming guide, nvidia corp, [online]. available: www.nvidia.com [8] j. sanders and e. kandrot, cuda by example, addison-wesley, 2010. [9] “wavelet properties browser”, [online]. available: http://wavelets.pybytes.com/ submitted 20.12.2013, accepted 05.03.2014. image denoising using wavelet transform and cuda 130 ä³ïï»ñáõù ³õùáõïç ñ»é³óáõù í»ûíé»ã ó¨³÷áëáõãû³ý ¨ cuda ï»ëýáéá·ç³ûç ïիñ³éù³ùµ ð. ´³ýïçïû³ý ²ù÷á÷áõù ¸çëïñ»ï í»ûíé»ã ó¨³÷áëáõãûáõýý áõýի ù»í ãíáí ïçñ³éáõãûáõýý»ñ ×³ñï³ñ³·çïáõãû³ý, ù³ã»ù³ïçï³ûç ¨ ñ³ù³ï³ñ·ã³ûçý ·çïáõãûáõýý»ñç µý³·³í³éý»ñáõù: ð³×³ë ³ûý û·ï³·áñííáõù ¿ ãí³ûçý ³½¹³ýß³ýá ³í»éóáõï³ûçý ï»ëùáí ý»ñï³û³óý»éáõ ¨ ïíû³éý»ñç ë»õùù³ýá ý³ë³å³ïñ³ëï»éáõ ñ³ù³ñ: 1990³ï³ýý»ñçó ի վեր í»ûíé»ãý»ñá ëïë»óçý û·ï³·áñíí»é áñå»ë ï³ñ³ï»ë³ï ³½¹³ýß³ýý»ñáõù ³õùáõïç ñ»é³óù³ý ·áñíçù: ðá¹í³íáõù ý»ñï³û³óí³í ¿ ³õùáõïç ñ»é³óù³ý ³é·áñçãùª ñçùýí³í ¹çëïñ»ï í»ûíé»ã ó¨³÷áëáõãû³ý íñ³, ¨ ½áõ·³ñ»é³óí³í cuda ï»ëýáéá·ç³ûç ùççáóáí: шумоподавление изображения с использованием вейвлет преобразования и cuda о. бантикян аннотация дискретное вейвлет преобразование имеет огромное количество применений в науке, инженерии, математике и информатике. в частности, он используется для кодирования сигнала, для представления дискретного сигнала в более избыточной форме, в качестве предварительной подготовки для сжатия данных. начиная с 1990-х годов, вейвлеты стали применяться как мощный инструмент для удаления шума из различных сигналов. в данной статье представлена реализация алгоритма шумоподавления на основе дискретного вейвлет преобразования, параллельно использующая технологию cuda. d:\sbornik\...\tigran_2014.dvi mathematical problems of computer science 41, 47{54, 2014. t he shannon cipher system with cor r elated sour ce outputs and wir etapper guessing subject to distor ion tig r a n m. ma r g a r ya n institute for informatics and automation problems of nas ra e-mail: tigran@ipia.sci.am abstract the shannon cipher system with discrete memoryless sources is considered. the wiretapper gains the cryptogram through the public noiseless channel and tries to guess the secret information which is related to the encrypted plaintext. it is assumed that at each step of sequential guesses the wiretapper has a testing mechanism to identify the secret message within the given distortion level. the security level of the encryption system is measured by the guessing rate which is the highest asymptotic exponential growth rate of the expected number of guesses. the estimations of guessing rate are obtained. keywords: shannon cipher system, correlated sources, guessing rate. 1 . in t r o d u c t io n th e t h e o r e t ic a l s e c r e c y o f s h a n n o n s̀ m o d e l o f c ip h e r s ys t e m ( s cs ) is t r a d it io n a lly m e a s u r e d b y equivocation [1 ]. in m o s t o f wo r ks in t h is a r e a it is s u p p o s e d t h a t wir e t a p p e r h a s e xa c t ly o n e c h a n c e t o e s t im a t e t h e p la in t e xt . s h a n n o n a ls o g a ve t h e id e a o f p r a c t ic a l s e c r e c y, wh ic h is t h e a ve r a g e a m o u n t o f wo r k r e qu ir e d t o b r e a k t h e ke y. h e llm a n t o o k o n e s t e p fo r wa r d in p r a c t ic a l ( o r c o m p u t a t io n a l) s e c r e c y, p r o p o s e d t o m e a s u r e t h e d e g r e e o f s e c u r it y o f t h e c ip h e r s ys t e m in t e r m s o f t h e e xp e c t e d n u m b e r o f ke y-p la in t e xt c o m b in a t io n s n e e d e d t o e xp la in t h e g ive n c ip h e r t e xt [2 ]. me r h a v a n d a r ika n s u g g e s t e d a n o t h e r s e c u r it y c r it e r io n fo r t h e s cs , guessing exponent, wh ic h is t h e h ig h e s t a s ym p t o t ic e xp o n e n t ia l g r o wt h r a t e o f t h e m o m e n t o f t h e n u m b e r o f g u e s s e s [3 ]. th e g u e s s in g e xp o n e n t fo r e xt e n d e d s cs m o d e ls wa s e xp lo r e d in t h e s e wo r ks : t h e s cs wit h c o r r e la t e d s o u r c e o u t p u t s wa s s t u d ie d b y h a ya s h i a n d y a m a m o t o [4 ] a n d wit h g e n e r a l s o u r c e s wa s s t u d ie d b y h a n a wa l a n d s u n d a r e s a n [5 ]. th e s cs wit h d is t o r t io n a n d r e lia b ilit y r e qu ir e m e n t s wa s in ve s t ig a t e d b y h a r o u t u n ia n a n d gh a z a r ya n [6 , 7 ]. w e h a ve e xa m in e d t h e guessing rate, wh ic h is t h e h ig h e s t a s ym p t o t ic e xp o n e n t ia l g r o wt h r a t e o f t h e e xp e c t e d n u m b e r o f g u e s s e s in m o d e ls : t h e s cs wit h a n o is y c h a n n e l t o t h e wir e t a p p e r [8 , 9 ], t h e s cs wit h t h e c o r r e la t e d s o u r c e o u t p u t s a n d t h e n o is y c h a n n e l [1 0 , 1 1 ], t h e s cs wit h t h e d is t o r t io n a n d t h e n o is y c h a n n e l [1 2 , 1 3 ]. in t h is p a p e r we s t u d y t h e c o m b in e d m o d e l o f t h e s cs c o n s id e r e d in t h e p a p e r s [6 ] a n d [4 ]. th e c r yp t o g r a p h ic s ys t e m d e p ic t e d in fig . 1 is t h e s cs wit h c o r r e la t e d s o u r c e s . th e 4 7 4 8 the shannon cipher system with correlated source outputs and wiretapper guessing subject m e m o r yle s s s o u r c e g e n e r a t e s m u t u a lly c o r r e la t e d m e s s a g e s o n e o f wh ic h is s e c r e t a n d t h e o t h e r m u s t b e t r a n s m it t e d t o t h e le g it im a t e r e c e ive r via a p u b lic c h a n n e l. th e a d ve r s a r y m a y g a in a t r a n s m it t e d m e s s a g e c o n t a in in g r e le va n t in fo r m a t io n a b o u t a s e c r e t m e s s a g e , t h e r e fo r e , it is d e s ir a b le t o t r a n s m it it a ft e r c ip h e r in g e ve n if it is n o t s e c r e t . th e t r a n s m it t e r e n c r yp t s t h e p la in t e xt u s in g t h e ke y g e n e r a t e d b y t h e m e m o r yle s s ke y s o u r c e . th e ke y is a ls o c o m m u n ic a t e d t o t h e d e c r yp t e r b y a s p e c ia l s e c u r e c h a n n e l. a ft e r c ip h e r in g t h e c r yp t o g r a m is s e n t o ve r a p u b lic c h a n n e l t o a le g it im a t e r e c e ive r , wh o c a n r e c o ve r t h e o r ig in a l p la in t e xt u s in g t h e c r yp t o g r a m a n d t h e s a m e ke y. n o t kn o win g t h e ke y, t h e wir e t a p p e r t h a t e a ve s d r o p s o n t h e p u b lic c h a n n e l a im s t o g u e s s t h e s e c r e t m e s s a g e r e la t e d t o t h e c r yp t o g r a m . it is a s s u m e d t h a t t h e wir e t a p p e r kn o ws t h e s o u r c e d is t r ib u t io n s a n d e n c r yp t io n fu n c t io n s a n d t r ie s t o r e c o n s t r u c t t h e s o u r c e s e c r e t m e s s a g e wit h in s o m e g ive n d is t o r t io n m e a s u r e a n d d is t o r t io n le ve l. x; y u x̂ z y s ou r ce key sou r ce c ip h er er fn (y; u) 6 decip h er er f ¡1 n (z; u) 6 legitim ate r eceiv er 6 gu esser w ir etap p er fig. 1. the shannon cipher system with correlated source outputs. fo r a n a p p r o xim a t ive r e c o n s t r u c t io n o f a s e c r e t m e s s a g e t h e wir e t a p p e r m a ke s s e qu e n t ia l g u e s s e s , e a c h t im e a p p lyin g a t e s t in g m e c h a n is m b y wh ic h h e c a n le a r n wh e t h e r t h e e s t im a t e is s u c c e s s fu l o r n o t a n d s t o p s wh e n t h e a n s we r is a ± r m a t ive . ou r g o a l is t o e s t im a t e t h e guessing rate, wh ic h c h a r a c t e r iz e t h e s e c r e c y le ve l o f t h e s ys t e m . 2 . s ys t e m mo d e l a n d d e ¯ n it io n s w e d e n o t e t h e r v b y c a p it a l le t t e r s , t h e r a n d o m ve c t o r b y b o ld c a p it a l le t t e r s , a n d t h e ir r e a liz a t io n s a r e d e n o t e d b y lo we r -c a s e le t t e r s , r e s p e c t ive ly. in t h e s ys t e m s h o wn in fig . 1 . t h e s o u r c e a n d t h e ke y-s o u r c e a r e s t a t io n a r y a n d m e m o r yle s s . th e s o u r c e is a s s u m e d t o g e n e r a t e d is c r e t e m u t u a lly c o r r e la t e d r a n d o m ve c t o r s x a n d y wh ic h c o n s is t o f d is c r e t e , in d e p e n d e n t , id e n t ic a lly d is t r ib u t e d ( i.i.d .) r a n d o m va r ia b le s ( r v s ) ( x1; x2; : : : ; xn ) a n d ( y1; y2; : : : ; yn ) . x is s e c r e t a n d y m u s t b e s e n t t o a le g it im a t e r e c e ive r . y h a s a p r o b a b ilit y d is t r ib u t io n ( p d ) p ¤ = fp ¤ ( y ) ; y 2 yg a n d x h a s a c o n d it io n a l p d v ¤ = fv ¤ ( xjy ) ; x 2 ^x ; y 2 yg wh e r e x a n d y a r e t h e ¯ n it e a lp h a b e t s o f s o u r c e . th e jo in t p d o f t h e p a ir ( x; y ) is p ¤ ± v ¤ = fp ¤ ± v ¤ ( x; y ) = p ¤ ( y ) v ¤ ( xjy ) ; y 2 y; x 2 x g a n d t h e m a r g in a l p d o f x is p ¤v ¤ = fp ¤v ¤ ( x ) = p y2y p ¤ ( y ) v ¤ ( xjy ) ; x 2 x g: th e ke y-s o u r c e g e n e r a t e s t h e r a n d o m ve c t o r u = ( u1; u2; : : : ; uk ) o f k p u r e ly r a n d o m b it s in d e p e n d e n t o f ( x; y ) . u is u s e d fo r e n c r yp t io n a n d a ls o is s e n t t o le g it im a t e r e c e ive r b y a s p e c ia l s e c u r e c h a n n e l. w e e n c r yp t o n ly y u s in g t h e ke y u b y t h e e n c r yp t io n fu n c t io n fn : yn £ u k ! z¤ wh e r e z is t h e c r yp t o g r a m a lp h a b e t . a ft e r c ip h e r in g we o b t a in a r a n d o m ve c t o r z ( m a y h a ve va r ia b le le n g t h d e p e n d in g o n n a n d k ) wh ic h is s e n t via a p u b lic c h a n n e l t o a le g it im a t e t. margaryan 4 9 r e c e ive r . th is e n c r yp t io n fu n c t io n is a s s u m e d t o b e c o n ve r t ib le p r o vid in g t h e g ive n ke y, i.e . t h e r e e xis t s t h e d e c r yp t io n fu n c t io n f¡1n : z¤ £ u k ! yn wh ic h a llo ws t h e le g it im a t e r e c e ive r t o r e c o ve r t h e o r ig in a l y. th e wir e t a p p e r g e t s t h e c r yp t o g r a m z b y t h e p u b lic n o is e le s s c h a n n e l. in t h is t a s k it is a llo we d fo r wir e t a p p e r t o r e c o ve r t h e o r ig in a l s e c r e t m a s s a g e wit h s o m e a c c e p t a b le d e via t io n . d e n o t e va lu e s o f t h e r v x̂ b y t h e x̂ r e p r e s e n t in g t h e r e c o n s t r u c t io n b y t h e wir e t a p p e r o f t h e s o u r c e s e c r e t m e s s a g e wit h va lu e s in t h e ¯ n it e wir e t a p p e r r e p r o d u c t io n a lp h a b e t ^x , in g e n e r a l, d i®e r e n t fr o m x . w e c o n s id e r a s in g le -le t t e r d is t o r t io n m e a s u r e b e t we e n t h e s o u r c e a n d t h e wir e t a p p e r r e p r o d u c t io n m e s s a g e s : d : x £ ^x ! [0 ; 1 ) : th e d is t o r t io n m e a s u r e b e t we e n t h e ve c t o r x 2 x n a n d a wir e t a p p e r r e p r o d u c t io n ve c t o r x̂ 2 ^x n is d e ¯ n e d a s t h e a ve r a g e o f t h e c o m p o n e n t d is t o r t io n s : d( x; x̂ ) 4 = n¡1 nx n=1 d( xn; x̂n ) : th e wir e t a p p e r g e t t in g t h e c r yp t o g r a m z p r o d u c e s s o m e g u e s s in g s t r a t e g y gn = fx̂1 ( z ) ; x̂2 ( z ) ; ¢ ¢ ¢g u n t il s o m e ve c t o r x̂ is fo u n d . w e s a y t h a t t h e g u e s s in g s t r a t e g y is ¢ a c h ie va b le if t h e r e e xis t s s o m e j s u c h t h a t p rfd( x; x̂j ( z ) ) · n ¢ g = 1 ( s e e [1 4 ]) . l e t gnf;g ( x̂jz) b e a n u m b e r o f g u e s s e s n e e d e d fo r t h e wir e t a p p e r t o r e p r o d u c e t h e x̂ b y t h e s t r a t e g y gn . de¯nition 1: the key rate rk of the key source is de¯ned by rk = n ¡1 lo g 2 k = k=n. de¯nition 2: the ¢ -achievable guessing rate r ( p ¤; v ¤; rk ; ¢ ) of this system is de¯ned by r( p ¤; v ¤; rk ; ¢ ) = lim n !1 s u p fn in f gn 1 n lo g e [gnf;g ( x̂jz) ]; where e [gnf;g ( x̂jz) ] is the expectation of gnf;g ( x̂jz) . w e a p p ly t h e m e t h o d o f t yp e s a n d c o ve r in g le m m a ( [1 5 ], [1 6 ]. th e t yp e p o f ve c t o r y = ( y1; : : : ; yn ) 2 yn is a p d p = fp ( y ) = n ( yjy ) =n; y 2 yg, wh e r e n ( yjy ) is t h e n u m b e r o f r e p e t it io n s o f t h e s ym b o l y a m o n g y1; : : : ; yn . th e s e t o f ve c t o r s y o f t yp e p is d e n o t e d b y t np ( y ) . th e s e t o f a ll p d o n y is d e n o t e d b y p ( y ) a n d t h e s u b s e t o f p ( y ) c o n s is t in g o f t h e p o s s ib le t yp e s o f s e qu e n c e s y 2 yn is d e n o t e d b y pn ( y ) . w e d e n o t e e n t r o p y o f r v y wit h p d p a n d , r e s p e c t ive ly, r e la t ive e n t r o p y b e t we e n p a n d p ¤ a s fo llo ws hp ( y ) 4 = ¡ x y2y p ( y ) lo g p ( y ) ; d ( p jjp ¤ ) 4= x y2y p ( y ) lo g p ( y ) p ¤ ( y ) : th e jo in t t yp e o f ve c t o r y 2 yn a n d z 2 zm is t h e p d fm ( y; zjy; z ) =m; y 2 y; z 2 zg, wh e r e m ( y; zjy; z ) is t h e n u m b e r o f o c c u r r e n c e s o f p a ir s ym b o ls ( y; z ) in t h e p a ir o f ve c t o r s ( y; z ) . w e s a y t h a t t h e c o n d it io n a l t yp e o f x fo r t h e g ive n y is p d v = fv ( xjy ) ; x 2 x ; y 2 yg if n ( y; xjy; x) = n ( yjy ) v ( xjy ) fo r a ll y 2 y; x 2 x . th e s e t o f a ll s e qu e n c e s x 2 x n o f t h e c o n d it io n a l t yp e v fo r t h e g ive n y 2 t np ( y ) is d e n o t e d b y t np;v ( xjy ) a n d c a lle d t h e v -s h e ll o f y . v n ( x ; p ) is t h e s e t o f a ll p o s s ib le v -s h e lls o f y o f t yp e p . 5 0 the shannon cipher system with correlated source outputs and wiretapper guessing subject fo r t h e g ive n p d s p ¤ a n d p o f y , c o n d it io n a l p d s v ¤ a n d v o f x fo r t h e g ive n y c o n d it io n a l e n t r o p y o f r v x fo r t h e g ive n r v y is d e ¯ n e d b y hp;v ( xjy ) 4 = ¡ x y2y;x2x p v ( x ) lo g v ( xjy ) ; t h e r e la t ive d ive r g e n c e b e t we e n jo in t p d s p ± v a n d p ¤ ± v ¤ is d e ¯ n e d b y d ( p ± v kp ¤ ± v ¤ ) 4= x y2y;x2x p ( y ) v ( xjy ) lo g p ( y ) v ( xjy ) p ¤ ( y ) v ¤ ( xjy ) : l e t p v = q = fq ( x) ; x 2 x g b e a p d o n x a n d w = fw ( x̂ j x ) ; x 2 x ; x̂ 2 ^x g b e a c o n d it io n a l p d o n ^x fo r t h e g ive n x, t h e n t h e m u t u a l in fo r m a t io n b e t we e n x a n d x̂ is d e ¯ n e d a s iq;w ( x ^ x̂ ) = x x;x̂ q ( x ) w ( x̂ j x) lo g w ( x̂ j x )p x q ( x) w ( x̂ j x ) : th e r a t e -d is t o r t io n fu n c t io n fo r a s o u r c e wit h p d q a n d d is t o r t io n le ve l ¢ is e qu a l t o r( q; ¢ ) = m in w 2w (q;¢ ) iq;w ( x ^ x̂ ) ; ( 1 ) wh e r e w ( q; ¢ ) is a s e t o f c o n d it io n a l p d s w s a t is fyin g t h e fo llo win g c o n d it io n e q; w d ( x ; x̂ ) = x x;x̂ q ( x ) w ( x̂ j x ) d ( x ; x̂ ) · ¢: w e will u s e t h e fo llo win g in e qu a lit ie s a n d c o ve r in g le m m a ( [1 5 ], [1 6 ]) jqn ( x ) j < ( n + 1 ) jx j; ( 2 ) jv n ( x ; p ) j < ( n + 1 ) jyjjx j; ( 3 ) jt np;v ( xjy ) j · e xp fnhp;v ( xjy ) g; ( 4 ) p ¤ ± v ¤ft np;v ( x; y ) g · e xp f¡nd ( p ± v kp ¤ ± v ¤ ) g: ( 5 ) w e wr it e f ( n ) = o( n ) a s n ! 1 t o m e a n t h a t lim n!1 f ( n ) =n = 0 : lemma 1: f or every type q 2 qn ( x ) and distortion level ¢ there exists a collection of vectors l ( n; q; ¢ ) 2 ^x n which covers t nq ( x ) , such that for every vector x 2 t nq ( x ) m in x̂2l (n ; q ;¢) d( x; x̂) · n ¢ ; and lo g jl ( n; q; ¢ ) j = nr ( q; ¢ ) + o( n ) : ( 6 ) w e a ls o u s e t h e fo llo win g n o t a t io n s h( p; v; rk ; ¢ ) = m in fr ( p; ¢ ) ; hp;v ( xjy ) + rkg; h( p; v; rk ) = m in fhp v ( x ) ; hp;v ( xjy ) + rkg a n d h( p; rk ; ¢ ) = m in fr ( p; ¢ ) ; rkg: t. margaryan 5 1 3 . fo r m u la t io n o f r e s u lt in t h e fo llo win g t h e o r e m t h e u p p e r a n d lo we r b o u n d s fo r t h e ¢ -a c h ie va b le g u e s s in g r a t e a r e p r e s e n t e d . t heor em 1: f or the given p d p ¤, conditional p d s v ¤, and any key rate rk , the following estimates are valid: r( p ¤; v ¤; rk ; ¢ ) · m a x p;v [h ( p; v; rk ; ¢ ) ¡ d ( p ± v kp ¤ ± v ¤ ) ]; r( p ¤; v ¤; rk ; ¢ ) ¸ m a x p [h( p; rk ; ¢ ) ¡ d ( p kp ¤ ) ]: cor ollar y 1: w hen the source generate only one message (x = y) we arrive at the result of haroutunian [7] if the reliability goes to in¯nity: r( p ¤; rk ; ¢ ) = m a x p [h( p; rk ; ¢ ) ¡ d ( p jjp ¤ ) ]: cor ollar y 2: w hen ¢ = 0 we arrive at the result hayashi and h. yamamoto [4]: r( p ¤; v ¤; rk ) = m a x p;v [h( p; v; rk ) ¡ d ( p ± v kp ¤ ± v ¤ ) ]: p r oof of t heor em l e t t h e p a ir o f ve c t o r s ( x; y ) b e g e n e r a t e d b y t h e s o u r c e h a vin g t h e jo in t t yp e p ± v ( ( x; y ) 2 tp;v ( x; y ) ) . to b u ild a s t r a t e g y fo r t h e wir e t a p p e r , we c o n s id e r t h e fo llo win g t wo s t r a t e g ie s gn1 a n d g n 2 . strategy gn1 : th e s e t x n c a n b e r e p r e s e n t e d a s t h e u n io n o f ve c t o r s o f va r io u s t yp e s x n = [ i=1;2;¢¢¢;jqn (x )j t nqi ( x ) : th e wir e t a p p e r s h o u ld r e c o n s t r u c t t h e ve c t o r x b y t h e g ive n d is t o r t io n le ve l ¢ . th e wir e t a p p e r s lig h t s t h e c r yp t o g r a m z a n d in t o e a c h t yp e q t r ie s t o ¯ n d s o m e x̂ s u c h t h a t d ( x; x̂ ) · n ¢ fo r e ve r y x 2 t nqi ( x ) . fo r s im p lic it y o f fo r m u la wr it in g we s h a ll n o t e o n ly i in s t e a d o f qi. w e c o n s id e r a g u e s s in g s t r a t e g y t h a t e n u m e r a t e s t h e t yp e s q fr o m a c c o r d in g t o n o n d e c r e a s in g va lu e s o f t h e c o r r e s p o n d in g r a t e -d is t o r t io n fu n c t io n s r( qi; ¢ ) : r( 1 ; ¢ ) · r ( 2 ; ¢ ) · : : :. a c c o r d in g t o t h e le m m a fo r t h e ¯ xe d i t h e wir e t a p p e r r e g a r d le s s o f a r r a n g e m e n t c h o o s e s u c h a c o lle c t io n o f ve c t o r s fx̂1;i; x̂2;i; :::x̂l;i l = 1 ; 2 ::jl( n; i; ¢ ) jg t h a t c o ve r s t ni ( x ) . th e ve c t o r x b e lo n g s t o t nq ( x ) a n d , t h e r e fo r e , it is c le a r t h a t in t h is s t r a t e g y gn1 t h e n u m b e r o f g u e s s e s is b o u n d e d wit h ( 2 ) a n d ( 6 ) in t h e fo llo win g wa y gnf;g1 ( x̂jz ) · x i:r(i;¢ )·r(q;¢ ) jl( n; i; ± ) j · ( n + 1 ) jx j e xp fn ( r( q; ¢ ) ) + o ( n ) g ( 7 ) = e xp fnr ( q; ¢ ) + o( n ) g: 5 2 the shannon cipher system with correlated source outputs and wiretapper guessing subject strategy gn2 : in t h is s u b -s t r a t e g y t h e wir e t a p p e r wa n t s t o ¯ n d x̂ e xa c t ly ( x̂ = x ) . x n c a n b e r e p r e s e n t e d a s a fa m ily o f ve c t o r s o f va r io u s c o n d it io n a l t yp e s fo r e a c h g ive n ve c t o r y 2 t np ( y ) . x n = [ i=1;2;¢¢¢jv n (x ;p )j t np;vi ( xjy ) : th e wir e t a p p e r u s in g a ll ke ys o n c r yp t o g r a m z o b t a in s a ll p o s s ib le fyi; i = 1 ; 2 ::: 2 kg, a ft e r t h a t h e t r ie s t o g u e s s x a s s u m in g c o n d it io n a l e n t r o p y fo r t h e g ive n ve c t o r yi in a s c e n d in g o r d e r ( hp;v1 ( xjy ) · hp;v2 ( xjy ) · ¢ ¢ ¢) t r ie s t o g u e s s x. th e n u m b e r o f g u e s s e s is b o u n d e d wit h t h e h e lp o f ( 3 ) a n d ( 4 ) a s fo llo ws gnf;g2 ( x̂jz ) · x vj :hp;vj (xjy )·hp;v (xjy ) jt np;vj ( xjy ) je xp fkg · ( n + 1 ) jx jjyj e xp fn hp;v ( xjy ) g e xp fn rkg = e xp fn ( hp;v ( xjy ) + rk ) + o ( n ) g: ( 8 ) strategy gn3 : co m b in in g t h e s t r a t e g ie s g n 1 a n d g n 2 , we d e ¯ n e a n e w g n 3 a s fo llo ws gn3 = ( x̂ 1 1; x̂ 2 1; x̂ 1 2; x̂ 2 2 ¢ ¢ ¢ ) : th e n , t h e n u m b e r o f g u e s s e s in t h e s t r a t e g y gn3 is n o t m o r e t h a n t wo fo ld t h e s m a lle r n u m b e r o f g u e s s e s in gn1 a n d g n 2 . w e c a n o b t a in fr o m ( 7 ) a n d ( 8 ) gnf;g3 ( x̂jz ) · 2 m in [e xp fn r( p; ¢ ) + o ( n ) g; e xp fn ( hp;v ( xjy ) + rk ) + o( n ) g] · e xp fnh( p; v; rk ; ¢ ) + o( n ) g: ( 9 ) a p p lyin g t h e in e qu a lit ie s ( 2 ) ,( 3 ) ,( 8 ) ,( 9 ) , t h e e xp e c t a t io n e [gnf;g3 ( x̂jz ) ] c a n b e b o u n d e d in t h e fo llo win g wa y e [gnf;g3 ( x̂jz ) ]· x p;v e xp f¡nd ( p ± v kp ¤ ± v ¤ ) ggnf;g3 ( x̂jz ) · ( n + 1 ) jyjjx j+jx j e xp f¡n d ( p ± v kp ¤ ± v ¤ ) ggnf;g3 ( x̂jz ) · e xp fn ( h ( p; v; rk ; ¢ ) ¡ d ( p ± v kp ¤ ± v ¤ ) ) + o ( n ) g: ( 1 0 ) s in c e o u r s t r a t e g y is va lid fo r a n y fu n c t io n fn , fr o m t h e in e qu a lit y ( 1 0 ) we o b t a in t h e u p p e r b o u n d fo r t h e g u e s s in g r a t e r( p ¤; v ¤; rk ; ¢ ) = lim n !1 s u p fn in f gn 1 n lo g e [gnf;g ( x̂jz ) ] · lim n !1 s u p 1 n lo g e [gnf;g3 ( x̂jz) ] · m a x p;v ( h( p; v; rk ; ¢ ) ¡ d ( p ± v kp ¤ ± v ¤ ) ) : it is o b vio u s t h a t t h e lo we r b o u n d fo r r( p ¤; rk ; ¢ ) ( [7 ]) is a ls o a lo we r b o u n d fo r r( p ¤; v ¤; rk ; ¢ ) r( p ¤; v ¤; rk ; ¢ ) ¸ m a x p [h( p; rk ; ¢ ) ¡ d ( p jjp ¤ ) ]: t heor em is pr oved. t. margaryan 5 3 refer ences [1 ] c. e . s h a n n o n , \ co m m u n ic a t io n t h e o r y o f s e c r e c y s ys t e m s " , b ell system thechnical j ournal, vo l.2 8 , n o . 3 , p p . 5 6 5 -7 1 5 , 1 9 4 9 . 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[1 2 ] e . a . h a r o u t u n ia n a n d t. m. ma r g a r ya n , \ on t h e s h a n n o n cip h e r s ys t e m wit h n o is y ch a n n e l t o t h e w ir e t a p p e r gu e s s in g s u b je c t t o d is t o r t io n cr it e r io n " , annual session d edicated to 90 anniversary of r afael alexandrian, y e r e va n , p p . 5 3 { 5 4 , 2 0 1 3 . [1 3 ] t. m. ma r g a r ya n a n d e . a . h a r o u t u n ia n , \ on t h e s h a n n o n c ip h e r s ys t e m wit h d is t o r t io n a n d g u e s s in g wir e t a p p e r e a ve s d r o p p in g t h r o u g h a n o is y c h a n n e l" , p roceedings of international conference computer science and information technologies, p p . 1 1 6 { 1 2 0 , 2 0 1 3 [1 4 ] e . a r ika n a n d n . me r h a v, \ gu e s s in g s u b je c t t o d is t o r t io n " , ie e e transactions on information theory, vo l. 4 4 , n o . 3 , p p . 1 0 4 1 -1 0 5 6 , 1 9 9 8 . [1 5 ] i. cs is z ¶a r a n d j. k äo r n e r , information theory: coding theorems for d iscrete m emoryless systems, n e w y o r k: a c a d e m ic , 1 9 8 1 . [1 6 ] t. m. co ve r a n d j. a . th o m a s , e lements of information theory,s e c o n d e d it io n , n e w y o r k: w ile y, 2 0 0 6 . 5 4 the shannon cipher system with correlated source outputs and wiretapper guessing subject submitted 10.01.2014, accepted 06.03.2014. îñí³í ß»õáõùáí ñ³ñ³µ»ñ³ïóí³í ñ³õáñ¹³·ñáõãûáõýý»ñáí þ»ýáýû³ý í³íï³·ñù³ý ñ³ù³ï³ñ·á ·áõß³ïáõ ·³õïý³í»ñéáõíáõç ³éï³ûáõãû³ùµ î. ø³ñ·³ñû³ý ²ù÷á÷áõù ¸çï³ñïí»é ¿ ñ³ñ³µ»ñ³ïóí³í ñ³õáñ¹³·ñáõãûáõýý»ñáí þ»ýáýû³ý í³íï³·ñù³ý ñ³ù³ï³ñ·á: ³õïý³·áõá ëï³ýáõù ¿ ·³õïý³·çñá ¨ ó·ïáõù ¿ ·áõß³ï»é ·³õïýç ï»õ»ïáõãûáõýá, áñá ï³åí³í ¿ ·³õïý³·ñç ñ»ï: ºýã³¹ñíáõù ¿, áñ ·áõß³ïù³ý ûáõñ³ù³ýãûáõñ ù³ûéáõù ·³õïý³í»ñéáõíáõá áõýç ëïáõ·ù³ý ù»ë³ýç½ù, áëï áñç ï³ñáõ ¿ çù³ý³é, ã» ³ñ¹ûáù, ·áõß³ïí³í ñ³õáñ¹³·ñáõãûáõýá µ³í³ñ³ñáõù ¿ ïñí³í ß»õù³ýá: ì³íï³·ñù³ý ñ³ù³ï³ñ·ç ·³õïýçáõãû³ý ³ëïç×³ýá ã³÷íáõù ¿ ·áõß³ïù³ý ³ñ³·áõãû³ùµ, áñá ·áõß³ïáõùý»ñç ù³ý³ïç ëå³ë»éç ³ù»ý³ù»í ³ëçùåïáïçï óáõóçãý ¿: ý³ñ³ïí»é ¿ ·³õïý³éëáõç ·áõß³ïù³ý ³ñ³·áõãûáõýá: øåííîíîâñêàÿ ñåêðåòíàÿ ñèñòåìà ñ êîððåëèðîâàííûìè ñîîáùåíèÿìè èñòî÷íèêà ñ çàäàííûì èñêàæåíèåì è óãàäûâàþùèì íàðóøèòåëåì ò. ìàðãàðÿí àííîòàöèÿ â ñòàòüå ðàññìàòðèâàåòñÿ øåííîíîâñêàÿ ñåêðåòíàÿ ñèñòåìà ñ äèñêðåòíûìè èñòî÷íèêàìè áåç ïàìÿòè. íàðóøèòåëü ïîëó÷àåò êðèïòîãðàììó è ñòðåìèòñÿ óãàäàòü ñåêðåòíóþ èíôîðìàöèþ, ñâÿçàííóþ ñ çàøèôðîâàííûì ñîîáùåíèåì. ïðåäïîëàãàòåñÿ, ÷òî íà êàæäîì øàãó íàðóøèòåëü âëàäååò òåñòèðóþùèì ìåõàíèçìîì, èñõîäÿ èç êîòîðîãî îí ìîæåò óçíàòü óäîâëåòâîðÿåò ëè êðèïòîãðàììà äàííîìó èñêàæåíèþ. óðîâåíü ñåêðåòíîñòè êðèïòîãðàôè÷åñêîé ñèñòåìû èçìåðÿåòñÿ ñêîðîñòüþ óãàäûâàíèÿ, êîòîðàÿ îïðåäåëÿåòñÿ íàèáîëüøèì àñèìòîòè÷åñêèì ïîêàçàòåëåì ìàòåìàòè÷åñêîãî îæèäàíèÿ ÷èñëà óãàäûâàíèé íàðóøèòåëÿ. îöåíåíà ñêîðîñòü óãàäûâàíèÿ íàðóøèòåëÿ. mathematical problems of computer science 53, 49–56, 2020. udc 004 a solution for preventing the rogue certificate attack sergey e. abrahamyan1 and arman g. zakaryan2 1institute for informatics and automation problems 2american university of armenia e-mail: sabrahamyan@sci.am, arman zakaryan20@alumni.aua.am abstract in today’s online world, internet security heavily relies on the trust in certificate authorities. modern browsers and operating systems provide a comprehensive list to their users, which includes all the cas they trust by default. this could turn into a serious problem when even one of the cas is compromised and/or goes rogue. it is especially relevant for enterprise applications, as they are more likely to be targeted for this kind of attack. in this paper, we propose a solution which can mitigate this kind of attack against large organizations. we also discuss the security of the proposed method, offering acceptable security/performance tradeoff. keywords: https, tls, digital certificates, masquerade attack, rogue certificate attack, security 1. introduction nowadays, security in digital space is of utmost importance. every day millions of people exchange messages, browse websites, perform banking transactions on the internet, exposing a significant amount of sensitive information to potential adversaries. that is why web service providers implement many techniques to provide sufficient protection to users in the online world. a variety of techniques are implemented to ensure the security of people on the internet, but this paper will primarily focus on authentication. user somehow has to be sure that he is communicating with the intended recipient and not someone who tries to impersonate him/her. for that reason, most of the web services now use digital certificates, which helps users to check if they are communicating with the right entity or not. digital certificates are like passports of web services, which leverage the power of public key cryptography to provide a method for server and client authentication. the current standard for authentication technologies on the internet is tls (transport layer security), a successor of now deprecated ssl (secure sockets layer). authentication is done by a process called tls/ssl handshake, during which the client and server exchange a couple of messages (including a signed x.509 certificate sent by the server) and thus verify each others identities. before the authentication technology became a security standard on the internet, the communication between the client and server was established by http (hypertext transfer 49 50 a solution for preventing rogue certificate attack protocol). data transaction under this protocol is implemented without any encryption. so this protocol is not secure against attacks like man-in-the-middle and eavesdropping. now that most of the web services use authentication, they utilize the protocol https, where s signifies that the communication channel is secure. any data sent through an https connection is encrypted, and the only two parties that can decrypt ciphertexts are the client and server, who exchanged the encryption key during the handshake phase. nowadays, the most common type of certificate used by web services is x.509. it is usually obtained from an entity called certificate authority (ca). it contains information about the owner, such as name, certificate version, validity date range, the encryption/signature algorithm supported by the server, owner’s public key, and most importantly, the signature of the ca. certificates are created as follows: web service requests a certificate from ca, by providing them the necessary information, and cas digitally sign it using their private keys. thus, whoever gets the signed certificate, can verify its validity by checking the signature on it using ca’s public key (which is stored in most modern browsers). after verifying the web service’s identity, user generates a pre-master key, encrypts it using tbe server’s public key and sends to the server. the server uses its private key to decrypt the premaster key. next, the client and server use the same pre-master key to generate a shared secret key and start communicating securely. this system inherently relies on people’s trust in the ca and most of the time it pays off. indeed, it is generally secure against third party intervention, which makes it very practical to use. however, the system is not secure against forged certificates, when the ca itself is involved. for a variety of reasons, such as political, economical or social, ca’s may have an incentive to distribute forged certificates to accomplish a masquerade attack, with the purpose of stealing users’ private information, spreading misinformation, etc. the probability of such attacks is considered to be very small, as it will diminish the credibility of ca, resulting in lawsuits and financial losses, so most of the modern browsers and os’s don’t implement any protection from this kind of attacks. however, if the outcome of the attack is significant, ca’s can take the risk and go rogue. although this can be relevant for individuals, most of the time the targets of this attack are computer networks of big organizations (e.g. banks), as they have access to very sensitive data. the proposed solution could theoretically be extended to individual use as well. in this paper, we consider a masquerade attack, where a malicious ca along with an intruder tries to obtain a legitimate server’s identity, using a forged certificate. we also propose a solution, which will prevent such kind of attack. 2. related work various attempts have been made to overcome forged certificate attacks. http public key pinning [1] is one such technique, which protects users from forged certificate attacks to some extent. it is based on trust on first use security model: the first time a user contacts the web service, it gets the certificate and a list of public keys, saying that these are the only public keys, which should be associated with their service in the future. if users receive a signed certificate with a public key that is not present in that list, then it’s most probably a rogue certificate and must be rejected. this technique was popular for some time among the major web browsers like chrome, firefox and opera. around 2018 chrome announced that they are planning to depreciate and then later remove the support for hpkp in the near future [2] because of multiple security and usability concerns. in the same manner, http strict transport security mechanism offered protection against protocol-downgrade attacks, s. abrahamyan and a. zakaryan 51 which are a part of the forged certificate attack family [3]. as a replacement for hpkp, google announced the certificate transparency project [4]. it is a public log of all issued certificates trusted by the browser. web service owners should regularly check the log for the domain names owned by them, to detect any misissuance of certificates. this provides some level of protection, but it is far from being a near-real time solution. it can take days for website owners to detect any wrongdoings and act appropriately. another solution is to bind certificates to dns records. that structure eliminates the need for ca’s altogether. this mechanism is known as dane [5], which is currently used with some websites, but mainly for smtps and not https. the reason it’s not widespread is because it requires dnssec to operate securely, otherwise dns lookups could be spoofed: man in the middle attacker can simply replace the certificate by another one and successfully carry out an attack. another scheme for protection, which is highly relevant to our case, is described in [6] perspectives. in their system, users submit hash values of the certificates they received to a network of notaries, which monitor them continuously. whenever a user tries to access a web service, it takes the certificate of that service and sends it over to the notary network. there, the certificate is checked with the already stored hash values and in case of a mismatch, notifies the user. this method is a significant improvement in the direction of mitigating the forged certificate attack, however, it has some drawbacks. first of all, it requires the notaries to be a network of trustworthy institutions, such as universities, governmental entities, etc. this really hurts the practicality of the scheme. secondly, it relies on the history of the submitted certificates. in case when a new certificate is issued for a web service, perspectives will trigger a false positive warning. 3. system design first, we fix the formal definition of the attack that our solution aims to mitigate. we assume that the network of the target organization is partially compromised. this can be performed in many ways: attackers could get access to the organization’s dns servers, routers etc. after the compromise, we assume that the attacker can control the traffic and redirect users at any time. once the traffic is captured and appropriately redirected, attackers initiate the rogue ssl certificate attack. the malicious server sends the user a rogue certificate, signed by the malicious ca. now the user’s browser tries to authenticate the server using the received certificate. in the field ”issuer name”, the user sees the name of the malicious server. so the user’s browser checks the received certificate’s signature using the public key of the malicious ca, which is trusted by the browser. in our case, the certificate is signed by the same or any other ca, the root of which is trusted by most browsers. now, when the user wants to access a web service, he is being redirected to a malicious web page impersonating the real one. as the root of the rogue certificate is a ca, which is trusted by the browser, no warnings will be triggered. the session key that is generated between the client and the web service will be encrypted with the attacker’s public key, so he can easily decrypt it and access all the information that is exchanged during the session. 3.1 definition of the rogue certificate attack 52 a solution for preventing rogue certificate attack fig. 1. attacker takes control of user’s dns and replaces the real ip of the requested website with an ip of a malicious website. this website has a forged certificate from a trusted root ca, so the user accepts it. as a solution, we present a system that could be integrated with all modern browsers. the system has two key roles: local caching and acting as a bridge between local and remote machines. caching is needed as a first step local validation of the certificate. to avoid the liabilities of trust-of-first-use approach, if the client is contacting a new web service with no prior mutual history, it naturally skips this first step. after validating it with the remote machine the first time, it stores the hash value of that certificate in a local cache. in the future, as long as the certificate provided by the web server matches the cached one, no further steps are needed. in case of a mismatch, it sends a validation request to the remote server. the remote server is a physical machine that is set up, preferably in a different geographic location (by minimizing the latency/security tradeoff), with reliable internet connection. inside, the machine contains a number of virtual machines, each using a vpn that reroutes the traffic to various geographic locations. when the remote machine gets a validation request, it checks the certificate with all the virtual machines, and then each vm votes on the consensus. if even one of the machines returns a different hash value of a certificate, the response of the server is negative. in case of a negative reply, depending on the end user’s role in the organization, different levels of warnings are issued. if the end user is in a tech savvy sphere, a full description of the issue is presented. otherwise, the system blocks the connection not letting the user access the potentially malicious resource. as the entire organization is going to use the same remote machine, a caching mechanism is also present there. similar to the 3.2 solution architecture s. abrahamyan and a. zakaryan 53 local machine, each validated certificate’s hash value is cached in the remote server as well. this way, once a certificate is marked as suspicious, for each subsequent request with that certificate a warning will be issued without all the checks, decreasing the validation time significantly. the handshake between local and remote machines relies on the prior knowledge about the remote machine. as it is set up by the organization itself, the public key and all the other information typically contained in the certificate are already known and stored on the user’s local machine. even with dns hijacking, attackers can’t impersonate the remote machine, as it would require a remote machine compromise, which is not a part of the considered attack. a proof of concept implementation of the solution is available in our github repository[10]. client side validation algorithm: input: url output: issecure 1. cert = getsslcertificatehash(url) 2. cachedcerts = getlocalcertificatescache() 3. if cert cachedcerts: (a) cachedcert = cachedcerts.find(cert) (b) if cert = cachedcert: i. return true 4. issecure = checkcertificatewithremoteserver(url, cert) 5. return issecure central server side validation algorithm: input: url, cert output: issecure 1. vms = getallavailablevms() 2. issecure = true 3. foreach vm in vms: (a) issecure = vm.checkcertificatewithvm(url, cert) 4. return issecure 54 a solution for preventing rogue certificate attack vm side validation algorithm: input: url, cert output: issecure 1. localized cert = getsslcertificatehash(url) 2. if cert = localized cert: (a) return true 3. return false fig. 2. server gets the certificate, and sends it to the validator server. validator server queries all the connected vms and based on their responses decides to validate or reject the certificate. 4. conclusion the proposed solution provides significant security overhead over the state of the art solutions. however, the practicality aspect of the solution, in this stage is not that strong. s. abrahamyan and a. zakaryan 55 it can easily be deployed and used by large organizations. theoretically, the system can be extended to individual use as well, but the maintenance efforts and costs would not be justified for most of the use cases. one direction for further study is extending our approach for individual use as well. furthermore, the service will eventually have enough labeled data on authentic/rogue certificates. this can be used to analyze and then further predict the probability of a certificate being rogue. methods like the one described by dong et al[7], can benefit from this acquired data, instead of using a generated set of rogue certificates. 5. acknowledgement the authors would like to thank various sponsors for supporting their research. references [1] c. evans, c. palmer and r. sleevi, public key pinning extension for http, ietf, rfc 7469, doi:10.17487/rfc7469, april 2015. [2] r. palmer, “http-based public key pinning (removed)”, retrieved from https://www.chromestatus.com/feature/5903385005916160, 2018. [3] j.hodges, c. jackson and a. barth, “hsts policy”, http strict transport security (hsts), ietf. doi:10.17487/rfc6797. rfc 6797. retrieved 31 january 2018. [4] b. laurie, a. langley and e. kasper, “certificate transparency”, ietf. doi:10.17487/rfc6962. issn 2070-1721. rfc 6962, june 2013. [5] r. barnes,“dane: taking tls authentication to the next level using dnssec”, ietf journal, october 6, 2011, retrieved august 5, 2018. [6] d. wendlandt, d. g. andersen and a. perrig, “perspectives: improving ssh-style host authentication with multi-path probing”, in proc. of usenix’08, vol. 200, pp. 321-334, 2008. [7] z. dong, k. kane and j. camp, “detection of rogue certificates from trusted certificate authorities using deep neural networks”, acm transactions on privacy and security, vol. 19 no. 2, article no. 5., https://doi.org/10.1145/2975591, september 2016. [8] d. fisher, “diginotar says its ca infrastructure was compromised”, retrieved from https:// threatpost.com/diginotar-says-its-ca-infrastructure-was-compromised083011/75594/, 2011. [9] z. durumeric, j. kasten, m. bailey and j. alex halderman, “analysis of the https certificate ecosystem”, in proc. of imc’13, acm, pp. 291-304, 2013. [10] arman zakaryan and sergey abrahamyan, online. [available]: https://github.com/armzak1/rogue certificate detector submitted 25.03.2020, accepted 18.06.2020. 5 6 a solution for preventing rogue certi¯cate attack î»õí ñ³í³ëï³·ñáí ñ³ñó³ïáõùá ï³ýë»éáõ éáõíáõù ê»ñ·»û º. ²µñ³ñ³ùû³ý1 ¨ ²ñù³ý ¶. ¼³ù³ñû³ý2 1ðð ¶²² æýýáñù³ïçï³ûç ¨ ³íïáù³ï³óù³ý åñáµé»ùý»ñç çýëïçïáõï 2 ð³û³ëï³ýç ³ù»ñçïû³ý ñ³ù³éë³ñ³ý e-mail: sabrahamyan@sci.am, arman zakaryan20@alumni.aua.am ²ù÷á÷áõù ø»ñûñû³ ³éó³ýó ³ßë³ññáõù ñ³ù³ó³ýóç ³ýíï³ý·áõãûáõýá ½·³éçáñ»ý ñçùýí³í ¿ ð³í³ëï³·ñ³ûçý ð»õçý³ïáõãûáõýý»ñç (ðð) ñ³ý¹»å íëï³ñáõãû³ý íñ³: ä³ù³ý³ï³ïçó çýï»ñý»ï ¹çï³ñïçãý»ñá ¨ ûå»ñ³óçáý ñ³ù³ï³ñ·»ñá û·ï³ï»ñ»ñçý ïñ³ù³¹ñáõù »ý ³ûý µáéáñ ðð-ý»ñç ó³ýïá, áñáýù ý³ë³å»ë ñ³ù³ñíáõù »ý íëï³ñ»éç: ê³ ï³ñáõ ¿ éáõñç ëý¹çñý»ñ ³é³ç³óý»é, »ñµ íëï³ñí³í ðð-ý»ñçó ýáõûýçëï ù»ïá ¹³éý³ ë³ñ¹³ëáõãû³ý ½áñ ï³ù ñ»ýó çýùá ¹çùç ë³ñ¹³ëáõãû³ý: üù³ý ëý¹çñý»ñç ñçùý³ï³ýáõù ³éýãíáõù »ý ù»í ï³½ù³ï»ñåáõãûáõýý»ñç íñ³·ñ³ûçý ó³ýó»ñá, ù³ýç áñ ¹ñ³ýù »ý ñ³×³ë³ïç ¹³éýáõù ñ³ñó³ïáõùý»ñç ãçñ³ë: ²ûë ³ßë³ï³ýùáõù ù»ýù ³é³ç³ñïáõù »ýù éáõíáõù, áñá ï³ñáõ ¿ ï³ýë»é ýù³ý³ïçå ñ³ñó³ïáõùý»ñá: ²ý¹ñ³¹³ñó ¿ ï³ï³ñí»é ý³¨ ³é³ç³ñïíáõ ù»ãá¹ç ³ýíï³ý·áõãû³ýá` ³é³ç³ñï»éáí ³ñ³·áõãû³ý ¨ ³ýíï³ý·áõãû³ý áý¹áõý»éç ÷áëñ³ñ³µ»ñáõãûáõý: ðåøåíèå äëÿ ïðåäîòâðàùåíèÿ àòàêè ñ ïîääåëüíûì ñåðòèôèêàòîì 1èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà 2àìåðèêàíñêèé óíèâåðñèòåò àðìåíèè e-mail: sabrahamyan@sci.am, arman zakaryan20@alumni.aua.am àííîòàöèÿ ñåðãåé å. àáðààìÿí1 è àðìàí ã. çàêàðÿí2 ´³ý³éç µ³é»ñ` https, tls, ãí³ûçý ñ³í³ëï³·ñ»ñ, ¹çù³ï³ñ³ý¹»ë³ûçý ñ³ñó³ïáõù, ï»õí ñ³í³ëï³·ñáí ñ³ñó³ïáõù, ³ýíï³ý·áõãûáõý: â ñîâðåìåííîì îíëàéí-ìèðå èíòåðíåò-áåçîïàñíîñòü âî ìíîãîì çàâèñèò îò äîâåðèÿ ê öåíòðàì ñåðòèôèêàöèè. ñîâðåìåííûå áðàóçåðû è îïåðàöèîííûå ñèñòåìû ïðåäîñòàâëÿþò ñâîèì ïîëüçîâàòåëÿì ïîëíûé ñïèñîê öåíòðîâ ñåðòèôèêàöèè, êîòîðûì îíè äîâåðÿþò ïî óìîë÷àíèþ. ýòî ìîæåò ïðåâðàòèòüñÿ â ñåðüåçíóþ ïðîáëåìó, åñëè õîòÿ áû îäèí èç öåíòðîâ ñåðòèôèêàöèè áóäåò âçëîìàí è/èëè ñòàíåò ìîøåííèêîì. ýòî îñîáåííî àêòóàëüíî äëÿ êîðïîðàòèâíûõ ïðèëîæåíèé, òàê êàê îíè ñ áîëüøåé âåðîÿòíîñòüþ ñòàíîâÿòñÿ ìèøåíÿìè äëÿ òàêîãî âèäà àòàê. â ýòîé ñòàòüå ìû ïðåäëàãàåì ðåøåíèå, êîòîðîå ìîæåò ñìÿã÷èòü äàííûé âèä àòàêè íà êðóïíûå îðãàíèçàöèè. ìû òàêæå îáñóæäàåì áåçîïàñíîñòü ïðåäëîæåííîãî ìåòîäà, ïðåäñòàâëÿÿ äîïóñòèìûé êîìïðîìèññ áåçîïàñíîñòè/ýôôåêòèâíîñòè. êëþ÷åâûå ñëîâà: https, tls, öèôðîâûå ñåðòèôèêàòû, ìàñêàðàäíàÿ àòàêà, ìîøåííè÷åñêàÿ àòàêà ñåðòèôèêàòà. 06_segrey_abrahamyan_53 06 d:\sbornik\...\article.dvi mathematical problems of computer science 29, 2007, 36{50. n ovel m athematical appr oach for random 3d spin system under the i n°uence of e xter nal field. gener alization of clausius-m ossotti e quation a s h o t s . ge vo r kya n a n d a r a x a . ge vo r kya n institue for informatics and automation problems of nas of ra e-mail: g ashot@sci.am abstract a dielectric medium consisting of roughly polarized molecules has been treated as a 3d disordered spin system. for investigation of statistical properties of this system on scales of space-time periods of standing electromagnetic wave a microscopic approach has been developed. using the birgo® ergodic hypothesis the initial 3d spin problem is reduced to two conditionally separate 1d problems along external electromagnetic ¯eld propagation. the ¯rst problem describes a quantum dynamics of disordered n -particles system with relaxation, while the second one describes statistical properties of steric disordered spin chain system. based on developed in both problems constructions, the coe±cient of polarizability related to collective orientational e®ects was calculated. the clausius-mossotti equation for dielectric constant was generalized on the micrometer space and nanosecond time scales. refer ences [1 ] ch . k it t e l, in t r o d u c t io n t o s o lid s t a t e p h ys ic s , j. w ile y a n d s o n s , in c ., n e w y o r k, l o n d o n , s yd n e y, to r o n t o , 1 9 6 2 . [2 ] d . j. gr i± t h , in t r o d u c t io n t o e le c t r o d yn a m ic s , p r e n t ic h a ll, n e w je r s y, 1 9 2 ( 1 9 8 9 ) . [3 ] r . b e c ke r , e le c t r o m a g n e t ic fie ld s a n d in t e r a c t io n s , d o ve r , n e w y o r k, 9 5 ( 1 9 7 2 ) . [4 ] y . tu , j. te r s o ® a n d g. gr in s t e in , p h ys . r e v. l e t t ., 81, 4 8 9 9 ( 1 9 9 8 ) . [5 ] i. m. l ifs h it s , s . a . gr e d e s ku l, l . a . p a s t u r , in t r o d u c t io n t o t h e t h e o r y o f d is o r d e r e d s ys t e m s , mo s c o w, n a u ka , ( in r u s s ia n ) 1 9 8 2 . [6 ] a . v . b o g d a n o v, a .s . ge vo r kya n a n d a . g. gr ig o r ya n , a ms / ip s t u d ie s in a d v. ma t h e m a t ic s , 13, 8 1 ( 1 9 9 9 ) . [7 ] a . s . ge vo r kya n a n d ch in -k u n h u , p r o c e e d in g s o f t h e is a a c co n f. o n a n a lys is , y e r e va n , a r m e n ia , e d s b y g. a . b a r s e g ia n e t a l, 1 6 4 ( 2 0 0 4 ) . [8 ] l . b e r t h ie r a n d a . p . y o u n g , tim e a n d le n g t h s c a le s in s p in g la s s , a r x iv;c o n d n a t / 0 3 1 0 7 2 1 v1 3 0 oc t 2 0 0 3 . 3 6 a. s. gevorkyan, ar. a. gevorkyan 3 7 [9 ] p . v . p a vlo v, a . f. k h o kh lo v, s o lid s t a t e p h ys ic s , h ig h s c h o o l b o o k co m p a n y, mo s c o w, ( in r u s s ia n ) 2 0 0 0 . [1 0 ] g. a . k o r n a n d t. m. k o r n , ma t h e m a t ic a l h a n d b o o k, fo r s c ie n t is t a n d e n g in e e r s , mc gr a w-h ill b o o k co m p a n y, n e w y o r k, 1 9 6 8 . [1 1 ] w . h . za c h a r ia s e n , j. a m . ch e m . s o c . 54, 3 8 4 1 ( 1 9 3 2 ) . [1 2 ] k . b in d e r a n d a .p . y o u n g , r e v. mo d . p h ys ic s , 58, 4 , 8 0 1 ( 1 9 8 6 ) . [1 3 ] ch . d a s g u r t a , s h a n g -ke n g ma , a n d ch in -k u n h u , p h ys . r e v. b 20, 3 8 3 7 ( 1 9 7 9 ) . [1 4 ] j. l . va n h e m m e n , in p r o c e e d in g s o f t h e h e id e lb e r g co lo qu iu m o n s p in gla s s e s , e d . b y j. l . va n h e m m e n a n d i. mo r g e n s t e r n , l e c t u r e n o t e s in p h ys ic s 192, ( s p in g e r , b e r lin , 1 9 8 3 ) . [1 5 ] b . a . b e r g a n d t. n e u h a u s ,, p h ys . r e v. l e t t . 