id	sid	tid	token	lemma	pos
ejpam-5569	1	1	european	european	PROPN
ejpam-5569	1	2	journal	journal	PROPN
ejpam-5569	1	3	of	of	ADP
ejpam-5569	1	4	pure	pure	ADJ
ejpam-5569	1	5	and	and	CCONJ
ejpam-5569	1	6	applied	applied	ADJ
ejpam-5569	1	7	mathematics	mathematic	NOUN
ejpam-5569	1	8	2025	2025	NUM
ejpam-5569	1	9	,	,	PUNCT
ejpam-5569	1	10	vol	vol	NOUN
ejpam-5569	1	11	.	.	PROPN
ejpam-5569	1	12	18	18	NUM
ejpam-5569	1	13	,	,	PUNCT
ejpam-5569	1	14	issue	issue	NOUN
ejpam-5569	1	15	2	2	NUM
ejpam-5569	1	16	,	,	PUNCT
ejpam-5569	1	17	article	article	NOUN
ejpam-5569	1	18	number	number	NOUN
ejpam-5569	1	19	5569	5569	NUM
ejpam-5569	1	20	issn	issn	PROPN
ejpam-5569	1	21	1307	1307	NUM
ejpam-5569	1	22	-	-	SYM
ejpam-5569	1	23	5543	5543	NUM
ejpam-5569	1	24	–	–	PUNCT
ejpam-5569	1	25	ejpam.com	ejpam.com	X
ejpam-5569	1	26	published	publish	VERB
ejpam-5569	1	27	by	by	ADP
ejpam-5569	1	28	new	new	PROPN
ejpam-5569	1	29	york	york	PROPN
ejpam-5569	1	30	business	business	PROPN
ejpam-5569	1	31	global	global	PROPN
ejpam-5569	1	32	upper	upper	ADJ
ejpam-5569	1	33	bound	bind	VERB
ejpam-5569	1	34	of	of	ADP
ejpam-5569	1	35	radio	radio	NOUN
ejpam-5569	1	36	span	span	NOUN
ejpam-5569	1	37	for	for	ADP
ejpam-5569	1	38	shadow	shadow	NOUN
ejpam-5569	1	39	-	-	PUNCT
ejpam-5569	1	40	path	path	NOUN
ejpam-5569	1	41	network	network	NOUN
ejpam-5569	1	42	and	and	CCONJ
ejpam-5569	1	43	its	its	PRON
ejpam-5569	1	44	mathematical	mathematical	ADJ
ejpam-5569	1	45	modeling	modeling	NOUN
ejpam-5569	1	46	nawal	nawal	PROPN
ejpam-5569	1	47	m.	m.	NOUN
ejpam-5569	1	48	noureldeen1,2	noureldeen1,2	PROPN
ejpam-5569	1	49	,	,	PUNCT
ejpam-5569	1	50	elsayed	elsaye	VERB
ejpam-5569	1	51	badr3,4,5,hanan	badr3,4,5,hanan	NOUN
ejpam-5569	1	52	m.	m.	NOUN
ejpam-5569	1	53	shabana6,∗	shabana6,∗	NOUN
ejpam-5569	1	54	,	,	PUNCT
ejpam-5569	1	55	m.	m.	PROPN
ejpam-5569	1	56	e.	e.	PROPN
ejpam-5569	1	57	abdel	abdel	PROPN
ejpam-5569	1	58	-	-	PUNCT
ejpam-5569	1	59	aal7	aal7	PROPN
ejpam-5569	1	60	1	1	NUM
ejpam-5569	1	61	department	department	NOUN
ejpam-5569	1	62	of	of	ADP
ejpam-5569	1	63	mathematics	mathematic	NOUN
ejpam-5569	1	64	,	,	PUNCT
ejpam-5569	1	65	college	college	NOUN
ejpam-5569	1	66	of	of	ADP
ejpam-5569	1	67	science	science	PROPN
ejpam-5569	1	68	,	,	PUNCT
ejpam-5569	1	69	taibah	taibah	PROPN
ejpam-5569	1	70	university	university	PROPN
ejpam-5569	1	71	,	,	PUNCT
ejpam-5569	1	72	madinah	madinah	PROPN
ejpam-5569	1	73	,	,	PUNCT
ejpam-5569	1	74	kingdom	kingdom	NOUN
ejpam-5569	1	75	of	of	ADP
ejpam-5569	1	76	saudi	saudi	PROPN
ejpam-5569	1	77	arabia	arabia	PROPN
ejpam-5569	1	78	2	2	NUM
ejpam-5569	1	79	department	department	NOUN
ejpam-5569	1	80	of	of	ADP
ejpam-5569	1	81	mathematics	mathematic	NOUN
ejpam-5569	1	82	,	,	PUNCT
ejpam-5569	1	83	women	woman	NOUN
ejpam-5569	1	84	’s	’s	PART
ejpam-5569	1	85	college	college	PROPN
ejpam-5569	1	86	of	of	ADP
ejpam-5569	1	87	arts	art	NOUN
ejpam-5569	1	88	,	,	PUNCT
ejpam-5569	1	89	science	science	NOUN
ejpam-5569	1	90	and	and	CCONJ
ejpam-5569	1	91	education	education	NOUN
ejpam-5569	1	92	,	,	PUNCT
ejpam-5569	1	93	ain	ain	PROPN
ejpam-5569	1	94	shams	shams	PROPN
ejpam-5569	1	95	university	university	PROPN
ejpam-5569	1	96	,	,	PUNCT
ejpam-5569	1	97	egypt	egypt	PROPN
ejpam-5569	1	98	3	3	NUM
ejpam-5569	1	99	department	department	PROPN
ejpam-5569	1	100	of	of	ADP
ejpam-5569	1	101	information	information	NOUN
ejpam-5569	1	102	systems	system	NOUN
ejpam-5569	1	103	,	,	PUNCT
ejpam-5569	1	104	college	college	NOUN
ejpam-5569	1	105	of	of	ADP
ejpam-5569	1	106	information	information	NOUN
ejpam-5569	1	107	technology	technology	NOUN
ejpam-5569	1	108	,	,	PUNCT
ejpam-5569	1	109	misr	misr	PROPN
ejpam-5569	1	110	university	university	PROPN
ejpam-5569	1	111	for	for	ADP
ejpam-5569	1	112	science	science	NOUN
ejpam-5569	1	113	and	and	CCONJ
ejpam-5569	1	114	technology	technology	NOUN
ejpam-5569	1	115	(	(	PUNCT
ejpam-5569	1	116	must	must	AUX
ejpam-5569	1	117	)	)	PUNCT
ejpam-5569	1	118	,	,	PUNCT
ejpam-5569	1	119	p.o	p.o	PROPN
ejpam-5569	1	120	.	.	PROPN
ejpam-5569	1	121	box	box	PROPN
ejpam-5569	1	122	77	77	NUM
ejpam-5569	1	123	,	,	PUNCT
ejpam-5569	1	124	giza	giza	PROPN
ejpam-5569	1	125	,	,	PUNCT
ejpam-5569	1	126	egypt	egypt	PROPN
ejpam-5569	1	127	4	4	NUM
ejpam-5569	1	128	scientific	scientific	ADJ
ejpam-5569	1	129	computing	compute	VERB
ejpam-5569	1	130	department	department	NOUN
ejpam-5569	1	131	,	,	PUNCT
ejpam-5569	1	132	faculty	faculty	NOUN
ejpam-5569	1	133	of	of	ADP
ejpam-5569	1	134	computers	computer	NOUN
ejpam-5569	1	135	and	and	CCONJ
ejpam-5569	1	136	artificial	artificial	ADJ
ejpam-5569	1	137	intelligence	intelligence	NOUN
ejpam-5569	1	138	,	,	PUNCT
ejpam-5569	1	139	benha	benha	VERB
ejpam-5569	1	140	university	university	NOUN
ejpam-5569	1	141	,	,	PUNCT
ejpam-5569	1	142	benha	benha	NOUN
ejpam-5569	1	143	,	,	PUNCT
ejpam-5569	1	144	egypt	egypt	PROPN
ejpam-5569	1	145	5	5	NUM
ejpam-5569	1	146	the	the	DET
ejpam-5569	1	147	egyptian	egyptian	ADJ
ejpam-5569	1	148	school	school	NOUN
ejpam-5569	1	149	of	of	ADP
ejpam-5569	1	150	data	data	PROPN
ejpam-5569	1	151	science	science	NOUN
ejpam-5569	1	152	(	(	PUNCT
ejpam-5569	1	153	esds	esds	NOUN
ejpam-5569	1	154	)	)	PUNCT
ejpam-5569	1	155	,	,	PUNCT
ejpam-5569	1	156	benha	benha	VERB
ejpam-5569	1	157	university	university	PROPN
ejpam-5569	1	158	,	,	PUNCT
ejpam-5569	1	159	egypt	egypt	PROPN
ejpam-5569	1	160	6	6	NUM
ejpam-5569	1	161	physics	physics	NOUN
ejpam-5569	1	162	and	and	CCONJ
ejpam-5569	1	163	engineering	engineering	NOUN
ejpam-5569	1	164	mathematics	mathematics	PROPN
ejpam-5569	1	165	department	department	NOUN
ejpam-5569	1	166	,	,	PUNCT
ejpam-5569	1	167	faculty	faculty	NOUN
ejpam-5569	1	168	of	of	ADP
ejpam-5569	1	169	electronic	electronic	ADJ
ejpam-5569	1	170	engineering	engineering	NOUN
ejpam-5569	1	171	,	,	PUNCT
ejpam-5569	1	172	menoufia	menoufia	PROPN
ejpam-5569	1	173	university	university	PROPN
ejpam-5569	1	174	,	,	PUNCT
ejpam-5569	1	175	menouf	menouf	PROPN
ejpam-5569	1	176	,	,	PUNCT
ejpam-5569	1	177	32952	32952	NUM
ejpam-5569	1	178	,	,	PUNCT
ejpam-5569	1	179	egypt	egypt	PROPN
ejpam-5569	1	180	7	7	NUM
ejpam-5569	1	181	department	department	NOUN
ejpam-5569	1	182	of	of	ADP
ejpam-5569	1	183	mathematics	mathematic	NOUN
ejpam-5569	1	184	,	,	PUNCT
ejpam-5569	1	185	faculty	faculty	NOUN
ejpam-5569	1	186	of	of	ADP
ejpam-5569	1	187	science	science	NOUN
ejpam-5569	1	188	,	,	PUNCT
ejpam-5569	1	189	benha	benha	VERB
ejpam-5569	1	190	university	university	NOUN
ejpam-5569	1	191	,	,	PUNCT
ejpam-5569	1	192	benha	benha	NOUN
ejpam-5569	1	193	13518	13518	NUM
ejpam-5569	1	194	,	,	PUNCT
ejpam-5569	1	195	egypt	egypt	PROPN
ejpam-5569	1	196	abstract	abstract	PROPN
ejpam-5569	1	197	.	.	PUNCT
ejpam-5569	2	1	motivated	motivate	VERB
ejpam-5569	2	2	by	by	ADP
ejpam-5569	2	3	the	the	DET
ejpam-5569	2	4	frequency	frequency	NOUN
ejpam-5569	2	5	assignment	assignment	NOUN
ejpam-5569	2	6	problem	problem	NOUN
ejpam-5569	2	7	,	,	PUNCT
ejpam-5569	2	8	we	we	PRON
ejpam-5569	2	9	investigate	investigate	VERB
ejpam-5569	2	10	radio	radio	NOUN
ejpam-5569	2	11	labeling	labeling	NOUN
ejpam-5569	2	12	of	of	ADP
ejpam-5569	2	13	graphs	graph	NOUN
ejpam-5569	2	14	.	.	PUNCT
ejpam-5569	3	1	in	in	ADP
ejpam-5569	3	2	graph	graph	NOUN
ejpam-5569	3	3	theory	theory	NOUN
ejpam-5569	3	4	and	and	CCONJ
ejpam-5569	3	5	discrete	discrete	ADJ
ejpam-5569	3	6	mathematics	mathematic	NOUN
ejpam-5569	3	7	,	,	PUNCT
ejpam-5569	3	8	radio	radio	NOUN
ejpam-5569	3	9	labeling	labeling	NOUN
ejpam-5569	3	10	of	of	ADP
ejpam-5569	3	11	graphs	graph	NOUN
ejpam-5569	3	12	has	have	AUX
ejpam-5569	3	13	received	receive	VERB
ejpam-5569	3	14	significant	significant	ADJ
ejpam-5569	3	15	attention	attention	NOUN
ejpam-5569	3	16	as	as	SCONJ
ejpam-5569	3	17	it	it	PRON
ejpam-5569	3	18	is	be	AUX
ejpam-5569	3	19	of	of	ADP
ejpam-5569	3	20	immense	immense	ADJ
ejpam-5569	3	21	importance	importance	NOUN
ejpam-5569	3	22	for	for	ADP
ejpam-5569	3	23	numerous	numerous	ADJ
ejpam-5569	3	24	applications	application	NOUN
ejpam-5569	3	25	to	to	ADP
ejpam-5569	3	26	a	a	DET
ejpam-5569	3	27	wide	wide	ADJ
ejpam-5569	3	28	range	range	NOUN
ejpam-5569	3	29	of	of	ADP
ejpam-5569	3	30	areas	area	NOUN
ejpam-5569	3	31	such	such	ADJ
ejpam-5569	3	32	as	as	ADP
ejpam-5569	3	33	circuit	circuit	NOUN
ejpam-5569	3	34	and	and	CCONJ
ejpam-5569	3	35	sensor	sensor	NOUN
ejpam-5569	3	36	network	network	NOUN
ejpam-5569	3	37	,	,	PUNCT
ejpam-5569	3	38	signal	signal	NOUN
ejpam-5569	3	39	processing	processing	NOUN
ejpam-5569	3	40	,	,	PUNCT
ejpam-5569	3	41	design	design	NOUN
ejpam-5569	3	42	,	,	PUNCT
ejpam-5569	3	43	frequency	frequency	NOUN
ejpam-5569	3	44	assignment	assignment	NOUN
ejpam-5569	3	45	in	in	ADP
ejpam-5569	3	46	mobile	mobile	ADJ
ejpam-5569	3	47	communication	communication	NOUN
ejpam-5569	3	48	systems	system	NOUN
ejpam-5569	3	49	,	,	PUNCT
ejpam-5569	3	50	etc	etc	X
ejpam-5569	3	51	.	.	X
ejpam-5569	4	1	an	an	DET
ejpam-5569	4	2	assignment	assignment	NOUN
ejpam-5569	4	3	of	of	ADP
ejpam-5569	4	4	labels	label	NOUN
ejpam-5569	4	5	satisfying	satisfy	VERB
ejpam-5569	4	6	specific	specific	ADJ
ejpam-5569	4	7	constraints	constraint	NOUN
ejpam-5569	4	8	to	to	ADP
ejpam-5569	4	9	the	the	DET
ejpam-5569	4	10	edges	edge	NOUN
ejpam-5569	4	11	,	,	PUNCT
ejpam-5569	4	12	vertices	vertex	NOUN
ejpam-5569	4	13	,	,	PUNCT
ejpam-5569	4	14	or	or	CCONJ
ejpam-5569	4	15	both	both	PRON
ejpam-5569	4	16	of	of	ADP
ejpam-5569	4	17	graph	graph	NOUN
ejpam-5569	4	18	g	g	PROPN
ejpam-5569	4	19	is	be	AUX
ejpam-5569	4	20	known	know	VERB
ejpam-5569	4	21	as	as	ADP
ejpam-5569	4	22	graph	graph	NOUN
ejpam-5569	4	23	labeling	labeling	NOUN
ejpam-5569	4	24	.	.	PUNCT
ejpam-5569	5	1	radio	radio	NOUN
ejpam-5569	5	2	labeling	labeling	NOUN
ejpam-5569	5	3	of	of	ADP
ejpam-5569	5	4	a	a	DET
ejpam-5569	5	5	graph	graph	NOUN
ejpam-5569	5	6	g	g	NOUN
ejpam-5569	5	7	is	be	AUX
ejpam-5569	5	8	a	a	DET
ejpam-5569	5	9	technique	technique	NOUN
ejpam-5569	5	10	of	of	ADP
ejpam-5569	5	11	labeling	labeling	NOUN
ejpam-5569	5	12	vertices	vertex	NOUN
ejpam-5569	5	13	of	of	ADP
ejpam-5569	5	14	g	g	NOUN
ejpam-5569	5	15	by	by	ADP
ejpam-5569	5	16	non	non	ADJ
ejpam-5569	5	17	-	-	ADJ
ejpam-5569	5	18	negative	negative	ADJ
ejpam-5569	5	19	integers	integer	NOUN
ejpam-5569	5	20	.	.	PUNCT
ejpam-5569	6	1	hence	hence	ADV
ejpam-5569	6	2	,	,	PUNCT
ejpam-5569	6	3	radio	radio	NOUN
ejpam-5569	6	4	labeling	labeling	NOUN
ejpam-5569	6	5	problem	problem	NOUN
ejpam-5569	6	6	presents	present	VERB
ejpam-5569	6	7	an	an	DET
ejpam-5569	6	8	efficient	efficient	ADJ
ejpam-5569	6	9	graph	graph	NOUN
ejpam-5569	6	10	modeling	modeling	NOUN
ejpam-5569	6	11	for	for	ADP
ejpam-5569	6	12	the	the	DET
ejpam-5569	6	13	frequency	frequency	NOUN
ejpam-5569	6	14	assignment	assignment	NOUN
ejpam-5569	6	15	problem	problem	NOUN
ejpam-5569	6	16	.	.	PUNCT
ejpam-5569	7	1	in	in	ADP
ejpam-5569	7	2	radio	radio	NOUN
ejpam-5569	7	3	labeling	labeling	NOUN
ejpam-5569	7	4	of	of	ADP
ejpam-5569	7	5	a	a	DET
ejpam-5569	7	6	graph	graph	NOUN
ejpam-5569	7	7	g	g	NOUN
ejpam-5569	7	8	,	,	PUNCT
ejpam-5569	7	9	the	the	DET
ejpam-5569	7	10	maximum	maximum	ADJ
ejpam-5569	7	11	label	label	NOUN
ejpam-5569	7	12	used	use	VERB
ejpam-5569	7	13	for	for	ADP
ejpam-5569	7	14	labeling	labeling	NOUN
ejpam-5569	7	15	vertices	vertex	NOUN
ejpam-5569	7	16	is	be	AUX
ejpam-5569	7	17	called	call	VERB
ejpam-5569	7	18	the	the	DET
ejpam-5569	7	19	span	span	NOUN
ejpam-5569	7	20	of	of	ADP
ejpam-5569	7	21	that	that	DET
ejpam-5569	7	22	radio	radio	NOUN
ejpam-5569	7	23	labeling	labeling	NOUN
ejpam-5569	7	24	.	.	PUNCT
ejpam-5569	8	1	the	the	DET
ejpam-5569	8	2	minimum	minimum	ADJ
ejpam-5569	8	3	span	span	NOUN
ejpam-5569	8	4	from	from	ADP
ejpam-5569	8	5	all	all	DET
ejpam-5569	8	6	radio	radio	NOUN
ejpam-5569	8	7	labelings	labeling	NOUN
ejpam-5569	8	8	of	of	ADP
ejpam-5569	8	9	g	g	PROPN
ejpam-5569	8	10	is	be	AUX
ejpam-5569	8	11	known	know	VERB
ejpam-5569	8	12	as	as	ADP
ejpam-5569	8	13	the	the	DET
ejpam-5569	8	14	radio	radio	NOUN
ejpam-5569	8	15	number	number	NOUN
ejpam-5569	8	16	of	of	ADP
ejpam-5569	8	17	g.	g.	PROPN
ejpam-5569	8	18	that	that	SCONJ
ejpam-5569	8	19	radio	radio	NOUN
ejpam-5569	8	20	number	number	NOUN
ejpam-5569	8	21	reflects	reflect	VERB
ejpam-5569	8	22	the	the	DET
ejpam-5569	8	23	efficient	efficient	ADJ
ejpam-5569	8	24	usage	usage	NOUN
ejpam-5569	8	25	of	of	ADP
ejpam-5569	8	26	the	the	DET
ejpam-5569	8	27	available	available	ADJ
ejpam-5569	8	28	frequencies	frequency	NOUN
ejpam-5569	8	29	in	in	ADP
ejpam-5569	8	30	frequency	frequency	NOUN
ejpam-5569	8	31	assignment	assignment	NOUN
ejpam-5569	8	32	for	for	ADP
ejpam-5569	8	33	a	a	DET
ejpam-5569	8	34	network	network	NOUN
ejpam-5569	8	35	modeled	model	VERB
ejpam-5569	8	36	by	by	ADP
ejpam-5569	8	37	the	the	DET
ejpam-5569	8	38	graph	graph	NOUN
ejpam-5569	8	39	g.	g.	NOUN
ejpam-5569	8	40	this	this	DET
ejpam-5569	8	41	paper	paper	NOUN
ejpam-5569	8	42	contributes	contribute	VERB
ejpam-5569	8	43	the	the	DET
ejpam-5569	8	44	mathematical	mathematical	ADJ
ejpam-5569	8	45	proof	proof	NOUN
ejpam-5569	8	46	(	(	PUNCT
ejpam-5569	8	47	theorems	theorem	NOUN
ejpam-5569	8	48	1	1	NUM
ejpam-5569	8	49	-	-	SYM
ejpam-5569	8	50	4	4	NUM
ejpam-5569	8	51	)	)	PUNCT
ejpam-5569	8	52	,	,	PUNCT
ejpam-5569	8	53	an	an	DET
ejpam-5569	8	54	integer	integer	NOUN
ejpam-5569	8	55	linear	linear	NOUN
ejpam-5569	8	56	programming	programming	NOUN
ejpam-5569	8	57	model	model	NOUN
ejpam-5569	8	58	for	for	ADP
ejpam-5569	8	59	finding	find	VERB
ejpam-5569	8	60	the	the	DET
ejpam-5569	8	61	upper	upper	ADJ
ejpam-5569	8	62	bound	bind	VERB
ejpam-5569	8	63	of	of	ADP
ejpam-5569	8	64	radio	radio	NOUN
ejpam-5569	8	65	number	number	NOUN
ejpam-5569	8	66	of	of	ADP
ejpam-5569	8	67	shadow	shadow	NOUN
ejpam-5569	8	68	graphs	graph	NOUN
ejpam-5569	8	69	.	.	PUNCT
ejpam-5569	9	1	it	it	PRON
ejpam-5569	9	2	also	also	ADV
ejpam-5569	9	3	introduces	introduce	VERB
ejpam-5569	9	4	an	an	DET
ejpam-5569	9	5	application	application	NOUN
ejpam-5569	9	6	of	of	ADP
ejpam-5569	9	7	radio	radio	NOUN
ejpam-5569	9	8	labeling	labeling	NOUN
ejpam-5569	9	9	of	of	ADP
ejpam-5569	9	10	graphs	graph	NOUN
ejpam-5569	9	11	in	in	ADP
ejpam-5569	9	12	cryptography	cryptography	NOUN
ejpam-5569	9	13	.	.	PUNCT
ejpam-5569	10	1	additionally	additionally	ADV
ejpam-5569	10	2	,	,	PUNCT
ejpam-5569	10	3	a	a	DET
ejpam-5569	10	4	computational	computational	ADJ
ejpam-5569	10	5	study	study	NOUN
ejpam-5569	10	6	has	have	AUX
ejpam-5569	10	7	been	be	AUX
ejpam-5569	10	8	conducted	conduct	VERB
ejpam-5569	10	9	wherein	wherein	SCONJ
ejpam-5569	10	10	it	it	PRON
ejpam-5569	10	11	demonstrates	demonstrate	VERB
ejpam-5569	10	12	that	that	SCONJ
ejpam-5569	10	13	our	our	PRON
ejpam-5569	10	14	results	result	NOUN
ejpam-5569	10	15	(	(	PUNCT
ejpam-5569	10	16	theorems	theorem	NOUN
ejpam-5569	10	17	1	1	NUM
ejpam-5569	10	18	-	-	SYM
ejpam-5569	10	19	4	4	NUM
ejpam-5569	10	20	)	)	PUNCT
ejpam-5569	10	21	outperform	outperform	NOUN
ejpam-5569	10	22	both	both	CCONJ
ejpam-5569	10	23	the	the	DET
ejpam-5569	10	24	mathematical	mathematical	ADJ
ejpam-5569	10	25	model	model	NOUN
ejpam-5569	10	26	,	,	PUNCT
ejpam-5569	10	27	and	and	CCONJ
ejpam-5569	10	28	the	the	DET
ejpam-5569	10	29	results	result	NOUN
ejpam-5569	10	30	published	publish	VERB
ejpam-5569	10	31	in	in	ADP
ejpam-5569	10	32	the	the	DET
ejpam-5569	10	33	literature	literature	NOUN
ejpam-5569	10	34	.	.	PUNCT
ejpam-5569	11	1	2020	2020	NUM
ejpam-5569	11	2	mathematics	mathematics	PROPN
ejpam-5569	11	3	subject	subject	NOUN
ejpam-5569	11	4	classifications	classification	NOUN
ejpam-5569	11	5	:	:	PUNCT
ejpam-5569	11	6	05c78	05c78	NUM
ejpam-5569	11	7	,	,	PUNCT
ejpam-5569	11	8	05c12	05c12	NOUN
ejpam-5569	11	9	,	,	PUNCT
ejpam-5569	11	10	05c15	05c15	NOUN
ejpam-5569	11	11	∗corresponding	∗corresponde	VERB
ejpam-5569	11	12	author	author	NOUN
ejpam-5569	11	13	.	.	PUNCT
ejpam-5569	12	1	doi	doi	NOUN
ejpam-5569	12	2	:	:	PUNCT
ejpam-5569	12	3	https://doi.org/10.29020/nybg.ejpam.v18i2.5569	https://doi.org/10.29020/nybg.ejpam.v18i2.5569	NOUN
ejpam-5569	12	4	email	email	NOUN
ejpam-5569	12	5	addresses	address	NOUN
ejpam-5569	12	6	:	:	PUNCT
ejpam-5569	12	7	neldeen@taibahu.edu.sa	neldeen@taibahu.edu.sa	PROPN
ejpam-5569	12	8	(	(	PUNCT
ejpam-5569	12	9	n.	n.	PROPN
ejpam-5569	12	10	m.	m.	PROPN
ejpam-5569	12	11	noureldeen	noureldeen	PROPN
ejpam-5569	12	12	)	)	PUNCT
ejpam-5569	12	13	,	,	PUNCT
ejpam-5569	12	14	elsayed.badr@must.edu.eg	elsayed.badr@must.edu.eg	PROPN
ejpam-5569	12	15	(	(	PUNCT
ejpam-5569	12	16	e.	e.	PROPN
ejpam-5569	12	17	badr	badr	PROPN
ejpam-5569	12	18	)	)	PUNCT
ejpam-5569	12	19	,	,	PUNCT
ejpam-5569	12	20	hananshabana22@gmail.com	hananshabana22@gmail.com	X
ejpam-5569	12	21	(	(	PUNCT
ejpam-5569	12	22	h.	h.	PROPN
ejpam-5569	12	23	m.	m.	PROPN
ejpam-5569	12	24	shabana	shabana	PROPN
ejpam-5569	12	25	)	)	PUNCT
ejpam-5569	12	26	,	,	PUNCT
ejpam-5569	12	27	mohamed.abdelghani@fsc.bu.edu.eg	mohamed.abdelghani@fsc.bu.edu.eg	PROPN
ejpam-5569	12	28	(	(	PUNCT
ejpam-5569	12	29	m.	m.	PROPN
ejpam-5569	12	30	e.	e.	PROPN
ejpam-5569	12	31	abdel	abdel	PROPN
ejpam-5569	12	32	-	-	PUNCT
ejpam-5569	12	33	aal	aal	PROPN
ejpam-5569	12	34	)	)	PUNCT
ejpam-5569	12	35	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5569	12	36	1	1	NUM
ejpam-5569	12	37	copyright	copyright	NOUN
ejpam-5569	12	38	:	:	PUNCT
ejpam-5569	13	1	©	©	PROPN
ejpam-5569	13	2	2025	2025	NUM
ejpam-5569	13	3	the	the	DET
ejpam-5569	13	4	author(s	author(s	NOUN
ejpam-5569	13	5	)	)	PUNCT
ejpam-5569	13	6	.	.	PUNCT
ejpam-5569	14	1	(	(	PUNCT
ejpam-5569	14	2	cc	cc	NOUN
ejpam-5569	14	3	by	by	ADP
ejpam-5569	14	4	-	-	PUNCT
ejpam-5569	14	5	nc	nc	PROPN
ejpam-5569	14	6	4.0	4.0	NUM
ejpam-5569	14	7	)	)	PUNCT
ejpam-5569	14	8	n.	n.	NOUN
ejpam-5569	14	9	m.	m.	NOUN
ejpam-5569	14	10	noureldeen	noureldeen	NOUN
ejpam-5569	14	11	et	et	PROPN
ejpam-5569	14	12	al	al	PROPN
ejpam-5569	14	13	.	.	PUNCT
ejpam-5569	14	14	/	/	SYM
ejpam-5569	14	15	eur	eur	PROPN
ejpam-5569	14	16	.	.	PUNCT
ejpam-5569	15	1	j.	j.	PROPN
ejpam-5569	15	2	pure	pure	PROPN
ejpam-5569	15	3	appl	appl	PROPN
ejpam-5569	15	4	.	.	PROPN
ejpam-5569	15	5	math	math	PROPN
ejpam-5569	15	6	,	,	PUNCT
ejpam-5569	15	7	18	18	NUM
ejpam-5569	15	8	(	(	PUNCT
ejpam-5569	15	9	2	2	NUM
ejpam-5569	15	10	)	)	PUNCT
ejpam-5569	15	11	(	(	PUNCT
ejpam-5569	15	12	2025	2025	NUM
ejpam-5569	15	13	)	)	PUNCT
ejpam-5569	15	14	,	,	PUNCT
ejpam-5569	15	15	5569	5569	NUM
ejpam-5569	15	16	2	2	NUM
ejpam-5569	15	17	of	of	ADP
ejpam-5569	15	18	25	25	NUM
ejpam-5569	15	19	key	key	ADJ
ejpam-5569	15	20	words	word	NOUN
ejpam-5569	15	21	and	and	CCONJ
ejpam-5569	15	22	phrases	phrase	NOUN
ejpam-5569	15	23	:	:	PUNCT
ejpam-5569	15	24	frequency(channel	frequency(channel	PRON
ejpam-5569	15	25	)	)	PUNCT
ejpam-5569	15	26	assignment	assignment	NOUN
ejpam-5569	15	27	problem	problem	NOUN
ejpam-5569	15	28	,	,	PUNCT
ejpam-5569	15	29	radio	radio	NOUN
ejpam-5569	15	30	labeling	labeling	NOUN
ejpam-5569	15	31	,	,	PUNCT
ejpam-5569	15	32	integer	integer	NOUN
ejpam-5569	15	33	linear	linear	PROPN
ejpam-5569	15	34	programing	programing	NOUN
ejpam-5569	15	35	,	,	PUNCT
ejpam-5569	15	36	radio	radio	NOUN
ejpam-5569	15	37	number	number	NOUN
ejpam-5569	15	38	,	,	PUNCT
ejpam-5569	15	39	cryptography	cryptography	NOUN
ejpam-5569	15	40	.	.	PUNCT
ejpam-5569	16	1	an	an	DET
ejpam-5569	16	2	overview	overview	NOUN
ejpam-5569	16	3	of	of	ADP
ejpam-5569	16	4	the	the	DET
ejpam-5569	16	5	notations	notation	NOUN
ejpam-5569	16	6	that	that	PRON
ejpam-5569	16	7	will	will	AUX
ejpam-5569	16	8	be	be	AUX
ejpam-5569	16	9	applied	apply	VERB
ejpam-5569	16	10	throughout	throughout	ADP
ejpam-5569	16	11	the	the	DET
ejpam-5569	16	12	paper	paper	NOUN
ejpam-5569	16	13	,	,	PUNCT
ejpam-5569	16	14	listed	list	VERB
ejpam-5569	16	15	in	in	ADP
ejpam-5569	16	16	table	table	NOUN
ejpam-5569	16	17	1	1	NUM
ejpam-5569	16	18	.	.	PUNCT
ejpam-5569	16	19	table	table	NOUN
ejpam-5569	16	20	1	1	NUM
ejpam-5569	16	21	:	:	PUNCT
ejpam-5569	16	22	table	table	NOUN
ejpam-5569	16	23	of	of	ADP
ejpam-5569	16	24	notations	notation	NOUN
ejpam-5569	16	25	.	.	PUNCT
ejpam-5569	17	1	parameter	parameter	NOUN
ejpam-5569	17	2	description	description	NOUN
ejpam-5569	17	3	v	v	ADP
ejpam-5569	17	4	(	(	PUNCT
ejpam-5569	17	5	g	g	NOUN
ejpam-5569	17	6	)	)	PUNCT
ejpam-5569	17	7	the	the	DET
ejpam-5569	17	8	set	set	NOUN
ejpam-5569	17	9	of	of	ADP
ejpam-5569	17	10	vertices	vertex	NOUN
ejpam-5569	17	11	of	of	ADP
ejpam-5569	17	12	the	the	DET
ejpam-5569	17	13	graph	graph	NOUN
ejpam-5569	17	14	g.	g.	PROPN
ejpam-5569	17	15	e	e	PROPN
ejpam-5569	17	16	(	(	PUNCT
ejpam-5569	17	17	g	g	NOUN
ejpam-5569	17	18	)	)	PUNCT
ejpam-5569	17	19	the	the	DET
ejpam-5569	17	20	set	set	NOUN
ejpam-5569	17	21	of	of	ADP
ejpam-5569	17	22	edges	edge	NOUN
ejpam-5569	17	23	of	of	ADP
ejpam-5569	17	24	the	the	DET
ejpam-5569	17	25	graph	graph	NOUN
ejpam-5569	17	26	g.	g.	PROPN
ejpam-5569	17	27	d	d	PROPN
ejpam-5569	17	28	(	(	PUNCT
ejpam-5569	17	29	u	u	NOUN
ejpam-5569	17	30	,	,	PUNCT
ejpam-5569	17	31	v	v	NOUN
ejpam-5569	17	32	)	)	PUNCT
ejpam-5569	17	33	the	the	DET
ejpam-5569	17	34	distance	distance	NOUN
ejpam-5569	17	35	between	between	ADP
ejpam-5569	17	36	two	two	NUM
ejpam-5569	17	37	vertices	vertex	NOUN
ejpam-5569	17	38	u	u	NOUN
ejpam-5569	17	39	,	,	PUNCT
ejpam-5569	17	40	v	v	NOUN
ejpam-5569	17	41	in	in	ADP
ejpam-5569	17	42	a	a	DET
ejpam-5569	17	43	graph	graph	NOUN
ejpam-5569	17	44	.	.	PUNCT
ejpam-5569	18	1	diam	diam	PROPN
ejpam-5569	18	2	(	(	PUNCT
ejpam-5569	18	3	g	g	NOUN
ejpam-5569	18	4	)	)	PUNCT
ejpam-5569	18	5	the	the	DET
ejpam-5569	18	6	diameter	diameter	NOUN
ejpam-5569	18	7	of	of	ADP
ejpam-5569	18	8	g.	g.	PROPN
ejpam-5569	18	9	pn	pn	PROPN
ejpam-5569	18	10	path	path	NOUN
ejpam-5569	18	11	of	of	ADP
ejpam-5569	18	12	n	n	PRON
ejpam-5569	18	13	vertices	vertex	NOUN
ejpam-5569	18	14	.	.	PUNCT
ejpam-5569	19	1	cn	cn	PROPN
ejpam-5569	19	2	cycle	cycle	NOUN
ejpam-5569	19	3	with	with	ADP
ejpam-5569	19	4	n	n	ADP
ejpam-5569	19	5	vertices	vertex	NOUN
ejpam-5569	19	6	.	.	PUNCT
ejpam-5569	20	1	rn	rn	PROPN
ejpam-5569	20	2	(	(	PUNCT
ejpam-5569	20	3	g	g	NOUN
ejpam-5569	20	4	)	)	PUNCT
ejpam-5569	20	5	the	the	DET
ejpam-5569	20	6	radio	radio	NOUN
ejpam-5569	20	7	number	number	NOUN
ejpam-5569	20	8	of	of	ADP
ejpam-5569	20	9	g.	g.	PROPN
ejpam-5569	20	10	fap	fap	PROPN
ejpam-5569	20	11	frequency	frequency	PROPN
ejpam-5569	20	12	assignment	assignment	NOUN
ejpam-5569	20	13	problem	problem	NOUN
ejpam-5569	20	14	d2	d2	PROPN
ejpam-5569	20	15	(	(	PUNCT
ejpam-5569	20	16	pn	pn	PROPN
ejpam-5569	20	17	)	)	PUNCT
ejpam-5569	20	18	the	the	DET
ejpam-5569	20	19	shadow	shadow	NOUN
ejpam-5569	20	20	path	path	PROPN
ejpam-5569	20	21	graph	graph	NOUN
ejpam-5569	20	22	ilpm	ilpm	PROPN
ejpam-5569	20	23	integer	integer	PROPN
ejpam-5569	20	24	linear	linear	PROPN
ejpam-5569	20	25	programming	programming	NOUN
ejpam-5569	20	26	model	model	NOUN
ejpam-5569	20	27	1	1	NUM
ejpam-5569	20	28	.	.	PUNCT
ejpam-5569	20	29	introduction	introduction	NOUN
ejpam-5569	20	30	the	the	DET
ejpam-5569	20	31	communication	communication	NOUN
ejpam-5569	20	32	in	in	ADP
ejpam-5569	20	33	wireless	wireless	ADJ
ejpam-5569	20	34	networks	network	NOUN
ejpam-5569	20	35	depends	depend	VERB
ejpam-5569	20	36	on	on	ADP
ejpam-5569	20	37	the	the	DET
ejpam-5569	20	38	frequencies	frequency	NOUN
ejpam-5569	20	39	(	(	PUNCT
ejpam-5569	20	40	channels	channel	NOUN
ejpam-5569	20	41	)	)	PUNCT
ejpam-5569	20	42	allotted	allot	VERB
ejpam-5569	20	43	to	to	ADP
ejpam-5569	20	44	them	they	PRON
ejpam-5569	20	45	.	.	PUNCT
ejpam-5569	21	1	for	for	ADP
ejpam-5569	21	2	an	an	DET
ejpam-5569	21	3	effective	effective	ADJ
ejpam-5569	21	4	communication	communication	NOUN
ejpam-5569	21	5	,	,	PUNCT
ejpam-5569	21	6	the	the	DET
ejpam-5569	21	7	frequencies	frequency	NOUN
ejpam-5569	21	8	should	should	AUX
ejpam-5569	21	9	assigned	assign	VERB
ejpam-5569	21	10	to	to	ADP
ejpam-5569	21	11	transmitters	transmitter	NOUN
ejpam-5569	21	12	in	in	ADP
ejpam-5569	21	13	such	such	DET
ejpam-5569	21	14	a	a	DET
ejpam-5569	21	15	way	way	NOUN
ejpam-5569	21	16	that	that	PRON
ejpam-5569	21	17	avoids	avoid	VERB
ejpam-5569	21	18	the	the	DET
ejpam-5569	21	19	interference	interference	NOUN
ejpam-5569	21	20	.	.	PUNCT
ejpam-5569	22	1	here	here	ADV
ejpam-5569	22	2	,	,	PUNCT
ejpam-5569	22	3	the	the	DET
ejpam-5569	22	4	interference	interference	NOUN
ejpam-5569	22	5	is	be	AUX
ejpam-5569	22	6	closely	closely	ADV
ejpam-5569	22	7	related	relate	VERB
ejpam-5569	22	8	to	to	ADP
ejpam-5569	22	9	the	the	DET
ejpam-5569	22	10	geographical	geographical	ADJ
ejpam-5569	22	11	position	position	NOUN
ejpam-5569	22	12	of	of	ADP
ejpam-5569	22	13	the	the	DET
ejpam-5569	22	14	transmitting	transmit	VERB
ejpam-5569	22	15	stations	station	NOUN
ejpam-5569	22	16	.	.	PUNCT
ejpam-5569	23	1	the	the	DET
ejpam-5569	23	2	interference	interference	NOUN
ejpam-5569	23	3	is	be	AUX
ejpam-5569	23	4	increasing	increase	VERB
ejpam-5569	23	5	whenever	whenever	SCONJ
ejpam-5569	23	6	the	the	DET
ejpam-5569	23	7	stations	station	NOUN
ejpam-5569	23	8	are	be	AUX
ejpam-5569	23	9	geographical	geographical	ADJ
ejpam-5569	23	10	close	close	ADV
ejpam-5569	23	11	.	.	PUNCT
ejpam-5569	24	1	to	to	PART
ejpam-5569	24	2	avoid	avoid	VERB
ejpam-5569	24	3	such	such	ADJ
ejpam-5569	24	4	interference	interference	NOUN
ejpam-5569	24	5	,	,	PUNCT
ejpam-5569	24	6	we	we	PRON
ejpam-5569	24	7	have	have	VERB
ejpam-5569	24	8	to	to	PART
ejpam-5569	24	9	assign	assign	VERB
ejpam-5569	24	10	frequencies	frequency	NOUN
ejpam-5569	24	11	to	to	ADP
ejpam-5569	24	12	stations	station	NOUN
ejpam-5569	24	13	such	such	ADJ
ejpam-5569	24	14	that	that	DET
ejpam-5569	24	15	difference	difference	NOUN
ejpam-5569	24	16	between	between	ADP
ejpam-5569	24	17	the	the	DET
ejpam-5569	24	18	assigned	assign	VERB
ejpam-5569	24	19	channels	channel	NOUN
ejpam-5569	24	20	must	must	AUX
ejpam-5569	24	21	be	be	AUX
ejpam-5569	24	22	large	large	ADJ
ejpam-5569	24	23	enough	enough	ADV
ejpam-5569	24	24	.	.	PUNCT
ejpam-5569	25	1	the	the	DET
ejpam-5569	25	2	process	process	NOUN
ejpam-5569	25	3	of	of	ADP
ejpam-5569	25	4	assigning	assign	VERB
ejpam-5569	25	5	a	a	DET
ejpam-5569	25	6	limited	limited	ADJ
ejpam-5569	25	7	number	number	NOUN
ejpam-5569	25	8	of	of	ADP
ejpam-5569	25	9	available	available	ADJ
ejpam-5569	25	10	frequencies	frequency	NOUN
ejpam-5569	25	11	to	to	ADP
ejpam-5569	25	12	transmitters	transmitter	NOUN
ejpam-5569	25	13	in	in	ADP
ejpam-5569	25	14	radio	radio	NOUN
ejpam-5569	25	15	networks	network	NOUN
ejpam-5569	25	16	with	with	ADP
ejpam-5569	25	17	avoiding	avoid	VERB
ejpam-5569	25	18	interference	interference	NOUN
ejpam-5569	25	19	is	be	AUX