68, 9 ( 1 9 9 2 ) . [1 6 ] m. a b r a m o wit z a n d i. s t e g u n , h a n d b o o k o f m a t h e m a t ic a l fu n c t io n s , n a t . b u r e a u o f s t a n d a r d s a p p lie d ma t h e m a t ic s s e r ie s -5 5 , 1 9 6 4 . [1 7 ] s . f. e d wa r d s a n d p . w . a n d e r s o n , j. p h ys . f 9, 9 6 5 ( 1 9 7 5 ) . [1 8 ] m. v . fe d o r yu k, me t h o d o f s a d d le -p o in t , mo s c o w, n a u ka , ( in r u s s ia n ) 1 9 7 7 . üáñ ù³ã»ù³ïçï³ï³ý ùáï»óáõù ãï³ñ·³íáñí³í 3d ëåçý³ûçý ñ³ù³ï³ñ·ç ñ³ù³ñ ³ñï³ùçý ¹³ßïç ³½¹»óáõãû³ý ï³ï: îé³áõ½çáõë-øáëáïçç ñ³í³ë³ñù³ý áý¹ñ³ýñ³óáõùá ². ê. ¶¨áñ·û³ý ¨ ². ². ¶¨áñ·û³ý ²ù÷á÷áõù îáßï µ¨»é³óù³ùµ ùáé»ïáõéý»ñ å³ñáõý³ïáõ ¹ç¿é»ïïñçï ùçç³í³ûñ»ñá ý»ñï³û³óíáõù ¿ áñå»ë ãï³ñ·³íáñí³í 3d ëåçý³ûçý ñ³ù³ï³ñ·: ¼³ñ·³óí³í ¿ ùçïñáëïáåçï å³ïï»ñ³óáõù ï³ý·áõý ¿é»ïïñáù³·ýçë³ï³ý ³éçùç ï³ñ³í³-å³ù³ý³ï³ûçý å³ñµ»ñáõãû³ý íñ³ ³û¹ ñ³ù³ï³ñ·ç íç×³ï³·ñ³ï³ý ñ³ïïáõãûáõýý»ñá áõëáõùý³ëçñ»éáõ ñ³ù³ñ: ú·ï³·áñíí³í ¿ ´çñ·áýýç »ñ·á¹çï ñçåáã»½á ¨ ëï½µý³ï³ý 3d ëåçý³ûçý ëý¹çñá, ³ñï³ùçý ¿é»ïïñáù³·ýçë³ï³ý ¹³ßïç ï³ñ³íù³ý áõõõáõãû³ùµ ó¨³÷áëí³í ¿ »ñïáõ å³ûù³ý³ï³ýáñ»ý ³ýï³ë 1d ëý¹çñý»ñç: ²é³ççý ëý¹çñá ýï³ñ³·ñáõù ¿ ãï³ñ·³íáñí³í n-ù³ëýçï³ýç ñ³ù³ï³ñ·ç ëý¹çñá ñ³ßíç ³éý»éáí ùçç³í³ûñç é»é³ïë³óç³ý, ³ûý ¹»åùáõù »ñµ »ñïñáñ¹áª ýï³ñ³·ñáõù ¿ ï³ñ³í³ï³ý ëåçý³ûçý ßõã³ûç íç×³ï³·ñ³ï³ý ñ³ïïáõãûáõýý»ñá: ðçùýí»éáí »ñïáõ ëý¹çñý»ñáõù ½³ñ·³óí³í ï³éáõóí³íùý»ñç íñ³ ñ³ßí³í ¿ ëùµ³ûçý áõõõáñ¹í³íáõãû³ý »ñ¨áõûãáí å³ûù³ý³íáñí³í µ¨»é³óí³íáõãû³ý ·áñí³ïçóá: îé³áõ½çáõë-øáëëáïïçç ñ³í³ë³ñáõùá ¹ç¿é»ïïñçï ñ³ëï³ïáõýç ñ³ù³ñ áý¹ñ³ýñ³óí³í ¿ ï³ñ³í³-å³ù³ý³ï³ûçý ý³ýáù³ëßï³µý»ñç íñ³: начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 50, 88--95, 2018. secure storage implementation for large files in ios environment levon m. hovsepyan and aren k. mayilyan national polytechnic university of armenia e-mail: lehovsepyan@gmail.com, mayilyan96@mail.ru abstract mobile device storage services are not secure by nature as all databases and files are stored on the client side. we cannot ignore that because nowadays mobile devices are made of hardware, which is capable for building and running applications, which cover almost every functionality that computers have. there is an inherent risk of data exposure (confidentiality) and data tampering (integrity). to avoid the risks mentioned above, software engineers use some approaches, which are provided by the ios sdk for securely storing sensitive data. however, those approaches currently work only for small amount of data (key-value pairs) and the issue still remains for large files. in this paper, we introduced an approach of securely storing large files in ios environment. keywords: ios, secure storage, data protection, encryption 1. introduction nowadays people use smartphones more often than computers in daily life. researches show that in 2016, an estimated 62.9 percent of the population worldwide already owned mobile phones [1], and in 2019 the number of mobile phone users is forecast to reach 4.68 billion. the tendency towards mobile devices leads to technological progress and hardware technologies already reached a level that giant manufacturers like apple, samsung, xiaomi produce smartphones with technical characteristics capable to compete with computers by their performance and run heavy cpu/gpu intensive applications like different games and ai powered apps. this mobile revolution brought almost every sphere to a digital platform, which raised data protection and privacy risks. for example, a lot of personal information, photos, videos and 88 mailto:lehovsepyan@gmail.com https://www.statista.com/statistics/274774/forecast-of-mobile-phone-users-worldwide/ l. hovsepyan and a. mayilyan 89 other files are now being published on social platforms, and it’s an urgent task for the software company to protect that data and make it accessible only for specified people. another example is that people store their passwords inside one software for remembering them easily, and the result of data leak here can be irreversible. those two examples mentioned above highlight the importance of data protection. in enterprise solutions, data protection is relatively easy than in mobile ones. in enterprise systems, data is being stored on server side, inside the specially designed databases or even data centers. this approach is more secure in contrast to mobile solutions for the following reasons:  server side hardware has more computational power and memory storage, and in addition to it, both of these parameters can be increased any time.  the access to server side storage is happening through network, and many penetration attempts can be blocked at network layer.  server side data storage can be isolated from network at any time to reduce the access in mobile platform, data is stored in embedded memory of mobile device, which gives full control to an intruder over it. even though ios has a secure file system divided into sandboxes [2], as shown in figure 1, it becomes accessible after device jailbreak. this means that the data can be observed as it is from the file directories. the only data protection method that can work here is to store everything in encrypted form, which will prevent data from being leaked even if it is accessible. fig.1 . an ios app operating within its own sandbox directory. as a solution, ios platform gives keychain [3] to developers, which is a shared key-value secure storage. it works fine for small data like passwords, cryptographic keys, certificates and notes, but when it comes to larger files, keychain becomes problematic to use. as it is shared https://developer.apple.com/library/archive/documentation/filemanagement/conceptual/filesystemprogrammingguide/filesystemoverview/filesystemoverview.html https://developer.apple.com/documentation/security/keychain_services secure storage implementation for large files in ios environment 90 storage between devices, which are connected to the same apple id, it syncs the stored files to icloud storage. for large files it will work slow and can cost a lot of money for network traffic. in this paper, we introduce an approach for securely storing large files in local memory. as the mobile devices have limitations in memory, the approach supports incremental usage as well. the security will be based on aes-256 encryption algorithm, and the encryption key will be securely stored in keychain storage. 2. related work as data protection is an actively studied subject, there are many suggested solutions in regard to how to securely store the data. particularly for ios platform, the most common solutions for storing data are core data and keychain. core data is apple’s persistence framework with an underlying sqlite database. this means that developers interact with core data methods, not the database directly. an important observation is that sqlite is not encrypted by default, when the device is unlocked. nevertheless, apple has a feature called “data protection” [4], which encrypts the sandbox while the device is locked with passcode. however, we cannot rely on the end user to passcodeprotect his/her device. the device can be jailbroken and passcodes can be easy to crack. thus, we see that this solution has vulnerabilities and contains risks for storing sensitive data. the second secure option that ios sdk gives is keychain. keychain api’s solve secure storage problems by giving the app a mechanism to store small bits of user data in an encrypted database. the keychain is not limited to passwords, as shown in figure 2. you can store other secrets that the user explicitly cares about, such as credit card information or even short notes. it is also a good place for storing items that the user needs, but may not be aware of, e.g., the cryptographic keys and certificates. fig. 2. securing the user's secrets in a keychain. keychain items are encrypted using two different aes-256-gcm keys: a table key (metadata) and per-row key (secret key). as a metadata, “created” and “last updated” timestamps are being generated, when storing data. keychain metadata is encrypted with the metadata key to speedsearch, while searching secret value is encrypted with the secret-key. the keychain is implemented as a sqlite database stored in the file system. there is only one database, and the “securityid” daemon determines which keychain items each process or app can access. keychain items can only be shared between apps from the same developer. this is managed by requiring third-party apps to use access groups with a prefix allocated to them https://developer.apple.com/documentation/uikit/core_app/protecting_the_user_s_privacy l. hovsepyan and a. mayilyan 91 through the apple developer program through application groups. the prefix requirement and application group uniqueness are enforced through code signing [5], provisioning profiles and apple developer program. even though keychain is a securely designed and easy-to-use solution: its disadvantage is the incapability to store large files. 3. overview of our approach as we saw in the previous section, native approaches have some disadvantages, and it is reasonable to come up with a solution, which will cover them. firstly, we need to choose the correct encryption algorithm. the most important characteristic to keep in mind when choosing an encryption algorithm is that you should select one that is widely used and accepted by the security community. the security of any encryption algorithm should never depend upon the secrecy of the algorithm itself. instead, the algorithm should be publicly disclosed and open to the cryptographic community for analysis. the true security of the algorithm always lies in the security of the keys used to decrypt message. after some researches and experiments aes-256 with cbc mode was chosen as the encryption algorithm. aes is advanced encryption standard, symmetric block encryption mechanism. 256 in aes-256 is a key size in bits. it is based on a design principle known as a substitution-permutation network, and is efficient in both software and hardware. aes-256 is a variant of rijndael, which has a fixed block size of 128 bits, and a key size of 256 bits. this cypher is considered among the tops and is widely used in ssl/tls over the internet. the only weakness of this algorithm is the key that we choose. as long as we choose a strong key for it, aes-256 will keep our files safe. as we are developing an approach for mobile side encryption, we consider that there is no way to retrieve encryption key from outside the phone. hence, the method of storing the key itself becomes a vulnerable point. if we hardcode the encryption key inside our source code, it can be easily hacked after jailbreak using reverse engineering [6] techniques. to avoid that issue, we suggest to generate an encryption key based on the advertising identifier (idfa), which is a unique id for each ios device’s application that mobile ad networks typically use to serve targeted ads. this unique id will not be visible in the source code even when you reverse engineer the app, therefore it will prevent the secret key exposure. for key generation, we use pbkdf2, which is a key derivation function [7], which derives a secret key from idfa using a pseudorandom function [8]. after successful generation, we store the generated secret key in the keychain where we can be sure it is safe. now when the secret key is generated, we are ready to encrypt any file and store it inside the “documents” directory of the application. there are various open-source cryptographic algorithm implementations available to use in ios, which are mostly based on apple’s corecrypto library. for our approach we chose rncryptor [9], because it supports incremental use [10]. incremental use of encryption is useful for cases where the data will no comfortably fit in memory, which is a common situation for mobile devices, or when the data is received via internet with chunks. receiving and encrypting data chunks asynchronously saves time. the overall process of file encryption is presented in figure 3. https://www.cisco.com/c/en/us/about/press/internet-protocol-journal/back-issues/table-contents-21/code-signing.html https://www.researchgate.net/publication/261199437_reverse_engineering_ios_mobile_applications https://www.researchgate.net/publication/220905226_on_the_security_of_key_derivation_functions https://dl.acm.org/citation.cfm?id=312213 https://github.com/rncryptor/rncryptor https://www.researchgate.net/lite.publication.publicationdownloadcitationmodal.downloadcitation.html?file-type-radio=ris&citation-radio=citation&publicationuid=316263841 secure storage implementation for large files in ios environment 92 fig. 3. overall process of file encryption. as the approach is designed for large files, it is important to have acceptable performance. performance measurements have been done on iphone 7, which has an average computing power. encryption experiments for relatively large files showed acceptable performance, as you can see in figure 4. the overall performance of data storage is presented in figure 5, which shows a comparison of data storing with and without encryption. the results can vary on different devices based on cpu power, but in average they are acceptable. fig. 4. file size/encryption time measurements. l. hovsepyan and a. mayilyan 93 fig. 5. file storage performance with/without encryption. 4. conclusion in this paper, we have shown an approach for securely storing large files in local memory of an ios device using cryptographic algorithms. we have shown how to generate the encryption key to protect it in case the application is reverse-engineered. this approach differs from the existing ones, because it still protects your data, even when the device is jailbroken and the access to applications file system is open to anyone. moreover, this approach increases the performance of data storage with the help of incremental encryption. references [1] [online]. available: https://www.statista.com/statistics/274774/forecast-of-mobilephone-users-worldwide/ [2] [online]. available: https://cocoacasts.com/what-is-application-sandboxing [3] apple’s ios security guide, pp 20-22, 2018 [4] https://developer.apple.com/documentation/uikit/core_app/protecting_the_user_s_privac [5] code signing the internet protocol journal, volume 5, number 1 by eric fleischman [6] e. mona and m. ali, “reverse engineering ios mobile applications”, proceedingsworking conference on reverse engineering, wcre, 10.1109/wcre.2012.27, pp. 177186, 2012. [7] c. adams, g. kramer, s. mister and r. zuccherato, “on the security of key derivation functions”, 7th internation conference on information security, lecture notes in computer science, vol. 3225, pp. 134-145, 2004. https://www.statista.com/statistics/274774/forecast-of-mobile-phone-users-worldwide/ https://www.statista.com/statistics/274774/forecast-of-mobile-phone-users-worldwide/ https://cocoacasts.com/what-is-application-sandboxing https://developer.apple.com/documentation/uikit/core_app/protecting_the_user_s_privac secure storage implementation for large files in ios environment 94 [8] j. hastad, r. impagliazzo, l. levin and m. luby, “a pseudorandom generator from any one-way function”, siam journal on computing. 28. 10.1137/s0097539793244708, vol.-28, no. 4, pp. 1364-1396, 1999. [9] [online]. available: https://github.com/rncryptor/rncryptor [10] i. mironov, o. pandey, o. reingold, and g. segev, “incremental deterministic publickey encryption”, journal of cryptology. 31. 10.1007/s00145-017-9252, vol.31, no. 1, pp. 134-161, 2017. submitted 05.06.2018, accepted 22.11.2018. ios միջավայրում մեծ ֆայլերի ապահով պահպանման մեթոդի իրականացում լ. հովսեփյան և ա․ մայիլյան ամփոփում բջջային սարքավորումներում տվյալների պահպանման ծառայություններն ապահով չեն իրենց բնույթով, քանի որ բոլոր տվյալների հենքերը և ֆայլերը պահպանվում են օգտագործողի կողմում։ մենք չենք կարող այս երևույթից խուսափել, որովհետև մեր օրերում բջջային սարքավորումները պատրաստվում են այնպիսի ապարատային սարքավորումներից, որոնք ի զորու են կառուցել և աշխատեցնել հավելվածներ, որոնք ծածկում են համակարգչին բնորոշ գրեթե բոլոր ֆունկցիոնալ հնարավորությունները, և շատ մեծ կիրառություն ունեն։ ապահով չլինելուց բխում են տվյալների բացահայտման (գաղտնիություն) և խարդախությամբ նրանց մեջ միջամտման (ամբողջականության) ռիսկեր։ վերոնշյալ ռիսկերից խուսափելու համար, ծրագրային ճարտարագետներն օգտագործում են ios sdk-ի կողմից տրամադրվող մոտեցումներ, նշանակալի կարևորության ֆայլերի ապահով պահպանման համար։ այնուամենայնիվ, այդ մոտեցումները ներկա պահին կիրառելի են միայն փոքր ծավալի տվյալների (բանալի-արժեք) դեպքում, և խնդիրը դեռևս առկա է մեծ ֆայլերի դեպքում։ այս հոդվածում մենք առաջարկում ենք մոտեցում, ios միջավայրում մեծ ֆայլերի ապահով պահպանման համար։ https://github.com/rncryptor/rncryptor l. hovsepyan and a. mayilyan 95 реализация безопасного способа хранения больших файлов в ios среде л. овсепян и а. маилян аннотация сервисы хранения данных для мобильных устройств по натуре не зашищены, так как все базы данных и файлы хранятся в стороне пользователя. мы не можем обойти это, потому что в наше время мобильные устройства изготовлены из апаратных средств, которые способны строить и привести в работу приложение, которые в себе содержат весь функционал, что имеет компьютер, и имеют массовое примемение. от этого существует неотъемлемый риск экспозиции (конфиденциальность) и фальсификации данных (целостность). чтобы избежать упомянутых выше рисков, разработчики используют подходы предоставляемые ios sdk для безопасного хранения конфиденциальных данных. тем не менее, эти подходы в насоящее время работают только для небольшого количества данных (пары ключ-значение), и проблема остается для больших файлов. в этой статье мы представили подход безопасного хранения больших файлов в среде ios. 1. introduction 2. related work 3. overview of our approach 4. conclusion ios միջավայրում մեծ ֆայլերի ապահով պահպանման մեթոդի իրականացում լ. հովսեփյան և ա․ մայիլյան ամփոփում аннотация mathematical problems of computer science 50, 96--103, 2018. experimenting with acquisition of and matching to systemic classifiers sedrak v. grigoryan and nairi p. hakobyan institute for informatics and automation problems of nas ra e-mail: addressforsd@gmail.com, hakobyannairi@gmail.com abstract we are modeling acquisition and classification abilities for the machine. the research line we follow, is based on the ideas of inventors of algorithms letting constructively model human computations and on some extension of those ideas aimed to model constructively other mental doings [1, 2]. we question the issues of acquisition of and matching to systemic classifiers and experimenting to prove the adequacy of our models. we experiment in the frame of rgt class of combinatorial problems for a rgt kernel problem, chess. keywords: systemic classifiers, matching, chess, rgt, cognitive systems, acquisition. 1. introduction we study the nature and the functions of cognition. there are different lines of researches in this area, e.g., machine learning solutions, such as neural networks concentrate on modeling of biological nature of human brain. we concentrate on studying of the nature of cognitive functions and the ways they are being processed. suggested in [1, 2] theory of mental doings provides ways for constructive and adequate models of various cognitive functions, e.g., classification, explanation, etc. in this work we conduct experiments on developed models of systemic classification theory addressing to described in [3] acquisition of systemic classifiers and matching to them. 1.1 human in universe problems human deals with realities, some of which are not classified, while he/she mainly deals with classified realities. classified realities can be divided into regularized and not regularized. a mighty way of enhancement of effectiveness of mental systems, and thus, cognizers, is the regularization of classifiers induced by mdoers and mental systems. 96 s. grigoryan and n. hakobyan 97 namely, classifiers cl of members x of communities c are regularized in c if accompanied by ontological in c methods, instructions allowing x regularly provide positive samples of inputs of cl as well as let the members of c do the same by communicating with x. [1]. 1.2 constructively regularized classifiers regularized classifiers can be considered in two spaces: constructively regularized and nonconstructively regularized. in constructive regularization those samples can be provided deterministically and without any involvement of cellular, while, otherwise, can be grown up from a priory given prototypes like cells or crystals, be the products of services to humans or machines. regularized classifiers induced can be equally represented, we assume, by (fuzzy) methods, and thus, following church thesis by (fuzzy) algorithms. regularly provided positives r of classifiers cl and cl themselves are interpreted as models of classifiers cl’ if r are classified as positives of cl` and cl are interpreted as adequate models of cl’ if positives r meet certain additional requirements focused for positives of cl. for example, algorithms are adequate models of deterministic methods if, following church, to any method by certain instructions equal algorithms can be corresponded. [1] 1.3 the line of our research interpreting the aims of algorithms to enhance the effectiveness of classifiers of deterministic methods other classifiers focusing the mental ones that extend algorithms are suggested and the following statements were provided [2]: i. algorithms are modeling and constructively regularize deterministic methods. ii. oo languages are constructively regularized and strongly expand algorithms. iii. mentals are constructively regularized and strongly expand ool. iv. mentals can consist of functional and connectivity mental models. v. for languages l of communities c allowing the members x of c to communicate, i.e., to explain and understand mental systems of each other expressed in l, it can be constructed communication algorithms lc letting computers communicate mental models mns of mss ms of c equally with respect to the members of c if mns and ms are equal to each other. natural languages contain a large number of constructively classified mentals, e.g., english has about 300 000 classifiers. [2] 1.4. current work following [1,2], algorithms are type of systems constructively modeling computational mental doings over numeric input ids of realities and oo languages expand them by adding attributing/have, parenting/be and do relations. mentals aim to expand oo languages to systems, exempted from the requirement of only numeric inputs, i.e., allowing with numeric input ids provided by experts the ones of any given sensors. while we need to provide ways to construct relationships we identify in natural languages, for now we concentrate on providing main have/be/do relations. we are going to experiment with mentals representing mental systems of rgt problems. experimenting with acquisition of and matching to systemic classifiers 98 1.5. rgt class of problems to provide a certain assessment, progress and statements we concentrate on a class of combinatorial problems, which is regularized and where space of solutions are reproducible game trees (rgt)[4,5]. rgt is a class of problems that satisfies the following conditions: 1. there are (a) interacting actors (players, competitors, etc.,) performing (b) identified types of actions in the (c) specified moments of time and (d) specified types of situations. 2. there are identified benefits for each of the actors. 3. there are descriptions of the situations the actors act in and transformed after actions. chess and chess-like games, network intrusion protection, management in oligopoly competition are considered as rgt class problems. ([4-6]). it has been shown [4] that the kernel of these problems is unique, which lets having unified framework for achievements and experiment solutions and achievements for a certain kernel problem (say chess) and then spread the solution to the whole class. thus, in the following work we conduct an experiment developed by rgt solver, which implements mentals for rgt problems and relevant knowledge. in the previous works models and algorithms for matching rgts situations were introduced [3], but their adequacy was not demonstrated, hence, in the following work we aim to provide experiments for rgt solver acquisition and matching functionalities, particularly we are going to experiment ‘mate’ chess concept acquisition and matching to situations. 2. acquisition of systemic classifiers oop is the most widely spread programming language now. it deals with objects and relations between them, as well as classes of objects. it is close to natural understanding of realities. we focus on the mental abilities that algorithms or oo languages model poorly or partially. we have a natural language, its description, we want to make the characteristics of a natural language that will be adequate and constructive to language classifiers. we must test whether we can do the same thing by mentals. we classify our mentals and mental systems in humanspace interaction and try to reflect the idea of a human being there. to narrow it down, we took rgt class, where the problem can be considered as a universe, and the solution can be interpreted as a human solution. for example, the described problem will sound for chess approximately as follows: can we do everything through the rgt class in solver that we can do through a natural fig. 1. peculiarities of rgt class. fig. 2. classifier ‘check’ by json. s. grigoryan and n. hakobyan 99 language? particularly, can we explain some idea to the solver, and then expect it to explain it back to us? here comes the acquisition problem. the knowledge taught by the teacher has been mastered through experiments. the expert is able to transfer his classifiers regularly to the solver. the knowledge is transmitted to solver by json format, which includes classifier id, type, possible parent id, its attributes, actions (if any) etc. here is an example of input for simple chess classifier ‘check’. classifier ‘check’ consists of two attributes/classifiers ‘field under attack’ and ‘king’. attributes for them are hidden in the picture for better understanding. here are inputs for both of them: ‘field under attack’ consists of an action attribute, i.e., ’move figure’ and ’field’ attribute (i.e., where that figure can be moved), and if the color of a figure in ’move figure’ and ’field’ mismatch (that condition is being checked in attributes of ’check.fieldunderattack.f), then we have the classifier of ’field under attack’ matched. classifier ‘king’ is a minimal classifier, so it has only nuclear classifiers as attributes. well, if we get an instance with the value of figure type equal to 6 (symbolic presentation of the king), ‘king’ is classified. here we have an additional condition checking in king.cx and king.cy, which means that the coordinates of king must be equal to the coordinates of a ‘field’ in classifier ‘field under attack’, which will activate matching of ‘check’. we develop structures similar to those of oop, including 3 dimensions of a natural language grammar, “have”, which is the relation between composers and attributes in oop classes, “be”, which is the inheritance of classes from each other and “do”, which is the ability to define methods in classes. other than hbd dimensions, we aim to provide the ability to define virtual classes, as it is in oop, e.g., “field under attack” concept of chess is virtual in its essence, so we need to provide the way to do it. we need to start definition of any problem from basic types, this is in parallel to oop can be compared with the declaration of built-in types. in comparison with human acquisition, the currently provided models in solver have some restrictions, particularly the human acquisition can be performed from examples, by inductive learning, while solver at the moment implements only one of acquisition ways, by certain exact models provided by the expert. transferring of expert models is adequate to the interaction of an expert and another person, where the expert starts passing mentals and classifiers from a certain level (background) and the human can acquire different levels and types of mentals, such as virtual concepts similar to oop abstract classes, simple rules and different composites. the acquisition is performed iteratively, level by level, starting from the level solver knows, in chess we consider the first level the set of 4 concepts (figure type, color and x/y coordinates). the rest of systemic classifiers acquired from the expert are classifiers that compose hbd relations and new rules. for example, the acquisition of a check requires an acquisition of chess concepts 'king', 'field under attack', which are acquired similar to the acquisition by a human. at the moment, we have transferred the main classifiers of chess to solver, and experiments show that each of them has been adequately acquired by the system. fig. 3. classifier ‘field under attack’ by json. experimenting with acquisition of and matching to systemic classifiers 100 3. matching to systemic classifiers fig. 4. chess example with classifier ‘check’ matching. in this chapter we will demonstrate the matching algorithm on the example of the chess complex concept “mate”. checkmate (often shortened to mate) is a game position in chess and other chess-like games, in which a player's king is in check (threatened with capture) and there is no way to remove the threat. checkmating the opponent wins the game [7]. we define ‘mate’ as a composite classifier with the following attributes: a. ‘check’, which is a composite classifier, b. ‘king has no move’, which is a dynamic classifier, c. ‘king has no defence’, which is also a dynamic classifier. let’s go deeper into the whole matching algorithm for this example. in figure 4, an example of chess concept ‘checkmate’ or ‘mate’ is shown. let's see which minimal classifiers are activated by our matching algorithm. the minimal classifier ‘white rook’ activates because its attributes match (figure colour is white; figure type is equal to the figure type of rook). then, by the bottom-up movement, it activates its parent ‘rook’, then ‘rook’ activates ‘figure’ and finally ‘figure’ activates ‘field’. from the left we see a similar example for the ‘empty field’. here ‘rook’ and ‘white rook’ are connected by the relationship have. other figures also activate similarly (‘black king’ -> ‘king’ -> ..., ‘white bishop’ -> ’bishop’). in figure 4, composite classifier ‘check’ matching is visualized. ‘check’ has the following attributes – 1. ‘field under attack’, which is activated by its child ‘field under attack of bishop’, the latter is activated by its attributes ‘field’ (min. classifier), ‘move bishop’ (action, activates with set ‘free diagonal’ and min. classifier ‘bishop’) and ‘bishop’ (min. classifier) and 2. ‘king’, which must be the same ‘king’ that activates ‘field’, which was mentioned in 1. for dynamic classifier ‘king has no move’ matching steps are as follows: 1. precondition matching of minimal classifier ‘king’ (parent of ‘black king’). 2. action – matching of action ‘move king’, which can (and should be applied for matching the dynamic) be applied to the situation. actions should be applied the number of times equal to the variable ‘depth’, which is 1 by default. 3. postcondition – after applying the action(s) (blue arrow) composite classifier ‘check’ should be matched (its matching was described previously) for every applied action (in this case, 3 possible moves for the king, after each ‘check’ will be matched). for dynamic classifier ‘king has no defence’ matching steps are as follows: 1. precondition matching of minimal classifier ‘king’ (parent of ‘black king’). s. grigoryan and n. hakobyan 101 2. action – matching of action ‘move figure' (in this case, figure should have the same figure colour as king in precondition), which can (and should be applied for matching the dynamic) be applied to the situation. actions should be applied the number of times equal to the variable ‘depth’, which is 1 by default. 3. postcondition – after applying the action(s) (blue arrow) composite classifier ‘check’ should be matched (its matching was described previously) for every applied action (in this case, 3 possible moves for black pawn, one for black rook, and 6 for black knight, after each ‘check’ will be matched). to sum up, we brought detailed visualization for matching composite classifier ‘mate’. as in previous figures, blue blocks are for min. classifiers, green for sets, yellow for actions, grey for dynamics and red for composites. white arrow is for action applying and colourful arrows are for derivative to parent relation (relation have). matching and classification algorithms, in comparison with human approaches, appear to be very similar. first level of matching is transforming input situations from their natural presentation to the numerical or understandable format. this step is currently in progress in solver and machine learning techniques are being used to transform situation from chess board images to a set of input instances. second level of classification and matching is processing the situation and finding the expected instances of classifiers (can be both do classifiers and systemic). this is done already and current experiments show the adequacy of the algorithms to human approaches. for systemic classifiers matching is done level by level, searching from top to bottom, e.g., to classify the situation and see if there is check or no it can search for check iteratively to bottom. first it will look for check itself and then match its components and related classifiers, 'king' and 'field under check' and so until it reaches to already classified instances. third level is using effectors to provide the output. currently solver simply names the matched classifiers as output and provides its instance in its presentation. as an improvement to it we will transform that output to situations on chess board images. as a summary to this section, adequacy of matching to systemic and do classifiers was demonstrated, which can be improved by providing sensors and effectors, which will let interact to solver more naturally. 4. conclusion from the urgent cognition modelling and understanding problems we consider the problem of knowledge acquisition and classification of realities by mentals, which extend object oriented programming language abilities. we also consider rgt class of combinatorial problems, particularly chess, as a kernel problem. solver is backed by systemic classification theory, and acquisition algorithms were implemented, and in the current work we have the following: 1. knowledge acquisition algorithms and structures developed in solver 18 are experimented chess. different chess concepts, including mate were acquired by solver adequately. in this paper, we have also shown how it is being modelled and how knowledge is being acquired by rgt solver 18. 2. matching algorithms for different types of acquired classifiers were experimented for chess and the adequacy was demonstrated, particularly chess concepts were matched to situations, including 'mate'. 3. provided experiments demonstrate the adequacy of acquisition and classification of situations from input by specific instances solver expects and it behaves like do classifiers experimenting with acquisition of and matching to systemic classifiers 102 described in [1, 2] while classifiers can be queried from their attributes thus showing their systemic essence and can be queried from not only expected certain instance based situations but also from natural presentation of realities and situations (it is in next steps of development). 4. plans for the next stage are the following: a) experimenting the performance of sensors implementation for chess, as for now we pass situations by instances of certain classifiers (i.e. transforming chess board images to rgt presentation); b) integration and experimenting of planning in chess endgames (e.g., rook vs king). references [1] e.pogossian, “towards adequate constructive models of mental systems”, proceedings of international conference of computer science and information technologies (csit), pp. 37-46, 2017. ieeexplore ttp://ieeexplore.ieee.org/xpl/mostrecentissue.jsp?punumber=8307363 [2] e. pogossian, “on the way to dominating cognition”, transactions of iiap nas ra, mathematical problems of computer sciences, vol. 48, pp. 79-91, 2018. [3] s. grigoryan, n. hakobyan and h.vrtanesyan “object –oriented modeling of matching to systemic classifiers”, transactions of iiap nas ra mathematical problems of computer sciences, vol. 48. pp. 115-121, 2018. [4] e. pogossian, “effectiveness enhancing knowledge based strategies for ssrgt class of defense problems”, nato asi 2011 prediction and recognition of piracy efforts using collaborative human-centric information systems, salamanca, spain, p. 16, 2011. [5] s. grigoryan “research and development of algorithms and programs of knowledge acquisition and their effective application to resistance problems”, phd, p 111, yerevan, armenia, 2016. [6] e. pogossian, “adaptation of combinatorial algorithms”, academy of sci. of armenia, p.293, 1983, yerevan [7] [online]. available: https://en.wikipedia.org/wiki/checkmate submitted 22.06.2018, accepted 04.12.2018. սիստեմիկ դասակարգիչների յուրացման և դասակարգման փորձարկումներ ս. գրիգորյան և ն. հակոբյան ամփոփում ուսումնասիրվում է յուրացման և դասակարգման կարողությունները մեքենայի համար մոդելավորելու հնարավորությունը: մեր հետազոտական ուղղվածությունը https://en.wikipedia.org/wiki/checkmate s. grigoryan and n. hakobyan 103 հիմնված է ալգորիթմների հիմնադիրների գաղափարների վրա, որը թույլ է տալիս մեզ կառուցողական կերպով մոդելավորել հաշվողականությունը՝ այլ մտավոր գործողությունները կառուցողական կերպով մոդելավորելու նպատակով ([1,2]): ներկայացվում են սիստեմիկ դասակարգիչների համակարգչային յուրացման օկծ մոդելներ և դրանց ճշտության փորձարկումներ շախմատի համար՝ rgt կոմբինատոր խնդիրների դասի շրջանակներում: экспериментирование компьютеризации и проверки системных классификаторов с. григорян и н. акопян аннотация исследуется возможность компьютерного моделирования приобретения системных классификаторов и проверки их правильной работы. исследовательское направление, которую мы придерживаемся, продолжает идеи изобретателей алгоритмов, позволяя конструктивно моделировать вычислимость, с целью конструктивного моделирования других ментальных функций [1,2]. представлены ооп модели компьютерного приобретения системных классификаторов и эксперименты проверки их правильной работы для шахмат в рамках класса rgt комбинаторных проблем. mathematical problems of computer science 40, 44---54, 2013. 44 time domain feature extraction and svm processing for activity recognition using smartphone signals sahak i. kaghyan1 and hakob g. sarukhanyan2 1armenian-russian (slavonic) university 2institute for informatics and automation problems of nas ra e-mail: sahak.kaghyan@gmail.com, hakop@ipia.sci.am abstract automatic classification of human movement is a feature that is desired for a multitude of applications and mobile phone technology continuously evolves and incorporates more and more sensors to enable advanced applications. combining these two concepts we can deal with “an activity recognition via smartphone sensors” problem where sensors of these devices play a core role when we deal with personalized activity tracking systems. in this paper we give an overview of the recent work in the field of activity recognition from mobile devices that can be attached to different parts of the body (pocket, wrist, and forearm). we focus on the technique of feature extraction from raw acceleration signal sequences of smartphone (mean, standard deviation, minimal and maximal signal values, correlation, median crossing). further processing of these data allowed to classify the activity performed by the user. the core classification stage of the current approach was based on the method of “learning with the teacher” where the features of signal sequences were analyzed using the support vector machines (svm) learning method. keywords: smartphone, accelerometer, activity recognition, time domain feature, svm. 1. introduction nowadays the internet and personal computer are the most common ways to connect people, allowing them to share information with each other. on the other hand, none of them is able to reach each person anywhere and anytime like the cell phone does. moreover, concerning the mobile technologies in general, now they are becoming ubiquitous all over the world, changing the way we communicate, conduct trade, and provide care and services. certainly, some of the most compelling benefits of mobile technologies are in the areas of disease prevention, chronic disease management and improving healthcare delivery. for all the advances that occur in mobile healthcare (mhealth), its full potential for a very large group of beneficiaries – the elderly and those who support them – is s. kaghyan, h. sarukhanyan 45 only starting to emerge. the ability to monitor the physical state of a person leads us to the concept of personalized healthcare system implementation. one of the ways to help people with diseases is to give the doctors an opportunity to remotely monitor their patients’ life activity via cellular phones and smartphones they care. human activity recognition (har) has matured in recent years which will enable many health promotions and intervention applications. there are no standardized performance evaluation strategies. recent efforts on designing public datasets might be one of the approaches to address this problem. generally, activity recognition (ar) aims to identify the actions taken by a person. three main classes of activity recognition are considered including coarse location tracking, video stream analysis and inertial navigation systems (ins) such as accelerometers. sensor data are typically communicated from sensors to servers for further processing. alternatively signal processing can be performed in mobile devices such as smart-phones. many authors usually don’t use standard tests for accuracy rate checks and validity of most re ported results depends on testing specifics. there is no consensus even on a standard list of activities, but most of the reports include “walking”, “sitting”, “jogging” and “standing” patterns. recognition can be accomplished, for example, by exploiting the information retrieved from inertial sensors such as accelerometers [4]. in some smartphones these sensors are embedded by default and we benefit from this to classify a set of physical activities (standing, walking, laying, , walking upstairs and walking downstairs) by processing inertial body signals through a supervised machine learning (ml) algorithm for hardware with limited resources. so, in general, activity recognition algorithms can be divided into two major categories. the first one is based on supervised and unsupervised machine learning methods. supervised learning requires the use of labeled data upon which an algorithm is trained. 2. related works recognizing a predefined set of activities is a recognition (classification) task: features are extracted from the space-time information collected by sensors and then used for classification. feature representations are used to map the data to another representation space with the intention of making the classification problem easier to solve. in most cases, a model of classification is used that relates the activity to sensor patterns. learning of such models is usually done in a supervised manner (human labeling) and requires a large annotated datasets recorded in different settings. smart phones include various sensors such as gyroscopes, accelerometers, proximity sensors and have become affordable and ubiquitous. convenient user interfaces make them attractive for all population groups. oner et al. in [1] presented an early work on a pedometer mobile application that was coupled with e-mail to notify medical assistants or family members. their purpose was to use a mobile smart phone to detect the fall event regardless of the phone position or orientation. the algorithm that was introduced in the article was based on the acceleration peak detection and was tested for different conditions. das et al in [3] introduced an attempt to recognize the activity using motorola droid smartphone. activity classification was done through several stages: data acquisition, signal processing, feature extraction and classification. using the nearest neighbor classifier the program could predict patterns or activities with 93% accuracy after it had been calibrated for a particular user. one of our early works [4] introduced a general method of classification that used the nearest neighbor method and showed 80% of accuracy. time domain feature extraction and svm processing for activity recognition using smartphone signals46 3. feature extraction concept there are many terms, used almost interchangeably, or the process of extracting the important features from a set of data, including “feature extraction,” “feature (subset) selection,” “data mining,” “information content” and “(feature) dimension reduction.” whatever the term used, the process of feature extraction can be defined as “a process of identifying valid, useful and understandable patterns in data”. the large amount of time-series data that is generated in a gait analysis study is called “high dimensional” data. high dimensional data can be thought of simply as lots of data (there are lots of dimensions to the data). fast developing gait measuring techniques and methodologies generate more and more data. this in turn results in what is known as the “curse of dimensionality” (more and more data to manage, more dimensions). the objective of feature extraction is to keep all the useful features of the data and discard all the redundant parts of the data. there are two required outcomes to this process: (1) reduce the amount of data to a manageable level (dimension reduction), and (2) keep the most important features of the data and eliminate all the redundant features of the data (feature selection). the idea is to provide a “summary” which can be used to give a meaningful interpretation of the data. the objective of the first step towards the feature selection is a dimension reduction, to reduce the search space to a lower, more manageable dimensionality and this has to be achieved in a way that retains relevant features of the data and removes irrelevant features. that is, the feature selection selects “m” relevant features from the entire set of “n” features such that m>>>>>< >>>>>>: p; wh e n p · 2 ±; p ¡ 1 ; wh e n 2 ± + 1 · p · 3 ± ¡ 2 ; q 2 p ¡ 1 0 + ( ± ¡ 7 2 ) 2 + ± + 1 2 ; wh e n p ¸ 3 ± ¡ 1 : s in c e g is 2 -c o n n e c t e d , we h a ve n ¸ 3 . if p · 2 ±, t h e n b y th e o r e m 1 , c ¸ p, im p lyin g t h a t c = p = n ( g is h a m ilt o n ia n ) a n d c = p > p 2 p. n e xt , if 2 ± + 1 · p · 3 ± ¡ 2 , t h e n b y th e o r e m 1 , c ¸ p ¡ 1 . s in c e p ¸ 2 ± + 1 ¸ 5 , we h a ve c ¸ p ¡ 1 ¸ 2 ± a n d c ¸ p ¡ 1 > p 2 p. fin a lly, if p ¸ 3 ± ¡ 1 , t h e n s 2 p ¡ 1 0 + µ ± ¡ 7 2 ¶2 ¸ s 2 ( 3 ± ¡ 1 ) ¡ 1 0 + µ ± ¡ 7 2 ¶2 = ± ¡ 1 2 ; ¤g.g. nicoghossian (up to 1997) 4 4 c. mosesyan and zh. nikoghosyan 4 5 im p lyin g t h a t c ¸ 2 ± ( b y th e o r e m 1 ) a n d µ ± + 1 2 ¶ s 2 p ¡ 1 0 + µ ± ¡ 7 2 ¶2 + ±2 ¡ 3 ± + 5 4 ¸ µ ± + 1 2 ¶ µ ± ¡ 1 2 ¶ + ±2 ¡ 3 ± + 5 4 = ( ± ¡ 1 ) ( 2 ± ¡ 1 ) > 0 : ob s e r vin g t h a t t h e in e qu a lit y µ ± + 1 2 ¶ s 2 p ¡ 1 0 + µ ± ¡ 7 2 ¶2 + ±2 ¡ 3 ± + 5 4 > 0 is e qu iva le n t t o s 2 p ¡ 1 0 + µ ± ¡ 7 2 ¶2 + ± + 1 2 > q 2 p; we c o n c lu d e ( b y th e o r e m 1 ) t h a t c > p 2 p. th u s , th e o r e m 1 yie ld s th e o r e m a a n d is s t r o n g e r t h a n th e o r e m b . to s h o w t h a t th e o r e m 1 is b e s t p o s s ib le in a s e n s e , o b s e r ve ¯ r s t t h a t in g e n e r a l, p ¸ c, t h a t is c = p wh e n p · 2 ±, im p lyin g t h a t t h e b o u n d c ¸ p in th e o r e m 1 c a n n o t b e r e p la c e d b y c ¸ p + 1 . on t h e o t h e r h a n d , t h e g r a p h k±;±+1, wh e r e p = 2 ± + 1 a n d c = 2 ± = p ¡ 1 s h o ws t h a t t h e c o n d it io n p · 2 ± c a n n o t b e r e la xe d t o p · 2 ± + 1 . in a d d it io n , t h e g r a p h k±;±+1, wh e r e c = p, s h o ws t h a t t h e b o u n d c ¸ p ¡ 1 ( wh e n 2 ± + 1 · p · 3 ± ¡ 2 ) c a n n o t b e r e p la c e d b y c ¸ p. fu r t h e r , t h e g r a p h k2 + 3 k±¡1, wh e r e n = p = 3 ± ¡ 1 a n d c = 2 ± · p ¡ 2 s h o ws t h a t t h e c o n d it io n p · 3 ± ¡ 2 c a n n o t b e r e la xe d t o p · 3 ± ¡ 1 . fin a lly, t h e s a m e g r a p h k2 + 3 k±¡1, wh e r e p = 3 ± ¡ 1 a n d c = 2 ± = s 2 p ¡ 1 0 + µ ± ¡ 7 2 ¶2 + ± + 1 2 ; s h o ws t h a t t h e b o u n d q 2 p ¡ 1 0 + ( ± ¡ 7 2 ) 2 + ± + 1 2 in th e o r e m 1 c a n n o t b e im p r o ve d t o q 2 p ¡ 1 0 + ( ± ¡ 7 2 ) 2 + ± + 1 . fo r a s p e c ia l c a s e wh e n 2 ± + 1 · p · 3 ± ¡ 2 , we u s e t h e r e s u lt o f oz e ki a n d y a m a s h it a [3 ]. t heor em c: [3 ]. l et g be a 2-connected graph. then either ( i ) c ¸ p ¡ 1 o r ( ii) c ¸ 3 ± ¡ 3 o r ( iii ) · = 2 a n d p ¸ 3 ± ¡ 1 . 2 . n o t a t io n a n d p r e lim in a r ie s th e s e t o f ve r t ic e s o f a g r a p h g is d e n o t e d b y v ( g ) a n d t h e s e t o f e d g e s b y e ( g ) . th e n e ig h b o r h o o d o f a ve r t e x x 2 v ( g ) will b e d e n o t e d b y n ( x ) . w e u s e d( x ) t o d e n o t e jn ( x ) j. p a t h s a n d c yc le s in a g r a p h g a r e c o n s id e r e d a s s u b g r a p h s o f g. if q is a p a t h o r a c yc le , t h e n t h e o r d e r o f q, d e n o t e d b y jqj, is jv ( q ) j. w e wr it e a c yc le q wit h a g ive n o r ie n t a t io n b y ¡! q . fo r x; y 2 v ( q ) , we d e n o t e b y x¡!q y t h e s u b p a t h o f q in t h e c h o s e n d ir e c t io n fr o m x t o y. fo r x 2 v ( q) , we d e n o t e t h e h-t h s u c c e s s o r a n d t h e h-t h p r e d e c e s s o r o f x o n ¡!q b y x+h a n d x¡h, r e s p e c t ive ly. w e a b b r e via t e x+1 a n d x¡1 b y x+ a n d x¡, r e s p e c t ive ly. fo r 4 6 a common generalization of dirac's two theorems u µ v ( q ) , u + = fu+ju 2 ug a n d u ¡ = fu¡ju 2 ug. w e s a y t h a t ve r t e x z1 p r e c e d e s ve r t e x z2 o n ¡! q if z1, z2 o c c u r o n ¡! q in t h is o r d e r , a n d in d ic a t e t h is r e la t io n s h ip b y z1 á z2. w e will wr it e z1 ¹ z2 wh e n e it h e r z1 = z2 o r z1 á z2. l e t p = x ¡! p y b e a p a t h . a vin e o n p is a s e t fli = xi ¡! l iyi : 1 · i · mg o f in t e r n a lly-d is jo in t p a t h s s u c h t h a t ( a ) v ( li ) \ v ( p ) = fxi; yig ( i = 1 ; :::; m) , ( b ) x = x1 á x2 á y1 ¹ x3 á y2 ¹ x4 á ::: ¹ xm á ym¡1 á ym = y o n p . t he vine lemma: [4 ]. l et g be a k-connected graph and p a path in g. then there are k ¡ 1 pairwise-disjoint vines on p . th e n e xt t h r e e le m m a s a r e c r u c ia l fo r t h e p r o o f o f th e o r e m 1 . lemma 1: l et g be a connected graph and p = x ¡! p y a longest path in g. ( i ) if xz; yz¡ 2 e ( g ) for some z 2 v ( x+¡!p y ) , then c = p = n, that is g is hamiltonian. ( ii) if d( x ) + d( y ) ¸ p, then c = p = n. ( iii ) l et yz1; xz2 2 e ( g ) for some z1; z2 2 v ( p ) with x á z1 á z2 á y and jz1 ¡! p z2j ¸ 3 . if xz; yz 62 e ( g) for each z 2 v ( z+1 ¡! p z¡2 ) and d ( x) + d ( y ) ¸ p + 3 ¡ jz1 ¡! p z2j, then c = p. lemma 2: l et g be a 2-connected graph and fl1; l2; :::; lmg be a vine on a longest path of g. then c ¸ 2 p ¡ 1 0 m + 1 + 4 : lemma 3: l et g be a connected graph and fl1; l2; :::; lmg be a vine on a longest path p = x ¡! p y of g. then either c = p or c ¸ d( x ) + d( y ) + m ¡ 2 . 3 . p r o o fs p r oof of lemma 1: ( i ) l e t xz; yz¡ 2 e ( g ) fo r s o m e z 2 v ( x+¡!p y ) . th e n c ¸ jxz¡!p yz¡ã¡p xj = p. if v ( g) = v ( p ) , t h e n c le a r ly c = p. ot h e r wis e , r e c a llin g t h a t g is c o n n e c t e d , we c a n fo r m a p a t h lo n g e r t h a t p , a c o n t r a d ic t io n . ( ii) l e t d( x ) + d( y ) ¸ p. if xz; yz¡ 2 e ( g) fo r s o m e z 2 v ( x+¡!p y ) , t h e n we c a n a r g u e a s in ( i) . ot h e r wis e n ( x ) \ n + ( y ) = ;. ob s e r vin g a ls o t h a t x 62 n ( x) [ n + ( y ) , we g e t p ¸ jn ( x ) j + jn + ( y ) j + jfx1gj = jn ( x) j + jn ( y ) j + 1 = d( x ) + d( y ) + 1 ; c o n t r a d ic t in g t h e h yp o t h e s is . ( iii ) a s s u m e t h e c o n t r a r y, t h a t is c · p ¡ 1 . th e n b y ( i ) , n ( x ) \ n + ( y ) = ;. cle a r ly, x 62 n ( x) [ n + ( y ) . fu r t h e r , b y t h e h yp o t h e s is , v ( z+21 ¡! p z¡2 ) \ ( n ( x ) [ n + ( y ) ) = ;; im p lyin g t h a t p ¸ jfxgj + jn ( x) j + jn + ( y ) j + jv ( z+21 ¡! p z¡2 ) j c. mosesyan and zh. nikoghosyan 4 7 = d( x ) + d( y ) + jz1 ¡! p z2j ¡ 2 ; c o n t r a d ic t in g t h e h yp o t h e s is . th u s , c = p. l e m m a 1 is p r o ve d . p r oof of lemma 2: l e t p = x ¡! p y b e a lo n g e s t p a t h in g. p u t li = xi ¡! l iyi ( i = 1 ; :::; m) ; a1 = x1 ¡! p x2; am = ym¡1 ¡! p ym; ai = yi¡1 ¡! p xi+1 ( i = 2 ; 3 ; :::; m ¡ 1 ) ; bi = xi+1 ¡! p yi ( i = 1 ; :::; m ¡ 1 ) ; jaij ¡ 1 = ai ( i = 1 ; :::; m ) ; jbij ¡ 1 = bi ( i = 1 ; :::; m ¡ 1 ) : b y c o m b in in g a p p r o p r ia t e li; ai; bi, we c a n fo r m t h e fo llo win g c yc le s : q1 = m[ i=1 ai [ m[ i=1 li; q2 = m¡1[ i=1 ai [ bm¡1 [ m¡1[ i=1 li; q3 = m[ i=2 ai [ b1 [ m[ i=2 li; ri = bi [ ai+1 [ bi+1 [ li+1 ( i = 1 ; :::; m ¡ 2 ) : s in c e jlij ¸ 2 ( i = 1 ; :::; m ) , we h a ve c ¸ jq1j = mx i=1 ai + mx i=1 ( jlij ¡ 1 ) ¸ mx i=1 ai + m; c ¸ jq2j = bm¡1 + m¡1x i=1 ai + m¡1x i=1 ( jlij ¡ 1 ) ¸ bm¡1 + m¡1x i=1 ai + m ¡ 1 ; c ¸ jq3j = b1 + mx i=2 ai + mx i=2 ( jlij ¡ 1 ) ¸ b1 + mx i=2 ai + m ¡ 1 ; c ¸ jrij = bi + ai+1 + bi+1 + jli+1j ¡ 1 ¸ bi + ai+1 + bi+1 + 1 ( i = 1 ; :::; m ¡ 2 ) : b y s u m m in g , we g e t ( m + 1 ) c ¸ ã 2 mx i=1 ai + 2 m¡1x i=1 bi ! + 2 m¡1x i=2 ai + 4 m ¡ 4 ¸ 2 ã mx i=1 ai + m¡1x i=1 bi + 1 ! + 4 m ¡ 6 = 2 p + 4 m ¡ 6 ; im p lyin g t h a t c ¸ 2 p ¡ 1 0 m + 1 + 4 : l e m m a 2 is p r o ve d . p r oof of lemma 3: if m = 1 , t h e n xy 2 e ( g ) a n d b y l e m m a 1 ( i ) , c = p. l e t m ¸ 2 . p u t li = xi ¡! l iyi ( i = 1 ; :::; m ) a n d le t ai; bi; ai; bi; qi 4 8 a common generalization of dirac's two theorems b e a s d e ¯ n e d in t h e p r o o f o f l e m m a 2 . case 1: m = 2 . a s s u m e wit h o u t lo s s o f g e n e r a lit y t h a t l1 a n d l2 a r e c h o s e n s o a s t o m in im iz e b1. th is m e a n s t h a t n ( x ) [ n ( y ) µ v ( a1 [ a2 ) . b y l e m m a 1 ( iii ) , e it h e r c = p o r d( x ) + d( y ) · p + 2 ¡ jz1 ¡! p z2j = p + 1 ¡ b1. if c = p, t h e n we a r e d o n e . l e t d ( x) + d( y ) · p + 1 ¡ b1, t h a t is p ¸ d ( x) + d( y ) + b1 ¡ 1 . th e n p = a1 + a2 + b1 + 1 ¸ d ( x) + d ( y ) + b1 ¡ 1 , im p lyin g t h a t c ¸ jq1j = a1 + a2 + 2 ¸ d( x ) + d ( y ) = d( x ) + d( y ) + m ¡ 2 : case 2: m = 3 . l e t xz1; yz2 2 e ( g ) fo r s o m e z1; z2 2 v ( p ) . if z2 á z1 t h e n fxz1; yz2g is a vin e c o n s is t in g o f t wo p a t h s ( e d g e s ) a n d we c a n a r g u e a s in ca s e 1 . ot h e r wis e we h a ve n ( x ) µ v ( a1 [ a2 ) ; n ( y ) µ v ( a2 [ a3 ) a n d z1 ¹ z2 fo r e a c h z1 2 n ( x) a n d z2 2 n ( y ) . th e r e fo r e , a1 + a2 + a3 ¸ d( x ) + d( y ) ¡ 2 a n d c ¸ jq1j = a1 + a2 + a3 + 3 ¸ d( x ) + d( y ) + 1 = d( x ) + d( y ) + m ¡ 2 : case 3: m ¸ 4 . ch o o s e fl1; :::; lmg s o a s t o m in im iz e m. th e n c le a r ly n ( x ) µ v ( a1 [ a2 ) ; n ( y ) µ v ( am¡1 [ am ) a n d z1 á z2 fo r e a c h z1 2 n ( x ) a n d z2 2 n ( y ) . ob s e r vin g a ls o t h a t a1 + a2 ¸ d( x ) ¡ 1 ; am¡1 + am ¸ d( y ) ¡ 1 ; we g e t c ¸ jq1j = mx i=1 ai + m = ( a1 + a2 + am¡1 + am ) + m¡2x i=3 ai + m ¸ d ( x ) + d( y ) ¡ 2 + m¡2x i=3 ai + m ¸ d ( x) + d ( y ) + m ¡ 2 : l e m m a 3 is p r o ve d . p r oof of t heor em 1: l e t p = x ¡! p y b e a lo n g e s t p a t h in g. case 1: p · 2 ±. if xy 2 e ( g) , t h e n b y l e m m a 1 ( i) , c = p. l e t xy 62 e ( g) . th e n d( x ) + d ( y ) ¸ 2 ± ¸ p a n d b y l e m m a 1 ( ii) , c = p. case 2: 2 ± + 1 · p · 3 ± ¡ 2 . if c ¸ 3 ± ¡ 3 , t h e n c ¸ p + 2 ¡ 3 = p ¡ 1 . n e xt , if · = 2 a n d p ¸ 3 ± ¡ 1 , t h e n p ¸ 3 ± ¡ 1 ¸ p + 1 , a c o n t r a d ic t io n . b y th e o r e m c, c ¸ p ¡ 1 . case 3: p ¸ 3 ± ¡ 1 . s in c e g is 2 -c o n n e c t e d , t h e r e is a vin e fl1; :::; lmg o n p . b y l e m m a 3 , m · c ¡ d ( x) ¡ d( y ) + 2 · c ¡ 2 ± + 2 . u s in g l e m m a 2 , we g e t c ¸ 2 p ¡ 1 0 m + 1 + 4 ¸ 2 p ¡ 1 0 c ¡ 2 ± + 3 + 4 ; im p lyin g t h a t c ¸ s 2 p ¡ 1 0 + µ ± ¡ 7 2 ¶2 + ± + 1 2 : th e o r e m 1 is p r o ve d . c. mosesyan and zh. nikoghosyan 4 9 refer ences [1 ] j. a . b o n d y a n d u . s . r . mu r t y, graph theory with applications, ma c m illa n , l o n d o n a n d e ls e vie r , n e w y o r k, 1 9 7 6 . [2 ] g. a . d ir a c , \ s o m e t h e o r e m s o n a b s t r a c t g r a p h s " , p roc. l ondon, m ath. soc., vo l. 2 , p p . 6 9 -8 1 , 1 9 5 2 . [3 ] k . oz e ki a n d t. y a m a s h it a , \ l e n g t h o f lo n g e s t c yc le s in a g r a p h wh o s e r e la t ive le n g t h is a t le a s t t wo " , graphs and combinatorics, vo l. 2 8 , p p . 8 5 9 -8 6 8 , 2 0 1 2 . [4 ] j. a . b o n d y, b asic graph theory: p aths and circuits, h a n d b o o k o f co m b in a t o r ic s , e ls e vie r , a m s t e r d a m , vo l. 1 ,2 , 1 9 9 0 . submitted 01.07.2016, accepted 02.11.2016. ¸çñ³ïç »ñïáõ ã»áñ»ùý»ñç áý¹ñ³ýñ³óáõù î. øáë»ëû³ý ¨ ä. üçïáõáëû³ý ²ù÷á÷áõù ²å³óáõóíáõù ¿ ùç ã»áñ»ù, áñý áý¹·ñïáõù ¿ 1952-çý ¸çñ³ïç ïáõùçó ëï³óí³í »ñïáõ ñ³ûïýç ã»áñ»ùý»ñá áñå»ë ù³ëý³íáñ ¹»åù»ñ: îáîáùåíèå äâóõ òåîðåì äèðàêà ê. ìîñåñÿí è æ. íèêîãîñÿàí àííîòàöèÿ äîêàçûâàåòñÿ îäíà òåîðåìà, êîòîðàÿ âêëþ÷àåò äâå èçâåñòíûå òåîðåìû äèðàêà, ïîëó÷åííûå â 1952 ã., êàê ÷àñòíûå ñëó÷àè. mathematical problems of computer science 51, 21–38, 2019. udc 519.1 on the manoussakis conjecture for a digraph to be hamiltonian samvel kh. darbinyan institute for informatics and automation problems of nas ra e-mail: samdarbin@ipia.sci.am abstract y. manoussakis (j. graph theory 16, 1992, 51-59) proposed the following conjecture. conjecture. let d be a 2-strongly connected digraph of order n such that for all distinct pairs of non-adjacent vertices x, y and w, z, we have d(x)+d(y)+d(w)+d(z) ≥ 4n − 3. then d is hamiltonian. in this note, we prove that if d satisfies the conditions of this conjecture, then (i) d has a cycle factor; (ii) if {x, y} is a pair of non-adjacent vertices of d such that d(x) + d(y) ≤ 2n − 2, then d is hamiltonian if and only if d contains a cycle passing through x and y; (iii) if {x, y} a pair of non-adjacent vertices of d such that d(x)+d(y) ≤ 2n−4, then d contains cycles of all lengths 3, 4, . . . , n−1; (iv) d contains a cycle of length at least n − 1. keywords: digraph, hamiltonian cycle, cycle factor, pancyclic digraph. 1. introduction in this paper, we consider finite digraphs (directed graphs) without loops and multiple arcs. every cycle and path are assumed simple and directed. a digraph d is hamiltonian if it contains a cycle passing through all the vertices of d. there are many conditions that guarantee that a digraph is hamiltonian (see, e. g., [1]-[5]). in [5], the following theorem was proved. theorem 1.1: (manoussakis [5]). let d be a strongly connected digraph of order n. suppose that d satisfies the following condition for every triple x, y, z ∈ v (d) such that x and y are non-adjacent: if there is no arc from x to z, then d(x) + d(y) + d+(x) + d−(z) ≥ 3n − 2. if there is no arc from z to x, then d(x)+d(y)+d−(x)+d+(z) ≥ 3n−2. then d is hamiltonian. definition 1.2: let d be a digraph of order n. we say that d satisfies condition (m ) when d(x) + d(y) + d(w) + d(z) ≥ 4n−3 for all distinct pairs of non-adjacent vertices x, y and w, z. 21 22 on the manoussakis conjecture for a digraph to be hamiltonian manoussakis [5] proposed the following conjecture. this conjecture is an extension of theorem 1.1 conjecture 1.3: (manoussakis [5]). let g be a 2-strongly connected digraph of order n such that for all distinct pairs of non-adjacent vertices x, y and w, z we have d(x) + d(y) + d(w) + d(z) ≥ 4n − 3. then d is hamiltonian. this conjecture seems quite difficult to prove. manoussakis [5] gave an example, which showed that if the conjecture is true, then the minimum degree condition is sharp. notice that another examples can be found in [6], where for any two integers k ≥ 2 and m ≥ 1, the author constructed a family of k-strongly connected digraphs of order 4k + m with minimum degree 4k + m − 1, which are not hamiltonian. this result improves a conjecture of thomassen [2] (conjecture 1.4.1). moreover, when m = 1, then from these digraphs we can obtain k-strongly connected non-hamiltonian digraphs of order n = 4k + 1 with minimum degree equal to n−1 and the minimal semi-degrees equal to (n−3)/2. thus, if in conjecture 1.3 we replace 4n−4 instead of 4n−3, then for every n there are many digraphs of order n with high connection and high semi-degrees, for which conjecture 1.3 is not true. the author [7] proved the following theorem. theorem 1.4: (darbinyan [7]). let d be a strongly connected digraph of order n ≥ 3. suppose that d(x) + d(y) ≥ 2n − 1 for every pair of non-adjacent vertices x, y ∈ v (d) \{z}, where z is some vertex of v (d). then either d is hamiltonian or contains a cycle of length n − 1. it is easy to see that if a digraph d satisfies the condition (m ), then it contains at most one pair of non-adjacent vertices x, y such that d(x) + d(y) ≤ 2n−2. from this and theorem 1.4 immediately follows the following corollary. corollary 1.5: let g be a strongly connected digraph of order n satisfying condition (m ). then d contains a cycle of length at least n − 1 (in particular, d contains a hamiltonian path). corollary 1.5 was also later proved by ning [8]. in this paper we investigate the properties those digraphs, which satisfy the condition of conjecture 1.3. let d be a 2-strongly connected digraph of order n satisfying the condition (m ) and let {x, y} be a pair of non-adjacent vertices of d. in section 4 we prove: (i) d has a cycle factor; (ii) if d(x) + d(y) ≤ 2n − 2, then d is hamiltonian if and only if d contains a cycle passing through x and y; (iii) if d(x) + d(y) ≤ 2n − 4, then d contains cycles of all lengths 3, 4, . . . , n − 1; (iv) suppose that x1x2 . . . xn−2yx1 is a cycle of length n−1 passing through y and avoiding x. if d(x) + d(y) ≤ 2n − 2 and xn−2 → x → x1, then d is hamiltonian. s. darbinyan 23 2. terminology and notation in this paper we consider finite digraphs without loops and multiple arcs. we shall assume that the reader is familiar with the standard terminology on digraphs and refer to [1] for terminology and notations not discussed here. the vertex set and the arc set of a digraph d are denoted by v (d) and a(d), respectively. the order of d is the number of its vertices. for any x, y ∈ v (d), we also write x → y if xy ∈ a(d). if xy ∈ a(d), y is an out-neighbour of x and x is an in-neighbour of y. if x → y and y → z, we write x → y → z. two distinct vertices x and y are adjacent if xy ∈ a(d) or yx ∈ a(d) (or both). if there is no arc from x to y, we shall use the notation xy /∈ a(d). we let n +(x), n−(x) denote the set of out-neighbours, respectively the set of inneighbours of a vertex x in a digraph d. if a ⊆ v (d), then n +(x, a) = a ∩ n +(x) and n−(x, a) = a ∩ n−(x). the out-degree of x is d+(x) = |n +(x)| and d−(x) = |n−(x)| is the in-degree of x. similarly, d+(x, a) = |n +(x, a)| and d−(x, a) = |n−(x, a)|. the degree of the vertex x in d is defined as d(x) = d+(x)+d−(x) (similarly, d(x, a) = d+(x, a)+d−(x, a)). the subdigraph of d induced by a subset a of v (d) is denoted by d〈a〉. if z is a vertex of a digraph d, then the subdigraph d〈v (d) \ {z}〉 is denoted by d − z. for integers a and b, a ≤ b, let [a, b] denote the set of all integers, which are not less than a and are not greater than b. the path (respectively, the cycle) consisting of the distinct vertices x1, x2, . . . , xm ( m ≥ 2) and the arcs xixi+1, i ∈ [1, m − 1] (respectively, xixi+1, i ∈ [1, m − 1], and xmx1), is denoted by x1x2 · · · xm (respectively, x1x2 · · · xmx1). the length of a cycle or path is the number of its arcs. we say that x1x2 · · · xm is a path from x1 to xm or is an (x1, xm)-path. let x and y be two distinct vertices of a digraph d. cycle that passing through x and y in d, we denote by c(x, y). a cycle (respectively, a path) that contains all the vertices of d, is a hamiltonian cycle (respectively, is a hamiltonian path). a digraph is hamiltonian if it contains a hamiltonian cycle. a digraph d of order n ≥ 3 is pancyclic if it contains cycles of all lengths m, 3 ≤ m ≤ n. for a cycle c = x1x2 · · · xkx1 of length k, the subscripts considered modulo k, i.e., xi = xs for every s and i such that i ≡ s (mod k). if p is a path containing a subpath from x to y, we let p [x, y] denote that subpath. similarly, if c is a cycle containing vertices x and y, c[x, y] denotes the subpath of c from x to y. a digraph d is strongly connected, if there exists a path from x to y and a path from y to x for every pair of distinct vertices x, y. a digraph d is k-strongly (k ≥ 1) connected if |v (d)| ≥ k + 1 and d〈v (d) \ a〉 is strongly connected for any subset a ⊂ v (d) of at most k − 1 vertices. let h be a non-trivial proper subdigraph of a digraph d. for the subdigraph h, a hbypass is a path of length at least two with both end-vertices in h and no other vertices in h. if c is a non-hamiltonian cycle in d and (x, y)-path p is a c-bypass with v (p ) ∩ v (c) = {x, y}, then we call the length of the path c[x, y] the gap of p with respect to c. a cycle factor in d is a collection of vertex disjoint cycles c1, . . . , cl such that v (c1) ∪ . . . ∪ v (cl) = v (d). for a pair of disjoint subsets a and b of v (d), we define a(a → b) = {xy ∈ a(d)|x ∈ a, y ∈ b} and a(a, b) = a(a → b) ∪ a(b → a). 24 on the manoussakis conjecture for a digraph to be hamiltonian 3. preliminaries lemma 3.1: (häggkvist, thomassen [9]). let d be a digraph of ordere n ≥ 3 containing a cycle c of length m, m ∈ [2, n − 1]. let x be a vertex not contained in this cycle. if d(x, v (c)) ≥ m + 1, then d contains a cycle of length k for all k ∈ [2, m + 1]. the following lemma is a modification of a lemma by bondy and thomassen [10], its proof is almost the same. lemma 3.2: let d be a digraph of order n ≥ 3 containing a path p := x1x2 . . . xm, m ∈ [2, n−1]. let x be a vertex not contained in this path. if one of the following statements holds: (i) d(x, v (p )) ≥ m + 2; (ii) d(x, v (p )) ≥ m + 1 and xx1 /∈ a(d) or xmx /∈ a(d); (iii) d(x, v (p )) ≥ m, xx1 /∈ a(d) and xmx /∈ a(d); then there is an i ∈ [1, m − 1] such that xix, xxi+1 ∈ a(d), i.e., d contains a path x1x2 . . . xixxi+1 . . . xm of length m (we say that x can be inserted into p or the path x1x2 . . . xixxi+1 . . . xm is an extended path obtained from p with x). it is not difficult to prove the following lemma. lemma 3.3: let d be a digraph of order n. assume that xy /∈ a(d) and the vertices x, y in d satisfy the degree condition d+(x) + d−(y) ≥ n − 2 + k, where k ≥ 1. then d contains at least k internally disjoint (x, y)-paths of length two. lemma 3.4: (bypass lemma, bondy [11]). let d be a strongly connected non-separable (i.e., u g(d) is 2-connected) digraph and let h be a non-trivial proper subdigraph of d. then d contains a h-bypass. theorem 3.4: (yeo [12]). let d be a digraph. then d has a cycle factor if and only if v (d) cannot be partitioned into subsets y , z, r1, r2 such that a(y → r1) = a(r2 → r1∪y ) = ∅, |y | > |z| and y is an independent set. theorem 3.5: (meyniel [4]). let d be a strongly connected digraph of order n ≥ 2. if d(x) + d(y) ≥ 2n − 1 for all pairs of non-adjacent vertices in d, then d is hamiltonian. before stating the main result of [14], we need to define a family of digraphs. definition 3.6: for any integers n and m, (n + 1)/2 < m ≤ n − 1, let φmn denote the set of digraphs d, which satisfy the following conditions: (i) v (d) = {x1, x2, . . . , xn}; (ii) xnxn−1 . . . x2x1xn is a hamiltonian cycle in d; (iii) for each k, 1 ≤ k ≤ n − m + 1, the vertices xk and xk+m−1 are not adjacent; (iv) xjxi /∈ a(d) whenever 2 ≤ i + 1 < j ≤ n and (v) the sum of degrees for any two distinct non-adjacent vertices at least 2n − 1. theorem 3.7: (darbinyan [13], [14]). let d be a strongly connected digraph of order n ≥ 3. suppose that d(x) + d(y) ≥ 2n − 1 for all pairs of distinct non-adjacent vertices x, y in d. then either (a) d is pancyclic or (b) n is even and d is isomorphic to one of k∗n/2,n/2, s. darbinyan 25 k∗n/2,n/2 \ {e}, where e is an arbitrary arc of k∗n/2,n/2, or (c) d ∈ φmn (in this case d does not contain a cycle of length m). later on, theorem 3.7 was also proved by benhocine [15]. 4. proofs of the results from the definition of condition (m ) the following lemma follows. lemma 4.1: let d be a digraph of order n satisfying condition (m ). then d contains at most one pair of non-adjacent vertices x, y such that d(x) + d(y) ≤ 2n − 2. theorem 4.2: let d be a 2-strongly connected digraph of order n ≥ 3 satisfying condition (m ). suppose that {x, y} is a pair of non-adjacent vertices of d such that d(x) + d(y) ≤ 2n−2. then d is hamiltonian if and only if d contains a cycle passing through the vertices x and y. proof. if d is hamiltonian, then obviously it contain a cycle passing through x and y. suppose that d contains a cycle passing through the vertices x and y but d is not hamiltonian. let c be a longest cycle, say of length m, passing through x and y. since d is not hamiltonian, we have that m ≤ n − 1. from 2-connectednees of d and bypasslemma it follows that there is a c-bypass, say p = uy1y2 . . . ykv, where u, v ∈ v (c) and y1, y2, . . . , yk ∈ v (d) \ v (c). without loss of generality, assume that the gap |c[u, v]| − 1 of p is the minimum among the gaps of all c-bypasses. then a(v (c[u, v]) \ {u, v}, v (p [y1, yk])) = ∅. (1) put f := |v (c[u, v]) \{u, v}|. since c is a longest cycle passing through x and y, it follows that f ≥ 1. now we extend the path c[v, u] with the vertices of v (c[u, v]) \{u, v} as mach as possible. we obtain a (v, u)-path, say r. then, since c is a longest cycle passing through x and y, r does not contain some vertices u1, u2, . . . , ud of v (c[u, v])\{u, v}. using (1) and lemma 3.2(i), for all yj and ui we obtain d(yj, v (c)) ≤ m−f + 1 and d(ui, v (c)) = d(ui, v (r)) + d(ui, {u1, . . . , ud}) ≤ m + d−1. by the minimality of the gap f + 1 we also have that d contains no path of the form yjzui and uizyj, where z ∈ b := v (d) \ (v (c) ∪ {y1, . . . , yk}). therefore, d(ui, b) + d(yj, b) ≤ 2|b|. now by a simple calculation we obtain d(ui)+d(yj) = d(ui, v (c))+d(yj, v (c))+d(ui, b)+d(yj, b)+d(ui, p [y1, yk])+d(yj, p [y1, yk]) ≤ m + d − 1 + m − f + 1 + 2|b| + 2k − 2 ≤ 2n − 2, a contradiction with lemma 4.1 since ui and yj are not adjacent and {ui, yj} 6= {x, y}. theorem 4.2 is proved. 26 on the manoussakis conjecture for a digraph to be hamiltonian clearly, the existence of a cycle factor is a necessary condition for a digraph to be hamiltonian. theorem 4.3: let d be a 2-strongly connected digraph of order n satisfying condition (m ). then d has a cycle factor. proof. suppose, on the contrary, that d has no cycle factor. by the yeo theorem, v (d) can be partitioned into subsets y , z, r1, r2 such that a(y → r1) = a(r2 → r1 ∪ y ) = ∅, |y | > |z| and y is an independent set. using these and 2-connectedness of d, we obtain that it follows that |z| ≥ 2 and hence, |y | ≥ 3. let x, y, z be three distinct vertices of y . since y is an independent set, we have that {x, y} and {x, z} are two distinct pairs of non-adjacent vertices of v (d). now using condition (m ), we obtain 4n−3 ≤ 2d(x) + d(y) + d(z) ≤ 8|z|+ 4|r1|+ 4|r2|+ 4|y |−4|y | = 4n−4(|y |−|z|) ≤ 4n−4, a contradiction. theorem 4.3 is proved. using yeo’s theorem, it is not difficult to show that in theorem 4.3 the minimum degree condition is sharp. theorem 4.4: let d be a 2-strongly connected digraph of order n ≥ 3. suppose that d contains at most one pair of non-adjacent vertices. then d is hamiltonian. proof. suppose, on the contrary, that d is not hamiltonian. therefore, d is not semicomplete and contains exactly one pair, say {x, y}, of non-adjacent vertices. then d(x) ≥ n − 2, d(y) ≥ n − 2 and d(z) ≥ n − 1 for all z ∈ v (d) −{x, y}. since d is 2-strongly connected, it follows that both subdigraphs d − x and d − y both are strongly connected semicomplete digraphs. therefore, d − x and d − y both are hamiltonian. let cn−1 := x1x2 . . . xn−2yx1 be a hamiltonian cycle in d − x. since d is not hamiltonian, from d(x) ≥ n − 2 and the fact that x is adjacent to every vertex of v (d) − {y} it follows that there exists an integer l ∈ [2, n − 3] such that {xl+1, xl+2, . . . , xn−2} → x → {x1, x2, . . . , xl}. (2) similarly, for some k ∈ [2, n − 3] we have {xk+1, xk+2, . . . , xn−2} → y → {x1, x2, . . . , xk}. (3) observe that for every pair of integers i, j, 1 ≤ i < j ≤ n − 2, in d there is no path of the types xi → y → xj and xi → z → xj. by symmetry between x and y, we may assume that k ≥ l. then by (2), (3) and our observation we have a({x, y} → {xk+1, xk+2, . . . , xn−2}) = ∅ and {xk+1, xk+2, . . . , xn−2} → {x, y}. (4) from (4) and 2-connectedness of d it follows that there exist i ∈ [1, k−1] and j ∈ [k+1, n−2] such that xi → xj (for otherwise, a({x1, x2, . . . , xk−1, x, y} → {xk+1, xk+2, . . . , xn−2}) = ∅, which means that d − xk is not strongly connected). note that y → xi+1. we choose j maximal with these properties. using (2) and (3), it is easy to see that if xj−1 → x, then x1 . . . xixj . . . xn−2yxi+1 . . . xj−1xx1 is a hamiltonian cycle, which contradicts our supposition. we may therefore assume that xj−1x /∈ a(d). then from k ≥ l, (4) and s. darbinyan 27 the maximality of j it follows that j = k + 1 and xkx /∈ a(d). hence, l = k and a({x1, x2, . . . , xk−1} → {xk+2, xk+3, . . . , xn−2}) = ∅. this together with (4) and 2connectedness of d implies that there is an integer s ∈ [k + 2, n − 2] such that xk → xs. therefore, x1 . . . xixk+1 . . . xs−1xxi+1 . . . xkxs . . . xn−2yx1 is a hamiltonian cycle, a contradiction. theorem 4.4 is proved. remark: there is a strongly connected non-hamiltonian digraph of order n ≥ 5, which is not 2-strongly connected and has exactly one pair of non-adjacent vertices. to see this, consider the following digraph d defined as follows: v (d) = {x1, x2, . . . , xn−2, y, z}, x1x2, . . . xn−2yx1 is a cycle of length n − 1 in d, n−(y) = n−(z) = {xk, xk+1, . . . , xn−2} and n +(y) = n +(z) = {x1, x2, . . . , xk}, where k ∈ [2, n − 3], d also contains all the arcs xjxi whenever 1 ≤ i < j ≤ n − 2 and it contain no other arcs. it is not difficult to check that d is neither 2-strongly connected nor hamiltonian. lemma 4.5: let d be a 2-strongly connected digraph of order n ≥ 3 and let u, v be two distinct vertices in d. if d contains no cycle passing through u and v, then u, v are not adjacent and there is no path of length two between them. in particular, d+(u) + d−(v) ≤ n − 2, d−(u) + d+(v) ≤ n − 2 and d(u) + d(v) ≤ 2n − 4. proof. it is obvious that u, v are not adjacent. suppose, on the contrary, that in d there is a path of length two between the vertices u and v, say u → z → v or v → z → u. since d is 2-strongly connected, it follows that d−z is strongly connected. therefore, in d−z there is an (u, v)and a (v, u)-path. it is easy to see that this (u, v)-path ((v, u)-path, respectively) together with v → z → u (u → z → v, respectively) forms a cycle passing through u and v. in both cases we have a contradiction, which proves that there is no path of length two between u and v. therefore, by lemma 3.3, d+(u)+d−(v) ≤ n−2 and d−(u)+d+(v) ≤ n−2. these imply that d(u) + d(v) ≤ 2n − 4. lemma 4.5 is proved. theorem 4.6: let d be a 2-strongly connected digraph of order n ≥ 3 satisfying condition (m ). suppose that {u, v} is a pair of non-adjacent vertices in d such that d(u) + d(v) ≤ 2n − 2. then d is hamiltonian or d contains a cycle of length n − 1 passing through u and avoiding v (passing through v and avoiding u). proof. suppose that d is not hamiltonian. from theorem 4.2 it follows that d contains no cycle passing through u and v. therefore, by lemma 4.5, d(u) + d(v) ≤ 2n − 4. since d is 2-strongly connected, it follows that d − u and d − v both are strongly connected. from the last inequality and condition (m ) it follows that if {x, y} is a pair of non-adjacent vertices in d − u (in d − v, respectively), then the following inequalities holds: d(x, v (d) \ {u}) + d(y, v (d) \ {u}) ≥ 2(n − 1) − 1, (d(x, v (d) \ {v}) + d(y, v (d) \ {v}) ≥ 2(n − 1) − 1, respectively). therefore, since d − u and d − v both are strongly connected, by meyniel’s theorem d − u and d − v both are hamiltonian, i.e., d contains a cycle of length n − 1 passing through u 28 on the manoussakis conjecture for a digraph to be hamiltonian and avoiding v (passing through v and avoiding u). theorem 4.6 is proved. as an immediate corollary of theorems 4.2 and 4.6 (respectively, theorem 4.6 and corollary 3.1), we obtain corollary 4.7 (respectively, corollary 4.8). corollary 4.7: let d be a 2-strongly connected non-hamiltonian digraph of order n ≥ 3 satisfying condition (m ). suppose that {u, v} is a pair of non-adjacent vertices in d such that d(u)+d(v) ≤ 2n−2. then d contains at most one cycle of length two passing through u (v). corollary 4.8: let d be a 2-strongly connected non-hamiltonian digraph of order n ≥ 3 satisfying condition (m ). suppose that {u, v} is a pair of non-adjacent vertices in d such that d(u) + d(v) ≤ 2n − 2. then d(u) ≤ n − 1 and d(v) ≤ n − 1 . theorem 4.9: let d be a 2-strongly connected digraph of order n ≥ 6 satisfying condition (m ). suppose that {x, y} is a pair of non-adjacent vertices in d such that d(x) + d(y) ≤ 2n − 4. then d contains cycles of all lengths 3, 4, . . . , n − 1. proof. suppose first that d contains exactly one pair of non-adjacent vertices, namely {x, y}. then d − x is a strongly connected semicomplete digraph. therefore, by the wellknown theorem of moser [16], d − x contains cycles of all lengths 3, 4, . . . , n − 1. suppose next that d contains at least two distinct pairs of non-adjacent vertices. let {u, v} be an arbitrary pair of non-adjacent vertices in v (d) \{x} (or in v (d) \{y}). from condition (m ) it follows that d(u) + d(v) ≥ 2n + 1. (5) now we consider the subdigraph h := d −x. for the digraph h we first prove the following claim. claim: if h ∼= k∗m,m − e, where e is an arbitrary arc of k∗m,m, then d contains cycles of all lengths 2, 3, . . . , n − 1. m,m − e. note that n = 2m + 1 ≥ 7. then d(u, v (d) \ {x}) + d(v, v (d) \ {x}) ≤ 4m = 2n − 2. therefore, by (5), d(x, {u, v}) ≥ 3. this, since m ≥ 3, in turn, implies that every partite set of h contains at least two vertices such that each of them together with x forms a 2-cycle. therefore, there exist two vertices z, w ∈ v (h) such that z ↔ w, z ↔ x and w ↔ x then, since for every k, k ∈ [1, m] there is a cycle of length 2k passing through the arc z → w, it follows that d contains cycles of all lengths 2, 3, . . . , n. the claim is proved. we now return to the proof of theorem 4.9. from (5) it also follows that d(u, v (d) \ {x}) + d(v, v (d) \ {x}) ≥ 2(n − 1) − 1. then, since h is strongly connected, from theorem 3.7 it follows that either h contains cycles of all lengths 3, 4, . . . , n − 1 or h ∈ {k∗m,m, k∗m,m − e}∪ φkn−1, where n/2 < k ≤ n − 2. in order to complete the proof of the theorem, by the above claim it suffices to consider only proof. let {u, v} be an arbitrary pair of non-adjacent vertices in k∗ s. darbinyan 29 the case when h ∈ φkn−1. from the definition of the set φkn−1 it follows that h contains cycles of all lengths 2, 3, . . . , n − 1 except the cycle of length k. let x1xn−1xn−2 . . . x2x1 be a hamiltonian cycle in h. since h ∈ φkn−1, it follows that {x1, xk}, {xn−k, xn−1} are two distinct pairs of non-adjacent vertices other than {x, y} and d(x1, v (h)) + d(xk, v (h)) = d(xn−k, v (h)) + d(xn−1, v (h)) = 2n − 3. this together with (5) implies that d(x, {x1, xn−k, xk, xn−1) = 8. if k 6= n − 3, then x1 → xn−3 and x1xn−3xn−4 . . . xn−kxx1 is a cycle of length k. assume that k = n − 3. then {x1, xn−3}, {x3, xn−1} are two pairs of non-adjacent vertices other than {x, y}. we have that d(x, {x1, x3, xn−3, xn−1) = 8, x1 → xn−4 and x3 → xn−1. if x2 → x, then x1xn−4...x3x2xx1 is a cycle of length n − 3. assume that x2x /∈ a(d). then, since the vertices x2 and xn−2 are not adjacent and d(x2) + d(xn−2) ≥ 2n + 1, it is not difficult to see that x2 → xn−3 and x → x2. therefore, xx2xn−3xn−4x3x is a cycle of length n − 3. this completes the proof of the theorem. in view of theorem 4.9, it is natural to set the following problem. problem: let d be a 2-strongly connected digraph of order n satisfying condition (m ). suppose that {x, y} is a pair of non-adjacent vertices in d such that 2n−3 ≤ d(x)+d(y) ≤ 2n−2. whether d contains cycles of all lengths 3, 4, . . . , n − 1? 5. remarks in the following, we suppose, further, that d is a 2-strongly connected digraph of order n satisfying condition (m ). moreover, d contains a pair {y, z} of non-adjacent distinct vertices y, z such that d(y)+d(z) ≤ 2n−4. in this section, we will prove a number of properties of d. lemma 5.1: let x1x2 . . . xn−2zx1 be a cycle of length n − 1 in d, which does not contain y. suppose that xa → xb, xq → y → xp and xt → y → xs, where 1 ≤ s ≤ a < p ≤ q < b ≤ t ≤ n − 2. then d is hamiltonian. proof. suppose, on the contrary, that d is not hamiltonian. by theorem 4.2, d contains no cycle passing through y and z. notice that there are no integers l and r, 1 ≤ l < r ≤ n−2, such that xl → y → xr (for otherwise, x1 . . . xlyxr . . . xn−2zx1 is a cycle passing through y and z. if z → xi with i ∈ [a + 1, q], then c(y, z) = xs . . . xaxb . . . xn−2zxi . . . xqyxs; if xj → z with j ∈ [p, b − 1], then c(y, z) = x1 . . . xaxb . . . xtyxp . . . xjzx1. thus, in both cases we have a contradiction. therefore, d+(z, {xa+1, . . . , xq}) = d−(z, {xp, . . . , xb−1}) = 0, in particular, d(z, {xp, . . . , xq}) = 0 and the vertices z and xp are not adjacent. the last equality together with the fact that d contains at most one cycle of length two passing through z (corollary 4.7) implies that d(z) = d(z, {x1, . . . , xp−1})+d(z, {xq+1, . . . , xn−2}) ≤ p−1+n−2−q +1 = n+p−q−2. (6) 30 on the manoussakis conjecture for a digraph to be hamiltonian now we consider the vertex xp. it is easy to see that if xi → xp with i ∈ [1, s − 1], then c(y, z) = x1 . . . xixp . . . xqyxs . . . xaxb . . . xn−2zx1, if xp → xj with j ∈ [t + 1, n − 2], then c(y, z) = x1 . . . xaxb . . . xtyxp xj . . . xn−2zx1. in both cases we have a contradiction. therefore, we may assume that d−(xp, {x1, . . . , xs−1}) = d+(xp, {xt+1, . . . , xn−2}) = 0. this implies that d(xp) = d +(xp, {x1, . . . , xs−1}) + d−(xp, {xt+1, . . . , xn−2}) + d(xp, {xs, . . . , xt}) + d(xp, {y}) ≤ s − 1 + n − 2 − t + 2(t − s + 1) = n + t − s − 1. (7) without loss of generality, we may assume that s, q are chosen as maximal as possible and p, t are chosen as minimal as possible. then d(y, {xs+1, . . . , xp−1}) = d(y, {xq+1, . . . , xt−1}) = 0. this, since d contains at most one cycle of length two passing through y, implies that d(y) = d(y, {x1, . . . , xs}) + d(y, {xp, . . . , xq}) + d(y, {xt, . . . , xn−2}) ≤ s + q − p + 1 + n − 2 − t + 1 + 1 = n + s + q − p − t + 1. (8) since {y, z} and {xp, z} are two distinct pairs of non-adjacent vertices, from (6), (7), (8) and condition (m ) it follows that 4n − 3 ≤ d(y) + 2d(z) + d(xp) ≤ 4n − 4 − (q − p) ≤ 4n − 4, which is a contradiction. lemma 5.1 is proved. the following claim is an immediate consequence of lemma 5.1. claim 1: let x1x2 . . . xn−2zx1 be a cycle of length n − 1 in d passing through z. if xn−2 → y → x1 and x1 → xn−2, then d is hamiltonian. the following claim will be very useful in the remaining proof. claim 2: let x1x2 . . . xn−2 be a hamiltonian path in d − {y, z}. suppose that for every pair of integers i and j, 1 ≤ i < j ≤ n − 2, if xi → y, then yxj /∈ a(d), and if xi → z, then zxj /∈ a(d). then either d is hamiltonian or for every k ∈ [2, n − 3], the following holds: a({x1, x2, . . . , xk−1} → {xk+1, xk+2, . . . , xn−2}) 6= ∅. proof. suppose, on the contrary, that d is not hamiltonian and there is an integer k ∈ [2, n − 3] such that a({x1, x2, . . . , xk−1} → {xk+1, xk+2, . . . , xn−2}) = ∅, (9) we can assume that the vertices xm and xl are chosen so that y → xm, z → xl and d+(y, {xm+1, . . . , xn−2}) = d+(z, {xl+1, . . . , xn−2}) = 0. s. darbinyan 31 without loss of generality, we assume that m ≤ l. since d is 2-strongly connected, it follows that 2 ≤ m ≤ l ≤ n − 3. from the supposition of this claim and (9) it follows that: (i) if k ≤ m or k ≥ l, then (respectively) a({x1, x2, . . . , xk−1} → {y, z, xk+1, xk+2, . . . , xn−2}) = ∅ or a({y, z, x1, x2, . . . , xk−1} → {xk+1, xk+2, . . . , xn−2}) = ∅, (ii) if m + 1 ≤ k ≤ l − 1, then a({y, x1, x2, . . . , xk−1} → {z, xk+1, xk+2, . . . , xn−2}) = ∅. thus, in each case we have that d − xk is not strongly connected, which contradicts that d is 2-strongly connected. claim 2 is proved. lemma 5.2: suppose that x1x2 . . . xn−2zx1 is a cycle in d passing through z and avoiding y. if xn−2 → y → x1, then d is hamiltonian. proof. suppose, on the contrary, that xn−2 → y → x1 but d is not hamiltonian. by theorem 4.2, d contains no cycle passing through y and z in d. it is easy to see that the conditions of claim 2 hold. let xk → y → xp, where 2 ≤ p ≤ k ≤ n − 3, k minimal and p maximal with this property. since d is 2-strongly connected, from lemma 5.1 it follows that p ≤ k − 1. this means that there is no cycle of length two passing through y. by symmetry between the vertices y and z, we may assume that also there is no cycle of length two passing through z. case 1. p = k − 1. by lemma 5.1 we have that a({x1, . . . , xp−1} → {xk+1, . . . , xn−2}) = ∅. then, by claim 2, there are some integers i ∈ [1, p − 1] and j ∈ [k + 1, n − 2] such that xi → xk and xk−1 → xj. therefore, c(y, z) = x1 . . . xixkyxk−1xj . . . xn−2zx1 is a cycle passing through y and z, which is a contradiction. case 2. p ≤ k − 2. then d(y, {xp+1, . . . , xk−1}) = 0. by lemma 5.1 we have that a({x1, . . . , xp−1} → {xk+1, . . . , xn−2}) = ∅. (10) therefore, by claim 2, there are s ∈ [1, p−1], a ∈ [p + 1, k], b ∈ [p, k−1] and t ∈ [k + 1, n−2] such that xs → xa and xb → xt. if a > b, then x1 . . . xsxa . . . xkyxp . . . xbxt . . . xn−2zx1 is a cycle passing through y and z, which is a contradiction. we may therefore assume that a ≤ b. then p + 1 ≤ a ≤ b ≤ k − 1 and the vertices y and xa are not adjacent. choose (i) a and t as maximal as possible and (ii) choose b and s as minimal as possible, subject to (i). this means that a({x1, . . . , xp−1} → {xa+1, . . . , xn−2}) = a({x1, . . . , xb−1} → {xk+1, . . . , xn−2}) = ∅. (11) from the minimality of s and the maximality of t we have that d(xa) = d +(xa, {x1, . . . , xs−1}) + d−(xa, {xt+1, . . . , xn−2}) + d(xa, {xs, . . . , xt}) + d(xa, {z}) ≤ s − 1 + n − 2 − t + 2t − 2s + 1 = n + t − s − 2. (12) 32 on the manoussakis conjecture for a digraph to be hamiltonian let m be the number of vertices of the set {xs+1, . . . , xp, xk, . . . , xt−1}, which are not adjacent to y. then, since y is not on the cycle of length two and d(y, {xp+1, . . . , xk−1}) = 0, it follows that d(y) ≤ n − 2 − m − (k − 1 − p) = n + p − m − k − 1. (13) assume first that d+(z, {xs+1, . . . , xp}) = d−(z, {xk, . . . , xt−1}) = 0. from this and taking into account lemma 4.5 (there is no path of length two between y and z) we obtain that d(z) ≤ s + n − 2 − t + 1 + m + k − 1 − p = n + k + m + s − t − p − 2. (14) combining (12)-(14), k − p ≥ 2 and m ≥ 0, we obtain 2d(y) + d(xa) + d(z) ≤ 4n − 6 − (k − p) − m ≤ 4n − 8, which is a contradiction to condition (m ), since {y, z} and {y, xa} are two distinct pairs of non-adjacent vertices. assume next that d+(z, {xs+1, . . . , xp}) 6= 0 or d−(z, {xk, . . . , xt−1}) 6= 0, i.e., there is a q ∈ [s + 1, p] such that z → xq or there is a r ∈ [k, t − 1] such that xr → z. using claim 2, we obtain that a({x1, . . . , xa−1} → {xa+1, . . . , xn−2}) 6= ∅. let xs1 → xt1 , where s1 ∈ [1, a − 1] and t1 ∈ [a + 1, n − 2]. from (11) it follows that s1 ∈ [p, a − 1] and t1 ∈ [a + 1, k]. choose t1 maximal with this property. then a({x1, . . . , xa−1} → {xt1+1, . . . , xn−2}) = ∅. (15) now using the facts that z → xq or xr → z, it is not difficult to check that: if t1 > b, then c(y, z) = x1 . . . xsxa . . . xbxt . . . xn−2zxq . . . xs1 xt1 . . . xkyx1 or c(y, z) = x1 . . . xsxa . . . xbxt . . . xn−2yxp . . . xs1 xt1 . . . xrzx1 is a cycle passing through y and z, when z → xq and xr → z, respectively. in both cases we have a contradiction. we may therefore assume that t1 ≤ b. from claim 2 we have that a({x1, . . . , xt1−1} → {xt1+1, . . . , xn−2}) 6= ∅. let xs2 → xt2 , where s2 ∈ [1, t1 −1] and t2 ∈ [t1 + 1, n−2]. from (11) and (15) it follows that s2 ∈ [a, t1 −1] and t2 ∈ [t1 + 1, k]. choose t2 maximal with this property. then a({x1, . . . , xt1−1} → {xt2+1, . . . , xn−2}) = ∅. (16) if t2 > b, then c(y, z) = x1 . . . xsxa . . . xs2 xt2 . . . xkyxp . . . xs1 xt1 . . . xbxt . . . xn−2zx1, a contradiction. we may therefore assume that t2 ≤ b. in particular, from t2 ≥ t1 + 1 it follows that t1 < b. using claim 2, (11) and (16), we obtain that there are some integers s3 ∈ [t1, t2 − 1] and t3 ∈ [t2 + 1, k] such that xs3 → xt3 . choose t3 maximal with this properties. then a({x1, . . . , xt2−1} → {xt3+1, . . . , xn−2}) = ∅. (17) s. darbinyan 33 if t3 > b, then c(y, z) = x1 . . . xsxa . . . xs2 xt2 . . . xbxt . . . xn−2zxq . . . xs1 xt1 . . . xs3 xt3 . . . xkyx1 or c(y, z) = x1 . . . xsxa . . . xs2 xt2 . . . xbxt . . . xn−2yxp . . . xs1 xt1 . . . xs3 xt3 . . . xrzx1, when z → xq or xr → z, respectively. we may therefore assume that t3 ≤ b. then t2 < b. again using claim 2, (11) and (17), we obtain that there are some integers s4 ∈ [t2, t3 − 1] and t4 ∈ [t3 + 1, k] such that xs4 → xt4 . if t4 > b, then c(x, y) = x1 . . . xs0 xt0 . . . xs2 xt2 . . . xs4 xt4 . . . xkyxp . . . xs1 xt1 . . . xs3 xt3 . . . xbxt . . . xn−2zx1, a contradiction. (here, xs := xs0 and xa = xt0 ). continuing this process, we finally conclude that for some l ≥ 0, tl > b since all the vertices xt0 , xt1 , . . . , xtl are distinct and in {xp, . . . , xk}. by the above arguments we have that: if tl is even, then c(y, z) = x1 . . . xs0 xt0 . . . xs2 xt2 . . . xs4 xt4 . . . xsl xtl . . . xkyxp . . . xs1 xt1 . . . xs3 xt3 . . . xsl−1 xtl−1 . . . xbxt . . . xn−2zx1, if l is odd, then c(y, z) = x1 . . . xs0 xt0 . . . xs2 xt2 . . . xsl−1 xtl−1 . . . xbxt . . . xn−2zxq . . . xs1 xt1 . . . xs3 xt3 . . . xsl xtl . . . xkyx1, or c(y, z) = x1 . . . xs0 xt0 . . . xs2 xt2 . . . xsl−1 xtl−1 . . . xbxt . . . xn−2yxp . . . xs1 xt1 . . . xs3 xt3 . . . xsl xtl . . . xrzx1, when z → xq or xr → z, respectively. in all cases we have a cycle passing through y and z, which contradicts our supposition. lemma 5.2 is proved. from theorem 4.4, lemma 5.2 and corollary 4.7 the following corollary follows. corollary 5.3: if d is not hamiltonian, then max{d(y), d(z)} ≤ n − 2. lemma 5.4: let c = x1x2 . . . xn−3x1 be a cycle of length n − 3 in d passing through y and avoiding z. let v (d) \ v (c) = {z, u, v}. if (i). d(y, {u, v}) = 0 and zuvz is a cycle of length 3 or (ii). d(y) ≤ n − 3 and z ↔ u, then d is hamiltonian. proof. suppose, on the contrary, that d is not hamiltonian. then, by theorem 4.2, d contains no cycle passing through y and z. (i). since d(y, {z, u, v}) = 0 and through y there is at most one cycle of length two, it follows that d(y) ≤ n − 3. it is easy to see that for every i ∈ [1, n − 3] the following holds: −→a [xi, z] + −→a [v, xi+1] ≤ 1, −→a [xi, u] + −→a [z, xi+1] ≤ 1 and −→a [xi, v] + −→a [u, xi+1] ≤ 1, (for otherwise, d is hamiltonian). therefore, d(z, v (c)) + d(u, v (c)) + d(v, v (c)) = n−3∑ i=1 (−→a [xi, z] + −→a [v, xi+1] + −→a [xi, u] + −→a [z, xi+1] + −→a [xi, v] + −→a [u, xi+1]) ≤ 3n − 9. then, since d(z, {u, v}) ≤ 3 and d(u, {z, v}) + d(v, {u, z}) ≤ 7 it follows that d(z) + d(u) + d(v) ≤ 3n + 1. therefore, since 2d(y) + d(z) + d(u) ≥ 4n − 3 and d(v) + d(y) ≥ 2n + 1, we have 6n − 2 ≤ 3d(y) + d(u) + d(v) + d(z) ≤ 6n − 8, which is a contradiction. (ii). then for every i ∈ [1, n − 3] we have −→a [xi, z] + −→a [u, xi+1] ≤ 1 and −→a [xi, u] +−→a [z, xi+1] ≤ 1. therefore, d(z, v (c)) + d(u, v (c)) = n−3∑ i=1 (−→a [xi, z] + −→a [u, xi+1] + −→a [xi, u] + −→a [z, xi+1]) ≤ 2n − 6. 