ejpam-5569	25	20	known	know	VERB
ejpam-5569	25	21	as	as	ADP
ejpam-5569	25	22	the	the	DET
ejpam-5569	25	23	frequency	frequency	NOUN
ejpam-5569	25	24	assignment	assignment	NOUN
ejpam-5569	25	25	problem	problem	NOUN
ejpam-5569	25	26	(	(	PUNCT
ejpam-5569	25	27	fap	fap	NOUN
ejpam-5569	25	28	)	)	PUNCT
ejpam-5569	25	29	.	.	PUNCT
ejpam-5569	26	1	wireless	wireless	ADJ
ejpam-5569	26	2	networks	network	NOUN
ejpam-5569	26	3	appear	appear	VERB
ejpam-5569	26	4	in	in	ADP
ejpam-5569	26	5	a	a	DET
ejpam-5569	26	6	variety	variety	NOUN
ejpam-5569	26	7	of	of	ADP
ejpam-5569	26	8	services	service	NOUN
ejpam-5569	26	9	such	such	ADJ
ejpam-5569	26	10	as	as	ADP
ejpam-5569	26	11	t.v	t.v	PROPN
ejpam-5569	26	12	.	.	PROPN
ejpam-5569	26	13	and	and	CCONJ
ejpam-5569	26	14	radio	radio	NOUN
ejpam-5569	26	15	broadcasting	broadcasting	NOUN
ejpam-5569	26	16	,	,	PUNCT
ejpam-5569	26	17	engineering	engineering	NOUN
ejpam-5569	26	18	,	,	PUNCT
ejpam-5569	26	19	economics	economic	NOUN
ejpam-5569	26	20	,	,	PUNCT
ejpam-5569	26	21	military	military	ADJ
ejpam-5569	26	22	services	service	NOUN
ejpam-5569	26	23	,	,	PUNCT
ejpam-5569	26	24	and	and	CCONJ
ejpam-5569	26	25	many	many	ADJ
ejpam-5569	26	26	other	other	ADJ
ejpam-5569	26	27	.	.	PUNCT
ejpam-5569	27	1	due	due	ADP
ejpam-5569	27	2	to	to	ADP
ejpam-5569	27	3	increasing	increase	VERB
ejpam-5569	27	4	popularity	popularity	NOUN
ejpam-5569	27	5	of	of	ADP
ejpam-5569	27	6	wireless	wireless	ADJ
ejpam-5569	27	7	services	service	NOUN
ejpam-5569	27	8	and	and	CCONJ
ejpam-5569	27	9	the	the	DET
ejpam-5569	27	10	limited	limited	ADJ
ejpam-5569	27	11	available	available	ADJ
ejpam-5569	27	12	frequencies	frequency	NOUN
ejpam-5569	27	13	,	,	PUNCT
ejpam-5569	27	14	fap	fap	NOUN
ejpam-5569	27	15	has	have	VERB
ejpam-5569	27	16	a	a	DET
ejpam-5569	27	17	lot	lot	NOUN
ejpam-5569	27	18	of	of	ADP
ejpam-5569	27	19	interest	interest	NOUN
ejpam-5569	27	20	from	from	ADP
ejpam-5569	27	21	the	the	DET
ejpam-5569	27	22	scientific	scientific	ADJ
ejpam-5569	27	23	and	and	CCONJ
ejpam-5569	27	24	the	the	DET
ejpam-5569	27	25	business	business	NOUN
ejpam-5569	27	26	communities	community	NOUN
ejpam-5569	27	27	.	.	PUNCT
ejpam-5569	28	1	as	as	SCONJ
ejpam-5569	28	2	the	the	DET
ejpam-5569	28	3	optimizing	optimize	VERB
ejpam-5569	28	4	usage	usage	NOUN
ejpam-5569	28	5	of	of	ADP
ejpam-5569	28	6	the	the	DET
ejpam-5569	28	7	available	available	ADJ
ejpam-5569	28	8	frequencies	frequency	NOUN
ejpam-5569	28	9	means	mean	VERB
ejpam-5569	28	10	higher	high	ADJ
ejpam-5569	28	11	traffic	traffic	NOUN
ejpam-5569	28	12	capacity	capacity	NOUN
ejpam-5569	28	13	,	,	PUNCT
ejpam-5569	28	14	more	more	ADJ
ejpam-5569	28	15	bandwidth	bandwidth	ADJ
ejpam-5569	28	16	and	and	CCONJ
ejpam-5569	28	17	bigger	big	ADJ
ejpam-5569	28	18	coverage	coverage	NOUN
ejpam-5569	28	19	for	for	ADP
ejpam-5569	28	20	the	the	DET
ejpam-5569	28	21	existing	exist	VERB
ejpam-5569	28	22	radio	radio	NOUN
ejpam-5569	28	23	networks	network	NOUN
ejpam-5569	28	24	.	.	PUNCT
ejpam-5569	29	1	as	as	ADP
ejpam-5569	29	2	a	a	DET
ejpam-5569	29	3	result	result	NOUN
ejpam-5569	29	4	,	,	PUNCT
ejpam-5569	29	5	many	many	ADJ
ejpam-5569	29	6	researchers	researcher	NOUN
ejpam-5569	29	7	and	and	CCONJ
ejpam-5569	29	8	a	a	DET
ejpam-5569	29	9	wide	wide	ADJ
ejpam-5569	29	10	variety	variety	NOUN
ejpam-5569	29	11	of	of	ADP
ejpam-5569	29	12	models	model	NOUN
ejpam-5569	29	13	have	have	AUX
ejpam-5569	29	14	investigated	investigate	VERB
ejpam-5569	29	15	faps	fap	NOUN
ejpam-5569	29	16	and	and	CCONJ
ejpam-5569	29	17	solution	solution	NOUN
ejpam-5569	29	18	techniques	technique	NOUN
ejpam-5569	29	19	have	have	AUX
ejpam-5569	29	20	been	be	AUX
ejpam-5569	29	21	proposed	propose	VERB
ejpam-5569	29	22	.	.	PUNCT
ejpam-5569	30	1	the	the	DET
ejpam-5569	30	2	labeling	labeling	NOUN
ejpam-5569	30	3	technique	technique	NOUN
ejpam-5569	30	4	in	in	ADP
ejpam-5569	30	5	graph	graph	NOUN
ejpam-5569	30	6	theory	theory	NOUN
ejpam-5569	30	7	has	have	AUX
ejpam-5569	30	8	played	play	VERB
ejpam-5569	30	9	an	an	DET
ejpam-5569	30	10	important	important	ADJ
ejpam-5569	30	11	role	role	NOUN
ejpam-5569	30	12	in	in	ADP
ejpam-5569	30	13	solving	solve	VERB
ejpam-5569	30	14	fap	fap	NOUN
ejpam-5569	30	15	;	;	PUNCT
ejpam-5569	30	16	thereby	thereby	ADV
ejpam-5569	30	17	the	the	DET
ejpam-5569	30	18	time	time	NOUN
ejpam-5569	30	19	and	and	CCONJ
ejpam-5569	30	20	cost	cost	NOUN
ejpam-5569	30	21	will	will	AUX
ejpam-5569	30	22	be	be	AUX
ejpam-5569	30	23	saved	save	VERB
ejpam-5569	30	24	.	.	PUNCT
ejpam-5569	31	1	for	for	ADP
ejpam-5569	31	2	modeling	model	VERB
ejpam-5569	31	3	fap	fap	NOUN
ejpam-5569	31	4	in	in	ADP
ejpam-5569	31	5	graph	graph	NOUN
ejpam-5569	31	6	theory	theory	NOUN
ejpam-5569	31	7	,	,	PUNCT
ejpam-5569	31	8	an	an	DET
ejpam-5569	31	9	interference	interference	NOUN
ejpam-5569	31	10	graph	graph	NOUN
ejpam-5569	31	11	g	g	PROPN
ejpam-5569	31	12	=	=	PUNCT
ejpam-5569	31	13	(	(	PUNCT
ejpam-5569	31	14	v	v	NOUN
ejpam-5569	31	15	(	(	PUNCT
ejpam-5569	31	16	g	g	NOUN
ejpam-5569	31	17	)	)	PUNCT
ejpam-5569	31	18	,	,	PUNCT
ejpam-5569	31	19	e	e	X
ejpam-5569	31	20	(	(	PUNCT
ejpam-5569	31	21	g))is	g))is	PROPN
ejpam-5569	31	22	constructed	construct	VERB
ejpam-5569	31	23	where	where	SCONJ
ejpam-5569	31	24	v	v	NOUN
ejpam-5569	31	25	(	(	PUNCT
ejpam-5569	31	26	g	g	NOUN
ejpam-5569	31	27	)	)	PUNCT
ejpam-5569	31	28	and	and	CCONJ
ejpam-5569	31	29	e	e	X
ejpam-5569	31	30	(	(	PUNCT
ejpam-5569	31	31	g	g	NOUN
ejpam-5569	31	32	)	)	PUNCT
ejpam-5569	31	33	are	be	AUX
ejpam-5569	31	34	the	the	DET
ejpam-5569	31	35	set	set	NOUN
ejpam-5569	31	36	of	of	ADP
ejpam-5569	31	37	vertices	vertex	NOUN
ejpam-5569	31	38	and	and	CCONJ
ejpam-5569	31	39	the	the	DET
ejpam-5569	31	40	set	set	NOUN
ejpam-5569	31	41	of	of	ADP
ejpam-5569	31	42	edges	edge	NOUN
ejpam-5569	31	43	of	of	ADP
ejpam-5569	31	44	g	g	NOUN
ejpam-5569	31	45	respectively	respectively	ADV
ejpam-5569	31	46	.	.	PUNCT
ejpam-5569	32	1	this	this	DET
ejpam-5569	32	2	graph	graph	NOUN
ejpam-5569	32	3	represents	represent	VERB
ejpam-5569	32	4	the	the	DET
ejpam-5569	32	5	interference	interference	NOUN
ejpam-5569	32	6	between	between	ADP
ejpam-5569	32	7	transmitters	transmitter	NOUN
ejpam-5569	32	8	.	.	PUNCT
ejpam-5569	33	1	each	each	DET
ejpam-5569	33	2	vertex	vertex	NOUN
ejpam-5569	33	3	belong	belong	VERB
ejpam-5569	33	4	to	to	ADP
ejpam-5569	33	5	v	v	NOUN
ejpam-5569	33	6	(	(	PUNCT
ejpam-5569	33	7	g	g	NOUN
ejpam-5569	33	8	)	)	PUNCT
ejpam-5569	33	9	represents	represent	VERB
ejpam-5569	33	10	a	a	DET
ejpam-5569	33	11	unique	unique	ADJ
ejpam-5569	33	12	transmitter	transmitter	NOUN
ejpam-5569	33	13	.	.	PUNCT
ejpam-5569	34	1	in	in	ADP
ejpam-5569	34	2	the	the	DET
ejpam-5569	34	3	event	event	NOUN
ejpam-5569	34	4	of	of	ADP
ejpam-5569	34	5	the	the	DET
ejpam-5569	34	6	broadcasting	broadcasting	NOUN
ejpam-5569	34	7	of	of	ADP
ejpam-5569	34	8	the	the	DET
ejpam-5569	34	9	two	two	NUM
ejpam-5569	34	10	transmitters	transmitter	NOUN
ejpam-5569	34	11	may	may	AUX
ejpam-5569	34	12	interfere	interfere	VERB
ejpam-5569	34	13	,	,	PUNCT
ejpam-5569	34	14	then	then	ADV
ejpam-5569	34	15	their	their	PRON
ejpam-5569	34	16	corresponding	corresponding	ADJ
ejpam-5569	34	17	vertices	vertex	NOUN
ejpam-5569	34	18	from	from	ADP
ejpam-5569	34	19	v	v	ADP
ejpam-5569	34	20	(	(	PUNCT
ejpam-5569	34	21	g	g	NOUN
ejpam-5569	34	22	)	)	PUNCT
ejpam-5569	34	23	are	be	AUX
ejpam-5569	34	24	joined	join	VERB
ejpam-5569	34	25	by	by	ADP
ejpam-5569	34	26	an	an	DET
ejpam-5569	34	27	n.	n.	NOUN
ejpam-5569	34	28	m.	m.	NOUN
ejpam-5569	34	29	noureldeen	noureldeen	NOUN
ejpam-5569	34	30	et	et	PROPN
ejpam-5569	34	31	al	al	PROPN
ejpam-5569	34	32	.	.	PUNCT
ejpam-5569	34	33	/	/	SYM
ejpam-5569	34	34	eur	eur	PROPN
ejpam-5569	34	35	.	.	PUNCT
ejpam-5569	35	1	j.	j.	PROPN
ejpam-5569	35	2	pure	pure	PROPN
ejpam-5569	35	3	appl	appl	PROPN
ejpam-5569	35	4	.	.	PROPN
ejpam-5569	35	5	math	math	PROPN
ejpam-5569	35	6	,	,	PUNCT
ejpam-5569	35	7	18	18	NUM
ejpam-5569	35	8	(	(	PUNCT
ejpam-5569	35	9	2	2	NUM
ejpam-5569	35	10	)	)	PUNCT
ejpam-5569	35	11	(	(	PUNCT
ejpam-5569	35	12	2025	2025	NUM
ejpam-5569	35	13	)	)	PUNCT
ejpam-5569	35	14	,	,	PUNCT
ejpam-5569	35	15	5569	5569	NUM
ejpam-5569	35	16	3	3	NUM
ejpam-5569	35	17	of	of	ADP
ejpam-5569	35	18	25	25	NUM
ejpam-5569	35	19	edge	edge	NOUN
ejpam-5569	35	20	.	.	PUNCT
ejpam-5569	36	1	positive	positive	ADJ
ejpam-5569	36	2	integers	integer	NOUN
ejpam-5569	36	3	are	be	AUX
ejpam-5569	36	4	used	use	VERB
ejpam-5569	36	5	as	as	ADP
ejpam-5569	36	6	labels	label	NOUN
ejpam-5569	36	7	for	for	ADP
ejpam-5569	36	8	the	the	DET
ejpam-5569	36	9	frequency	frequency	NOUN
ejpam-5569	36	10	channels	channel	NOUN
ejpam-5569	36	11	.	.	PUNCT
ejpam-5569	37	1	therefore	therefore	ADV
ejpam-5569	37	2	,	,	PUNCT
ejpam-5569	37	3	fap	fap	NOUN
ejpam-5569	37	4	[	[	X
ejpam-5569	37	5	1	1	X
ejpam-5569	37	6	]	]	PUNCT
ejpam-5569	37	7	is	be	AUX
ejpam-5569	37	8	equivalent	equivalent	ADJ
ejpam-5569	37	9	to	to	ADP
ejpam-5569	37	10	the	the	DET
ejpam-5569	37	11	vertex	vertex	NOUN
ejpam-5569	37	12	coloring	coloring	NOUN
ejpam-5569	37	13	(	(	PUNCT
ejpam-5569	37	14	labeling	labeling	NOUN
ejpam-5569	37	15	)	)	PUNCT
ejpam-5569	37	16	issue	issue	NOUN
ejpam-5569	37	17	of	of	ADP
ejpam-5569	37	18	the	the	DET
ejpam-5569	37	19	graph	graph	NOUN
ejpam-5569	37	20	g	g	NOUN
ejpam-5569	37	21	with	with	ADP
ejpam-5569	37	22	certain	certain	ADJ
ejpam-5569	37	23	labeling	labeling	NOUN
ejpam-5569	37	24	constraints	constraint	NOUN
ejpam-5569	37	25	.	.	PUNCT
ejpam-5569	38	1	in	in	ADP
ejpam-5569	38	2	[	[	X
ejpam-5569	38	3	2	2	NUM
ejpam-5569	38	4	]	]	PUNCT
ejpam-5569	38	5	author	author	NOUN
ejpam-5569	38	6	examined	examine	VERB
ejpam-5569	38	7	a	a	DET
ejpam-5569	38	8	comprehensive	comprehensive	ADJ
ejpam-5569	38	9	overview	overview	NOUN
ejpam-5569	38	10	of	of	ADP
ejpam-5569	38	11	graph	graph	NOUN
ejpam-5569	38	12	labeling	labeling	NOUN
ejpam-5569	38	13	.	.	PUNCT
ejpam-5569	39	1	griggs	griggs	PROPN
ejpam-5569	39	2	and	and	CCONJ
ejpam-5569	39	3	yeh	yeh	NOUN
ejpam-5569	40	1	[	[	X
ejpam-5569	40	2	3	3	X
ejpam-5569	40	3	]	]	PUNCT
ejpam-5569	40	4	presented	present	VERB
ejpam-5569	40	5	the	the	DET
ejpam-5569	40	6	first	first	ADJ
ejpam-5569	40	7	graph	graph	NOUN
ejpam-5569	40	8	labeling	labeling	NOUN
ejpam-5569	40	9	technique	technique	NOUN
ejpam-5569	40	10	,	,	PUNCT
ejpam-5569	40	11	known	know	VERB
ejpam-5569	40	12	as	as	ADP
ejpam-5569	40	13	l(2	l(2	PROPN
ejpam-5569	40	14	,	,	PUNCT
ejpam-5569	40	15	1	1	X
ejpam-5569	40	16	)	)	PUNCT
ejpam-5569	40	17	labeling	labeling	NOUN
ejpam-5569	40	18	or	or	CCONJ
ejpam-5569	40	19	distance	distance	NOUN
ejpam-5569	40	20	two	two	NUM
ejpam-5569	40	21	labeling	labeling	NOUN
ejpam-5569	40	22	to	to	PART
ejpam-5569	40	23	address	address	VERB
ejpam-5569	40	24	the	the	DET
ejpam-5569	40	25	frequency	frequency	NOUN
ejpam-5569	40	26	assignment	assignment	NOUN
ejpam-5569	40	27	problem	problem	NOUN
ejpam-5569	40	28	.	.	PUNCT
ejpam-5569	41	1	the	the	DET
ejpam-5569	41	2	graph	graph	NOUN
ejpam-5569	41	3	labeling	labeling	PROPN
ejpam-5569	41	4	l(2	l(2	PROPN
ejpam-5569	41	5	,	,	PUNCT
ejpam-5569	41	6	1	1	NUM
ejpam-5569	41	7	)	)	PUNCT
ejpam-5569	41	8	of	of	ADP
ejpam-5569	41	9	a	a	DET
ejpam-5569	41	10	simple	simple	ADJ
ejpam-5569	41	11	graph	graph	NOUN
ejpam-5569	41	12	g	g	PROPN
ejpam-5569	41	13	is	be	AUX
ejpam-5569	41	14	a	a	DET
ejpam-5569	41	15	non	non	ADJ
ejpam-5569	41	16	-	-	ADJ
ejpam-5569	41	17	negative	negative	ADJ
ejpam-5569	41	18	real	real	ADV
ejpam-5569	41	19	-	-	PUNCT
ejpam-5569	41	20	valued	value	VERB
ejpam-5569	41	21	function	function	NOUN
ejpam-5569	41	22	l	l	NOUN
ejpam-5569	41	23	:	:	PUNCT
ejpam-5569	41	24	v	v	X
ejpam-5569	41	25	(	(	PUNCT
ejpam-5569	41	26	g	g	NOUN
ejpam-5569	41	27	)	)	PUNCT
ejpam-5569	41	28	→	→	PUNCT
ejpam-5569	42	1	[	[	X
ejpam-5569	42	2	0	0	NUM
ejpam-5569	42	3	,	,	PUNCT
ejpam-5569	42	4	∞	∞	PROPN
ejpam-5569	42	5	)	)	PUNCT
ejpam-5569	42	6	where	where	SCONJ
ejpam-5569	42	7	v	v	X
ejpam-5569	42	8	(	(	PUNCT
ejpam-5569	42	9	g	g	NOUN
ejpam-5569	42	10	)	)	PUNCT
ejpam-5569	42	11	is	be	AUX
ejpam-5569	42	12	the	the	DET
ejpam-5569	42	13	vertex	vertex	NOUN
ejpam-5569	42	14	set	set	NOUN
ejpam-5569	42	15	of	of	ADP
ejpam-5569	42	16	the	the	DET
ejpam-5569	42	17	graph	graph	NOUN
ejpam-5569	42	18	g.	g.	NOUN
ejpam-5569	42	19	whenever	whenever	SCONJ
ejpam-5569	42	20	u	u	PROPN
ejpam-5569	42	21	and	and	CCONJ
ejpam-5569	42	22	v	v	NOUN
ejpam-5569	42	23	are	be	AUX
ejpam-5569	42	24	two	two	NUM
ejpam-5569	42	25	vertices	vertex	NOUN
ejpam-5569	42	26	in	in	ADP
ejpam-5569	42	27	g	g	PROPN
ejpam-5569	42	28	such	such	ADJ
ejpam-5569	42	29	that	that	SCONJ
ejpam-5569	42	30	the	the	DET
ejpam-5569	42	31	distance	distance	NOUN
ejpam-5569	42	32	between	between	ADP
ejpam-5569	42	33	u	u	NOUN
ejpam-5569	42	34	and	and	CCONJ
ejpam-5569	42	35	v	v	NOUN
ejpam-5569	42	36	is	be	AUX
ejpam-5569	42	37	1	1	NUM
ejpam-5569	42	38	,	,	PUNCT
ejpam-5569	42	39	then	then	ADV
ejpam-5569	42	40	|	|	ADV
ejpam-5569	42	41	l(u	l(u	PROPN
ejpam-5569	42	42	)	)	PUNCT
ejpam-5569	42	43	−	−	PROPN
ejpam-5569	42	44	l(v	l(v	NOUN
ejpam-5569	42	45	)	)	PUNCT
ejpam-5569	43	1	|	|	ADV
ejpam-5569	43	2	≥	≥	NOUN
ejpam-5569	43	3	2	2	NUM
ejpam-5569	43	4	,	,	PUNCT
ejpam-5569	43	5	and	and	CCONJ
ejpam-5569	43	6	whenever	whenever	SCONJ
ejpam-5569	43	7	the	the	DET
ejpam-5569	43	8	vertices	vertex	NOUN
ejpam-5569	43	9	u	u	NOUN
ejpam-5569	43	10	and	and	CCONJ
ejpam-5569	43	11	v	v	NOUN
ejpam-5569	43	12	are	be	AUX
ejpam-5569	43	13	of	of	ADP
ejpam-5569	43	14	distance	distance	NOUN
ejpam-5569	43	15	2	2	NUM
ejpam-5569	43	16	from	from	ADP
ejpam-5569	43	17	each	each	DET
ejpam-5569	43	18	other	other	ADJ
ejpam-5569	43	19	,	,	PUNCT
ejpam-5569	43	20	then	then	ADV
ejpam-5569	43	21	|	|	ADV
ejpam-5569	43	22	l(u	l(u	PROPN
ejpam-5569	43	23	)	)	PUNCT
ejpam-5569	43	24	−	−	PROPN
ejpam-5569	43	25	l(v	l(v	NOUN
ejpam-5569	43	26	)	)	PUNCT
ejpam-5569	44	1	|	|	ADV
ejpam-5569	44	2	≥	≥	X
ejpam-5569	44	3	1|	1|	NUM
ejpam-5569	44	4	.	.	PUNCT
ejpam-5569	45	1	another	another	DET
ejpam-5569	45	2	graph	graph	NOUN
ejpam-5569	45	3	labeling	labeling	NOUN
ejpam-5569	45	4	technique	technique	NOUN
ejpam-5569	45	5	called	call	VERB
ejpam-5569	45	6	radio	radio	NOUN
ejpam-5569	45	7	labeling	labeling	NOUN
ejpam-5569	45	8	was	be	AUX
ejpam-5569	45	9	introduced	introduce	VERB
ejpam-5569	45	10	by	by	ADP
ejpam-5569	45	11	chartrand	chartrand	PROPN
ejpam-5569	45	12	et	et	PROPN
ejpam-5569	45	13	al	al	PROPN
ejpam-5569	45	14	.	.	PUNCT
ejpam-5569	46	1	[	[	X
ejpam-5569	46	2	4	4	NUM
ejpam-5569	46	3	]	]	PUNCT
ejpam-5569	46	4	.	.	PUNCT
ejpam-5569	47	1	the	the	DET
ejpam-5569	47	2	radio	radio	NOUN
ejpam-5569	47	3	labeling	labeling	NOUN
ejpam-5569	47	4	problem	problem	NOUN
ejpam-5569	47	5	of	of	ADP
ejpam-5569	47	6	a	a	DET
ejpam-5569	47	7	connected	connected	ADJ
ejpam-5569	47	8	graph	graph	NOUN
ejpam-5569	47	9	g	g	PROPN
ejpam-5569	47	10	=	=	PUNCT
ejpam-5569	47	11	(	(	PUNCT
ejpam-5569	47	12	v	v	NOUN
ejpam-5569	47	13	(	(	PUNCT
ejpam-5569	47	14	g	g	NOUN
ejpam-5569	47	15	)	)	PUNCT
ejpam-5569	47	16	,	,	PUNCT
ejpam-5569	47	17	e	e	X
ejpam-5569	47	18	(	(	PUNCT
ejpam-5569	47	19	g))is	g))is	PROPN
ejpam-5569	47	20	denoted	denote	VERB
ejpam-5569	47	21	as	as	SCONJ
ejpam-5569	47	22	follows	follow	VERB
ejpam-5569	47	23	.	.	PUNCT
ejpam-5569	48	1	let	let	VERB
ejpam-5569	48	2	u	u	NOUN
ejpam-5569	48	3	,	,	PUNCT
ejpam-5569	48	4	v	v	PROPN
ejpam-5569	48	5	∈	∈	PROPN
ejpam-5569	48	6	v	v	NOUN
ejpam-5569	48	7	(	(	PUNCT
ejpam-5569	48	8	g	g	NOUN
ejpam-5569	48	9	)	)	PUNCT
ejpam-5569	48	10	.	.	PUNCT
ejpam-5569	49	1	the	the	DET
ejpam-5569	49	2	distance	distance	NOUN
ejpam-5569	49	3	d	d	X
ejpam-5569	49	4	(	(	PUNCT
ejpam-5569	49	5	u	u	NOUN
ejpam-5569	49	6	,	,	PUNCT
ejpam-5569	49	7	v	v	NOUN
ejpam-5569	49	8	)	)	PUNCT
ejpam-5569	49	9	denote	denote	VERB
ejpam-5569	49	10	the	the	DET
ejpam-5569	49	11	length	length	NOUN
ejpam-5569	49	12	of	of	ADP
ejpam-5569	49	13	the	the	DET
ejpam-5569	49	14	shortest	short	ADJ
ejpam-5569	49	15	path	path	NOUN
ejpam-5569	49	16	between	between	ADP
ejpam-5569	49	17	u	u	PROPN
ejpam-5569	49	18	,	,	PUNCT
ejpam-5569	49	19	v.	v.	ADP
ejpam-5569	49	20	the	the	DET
ejpam-5569	49	21	diameter	diameter	NOUN
ejpam-5569	49	22	diam	diam	PROPN
ejpam-5569	49	23	(	(	PUNCT
ejpam-5569	49	24	g	g	NOUN
ejpam-5569	49	25	)	)	PUNCT
ejpam-5569	49	26	of	of	ADP
ejpam-5569	49	27	g	g	PROPN
ejpam-5569	49	28	is	be	AUX
ejpam-5569	49	29	defined	define	VERB
ejpam-5569	49	30	as	as	ADP
ejpam-5569	49	31	the	the	DET
ejpam-5569	49	32	maximum	maximum	ADJ
ejpam-5569	49	33	distance	distance	NOUN
ejpam-5569	49	34	between	between	ADP
ejpam-5569	49	35	any	any	DET
ejpam-5569	49	36	two	two	NUM
ejpam-5569	49	37	vertices	vertex	NOUN
ejpam-5569	49	38	in	in	ADP
ejpam-5569	49	39	g.	g.	PROPN
ejpam-5569	49	40	thus	thus	ADV
ejpam-5569	49	41	,	,	PUNCT
ejpam-5569	49	42	diam	diam	PROPN
ejpam-5569	49	43	(	(	PUNCT
ejpam-5569	49	44	g	g	NOUN
ejpam-5569	49	45	)	)	PUNCT
ejpam-5569	50	1	=	=	SYM
ejpam-5569	50	2	max	max	X
ejpam-5569	50	3	{	{	PUNCT
ejpam-5569	50	4	d	d	X
ejpam-5569	50	5	(	(	PUNCT
ejpam-5569	50	6	u	u	NOUN
ejpam-5569	50	7	,	,	PUNCT
ejpam-5569	50	8	v	v	NOUN
ejpam-5569	50	9	)	)	PUNCT
ejpam-5569	50	10	:	:	PUNCT
ejpam-5569	50	11	u	u	NOUN
ejpam-5569	50	12	,	,	PUNCT
ejpam-5569	50	13	v	v	PROPN
ejpam-5569	50	14	∈	∈	PROPN
ejpam-5569	50	15	v	v	NOUN
ejpam-5569	50	16	(	(	PUNCT
ejpam-5569	50	17	g	g	NOUN
ejpam-5569	50	18	)	)	PUNCT
ejpam-5569	50	19	}	}	PUNCT
ejpam-5569	50	20	.	.	PUNCT
ejpam-5569	51	1	a	a	DET
ejpam-5569	51	2	radio	radio	NOUN
ejpam-5569	51	3	labeling	labeling	NOUN
ejpam-5569	51	4	of	of	ADP
ejpam-5569	51	5	g	g	PROPN
ejpam-5569	51	6	is	be	AUX
ejpam-5569	51	7	an	an	DET
ejpam-5569	51	8	injective	injective	ADJ
ejpam-5569	51	9	function	function	NOUN
ejpam-5569	51	10	f	f	NOUN
ejpam-5569	51	11	from	from	ADP
ejpam-5569	51	12	v	v	NUM
ejpam-5569	51	13	(	(	PUNCT
ejpam-5569	51	14	g	g	NOUN
ejpam-5569	51	15	)	)	PUNCT
ejpam-5569	51	16	to	to	ADP
ejpam-5569	51	17	n	n	NOUN
ejpam-5569	51	18	=	=	PUNCT
ejpam-5569	51	19	{	{	PUNCT
ejpam-5569	51	20	0	0	NUM
ejpam-5569	51	21	,	,	PUNCT
ejpam-5569	51	22	1	1	NUM
ejpam-5569	51	23	,	,	PUNCT
ejpam-5569	51	24	2	2	NUM
ejpam-5569	51	25	,	,	PUNCT
ejpam-5569	51	26	3	3	NUM
ejpam-5569	51	27	,	,	PUNCT
ejpam-5569	51	28	·	·	PUNCT
ejpam-5569	51	29	·	·	PUNCT
ejpam-5569	51	30	·	·	PUNCT
ejpam-5569	52	1	}	}	PUNCT
ejpam-5569	52	2	,	,	PUNCT
ejpam-5569	52	3	satisfying	satisfy	VERB
ejpam-5569	52	4	the	the	DET
ejpam-5569	52	5	following	follow	VERB
ejpam-5569	52	6	constraints	constraint	NOUN
ejpam-5569	52	7	|f	|f	PROPN
ejpam-5569	52	8	(	(	PUNCT
ejpam-5569	52	9	u)−	u)−	PROPN
ejpam-5569	52	10	f	f	PROPN
ejpam-5569	52	11	(	(	PUNCT
ejpam-5569	52	12	v)|	v)|	X
ejpam-5569	52	13	≥	≥	X
ejpam-5569	52	14	diam	diam	PROPN
ejpam-5569	52	15	(	(	PUNCT
ejpam-5569	52	16	g	g	NOUN
ejpam-5569	52	17	)	)	PUNCT
ejpam-5569	53	1	+	+	CCONJ
ejpam-5569	54	1	1−	1−	NUM
ejpam-5569	54	2	d	d	NOUN
ejpam-5569	54	3	(	(	PUNCT
ejpam-5569	54	4	u	u	NOUN
ejpam-5569	54	5	,	,	PUNCT
ejpam-5569	54	6	v	v	NOUN
ejpam-5569	54	7	)	)	PUNCT
ejpam-5569	54	8	.	.	PUNCT
ejpam-5569	55	1	for	for	ADP
ejpam-5569	55	2	all	all	DET
ejpam-5569	55	3	u	u	NOUN
ejpam-5569	55	4	,	,	PUNCT
ejpam-5569	55	5	v	v	NOUN
ejpam-5569	55	6	∈	∈	PROPN
ejpam-5569	55	7	v	v	NOUN
ejpam-5569	55	8	(	(	PUNCT
ejpam-5569	55	9	g	g	NOUN
ejpam-5569	55	10	)	)	PUNCT
ejpam-5569	55	11	.	.	PUNCT
ejpam-5569	56	1	the	the	DET
ejpam-5569	56	2	integer	integer	NOUN
ejpam-5569	56	3	f	f	PROPN
ejpam-5569	56	4	(	(	PUNCT
ejpam-5569	56	5	u	u	NOUN
ejpam-5569	56	6	)	)	PUNCT
ejpam-5569	56	7	is	be	AUX
ejpam-5569	56	8	said	say	VERB
ejpam-5569	56	9	to	to	PART
ejpam-5569	56	10	be	be	AUX
ejpam-5569	56	11	the	the	DET
ejpam-5569	56	12	color	color	NOUN
ejpam-5569	56	13	(	(	PUNCT
ejpam-5569	56	14	label	label	NOUN
ejpam-5569	56	15	)	)	PUNCT
ejpam-5569	56	16	of	of	ADP
ejpam-5569	56	17	u	u	PROPN
ejpam-5569	56	18	under	under	ADP
ejpam-5569	56	19	f	f	PROPN
ejpam-5569	56	20	and	and	CCONJ
ejpam-5569	56	21	,	,	PUNCT
ejpam-5569	56	22	the	the	DET
ejpam-5569	56	23	span	span	NOUN
ejpam-5569	56	24	of	of	ADP
ejpam-5569	56	25	f	f	PROPN
ejpam-5569	56	26	is	be	AUX
ejpam-5569	56	27	denoted	denote	VERB
ejpam-5569	56	28	by	by	ADP
ejpam-5569	56	29	span	span	NOUN
ejpam-5569	56	30	(	(	PUNCT
ejpam-5569	56	31	f	f	X
ejpam-5569	56	32	)	)	PUNCT
ejpam-5569	56	33	=	=	SYM
ejpam-5569	56	34	max	max	PROPN
ejpam-5569	56	35	{	{	PUNCT
ejpam-5569	56	36	|f	|f	PROPN
ejpam-5569	56	37	(	(	PUNCT
ejpam-5569	56	38	u	u	NOUN
ejpam-5569	56	39	)	)	PUNCT
ejpam-5569	57	1	−	−	PROPN
ejpam-5569	57	2	f	f	NOUN
ejpam-5569	57	3	(	(	PUNCT
ejpam-5569	57	4	v)|	v)|	NOUN
ejpam-5569	57	5	:	:	PUNCT
ejpam-5569	57	6	u	u	NOUN
ejpam-5569	57	7	,	,	PUNCT
ejpam-5569	57	8	v	v	PROPN
ejpam-5569	57	9	∈	∈	PROPN
ejpam-5569	57	10	v	v	NOUN
ejpam-5569	57	11	(	(	PUNCT
ejpam-5569	57	12	g	g	NOUN
ejpam-5569	57	13	)	)	PUNCT
ejpam-5569	57	14	}	}	PUNCT
ejpam-5569	57	15	.	.	PUNCT
ejpam-5569	58	1	the	the	DET
ejpam-5569	58	2	radio	radio	NOUN
ejpam-5569	58	3	number	number	NOUN
ejpam-5569	58	4	of	of	ADP
ejpam-5569	58	5	g	g	NOUN
ejpam-5569	58	6	,	,	PUNCT
ejpam-5569	58	7	or	or	CCONJ
ejpam-5569	58	8	(	(	PUNCT
ejpam-5569	58	9	rn	rn	PROPN
ejpam-5569	58	10	(	(	PUNCT
ejpam-5569	58	11	g	g	NOUN
ejpam-5569	58	12	)	)	PUNCT
ejpam-5569	58	13	)	)	PUNCT
ejpam-5569	58	14	,	,	PUNCT
ejpam-5569	58	15	is	be	AUX
ejpam-5569	58	16	defined	define	VERB
ejpam-5569	58	17	as	as	ADP
ejpam-5569	58	18	rn	rn	PROPN
ejpam-5569	58	19	(	(	PUNCT
ejpam-5569	58	20	g	g	NOUN
ejpam-5569	58	21	)	)	PUNCT
ejpam-5569	58	22	=	=	SYM
ejpam-5569	58	23	min	min	PROPN
ejpam-5569	58	24	{	{	PUNCT
ejpam-5569	58	25	span	span	NOUN
ejpam-5569	58	26	(	(	PUNCT
ejpam-5569	58	27	f	f	X
ejpam-5569	58	28	)	)	PUNCT
ejpam-5569	58	29	}	}	PUNCT
ejpam-5569	58	30	for	for	ADP
ejpam-5569	58	31	all	all	DET
ejpam-5569	58	32	radio	radio	NOUN
ejpam-5569	58	33	labeling	labeling	NOUN
ejpam-5569	58	34	f	f	PROPN
ejpam-5569	58	35	of	of	ADP
ejpam-5569	58	36	g.	g.	PROPN
ejpam-5569	58	37	the	the	DET
ejpam-5569	58	38	objective	objective	NOUN
ejpam-5569	58	39	of	of	ADP
ejpam-5569	58	40	radio	radio	NOUN
ejpam-5569	58	41	labeling	labeling	NOUN
ejpam-5569	58	42	problem	problem	NOUN
ejpam-5569	58	43	of	of	ADP
ejpam-5569	58	44	g	g	PROPN
ejpam-5569	58	45	is	be	AUX
ejpam-5569	58	46	finding	find	VERB
ejpam-5569	58	47	the	the	DET
ejpam-5569	58	48	value	value	NOUN
ejpam-5569	58	49	of	of	ADP
ejpam-5569	58	50	rn	rn	PROPN
ejpam-5569	58	51	(	(	PUNCT
ejpam-5569	58	52	g	g	NOUN
ejpam-5569	58	53	)	)	PUNCT
ejpam-5569	58	54	.	.	PUNCT
ejpam-5569	59	1	2	2	X
ejpam-5569	59	2	.	.	X
ejpam-5569	59	3	related	relate	VERB
ejpam-5569	59	4	work	work	NOUN
ejpam-5569	59	5	from	from	ADP
ejpam-5569	59	6	viewpoint	viewpoint	NOUN
ejpam-5569	59	7	of	of	ADP
ejpam-5569	59	8	complexity	complexity	NOUN
ejpam-5569	59	9	,	,	PUNCT
ejpam-5569	59	10	the	the	DET
ejpam-5569	59	11	problem	problem	NOUN
ejpam-5569	59	12	of	of	ADP
ejpam-5569	59	13	getting	get	VERB
ejpam-5569	59	14	the	the	DET
ejpam-5569	59	15	radio	radio	NOUN
ejpam-5569	59	16	number	number	NOUN
ejpam-5569	59	17	of	of	ADP
ejpam-5569	59	18	a	a	DET
ejpam-5569	59	19	given	give	VERB
ejpam-5569	59	20	graph	graph	NOUN
ejpam-5569	59	21	is	be	AUX
ejpam-5569	59	22	np	np	INTJ
ejpam-5569	59	23	complete	complete	ADJ
ejpam-5569	59	24	[	[	X
ejpam-5569	59	25	5	5	NUM
ejpam-5569	59	26	]	]	PUNCT
ejpam-5569	59	27	.	.	PUNCT
ejpam-5569	60	1	saha	saha	PROPN
ejpam-5569	60	2	and	and	CCONJ
ejpam-5569	60	3	panigrahi	panigrahi	NOUN
ejpam-5569	60	4	[	[	X
ejpam-5569	60	5	6	6	NUM
ejpam-5569	60	6	]	]	PUNCT
ejpam-5569	60	7	presented	present	VERB
ejpam-5569	60	8	an	an	DET
ejpam-5569	60	9	algorithm	algorithm	NOUN
ejpam-5569	60	10	which	which	PRON
ejpam-5569	60	11	finds	find	VERB
ejpam-5569	60	12	the	the	DET
ejpam-5569	60	13	upper	upper	ADJ
ejpam-5569	60	14	bounds	bound	NOUN
ejpam-5569	60	15	of	of	ADP
ejpam-5569	60	16	rn	rn	PROPN
ejpam-5569	60	17	(	(	PUNCT
ejpam-5569	60	18	g	g	NOUN
ejpam-5569	60	19	)	)	PUNCT
ejpam-5569	60	20	for	for	ADP
ejpam-5569	60	21	a	a	DET
ejpam-5569	60	22	given	give	VERB
ejpam-5569	60	23	graph	graph	NOUN
ejpam-5569	60	24	g.	g.	NOUN
ejpam-5569	60	25	in	in	ADP
ejpam-5569	60	26	[	[	X
ejpam-5569	60	27	7	7	X
ejpam-5569	60	28	]	]	X
ejpam-5569	60	29	badr	badr	PROPN
ejpam-5569	60	30	and	and	CCONJ
ejpam-5569	60	31	moussa	moussa	PROPN
ejpam-5569	60	32	suggested	suggest	VERB
ejpam-5569	60	33	a	a	DET
ejpam-5569	60	34	developing	develop	VERB
ejpam-5569	60	35	algorithm	algorithm	NOUN
ejpam-5569	60	36	for	for	ADP
ejpam-5569	60	37	one	one	NUM
ejpam-5569	60	38	presented	present	VERB
ejpam-5569	60	39	in	in	ADP
ejpam-5569	60	40	[	[	X
ejpam-5569	60	41	6	6	NUM
ejpam-5569	60	42	]	]	PUNCT
ejpam-5569	60	43	.	.	PUNCT
ejpam-5569	61	1	besides	besides	SCONJ
ejpam-5569	61	2	that	that	PRON
ejpam-5569	61	3	they	they	PRON
ejpam-5569	61	4	gave	give	VERB
ejpam-5569	61	5	an	an	DET
ejpam-5569	61	6	approach	approach	NOUN
ejpam-5569	61	7	for	for	ADP
ejpam-5569	61	8	solving	solve	VERB
ejpam-5569	61	9	the	the	DET
ejpam-5569	61	10	radio	radio	NOUN
ejpam-5569	61	11	labeling	labeling	NOUN
ejpam-5569	61	12	problem	problem	NOUN
ejpam-5569	61	13	using	use	VERB
ejpam-5569	61	14	mathematical	mathematical	ADJ
ejpam-5569	61	15	modeling	modeling	NOUN
ejpam-5569	61	16	.	.	PUNCT
ejpam-5569	62	1	recently	recently	ADV
ejpam-5569	62	2	radio	radio	NOUN
ejpam-5569	62	3	labeling	labeling	NOUN
ejpam-5569	62	4	of	of	ADP
ejpam-5569	62	5	graphs	graph	NOUN
ejpam-5569	62	6	and	and	CCONJ
ejpam-5569	62	7	finding	find	VERB
ejpam-5569	62	8	the	the	DET
ejpam-5569	62	9	exact	exact	NOUN
ejpam-5569	62	10	or	or	CCONJ
ejpam-5569	62	11	the	the	DET
ejpam-5569	62	12	upper	upper	ADJ
ejpam-5569	62	13	bounds	bound	NOUN
ejpam-5569	62	14	of	of	ADP
ejpam-5569	62	15	radio	radio	NOUN
ejpam-5569	62	16	number	number	NOUN
ejpam-5569	62	17	of	of	ADP
ejpam-5569	62	18	graphs	graph	NOUN
ejpam-5569	62	19	has	have	VERB
ejpam-5569	62	20	a	a	DET
ejpam-5569	62	21	lot	lot	NOUN
ejpam-5569	62	22	of	of	ADP
ejpam-5569	62	23	attention	attention	NOUN
ejpam-5569	62	24	.	.	PUNCT
ejpam-5569	63	1	in	in	ADP
ejpam-5569	63	2	[	[	X
ejpam-5569	63	3	8	8	NUM
ejpam-5569	63	4	]	]	PUNCT
ejpam-5569	63	5	an	an	DET
ejpam-5569	63	6	assignment	assignment	NOUN
ejpam-5569	63	7	of	of	ADP
ejpam-5569	63	8	radio	radio	NOUN
ejpam-5569	63	9	numbers	number	NOUN
ejpam-5569	63	10	to	to	ADP
ejpam-5569	63	11	triangular	triangular	NOUN
ejpam-5569	63	12	networks	network	NOUN
ejpam-5569	63	13	and	and	CCONJ
ejpam-5569	63	14	rhombic	rhombic	ADJ
ejpam-5569	63	15	honeycomb	honeycomb	NOUN
ejpam-5569	63	16	networks	network	NOUN
ejpam-5569	63	17	is	be	AUX
ejpam-5569	63	18	discussed	discuss	VERB