34 on the manoussakis conjecture for a digraph to be hamiltonian hence, d(z) + d(u) ≤ 2n + 1. this together with d(y) ≤ n − 3 and condition (m ) gives 4n − 3 ≤ d(z) + d(u) + 2d(y) ≤ 2n + 1 + 2n − 6 = 2n − 5, which is a contradiction. lemma 5.4 is proved. lemma 5.5: let c := x1x2 . . . xn−4x1 be a cycle of length n− 4 in d passing through y and avoiding z. let v (d) \ v (c)) = {z, u1, u2, u3}. if one of the following conditions holds (i). d(y) ≤ n − 3 and z ↔ u1 , ii). zu1u2z is a cycle of length 3 and d(y, {u1, u2}) = 0, (iii). zu1u2u3z is a cycle of length 4 and d(y, {u1, u2, u3}) = 0, then d is hamiltonian. proof. suppose, on the contrary, that d is not hamiltonian. by theorem 4.2, d contains no cycle passing through y and z. (i). note that y and u1 are not adjacent by lemma 4.5. since z ↔ u1, it is easy to see that −→a [xi, z] + −→a [u1, xi+1] ≤ 1 and −→a [xi, u1] + −→a [z, xi+1] ≤ 1. hence, d(z, v (c)) + d(u1, v (c)) ≤ 2n − 8. therefore, since, d(u1, {z, u2, u3}) ≤ 6 and d(z, {u1, u2, u3}) ≤ 4, we have d(z) + d(u1) ≤ 2n + 2. this together with d(y) ≤ n − 3 and condition (m ) implies that 4n − 3 ≤ d(z) + 2d(y) + d(u1) ≤ 4n − 4, which is a contradiction. (ii). then it is easy to see that −→a [xi, z] + −→a [u2, xi+1] ≤ 1, −→a [xi, u1] + −→a [z, xi+1] ≤ 1 and −→a [xi, u2] + −→a [u1, xi+1] ≤ 1. hence, d(z, v (c)) + d(u1, v (c)) + d(u2, v (c)) ≤ 3n − 12. therefore, since d(z, {u1, u2, u3}) ≤ 4 and d(u1, {z, u2, u3}) + d(u2, {z, u1, u3}) ≤ 11, we obtain that d(z) + d(u1) + d(u2) ≤ 3n + 3. this together with d(y, {z, u1, u2}) = 0 implies that d(y) ≤ n − 3 and d(z) + d(u1) + d(u2) + 3d(y) ≤ 6n − 6. on the other hand, since d(z) + d(u1) + 2d(y) ≥ 4n − 3 and d(y) + d(u2) ≥ 2n + 1, we have 6n − 2 ≤ d(z) + d(u1) + 3d(y) + d(u2) ≤ 6n − 6, which is a contradiction. (iii). first, notice that d(y) ≤ n − 4. by an argument similar to that in the proof of (ii), we can show that d(z, v (cn−4)) + d(u1, v (cn−4)) + d(u2, v (cn−4)) + d(u3, v (cn−4)) ≤ 4n − 16. then, since d(z, {u1, u2, u3}) + d(u1, {z, u2, u3}) + d(u2, {u1, u2, z}) + d(u3, {u1, u2, z}) ≤ 20, we have d(z) + d(u1) + d(u2) + d(u3) ≤ 4n + 4. besides, from condition (m ) and d(y) ≤ n−4 it follows that 8n − 6 ≤ d(z) + 4d(y) + d(u1) + d(u2) + d(u3) ≤ 8n − 12, s. darbinyan 35 which is a contradiction. lemma 5.5 is proved. lemma 5.6: let c := x1x2 . . . xn−5x1 be a cycle of length n− 5 in d passing through y and avoiding z. let a = v (d) \ v (c)) = {z, u1, u2, u3, u4}. if d(y, {u1, u2}) = 0 and zu1u2z is a cycle of length three, then d is hamiltonian. proof. suppose, on the contrary, that d is not hamiltohian. by theorem 4.2, d contains no cycle passing through y and z. then d(y) ≤ n−3. it is easy to see that for all i ∈ [1, n−5], −→a [xi, z] + −→a [u2, xi+1] ≤ 1, −→a [xi, u1] + −→a [z, xi+1] ≤ 1 and −→a [xi, u2] + −→a [u1, xi+1] ≤ 1. therefore, d(z, v (c)) + d(u1, v (c)) + d(u2, v (c)) = n−5∑ i=1 (−→a [xi, z] +−→a [u2, xi+1] +−→a [xi, u1] +−→a [z, xi+1] +−→a [xi, u2] + −→a [u1, xi+1]) ≤ 3n − 15. since d(z, a) ≤ 5 and d(u1, a) + d(u2, a) ≤ 15, it follows that d(z) + d(u1) + d(u2) ≤ 3n + 5. this together with d(y) ≤ n − 3, d(y) + d(u2) ≥ 2n + 1 and condition (m ) implies that 6n − 2 ≤ d(z) + 3d(y) + d(u1) + d(u2) ≤ 6n − 4, which is a contradiction. this proves lemma 5.6. lemma 5.7: suppose that c := x1x2 . . . xn−2x1 is a cycle of length n − 2 in d passing through y and avoiding z. let v (d) \ v (c)) = {z, u}. if u ↔ z, then d is hamiltonian. proof. suppose, on the contrary, that z ↔ x but d is not hamiltonian. then, by lemma 4.5, the vertices y and x are not adjacent. hence, d(y) ≤ n− 2. since d is not hamiltonian, it follows that for every i ∈ [1, n − 2] we have −→a [xi, z] + −→a [x, xi+1] ≤ 1 and −→a [xi, x] +−→a [z0, xi+1] ≤ 1. these imply that d(z) + d(x) ≤ 2n. therefore, by condition (m ), we have 4n − 3 ≤ d(z) + d(x) + 2d(y) ≤ 4n − 4, which is a contradiction. lemma 5.7 is proved. 6. conclusion in the current article, we have examined the manoussakis conjecture for a digraph to be hamiltonian. for a digraph with the conditions of the manoussakis conjecture, a number of theorems and lemmas are proved. found results may be the first step towards confirming the manoussakis conjecture. added in proof. recently, using some results of this paper, the author confirmed the manoussakis conjecture. acknowledgement the author would like to thank the referees for careful reading and many helpful remarks and suggestions. 36 on the manoussakis conjecture for a digraph to be hamiltonian references [1] j. bang-jensen and g. gutin, digraphs: theory, algorithms and applications, springerverlag, london, 2000. [2] j.-c. bermond and c. thomassen, “cycles in digraphsa survey”, j. graph theory, vol. 5, pp.1-43, 1981. [3] d. küh and d. ostus, “a survey on hamilton cycles in directed graphs”, european j. combin., vol. 33, pp. 750-766, 2012. [4] m. meyniel, “une condition suffisante d’existence d’un circuit hamiltonien dans un graphe oriente”, j. combinatorial theory b, vol. 14, pp. 137-147, 1973. [5] y. manoussakis, “directed hamiltonian graphs”, j. graph theory, vol. 16, no. 1, pp. 51-59, 1992. [6] s. kh. darbinyan, “disproof of a conjecture of thomassen”, akad. nauk armyan. ssr dokl., vol. 76, no. 2, pp. 51-54, 1983. [7] s. kh. darbinyan, “on hamiltonian and hamilton-connected digraphs”, akad. nauk armyan. ssr dokl., vol. 91, no. 1, pp. 3-6, 1990 (for a detailed proof see arxiv: 1801.05166v1, 16 jan 2018. [8] b. ning, “notes on a conjecture of manoussakis concerning hamilton cycles in digraphs, electronic preprint”, arxiv:1404.5013v1 [math.co] 20 apr 2014, pp. 8. [9] r. häggvist and c. thomassen, “on pancyclic digraphs”,j. combin. theory b, vol. 20, pp. 20-40, 1976. [10] j. a. bondy and c. thomassen, “a short proof of meyniel’s theorem’, discrete math., vol. 19, pp. 195-197, 1977. [11] j. a. bondy, “basic graph theory: paths and circuits”, in handbook of combinatorics, vol. 1-2, elsevier, amsterdam, 1995. [12] a.yeo, “how close to regular must a semicomplete multipartite digraph to be secure hamiltonisity?”, graphs combin., vol. 15, pp. 481-493, 1999. [13] s. kh. darbinyan, “on pancyclic digraphs”, preprint of the computing center of akademy of sciences of armenia, 21 pages, 1979. [14] s. kh. darbinyan, “pancyclicity of digraphs with the meyniel condition”, studia sci. math. hungar., vol. 20, no. 1-4, pp. 95-117, 1985. (ph.d. thesis, institute mathematici akad. nauk bssr, minsk, 1981). [15] a. benhocine, “pancyclism and meyniel’s conditions”, discrete math., vol. 58, pp. 113120, 1986. [16] f. harary and l. moser, “the theory of round robin tournaments”, amer. math. monthly, vol. 73, pp. 231-246, 1966. submitted 14.12.2018, accepted 18.04.2019. s. darbinyan 3 7 îáõùýáñáßí³í ·ñ³ýç ñ³ùçéïáýû³ýáõãû³ý í»ñ³µ»ñû³é ø³ýááõë³ïçëç í³ñï³íç ù³ëçý ê³ùí»é ê. ¸³ñµçýû³ý ðð ¶²² æýýáñù³ïçï³ûç ¨ ³íïáù³ï³óù³ý åñáµé»ùý»ñç çýëïçïáõï e-mail: samdarbin@ipia.sci.am ²ù÷á÷áõù ø³ýááõë³ïçëá (j. of graph theory, vol. 16, pp. 51-59, 1992) ³é³ç³ñï»é ¿ ñ»ï¨û³é í³ñï³íá: ì³ñï³í: ¸çóáõù d-ý 2-áõå»õ ï³å³ïóí³í n-·³·³ã³ýç ïáõùýáñáßí³í ·ñ³ý ¿: ºã» d-ç ó³ýï³ó³í áã ïçó ·³·³ãý»ñç ó³ýï³ó³í »ñïáõ ï³ñµ»ñ fx; yg ¨ fu; vg ½áõû·»ñç ñ³ù³ñ ï»õç áõýç ñ»ï¨û³é d( x ) + d ( y ) + d ( w ) + d( z ) ¸ 4 n ¡ 3 ³ýñ³í³ë³ñáõãûáõýá, ³å³ d-ý ñ³ý¹çë³ýáõù ¿ ñ³ùçéïáýû³ý: ü»ñï³ ³ßë³ï³ýùáõù ³å³óáõóí»é ¿, áñ »ã» d ïáõùýáñáßí³í ·ñ³ýá µ³í³ñ³ñáõù ¿ ø³ýááõë³ïçëç í³ñï³íç å³ûù³ýý»ñçý, ³å³ (1). d ·ñ³ýá å³ñáõý³ïáõù ¿ óçïé-ý³ïïáñ; (2). ºã» d-ç áã ïçó ·³·³ãý»ñç áñ¨¿ fx; yg ½áõû·ç ñ³ù³ñ d( x ) + d( y ) · 2 n ¡ 2 , ³å³ (i) d-ý ñ³ùçéïáýû³ý ¿ ³ûý ¨ ùç³ûý ³ûý å³ù³ý³ï, »ñµ d-ý å³ñáõý³ïáõù x ¨ y ·³·³ãý»ñáí ³ýóýáõ ïáõùýáñáßí³í óçïé; (ii) d-ý ñ³ùçéïáýû³ý ¿ ï³ù å³ñáõý³ïáõù ¿ x ( y ) ·³·³ãáí ³ýóýáõ n ¡ 1 »ñï³ñáõãû³ý ïáõùýáñáßí³í óçïé, áñá ãç ³ýóýáõù y ( x ) ·³·³ãáí (ù³ëý³íáñ³å»ë, d-ý å³ñáõý³ïáõù ¿ ³éýí³½ý n ¡ 1 »ñï³ñáõãû³ý óçïé) ; (3). ºã» d-ç áã ïçó ·³·³ãý»ñç áñ¨¿ fx; yg ½áõû·ç ñ³ù³ñ d ( x) + d( y ) · 2 n ¡ 4 , ³å³ ó³ýï³ó³í k, 2 · k · n ¡ 1 , ³ýµáõç ãíç ñ³ù³ñ d-ý å³ñáõý³ïáõù ¿ k »ñï³ñáõãû³ý ïáõùýáñáßí³í óçïé; (4). d ·ñ³ýç áñáß³ïç »ñï³ñáõãûáõýý»ñ ( n¡ 5 ) -çó ùçýã¨ ( n¡ 1 ) áõý»óáõ ïáõùýáñáßí³í ïáõùýáñáßí³í ·ñ³ý: î ãèïîòåçå ìàíîóññàêèñà î ãàìèëüòîíîâîñòè îðãðàôîâ ñàìâåë õ. äàðáèíÿí èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà e-mail: samdarbin@ipia.sci.am àííîòàöèÿ ìàíîóññàêèñ (j. of graph theory, vol. 16, pp. 51-59, 1992) ïðåäëîæèë ñëåäóþùóþ ãèïîòåçó. ãèïîòåçà: ïóñòü d ÿâëÿåòñÿ 2-ñèëüíî ñâÿçíûì n-âåðøèííûì îðãðàôîì, â êîòîðîì äëÿ ëþáûõ ðàçëè÷íûõ ïàð fx; yg, fu; vg íåñìåæíûõ âåðøèí èìååò ìåñòî óçïé»ñç ñ³ù³ñ ³å³óáõóí»é »ý ùç ß³ñù åý¹áõùý»ñ: ´³ý³éç µ³é»ñ՝ ïáõùýáñáßí³í ·ñ³ý, ñ³ùçéïáýû³ý óçïé, ý³ïïáñ óçïé, ñ³ùóçïéçï 3 8 on the manoussakis conjecture for a digraph to be hamiltonian d( x ) + d( y ) + d( w ) + d ( z ) ¸ 4 n ¡ 3 . òîãäà d ÿâëÿåòñÿ ãàìèëüòîíîâûì. â íàñòîÿùåé ðàáîòå äîêàçàíî, ÷òî åñëè îðãðàô d óäîâëåòâîðÿåò óñëîâèÿì ãèïîòåçà ìàíîóññàêèñà, òî (1). d ñîäåðæèò öèêë-ôàêòîð; (2). åñëè äëÿ íåêîòîðîé ïàðû íåñìåæíûõ âåðøèí x è y èìååò ìåñòî d( x ) + d( y ) · 2 n ¡ 2 , òî èìåþò ìåñòî: (i) d ÿâëÿåòñÿ ãàìèëüòîíîâûì òîãäà è òîëüêî òîãäà, êîãäà d ñîäåðæèò êîíòóð ïðîõîäÿùèé ÷åðåç âåðøèí x è y, (ii) d ÿâëÿåòñÿ ãàìèëüòîíîâûì èëè ñîäåðæèò êîíòóð äëèíû n ¡ 1 , êîòîðûé ïðîõîäèò ÷åðåç âåðøèíó x ( y ) (â ÷àñòíîñòè, d ñîäåðæèò êîíòóð äëèíû ïî êðàéíåé ìåðå n ¡ 1 ); (3). åñëè äëÿ íåêîòîðîé ïàðû íåñìåæíûõ âåðøèí x è y èìååò ìåñòî d( x ) + d( y ) · 2 n ¡ 4 , òî d ñîäåðæèò êîíòóð ëþáîé äëèíû k, 3 · k · n ¡ 1 ; (4). äîêàçàíû ðÿä ñâîéñòâ äëÿ êîíòóðîâ äëèíû îò n ¡ 5 äî n ¡ 1 . êëþ÷åâûå ñëîâà: îðãðàô, ãàìèëüòîíîâûé öèêë, ôàêòîð öèêë, ïàíöèêëè÷åñêèé îðãðàô. 2_ 2_darbinyan_article начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 42, 85--96, 2014. 85 palm vein minutiae feature extraction for human identification sergey s. chidemyan 1 , aram h. jivanyan 2 and gurgen h. khachatryan 2 1 russian-armenian (slavonic) university 2 american university of armenia e-mail: serchch@gmail.com, ajivanyan@aua.am, gurgenkh@aua.am abstract this paper presents the approach for extracting the minutiae features from palmvein images that can be useful for biometric purposes. in this paper we will show how the extraction of palm-vein features can be made efficiently and accurately using this approach, particularly, how problems of potential deformations, rotational and translational changes are accommodated by this approach. as minutiae features extracted from palm-veins, bifurcation and ending points are chosen. analysis of a database shows that there are 25 minutiae features in each palm-vein image. the experimental results show that the quantity of minutiae features in each vein pattern is enough to perform the personal identification task. keywords: vein pattern, palm vein, biometrics, minutiae features, personal identification 1. introduction palm vein human identification technology is based on pictures of palms that are made by infrared cameras. vein pattern is formed because hemoglobin absorbs infrared radiation. as a result, the reflectance is reduced and the veins are visible as black lines. the vein pattern is a network of vessels underneath human skin. like fingerprints, it is considered that the location of vein patterns in the same part of palm is distinct for each person. taking this fact into consideration and the facts that vein patterns are stable over a long period of time and, what is more important, are invisible to the human eyes (what makes it harder, if not impossible, to copy these features), vein patterns can be considered as very useful biometric features for human identification problem. there are four processing steps to extract the final features: palm-vein database acquisition, image enhancement, vein pattern extraction and feature extraction (fig.1). during the image acquisition stage, casia multi-spectral palmprint image database v1.0 (casia database) is used. this casia database has been acquired using a contactless imaging device and has images from 100 users. six images were acquired from each user and these images were acquired in two different data acquisition sessions (three images in each mailto:serchch@gmail.com mailto:ajivanyan@aua.am mailto:gurgenkh@aua.am palm vein minutiae feature extraction for human identification 86 session) with a minimum interval of one month. since palm veins are most visible under 850 nm wavelength illuminations, only the images under these wavelength illuminations are chosen. in this paper an approach of image enhancement, vein pattern extraction and features extraction is presented. the rest of the paper is organized in the following manner. in section 2 the problems that arise during preprocessing of infrared images and the methods to solve these problems are discussed. this includes roi (region of interest) selection, image quality enhancement and segmentation of vein pattern. in section 3 feature extraction problem is discussed and the method used to solve this problem isdescribed. in section 4 some experimental results are introduced. finally, in section 5, conclusions are given with some discussions. fig .1 feature extraction steps. 2. preprocessing methods of infrared images 2.1 region of interest extraction at this step our purpose is to segment the rectangular region in hand images that defines the region of interest (roi) which contains the main vein pattern on the palm. the main problem that arises during the process of roi segmentation is how automatically to normalize the image in such a way that potential deformations, rotational and translational changes, caused by interactions of user with the device (nir camera), can be minimized. to solve this problem it is necessary to create a coordinate system that is invariant to each of variations mentioned above. it is a good idea to associate the roi with palm itself, so the two reference points are chosen for building such a coordinate system: the web between the index finger and middle finger together with the web between the ring finger and the little finger (fig. 2(c)). to locate the roi the contour of the hand is extracted using any of the filters suitable for this purpose, like sobel, canny, prewitt or roberts filters (fig. 2(a)). then the mid-point of the hand is defined and the distance between each point on hand contour and mid-point is calculated. reference points can be found according to distance profile based on information about these distances (fig. 2(b)). having reference points p1 and p2, the rectangular region of interest can be defined as follows: 𝐿𝑝1𝑝3 = 𝐿𝑝1𝑝2 ∗ 𝛼, where 𝐿𝑝1𝑝2– a distance between 𝑃1 and 𝑃2 and 𝛼 is defined experimentally from database of acquired images. our experiments show that the common value of α is 1.36. (fig. 2(d)). vein feature extraction unprocessed images vein pattern extraction roi selection image enhancement enhanced images casia db feature extraction roi selection image enhancement s. chidemyan, a. jivanyan and g. khachatryan 87 a c b d fig. 2. roi extraction. (a) image after edge detection using sobel filter. (b) distance profile between the mid-point and contour points. (c) valley points detecting. (d) final roi. 2.2 image enhancement images in our database are under near-infrared illumination, which affects on their quality. particularly, they appear darker with low contrast and need to be adjusted. for this purpose a 5 × 5 median filter is used to remove the speckling noise from the image. median filtering is a standard procedure for denoising of images. the main idea behind it is to run through the image pixel by pixel, replacing each pixel with the median of neighboring pixels. to solve the problem with high frequency noise 10 × 10 wiener filter is used. for image i by estimating local mean and variance in 𝑀 × 𝑁 window around each pixel, the wiener filter can be defined as follows: 𝑊𝑖𝑒𝑛𝑒𝑟(𝑖 , 𝑗) = 𝜇 + 𝜎2−𝑣2 𝜎2 (𝐼(𝑖, 𝑗) − 𝜇), (1) where 𝜇 = 1 𝑀𝑁 ∑ ∑ 𝐼(𝑖, 𝑗)𝑁𝑗 𝑀 𝑖 , 𝜎2 = 1 𝑀𝑁 ∑ ∑ (𝐼(𝑖, 𝑗)2 − 𝜇2)𝑁𝑗 𝑀 𝑖 and 𝑣 2 is the average of all the variances. after removing the high frequency noise the last noise removal filter is used: anisotropic diffusion filter. we use this filter to preserve the edges of vein pattern that should be extracted. anisotropic diffusion is a technique aiming at reducing the image noise without removing significant parts of the image content, typically the details that are important for the d is ta n c e f ro m m id -p o in t the position of the pixel in the hand ontour http://en.wikipedia.org/wiki/median http://en.wikipedia.org/wiki/image_noise palm vein minutiae feature extraction for human identification 88 interpretation of the image. anisotropic diffusion resembles the process that creates a scale space, where an image generates a parameterized family of successively more and more blurred images based on a diffusion process. each of the resulting images in this family is given as a convolution between the image and a 2d isotropic gaussian filter, where the width of the filter increases with the parameter. this diffusion process is a linear and spaceinvariant transformation of the original image. anisotropic diffusion is a generalization of this diffusion process: it produces a family of parameterized images, but each resulting image is a combination between the original image and a filter that depends on the local content of the original image. as a consequence, anisotropic diffusion is a non-linear and spacevariant transformation of the original image. formally, let ω ⊂ ℝ2 denote a subset of the plane and 𝐼(. , 𝑡): ω → ℝ be a family of gray scale images, then the anisotropic diffusion is defined as follows: 𝜕𝐼 𝜕𝑡 = 𝑑𝑖𝑣(𝑐(𝑥, 𝑦, 𝑡)𝛻𝐼) = 𝛻𝑐𝛻𝐼 + 𝑐(𝑥, 𝑦, 𝑡)𝛥𝐼, (2) where δ denotes the laplacian, ∇ denotes the gradient, 𝑑𝑖𝑣(… ) is the divergence operator and c(x,y,t) is the diffusion coefficient. c(x,y,t) controls the rate of diffusion and is usually chosen as a function of the image gradient so as to preserve edges in the image. as a diffusion coefficient we used the function: 𝑐(‖𝛻𝐼‖) = 𝑒 −( ‖𝛻𝐼‖ 𝐾 )2 , where the constant k controls the sensitivity to edges and its value is chosen experimentally equal to 20. finally, to adjust the contrast of the images, simple histogram equalization is used. fig.3 shows the extracted roi image after median, wiener and anisotropic diffusion filter and after image enhancement with histogram equalization. it can be seen that the quality of the image has been improved significantly. (a) (b) fig. 3. illustration of image enhancement: (a) original roi images, (b) correspondingly enhanced roi images of (a). http://en.wikipedia.org/wiki/anisotropic http://en.wikipedia.org/wiki/scale_space http://en.wikipedia.org/wiki/scale_space http://en.wikipedia.org/wiki/diffusion_process http://en.wikipedia.org/wiki/convolution http://en.wikipedia.org/wiki/isotropic http://en.wikipedia.org/wiki/gaussian_filter http://en.wikipedia.org/wiki/laplacian http://en.wikipedia.org/wiki/gradient http://en.wikipedia.org/wiki/divergence http://en.wikipedia.org/wiki/anisotropic s. chidemyan, a. jivanyan and g. khachatryan 89 2.3 segmentation of vein pattern a. multiscale vessel enhancement by observing our database we found that multiscale vessel enhancement scheme proposed in [1] can be used to segment a vein pattern. thus, in this approach a vessel enhancement is conceived as a filtering process that searches for geometrical structures which can be regarded as tubular. since vessels appear in different sizes it is important to introduce a measurement scale which varies within a certain range. to analyze the local behavior of an image l the taylor expansion in the neighborhood of a point 𝑥0 is considered. 𝐿 (𝑥0 + ∆𝑥0 , 𝑠) ≈ 𝐿(𝑥0 , 𝑠) + ∆𝑥0 𝑇 ∇𝑜,𝑠 + ∆𝑥0 𝑇 𝐻𝑜,𝑠∆𝑥0. (3) this expansion approximates the structure of the image up to second order. ∇o,s and ho,s are the gradient vector and hessian matrix of the image computed in xo at scale s. the differentiation here is defined as a convolution with the derivatives of the gaussian functions: 𝜕 𝜕𝑥 𝐿(𝑥, 𝑠) = 𝑠𝛾 𝐿(𝑥) ∗ 𝜕 𝜕𝑥 𝐺(𝑥, 𝑠), (4) where the d-dimensional gaussian function at scales s is defined as follows: 𝐺(𝑥, 𝑠) = 1 √2𝜋𝑠2 𝐷 𝑒 −‖𝑥2‖ 2𝑠2 , (5) and * is the convolution operator. the parameter 𝛾 was introduced by lindeberg [2] to define a family of normalized derivatives. for our purpose it is set equal to 1. the idea behind the eigenvalue analysis of the hessian is to extract the principal directions in which the local second order structure of the image can be decomposed. since this directly gives the direction of smallest curvature (along the vessel) the application of several filters in multiple orientations is avoided. let 𝜆𝑠,𝑘denote the eigenvalue corresponding to the k-th normalized eigenvector 𝑢𝑠,𝑘of the hessian 𝐻𝑜,𝑠, all computed at scale s. from the definition of eigenvalues: 𝐻𝑜,𝑠𝑢𝑠,𝑘 = 𝜆𝑠,𝑘 𝑢𝑠,𝑘 , (6) and it follows that: 𝑢𝑠,𝑘 𝑇 𝐻𝑜.𝑠𝑢𝑠,𝑘 = 𝜆𝑠,𝑘. (7) let 𝜆𝑘denote the eigenvalue with the k-th smallest magnitude (|𝜆1| ≤ |𝜆2| ≤ ⋯). in particular, a pixel belonging to a vessel region will be signaled by 𝜆1being small (ideally zero): 𝜆1 ≈ 0 and|𝜆1| ≪ |𝜆2|. two local characteristics of image can be measured by analyzing the above two equations. first, the norm of the eigenvalues will be small at the location where no structure information is shown since the contrast difference is low, and it will become larger when the region occupies a higher contrast since at least one of the eigenvalues will be large. second, the ratio between |𝜆1| 𝑎𝑛𝑑 |𝜆2| will be large when the blob-like structure appears in the local area, and will be very close to zero when the structure shown is line-like. mathematically, the two measures are represented as follows: palm vein minutiae feature extraction for human identification 90 𝐿𝑠 = √∑ 𝜆𝑖 2,𝑖≤𝐷 (8) 𝐿𝑙 = |𝜆1| |𝜆2| , (9) where 𝐿𝑠 and 𝐿𝑙 are measurement scales mentioned above and d is the dimension of the image. for 2d images the proposed local vesselness measure of position 𝑥0 in scale s will have the following form: 𝑣𝑜.𝑠 = { 0, 𝑖𝑓𝜆2 > 0, 𝑒 − 𝐿𝑙 2 2𝛼2 (1 − 𝑒 − 𝐿𝑠 2 2𝛽2 ) . (10) the vesselness measure in equation (10) is analyzed at different scales s. the response of the line filter will be maximum at a scale that approximately matches the size of the vessel to detect. the vesselness measure provided by the filter response at different scales is integrated to obtain a final estimate of vesselness: 𝜈0 (𝜁) = max𝑠𝑚𝑖𝑛<𝑠<𝑠𝑚𝑎𝑥 𝜈0(𝑠, 𝜁), (11) where 𝑠𝑚𝑖𝑛 and 𝑠𝑚𝑎𝑥 are the maximum and minimum scales at which relevant structures are expected to be found. they can be chosen so that they will cover the range of vessel widths. experimentally we find out that for our purpose it is enough to take 𝛼 equal to 0.5 and 𝛽 equal to 8. fig. 4 shows enhanced roi images and corresponded images after multiscale vessel enhancement. (a) (b) fig. 4. illustration of image multiscale vessel enhancement: (a) enhanced roi images, (b) correspondingly roi images after multiscale vessel filter. b. local thresholding with global reduction as it can be seen from fig. 4, the quality of images improves after the noise reduction and enhancement, however there is still many faint white regions near the vein pattern. to obtain a s. chidemyan, a. jivanyan and g. khachatryan 91 better representation of a vein pattern, it is necessary to separate the vein pattern from the background. since the global thresholding techniques alone do not provide satisfactory results local thresholding with global reduction (ltgr) is used. ltgr is an adaptive algorithm that combines both local and global thresholding. the algorithm chooses different threshold values for every pixel in the image based on the values of its surrounding neighbors. mathematically it can be expressed as follows: 𝐵𝑖𝑛𝑎𝑟𝑖𝑧𝑒𝑑𝐼𝑚𝑎𝑔𝑒(𝑖, 𝑗) = { 1, 𝑖𝑓𝐼𝑚𝑎𝑔𝑒(𝑖, 𝑗) ≥ 𝜇𝑖,𝑗 − 𝑇, 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒, (12) where 𝜇𝑖,𝑗 is the mean value for (𝑖, 𝑗) pixel’s 𝑁 × 𝑁 neighborhood and t is a common offset for all the pixels. parameters n and t are found experimentally equal to 10 and 6, correspondingly. fig. 5 illustrates the final extracted vein pattern after ltgr. it can be noticed that the true vein structures are quite well preserved in the resulting binarized image. (a) (b) fig. 5. illustration of vein pattern extraction after ltgr algorithm: (a) roi images after multiscale vessel enhancement, (b) roi images after ltgr algorithm is used. 3. minutiae feature extraction 3.1 skeletonization to obtain minutiae features from binarized vein pattern the skeleton representation of it firstly should be obtained. any of skeletonization algorithms can be used for this purpose. we use a thinning algorithm proposed in [3]. fig. 6 shows the skeleton of vein patterns extracted from the images from a previous step. as it can be seen from fig. 6, vein patterns are now well preserved. palm vein minutiae feature extraction for human identification 92 (a) (b) fig. 6. roi images after thinning algorithm. 3.2 extraction of minutiae points the vein pattern can be well represented by a number of critical points referred as minutiae points. the branching points and the ending points in the vein pattern skeleton image are the two types of critical points to be extracted. ending points here are mainly ending points of vein skeleton curves that placed at the edge of roi and resulted from the cropping of hand image while obtaining roi. although these ending points are not real ending points of vein on palm, they are taken because they contain geometrical information about the shape of the skeletons of the vein pattern. as for bifurcation points, they are the junction points of three curves. fig. 7 illustrates some of bifurcation and ending points on vein pattern’s skeleton representation. fig. 7. some of bifurcation points are marked by red circle to obtain the junction points the method used in [4] is taken. we run a pixel-wise operation for 3 × 3 region: p1 p2 p3 p8 p0 p4 p7 p6 p5 s. chidemyan, a. jivanyan and g. khachatryan 93 and define the number of transition between 0 and 1 (and vice versa) from 𝑃1 to 𝑃8 as follows: 𝑁𝑡𝑟𝑎𝑛𝑠 = ∑ |𝑃𝑖+1 − 𝑃𝑖 | 8 𝑖=1 , where 𝑃1 = 𝑃9. then 𝑃0 𝑖𝑠 { 𝑒𝑛𝑑𝑖𝑛𝑔 𝑝𝑜𝑖𝑛𝑡, 𝑖𝑓𝑃0 = 1 𝑎𝑛𝑑 𝑁𝑡𝑟𝑎𝑛𝑠 = 2, 𝑏𝑟𝑎𝑛𝑐ℎ 𝑝𝑜𝑖𝑛𝑡, 𝑖𝑓𝑃0 = 1 𝑎𝑛𝑑 𝑁𝑡𝑟𝑎𝑛𝑠 ≥ 6. fig. 8 illustrates the extracted bifurcation and ending points after merging these two sets of points for one of the extracted palm vein pattern’s skeleton.these minutiae points are widely used as a feature to match pair of palm veins, so to identify a person, too. fig. 8. union of sets of bifurcation and ending points for some vein pattern’s skeleton. 4. experimental results different techniques can be used to compare the similarity of extracted minutiae points for two different persons. for our purpose we used the modified hausdorff distance, proposed in [5]. for two point sets a and b, the modified hausdorff distance (mhd) can be defined as follows: ℎ(𝐴, 𝐵) = 1 𝑁 ∑ 𝑚𝑖𝑛𝑏𝑗𝜖𝐵𝑎𝑖∈𝐴 ∥ 𝑎𝑖 − 𝑏𝑗 ∥, (13) where n is the number of points in a. fig. 9. error rate curves for minutiae recognition using mhd. palm vein minutiae feature extraction for human identification 94 fig. 9 shows false acceptance rate and false rejection rate curves from which it can be seen mhd algorithm achieves approximately 1% equal error rate (eer), when the threshold value was set equal to 11, which is quite good for this kind of systems. 5. conclusion in this paper we have attempted to present the approach and some experimental results to show how problems that arise during hand palm segmentation, roi image enhancement, palm vein extraction and minutiae feature extraction can be solved efficiently and quickly. we proposed different techniques of image enhancement including some morphological techniques. after each step of preprocessing, as it can be seen, the images are enhanced significantly. experiments on casia database show that we can extract on average 25 minutiae points from each vein pattern, including 10 bifurcation points and 15 ending points on average for each vein pattern. this quantity of minutiae point is quite enough for the purpose of human identification. since the database is big enough (6 hand image for each of 100 persons) it can be argued according to our experiments that these minutiae features are discriminating features in the hand vein images. references [1] a. f. frangi, w. j. niessen, k. l. vincken and m. a. viergever,“multiscale vessel enhancement filtering”, miccai, springer, lncs 1496, pp. 130-137, 1998. [2] t. lindeberg,“edge detection and ridge detection with automatic scale selection”, proc. conf. on comp. vis. and pat.recog.,san francisco, ca, june, pp. 465–470, 1996. [3] z. guo and r.w. hall, “fast fully parallel thinningalgorithms”,computer vision, graphics and image processing, vol. 55, no.3, pp.317-328, 1992. [4] l.y. wanga, g. leedhamb, and d. s. y. choa, “minutiae feature analysis for infrared hand vein pattern biometrics”, pattern recognition society, published by elsevier ltd, all rights reserved, 2007. [5] m.p. dubuisson and a.k. jain, “a modified hausdorff distance for object matching”, proceedings of the 12 th international conference on pattern recognition, pp. 566-568, jerusalem, israel, 1994. [6] h.soliman, a. saber mohamed and ahmed atwan, “feature level fusion of palm veins and signature biometrics”, international journal of video & image processing and network security ijvipns-ijens,vol. 12, no. 01 28, pp.28--39, 2012. [7] y. zhou and a. kumar, “human identification using palm-vein images”,ieee transactions on information forensics and security, vol. 6, no. 4, pp. 1259--1274, 2011. [8] gohkahong michael, t. connie and a.bengjinteoh,”touch-less palm print biometrics: novel design and implementation”, image and vision computing,vol. 26,pp. 1551– 1560, 2008. s. chidemyan, a. jivanyan and g. khachatryan 95 [9] m. h.-m. khan, r. k. subramanian and n. a. mamode khan, “low dimensional representation of dorsal hand vein features using principle component analysis (pca)”, world academy of science, engineering and technology 49,vol.3, pp. 848--854, 2009. [10] l. mirmohamadsadeghi and a. drygajlo, “palm vein recognition with local binary patterns and local derivative patterns”, proceedings of the 2011 international joint conference on biometrics, october 11-13, p.1--6, 2011. [11] a. kumar and k. v. prathyusha, “personal authentication using hand vein triangulation and knuckle shape”, ieee trans. image process., vol. 38, pp. 2127--2136, sep. 2009. [12] yi-bo zhang, qin li and j. you and p. bhattacharya, “palmveinextraction and matching for personal authentication”, g. qiu et al. (eds.): visual 2007, lncs 4781, pp. 154– 164, 2007. [13] d. zhang, w.-k. kong, j. you and m. wong, “online palmprintidentification”, ieee transactions on pattern analysis and machine intelligence, vol. 25, no. 9, pp. 10411050, september, 2003. [14] a.shrotri, s.c.rethrekar, m.h.patil, d. bhattacharyya and tai-hoon kim, ”infrared imaging of hand vein patterns for biometric purposes”, journal of security engineering, pp.57-66, 2009. [15] m. deepamalar and m. madheswaran, “an enhanced palm vein recognition system using multi-level fusion of multimodal features and adaptive resonance theory”, international journal of computer applications (0975 – 8887),vol. 1,no. 20, pp. 95-101, 2010. [16] casia ms palmprint v1 database, [online]. available: http://biometrics.idealtest.org/dbdetailforuser.do?id=5. submitted 05.08.2014, accepted 28.11.2014. ձեռքի ափի երակներից բնութագրերի առանձնացում մարդկանց նույնականացման համար ս. չիդեմյան, ա. ջիվանյան և գ. խաչատրյան ամփոփում այս հոդվածում ներկայացված է ձեռքի ափի երակների պատկերներից բնութագրերի առանձնացման մեթոդը, որը կարող է օգտակար լինել բիոմետրիկ խնդիրների լուծման ժամանակ: այստեղ նաև ցույց է տրվել, թե ինչպես ափի երակներից բնութագրերի առանձնացումը կարող է իրականացվել հստակ և արդյունավետ` մեր առաջարկած եղանակով, մասնավորապես՝ ինչպես հնարավոր աղավաղումների և նկարի պտտելու հետ կապված խնդիրները լուծելի են մեր առաջարկած եղանակով: որպես բնութագիր ընտրվել են ճյուղավորման և եզրային կետերը: պատկերների http://biometrics.idealtest.org/dbdetailforuser.do?id=5 palm vein minutiae feature extraction for human identification 96 բազայի վերլուծությունը ցույց տվեց, որ յուրաքանչյուր ափի երակների պատկերից կարելի է առանձնացնել մոտավորապես 25 բնութագիր կետեր: փորձերի արդյունքում եկանք այն եզրահանգմանը, որ յուրաքանչյուր երակների ցանցի համար բնութագիր կետերի քանակը բավական է անձի նույնականացման խնդիրը լուծելու համար: извлечение минуций вен ладони для распознавания людей с. чидемян, а. дживанян и г. хачатрян. аннотация в этой статье представлен способ извлечения минуций из изображения вен ладони, которые могут быть полезными при решении биометрических задач. также было показано как выделение характеристик из вен ладони может быть произведено эффективно и точно при помощи нашего метода, в частности, как проблемы, связанные с потенциальными искажениями, а также с вращениями изображения, могут быть решены описанным способом. в качестве характеристик, извлекаемых из вен ладони, были выбраны точки ветвления и краевые точки. анализ базы изображений показал, что каждое изображение вен ладони содержит около 25 минуций. по результатам экспериментов можно заключить, что количество минуций для каждой сетки вен достаточно для решения задач персональной идентификации. mathematical problems of computer science 57, 39–46, 2022. doi: 10.51408/1963-0085 udc 004.05 electronic voting system essentials and problems arman a. avetisyan russian-armenian university e-mail: armanavetisyan1997@gmail.com abstract the development of reliable and safe e-voting systems is relevant because of the wide range of applications. this paper provides an analysis of modern electronic voting systems based on security criteria. an analysis was conducted based on the most popular modern e-voting system architectures. the analysis provides a baseline for developing a secure e-voting system. keywords: electronic voting, internet voting, information security, elections, system architecture, voting systems. 1. introduction electronic voting (e-voting) is a term that encompasses several different types of voting methods and electronic means of counting votes. e-voting systems include punched cards, optical voting systems and specialized voting kiosks (including stand-alone electronic systems for direct voting), as well as means for the transmission of ballots and votes by telephone, via a private computer network or via the internet [1]. such systems would speed up the counting of votes and make voting more accessible and transparent. however, weak e-voting systems could encourage electoral fraud. the advantages and disadvantages of modern e-voting solutions and technologies should be explored in order to create a secure system. this study focuses on the electronic systems through which the entire electoral process (voter registration, voting and counting votes) is conducted. the study distinguishes the standard functionality of e-voting systems. standard e-voting systems include the following modules [2]: • electronic voter lists and a method of voter identification, • interface for polling station staff, • interface for voters, • system for sending votes to count, • interface to show results. 39 40 electronic voting system essentials and problems the e-voting system should correspond to a series of criteria, which can be divided into two important groups: primary based on the security and safety of the system, and secondary based on the user friendliness and accessibility. system safety requirements are: • integrity of elections (ensuring the accuracy of the elections, all the ballots should be accounted for and no changes must be made to them), • privacy of the vote (make ballots indistinguishable from one another, as well as protect any information about the voter ), • authenticity of the voters (only eligable voters can take part in elections), • verifiability of the votes (a person should be able to verify that his vote has been cast and counted), • protection against attacks (the system should be secure against attacks in any phase of the elections, and the voters should be able to alert the committee if any fraud has been detected), • ensuring the confidentiality of personal data (no voter should be able to prove to a third party that he voted for a particular person). at the international level, the systems developed and tested today have some security problems. a great deal of scientific literature has been devoted to this study [2] [5], but a number of questions still remain. even the best e-voting systems today have some drawbacks. in [6], the author reviewed electoral systems in some countries, where e-voting was used during elections. the comparative analysis was carried out on the basis of the main safety criteria of the most popular modern e-voting systems used in several countries. the study showed that the systems used nowadays are insecure against external attacks and thus raise questions about the integrity of elections they are used in. most of the systems were implemented more than a decade ago, and the security protocols used have gone obsolete since then. even the best systems that are steadily replacing paper voting, have been criticized by third-party studies due to massive vulnerabilities. the need for secure and reliable e-voting systems remains a relevant problem nowadays. this paper proposes a model based on the most popular practices in modern e-voting systems. 2. e-voting system architecture in the last twenty years there has been active research into the creation of secure voting systems. these systems are based on public-key cryptography and on the approach that the voter’s vote is encrypted with a public key that corresponds to it. the private key is distributed among the members of the electoral commission, so the members of the electoral commission will be able to decrypt and count the votes together. in addition, special methods are used to ensure the secrecy of the ballots (mix network, additive homomorphic encryption systems...). this study proposes a baseline system architecture based on e-voting systems and protocols used nowadays, as well as their known vulnerabilities. a. avetisyan 41 first, we outline the basic voting procedure, divided into 5 key phases: • setup phase setting up the election system architecture, • e-ballot filling phase filling out an e-ballot on a device and casting the vote, • e-ballot registration phase checking if the e-ballot is eligible and storing it on a server, • anonymity phase making sure all the voter info is stripped away from the e-ballot, • counting phase counting all the anonymous ballots and providing the results to public. 2.1 3-server architecture the proposed e-voting system architecture consists of 3 main servers (see fig. 1): • vote forwarding server is the only publicly accessible server. it verifies the eligibility of the voter and acts as an intermediary to the backend vote storage server. • vote storage server is a backend server that stores signed, encrypted votes during the online voting period. upon receiving a vote from the forwarding server, it confirms that the vote is formatted correctly and verifies the voter’s digital signature. • vote counting server is never connected to a network and is only used during the final stage of the election to count the votes received from the storage server. fig. 1. 3-server architecture. 42 electronic voting system essentials and problems 2.2 voting process during the initial voting stage, the voter uses client software to cast a vote. the software has a connection with a forwarding server, which is used to authenticate the voter and check their eligibility. all communication with elections servers is done via the vote forwarding server which is the only server accessible from the voter’s device. the vote forwarding server is an intermediary between the client device and the storage server. after the voter is authenticated, the client receives a package with a set of candidates. when the voter picks a candidate, the software encrypts the information about the candidate and signs the data with the unique key of the voter.the software then sends the encrypted data package to the forwarding server which returns an id of the package meaning the vote has been successfully casted. it is important to note that no information about the voter is sent to the election server other than their unique signature which can only be used to check the eligibility of the vote and not the identity of the voter. the transfer of this data between the forwarding server and storage server remains a huge issue even nowadays because each part of the system trusts the channels through which the vote data is transferred. the transfer from client to storage is done via the internet using secure protocols (see fig. 2). fig. 2. casting vote. when the vote data arrives at the vote storage server, it is once again checked and verified. then the sensitive data like the unique signature is stripped from the votes to make them completely anonymous before sending to the counting server where the data (which contains only information about who the vote is for) is decrypted and tabulated (see fig. 3). counting server then writes all the data into an accessible database from which the final results can be gathered. a. avetisyan 43 fig. 3. vote counting. the discussed system is a basic model of e-voting systems being used nowadays [7] [10], but there are vulnerabilities and areas for improvement. 3. discussion about vulnerabilities although the system may seem straightforward and secure, its current implementations have raised serious concerns [11] [13]. the main concern is the secure transfer of the vote data. usage of client-side software is essential in e-voting systems so the process of transferring votes from the client device to the election server should be handled carefully. the inclusion of transfer server as a buffer between the client and the storage server is integral. we can differentiate two types of attacks: client-side and server-side. client-side attacks target the client device and exploit its vulnerabilities. the client software needs to be as secure as possible against this kind of attacks by not having any unwarranted communication with the device. server-side attacks target server architecture. the system must be minimally dependent on the person or people controlling it to be secure. the human factor plays a huge role in exploiting the election systems. another major vulnerability is the code vulnerability. due to the complex nature of the system, current implementations have been proven to have significant oversight in security against attacks like denial of service or shell injection. attacks target client software or server architecture and try to achieve the following: • find out confidential information about voters like the candidate they voted for, • try to alter the results of the election by changing or adding fake votes, • altering/destroying enough votes to create mistrust in the election results. to find out information about voters, the attack needs to take place before the anonymity phase, so client-side attacks mainly take place for this purpose. if the channel between the 44 electronic voting system essentials and problems client software and the transfer server is not secure, the data may be intercepted and even altered. modern cryptography methods help to encrypt data, so it is hard to decode, but adding noise to an insecure channel to alter or ruin the vote is easy if someone already has access. these attacks are generally low-scale as user devices need to be exploited one by one, even with techniques like botnets it is unlikely to cause too much harm to the overall security of elections. the main damage to elections is caused by attacking storage and counting servers and the channel between them. as all the important vote databases are in those servers, if access is received by an attacker, the damage to elections will be massive. having a good and secure architecture is the key to preventing that from happening. currently, the systems use physical data transfer from one server to another, as well as a physical decryption device that holds the key. this relies too much on actual human beings to securely transfer a lot of essential data from one point to another, and the whole point of implementing e-voting is to get rid of the human factor. this also means that the storage server must be easily accessible for humans to put in data, which is not ideal. we see that modern systems in use still use ”hybrid” systems partly run by people to make up for software vulnerabilities that need to be addressed in the future works when it comes to developing better e-voting systems. 4. conclusion and future work the paper discussed the basic architecture of the e-voting system, which is the baseline of systems used nowadays. understanding the potential problems and vulnerabilities is essential to creating a secure e-voting system. the 3-server architecture is a good starting point to build a new system. it is evident that the physical transportation of data in modern systems is the key problem that has to be addressed first. the use of various steganographic models can help to reduce the risks of data corruption and tampering. more specifically, the steganography models with active adversary are very close to imitating attacks that can happen during elections. one of the main focus points of more secure systems will be the testability, it is essential to have an ability to quantify the level of security of the model. to achieve this goal, research has to be conducted in various fields of security, specifically steganography and differential privacy. 5. acknowledgment the work was supported by the science committee of ra, within the framework of the research project 21t-1b151. references [1] m. stenbro, “survey of modern electronic voting technologies”, the norwegian university of science and technology, master thesis, 2010. [2] international idea “introducing electronic voting: essential considerations”, https://www.corteidh.or.cr/tablas/28047, 2011. a. avetisyan 45 [3] k. sanjay and e. walia, “analysis of electronic voting system in various countries”, international journal on computer science and engineering, vol. 3, no. 5, pp. 1825– 1830, may 2011. [4] v. martin, “evaluation of internet voting systems based on requirements satisfaction”, international review of social sciences and humanities, vol. 6, no.1 , pp. 41-52, 2013. [5] a. t. sherman, r. a. fink, r. carback and d. chaum, “scantegrity iii: automatic trustworthy receipts, highlighting over/under votes, and full voter verifiability”, in proceedings of the electronic voting technology/workshop on trustworthy elections, evt/wote, 2011. [6] a. avetisyan, “comparative analysis of modern e-voting systems based on security criteria”,proceedings of international conference csit 2021, yerevan, armenia, pp. 81-84, 2021. [7] n. goodman, j.h. pammett and j. de bardeleben, “a comparative assessment of electronic voting”, report prepared for elections canada, 2010. [8] l. loeber, “e-voting in the netherlands: from general acceptance to general doubt in two years, 3rd international conference on electronic voting, pp. 2130, 2008. [9] d. f. aranha and j. van de graaf, “the good, the bad, and the ugly: two decades of e-voting in brazil”, ieee security and privacy, vol. 16, no. 6, pp. 22-30, nov.