ejpam-5569	63	19	.	.	PUNCT
ejpam-5569	64	1	the	the	DET
ejpam-5569	64	2	study	study	NOUN
ejpam-5569	64	3	of	of	ADP
ejpam-5569	64	4	the	the	DET
ejpam-5569	64	5	radio	radio	NOUN
ejpam-5569	64	6	labeling	labeling	NOUN
ejpam-5569	64	7	of	of	ADP
ejpam-5569	64	8	fuzzy	fuzzy	ADJ
ejpam-5569	64	9	graphs	graph	NOUN
ejpam-5569	64	10	is	be	AUX
ejpam-5569	64	11	n.	n.	NOUN
ejpam-5569	64	12	m.	m.	NOUN
ejpam-5569	65	1	noureldeen	noureldeen	NOUN
ejpam-5569	65	2	et	et	PROPN
ejpam-5569	65	3	al	al	PROPN
ejpam-5569	65	4	.	.	PUNCT
ejpam-5569	65	5	/	/	SYM
ejpam-5569	65	6	eur	eur	PROPN
ejpam-5569	65	7	.	.	PUNCT
ejpam-5569	66	1	j.	j.	PROPN
ejpam-5569	66	2	pure	pure	PROPN
ejpam-5569	66	3	appl	appl	PROPN
ejpam-5569	66	4	.	.	PROPN
ejpam-5569	66	5	math	math	PROPN
ejpam-5569	66	6	,	,	PUNCT
ejpam-5569	66	7	18	18	NUM
ejpam-5569	66	8	(	(	PUNCT
ejpam-5569	66	9	2	2	NUM
ejpam-5569	66	10	)	)	PUNCT
ejpam-5569	66	11	(	(	PUNCT
ejpam-5569	66	12	2025	2025	NUM
ejpam-5569	66	13	)	)	PUNCT
ejpam-5569	66	14	,	,	PUNCT
ejpam-5569	66	15	5569	5569	NUM
ejpam-5569	66	16	4	4	NUM
ejpam-5569	66	17	of	of	ADP
ejpam-5569	66	18	25	25	NUM
ejpam-5569	66	19	studied	study	VERB
ejpam-5569	66	20	in	in	ADP
ejpam-5569	66	21	[	[	X
ejpam-5569	66	22	9	9	NUM
ejpam-5569	66	23	]	]	PUNCT
ejpam-5569	66	24	.	.	PUNCT
ejpam-5569	67	1	in	in	ADP
ejpam-5569	67	2	addition	addition	NOUN
ejpam-5569	67	3	,	,	PUNCT
ejpam-5569	67	4	the	the	DET
ejpam-5569	67	5	study	study	NOUN
ejpam-5569	67	6	of	of	ADP
ejpam-5569	67	7	the	the	DET
ejpam-5569	67	8	radio	radio	NOUN
ejpam-5569	67	9	labeling	labeling	NOUN
ejpam-5569	67	10	for	for	ADP
ejpam-5569	67	11	many	many	ADJ
ejpam-5569	67	12	different	different	ADJ
ejpam-5569	67	13	types	type	NOUN
ejpam-5569	67	14	of	of	ADP
ejpam-5569	67	15	graphs	graph	NOUN
ejpam-5569	67	16	has	have	AUX
ejpam-5569	67	17	been	be	AUX
ejpam-5569	67	18	the	the	DET
ejpam-5569	67	19	subject	subject	NOUN
ejpam-5569	67	20	of	of	ADP
ejpam-5569	67	21	intellectual	intellectual	ADJ
ejpam-5569	67	22	research	research	NOUN
ejpam-5569	67	23	by	by	ADP
ejpam-5569	67	24	several	several	ADJ
ejpam-5569	67	25	authors	author	NOUN
ejpam-5569	67	26	[	[	X
ejpam-5569	67	27	10–23	10–23	NUM
ejpam-5569	67	28	]	]	PUNCT
ejpam-5569	67	29	.	.	PUNCT
ejpam-5569	68	1	many	many	ADJ
ejpam-5569	68	2	authors	author	NOUN
ejpam-5569	68	3	have	have	AUX
ejpam-5569	68	4	found	find	VERB
ejpam-5569	68	5	the	the	DET
ejpam-5569	68	6	radio	radio	NOUN
ejpam-5569	68	7	labeling	labeling	NOUN
ejpam-5569	68	8	and	and	CCONJ
ejpam-5569	68	9	radio	radio	NOUN
ejpam-5569	68	10	labeling	labeling	NOUN
ejpam-5569	68	11	of	of	ADP
ejpam-5569	68	12	path	path	NOUN
ejpam-5569	68	13	graphs	graph	NOUN
ejpam-5569	68	14	,	,	PUNCT
ejpam-5569	68	15	the	the	DET
ejpam-5569	68	16	middle	middle	ADJ
ejpam-5569	68	17	graph	graph	NOUN
ejpam-5569	68	18	of	of	ADP
ejpam-5569	68	19	the	the	DET
ejpam-5569	68	20	path	path	NOUN
ejpam-5569	68	21	,	,	PUNCT
ejpam-5569	68	22	a	a	DET
ejpam-5569	68	23	strong	strong	ADJ
ejpam-5569	68	24	product	product	NOUN
ejpam-5569	68	25	of	of	ADP
ejpam-5569	68	26	the	the	DET
ejpam-5569	68	27	path	path	NOUN
ejpam-5569	68	28	graph	graph	NOUN
ejpam-5569	68	29	,	,	PUNCT
ejpam-5569	68	30	the	the	DET
ejpam-5569	68	31	cross	cross	NOUN
ejpam-5569	68	32	product	product	NOUN
ejpam-5569	68	33	of	of	ADP
ejpam-5569	68	34	the	the	DET
ejpam-5569	68	35	path	path	NOUN
ejpam-5569	68	36	,	,	PUNCT
ejpam-5569	68	37	and	and	CCONJ
ejpam-5569	68	38	the	the	DET
ejpam-5569	68	39	corona	corona	NOUN
ejpam-5569	68	40	product	product	NOUN
ejpam-5569	68	41	of	of	ADP
ejpam-5569	68	42	the	the	DET
ejpam-5569	68	43	path	path	NOUN
ejpam-5569	68	44	sub	sub	NOUN
ejpam-5569	68	45	division	division	NOUN
ejpam-5569	68	46	of	of	ADP
ejpam-5569	68	47	paths	path	NOUN
ejpam-5569	68	48	,	,	PUNCT
ejpam-5569	68	49	etc	etc	X
ejpam-5569	68	50	.	.	X
ejpam-5569	69	1	this	this	DET
ejpam-5569	69	2	study	study	NOUN
ejpam-5569	69	3	aims	aim	VERB
ejpam-5569	69	4	to	to	PART
ejpam-5569	69	5	fill	fill	VERB
ejpam-5569	69	6	the	the	DET
ejpam-5569	69	7	research	research	NOUN
ejpam-5569	69	8	gaps	gap	NOUN
ejpam-5569	69	9	by	by	ADP
ejpam-5569	69	10	determining	determine	VERB
ejpam-5569	69	11	the	the	DET
ejpam-5569	69	12	radio	radio	NOUN
ejpam-5569	69	13	number	number	NOUN
ejpam-5569	69	14	for	for	ADP
ejpam-5569	69	15	the	the	DET
ejpam-5569	69	16	shadow	shadow	NOUN
ejpam-5569	69	17	of	of	ADP
ejpam-5569	69	18	path	path	NOUN
ejpam-5569	69	19	graph	graph	NOUN
ejpam-5569	69	20	.	.	PUNCT
ejpam-5569	70	1	in	in	ADP
ejpam-5569	70	2	this	this	DET
ejpam-5569	70	3	paper	paper	NOUN
ejpam-5569	70	4	,	,	PUNCT
ejpam-5569	70	5	we	we	PRON
ejpam-5569	70	6	are	be	AUX
ejpam-5569	70	7	concerned	concerned	ADJ
ejpam-5569	70	8	with	with	ADP
ejpam-5569	70	9	finding	find	VERB
ejpam-5569	70	10	an	an	DET
ejpam-5569	70	11	upper	upper	ADJ
ejpam-5569	70	12	bound	bind	VERB
ejpam-5569	70	13	for	for	ADP
ejpam-5569	70	14	radio	radio	NOUN
ejpam-5569	70	15	number	number	NOUN
ejpam-5569	70	16	rn	rn	PROPN
ejpam-5569	70	17	(	(	PUNCT
ejpam-5569	70	18	g	g	NOUN
ejpam-5569	70	19	)	)	PUNCT
ejpam-5569	70	20	of	of	ADP
ejpam-5569	70	21	networks	network	NOUN
ejpam-5569	70	22	modeled	model	VERB
ejpam-5569	70	23	by	by	ADP
ejpam-5569	70	24	the	the	DET
ejpam-5569	70	25	shadow	shadow	NOUN
ejpam-5569	70	26	of	of	ADP
ejpam-5569	70	27	path	path	NOUN
ejpam-5569	70	28	graph	graph	NOUN
ejpam-5569	70	29	.	.	PUNCT
ejpam-5569	71	1	we	we	PRON
ejpam-5569	71	2	present	present	VERB
ejpam-5569	71	3	a	a	DET
ejpam-5569	71	4	theoretical	theoretical	ADJ
ejpam-5569	71	5	approach	approach	NOUN
ejpam-5569	71	6	for	for	ADP
ejpam-5569	71	7	getting	get	VERB
ejpam-5569	71	8	an	an	DET
ejpam-5569	71	9	upper	upper	ADJ
ejpam-5569	71	10	bound	bind	VERB
ejpam-5569	71	11	of	of	ADP
ejpam-5569	71	12	radio	radio	NOUN
ejpam-5569	71	13	number	number	NOUN
ejpam-5569	71	14	of	of	ADP
ejpam-5569	71	15	the	the	DET
ejpam-5569	71	16	shadow	shadow	NOUN
ejpam-5569	71	17	of	of	ADP
ejpam-5569	71	18	path	path	NOUN
ejpam-5569	71	19	graph	graph	NOUN
ejpam-5569	71	20	.	.	PUNCT
ejpam-5569	72	1	moreover	moreover	ADV
ejpam-5569	72	2	,	,	PUNCT
ejpam-5569	72	3	modeling	model	VERB
ejpam-5569	72	4	the	the	DET
ejpam-5569	72	5	problem	problem	NOUN
ejpam-5569	72	6	of	of	ADP
ejpam-5569	72	7	radio	radio	NOUN
ejpam-5569	72	8	labeling	labeling	NOUN
ejpam-5569	72	9	of	of	ADP
ejpam-5569	72	10	the	the	DET
ejpam-5569	72	11	shadow	shadow	NOUN
ejpam-5569	72	12	path	path	NOUN
ejpam-5569	72	13	graph	graph	NOUN
ejpam-5569	72	14	using	use	VERB
ejpam-5569	72	15	integer	integer	NOUN
ejpam-5569	72	16	linear	linear	PROPN
ejpam-5569	72	17	programing	programing	NOUN
ejpam-5569	72	18	is	be	AUX
ejpam-5569	72	19	proposed	propose	VERB
ejpam-5569	72	20	.	.	PUNCT
ejpam-5569	73	1	besides	besides	SCONJ
ejpam-5569	73	2	that	that	PRON
ejpam-5569	73	3	,	,	PUNCT
ejpam-5569	73	4	an	an	DET
ejpam-5569	73	5	experimental	experimental	ADJ
ejpam-5569	73	6	study	study	NOUN
ejpam-5569	73	7	is	be	AUX
ejpam-5569	73	8	carried	carry	VERB
ejpam-5569	73	9	out	out	ADP
ejpam-5569	73	10	in	in	ADP
ejpam-5569	73	11	order	order	NOUN
ejpam-5569	73	12	to	to	PART
ejpam-5569	73	13	demonstrate	demonstrate	VERB
ejpam-5569	73	14	our	our	PRON
ejpam-5569	73	15	approach	approach	NOUN
ejpam-5569	73	16	’s	’s	PART
ejpam-5569	73	17	effectiveness	effectiveness	NOUN
ejpam-5569	73	18	in	in	ADP
ejpam-5569	73	19	comparison	comparison	NOUN
ejpam-5569	73	20	to	to	ADP
ejpam-5569	73	21	previously	previously	ADV
ejpam-5569	73	22	published	publish	VERB
ejpam-5569	73	23	results	result	NOUN
ejpam-5569	73	24	.	.	PUNCT
ejpam-5569	74	1	overall	overall	ADV
ejpam-5569	74	2	,	,	PUNCT
ejpam-5569	74	3	our	our	PRON
ejpam-5569	74	4	investigation	investigation	NOUN
ejpam-5569	74	5	demonstrates	demonstrate	VERB
ejpam-5569	74	6	that	that	SCONJ
ejpam-5569	74	7	,	,	PUNCT
ejpam-5569	74	8	in	in	ADP
ejpam-5569	74	9	terms	term	NOUN
ejpam-5569	74	10	of	of	ADP
ejpam-5569	74	11	the	the	DET
ejpam-5569	74	12	running	running	ADJ
ejpam-5569	74	13	time	time	NOUN
ejpam-5569	74	14	and	and	CCONJ
ejpam-5569	74	15	radio	radio	NOUN
ejpam-5569	74	16	numbers	number	NOUN
ejpam-5569	74	17	’	'	PUNCT
ejpam-5569	74	18	upper	upper	ADJ
ejpam-5569	74	19	bound	bind	VERB
ejpam-5569	74	20	,	,	PUNCT
ejpam-5569	74	21	the	the	DET
ejpam-5569	74	22	suggested	suggest	VERB
ejpam-5569	74	23	results	result	NOUN
ejpam-5569	74	24	perform	perform	VERB
ejpam-5569	74	25	better	well	ADJ
ejpam-5569	74	26	than	than	ADP
ejpam-5569	74	27	the	the	DET
ejpam-5569	74	28	earlier	early	ADJ
ejpam-5569	74	29	findings	finding	NOUN
ejpam-5569	74	30	.	.	PUNCT
ejpam-5569	75	1	the	the	DET
ejpam-5569	75	2	rest	rest	NOUN
ejpam-5569	75	3	of	of	ADP
ejpam-5569	75	4	the	the	DET
ejpam-5569	75	5	paper	paper	NOUN
ejpam-5569	75	6	is	be	AUX
ejpam-5569	75	7	structured	structure	VERB
ejpam-5569	75	8	as	as	SCONJ
ejpam-5569	75	9	follows	follow	VERB
ejpam-5569	75	10	.	.	PUNCT
ejpam-5569	76	1	section	section	NOUN
ejpam-5569	76	2	3	3	NUM
ejpam-5569	76	3	introduces	introduce	VERB
ejpam-5569	76	4	our	our	PRON
ejpam-5569	76	5	theoretical	theoretical	ADJ
ejpam-5569	76	6	approach	approach	NOUN
ejpam-5569	76	7	for	for	ADP
ejpam-5569	76	8	finding	find	VERB
ejpam-5569	76	9	an	an	DET
ejpam-5569	76	10	upper	upper	ADJ
ejpam-5569	76	11	bound	bind	VERB
ejpam-5569	76	12	of	of	ADP
ejpam-5569	76	13	radio	radio	NOUN
ejpam-5569	76	14	number	number	NOUN
ejpam-5569	76	15	of	of	ADP
ejpam-5569	76	16	the	the	DET
ejpam-5569	76	17	shadow	shadow	NOUN
ejpam-5569	76	18	of	of	ADP
ejpam-5569	76	19	path	path	NOUN
ejpam-5569	76	20	graph	graph	NOUN
ejpam-5569	76	21	.	.	PUNCT
ejpam-5569	77	1	the	the	DET
ejpam-5569	77	2	modeling	modeling	NOUN
ejpam-5569	77	3	of	of	ADP
ejpam-5569	77	4	radio	radio	NOUN
ejpam-5569	77	5	labeling	labeling	NOUN
ejpam-5569	77	6	of	of	ADP
ejpam-5569	77	7	shadow	shadow	NOUN
ejpam-5569	77	8	of	of	ADP
ejpam-5569	77	9	path	path	NOUN
ejpam-5569	77	10	graph	graph	NOUN
ejpam-5569	77	11	using	use	VERB
ejpam-5569	77	12	integer	integer	NOUN
ejpam-5569	77	13	linear	linear	PROPN
ejpam-5569	77	14	programing	programing	NOUN
ejpam-5569	77	15	is	be	AUX
ejpam-5569	77	16	presented	present	VERB
ejpam-5569	77	17	in	in	ADP
ejpam-5569	77	18	section	section	NOUN
ejpam-5569	77	19	4	4	NUM
ejpam-5569	77	20	.	.	PUNCT
ejpam-5569	77	21	section	section	NOUN
ejpam-5569	77	22	5	5	NUM
ejpam-5569	77	23	,	,	PUNCT
ejpam-5569	77	24	gives	give	VERB
ejpam-5569	77	25	the	the	DET
ejpam-5569	77	26	computational	computational	ADJ
ejpam-5569	77	27	study	study	NOUN
ejpam-5569	77	28	for	for	ADP
ejpam-5569	77	29	comparing	compare	VERB
ejpam-5569	77	30	our	our	PRON
ejpam-5569	77	31	theoretical	theoretical	ADJ
ejpam-5569	77	32	approach	approach	NOUN
ejpam-5569	77	33	,	,	PUNCT
ejpam-5569	77	34	integer	integer	NOUN
ejpam-5569	77	35	programing	programing	NOUN
ejpam-5569	77	36	model	model	NOUN
ejpam-5569	77	37	and	and	CCONJ
ejpam-5569	77	38	previous	previous	ADJ
ejpam-5569	77	39	algorithms	algorithm	NOUN
ejpam-5569	77	40	known	know	VERB
ejpam-5569	77	41	in	in	ADP
ejpam-5569	77	42	the	the	DET
ejpam-5569	77	43	literature	literature	NOUN
ejpam-5569	77	44	.	.	PUNCT
ejpam-5569	78	1	the	the	DET
ejpam-5569	78	2	application	application	NOUN
ejpam-5569	78	3	of	of	ADP
ejpam-5569	78	4	radio	radio	NOUN
ejpam-5569	78	5	labeling	labeling	NOUN
ejpam-5569	78	6	technique	technique	NOUN
ejpam-5569	78	7	in	in	ADP
ejpam-5569	78	8	cryptography	cryptography	NOUN
ejpam-5569	78	9	is	be	AUX
ejpam-5569	78	10	introduced	introduce	VERB
ejpam-5569	78	11	in	in	ADP
ejpam-5569	78	12	section	section	NOUN
ejpam-5569	78	13	6	6	NUM
ejpam-5569	78	14	.	.	PUNCT
ejpam-5569	79	1	the	the	DET
ejpam-5569	79	2	conclusion	conclusion	NOUN
ejpam-5569	79	3	of	of	ADP
ejpam-5569	79	4	work	work	NOUN
ejpam-5569	79	5	is	be	AUX
ejpam-5569	79	6	presented	present	VERB
ejpam-5569	79	7	in	in	ADP
ejpam-5569	79	8	section	section	NOUN
ejpam-5569	79	9	7	7	NUM
ejpam-5569	79	10	.	.	NOUN
ejpam-5569	79	11	3	3	NUM
ejpam-5569	79	12	.	.	X
ejpam-5569	79	13	theoretical	theoretical	ADJ
ejpam-5569	79	14	approach	approach	NOUN
ejpam-5569	79	15	for	for	ADP
ejpam-5569	79	16	upper	upper	ADJ
ejpam-5569	79	17	bound	bind	VERB
ejpam-5569	79	18	of	of	ADP
ejpam-5569	79	19	radio	radio	NOUN
ejpam-5569	79	20	number	number	NOUN
ejpam-5569	79	21	of	of	ADP
ejpam-5569	79	22	shadow	shadow	NOUN
ejpam-5569	79	23	networks	network	NOUN
ejpam-5569	79	24	hereafter	hereafter	ADV
ejpam-5569	79	25	,	,	PUNCT
ejpam-5569	79	26	we	we	PRON
ejpam-5569	79	27	are	be	AUX
ejpam-5569	79	28	interested	interested	ADJ
ejpam-5569	79	29	in	in	ADP
ejpam-5569	79	30	getting	get	VERB
ejpam-5569	79	31	an	an	DET
ejpam-5569	79	32	upper	upper	ADJ
ejpam-5569	79	33	bound	bind	VERB
ejpam-5569	79	34	of	of	ADP
ejpam-5569	79	35	radio	radio	NOUN
ejpam-5569	79	36	number	number	NOUN
ejpam-5569	79	37	of	of	ADP
ejpam-5569	79	38	shadow	shadow	NOUN
ejpam-5569	79	39	networks	network	NOUN
ejpam-5569	79	40	.	.	PUNCT
ejpam-5569	80	1	definition	definition	NOUN
ejpam-5569	80	2	1	1	NUM
ejpam-5569	80	3	.	.	PUNCT
ejpam-5569	81	1	let	let	VERB
ejpam-5569	81	2	pn	pn	PART
ejpam-5569	81	3	be	be	AUX
ejpam-5569	81	4	the	the	DET
ejpam-5569	81	5	path	path	NOUN
ejpam-5569	81	6	with	with	ADP
ejpam-5569	81	7	n	n	SYM
ejpam-5569	81	8	vertices	vertex	NOUN
ejpam-5569	81	9	.	.	PUNCT
ejpam-5569	82	1	the	the	DET
ejpam-5569	82	2	shadow	shadow	NOUN
ejpam-5569	82	3	graph	graph	VERB
ejpam-5569	82	4	d2(pn	d2(pn	PROPN
ejpam-5569	82	5	)	)	PUNCT
ejpam-5569	82	6	is	be	AUX
ejpam-5569	82	7	constructed	construct	VERB
ejpam-5569	82	8	by	by	ADP
ejpam-5569	82	9	taking	take	VERB
ejpam-5569	82	10	two	two	NUM
ejpam-5569	82	11	copies	copy	NOUN
ejpam-5569	82	12	of	of	ADP
ejpam-5569	82	13	pn	pn	PROPN
ejpam-5569	82	14	.	.	PROPN
ejpam-5569	82	15	join	join	VERB
ejpam-5569	82	16	each	each	DET
ejpam-5569	82	17	vertex	vertex	NOUN
ejpam-5569	82	18	u	u	NOUN
ejpam-5569	82	19	in	in	ADP
ejpam-5569	82	20	the	the	DET
ejpam-5569	82	21	first	first	ADJ
ejpam-5569	82	22	path	path	NOUN
ejpam-5569	82	23	to	to	ADP
ejpam-5569	82	24	the	the	DET
ejpam-5569	82	25	neighbors	neighbor	NOUN
ejpam-5569	82	26	of	of	ADP
ejpam-5569	82	27	the	the	DET
ejpam-5569	82	28	corresponding	corresponding	ADJ
ejpam-5569	82	29	vertex	vertex	NOUN
ejpam-5569	82	30	v	v	NOUN
ejpam-5569	82	31	in	in	ADP
ejpam-5569	82	32	the	the	DET
ejpam-5569	82	33	second	second	ADJ
ejpam-5569	82	34	path	path	NOUN
ejpam-5569	82	35	as	as	SCONJ
ejpam-5569	82	36	shown	show	VERB
ejpam-5569	82	37	in	in	ADP
ejpam-5569	82	38	figure	figure	NOUN
ejpam-5569	82	39	1	1	NUM
ejpam-5569	82	40	.	.	PUNCT
ejpam-5569	83	1	figure	figure	NOUN
ejpam-5569	83	2	1	1	NUM
ejpam-5569	83	3	:	:	PUNCT
ejpam-5569	83	4	the	the	DET
ejpam-5569	83	5	graph	graph	NOUN
ejpam-5569	83	6	d2(pn	d2(pn	PROPN
ejpam-5569	83	7	)	)	PUNCT
ejpam-5569	83	8	.	.	PUNCT
ejpam-5569	84	1	in	in	ADP
ejpam-5569	84	2	d2(pn	d2(pn	PROPN
ejpam-5569	84	3	)	)	PUNCT
ejpam-5569	84	4	,	,	PUNCT
ejpam-5569	84	5	the	the	DET
ejpam-5569	84	6	set	set	NOUN
ejpam-5569	84	7	of	of	ADP
ejpam-5569	84	8	vertices	vertex	NOUN
ejpam-5569	84	9	is	be	AUX
ejpam-5569	84	10	v	v	NOUN
ejpam-5569	84	11	(	(	PUNCT
ejpam-5569	84	12	d2(pn	d2(pn	PROPN
ejpam-5569	84	13	)	)	PUNCT
ejpam-5569	84	14	)	)	PUNCT
ejpam-5569	85	1	=	=	PRON
ejpam-5569	85	2	{	{	PUNCT
ejpam-5569	85	3	ui	ui	PROPN
ejpam-5569	85	4	,	,	PUNCT
ejpam-5569	85	5	vi	vi	NOUN
ejpam-5569	85	6	:	:	PUNCT
ejpam-5569	85	7	1	1	NUM
ejpam-5569	85	8	≤	≤	NUM
ejpam-5569	85	9	i	i	PRON
ejpam-5569	85	10	≤	≤	NOUN
ejpam-5569	85	11	n	n	CCONJ
ejpam-5569	85	12	}	}	PUNCT
ejpam-5569	85	13	and	and	CCONJ
ejpam-5569	85	14	the	the	DET
ejpam-5569	85	15	set	set	NOUN
ejpam-5569	85	16	of	of	ADP
ejpam-5569	85	17	edges	edge	NOUN
ejpam-5569	85	18	n.	n.	PROPN
ejpam-5569	85	19	m.	m.	NOUN
ejpam-5569	85	20	noureldeen	noureldeen	PROPN
ejpam-5569	85	21	et	et	PROPN
ejpam-5569	85	22	al	al	PROPN
ejpam-5569	85	23	.	.	PUNCT
ejpam-5569	85	24	/	/	SYM
ejpam-5569	85	25	eur	eur	PROPN
ejpam-5569	85	26	.	.	PUNCT
ejpam-5569	86	1	j.	j.	PROPN
ejpam-5569	86	2	pure	pure	PROPN
ejpam-5569	86	3	appl	appl	PROPN
ejpam-5569	86	4	.	.	PROPN
ejpam-5569	86	5	math	math	PROPN
ejpam-5569	86	6	,	,	PUNCT
ejpam-5569	86	7	18	18	NUM
ejpam-5569	86	8	(	(	PUNCT
ejpam-5569	86	9	2	2	NUM
ejpam-5569	86	10	)	)	PUNCT
ejpam-5569	86	11	(	(	PUNCT
ejpam-5569	86	12	2025	2025	NUM
ejpam-5569	86	13	)	)	PUNCT
ejpam-5569	86	14	,	,	PUNCT
ejpam-5569	86	15	5569	5569	NUM
ejpam-5569	86	16	5	5	NUM
ejpam-5569	86	17	of	of	ADP
ejpam-5569	86	18	25	25	NUM
ejpam-5569	86	19	is	be	AUX
ejpam-5569	86	20	defined	define	VERB
ejpam-5569	86	21	as	as	ADP
ejpam-5569	86	22	:	:	PUNCT
ejpam-5569	86	23	e(d2(pn	e(d2(pn	ADJ
ejpam-5569	86	24	)	)	PUNCT
ejpam-5569	86	25	)	)	PUNCT
ejpam-5569	87	1	=	=	PRON
ejpam-5569	87	2	{	{	PUNCT
ejpam-5569	87	3	uiui+1	uiui+1	PROPN
ejpam-5569	87	4	,	,	PUNCT
ejpam-5569	87	5	vivi+1	vivi+1	PROPN
ejpam-5569	87	6	vi	vi	NOUN
ejpam-5569	87	7	:	:	PUNCT
ejpam-5569	87	8	1	1	NUM
ejpam-5569	87	9	≤	≤	NUM
ejpam-5569	87	10	i	i	PRON
ejpam-5569	87	11	≤	≤	ADJ
ejpam-5569	87	12	n−	n−	NOUN
ejpam-5569	87	13	1	1	NUM
ejpam-5569	87	14	}	}	PUNCT
ejpam-5569	87	15	∪	∪	ADJ
ejpam-5569	87	16	{	{	PUNCT
ejpam-5569	87	17	uivi+1	uivi+1	PROPN
ejpam-5569	87	18	,	,	PUNCT
ejpam-5569	87	19	viui+1	viui+1	NOUN
ejpam-5569	87	20	vi	vi	NOUN
ejpam-5569	87	21	:	:	PUNCT
ejpam-5569	87	22	1	1	NUM
ejpam-5569	87	23	≤	≤	NUM
ejpam-5569	87	24	i	i	PRON
ejpam-5569	87	25	≤	≤	ADJ
ejpam-5569	87	26	n−	n−	NOUN
ejpam-5569	87	27	1	1	NUM
ejpam-5569	87	28	}	}	PUNCT
ejpam-5569	87	29	.	.	PUNCT
ejpam-5569	88	1	for	for	ADP
ejpam-5569	88	2	convenience	convenience	NOUN
ejpam-5569	88	3	,	,	PUNCT
ejpam-5569	88	4	we	we	PRON
ejpam-5569	88	5	divided	divide	VERB
ejpam-5569	88	6	all	all	DET
ejpam-5569	88	7	vertices	vertex	NOUN
ejpam-5569	88	8	of	of	ADP
ejpam-5569	88	9	d2(pn	d2(pn	PROPN
ejpam-5569	88	10	)	)	PUNCT
ejpam-5569	88	11	into	into	ADP
ejpam-5569	88	12	two	two	NUM
ejpam-5569	88	13	subsets	subset	NOUN
ejpam-5569	88	14	vu	vu	NOUN
ejpam-5569	88	15	and	and	CCONJ
ejpam-5569	88	16	vv	vv	INTJ
ejpam-5569	88	17	.	.	PUNCT
ejpam-5569	88	18	where	where	SCONJ
ejpam-5569	88	19	vu	vu	X
ejpam-5569	88	20	=	=	PRON
ejpam-5569	89	1	{	{	PUNCT
ejpam-5569	89	2	ui	ui	NOUN
ejpam-5569	89	3	:	:	PUNCT
ejpam-5569	89	4	1	1	NUM
ejpam-5569	89	5	≤	≤	NUM
ejpam-5569	89	6	i	i	PRON
ejpam-5569	89	7	≤	≤	NOUN
ejpam-5569	89	8	n	n	CCONJ
ejpam-5569	89	9	}	}	PUNCT
ejpam-5569	89	10	and	and	CCONJ
ejpam-5569	89	11	vv	vv	NOUN
ejpam-5569	89	12	=	=	SYM
ejpam-5569	89	13	{	{	PUNCT
ejpam-5569	89	14	vi	vi	NOUN
ejpam-5569	89	15	:	:	PUNCT
ejpam-5569	89	16	1	1	NUM
ejpam-5569	89	17	≤	≤	NUM
ejpam-5569	89	18	i	i	PRON
ejpam-5569	89	19	≤	≤	NOUN
ejpam-5569	89	20	n	n	CCONJ
ejpam-5569	89	21	}	}	PUNCT
ejpam-5569	89	22	.	.	PUNCT
ejpam-5569	90	1	hence	hence	ADV
ejpam-5569	90	2	,	,	PUNCT
ejpam-5569	90	3	the	the	DET
ejpam-5569	90	4	graph	graph	NOUN
ejpam-5569	90	5	d2(pn	d2(pn	PROPN
ejpam-5569	90	6	)	)	PUNCT
ejpam-5569	90	7	has	have	VERB
ejpam-5569	90	8	a	a	DET
ejpam-5569	90	9	set	set	NOUN
ejpam-5569	90	10	of	of	ADP
ejpam-5569	90	11	2n	2n	NUM
ejpam-5569	90	12	vertices	vertex	NOUN
ejpam-5569	90	13	;	;	PUNCT
ejpam-5569	90	14	that	that	PRON
ejpam-5569	90	15	is	be	AUX
ejpam-5569	90	16	|vu|	|vu|	VERB
ejpam-5569	90	17	=	=	SYM
ejpam-5569	90	18	|vv|	|vv|	NOUN
ejpam-5569	90	19	=	=	SYM
ejpam-5569	90	20	n	n	NOUN
ejpam-5569	90	21	and	and	CCONJ
ejpam-5569	90	22	the	the	DET
ejpam-5569	90	23	diameter	diameter	NOUN
ejpam-5569	90	24	of	of	ADP
ejpam-5569	90	25	d2(pn	d2(pn	PROPN
ejpam-5569	90	26	)	)	PUNCT
ejpam-5569	90	27	equals	equal	VERB
ejpam-5569	90	28	n−	n−	NOUN
ejpam-5569	90	29	1	1	NUM
ejpam-5569	90	30	,	,	PUNCT
ejpam-5569	90	31	n	n	CCONJ
ejpam-5569	90	32	>	>	X
ejpam-5569	90	33	2	2	X
ejpam-5569	90	34	.	.	PUNCT
ejpam-5569	91	1	in	in	ADP
ejpam-5569	91	2	the	the	DET
ejpam-5569	91	3	following	following	NOUN
ejpam-5569	91	4	,	,	PUNCT
ejpam-5569	91	5	we	we	PRON
ejpam-5569	91	6	investigate	investigate	VERB
ejpam-5569	91	7	the	the	DET
ejpam-5569	91	8	radio	radio	NOUN
ejpam-5569	91	9	number	number	NOUN
ejpam-5569	91	10	of	of	ADP
ejpam-5569	91	11	d2(pn	d2(pn	PROPN
ejpam-5569	91	12	)	)	PUNCT
ejpam-5569	91	13	.	.	PUNCT
ejpam-5569	92	1	lemma	lemma	PROPN
ejpam-5569	92	2	1	1	NUM
ejpam-5569	92	3	.	.	PUNCT
ejpam-5569	93	1	for	for	ADP
ejpam-5569	93	2	a	a	DET
ejpam-5569	93	3	positive	positive	ADJ
ejpam-5569	93	4	integer	integer	NOUN
ejpam-5569	93	5	n	n	NOUN
ejpam-5569	93	6	=	=	SYM
ejpam-5569	93	7	2	2	NUM
ejpam-5569	93	8	,	,	PUNCT
ejpam-5569	93	9	rn(d2(pn	rn(d2(pn	NOUN
ejpam-5569	93	10	)	)	PUNCT
ejpam-5569	93	11	)	)	PUNCT
ejpam-5569	93	12	≤	≤	NUM
ejpam-5569	93	13	4	4	NUM
ejpam-5569	93	14	.	.	PUNCT
ejpam-5569	94	1	proof	proof	NOUN
ejpam-5569	94	2	.	.	PUNCT
ejpam-5569	95	1	let	let	VERB
ejpam-5569	95	2	(	(	PUNCT
ejpam-5569	95	3	f(u1	f(u1	NOUN
ejpam-5569	95	4	)	)	PUNCT
ejpam-5569	95	5	,	,	PUNCT
ejpam-5569	95	6	f(u2	f(u2	PROPN
ejpam-5569	95	7	)	)	PUNCT
ejpam-5569	95	8	,	,	PUNCT
ejpam-5569	95	9	f(v1	f(v1	NOUN
ejpam-5569	95	10	)	)	PUNCT
ejpam-5569	95	11	,	,	PUNCT
ejpam-5569	95	12	f(v2	f(v2	NOUN
ejpam-5569	95	13	)	)	PUNCT
ejpam-5569	95	14	)	)	PUNCT
ejpam-5569	96	1	=	=	PUNCT
ejpam-5569	96	2	(	(	PUNCT
ejpam-5569	96	3	0	0	NUM
ejpam-5569	96	4	,	,	PUNCT
ejpam-5569	96	5	3	3	NUM
ejpam-5569	96	6	,	,	PUNCT
ejpam-5569	96	7	1	1	NUM
ejpam-5569	96	8	,	,	PUNCT
ejpam-5569	96	9	4	4	NUM
ejpam-5569	96	10	)	)	PUNCT
ejpam-5569	96	11	.	.	PUNCT
ejpam-5569	97	1	lemma	lemma	PROPN
ejpam-5569	97	2	2	2	NUM
ejpam-5569	97	3	.	.	X
ejpam-5569	98	1	for	for	ADP
ejpam-5569	98	2	a	a	DET
ejpam-5569	98	3	positive	positive	ADJ
ejpam-5569	98	4	integer	integer	NOUN
ejpam-5569	98	5	n	n	NOUN
ejpam-5569	98	6	=	=	SYM
ejpam-5569	98	7	3	3	NUM
ejpam-5569	98	8	,	,	PUNCT
ejpam-5569	98	9	rn(d2(pn	rn(d2(pn	NOUN
ejpam-5569	98	10	)	)	PUNCT
ejpam-5569	98	11	)	)	PUNCT
ejpam-5569	98	12	≤	≤	NUM
ejpam-5569	98	13	6	6	NUM
ejpam-5569	98	14	.	.	PUNCT
ejpam-5569	99	1	proof	proof	NOUN
ejpam-5569	99	2	.	.	PUNCT
ejpam-5569	100	1	let	let	VERB
ejpam-5569	100	2	(	(	PUNCT
ejpam-5569	100	3	f(u1	f(u1	NOUN
ejpam-5569	100	4	)	)	PUNCT
ejpam-5569	100	5	,	,	PUNCT
ejpam-5569	100	6	f(u2	f(u2	PROPN
ejpam-5569	100	7	)	)	PUNCT
ejpam-5569	100	8	,	,	PUNCT
ejpam-5569	100	9	f(u3	f(u3	NOUN
ejpam-5569	100	10	)	)	PUNCT
ejpam-5569	100	11	,	,	PUNCT
ejpam-5569	100	12	f(v1	f(v1	ADJ
ejpam-5569	100	13	)	)	PUNCT
ejpam-5569	100	14	,	,	PUNCT
ejpam-5569	100	15	f(v2	f(v2	NOUN
ejpam-5569	100	16	)	)	PUNCT
ejpam-5569	100	17	,	,	PUNCT
ejpam-5569	100	18	f(v3	f(v3	NOUN
ejpam-5569	100	19	)	)	PUNCT
ejpam-5569	100	20	)	)	PUNCT
ejpam-5569	101	1	=	=	SYM
ejpam-5569	101	2	(	(	PUNCT
ejpam-5569	101	3	0	0	NUM
ejpam-5569	101	4	,	,	PUNCT
ejpam-5569	101	5	5	5	NUM
ejpam-5569	101	6	,	,	PUNCT
ejpam-5569	101	7	1	1	NUM
ejpam-5569	101	8	,	,	PUNCT
ejpam-5569	101	9	2	2	NUM
ejpam-5569	101	10	,	,	PUNCT
ejpam-5569	101	11	6	6	NUM
ejpam-5569	101	12	,	,	PUNCT
ejpam-5569	101	13	3	3	NUM
ejpam-5569	101	14	)	)	PUNCT
ejpam-5569	101	15	.	.	PUNCT
ejpam-5569	102	1	lemma	lemma	PROPN
ejpam-5569	102	2	3	3	X
ejpam-5569	102	3	.	.	PUNCT
ejpam-5569	103	1	for	for	ADP
ejpam-5569	103	2	a	a	DET
ejpam-5569	103	3	positive	positive	ADJ
ejpam-5569	103	4	integer	integer	NOUN
ejpam-5569	103	5	n	n	NOUN
ejpam-5569	103	6	=	=	SYM
ejpam-5569	103	7	4	4	NUM
ejpam-5569	103	8	,	,	PUNCT
ejpam-5569	103	9	rn(d2(pn	rn(d2(pn	NOUN
ejpam-5569	103	10	)	)	PUNCT
ejpam-5569	103	11	)	)	PUNCT
ejpam-5569	103	12	≤	≤	NUM
ejpam-5569	103	13	12	12	NUM
ejpam-5569	103	14	.	.	PUNCT
ejpam-5569	103	15	proof	proof	NOUN
ejpam-5569	103	16	.	.	PUNCT
ejpam-5569	104	1	for	for	ADP
ejpam-5569	104	2	n	n	NOUN
ejpam-5569	104	3	=	=	SYM
ejpam-5569	104	4	4	4	NUM
ejpam-5569	104	5	,	,	PUNCT
ejpam-5569	104	6	let	let	VERB
ejpam-5569	104	7	(	(	PUNCT
ejpam-5569	104	8	f(u1	f(u1	NOUN
ejpam-5569	104	9	)	)	PUNCT
ejpam-5569	104	10	,	,	PUNCT
ejpam-5569	104	11	f(u2	f(u2	PROPN
ejpam-5569	104	12	)	)	PUNCT
ejpam-5569	104	13	,	,	PUNCT
ejpam-5569	104	14	f(u3	f(u3	NOUN
ejpam-5569	104	15	)	)	PUNCT
ejpam-5569	104	16	,	,	PUNCT
ejpam-5569	104	17	f(u4	f(u4	NOUN
ejpam-5569	104	18	)	)	PUNCT
ejpam-5569	104	19	,	,	PUNCT
ejpam-5569	104	20	f(v1	f(v1	ADJ
ejpam-5569	104	21	)	)	PUNCT
ejpam-5569	104	22	,	,	PUNCT
ejpam-5569	104	23	f(v2	f(v2	NOUN
ejpam-5569	104	24	)	)	PUNCT
ejpam-5569	104	25	,	,	PUNCT
ejpam-5569	104	26	f(v3	f(v3	NOUN
ejpam-5569	104	27	)	)	PUNCT
ejpam-5569	104	28	,	,	PUNCT
ejpam-5569	104	29	f(v4	f(v4	NUM
ejpam-5569	104	30	)	)	PUNCT
ejpam-5569	104	31	)	)	PUNCT
ejpam-5569	105	1	=	=	SYM
ejpam-5569	105	2	(	(	PUNCT
ejpam-5569	105	3	0	0	NUM
ejpam-5569	105	4	,	,	PUNCT
ejpam-5569	105	5	5	5	NUM
ejpam-5569	105	6	,	,	PUNCT
ejpam-5569	105	7	10	10	NUM
ejpam-5569	105	8	,	,	PUNCT
ejpam-5569	105	9	1	1	NUM
ejpam-5569	105	10	,	,	PUNCT
ejpam-5569	105	11	2	2	NUM
ejpam-5569	105	12	,	,	PUNCT
ejpam-5569	105	13	7	7	NUM
ejpam-5569	105	14	,	,	PUNCT
ejpam-5569	105	15	12	12	NUM
ejpam-5569	105	16	,	,	PUNCT
ejpam-5569	105	17	3	3	NUM
ejpam-5569	105	18	)	)	PUNCT
ejpam-5569	105	19	.	.	PUNCT
ejpam-5569	106	1	lemma	lemma	PROPN
ejpam-5569	106	2	4	4	NUM
ejpam-5569	106	3	.	.	PUNCT
ejpam-5569	107	1	for	for	ADP
ejpam-5569	107	2	a	a	DET
ejpam-5569	107	3	positive	positive	ADJ
ejpam-5569	107	4	integer	integer	NOUN
ejpam-5569	107	5	n	n	NOUN
ejpam-5569	107	6	=	=	SYM
ejpam-5569	107	7	5	5	NUM
ejpam-5569	107	8	,	,	PUNCT
ejpam-5569	107	9	rn(d2(pn	rn(d2(pn	NOUN
ejpam-5569	107	10	)	)	PUNCT
ejpam-5569	107	11	)	)	PUNCT
ejpam-5569	107	12	≤	≤	NUM
ejpam-5569	107	13	22	22	NUM
ejpam-5569	107	14	.	.	PUNCT
ejpam-5569	108	1	proof	proof	NOUN
ejpam-5569	108	2	.	.	PUNCT
ejpam-5569	109	1	for	for	ADP
ejpam-5569	109	2	n	n	NOUN
ejpam-5569	109	3	=	=	SYM
ejpam-5569	109	4	5	5	NUM
ejpam-5569	109	5	let	let	VERB
ejpam-5569	109	6	(	(	PUNCT
ejpam-5569	109	7	f(u1	f(u1	NOUN
ejpam-5569	109	8	)	)	PUNCT
ejpam-5569	109	9	,	,	PUNCT
ejpam-5569	109	10	f(u2	f(u2	PROPN
ejpam-5569	109	11	)	)	PUNCT
ejpam-5569	109	12	,	,	PUNCT
ejpam-5569	109	13	f(u3	f(u3	NOUN
ejpam-5569	109	14	)	)	PUNCT
ejpam-5569	109	15	,	,	PUNCT
ejpam-5569	109	16	f(u4	f(u4	NOUN
ejpam-5569	109	17	)	)	PUNCT
ejpam-5569	109	18	,	,	PUNCT
ejpam-5569	109	19	f(u5	f(u5	NOUN
ejpam-5569	109	20	)	)	PUNCT
ejpam-5569	109	21	,	,	PUNCT
ejpam-5569	109	22	f(v1	f(v1	ADJ
ejpam-5569	109	23	)	)	PUNCT
ejpam-5569	109	24	,	,	PUNCT
ejpam-5569	109	25	f(v2	f(v2	NOUN
ejpam-5569	109	26	)	)	PUNCT
ejpam-5569	109	27	,	,	PUNCT
ejpam-5569	109	28	f(v3	f(v3	NOUN
ejpam-5569	109	29	)	)	PUNCT
ejpam-5569	109	30	,	,	PUNCT
ejpam-5569	109	31	f(v4	f(v4	NUM
ejpam-5569	109	32	)	)	PUNCT
ejpam-5569	109	33	,	,	PUNCT
ejpam-5569	109	34	f(v5	f(v5	NOUN
ejpam-5569	109	35	)	)	PUNCT
ejpam-5569	109	36	)	)	PUNCT
ejpam-5569	110	1	=	=	PUNCT