-dec. 2018. [10] m. hapsara, a. imran and t. turner, ”e-voting in developing countries”, electronic voting. e-vote-id 2016. lecture notes in computer science, vol. 10141, pp. 3655, 2017. [11] d. springall, t. finkenauer, z. durumeric, j. kitcat, h. hursti, m. macalpine and j. halderman, ”security analysis of the estonian internet voting system”, proceedings of the 21st acm conference on computer and communications security, pp. 703-715, 2014. [12] t. haines, s. j. lewis, o. pereira and v. teague, “how not to prove your election outcome,” ieee symposium on security and privacy (sp), pp. 644-660, 2020. [13] m. a. specter, j. koppel and d. weitzner, “the ballot is busted before the blockchain: a security analysis of voatz, the first internet voting application used in u.s. federal elections”, proceedings of the 29th usenix conference on security symposium. usenix association, pp. 1535-1552, 2020. submitted 18.02.2022, accepted 27.05.2022. 4 6 electronic voting system essentials and problems ¾é»ïïñáý³ûçý ùí»³ñïáõãû³ý ñçùáõýùý»ñá ¨ ëý¹çñý»ñá ²ñù³ý ². ²í»ïçëû³ý èáõë-ñ³ûï³ï³ý ñ³ù³éë³ñ³ý, ºñ¨³ý, ð³û³ëï³ý e-mail: armanavetisyan1997@gmail.com ²ù÷á÷áõù îñíîâû è ïðîáëåìû ýëåêòðîííîãî ãîëîñîâàíèÿ àðìàí à. àâåòèñÿí ðîññèéñêî-àðìÿíñêèé óíèâåðñèòåò, åðåâàí, àðìåíèÿ e-mail: armanavetisyan1997@gmail.com àííîòàöèÿ ðàçðàáîòêà íàäåæíûõ è áåçîïàñíûõ ñèñòåì ýëåêòðîííîãî ãîëîñîâàíèÿ àêòóàëüíà èç-çà øèðîêîãî ñïåêòðà ïðèìåíåíèé. â äàííîé ðàáîòå áûë ïðîâåäåí àíàëèç ñîâðåìåííûõ ñèñòåì ýëåêòðîííîãî ãîëîñîâàíèÿ íà îñíîâå êðèòåðèåâ áåçîïàñíîñòè. àíàëèç ïðîâîäèëñÿ íà îñíîâå íàèáîëåå ïîïóëÿðíûõ ñîâðåìåííûõ àðõèòåêòóð ñèñòåì ýëåêòðîííîãî ãîëîñîâàíèÿ. äàííûé àíàëèç ÿâëÿåòñÿ îñíîâîé äëÿ ðàçðàáîòêè áåçîïàñíîé ñèñòåìû ýëåêòðîííîãî ãîëîñîâàíèÿ. êëþ÷åâûå ñëîâà: ýëåêòðîííîå ãîëîñîâàíèå, èíòåðíåò-ãîëîñîâàíèå, èíôîðìàöèîííàÿ áåçîïàñíîñòü, âûáîðû, àðõèòåêòóðà ñèñòåìû, ñèñòåìû ãîëîñîâàíèÿ. ðáõë³éç ¨ ³ýíï³ý· ¿é»ïïñáý³ûçý ùí»³ñïáõãû³ý ñ³ù³ï³ñ·»ñç ùß³ïáõùý ³ñ¹ç³ï³ý ¿ ïçñ³éáõãûáõýý»ñç é³ûý ßñç³ý³ïç å³ï×³éáí: ²ßë³ï³ýùáõù ý»ñï³û³óíáõù ¿ å³ù³ý³ï³ïçó ¿é»ïïñáý³ûçý ùí»³ñïáõãû³ý ñ³ù³ï³ñ·»ñç í»ñéáõíáõãûáõý` ñçùýí³í ³ýíï³ý·áõãû³ý ã³÷³ýçßý»ñç íñ³: ì»ñéáõíáõãûáõýý çñ³ï³ý³óí»é ¿` ñçùýí»éáí ¿é»ïïñáý³ûçý ùí»³ñïáõãû³ý ³ù»ý³ñ³ûïýç å³ù³ý³ï³ïçó ñ³ù³ï³ñ·»ñç íñ³: ì»ñéáõíáõãûáõýá ñçùù ¿ ñ³ý¹çë³ýáõù ³ýíï³ý· ¿é»ïïñáý³ûçý ùí»³ñïáõãû³ý ñ³ù³ï³ñ·»ñç ùß³ïù³ý ñ³ù³ñ: ´³ý³éç µ³é»ñ` ¿é»ïïñáý³ûçý ùí»³ñïáõãûáõý, çýï»ñý»ï ùí»³ñïáõãûáõý, ï»õ»ï³ïí³ï³ý ³ýíï³ý·áõãûáõý, áýïñáõãûáõýý»ñ, ñ³ù³ï³ñ·ç ×³ñï³ñ³å»ïáõãûáõý, ùí»³ñïáõãû³ý ñ³ù³ï³ñ·»ñ: 04_avetisyan_39_46 04a mathematical problems of computer science 56, 56–64, 2021. udc 519.688 determining the degree of fuzzy regularity of a string armen h. kostanyan it educational and research center yerevan state university e-mail: armko@gmail.com abstract the paper deals with the issue of determining the degree of fuzzy regularity of a crisp string. it is assumed that the concept of fuzzy regularity is formalized by a pattern given as a finite automaton with fuzzy properties of alphabet characters on transitions. proceeding from this, we replace the problem of determining the degree of fuzzy regularity of a crisp string with the problem of determining the degree of belonging of such a string to the language of the corresponding automaton and propose an effective method for solving it using the dynamic programming approach. the solution to the considered problem makes it possible to fuzzify the set of strings in a given alphabet based on a pattern defining fuzzy regularity. this work is a continuation of the author’s previous works related to finding occurrences of a fuzzy pattern in the text. it may have applications in the field of pattern recognition, data clustering, bio-informatics, etc. keywords: fuzzy string matching, pattern recognition, fuzzy automaton. 1. introduction this paper refers to the definition of a degree of fuzzy regularity of a crisp string in a given alphabet. to formalize the concept of fuzzy regularity, we use the highest degree of matching of such a string to the elements of a given set of periodical sequences of fuzzy properties of the alphabetic characters. we treat this set of sequences as a matching pattern. as we noted, the sequences of fuzzy properties included in the pattern must have a certain periodicity. to achieve this goal, we propose to define the matching pattern as a finite automaton with fuzzy properties of alphabetic characters on transitions. as the pumping lemma states, any sufficiently large sequence of fuzzy properties accepted by such an automaton will be periodic with a period size not exceeding the number of states of the automaton. thus, in the concept we propose, the problem of determining the fuzzy regularity of a crisp string is defined as a problem of determining the degree of belonging such a string to the language of a fuzzy automaton. 56 a. kostanyan 57 it should be noted that the concept of a fuzzy automaton, which we use as a pattern for determining the regularity, is somewhat different from the one generally accepted in the literature. for example, in the concept of a fuzzy automaton used in [1] and [2], the automaton states and the alphabetic characters are assumed to be crisp, while the start and final states as well as the transitions of the automaton are assumed to be fuzzy. a review of the results on fuzzy automata in their various interpretations, as well as the fuzzy languages they accept (including automaton analysis and synthesis, minimization, closure properties, etc.) is given in [3]. the problem of determining the regularity of a string is widely used in pattern recognition, where it is often necessary to detect regularities in strings encoding images [4]. such regularity is not always specified precisely, so we have to deal with a fuzzy regularity. the concept of fuzzy regularity can also be used in the area of fuzzy data clustering [5], which deals with grouping elements into fuzzy clusters, a particular case of which is fuzzy sequence labeling [6]. the investigation on fuzzy regularity of a string that we propose in this paper continues our previous research in the field of fuzzy string matching [7], [8], [9]. the problem of splitting a string into adjacent segments to best match the pattern, which is a sequence of fuzzy properties of substrings, was considered in [10], [11]. in this paper, we concretize the concept of a fuzzy property of a substring, defining it as the degree of belonging the substring to the language of a fuzzy automaton. the paper is organized as follows. section 2 presents the concepts of a fuzzy symbol and a finite automaton over a set of fuzzy symbols, called a fuzzy automaton. section 3 considers the problem of matching a crisp string with a pattern given by a fuzzy automaton and provides an efficient algorithm for determining the degree of matching based on the dynamic programming approach. finally, the conclusion summarizes the obtained results. 2. preliminaries 2.1 fuzzy symbols suppose that (l, ≤, ⊗, 0, 1) is a finite linearly ordered set of measures with the smallest element 0, the largest element 1, and the monotonic accumulation operation ⊗ such that l is a commutative monoid with unit element 1 and zero element 0. that is, for all a, b, c ∈ l a ⊗ 0 = 0, a ⊗ 1 = a, a ≤ b ⇒ a ⊗ c ≤ b ⊗ c. in our further considerations, we will assume that the maximum over the empty set of l values is equal to 0. according to [12], the fuzzy subset x of the universal set u is defined by the membership function µx : u → l that associates with each element u ∈ u the value µx(u) ∈ l, representing the degree of belonging u to x. a fuzzy subset x of u can be represented by the additive form x = ∑ u∈u u/µx(u). we say that an element u ∈ u certainly belongs to x if µx(u) = 1, and it certainly does not belong to x if µx(u) = 0. conversely, if 0 < µx(u) < 1, then we say that u belongs to x with degree µx(u). 58 determining the degree of fuzzy regularity of a string given an alphabet σ of characters, we define a fuzzy symbol α over σ as a fuzzy subset of σ. given a character x ∈ σ and a fuzzy symbol α over σ, we say that x matches α with degree µα(x). this definition can be extended in the usual way to equal length sequences of characters and fuzzy symbols, respectively. that is, for a set of fuzzy symbols ξ, x = x1...xn ∈ σ∗ and ω = ω1...ωn ∈ ξ∗, we define the matching degree of x to ω as the l-value µω(x) = { µω1(x1) ⊗ ... ⊗ µωn(xn), if x ̸= ϵ, 1, if x = ϵ. • s = 1/1 + 2/0.9 + 3/0.6 + 4/0.3 + 5/0.1 + 6/0 + 7/0, • m = 1/0 + 2/0.25 + 3/0.75 + 4/1 + 5/0.75 + 6/0.25 + 7/0, • l = 1/0 + 2/0 + 3/0.1 + 4/0.3 + 5/0.6 + 6/0.9 + 7/1. let x = 35634, ω1 = smlsm, ω2 = mlmsl. then • µω1(x) = µs(3) ⊗ µm(5) ⊗ µl(6) ⊗ µs(3) ⊗ µm(4) = 3 5 · 3 4 · 9 10 · 3 5 · 1 = 243 1000 , • µω2(x) = µm(3) ⊗ µl(5) ⊗ µm(6) ⊗ µs(3) ⊗ µl(4) = 3 4 · 3 5 · 1 4 · 3 5 · 3 10 = 81 4000 . 2.2 fuzzy automaton given a finite set ξ of fuzzy symbols over the alphabet σ, we define a fuzzy automaton as a deterministic finite automaton using symbols from ξ on transitions. that is, a fuzzy automaton is a 5-tuple a = (q, ξ, δ, qin, f), where • q is the finite non-empty set of states, • δ ⊆ q × ξ → q is the transition function, • qin ∈ q is the initial state, • f ⊆ q is the set of final states. let h = p0−→α1 p1−→α2 ... −→αn pn, n ≥ 0, be a path leading from the state p0 ∈ q to the state pn ∈ q in the diagram of the fuzzy automaton a. we say that the path h generates a sequence α = α1...αn ∈ ξ∗ of fuzzy symbols. we define the language l(a) ⊆ ξ∗ of the automaton a as the set of all sequences of fuzzy symbols generated by all paths leading from the initial state to a final state. for k ≥ 0 we denote by l(a)/k the set of all k-length strings in l(a). given a string x ∈ σ∗ and a fuzzy automaton a, we say that x matches a with degree µa(x), if µa(x) = max{µω(x)| for all ω ∈ l(a)/|x|} example 1. suppose σ = {1, 2, 3, 4, 5, 6, 7} and the measures are rational numbers that belong to the segment [0, 1] with the accumulation operation defined as multiplication. define the fuzzy symbols s (small), m (middle) and l(large) to be the following fuzzy subsets of σ: a. kostanyan 59 fig. 1. a fuzzy automaton listed below are all the strings in l(a)/5: ω1 = smmmm, ω2 = smlms, ω3 = slmsm, ω4 = lmsmm, ω5 = lmlms. note that µa(x) = max{µω1(x), µω2(x)µω3(x), µω4(x), µω5(x)} = max { µsmmmm(56364), µsmlms(56364), µslmsm(56364), µlmsmm(56364), µlmlms(56364) } = max {( 1 10 · 1 4 · 3 4 · 1 4 · 1 ) ,( 1 10 · 1 4 · 1 10 · 1 4 · 3 10 ) , ( 1 10 · 9 10 · 3 4 · 0 · 1 ) , ( 3 5 · 1 4 · 3 5 · 1 4 · 1 ) , ( 3 5 · 1 4 · 1 10 · 1 4 · 3 10 )} = = max { 3 640 , 3 16000 , 0, 9 400 , 9 8000 } = 9 400 . 3. calculation of fuzzy regularity 3.1 the fuzzy regularity determination problem let σ be a finite alphabet of characters, ξ be a finite set of fuzzy symbols over σ, p be a fuzzy automaton over ξ called a fuzzy regularity pattern (or, in short, a regularity pattern). for a given x ∈ σ∗, we define the (x, p) matching problem as the problem of determining the value µp (x). we assume that this value represents the degree of fuzy regularity of the crisp string x according to the regularity pattern p . 3.2 recursive solution let p = (q, ξ, δ, qin, f) be a regularity pattern, q ∈ q. let us denote pq = (q, ξ, δ, q, f) the regularity pattern obtained from p by replacing the initial state qin with the state q. the value µpq(x), representing the solution to the (x, pq) matching problem for the string x and the regularity pattern pq, let us denote µp (q, x). in particular, we have that µp (qin, x) = µp (x). example 2. let x = 56364, |x| = 5, ξ = {s, m, l}, a is the fuzzy automaton in fig. 1. 60 determining the degree of fuzzy regularity of a string theorem 1: (optimal substructure of the (x, pq) matching problem) 1. x = ϵ ⇒ [ if q ∈ f then µp (q, x) = 1 else µp (q, x) = 0]. 2. x = ax′ ⇒ [µp (q, x) = max{µα(a) ⊗ µp (q′, x′)| for all δ(q, α) = q′}]. proof: the first statement is obvious. the second statement follows from [x = ax′] ⇒ [µp (q, x) = max{µα(a) ⊗ µω(x′)| δ(q, α) = q′, α ∈ ξ, ω ∈ l(pq′)] ⇒ [µp (q, x) = max{µα(a) ⊗ µp (q′, x′)| for all δ(q, α) = q′}]. theorem 1 implies the following recurrent equation for calculation µp (q, x): µp (q, x) = { (q ∈ f)?1 : 0, if x = ϵ, max{µα(a) ⊗ µp (q′, x′)| for all δ(q, α) = q′}, if x = ax′. direct calculation of the value of µp (q, x) using this formula will be inefficient due to overlapping subproblems. in order to get a more efficient solution, let us apply the dynamic programming approach. 3.3 dynamic programming solution suppose q is represented as an m-tuple < q1, ..., qm >, so that qin = q1. for 1 ≤ i ≤ m, 1 ≤ j ≤ n + 1, let us denote s[i, j] = µp (qi, x[j..n]) (we assume that x[n + 1..n] = ϵ). the optimal substructure of the (x, pq) matching problem dictates the following recurrent equation for calculating s[i, j]: s[i, j] = { qi ∈ f)?1 : 0, if j = n + 1, max{µα(x[j]) ⊗ s[k, j + 1]| for all δ(qi, α) = qk}, if 1 ≤ j ≤ n. (1) note that the (x, p) matching problem can be represented as the problem of determining the value s[1, 1] = µp (x). for q ∈ q, let us denote out(q) = {(α, p)|δ(q, α) = p }, which is the set of pairs consisting of states and fuzzy symbols corresponding to transitions outgoing from q. the algorithm in fig. 2 presents the process of calculating the matrix {s[i, j]|, 1 ≤ i ≤ m, 1 ≤ j ≤ n + 1} of matching degrees according to formula (1): a. kostanyan 61 fig. 2. building the matrix of matching degrees the solution to the (x, p) matching problem, presented in fig. 3, is simply reduced to extracting the value s[1, 1] from the matrix s. fig. 3. determination of fuzzy regularity example 3 . the matrix s of matching degrees, constructed by the calculate−matching− degrees algorithm on the input x = 56364 and the regularity pattern in fig. 1, is presented in fig. 4. fig. 4. memoization results 62 determining the degree of fuzzy regularity of a string in line with determine − fuzzy − regularity algorithm, the value s[1, 1] = 9 400 is the solution to the (x, p) matching problem for the string x = 56364 and regularity pattern in fig. 1, which is consistent with the result obtained in example 2. according to our approach, this value determines the degree of fuzzy regularity of the crisp string x based on the pattern p . 3.4 analysis to estimate the complexity of the proposed solution to the (x, p)-matching problem, let us assume that there are k transitions in the graph of the m-state fuzzy automaton p , and that this graph is represented as an array a[1..m] such that a[i] = out(qi), 1 ≤ i ≤ m. in this case, the construction of the previous column of the matrix s in lines 5-12 of the calculate − matching − degrees algorithm takes time o(1 + k) and, consequently, the construction of the entire matrix s for a string of length n takes time o(n(1 + k)). as a result, the instruction in line 1 of the determine − fuzzy − regularity algorithm runs in o(n(1 + k)) time. the instruction in line 2 of this algorithm obviously runs in o(1) time, which makes the time complexity of the determine − fuzzy − regularity algorithm equal to o(n(1 + k)). for an n-length string x and an m-state regularity pattern p , the algorithm uses o(mn) extra memory to represent the matrix s[1..m, 1..n + 1] of matching degrees. 4. conclusion the problem of determining the fuzzy regularity of a crisp string has been considered in this paper, where the concept of fuzzy regularity is formalized by means of a finite automaton with fuzzy properties of alphabet characters on transitions. using the dynamic programming approach, we propose a solution to this problem with o(n(k + 1)) time complexity, and o(mn) space complexity, where n is the length of the input string; m and k are the number of states and the number of transitions of the automaton, respectively. the proposed algorithm can be used in the field of pattern recognition, data clustering, dna analysis, etc. acknowledgments this work was supported by the ministry of education, science, culture and sports of the republic of armenia, project 21t-1b326. a. kostanyan 63 references [1] y. cao and y. ezawa, “non-determinstic fuzzy automata”, information sciences, vol. 191, pp. 86-97, 2012. [2] j. n. mordeson and d.s. malik, fuzzy automata and languages: theory and applications, chapman & hall, crc, boca raton, london 2002. [3] r. k. singh, a. rani and m.k. sachan, ”fuzzy automata: a quantitative review”, international journal on future revolution in computer science & communication engineering, vol 3, no. 7, pp. 11-17, 2017. [4] c. m. bishop, pattern recognition and machine learning, springer, 2006. [5] wang zhen zhou, ”image segmentation by combining the global and local properties”, expert systems with applications, vol. 87, pp. 30-40, 2017. [6] bezdek, james c, pattern recognition with fuzzy objective function algorithms, 1981. [7] a. kostanyan, fuzzy string matching with finite automata, proc. on 2017 ieee conference csit-2017, yerevan, armenia, pp. 25-29. ieee press, usa 2018. [8] a. kostanyan, fuzzy string matching using prefix table, transactions of iiap nas ra, mathematical problems of computer science, vol. 54, 2020, pp.116 -121. [9] a. kostanyan and a. karapetyan, string matching in case of periodicity in the pattern, in: dolinina o., brovko a., pechenkin v., lvov a., zhmud v., kreinovich v. (eds) recent research in control engineering and decision making. icit 2019. studies in systems, decision and control, vol 199. springer, 2019. [10] a. kostanyan and a. harmandayan, segmentation of string to match a fuzzy pattern, in proc. of computer science & information technologies (csit) conference, yerevan, armenia, september, pp. 17-19, 2019. [11] a. kostanyan and a. harmandayan, mapping a fuzzy pattern onto a string, in proc. on 2019 ieee conference ”2019 computer science and information technologies (csit), yerevan, armenia, 23-27 sep., 2019, ieee press, usa, pp. 5-8, 2019. [12] l. a. zadeh, the concept of a linguistic variable and its application to approximate reasoning-i, information sciences, vol. 8, pp. 199-249, 1975. submitted 04.10.2021, accepted 06.12.2021. 6 4 determining the degree of fuzzy regularity of a string îáõç áã ñëï³ï ï³ýáý³íáñáõãû³ý ³ëïç×³ýç áñáßáõù ²ñù»ý ð. îáëï³ýû³ý îî ïñã³ï³ý ¨ ñ»ï³½áï³ï³ý ï»ýïñáý ºñ¨³ýç å»ï³ï³ý ñ³ù³éë³ñ³ý e-mail: armhko@gmail.com ²ù÷á÷áõù ðá¹í³íáõù ¹çï³ñïíáõù ¿ ïáõç áã ñëï³ï ï³ýáý³íáñáõãû³ý ã³÷ç áñáßù³ý ëý¹çñá: ºýã³¹ñíáõù ¿, áñ áã ñëï³ï ï³ýáý³íáñáõãû³ý ñ³ëï³óáõãûáõýá ýáñù³éç½³óíáõù ¿ í»ñç³íáñ ³íïáù³ïç (ß³µéáýç) ùççáóáí, áñç ³ýóáõùý»ñçý í»ñ³·ñí³í »ý ³ûµáõµ»ýç ýß³ýý»ñç áã ñëï³ï ñ³ïïáõãûáõýý»ñ: ²ñ¹ûáõýùáõù, ñëï³ï ïáõç áã ñëï³ï ï³ýáý³íáñáõãû³ý ã³÷ç áñáßù³ý ëý¹çñá ÷áë³ñçýíáõù ¿ ñ³ù³å³ï³ëë³ý ³íïáù³ïç é»½íçý ýßí³í ïáõç å³ïï³ý»éçáõãû³ý ã³÷ç áñáßù³ý ëý¹ñáí, áñç éáõíù³ý ñ³ù³ñ ³é³ç³ñïíáõù ¿ û·ï³·áñí»é ¹çý³ùçï íñ³·ñ³íáñù³ý ù»ãá¹á: îïðåäåëåíèå ñòåïåíè íå÷åòêîé ðåãóëÿðíîñòè ñòðîêè àðìåí ã. êîñòàíÿí îáðàçîâàòåëüíûé è íàó÷íûé öåíòð èíôîðìàöèîííûõ òåõíîëîãèé åðåâàíñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò e-mail: armhko@gmail.com àííîòàöèÿ â ñòàòüå ðàññìàòðèâàåòñÿ çàäà÷à îïðåäåëåíèÿ ñòåïåíè íå÷åòêîé ðåãóëÿðíîñòè äàííîé ñòðîêè. ïðåäïîëàãàåòñÿ, ÷òî íå÷åòêàÿ ðåãóëÿðíîñòü ôîðìàëèçóåòñÿ ïîñðåäñòâîì ïàòåðíà, ïðåäñòàâëåííîãî â âèäå êîíå÷íîãî àâòîìàòà ñ íå÷åòêèìè ñâîéñòâàìè ñèìâîëîâ àëôàâèòà íà ïåðåõîäàõ. â ðåçóëüòàòå, çàäà÷à îïðåäåëåíèÿ ñòåïåíè íå÷åòêîé ðåãóëÿðíîñòè ñòðîêè çàìåíÿåòñÿ çàäà÷åé îïðåäåëåíèÿ ñòåïåíè åå ïðèíàäëåæíîñòè ÿçûêó íå÷åòêîãî àâòîìàòà, äëÿ ðåøåíèÿ êîòîðîé ïðåäëàãàåòñÿ èñïîëüçîâàòü ìåòîä äèíàìè÷åñêîãî ïðîãðàììèðîâàíèÿ. ðåøåíèå ðàññìàòðèâàåìîé çàäà÷è ïîçâîëÿåò ôàççèôèöèðîâàòü ìíîæåñòâî ñëîâ â äàííîì àëôàâèòå íà îñíîâå ïàòòåðíà îïðåäåëåíèÿ íå÷åòêîé ðåãóëÿðíîñòè. äàííàÿ ðàáîòà ÿâëÿåòñÿ ïðîäîëæåíèåì ðÿäà ïðåäûäóùèõ ðàáîò àâòîðà ïî ïîèñêó íå÷åòêîãî ïàòòåðíà â ñòðîêå. îíà ìîæåò èìåòü ïðèìåíåíèÿ â òàêèõ îáëàñòÿõ, êàê ðàñïîçíàâàíèå ïàòòåðíîâ, êëàñòåðèçàöèÿ äàííûõ, áîèíôîðìàòèêà, è ò. ä. êëþ÷åâûå ñëîâà: íå÷åòêîå ñîïîñòàâëåíèå ñ îáðàçöîì, ðàñïîçíàâàíèå ïàòòåðíîâ, íå÷åòêèé àâòîìàò. üßí³í ëý¹ñç éáõíáõùá ñý³ñ³íáñáõãûáõý ¿ ï³éçë ï³éáõó»é ïñí³í ³ûµáõµ»ýç µ³é»ñç áã ñëï³ï µ³½ùáõãûáõý՝ áã ñëï³ï ï³ýáý³íáñáõãû³ý áñáßù³ý ß³µéáýç ñçù³ý íñ³: îíû³é ³ßë³ï³ýùá ß³ñáõý³ïáõãûáõýý ¿ ñ»õçý³ïç ý³ëáñ¹ ³ßë³ï³ýùý»ñç՝ ïáõáõù áã ñëï³ï »ýã³ïáõç (ß³µéáýç) áñáýù³ý áõõõáõãû³ùµ: ²ûý ï³ñáõ ¿ ïçñ³éí»é ß³µéáýý»ñç ×³ý³ãù³ý, ïíû³éý»ñç ëùµ³íáñù³ý, µçáçýýáñù³ïçï³ûç ¨ ³ûé µý³·³í³éý»ñáõù: ´³ý³éç µ³é»ñ՝ ýùáõßç áã ñëï³ï ñ³ù³¹ñáõù, ß³µéáýý»ñç ×³ý³ãáõù, áã ñëï³ï ³íïáù³ï: 05_kostanyan_56 kostanyan d:\sbornik\...\tpel.dvi mathematical problems of computer science 32, 35{38, 2009. t he clique cover ing n umber for the str ong p r oduct of gener alized cycles s e va k h . b a d a lya n yerevan state university sevak badalyan@yahoo.com abstract in this paper the clique covering number for the strong product of generalized cycles is investigated. a method is given to construct a minimal clique cover in case some conditions hold. refer ences [1 ] m. r o s e n fe ld ,\ on a p r o b le m o f c. e . s h a n n o n in g r a p h t h e o r y" , p roc. amer. m ath. soc. 18, 3 1 5 -3 1 9 , 1 9 6 7 . [2 ] r . s . h a le s , \ n u m e r ic a l in va r ia n t s a n d t h e s t r o n g p r o d u c t o f g r a p h s " , combin. theory (b ) 15, 1 4 6 -1 5 5 , 1 9 7 3 . ì³íïáõûãç ãçíá áý¹ñ³ýñ³óí³í óçïé»ñç áõå»õ ³ñï³¹ñû³éç ñ³ù³ñ ê. ´³¹³éû³ý ²ù÷á÷áõù êáõûý ³ßë³ï³ýùáõù áõëáõùý³ëçñí³í ¿ áý¹ñ³ýñ³óí³í óçïé»ñç áõå»õ ³ñï³¹ñû³éç í³íïáõûãç ãçíá: àñáß å³ûù³ýý»ñç ³éï³ûáõãû³ý ¹»åùáõù ïñí³í ¿ »õ³ý³ï ùçýçù³é í³íïáõûãá ï³éáõó»éáõ ñ³ù³ñ: 3 5 d:\user\sbornik_38_pdf\7.dvi ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 38, 17, 2012. îáùåå ðåøåíèå ñèììåòðè÷åñêèõ óðàâíåíèé ñ ÷åòíîé ñòåïåíüþ â ñâîáîäíîì ìîíîèäå à. ø. ìàëõàñÿí e-mail: amalkhasyan@mail.ru â ðÿäå ðàáîò ã.ñ.ìàêàíèíà îïèñûâàþòñÿ îáùèå ðåøåíèÿ óðàâíåíèé â ñâîáîäíîé ãðóïïå è â ñâîáîäíîì ìîíîèäå. òàê, â ðàáîòå [1] äàíî îïèñàíèå îáùèõ ðåøåíèé ñèììåòðè÷åñêèõ óðàâíåíèé âèäà x®1 x ® 2 :::x ® n¡1x ® n = x ® nx ® n¡1:::x ® 2 x ® 1 ïðè ® = 1 è ® = 2 â ñâîáîäíîì ìîíîèäå. íàìè ïîëó÷åíî, ÷òî èñïîëüçîâàíèå ôóíêöèé ìàêàíèíà, ïîñòðîåííûõ â [1] è [2], ïîçâîëÿåò ïîëó÷èòü îïèñàíèå áîëåå îáùåãî âèäà óðàâíåíèé. ñïðàâåäëèâà òåîðåìà 1. ñóùåñòâóþò ðåêóðñèâíûå ôóíêöèè, ñ ïîìîùüþ êîòîðûõ îïèñûâàåòñÿ îáùåå ðåøåíèå óðàâíåíèÿ âèäà x2k1 x 2k 2 :::x 2k n¡1x 2k n = x 2k n x 2k n¡1:::x 2k 2 x 2k 1 : ñïèñîê ëèòåðàòóðû [1] ã. ñ. ìàêàíèí, “êîíå÷íàÿ ïàðàìåòðèçàöèÿ ðåøåíèé óðàâíåíèé â ñâîáîäíîì ìîíîèäå. ii”, ìàòåì. ñá., 195:4 (2004), 65–96. [2] ã. ñ. ìàêàíèí, “ïàðàìåòðèçàöèÿ ðåøåíèé óðàâíåíèÿ x1x2xn¡1xn = xnxn¡1x2x1 â ñâîáîäíîì ìîíîèäå”, ìàòåì. çàìåòêè, 89:6 (2011), 879–884. 1 7 d:\sbornik\...\article.dvi mathematical problems of computer science 28, 2007, 60{64. on e dge-disjoint p air s of m atchings¤ v a h a n v . mkr t c h ya n y, v a h e l . mu s o ya n z, a n u s h v . ts e r u n ya n z y institue for informatics and automation problems of nas of ra e-mail: vahanmkrtchyan2002@fysu.am, ipia.sci.am, yahoo.comg z department of informatics and applied mathematics, yerevan state university, armenia e-mail: vahe musoyan@ysu.am, anush tserunyan@fysu.am, yahoo.comg abstract for a given graph consider the pairs of edge-disjoint matchings whose union contains as many edges as possible, and consider the relation of the cardinality of a maximum matching to the cardinality of the largest matching among such pairs. we show that 5=4 is a tight upper bound for this relation. refer ences [1 ] f. h a r a r y, gr a p h th e o r y, a d d is o n -w e s le y, r e a d in g , ma , 1 9 6 9 . [2 ] f. h a r a r y, m.d . p lu m m e r , " on t h e c o r e o f a g r a p h " , p roc. l ondon m ath. soc. 17, p p . 3 0 5 -3 1 4 , 1 9 6 7 . [3 ] l . l o va s z , m.d . p lu m m e r , ma t c h in g t h e o r y, a n n . d is c r e t e ma t h . 2 9 , 1 9 8 6 . [4 ] v . v . mkr t c h ya n , " on t r e e s wit h a m a xim u m p r o p e r p a r t ia l 0 -1 c o lo r in g c o n t a in in g a m a xim u m m a t c h in g " , d iscrete m athematics , vo l. 3 0 6 , p p . 4 5 6 -4 5 9 , 2 0 0 6 . [5 ] v . v . mkr t c h ya n , " a n o t e o n m in im a l m a t c h in g c o ve r e d g r a p h s " , d iscrete m athematics , vo l. 3 0 6 , p p . 4 5 2 -4 5 5 , 2 0 0 6 . [6 ] d . b . w e s t , in t r o d u c t io n t o gr a p h th e o r y, p r e n t ic e -h a ll, e n g le wo o d cli®s , 1 9 9 6 . ¶ñ³ýáõù áý¹ñ³ýáõñ ïáõ ãáõý»óáõ ½áõ·³ïóáõùý»ñç ½áõû·»ñç ù³ëçý ì. øïñïãû³ý, ì. øáõëáû³ý, ². ì»ñáõýû³ý ²ù÷á÷áõù îñí³í ·ñ³ýç ñ³ù³ñ ¹çï³ñï»ýù áý¹ñ³ýáõñ ïáõ ãáõý»óáõ ½áõ·³ïóáõùý»ñç ·áõû·»ñá, áñáýó ùç³íáñáõùá å³ñáõý³ïáõù ¿ ³ù»ý³ß³ï ãíáí ïáõ»ñ, ¨ ¹çï³ñï»ýù ·ñ³ýç ù³ùëçù³é ½áõ·³ïóù³ý ñ½áñáõãû³ý ñ³ñ³µ»ñáõãûáõýá ³û¹åçëç ½áõû·»ñáõù ³ù»ý³ß³ï ïáõ»ñ å³ñáõý³ïáõ ½áõ·³ïóù³ý ñõáñáõãû³ýá: ø»ýù óáõûó »ýù ïí»é, áñ 5/4-á ×çßï í»ñçý ·ý³ñ³ï³ï³ý ¿ ³ûë ñ³ñ³µ»ñáõãû³ý ñ³ù³ñ: ¤the authors are supported by a grant of the armenian national science and educational fund 6 0 analyzereductionlengthsmpcs.dvi mathematical problems of computer science 29, 2007, 5{15. analysis of b ounds for lengths of reductions in t yped ¸-calculus tig r a n m. ga lo ya n institue for informatics and automation problems of nas of ra e-mail: tigran.galoyan@gmail.com abstract we analyze bounds for the lengths of arbitrary reduction sequences of terms in typed ¸-calculus, consider some estimates obtained by the other authors and compare these estimates. the cut elimination and normalization algorithms are also investigated in this paper. thereafter we re¯ne the estimates achieved in [3] (for pure implicational logic only) by supplement of ´-conversion and then we extend evaluations to ¯rst-order logic. refer ences [1 ] b a a z , m., l e it s c h , a .: cu t -e lim in a t io n a n d r e d u n d a n c y-e lim in a t io n b y r e s o lu t io n . jo u r n a l o f s ym b o lic co m p u t a t io n , 2 9 :1 4 9 { 1 7 6 , 2 0 0 0 . [2 ] b e c km a n n , a ., w e ie r m a n n , a .: a n a lyz in g gäo d e l's t via e xp a n d e d h e a d r e d u c t io n t r e e s . ma t h e m a t ic a l l o g ic s qu a r t e r ly, 4 6 :5 1 7 { 5 3 6 , 2 0 0 0 . [3 ] b e c km a n n , a .: e xa c t b o u n d s fo r le n g t h s o f r e d u c t io n s in t yp e d la m b d a -c a lc u lu s . j. s ym b o lic l o g ic , 6 6 :1 2 7 7 -1 2 8 5 , 2 0 0 1 . [4 ] ga lo ya n , t.: s t r o n g n o r m a liz a t io n fo r ¯ r s t -o r d e r lo g ic . tr a n s a c t io n s o f t h e in s t it u t e fo r in fo r m a t ic s a n d a u t o m a t io n p r o b le m s o f t h e n a s o f r a . ma t h e m a t ic a l p r o b le m s o f co m p u t e r s c ie n c e 2 8 , p p . 4 5 -5 0 , 2 0 0 7 . [5 ] k le e n e , s .: in t r o d u c t io n t o me t a m a t h e m a t ic s . d . va n n o s t r a n d , n e w y o r k, 1 9 5 2 . [6 ] s c h wic h t e n b e r g , h .: n o r m a liz a t io n . in f.l . b a u e r , e d it o r , l o g ic , a lg e b r a a n d co m p u t a t io n . p r o c e e d in g s o f t h e in t e r n a t io n a l s u m m e r s c h o o l ma r kt o b e r d o r f, ge r m a n y, ju ly 2 5 a u g u s t 6 , 1 9 8 9 , s e r ie s f: co m p u t e r a n d s ys t e m s s c ie n c e s , v o l. 7 9 , p a g e s 2 0 1 -2 3 7 . n a to a d va n c e d s t u d y in s t it u t e , s p r in g e r v e r la g , b e r lin , h e id e lb e r g , n e w y o r k, 1 9 9 1 . [7 ] s c h wic h t e n b e r g , h .: a n u p p e r b o u n d fo r r e d u c t io n s e qu e n c e s in t h e t yp e d la m b d a c a lc u lu s . a r c h ive fo r ma t h e m a t ic a l l o g ic , 3 0 :4 0 5 -4 0 8 , 1 9 9 1 . [8 ] s c h wic h t e n b e r g , h .: p r o o f t h e o r y. l e c t u r e n o t e s , ma t h e m a t is c h e s in s t it u t d e r u n ive r s it äa t mäu n c h e n , ge r m a n y, 1 9 9 4 . 5 6 analysis of bounds for lengths of reductions in typed ¸-calculus è»¹áõïóçáý ñ³çáñ¹³ï³ýáõãûáõýý»ñç »ñï³ñáõãû³ý ë³ñù³ýý»ñç í»ñéáõíáõãûáõýá ïçå³ï³ý³óí³í ¸-ñ³ßíáõù î. ø. ¶³éáû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù áõëáõùý³ëçñíáõù »ý é»¹áõïóçáý ñ³çáñ¹³ï³ýáõãûáõýý»ñç »ñï³ñáõãû³ý ·ý³ñ³ï³ï³ýý»ñá ïçå³ï³ý³óí³í ¸-ñ³ßíç ã»ñù»ñç ñ³ù³ñ£ ¸çï³ñïíáõù ¨ ñ³ù»ù³ïíáõù »ý ³ûé ñ»õçý³ïý»ñç ïáõùçó ïñí³í ·ý³ñ³ï³ï³ýý»ñá£ ò¨³÷áëíáõù ¿ [3]-áõù ïñí³í ·ý³ñ³ï³ï³ýç ³å³óáõûóá (ù³ùáõñ çùåéçï³óçáý ïñ³ù³µ³ýáõãû³ý ñ³ù³ñ)ª áý¹·ñï»éáí ý³¨ ´-é»¹áõïóç³ûç ï³ýáýá£ ²ûýáõñ»ï¨ ëï³óí³í ³ñ¹ûáõýùá áý¹ñ³ýñ³óíáõù ¿ ³é³ççý ï³ñ·ç ïñ³ù³µ³ýáõãû³ý ñ³ù³ñ£ ²ßë³ï³ýùáõù ñ»ï³½áïíáõù »ý ý³¨ ïñ×³ïù³ý ·áñíáõáõãû³ý ³ñï³ùëù³ý ¨ ýáñù³é³óù³ý ³é·áñçãùý»ñá£ d:\sbornik\...\doc11.dvi mathematical problems of computer science 33, 111{120, 2010. on the p r oblem of wir eless scheduling with linear p ower levels tig r a n to n o ya n yerevan state university, faculty of informatics and applied mathematics, department of algebra and discrete mathematics abstract in this work we consider the problem of communication scheduling in wireless networks with respect to the sinr(signal to interference plus noise ratio) constraint in metric spaces. for the powers of sender nodes we consider the linear power assignment, which is one of commonly considered power schemes. we give a constant factor deterministic approximation algorithm for scheduling in wireless networks, which are given in some special class of metric spaces, which contains the euclidean spaces. to the best of our knowledge, this is the ¯rst constant factor approximation algorithm for this problem. simultaneously we obtain the approximate value of the optimal schedule length with error at most a constant factor. refer ences [1 ] a . fa n g h äa n e l, t. k e ¼ e lh e im , h . r äa c ke , b . v äo kin g , \ ob livio u s in t e r fe r e n c e s c h e d u lin g " , p roc. 28th symposium on p rinciples of d istributed computing (p od c), 2 0 0 9 . [2 ] a . fa n g h äa n e l, t. k e ¼ e lh e im a n d b . v äo kin g , \ im p r o ve d a lg o r it h m s o n l a t e n c y min im iz a t io n in w ir e le s s n e t wo r ks " , p roc. 36th international colloqium on automata, l anguages and p rogramming (ical p ), 2 0 0 9 . [3 ] o. go u s s e vs ka ia , m.m. h a lld ¶o r s s o n , r . w a t t e n h o fe r a n d e m o w e lz l. \ ca p a c it y o f a r b it r a r y w ir e le s s n e t wo r ks " , 28th annual ie e e conference on computer communications (inf ocom ), 2 0 0 9 . [4 ] m.m. h a lld ¶o r s s o n , \ w ir e le s s s c h e d u lin g wit h p o we r co n t r o l" p roc. 17th annual e uropean symposium on algorithms (e sa), 2 0 0 9 . [5 ] m.m. h a lld ¶o r s s o n a n d r . w a t t e n h o fe r , \ w ir e le s s co m m u n ic a t io n is in a p x " , p roc. 36th international colloqium on automata, l anguages and p rogramming (ical p ), 2 0 0 9 . [6 ] g.h . h a r d y, j.e . l it t le wo o d , g. p ¶o lya , in e qu a lit ie s , ca m b r id g e u n ive r s it y p r e s s , 1 9 3 4 . [7 ] s .o. k r u m ke , m.v . ma r a t h e a n d s . s . r a vi, \ mo d e ls a n d a p p r o xim a t io n a lg o r it h m s fo r c h a n n e l a s s ig n m e n t in r a d io n e t wo r ks " , w ir e le s s n e t wo r ks 7 , 2 0 0 1 . [8 ] t. mo s c ib r o d a , r . w a t t e n h o fe r , \ th e c o m p le xit y o f c o n n e c t ivit y in wir e le s s n e t wo r ks " , 25th annual ie e e conference on computer communications (inf ocom ), 2 0 0 6 . [9 ] t. mo s c ib r o d a , r . w a t t e n h o fe r a n d y . w e b e r . \ p r o t o c o l d e s ig n b e yo n d gr a p h -b a s e d mo d e ls " , h o t to p ic s in n e t wo r ks ( h o t n e t s ) , 2 0 0 6 . 1 1 1 1 1 2 on the problem of wireless scheduling with linear power levels [1 0 ] t. mo s c ib r o d a , r . w a t t e n h o fe r a n d a . zo llin g e r , \ to p o lo g y c o n t r o l m e e t s s in r : th e s c h e d u lin g c o m p le xit y o f a r b it r a r y t o p o lo g ie s " , acm international symposium on m obile ad hoc networking and computing (m ob ihoc), 2 0 0 7 . [1 1 ] v . s . a . k u m a r , m. v . ma r a t h e , s . p a r t h a s a r a t h y a n d a . s r in iva s a n , \ a lg o r it h m ic a s p e c t s o f ca p a c it y in w ir e le s s n e t wo r ks " , acm international conference on m easurement and m odeling of computer systems (sigm e tr ics), 2 0 0 5 . ²é³ýó é³ñç ó³ýó»ñáõù ·í³ûçý ñ½áñáõãûáõýý»ñáí ñ»ñã³·ñù³ý ëý¹ñç ù³ëçý î. îáýáû³ý ²ù÷á÷áõù ²ûë ³ßë³ï³ýùáõù ¹çï³ñïíáõù ¿ ³é³ýó é³ñç ó³ýó»ñáõù ñ»é³ñó³ïáõùý»ñç ñ»ñã³·ñù³ý ëý¹çñá ù»ïñçï³ï³ý ï³ñ³íáõãûáõýý»ñáõù‘ sinr(signal to interference plus noise ratio) ë³ñù³ý³÷³ïáõùý»ñç ýï³ïù³ùµ: ò³ýóç ñ³ý·áõûóý»ñç ñ½áñáõãûáõýý»ñç ñ³ù³ñ ¹çï³ñïíáõù ¿ ·í³ûçý ëë»ù³ý, áñá ñ³×³ë ¹çï³ñïíáõ ñ½áñáõãû³ý ëë»ù³ý»ñçó ¿: ø»ýù µ»ñáõù »ýù ñ³ëï³ïáõý ·áñí³ïóáí ùáï³ñïáõ ³é·áñçãù ñ»ñã³·ñù³ý ëý¹ñç ñ³ù³ñ, ¹çï³ñï»éáí ëý¹çñá áñáß³ïç ïçåç ù»ïñçï³ï³ý ï³ñ³íáõãûáõýý»ñáõù, ý»ñ³éû³é ¾íïéç¹û³ý ï³ñ³íáõãûáõýý»ñá: àëï ù»ñ ï»õ»ïáõãûáõýý»ñç‘ ë³ ³é³ççý ñ³ëï³ïáõý ·áñí³ïóáí ùáï³ñïáõ ³é·áñçãùý ¿ ³ûë ëý¹ñç ñ³ù³ñ: øç¨ýáõûý å³ù³ý³ï ù»ýù ·ïýáõù »ýù ûåïçù³é ñ»ñãç »ñï³ñáõãû³ý ùáï³íáñ ³ñå»ùá, ³é³í»é³·áõûýá ñ³ëï³ïáõý ·áñí³ïóç ëë³é³ýùáí: microsoft word tpel.doc mathematical problems of computer science 33, 150—161, 2010. 150 effective digital image watermarking in ycbcr color space accompanied by presenting a novel technique using dwt mehdi khalili 1 and david asatryan 2 institute for informatics and automation problems of nas of ra e-mail: khalili@ipia.sci.am 1 , dasat@ipia.sci.am 2 abstract in this paper, a quantization based watermark casting and blind watermark retrieval algorithm operating in ycbcr color space using discrete wavelet transform (dwt), for ownership verification and image authentication applications is implemented. this method uses implicit visual masking by inserting watermark bits into only the wavelet coefficients of high magnitude, in y channel of ycbcr color space. a blind watermark retrieval technique that can detect the embedded watermark without the help from the original uncorrupted image is devised which is computationally efficient. the new watermarking algorithm combines and adapts various aspects from existing watermarking methods. experimental results show that the proposed technique to embed watermark provides extra imperceptibility and robustness against various signal processing attacks in comparison with the same technique in rgb color space. references [1] s. h. wang, y.p. lin, “wavelet tree quantization for copyright protection watermarking”, ieee transactions on image processing, vol. 13, no. 2, pp. 154-165, 2004. [2] s. joo, y. suh, j. shin, h. kikuchi, s.j.cho, “a new robust watermark embedding into wavelet dc components”, etri journal, vol. 24, no. 5, pp. 401-404, 2002. [3] y. wang, a. pearmain, “blind image data hiding based on self reference”, pattern recognition letters, pp. 1681-1689, 2004. [4] x.d. zhang, j. feng, k.t. lo, “image watermarking using tree-based spatial-frequency feature of wavelet transform”, journal visual and communication image representation, vol. 14, pp. 474–491, 2003. [5] h.w. wong, y.m. yeung, c. au, “capacity for jpeg2000-to-jpeg2000 images watermarking”, proc. of 2003 ieee international conference on multimedia and expo, vol. 2, pp. 485-488, 2003. [6] v.v.h. guzmán, m.n. miyatake, h.m.p. meana, “analysis of a wavelet-based watermarking algorithm”, ieee, proceedings of the 14th international conference on electronics, communications and computers (conielecomp’04), 2004. [7] r. dugad, k. ratakonda, n. ahuja, “a new wavelet-based scheme for watermarking images”, in proceedings of the ieee international conference on image processing, icip'98, chicago, volume il, 1998. m. khalili, d. asatryan 151 [8] h. inoue, a. miyazaki, a. yamamoto, t. katsura, “a digital watermark based on the wavelet transform and its robustness on image compression and transformation”, ieice trans, special section on cryptography and information security, e82-a, no.1, pp. 2-10, 1999. [9] m. khalili, “a comparison between digital images watermarking in two different color spaces using dwt2”, csit, proceedings of the 7th international conference on computer science and information technologies, yerevan, armenia, pp. 158-162, 2009. [10] x.g. xia, c.g. boncelet, g.r. arce, “wavelet transform based watermark for digital images”, image processing, optics express, 497, vol. 3, no. 12, 1998. [11] s.e. el-khamy, m.i. lotfy, r.a. sadek, “a block based wavelet watermarking technique for copyright protection and authentication”, ieee, pp. 90-93, 2004. [12] d. kundur, d. hatzinakos, “a robust digital image watermarking scheme using the wavelet-based fusion”, ieee intern conf on image processing (icip'97), california, pp. 544-547, 1997. [3] p. meerwald, a. uhl, “a survey of wavelet-domain watermarking algorithms”, ieee int. conf. on information technology: coding and computing, las vegas, nv, 2001. [14] a.b. watson, g.y. yang, j.a. solomon, j. villasenor, “visibility of wavelet quantization noise”, ieee trans. image processing, vol. 6, pp.1164-1175, 1997. [15] m.s. hsieh, “wavelet-based image watermarking and compression”. ph-d thesis, institute of computer science and information engineering national central university, taiwan, 2001. [16] s.c.b. lo, h. li, m.t. freedman, “optimization of wavelet decomposition for image compression and feature preservation”, ieee transactions on medical imaging, vol. 22, no. 9, pp. 1141-1151, 2003. [17] m. kokare, p.k. biswas, b.n. chatterji, “texture image retrieval using new rotated complex wavelet filters”, ieee transactions on systems, vol. 35, no. 6, pp. 1168-1178, 2005. [18] h.j.m. wang, p.c. su, c.c.j. kuo, “wavelet-based digital image watermarking”, optics express, pp. 491-496, vol. 3, no. 12, 1998. [19] h. hui zha, “progressive lossless image compression using image decomposition and context quantization”, m.s. thesis, university of waterloo, 2007. [20] c. lin, “face detection in complicated backgrounds and different illumination conditions by using ycbcr color space and neural network”, elsevier, pattern recognition letters 28, pp. 2190–2200, 2007. [21] u. ahlvers, u. zölzer, r. rajagopalan, “model-free face detection and head tracking with morphological hole mapping”, eusipco'05, antalya, turkey, 2005. [22] i. cox, j. bloom, m. miller, “digital watermarking: principle & practice”. morgan kaufinann publishers; 1st edition, 2001. [23] s. kumbadakon, “a dwt/svd multiple description digital image watermarking scheme”. m.s. thesis, faculty of the graduate school of the university of texas, 2005. [24] j. j. chane, “robust techniques for hiding data in images and video”, ph-d thesis, santa barbara, university of california, 2000. 152 effective digital image watermarking in ycbcr color space ¶áõý³íáñ å³ïï»ñç` ycbcr ï³ñ³íáõãû³ý ù»ç dwt ïçñ³éù³ùµ ãí³ûçý çñ³ýßù³ý ýáñ ³ñ¹ûáõý³í»ï ù»ãá¹ ø. ê³éçéç ¨ ¸. ²ë³ïñû³ý ²ù÷á÷áõù ðá¹í³íáõù ³é³ç³ñïí»é ¿ ã³ñïáýí³í û·ï³·áñíáõùçó ãí³ûý³óí³í å³ïï»ñç å³ßïå³ýáõãû³ý ³é·áñçãù` »ñïã³÷³ýç áý¹ñ³ï í»ûíé»ï-ó¨³÷áëáõãûáõýý»ñç (dwt) ïçñ³éù³ùµ: ²é·áñçãùá ùß³ïí»é ¿ ·áõý³íáñ å³ïï»ñç ycbcr ï³ñ³íáõãû³ý ñ³ù³ñ, áý¹ áñáõù çñ³ýçßç µçïá ý»ñ¹ñíáõù ¿ ³û¹ ï³ñ³íáõãû³ý y µ³õ³¹ñçãç ³é³í»é ù»í ù³·ýçïáõ¹áí í»ûíé»ï-·áñí³ïçóý»ñç ù»ç: òáõûó ¿ ïñí»é, áñ ³é³ç³ñïí³í ù»ãá¹á ³å³ñáíáõù ¿ çñ³ýçßç ³í»éç µ³ñóñ ³ýï»ë³ý»éçáõãûáõý ¨ ï³ûáõýáõãûáõý ï³ñµ»ñ ï»ë³ïç ñ³ñó³ïáõùý»ñç ýï³ïù³ùµ, ù³ý ·áõý³íáñ å³ïï»ñç rgb ï³ñ³íáõãû³ý ù»ç ·áñíáõ ýáõûý³ïçå ³é·áñçãùý»ñá: microsoft word d. asatryan_4.doc mathematical problems of computer science 34, 2010. 15 on some problems of signal and image processind, analysing and protection david g. asatryan institute for informatics and automation problems, nas of armenia establishing the computing center of the national academy of armenia and yerevan state university in fifties of the last century initiated the deployment of intensive scientific investigations in the area of computer science and information technologies in armenia. now it can be stated that the institute is an important component of e-infrastructure of armenia, providing high level of scientific investigations and supplying numerous organizations of the republic with highly qualified specialists in the specified area. the research area of the institute includes all basic fields of computer science and information technologies. in this report we consider briefly three subjects of research and technologies, related to the signal and image processing, digital analysis and protection, carried out in the institute for last decades. 1. structural analyzing and recovering of signal and image. this subject includes following problems: a. determination of signal change-points. some versions of this problem named “changepoint estimation of a random sequence” were stated and solved in sixties of last century [1]. in what follows these methods were developed by other scientists of the institute and now they are in the developing process. b. edge detection in an image. there are many approaches to solving this problem, proposed in the scientific literature, which are based on using the algorithms in the spatial or frequency domain. recently a new approach named “continuous extension of dct” was proposed by j. patera and a. atoyan [2-3], which allows considering the discrete image as a continuous one. it was shown that usage of this method decreases the spectrum aliasing. we propose a new method for image edge detection [4], which also has some advantages against those of other methods. c. segmentation and simplification of an image. segmentation is an important task at an image analysis. it has a huge amount of applications to various problems in the image processing area. we have proposed a hierarchical coherent segmentation algorithm, as well as a software tool to solve many standard problems, which can arise during image analysis and interpretation [5]. particularly, the problem of image “simplification” (piecewise-linear approximation) is solved quite effectively. stop-rule is proposed to estimate the quality of segmentation at any step of hierarchical segmentation, which helps to make adequate decisions. d. restoration of a damaged or distorted image. the complex of algorithm and software tools mentioned in the previous section allows effectively restoring the damaged or distorted images. we suppose that distortion of an image has a local character (i.e. the damaged part is a segment). a special additional tool allows the transfer of the content of any segment to any other 16 segment. in such manner we “recover” the image. as examples of important application of this complex we consider the restoration of distorted images from the ancient manuscripts, distorted artworks, photos etc. [6]. 2. audio and image watermarking. the history of developing of methods to protect a digital content from an unauthorized access, usage and tampering numbers accounts about two decades. we have proposed and investigated a few new algorithms, which have some advantages against the other methods described in the literature. particularly, we have proposed the following algorithms. a. audio watermarking algorithms. these audio watermarking algorithms are based on the properties of the human auditory system, so they have enough robustness to various attacks of well known types [7]. the algorithm is applied to the protecting of musical compositions. b. watermarking algorithms based on embedding a binary image into a host image. these algorithms are based on human visual system, so they have enough robustness to various attacks of well known types [8]. an analytical method for the quality assessment of watermarking procedure is proposed. c. high payload watermarking algorithms. a class of combined watermarking algorithms using both spatial and spectral methods is proposed and investigated [9]. it is shown that the payload of the watermarking procedure can be significantly larger than that of the host image size with keeping other parameters of the procedure quality on the acceptable level. c. watermarking algorithm using wavelet analysis. this algorithm heightened robustness to various types of attack [10]. 3. new method for image quality assessment and applications. it is well known that the results of the evaluation of image quality by widely used criteria based on the mean square discrepancy of images, often does not match the quality assessment in visual manner. therefore a new philosophy based on conception that the human visual system retrieves mainly the structural information from the images, is developing last decades. in accordance with this concept, a series of papers were published in the scientific literature, in which the authors proposed criteria, which are in some sense adapted to the structure of the image. we have proposed an approach based on acceptance as structural information of the distribution of gradient of the image. an image quality measure which is invariant to the size and rotation of images is proposed [11]. references 1. d.g. asatryan, b.e. brodsky, i.a. safaryan. detection of structural changes in the multivariate data using change-point models. advances and challenges in multisensor data and information processing. nato security through science series. d: information and communication security – vol. 8. , ios press, 2007, pp. 106-113. (proc. of the nato asi2005, (albena, bulgaria). 2. a. atoyan, j. patera, application of continuous extension of dct to flir images, in data fusion for situation monitoring, incident detection, alert and response management, e. shahbazian et al. (eds.), ios press (2005), pp. 417-425 3. a. atoyan, j. patera, properties of continuous fourier extension of the discrete cosine transform and its multidimensional generalization. journal of mathematical physics, vol. 45, n 6, june 2004, pp. 2468-2491. 4. d. asatryan, j. patera. edge detection algorithm based on dct continuous extension technique. physics of atomic nuclei, vol.71, no.5, pp.795-799, 2008. 5. d.g. asatryan, g.s. sazhumyan, h.s. shahverdyan. technique for coherent segmentation of image and applications. mathematical problems of computer science, iiap, yerevan, armenia, vol. 28, 2007, pp. 88-93. 17 6. d.g. asatryan, g.s. sazhumyan. coherent segmentation method and its application to the distorted image restoration. vestnik giua, modelling, optimization and control, vol. 2, № 9, pp. 15-21, 2006 (in russian). 7. d.g.asatryan, s.v. tairyan. robust audio watermarking algorithm. mathematical problems of computer science, iiap, 32, pp. 96-100, 2009. 