ejpam-5569	110	2	=	=	PUNCT
ejpam-5569	110	3	(	(	PUNCT
ejpam-5569	110	4	6	6	NUM
ejpam-5569	110	5	,	,	PUNCT
ejpam-5569	110	6	13	13	NUM
ejpam-5569	110	7	,	,	PUNCT
ejpam-5569	110	8	0	0	NUM
ejpam-5569	110	9	,	,	PUNCT
ejpam-5569	110	10	16	16	NUM
ejpam-5569	110	11	,	,	PUNCT
ejpam-5569	110	12	7	7	NUM
ejpam-5569	110	13	,	,	PUNCT
ejpam-5569	110	14	9	9	NUM
ejpam-5569	110	15	,	,	PUNCT
ejpam-5569	110	16	19	19	NUM
ejpam-5569	110	17	,	,	PUNCT
ejpam-5569	110	18	3	3	NUM
ejpam-5569	110	19	,	,	PUNCT
ejpam-5569	110	20	22	22	NUM
ejpam-5569	110	21	,	,	PUNCT
ejpam-5569	110	22	10	10	NUM
ejpam-5569	110	23	)	)	PUNCT
ejpam-5569	110	24	in	in	ADP
ejpam-5569	110	25	the	the	DET
ejpam-5569	110	26	next	next	ADJ
ejpam-5569	110	27	theorems	theorem	NOUN
ejpam-5569	110	28	,	,	PUNCT
ejpam-5569	110	29	we	we	PRON
ejpam-5569	110	30	find	find	VERB
ejpam-5569	110	31	upper	upper	ADJ
ejpam-5569	110	32	bound	bind	VERB
ejpam-5569	110	33	of	of	ADP
ejpam-5569	110	34	rn(d2(pn	rn(d2(pn	NOUN
ejpam-5569	110	35	)	)	PUNCT
ejpam-5569	110	36	)	)	PUNCT
ejpam-5569	110	37	for	for	ADP
ejpam-5569	110	38	n	n	X
ejpam-5569	110	39	≥	≥	NUM
ejpam-5569	110	40	6	6	NUM
ejpam-5569	110	41	.	.	PUNCT
ejpam-5569	111	1	theorem	theorem	NOUN
ejpam-5569	111	2	1	1	NUM
ejpam-5569	111	3	.	.	PUNCT
ejpam-5569	112	1	let	let	VERB
ejpam-5569	112	2	n	n	PRON
ejpam-5569	112	3	≥	≥	NUM
ejpam-5569	112	4	6	6	NUM
ejpam-5569	112	5	be	be	AUX
ejpam-5569	112	6	a	a	DET
ejpam-5569	112	7	positive	positive	ADJ
ejpam-5569	112	8	integer	integer	NOUN
ejpam-5569	112	9	such	such	ADJ
ejpam-5569	112	10	that	that	SCONJ
ejpam-5569	112	11	n	n	NUM
ejpam-5569	112	12	≡	≡	PROPN
ejpam-5569	112	13	2(mod4	2(mod4	NUM
ejpam-5569	112	14	)	)	PUNCT
ejpam-5569	112	15	.	.	PUNCT
ejpam-5569	113	1	then	then	ADV
ejpam-5569	113	2	rn(d2(pn	rn(d2(pn	PROPN
ejpam-5569	113	3	)	)	PUNCT
ejpam-5569	113	4	)	)	PUNCT
ejpam-5569	114	1	≤	≤	NUM
ejpam-5569	114	2	n2	n2	NOUN
ejpam-5569	114	3	−	−	PROPN
ejpam-5569	114	4	n	n	PRON
ejpam-5569	114	5	2	2	NUM
ejpam-5569	114	6	.	.	PUNCT
ejpam-5569	115	1	proof	proof	NOUN
ejpam-5569	115	2	.	.	PUNCT
ejpam-5569	116	1	for	for	ADP
ejpam-5569	116	2	k	k	PROPN
ejpam-5569	116	3	≥	≥	NUM
ejpam-5569	116	4	1	1	NUM
ejpam-5569	116	5	,	,	PUNCT
ejpam-5569	116	6	n	n	PRON
ejpam-5569	116	7	≡	≡	PROPN
ejpam-5569	116	8	2(mod4	2(mod4	NUM
ejpam-5569	116	9	)	)	PUNCT
ejpam-5569	116	10	such	such	ADJ
ejpam-5569	116	11	that	that	SCONJ
ejpam-5569	116	12	n	n	NOUN
ejpam-5569	116	13	=	=	SYM
ejpam-5569	116	14	4k+2	4k+2	PROPN
ejpam-5569	116	15	,	,	PUNCT
ejpam-5569	116	16	let	let	VERB
ejpam-5569	116	17	f	f	PRON
ejpam-5569	116	18	:	:	PUNCT
ejpam-5569	116	19	v	v	PROPN
ejpam-5569	116	20	(	(	PUNCT
ejpam-5569	116	21	d2(pn	d2(pn	PROPN
ejpam-5569	116	22	)	)	PUNCT
ejpam-5569	116	23	)	)	PUNCT
ejpam-5569	116	24	→	→	SYM
ejpam-5569	116	25	n	n	CCONJ
ejpam-5569	116	26	define	define	VERB
ejpam-5569	116	27	as	as	SCONJ
ejpam-5569	116	28	follows	follow	VERB
ejpam-5569	116	29	:	:	PUNCT
ejpam-5569	116	30	f(ui	f(ui	PROPN
ejpam-5569	116	31	)	)	PUNCT
ejpam-5569	116	32	=	=	SYM
ejpam-5569	116	33			PROPN
ejpam-5569	116	34	(	(	PUNCT
ejpam-5569	116	35	2n+	2n+	NUM
ejpam-5569	116	36	3)(i−	3)(i−	NUM
ejpam-5569	116	37	1	1	NUM
ejpam-5569	116	38	)	)	PUNCT
ejpam-5569	116	39	+	+	CCONJ
ejpam-5569	117	1	2(i−	2(i−	NUM
ejpam-5569	117	2	1)(i−	1)(i−	NUM
ejpam-5569	117	3	2	2	NUM
ejpam-5569	117	4	)	)	PUNCT
ejpam-5569	117	5	,	,	PUNCT
ejpam-5569	117	6	if	if	SCONJ
ejpam-5569	117	7	1	1	NUM
ejpam-5569	117	8	≤	≤	NUM
ejpam-5569	117	9	i	i	NOUN
ejpam-5569	117	10	≤	≤	NOUN
ejpam-5569	118	1	k	k	X
ejpam-5569	119	1	+	+	CCONJ
ejpam-5569	119	2	1	1	NUM
ejpam-5569	119	3	f(uk+1)−	f(uk+1)−	ADP
ejpam-5569	119	4	(	(	PUNCT
ejpam-5569	119	5	2n−	2n−	PROPN
ejpam-5569	119	6	1)−	1)−	PROPN
ejpam-5569	119	7	(	(	PUNCT
ejpam-5569	119	8	3n−	3n−	NUM
ejpam-5569	119	9	5)(j	5)(j	NUM
ejpam-5569	119	10	−	−	NUM
ejpam-5569	119	11	1	1	NUM
ejpam-5569	119	12	)	)	PUNCT
ejpam-5569	119	13	+	+	CCONJ
ejpam-5569	119	14	2(j	2(j	NUM
ejpam-5569	120	1	−	−	PROPN
ejpam-5569	120	2	1)(j	1)(j	NUM
ejpam-5569	120	3	−	−	PROPN
ejpam-5569	120	4	2	2	NUM
ejpam-5569	120	5	)	)	PUNCT
ejpam-5569	120	6	,	,	PUNCT
ejpam-5569	120	7	if	if	SCONJ
ejpam-5569	120	8	k	k	PROPN
ejpam-5569	120	9	+	+	CCONJ
ejpam-5569	120	10	2	2	X
ejpam-5569	120	11	≤	≤	NUM
ejpam-5569	120	12	i	i	NOUN
ejpam-5569	120	13	≤	≤	NOUN
ejpam-5569	120	14	2k	2k	NOUN
ejpam-5569	120	15	+	+	CCONJ
ejpam-5569	120	16	1	1	NUM
ejpam-5569	120	17	where	where	SCONJ
ejpam-5569	120	18	1	1	NUM
ejpam-5569	120	19	≤	≤	NUM
ejpam-5569	120	20	j	j	PROPN
ejpam-5569	120	21	≤	≤	PROPN
ejpam-5569	120	22	k	k	PROPN
ejpam-5569	120	23	,	,	PUNCT
ejpam-5569	120	24	and	and	CCONJ
ejpam-5569	120	25	i	i	PRON
ejpam-5569	120	26	=	=	PUNCT
ejpam-5569	121	1	k	k	PROPN
ejpam-5569	122	1	+	+	CCONJ
ejpam-5569	122	2	1	1	NUM
ejpam-5569	122	3	+	+	CCONJ
ejpam-5569	122	4	j	j	PROPN
ejpam-5569	122	5	f(uk+1+j	f(uk+1+j	PROPN
ejpam-5569	122	6	)	)	PUNCT
ejpam-5569	123	1	+	+	CCONJ
ejpam-5569	123	2	(	(	PUNCT
ejpam-5569	123	3	6k	6k	NOUN
ejpam-5569	123	4	+	+	X
ejpam-5569	123	5	1)−	1)−	NUM
ejpam-5569	123	6	2(j	2(j	NUM
ejpam-5569	123	7	−	−	NOUN
ejpam-5569	123	8	1	1	NUM
ejpam-5569	123	9	)	)	PUNCT
ejpam-5569	123	10	,	,	PUNCT
ejpam-5569	123	11	if	if	SCONJ
ejpam-5569	123	12	2k	2k	NUM
ejpam-5569	123	13	+	+	CCONJ
ejpam-5569	123	14	2	2	NUM
ejpam-5569	123	15	≤	≤	NUM
ejpam-5569	124	1	i	i	PRON
ejpam-5569	124	2	≤	≤	NOUN
ejpam-5569	124	3	3k	3k	X
ejpam-5569	124	4	+	+	CCONJ
ejpam-5569	124	5	1	1	NUM
ejpam-5569	124	6	,	,	PUNCT
ejpam-5569	124	7	where	where	SCONJ
ejpam-5569	124	8	1	1	NUM
ejpam-5569	124	9	≤	≤	NUM
ejpam-5569	124	10	j	j	PROPN
ejpam-5569	124	11	≤	≤	PROPN
ejpam-5569	124	12	k	k	PROPN
ejpam-5569	124	13	,	,	PUNCT
ejpam-5569	124	14	and	and	CCONJ
ejpam-5569	124	15	,	,	PUNCT
ejpam-5569	124	16	i	i	PRON
ejpam-5569	124	17	=	=	PUNCT
ejpam-5569	124	18	3k	3k	X
ejpam-5569	124	19	+	+	CCONJ
ejpam-5569	124	20	2−	2−	NUM
ejpam-5569	124	21	j	j	NOUN
ejpam-5569	124	22	f(uj	f(uj	PROPN
ejpam-5569	124	23	)	)	PUNCT
ejpam-5569	125	1	+	+	CCONJ
ejpam-5569	125	2	2j	2j	NUM
ejpam-5569	125	3	−	−	NOUN
ejpam-5569	125	4	1	1	NUM
ejpam-5569	125	5	,	,	PUNCT
ejpam-5569	125	6	if	if	SCONJ
ejpam-5569	125	7	3k	3k	PRON
ejpam-5569	125	8	+	+	CCONJ
ejpam-5569	125	9	2	2	NUM
ejpam-5569	125	10	≤	≤	NUM
ejpam-5569	125	11	i	i	PRON
ejpam-5569	125	12	≤	≤	NOUN
ejpam-5569	125	13	4k	4k	NUM
ejpam-5569	125	14	+	+	CCONJ
ejpam-5569	125	15	2	2	NUM
ejpam-5569	125	16	=	=	SYM
ejpam-5569	125	17	n	n	CCONJ
ejpam-5569	125	18	,	,	PUNCT
ejpam-5569	125	19	where	where	SCONJ
ejpam-5569	125	20	1	1	NUM
ejpam-5569	125	21	≤	≤	NUM
ejpam-5569	125	22	j	j	PROPN
ejpam-5569	125	23	≤	≤	PROPN
ejpam-5569	125	24	k	k	PROPN
ejpam-5569	126	1	+	+	CCONJ
ejpam-5569	126	2	1	1	NUM
ejpam-5569	126	3	,	,	PUNCT
ejpam-5569	126	4	and	and	CCONJ
ejpam-5569	126	5	i	i	PRON
ejpam-5569	126	6	=	=	VERB
ejpam-5569	126	7	4k	4k	PRON
ejpam-5569	126	8	+	+	CCONJ
ejpam-5569	126	9	3−	3−	NUM
ejpam-5569	126	10	j	j	PROPN
ejpam-5569	126	11	n.	n.	PROPN
ejpam-5569	126	12	m.	m.	NOUN
ejpam-5569	126	13	noureldeen	noureldeen	INTJ
ejpam-5569	126	14	et	et	PROPN
ejpam-5569	127	1	al	al	PROPN
ejpam-5569	127	2	.	.	PUNCT
ejpam-5569	127	3	/	/	SYM
ejpam-5569	127	4	eur	eur	PROPN
ejpam-5569	127	5	.	.	PUNCT
ejpam-5569	128	1	j.	j.	PROPN
ejpam-5569	128	2	pure	pure	PROPN
ejpam-5569	128	3	appl	appl	PROPN
ejpam-5569	128	4	.	.	PROPN
ejpam-5569	128	5	math	math	PROPN
ejpam-5569	128	6	,	,	PUNCT
ejpam-5569	128	7	18	18	NUM
ejpam-5569	128	8	(	(	PUNCT
ejpam-5569	128	9	2	2	NUM
ejpam-5569	128	10	)	)	PUNCT
ejpam-5569	128	11	(	(	PUNCT
ejpam-5569	128	12	2025	2025	NUM
ejpam-5569	128	13	)	)	PUNCT
ejpam-5569	128	14	,	,	PUNCT
ejpam-5569	128	15	5569	5569	NUM
ejpam-5569	128	16	6	6	NUM
ejpam-5569	128	17	of	of	ADP
ejpam-5569	128	18	25	25	NUM
ejpam-5569	128	19	and	and	CCONJ
ejpam-5569	128	20	for	for	ADP
ejpam-5569	128	21	the	the	DET
ejpam-5569	128	22	vertices	vertex	NOUN
ejpam-5569	128	23	belongs	belong	VERB
ejpam-5569	128	24	to	to	ADP
ejpam-5569	128	25	vv	vv	NOUN
ejpam-5569	128	26	,	,	PUNCT
ejpam-5569	128	27	f(vi	f(vi	NOUN
ejpam-5569	128	28	)	)	PUNCT
ejpam-5569	128	29	=	=	SYM
ejpam-5569	128	30			PROPN
ejpam-5569	128	31	(	(	PUNCT
ejpam-5569	128	32	n+	n+	NOUN
ejpam-5569	128	33	2	2	NUM
ejpam-5569	128	34	)	)	PUNCT
ejpam-5569	129	1	+	+	CCONJ
ejpam-5569	129	2	(	(	PUNCT
ejpam-5569	129	3	2n+	2n+	NUM
ejpam-5569	129	4	7)(i−	7)(i−	NUM
ejpam-5569	129	5	1	1	NUM
ejpam-5569	129	6	)	)	PUNCT
ejpam-5569	129	7	+	+	CCONJ
ejpam-5569	130	1	2(i−	2(i−	NUM
ejpam-5569	130	2	1)(i−	1)(i−	NUM
ejpam-5569	130	3	2	2	NUM
ejpam-5569	130	4	)	)	PUNCT
ejpam-5569	130	5	,	,	PUNCT
ejpam-5569	130	6	if	if	SCONJ
ejpam-5569	130	7	1	1	NUM
ejpam-5569	130	8	≤	≤	NUM
ejpam-5569	130	9	i	i	NOUN
ejpam-5569	130	10	≤	≤	NOUN
ejpam-5569	130	11	k	k	PROPN
ejpam-5569	130	12	,	,	PUNCT
ejpam-5569	130	13	f(vk	f(vk	PROPN
ejpam-5569	130	14	)	)	PUNCT
ejpam-5569	130	15	+	+	CCONJ
ejpam-5569	130	16	(	(	PUNCT
ejpam-5569	130	17	2n+	2n+	NUM
ejpam-5569	130	18	1	1	NUM
ejpam-5569	130	19	)	)	PUNCT
ejpam-5569	130	20	+	+	CCONJ
ejpam-5569	130	21	(	(	PUNCT
ejpam-5569	130	22	n+	n+	NOUN
ejpam-5569	130	23	1)(j	1)(j	NUM
ejpam-5569	130	24	−	−	NOUN
ejpam-5569	130	25	1	1	NUM
ejpam-5569	130	26	)	)	PUNCT
ejpam-5569	130	27	+	+	CCONJ
ejpam-5569	130	28	2(j	2(j	NUM
ejpam-5569	130	29	−	−	PROPN
ejpam-5569	130	30	1)(j	1)(j	NUM
ejpam-5569	130	31	−	−	PROPN
ejpam-5569	130	32	2	2	NUM
ejpam-5569	130	33	)	)	PUNCT
ejpam-5569	130	34	,	,	PUNCT
ejpam-5569	130	35	if	if	SCONJ
ejpam-5569	130	36	k	k	PROPN
ejpam-5569	130	37	+	+	PROPN
ejpam-5569	130	38	1	1	X
ejpam-5569	130	39	≤	≤	NUM
ejpam-5569	130	40	i	i	NOUN
ejpam-5569	130	41	≤	≤	NOUN
ejpam-5569	130	42	2k	2k	NOUN
ejpam-5569	130	43	+	+	CCONJ
ejpam-5569	130	44	1	1	NUM
ejpam-5569	130	45	,	,	PUNCT
ejpam-5569	130	46	where	where	SCONJ
ejpam-5569	130	47	1	1	NUM
ejpam-5569	130	48	≤	≤	NUM
ejpam-5569	130	49	j	j	PROPN
ejpam-5569	130	50	≤	≤	PROPN
ejpam-5569	131	1	k	k	PROPN
ejpam-5569	132	1	+	+	CCONJ
ejpam-5569	132	2	1	1	NUM
ejpam-5569	132	3	,	,	PUNCT
ejpam-5569	132	4	andi	andi	PROPN
ejpam-5569	132	5	=	=	PROPN
ejpam-5569	132	6	k	k	PROPN
ejpam-5569	133	1	+	+	CCONJ
ejpam-5569	133	2	j	j	PROPN
ejpam-5569	133	3	f(vk+j	f(vk+j	PROPN
ejpam-5569	133	4	)	)	PUNCT
ejpam-5569	134	1	+	+	CCONJ
ejpam-5569	134	2	(	(	PUNCT
ejpam-5569	134	3	2k	2k	NUM
ejpam-5569	134	4	+	+	CCONJ
ejpam-5569	134	5	1	1	NUM
ejpam-5569	134	6	)	)	PUNCT
ejpam-5569	134	7	+	+	CCONJ
ejpam-5569	135	1	2(j	2(j	NUM
ejpam-5569	135	2	−	−	NOUN
ejpam-5569	135	3	1	1	NUM
ejpam-5569	135	4	)	)	PUNCT
ejpam-5569	135	5	,	,	PUNCT
ejpam-5569	135	6	if	if	SCONJ
ejpam-5569	135	7	2k	2k	NUM
ejpam-5569	135	8	+	+	CCONJ
ejpam-5569	135	9	2	2	NUM
ejpam-5569	135	10	≤	≤	NUM
ejpam-5569	135	11	i	i	PRON
ejpam-5569	135	12	≤	≤	NOUN
ejpam-5569	135	13	3k	3k	X
ejpam-5569	135	14	+	+	CCONJ
ejpam-5569	135	15	2	2	NUM
ejpam-5569	135	16	,	,	PUNCT
ejpam-5569	135	17	where	where	SCONJ
ejpam-5569	135	18	1	1	NUM
ejpam-5569	135	19	≤	≤	NUM
ejpam-5569	135	20	j	j	PROPN
ejpam-5569	135	21	≤	≤	PROPN
ejpam-5569	136	1	k	k	PROPN
ejpam-5569	137	1	+	+	CCONJ
ejpam-5569	137	2	1	1	NUM
ejpam-5569	137	3	,	,	PUNCT
ejpam-5569	137	4	and	and	CCONJ
ejpam-5569	137	5	i	i	PRON
ejpam-5569	137	6	=	=	NOUN
ejpam-5569	138	1	3k	3k	X
ejpam-5569	139	1	+	+	CCONJ
ejpam-5569	139	2	3−	3−	NUM
ejpam-5569	139	3	j	j	PROPN
ejpam-5569	139	4	f(vj)−	f(vj)−	NOUN
ejpam-5569	139	5	2j	2j	NUM
ejpam-5569	140	1	+	+	CCONJ
ejpam-5569	140	2	1	1	NUM
ejpam-5569	140	3	,	,	PUNCT
ejpam-5569	140	4	if	if	SCONJ
ejpam-5569	140	5	3k	3k	PRON
ejpam-5569	140	6	+	+	CCONJ
ejpam-5569	140	7	3	3	NUM
ejpam-5569	140	8	≤	≤	NUM
ejpam-5569	141	1	i	i	PRON
ejpam-5569	141	2	≤	≤	NOUN
ejpam-5569	141	3	4k	4k	NUM
ejpam-5569	141	4	+	+	CCONJ
ejpam-5569	141	5	2	2	NUM
ejpam-5569	141	6	=	=	SYM
ejpam-5569	141	7	n	n	CCONJ
ejpam-5569	141	8	,	,	PUNCT
ejpam-5569	141	9	where	where	SCONJ
ejpam-5569	141	10	1	1	NUM
ejpam-5569	141	11	≤	≤	NUM
ejpam-5569	141	12	j	j	PROPN
ejpam-5569	141	13	≤	≤	PROPN
ejpam-5569	141	14	k	k	PROPN
ejpam-5569	141	15	,	,	PUNCT
ejpam-5569	141	16	and	and	CCONJ
ejpam-5569	141	17	i	i	PRON
ejpam-5569	141	18	=	=	VERB
ejpam-5569	142	1	4k	4k	PRON
ejpam-5569	142	2	+	+	CCONJ
ejpam-5569	143	1	3−	3−	NUM
ejpam-5569	143	2	j	j	NOUN
ejpam-5569	143	3	now	now	ADV
ejpam-5569	143	4	we	we	PRON
ejpam-5569	143	5	claim	claim	VERB
ejpam-5569	143	6	to	to	PART
ejpam-5569	143	7	prove	prove	VERB
ejpam-5569	143	8	that	that	SCONJ
ejpam-5569	143	9	the	the	DET
ejpam-5569	143	10	function	function	NOUN
ejpam-5569	143	11	f	f	NOUN
ejpam-5569	143	12	given	give	VERB
ejpam-5569	143	13	above	above	ADV
ejpam-5569	143	14	is	be	AUX
ejpam-5569	143	15	a	a	DET
ejpam-5569	143	16	radio	radio	NOUN
ejpam-5569	143	17	labelling	labelling	NOUN
ejpam-5569	143	18	of	of	ADP
ejpam-5569	143	19	d2(pn	d2(pn	PROPN
ejpam-5569	143	20	)	)	PUNCT
ejpam-5569	143	21	consequently	consequently	ADV
ejpam-5569	143	22	,	,	PUNCT
ejpam-5569	143	23	it	it	PRON
ejpam-5569	143	24	is	be	AUX
ejpam-5569	143	25	required	require	VERB
ejpam-5569	143	26	that	that	SCONJ
ejpam-5569	143	27	f	f	PROPN
ejpam-5569	143	28	verifies	verify	VERB
ejpam-5569	143	29	the	the	DET
ejpam-5569	143	30	following	follow	VERB
ejpam-5569	143	31	condition	condition	NOUN
ejpam-5569	143	32	|f(x)−	|f(x)−	NOUN
ejpam-5569	143	33	f(y)|	f(y)|	PROPN
ejpam-5569	144	1	≥	≥	NUM
ejpam-5569	144	2	n−	n−	PROPN
ejpam-5569	144	3	d(x	d(x	PROPN
ejpam-5569	144	4	,	,	PUNCT
ejpam-5569	144	5	y	y	NOUN
ejpam-5569	144	6	)	)	PUNCT
ejpam-5569	144	7	for	for	ADP
ejpam-5569	144	8	any	any	DET
ejpam-5569	144	9	two	two	NUM
ejpam-5569	144	10	distinct	distinct	ADJ
ejpam-5569	144	11	vertices	vertex	NOUN
ejpam-5569	144	12	x	x	X
ejpam-5569	144	13	,	,	PUNCT
ejpam-5569	144	14	y	y	PROPN
ejpam-5569	144	15	∈	∈	PROPN
ejpam-5569	144	16	v	v	PROPN
ejpam-5569	144	17	(	(	PUNCT
ejpam-5569	144	18	d2(pn	d2(pn	PROPN
ejpam-5569	144	19	)	)	PUNCT
ejpam-5569	144	20	)	)	PUNCT
ejpam-5569	144	21	.	.	PUNCT
ejpam-5569	145	1	case	case	NOUN
ejpam-5569	145	2	1	1	NUM
ejpam-5569	145	3	x	x	NOUN
ejpam-5569	145	4	,	,	PUNCT
ejpam-5569	145	5	y	y	PROPN
ejpam-5569	145	6	∈	∈	PROPN
ejpam-5569	145	7	vu	vu	X
ejpam-5569	145	8	.	.	PUNCT
ejpam-5569	146	1	if	if	SCONJ
ejpam-5569	146	2	{	{	PUNCT
ejpam-5569	146	3	x	x	NOUN
ejpam-5569	146	4	,	,	PUNCT
ejpam-5569	146	5	y	y	PROPN
ejpam-5569	146	6	}	}	PUNCT
ejpam-5569	146	7	=	=	SYM
ejpam-5569	146	8	{	{	PUNCT
ejpam-5569	146	9	u1	u1	NOUN
ejpam-5569	146	10	,	,	PUNCT
ejpam-5569	146	11	un	un	ADJ
ejpam-5569	146	12	}	}	PUNCT
ejpam-5569	146	13	then	then	ADV
ejpam-5569	146	14	|f(x)−	|f(x)−	NOUN
ejpam-5569	146	15	f(y)|	f(y)|	PROPN
ejpam-5569	146	16	=	=	SYM
ejpam-5569	146	17	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	146	18	f(un)|	f(un)|	PROPN
ejpam-5569	147	1	=	=	SYM
ejpam-5569	147	2	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	148	1	[	[	X
ejpam-5569	148	2	f(u1	f(u1	NOUN
ejpam-5569	148	3	)	)	PUNCT
ejpam-5569	149	1	+	+	CCONJ
ejpam-5569	149	2	2−	2−	NUM
ejpam-5569	149	3	1]|	1]|	NUM
ejpam-5569	149	4	=	=	SYM
ejpam-5569	149	5	1	1	NUM
ejpam-5569	149	6	≥	≥	NOUN
ejpam-5569	149	7	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	149	8	)	)	PUNCT
ejpam-5569	149	9	)	)	PUNCT
ejpam-5569	150	1	+	+	CCONJ
ejpam-5569	150	2	1−	1−	NUM
ejpam-5569	150	3	d(x	d(x	PROPN
ejpam-5569	150	4	,	,	PUNCT
ejpam-5569	150	5	y	y	NOUN
ejpam-5569	150	6	)	)	PUNCT
ejpam-5569	150	7	,	,	PUNCT
ejpam-5569	150	8	where	where	SCONJ
ejpam-5569	150	9	d(x	d(x	PROPN
ejpam-5569	150	10	,	,	PUNCT
ejpam-5569	150	11	y	y	NOUN
ejpam-5569	150	12	)	)	PUNCT
ejpam-5569	150	13	=	=	VERB
ejpam-5569	150	14	n−	n−	NOUN
ejpam-5569	150	15	1	1	NUM
ejpam-5569	150	16	.	.	PUNCT
ejpam-5569	150	17	case	case	NOUN
ejpam-5569	150	18	2	2	NUM
ejpam-5569	150	19	if	if	SCONJ
ejpam-5569	150	20	{	{	PUNCT
ejpam-5569	150	21	x	x	NOUN
ejpam-5569	150	22	,	,	PUNCT
ejpam-5569	150	23	y	y	PROPN
ejpam-5569	150	24	}	}	PUNCT
ejpam-5569	150	25	=	=	SYM
ejpam-5569	150	26	{	{	PUNCT
ejpam-5569	150	27	u1	u1	NOUN
ejpam-5569	150	28	,	,	PUNCT
ejpam-5569	150	29	uk+1	uk+1	VERB
ejpam-5569	150	30	}	}	PUNCT
ejpam-5569	150	31	then	then	ADV
ejpam-5569	150	32	|f(x)−	|f(x)−	NOUN
ejpam-5569	150	33	f(y)|	f(y)|	PROPN
ejpam-5569	150	34	=	=	SYM
ejpam-5569	150	35	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	151	1	f(uk+1)|	f(uk+1)|	X
ejpam-5569	151	2	=	=	SYM
ejpam-5569	151	3	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	152	1	[	[	X
ejpam-5569	152	2	(	(	PUNCT
ejpam-5569	152	3	2n+	2n+	NUM
ejpam-5569	152	4	3)(k	3)(k	NUM
ejpam-5569	152	5	)	)	PUNCT
ejpam-5569	153	1	+	+	NUM
ejpam-5569	153	2	2k(k	2k(k	NUM
ejpam-5569	153	3	−	−	NOUN
ejpam-5569	153	4	1)]|	1)]|	NOUN
ejpam-5569	153	5	=	=	NOUN
ejpam-5569	154	1	|	|	ADV
ejpam-5569	154	2	−	−	PROPN
ejpam-5569	155	1	[	[	X
ejpam-5569	155	2	(	(	PUNCT
ejpam-5569	155	3	2n+	2n+	NUM
ejpam-5569	155	4	3)(k	3)(k	NUM
ejpam-5569	155	5	)	)	PUNCT
ejpam-5569	156	1	+	+	NUM
ejpam-5569	157	1	2k(k	2k(k	NUM
ejpam-5569	157	2	−	−	NOUN
ejpam-5569	157	3	1)]|	1)]|	NOUN
ejpam-5569	157	4	=	=	SYM
ejpam-5569	157	5	(	(	PUNCT
ejpam-5569	157	6	2n+	2n+	NUM
ejpam-5569	157	7	3)(k	3)(k	NUM
ejpam-5569	157	8	)	)	PUNCT
ejpam-5569	158	1	+	+	CCONJ
ejpam-5569	158	2	2k(k	2k(k	NUM
ejpam-5569	158	3	−	−	NOUN
ejpam-5569	158	4	1	1	NUM
ejpam-5569	158	5	)	)	PUNCT
ejpam-5569	158	6	≥	≥	NOUN
ejpam-5569	158	7	n−	n−	PROPN
ejpam-5569	158	8	k	k	PROPN
ejpam-5569	158	9	,	,	PUNCT
ejpam-5569	158	10	≥	≥	NOUN
ejpam-5569	158	11	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	158	12	)	)	PUNCT
ejpam-5569	158	13	)	)	PUNCT
ejpam-5569	159	1	+	+	CCONJ
ejpam-5569	159	2	1−	1−	NUM
ejpam-5569	159	3	d(x	d(x	PROPN
ejpam-5569	159	4	,	,	PUNCT
ejpam-5569	159	5	y	y	NOUN
ejpam-5569	159	6	)	)	PUNCT
ejpam-5569	159	7	,	,	PUNCT
ejpam-5569	159	8	where	where	SCONJ
ejpam-5569	159	9	d(x	d(x	PROPN
ejpam-5569	159	10	,	,	PUNCT
ejpam-5569	159	11	y	y	NOUN
ejpam-5569	159	12	)	)	PUNCT
ejpam-5569	159	13	=	=	SYM
ejpam-5569	159	14	k	k	PROPN
ejpam-5569	159	15	case	case	NOUN
ejpam-5569	159	16	3	3	NUM
ejpam-5569	159	17	if	if	SCONJ
ejpam-5569	159	18	{	{	PUNCT
ejpam-5569	159	19	x	x	NOUN
ejpam-5569	159	20	,	,	PUNCT
ejpam-5569	159	21	y	y	PROPN
ejpam-5569	159	22	}	}	PUNCT
ejpam-5569	159	23	=	=	SYM
ejpam-5569	159	24	{	{	PUNCT
ejpam-5569	159	25	u1	u1	NOUN
ejpam-5569	159	26	,	,	PUNCT
ejpam-5569	159	27	u2k+1	u2k+1	ADP
ejpam-5569	159	28	}	}	PUNCT
ejpam-5569	159	29	then	then	ADV
ejpam-5569	159	30	d(x	d(x	PROPN
ejpam-5569	159	31	,	,	PUNCT
ejpam-5569	159	32	y	y	NOUN
ejpam-5569	159	33	)	)	PUNCT
ejpam-5569	159	34	=	=	SYM
ejpam-5569	160	1	2k	2k	NUM
ejpam-5569	160	2	|f(x)−	|f(x)−	NOUN
ejpam-5569	160	3	f(y)|	f(y)|	PROPN
ejpam-5569	160	4	=	=	SYM
ejpam-5569	160	5	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	161	1	f(u2k+1)|	f(u2k+1)|	NUM
ejpam-5569	161	2	=	=	SYM
ejpam-5569	161	3	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	162	1	[	[	X
ejpam-5569	162	2	f(uk+1)−	f(uk+1)−	X
ejpam-5569	162	3	(	(	PUNCT
ejpam-5569	162	4	2n−	2n−	PROPN
ejpam-5569	162	5	1)−	1)−	PROPN
ejpam-5569	162	6	(	(	PUNCT
ejpam-5569	162	7	3n−	3n−	PROPN
ejpam-5569	162	8	5)(k	5)(k	NUM
ejpam-5569	162	9	−	−	NOUN
ejpam-5569	162	10	1	1	NUM
ejpam-5569	162	11	)	)	PUNCT
ejpam-5569	162	12	+	+	CCONJ
ejpam-5569	163	1	2(k	2(k	NUM
ejpam-5569	163	2	−	−	NOUN
ejpam-5569	163	3	1)(k	1)(k	NUM
ejpam-5569	163	4	−	−	PROPN
ejpam-5569	163	5	2)]|	2)]|	NOUN
ejpam-5569	163	6	=	=	SYM
ejpam-5569	163	7	|f(uk+1)−	|f(uk+1)−	NOUN
ejpam-5569	163	8	(	(	PUNCT
ejpam-5569	163	9	2n−	2n−	PROPN
ejpam-5569	163	10	1)−	1)−	PROPN
ejpam-5569	163	11	(	(	PUNCT
ejpam-5569	163	12	3n−	3n−	PROPN
ejpam-5569	163	13	5)(k	5)(k	NUM
ejpam-5569	163	14	−	−	NOUN
ejpam-5569	163	15	1	1	NUM
ejpam-5569	163	16	)	)	PUNCT
ejpam-5569	163	17	+	+	CCONJ
ejpam-5569	163	18	2(k	2(k	NUM
ejpam-5569	163	19	−	−	NOUN
ejpam-5569	163	20	1)(k	1)(k	NUM
ejpam-5569	163	21	−	−	PROPN
ejpam-5569	163	22	2)|	2)|	NUM
ejpam-5569	164	1	=	=	SYM
ejpam-5569	164	2	|(2n+	|(2n+	PROPN
ejpam-5569	164	3	3)(k	3)(k	NUM
ejpam-5569	164	4	)	)	PUNCT
ejpam-5569	165	1	+	+	NUM
ejpam-5569	166	1	2k(k	2k(k	NUM
ejpam-5569	166	2	−	−	PROPN
ejpam-5569	166	3	1)−	1)−	NUM
ejpam-5569	166	4	(	(	PUNCT
ejpam-5569	166	5	2n−	2n−	PROPN
ejpam-5569	166	6	1)−	1)−	PROPN
ejpam-5569	166	7	(	(	PUNCT
ejpam-5569	166	8	3n−	3n−	PROPN
ejpam-5569	166	9	5)(k	5)(k	NUM
ejpam-5569	166	10	−	−	NOUN
ejpam-5569	166	11	1	1	NUM
ejpam-5569	166	12	)	)	PUNCT
ejpam-5569	166	13	+	+	CCONJ
ejpam-5569	166	14	2(k	2(k	NUM
ejpam-5569	166	15	−	−	NOUN
ejpam-5569	166	16	1)(k	1)(k	NUM
ejpam-5569	166	17	−	−	PROPN
ejpam-5569	166	18	2)|	2)|	NUM
ejpam-5569	166	19	≥	≥	NUM
ejpam-5569	166	20	n−	n−	NOUN
ejpam-5569	166	21	2k	2k	NUM
ejpam-5569	166	22	case	case	NOUN
ejpam-5569	166	23	4	4	NUM
ejpam-5569	166	24	if	if	SCONJ
ejpam-5569	166	25	{	{	PUNCT
ejpam-5569	166	26	x	x	NOUN
ejpam-5569	166	27	,	,	PUNCT
ejpam-5569	166	28	y	y	PROPN
ejpam-5569	166	29	}	}	PUNCT
ejpam-5569	166	30	=	=	SYM
ejpam-5569	166	31	{	{	PUNCT
ejpam-5569	166	32	u1	u1	NOUN
ejpam-5569	166	33	,	,	PUNCT
ejpam-5569	166	34	u3k+1	u3k+1	NOUN
ejpam-5569	166	35	}	}	PUNCT
ejpam-5569	166	36	then	then	ADV
ejpam-5569	166	37	d(x	d(x	PROPN
ejpam-5569	166	38	,	,	PUNCT
ejpam-5569	166	39	y	y	NOUN
ejpam-5569	166	40	)	)	PUNCT
ejpam-5569	166	41	=	=	PUNCT
ejpam-5569	167	1	3k	3k	PRON
ejpam-5569	167	2	|f(x)−	|f(x)−	NOUN
ejpam-5569	167	3	f(y)|	f(y)|	PROPN
ejpam-5569	167	4	=	=	SYM
ejpam-5569	167	5	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	168	1	f(u3k+1)|	f(u3k+1)|	PROPN
ejpam-5569	168	2	=	=	SYM
ejpam-5569	168	3	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	169	1	[	[	X
ejpam-5569	169	2	f(uk+2	f(uk+2	NOUN
ejpam-5569	169	3	)	)	PUNCT
ejpam-5569	169	4	+	+	CCONJ
ejpam-5569	169	5	(	(	PUNCT
ejpam-5569	169	6	6k	6k	NOUN
ejpam-5569	169	7	+	+	CCONJ
ejpam-5569	169	8	1)]|	1)]|	ADJ
ejpam-5569	169	9	=	=	SYM
ejpam-5569	169	10	|f(uk+2	|f(uk+2	NOUN
ejpam-5569	169	11	)	)	PUNCT
ejpam-5569	170	1	+	+	CCONJ
ejpam-5569	170	2	(	(	PUNCT
ejpam-5569	170	3	6k	6k	NOUN
ejpam-5569	170	4	+	+	CCONJ
ejpam-5569	170	5	1)|	1)|	NUM
ejpam-5569	170	6	n.	n.	NOUN
ejpam-5569	170	7	m.	m.	NOUN
ejpam-5569	170	8	noureldeen	noureldeen	NOUN
ejpam-5569	170	9	et	et	PROPN
ejpam-5569	170	10	al	al	PROPN
ejpam-5569	170	11	.	.	PUNCT
ejpam-5569	170	12	/	/	SYM
ejpam-5569	170	13	eur	eur	PROPN
ejpam-5569	170	14	.	.	PUNCT
ejpam-5569	171	1	j.	j.	PROPN
ejpam-5569	171	2	pure	pure	PROPN
ejpam-5569	171	3	appl	appl	PROPN
ejpam-5569	171	4	.	.	PROPN
ejpam-5569	171	5	math	math	PROPN
ejpam-5569	171	6	,	,	PUNCT
ejpam-5569	171	7	18	18	NUM
ejpam-5569	171	8	(	(	PUNCT
ejpam-5569	171	9	2	2	NUM
ejpam-5569	171	10	)	)	PUNCT
ejpam-5569	171	11	(	(	PUNCT
ejpam-5569	171	12	2025	2025	NUM
ejpam-5569	171	13	)	)	PUNCT
ejpam-5569	171	14	,	,	PUNCT
ejpam-5569	171	15	5569	5569	NUM
ejpam-5569	171	16	7	7	NUM
ejpam-5569	171	17	of	of	ADP
ejpam-5569	171	18	25	25	NUM
ejpam-5569	171	19	=	=	SYM
ejpam-5569	171	20	|f(uk+1)−	|f(uk+1)−	NOUN
ejpam-5569	171	21	(	(	PUNCT
ejpam-5569	171	22	2n−	2n−	PROPN
ejpam-5569	171	23	1	1	NUM
ejpam-5569	171	24	)	)	PUNCT
ejpam-5569	171	25	+	+	CCONJ
ejpam-5569	171	26	(	(	PUNCT
ejpam-5569	171	27	6k	6k	NOUN
ejpam-5569	171	28	+	+	NOUN
ejpam-5569	171	29	1)|	1)|	NUM
ejpam-5569	171	30	=	=	SYM
ejpam-5569	171	31	|(2n+	|(2n+	PROPN
ejpam-5569	171	32	3)(k	3)(k	NUM
ejpam-5569	171	33	)	)	PUNCT
ejpam-5569	172	1	+	+	NUM
ejpam-5569	173	1	2k(k	2k(k	NUM
ejpam-5569	173	2	−	−	PROPN
ejpam-5569	173	3	1)−	1)−	NUM
ejpam-5569	173	4	(	(	PUNCT
ejpam-5569	173	5	2n−	2n−	PROPN
ejpam-5569	173	6	1	1	NUM
ejpam-5569	173	7	)	)	PUNCT
ejpam-5569	173	8	+	+	CCONJ
ejpam-5569	173	9	(	(	PUNCT
ejpam-5569	173	10	6k	6k	NOUN
ejpam-5569	173	11	+	+	NOUN
ejpam-5569	173	12	1)|	1)|	NUM
ejpam-5569	173	13	≥	≥	NOUN
ejpam-5569	173	14	n−	n−	NOUN
ejpam-5569	173	15	3k	3k	PART
ejpam-5569	173	16	case	case	NOUN
ejpam-5569	173	17	5	5	NUM
ejpam-5569	173	18	x	x	NOUN
ejpam-5569	173	19	,	,	PUNCT
ejpam-5569	173	20	y	y	PROPN
ejpam-5569	173	21	∈	∈	PROPN
ejpam-5569	173	22	vv	vv	INTJ
ejpam-5569	173	23	.	.	PUNCT
ejpam-5569	174	1	if	if	SCONJ
ejpam-5569	174	2	{	{	PUNCT
ejpam-5569	174	3	x	x	NOUN
ejpam-5569	174	4	,	,	PUNCT
ejpam-5569	174	5	y	y	PROPN
ejpam-5569	174	6	}	}	PUNCT
ejpam-5569	174	7	=	=	SYM
ejpam-5569	174	8	{	{	PUNCT
ejpam-5569	174	9	v1	v1	PROPN
ejpam-5569	174	10	,	,	PUNCT
ejpam-5569	174	11	vn	vn	VERB
ejpam-5569	174	12	}	}	PUNCT
ejpam-5569	174	13	then	then	ADV
ejpam-5569	174	14	|f(x)−	|f(x)−	NOUN
ejpam-5569	174	15	f(y)|	f(y)|	PROPN
ejpam-5569	174	16	=	=	PROPN
ejpam-5569	175	1	|f(v1)−	|f(v1)−	ADJ
ejpam-5569	175	2	f(vn)|	f(vn)|	NOUN
ejpam-5569	175	3	=	=	X
ejpam-5569	176	1	|f(v1)−	|f(v1)−	PROPN
ejpam-5569	176	2	[	[	X
ejpam-5569	176	3	f(v1)−	f(v1)−	PROPN
ejpam-5569	176	4	1]|	1]|	NUM
ejpam-5569	176	5	=	=	SYM
ejpam-5569	176	6	1	1	NUM
ejpam-5569	176	7	≥	≥	NOUN
ejpam-5569	176	8	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	176	9	)	)	PUNCT
ejpam-5569	176	10	)	)	PUNCT
ejpam-5569	177	1	+	+	CCONJ
ejpam-5569	177	2	1−	1−	NUM
ejpam-5569	177	3	d(x	d(x	PROPN
ejpam-5569	177	4	,	,	PUNCT
ejpam-5569	177	5	y	y	NOUN
ejpam-5569	177	6	)	)	PUNCT
ejpam-5569	177	7	,	,	PUNCT
ejpam-5569	177	8	where	where	SCONJ
ejpam-5569	177	9	d(x	d(x	PROPN
ejpam-5569	177	10	,	,	PUNCT
ejpam-5569	177	11	y	y	NOUN
ejpam-5569	177	12	)	)	PUNCT
ejpam-5569	177	13	=	=	VERB
ejpam-5569	177	14	n−	n−	NOUN
ejpam-5569	177	15	1	1	NUM
ejpam-5569	177	16	.	.	PUNCT
ejpam-5569	177	17	case	case	NOUN
ejpam-5569	177	18	6	6	NUM
ejpam-5569	177	19	if	if	SCONJ
ejpam-5569	177	20	{	{	PUNCT
ejpam-5569	177	21	x	x	NOUN
ejpam-5569	177	22	,	,	PUNCT
ejpam-5569	177	23	y	y	PROPN
ejpam-5569	177	24	}	}	PUNCT
ejpam-5569	177	25	=	=	SYM
ejpam-5569	177	26	{	{	PUNCT
ejpam-5569	177	27	v1	v1	NOUN
ejpam-5569	177	28	,	,	PUNCT
ejpam-5569	177	29	vk	vk	ADP
ejpam-5569	177	30	}	}	PUNCT
ejpam-5569	177	31	then	then	ADV
ejpam-5569	177	32	|f(x)−	|f(x)−	NOUN
ejpam-5569	177	33	f(y)|	f(y)|	PROPN
ejpam-5569	177	34	=	=	PROPN
ejpam-5569	178	1	|f(v1)−	|f(v1)−	ADJ
ejpam-5569	178	2	f(vk)|	f(vk)|	NOUN
ejpam-5569	178	3	=	=	PUNCT
ejpam-5569	179	1	|(n+	|(n+	VERB
ejpam-5569	179	2	2)−	2)−	NUM
ejpam-5569	179	3	[	[	X
ejpam-5569	179	4	(	(	PUNCT
ejpam-5569	179	5	n+	n+	NOUN
ejpam-5569	179	6	2	2	NUM
ejpam-5569	179	7	)	)	PUNCT
ejpam-5569	179	8	+	+	CCONJ
ejpam-5569	179	9	(	(	PUNCT
ejpam-5569	179	10	2n+	2n+	NUM
ejpam-5569	179	11	7)(k	7)(k	NUM
ejpam-5569	179	12	−	−	NOUN
ejpam-5569	179	13	1	1	NUM
ejpam-5569	179	14	)	)	PUNCT
ejpam-5569	179	15	+	+	CCONJ
ejpam-5569	179	16	2(k	2(k	NUM
ejpam-5569	179	17	−	−	NOUN
ejpam-5569	179	18	1)(k	1)(k	NUM
ejpam-5569	179	19	−	−	NOUN
ejpam-5569	179	20	2)]|	2)]|	NOUN
ejpam-5569	179	21	=	=	SYM
ejpam-5569	179	22	|(2n+	|(2n+	X
ejpam-5569	180	1	7)(k	7)(k	NUM
ejpam-5569	180	2	−	−	NOUN
ejpam-5569	180	3	1	1	NUM
ejpam-5569	180	4	)	)	PUNCT
ejpam-5569	180	5	+	+	CCONJ
ejpam-5569	180	6	2(k	2(k	NUM
ejpam-5569	180	7	−	−	NOUN
ejpam-5569	180	8	1)(k	1)(k	NUM
ejpam-5569	180	9	−	−	PROPN
ejpam-5569	180	10	2)|	2)|	NUM
ejpam-5569	180	11	≥	≥	NOUN
ejpam-5569	180	12	n	n	CCONJ
ejpam-5569	180	13	,	,	PUNCT
ejpam-5569	180	14	where	where	SCONJ
ejpam-5569	180	15	n	n	NOUN
ejpam-5569	180	16	=	=	SYM
ejpam-5569	180	17	4k	4k	NUM
ejpam-5569	180	18	+	+	CCONJ
ejpam-5569	180	19	2	2	NUM
ejpam-5569	180	20	≥	≥	NOUN
ejpam-5569	180	21	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	180	22	)	)	PUNCT
ejpam-5569	180	23	)	)	PUNCT
ejpam-5569	181	1	+	+	CCONJ
ejpam-5569	181	2	1−	1−	NUM
ejpam-5569	181	3	d(x	d(x	PROPN
ejpam-5569	181	4	,	,	PUNCT
ejpam-5569	181	5	y	y	NOUN
ejpam-5569	181	6	)	)	PUNCT
ejpam-5569	181	7	case	case	NOUN
ejpam-5569	181	8	7	7	NUM
ejpam-5569	181	9	if	if	SCONJ
ejpam-5569	181	10	{	{	PUNCT
ejpam-5569	181	11	x	x	NOUN
ejpam-5569	181	12	,	,	PUNCT
ejpam-5569	181	13	y	y	PROPN
ejpam-5569	181	14	}	}	PUNCT
ejpam-5569	181	15	=	=	SYM
ejpam-5569	181	16	{	{	PUNCT
ejpam-5569	181	17	v1	v1	NOUN
ejpam-5569	181	18	,	,	PUNCT
ejpam-5569	181	19	v2k+1	v2k+1	ADP
ejpam-5569	181	20	}	}	PUNCT
ejpam-5569	181	21	then	then	ADV
ejpam-5569	181	22	|f(x)−	|f(x)−	NOUN
ejpam-5569	181	23	f(y)|	f(y)|	PROPN
ejpam-5569	181	24	=	=	PROPN
ejpam-5569	182	1	|f(v1)−	|f(v1)−	PROPN
ejpam-5569	182	2	f(v2k+1)|	f(v2k+1)|	X
ejpam-5569	183	1	=	=	PUNCT
ejpam-5569	183	2	|(n+	|(n+	NOUN
ejpam-5569	183	3	2)−	2)−	NUM
ejpam-5569	183	4	[	[	X
ejpam-5569	183	5	(	(	PUNCT
ejpam-5569	183	6	n+	n+	NOUN
ejpam-5569	183	7	2	2	NUM
ejpam-5569	183	8	)	)	PUNCT
ejpam-5569	183	9	+	+	CCONJ
ejpam-5569	183	10	(	(	PUNCT
ejpam-5569	183	11	2n+	2n+	NUM
ejpam-5569	183	12	7)(k	7)(k	NUM
ejpam-5569	183	13	−	−	NOUN
ejpam-5569	183	14	1	1	NUM
ejpam-5569	183	15	)	)	PUNCT
ejpam-5569	183	16	+	+	CCONJ
ejpam-5569	183	17	2(k	2(k	NUM
ejpam-5569	183	18	−	−	NOUN
ejpam-5569	183	19	1)(k	1)(k	NUM
ejpam-5569	183	20	−	−	PROPN
ejpam-5569	183	21	2)+	2)+	NUM
ejpam-5569	183	22	+	+	CCONJ
ejpam-5569	183	23	(	(	PUNCT
ejpam-5569	183	24	2n+	2n+	NUM
ejpam-5569	183	25	1	1	NUM
ejpam-5569	183	26	)	)	PUNCT
ejpam-5569	183	27	+	+	CCONJ
ejpam-5569	183	28	(	(	PUNCT
ejpam-5569	183	29	n+	n+	NOUN
ejpam-5569	183	30	1)(k	1)(k	NUM
ejpam-5569	183	31	)	)	PUNCT
ejpam-5569	183	32	+	+	CCONJ
ejpam-5569	183	33	2(k)(k	2(k)(k	NUM
ejpam-5569	183	34	−	−	PROPN
ejpam-5569	183	35	1)]|	1)]|	NUM
ejpam-5569	183	36	=	=	SYM
ejpam-5569	183	37	|(2n+	|(2n+	X
ejpam-5569	184	1	7)(k	7)(k	NUM
ejpam-5569	184	2	−	−	NOUN
ejpam-5569	184	3	1	1	NUM
ejpam-5569	184	4	)	)	PUNCT
ejpam-5569	184	5	+	+	CCONJ
ejpam-5569	184	6	2(k	2(k	NUM
ejpam-5569	184	7	−	−	NOUN
ejpam-5569	184	8	1)(k	1)(k	NUM
ejpam-5569	184	9	−	−	PROPN
ejpam-5569	184	10	2	2	NUM
ejpam-5569	184	11	)	)	PUNCT
ejpam-5569	184	12	+	+	CCONJ
ejpam-5569	184	13	(	(	PUNCT
ejpam-5569	184	14	2n+	2n+	NUM
ejpam-5569	184	15	1	1	NUM
ejpam-5569	184	16	)	)	PUNCT
ejpam-5569	184	17	+	+	CCONJ
ejpam-5569	184	18	(	(	PUNCT
ejpam-5569	184	19	n+	n+	NOUN
ejpam-5569	184	20	1)(k	1)(k	NUM
ejpam-5569	184	21	)	)	PUNCT
ejpam-5569	184	22	+	+	CCONJ
ejpam-5569	184	23	2(k)(k	2(k)(k	NUM
ejpam-5569	184	24	−	−	PROPN
ejpam-5569	184	25	1)|	1)|	NUM
ejpam-5569	184	26	≥	≥	NOUN
ejpam-5569	184	27	n	n	CCONJ
ejpam-5569	184	28	,	,	PUNCT
ejpam-5569	184	29	where	where	SCONJ
ejpam-5569	184	30	n	n	NOUN
ejpam-5569	184	31	=	=	SYM
ejpam-5569	184	32	4k	4k	X
ejpam-5569	184	33	+	+	NOUN
ejpam-5569	184	34	2	2	NUM
ejpam-5569	184	35	,	,	PUNCT
ejpam-5569	184	36	d(x	d(x	PROPN
ejpam-5569	184	37	,	,	PUNCT
ejpam-5569	184	38	y	y	PROPN
ejpam-5569	184	39	)	)	PUNCT
ejpam-5569	184	40	≥	≥	NOUN
ejpam-5569	184	41	1	1	NUM
ejpam-5569	184	42	≥	≥	NOUN
ejpam-5569	184	43	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	184	44	)	)	PUNCT
ejpam-5569	184	45	)	)	PUNCT
ejpam-5569	185	1	+	+	CCONJ
ejpam-5569	185	2	1−	1−	NUM
ejpam-5569	185	3	d(x	d(x	PROPN
ejpam-5569	185	4	,	,	PUNCT
ejpam-5569	185	5	y	y	NOUN
ejpam-5569	185	6	)	)	PUNCT
ejpam-5569	185	7	case	case	NOUN
ejpam-5569	185	8	8	8	NUM
ejpam-5569	185	9	if	if	SCONJ
ejpam-5569	185	10	{	{	PUNCT
ejpam-5569	185	11	x	x	NOUN
ejpam-5569	185	12	,	,	PUNCT
ejpam-5569	185	13	y	y	PROPN
ejpam-5569	185	14	}	}	PUNCT
ejpam-5569	185	15	=	=	SYM
ejpam-5569	185	16	{	{	PUNCT
ejpam-5569	185	17	v1	v1	NOUN
ejpam-5569	185	18	,	,	PUNCT
ejpam-5569	185	19	v3k+2	v3k+2	NOUN
ejpam-5569	185	20	}	}	PUNCT
ejpam-5569	185	21	then	then	ADV
ejpam-5569	185	22	|f(x)−	|f(x)−	NOUN
ejpam-5569	185	23	f(y)|	f(y)|	PROPN
ejpam-5569	185	24	=	=	PROPN
ejpam-5569	186	1	|f(v1)−	|f(v1)−	NOUN
ejpam-5569	186	2	f(v3k+2)|	f(v3k+2)|	PROPN
ejpam-5569	187	1	=	=	PUNCT
ejpam-5569	188	1	|f(v1)−	|f(v1)−	ADJ
ejpam-5569	188	2	[	[	X
ejpam-5569	188	3	f(vk+1	f(vk+1	X
ejpam-5569	188	4	)	)	PUNCT
ejpam-5569	188	5	+	+	CCONJ
ejpam-5569	188	6	(	(	PUNCT
ejpam-5569	188	7	2k	2k	NUM
ejpam-5569	188	8	+	+	CCONJ
ejpam-5569	188	9	1)]|	1)]|	ADJ
ejpam-5569	188	10	=	=	NOUN
ejpam-5569	188	11	|f(v1)−	|f(v1)−	PROPN
ejpam-5569	188	12	[	[	X
ejpam-5569	188	13	f(vk	f(vk	X
ejpam-5569	188	14	)	)	PUNCT
ejpam-5569	188	15	+	+	CCONJ
ejpam-5569	188	16	(	(	PUNCT
ejpam-5569	188	17	2n+	2n+	NUM
ejpam-5569	188	18	1	1	NUM
ejpam-5569	188	19	)	)	PUNCT
ejpam-5569	188	20	+	+	CCONJ
ejpam-5569	188	21	(	(	PUNCT
ejpam-5569	188	22	2k	2k	NUM
ejpam-5569	188	23	+	+	CCONJ
ejpam-5569	188	24	1)]|	1)]|	NUM
ejpam-5569	188	25	=	=	NOUN
ejpam-5569	188	26	|(n+	|(n+	NOUN
ejpam-5569	188	27	2)−	2)−	NUM
ejpam-5569	189	1	[	[	X
ejpam-5569	189	2	(	(	PUNCT
ejpam-5569	189	3	n+	n+	NOUN
ejpam-5569	189	4	2	2	NUM
ejpam-5569	189	5	)	)	PUNCT
ejpam-5569	189	6	+	+	CCONJ
ejpam-5569	189	7	(	(	PUNCT
ejpam-5569	189	8	2n+	2n+	NUM
ejpam-5569	189	9	7)(k	7)(k	NUM
ejpam-5569	189	10	−	−	NOUN
ejpam-5569	189	11	1	1	NUM
ejpam-5569	189	12	)	)	PUNCT
ejpam-5569	189	13	+	+	CCONJ
ejpam-5569	189	14	2(k	2(k	NUM
ejpam-5569	189	15	−	−	NOUN
ejpam-5569	189	16	1)(k	1)(k	NUM
ejpam-5569	189	17	−	−	PROPN
ejpam-5569	189	18	2)+	2)+	NUM
ejpam-5569	190	1	+	+	CCONJ
ejpam-5569	190	2	(	(	PUNCT
ejpam-5569	190	3	2n+	2n+	NUM
ejpam-5569	190	4	1	1	NUM
ejpam-5569	190	5	)	)	PUNCT
ejpam-5569	190	6	+	+	CCONJ
ejpam-5569	190	7	(	(	PUNCT
ejpam-5569	190	8	2k	2k	NUM
ejpam-5569	190	9	+	+	CCONJ
ejpam-5569	190	10	1)]|	1)]|	NUM
ejpam-5569	190	11	=	=	SYM
ejpam-5569	190	12	|(2n+	|(2n+	PROPN
ejpam-5569	191	1	7)(k	7)(k	NUM
ejpam-5569	191	2	−	−	NOUN
ejpam-5569	191	3	1	1	NUM
ejpam-5569	191	4	)	)	PUNCT
ejpam-5569	191	5	+	+	CCONJ
ejpam-5569	191	6	2(k	2(k	NUM
ejpam-5569	191	7	−	−	NOUN
ejpam-5569	191	8	1)(k	1)(k	NUM
ejpam-5569	191	9	−	−	PROPN
ejpam-5569	191	10	2	2	NUM
ejpam-5569	191	11	)	)	PUNCT
ejpam-5569	191	12	+	+	CCONJ
ejpam-5569	191	13	(	(	PUNCT
ejpam-5569	191	14	2n+	2n+	NUM
ejpam-5569	191	15	1	1	NUM
ejpam-5569	191	16	)	)	PUNCT
ejpam-5569	191	17	+	+	CCONJ
ejpam-5569	191	18	(	(	PUNCT
ejpam-5569	191	19	2k	2k	NUM
ejpam-5569	191	20	+	+	CCONJ
ejpam-5569	191	21	1)|	1)|	NUM
ejpam-5569	191	22	≥	≥	NOUN
ejpam-5569	191	23	n−	n−	PROPN
ejpam-5569	191	24	(	(	PUNCT
ejpam-5569	191	25	3k	3k	X
ejpam-5569	191	26	+	+	CCONJ
ejpam-5569	191	27	1	1	NUM
ejpam-5569	191	28	)	)	PUNCT
ejpam-5569	191	29	,	,	PUNCT
ejpam-5569	191	30	where	where	SCONJ
ejpam-5569	191	31	n	n	NOUN
ejpam-5569	191	32	=	=	SYM
ejpam-5569	191	33	4k	4k	X
ejpam-5569	191	34	+	+	NOUN
ejpam-5569	191	35	2	2	NUM
ejpam-5569	191	36	,	,	PUNCT
ejpam-5569	191	37	d(x	d(x	PROPN
ejpam-5569	191	38	,	,	PUNCT
ejpam-5569	191	39	y	y	NOUN
ejpam-5569	191	40	)	)	PUNCT
ejpam-5569	191	41	=	=	PUNCT
ejpam-5569	192	1	3k	3k	NOUN
ejpam-5569	192	2	+	+	CCONJ
ejpam-5569	192	3	1	1	NUM
ejpam-5569	192	4	≥	≥	NOUN
ejpam-5569	192	5	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	192	6	)	)	PUNCT
ejpam-5569	192	7	)	)	PUNCT
ejpam-5569	193	1	+	+	CCONJ
ejpam-5569	193	2	1−	1−	NUM
ejpam-5569	193	3	d(x	d(x	PROPN
ejpam-5569	193	4	,	,	PUNCT
ejpam-5569	193	5	y	y	NOUN
ejpam-5569	193	6	)	)	PUNCT
ejpam-5569	193	7	case	case	NOUN
ejpam-5569	193	8	9	9	NUM
ejpam-5569	193	9	x	x	SYM
ejpam-5569	193	10	∈	∈	NOUN
ejpam-5569	193	11	vu	vu	NOUN
ejpam-5569	193	12	and	and	CCONJ
ejpam-5569	193	13	y	y	PROPN
ejpam-5569	193	14	∈	∈	PROPN
ejpam-5569	193	15	vv	vv	INTJ
ejpam-5569	193	16	let	let	VERB
ejpam-5569	193	17	x	x	PUNCT
ejpam-5569	193	18	=	=	PUNCT
ejpam-5569	193	19	ui	ui	PROPN
ejpam-5569	193	20	and	and	CCONJ
ejpam-5569	193	21	y	y	PROPN
ejpam-5569	193	22	=	=	PUNCT
ejpam-5569	193	23	vl	vl	PROPN
ejpam-5569	193	24	,	,	PUNCT
ejpam-5569	193	25	where	where	SCONJ
ejpam-5569	193	26	1	1	NUM
ejpam-5569	193	27	≤	≤	PUNCT
ejpam-5569	193	28	i	i	PRON
ejpam-5569	193	29	,	,	PUNCT
ejpam-5569	193	30	l	l	PROPN
ejpam-5569	193	31	≤	≤	X
ejpam-5569	193	32	n	n	CCONJ
ejpam-5569	193	33	in	in	ADP
ejpam-5569	193	34	this	this	DET
ejpam-5569	193	35	case	case	NOUN
ejpam-5569	193	36	we	we	PRON
ejpam-5569	193	37	have	have	VERB
ejpam-5569	193	38	several	several	ADJ
ejpam-5569	193	39	subcases	subcase	NOUN
ejpam-5569	193	40	:	:	PUNCT
ejpam-5569	193	41	n.	n.	PROPN
ejpam-5569	193	42	m.	m.	NOUN
ejpam-5569	194	1	noureldeen	noureldeen	NOUN
ejpam-5569	194	2	et	et	PROPN
ejpam-5569	194	3	al	al	PROPN
ejpam-5569	194	4	.	.	PUNCT
ejpam-5569	194	5	/	/	SYM
ejpam-5569	194	6	eur	eur	PROPN
ejpam-5569	194	7	.	.	PUNCT
ejpam-5569	195	1	j.	j.	PROPN
ejpam-5569	195	2	pure	pure	PROPN
ejpam-5569	195	3	appl	appl	PROPN
ejpam-5569	195	4	.	.	PROPN
ejpam-5569	195	5	math	math	PROPN
ejpam-5569	195	6	,	,	PUNCT
ejpam-5569	195	7	18	18	NUM
ejpam-5569	195	8	(	(	PUNCT
ejpam-5569	195	9	2	2	NUM
ejpam-5569	195	10	)	)	PUNCT
ejpam-5569	195	11	(	(	PUNCT
ejpam-5569	195	12	2025	2025	NUM
ejpam-5569	195	13	)	)	PUNCT
ejpam-5569	195	14	,	,	PUNCT
ejpam-5569	195	15	5569	5569	NUM
ejpam-5569	195	16	8	8	NUM
ejpam-5569	195	17	of	of	ADP
ejpam-5569	195	18	25	25	NUM
ejpam-5569	195	19	subcase	subcase	NOUN
ejpam-5569	195	20	9.1	9.1	NUM
ejpam-5569	195	21	if	if	SCONJ
ejpam-5569	195	22	i	i	PRON
ejpam-5569	195	23	=	=	NOUN
ejpam-5569	195	24	1	1	NUM
ejpam-5569	195	25	,	,	PUNCT
ejpam-5569	195	26	j	j	PROPN
ejpam-5569	196	1	=	=	SYM
ejpam-5569	196	2	k	k	PROPN
ejpam-5569	196	3	then	then	ADV
ejpam-5569	196	4	|f(x)−	|f(x)−	NOUN
ejpam-5569	196	5	f(y)|	f(y)|	PROPN
ejpam-5569	196	6	=	=	PUNCT
ejpam-5569	196	7	|f(u1)−	|f(u1)−	ADJ
ejpam-5569	196	8	f(vk)|	f(vk)|	NOUN
ejpam-5569	197	1	=	=	SYM
ejpam-5569	197	2	|0−	|0−	NOUN
ejpam-5569	197	3	[	[	X
ejpam-5569	197	4	(	(	PUNCT
ejpam-5569	197	5	n+	n+	NOUN
ejpam-5569	197	6	2	2	NUM
ejpam-5569	197	7	)	)	PUNCT
ejpam-5569	197	8	+	+	CCONJ
ejpam-5569	197	9	(	(	PUNCT
ejpam-5569	197	10	2n+	2n+	NUM
ejpam-5569	197	11	7)(k	7)(k	NUM
ejpam-5569	197	12	−	−	NOUN
ejpam-5569	197	13	1	1	NUM
ejpam-5569	197	14	)	)	PUNCT
ejpam-5569	197	15	+	+	CCONJ
ejpam-5569	198	1	2(k	2(k	NUM
ejpam-5569	198	2	−	−	NOUN
ejpam-5569	198	3	1)(k	1)(k	NUM
ejpam-5569	198	4	−	−	PROPN
ejpam-5569	198	5	2)]|	2)]|	NOUN
ejpam-5569	198	6	=	=	SYM
ejpam-5569	198	7	|(n+	|(n+	NOUN
ejpam-5569	198	8	2	2	NUM
ejpam-5569	198	9	)	)	PUNCT
ejpam-5569	198	10	+	+	CCONJ
ejpam-5569	198	11	(	(	PUNCT
ejpam-5569	198	12	2n+	2n+	NUM
ejpam-5569	198	13	7)(k	7)(k	NUM
ejpam-5569	198	14	−	−	NOUN
ejpam-5569	198	15	1	1	NUM
ejpam-5569	198	16	)	)	PUNCT
ejpam-5569	198	17	+	+	CCONJ
ejpam-5569	198	18	2(k	2(k	NUM
ejpam-5569	198	19	−	−	NOUN
ejpam-5569	198	20	1)(k	1)(k	NUM
ejpam-5569	198	21	−	−	PROPN
ejpam-5569	198	22	2)|	2)|	NUM
ejpam-5569	198	23	≥	≥	NOUN
ejpam-5569	198	24	n−	n−	PROPN
ejpam-5569	198	25	(	(	PUNCT
ejpam-5569	198	26	k	k	NOUN
ejpam-5569	198	27	−	−	PROPN
ejpam-5569	198	28	1	1	NUM
ejpam-5569	198	29	)	)	PUNCT
ejpam-5569	198	30	,	,	PUNCT
ejpam-5569	198	31	where	where	SCONJ
ejpam-5569	198	32	,	,	PUNCT
ejpam-5569	198	33	d(x	d(x	PROPN
ejpam-5569	198	34	,	,	PUNCT
ejpam-5569	198	35	y	y	NOUN
ejpam-5569	198	36	)	)	PUNCT
ejpam-5569	198	37	=	=	PUNCT
ejpam-5569	199	1	k	k	NOUN
ejpam-5569	199	2	−	−	PROPN
ejpam-5569	199	3	1	1	NUM
ejpam-5569	199	4	≥	≥	NOUN
ejpam-5569	199	5	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	199	6	)	)	PUNCT
ejpam-5569	199	7	)	)	PUNCT
ejpam-5569	200	1	+	+	CCONJ
ejpam-5569	200	2	1−	1−	NUM
ejpam-5569	200	3	d(x	d(x	PROPN
ejpam-5569	200	4	,	,	PUNCT
ejpam-5569	200	5	y	y	NOUN
ejpam-5569	200	6	)	)	PUNCT
ejpam-5569	200	7	.	.	PUNCT
ejpam-5569	201	1	otherwise	otherwise	ADV
ejpam-5569	201	2	,	,	PUNCT
ejpam-5569	201	3	|f(x)−	|f(x)−	PROPN
ejpam-5569	201	4	f(y)|	f(y)|	PROPN
ejpam-5569	201	5	≥	≥	PROPN
ejpam-5569	201	6	n.	n.	PROPN
ejpam-5569	201	7	subcase	subcase	PROPN
ejpam-5569	201	8	9.2	9.2	NUM
ejpam-5569	201	9	,	,	PUNCT
ejpam-5569	201	10	1	1	NUM
ejpam-5569	201	11	≤	≤	NUM
ejpam-5569	201	12	i	i	NOUN
ejpam-5569	201	13	≤	≤	PUNCT
ejpam-5569	202	1	k	k	X
ejpam-5569	203	1	+	+	CCONJ
ejpam-5569	203	2	1	1	NUM
ejpam-5569	203	3	and	and	CCONJ
ejpam-5569	203	4	k	k	PROPN
ejpam-5569	204	1	+	+	CCONJ
ejpam-5569	204	2	1	1	NUM
ejpam-5569	204	3	≤	≤	NUM
ejpam-5569	204	4	l	l	NOUN
ejpam-5569	204	5	≤	≤	NOUN
ejpam-5569	204	6	2k	2k	NOUN
ejpam-5569	204	7	+	+	CCONJ
ejpam-5569	204	8	1	1	NUM
ejpam-5569	204	9	|f(x)−	|f(x)−	NOUN
ejpam-5569	204	10	f(y)|	f(y)|	PROPN
ejpam-5569	204	11	=	=	NUM
ejpam-5569	204	12	|f(ui)−	|f(ui)−	ADJ
ejpam-5569	204	13	f(vl)|	f(vl)|	NOUN
ejpam-5569	204	14	=	=	SYM
ejpam-5569	204	15	|f(ui)−	|f(ui)−	ADJ
ejpam-5569	204	16	f(vk+j)|	f(vk+j)|	PROPN
ejpam-5569	204	17	,	,	PUNCT
ejpam-5569	204	18	where	where	SCONJ
ejpam-5569	204	19	1	1	NUM
ejpam-5569	204	20	≤	≤	NUM
ejpam-5569	204	21	j	j	PROPN
ejpam-5569	204	22	≤	≤	PROPN
ejpam-5569	204	23	k	k	X
ejpam-5569	205	1	=	=	PUNCT
ejpam-5569	205	2	|(2n+	|(2n+	PROPN
ejpam-5569	205	3	3)(i−	3)(i−	NUM
ejpam-5569	205	4	1	1	NUM
ejpam-5569	205	5	)	)	PUNCT
ejpam-5569	205	6	+	+	CCONJ
ejpam-5569	206	1	2(i−	2(i−	NUM
ejpam-5569	206	2	1)(i−	1)(i−	NUM
ejpam-5569	206	3	2)−	2)−	NUM
ejpam-5569	207	1	[	[	X
ejpam-5569	207	2	f(vk	f(vk	X
ejpam-5569	207	3	)	)	PUNCT
ejpam-5569	207	4	+	+	CCONJ
ejpam-5569	207	5	(	(	PUNCT
ejpam-5569	207	6	2n+	2n+	NUM
ejpam-5569	207	7	1	1	NUM
ejpam-5569	207	8	)	)	PUNCT
ejpam-5569	207	9	+	+	CCONJ
ejpam-5569	207	10	(	(	PUNCT
ejpam-5569	207	11	n+	n+	ADP
ejpam-5569	207	12	1)(j	1)(j	NUM
ejpam-5569	207	13	−	−	NOUN
ejpam-5569	207	14	1)+	1)+	NUM
ejpam-5569	208	1	+	+	CCONJ
ejpam-5569	208	2	2(j	2(j	NUM
ejpam-5569	208	3	−	−	PROPN
ejpam-5569	208	4	1)(j	1)(j	NUM
ejpam-5569	208	5	−	−	PROPN
ejpam-5569	208	6	2)]|	2)]|	NOUN
ejpam-5569	208	7	=	=	SYM
ejpam-5569	208	8	|(2n+	|(2n+	PROPN
ejpam-5569	209	1	3)(i−	3)(i−	NUM
ejpam-5569	209	2	1	1	NUM
ejpam-5569	209	3	)	)	PUNCT
ejpam-5569	209	4	+	+	CCONJ
ejpam-5569	210	1	2(i−	2(i−	NUM
ejpam-5569	210	2	1)(i−	1)(i−	NUM
ejpam-5569	210	3	2)−	2)−	NUM
ejpam-5569	211	1	[	[	X
ejpam-5569	211	2	(	(	PUNCT
ejpam-5569	211	3	n+	n+	NOUN
ejpam-5569	211	4	2	2	NUM
ejpam-5569	211	5	)	)	PUNCT
ejpam-5569	211	6	+	+	CCONJ
ejpam-5569	211	7	(	(	PUNCT
ejpam-5569	211	8	2n+	2n+	NUM
ejpam-5569	211	9	7)(k	7)(k	NUM
ejpam-5569	211	10	−	−	PROPN
ejpam-5569	211	11	1)+	1)+	NUM
ejpam-5569	212	1	+	+	CCONJ
ejpam-5569	213	1	2(k	2(k	NUM
ejpam-5569	213	2	−	−	NOUN
ejpam-5569	213	3	1)(k	1)(k	NUM
ejpam-5569	213	4	−	−	PROPN
ejpam-5569	213	5	2	2	NUM
ejpam-5569	213	6	)	)	PUNCT
ejpam-5569	213	7	+	+	CCONJ
ejpam-5569	213	8	(	(	PUNCT
ejpam-5569	213	9	2n+	2n+	NUM
ejpam-5569	213	10	1	1	NUM
ejpam-5569	213	11	)	)	PUNCT
ejpam-5569	213	12	+	+	CCONJ
ejpam-5569	213	13	(	(	PUNCT
ejpam-5569	213	14	n+	n+	NOUN
ejpam-5569	213	15	1)(j	1)(j	NUM
ejpam-5569	213	16	−	−	NOUN
ejpam-5569	213	17	1	1	NUM
ejpam-5569	213	18	)	)	PUNCT
ejpam-5569	213	19	+	+	CCONJ
ejpam-5569	213	20	2(j	2(j	NUM
ejpam-5569	213	21	−	−	DET
ejpam-5569	213	22	1)(j	1)(j	NUM
ejpam-5569	213	23	−	−	NOUN
ejpam-5569	213	24	2)]|	2)]|	NOUN
ejpam-5569	213	25	.	.	PUNCT
ejpam-5569	214	1	if	if	SCONJ
ejpam-5569	214	2	i	i	PRON
ejpam-5569	214	3	=	=	NOUN
ejpam-5569	214	4	1	1	NUM
ejpam-5569	214	5	,	,	PUNCT
ejpam-5569	214	6	l	l	NOUN
ejpam-5569	214	7	=	=	PUNCT
ejpam-5569	215	1	k	k	X
ejpam-5569	216	1	+	+	CCONJ
ejpam-5569	216	2	1	1	NUM
ejpam-5569	216	3	i.e.	i.e.	X
ejpam-5569	216	4	j	j	X
ejpam-5569	216	5	=	=	SYM
ejpam-5569	216	6	1	1	NUM
ejpam-5569	216	7	then	then	ADV
ejpam-5569	216	8	|f(x)−	|f(x)−	NOUN
ejpam-5569	216	9	f(y)|	f(y)|	PROPN
ejpam-5569	216	10	=	=	NUM
ejpam-5569	216	11	|f(ui)−	|f(ui)−	ADJ
ejpam-5569	216	12	f(vl)|	f(vl)|	NOUN
ejpam-5569	216	13	=	=	SYM
ejpam-5569	216	14	|f(ui)−	|f(ui)−	ADJ
ejpam-5569	216	15	f(vk+j)|	f(vk+j)|	PROPN
ejpam-5569	216	16	,	,	PUNCT
ejpam-5569	216	17	where	where	SCONJ
ejpam-5569	216	18	1	1	NUM
ejpam-5569	216	19	≤	≤	NUM
ejpam-5569	216	20	j	j	PROPN
ejpam-5569	216	21	≤	≤	PROPN
ejpam-5569	217	1	k	k	NOUN
ejpam-5569	217	2	=	=	PUNCT
ejpam-5569	217	3	|0−	|0−	NOUN
ejpam-5569	217	4	[	[	X
ejpam-5569	217	5	f(vk	f(vk	X
ejpam-5569	217	6	)	)	PUNCT
ejpam-5569	217	7	+	+	CCONJ
ejpam-5569	217	8	(	(	PUNCT
ejpam-5569	217	9	2n+	2n+	NUM
ejpam-5569	217	10	1	1	NUM
ejpam-5569	217	11	)	)	PUNCT
ejpam-5569	217	12	+	+	CCONJ
ejpam-5569	217	13	(	(	PUNCT
ejpam-5569	217	14	n+	n+	NOUN
ejpam-5569	217	15	1)(j	1)(j	NUM
ejpam-5569	217	16	−	−	NOUN
ejpam-5569	217	17	1	1	NUM
ejpam-5569	217	18	)	)	PUNCT
ejpam-5569	217	19	+	+	CCONJ
ejpam-5569	217	20	2(j	2(j	NUM
ejpam-5569	217	21	−	−	PROPN
ejpam-5569	217	22	1)(j	1)(j	NUM
ejpam-5569	217	23	−	−	PROPN
ejpam-5569	217	24	2)]|	2)]|	NOUN
ejpam-5569	217	25	=	=	NOUN
ejpam-5569	218	1	|	|	ADV
ejpam-5569	218	2	−	−	NOUN
ejpam-5569	219	1	[	[	X
ejpam-5569	219	2	(	(	PUNCT
ejpam-5569	219	3	n+	n+	NOUN
ejpam-5569	219	4	2	2	NUM
ejpam-5569	219	5	)	)	PUNCT
ejpam-5569	219	6	+	+	CCONJ
ejpam-5569	219	7	(	(	PUNCT
ejpam-5569	219	8	2n+	2n+	NUM
ejpam-5569	219	9	7)(k	7)(k	NUM
ejpam-5569	219	10	−	−	NOUN
ejpam-5569	219	11	1	1	NUM
ejpam-5569	219	12	)	)	PUNCT
ejpam-5569	219	13	+	+	CCONJ
ejpam-5569	219	14	2(k	2(k	NUM
ejpam-5569	219	15	−	−	NOUN
ejpam-5569	219	16	1)(k	1)(k	NUM
ejpam-5569	219	17	−	−	PROPN
ejpam-5569	219	18	2	2	NUM
ejpam-5569	219	19	)	)	PUNCT
ejpam-5569	219	20	+	+	CCONJ
ejpam-5569	219	21	(	(	PUNCT
ejpam-5569	219	22	2n+	2n+	NUM
ejpam-5569	219	23	1)|	1)|	NUM
ejpam-5569	219	24	=	=	PUNCT
ejpam-5569	219	25	|(n+	|(n+	NOUN
ejpam-5569	219	26	2	2	NUM
ejpam-5569	219	27	)	)	PUNCT
ejpam-5569	219	28	+	+	CCONJ
ejpam-5569	219	29	(	(	PUNCT
ejpam-5569	219	30	2n+	2n+	NUM
ejpam-5569	219	31	7)(k	7)(k	NUM
ejpam-5569	219	32	−	−	NOUN
ejpam-5569	219	33	1	1	NUM
ejpam-5569	219	34	)	)	PUNCT
ejpam-5569	219	35	+	+	CCONJ
ejpam-5569	220	1	2(k	2(k	NUM
ejpam-5569	220	2	−	−	NOUN
ejpam-5569	220	3	1)(k	1)(k	NUM
ejpam-5569	220	4	−	−	PROPN
ejpam-5569	220	5	2	2	NUM
ejpam-5569	220	6	)	)	PUNCT
ejpam-5569	220	7	+	+	CCONJ
ejpam-5569	220	8	(	(	PUNCT
ejpam-5569	220	9	2n+	2n+	NUM
ejpam-5569	220	10	1)|	1)|	NUM
ejpam-5569	220	11	≥	≥	NOUN
ejpam-5569	220	12	n	n	CCONJ
ejpam-5569	220	13	,	,	PUNCT
ejpam-5569	220	14	where	where	SCONJ
ejpam-5569	220	15	n	n	NOUN
ejpam-5569	220	16	=	=	SYM
ejpam-5569	220	17	4k	4k	X
ejpam-5569	220	18	+	+	NOUN
ejpam-5569	220	19	2	2	NUM
ejpam-5569	220	20	,	,	PUNCT
ejpam-5569	220	21	d(x	d(x	PROPN
ejpam-5569	220	22	,	,	PUNCT
ejpam-5569	220	23	y	y	PROPN
ejpam-5569	220	24	)	)	PUNCT
ejpam-5569	220	25	≥	≥	NOUN
ejpam-5569	220	26	1	1	NUM
ejpam-5569	220	27	≥	≥	NOUN
ejpam-5569	220	28	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	220	29	)	)	PUNCT
ejpam-5569	220	30	)	)	PUNCT
ejpam-5569	221	1	+	+	CCONJ
ejpam-5569	221	2	1−	1−	NUM
ejpam-5569	221	3	d(x	d(x	PROPN
ejpam-5569	221	4	,	,	PUNCT
ejpam-5569	221	5	y	y	NOUN
ejpam-5569	221	6	)	)	PUNCT
ejpam-5569	221	7	.	.	PUNCT
ejpam-5569	222	1	subcase	subcase	PROPN
ejpam-5569	222	2	9.3	9.3	NUM
ejpam-5569	222	3	.	.	PUNCT
ejpam-5569	223	1	suppose	suppose	VERB
ejpam-5569	223	2	,	,	PUNCT
ejpam-5569	223	3	1	1	NUM
ejpam-5569	223	4	≤	≤	NUM
ejpam-5569	224	1	i	i	NOUN
ejpam-5569	224	2	≤	≤	PUNCT
ejpam-5569	225	1	k	k	X
ejpam-5569	226	1	+	+	CCONJ
ejpam-5569	226	2	1	1	NUM
ejpam-5569	226	3	and	and	CCONJ
ejpam-5569	226	4	2k	2k	NUM
ejpam-5569	226	5	+	+	CCONJ
ejpam-5569	226	6	2	2	NUM
ejpam-5569	226	7	≤	≤	NUM
ejpam-5569	226	8	l	l	NOUN
ejpam-5569	226	9	≤	≤	NOUN
ejpam-5569	226	10	3k	3k	NOUN
ejpam-5569	227	1	+	+	CCONJ
ejpam-5569	227	2	2	2	NUM
ejpam-5569	227	3	if	if	SCONJ
ejpam-5569	227	4	i	i	PRON
ejpam-5569	227	5	=	=	NOUN
ejpam-5569	227	6	1	1	NUM
ejpam-5569	227	7	,	,	PUNCT
ejpam-5569	227	8	l	l	NOUN
ejpam-5569	227	9	=	=	PUNCT
ejpam-5569	228	1	3k	3k	X
ejpam-5569	229	1	+	+	CCONJ
ejpam-5569	229	2	2	2	NUM
ejpam-5569	229	3	i.e.	i.e.	X
ejpam-5569	229	4	j	j	X
ejpam-5569	229	5	=	=	SYM
ejpam-5569	229	6	1	1	NUM
ejpam-5569	229	7	then	then	ADV
ejpam-5569	229	8	|f(x)−	|f(x)−	NOUN
ejpam-5569	229	9	f(y)|	f(y)|	PROPN
ejpam-5569	230	1	=	=	PROPN
ejpam-5569	230	2	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	230	3	f(v3k+2)|	f(v3k+2)|	PROPN
ejpam-5569	230	4	=	=	PUNCT
ejpam-5569	230	5	|f(ui)−	|f(ui)−	ADJ
ejpam-5569	230	6	f(vn−k)|	f(vn−k)|	NOUN
ejpam-5569	230	7	=	=	PUNCT
ejpam-5569	230	8	|0−	|0−	NOUN
ejpam-5569	230	9	f(vk+1	f(vk+1	NOUN
ejpam-5569	230	10	)	)	PUNCT
ejpam-5569	230	11	+	+	CCONJ
ejpam-5569	230	12	2k	2k	NUM
ejpam-5569	230	13	+	+	CCONJ
ejpam-5569	230	14	1|	1|	NUM
ejpam-5569	230	15	=	=	SYM
ejpam-5569	230	16	|(n+	|(n+	NOUN
ejpam-5569	230	17	2	2	NUM
ejpam-5569	230	18	)	)	PUNCT
ejpam-5569	230	19	+	+	CCONJ
ejpam-5569	230	20	(	(	PUNCT
ejpam-5569	230	21	2n+	2n+	NUM
ejpam-5569	230	22	7)(k	7)(k	NUM
ejpam-5569	230	23	−	−	NOUN
ejpam-5569	230	24	1	1	NUM
ejpam-5569	230	25	)	)	PUNCT
ejpam-5569	230	26	+	+	CCONJ
ejpam-5569	230	27	2(k	2(k	NUM
ejpam-5569	230	28	−	−	NOUN
ejpam-5569	230	29	1)(k	1)(k	NUM
ejpam-5569	230	30	−	−	PROPN
ejpam-5569	230	31	2	2	NUM
ejpam-5569	230	32	)	)	PUNCT
ejpam-5569	230	33	+	+	NOUN
ejpam-5569	230	34	2k	2k	NUM
ejpam-5569	230	35	+	+	CCONJ
ejpam-5569	230	36	1|	1|	NUM
ejpam-5569	230	37	≥	≥	NOUN
ejpam-5569	230	38	n	n	CCONJ
ejpam-5569	230	39	,	,	PUNCT
ejpam-5569	230	40	where	where	SCONJ
ejpam-5569	230	41	n	n	NOUN
ejpam-5569	230	42	=	=	SYM
ejpam-5569	230	43	4k	4k	X
ejpam-5569	230	44	+	+	NOUN
ejpam-5569	230	45	2	2	NUM
ejpam-5569	230	46	,	,	PUNCT
ejpam-5569	230	47	d(x	d(x	PROPN
ejpam-5569	230	48	,	,	PUNCT
ejpam-5569	230	49	y	y	PROPN
ejpam-5569	230	50	)	)	PUNCT
ejpam-5569	230	51	≥	≥	NOUN
ejpam-5569	230	52	1	1	NUM
ejpam-5569	230	53	≥	≥	NOUN
ejpam-5569	230	54	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	230	55	)	)	PUNCT
ejpam-5569	230	56	)	)	PUNCT
ejpam-5569	231	1	+	+	CCONJ
ejpam-5569	231	2	1−	1−	NUM
ejpam-5569	231	3	d(x	d(x	PROPN
ejpam-5569	231	4	,	,	PUNCT
ejpam-5569	231	5	y	y	NOUN
ejpam-5569	231	6	)	)	PUNCT
ejpam-5569	231	7	.	.	PUNCT
ejpam-5569	232	1	otherwise	otherwise	ADV
ejpam-5569	232	2	,	,	PUNCT
ejpam-5569	232	3	|f(x)−	|f(x)−	PROPN
ejpam-5569	232	4	f(y)|	f(y)|	PROPN
ejpam-5569	232	5	≥	≥	AUX
ejpam-5569	232	6	n.	n.	VERB
ejpam-5569	232	7	for	for	ADP
ejpam-5569	232	8	more	more	ADJ
ejpam-5569	232	9	illustration	illustration	NOUN
ejpam-5569	232	10	,	,	PUNCT
ejpam-5569	232	11	figure	figure	NOUN
ejpam-5569	232	12	2	2	NUM
ejpam-5569	232	13	shows	show	VERB
ejpam-5569	232	14	the	the	DET
ejpam-5569	232	15	radio	radio	NOUN
ejpam-5569	232	16	labeling	labeling	NOUN
ejpam-5569	232	17	of	of	ADP
ejpam-5569	232	18	d2(p14	d2(p14	NOUN
ejpam-5569	232	19	)	)	PUNCT
ejpam-5569	232	20	according	accord	VERB
ejpam-5569	232	21	to	to	ADP
ejpam-5569	232	22	theorem	theorem	ADJ
ejpam-5569	232	23	1	1	NUM
ejpam-5569	232	24	.	.	PUNCT
ejpam-5569	232	25	n.	n.	PROPN
ejpam-5569	232	26	m.	m.	PROPN
ejpam-5569	233	1	noureldeen	noureldeen	INTJ
ejpam-5569	233	2	et	et	PROPN
ejpam-5569	233	3	al	al	PROPN
ejpam-5569	233	4	.	.	PUNCT
ejpam-5569	233	5	/	/	SYM
ejpam-5569	233	6	eur	eur	PROPN
ejpam-5569	233	7	.	.	PUNCT
ejpam-5569	234	1	j.	j.	PROPN
ejpam-5569	234	2	pure	pure	PROPN
ejpam-5569	234	3	appl	appl	PROPN
ejpam-5569	234	4	.	.	PROPN
ejpam-5569	234	5	math	math	PROPN
ejpam-5569	234	6	,	,	PUNCT
ejpam-5569	234	7	18	18	NUM
ejpam-5569	234	8	(	(	PUNCT
ejpam-5569	234	9	2	2	NUM
ejpam-5569	234	10	)	)	PUNCT
ejpam-5569	234	11	(	(	PUNCT
ejpam-5569	234	12	2025	2025	NUM
ejpam-5569	234	13	)	)	PUNCT
ejpam-5569	234	14	,	,	PUNCT
ejpam-5569	234	15	5569	5569	NUM
ejpam-5569	234	16	9	9	NUM
ejpam-5569	234	17	of	of	ADP
ejpam-5569	234	18	25	25	NUM
ejpam-5569	234	19	figure	figure	NOUN
ejpam-5569	234	20	2	2	NUM
ejpam-5569	234	21	:	:	PUNCT
ejpam-5569	234	22	radio	radio	NOUN
ejpam-5569	234	23	labeling	labeling	NOUN
ejpam-5569	234	24	of	of	ADP
ejpam-5569	234	25	d2(p14	d2(p14	NOUN
ejpam-5569	234	26	)	)	PUNCT
ejpam-5569	234	27	following	follow	VERB
ejpam-5569	234	28	theorem	theorem	VERB
ejpam-5569	234	29	1	1	NUM
ejpam-5569	234	30	.	.	PUNCT
ejpam-5569	234	31	theorem	theorem	NOUN
ejpam-5569	234	32	2	2	NUM
ejpam-5569	234	33	.	.	PUNCT
ejpam-5569	235	1	let	let	VERB
ejpam-5569	235	2	n	n	PRON
ejpam-5569	235	3	>	>	X
ejpam-5569	235	4	6	6	NUM
ejpam-5569	235	5	be	be	AUX
ejpam-5569	235	6	a	a	DET
ejpam-5569	235	7	positive	positive	ADJ
ejpam-5569	235	8	integer	integer	NOUN
ejpam-5569	235	9	such	such	ADJ
ejpam-5569	235	10	that	that	SCONJ
ejpam-5569	235	11	n	n	NUM
ejpam-5569	235	12	≡	≡	PROPN
ejpam-5569	235	13	3(mod4	3(mod4	NUM
ejpam-5569	235	14	)	)	PUNCT
ejpam-5569	235	15	.	.	PUNCT
ejpam-5569	236	1	then	then	ADV
ejpam-5569	236	2	rn(d2(pn	rn(d2(pn	PROPN
ejpam-5569	236	3	)	)	PUNCT
ejpam-5569	236	4	)	)	PUNCT
ejpam-5569	237	1	≤	≤	NUM
ejpam-5569	237	2	n2	n2	NOUN
ejpam-5569	237	3	−	−	PROPN
ejpam-5569	237	4	3	3	X
ejpam-5569	237	5	.	.	PUNCT
ejpam-5569	237	6	proof	proof	NOUN
ejpam-5569	237	7	.	.	PUNCT
ejpam-5569	238	1	let	let	VERB
ejpam-5569	238	2	k	k	PROPN
ejpam-5569	238	3	≥	≥	NUM
ejpam-5569	238	4	1	1	NUM
ejpam-5569	238	5	,	,	PUNCT
ejpam-5569	238	6	n	n	PRON
ejpam-5569	238	7	≡	≡	PROPN
ejpam-5569	238	8	3(mod4	3(mod4	NUM
ejpam-5569	238	9	)	)	PUNCT
ejpam-5569	238	10	such	such	ADJ
ejpam-5569	238	11	that	that	SCONJ
ejpam-5569	238	12	n	n	NOUN
ejpam-5569	238	13	=	=	SYM
ejpam-5569	238	14	4k+3	4k+3	PROPN
ejpam-5569	238	15	,	,	PUNCT
ejpam-5569	238	16	we	we	PRON
ejpam-5569	238	17	define	define	VERB
ejpam-5569	238	18	f	f	X
ejpam-5569	238	19	:	:	PUNCT
ejpam-5569	238	20	v	v	PROPN
ejpam-5569	238	21	(	(	PUNCT
ejpam-5569	238	22	d2(pn	d2(pn	PROPN
ejpam-5569	238	23	)	)	PUNCT
ejpam-5569	238	24	)	)	PUNCT
ejpam-5569	238	25	→	→	SYM
ejpam-5569	239	1	n	n	CCONJ
ejpam-5569	239	2	as	as	SCONJ
ejpam-5569	239	3	follows	follow	VERB
ejpam-5569	239	4	:	:	PUNCT
ejpam-5569	239	5	f(ui	f(ui	PROPN
ejpam-5569	239	6	)	)	PUNCT
ejpam-5569	240	1	=	=	SYM
ejpam-5569	240	2			NUM
ejpam-5569	240	3	(	(	PUNCT
ejpam-5569	240	4	2n+	2n+	NUM
ejpam-5569	240	5	3)(i−	3)(i−	NUM
ejpam-5569	240	6	1	1	NUM
ejpam-5569	240	7	)	)	PUNCT
ejpam-5569	240	8	+	+	CCONJ
ejpam-5569	240	9	2(i−	2(i−	NUM
ejpam-5569	240	10	1)(i−	1)(i−	NUM
ejpam-5569	240	11	2	2	NUM
ejpam-5569	240	12	)	)	PUNCT
ejpam-5569	240	13	,	,	PUNCT
ejpam-5569	240	14	if	if	SCONJ
ejpam-5569	240	15	1	1	NUM
ejpam-5569	240	16	≤	≤	NUM
ejpam-5569	240	17	i	i	NOUN