8. d.g.asatryan, n.s. lanina. adaptive robust watermarking algorithm for image protection. vestnik rau (herald of the rau), armenia, yerevan, pp. 50-56, 2009. 9. d.g. asatryan, n.s. asatryan. combined spatial and frequency watermarking. proc. of 7th int. conf. on computer science and information technologies csit'2009, yerevan, pp. 323-326, 2009. 10. m. khalili, d. asatryan. effective digital image watermarking in ycbcr color space accompanied by presenting a novel technique using dwt. mathematical problems of computer science, iiap, vol. 33, pp. 150-161, 2010. 11. d. asatryan, k. egiazarian quality assessment measure based on image structural properties. proc. of the international workshop on local and non-local approximation in image processing, finland, helsinki, pp. 70-73, 2009. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 50, 76--80, 2018. use of info-communicational resources in software systems andranik e. mkhitaryan, arthur s. petrosyan and aram s. nanassian institute for informatics and automation problems of nas ra e-mail: and.mkhitaryan@gmail.com, arthur@sci.am, ananas@sci.am abstract the use of info-communicational resources in software systems is considered. the integration details and mechanisms of info-communicational resource unimail into other systems are considered. examples of software systems types, which could use the services of unimail resource are discussed. keywords: notification, alert, e-mail, sms. 1. introduction nowadays, the creation of software systems considers the use of many different independent modules. each of the modules has its own part of work to complete. this helps to economize the resources by sharing them and the time that would be spent for implementation of all components by using already implemented items. all software systems, which should send notifications by any network should have an info-communicational module. such systems are being used to pass messages from one resource to another. the most popular infocommunicational platforms are based on the internet, gsm, etc., networks. the software could be designed by different ways depending on the requirements. the component could be either embedded or independent. each variation has specific advantages and disadvantages, which should be considered in the architecture of the software. 2. solution the module responsible for sending operative notifications is one of the most important and common components of software systems. the module could be either embedded or separated. nowadays, both techniques are widespread because of their specific advantages. 76 mailto:and.mkhitaryan@gmail.com mailto:arthur@sci.am mailto:ananas@sci.am a. mkhitaryan, a. petrosyan and a. nanassian 77 fig. 1. embedded module in software system. the embedded case considers that the system has its own software and hardware equipment. for instance, a connection to gsm network requires gsm modem and the appropriate software component. for such cases the module is being used only for that system (figure 1). one of the advantages of this structure is that the system is able to fully control the module’s resources because it is the component’s only client. such technique is preferable in case the module is used intensively on heavy load by the system. sharing its resources could lead to critical health issues and timeouts. the other advantage is that the implementation will be more specific to the software needs because it is not mandatory for the module to be generic. the other technique is about to usage of independent modules in common inn case of possibility. for this type of systems some of the resources will be shared with other clients. on this concept the module is a detached and self-determined resource, which provides already defined public application programming interface (api) to access the listed by itself functionality. each permitted software system has an ability to access and use the functionalities provided by the component. such systems are much more economical than the above described one. the main advantage is that the software and hardware of the module are being shared by many applications with api. the module can serve as many clients as its equipment allows. unimail is a software system, which provides info-communicational services [1],[2]. the system is connected to both internet and gsm networks. unimail is able to send sms messages to the mobile phone subscribers by gsm network. it provides a public interface to access the available services. as a public interface it uses email technology. there is a specified list of commands to configure and use the provided services. in general, animal could act as a notification module for other systems. usually, the same information is being sent by e-mail and by sms. for such cases unimail services become much more practical. the user just needs to add some additional configurations during the email message preparation process. the integration of unimail is suitable for any type of software. some corresponding commands of unimail with descriptions:  ** command in subject sending to more than one email addresses parallel to the letter send sms notification to the addressee(s)  ** command in subject and sending to the unimail’s email address only send an sms to the addressee(s) here are some examples of systems, which could use its services. fig. 2. independent components of software system. use of info-communicational resources in software systems 78 2.1 monitoring systems monitoring systems are a typical example of a complex software system. they have their important role in stability and health of other software resources. such systems are being used for analyzing the state of the resource (i.e., web server) and reporting about the critical issues. the reporting part of systems is based on the info-communicational mechanisms, such as e-mail, sms, internet, etc. nowadays, one of the widespread mechanisms of info-communication is sms. the undeniable advantage of this mechanism is that the messages are being directly delivered to the addressee, which is very important for monitoring systems. usually in emergency situations a quick action should be taken to avoid further problems, thus the responsible stuff should be aware about the issues as soon as possible. unimail could be used for sending operative notifications by sms messages. in such systems usually e-mail and sms messages are used for alerting. in case the emailing mechanism is already configured, the minimal effort will be required to send an sms notification as well (figure 3). unimail is a part of monitoring process of a mail server at the institute for informatics and automation problems of nas ra. there are types of issues, which affect critically the workflow of the server, i.e., network or queue size related problems [5].  network related issues are one of the most critical categories of problems that could appear on mail servers. in such cases the server could become unavailable for users and other mail providers.  another category of issues is related to the queue of e-mails. as a result of attack one or more accounts of a mail provider could be hacked. the next step of such attacks is sending spam messages to mail addresses of other providers (sum-outbreaking issues). well-known mail providers like gmail, yahoo will block the mail account, from which lots of spam e-mails are being received as well as the mail provider. such issues should be detected and resolved as soon as possible. otherwise, e-mails sent from the accounts of that provider will be considered as a spam and will not be received by the addressee. there is a deployed application on the mail provider’s server, which is periodically checking for network connectivity and sum-outbreaking issues. if any issue has been detected, the application combines an email message with the corresponding content (figure 3) and sends it to the admin’s and unimail’s addresses. then the administrator will receive both e-mail and sms alerts. in case the unimail server and mail server are located at the same local network the aroused issues will be reported even if there are connection problems with global network. in this case the email message could not be sent as mail providers communicate p2p by the internet protocols. however, the sms alert will be delivered because it is transmitted through gsm, which is a completely different communicational network. fig. 3. format of e-mail components. a. mkhitaryan, a. petrosyan and a. nanassian 79 2.2 authentication module sms technology leaves its footprint on the authentication process of the multiuser services. for most known multiuser services, the user should be registered and authenticated by phone number (or by email address, but most probably, the user has been authenticated by phone number during the creation of mail account). to be able to identify the person with his phone number an sms message with randomly generated verification code is being sent to him. the code should be reentered into the system, where the user is trying to register, then the system compares and checks the correctness of the sent (by sms) and entered (by user) codes. unimail as an infocommunicational resource could make the registration process of multiuser applications easier. any permitted app could send an sms message with verification code to the user by email interface. a similar technique is being used for password resetting process of already registered users. some systems require users to get verified by phone number, which they used for registration to permit password change. this way they prevent unauthorized users (i.e., hackers) who somehow could access the account, change the password. a similar code being generated by unimail as an info-communicational resource could make the registration and password reset processes of multiuser applications easier. any permitted app could send an sms message with a verification code to the user via email interface. conclusion the applications, which need to send messages by gsm network are required to have a module for it. in general, the module could be either embedded or separated. separated module is more economical taking into account resources and time. for complex software systems the unimail resource can serve as an info-communicational module. it is providing an easy to use public interface by email technology. the use of unimail can make the creation process of systems easier, which needs to get to gsm network and send operative sms messages. references [1] д. геворкян, а. нанасян и к. хачатрян «новые web ресурсы asnet.am», материалы конф. csit-2011, ереван, сс. 311-313, 2011. [2] d. gevorkyan, k. khachatryan, a. nanassian, a. petrosyan, g. petrosyan, v. sahakyan and e. vardanyan, “mail informerselective incoming instand phone notification system”, proceedings of international conference computer science and information technologies, csit, yerevan, armenia, pp. 466-467, 2009. [3] а. нанасян и к. хачатрян, «mail2sms.asnet.am – система оповещения о входящих письмах», материалы конф. csit-2013. ереван, 2013. [4] e. mateev, a. mkhitaryan, a. nanasyan, v. sahakyan and a. petrosyan, “hybrid infocommunication email sms unimail system”, proceedings of international conference computer science and information technologies, yerevan, armenia, pp. 389-391, 2009. [5] natale maria bianchi, “a survival guide for the small mail server” spamhaus 2015. use of info-communicational resources in software systems 80 submitted 20.07.2018, accepted 02.12.2018. ծրագրային համակարգերում ինֆոկոմունիկացիոն ռեսուրսների օգտագործումը ա. մխիթարյան, ա․ պետրոսյան և ա․նանասյան ամփոփում աշխատանքում դիտարկվել է ինֆո-կոմունիկացիոն ռեսուրսների կիրառությունը ծրագրային համալիրներում: քննարկվել են unimail ինֆո-կոմունիկացիոն ռեսուրսը այլ ծրագրային փաթեթներում ներառելու մանրամասներն ու առանձնահատկությունները։ թվարկվել են unimail ռեսուրսի կողմից առաջարկված ծառայությունների կիրառման օրինակներ։ использование информационнокоммуникационных ресурсов в программных системах а․ мхитарян, а. петросян и а․ нанасян аннотация рассмотрено использование информационно-коммуникационных ресурсов в программных системах. рассмотрены детали и механизмы интеграции информационно-коммуникативного ресурса unimail в других системax. обсуждаются примеры типов программных систем, которые могут использовать сервисы ресурса unimail. mathematical problems of computer science 55, 9–25, 2021. udc 519.1 a theorem on even pancyclic bipartite digraphs samvel kh. darbinyan institute for informatics and automation problems of nas ra e-mail: samdarbin@iiap.am abstract we prove a meyniel-type condition and a bang-jensen, gutin and li-type condition for a strongly connected balanced bipartite digraph to be even pancyclic. let d be a balanced bipartite digraph of order 2a ≥ 6. we prove that (i) if d(x)+d(y) ≥ 3a for every pair of vertices x, y from the same partite set, then d contains cycles of all even lengths 2, 4, . . . , 2a, in particular, d is hamiltonian. (ii) if d is other than a directed cycle of length 2a and d(x) + d(y) ≥ 3a for every pair of vertices x, y with a common out-neighbor or in-neighbor, then either d contains cycles of all even lengths 2, 4, . . . , 2a or d(u) + d(v) ≥ 3a for every pair of vertices u, v from the same partite set. moreover, by (i), d contains cycles of all even lengths 2, 4, . . . , 2a, in particular, d is hamiltonian. keywords: digraphs, hamiltonian cycles, bipartite digraphs, pancyclic, even pancyclic. 1. introduction in this paper, we consider finite digraphs without loops and multiple arcs. we assume that the reader is familiar with the standard terminology on digraphs and refer the reader to [1]. every cycle and path is assumed simple and directed. a cycle in a digraph d is called hamiltonian if it includes all the vertices of d. a digraph d is hamiltonian if it contains a hamiltonian cycle. a digraph d of order n ≥ 3 is pancyclic if it contains cycles of every length k, 3 ≤ k ≤ n. there are numerous sufficient conditions for the existence of a hamiltonian cycle in a digraph (see, e.g., [1] [10]). it was proved (see, e.g., [1], [6], [8], [9], [11] [14]) that a number of sufficient conditions for a digraph (undirected graph) to be hamiltonian are also sufficient for the digraph to be pancyclic (with some exceptions). for hamiltonicity, the more general and classical one is the following theorem due to m. meyniel. theorem 1: (meyniel [10]). let d be a strong digraph of order n ≥ 2. if d(x)+d(y) ≥ 2n−1 for all pairs of non-adjacent vertices in d, then d is hamiltonian. notice that meyniel’s theorem is a generalization of ghouila-houri’s and woodall’s theorems. 9 10 a theorem on even pancyclic bipartite digraphs a digraph d is a bipartite if there exists a partition x, y of its vertex set into two partite sets such that every arc of d has its end-vertices in different partite sets. it is called balanced if |x| = |y |. following [1], we will say that a balanced bipartite digraph d of order 2a is even pancyclic (note that a number of authors use the term ”bipancyclic” instead of ”even pancyclic”) if it contains cycles of all even lengths 4, 6, . . . , 2a. an analogue of meyniel’s theorem for the hamiltonicity of balanced bipartite digraphs was given by adamus et al. [3]. theorem 2: (adamus et al. [3]). let d be a balanced bipartite digraph of order 2a ≥ 4. then d is hamiltonian provided one of the following holds: (a) d(x) + d(y) ≥ 3a + 1 for each pair of non-adjacent vertices x, y ∈ v (d); (b) d is strong and d(x) + d(y) ≥ 3a for each pair of non-adjacent vertices x, y ∈ v (d); (c) the minimal degree of d is at least (3a + 1)/2; (d) d is strong, and the minimal degree of d is at least 3a/2. meszka [15] investigated the even pancyclicity of a balanced bipartite digraph satisfying a weaker condition than those in theorem 2(a). he proved the following theorem. theorem 3: (meszka [15]). let d be a balanced bipartite digraph of order 2a ≥ 4. suppose that d(x) + d(y) ≥ 3a + 1 for each two distinct vertices x, y from the same partite set. then d contains cycles of all even lengths 4, 6, . . . , 2a. let x, y be a pair of distinct vertices in a digraph d. the pair {x, y} is a dominated pair (respectively, dominating pair) if there is a vertex z ∈ v (d) \ {x, y} such that z → {x, y} (respectively, {x, y} → z). we will say that a pair of vertices {u, v} is a good pair if it is dominated or dominating. in this case we will say that u (respectively, v) is a partner of v (respectively, u). in [5], bang-jensen et al. gave a new type condition for a digraph to be hamiltonian. in the same paper, they also conjectured the following strengthening of meyniel’s theorem. conjecture 1: let d be a strong digraph of order n. suppose that d(x) + d(y) ≥ 2n − 1 for every good pair of non-adjacent distinct vertices x, y. then d is hamiltonian. they also conjectured that this can even be generalized to the following: conjecture 2:. let d be a strong digraph of order n. suppose that d(x) + d(y) ≥ 2n − 1 for every pair of non-adjacent distinct vertices x, y with a common in-neighbor. then d is hamiltonian. in [5] and [4], it was proved that conjecture 1 (2) is true if we also require an additional condition. theorem 4: (bang-jensen et al. [5]). let d be a strong digraph of order n ≥ 2. suppose that min{d(x), d(y)} ≥ n − 1 and d(x) + d(y) ≥ 2n − 1 for any pair of non-adjacent vertices x, y with a common in-neighbor. then d is hamiltonian. in [4], it was proved that if in conjecture 1 we replace the degree condition d(x)+d(y) ≥ s. darbinyan 11 2n − 1 with d(x) + d(y) ≥ 5n/2 − 4, then conjecture 1 is true. there are some versions of conjecture 1 and 2 for balanced bipartite digraphs. (see, e.g., theorems 5, 6 and 7). theorem 5: (adamus [2]). let d be a strong balanced bipartite digraph of order 2a ≥ 6. if d(x) + d(y) ≥ 3a for every good pair of distinct vertices x, y, then d is hamiltonian. an analogue of theorem 4 was given by wang [16], and recently strengthened by the author [17]. theorem 6: (wang [16]). let d be a strong balanced bipartite digraph of order 2a ≥ 4. suppose that, for every dominating pair of vertices {x, y}, either d(x) ≥ 2a−1 and d(y) ≥ a+1 or d(y) ≥ 2a − 1 and d(x) ≥ a + 1. then d is hamiltonian. before stating the next theorem we need to define a digraph of order eight. example 1: let d(8) be the bipartite digraph with partite sets x = {x0, x1, x2, x3} and y = {y0, y1, y2, y3}, and a(d(8)) contains exactly the arcs y0x1, y1x0, x2y3, x3y2 and all the arcs of the following 2-cycles: xi ↔ yi, i ∈ [0, 3], y0 ↔ x2, y0 ↔ x3, y1 ↔ x2 and y1 ↔ x3. it is not difficult to check that d(8) is strongly connected, max{d(x), d(y)} ≥ 2a − 1 for every pair of vertices {x, y} with a common out-neighbor, but it is not hamiltonian. indeed, if c is a hamiltonian cycle in d(8), then c would contain the arcs x1y1 and x0y0 and therefore, the path x1y1x0y0 or the path x0y0x1y1, which is impossible since n−(x0) = n −(x1) = {y0, y1}. theorem 7: (darbinyan [17]). let d be a strong balanced bipartite digraph of order 2a ≥ 8. suppose that max{d(x), d(y)} ≥ 2a − 1 for every pair of distinct vertices {x, y} with a common out-neighbor. then d is hamiltonian unless d is isomorphic to the digraph d(8). motivated by the bondy famous metaconjecture, the author, together with karapetyan [20],proposed the following problem: problem 1: characterize those digraphs, which satisfy the conditions of theorem 5 (or 6 or 7) but are not even pancyclic. this problem for theorems 6 and 7 was solved by the author [18] (theorem 8(ii)), and for theorem 5 by adamus [19] (theorem 9). theorem 8: let d be a strong balanced bipartite digraph of order 2a. (i). (darbinyan [18]). if d contains a cycle of length 2a − 2 and max{d(x), d(y)} ≥ 2a − 2 ≥ 6 for every pair of distinct vertices {x, y} with a common out-neighbor, then for every k, 1 ≤ k ≤ a − 1, d contains a cycle of length 2k. (ii). (darbinyan [18]). if d is not a directed cycle of length 2a ≥ 8 and max{d(x), d(y)} ≥ 2a − 1 for every pair of distinct vertices {x, y} with a common outneighbor, then for every k, 1 ≤ k ≤ a, d contains a cycle of length 2k (in particular, d is even pancyclic) unless d is isomorphic to the digraph d(8). 12 a theorem on even pancyclic bipartite digraphs (iii). (darbinyan and karapetyan [20]). suppose that d is not a directed cycle of length 2a ≥ 10 and max{d(x), d(y)} ≥ 2a − 2 for every pair of distinct vertices {x, y} with a common out-neighbor. then d contains a cycle of length 2a − 2 unless d is isomorphic to a digraph of order ten, which we specify. the following theorem by adamus (theorem 9) and the main result of this paper (theorem 10) were proved simultaneously and independently. theorem 9: (adamus [19]). let d be a balanced bipartite hamiltonian digraph of order 2a ≥ 6 other than a directed cycle of length 2a. suppose that d(x)+d(y) ≥ 3a for every good pair of distinct vertices x, y. then d contains cycles of all even lengths 2, 4, . . . , 2a. theorem 10: let d be a strong balanced bipartite digraph of order 2a ≥ 6 with partite sets x and y . if d(x) + d(y) ≥ 3a for every pair of distinct vertices {x, y} either both in x or both in y , then d contains cycles of all even lengths less than or equal to 2a (in particular, d is hamiltonian). the last result (theorem 10) was presented at the ”international conference dedicated to 90th anniversary of sergey mergelyan”, 20-25 may, 2018, yerevan, armenia. using some arguments of [2] by adamus, we can prove the following lemma. lemma 1: let d be a balanced bipartite digraph of order 2a ≥ 6 with partite sets x and y . suppose that d is not a directed cycle of length 2a and d(u) + d(v) ≥ 3a for every good pair of distinct vertices u, v. then d either is even pancyclic or every pair of distinct vertices {x, y} from the same partite set is a good pair. the following theorem follows from theorem 10 and lemma 1. theorem 11: let d be a strong balanced bipartite digraph of order 2a ≥ 6 other than a directed cycle of length 2a. suppose that d(x) + d(y) ≥ 3a for every good pair of distinct vertices x, y. then d contains cycles of all even lengths 2, 4, . . . , 2a. it is worth to noting that in the proof of theorem 10 does not use the fact that d is hamiltonian. thus, we have a common alternative proof for theorems 2, 3, 5 and 9. note that if a balanced bipartite digraph satisfies the condition of theorem 2(a) (or theorem 2(c)), then d is strong. example 2: for any even integer a ≥ 2 there is a non-strongly connected balanced bipartite digraph d of order 2a with partite sets x and y , such that d(x)+d(y) ≥ 3a for every pair of distinct vertices {x, y} either both in x or both in y ,i.e., if d is not strong, then theorem 10 is not true. to see this, we take two balanced bipartite complete digraphs both of order a (a is even) with partite sets u, v and z, w , respectively. by adding all the possible arcs from z to v and from w to u we obtain a digraph d. it is easy to check that d(x) + d(y) ≥ 3a for every pair of non-adjacent distinct vertices {x, y} of d, but d is not strongly connected and hence, d is not hamiltonian. s. darbinyan 13 2. terminology and notations in this paper, we consider finite digraphs without loops and multiple arcs. terminology and notations not defined here or above are consistent with [1]. the vertex set and the arc set of a digraph d are denoted by v (d) and a(d), respectively. the order of d is the number of its vertices. if xy ∈ a(d), then we also write x → y and say that x dominates y or y is an out-neighbor of x and x is an in-neighbor of y. if x → y and y → x we shall use the notation x ↔ y (x ↔ y is called 2-cycle). we set −→a [x, y] = 1 if xy ∈ a(d) and −→a [x, y] = 0 if xy /∈ a(d). if a and b are two disjoint subsets of v (d) such that every vertex of a dominates every vertex of b, then we say that a dominates b, denoted by a → b. similarly, a ↔ b means that a → b and b → a. if x ∈ v (d) and a = {x} we sometimes write x instead of {x}. let n+d(x), n − d(x) denote the set of out-neighbors, respectively the set of in-neighbors of a vertex x in a digraph d. if a ⊆ v (d), then n+d(x, a) = a ∩ n + d(x) and n − d(x, a) = a ∩ n − d(x). the out-degree of x is d+d(x) = |n + d(x)| and d − d(x) = |n − d(x)| is the in-degree of x. similarly, d+d(x, a) = |n + d(x, a)| and d − d(x, a) = |n − d(x, a)|. the degree of the vertex x in d is defined as dd(x) = d + d(x)+d − d(x) (similarly, dd(x, a) = d + d(x, a)+d − d(x, a)). we omit the subscript if the digraph is clear from the context. the subdigraph of d induced by a subset a of v (d) is denoted by d[a]. for integers a and b, a ≤ b, let [a, b] denote the set of all the integers, which are not less than a and are not greater than b. the path (respectively, the cycle) consisting of the distinct vertices x1, x2, . . . , xm (m ≥ 2) and the arcs xixi+1, i ∈ [1, m − 1] (respectively, xixi+1, i ∈ [1, m − 1], and xmx1), is denoted by x1x2 · · · xm (respectively, x1x2 · · · xmx1). the length of a cycle or a path is the number of its arcs. we say that x1x2 · · · xm is a path from x1 to xm or is an (x1, xm)-path. if a digraph d contains a path from a vertex x to a vertex y we say that y is reachable from x in d. in particular, x is reachable from itself. we denote by k∗a,b the complete bipartite digraph with partite sets of cardinalities a and b. a digraph d is strongly connected (or, just, strong) if there exists a path from x to y and a path from y to x for every pair of distinct vertices x, y. two distinct vertices x and y are adjacent if xy ∈ a(d) or yx ∈ a(d) (or both). let d be a bipartite digraph with partite sets x and y . a matching from x to y (from y to x) is an independent set of arcs with origin in x and terminus in y (origin in y and terminus in x). (a set of arcs with no common end-vertices is called independent). if d is balanced, one says that such a matching is perfect if it consists of precisely |x| arcs. 3. preliminaries in [21] and [11], the author studied pancyclicity of a digraph with the condition of the meyniel theorem. before stating the main result of [11] we need to define a family of digraphs. definition 1: for any integers n and m, (n + 1)/2 < m ≤ n − 1, let φmn denote the set of digraphs d, which satisfy the following conditions: (i) v (d) = {x1, x2, . . . , xn}; (ii) xnxn−1 . . . x2x1xn is a hamiltonian cycle in d; (iii) for each k, 1 ≤ k ≤ n − m + 1, the vertices xk and xk+m−1 are not adjacent; (iv) xjxi /∈ a(d) whenever 2 ≤ i + 1 < j ≤ n and (v) the sum of degrees for any two distinct non-adjacent vertices is at least 2n − 1. 14 a theorem on even pancyclic bipartite digraphs theorem 12: (darbinyan [11]). let d be a strong digraph of order n ≥ 3. suppose that d(x) + d(y) ≥ 2n − 1 for all pairs of distinct non-adjacent vertices x, y in d. then either (a) d is pancyclic or (b) n is even and d is isomorphic to one of digraphs k∗n/2,n/2, k∗n/2,n/2 \ {e}, where e is an arbitrary arc of k ∗ n/2,n/2, or (c) d ∈ φ m n (in this case d does not contain only a cycle of length m). later, theorem 12, was also proved independently by benhocine [22]. lemma 2: (adamus et al. [3]). let d be a strong balanced bipartite digraph of order 2a ≥ 4 with partite sets x and y . if d(x) + d(y) ≥ 3a for every pair of distinct vertices x, y from the same partite set, then d contains a perfect matching from y to x and a perfect matching from x to y . following [15], we give the following definition. definition 2: let d be a balanced bipartite digraph of order 2a ≥ 4 with partite sets x and y . let my,x = {yixi ∈ a(d) | i = 1, 2, . . . , a} be a perfect matching from y to x. we define a digraph d∗[my,x] with vertex set {v1, v2, . . . , va} as follows: each vertex vi corresponds to a pair {xi, yi} of vertices in d and for each pair of distinct vertices vl, vj, vlvj ∈ a(d∗[my,x]) if and only if xlyj ∈ a(d). let d be a balanced bipartite digraph with partite sets x and y . let my,x be a perfect matching from y to x in d and d∗[my,x] be its corresponding digraph. further, in this paper, we will denote the vertices of d (respectively, of d∗[my,x]) by letters x, y (respectively, u, v) with subscripts or without them. the size of a perfect matching my,x = {yixi ∈ a(d) | i = 1, 2, . . . , a} from y to x in d (denoted by s(my,x)) is the number of arcs yixi such that xiyi /∈ a(d). using the arguments of [15] by meszka, we can formulate the following lemma. lemma 3: let d be a balanced bipartite digraph of order 2a ≥ 6 with partite sets x and y . let my,x = {yixi ∈ a(d) | i = 1, 2, . . . , a} be a perfect matching from y to x. then the following hold: (i). d+(vi) = d +(xi) − −→a [xi, yi] and d−(vi) = d−(yi) − −→a [xi, yi]. (ii). if d∗[my,x] contains a cycle of length k, where k ∈ [2, a], then d contains a cycle of length 2k. (iii). suppose that a is even, and d∗[my,x] is isomorphic to k ∗ a/2,a/2 with partite sets {v1, v2, . . . , va/2} and {va/2+1, va/2+2, . . . , va}. if d contains an arc from {y1, y2, . . . , ya/2} to {xa/2+1, xa/2+2, . . . , xa}, say ya/2xa ∈ a(d), then d contains a cycle of length 2k for all k = 2, 3, . . . , a. proof. the proof of lemma 3 can be found in [15], but we give it here for completeness. (i). it follows immediately from the definition of d∗[my,x]. (ii). indeed, if vi1vi2 . . . vikvi1 is a cycle of length k in d ∗[my,x], then yi1xi1yi2xi2yi3 . . . yikxikyi1 is a cycle of length 2k in d. (iii). by (ii), it is clear that d contains cycles of every length 4k, k = 1, 2, . . . , a/2. it remains to show that d also contains cycles of every length 4k+2, k = 1, 2, . . . , a/2−1. indeed, since xiyj ∈ a(d) and xjyi ∈ a(d) for all i ∈ [1, a/2], j ∈ [a/2 + 1, a] and ya/2xa ∈ a(d), s. darbinyan 15 from the definition of d∗[my,x] it follows that y1x1ya/2+1xa/2+1y2x2ya/2+2xa/2+2y3 x3 . . . xkya/2+kxa/2+kya/2xay1 is a cycle of length 4k + 2 in d. lemma 4: (adamus [2]). let d be a balanced bipartite digraph of order 2a ≥ 6 other than a directed cycle of length 2a. suppose that d(x) + d(y) ≥ 3a for every good pair {x, y} of distinct vertices in d. then d(u) ≥ a for all u ∈ v (d). now let us prove lemma 1. for convenience, we will restate it here. lemma 1: let d be a balanced bipartite digraph of order 2a ≥ 6 with partite sets x and y . suppose that d is not a directed cycle of length 2a and d(u) + d(v) ≥ 3a for every good pair of distinct vertices u, v. then d either is even pancyclic or every pair of distinct vertices {x, y} from the same partite set is a good pair. proof: let x = {x1, x2, . . . , xa} and y = {y1, y2, . . . , ya}. suppose that v (d) contains a pair of vertices from the same partite set, which is not a good pair. without loss of generality, assume that {x1, x2} is not a good pair. then n+(x1) ∩ n+(x2) = n−(x1) ∩ n−(x2) = ∅, d+(x1) + d+(x2) ≤ a , d−(x1) + d−(x2) ≤ a. hence, d(x1) + d(x2) ≤ 2a. this together with d(x1) ≥ a and d(x2) ≥ a (lemma 4) implies that d(x1) = d(x2) = d +(x1) + d +(x2) = d −(x1) + d −(x2) = a. now we obtain that n+(x1) ∪ n+(x2) = n−(x1) ∪ n−(x2) = y . let xi ∈ x \ {x1, x2} be an arbitrary vertex. we claim that {x1, xi} or {x2, xi} is a good pair. assume that this is not the case. then (n+(x1) ∪ n+(x2)) ∩ n+(xi) = ∅, which contradicts the facts that d is strong and n+(x1) ∪ n+(x2) = y . thus, {x1, xi} or {x2, xi} is a good pair for all i, 3 ≤ i ≤ a. therefore, from condition (a) and d(x1) = d(x2) = a it follows that d(xi) = 2a for all i, 3 ≤ i ≤ a, i.e., d[x ∪ y \ {x1, x2}] is a complete bipartite digraph with partite sets x \ {x1, x2} and y . from d(x3) = 2a it follows that d +(x3) = d −(x3) = a. therefore, if d contains a hamiltonian cycle, then d contains cycles of all even lengths 2, 4, . . . , 2a. now we will show that d contains a hamiltonian cycle. assume first that there is an (x1, x2)-path of length two. let x1y1x2 be an (x1, x2)-path of length two. then y1x1 /∈ a(d) and x2y1 /∈ a(d) as {x1, x2} is not a good pair. now, since x2y1 /∈ a(d) and d+(x2) ≥ 1, we may assume that x2y2 ∈ a(d). from d−(x1)+d−(x2) = a ≥ 3 it follows that d−(x1) ≥ 2 or d−(x2) ≥ 2. assume that d−(x1, {y3, y4, . . . , ya}) ≥ 1. we may assume that y3x1 ∈ a(d). now using the fact that d[x ∪y \{x1, x2}] is a complete bipartite digraph, we see that y3x1y1x2y2x3y4x4 . . . yaxay3 is a hamiltonian cycle in d. assume now that d−(x1, {y3, y4, . . . , ya}) = 0. then from y1x1 /∈ a(d) and d−(x1) = 1 it follows that y2x1 ∈ a(d). then x2y2x1 is an (x2, x1)-path of length two and d−(x2) ≥ 2. now, we have that y2x2 /∈ a(d) since {x1, x2} is not a good pair. therefore, d−(x2, {y3, y4, . . . , ya}) ≥ 1. now, by repeating the above argument, we conclude that d is hamiltonian. similarly, one can show that if there is an (x2, x1)-path of length two, then again d is hamiltonian. assume next that there is no path of length two between x1 and x2. then d−(x1, n +(x2)) = d −(x2, n +(x1)) = 0, and from n −(x1) ∪ n−(x2) = y it follows that n−(x1) = n +(x1) and n −(x2) = n +(x2). this together with d(x1) = d(x2) = a implies that |n+(x1)| = |n+(x2)| = a/2, a is even and a ≥ 4. without loss of generality, we assume that x1 ↔ {y1, y2} and x2 ↔ {ya−1, ya}. now, since d[x ∪ y \ {x1, x2}] is a complete 16 a theorem on even pancyclic bipartite digraphs bipartite digraph, it is not difficult to check that x3y1x1y2x4y3x5y4 . . . xa−1ya−2xaya−1x2 yax3 is a hamiltonian cycle in d. thus, in all possible cases, d is hamiltonian. lemma 1 is proved. 4. proof of the main result let d be a strong balanced bipartite digraph of order 2a. we say that d satisfies condition (a) when d(x) + d(y) ≥ 3a for all distinct vertices x, y from the same partite set. the proof of theorem 10 will be based on the following three lemmas below. lemma 5: let d be a strong balanced bipartite digraph of order 2a ≥ 6 with partite sets x and y . if d satisfies condition (a), then d contains cycles of lengths 2 and 4. proof: from condition (a) immediately follows that d contains a cycle of length 2. we will prove that d also contains a cycle of length 4. by lemma 2, d contains a perfect matching from y to x. let my,x = {yixi ∈ a(d) | i = 1, 2, . . . , a} be a perfect matching from y to x. if for some integers i, j, 1 ≤ i ̸= j ≤ a, the arcs xiyj, xjyi are in d, then xiyjxjyixi is a cycle of length 4. we may, therefore, assume that for every pair of integers i, j, 1 ≤ i ̸= j ≤ a, −→a [xi, yj] + −→a [xj, yi] ≤ 1. therefore, for all i ∈ [1, a], d−(yi) ≤ a − d+(xi) − 1, if −→a [xi, yi] = 0 and d−(yi) ≤ a − d+(xi) + 1, if −→a [xi, yi] = 1. (1) assume that there are two distinct integers i, j, 1 ≤ i, j ≤ a, such that −→a [xi, yi] = −→a [xj, yj] = 0. then, by (1), d−(yi) + d+(xi) ≤ a − 1 and d−(yj) + d+(xj) ≤ a − 1. these together with condition (a) and the fact that the semi-degrees of every vertex in d are bounded above by a thus implies that 6a ≤ d(xi) + d(xj) + d(yi) + d(yj) = d−(yi) + d+(xi) + d−(yj) + d+(xj) +d+(yi) + d +(yj) + d −(xi) + d −(xj) ≤ 6a − 2, which is a contradiction. assume now that for some i ∈ [1, a], −→a [xi, yi] = 0 and for all j ∈ [1, a]\{i}, −→a [xj, yj] = 1. without loss of generality, we may assume that i = 1. by (1), d−(y1) + d +(x1) ≤ a − 1 and d−(y2)+d +(x2) ≤ a+1. if for some k ∈ [3, a], y2xk ∈ a(d) and ykx2 ∈ a(d), then x2y2xkykx2 is a cycle of length 4 in d. we may, therefore, assume that −→a [y2, xk] + −→a [yk, x2] ≤ 1 for all k ∈ [3, a]. this implies that d−(x2) + d +(y2) = d −(x2, {y1, y2}) + d+(y2, {x1, x2}) + d−(x2, y \ {y1, y2}) +d+(y2, x \ {x1, x2}) ≤ 4 + a − 2 = a + 2. using the above inequalities and condition (a), we obtain 6a ≤ d(x1) + d(x2) + d(y1) + d(y2) = d−(y1) + d+(x1) + d−(y2) + d+(x2) +d−(x2) + d +(y2) + d −(x1) + d +(y1) ≤ 5a + 2, which is a contradiction since a ≥ 3. s. darbinyan 17 assume finally that xiyi ∈ a(d) for all i ∈ [1, a]. in this case, by the symmetry between the vertices xi and yi, similar to (1), we obtain that d −(xi) + d +(yi) ≤ a + 1. this together with (1) implies that for any i, j (1 ≤ i ̸= j ≤ a), 6a ≤ d(xi) + d(xj) + d(yi) + d(yj) ≤ 4a + 4, a contradiction since a ≥ 3. lemma 5 is proved. remark 1: there is a strong balanced bipartite digraph of order 4, which satisfies condition (a), but contains no cycle of length 4. to see this, we consider the following digraph with vertex set v (d) = {x1, x2, y1, y2} and arc set d(a) = {x1y2, y2x2, x2y2, x2y1, y1x1}. lemma 6: let d be a strong balanced bipartite digraph of order 2a ≥ 6 with partite sets x and y . let my,x = {yixi ∈ a(d) | i = 1, 2, . . . , a} be a perfect matching from y to x in d such that the size s(my,x) of my,x is maximum among the sizes of all the perfect matching from y to x in d. if d satisfies condition (a), then the digraph d∗[my,x] either is strong or d contains cycles of all lengths 2, 4, . . . , 2a. proof: notice that, by lemma 5, d contains cycles of lengths 2 and 4. suppose that the digraph d∗[my,x] is not strong. then in d ∗[my,x] there are two distinct vertices, say v1 and vj, such that there is no path from v1 to vj in d ∗[my,x]. let u be the set of all vertices reachable from v1 and w be the set of all vertices from which vj is reachable. notice that v1 ∈ u, vj ∈ w and u ∩ w = ∅. case 1. d+(v1) ≥ 1 and d−(vj) ≥ 1. then |u| ≥ 2 and |w| ≥ 2. let vl, vk be two distinct vertices in u and vp, vq be two distinct vertices in w. from condition (a) and the fact that the semi-degrees of every vertex in d are bounded above by a it follows that d+(xl) + d +(xk) ≥ a and d−(xp) + d−(xq) ≥ a. (2) by lemma 3(i), d+(vl) + d +(vk) = d +(xl) + d +(xk) − −→a [xl, yl] − −→a [xk, yk], and d−(vp) + d −(vq) = d −(yp) + d −(yq) − −→a [xp, yp] − −→a [xq, yq]. (3) it follows from them and (2) that d+(vl) + d +(vk) ≥ a − 2 and d−(vp) + d−(vq) ≥ a − 2. without loss of generality we may assume that d+(vl) ≥ (d+(vl) + d+(vk))/2 and d−(vp) ≥ (d−(vp) + d −(vp))/2. these imply that d +(vl) ≥ (a − 2)/2 and d−(vp) ≥ (a − 2)/2, which in turn imply that |u| ≥ a/2 and |w| ≥ a/2. if d+(vl)+d +(vk) ≥ a−1 or d−(vp)+d−(vq) ≥ a−1, then |u| ≥ (a+1)/2 or |w| ≥ (a+1)/2, respectively. hence |u| + |w| ≥ (2a + 1)/2, which is a contradiction since |u| + |w| ≤ a. using (2) and (3), we may therefore assume that d+(vl) + d +(vk) = d +(xl) + d +(xk) − 2 = d−(vp) + d−(vq) = d−(yp) + d−(yq) − 2 = a − 2. 18 a theorem on even pancyclic bipartite digraphs then it is easy to see that the arcs xlyl, xkyk, xpyp and xqyq are in d, |u| = |w| = a/2 and v (d∗[my,x]) = u ∪ w . in particular, a is even. without loss of generality, we assume that u = {v1, v2, . . . , va/2} and w = {va/2+1, va/2+2, . . . , va}. since there is no arc from a vertex in u to a vertex in w, the following holds: a({x1, x2, . . . , xa/2} → {ya/2+1, ya/2+2, . . . , ya}) = ∅. (4) therefore, if i ∈ [1, a/2] and j ∈ [a/2 + 1, a], then d+(xi) ≤ a/2 and d−(yj) ≤ a/2. together with (2) they imply that d+(xi) = d −(yj) = a/2 and xi → {y1, y2, . . . , ya/2} and {xa/2+1, xa/2+2, . . . , xa} → yj (5) for all i ∈ [1, a/2] and j ∈ [a/2 + 1, a], respectively. therefore, by condition (a), 3a ≤ d(xi) + d(xk) ≤ a + d−(xi) + d−(xk), for every pair of i, k ∈ [1, a/2]. this implies that d−(xi) = d−(xk) = a, which means that {y1, y2, . . . , ya} → {xi, xk}. similarly, yj → {x1, x2 . . . , xa}, for all j ∈ [a/2 + 1, a]. from this and (5) it follows that the induced subdigraphs d[{x1, x2, . . . , xa/2, y1, y2, . . . , ya/2}] and d[{xa/2+1, xa/2+2, . . . , xa, ya/2+1, ya/2+2, . . . , ya}] both are balanced bipartite complete digraphs. therefore, d contains cycles of all lengths 2, 4, . . . , a. it remains to show that d also contains cycles of every length a + 2b, b ∈ [1, a/2]. since d is strong and (4), it follows that there is an arc from a vertex in {y1, y2, . . . , ya/2} to a vertex in {xa/2+1, xa/2+2, . . . , xa}. without loss of generality, we may assume that ya/2xa/2+1 ∈ a(d). then x1y1x2y2 . . . xa/2ya/2xa/2+1 ya/2+1xa/2+2 . . . xa/2+bya/2+bx1 is a cycle of length a + 2b. thus, d contains cycles of all lengths 2, 4, . . . , 2a. this completes the discussion of case 1. case 2. d+(v1) = 0. then d+(x1) = 1 and x1y1 ∈ a(d), since d is strong. hence d(x1) ≤ a + 1. together with condition a this implies that a ≤ d(x1) ≤ a + 1. we distinguish two subcases depending on d(x1). case 2.1. d(x1) = a. then d(xi) ≥ 2a for all i ∈ [2, a] because of condition a. therefore, the induced subdigraph d⟨y ∪x \{x1}⟩ is a complete bipartite digraph with partite sets y and x \{x1}. it is clear that d contains cycles of every lengths 2, 4, . . . , 2a − 2. since d(x1) = a, d+(x1) = 1 and a ≥ 3, we have that d−(x1) = a − 1 ≥ 2. without loss of generality we may assume that y2x1 ∈ a(d). then y2x1y1x3y3 . . . xayax2y2 is a cycle of length 2a. case 2.2. d(x1) = a + 1. then {y1, y2, . . . , ya} → x1 because of d+(x1) = 1, and, by condition (a), d(xi) ≥ 2a − 1 for all i ∈ [2, a]. observe that if for some i ∈ [2, a], y1xi ∈ a(d), then miy,x := {yix1, y1xi} ∪ {yjxj | j ∈ [1, a] \ {1, i}} is a perfect matching from y to x in d. assume that for some i ∈ [2, a], xiy1 /∈ a(d). then y1xi ∈ a(d) because of d(xi) ≥ 2a−1. since x1y1 ∈ a(d), xiy1 /∈ a(d) and x1yi /∈ a(d), it follows that s(miy,x) > s(my,x), which contradicts the choice of my,x. we may therefore assume that {x2, x3, . . . , xa} → y1. if s. darbinyan 19 y1xi ∈ a(d) and xiyi ∈ a(d), where i ∈ [2, a], then again we have s(miy,x) > s(my,x), since the arcs x1y1, xiyi are in d and x1yi /∈ a(d). we may therefore assume that −→a [y1, xi] + −→a [xi, yi] ≤ 1. this together with d(xi) ≥ 2a − 1, i ∈ [2, a], implies that {y2, y3, . . . , ya} → xi → {y2, y3, . . . , ya} \ {yi}. (6) since d is strong and d+(x1, {y2, y3, . . . , ya}) = 0, it follows that d+(y1, {x2, x3, . . . , xa}) ≥ 1. without loss of generality, we assume that y1x2 ∈ a(d). then, since y2x1 ∈ a(d) and (6), x1y1x2y3x3 . . . xk−1ykxky2x1 is a cycle of length 2k for every k ∈ [3, a]. lemma 6 is proved. lemma 7: let d be a strong balanced bipartite digraph of order 2a ≥ 6 with partite sets x and y . let my,x = {yixi ∈ a(d) | i = 1, 2, . . . , a} be a perfect matching from y to x in d such that the size s(my,x) of my,x is maximum among the sizes of all the perfect matching from y to x in d. if d satisfies condition (a), then either d(u) + d(v) ≥ 2a − 1 for every pair of non-adjacent vertices u, v in d∗[my,x] or d contains cycles of all lengths 2, 4, . . . , 2a. proof: suppose that d is not even pancyclic. then by lemma 6, d∗[my,x] is strong. let vi and vj be two arbitrary distinct vertices in d ∗[my,x]. write g(i, j) := d+(xi) + d +(xj) + d −(yi) + d −(yj) and f(i, j) := d −(xi) + d −(xj) + d +(yi) + d +(yj). by lemma 3(i), we have d(vi) + d(vj) = g(i, j) − 2−→a [xi, yi] − 2−→a [xj, yj]. (7) by condition (a), we have 6a ≤ d(xi) + d(xj) + d(yi) + d(yj) = f(i, j) + g(i, j). hence, g(i, j) ≥ 2a and 4a ≥ f(i, j) ≥ 6a − g(i, j). (8) since the semi-degrees of every vertex of d are bounded above by a. now we prove the following claim. claim 1: assume that the vertices vi and vj in d ∗[my,x] are not adjacent. then the following hold: (i). if xiyi ∈ a(d) or xjyj ∈ a(d), then −→a [yi, xj] + −→a [yj, xi] ≤ 1. (ii). if xiyi /∈ a(d) or xjyj /∈ a(d), then d(vi) + d(vj) ≥ 2a − 1 in d∗[my,x]. proof: since the vertices vi and vj in d ∗[my,x] are not adjacent, it follows that xiyj /∈ a(d) and xjyi /∈ a(d). (i). suppose, to the contrary, that xiyi ∈ a(d) or xjyj ∈ a(d), but −→a [yi, xj] + −→a [yj, xi] = 2. then m ′y,x := {yixj, yjxi}∪{ykxk | k ∈ [1, a]\{i, j}} is a new perfect matching from y to x in d. since xjyi /∈ a(d), xiyj /∈ a(d) and xiyi ∈ a(d) or xjyj ∈ a(d), it follows that s(m ′y,x) > s(my,x), which contradicts the choice of my,x. (ii). if −→a [xi, yi] = −→a [xj, yj] = 0, then from (7) and g(i, j) ≥ 2a it follows that d(vi) + d(vj) ≥ 2a in d∗[my,x]. we may therefore assume that xiyi ∈ a(d). then xjyj /∈ a(d) by the assumption of claim 1(ii). if g(i, j) ≥ 2a+1, then, by (7), d(vi)+d(vj) ≥ 2a−1. 