ejpam-5569	240	18	≤	≤	NOUN
ejpam-5569	241	1	k	k	X
ejpam-5569	242	1	+	+	CCONJ
ejpam-5569	242	2	1	1	NUM
ejpam-5569	242	3	f(uk+1)−	f(uk+1)−	NUM
ejpam-5569	242	4	(	(	PUNCT
ejpam-5569	242	5	2n−	2n−	PROPN
ejpam-5569	242	6	2)−	2)−	PROPN
ejpam-5569	242	7	(	(	PUNCT
ejpam-5569	242	8	3n−	3n−	NUM
ejpam-5569	242	9	6)(j	6)(j	NUM
ejpam-5569	242	10	−	−	NUM
ejpam-5569	242	11	1	1	NUM
ejpam-5569	242	12	)	)	PUNCT
ejpam-5569	242	13	+	+	CCONJ
ejpam-5569	242	14	2(j	2(j	NUM
ejpam-5569	243	1	−	−	PROPN
ejpam-5569	243	2	1)(j	1)(j	NUM
ejpam-5569	243	3	−	−	PROPN
ejpam-5569	243	4	2	2	NUM
ejpam-5569	243	5	)	)	PUNCT
ejpam-5569	243	6	,	,	PUNCT
ejpam-5569	243	7	if	if	SCONJ
ejpam-5569	243	8	k	k	PROPN
ejpam-5569	243	9	+	+	CCONJ
ejpam-5569	243	10	2	2	X
ejpam-5569	243	11	≤	≤	NUM
ejpam-5569	243	12	i	i	NOUN
ejpam-5569	243	13	≤	≤	NOUN
ejpam-5569	243	14	2k	2k	NOUN
ejpam-5569	243	15	+	+	CCONJ
ejpam-5569	243	16	1	1	NUM
ejpam-5569	243	17	,	,	PUNCT
ejpam-5569	243	18	where	where	SCONJ
ejpam-5569	243	19	1	1	NUM
ejpam-5569	243	20	≤	≤	NUM
ejpam-5569	243	21	j	j	PROPN
ejpam-5569	243	22	≤	≤	PROPN
ejpam-5569	243	23	k	k	PROPN
ejpam-5569	243	24	,	,	PUNCT
ejpam-5569	243	25	and	and	CCONJ
ejpam-5569	243	26	i	i	PRON
ejpam-5569	243	27	=	=	PUNCT
ejpam-5569	244	1	k	k	PROPN
ejpam-5569	245	1	+	+	CCONJ
ejpam-5569	245	2	1	1	NUM
ejpam-5569	245	3	+	+	CCONJ
ejpam-5569	245	4	j	j	PROPN
ejpam-5569	245	5	n2	n2	NOUN
ejpam-5569	245	6	−	−	PROPN
ejpam-5569	245	7	n−	n−	NOUN
ejpam-5569	245	8	1	1	NUM
ejpam-5569	245	9	,	,	PUNCT
ejpam-5569	245	10	if	if	SCONJ
ejpam-5569	245	11	i	i	PRON
ejpam-5569	245	12	=	=	SYM
ejpam-5569	245	13	2k	2k	NUM
ejpam-5569	245	14	+	+	CCONJ
ejpam-5569	245	15	2	2	NUM
ejpam-5569	245	16	f(uk+1+j	f(uk+1+j	NUM
ejpam-5569	245	17	)	)	PUNCT
ejpam-5569	246	1	+	+	CCONJ
ejpam-5569	246	2	(	(	PUNCT
ejpam-5569	246	3	6k	6k	NOUN
ejpam-5569	246	4	+	+	X
ejpam-5569	246	5	2)−	2)−	NUM
ejpam-5569	246	6	2(j	2(j	NUM
ejpam-5569	246	7	−	−	NOUN
ejpam-5569	246	8	1	1	NUM
ejpam-5569	246	9	)	)	PUNCT
ejpam-5569	246	10	,	,	PUNCT
ejpam-5569	246	11	if	if	SCONJ
ejpam-5569	246	12	2k	2k	NUM
ejpam-5569	246	13	+	+	CCONJ
ejpam-5569	246	14	3	3	NUM
ejpam-5569	246	15	≤	≤	NUM
ejpam-5569	247	1	i	i	PRON
ejpam-5569	247	2	≤	≤	NOUN
ejpam-5569	247	3	3k	3k	X
ejpam-5569	247	4	+	+	CCONJ
ejpam-5569	247	5	2	2	NUM
ejpam-5569	247	6	where	where	SCONJ
ejpam-5569	247	7	1	1	NUM
ejpam-5569	247	8	≤	≤	NUM
ejpam-5569	247	9	j	j	PROPN
ejpam-5569	247	10	≤	≤	PROPN
ejpam-5569	247	11	k	k	PROPN
ejpam-5569	247	12	,	,	PUNCT
ejpam-5569	247	13	and	and	CCONJ
ejpam-5569	247	14	i	i	PRON
ejpam-5569	247	15	=	=	NOUN
ejpam-5569	248	1	3k	3k	X
ejpam-5569	249	1	+	+	CCONJ
ejpam-5569	249	2	3−	3−	NUM
ejpam-5569	249	3	j	j	NOUN
ejpam-5569	249	4	)	)	PUNCT
ejpam-5569	249	5	f(uj	f(uj	PROPN
ejpam-5569	249	6	)	)	PUNCT
ejpam-5569	250	1	+	+	CCONJ
ejpam-5569	250	2	2j	2j	NUM
ejpam-5569	250	3	−	−	NOUN
ejpam-5569	250	4	1	1	NUM
ejpam-5569	250	5	,	,	PUNCT
ejpam-5569	250	6	if	if	SCONJ
ejpam-5569	250	7	3k	3k	PRON
ejpam-5569	250	8	+	+	CCONJ
ejpam-5569	250	9	3	3	NUM
ejpam-5569	250	10	≤	≤	NUM
ejpam-5569	251	1	i	i	PRON
ejpam-5569	251	2	≤	≤	NOUN
ejpam-5569	251	3	4k	4k	NUM
ejpam-5569	251	4	+	+	CCONJ
ejpam-5569	251	5	3	3	NUM
ejpam-5569	251	6	=	=	SYM
ejpam-5569	251	7	n	n	X
ejpam-5569	251	8	where	where	SCONJ
ejpam-5569	251	9	1	1	NUM
ejpam-5569	251	10	≤	≤	NUM
ejpam-5569	251	11	j	j	PROPN
ejpam-5569	251	12	≤	≤	PROPN
ejpam-5569	252	1	k	k	PROPN
ejpam-5569	253	1	+	+	CCONJ
ejpam-5569	253	2	1	1	NUM
ejpam-5569	253	3	,	,	PUNCT
ejpam-5569	253	4	and	and	CCONJ
ejpam-5569	253	5	i	i	PRON
ejpam-5569	253	6	=	=	VERB
ejpam-5569	253	7	4k	4k	NUM
ejpam-5569	253	8	+	+	CCONJ
ejpam-5569	253	9	4−	4−	NUM
ejpam-5569	253	10	j	j	NOUN
ejpam-5569	253	11	and	and	CCONJ
ejpam-5569	253	12	for	for	ADP
ejpam-5569	253	13	the	the	DET
ejpam-5569	253	14	vertices	vertex	NOUN
ejpam-5569	253	15	belongs	belong	VERB
ejpam-5569	253	16	to	to	ADP
ejpam-5569	253	17	vv	vv	NOUN
ejpam-5569	253	18	,	,	PUNCT
ejpam-5569	253	19	f(vi	f(vi	NOUN
ejpam-5569	253	20	)	)	PUNCT
ejpam-5569	254	1	=	=	SYM
ejpam-5569	254	2			X
ejpam-5569	254	3	(	(	PUNCT
ejpam-5569	254	4	n+	n+	NOUN
ejpam-5569	254	5	3	3	NUM
ejpam-5569	254	6	)	)	PUNCT
ejpam-5569	254	7	+	+	CCONJ
ejpam-5569	254	8	(	(	PUNCT
ejpam-5569	254	9	2n+	2n+	NUM
ejpam-5569	254	10	7)(i−	7)(i−	NUM
ejpam-5569	254	11	1	1	NUM
ejpam-5569	254	12	)	)	PUNCT
ejpam-5569	254	13	+	+	CCONJ
ejpam-5569	254	14	2(i−	2(i−	NUM
ejpam-5569	254	15	1)(i−	1)(i−	NUM
ejpam-5569	254	16	2	2	NUM
ejpam-5569	254	17	)	)	PUNCT
ejpam-5569	254	18	,	,	PUNCT
ejpam-5569	254	19	if	if	SCONJ
ejpam-5569	254	20	1	1	NUM
ejpam-5569	254	21	≤	≤	NUM
ejpam-5569	254	22	i	i	NOUN
ejpam-5569	254	23	≤	≤	NOUN
ejpam-5569	254	24	k	k	PROPN
ejpam-5569	254	25	f(vk	f(vk	PROPN
ejpam-5569	254	26	)	)	PUNCT
ejpam-5569	254	27	+	+	CCONJ
ejpam-5569	254	28	(	(	PUNCT
ejpam-5569	254	29	2n−	2n−	NUM
ejpam-5569	254	30	1	1	NUM
ejpam-5569	254	31	)	)	PUNCT
ejpam-5569	254	32	+	+	CCONJ
ejpam-5569	254	33	n(j	n(j	PROPN
ejpam-5569	254	34	−	−	NUM
ejpam-5569	254	35	1	1	NUM
ejpam-5569	254	36	)	)	PUNCT
ejpam-5569	254	37	+	+	CCONJ
ejpam-5569	254	38	2(j	2(j	NUM
ejpam-5569	254	39	−	−	PROPN
ejpam-5569	254	40	1)(j	1)(j	NUM
ejpam-5569	254	41	−	−	PROPN
ejpam-5569	254	42	2	2	NUM
ejpam-5569	254	43	)	)	PUNCT
ejpam-5569	254	44	,	,	PUNCT
ejpam-5569	254	45	if	if	SCONJ
ejpam-5569	254	46	k	k	PROPN
ejpam-5569	254	47	+	+	PROPN
ejpam-5569	254	48	1	1	X
ejpam-5569	254	49	≤	≤	NUM
ejpam-5569	254	50	i	i	NOUN
ejpam-5569	254	51	≤	≤	NOUN
ejpam-5569	254	52	2k	2k	NOUN
ejpam-5569	254	53	+	+	CCONJ
ejpam-5569	254	54	1,where	1,where	NUM
ejpam-5569	254	55	1	1	NUM
ejpam-5569	254	56	≤	≤	NUM
ejpam-5569	254	57	j	j	PROPN
ejpam-5569	254	58	≤	≤	PROPN
ejpam-5569	255	1	k	k	PROPN
ejpam-5569	256	1	+	+	CCONJ
ejpam-5569	256	2	1	1	NUM
ejpam-5569	256	3	,	,	PUNCT
ejpam-5569	256	4	and	and	CCONJ
ejpam-5569	256	5	i	i	PRON
ejpam-5569	256	6	=	=	PUNCT
ejpam-5569	257	1	k	k	PROPN
ejpam-5569	257	2	+	+	CCONJ
ejpam-5569	257	3	j	j	PROPN
ejpam-5569	257	4	n2	n2	NOUN
ejpam-5569	257	5	−	−	PROPN
ejpam-5569	257	6	3	3	NUM
ejpam-5569	257	7	,	,	PUNCT
ejpam-5569	257	8	if	if	SCONJ
ejpam-5569	257	9	i	i	PRON
ejpam-5569	257	10	=	=	SYM
ejpam-5569	257	11	2k	2k	NUM
ejpam-5569	257	12	+	+	CCONJ
ejpam-5569	257	13	2	2	NUM
ejpam-5569	257	14	f(vk+j	f(vk+j	NUM
ejpam-5569	257	15	)	)	PUNCT
ejpam-5569	258	1	+	+	CCONJ
ejpam-5569	258	2	(	(	PUNCT
ejpam-5569	258	3	2k	2k	NUM
ejpam-5569	258	4	+	+	CCONJ
ejpam-5569	258	5	1	1	NUM
ejpam-5569	258	6	)	)	PUNCT
ejpam-5569	258	7	+	+	CCONJ
ejpam-5569	259	1	2(j	2(j	NUM
ejpam-5569	259	2	−	−	NOUN
ejpam-5569	259	3	1	1	NUM
ejpam-5569	259	4	)	)	PUNCT
ejpam-5569	259	5	,	,	PUNCT
ejpam-5569	259	6	if	if	SCONJ
ejpam-5569	259	7	2k	2k	NUM
ejpam-5569	259	8	+	+	CCONJ
ejpam-5569	259	9	3	3	NUM
ejpam-5569	259	10	≤	≤	NUM
ejpam-5569	259	11	i	i	PRON
ejpam-5569	259	12	≤	≤	NOUN
ejpam-5569	259	13	3k	3k	X
ejpam-5569	259	14	+	+	CCONJ
ejpam-5569	259	15	3	3	NUM
ejpam-5569	259	16	where	where	SCONJ
ejpam-5569	259	17	1	1	NUM
ejpam-5569	259	18	≤	≤	NUM
ejpam-5569	259	19	j	j	PROPN
ejpam-5569	259	20	≤	≤	PROPN
ejpam-5569	259	21	k	k	PROPN
ejpam-5569	260	1	+	+	CCONJ
ejpam-5569	260	2	1	1	NUM
ejpam-5569	260	3	,	,	PUNCT
ejpam-5569	260	4	and	and	CCONJ
ejpam-5569	260	5	i	i	PRON
ejpam-5569	260	6	=	=	NOUN
ejpam-5569	261	1	3k	3k	X
ejpam-5569	261	2	+	+	CCONJ
ejpam-5569	261	3	4−	4−	NUM
ejpam-5569	261	4	j	j	NOUN
ejpam-5569	261	5	f(vn+1−j	f(vn+1−j	PROPN
ejpam-5569	261	6	)	)	PUNCT
ejpam-5569	262	1	=	=	SYM
ejpam-5569	262	2	f(vj)−	f(vj)−	NOUN
ejpam-5569	262	3	2j	2j	NUM
ejpam-5569	262	4	,	,	PUNCT
ejpam-5569	262	5	if	if	SCONJ
ejpam-5569	262	6	3k	3k	PRON
ejpam-5569	262	7	+	+	CCONJ
ejpam-5569	262	8	4	4	NUM
ejpam-5569	262	9	≤	≤	NUM
ejpam-5569	262	10	i	i	PRON
ejpam-5569	262	11	≤	≤	NOUN
ejpam-5569	262	12	4k	4k	NUM
ejpam-5569	262	13	+	+	CCONJ
ejpam-5569	262	14	3	3	NUM
ejpam-5569	262	15	=	=	SYM
ejpam-5569	262	16	n	n	X
ejpam-5569	262	17	where	where	SCONJ
ejpam-5569	262	18	1	1	NUM
ejpam-5569	262	19	≤	≤	NUM
ejpam-5569	262	20	j	j	PROPN
ejpam-5569	262	21	≤	≤	PROPN
ejpam-5569	262	22	k	k	PROPN
ejpam-5569	262	23	,	,	PUNCT
ejpam-5569	262	24	and	and	CCONJ
ejpam-5569	262	25	i	i	PRON
ejpam-5569	262	26	=	=	VERB
ejpam-5569	262	27	4k	4k	NUM
ejpam-5569	263	1	+	+	CCONJ
ejpam-5569	264	1	4−	4−	NUM
ejpam-5569	264	2	j	j	NOUN
ejpam-5569	264	3	now	now	ADV
ejpam-5569	264	4	we	we	PRON
ejpam-5569	264	5	claim	claim	VERB
ejpam-5569	264	6	to	to	PART
ejpam-5569	264	7	prove	prove	VERB
ejpam-5569	264	8	that	that	SCONJ
ejpam-5569	264	9	the	the	DET
ejpam-5569	264	10	function	function	NOUN
ejpam-5569	264	11	f	f	NOUN
ejpam-5569	264	12	given	give	VERB
ejpam-5569	264	13	above	above	ADV
ejpam-5569	264	14	is	be	AUX
ejpam-5569	264	15	a	a	DET
ejpam-5569	264	16	radio	radio	NOUN
ejpam-5569	264	17	labelling	labelling	NOUN
ejpam-5569	264	18	of	of	ADP
ejpam-5569	264	19	d2(pn	d2(pn	PROPN
ejpam-5569	264	20	)	)	PUNCT
ejpam-5569	264	21	consequently	consequently	ADV
ejpam-5569	264	22	,	,	PUNCT
ejpam-5569	264	23	it	it	PRON
ejpam-5569	264	24	is	be	AUX
ejpam-5569	264	25	required	require	VERB
ejpam-5569	264	26	that	that	SCONJ
ejpam-5569	264	27	f	f	PROPN
ejpam-5569	264	28	verifies	verify	VERB
ejpam-5569	264	29	the	the	DET
ejpam-5569	264	30	following	follow	VERB
ejpam-5569	264	31	condition	condition	NOUN
ejpam-5569	264	32	with	with	ADP
ejpam-5569	264	33	a	a	DET
ejpam-5569	264	34	span	span	NOUN
ejpam-5569	264	35	equal	equal	ADJ
ejpam-5569	264	36	to	to	ADP
ejpam-5569	264	37	the	the	DET
ejpam-5569	264	38	desired	desire	VERB
ejpam-5569	264	39	number	number	NOUN
ejpam-5569	264	40	.	.	PUNCT
ejpam-5569	265	1	so	so	ADV
ejpam-5569	265	2	,	,	PUNCT
ejpam-5569	265	3	we	we	PRON
ejpam-5569	265	4	need	need	VERB
ejpam-5569	265	5	only	only	ADV
ejpam-5569	265	6	to	to	PART
ejpam-5569	265	7	check	check	VERB
ejpam-5569	265	8	it	it	PRON
ejpam-5569	265	9	for	for	ADP
ejpam-5569	265	10	n	n	X
ejpam-5569	265	11	≥	≥	NUM
ejpam-5569	265	12	6	6	NUM
ejpam-5569	265	13	,	,	PUNCT
ejpam-5569	265	14	n	n	NOUN
ejpam-5569	265	15	=	=	SYM
ejpam-5569	265	16	4k	4k	NOUN
ejpam-5569	265	17	+	+	NOUN
ejpam-5569	265	18	3	3	NUM
ejpam-5569	265	19	where	where	SCONJ
ejpam-5569	265	20	k	k	PROPN
ejpam-5569	265	21	≥	≥	NUM
ejpam-5569	265	22	1	1	NUM
ejpam-5569	265	23	.	.	PUNCT
ejpam-5569	266	1	it	it	PRON
ejpam-5569	266	2	remains	remain	VERB
ejpam-5569	266	3	to	to	PART
ejpam-5569	266	4	verify	verify	VERB
ejpam-5569	266	5	that	that	SCONJ
ejpam-5569	266	6	|f(x)−	|f(x)−	PROPN
ejpam-5569	266	7	f(y)|	f(y)|	PROPN
ejpam-5569	266	8	≥	≥	NUM
ejpam-5569	266	9	diam(d2(pn	diam(d2(pn	PROPN
ejpam-5569	266	10	)	)	PUNCT
ejpam-5569	266	11	)	)	PUNCT
ejpam-5569	267	1	+	+	CCONJ
ejpam-5569	267	2	1−	1−	NUM
ejpam-5569	267	3	d(x	d(x	PROPN
ejpam-5569	267	4	,	,	PUNCT
ejpam-5569	267	5	y	y	NOUN
ejpam-5569	267	6	)	)	PUNCT
ejpam-5569	267	7	for	for	ADP
ejpam-5569	267	8	any	any	DET
ejpam-5569	267	9	two	two	NUM
ejpam-5569	267	10	distinct	distinct	ADJ
ejpam-5569	267	11	vertices	vertex	NOUN
ejpam-5569	267	12	x	x	X
ejpam-5569	267	13	,	,	PUNCT
ejpam-5569	267	14	y	y	PROPN
ejpam-5569	267	15	of	of	ADP
ejpam-5569	267	16	d2(pn	d2(pn	PROPN
ejpam-5569	267	17	)	)	PUNCT
ejpam-5569	267	18	.	.	PUNCT
ejpam-5569	268	1	we	we	PRON
ejpam-5569	268	2	distinguish	distinguish	VERB
ejpam-5569	268	3	several	several	ADJ
ejpam-5569	268	4	cases	case	NOUN
ejpam-5569	268	5	:	:	PUNCT
ejpam-5569	268	6	case	case	NOUN
ejpam-5569	268	7	1	1	NUM
ejpam-5569	268	8	x	x	NOUN
ejpam-5569	268	9	,	,	PUNCT
ejpam-5569	268	10	y	y	PROPN
ejpam-5569	268	11	∈	∈	PROPN
ejpam-5569	268	12	vu	vu	X
ejpam-5569	268	13	.	.	PUNCT
ejpam-5569	269	1	if	if	SCONJ
ejpam-5569	269	2	{	{	PUNCT
ejpam-5569	269	3	x	x	NOUN
ejpam-5569	269	4	,	,	PUNCT
ejpam-5569	269	5	y	y	PROPN
ejpam-5569	269	6	}	}	PUNCT
ejpam-5569	269	7	=	=	SYM
ejpam-5569	269	8	{	{	PUNCT
ejpam-5569	269	9	u1	u1	NOUN
ejpam-5569	269	10	,	,	PUNCT
ejpam-5569	269	11	un	un	ADJ
ejpam-5569	269	12	}	}	PUNCT
ejpam-5569	269	13	then	then	ADV
ejpam-5569	269	14	|f(x)−	|f(x)−	NOUN
ejpam-5569	269	15	f(y)|	f(y)|	PROPN
ejpam-5569	269	16	=	=	SYM
ejpam-5569	269	17	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	269	18	f(un)|	f(un)|	PROPN
ejpam-5569	270	1	=	=	SYM
ejpam-5569	270	2	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	271	1	[	[	X
ejpam-5569	271	2	f(u1	f(u1	NOUN
ejpam-5569	271	3	)	)	PUNCT
ejpam-5569	272	1	+	+	CCONJ
ejpam-5569	272	2	2−	2−	NUM
ejpam-5569	272	3	1]|	1]|	NUM
ejpam-5569	272	4	=	=	SYM
ejpam-5569	272	5	1	1	NUM
ejpam-5569	272	6	≥	≥	NOUN
ejpam-5569	272	7	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	272	8	)	)	PUNCT
ejpam-5569	272	9	)	)	PUNCT
ejpam-5569	273	1	+	+	CCONJ
ejpam-5569	273	2	1−	1−	NUM
ejpam-5569	273	3	d(x	d(x	PROPN
ejpam-5569	273	4	,	,	PUNCT
ejpam-5569	273	5	y	y	NOUN
ejpam-5569	273	6	)	)	PUNCT
ejpam-5569	273	7	,	,	PUNCT
ejpam-5569	273	8	where	where	SCONJ
ejpam-5569	273	9	d(x	d(x	PROPN
ejpam-5569	273	10	,	,	PUNCT
ejpam-5569	273	11	y	y	NOUN
ejpam-5569	273	12	)	)	PUNCT
ejpam-5569	273	13	=	=	VERB
ejpam-5569	274	1	n−	n−	NOUN
ejpam-5569	274	2	1	1	NUM
ejpam-5569	274	3	.	.	PUNCT
ejpam-5569	275	1	n.	n.	PROPN
ejpam-5569	275	2	m.	m.	PROPN
ejpam-5569	275	3	noureldeen	noureldeen	INTJ
ejpam-5569	275	4	et	et	PROPN
ejpam-5569	275	5	al	al	PROPN
ejpam-5569	275	6	.	.	PUNCT
ejpam-5569	275	7	/	/	SYM
ejpam-5569	275	8	eur	eur	PROPN
ejpam-5569	275	9	.	.	PUNCT
ejpam-5569	276	1	j.	j.	PROPN
ejpam-5569	276	2	pure	pure	PROPN
ejpam-5569	276	3	appl	appl	PROPN
ejpam-5569	276	4	.	.	PROPN
ejpam-5569	276	5	math	math	PROPN
ejpam-5569	276	6	,	,	PUNCT
ejpam-5569	276	7	18	18	NUM
ejpam-5569	276	8	(	(	PUNCT
ejpam-5569	276	9	2	2	NUM
ejpam-5569	276	10	)	)	PUNCT
ejpam-5569	276	11	(	(	PUNCT
ejpam-5569	276	12	2025	2025	NUM
ejpam-5569	276	13	)	)	PUNCT
ejpam-5569	276	14	,	,	PUNCT
ejpam-5569	276	15	5569	5569	NUM
ejpam-5569	276	16	10	10	NUM
ejpam-5569	276	17	of	of	ADP
ejpam-5569	276	18	25	25	NUM
ejpam-5569	276	19	case	case	NOUN
ejpam-5569	276	20	2	2	NUM
ejpam-5569	276	21	if	if	SCONJ
ejpam-5569	276	22	{	{	PUNCT
ejpam-5569	276	23	x	x	NOUN
ejpam-5569	276	24	,	,	PUNCT
ejpam-5569	276	25	y	y	PROPN
ejpam-5569	276	26	}	}	PUNCT
ejpam-5569	276	27	=	=	SYM
ejpam-5569	276	28	{	{	PUNCT
ejpam-5569	276	29	u1	u1	NOUN
ejpam-5569	276	30	,	,	PUNCT
ejpam-5569	276	31	uk+1	uk+1	VERB
ejpam-5569	276	32	}	}	PUNCT
ejpam-5569	276	33	then	then	ADV
ejpam-5569	276	34	|f(x)−	|f(x)−	NOUN
ejpam-5569	276	35	f(y)|	f(y)|	PROPN
ejpam-5569	277	1	=	=	SYM
ejpam-5569	277	2	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	278	1	f(uk+1)|	f(uk+1)|	X
ejpam-5569	278	2	=	=	SYM
ejpam-5569	278	3	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	279	1	[	[	X
ejpam-5569	279	2	(	(	PUNCT
ejpam-5569	279	3	2n+	2n+	NUM
ejpam-5569	279	4	3)(k	3)(k	NUM
ejpam-5569	279	5	)	)	PUNCT
ejpam-5569	280	1	+	+	NUM
ejpam-5569	280	2	2k(k	2k(k	NUM
ejpam-5569	280	3	−	−	NOUN
ejpam-5569	280	4	1)]|	1)]|	NOUN
ejpam-5569	280	5	=	=	NOUN
ejpam-5569	281	1	|	|	ADV
ejpam-5569	281	2	−	−	PROPN
ejpam-5569	282	1	[	[	X
ejpam-5569	282	2	(	(	PUNCT
ejpam-5569	282	3	2n+	2n+	NUM
ejpam-5569	282	4	3)(k	3)(k	NUM
ejpam-5569	282	5	)	)	PUNCT
ejpam-5569	283	1	+	+	NUM
ejpam-5569	284	1	2k(k	2k(k	NUM
ejpam-5569	284	2	−	−	NOUN
ejpam-5569	284	3	1)]|	1)]|	NOUN
ejpam-5569	284	4	=	=	SYM
ejpam-5569	284	5	(	(	PUNCT
ejpam-5569	284	6	2n+	2n+	NUM
ejpam-5569	284	7	3)(k	3)(k	NUM
ejpam-5569	284	8	)	)	PUNCT
ejpam-5569	285	1	+	+	CCONJ
ejpam-5569	285	2	2k(k	2k(k	NUM
ejpam-5569	285	3	−	−	NOUN
ejpam-5569	285	4	1	1	NUM
ejpam-5569	285	5	)	)	PUNCT
ejpam-5569	285	6	≥	≥	NOUN
ejpam-5569	285	7	n	n	CCONJ
ejpam-5569	285	8	≥	≥	NOUN
ejpam-5569	285	9	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	285	10	)	)	PUNCT
ejpam-5569	285	11	)	)	PUNCT
ejpam-5569	286	1	+	+	CCONJ
ejpam-5569	286	2	1−	1−	NUM
ejpam-5569	286	3	d(x	d(x	PROPN
ejpam-5569	286	4	,	,	PUNCT
ejpam-5569	286	5	y	y	NOUN
ejpam-5569	286	6	)	)	PUNCT
ejpam-5569	286	7	case	case	NOUN
ejpam-5569	286	8	3	3	NUM
ejpam-5569	286	9	if	if	SCONJ
ejpam-5569	286	10	{	{	PUNCT
ejpam-5569	286	11	x	x	NOUN
ejpam-5569	286	12	,	,	PUNCT
ejpam-5569	286	13	y	y	PROPN
ejpam-5569	286	14	}	}	PUNCT
ejpam-5569	286	15	=	=	SYM
ejpam-5569	286	16	{	{	PUNCT
ejpam-5569	286	17	u1	u1	NOUN
ejpam-5569	286	18	,	,	PUNCT
ejpam-5569	286	19	u2k+1	u2k+1	ADP
ejpam-5569	286	20	}	}	PUNCT
ejpam-5569	286	21	then	then	ADV
ejpam-5569	286	22	|f(x)−	|f(x)−	NOUN
ejpam-5569	286	23	f(y)|	f(y)|	PROPN
ejpam-5569	286	24	=	=	SYM
ejpam-5569	286	25	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	287	1	f(u2k+1)|	f(u2k+1)|	NUM
ejpam-5569	287	2	=	=	SYM
ejpam-5569	287	3	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	288	1	[	[	X
ejpam-5569	288	2	f(uk+1)−	f(uk+1)−	X
ejpam-5569	288	3	(	(	PUNCT
ejpam-5569	288	4	2n−	2n−	PROPN
ejpam-5569	288	5	2)−	2)−	PROPN
ejpam-5569	288	6	(	(	PUNCT
ejpam-5569	288	7	3n−	3n−	NUM
ejpam-5569	288	8	6)(k	6)(k	NUM
ejpam-5569	288	9	−	−	NOUN
ejpam-5569	288	10	1	1	NUM
ejpam-5569	288	11	)	)	PUNCT
ejpam-5569	288	12	+	+	CCONJ
ejpam-5569	289	1	2(k	2(k	NUM
ejpam-5569	289	2	−	−	NOUN
ejpam-5569	289	3	1)(k	1)(k	NUM
ejpam-5569	289	4	−	−	PROPN
ejpam-5569	289	5	2)]|	2)]|	NOUN
ejpam-5569	289	6	=	=	SYM
ejpam-5569	289	7	|f(uk+1)−	|f(uk+1)−	NOUN
ejpam-5569	289	8	(	(	PUNCT
ejpam-5569	289	9	2n−	2n−	PROPN
ejpam-5569	289	10	2)−	2)−	PROPN
ejpam-5569	289	11	(	(	PUNCT
ejpam-5569	289	12	3n−	3n−	NUM
ejpam-5569	289	13	6)(k	6)(k	NUM
ejpam-5569	289	14	−	−	NOUN
ejpam-5569	289	15	1	1	NUM
ejpam-5569	289	16	)	)	PUNCT
ejpam-5569	289	17	+	+	CCONJ
ejpam-5569	290	1	2(k	2(k	NUM
ejpam-5569	290	2	−	−	NOUN
ejpam-5569	290	3	1)(k	1)(k	NUM
ejpam-5569	290	4	−	−	PROPN
ejpam-5569	290	5	2)|	2)|	NUM
ejpam-5569	290	6	=	=	SYM
ejpam-5569	291	1	|(2n+	|(2n+	PROPN
ejpam-5569	291	2	3)(k	3)(k	NUM
ejpam-5569	291	3	)	)	PUNCT
ejpam-5569	292	1	+	+	NUM
ejpam-5569	293	1	2k(k	2k(k	NUM
ejpam-5569	293	2	−	−	PROPN
ejpam-5569	293	3	1)−	1)−	NUM
ejpam-5569	293	4	(	(	PUNCT
ejpam-5569	293	5	2n−	2n−	PROPN
ejpam-5569	293	6	2)−	2)−	PROPN
ejpam-5569	293	7	(	(	PUNCT
ejpam-5569	293	8	3n−	3n−	NUM
ejpam-5569	293	9	6)(k	6)(k	NUM
ejpam-5569	293	10	−	−	NOUN
ejpam-5569	293	11	1	1	NUM
ejpam-5569	293	12	)	)	PUNCT
ejpam-5569	293	13	+	+	CCONJ
ejpam-5569	293	14	2(k	2(k	NUM
ejpam-5569	293	15	−	−	NOUN
ejpam-5569	293	16	1)(k	1)(k	NUM
ejpam-5569	293	17	−	−	PROPN
ejpam-5569	293	18	2)|	2)|	NUM
ejpam-5569	293	19	≥	≥	NUM
ejpam-5569	293	20	n−	n−	PROPN
ejpam-5569	293	21	2k	2k	NUM
ejpam-5569	293	22	≥	≥	NUM
ejpam-5569	293	23	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	293	24	)	)	PUNCT
ejpam-5569	293	25	)	)	PUNCT
ejpam-5569	294	1	+	+	CCONJ
ejpam-5569	294	2	1−	1−	NUM
ejpam-5569	294	3	d(x	d(x	PROPN
ejpam-5569	294	4	,	,	PUNCT
ejpam-5569	294	5	y	y	NOUN
ejpam-5569	294	6	)	)	PUNCT
ejpam-5569	294	7	case	case	NOUN
ejpam-5569	294	8	4	4	NUM
ejpam-5569	294	9	if	if	SCONJ
ejpam-5569	294	10	{	{	PUNCT
ejpam-5569	294	11	x	x	NOUN
ejpam-5569	294	12	,	,	PUNCT
ejpam-5569	294	13	y	y	PROPN
ejpam-5569	294	14	}	}	PUNCT
ejpam-5569	294	15	=	=	SYM
ejpam-5569	294	16	{	{	PUNCT
ejpam-5569	294	17	u1	u1	NOUN
ejpam-5569	294	18	,	,	PUNCT
ejpam-5569	294	19	u2k+2	u2k+2	NOUN
ejpam-5569	294	20	}	}	PUNCT
ejpam-5569	294	21	then	then	ADV
ejpam-5569	294	22	|f(x)−	|f(x)−	NOUN
ejpam-5569	294	23	f(y)|	f(y)|	PROPN
ejpam-5569	295	1	=	=	SYM
ejpam-5569	295	2	|f(u1)−	|f(u1)−	ADJ
ejpam-5569	295	3	f(u2k+2)|	f(u2k+2)|	PROPN
ejpam-5569	295	4	=	=	SYM
ejpam-5569	295	5	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	296	1	[	[	X
ejpam-5569	296	2	n2	n2	ADJ
ejpam-5569	296	3	−	−	PROPN
ejpam-5569	296	4	n−	n−	NOUN
ejpam-5569	296	5	1]|	1]|	NUM
ejpam-5569	296	6	=	=	PUNCT
ejpam-5569	296	7	|n2	|n2	X
ejpam-5569	296	8	−	−	PROPN
ejpam-5569	296	9	n−	n−	NOUN
ejpam-5569	296	10	1|	1|	NUM
ejpam-5569	296	11	≥	≥	NOUN
ejpam-5569	296	12	n	n	CCONJ
ejpam-5569	296	13	≥	≥	NOUN
ejpam-5569	296	14	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	296	15	)	)	PUNCT
ejpam-5569	296	16	)	)	PUNCT
ejpam-5569	297	1	+	+	CCONJ
ejpam-5569	297	2	1−	1−	NUM
ejpam-5569	297	3	d(x	d(x	PROPN
ejpam-5569	297	4	,	,	PUNCT
ejpam-5569	297	5	y	y	NOUN
ejpam-5569	297	6	)	)	PUNCT
ejpam-5569	297	7	case	case	NOUN
ejpam-5569	297	8	5	5	NUM
ejpam-5569	297	9	if	if	SCONJ
ejpam-5569	297	10	{	{	PUNCT
ejpam-5569	297	11	x	x	NOUN
ejpam-5569	297	12	,	,	PUNCT
ejpam-5569	297	13	y	y	PROPN
ejpam-5569	297	14	}	}	PUNCT
ejpam-5569	297	15	=	=	SYM
ejpam-5569	297	16	{	{	PUNCT
ejpam-5569	297	17	u1	u1	NOUN
ejpam-5569	297	18	,	,	PUNCT
ejpam-5569	297	19	u3k+2	u3k+2	NOUN
ejpam-5569	297	20	}	}	PUNCT
ejpam-5569	297	21	then	then	ADV
ejpam-5569	297	22	|f(x)−	|f(x)−	NOUN
ejpam-5569	297	23	f(y)|	f(y)|	PROPN
ejpam-5569	297	24	=	=	PROPN
ejpam-5569	297	25	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	298	1	f(u3k+2)|	f(u3k+2)|	PROPN
ejpam-5569	298	2	=	=	SYM
ejpam-5569	298	3	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	299	1	[	[	X
ejpam-5569	299	2	f(uk+2	f(uk+2	NOUN
ejpam-5569	299	3	)	)	PUNCT
ejpam-5569	299	4	+	+	CCONJ
ejpam-5569	299	5	(	(	PUNCT
ejpam-5569	299	6	6k	6k	NOUN
ejpam-5569	299	7	+	+	CCONJ
ejpam-5569	299	8	2)]|	2)]|	NOUN
ejpam-5569	299	9	=	=	SYM
ejpam-5569	299	10	|f(uk+2	|f(uk+2	NOUN
ejpam-5569	299	11	)	)	PUNCT
ejpam-5569	300	1	+	+	CCONJ
ejpam-5569	300	2	(	(	PUNCT
ejpam-5569	300	3	6k	6k	NOUN
ejpam-5569	300	4	+	+	NOUN
ejpam-5569	300	5	2)|	2)|	NUM
ejpam-5569	300	6	=	=	NOUN
ejpam-5569	300	7	|(n2	|(n2	PRON
ejpam-5569	300	8	−	−	PROPN
ejpam-5569	300	9	n−	n−	NOUN
ejpam-5569	300	10	1	1	NUM
ejpam-5569	300	11	)	)	PUNCT
ejpam-5569	300	12	+	+	CCONJ
ejpam-5569	300	13	(	(	PUNCT
ejpam-5569	300	14	6k	6k	NOUN
ejpam-5569	300	15	+	+	NOUN
ejpam-5569	300	16	2)|	2)|	NUM
ejpam-5569	300	17	≥	≥	NOUN
ejpam-5569	300	18	n	n	CCONJ
ejpam-5569	300	19	≥	≥	NOUN
ejpam-5569	300	20	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	300	21	)	)	PUNCT
ejpam-5569	300	22	)	)	PUNCT
ejpam-5569	301	1	+	+	CCONJ
ejpam-5569	301	2	1−	1−	NUM
ejpam-5569	301	3	d(x	d(x	PROPN
ejpam-5569	301	4	,	,	PUNCT
ejpam-5569	301	5	y	y	NOUN
ejpam-5569	301	6	)	)	PUNCT
ejpam-5569	301	7	case	case	NOUN
ejpam-5569	301	8	6	6	NUM
ejpam-5569	301	9	x	x	NOUN
ejpam-5569	301	10	,	,	PUNCT
ejpam-5569	301	11	y	y	PROPN
ejpam-5569	301	12	∈	∈	PROPN
ejpam-5569	301	13	vv	vv	INTJ
ejpam-5569	301	14	.	.	PUNCT
ejpam-5569	302	1	if	if	SCONJ
ejpam-5569	302	2	{	{	PUNCT
ejpam-5569	302	3	x	x	NOUN
ejpam-5569	302	4	,	,	PUNCT
ejpam-5569	302	5	y	y	PROPN
ejpam-5569	302	6	}	}	PUNCT
ejpam-5569	302	7	=	=	SYM
ejpam-5569	302	8	{	{	PUNCT
ejpam-5569	302	9	v1	v1	PROPN
ejpam-5569	302	10	,	,	PUNCT
ejpam-5569	302	11	vn	vn	VERB
ejpam-5569	302	12	}	}	PUNCT
ejpam-5569	302	13	then	then	ADV
ejpam-5569	302	14	|f(x)−	|f(x)−	NOUN
ejpam-5569	302	15	f(y)|	f(y)|	PROPN
ejpam-5569	302	16	=	=	PROPN
ejpam-5569	303	1	|f(v1)−	|f(v1)−	ADJ
ejpam-5569	303	2	f(vn)|	f(vn)|	NOUN
ejpam-5569	303	3	=	=	X
ejpam-5569	304	1	|f(v1)−	|f(v1)−	PROPN
ejpam-5569	304	2	[	[	X
ejpam-5569	304	3	f(v1)−	f(v1)−	PROPN
ejpam-5569	304	4	2]|	2]|	NOUN
ejpam-5569	304	5	=	=	SYM
ejpam-5569	304	6	2	2	NUM
ejpam-5569	304	7	≥	≥	NOUN
ejpam-5569	304	8	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	304	9	)	)	PUNCT
ejpam-5569	304	10	)	)	PUNCT
ejpam-5569	305	1	+	+	CCONJ
ejpam-5569	305	2	1−	1−	NUM
ejpam-5569	305	3	d(x	d(x	PROPN
ejpam-5569	305	4	,	,	PUNCT
ejpam-5569	305	5	y	y	NOUN
ejpam-5569	305	6	)	)	PUNCT
ejpam-5569	305	7	,	,	PUNCT
ejpam-5569	305	8	where	where	SCONJ
ejpam-5569	305	9	d(x	d(x	PROPN
ejpam-5569	305	10	,	,	PUNCT
ejpam-5569	305	11	y	y	NOUN
ejpam-5569	305	12	)	)	PUNCT
ejpam-5569	305	13	=	=	VERB
ejpam-5569	306	1	n−	n−	NOUN
ejpam-5569	306	2	1	1	NUM
ejpam-5569	306	3	.	.	PUNCT
ejpam-5569	307	1	n.	n.	PROPN
ejpam-5569	307	2	m.	m.	PROPN
ejpam-5569	307	3	noureldeen	noureldeen	INTJ
ejpam-5569	307	4	et	et	PROPN
ejpam-5569	307	5	al	al	PROPN
ejpam-5569	307	6	.	.	PUNCT
ejpam-5569	307	7	/	/	SYM
ejpam-5569	307	8	eur	eur	PROPN
ejpam-5569	307	9	.	.	PUNCT
ejpam-5569	308	1	j.	j.	PROPN
ejpam-5569	308	2	pure	pure	PROPN
ejpam-5569	308	3	appl	appl	PROPN
ejpam-5569	308	4	.	.	PROPN
ejpam-5569	308	5	math	math	PROPN
ejpam-5569	308	6	,	,	PUNCT
ejpam-5569	308	7	18	18	NUM
ejpam-5569	308	8	(	(	PUNCT
ejpam-5569	308	9	2	2	NUM
ejpam-5569	308	10	)	)	PUNCT
ejpam-5569	308	11	(	(	PUNCT
ejpam-5569	308	12	2025	2025	NUM
ejpam-5569	308	13	)	)	PUNCT
ejpam-5569	308	14	,	,	PUNCT
ejpam-5569	308	15	5569	5569	NUM
ejpam-5569	308	16	11	11	NUM
ejpam-5569	308	17	of	of	ADP
ejpam-5569	308	18	25	25	NUM
ejpam-5569	308	19	case	case	NOUN
ejpam-5569	308	20	7	7	NUM
ejpam-5569	308	21	if	if	SCONJ
ejpam-5569	308	22	{	{	PUNCT
ejpam-5569	308	23	x	x	NOUN
ejpam-5569	308	24	,	,	PUNCT
ejpam-5569	308	25	y	y	PROPN
ejpam-5569	308	26	}	}	PUNCT
ejpam-5569	308	27	=	=	SYM
ejpam-5569	308	28	{	{	PUNCT
ejpam-5569	308	29	v1	v1	NOUN
ejpam-5569	308	30	,	,	PUNCT
ejpam-5569	308	31	vk	vk	ADP
ejpam-5569	308	32	}	}	PUNCT
ejpam-5569	308	33	then	then	ADV
ejpam-5569	308	34	|f(x)−	|f(x)−	NOUN
ejpam-5569	308	35	f(y)|	f(y)|	PROPN
ejpam-5569	308	36	=	=	PROPN
ejpam-5569	308	37	|f(v1)−	|f(v1)−	ADJ
ejpam-5569	308	38	f(vk)|	f(vk)|	NOUN
ejpam-5569	308	39	=	=	PUNCT
ejpam-5569	309	1	|(n+	|(n+	NOUN
ejpam-5569	309	2	3)−	3)−	NUM
ejpam-5569	309	3	[	[	X
ejpam-5569	309	4	(	(	PUNCT
ejpam-5569	309	5	n+	n+	NOUN
ejpam-5569	309	6	3	3	NUM
ejpam-5569	309	7	)	)	PUNCT
ejpam-5569	309	8	+	+	CCONJ
ejpam-5569	309	9	(	(	PUNCT
ejpam-5569	309	10	2n+	2n+	NUM
ejpam-5569	309	11	7)(k	7)(k	NUM
ejpam-5569	309	12	−	−	NOUN
ejpam-5569	309	13	1	1	NUM
ejpam-5569	309	14	)	)	PUNCT
ejpam-5569	309	15	+	+	CCONJ
ejpam-5569	309	16	2(k	2(k	NUM
ejpam-5569	309	17	−	−	NOUN
ejpam-5569	309	18	1)(k	1)(k	NUM
ejpam-5569	309	19	−	−	NOUN
ejpam-5569	309	20	2)]|	2)]|	NOUN
ejpam-5569	309	21	=	=	SYM
ejpam-5569	309	22	|(2n+	|(2n+	X
ejpam-5569	310	1	7)(k	7)(k	NUM
ejpam-5569	310	2	−	−	NOUN
ejpam-5569	310	3	1	1	NUM
ejpam-5569	310	4	)	)	PUNCT
ejpam-5569	310	5	+	+	CCONJ
ejpam-5569	310	6	2(k	2(k	NUM
ejpam-5569	310	7	−	−	NOUN
ejpam-5569	310	8	1)(k	1)(k	NUM
ejpam-5569	310	9	−	−	PROPN
ejpam-5569	310	10	2)|	2)|	NUM
ejpam-5569	310	11	≥	≥	NOUN
ejpam-5569	310	12	n	n	CCONJ
ejpam-5569	310	13	,	,	PUNCT
ejpam-5569	310	14	where	where	SCONJ
ejpam-5569	310	15	n	n	NOUN
ejpam-5569	310	16	=	=	SYM
ejpam-5569	310	17	4k	4k	X
ejpam-5569	310	18	+	+	NOUN
ejpam-5569	310	19	3	3	NUM
ejpam-5569	310	20	,	,	PUNCT
ejpam-5569	310	21	d(x	d(x	PROPN
ejpam-5569	310	22	,	,	PUNCT
ejpam-5569	310	23	y	y	PROPN
ejpam-5569	310	24	)	)	PUNCT
ejpam-5569	310	25	≥	≥	NOUN
ejpam-5569	310	26	1	1	NUM
ejpam-5569	310	27	≥	≥	NOUN
ejpam-5569	310	28	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	310	29	)	)	PUNCT
ejpam-5569	310	30	)	)	PUNCT
ejpam-5569	311	1	+	+	CCONJ
ejpam-5569	311	2	1−	1−	NUM
ejpam-5569	311	3	d(x	d(x	PROPN
ejpam-5569	311	4	,	,	PUNCT
ejpam-5569	311	5	y	y	NOUN