20 a theorem on even pancyclic bipartite digraphs thus, we may assume that g(i, j) = 2a. then f(i, j) ≥ 4a by (8). the last inequality implies that the arcs yixj, yjxi are in d. therefore, m ′ y,x := {yixj, yjxi}∪{ykxk | k ∈ [1, a]\{i, j}} is a new perfect matching from y to x in d. since xiyi ∈ a(d), xiyj /∈ a(d) and xjyi /∈ a(d), it follows that s(m ′y,x) > s(my,x), which contradicts the choice of my,x. the claim is proved. we now return to the proof of lemma 7. suppose that there exist two distinct nonadjacent vertices, say v1 and v2, in d ∗[my,x] such that d(v1) + d(v2) ≤ 2a − 2. (9) this together with (7), −→a [x1, y1] ≤ 1 and −→a [x2, y2] ≤ 1 implies that g(1, 2) ≤ 2a + 2. therefore, 2a ≤ g(1, 2) ≤ 2a + 2. case 1. −→a [x1, y1] = 0. then from (7), (9) and the fact that g(1, 2) ≥ 2a, it follows that −→a [x2, y2] = 1 (i.e., x2y2 ∈ a(d)) and g(1, 2) = 2a. from this and (8) it follows that f(1, 2) ≥ 4a, which in turn implies that y1x2 ∈ a(d) and y2x1 ∈ a(d). the aforementioned contradicts claim 1(i) since x2y2 ∈ a(d). case 2. −→a [x1, y1] = −→a [x2, y2] = 1, i.e., x1y1 ∈ a(d) and x2y2 ∈ a(d). from claim 1(i) it follows that y1x2 /∈ a(d) or y2x1 /∈ a(d). if 2a ≤ g(1, 2) ≤ 2a + 1, then from (8) it follows that f(1, 2) ≥ 4a − 1, which in turn implies that y1x2 ∈ a(d) and y2x1 ∈ a(d), which is a contradiction. we may therefore assume that g(1, 2) = 2a+2. this and (8) imply that f(1, 2) ≥ 4a−2. then, since y1x2 /∈ a(d) or y2x1 /∈ a(d), it follows that y1x2 ∈ a(d) or y2x1 ∈ a(d). without loss of generality, we may assume that y1x2 /∈ a(d) and y2x1 ∈ a(d). note that the vertices y1 and x2 are not adjacent. then f(1, 2) = 4a − 2, which in turn implies that d−(x1) = d +(y2) = a and d −(x2) = d +(y1) = a − 1. therefore, y2 → {x1, x2, . . . , xa}; {y1, y2, . . . , ya} → x1; y1 → {x1, x3, x4, . . . , xa}; {y2, y3, . . . , ya} → x2. (10) since y1x2 /∈ a(d). using (10), it is easy to see that for all i ∈ [3, a], miy,x := {y2x1, yix2, y1xi} ∪ {ykxk | k ∈ [3, a] \ {i}} is a perfect matching from y to x in d. using the facts that the arcs x1y1, x2y2 are in d, it is not difficult to see that if for some i ∈ [3, a], either x2yi /∈ a(d) or xiy1 /∈ a(d) or xiyi ∈ a(d), then s(miy,x) > s(my,x), which contradicts the choice of my,x. we may therefore assume that xiyi /∈ a(d) for all i ∈ [3, a], and x2 → {y2, y3, . . . , ya} and {x3, x4, . . . , xa} → y1. together with (10) they imply that x2 ↔ {y2, y3, . . . , ya} and y1 ↔ {x1, x3, x4, . . . , xa}. (11) since the vertices y1, x2 are not adjacent, from (11) and lemma 3(i) it follows that d−(y1) = d +(x2) = a − 1, d−(v1) = d+(v2) = a − 2. (12) from g(1, 2) = 2a + 2, (7), x1y1 ∈ a(d) and x2y2 ∈ a(d) it follows that d(v1) + d(v2) = g(1, 2)−4 = 2a−2. this together with (12) and the fact that d∗[my,x] is strong implies that s. darbinyan 21 d+(v1) = d −(v2) = 1. this means that d +(x1) = d −(y2) = 2. therefore, d(x1) = d(y2) = a+2 by (10). now for every i ∈ [3, a] we consider the perfect matching miy,x and its corresponding digraph d∗[miy,x]. notice that s(my,x) = s(m i y,x) = a−2, the vertices y1, x2 are not adjacent and the arcs xiyi, x1y2 are not in a(d). hence, the vertices v i 1 = {y1, xi}, vi2 = {yi, x2} in d∗[miy,x] are not adjacent. from claim 1(ii) it follows that in d ∗[miy,x] the degree sum of every pair of two distinct non-adjacent vertices, other than {vi1, vi2}, is at least 2a − 1. if in d∗[miy,x], d(v i 1) + d(v i 2) ≤ 2a − 2, then by the arguments to that in the proof of d(x1) = d(y2) = a + 2, we deduce that d(xi) = d(yi) = a + 2 for all i ∈ [3, a]. therefore, for all i ∈ [3, a], 3a ≤ d(x1) + d(xi) ≤ 2a + 4. this means that a ≤ 4, i.e., a = 3 or a = 4. let a = 3. by lemma 5, it suffices to show that d contains a cycle of length 6. using (10) and (11), it is easy to check that x3y2x2y3x1y1x3 is a cycle of length 6 in d. let now a = 4. by lemma 5, we need to show that d contains cycles of lengths 6 and 8. from d(x4) = 6 and x4y4 /∈ a(d) it follows that x4y2 ∈ a(d) or x4y3 ∈ a(d). assume that x3y4 ∈ a(d). then using (10) and (11) it is not difficult to see that x3y4x2y2x1y1x3 is a cycle of length 6, and x3y4x4y2x2y3x1y1x3 (respectively, x3y4x4y3x2y2 x1y1x3) is a cycle of length 8, when x4y2 ∈ a(d) (respectively, when x4y3 ∈ a(d)). assume now that x3y4 /∈ a(d). then from x4y4 /∈ a(d) and d(y4) = 6 it follows that x1y4 ∈ a(d). now again using (10) and (11), we see that x1y4x2y2x3y1x1 is a cycle of length 6, and x1y4x4y2x2y3x3y1x1 (respectively, x1y4x4y3x2y2x3y1x1) is a cycle length 8, when x4y2 ∈ a(d) (respectively, when x4y3 ∈ a(d)). thus, we have shown that if a = 3 or a = 4, then d contains cycles of all lengths 2, 4, . . . , 2a, which contradicts our supposition that d is not even pancyclic. this completes the proof of lemma 7. we now ready to complete the proof of theorem 10. proof of theorem 10: let d be a digraph satisfying the conditions of theorem 10. by lemma 5, d contains cycles of lengths 2 and 4. by lemma 2, d contains a perfect matching from y to x. let my,x = {yixi ∈ a(d) | i = 1, 2, . . . , a} be a perfect matching from y to x in d with the maximum size among the sizes of all the perfect matching from y to x in d. by lemma 6, the digraph d∗[my,x] either contains cycles of all lengths 2, 4, . . . , 2a or is strongly connected. in the former case we are done. assume that d∗[my,x] is strongly connected. by lemma 7, d either contains cycles of all lengths 2, 4, . . . , 2a or (ii) d(u) + d(v) ≥ 2a − 1 for every pair of non-adjacent vertices u, v in d∗[my,x]. assume that the second case holds. therefore, by theorem 12, either (a) d∗[my,x] contains cycles of every length k, k ∈ [3, a] or (b) a is even and d∗[my,x] is isomorphic to one of digraphs k ∗ a/2,a/2, k ∗ a/2,a/2 \ {e} or (c) d∗[my,x] ∈ φma , where (a + 1)/2 < m ≤ a − 1. (a). in this case, by lemma 5 and lemma 3(ii), d contains cycles of every length 2k, k ∈ [1, a]. (b). d∗[my,x] is isomorphic to k ∗ a/2,a/2 or k ∗ a/2,a/2 \ {e} with partite sets {v1, v2, . . . , va/2} and {va/2+1, va/2+2, . . . , va}. notice that a ≥ 4 and d∗[my,x] contains cycles of every length 2k, k ∈ [1, a/2]. therefore, by lemma 3(ii), d contains cycles of every length 4k, k ∈ [1, a/2]. it remains to show that for any k ∈ [1, a/2 − 1], d also contains a cycle of length 4k + 2. we claim that there exist p ∈ [1, a/2] and q ∈ [a/2+1, a] such that ypxq ∈ a(d). assume that this is not the case, i.e., there is no arc from a vertex of {y1, y2, . . . , ya/2} to a vertex of {xa/2+1, xa/2+2, ldots, xa}. then, since d∗[my,x] is isomorphic to k∗a/2,a/2 or k ∗ a/2,a/2 \ {e}, 22 a theorem on even pancyclic bipartite digraphs from the definition of d∗[my,x] it follows that d +(y1) ≤ a/2, d+(ya/2) ≤ a/2, d−(y1) ≤ a/2+1 and d−(ya/2) ≤ a/2+1. combining these inequalities, we obtain that d(y1)+d(ya/2) ≤ 2a+2, which contradicts condition (a) since a ≥ 4. it suffices to consider the case when d∗[my,x] is isomorphic to k ∗ a/2,a/2 \ {e}. without loss of generality, we may assume that e = vava/2. from the definition of d ∗[my,x] it follows that {x1, x2, . . . , xa/2} → {ya/2+1, ya/2+2, . . . , ya} and d contains all possible arcs from {xa/2+1, xa/2+2, . . . , xa} to {y1, y2, . . . , ya/2} except xaya/2. if p = a/2 and q = a (i.e., ya/2xa ∈ a(d)), then y1x1ya/2+1xa/2+1y2x2ya/2+2xa/2+2 . . . ykxkya/2+k xa/2+kya/2xay1 is a cycle of length 4k + 2, where k ∈ [1, a/2 − 1]. thus, we may assume that ya/2xa /∈ a(d). then the vertices xa, ya/2 are not adjacent since xaya/2 /∈ a(d). this together with d−(ya/2, {x1, x2, . . . , xa/2}) ≤ 1 implies that d(ya/2) ≤ 3a/2 − 1. therefore, by condition (a), d(ya/2−1) ≥ 3a/2 + 1 and hence, ya/2−1xa ∈ a(d) since d−(ya/2−1, {x1, x2, . . . , xa/2}) ≤ 1. now it is not difficult to check that if a ≥ 6, then y1x1ya/2+1xa/2+1y2x2ya/2+2xa/2+2 . . . ykxkya/2+kxa/2+kya/2−1xay1 is a cycle of length 4k + 2 when k ∈ [1, a/2 − 2], and y1x1ya/2+1xa/2+1y2x2 ya/2+2xa/2+2 . . . xa/2−2ya−2xa−2ya/2xa/2 ya−1xa−1ya/2−1 xay1 is a cycle of length 2a − 2. if a = 4, then y2x2y4x4y1x3y2 is a cycle of length 6 = 2a − 2. (c). d∗[my,x] ∈ φma . since d contains cycles of lengths 2, 4 (lemma 5) and every digraph in φma is hamiltonian, we can assume that a ≥ 4. let v (d∗[my,x]) = {v1, v2, . . . , va} and vava−1 . . . v2v1va be a hamiltonian cycle in d ∗[my,x]. therefore, by the definition of d ∗[my,x], for all i ∈ [2, a], xiyi−1 ∈ a(d) and x1ya ∈ a(d). from the definition of φma we have d+(va) = 1 and d +(va−1) ≤ 2. this means that d+(xa) ≤ 2 and d+(xa−1) ≤ 3. these together with d−(xa) ≤ a, d−(xa−1) ≤ a and condition (a) implies that d(xa) ≤ a + 2, d(xa−1) ≤ a + 3 and 3a ≤ d(xa) + d(xa−1) ≤ 2a + 5. (13) the last inequality of (13) implies that a ≤ 5, i.e., a = 4 or a = 5. let a = 5. then from (13) it follows that d(xa) + d(xa−1) = 2a + 5, d −(xa) = d−(xa−1) = a, i.e., {y1, y2, . . . , ya} → {xa, xa−1}. therefore, y2x5y4x4y3x3y2 (respectively, y1x5y4x4y3x3y2x2y1) is a cycle of length 6 (respectively, of length 8). let a = 4. in this case, we need to show that d contains a cycle of length 6. if x1y3 ∈ a(d) (or y2x1 ∈ a(d)), then x1y3x3y2x2y1x1 (respectively, x1y4x4y3x3y2x1) is a cycle of length 6. we may therefore assume that x1y3 /∈ a(d) and y2x1 /∈ a(d). then d(x1) = d(x4) = 6 since d(x4) ≤ a + 2, d+(xa) ≤ 2 and d(x1) + d(x4) ≥ 12. therefore, d−(x4) = 4, which in turn implies that y1x4 ∈ a(d). hence, y1x4y3x3y2x2y1 is a cycle of length 6. thus, we have shown that if d∗[my,x] ∈ φma , then a = 4 or a = 5 and d contains cycles of all lengths 2, 4, . . . , 2a. this completes the proof of the theorem. 5. conclusion in the current article, we prove a meyniel-type condition and a bang-jensen, gutin and li-type condition for a strong balanced bipartite digraph of order 2a ≥ 6 to have cycles of all even lengths less than equal to 2a. it is worth to noting that over the past three years, various authors have received a number of sufficient conditions for the existence of cycles with certain properties in bipartite digraphs. in particular, several sufficient conditions for a balanced bipartite digraph to be s. darbinyan 23 hamiltonian or be even pancyclic were obtained (see, e.g., [23] by wang and wu, [24] by adamus, [25] by wang, [26] by wang et al.). a hamiltonian path in a digraph d in which the initial vertex dominates the terminal vertex is called a hamiltonian bypass in d. it was proved that a number of sufficient conditions for a digraph to be hamiltonian is also sufficient for a digraph to contain a hamiltonian bypass with some exceptions, which are characterized in [27], and the papers cited there. it is not difficult to show that, if a balanced bipartite digraph of order 2a ≥ 4 satisfies the conditions of theorem 2(a) (or 2(b)), then d has a hamiltonian bypass. in this regard, we believe that the the following conjecture is true. conjecture 3: d be a strong balanced bipartite digraph of order 2a ≥ 6. if d satisfies the conditions one of theorems 2, 5 and 7, then d contains a hamiltonian bypass, with some exceptions. to conclude this section, we mention that wang et al. [28] constructed an infinite family of counterexamples to conjecture 2. note that each of these counterexamples contains a vertex, which has degree equal to three. thus, conjecture 2 remains open for digraphs with the minimum degree is at least four and for k-strong digraphs, where k ≥ 2. references [1] j. bang-jensen and g. gutin, digraphs: theory, algorithms and applications, springer, 2000. [2] j. adamus, “a degree sum condition for hamiltonicity in balanced bipartite digraphs”, graphs and combinatorics, vol. 33, no. 1, pp. 43-51, 2017. [3] j. adamus, l. adamus and a. yeo, “on the meyniel condition for hamiltonicity in bipartite digraphs”, discrete mathematics and theoretical computer science, vol. 16 no. 1, pp. 293-302, 2014. [4] j. bang-jensen, y. guo and a. yeo, “a new sufficient condition for a digraph to be hamiltonian”, discrete applied mathematics, vol. 95, pp. 61-72, 1999. [5] j. bang-jensen, g. gutin and h. li, “sufficient conditions for a digraph to be hamiltonian”, journal of graph theory, vol. 22, no. 2, pp. 181-187, 1996. [6] j.-c. bermond and c. thomassen, “cycles in digraphs a survey”, journal of graph theory, vol. 5, no. 1, pp. 1-43, 1981. [7] a. ghouila-houri, “une condition suffisante d’existence d’un circuit hamiltonien”, comptes rendus de i’ academie des science paris ser. a-b, vol. 25, pp. 495-497, 1960. [8] g. gutin, “cycles and paths in semicomplete multipartite digraphs, theorems and algorithms: a survey”, journal of graph theory, vol. 19, no. 4, pp. 481-505, 1995. [9] d. kühn and d. osthus, “a survey on hamilton cycles in directed graphs”, european journal of combinatorics, vol. 33, no. 5, pp. 750-766, 2012. [10] m. meyniel, “une condition suffisante d’existence d’un circuit hamiltonien dans un graphe oriente”, journal combinatorial theory ser. b, vol. 14, pp. 137-147, 1973. 24 a theorem on even pancyclic bipartite digraphs [11] s.kh. darbinyan, “pancyclicity of digraphs with the meyniel condition”, studia scientiarum mathematicarum hungarica, vol. 20, no. 1-4, pp. 97-117, 1985 (ph.d. thesis, institute mathematici akad. nauk bssr, minsk, 1981) (in russian). [12] s.kh. darbinyan, “on the pancyclicity of digraphs with large semidegrees”, akademy nauk armyan ssr dokllady, vol. 83, no. 3, pp. 99-101, 1986 (arxiv: 1111.1841v1). [13] r. häggkvist and c. thomassen, “on pancyclic digraphs”, journal combinatorial theory ser. b, vol. 20, no. 1, pp. 20-40, 1976. [14] c. thomassen, “an ore-type condition implying a digraph to be pancyclic”, discrete mathematics, vol. 19, pp. 85-92, 1977. [15] m. meszka, “new sufficient conditions for bipancyclicity of balanced bipartite digraphs”, discrete mathematics, vol. 341, no. 11, pp. 3237-3240, 2018. [16] r. wang, “a sufficient condition for a balanced bipartite digraph to be hamiltonian”, discrete mathematics and theoretical computer science, vol. 19, no. 3, no. 11, 2017. [17] s.kh. darbinyan, “sufficient conditions for hamiltonian cycles in bipartite digraphs”, discrete applied mathematics, vol. 258, pp. 87-96, 2019. [18] s.kh. darbinyan, “sufficient conditions for a balanced bipartite digraph to be even pancyclic”, discrete applied mathematics, vol. 238, pp. 70-76, 2018. [19] j. adamus, “a meyniel-type condition for bipancyclicity in balanced bipartite digraphs”, graphs and combinatorics, vol. 34, no. 4, pp. 703-709, 2018. [20] s.kh. darbinyan and i.a. karapetyan, “a sufficient condition for pre-hamiltonian cycles in bipartite digraphs”, ”2017 computer science and information technologies (csit)”, yerevan, doi:10.1109/csitechnol. 2017.8312150, pp. 101-109, 2017. [21] s.kh. darbinyan, “on pancyclic digraphs”, preprint of the computing center of akademy nauk armyan. ssr, 21 pp.,1979. [22] a. benhocine, “pancyclism and meyniel’s conditions”, discrete mathematics, vol. 58, pp. 113-120, 1986. [23] r. wang and l. wu, “a dominating pair condition for a balanced bipartite digraph to be hamiltonian” , australas. j. combin. vol. 77, no. 1, pp. 136-143, 2020. [24] j. adamus, “on dominating pair degree conditions for hamiltonicity in balanced bipartite digraphs”, discrete mathematics, vol. 344, no.3, article 112240, 2021. [25] r. wang, “extremal digraphs on woodall-type condition for hamiltonian cycles in balanced bipartite digraphs”, journal of graph theory, https://doi.org/10.1002/jgt.22649, 25 november, 2020. [26] r. wang, l. wu and w. meng, “extremal digraphs on meyniel-type condition for hamiltonian cycles in balanced bipartite digraphs”, arxiv:1910.05542v1. [27] s.kh. darbinyan, “on hamiltonian bypasses with the condition of y. manoussakis”, ”2015 computer science and information technologies (csit)” yerevan, doi:10.1109/csittechnol.2015.7358250 pp. 53-63, 2015. [28] r. wang, j. chang and l. wu, “a dominated pair condition for a digraph to be hamiltonian”, discrete mathematics, vol. 343, no. 5, 111794, 2020. submitted 03.12.2020, accepted 04.03.2021. s. darbinyan 2 5 â»áñ»ù »ñïù³ëýû³ ïáõùýáñáßí³í ·ñ³ýý»ñç ½áõû· ñ³ù³óçïéçïáõãû³ý ù³ëçý ê³ùí»é ê. ¸³ñµçýû³ý ðð ¶²² æýýáñù³ïçï³ûç ¨ ³íïáù³ï³óù³ý åñáµé»ùý»ñç çýëïçïáõï e-mail: samdarbin@iiap.sci.am ²ù÷á÷áõù ü»ñï³ ³ßë³ï³ýùáõù ³å³óáõóí»é ¿ ñ»ï¨û³é ã»áñ»ùá: â»áñ»ù: ¸çóáõù d-ý áõå»õ ï³å³ïóí³í 2 a ¸ 6 -·³·³ã³ýç »ñïù³ëýû³ ñ³í³ ë³ñ³ïßéí³í ïáõùýáñáßí³í ·ñ³ý ¿, çëï fx; yg-á d ·ñ³ýç ·³·³ãý»ñç ó³ýï³ó³í å³ñïíáõ ï³ù ñ³õãáõ ½áõû· ¿: ºã» ï»õç áõýç d ( x ) + d( y ) ¸ 3 a å³ûù³ýá, ³å³ d -ý å³ñáõý³ïáõù 2 a-çó ÷áùñ ï³ù ñ³í³ë³ñ ó³ýï³ó³í ½áõû· »ñï³ñáõãû³ý óçïé, µ³óç ³ûý ¹»åùçó, »ñµ d-ý 2 a »ñï³ñáõãû³ý óçïé ¿: òåîðåìà î ÷åòíûõ ïàíöèêëè÷åñêèõ äâóäîëüíûõ îðãðàôàõ ñàìâåë õ. äàðáèíÿí èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà e-mail: samdarbin@iiap.sci.am àííîòàöèÿ â íàñòîÿùåé ðàáîòå äîêàçàíà ñëåäóþùàÿ òåîðåìà: òåîðåìà: ïóñòü d åñòü ñèëüíî ñâÿçíûé 2 a ¸ 6 âåðøèííûé áàëàíñèðîâàííûé äâóäîëüíûé îðãðàô. ïðåäïîëîæèì, ÷òî äëÿ êàæäîé äîìèíèðóþùåé è êàæäîé äîìèíèðóìîé ïàðû fx; yg ðàçëè÷íûõ âåðøèí èìååò ìåñòî d( x ) + d( y ) ¸ 3 a. òîãäà d ñîäåðæèò êîíòóð ëþáîé ÷åòíîé äëèíû 2 k, 0 · k · a , êðîìå ñëó÷àÿ êîêäà d êèé îðãðàô, ÷åòíûé ïàíöèêëè÷åñêèé îðãðàô. ´³ý³éç µ³é»ñ` îáõùýáñáßí³í ·ñ³ý, ñ³ùçéïáýû³ý óçïé, »ñïù³ëýû³ ïáõùýáñáßí³í ·ñ³ý, ñ³ù³óçïéçï, ½áõû· ñ³ù³óçïéçï: ÿâëÿåòñÿ êîíòóðîì äëèíû 2 a. êëþ÷åâûå ñëîâà: îðãðàô, ãàìèëüòîíîâ öèêë, äâóäîëüíèé îðãðàô, ïàíöèêëè÷åñ 01_darbinyan_55_mpcs_55.pdf (p.1-16) 01.pdf (p.17) начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 44, 154--160, 2015. modeling of irregularly shaped two – dimensional objects arthur j. keyan institute for informatics and automation problems of nas ra e-mail: artur.keyan@gmail.com abstract in this paper the description of the two dimensional modeling program of irregularly shaped objects is given. the program is very easy to use, it works fast and has many other advantages. keywords: optimal cutting of materials, two dimensional modeling, packing problem. 1. introduction the problem of two-dimensional modeling is the first step of achieving the optimal arrangement of any irregular shape details on the surface. the applications of this optimization problem are related to the real life issues in the field of optimal cutting from any limited material, for example, wood, leather, metal or paper [1-4]. actually there are some existing softs that can solve this issue [5], but they do not work with irregular shapes. some of them are working with rectangles only [6,7], the others are private or very expensive. the description of the objects usually begins with the design of digital model. for this purpose a software environment is implemented in which the two dimensional digital models are created. the interface of this tool is similar to already existing three-dimensional editors like autocad, 3ds max, etc. but today’s existing tools are designed for three-dimension objects therefore, they contain a very huge set of unnecessary tools and are hard to use. meanwhile the approach of creating a two-dimensional editor will simplify and speed up the process multiple times, and will need less productive capacity. the main principle of making an irregular shape object is manipulation with simple geometrical shapes, because every complex shape consists of more or less amount of simple shapes. these shapes can be designed manually, by mouse and with coordinates. the paper is organized as follows. the description of simple geometrical shapes modeling is given in the second section. how to manipulate with the existing shapes is given in section three. making of any shapes from simple ones is described in section four. 154 a. keyan 155 2. modeling of simple geometrical shapes in our program there are corresponding tools for creating simple shapes, the main shapes supported by the soft are: 1. rectangle 2. ellipse 3. polyline they are automatically including the square, circle and other shapes. for creating complex shapes from simple ones they have to be transformed, rotated and changed in dimensions. when soft works only with these simple shapes, it needs only to remove the existed shape and create some new shape with new given properties. but when it needs to work with complex shapes the program has to remember all the simple shapes that the complex shape is made of, and after transformation recreate all simple shapes, then transform them with the given way and then again create a complex shape from them. each of the above-mentioned elements has specified tools. it is made for even more flexibility of the soft, and the modeling of the shapes can be done without any limitations and deep knowledge of modeling. here is the list of the tools and their working modes: 1. rectangle i) to give the diagonal points: it can be done by mouse or by giving x and y coordinates of that points in the specified fields, or with combined way, by giving the x point by mouse and y by coordinates. ii) to give a one diagonal point with width and length of the rectangle: it can be done by the same three ways. 2. ellipse i) to give the diagonal points of the circumscribed rectangle: this is very similar to the first mode of the rectangle. ii) in the special case to give the circle: it can be done by giving the center and r radius, it also can be done by multiple ways, by mouse and by coordinates. 3. polyline i) to give the coordinates of the corner points: after giving the last point the soft will automatically close the chain by linking the first and the last points. 4. contour i) it is the special case of the polyline with very small distances, it has been included for easier work with irregular lines and curves: it is convenient to give by mouse, it continuously takes the points and creates a very deep cornered polyline. if it is not necessary to achieve a very high level of accuracy, this tool can be used (see image 1). 5. point i) it is not an independent object, but it will help to use all the above described tools. modeling of irregularly shaped two – dimentional objects 156 fig. 1. object created by the program. 3. manipulating with existing shapes the movement of the objects is realized by the geometric center. the coordinates of that point can be calculated with the help of the circumscribed rectangle: 𝑥𝑥 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇. 𝑥𝑥 + 𝑊𝑊𝑊𝑊𝑊𝑊𝑇𝑇ℎ/2, (1) 𝑦𝑦 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇. 𝑦𝑦 + 𝐻𝐻𝑇𝑇𝑊𝑊𝐻𝐻𝑇𝑇ℎ/2, where topleft.x and topleft.y are x and y coordinates of the top left corner point of the rectangle. after movement the coordinates of the new point will be 𝑥𝑥′ = 𝑥𝑥 + 𝑊𝑊𝑥𝑥, (2) 𝑦𝑦′ = 𝑦𝑦 + 𝑊𝑊𝑦𝑦, where 𝑥𝑥′ and 𝑦𝑦′ are new coordinates of the point and 𝑊𝑊𝑥𝑥 and 𝑊𝑊𝑦𝑦 are the movement distance. the rotation of the objects can be realized by axis or by the center point or by any given point. the rotation angle can be given either by the mouse wheel or by the given quantity. the new coordinates are calculated by the following formulas: 𝑥𝑥′ = 𝑥𝑥 + tan(𝐴𝐴) − (𝑦𝑦 − 𝑐𝑐𝑐𝑐𝑥𝑥), (3) 𝑦𝑦′ = 𝑦𝑦 + tan(𝐴𝐴) − �𝑥𝑥 − 𝑐𝑐𝑐𝑐𝑦𝑦�, where a is the rotation angle, 𝑐𝑐𝑐𝑐𝑥𝑥 and 𝑐𝑐𝑐𝑐𝑦𝑦 are 𝑥𝑥 and 𝑦𝑦 coordinates of the central point. it is possible also to make size modifications. for that again the coordinates of the central point are needed, the transformation will be done by the following formulas: 𝑥𝑥′ = 𝑠𝑠𝑥𝑥 ∗ 𝑥𝑥 + (𝑐𝑐𝑠𝑠𝑥𝑥 − 𝑠𝑠𝑥𝑥 ∗ 𝑐𝑐𝑠𝑠𝑥𝑥), (4) 𝑦𝑦′ = 𝑠𝑠𝑦𝑦 ∗ 𝑦𝑦 + �𝑐𝑐𝑠𝑠𝑥𝑥 − 𝑠𝑠𝑦𝑦 ∗ 𝑐𝑐𝑠𝑠𝑦𝑦�, where 𝑠𝑠𝑥𝑥 and 𝑠𝑠𝑦𝑦 are horizontal and vertical zoom coefficients and 𝑐𝑐𝑠𝑠𝑥𝑥 and 𝑐𝑐𝑠𝑠𝑦𝑦 are coordinates of central point. a. keyan 157 4. making complex shapes from simple ones of course, the contur tool can be used for creating irregular shapes but in the more presice way it can be made from simple shapes. there are 4 kinds of combination tools (see fig. 2): 1. union creates an object that contains all the points of component objects, 2. intersect creates an object, that contains only the common points of component objects, 3. xor creates anobject, that contains points, that belong only to one of objects, 4. exclude creates an object that contains all the point that belong to the first object but not to the second one. fig. 2. all tools in action. the last two tools work only for two objects, but for multiple objects the recursive approach can be used. to select an object to be manipulated, there are three conditions: 1. the selection area will be completely in the object (see fig. 3). 2. the object will be completely in the selection area (see fig. 4). 3. the selected area must be crossed with the object (see fig. 5). modeling of irregularly shaped two – dimentional objects 158 fig. 3 fig. 4 fig. 5 an example of selection of multiple objects is given in fig. 6. fig. 6. example of selection of multiple objects. a. keyan 159 5. conclusion we have implemented a program which is modeling two dimensional irregularly shaped objects. it includes tools for creating simple geometrical objects, for manipulating with them and for creating complex shapes. references [1] v. a. skaternoy, optimization of material cutting in light industry, 1989. [2] g. belov, problems, models and algorithms in oneand two-dimensional cutting, dresden, 2004. [3] a. lodi, s. martello, m. monaci, "two-dimensional packing problems: a survey". european journal of operational research (elsevier), vol. 141, pp. 241–252, 2002. [4] h. dyckhoff, “a typology of cutting and packing problems”, european journal of operational research, vol. 44, pp. 145-159, 1990. [5] f. v. babaev, optimal cutting of materials by computer, moscow, 1982. [6] b. e. bengtsson, “packing rectangular pieces. a heuristic approach”, the computer journal, vol. 25, no. 3, pp. 353-357, 1982. [7] p. wang, l. valenzeva, “data set generation for rectangular placement problems”, european journal of operational research, vol. 134, no. 2, pp.378-391, 2001. submitted 28.07.2015, accepted 27.11.2015 կամայական ուրվագծով երկչափ օբյեկտների մոդելավորում ա. քեյան ամփոփում աշխատանքում նկարագրված է կամայական ուրվագծով օբյեկտների երկչափ մոդելավորման մշակված ծրագիրը: ծրագիրը շատ հեշտ է օգտագործել, այն աշխատում է արագ և ունի մի շարք այլ առավելություններ: այն ներառում է գործիքներ պարզ երկրաչափական օբյեկտների ստեղծման համար, դրանց հետ տարբեր գործողություններ կատարելու համար, ինչպես նաև բարդ ուրվագծեր ստեղծելու համար: modeling of irregularly shaped two – dimentional objects 160 моделирование двумерных объектов произвольного очертания а. кеян аннотация в данной статье приведено описание программы двух мерного моделирования объектов неправильной формы. программа очень проста в использовании, работает быстро и имеет много других преимуществ. она включает в себя инструменты для создания простых геометрических объектов, для манипуляции с ними и для создания сложных форм. 2. modeling of simple geometrical shapes mathematical problems of computer science 55, 35–43, 2021. udc 519.72 the role of information theory in the field of big data privacy mariam e. haroutunian1 and karen a. mastoyan2 1institute for informatics and automation problems of nas ra 2gavar state university e-mail: armar@sci.am, kmastoyan@yandex.com abstract protecting privacy in big data is a rapidly growing research area. the first approach towards privacy assurance was the anonymity method. however, recent research indicated that simply anonymized data sets can be easily attacked. later, differential privacy was proposed, which proved to be the most promising approach. the trade-off between privacy and the usefulness of published data, as well as other problems, such as the availability of metrics to compare different ways of achieving anonymity, are in the realm of information theory. although a number of review articles are available in literature, the information theoretic methods capacities haven’t been paid due attention. in the current article an overview of state-of-the-art methods from information theory to ensure privacy are provided. keywords: big data, anonymization, differential privacy, entropy, mutual information, distortion 1. introduction in recent years, big data has become a hot research topic, because it helps businesses and organizations to improve the decision making power and provides new opportunities with data analysis. big data life cycle can be divided into the following stages: data generation, storage and processing. multiple parties are involved in these stages, hence, the privacy violation risks are increased. a number of privacy preserving mechanisms have been developed [1], [2], however, the study on big data privacy issues are at a very early stage [3]. modern technologies and tools, such as social networks, search engines, hacking packages, data mining and machine learning tools, cause a lot of problems to individual privacy. in general, it is very hard to find a clear definition or a global measurement on privacy. the studies on privacy can be separated into two classes: content privacy and interaction privacy. so far, the majority of research on privacy protection is conducted in the context 35 36 the role of information theory in the field of big data privacy of databases. the goal of a privacy preserving statistical database is to enable the user to learn properties of the population while securing the personal information. the practically dominant privacy protection strategy is the use of cryptography. one way to protect data is to encrypt it in such a way that only the owner can decrypt it. the task of machine learning is to find the dependency in the data. an idea proceeds: why not to train the model on encrypted data? the problem with this approach is that when we encrypt data, the dependencies in it are lost, because this is the aim of encryption to change the data so that the dependencies cannot be discovered. the degree of entropy in the data after encryption prevents models from capturing these dependencies. therefore, encrypting data and training models on them do not work. the other disadvantage of cryptography for privacy is the limited computing power of mobile devices for safe encryption and decryption algorithms. that is why other methods for privacy preserving are required. the main research categories of privacy are the data clustering and the theoretical frameworks. anonymization is a key component of data clustering. anonymization is the process of removing personal identifiers, both direct and indirect, that can lead to a person’s identification. a person can be directly identified by name, address, zip code, telephone number, photograph or image, or other unique personal characteristics. a person can be indirectly identified if certain information is linked to other sources of information, including workplace, job title, salary, zip code, or even the fact of having a specific diagnosis or condition. data cannot be completely anonymous and useful. generally speaking, the richer the data, the more interesting and useful it is. this has led to the concepts of anonymization and removal of personally identifiable information, by which it is hoped that sensitive parts of the data can be suppressed to maintain the confidentiality of the records, while the rest can be published and used for analysis. the early approach in data clustering direction is the k-anonymity method (1998), then its extension as l-diversity was suggested in 2007 and later the t-closeness method was developed in 2010. in the second category of privacy frameworks the differential privacy (dp) and its developments are included (fig. 1). dp neutralizes linkage attacks, since it is a property of the data access mechanism and is not related to the presence or absence of auxiliary information available to the attacker. fig. 1. the categories of privacy study [3]. when comparing syntactic and dp approaches, one can recall the trade-off between privacy and data utility. finally, databases are passed on to provide certain benefits (for example, research knowledge, such as the effectiveness of a medical procedure). on the one m. haroutunian and k. mastoyan 37 hand, in order to maximize the usefulness of the data, all data can be published untouched, thus completely breaching privacy. on the other hand, not publishing any of the data will lead to maximum privacy, but this empty database will be useless. therefore, the organization should seriously consider both its interests in data exchange and the risks they are willing to accept. an organization seeking to exchange sensitive data should consider the logistics issues involved in the exchange process and carefully balance the confidentiality and usefulness of the published data. an important step in one of these considerations is the choice of syntactic and dp. such problems can be solved by information theoretic methods and tools. although, there are some survey/review type papers introducing the concept of privacy protection in big data, none of them provide detailed discussion regarding privacy with respect to information theory. in the current paper we introduce a summary of up-to-date methods on privacy assurance from information theory standpoints. the rest of the paper is structured as follows. the dp is discussed in section 2. the developments of privacy based on information theoretic methods are presented in section 3. the paper is summarized in section 4. 2. differential privacy the dp framework was suggested in 2006 [4], that offers privacy protection in the sense of information theory. in recent years this topic has attracted attention and has been researched in literature. main results, among many others, are surveyed in [5] [8]. dp examines impossibility paradox of obtaining any information about a specific person by studying useful information about a multitude of people. suppose the trusted party contains a set of sensitive personal data (eg email usage data, movie watching data, medical records) and wants to provide global statistical information about it. such a system is called a statistical database. by providing such aggregated statistical information about the data, it is possible to disclose some information about individuals. dp ensures that data about individuals from such a database cannot be retrieved, no matter what additional datasets or sources of information are available to the attacker. such a guarantee is achieved due to the fact that the owner of the database uses such a mechanism (algorithm) for providing data, in which the presence or absence of information about a person in the database will not significantly affect the result of the request to it. dp is designed to maximize the accuracy of queries from statistical databases while minimizing the possibility of disclosing the anonymity of records. the problem of analyzing sensitive data has a long history spanning many areas. as data about people become more and more detailed and technology allows more and more of this data to be collected and analyzed, there is an increasing need for a reliable, mathematically rigorous definition of privacy, as well as a class of algorithms that satisfy this definition. various approaches to anonymizing data have failed when researchers have been able to identify personal information by combining two or more separate statistical databases. dp is the basis for the formalization of confidentiality in statistical databases and was introduced in order to protect against such methods of disclosing anonymity (deanonymization). dp allows users to protect and maintain privacy when their data is in a specific database, just as they would be safe, if the data were not in some database. after the publication of human data in the database, in accordance with the differentiated confidentiality, the likelihood of violation of the confidentiality of people should not increase. that is, the 38 the role of information theory in the field of big data privacy degree of secrecy can be assessed by the likelihood of damage. this is one of the practical definitions of privacy. dp can provide extremely strong guarantees of user privacy, but it does not guarantee unconditional relief from all damages. and it doesn’t provide privacy where it didn’t exist before. in general, dp does not guarantee that what a person considers his secret will remain secret. it simply ensures that participation in the survey is not disclosed by itself, that participation does not reveal any of the characteristics included in the survey. it is possible that the results of the survey may reflect statistical data about a person. health screening for early signs of illness can produce strong, even convincing results. the fact that these findings are valid for humans does not imply a breach of confidentiality. the person may not even participate in the survey (dp ensures that these results are equally likely, regardless of whether the person participated in the survey or not). it is desirable that dp be endowed with the following qualities: protection against arbitrary risks, automatic neutralization of linkage attacks, quantification of privacy loss. dp is based on introducing randomness into data. to realize this, there are different mechanisms, e.g. the laplace mechanism, the exponential mechanism, mechanisms via α -nets, etc. due to the fact that differential privacy is a probabilistic concept, any of its methods necessarily has a random component. some of them, like laplace’s method, use the addition of controlled noise to the function to be calculated. laplace’s method adds laplace noise, i.e. the noise from the laplace distribution. dp works by adding statistical noise to data (or its inputs or outputs). depending on the location of the noise, dp is classified into two types: local dp and global dp (fig. 2). the most commonly used threat model in differential privacy is the global dp model. the main component is a trusted data curator. each source sends him his confidential data, and it collects them in one place (for example, on a server). a repository is trusted if we assume that it processes our sensitive data on its own, does not transfer it to anyone, and cannot be compromised by anyone. in other words, we believe that a server with sensitive data cannot be hacked. within the central model, we usually add noise to query responses. the advantage of this model is the ability to add the lowest possible noise value, thus maintaining the maximum accuracy allowed by the principles of dp. the disadvantage of the central model is that it requires a trusted store, and many of them are not. in fact, the lack of trust in the consumer of the data is usually the main reason for using dp principles. the local dp model allows you to get rid of the trusted data store: each data source (or data owner) adds noise to their data before transferring it to the store. this means that the storage will never contain sensitive information, implying there is no need for its power of attorney. the local model of dp avoids the main problem of the central model: if the data warehouse is compromised, then hackers will only have access to noisy data that already meets the requirements of dp. the local model is less accurate than the central one. in the local model, each source independently adds noise to satisfy its own differential privacy conditions, so that the total noise from all participants is much greater than the noise in the central model. ultimately, this approach is only justified for queries with a very persistent trend (signal). apple, for example, uses a local model to estimate the popularity of emoji, but the result is only useful for the most popular emoji (where the trend is most pronounced). typically, this model is not used for more complex queries, such as those used by the us census bureau or machine learning. the central and local models have both advantages and disadvantages, and now the main effort is to get the best of them. m. haroutunian and k. mastoyan 39 fig. 2 global differential privacy and local differential privacy 3. review of information theory based results in privacy the first problem is to have a metric to compare various ways of achieving anonymity. the initial focus on analyzing the anonymity of messaging through mixed based anonymity systems in which all network communication is available for the attacker is given in [9]. an information theoretic metric based on the idea of anonymity probability distributions is introduced. in the same paper it is demonstrated that if maximum route length in the mix system exists, it is known to the attacker and can be used to extract additional information. it means that more advanced probabilistic metrics of anonymity are needed. an analytical measure of anonymity of routs in eavesdropped networks is proposed in [10] using the information-theoretic equivocation. cryptographic techniques prevent analysis of packet content, however, information can be gained by analyzing the correlation of transmission schedules of multiple nodes, as the packet timing information is easy to obtain in wireless networks. for anonymity it is necessary for the routes to be undetectable using the correlation across the transmission schedules, which results in a tradeoff between anonymity and network performance. for this purpose a quantifiable metric is defined in [10] using the uncertainty in networking information. the key result shows the equivalence between anonymity performance tradeoff and information theoretic rate distortion. it is often important to allow researchers to analyze data without compromising the privacy of individuals or leaking confidential information outside the organization. in [11] it is shown that sparse regression for high dimensional data can be carried out directly on a compressed form of the data, in a manner to guard privacy in information theoretic sense. the suggested compression reduces the number of data records exponentially preserving the number of input variables. these compressed data can then be made available for 40 the role of information theory in the field of big data privacy statistical analyses with the same accuracy as the original data. in this case the original data are not recoverable from the compressed data, and the algorithms run faster, requiring fewer resources. the privacy (the problem of recovering the uncompressed data from the compressed one) is evaluated in information theoretic terms by bounding the average mutual information, which is connected with the problem of computing the channel capacity of certain systems. a new privacy measure in terms of information theory, similar to t-closeness is defined in [12], which can be achieved by the postrandomization method in the descrete case and by noise addition in the general case. the privacy criterion here is an average measure over a divergence, and the privacy distortion problem is strongly related to the rate distortion problem in the field of information theory, namely, the problem of lossy compression of source data subject to a distortion criterion. a new measure for privacy of votes is proposed in [13], that relies on the notion of entropy. entropy is a natural choice to measure privacy in an information theoretic setting, and authors demonstrate how different formulations of conditional entropy answer different questions about vote privacy. a theorem has been established that enables accurate analysis of privacy offered by complex cryptographic voting protocols. connections between two existing privacy notions for votes have been established. the study of dp from a rate distortion perspective has been initiated in [14]. rate distortion is applicable when the goal of the data collector is to publish an approximation of the data itself. the case when the data collector is not trusted is considered, which leads to using the local dp as a privacy measure. a robust rate-distortion setting is considered, in which the source distribution is unknown, but comes from some class. the goal is to look for a locally differentially private channel, that achieves minimum privacy risks while guaranteeing distortion of the given level. in [15] the relation between three different notions of privacy: identifiability, differential privacy and mutual information privacy is investigated. identifiability guarantees indistinguishability between probabilities, dp guarantees limited additional disclosures, and mutual information is the information theoretic notion. under a unified privacy distortion framework, where the distortion is the hamming distance between the input and output databases, some connections between these three privacy notions have been established. guaranteeing a tight bound on privacy risk often incurs a significant penalty in terms of the usefulness of the published result. this privacy-utility tradeoff is studied in [16] in the context of publishing a differentially private approximation of the full data set and measure utility via a distortion measure. 4. conclusion in this article a general outlook on the current methods for estimating privacy of databases from information theory perspectives is provided. a series of publications devoted to various problems of privacy solved by information theoretic tools and methods is analyzed. research has shown that information-theoretic methods are effective for a wide range of tasks ranging from anonymity to differential privacy. m. haroutunian and k. mastoyan 41 references [1] a. mehmood, i. natgunanathan, y. xiang, g. hua and s. guo, “protection of big data privacy”, ieee access, vol. 4, pp. 1821–1834, 2016, doi: 10.1109/access.2016.2558446. 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[12] d. rebollo-monedero, j. forne, and j. domingo-ferrer, “from t-closeness-like privacy to postrandomization via information theory”, ieee trans. on knowl. and data eng., vol. 22, no. 11, pp. 1623-1636, 2010. doi:https://doi.org/10.1109/tkde.2009.190 [13] d. bernhard, v. cortier, o. pereira, and b. warinschi, “measuring vote privacy, revisited”, proc. of acm conf. on computer and communications security, association for computing machinery, new york, ny, usa, pp. 941952, 2012. doi:https://doi.org/10.1145/2382196.2382295 4 2 the role of information theory in the field of big data privacy [1 4 ] a . s a r wa t e a n d l . s a n ka r , " a r a t e -d is t o r t io n p e r s p e c t ive o n lo c a l d i®e r e n t ia l p r iva c y" , 52 annual allerton conf., uiuc, illin o is , u s a , p p . 9 0 3 { 9 0 8 , 2 0 1 4 . [1 5 ] w . w a n g , l . y in g a n d j. zh a n g , " on t h e r e la t io n b e t we e n id e n t i¯ a b ilit y, d i®e r e n t ia l p r iva c y, a n d m u t u a l-in fo r m a t io n p r iva c y," ie e e trans. on information theory, vo l. 6 2 , n o . 9 , p p . 5 0 1 8 { 5 0 2 9 , s e p t . 2 0 1 6 . d o i: 1 0 .1 1 0 9 / tit.2 0 1 6 .2 5 8 4 6 1 0 . [1 6 ] k . k a la n t a r i, l . s a n ka r a n d a . d . s a r wa t e , " op t im a l d i®e r e n t ia l p r iva c y m e c h a n is m s u n d e r h a m m in g d is t o r t io n fo r s t r u c t u r e d s o u r c e c la s s e s ," ie e e intern. symp. on information theory, b a r c e lo n a , s p a in , p p . 2 0 6 9 { 2 0 7 3 , 2 0 1 6 . d o i: 1 0 .1 1 0 9 / is it.2 0 1 6 .7 5 4 1 6 6 3 . submitted 20.02.2021, accepted 18.04.2021. æýýáñù³óç³ûç ï»ëáõãû³ý ¹»ñá ù»í ïíû³éý»ñç ·³õïýçáõãû³ý áéáñïáõù ø³ñç³ù º. ð³ñáõãûáõýû³ý1 ¨ î³ñ»ý ². ø³ëïáû³ý2 1ðð ¶²² æýýáñù³ïçï³ûç ¨ ³íïáù³ï³óù³ý åñáµé»ùý»ñç çýëïçïáõï 2¶³í³éç å»ï³ï³ý ñ³ù³éë³ñ³ý e-mail: earmar@sci.am, kmastoyan@yandex.com ²ù÷á÷áõù ø»í ïíû³éý»ñç ñ»ï ³ßë³ï»éçë ·³õïýçáõãû³ý å³ñå³ýáõùá ñ»ï³½áïáõãû³ý ³ñ³· ³×áõ áéáñï ¿: ¶³õïýçáõãû³ý ³é³ççý ùáï»óáõùá ³ý³ýáõýáõãû³ý ù»ãá¹ý ¿ñ: ì»ñççý áõëáõùý³ëçñáõãûáõýý»ñá óáõûó »ý ïí»é, áñ å³ñ½³å»ë ³ý³ýáõý ïíû³éý»ñç ßï»ù³ñ³ýý»ñá ï³ñáõ »ý ñ»ßïáõãû³ùµ »ýã³ñïí»é ñ³ñó³ïù³ý ·³õïýçáõãû³ý ï»ë³ýïûáõýçó: ð»ï³·³ûáõù ³é³ç³ñïí»ó ¹çý»ñ»ýóç³é ·³õïýçáõãûáõýá, áñý ³å³óáõó»ó, áñ ³ù»ý³ñ»é³ýï³ñ³ûçýý ¿: ¶³õïýçáõãû³ý ¨ ññ³å³ñ³ïí³í ïíû³éý»ñç û·ï³ï³ñáõãû³ý ùçç¨ ÷áë½ççáõùá, çýãå»ë ý³¨ ³ûé ñ³ñó»ñç, çýãåçëçù »ý ³ý³ýáõýáõãû³ý ñ³ëý»éáõ ï³ñµ»ñ »õ³ý³ïý»ñá ñ³ù»ù³ï»éáõ ñ³ù³ñ ã³÷ç ³éï³ûáõãûáõýá, áýïýáõù ¿ çýýáñù³óç³ûç ï»ëáõãû³ý ïçñáõûãáõù: âý³û³í ·ñ³ï³ýáõãû³ý ù»ç ùç ß³ñù ³ïý³ñï³ûçý ñá¹í³íý»ñç ³éï³ûáõãû³ýá, çýýáñù³óç³ûç ï»ëáõãû³ý ù»ãá¹ý»ñç ñý³ñ³íáñáõãûáõýý»ñá å³ïß³× áõß³¹ñáõãû³ý ã»ý ³ñå³ý³ó»é: ²ûë ñá¹í³íáõù ù»ýù ý»ñï³û³óýáõù »ýù çýýáñù³óç³ûç ï»ëáõãû³ý å³ù³ý³ï³ïçó ù»ãá¹ý»ñç ³ïý³ñï` ·³õïýçáõãûáõýý ³å³ñáí»éáõ ñ³ù³ñ: ì»ñéáõííáõù »ý çýýáñù³óç³ûç ï»ëáõãû³ý ·áñíçùý»ñç ¨ ù»ãá¹ý»ñç ùççáóáí éáõíí³í ·³õïýçáõãû³ý ï³ñ³µýáõûã ëý¹çñý»ñçý ýíçñí³í ùç ß³ñù ññ³å³ñ³ïáõùý»ñ: ð»ï³½áïáõãûáõýý»ñá óáõûó »ý ïí»é, áñ çýýáñù³óçáý ï»ë³ï³ý ù»ãá¹ý»ñý ³ñ¹ûáõý³í»ï »ý ëý¹çñý»ñç é³ûý ßñç³ý³ïç ñ³ù³ñª ³ý³ýáõýáõãûáõýçó ùçýã¨ ¹çý»ñ»ýóç³é ·³õïýçáõãûáõý: ´³ý³éç µ³é»ñ` ø»í ïíû³éý»ñ, ³ý³ýáõý³óáõù, ¹çý»ñ»ýóç³é ·³õïýçáõãûáõý, ¾ýïñáåç³, ÷áë³¹³ñó çýýáñù³óç³, ß»õáõù: m. haroutunian and k. mastoyan 4 3 ðîëü òåîðèè èíôîðìàöèè â îáëàñòè êîíôèäåíöèàëüíîñòè áîëüøèõ äàííûõ ìàðèàì à. àðóòóíÿí1 è êàðåí à, ìàñòîÿí2 1èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà 2ãàâàðñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò e-mail: armar@sci.am, kmastoyan@yandex.com àííîòàöèÿ çàùèòà êîíôèäåíöèàëüíîñòè ïðè ðàáîòå ñ áîëüøèìè äàííûìè áûñòðîðàñòóùàÿ îáëàñòü èññëåäîâàíèé. ïåðâûì ïîäõîäîì ê êîíôèäåíöèàëüíîñòè áûë ìåòîä àíîíèìíîñòè. íåäàâíèå èññëåäîâàíèÿ ïîêàçàëè, ÷òî ïðîñòî àíîíèìíûå íàáîðû äàííûõ ìîãóò áûòü ëåãêî àòàêîâàíû ñ òî÷êè çðåíèÿ êîíôèäåíöèàëüíîñòè. ïîçæå áûëà ïðåäëîæåíà äèôôåðåíöèàëüíàÿ êîíôèäåíöèàëüíîñòü, êîòîðàÿ îêàçàëàñü íàèáîëåå ìíîãîîáåùàþùåé. êîìïðîìèññ ìåæäó êîíôèäåíöèàëüíîñòüþ è ïîëåçíîñòüþ îïóáëèêîâàííûõ äàííûõ, à òàêæå äðóãèå ïðîáëåìû, òàêèå êàê íàëè÷èå ìåòðèê äëÿ ñðàâíåíèÿ ðàçëè÷íûõ ñïîñîáîâ äîñòèæåíèÿ àíîíèìíîñòè, îòíîñÿòñÿ ê ñôåðå òåîðèè èíôîðìàöèè. íåñìîòðÿ íà íàëè÷èå â ëèòåðàòóðå ðÿäà îáçîðíûõ ñòàòåé, âîçìîæíîñòÿì ìåòîäîâ òåîðèè èíôîðìàöèè íå óäåëÿëîñü äîëæíîãî âíèìàíèÿ. â ýòîé ñòàòüå ìû äàåì îáçîð íîâåéøèõ ìåòîäîâ òåîðèè èíôîðìàöèè äëÿ îáåñïå÷åíèÿ êîíôèäåíöèàëüíîñòè. àíàëèçèðóåòñÿ ñåðèÿ ïóáëèêàöèé, ïîñâÿùåííûõ ðàçëè÷íûì ïðîáëåìàì êîíôèäåíöèàëüíîñòè, ðåøàåìûì ñ ïîìîùüþ èíñòðóìåíòîâ è ìåòîäîâ òåîðèè èíôîðìàöèè. èññëåäîâàíèÿ ïîêàçàëè, ÷òî òåîðåòèêî-èíôîðìàöèîííûå ìåòîäû ýôôåêòèâíû äëÿ øèðîêîãî êðóãà çàäà÷, îò àíîíèìíîñòè äî äèôôåðåíöèàëüíîé êîíôèäåíöèàëüíîñòè. êëþ÷åâûå ñëîâà: áîëüøèå äàííûå, àíîíèìèçàöèÿ, äèôôåðåíöèàëüíàÿ êîíôèäåíöèàëüíîñòü, ýíòðîïèÿ, âçàèìíàÿ èíôîðìàöèÿ, èñêàæåíèå 03_mariam_haroutunian_mpcs_55 03 d:\sbornik\...\rev.dvi mathematical problems of computer science 28, 2007, 18{33. random coding b ound of rever sible i nfor mation h iding e-capacity¤ ma r ia m h a r o u t u n ia n y, s m b a t to n o ya n y, ole ks iy k o va lz, s vya t o s la v v o lo s h yn o vs kiyz y institue for informatics and automation problems of nas of ra e-mail: farmar, smbattg@ipia.sci.am zuniversity of geneva, switzerland e-mail: foleksiy.koval, svolosg@cui.unige.ch abstract in this paper we consider the problem of reversible information hiding in the case when the attacker uses only discrete memoryless channels (dmc), the decoder knows only the class of channels, but not the dmc chosen by the attacker, the attacker knows the information-hiding strategy, probability distributions of all random variables, but not the side information. we introduce the notion of reversible information hiding e-capacity, which expresses the dependence of the information hiding rate on the error probability exponent and the distortion levels for information hider, for attacker and for the host data approximation. the random coding bound for reversible information hiding e-capacity is found. when e ! 0 we obtain the lower bound for reversibility information hiding capacity. in particular, we have analyzed two special cases of the general problem formulation, pure reversibility and pure message communications. refer ences [1 ] a . v . k u s n e t s o v a n d b . s . ts yb a ko v, " co d in g in a m e m o r y wit h d e fe c t ive c e lls " , ( in r u s s ia n ) , p robl. p eredachi informacii, vo l. 1 0 , n u m . 2 , p . 5 2 -6 0 , 1 9 7 4 . 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[1 4 ] m. e . h a r o u t u n ia n , " e s t im a t e s o f e-c a p a c it y a n d c a p a c it y r e g io n s fo r m u lt ip le -a c c e s s c h a n n e l wit h r a n d o m p a r a m e t e r " , e lectronic notes in d iscrete m athematics, v.2 1 , ge n e r a l th e o r y o f in fo r m a t io n tr a n s fe r a n d co m b in a t o r ic s , p p . 3 0 3 -3 0 8 , 2 0 0 5 . a va ila b le a t h t t p :/ / www.s c ie n c e d ir e c t .c o m / s c ie n c e . [1 5 ] i. cs is z ¶a r a n d j. k äo r n e r , information theory: coding theorems for discrete memoryless systems, a c a d e m ic p r e s s , n e w y o r k, 1 9 8 1 . [1 6 ] i. cs is z ¶a r , " th e m e t h o d o f t yp e s " , ie e e trans. inform. theory, vo l. 4 4 , n o . 6 , p p . 2 5 0 5 { 2 5 2 3 , 1 9 9 8 . [1 7 ] p . mo u lin a n d j. a . o's u lliva n , " in fo r m a t io n -t h e o r e t ic a n a lys is o f in fo r m a t io n h id in g " , ie e e trans. inform. theory, vo l. 4 9 , n o . 3 , p p . 5 6 3 -5 9 3 , ma r . 2 0 0 3 . [1 8 ] t. m. co ve r a n d j. a . th o m a s , e lements of information theory, w ile y, n e w y o r k, 1 9 9 1 . [1 9 ] e . a . h a r o u t u n ia n a n d h . b e lb a s h ir , " l o we r b o u n d o f t h e o p t im a l t r a n s m is s io n r a t e d e p e n d in g o n g ive n e r r o r p r o b a b ilit y e xp o n e n t fo r d is c r e t m e m o r yle s s c h a n n e l a n d fo r 2 0 random coding bound of reversible information hiding e-capacity a s ym m e t r ic b r o a d c a s t c h a n n e l" , ( in r u s s ia n ) , abstracts of papers of sixth international symposium on information theory, tashkent, ussr , vo l. 1 , p p . 1 9 -2 1 , 1 9 8 4 . î»õ»ïáõãûáõýý»ñ ã³ùóýáõ ßñç»éç ñ³ù³ï³ñ·ç e-áõý³ïáõãû³ý å³ï³ñ³ï³ý ïá¹³íáñù³ý ·ý³ñ³ï³ï³ýá ø. ð³ñáõãûáõýû³ý, ê. îáýáû³ý, ú. îáí³é, ê. ìáéáßçýáíëïç ²ù÷á÷áõù ²ßë³ï³ýùáõù ñ»ï³½áïí³í ¿ ï»õ»ïáõãûáõýý»ñç ßñç»éç ã³ùóù³ý çýýáñù³óçáýï»ë³ï³ý ëý¹çñá: ¸çï³ñïí³í ñ³ù³ï³ñ·ç ñ³ù³ñ ñ»ï³½áïí»é ¿ e-áõý³ïáõãûáõý ýáõýïóç³ý: ²ûý çñ»ýçó ý»ñï³û³óýáõù ¿ ï»õ»ïáõãûáõýý»ñç ã³ùóù³ý ³ñ³·áõãû³ý ï³ëí³íáõãûáõýá ñáõë³éçáõãûáõýçó, ï»õ»ïáõãûáõýý»ñ ã³ùóýáõç áõ ñ³ñó³ïíáõç ñ³ù³ñ ãáõûé³ïñ»éç ß»õù³ý ù³ï³ñ¹³ïý»ñçó ¨ ý³ëý³ï³ý ïíû³éý»ñç í»ñ³ï³ý·ýù³ý ñ³ù³ñ ãáõûé³ïñ»éç ß»õù³ý ù³ï³ñ¹³ïçó: e-áõý³ïáõãû³ý ñ³ù³ñ ï³éáõóí»é ¿ å³ï³ñ³ï³ý ïá¹³íáñù³ý ·ý³ñ³ï³ï³ý: ð³ù³å³ï³ëë³ý ·ý³ñ³ï³ï³ý ¿ ëï³óí»é ñ³ù³ï³ñ·ç áõý³ïáõãû³ý ñ³ù³ñ: àõëáõùý³ëçñí»é »ý ý³¨ »ñïáõ ù³ëý³íáñ ¹»åù»ñª ù³ùáõñ ßñç»éçáõãû³ý ¨ ñ³õáñ¹³·ñáõãûáõýý»ñç ù³ùáõñ ñ³õáñ¹ù³ý ¹»åù»ñá: artur_article.dvi mathematical problems of computer science 50, 104{106, 2018. i mpr oved m odel of scheduling algor ithm a r t u r p . v a r d a n ya n institute for informatics and automation problems of nas ra e-mail: artvardanyan@asnet.am abstract cluster computing is becoming increasingly practical for high performance computing research and development. a computer cluster is a set of connected computers that work together so that, they can be viewed as a single system. clusters o®er a scalable means of linking computers together to provide an expansive environment for hosting enterprise applications. as the number of nodes in cluster con¯gurations grows, the cluster administration becomes more challenging. we need to study the challenges of cluster management and to provide a solution. to have an e®ective cluster management we need to have an e®ective task scheduling algorithm. with the explosive growth of information, the demand on computing is sharply increasing. due to a large number of computing tasks, the scheduling algorithm is an important part of cluster computing and has a great in°uence on the quality of claster service. in cluster computing, some large tasks may occupy too many resources and some small tasks may wait for a long time based on first-in-first-out (fifo) scheduling algorithm. this paper provides an overview of an improved scheduling algorithm that shortens the execution time of tasks and increases the resource utilization. keywords: cluster computing, task scheduling, scheduling algorithm. 1 . in t r o d u c t io n ta s k s c h e d u lin g is t o a s s ig n e d a t a s k t o a n a va ila b le r e s o u r c e fo r it s e xe c u t io n . ta s k s c h e d u lin g is a n im p o r t a n t c o m p o n e n t o f c lu s t e r c o m p u t in g a n d h a s a g r e a t im p a c t o n t h e qu a lit y o f c lu s t e r s e r vic e . th e b a s ic p r o b le m o f t a s k s c h e d u lin g t h a t n e e d s t o b e s o lve d is a s s ig n in g t a s ks t o r e s o u r c e s . a s c h e d u le r is wh a t c a r r ie s o u t t h e s c h e d u lin g a c t ivit y. s c h e d u le r s a r e o ft e n im p le m e n t e d s o t h e y ke e p a ll c lu s t e r r e s o u r c e s b u s y ( a s in lo a d b a la n c in g ) , a llo w m u lt ip le u s e r s t o s h a r e c lu s t e r r e s o u r c e s e ®e c t ive ly, o r t o a c h ie ve a t a r g e t qu a lit y o f s e r vic e . fifo s c h e d u le r is a c o m m o n s c h e d u le r , a n d it c h o o s e s t h e t a s k t o e xe c u t e a c c o r d in g t o t a s k a r r iva l t im e . b y d e fa u lt , fifo s c h e d u le r d o e s n o t c o n s id e r a t a s ks p r io r it y. u s u a lly, in t a s k s c h e d u lin g a lg o r it h m s , t a s ks a r e d e s c r ib e d b y t wo p a r a m e t e r s : t h e r e s o u r c e u t iliz a t io n p a r a m e t e r a n d t a s k's e xe c u t io n t im e . ta s k's r e s o u r c e u t iliz a t io n p a r a m e t e r is t h e n u m b e r o f c lu s t e r n o d e s r e qu ir e d t o p e r fo r m t h is t a s k. ta s k's e xe c u t io n t im e is t h e m a xim u m t im e n e c e s s a r y t o c o m p le t e t h is t a s k. b u t in t h is p a p e r we p r o p o s e a t a s k s c h e d u lin g a lg o r it h m , in wh ic h t a s ks a r e d e s c r ib e d b y t h r e e p a r a m e t e r s : t h e r e s o u r c e u t iliz a t io n p a r a m e t e r , t a s k's e xe c u t io n t im e a n d t a s k's wa it in g t im e . w h a t is a t a s ks wa it in g t im e ? ta s ks wa it in g t im e is t h e m a xim u m t im e t h a t t h is t a s k c a n wa it b e fo r e a s s ig n in g t o r u n . to d e s c r ib e e a c h 1 0 4 a. vardanyan 1 0 5 t a s k we u s e d ( º; ¯; ! ) t r ip le t , wh e r e º is t h e n u m b e r o f c lu s t e r n o d e s r e qu ir e d t o p e r fo r m t h is t a s k, ¯ is t h e m a xim u m t im e n e c e s s a r y t o c o m p le t e t h is t a s k a n d ! is t h e m a xim u m t im e t h a t t h is t a s k c a n wa it b e fo r e a s s ig n in g t o r u n . fo r t h e va lu e t o t h e n u m b e r o f t h e c lu s t e r n o d e s we t o o k m . th e ¯ r s t m o m e n t o f t h e t im e we h a ve a qu e u e , in wh ic h t h e r e a r e n t a s ks , t h a t is we h a ve a qu e u e wit h n t h r e e -d im e n s io n a l ve c t o r s . 2 . mo d e l o f s c h e d u lin g a lg o r it h m fo r fifo im p r o ve d a lg o r it h m d e s c r ip t io n : in p u t : n tree-dementional vectors: f ( ºi; ¯i; !i ) ; i = 1 ; 2 ; :::; ng; the number of cluster nodes: m; p r o c e s s : s t e p 1 : declare the variables the number of assigned tasks: k = 0 ; the number of cluster occupied nodes:s = 0 the number of cluster idle nodes: f; s t e p 2 : while s + ºk+1 · 0 then k = k + 1 ; s = s + ºk and serve kth task; s t e p 3 : set f = m ¡ s; s t e p 4 : sort ¯1; ¯2; :::; ¯k in non-decreasing order: ¯i1; ¯i2 ; :::; ¯ik ; s t e p 5 : declare the variables ¿i th moment of the time: ¿i; i = 1 ; 2 ; :::; k; the number of cluster idle nodes at ¿i th moment of the time: mi; i = 1 ; 2 ; :::; k; s t e p 6 : for j = 1 to k set ¿i = ¯ij and mj = f + pj i=1 mi; s t e p 7 : go through the queue and ¯nd which ( ºj ; ¯j; !j ) so that ºj · f; ¯j < ¿1 for j = k + 1 to n if ºj · f; ¯j < ¿1 then serve jth task and break for; s t e p 8 : check if there are tasks in queue that have the waiting time less than ¿1 for j = k + 1 to n if !j < ¿1 then delete j th task from the queue and rearrange the queue; s t e p 9 : while the queue is not empty at each ¿i; i = 1 ; :::; k point do these cheeks for j = k + i + 1 to n if !j < ¿1 then delete j th task from the queue and rearrange the queue; if ºk+i · mi then serve ( k + i) th task; else for j = k + i + 1 to n if ºj · mi and ¿i + ¯j < ¿i+1 then serve jth task and break for. 3 . co n c lu s io n w e p r e s e n t e d t h e im p r o ve d m o d e l o f t h e s c h e d u lin g a lg o r it h m , wh ic h d e c r e a s e s t h e e xe c u t io n t im e o f t a s ks a n d in c r e a s e s t h e r e s o u r c e u t iliz a t io n o f t h e s ys t e m . in t h e a b o ve m e n t io n e d a lg o r it h m e a c h t a s k is c h a r a c t e r iz e d b y p a r a m e t e r s ( º; ¯; ! ) , wh e r e º is t h e n u m b e r o f c lu s t e r n o d e s r e qu ir e d t o p e r fo r m t h is t a s k, ¯ is t h e m a xim u m t im e n e c e s s a r y t o c o m p le t e t h is t a s k a n d ! is t h e m a xim u m t im e t h a t t h is t a s k c a n wa it b e fo r e a s s ig n in g t o r u n . n o t e t h a t in 1 0 6 improved model of scheduling algorithm t h e im p r o ve d a lg o r it h m we h a ve u s e d t h e b a c k¯ ll s c h e d u lin g m o d e l. b a c k¯ ll is a s c h e d u lin g o p t im iz a t io n , wh ic h a llo ws t h e s c h e d u le r t o m a ke b e t t e r u s e o f a va ila b le r e s o u r c e s b y r u n n in g jo b s o u t o f o r d e r . refer ences [1 ] v . g. s a h a kya n , y . h . s h o u ko u r ia n , h . v . a s t s a t r ya n , " a b o u t s o m e qu e u e in g m o d e ls fo r c o m p u t a t io n a l g r id s ys t e m s " , m athematical p roblems of computer science, y e r e va n , a r m e n ia , 2 0 1 6 . [2 ] m. l . p in e d o , s c h e d u lin g : th e o r y, a lg o r it h m s , a n d s ys t e m s , springer international p ublishing, 5 t h e d ., 2 0 1 6 . [3 ] p . b r u c ke r , s c h e d u lin g a lg o r it h m s , springer international p ublishing, 5 t h e d ., 2 0 0 7 . submitted 10.04.2018, accepted 28.11.2018. äé³ý³íáñù³ý ³é·áñçãùç µ³ñ»é³íí³í ùá¹»é ². ì³ñ¹³ýû³ý ²ù÷á÷áõù ðá¹í³íáõù ý»ñï³û³óíáõù åé³ý³íáñù³ý ³é·áñçãùç ùá¹»é, áñá ýí³½»óýáõù ¿ ³é³ç³¹áñ³ýùç ï³ï³ñù³ý å³ù³ý³ïá ¨ ù»í³óýáõù ñ³ù³ï³ñ·ç é»ëáõñëý»ñç û·ï³·áñíáõùá: ²é·áñçãùáõù ûáõñ³ù³ýãûáõñ ³é³ç³¹ñ³ýù µýáõã³·ñíáõù ¿ ( º; ¯; ! ) å³ñ³ù»ïñ»ñáí, áñï»õ º-ý ³û¹ ëý¹çñá ëå³ë³ñï»éáõ ñ³ù³ñ å³ñ³ýçíáõ ïé³ëï»ñ³ûçý ñ³ý·áõûóý»ñç ù³ý³ïý ¿, ¯-ý ³û¹ ëý¹çñá ëå³ë³ñï»éáõ ñ³ù³ñ å³ñ³ýçíáõ ³é³í»é³·áõûý å³ù³ý³ïý ¿, çëï ! ý ³ûý ³é³í»é³·áõûý å³ù³ý³ïý ¿, áñ ³û¹ ³é³ç³¹ñ³ýùá ï³ñáõ ¿ ëå³ë»é ùçýã¨ ëå³ë³ñïù³ý áõõ³ñïí»éá: óëó÷øåííàÿ ìîäåëü àëãîðèòìà ïëàíèðîâàíèÿ à. âàðäàíÿí àííîòàöèÿ â ñòàòüå ââîäèòñÿ óëó÷øåííóþ ìîäåëü àëãîðèòìà ïëàíèðîâàíèÿ, êîòîðàÿ óìåíüøàåò âðåìÿ âûïîëíåíèÿ çàäà÷ è óâåëè÷èâàåò èñïîëüçîâàíèå ðåñóðñîâ ñèñòåìû. â âûøåóïîìÿíóòîì àëãîðèòìå êàæäàÿ çàäà÷à õàðàêòåðèçóåòñÿ ïàðàìåòðàìè ( º; ¯; ! ) , ãäå º-êîëè÷åñòâî óçëîâ êëàñòåðà, íåîáõîäèìûõ äëÿ âûïîëíåíèÿ ýòîé çàäà÷è, ¯ìàêñèìàëüíîå âðåìÿ, íåîáõîäèìîå äëÿ âûïîëíåíèÿ ýòîé çàäà÷è, à ! -ìàêñèìàëüíîå âðåìÿ, ÷òî ýòà çàäà÷à ìîæåò ïîäîæäàòü äî íàçíà÷åíèÿ äëÿ çàïóñêà. mathematical problems of computer science 57, 18–29, 2022. doi: 10.51408/1963-0083 udc 004.932 a new image decolorization evaluation quality metric hrach y. ayunts1 and sos s. agaian2 1yerevan state university, yerevan, armenia 2college of staten island (csi), city university of new york, new york, usa e-mail: hrach.ayunc@gmail.com, sos.agaian@csi.cuny.edu abstract image decolorization, the process of color-to-gray conversion, plays a crucial role in single-channel processing, computer vision, digital printing, and monochrome visualization. this process induces new artifacts, the impact of which on visual quality has to be identified. while visual quality assessment has been the subject of many studies, there are still some open questions regarding new color-to-gray conversion quality metrics. for example, computer simulations show that the commonly used grayscale conversion quality metrics such as ccpr, ccfr, and e-score depend on parameters and may pick different best decolorization methods by changing the parameters. this paper proposes a new quality metric to evaluate image decolorization methods. it uses the human visual properties information and regression method. experimental results also show (i) strong correlations between the presented image decolorization quality metric and the mean opinion score (mos), (ii) more robust than the existing quality metrics, and (iii) help to choose the best state-of-the-art decolorization methods using the presented metric and existing quality metrics. keywords: color-to-gray conversion, decolorization, grayscale, regression, quality metric. 1. introduction image decolorization aims to convert a color image into a grayscale image to improve the image’s visual appearance or provide a “better” gray-level representation for the future automated image. the overall purpose of image decolorization is to preserve the visible color contrast, which usually suffers from information loss. it plays an essential role in single-channel image processing (analysis, detection, segmentation, and recognition), computer vision, monochrome printing, e-ink display, etc. [1]. the analysis of the existing image decolorization techniques shows the common problems that need to be solved because such methods introduce certain artifacts. it isn’t easy to evaluate decolorization methods and select their optimal parameters. there are various quality metrics for color images [2, 3]. 18 h. ayunts and s. agaian 19 thus, these types of metrics are not suitable for the evaluation of color-to-gray conversion. there is also no efficient measure that can be served as a building criterion for image decolorization. practically, all quality metrics for image decolorization are based on the fact that the human visual system cannot perceive color differences smaller than a certain threshold [4, 5, 6]. extensive computer simulations show that (i) commonly used grayscale conversion quality metrics such as ccpr, ccfr and e-score depend on the color difference parameter, and (ii) by changing the parameter, we pick a different decolorization method. thus, one needs to develop a new robust threshold-independent quality metric that does not require a reference image. this paper makes several key contributions: 1. propose a non-parametric, robust, monotonic, and non-reference quality metric for image decolorization. 2. present extensive computer simulation results. 3. present qualitative and perceptual evaluation of state-of-the-art decolorization methods. the structure of this paper is organized as follows. section 2 discusses the existing image decolorization methods and quality metrics. section 3 presents a new non-parametric quality metric. section 4 provides the results of extensive computer simulation. section 5 validates the new metric using preference scores from the user study. finally, section 6 concludes the work. 2. background this section presents the existing color-to-gray conversion methods and quality metrics. traditional color-to-gray conversion methods usually use a linear combination of r (red), g(green), b (blue) channels of a color image. it is based on the theory of t. young (1802), which states that any color can be created by combining three primary colors: r, g, and b. gray = ar + bg + cb [7], were a, b, c coefficients are calculated as (i) lightness method: gray = max(r,g,b)+min(r,g,b) 2 (ii) average method: a = b = c = 1/3, or gray = r+g+b 3 (iii) luminosity method: a = 0.21, b = 0.72, c = 0.07, or gray = 0.21r + 0.72g + 0.07b. these are the most popular and straightforward conversions used in electronic displays, printers, computer vision, image processing, and many other algorithms as a preprocessing step. however, fig. 1 shows that the grayscale conversion suffers from information loss (many details didn’t preserve, and the color contrast was lost in the grayscale images). it is natural to ask. (a) can we have a better decolorization algorithm? (b) how to quantitatively evaluate the performance of different methods or choose the parameters such as a, b, and c? (c) how do you improve the quality of a color image using decolorized images? more advanced decolorization methods use the values of other pixels to specify color orders to preserve the color contrast. local methods rely on local chrominance edges to enhance the contrast [8, 9]. most recent notable decolorization methods are based on the parametric decolorization model and its modification [5, 4, 10]. 20 a new image decolorization evaluation quality metric source lightness average luminosity we need this fig. 1. comparison of traditional grayscale conversion methods. decolorized images can lose can lose the contrast and become hardly visible. parametric decolorization model(pdm). the basic idea here is to convert a color image into gray using a combination of a polynomial of r, g, and b components: {r, g, b, rg, rb, gb, r2, g2, b2}. it generalizes commonly used linear and nonlinear color-to-gray conversion/mapping systems. more details on this method one can find in [5]. there are also neural network solutions to this problem [11]. decolorization needs quantitative evaluation to understand the performance of different methods. exiting decolorization quality metrics. the most commonly used decolorization quality metrics are based on the fact that the human visual system cannot perceive color difference δ smaller than a certain threshold. for example, the color contrast preserving ratio (ccpr) (suggested by lu et al. [4]), defined as ccpr = #{(x, y)|(x, y) ∈ ω, |gx − gy| ≥ τ} ||ω|| , (1) where ω is the set of all pixel pairs with δx,y ≥ τ, and gx, is the value of the x pixel after decolorization. ccpr shows the percentage of distinctive pixel pairs after the conversion, but it does not necessarily indicate if the grayscale image was “distorted” after conversion. to complement ccpr, lu et al. [4] suggested color content fidelity ratio (ccfr). it is defined as ccfr = 1 − #{(x, y)|(x, y) ∈ θ, δx,y ≤ τ} ||θ|| , (2) where θ is the set of all pixel pairs with |gx − gy| > τ. this metric shows how much the converted image has changed in terms of structure. finally, the combination of ccpr and ccfr, e-score [4], is defined as e-score = 2 · ccpr · ccfr ccpr + ccfr . (3) kind of quality h. ayunts and s. agaian 21 3. proposed quality metric this section shows the shortcomings of the existing decolorization metrics and suggests a better quality metric for quantitative evaluations. table 1: e-score metric for some threshold values for different decolorization methods image method τ = 3 τ = 5 τ = 7 τ = 9 τ = 15 τ = 25 pdm 0.9934 0.9776 0.9759 0.9737 0.9644 0.9551 lum 0.9613 0.9174 0.8526 0.7990 0.5956 0.3997 spd 0.9862 0.9769 0.9751 0.9713 0.9475 0.9192 svd 0.9896 0.9821 0.9765 0.9744 0.9279 0.8514 pdm 0.9726 0.9502 0.9272 0.9035 0.8298 0.6647 lum 0.9646 0.9356 0.9046 0.8704 0.7447 0.4993 spd 0.9777 0.9550 0.9275 0.8956 0.7823 0.5662 svd 0.9745 0.9525 0.9279 0.9003 0.7987 0.5965 the commonly used grayscale conversion quality metrics such as ccpr, ccfr, and e-score depend on the color difference parameter τ. computer simulations show that by changing the parameter τ, we pick a different decolorization method. to verify this statement, we compare different decolorization methods on a couple of images from ĉad́ık’s dataset [12]. we calculate the e-score quality metric for different values of threshold. we use three state-of-the-art methods (lu et al. [5], sowmya et al. [13], liu et al. [10]) and the luminosity method for comparison. the results are listed in table 1. obviously, the best method differs depending on the threshold value. for example, we can pick three different best methods by changing parameter τ in the case of the second image. the visual results of decolorization on these images are shown in fig. 4. in the previous work, the quantitative evaluation of color-to-gray conversion was performed using e-score for fixed values of threshold [4, 5]) or the average of several threshold values [10]. therefore, there is a need for more independent metrics to investigate the conversion process for each image. we introduce a new quality metric called threshold-independent slope (tis), which shows the decreasing speed of the e-score as the threshold value grows. we calculate the e-score metric for different τ values (τ = 1, 2, ..15) and choose the slope of the linear regression of this data as a new metric. the main advantage of the new metric is that it is not dependent on the τ parameter. linear regression can be solved using several linear models. a simple linear model function is defined as y = α + βx, (4) which describes a line with a slope β and y-intercept α. one of the easiest ways to estimate the slope is to use the least squares method: β̂ls = arg min β ||y − βx||22. (5) 22 a new image decolorization evaluation quality metric fig. 2. simple linear function estimation using the least squares method, ridge regression, and lasso regression. another method for coefficient estimation of (4) is ridge regression [14]. it is most suitable when data contains a higher number of predictor variables than the number of observations. the ridge regression estimator solves the regression problem using l2 penalized least squares: β̂ridge = arg min β ||y − βx||22 + λ||β|| 2 2, (6) where λ > 0 is a tuning parameter that controls the strength of the penalty term. similar to ridge regression, lasso regression can be used for slope estimation [15]. the lasso estimator uses l1 penalized least squares for solving the following optimization problem with λ tuning parameter: β̂lasso = arg min β ||y − βx||22 + λ||β||1. (7) fig. 2 compares these three regression models on a sample image from ĉad́ık’s dataset [12]. each of these models is used to calculate the tis metric. to find the best model for our case, we calculate the fitting scores of each model on every image from the dataset. the least squares method has the best average fitting score: thus, we use it for further evaluations. therefore, our tis metric is defined as tis = max(1 − |αβ|, 0), (8) where α and β are coefficients of a simple linear function (4) estimated with the least squares method (5). tis ranges in [0, 1], and higher values mean a lower decreasing speed of the e-score metric when the threshold is increased. fig. 3 shows the decolorization result on a sample image with four different levels of visibility. the value of our tis metric grows with better visibility and contrast in the result. therefore, the tis is also a monotonic metric. h. ayunts and s. agaian 23 fig. 3. the tis metric grows with a contrast and visibility increase. 4. computer simulation this section evaluates four decolorization methods using our tis metric and the existing quality metrics. we also show the usefulness of our metric in picking the best parameters for grayscale conversion. evaluation of decolorization methods. we chose one traditional conversion method: the luminosity method (denoted as lum in tables) is the most popular conversion used in many image processing algorithms and electronic devices. in many cases, it fails to preserve the contrast because the conversion considers only current pixel information. we also chose three state-of-the-art contrast preserving decolorization methods for evaluation. these methods are suggested by lu et al. [5], liu et al. [10], and sowmya et al. [13] (we use pdm, spd, and svd acronyms in the tables, respectively). these methods consider global pixel information and color differences in the image for better conversion. fig. 4. visual results of different decolorization algorithms (from left to right: source image, pdm, lum, spd, svd) tis 0.53565 0.72862 0.82799 0.87882 24 a new image decolorization evaluation quality metric fig. 5. visual results of different decolorization algorithms (from left to right: source image, pdm, lum, spd, svd) figs. 4 and 5 show the visual results of four decolorization methods on several images. the simple luminosity method usually fails to preserve the color contrast, while the other three methods produce better visual outputs. we use ĉad́ık’s dataset [12] for performance evaluation. it contains 24 png images and mainly consists of synthetically generated images and some colorful real-life photos. most of these images are challenging for traditional color-to-gray conversion methods. that’s why ĉad́ık’s dataset is the most popular in this field and can be beneficial for the evaluation of decolorization methods. table 2: average tis and e-score for different thresholds on ĉad́ık’s dataset. method τ = 3 τ = 5 τ = 7 τ = 9 τ = 15 τ = 25 tis pdm 0.98222 0.97009 0.95866 0.94697 0.90971 0.84409 0.91635 lum 0.96340 0.93755 0.91399 0.89556 0.83167 0.71761 0.84992 spd 0.98241 0.97060 0.95922 0.94810 0.90966 0.83835 0.91560 svd 0.98045 0.96651 0.95334 0.94040 0.89324 0.81638 0.90121 the quantitative evaluation of four decolorization methods using the e-score metric for different thresholds and our new tis metric on ĉad́ık’s dataset are presented in table 2. it presents the performance of each metric (average value) of all images from ĉad́ık’s dataset. it also shows that the presented tis metric is more stable and picks only the best method for this dataset. so it can be helpful in both individual and large-scale evaluations of the grayscale conversion methods. picking the best parameter for the simple grayscale conversion. image decolorization quality metrics can not only be useful in method evaluation, but they can also h. ayunts and s. agaian 25 help pick the best parameters for an algorithm. for example, in simple grayscale conversion, coefficients can be changed to get a “better” conversion. fig. 6. comparison of the “best” linear conversion with the luminosity method (from left to right: source image, luminosity, the best conversion). we pick the best parameters of the linear grayscale conversion by maximizing the value of the quality metric for each image individually. fig. 6 shows the results corresponding to the highest values of the tis for two images (a = 0.02, b = 0, c = 0.98 for the first image, and a = 0, b = 0.06, c = 0.94 for the second one). 5. perceptual validation this section validates our tis metric using the preference scores. we invited 20 users to participate in a survey to show the effectiveness and importance of our metric. after a small introduction to decolorization, they were asked to rate the color-to-gray conversion for ten random images from the ĉad́ık’s dataset on a scale of one to three. to facilitate the scoring process, we use the three-scale modification of the mean opinion score (mos). one means the conversion is bad, and it failed to preserve the contrast. score two corresponds to mediocre conversion. finally, three is for the best conversion with contrast preservation and the most visually pleasing result. to validate our metric, we use the kendall rank correlation coefficient [16]. it is defined as r = #{concordant pair} − #{disconcordant pair} 1 2 n(n − 1) , (9) where n = 4 denotes the number of methods. let si be the score for the result produced by the ith method, and pi be the preference score for the same result. if two pairs (si, sj) and (pi, pj) are with the same order (i.e., (si − sj)(pi − pj) > 0), the pair (i, j) is concordant. 26 a new image decolorization evaluation quality metric otherwise, it is discordant. r ranges in [−1, 1]. we get r > 0 if the two rankings agree with each other and r < 0 otherwise. we calculate the kendall rank correlation coefficient (r) for the existing metrics and our tis metric. the tis metric has a high correlation with the user preference scores and can easily replace the existing quality metrics for quantitative evaluations of decolorization. the ranks for several images presented in the survey are listed in table 3. table 3: kendall rank correlation coefficient for the e-score metric with different thresholds, and our tis metric image τ = 5 τ = 7 τ = 9 τ = 15 τ = 25 tis 0.333 0.333 0.333 0.667 0.667 0.667 0.333 0.333 0.333 1 1 1 0.667 0.333 0.333 0.333 0 0.333 6. conclusion this paper proposes a new tis image quality metric for accurately evaluating image decolorization methods. the tis quality metric is a blind, robust, monotonic, non-parametric metric and correlates with subjective preference scores. the quantitative and qualitative computer simulations on the ĉad́ık’s dataset demonstrate that the proposed metric outperforms the current state-of-the-art metrics. the tis metric is also helpful in picking the best parameters of the grayscale algorithm. our future work will extend the proposed work to other types of distortion, generate new decolorization methods, and evaluate them on other databases. 7. acknowledgment this work was supported by the professors of the department of informatics and applied mathematics of yerevan state university (s. sargsyan, e. danoyan, yu. hakopian) and the armenian engineers and scientists of america. the authors would like to thank the picsart employees for participating in the user study. the authors would also like to thank sowmya et al. [13] and lu et al. [4] for sharing source codes and experiment materials. h. ayunts and s. agaian 27 references [1] c. saravanan, “color image to grayscale image conversion”, second inter. conference on computer engineering and applications, ieee, vol. 2, pp. 196–199, march 2010. [2] k. panetta, c. gao, and s. agaian, “no reference color image contrast and quality measures”, ieee trans. on consumer electronics, vol. 59, no. 3, pp. 643–651, 2013. [3] k. panetta, c. gao, and s. agaian, “human-visual-system-inspired underwater image quality measures”, ieee journal of oceanic engineering, vol. 41, no. 3, pp. 541–551, 2015. [4] c. lu, l. xu, and j. jia, “contrast preserving decolorization with perception-based quality metrics”, inter. journal of computer vision, vol. 110, no. 2, pp. 222–239, 2014. [5] c. lu, l. xu, and j. jia, “contrast preserving decolorization”, proc. of ieee inter. conference on computational photography (iccp), pp. 1–7, 2012. [6] s. agaian, “visual morphology”, proc. of spie, nonlinear image processing x, san jose ca, vol. 3646, pp. 139–150, 1999. [7] j. cook, (2009) three algorithms for converting color to grayscale. [online]. available: https://www.johndcook.com/blog/2009/08/24/algorithms-convert-color-grayscale/ [8] r. bala, and r. eschbach, “spatial color-to-grayscale transform preserving chrominance edge information”, color and imaging conference, society for imaging science and technology, vol. 2004, no. 1, pp. 82–86, 2004. [9] l. neumann, m. ĉad́ık, and a. nemcsics, “an efficient perception-based adaptive color to gray transformation”, proc. of the third eurographics conference on computational aesthetics in graphics, visualization and imaging pp. 73–80, 2007. [10] q. liu, p. liu, y. wang, and h. leung, “semiparametric decolorization with laplacianbased perceptual quality metric”, ieee trans. on circuits and systems for video technology, vol. 27, no. 9, pp. 1856–1868, 2016. 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[16] h. abdi, “the kendall rank correlation coefficient”, encyclopedia of measurement and statistics. sage, thousand oaks, ca, pp. 508–510, 2007. submitted 18.04.2022, accepted 27.05.2022. 2 8 a new image decolorization evaluation quality metric ä³ïï»ñç ·áõý³½ñïù³ý áñ³ïç ·ý³ñ³ïù³ý ýáñ ã³÷áñáßçã ðñ³ã úáõ. ²ûáõýó1 ¨ êáë ê. ²õ³û³ý2 1ºñ¨³ýç å»ï³ï³ý ñ³ù³éë³ñ³ý, ºñ¨³ý, ð³û³ëï³ý 2êã»ûã»ý ²ûé»ý¹ç ùáé»ç, üûáõ úáñùç ù³õ³ù³ûçý ñ³ù³éë³ñ³ý, üûáõ úáñù, ²øü e-mail: hrach.ayunc@gmail.com, sos.agaian@csi.cuny.edu ²ù÷á÷áõù íîâàÿ ìåòðèêà êà÷åñòâà îöåíêè îáåñöâå÷èâàíèÿ èçîáðàæåíèÿ ãðà÷ þ. àþíö1 è ñîñ ñ. àãàÿí2 1åðåâàíñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, åðåâàí, àðìåíèÿ 2êîëëåäæ ñòàòåí-àéëåíäà, ãîðîäñêîé óíèâåðñèòåò íüþ-éîðêà, íüþ-éîðê, ñøà e-mail: hrach.ayunc@gmail.com, sos.agaian@csi.cuny.edu àííîòàöèÿ îáåñöâå÷èâàíèå èçîáðàæåíèÿ, ïðîöåññ ïðåîáðàçîâàíèÿ öâåòíîãî èçîáðàæåíèÿ â ìîíîõðîìíîå, èãðàåò ðåøàþùóþ ðîëü â îäíîêàíàëüíîé îáðàáîòêå, êîìïüþòåðíîì çðåíèè, öèôðîâîé ïå÷àòè è ìîíîõðîìíîé âèçóàëèçàöèè. ýòîò ïðîöåññ âûçûâàåò íîâûå àðòåôàêòû, âëèÿíèå êîòîðûõ íà âèçóàëüíîå êà÷åñòâî äîëæíî áûòü îïðåäåëåíî. íåñìîòðÿ íà òî, ÷òî îöåíêà âèçóàëüíîãî êà÷åñòâà ä³ïï»ñç ·áõý³½ñïáõùá` ·áõý³íáñ å³ïï»ñçó ùáýáëñáù å³ïï»ñç ÷áë³ï»ñåù³ý ·áñíáýã³óá, ï³ñ¨áñ ¹»ñ ¿ ë³õáõù ù»ï µ³õ³¹ñçãáí å³ïï»ñý»ñç ùß³ïù³ý, ñ³ù³ï³ñ·ã³ûçý ï»ëáõáõãû³ý, ãí³ûçý ïå³·ñáõãû³ý ¨ ùáýáëñáù íç½áõ³éç½³óç³ûç ù»ç: ²ûë ·áñíáýã³óá ³é³ç³óýáõù ¿ ýáñ ³õ³í³õáõùý»ñ, áñáýó ³½¹»óáõãûáõýá ï»ëáõ³ï³ý áñ³ïç íñ³ å»ïù ¿ µ³ó³ñ³ûïíç: â»¨ ï»ëáõ³ï³ý áñ³ïç ·ý³ñ³ïáõùá »õ»é ¿ µ³½ù³ãçí áõëáõùý³ëçñáõãûáõýý»ñç ³é³ñï³, ¹»é¨ë ï³ý áñáß µ³ó ñ³ñó»ñ ï³åí³í ÷áë³ï»ñåù³ý áñ³ïç ýáñ ù»ïñçï³ý»ñç ñ»ï: úñçý³ï` ñ³ù³ï³ñ·ã³ûçý ùá¹»é³íáñáõùá óáõûó ¿ ï³éçë, áñ ñ³×³ë û·ï³·áñííáõ áñ³ïç ã³÷áñáßçãý»ñá, çýãåçëçù »ý` ccpr-á, ccfr-á ¨ e-score-á, ï³ëí³í »ý å³ñ³ù»ïñ»ñçó ¨ ï³ñáõ »ý áýïñ»é ï³ñµ»ñ é³í³·áõûý ù»ãá¹ý»ñ‘ ÷á÷áë»éáí å³ñ³ù»ïñ»ñá: ðá¹í³íáõù ³é³ç³ñïíáõù ¿ å³ïï»ñý»ñç ·áõý³½ñïù³ý áñ³ïç ·ý³ñ³ïù³ý ýáñ ã³÷áñáßçã, áñá ñçùýí³í ¿ ù³ñ¹áõ ï»ëáõ³ï³ý ñ³ïïáõãûáõýý»ñç íñ³ ¨ ñ³ßííáõù ¿ é»·ñ»ëç³ûç ù»ãá¹ç ùççáóáí: öáñó³ñ³ñ³ï³ý ³ñ¹ûáõýùý»ñá óáõûó »ý ï³éçë, áñ ³é³ç³ñïíáõ ù»ïñçï³ý ³í»éç ï³ûáõý ¿, ù³ý ·áûáõãûáõý áõý»óáõý»ñá, ³ûý ý³¨ áõýç µ³ñóñ ïáñ»é³óç³ ùçççý ï³ñíçùç ·ý³ñ³ï³ï³ýç (mos) ñ»ï, ¨ ¹ñ³ û·ýáõãû³ùµ ñý³ñ³íáñ ¿ áýïñ»é é³í³·áõûý ·áõý³½»ñíù³ý ù»ãá¹ý»ñá: ´³ý³éç µ³é»ñ` ·áõý³íáñ å³ïï»ñý»ñç ÷áë³ï»ñåáõù ùáýáëñáù å³ïï»ñý»ñç, ·áõý³½ñïáõù, ùáýáëñáù å³ïï»ñ, é»·ñ»ëç³, áñ³ïç ã³÷áñáßçã: h. ayunts and s. agaian 2 9 áûëà ïðåäìåòîì ìíîãèõ èññëåäîâàíèé, âñå åùå îñòàåòñÿ íåñêîëüêî îòêðûòûõ âîïðîñîâ, êàñàþùèõñÿ íîâûõ ïîêàçàòåëåé êà÷åñòâà ïðåîáðàçîâàíèÿ öâåòíîãî èçîáðàæåíèÿ â ñåðûé. íàïðèìåð, êîìïüþòåðíîå ìîäåëèðîâàíèå ïîêàçûâàåò, ÷òî îáû÷íî èñïîëüçóåìûå ïîêàçàòåëè êà÷åñòâà ïðåîáðàçîâàíèè, òàêèå êàê ccpr, ccfr è e-score, çàâèñÿò îò ïàðàìåòðîâ è ìîãóò âûáèðàòü ðàçëè÷íûå íàèëó÷øèå ìåòîäû ïóòåì èçìåíåíèÿ ïàðàìåòðîâ. â ýòîé ñòàòüå ïðåäëàãàåòñÿ íîâàÿ ìåòðèêà êà÷åñòâà äëÿ îöåíêè ìåòîäîâ îáåñöâå÷èâàíèÿ èçîáðàæåíèÿ. îíà èñïîëüçóåò èíôîðìàöèþ î çðèòåëüíûõ ñâîéñòâàõ ÷åëîâåêà è ìåòîä ðåãðåññèè. ýêñïåðèìåíòàëüíûå ðåçóëüòàòû òàêæå ïîêàçûâàþò ñèëüíóþ êîððåëÿöèþ ìåæäó ïðåäñòàâëåííîé ìåòðèêîé êà÷åñòâà îáåñöâå÷èâàíèÿ èçîáðàæåíèÿ è ñðåäíåé îöåíêîé ìíåíèé (mos), áîëåå íàäåæíóþ, ÷åì ñóùåñòâóþùèå ìåòðèêè êà÷åñòâà, è ïîìîãàþò âûáðàòü ëó÷øèé èç ñîâðåìåííûõ ìåòîäîâ îáåñöâå÷èâàíèÿ ñ èñïîëüçîâàíèåì ïðåäñòàâëåííîé ìåòðèêè è ñóùåñòâóþùèõ ìåòðèê êà÷åñòâà. êëþ÷åâûå ñëîâà: ïðåîáðàçîâàíèÿ öâåòíîãî èçîáðàæåíèÿ â ìîíîõðîìíîå, îáåñöâå÷èâàíèå, ìîíîõðîìíîå èçîáðàæåíèå, ðåãðåññèÿ, ìåòðèêà êà÷åñòâà. 02_ayunts_57_18_29 (1) 02a d:\sbornik\...\stgy.dvi mathematical problems of computer science 32, 5{13, 2009. t he b asic semantics of untyped functional p r ogr ams and reduction str ategies gu r g e n g. h r a c h ya n yerevan state university gurgen.hrachyan@gmail.com abstract the paper is devoted to untyped functional programs, which are de¯ned as equation systems with separating variables in the untyped ¸-calculus. the semantics of such programs is usually de¯ned by means of the ¯x-point combinator y . previously, it was proved that the semantics of such programs is invariant with respect to the ¯x-point combinator. however, in this paper, we prove that this invariance is no longer valid when the reduction strategy is ¯xed. refer ences [1 ] h . p . b a r e n d r e g t , \ th e l a m b d a ca lc u lu s , it s s yn t a x a n d s e m a n t ic s " , north-holland p ub. comp., 1 9 8 1 . [2 ] s . a . n ig iya n a n d s . a . a ve t is ya n , \ on t h e s e m a n t ic s o f u n t yp e d fu n c t io n a l p r o g r a m s " , p rogramming and computer software, v o l. 2 8 , n 3 , p p . 1 1 9 -1 2 6 , 2 0 0 2 . [3 ] g. g. h r a c h ya n , \ th e in va r ia n c e o f t h e ma in s e m a n t ic s o f typ e -fr e e fu n c t io n a l p r o g r a m s r e la t ive ly t o t h e fixp o in t co m b in a t o r " , p roceedings of the conference csit-2007, y e r e va n , p p . 4 8 -5 0 . [4 ] s . a . a ve t is ya n , \ on t h e s e m a n t ic s o f u n t yp e d fu n c t io n a l p r o g r a m s " , p hd thesis, y e r e va n , 2 0 0 5 . ²é³ýó ïçå»ñç ýáõýïóçáý³é íñ³·ñ»ñç ñçùý³ï³ý ë»ù³ýïçï³ý ¨ é»¹áõïóçáý ëïñ³ï»¹ç³ý»ññ ¶. ðñ³ãû³ý ²ù÷á÷áõù ²ßë³ï³ýùá ýíçñí³í ¿ ³é³ýó ïçå»ñç ýáõýïçáý³ï³é íñ³·ñ»ñçý, áñáýù çñ»ýóçó ý»ñï³û³óýáõù »ý ñ³í³ë³ñáõùý»ñç ñ³ù³ï³ñ·»ñ ³é³ýó ïçå»ñç ¸ ñ³ßíáõù: üù³ý íñ³·ñ»ñç ñçùý³ï³ý ë»ù³ýïçï³ý çýãå»ë ï³ýáý ë³ñù³ýíáõù ¿ y ³ýß³ñå ï»ïç ïáùµçý³áñç ùççáóáí: ü³ëïçýáõù ³å³óáõóí»é ¿ , áñ ñçùý³ï³ý ë»ù³ýïçï³ý çýí³ñç³ýï ¿ ³ýß³ñå ï»ïç ïáùµçý³ïáñç ýï³ïù³ùµ: ê³ï³ûý, ³ûë ³ßë³ï³ýùçó ñ»ï¨áõù ¿ , áñ ïñí³í é»¹áõïóçáý ëïñ³ï»·ç³ûç ñ³ù³ñ ýù³ý çýí³ñç³ýïáõãûáõý ï»õç ãáõýç: 5 nellysbornik0107.dvi mathematical problems of computer science 29, 2007, 58{65. stability and oscillations in spatially-extended m odels of p opulation i nter action n e lli a . a ja b ya n institue for informatics and automation problems of nas of ra e-mail: najabyan@ipia.sci.am abstract conditions for species coexistence in a simple patch-occupancy metapopulation model are derived. the model is described by a system of ordinary di®erential equations. the ecological stability of the community is interpreted in terms of conservation of species composition in the model. the limits for the stability are related to the boundness of the model solutions in phase space. analytical results show that there are regions in the parameter space where the species coexistence can occur depending on characteristics of competing species interaction and structure of the links connecting isolated patches. refer ences [1 ] d e -a n g e lis d . l . d yn a m ic s o f n u t r ie n t cyc lin g a n d fo o d we b s . chapman and hall p ublish. 1 9 9 2 . [2 ] a u g e r p . b e h a vio r a l ch o ic e s b a s e d o n p a t c h s e le c t io n : a mo d e l u s in g a g g r e g a t io n me t h o d s . in : m athematical b iosciences, 1 5 7 , 1 9 9 9 , p p . 1 8 6 -2 1 6 . [3 ] me d vin s ky a . b . e t a l. s p a t ia lly s t r u c t u r e d p la n kt o n d yn a m ic s , in :nonlinear d ynamics in the l ife and social sciences. science series, v. 3 2 0 , 2 0 0 0 , p p . 3 8 3 -3 9 3 . [4 ] fie d le r b . glo b a l b ifu r c a t io n o f p e r io d ic s o lu t io n s wit h s ym m e t r y. l ect. notes in m athematics. s p r in g e r -v e r la g ., b e r lin , n e w-y o r k, to kyo , 1 9 8 8 , v. 1 3 0 9 , p p . 1 -1 4 0 . [5 ] l o g o fe t d . o., s vir e z e v y u . m. e c o lo g ic a l s t a b ilit y a n d l a g r a n g e s t a b ilit y. a n e w a p p r o a c h t o t h e p r o b le m . in : p roblems of e cological m onitoring and e cosystems m odeling. s t p e t e r s b u r g , gid r o m e t iz d a t , v. 7 , 1 9 8 5 , p p . 2 5 3 -2 5 8 . ( in r u s s ia n ) . [6 ] a ja b ja n n . a ., l o g o fe t d . o. p o p u la t io n s iz e d yn a m ic s in tr o p h ic ch a in s ., in : p roblems of e cological m onitoring and e cosystems m odeling. s t . p e t e r s b u r g , gid r o m e t iz d a t , v. 1 4 , 1 9 9 1 , p p . 1 3 5 -1 5 2 ( in r u s s ia n ) [7 ] s vir e z e v y u . m., l o g o fe t d . o. stability of b iological communities. mo s c o w, 1 9 7 8 ., p p . 3 5 2 ( in r u s s ia n ) . 5 8 n. a. ajabyan 5 9 [8 ] a le xa n d e r j. c., a u c h m u t y g. glo b a l b ifu r c a t io n o f p h a s e -l o c ke d os c illa t o r s . archive for rational m echanics and analysis. springer-velag., b e r lin , 1 9 8 8 , v. 9 3 , p p . 2 5 3 -2 7 0 . î»ýë³µ³ý³ï³ý ñ³ù³ï»óáõãûáõýý»ñç ùá¹»éý»ñç ï³ûáõýáõãûáõýá ³ýñ³ù³ë»é ùçç³í³ûñáõù ü. ². ²ç³µû³ý ²ù÷á÷áõù ²ñï³íí³í »ý å³ûù³ýý»ñ áã ñ³ù³ë»é ùçç³í³ûñáõù å³ñ½ áý¹ñ³ýñ³óí³í ï»ýë³ñ³ýñáõãû³ý ï»ë³ïý»ñç ·áû³ï¨ù³ý ñ³ù³ñ: ü³½³ûçý ï³ñ³íáõãûáõýáõù ùá¹»éç éáõíáõùý»ñç ë³ñù³ý³÷³ïáõãû³ý å³ûù³ýý»ñáí »ý ³ñï³ñ³ûïíáõù ¿ïáéá·ç³ï³ý ï³ûáõýáõãû³ý ë³ñù³ýý»ñá: microsoft word article.doc ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 29, 2007, 85–88. 85 î áëèçîñòè ìàêñèìàëüíûõ òóïèêîâûõ ðàñïîçíà þùèõ ñèñòåì â êëàññå äâóõýëåìåíòíûõ ïîäìíîæåñòâ ñåéðàí ì. âàðäàíÿí институт проблем информатики и автоматизации нан ра e-mail: seyranv@ipia.sci.am аннотация â ñòàòüå îïðåäåëÿåòñÿ ïîíÿòèå áëèçîñòè ðàñïîçíà þùèõ ñèñòåì è äîêàçûâà þòñÿ íåêîòîðûå òåîðåìû î êîëè÷åñòâå ìàêñèìàëüíûõ òóïèêîâûõ ðàñïîçíà þùèõ ñèñòåì äàííîé áëèçîñòè. ëèòåðàòóðà [1] ñ. ì. âàðäàíÿí. ðàñïîçíàâàíèå ïî ïåðåñå÷åíèÿì ìíîæåñòâ, òðóäû èíñòèòóòà ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè, “ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè”, òîì 24, ñ. 144-146, 2005ã. [2] s. m. vardanyan. “recognizing sets (systems)”. computer science and information technologies. proceedings of the conference csit, yerevan, armenia 2005, pp 161162. [3] ñ. ì. âàðäàíÿí. î äëèíàõ òóïèêîâûõ ðàñïîçíà þùèõ ñèñòåì â êëàññå äâóõýëåìåíòíûõ ïîäìíîæåñòâ. äàí ðà, òîì 107, âûï. 1, ñòð. 37-43. 2007. [4] f. harary, ”graph theory”. addison-wesley, reading, ma, 1969. ºñïáõ ï³ññ³ýáó »ýã³µ³½ùáõãûáõýý»ñç ¹³ëáõù ù³ùëçù³é ÷³ïáõõ³ûçý ×³ý³ãáõ ñ³ù³ï³ñ·»ñç ùáïçïáõãû³ý ù³ëçý ê. ø. ì³ñ¹³ýû³ý ²ù÷á÷áõù ê³ñù³ýíáõù ¿ »ñïáõ ×³ý³ãáõ ñ³ù³ï³ñ·»ñç ùáïçïáõãû³ý ·³õ³÷³ñá ¨ ³å³óáõóíáõù »ý ã»áñ»ùý»ñ ïñí³í ùáïçïáõãû³ý ù³ùëçù³é ÷³ïáõõ³ûçý ×³ý³ãáõ ñ³ù³ï³ñ·»ñç ù³ý³ïý»ñç í»ñ³µ»ñû³é: d:\sbornik\...\idenm2.dvi mathematical problems of computer science 28, 2007, 114{119. on i denti¯cation of distr ibutions of t wo i ndependent objects e vg u e n i a . h a r o u t u n ia n a n d p a r a n d z e m m. h a ko b ya n institue for informatics and automation problems of nas of ra e-mail: evhar@ipia.sci.am abstract ahlswede and haroutunian formulated new problems on multiple hypotheses testing and on identi¯cation of hypotheses. the problems of identi¯cation of distribution and of distributions ranking for one object were solved completely. in this paper we study the problem of identi¯cation of distributions for two independent objects. refer ences [1 ] e . a . h a r o u t u n ia n , \ r e lia b ilit y in m u lt ip le h yp o t h e s e s t e s t in g a n d id e n t i¯ c a t io n p r o b le m s " . p r o c e e d in g s o f t h e n a to a s i, y e r e va n , 2 0 0 3 . n a to s c ie n c e s e r ie s iii: co m p u t e r a n d s ys t e m s s c ie n c e s { vo l. 1 9 8 , p p . 1 8 9 { 2 0 1 . ios p r e s s , 2 0 0 5 . [2 ] r . f. a h ls we d e a n d e . a . h a r o u t u n ia n , \ on lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f h yp o t h e s e s a n d id e n t ī c a t io n " . l e c t u r e n o t e s in co m p u t e r s c ie n c e , vo l. 4 1 2 3 ,\ ge n e r a l th e o r y o f in fo r m a t io n tr a n s fe r a n d co m b in a t o r ic s " s p r in g e r , p p . 4 6 2 { 4 7 8 , 2 0 0 6 . [3 ] r . e . b e c h h o fe r , j. k ie fe r , a n d m. s o b e l, sequential identi¯cation and ranking procedures. th e u n ive r s it y o f ch ic a g o p r e s s , ch ic a g o , 1 9 6 8 . [4 ] r . f. a h ls we d e a n d i. w e g e n e r , search problems. w ile y, n e w y o r k, 1 9 8 7 . [5 ] e . a . h a r o u t u n ia n a n d p . m. h a ko b ya n , \ on l a o t e s t in g o f m u lt ip le h yp o t h e s is fo r p a ir o f o b je c t s " , m athematical p roblems of computer science vol. 26, p p . 9 2 { 1 0 0 , 2 0 0 5 . [6 ] e . a . h a r o u t u n ia n a n d p . m. h a ko b ya n , \ on m u lt ip le h yp o t h e s e s l a o t e s t in g fo r m a n y in d e p e n d e n t o b je c t s " . ( a c c e p t t e d t o ie e e in t e r n a t io n a l s ym p o s iu m o n in fo r m a t io n th e o r y, fr a n c e , n ic e , 2 0 0 7 ) . [7 ] e . a . h a r o u t u n ia n , \ l o g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f m u lt ip le s t a t is t ic a l h yp o t h e s e s " , p roblems of control and information theory, vo l. 1 9 ( 5 -6 ) , p p . 4 1 3 { 4 2 1 , 1 9 9 0 . [8 ] i. cs is z ¶a r a n d j. k äo r n e r , information theory: coding theorems for discrete memoryless systems, a c a d e m ic p r e s s , n e w y o r k, 1 9 8 1 . 1 1 4 e. a. haroutunian and p. m. hakobyan 1 1 5 ºñïáõ ³ýï³ë ûµû»ïïý»ñç µ³ßëáõùý»ñç ýáõûý³ï³ý³óù³ý ù³ëçý º. ². ð³ñáõãûáõýû³ý ¨ ö. ø. ð³