ejpam-5569	311	6	)	)	PUNCT
ejpam-5569	311	7	case	case	NOUN
ejpam-5569	311	8	8	8	NUM
ejpam-5569	311	9	if	if	SCONJ
ejpam-5569	311	10	{	{	PUNCT
ejpam-5569	311	11	x	x	NOUN
ejpam-5569	311	12	,	,	PUNCT
ejpam-5569	311	13	y	y	PROPN
ejpam-5569	311	14	}	}	PUNCT
ejpam-5569	311	15	=	=	SYM
ejpam-5569	311	16	{	{	PUNCT
ejpam-5569	311	17	v1	v1	NOUN
ejpam-5569	311	18	,	,	PUNCT
ejpam-5569	311	19	v2k+1	v2k+1	ADP
ejpam-5569	311	20	}	}	PUNCT
ejpam-5569	311	21	then	then	ADV
ejpam-5569	311	22	|f(x)−	|f(x)−	NOUN
ejpam-5569	311	23	f(y)|	f(y)|	PROPN
ejpam-5569	311	24	=	=	PROPN
ejpam-5569	312	1	|f(v1)−	|f(v1)−	PROPN
ejpam-5569	312	2	f(v2k+1)|	f(v2k+1)|	X
ejpam-5569	313	1	=	=	PUNCT
ejpam-5569	313	2	|(n+	|(n+	NOUN
ejpam-5569	313	3	3)−	3)−	NUM
ejpam-5569	314	1	[	[	X
ejpam-5569	314	2	(	(	PUNCT
ejpam-5569	314	3	n+	n+	NOUN
ejpam-5569	314	4	3	3	NUM
ejpam-5569	314	5	)	)	PUNCT
ejpam-5569	314	6	+	+	CCONJ
ejpam-5569	314	7	(	(	PUNCT
ejpam-5569	314	8	2n+	2n+	NUM
ejpam-5569	314	9	7)(k	7)(k	NUM
ejpam-5569	314	10	−	−	NOUN
ejpam-5569	314	11	1	1	NUM
ejpam-5569	314	12	)	)	PUNCT
ejpam-5569	314	13	+	+	CCONJ
ejpam-5569	314	14	2(k	2(k	NUM
ejpam-5569	314	15	−	−	NOUN
ejpam-5569	314	16	1)(k	1)(k	NUM
ejpam-5569	314	17	−	−	PROPN
ejpam-5569	314	18	2)+	2)+	NUM
ejpam-5569	315	1	+	+	CCONJ
ejpam-5569	315	2	(	(	PUNCT
ejpam-5569	315	3	2n−	2n−	NUM
ejpam-5569	315	4	1	1	NUM
ejpam-5569	315	5	)	)	PUNCT
ejpam-5569	315	6	+	+	CCONJ
ejpam-5569	315	7	(	(	PUNCT
ejpam-5569	315	8	n)(k	n)(k	NOUN
ejpam-5569	315	9	)	)	PUNCT
ejpam-5569	315	10	+	+	CCONJ
ejpam-5569	315	11	2(k)(k	2(k)(k	NUM
ejpam-5569	315	12	−	−	PROPN
ejpam-5569	315	13	1)]|	1)]|	NUM
ejpam-5569	315	14	=	=	SYM
ejpam-5569	315	15	|(2n+	|(2n+	X
ejpam-5569	316	1	7)(k	7)(k	NUM
ejpam-5569	316	2	−	−	NOUN
ejpam-5569	316	3	1	1	NUM
ejpam-5569	316	4	)	)	PUNCT
ejpam-5569	316	5	+	+	CCONJ
ejpam-5569	316	6	2(k	2(k	NUM
ejpam-5569	316	7	−	−	NOUN
ejpam-5569	316	8	1)(k	1)(k	NUM
ejpam-5569	316	9	−	−	PROPN
ejpam-5569	316	10	2	2	NUM
ejpam-5569	316	11	)	)	PUNCT
ejpam-5569	316	12	+	+	CCONJ
ejpam-5569	316	13	(	(	PUNCT
ejpam-5569	316	14	2n−	2n−	NUM
ejpam-5569	316	15	1	1	NUM
ejpam-5569	316	16	)	)	PUNCT
ejpam-5569	316	17	+	+	CCONJ
ejpam-5569	316	18	(	(	PUNCT
ejpam-5569	316	19	n)(k	n)(k	NOUN
ejpam-5569	316	20	)	)	PUNCT
ejpam-5569	317	1	+	+	CCONJ
ejpam-5569	317	2	2(k)(k	2(k)(k	NUM
ejpam-5569	317	3	−	−	PROPN
ejpam-5569	317	4	1)|	1)|	NUM
ejpam-5569	317	5	≥	≥	NOUN
ejpam-5569	317	6	n	n	CCONJ
ejpam-5569	317	7	,	,	PUNCT
ejpam-5569	317	8	where	where	SCONJ
ejpam-5569	317	9	n	n	NOUN
ejpam-5569	317	10	=	=	SYM
ejpam-5569	317	11	4k	4k	X
ejpam-5569	317	12	+	+	NOUN
ejpam-5569	317	13	3	3	NUM
ejpam-5569	317	14	,	,	PUNCT
ejpam-5569	317	15	d(x	d(x	PROPN
ejpam-5569	317	16	,	,	PUNCT
ejpam-5569	317	17	y	y	PROPN
ejpam-5569	317	18	)	)	PUNCT
ejpam-5569	317	19	≥	≥	NOUN
ejpam-5569	317	20	1	1	NUM
ejpam-5569	317	21	≥	≥	NOUN
ejpam-5569	317	22	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	317	23	)	)	PUNCT
ejpam-5569	317	24	)	)	PUNCT
ejpam-5569	318	1	+	+	CCONJ
ejpam-5569	318	2	1−	1−	NUM
ejpam-5569	318	3	d(x	d(x	PROPN
ejpam-5569	318	4	,	,	PUNCT
ejpam-5569	318	5	y	y	NOUN
ejpam-5569	318	6	)	)	PUNCT
ejpam-5569	318	7	case	case	NOUN
ejpam-5569	318	8	9	9	NUM
ejpam-5569	318	9	if	if	SCONJ
ejpam-5569	318	10	{	{	PUNCT
ejpam-5569	318	11	x	x	NOUN
ejpam-5569	318	12	,	,	PUNCT
ejpam-5569	318	13	y	y	PROPN
ejpam-5569	318	14	}	}	PUNCT
ejpam-5569	318	15	=	=	SYM
ejpam-5569	318	16	{	{	PUNCT
ejpam-5569	318	17	v1	v1	NOUN
ejpam-5569	318	18	,	,	PUNCT
ejpam-5569	318	19	v2k+2	v2k+2	NOUN
ejpam-5569	318	20	}	}	PUNCT
ejpam-5569	318	21	then	then	ADV
ejpam-5569	318	22	|f(x)−	|f(x)−	NOUN
ejpam-5569	318	23	f(y)|	f(y)|	PROPN
ejpam-5569	318	24	=	=	PROPN
ejpam-5569	319	1	|f(v1)−	|f(v1)−	ADJ
ejpam-5569	319	2	f(v2k+2)|	f(v2k+2)|	NOUN
ejpam-5569	319	3	=	=	PUNCT
ejpam-5569	320	1	|f(v1)−	|f(v1)−	PROPN
ejpam-5569	321	1	[	[	X
ejpam-5569	321	2	n2	n2	NOUN
ejpam-5569	321	3	−	−	NUM
ejpam-5569	321	4	3]|	3]|	NUM
ejpam-5569	321	5	=	=	PUNCT
ejpam-5569	321	6	|n2	|n2	PROPN
ejpam-5569	321	7	−	−	PROPN
ejpam-5569	321	8	3|	3|	NUM
ejpam-5569	321	9	≥	≥	NUM
ejpam-5569	321	10	n	n	CCONJ
ejpam-5569	321	11	≥	≥	NOUN
ejpam-5569	321	12	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	321	13	)	)	PUNCT
ejpam-5569	321	14	)	)	PUNCT
ejpam-5569	322	1	+	+	CCONJ
ejpam-5569	322	2	1−	1−	NUM
ejpam-5569	322	3	d(x	d(x	PROPN
ejpam-5569	322	4	,	,	PUNCT
ejpam-5569	322	5	y	y	NOUN
ejpam-5569	322	6	)	)	PUNCT
ejpam-5569	322	7	case	case	NOUN
ejpam-5569	322	8	10	10	NUM
ejpam-5569	323	1	if	if	SCONJ
ejpam-5569	323	2	{	{	PUNCT
ejpam-5569	323	3	x	x	NOUN
ejpam-5569	323	4	,	,	PUNCT
ejpam-5569	323	5	y	y	PROPN
ejpam-5569	323	6	}	}	PUNCT
ejpam-5569	323	7	=	=	SYM
ejpam-5569	323	8	{	{	PUNCT
ejpam-5569	323	9	v1	v1	NOUN
ejpam-5569	323	10	,	,	PUNCT
ejpam-5569	323	11	v3k+3	v3k+3	PROPN
ejpam-5569	323	12	}	}	PUNCT
ejpam-5569	323	13	then	then	ADV
ejpam-5569	323	14	|f(x)−	|f(x)−	NOUN
ejpam-5569	323	15	f(y)|	f(y)|	PROPN
ejpam-5569	323	16	=	=	PUNCT
ejpam-5569	324	1	|f(v1)−	|f(v1)−	NOUN
ejpam-5569	324	2	f(v3k+3)|	f(v3k+3)|	NUM
ejpam-5569	325	1	=	=	PUNCT
ejpam-5569	326	1	|f(v1)−	|f(v1)−	ADJ
ejpam-5569	326	2	[	[	X
ejpam-5569	326	3	f(vk+1	f(vk+1	X
ejpam-5569	326	4	)	)	PUNCT
ejpam-5569	326	5	+	+	CCONJ
ejpam-5569	326	6	(	(	PUNCT
ejpam-5569	326	7	2k	2k	NUM
ejpam-5569	326	8	+	+	CCONJ
ejpam-5569	326	9	1)]|	1)]|	ADJ
ejpam-5569	326	10	=	=	NOUN
ejpam-5569	326	11	|f(v1)−	|f(v1)−	PROPN
ejpam-5569	326	12	[	[	X
ejpam-5569	326	13	f(vk	f(vk	X
ejpam-5569	326	14	)	)	PUNCT
ejpam-5569	326	15	+	+	CCONJ
ejpam-5569	326	16	(	(	PUNCT
ejpam-5569	326	17	2n−	2n−	NUM
ejpam-5569	326	18	1	1	NUM
ejpam-5569	326	19	)	)	PUNCT
ejpam-5569	326	20	+	+	CCONJ
ejpam-5569	326	21	(	(	PUNCT
ejpam-5569	326	22	2k	2k	NUM
ejpam-5569	326	23	+	+	CCONJ
ejpam-5569	326	24	1)]|	1)]|	NUM
ejpam-5569	326	25	=	=	NOUN
ejpam-5569	326	26	|(n+	|(n+	NOUN
ejpam-5569	326	27	2)−	2)−	NUM
ejpam-5569	327	1	[	[	X
ejpam-5569	327	2	(	(	PUNCT
ejpam-5569	327	3	n+	n+	NOUN
ejpam-5569	327	4	3	3	NUM
ejpam-5569	327	5	)	)	PUNCT
ejpam-5569	327	6	+	+	CCONJ
ejpam-5569	327	7	(	(	PUNCT
ejpam-5569	327	8	2n+	2n+	NUM
ejpam-5569	327	9	7)(k	7)(k	NUM
ejpam-5569	327	10	−	−	NOUN
ejpam-5569	327	11	1	1	NUM
ejpam-5569	327	12	)	)	PUNCT
ejpam-5569	327	13	+	+	CCONJ
ejpam-5569	327	14	2(k	2(k	NUM
ejpam-5569	327	15	−	−	NOUN
ejpam-5569	327	16	1)(k	1)(k	NUM
ejpam-5569	327	17	−	−	PROPN
ejpam-5569	327	18	2	2	NUM
ejpam-5569	327	19	)	)	PUNCT
ejpam-5569	327	20	+	+	CCONJ
ejpam-5569	327	21	(	(	PUNCT
ejpam-5569	327	22	2n+	2n+	NUM
ejpam-5569	327	23	1)+	1)+	NUM
ejpam-5569	327	24	+	+	CCONJ
ejpam-5569	327	25	(	(	PUNCT
ejpam-5569	327	26	2k	2k	NUM
ejpam-5569	327	27	+	+	CCONJ
ejpam-5569	328	1	1)]|	1)]|	NUM
ejpam-5569	328	2	=	=	SYM
ejpam-5569	328	3	|(2n+	|(2n+	PROPN
ejpam-5569	329	1	7)(k	7)(k	NUM
ejpam-5569	329	2	−	−	NOUN
ejpam-5569	329	3	1	1	NUM
ejpam-5569	329	4	)	)	PUNCT
ejpam-5569	329	5	+	+	CCONJ
ejpam-5569	329	6	2(k	2(k	NUM
ejpam-5569	329	7	−	−	NOUN
ejpam-5569	329	8	1)(k	1)(k	NUM
ejpam-5569	329	9	−	−	PROPN
ejpam-5569	329	10	2	2	NUM
ejpam-5569	329	11	)	)	PUNCT
ejpam-5569	329	12	+	+	CCONJ
ejpam-5569	329	13	(	(	PUNCT
ejpam-5569	329	14	2n+	2n+	NUM
ejpam-5569	329	15	1	1	NUM
ejpam-5569	329	16	)	)	PUNCT
ejpam-5569	329	17	+	+	CCONJ
ejpam-5569	329	18	(	(	PUNCT
ejpam-5569	329	19	2k	2k	NUM
ejpam-5569	329	20	+	+	CCONJ
ejpam-5569	329	21	1	1	NUM
ejpam-5569	329	22	)	)	PUNCT
ejpam-5569	329	23	+	+	CCONJ
ejpam-5569	329	24	1|	1|	NUM
ejpam-5569	329	25	≥	≥	NOUN
ejpam-5569	329	26	n	n	CCONJ
ejpam-5569	329	27	≥	≥	NOUN
ejpam-5569	329	28	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	329	29	)	)	PUNCT
ejpam-5569	329	30	)	)	PUNCT
ejpam-5569	330	1	+	+	CCONJ
ejpam-5569	330	2	1−	1−	NUM
ejpam-5569	330	3	d(x	d(x	PROPN
ejpam-5569	330	4	,	,	PUNCT
ejpam-5569	330	5	y	y	NOUN
ejpam-5569	330	6	)	)	PUNCT
ejpam-5569	330	7	case	case	NOUN
ejpam-5569	330	8	11	11	NUM
ejpam-5569	330	9	x	x	SYM
ejpam-5569	330	10	∈	∈	NOUN
ejpam-5569	330	11	vu	vu	NOUN
ejpam-5569	330	12	and	and	CCONJ
ejpam-5569	330	13	y	y	PROPN
ejpam-5569	330	14	∈	∈	PROPN
ejpam-5569	330	15	vv	vv	AUX
ejpam-5569	330	16	let	let	VERB
ejpam-5569	330	17	x	x	PUNCT
ejpam-5569	330	18	=	=	PUNCT
ejpam-5569	330	19	ui	ui	PROPN
ejpam-5569	330	20	and	and	CCONJ
ejpam-5569	330	21	y	y	PROPN
ejpam-5569	330	22	=	=	PUNCT
ejpam-5569	330	23	vl	vl	PROPN
ejpam-5569	330	24	,	,	PUNCT
ejpam-5569	330	25	where	where	SCONJ
ejpam-5569	330	26	1	1	NUM
ejpam-5569	330	27	≤	≤	PUNCT
ejpam-5569	330	28	i	i	PRON
ejpam-5569	330	29	,	,	PUNCT
ejpam-5569	330	30	l	l	PROPN
ejpam-5569	330	31	≤	≤	X
ejpam-5569	330	32	n	n	CCONJ
ejpam-5569	330	33	in	in	ADP
ejpam-5569	330	34	this	this	DET
ejpam-5569	330	35	case	case	NOUN
ejpam-5569	330	36	we	we	PRON
ejpam-5569	330	37	have	have	VERB
ejpam-5569	330	38	several	several	ADJ
ejpam-5569	330	39	subcases	subcase	NOUN
ejpam-5569	330	40	:	:	PUNCT
ejpam-5569	330	41	subcase	subcase	VERB
ejpam-5569	330	42	11.1	11.1	NUM
ejpam-5569	330	43	if	if	SCONJ
ejpam-5569	330	44	i	i	PRON
ejpam-5569	330	45	=	=	SYM
ejpam-5569	330	46	l	l	NOUN
ejpam-5569	330	47	,	,	PUNCT
ejpam-5569	330	48	1	1	NUM
ejpam-5569	330	49	≤	≤	NUM
ejpam-5569	330	50	i	i	PRON
ejpam-5569	330	51	,	,	PUNCT
ejpam-5569	330	52	l	l	PROPN
ejpam-5569	330	53	≤	≤	PUNCT
ejpam-5569	331	1	k	k	NOUN
ejpam-5569	331	2	then	then	ADV
ejpam-5569	331	3	|f(x)−	|f(x)−	NOUN
ejpam-5569	331	4	f(y)|	f(y)|	PROPN
ejpam-5569	331	5	=	=	NUM
ejpam-5569	331	6	|f(ui)−	|f(ui)−	ADJ
ejpam-5569	331	7	f(vi)|	f(vi)|	NOUN
ejpam-5569	331	8	,	,	PUNCT
ejpam-5569	331	9	1	1	NUM
ejpam-5569	331	10	≤	≤	NUM
ejpam-5569	331	11	i	i	NOUN
ejpam-5569	331	12	≤	≤	NOUN
ejpam-5569	331	13	k	k	PROPN
ejpam-5569	331	14	n.	n.	PROPN
ejpam-5569	331	15	m.	m.	NOUN
ejpam-5569	331	16	noureldeen	noureldeen	NOUN
ejpam-5569	331	17	et	et	PROPN
ejpam-5569	331	18	al	al	PROPN
ejpam-5569	331	19	.	.	PUNCT
ejpam-5569	331	20	/	/	SYM
ejpam-5569	331	21	eur	eur	PROPN
ejpam-5569	331	22	.	.	PUNCT
ejpam-5569	332	1	j.	j.	PROPN
ejpam-5569	332	2	pure	pure	PROPN
ejpam-5569	332	3	appl	appl	PROPN
ejpam-5569	332	4	.	.	PROPN
ejpam-5569	332	5	math	math	PROPN
ejpam-5569	332	6	,	,	PUNCT
ejpam-5569	332	7	18	18	NUM
ejpam-5569	332	8	(	(	PUNCT
ejpam-5569	332	9	2	2	NUM
ejpam-5569	332	10	)	)	PUNCT
ejpam-5569	332	11	(	(	PUNCT
ejpam-5569	332	12	2025	2025	NUM
ejpam-5569	332	13	)	)	PUNCT
ejpam-5569	332	14	,	,	PUNCT
ejpam-5569	332	15	5569	5569	NUM
ejpam-5569	332	16	12	12	NUM
ejpam-5569	332	17	of	of	ADP
ejpam-5569	332	18	25	25	NUM
ejpam-5569	332	19	=	=	SYM
ejpam-5569	332	20	|(2n+	|(2n+	PROPN
ejpam-5569	332	21	3)(i−	3)(i−	NUM
ejpam-5569	332	22	1	1	NUM
ejpam-5569	332	23	)	)	PUNCT
ejpam-5569	332	24	+	+	CCONJ
ejpam-5569	333	1	2(i−	2(i−	NUM
ejpam-5569	333	2	1)(i−	1)(i−	NUM
ejpam-5569	333	3	2)−	2)−	NUM
ejpam-5569	334	1	[	[	X
ejpam-5569	334	2	(	(	PUNCT
ejpam-5569	334	3	n+	n+	NOUN
ejpam-5569	334	4	3	3	NUM
ejpam-5569	334	5	)	)	PUNCT
ejpam-5569	334	6	+	+	CCONJ
ejpam-5569	334	7	(	(	PUNCT
ejpam-5569	334	8	2n+	2n+	NUM
ejpam-5569	334	9	7)(i−	7)(i−	NUM
ejpam-5569	334	10	1)+	1)+	NUM
ejpam-5569	334	11	+	+	CCONJ
ejpam-5569	334	12	2(i−	2(i−	NUM
ejpam-5569	334	13	1)(i−	1)(i−	NUM
ejpam-5569	334	14	2)]|	2)]|	NOUN
ejpam-5569	334	15	=	=	NOUN
ejpam-5569	335	1	|	|	ADV
ejpam-5569	335	2	−	−	NOUN
ejpam-5569	336	1	[	[	X
ejpam-5569	336	2	(	(	PUNCT
ejpam-5569	336	3	n+	n+	NOUN
ejpam-5569	336	4	3	3	NUM
ejpam-5569	336	5	)	)	PUNCT
ejpam-5569	336	6	+	+	CCONJ
ejpam-5569	336	7	(	(	PUNCT
ejpam-5569	336	8	4)(i−	4)(i−	NUM
ejpam-5569	336	9	1)]|	1)]|	NUM
ejpam-5569	336	10	=	=	SYM
ejpam-5569	336	11	|(n+	|(n+	NOUN
ejpam-5569	336	12	3	3	NUM
ejpam-5569	336	13	)	)	PUNCT
ejpam-5569	336	14	+	+	CCONJ
ejpam-5569	336	15	(	(	PUNCT
ejpam-5569	336	16	4)(i−	4)(i−	NUM
ejpam-5569	336	17	1)|	1)|	NUM
ejpam-5569	336	18	,	,	PUNCT
ejpam-5569	336	19	1	1	NUM
ejpam-5569	336	20	≤	≤	NUM
ejpam-5569	336	21	i	i	NOUN
ejpam-5569	336	22	≤	≤	PROPN
ejpam-5569	337	1	k	k	PROPN
ejpam-5569	337	2	≥	≥	PROPN
ejpam-5569	337	3	n	n	CCONJ
ejpam-5569	337	4	≥	≥	NOUN
ejpam-5569	337	5	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	337	6	)	)	PUNCT
ejpam-5569	337	7	)	)	PUNCT
ejpam-5569	338	1	+	+	CCONJ
ejpam-5569	338	2	1−	1−	NUM
ejpam-5569	338	3	d(x	d(x	PROPN
ejpam-5569	338	4	,	,	PUNCT
ejpam-5569	338	5	y	y	NOUN
ejpam-5569	338	6	)	)	PUNCT
ejpam-5569	338	7	.	.	PUNCT
ejpam-5569	339	1	subcase	subcase	PROPN
ejpam-5569	339	2	11.2	11.2	NUM
ejpam-5569	340	1	if	if	SCONJ
ejpam-5569	340	2	i	i	PRON
ejpam-5569	340	3	=	=	SYM
ejpam-5569	340	4	l	l	NOUN
ejpam-5569	340	5	=	=	PUNCT
ejpam-5569	340	6	2k	2k	NUM
ejpam-5569	340	7	+	+	CCONJ
ejpam-5569	340	8	2	2	NUM
ejpam-5569	340	9	|f(x)−	|f(x)−	NOUN
ejpam-5569	340	10	f(y)|	f(y)|	PROPN
ejpam-5569	340	11	=	=	NUM
ejpam-5569	340	12	|f(ui)−	|f(ui)−	ADJ
ejpam-5569	340	13	f(vi)|	f(vi)|	X
ejpam-5569	340	14	,	,	PUNCT
ejpam-5569	340	15	i	i	NOUN
ejpam-5569	340	16	=	=	PUNCT
ejpam-5569	340	17	2k	2k	NUM
ejpam-5569	340	18	+	+	CCONJ
ejpam-5569	340	19	2	2	NUM
ejpam-5569	340	20	=	=	SYM
ejpam-5569	340	21	|n2	|n2	X
ejpam-5569	340	22	−	−	PROPN
ejpam-5569	340	23	n−	n−	PROPN
ejpam-5569	340	24	1−	1−	NUM
ejpam-5569	341	1	[	[	X
ejpam-5569	341	2	n2	n2	NOUN
ejpam-5569	341	3	−	−	NUM
ejpam-5569	341	4	3]|	3]|	NUM
ejpam-5569	341	5	=	=	PUNCT
ejpam-5569	342	1	|	|	ADV
ejpam-5569	342	2	−	−	NOUN
ejpam-5569	343	1	[	[	X
ejpam-5569	343	2	(	(	PUNCT
ejpam-5569	343	3	n−	n−	NOUN
ejpam-5569	343	4	2)]|	2)]|	NOUN
ejpam-5569	343	5	=	=	NOUN
ejpam-5569	343	6	|(n−	|(n−	ADJ
ejpam-5569	343	7	2)|,where	2)|,where	ADJ
ejpam-5569	343	8	d(x	d(x	NOUN
ejpam-5569	343	9	,	,	PUNCT
ejpam-5569	343	10	y	y	NOUN
ejpam-5569	343	11	)	)	PUNCT
ejpam-5569	343	12	=	=	SYM
ejpam-5569	343	13	2	2	NUM
ejpam-5569	343	14	≥	≥	NOUN
ejpam-5569	343	15	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	343	16	)	)	PUNCT
ejpam-5569	343	17	)	)	PUNCT
ejpam-5569	344	1	+	+	CCONJ
ejpam-5569	344	2	1−	1−	NUM
ejpam-5569	344	3	d(x	d(x	PROPN
ejpam-5569	344	4	,	,	PUNCT
ejpam-5569	344	5	y	y	NOUN
ejpam-5569	344	6	)	)	PUNCT
ejpam-5569	344	7	.	.	PUNCT
ejpam-5569	345	1	subcase	subcase	PROPN
ejpam-5569	345	2	11.3	11.3	NUM
ejpam-5569	346	1	if	if	SCONJ
ejpam-5569	346	2	i	i	PRON
ejpam-5569	346	3	=	=	SYM
ejpam-5569	346	4	l	l	NOUN
ejpam-5569	346	5	,	,	PUNCT
ejpam-5569	346	6	3k	3k	X
ejpam-5569	346	7	+	+	CCONJ
ejpam-5569	346	8	4	4	NUM
ejpam-5569	346	9	≤	≤	NOUN
ejpam-5569	346	10	i	i	PRON
ejpam-5569	346	11	,	,	PUNCT
ejpam-5569	346	12	l	l	NOUN
ejpam-5569	346	13	≤	≤	X
ejpam-5569	346	14	4k	4k	NUM
ejpam-5569	346	15	+	+	CCONJ
ejpam-5569	346	16	3	3	NUM
ejpam-5569	346	17	=	=	SYM
ejpam-5569	346	18	n	n	NOUN
ejpam-5569	346	19	then	then	ADV
ejpam-5569	346	20	|f(x)−	|f(x)−	NOUN
ejpam-5569	346	21	f(y)|	f(y)|	PROPN
ejpam-5569	346	22	=	=	NUM
ejpam-5569	346	23	|f(ui)−	|f(ui)−	ADJ
ejpam-5569	346	24	f(vi)|	f(vi)|	X
ejpam-5569	346	25	,	,	PUNCT
ejpam-5569	346	26	3k	3k	X
ejpam-5569	346	27	+	+	CCONJ
ejpam-5569	346	28	4	4	NUM
ejpam-5569	346	29	≤	≤	NOUN
ejpam-5569	346	30	i	i	PRON
ejpam-5569	346	31	,	,	PUNCT
ejpam-5569	346	32	l	l	NOUN
ejpam-5569	346	33	≤	≤	X
ejpam-5569	346	34	4k	4k	NUM
ejpam-5569	346	35	+	+	CCONJ
ejpam-5569	346	36	3	3	NUM
ejpam-5569	346	37	=	=	SYM
ejpam-5569	346	38	n	n	PROPN
ejpam-5569	346	39	=	=	SYM
ejpam-5569	346	40	|f(uj	|f(uj	PROPN
ejpam-5569	346	41	)	)	PUNCT
ejpam-5569	347	1	+	+	CCONJ
ejpam-5569	347	2	2j	2j	NUM
ejpam-5569	347	3	−	−	NOUN
ejpam-5569	347	4	1−	1−	NUM
ejpam-5569	348	1	[	[	X
ejpam-5569	348	2	f(vj)−	f(vj)−	NOUN
ejpam-5569	348	3	2j]|	2j]|	NUM
ejpam-5569	348	4	,	,	PUNCT
ejpam-5569	348	5	1	1	NUM
ejpam-5569	348	6	≤	≤	NUM
ejpam-5569	349	1	j	j	PROPN
ejpam-5569	349	2	≤	≤	PROPN
ejpam-5569	350	1	k	k	NOUN
ejpam-5569	350	2	=	=	SYM
ejpam-5569	350	3	|f(uj)−	|f(uj)−	NOUN
ejpam-5569	350	4	f(vj	f(vj	PROPN
ejpam-5569	350	5	)	)	PUNCT
ejpam-5569	351	1	+	+	CCONJ
ejpam-5569	351	2	4j	4j	NOUN
ejpam-5569	351	3	−	−	ADP
ejpam-5569	351	4	1|	1|	NUM
ejpam-5569	351	5	,	,	PUNCT
ejpam-5569	351	6	1	1	NUM
ejpam-5569	351	7	≤	≤	NUM
ejpam-5569	351	8	j	j	PROPN
ejpam-5569	351	9	≤	≤	PROPN
ejpam-5569	351	10	k	k	PROPN
ejpam-5569	351	11	≥	≥	NUM
ejpam-5569	351	12	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	351	13	)	)	PUNCT
ejpam-5569	351	14	)	)	PUNCT
ejpam-5569	352	1	+	+	CCONJ
ejpam-5569	352	2	1−	1−	NUM
ejpam-5569	352	3	d(x	d(x	PROPN
ejpam-5569	352	4	,	,	PUNCT
ejpam-5569	352	5	y	y	NOUN
ejpam-5569	352	6	)	)	PUNCT
ejpam-5569	352	7	.	.	PUNCT
ejpam-5569	353	1	otherwise	otherwise	ADV
ejpam-5569	353	2	,	,	PUNCT
ejpam-5569	353	3	|f(x)−	|f(x)−	PROPN
ejpam-5569	353	4	f(y)|	f(y)|	PROPN
ejpam-5569	353	5	≥	≥	PROPN
ejpam-5569	353	6	n.	n.	PROPN
ejpam-5569	353	7	subcase	subcase	PROPN
ejpam-5569	353	8	11.4	11.4	NUM
ejpam-5569	353	9	assume	assume	VERB
ejpam-5569	353	10	i	i	PRON
ejpam-5569	353	11	̸=	̸=	PROPN
ejpam-5569	353	12	l.	l.	NOUN
ejpam-5569	353	13	let	let	VERB
ejpam-5569	353	14	,	,	PUNCT
ejpam-5569	353	15	1	1	NUM
ejpam-5569	353	16	≤	≤	NUM
ejpam-5569	354	1	i	i	NOUN
ejpam-5569	354	2	≤	≤	PUNCT
ejpam-5569	355	1	k	k	X
ejpam-5569	356	1	+	+	CCONJ
ejpam-5569	356	2	1	1	NUM
ejpam-5569	356	3	and	and	CCONJ
ejpam-5569	356	4	k	k	PROPN
ejpam-5569	357	1	+	+	CCONJ
ejpam-5569	357	2	1	1	NUM
ejpam-5569	357	3	≤	≤	NUM
ejpam-5569	357	4	l	l	NOUN
ejpam-5569	357	5	≤	≤	NOUN
ejpam-5569	357	6	2k	2k	NOUN
ejpam-5569	357	7	+	+	CCONJ
ejpam-5569	357	8	1	1	NUM
ejpam-5569	357	9	|f(x)−	|f(x)−	NOUN
ejpam-5569	357	10	f(y)|	f(y)|	PROPN
ejpam-5569	357	11	=	=	NUM
ejpam-5569	357	12	|f(ui)−	|f(ui)−	ADJ
ejpam-5569	357	13	f(vl)|	f(vl)|	NOUN
ejpam-5569	357	14	=	=	SYM
ejpam-5569	357	15	|f(ui)−	|f(ui)−	ADJ
ejpam-5569	357	16	f(vk+j)|	f(vk+j)|	PROPN
ejpam-5569	357	17	,	,	PUNCT
ejpam-5569	357	18	where	where	SCONJ
ejpam-5569	357	19	1	1	NUM
ejpam-5569	357	20	≤	≤	NUM
ejpam-5569	357	21	j	j	PROPN
ejpam-5569	357	22	≤	≤	PROPN
ejpam-5569	357	23	k	k	X
ejpam-5569	358	1	=	=	PUNCT
ejpam-5569	358	2	|(2n+	|(2n+	PROPN
ejpam-5569	358	3	3)(i−	3)(i−	NUM
ejpam-5569	358	4	1	1	NUM
ejpam-5569	358	5	)	)	PUNCT
ejpam-5569	358	6	+	+	CCONJ
ejpam-5569	359	1	2(i−	2(i−	NUM
ejpam-5569	359	2	1)(i−	1)(i−	NUM
ejpam-5569	359	3	2)−	2)−	NUM
ejpam-5569	360	1	[	[	X
ejpam-5569	360	2	f(vk	f(vk	X
ejpam-5569	360	3	)	)	PUNCT
ejpam-5569	360	4	+	+	CCONJ
ejpam-5569	361	1	(	(	PUNCT
ejpam-5569	361	2	2n−	2n−	NUM
ejpam-5569	361	3	1)+	1)+	NUM
ejpam-5569	362	1	+	+	CCONJ
ejpam-5569	362	2	(	(	PUNCT
ejpam-5569	362	3	n)(j	n)(j	X
ejpam-5569	362	4	−	−	NOUN
ejpam-5569	362	5	1	1	NUM
ejpam-5569	362	6	)	)	PUNCT
ejpam-5569	362	7	+	+	CCONJ
ejpam-5569	362	8	2(j	2(j	NUM
ejpam-5569	362	9	−	−	DET
ejpam-5569	362	10	1)(j	1)(j	NUM
ejpam-5569	362	11	−	−	NOUN
ejpam-5569	362	12	2)]|	2)]|	NOUN
ejpam-5569	362	13	,	,	PUNCT
ejpam-5569	362	14	=	=	PUNCT
ejpam-5569	362	15	|(2n+	|(2n+	PROPN
ejpam-5569	363	1	3)(i−	3)(i−	NUM
ejpam-5569	363	2	1	1	NUM
ejpam-5569	363	3	)	)	PUNCT
ejpam-5569	363	4	+	+	CCONJ
ejpam-5569	364	1	2(i−	2(i−	NUM
ejpam-5569	364	2	1)(i−	1)(i−	NUM
ejpam-5569	364	3	2)−	2)−	NUM
ejpam-5569	365	1	[	[	X
ejpam-5569	365	2	(	(	PUNCT
ejpam-5569	365	3	n+	n+	NOUN
ejpam-5569	365	4	3	3	NUM
ejpam-5569	365	5	)	)	PUNCT
ejpam-5569	365	6	+	+	CCONJ
ejpam-5569	365	7	(	(	PUNCT
ejpam-5569	365	8	2n+	2n+	NUM
ejpam-5569	365	9	7)(k	7)(k	NUM
ejpam-5569	365	10	−	−	PROPN
ejpam-5569	365	11	1)+	1)+	NUM
ejpam-5569	366	1	+	+	CCONJ
ejpam-5569	367	1	2(k	2(k	NUM
ejpam-5569	367	2	−	−	NOUN
ejpam-5569	367	3	1)(k	1)(k	NUM
ejpam-5569	367	4	−	−	PROPN
ejpam-5569	367	5	2	2	NUM
ejpam-5569	367	6	)	)	PUNCT
ejpam-5569	367	7	+	+	CCONJ
ejpam-5569	367	8	(	(	PUNCT
ejpam-5569	367	9	2n−	2n−	NUM
ejpam-5569	367	10	1	1	NUM
ejpam-5569	367	11	)	)	PUNCT
ejpam-5569	367	12	+	+	CCONJ
ejpam-5569	367	13	(	(	PUNCT
ejpam-5569	367	14	n)(j	n)(j	X
ejpam-5569	367	15	−	−	NOUN
ejpam-5569	367	16	1	1	NUM
ejpam-5569	367	17	)	)	PUNCT
ejpam-5569	367	18	+	+	CCONJ
ejpam-5569	367	19	2(j	2(j	NUM
ejpam-5569	367	20	−	−	PROPN
ejpam-5569	367	21	1)(j	1)(j	NUM
ejpam-5569	367	22	−	−	NOUN
ejpam-5569	367	23	2)]|	2)]|	NOUN
ejpam-5569	367	24	if	if	SCONJ
ejpam-5569	367	25	i	i	PRON
ejpam-5569	367	26	=	=	NOUN
ejpam-5569	367	27	1	1	NUM
ejpam-5569	367	28	,	,	PUNCT
ejpam-5569	367	29	l	l	NOUN
ejpam-5569	367	30	=	=	PUNCT
ejpam-5569	367	31	2k	2k	NOUN
ejpam-5569	367	32	+	+	CCONJ
ejpam-5569	367	33	1	1	NUM
ejpam-5569	367	34	i.e.	i.e.	X
ejpam-5569	367	35	j	j	X
ejpam-5569	368	1	=	=	SYM
ejpam-5569	368	2	k	k	PROPN
ejpam-5569	369	1	+	+	CCONJ
ejpam-5569	369	2	1	1	NUM
ejpam-5569	369	3	then	then	ADV
ejpam-5569	369	4	|f(x)−	|f(x)−	NOUN
ejpam-5569	369	5	f(y)|	f(y)|	PROPN
ejpam-5569	369	6	=	=	NUM
ejpam-5569	369	7	|f(ui)−	|f(ui)−	ADJ
ejpam-5569	369	8	f(vl)|	f(vl)|	NOUN
ejpam-5569	369	9	=	=	SYM
ejpam-5569	369	10	|f(u1)−	|f(u1)−	ADJ
ejpam-5569	370	1	f(v2k+1)|	f(v2k+1)|	VERB
ejpam-5569	370	2	=	=	PUNCT
ejpam-5569	370	3	|0−	|0−	NOUN
ejpam-5569	370	4	[	[	X
ejpam-5569	370	5	f(vk	f(vk	X
ejpam-5569	370	6	)	)	PUNCT
ejpam-5569	370	7	+	+	CCONJ
ejpam-5569	370	8	(	(	PUNCT
ejpam-5569	370	9	2n−	2n−	NUM
ejpam-5569	370	10	1	1	NUM
ejpam-5569	370	11	)	)	PUNCT
ejpam-5569	370	12	+	+	CCONJ
ejpam-5569	370	13	(	(	PUNCT
ejpam-5569	370	14	n)(j	n)(j	X
ejpam-5569	370	15	−	−	NOUN
ejpam-5569	370	16	1	1	NUM
ejpam-5569	370	17	)	)	PUNCT
ejpam-5569	370	18	+	+	CCONJ
ejpam-5569	370	19	2(j	2(j	NUM
ejpam-5569	370	20	−	−	PROPN
ejpam-5569	370	21	1)(j	1)(j	NUM
ejpam-5569	370	22	−	−	PROPN
ejpam-5569	370	23	2)]|	2)]|	NOUN
ejpam-5569	370	24	=	=	NOUN
ejpam-5569	370	25	|	|	ADV
ejpam-5569	370	26	−	−	PROPN
ejpam-5569	370	27	(	(	PUNCT
ejpam-5569	370	28	n+	n+	NOUN
ejpam-5569	370	29	3	3	NUM
ejpam-5569	370	30	)	)	PUNCT
ejpam-5569	370	31	+	+	CCONJ
ejpam-5569	370	32	(	(	PUNCT
ejpam-5569	370	33	2n+	2n+	NUM
ejpam-5569	370	34	7)(k	7)(k	NUM
ejpam-5569	370	35	−	−	NOUN
ejpam-5569	370	36	1	1	NUM
ejpam-5569	370	37	)	)	PUNCT
ejpam-5569	370	38	+	+	CCONJ
ejpam-5569	371	1	2(k	2(k	NUM
ejpam-5569	371	2	−	−	NOUN
ejpam-5569	371	3	1)(k	1)(k	NUM
ejpam-5569	371	4	−	−	PROPN
ejpam-5569	371	5	2)+	2)+	NUM
ejpam-5569	371	6	=	=	SYM
ejpam-5569	371	7	|(n+	|(n+	NOUN
ejpam-5569	371	8	3	3	NUM
ejpam-5569	371	9	)	)	PUNCT
ejpam-5569	371	10	+	+	CCONJ
ejpam-5569	371	11	(	(	PUNCT
ejpam-5569	371	12	2n+	2n+	NUM
ejpam-5569	371	13	7)(k	7)(k	NUM
ejpam-5569	371	14	−	−	NOUN
ejpam-5569	371	15	1	1	NUM
ejpam-5569	371	16	)	)	PUNCT
ejpam-5569	371	17	+	+	CCONJ
ejpam-5569	372	1	2(k	2(k	NUM
ejpam-5569	372	2	−	−	NOUN
ejpam-5569	372	3	1)(k	1)(k	NUM
ejpam-5569	372	4	−	−	PROPN
ejpam-5569	372	5	2)+	2)+	NUM
ejpam-5569	373	1	+	+	CCONJ
ejpam-5569	373	2	(	(	PUNCT
ejpam-5569	373	3	2n−	2n−	NUM
ejpam-5569	373	4	1	1	NUM
ejpam-5569	373	5	)	)	PUNCT
ejpam-5569	373	6	+	+	CCONJ
ejpam-5569	373	7	(	(	PUNCT
ejpam-5569	373	8	n)(k	n)(k	NOUN
ejpam-5569	373	9	)	)	PUNCT
ejpam-5569	373	10	+	+	CCONJ
ejpam-5569	373	11	2(k)(k	2(k)(k	NUM
ejpam-5569	373	12	−	−	PROPN
ejpam-5569	373	13	1)|	1)|	NUM
ejpam-5569	373	14	n.	n.	NOUN
ejpam-5569	373	15	m.	m.	NOUN
ejpam-5569	373	16	noureldeen	noureldeen	NOUN
ejpam-5569	373	17	et	et	PROPN
ejpam-5569	373	18	al	al	PROPN
ejpam-5569	373	19	.	.	PUNCT
ejpam-5569	373	20	/	/	SYM
ejpam-5569	373	21	eur	eur	PROPN
ejpam-5569	373	22	.	.	PUNCT
ejpam-5569	374	1	j.	j.	PROPN
ejpam-5569	374	2	pure	pure	PROPN
ejpam-5569	374	3	appl	appl	PROPN
ejpam-5569	374	4	.	.	PROPN
ejpam-5569	374	5	math	math	PROPN
ejpam-5569	374	6	,	,	PUNCT
ejpam-5569	374	7	18	18	NUM
ejpam-5569	374	8	(	(	PUNCT
ejpam-5569	374	9	2	2	NUM
ejpam-5569	374	10	)	)	PUNCT
ejpam-5569	374	11	(	(	PUNCT
ejpam-5569	374	12	2025	2025	NUM
ejpam-5569	374	13	)	)	PUNCT
ejpam-5569	374	14	,	,	PUNCT
ejpam-5569	374	15	5569	5569	NUM
ejpam-5569	374	16	13	13	NUM
ejpam-5569	374	17	of	of	ADP
ejpam-5569	374	18	25	25	NUM
ejpam-5569	374	19	≥	≥	NOUN
ejpam-5569	374	20	n	n	CCONJ
ejpam-5569	374	21	,	,	PUNCT
ejpam-5569	374	22	where	where	SCONJ
ejpam-5569	374	23	n	n	NOUN
ejpam-5569	374	24	=	=	SYM
ejpam-5569	374	25	4k	4k	X
ejpam-5569	374	26	+	+	NOUN
ejpam-5569	374	27	3	3	NUM
ejpam-5569	374	28	,	,	PUNCT
ejpam-5569	374	29	d(x	d(x	PROPN
ejpam-5569	374	30	,	,	PUNCT
ejpam-5569	374	31	y	y	PROPN
ejpam-5569	374	32	)	)	PUNCT
ejpam-5569	374	33	≥	≥	NOUN
ejpam-5569	374	34	1	1	NUM
ejpam-5569	374	35	≥	≥	NOUN
ejpam-5569	374	36	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	374	37	)	)	PUNCT
ejpam-5569	374	38	)	)	PUNCT
ejpam-5569	375	1	+	+	CCONJ
ejpam-5569	375	2	1−	1−	NUM
ejpam-5569	375	3	d(x	d(x	PROPN
ejpam-5569	375	4	,	,	PUNCT
ejpam-5569	375	5	y	y	NOUN
ejpam-5569	375	6	)	)	PUNCT
ejpam-5569	375	7	.	.	PUNCT
ejpam-5569	376	1	otherwise	otherwise	ADV
ejpam-5569	376	2	,	,	PUNCT
ejpam-5569	376	3	|f(x)−	|f(x)−	PROPN
ejpam-5569	376	4	f(y)|	f(y)|	PROPN
ejpam-5569	376	5	≥	≥	AUX
ejpam-5569	376	6	n.	n.	VERB
ejpam-5569	376	7	if	if	SCONJ
ejpam-5569	376	8	i	i	PRON
ejpam-5569	376	9	=	=	VERB
ejpam-5569	376	10	4k	4k	NUM
ejpam-5569	376	11	+	+	CCONJ
ejpam-5569	376	12	3	3	NUM
ejpam-5569	376	13	and	and	CCONJ
ejpam-5569	376	14	l	l	NOUN
ejpam-5569	376	15	=	=	SYM
ejpam-5569	377	1	2k	2k	NUM
ejpam-5569	377	2	+	+	CCONJ
ejpam-5569	377	3	2	2	NUM
ejpam-5569	377	4	then	then	ADV
ejpam-5569	377	5	|f(x)−	|f(x)−	NOUN
ejpam-5569	377	6	f(y)|	f(y)|	PROPN
ejpam-5569	377	7	=	=	NUM
ejpam-5569	378	1	|f(ui)−	|f(ui)−	ADJ
ejpam-5569	378	2	f(vl)|	f(vl)|	NOUN
ejpam-5569	378	3	=	=	SYM
ejpam-5569	378	4	|f(u4k+3)−	|f(u4k+3)−	PROPN
ejpam-5569	378	5	f(v2k+2)|	f(v2k+2)|	PROPN
ejpam-5569	378	6	=	=	SYM
ejpam-5569	378	7	|f(u1	|f(u1	NOUN
ejpam-5569	378	8	)	)	PUNCT
ejpam-5569	378	9	+	+	CCONJ
ejpam-5569	378	10	1−	1−	NUM
ejpam-5569	378	11	f(v2k+2)|	f(v2k+2)|	NOUN
ejpam-5569	378	12	=	=	PUNCT
ejpam-5569	378	13	|0	|0	NUM
ejpam-5569	378	14	+	+	SYM
ejpam-5569	378	15	1−	1−	NUM
ejpam-5569	378	16	[	[	X
ejpam-5569	378	17	n2	n2	NOUN
ejpam-5569	378	18	−	−	NUM
ejpam-5569	378	19	3]|	3]|	NUM
ejpam-5569	378	20	=	=	PUNCT
ejpam-5569	379	1	|	|	ADV
ejpam-5569	379	2	−	−	PROPN
ejpam-5569	380	1	[	[	X
ejpam-5569	380	2	n2	n2	NOUN
ejpam-5569	380	3	−	−	PROPN
ejpam-5569	380	4	3−	3−	NUM
ejpam-5569	380	5	1]|	1]|	NUM
ejpam-5569	381	1	=	=	PUNCT
ejpam-5569	381	2	|n2	|n2	X
ejpam-5569	381	3	−	−	PROPN
ejpam-5569	381	4	4|	4|	NUM
ejpam-5569	381	5	≥	≥	NOUN
ejpam-5569	381	6	n	n	CCONJ
ejpam-5569	381	7	,	,	PUNCT
ejpam-5569	381	8	where	where	SCONJ
ejpam-5569	381	9	n	n	NOUN
ejpam-5569	381	10	=	=	SYM
ejpam-5569	381	11	4k	4k	X
ejpam-5569	381	12	+	+	NOUN
ejpam-5569	381	13	3	3	NUM
ejpam-5569	381	14	,	,	PUNCT
ejpam-5569	381	15	k	k	X
ejpam-5569	381	16	≥	≥	NUM
ejpam-5569	381	17	1	1	NUM
ejpam-5569	381	18	,	,	PUNCT
ejpam-5569	381	19	d(x	d(x	PROPN
ejpam-5569	381	20	,	,	PUNCT
ejpam-5569	381	21	y	y	PROPN
ejpam-5569	381	22	)	)	PUNCT
ejpam-5569	381	23	≥	≥	NOUN
ejpam-5569	381	24	1	1	NUM
ejpam-5569	381	25	≥	≥	NOUN
ejpam-5569	381	26	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	381	27	)	)	PUNCT
ejpam-5569	381	28	)	)	PUNCT
ejpam-5569	382	1	+	+	CCONJ
ejpam-5569	382	2	1−	1−	NUM
ejpam-5569	382	3	d(x	d(x	PROPN
ejpam-5569	382	4	,	,	PUNCT
ejpam-5569	382	5	y	y	NOUN
ejpam-5569	382	6	)	)	PUNCT
ejpam-5569	382	7	.	.	PUNCT
ejpam-5569	383	1	otherwise	otherwise	ADV
ejpam-5569	383	2	,	,	PUNCT
ejpam-5569	383	3	|f(x	|f(x	PROPN
ejpam-5569	383	4	)	)	PUNCT
ejpam-5569	383	5	−	−	PROPN
ejpam-5569	384	1	f(y)|	f(y)|	PROPN
ejpam-5569	384	2	≥	≥	AUX
ejpam-5569	384	3	n.	n.	VERB
ejpam-5569	384	4	the	the	DET
ejpam-5569	384	5	radio	radio	NOUN
ejpam-5569	384	6	condition	condition	NOUN
ejpam-5569	384	7	is	be	AUX
ejpam-5569	384	8	|f(x	|f(x	PROPN
ejpam-5569	384	9	)	)	PUNCT
ejpam-5569	384	10	−	−	PROPN
ejpam-5569	385	1	f(y)|	f(y)|	PROPN
ejpam-5569	385	2	≥	≥	PRON
ejpam-5569	385	3	n	n	CCONJ
ejpam-5569	385	4	satisfied	satisfied	ADJ
ejpam-5569	385	5	by	by	ADP
ejpam-5569	385	6	every	every	DET
ejpam-5569	385	7	pair	pair	NOUN
ejpam-5569	385	8	of	of	ADP
ejpam-5569	385	9	vertices	vertex	NOUN
ejpam-5569	385	10	figure	figure	VERB
ejpam-5569	385	11	3	3	NUM
ejpam-5569	385	12	.	.	NUM
ejpam-5569	385	13	,	,	PUNCT
ejpam-5569	385	14	illustrates	illustrate	VERB
ejpam-5569	385	15	the	the	DET
ejpam-5569	385	16	proof	proof	NOUN
ejpam-5569	385	17	of	of	ADP
ejpam-5569	385	18	this	this	DET
ejpam-5569	385	19	case	case	NOUN
ejpam-5569	385	20	,	,	PUNCT
ejpam-5569	385	21	presents	present	VERB
ejpam-5569	385	22	the	the	DET
ejpam-5569	385	23	labeling	labeling	NOUN
ejpam-5569	385	24	d2(p15	d2(p15	PROPN
ejpam-5569	385	25	)	)	PUNCT
ejpam-5569	385	26	according	accord	VERB
ejpam-5569	385	27	theorem	theorem	ADJ
ejpam-5569	385	28	2	2	NUM
ejpam-5569	385	29	.	.	PUNCT
ejpam-5569	385	30	figure	figure	NOUN
ejpam-5569	385	31	3	3	NUM
ejpam-5569	385	32	:	:	PUNCT
ejpam-5569	385	33	radio	radio	NOUN
ejpam-5569	385	34	labeling	labeling	NOUN
ejpam-5569	385	35	of	of	ADP
ejpam-5569	385	36	d2(p15	d2(p15	PROPN
ejpam-5569	385	37	)	)	PUNCT
ejpam-5569	385	38	following	follow	VERB
ejpam-5569	385	39	theorem	theorem	VERB
ejpam-5569	385	40	2	2	NUM
ejpam-5569	385	41	.	.	PUNCT
ejpam-5569	385	42	theorem	theorem	NOUN
ejpam-5569	385	43	3	3	X
ejpam-5569	385	44	.	.	PUNCT
ejpam-5569	386	1	let	let	VERB
ejpam-5569	386	2	n	n	PRON
ejpam-5569	386	3	>	>	X
ejpam-5569	386	4	6	6	NUM
ejpam-5569	386	5	be	be	AUX
ejpam-5569	386	6	a	a	DET
ejpam-5569	386	7	positive	positive	ADJ
ejpam-5569	386	8	integer	integer	NOUN
ejpam-5569	386	9	such	such	ADJ
ejpam-5569	386	10	that	that	SCONJ
ejpam-5569	386	11	n	n	NUM
ejpam-5569	386	12	≡	≡	PROPN
ejpam-5569	386	13	0(mod4	0(mod4	NUM
ejpam-5569	386	14	)	)	PUNCT
ejpam-5569	386	15	.	.	PUNCT
ejpam-5569	387	1	then	then	ADV
ejpam-5569	387	2	rn(d2(pn	rn(d2(pn	PROPN
ejpam-5569	387	3	)	)	PUNCT
ejpam-5569	387	4	)	)	PUNCT
ejpam-5569	388	1	≤	≤	NUM
ejpam-5569	388	2	n2	n2	NOUN
ejpam-5569	388	3	−	−	PROPN
ejpam-5569	388	4	n	n	PRON
ejpam-5569	388	5	2	2	NUM
ejpam-5569	388	6	.	.	PUNCT
ejpam-5569	389	1	proof	proof	NOUN
ejpam-5569	389	2	.	.	PUNCT
ejpam-5569	390	1	for	for	ADP
ejpam-5569	390	2	any	any	DET
ejpam-5569	390	3	positive	positive	ADJ
ejpam-5569	390	4	integer	integer	NOUN
ejpam-5569	390	5	n	n	PRON
ejpam-5569	390	6	≡	≡	PROPN
ejpam-5569	390	7	0(mod4	0(mod4	NUM
ejpam-5569	390	8	)	)	PUNCT
ejpam-5569	390	9	,	,	PUNCT
ejpam-5569	390	10	such	such	ADJ
ejpam-5569	390	11	that	that	SCONJ
ejpam-5569	390	12	n	n	NOUN
ejpam-5569	390	13	=	=	SYM
ejpam-5569	390	14	4k	4k	NUM
ejpam-5569	390	15	,	,	PUNCT
ejpam-5569	390	16	k	k	PROPN
ejpam-5569	390	17	≥	≥	NUM
ejpam-5569	390	18	2	2	NUM
ejpam-5569	390	19	,	,	PUNCT
ejpam-5569	390	20	let	let	VERB
ejpam-5569	390	21	f	f	PRON
ejpam-5569	390	22	:	:	PUNCT
ejpam-5569	390	23	v	v	PROPN
ejpam-5569	390	24	(	(	PUNCT
ejpam-5569	390	25	d2(pn	d2(pn	PROPN
ejpam-5569	390	26	)	)	PUNCT
ejpam-5569	390	27	)	)	PUNCT
ejpam-5569	390	28	→	→	SYM
ejpam-5569	390	29	n	n	CCONJ
ejpam-5569	390	30	define	define	VERB
ejpam-5569	390	31	as	as	SCONJ
ejpam-5569	390	32	follows	follow	VERB
ejpam-5569	390	33	:	:	PUNCT
ejpam-5569	390	34	f(ui	f(ui	PROPN
ejpam-5569	390	35	)	)	PUNCT
ejpam-5569	391	1	=	=	SYM
ejpam-5569	391	2			PROPN
ejpam-5569	391	3	(	(	PUNCT
ejpam-5569	391	4	2n+	2n+	NUM
ejpam-5569	391	5	3)(i−	3)(i−	NUM
ejpam-5569	391	6	1	1	NUM
ejpam-5569	391	7	)	)	PUNCT
ejpam-5569	391	8	+	+	CCONJ
ejpam-5569	391	9	2(i−	2(i−	NUM
ejpam-5569	391	10	1)(i−	1)(i−	NUM
ejpam-5569	391	11	2	2	NUM
ejpam-5569	391	12	)	)	PUNCT
ejpam-5569	391	13	,	,	PUNCT
ejpam-5569	391	14	if	if	SCONJ
ejpam-5569	391	15	1	1	NUM
ejpam-5569	391	16	≤	≤	NUM
ejpam-5569	391	17	i	i	NOUN
ejpam-5569	391	18	≤	≤	NOUN
ejpam-5569	391	19	k	k	X
ejpam-5569	391	20	f(uk	f(uk	PROPN
ejpam-5569	391	21	)	)	PUNCT
ejpam-5569	391	22	+	+	PUNCT
ejpam-5569	391	23	2n−	2n−	NUM
ejpam-5569	391	24	3	3	NUM
ejpam-5569	391	25	if	if	SCONJ
ejpam-5569	391	26	i	i	PRON
ejpam-5569	391	27	=	=	PUNCT
ejpam-5569	391	28	k	k	PROPN
ejpam-5569	392	1	+	+	NOUN
ejpam-5569	392	2	1	1	NUM
ejpam-5569	392	3	f(uk)−	f(uk)−	NOUN
ejpam-5569	392	4	2n+	2n+	NUM
ejpam-5569	392	5	2−	2−	NUM
ejpam-5569	392	6	(	(	PUNCT
ejpam-5569	392	7	3n−	3n−	NUM
ejpam-5569	392	8	7)(j	7)(j	NUM
ejpam-5569	392	9	−	−	NOUN
ejpam-5569	392	10	1	1	NUM
ejpam-5569	392	11	)	)	PUNCT
ejpam-5569	392	12	+	+	CCONJ
ejpam-5569	392	13	2(j	2(j	NUM
ejpam-5569	392	14	−	−	PROPN
ejpam-5569	392	15	1)(j	1)(j	NUM
ejpam-5569	392	16	−	−	PROPN
ejpam-5569	392	17	2	2	NUM
ejpam-5569	392	18	)	)	PUNCT
ejpam-5569	392	19	,	,	PUNCT
ejpam-5569	392	20	if	if	SCONJ
ejpam-5569	392	21	k	k	PROPN
ejpam-5569	392	22	+	+	CCONJ
ejpam-5569	392	23	2	2	X
ejpam-5569	392	24	≤	≤	NUM
ejpam-5569	392	25	i	i	NOUN
ejpam-5569	392	26	≤	≤	ADJ
ejpam-5569	392	27	2k	2k	NUM
ejpam-5569	392	28	,	,	PUNCT
ejpam-5569	392	29	where	where	SCONJ
ejpam-5569	392	30	1	1	NUM
ejpam-5569	392	31	≤	≤	NUM
ejpam-5569	392	32	j	j	PROPN
ejpam-5569	392	33	≤	≤	PROPN
ejpam-5569	392	34	k	k	NOUN
ejpam-5569	393	1	−	−	PROPN
ejpam-5569	393	2	1	1	NUM
ejpam-5569	393	3	,	,	PUNCT
ejpam-5569	393	4	and	and	CCONJ
ejpam-5569	393	5	i	i	PRON
ejpam-5569	393	6	=	=	PUNCT
ejpam-5569	394	1	k	k	PROPN
ejpam-5569	395	1	+	+	CCONJ
ejpam-5569	395	2	1	1	NUM
ejpam-5569	395	3	+	+	CCONJ
ejpam-5569	395	4	j	j	PROPN
ejpam-5569	395	5	f(uk+1+j	f(uk+1+j	PROPN
ejpam-5569	395	6	)	)	PUNCT
ejpam-5569	396	1	+	+	CCONJ
ejpam-5569	396	2	(	(	PUNCT
ejpam-5569	396	3	6k	6k	NOUN
ejpam-5569	396	4	−	−	PROPN
ejpam-5569	396	5	3)−	3)−	NUM
ejpam-5569	396	6	2(j	2(j	NUM
ejpam-5569	396	7	−	−	NOUN
ejpam-5569	396	8	1	1	NUM
ejpam-5569	396	9	)	)	PUNCT
ejpam-5569	396	10	,	,	PUNCT
ejpam-5569	396	11	if	if	SCONJ
ejpam-5569	396	12	2k	2k	NUM
ejpam-5569	396	13	+	+	CCONJ
ejpam-5569	396	14	1	1	NUM
ejpam-5569	396	15	≤	≤	NUM
ejpam-5569	397	1	i	i	PRON
ejpam-5569	397	2	≤	≤	NOUN
ejpam-5569	397	3	3k	3k	NUM
ejpam-5569	397	4	−	−	NOUN
ejpam-5569	397	5	1	1	NUM
ejpam-5569	398	1	where	where	SCONJ
ejpam-5569	398	2	1	1	NUM
ejpam-5569	398	3	≤	≤	NUM
ejpam-5569	398	4	j	j	PROPN
ejpam-5569	398	5	≤	≤	PROPN
ejpam-5569	399	1	k	k	NOUN
ejpam-5569	400	1	−	−	PROPN
ejpam-5569	400	2	1	1	NUM
ejpam-5569	400	3	,	,	PUNCT
ejpam-5569	400	4	and	and	CCONJ
ejpam-5569	400	5	i	i	PRON
ejpam-5569	400	6	=	=	NOUN
ejpam-5569	401	1	3k	3k	PRON
ejpam-5569	401	2	−	−	PROPN
ejpam-5569	401	3	j	j	PROPN
ejpam-5569	401	4	f(uj	f(uj	PROPN
ejpam-5569	401	5	)	)	PUNCT
ejpam-5569	402	1	+	+	CCONJ
ejpam-5569	402	2	2j	2j	NUM
ejpam-5569	402	3	−	−	NOUN
ejpam-5569	402	4	1	1	NUM
ejpam-5569	402	5	,	,	PUNCT
ejpam-5569	402	6	then	then	ADV
ejpam-5569	402	7	3k	3k	X
ejpam-5569	402	8	≤	≤	X
ejpam-5569	403	1	i	i	PRON
ejpam-5569	403	2	≤	≤	NOUN
ejpam-5569	403	3	4k	4k	NUM
ejpam-5569	403	4	=	=	SYM
ejpam-5569	403	5	n	n	CCONJ
ejpam-5569	403	6	where	where	SCONJ
ejpam-5569	403	7	1	1	NUM
ejpam-5569	403	8	≤	≤	NUM
ejpam-5569	403	9	j	j	PROPN
ejpam-5569	403	10	≤	≤	PROPN
ejpam-5569	404	1	k	k	PROPN
ejpam-5569	405	1	+	+	CCONJ
ejpam-5569	405	2	1	1	NUM
ejpam-5569	405	3	and	and	CCONJ
ejpam-5569	405	4	i	i	PRON
ejpam-5569	405	5	=	=	PUNCT
ejpam-5569	405	6	4k	4k	PRON
ejpam-5569	405	7	+	+	NOUN
ejpam-5569	406	1	1−	1−	NUM
ejpam-5569	406	2	j	j	PROPN
ejpam-5569	406	3	n.	n.	PROPN
ejpam-5569	406	4	m.	m.	PROPN
ejpam-5569	406	5	noureldeen	noureldeen	INTJ
ejpam-5569	406	6	et	et	PROPN
ejpam-5569	406	7	al	al	PROPN
ejpam-5569	406	8	.	.	PUNCT
ejpam-5569	406	9	/	/	SYM
ejpam-5569	406	10	eur	eur	PROPN
ejpam-5569	406	11	.	.	PUNCT
ejpam-5569	407	1	j.	j.	PROPN
ejpam-5569	407	2	pure	pure	PROPN
ejpam-5569	407	3	appl	appl	PROPN
ejpam-5569	407	4	.	.	PROPN
ejpam-5569	407	5	math	math	PROPN
ejpam-5569	407	6	,	,	PUNCT
ejpam-5569	407	7	18	18	NUM
ejpam-5569	407	8	(	(	PUNCT
ejpam-5569	407	9	2	2	NUM
ejpam-5569	407	10	)	)	PUNCT
ejpam-5569	407	11	(	(	PUNCT
ejpam-5569	407	12	2025	2025	NUM
ejpam-5569	407	13	)	)	PUNCT
ejpam-5569	407	14	,	,	PUNCT
ejpam-5569	407	15	5569	5569	NUM
ejpam-5569	407	16	14	14	NUM
ejpam-5569	407	17	of	of	ADP
ejpam-5569	407	18	25	25	NUM
ejpam-5569	407	19	and	and	CCONJ
ejpam-5569	407	20	for	for	ADP
ejpam-5569	407	21	the	the	DET
ejpam-5569	407	22	vertices	vertex	NOUN
ejpam-5569	407	23	belongs	belong	VERB
ejpam-5569	407	24	to	to	ADP
ejpam-5569	407	25	vv	vv	NOUN
ejpam-5569	407	26	,	,	PUNCT
ejpam-5569	407	27	f(vi	f(vi	NOUN
ejpam-5569	407	28	)	)	PUNCT
ejpam-5569	407	29	=	=	SYM
ejpam-5569	407	30			X
ejpam-5569	407	31	(	(	PUNCT
ejpam-5569	407	32	n+	n+	NOUN
ejpam-5569	407	33	2	2	NUM
ejpam-5569	407	34	)	)	PUNCT
ejpam-5569	407	35	+	+	CCONJ
ejpam-5569	407	36	(	(	PUNCT
ejpam-5569	407	37	2n+	2n+	NUM
ejpam-5569	407	38	7)(i−	7)(i−	NUM
ejpam-5569	407	39	1	1	NUM
ejpam-5569	407	40	)	)	PUNCT
ejpam-5569	407	41	+	+	CCONJ
ejpam-5569	407	42	2(i−	2(i−	NUM
ejpam-5569	407	43	1)(i−	1)(i−	NUM
ejpam-5569	407	44	2	2	NUM
ejpam-5569	407	45	)	)	PUNCT
ejpam-5569	407	46	,	,	PUNCT
ejpam-5569	407	47	if	if	SCONJ
ejpam-5569	407	48	1	1	NUM
ejpam-5569	407	49	≤	≤	NUM
ejpam-5569	407	50	i	i	NOUN
ejpam-5569	407	51	≤	≤	NOUN
ejpam-5569	408	1	k	k	PRON
ejpam-5569	408	2	−	−	PROPN
ejpam-5569	408	3	1	1	NUM
ejpam-5569	408	4	f(vk−1	f(vk−1	NOUN
ejpam-5569	408	5	)	)	PUNCT
ejpam-5569	409	1	+	+	CCONJ
ejpam-5569	409	2	2n−	2n−	NUM
ejpam-5569	409	3	1	1	NUM
ejpam-5569	409	4	if	if	SCONJ
ejpam-5569	409	5	i	i	PRON
ejpam-5569	409	6	=	=	SYM
ejpam-5569	409	7	k	k	X
ejpam-5569	409	8	f(vk	f(vk	PROPN
ejpam-5569	409	9	)	)	PUNCT
ejpam-5569	409	10	+	+	CCONJ
ejpam-5569	409	11	(	(	PUNCT
ejpam-5569	409	12	2n+	2n+	NUM
ejpam-5569	409	13	1	1	NUM
ejpam-5569	409	14	)	)	PUNCT
ejpam-5569	409	15	+	+	CCONJ
ejpam-5569	409	16	(	(	PUNCT
ejpam-5569	409	17	n+	n+	NUM
ejpam-5569	409	18	3)(j	3)(j	NUM
ejpam-5569	409	19	−	−	NOUN
ejpam-5569	409	20	1	1	NUM
ejpam-5569	409	21	)	)	PUNCT
ejpam-5569	409	22	+	+	CCONJ
ejpam-5569	409	23	2(j	2(j	NUM
ejpam-5569	410	1	−	−	PROPN
ejpam-5569	410	2	1)(j	1)(j	NUM
ejpam-5569	410	3	−	−	PROPN
ejpam-5569	410	4	2	2	NUM
ejpam-5569	410	5	)	)	PUNCT
ejpam-5569	410	6	,	,	PUNCT
ejpam-5569	410	7	if	if	SCONJ
ejpam-5569	410	8	k	k	PROPN
ejpam-5569	410	9	≤	≤	X
ejpam-5569	411	1	i	i	NOUN
ejpam-5569	411	2	≤	≤	ADJ
ejpam-5569	411	3	2k	2k	NUM
ejpam-5569	411	4	where	where	SCONJ
ejpam-5569	411	5	1	1	NUM
ejpam-5569	411	6	≤	≤	NUM
ejpam-5569	411	7	j	j	PROPN
ejpam-5569	411	8	≤	≤	PROPN
ejpam-5569	411	9	k	k	PROPN
ejpam-5569	411	10	,	,	PUNCT
ejpam-5569	411	11	and	and	CCONJ
ejpam-5569	411	12	i	i	PRON
ejpam-5569	411	13	=	=	PUNCT
ejpam-5569	411	14	k	k	PROPN
ejpam-5569	412	1	+	+	CCONJ
ejpam-5569	412	2	j	j	PROPN
ejpam-5569	412	3	f(vk−1+j	f(vk−1+j	NOUN
ejpam-5569	412	4	)	)	PUNCT
ejpam-5569	412	5	+	+	CCONJ
ejpam-5569	412	6	(	(	PUNCT
ejpam-5569	412	7	2k	2k	NOUN
ejpam-5569	412	8	−	−	NOUN
ejpam-5569	412	9	1	1	NUM
ejpam-5569	412	10	)	)	PUNCT
ejpam-5569	413	1	+	+	CCONJ
ejpam-5569	413	2	2(j	2(j	NUM
ejpam-5569	413	3	−	−	NOUN
ejpam-5569	413	4	1	1	NUM
ejpam-5569	413	5	)	)	PUNCT
ejpam-5569	413	6	,	,	PUNCT
ejpam-5569	413	7	if	if	SCONJ
ejpam-5569	413	8	2k	2k	NUM
ejpam-5569	413	9	+	+	CCONJ
ejpam-5569	413	10	1	1	NUM
ejpam-5569	413	11	≤	≤	NUM
ejpam-5569	413	12	i	i	PRON
ejpam-5569	413	13	≤	≤	NOUN
ejpam-5569	413	14	3k	3k	X
ejpam-5569	413	15	+	+	CCONJ
ejpam-5569	413	16	1	1	NUM
ejpam-5569	414	1	where	where	SCONJ
ejpam-5569	414	2	1	1	NUM
ejpam-5569	414	3	≤	≤	NUM
ejpam-5569	414	4	j	j	NOUN
ejpam-5569	414	5	≤	≤	PROPN
ejpam-5569	414	6	k	k	PROPN
ejpam-5569	415	1	+	+	PROPN
ejpam-5569	416	1	1and	1and	NUM
ejpam-5569	416	2	i	i	NOUN
ejpam-5569	416	3	=	=	PUNCT
ejpam-5569	417	1	3k	3k	NUM
ejpam-5569	417	2	+	+	CCONJ
ejpam-5569	417	3	2−	2−	NUM
ejpam-5569	417	4	j	j	PROPN
ejpam-5569	417	5	f(vj)−	f(vj)−	NOUN
ejpam-5569	417	6	(	(	PUNCT
ejpam-5569	417	7	2j	2j	NUM
ejpam-5569	417	8	−	−	NOUN
ejpam-5569	417	9	1	1	NUM
ejpam-5569	417	10	)	)	PUNCT
ejpam-5569	417	11	,	,	PUNCT
ejpam-5569	417	12	if	if	SCONJ
ejpam-5569	417	13	3k	3k	PRON
ejpam-5569	417	14	+	+	CCONJ
ejpam-5569	417	15	2	2	NUM
ejpam-5569	417	16	≤	≤	NUM
ejpam-5569	417	17	i	i	PRON
ejpam-5569	417	18	≤	≤	NOUN
ejpam-5569	417	19	4k	4k	NUM
ejpam-5569	417	20	=	=	SYM
ejpam-5569	417	21	n	n	CCONJ
ejpam-5569	417	22	where	where	SCONJ
ejpam-5569	417	23	1	1	NUM
ejpam-5569	417	24	≤	≤	NUM
ejpam-5569	417	25	j	j	PROPN
ejpam-5569	417	26	≤	≤	PROPN
ejpam-5569	418	1	k	k	NOUN
ejpam-5569	419	1	−	−	PROPN
ejpam-5569	419	2	1	1	NUM
ejpam-5569	419	3	,	,	PUNCT
ejpam-5569	419	4	and	and	CCONJ
ejpam-5569	419	5	i	i	PRON
ejpam-5569	419	6	=	=	SYM
ejpam-5569	419	7	4k	4k	PRON
ejpam-5569	419	8	+	+	SYM
ejpam-5569	419	9	1−	1−	NUM
ejpam-5569	419	10	j	j	PROPN
ejpam-5569	419	11	hereafter	hereafter	ADV
ejpam-5569	419	12	,	,	PUNCT
ejpam-5569	419	13	we	we	PRON
ejpam-5569	419	14	show	show	VERB
ejpam-5569	419	15	that	that	SCONJ
ejpam-5569	419	16	the	the	DET
ejpam-5569	419	17	function	function	NOUN
ejpam-5569	419	18	f	f	PROPN
ejpam-5569	419	19	satisfies	satisfy	VERB
ejpam-5569	419	20	the	the	DET
ejpam-5569	419	21	radio	radio	NOUN
ejpam-5569	419	22	labeling	labeling	NOUN
ejpam-5569	419	23	condition	condition	NOUN
ejpam-5569	419	24	for	for	ADP
ejpam-5569	419	25	any	any	DET
ejpam-5569	419	26	two	two	NUM
ejpam-5569	419	27	distinct	distinct	ADJ
ejpam-5569	419	28	vertices	vertex	NOUN
ejpam-5569	419	29	x	x	X
ejpam-5569	419	30	,	,	PUNCT
ejpam-5569	419	31	y	y	PROPN
ejpam-5569	419	32	of	of	ADP
ejpam-5569	419	33	d2(pn	d2(pn	PROPN
ejpam-5569	419	34	)	)	PUNCT
ejpam-5569	419	35	for	for	ADP
ejpam-5569	419	36	n	n	PROPN
ejpam-5569	419	37	>	>	X
ejpam-5569	419	38	6	6	NUM
ejpam-5569	419	39	,	,	PUNCT
ejpam-5569	419	40	n	n	NOUN
ejpam-5569	419	41	=	=	SYM
ejpam-5569	419	42	4k	4k	NUM
ejpam-5569	419	43	where	where	SCONJ
ejpam-5569	419	44	k	k	PROPN
ejpam-5569	419	45	≥	≥	NUM
ejpam-5569	419	46	2	2	NUM
ejpam-5569	419	47	.	.	PUNCT
ejpam-5569	419	48	case	case	NOUN
ejpam-5569	419	49	1	1	NUM
ejpam-5569	419	50	x	x	NOUN
ejpam-5569	419	51	,	,	PUNCT
ejpam-5569	419	52	y	y	PROPN
ejpam-5569	419	53	∈	∈	PROPN
ejpam-5569	419	54	vu	vu	X
ejpam-5569	419	55	.	.	PUNCT
ejpam-5569	420	1	if	if	SCONJ
ejpam-5569	420	2	{	{	PUNCT
ejpam-5569	420	3	x	x	NOUN
ejpam-5569	420	4	,	,	PUNCT
ejpam-5569	420	5	y	y	PROPN
ejpam-5569	420	6	}	}	PUNCT
ejpam-5569	420	7	=	=	SYM
ejpam-5569	420	8	{	{	PUNCT
ejpam-5569	420	9	u1	u1	NOUN
ejpam-5569	420	10	,	,	PUNCT
ejpam-5569	420	11	un	un	ADJ
ejpam-5569	420	12	}	}	PUNCT
ejpam-5569	420	13	then	then	ADV
ejpam-5569	420	14	|f(x)−	|f(x)−	NOUN
ejpam-5569	420	15	f(y)|	f(y)|	PROPN
ejpam-5569	420	16	=	=	SYM
ejpam-5569	420	17	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	420	18	f(un)|	f(un)|	PROPN
ejpam-5569	421	1	=	=	SYM
ejpam-5569	421	2	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	422	1	[	[	X
ejpam-5569	422	2	f(u1	f(u1	NOUN
ejpam-5569	422	3	)	)	PUNCT
ejpam-5569	423	1	+	+	CCONJ
ejpam-5569	423	2	2−	2−	NUM
ejpam-5569	423	3	1]|	1]|	NUM
ejpam-5569	423	4	=	=	SYM
ejpam-5569	423	5	1	1	NUM
ejpam-5569	423	6	≥	≥	NOUN
ejpam-5569	423	7	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	423	8	)	)	PUNCT
ejpam-5569	423	9	)	)	PUNCT
ejpam-5569	424	1	+	+	CCONJ
ejpam-5569	424	2	1−	1−	NUM
ejpam-5569	424	3	d(x	d(x	PROPN
ejpam-5569	424	4	,	,	PUNCT
ejpam-5569	424	5	y	y	NOUN
ejpam-5569	424	6	)	)	PUNCT
ejpam-5569	424	7	,	,	PUNCT
ejpam-5569	424	8	where	where	SCONJ
ejpam-5569	424	9	d(x	d(x	PROPN
ejpam-5569	424	10	,	,	PUNCT
ejpam-5569	424	11	y	y	NOUN
ejpam-5569	424	12	)	)	PUNCT
ejpam-5569	424	13	=	=	VERB
ejpam-5569	424	14	n−	n−	NOUN
ejpam-5569	424	15	1	1	NUM
ejpam-5569	424	16	.	.	PUNCT
ejpam-5569	424	17	case	case	NOUN
ejpam-5569	424	18	2	2	NUM
ejpam-5569	424	19	if	if	SCONJ
ejpam-5569	424	20	{	{	PUNCT
ejpam-5569	424	21	x	x	NOUN
ejpam-5569	424	22	,	,	PUNCT
ejpam-5569	424	23	y	y	PROPN
ejpam-5569	424	24	}	}	PUNCT
ejpam-5569	424	25	=	=	SYM
ejpam-5569	424	26	{	{	PUNCT
ejpam-5569	424	27	u1	u1	NOUN
ejpam-5569	424	28	,	,	PUNCT
ejpam-5569	424	29	uk	uk	PROPN
ejpam-5569	424	30	}	}	PUNCT
ejpam-5569	424	31	then	then	ADV
ejpam-5569	424	32	|f(x)−	|f(x)−	NOUN
ejpam-5569	424	33	f(y)|	f(y)|	PROPN
ejpam-5569	424	34	=	=	PUNCT
ejpam-5569	424	35	|f(u1)−	|f(u1)−	ADJ
ejpam-5569	424	36	f(uk)|	f(uk)|	NOUN
ejpam-5569	425	1	=	=	PUNCT
ejpam-5569	425	2	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	426	1	[	[	X
ejpam-5569	426	2	(	(	PUNCT
ejpam-5569	426	3	2n+	2n+	NUM
ejpam-5569	426	4	3)(k	3)(k	NUM
ejpam-5569	426	5	−	−	NOUN
ejpam-5569	426	6	1	1	NUM
ejpam-5569	426	7	)	)	PUNCT
ejpam-5569	426	8	+	+	CCONJ
ejpam-5569	427	1	2(k	2(k	NUM
ejpam-5569	427	2	−	−	NOUN
ejpam-5569	427	3	1)(k	1)(k	NUM
ejpam-5569	427	4	−	−	PROPN
ejpam-5569	427	5	2)]|	2)]|	NOUN
ejpam-5569	427	6	=	=	NOUN
ejpam-5569	428	1	|	|	ADV
ejpam-5569	428	2	−	−	PROPN
ejpam-5569	429	1	[	[	X
ejpam-5569	429	2	(	(	PUNCT
ejpam-5569	429	3	2n+	2n+	NUM
ejpam-5569	429	4	3)(k	3)(k	NUM
ejpam-5569	429	5	−	−	NOUN
ejpam-5569	429	6	1	1	NUM
ejpam-5569	429	7	)	)	PUNCT
ejpam-5569	429	8	+	+	CCONJ
ejpam-5569	430	1	2(k	2(k	NUM
ejpam-5569	430	2	−	−	NOUN
ejpam-5569	430	3	1)(k	1)(k	NUM
ejpam-5569	430	4	−	−	NOUN
ejpam-5569	430	5	2)]|	2)]|	NOUN
ejpam-5569	430	6	=	=	SYM
ejpam-5569	430	7	(	(	PUNCT
ejpam-5569	430	8	2n+	2n+	NUM
ejpam-5569	430	9	3)(k	3)(k	NUM
ejpam-5569	430	10	−	−	NOUN
ejpam-5569	430	11	1	1	NUM
ejpam-5569	430	12	)	)	PUNCT
ejpam-5569	430	13	+	+	CCONJ
ejpam-5569	430	14	2(k	2(k	NUM
ejpam-5569	430	15	−	−	NOUN
ejpam-5569	430	16	1)(k	1)(k	NUM
ejpam-5569	430	17	−	−	PROPN
ejpam-5569	430	18	2	2	NUM
ejpam-5569	430	19	)	)	PUNCT
ejpam-5569	430	20	≥	≥	NOUN
ejpam-5569	430	21	n	n	CCONJ
ejpam-5569	430	22	,	,	PUNCT
ejpam-5569	430	23	where	where	SCONJ
ejpam-5569	430	24	n	n	NOUN
ejpam-5569	430	25	=	=	SYM
ejpam-5569	430	26	4k	4k	X
ejpam-5569	430	27	≥	≥	NOUN
ejpam-5569	430	28	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	430	29	)	)	PUNCT
ejpam-5569	430	30	)	)	PUNCT
ejpam-5569	431	1	+	+	CCONJ
ejpam-5569	431	2	1−	1−	NUM
ejpam-5569	431	3	d(x	d(x	PROPN
ejpam-5569	431	4	,	,	PUNCT
ejpam-5569	431	5	y	y	NOUN
ejpam-5569	431	6	)	)	PUNCT
ejpam-5569	431	7	case	case	NOUN
ejpam-5569	431	8	3	3	NUM
ejpam-5569	431	9	if	if	SCONJ
ejpam-5569	431	10	{	{	PUNCT
ejpam-5569	431	11	x	x	NOUN
ejpam-5569	431	12	,	,	PUNCT
ejpam-5569	431	13	y	y	PROPN
ejpam-5569	431	14	}	}	PUNCT
ejpam-5569	431	15	=	=	SYM
ejpam-5569	431	16	{	{	PUNCT
ejpam-5569	431	17	u1	u1	NOUN
ejpam-5569	431	18	,	,	PUNCT
ejpam-5569	431	19	uk+1	uk+1	VERB
ejpam-5569	431	20	}	}	PUNCT
ejpam-5569	431	21	then	then	ADV
ejpam-5569	431	22	|f(x)−	|f(x)−	NOUN
ejpam-5569	431	23	f(y)|	f(y)|	PROPN
ejpam-5569	431	24	=	=	SYM
ejpam-5569	431	25	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	432	1	f(uk+1)|	f(uk+1)|	X
ejpam-5569	432	2	=	=	SYM
ejpam-5569	432	3	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	433	1	[	[	X
ejpam-5569	433	2	f(uk	f(uk	PROPN
ejpam-5569	433	3	)	)	PUNCT
ejpam-5569	433	4	+	+	PUNCT
ejpam-5569	434	1	2n−	2n−	NUM
ejpam-5569	434	2	3]|	3]|	NUM
ejpam-5569	434	3	=	=	SYM
ejpam-5569	434	4	|[f(uk	|[f(uk	PROPN
ejpam-5569	434	5	)	)	PUNCT
ejpam-5569	435	1	+	+	CCONJ
ejpam-5569	436	1	2n−	2n−	NUM
ejpam-5569	436	2	3]|	3]|	NUM
ejpam-5569	436	3	=	=	PUNCT
ejpam-5569	436	4	|(2n+	|(2n+	PROPN
ejpam-5569	436	5	3)(k	3)(k	NUM
ejpam-5569	436	6	−	−	NOUN
ejpam-5569	436	7	1	1	NUM
ejpam-5569	436	8	)	)	PUNCT
ejpam-5569	436	9	+	+	CCONJ
ejpam-5569	436	10	2(k	2(k	NUM
ejpam-5569	436	11	−	−	NOUN
ejpam-5569	436	12	1)(k	1)(k	NUM
ejpam-5569	436	13	−	−	PROPN
ejpam-5569	436	14	2	2	NUM
ejpam-5569	436	15	)	)	PUNCT
ejpam-5569	436	16	+	+	CCONJ
ejpam-5569	437	1	2n−	2n−	PROPN
ejpam-5569	437	2	3|	3|	NUM
ejpam-5569	437	3	≥	≥	NOUN
ejpam-5569	437	4	n	n	CCONJ
ejpam-5569	437	5	,	,	PUNCT
ejpam-5569	437	6	where	where	SCONJ
ejpam-5569	437	7	n	n	NOUN
ejpam-5569	437	8	=	=	SYM
ejpam-5569	437	9	4k	4k	NOUN
ejpam-5569	437	10	,	,	PUNCT
ejpam-5569	437	11	d(x	d(x	PROPN
ejpam-5569	437	12	,	,	PUNCT
ejpam-5569	437	13	y	y	PROPN
ejpam-5569	437	14	)	)	PUNCT
ejpam-5569	437	15	≥	≥	NOUN
ejpam-5569	437	16	1	1	NUM
ejpam-5569	437	17	≥	≥	NOUN
ejpam-5569	437	18	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	437	19	)	)	PUNCT
ejpam-5569	437	20	)	)	PUNCT
ejpam-5569	438	1	+	+	CCONJ
ejpam-5569	438	2	1−	1−	NUM
ejpam-5569	438	3	d(x	d(x	PROPN
ejpam-5569	438	4	,	,	PUNCT
ejpam-5569	438	5	y	y	NOUN
ejpam-5569	438	6	)	)	PUNCT
ejpam-5569	438	7	case	case	NOUN
ejpam-5569	438	8	4	4	NUM
ejpam-5569	438	9	if	if	SCONJ
ejpam-5569	438	10	{	{	PUNCT
ejpam-5569	438	11	x	x	NOUN
ejpam-5569	438	12	,	,	PUNCT
ejpam-5569	438	13	y	y	PROPN
ejpam-5569	438	14	}	}	PUNCT
ejpam-5569	438	15	=	=	SYM
ejpam-5569	438	16	{	{	PUNCT
ejpam-5569	438	17	u1	u1	NOUN
ejpam-5569	438	18	,	,	PUNCT
ejpam-5569	438	19	u2k	u2k	PROPN
ejpam-5569	438	20	}	}	PUNCT
ejpam-5569	438	21	then	then	ADV
ejpam-5569	438	22	|f(x)−	|f(x)−	NOUN
ejpam-5569	438	23	f(y)|	f(y)|	PROPN
ejpam-5569	438	24	=	=	SYM
ejpam-5569	438	25	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	438	26	f(u2k)|	f(u2k)|	NOUN
ejpam-5569	438	27	=	=	PUNCT
ejpam-5569	439	1	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	440	1	[	[	X
ejpam-5569	440	2	f(uk)−	f(uk)−	NOUN
ejpam-5569	440	3	2n+	2n+	NUM
ejpam-5569	440	4	2−	2−	NUM
ejpam-5569	440	5	(	(	PUNCT
ejpam-5569	440	6	3n−	3n−	NUM
ejpam-5569	440	7	7)(k	7)(k	NUM
ejpam-5569	440	8	−	−	NOUN
ejpam-5569	440	9	2	2	NUM
ejpam-5569	440	10	)	)	PUNCT
ejpam-5569	440	11	+	+	CCONJ
ejpam-5569	440	12	2(k	2(k	NUM
ejpam-5569	441	1	−	−	NOUN
ejpam-5569	441	2	2)(k	2)(k	NUM
ejpam-5569	441	3	−	−	NOUN
ejpam-5569	442	1	3)]|	3)]|	ADJ
ejpam-5569	442	2	=	=	NOUN
ejpam-5569	442	3	|f(uk)−	|f(uk)−	ADJ
ejpam-5569	442	4	2n+	2n+	NUM
ejpam-5569	442	5	2−	2−	NUM
ejpam-5569	442	6	(	(	PUNCT
ejpam-5569	442	7	3n−	3n−	NUM
ejpam-5569	442	8	7)(k	7)(k	NUM
ejpam-5569	442	9	−	−	NOUN
ejpam-5569	442	10	2	2	NUM
ejpam-5569	442	11	)	)	PUNCT
ejpam-5569	442	12	+	+	CCONJ
ejpam-5569	442	13	2(k	2(k	NUM
ejpam-5569	443	1	−	−	NOUN
ejpam-5569	443	2	2)(k	2)(k	NUM
ejpam-5569	443	3	−	−	PROPN
ejpam-5569	444	1	3)|	3)|	NUM
ejpam-5569	444	2	=	=	PUNCT
ejpam-5569	445	1	|(2n+	|(2n+	PROPN
ejpam-5569	446	1	3)(k	3)(k	NUM
ejpam-5569	446	2	−	−	NOUN
ejpam-5569	446	3	1	1	NUM
ejpam-5569	446	4	)	)	PUNCT
ejpam-5569	447	1	+	+	CCONJ
ejpam-5569	448	1	2(k	2(k	NUM
ejpam-5569	448	2	−	−	NOUN
ejpam-5569	448	3	1)(k	1)(k	NUM
ejpam-5569	448	4	−	−	PROPN
ejpam-5569	448	5	2)−	2)−	NUM
ejpam-5569	448	6	2n+	2n+	NUM
ejpam-5569	448	7	2−	2−	NUM
ejpam-5569	448	8	(	(	PUNCT
ejpam-5569	448	9	3n−	3n−	NUM
ejpam-5569	448	10	7)(k	7)(k	NUM
ejpam-5569	448	11	−	−	PROPN
ejpam-5569	448	12	2)+	2)+	NUM
ejpam-5569	448	13	n.	n.	NOUN
ejpam-5569	448	14	m.	m.	NOUN
ejpam-5569	448	15	noureldeen	noureldeen	NOUN
ejpam-5569	448	16	et	et	PROPN
ejpam-5569	448	17	al	al	PROPN
ejpam-5569	448	18	.	.	PUNCT
ejpam-5569	448	19	/	/	SYM
ejpam-5569	448	20	eur	eur	PROPN
ejpam-5569	448	21	.	.	PUNCT
ejpam-5569	449	1	j.	j.	PROPN
ejpam-5569	449	2	pure	pure	PROPN
ejpam-5569	449	3	appl	appl	PROPN
ejpam-5569	449	4	.	.	PROPN
ejpam-5569	449	5	math	math	PROPN
ejpam-5569	449	6	,	,	PUNCT
ejpam-5569	449	7	18	18	NUM
ejpam-5569	449	8	(	(	PUNCT
ejpam-5569	449	9	2	2	NUM
ejpam-5569	449	10	)	)	PUNCT
ejpam-5569	449	11	(	(	PUNCT
ejpam-5569	449	12	2025	2025	NUM
ejpam-5569	449	13	)	)	PUNCT
ejpam-5569	449	14	,	,	PUNCT
ejpam-5569	449	15	5569	5569	NUM
ejpam-5569	449	16	15	15	NUM
ejpam-5569	449	17	of	of	ADP
ejpam-5569	449	18	25	25	NUM
ejpam-5569	449	19	+	+	CCONJ
ejpam-5569	449	20	2(k	2(k	NUM
ejpam-5569	450	1	−	−	NOUN
ejpam-5569	450	2	2)(k	2)(k	NUM
ejpam-5569	450	3	−	−	PROPN
ejpam-5569	450	4	3)|	3)|	NUM
ejpam-5569	450	5	≥	≥	NOUN
ejpam-5569	450	6	n	n	CCONJ
ejpam-5569	450	7	,	,	PUNCT
ejpam-5569	450	8	where	where	SCONJ
ejpam-5569	450	9	n	n	NOUN
ejpam-5569	450	10	=	=	SYM
ejpam-5569	450	11	4k	4k	NOUN
ejpam-5569	450	12	,	,	PUNCT
ejpam-5569	450	13	d(x	d(x	PROPN
ejpam-5569	450	14	,	,	PUNCT
ejpam-5569	450	15	y	y	PROPN
ejpam-5569	450	16	)	)	PUNCT
ejpam-5569	450	17	≥	≥	NOUN
ejpam-5569	450	18	1	1	NUM
ejpam-5569	450	19	≥	≥	NOUN
ejpam-5569	450	20	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	450	21	)	)	PUNCT
ejpam-5569	450	22	)	)	PUNCT
ejpam-5569	451	1	+	+	CCONJ
ejpam-5569	451	2	1−	1−	NUM
ejpam-5569	451	3	d(x	d(x	PROPN
ejpam-5569	451	4	,	,	PUNCT
ejpam-5569	451	5	y	y	NOUN
ejpam-5569	451	6	)	)	PUNCT
ejpam-5569	451	7	case	case	NOUN
ejpam-5569	451	8	5	5	NUM
ejpam-5569	451	9	if	if	SCONJ
ejpam-5569	451	10	{	{	PUNCT
ejpam-5569	451	11	x	x	NOUN
ejpam-5569	451	12	,	,	PUNCT
ejpam-5569	451	13	y	y	PROPN
ejpam-5569	451	14	}	}	PUNCT
ejpam-5569	451	15	=	=	SYM
ejpam-5569	451	16	{	{	PUNCT
ejpam-5569	451	17	u1	u1	NOUN
ejpam-5569	451	18	,	,	PUNCT
ejpam-5569	451	19	u3k+1	u3k+1	NOUN
ejpam-5569	451	20	}	}	PUNCT
ejpam-5569	451	21	then	then	ADV
ejpam-5569	451	22	|f(x)−	|f(x)−	NOUN
ejpam-5569	451	23	f(y)|	f(y)|	PROPN
ejpam-5569	451	24	=	=	PUNCT
ejpam-5569	451	25	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	452	1	f(u3k−1)|	f(u3k−1)|	NOUN
ejpam-5569	452	2	=	=	SYM
ejpam-5569	452	3	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	453	1	[	[	X
ejpam-5569	453	2	f(uk+2	f(uk+2	NOUN
ejpam-5569	453	3	)	)	PUNCT
ejpam-5569	453	4	+	+	CCONJ
ejpam-5569	453	5	(	(	PUNCT
ejpam-5569	453	6	6k	6k	NOUN
ejpam-5569	453	7	−	−	NOUN
ejpam-5569	453	8	3)]|	3)]|	NOUN
ejpam-5569	453	9	=	=	SYM
ejpam-5569	453	10	|f(uk+2	|f(uk+2	NOUN
ejpam-5569	453	11	)	)	PUNCT
ejpam-5569	454	1	+	+	CCONJ
ejpam-5569	454	2	(	(	PUNCT
ejpam-5569	454	3	6k	6k	NOUN
ejpam-5569	454	4	−	−	PROPN
ejpam-5569	454	5	3)|	3)|	NUM
ejpam-5569	454	6	=	=	PUNCT
ejpam-5569	454	7	|f(uk)−	|f(uk)−	ADJ
ejpam-5569	454	8	2n+	2n+	NUM
ejpam-5569	454	9	2	2	NUM
ejpam-5569	454	10	+	+	CCONJ
ejpam-5569	454	11	(	(	PUNCT
ejpam-5569	454	12	6k	6k	NOUN
ejpam-5569	454	13	−	−	PROPN
ejpam-5569	454	14	3)|	3)|	NUM
ejpam-5569	454	15	=	=	PUNCT
ejpam-5569	454	16	|(2n+	|(2n+	PROPN
ejpam-5569	455	1	3)(k	3)(k	NUM
ejpam-5569	455	2	−	−	NOUN
ejpam-5569	455	3	1	1	NUM
ejpam-5569	455	4	)	)	PUNCT
ejpam-5569	456	1	+	+	CCONJ
ejpam-5569	457	1	2(k	2(k	NUM
ejpam-5569	457	2	−	−	NOUN
ejpam-5569	457	3	1)(k	1)(k	NUM
ejpam-5569	457	4	−	−	PROPN
ejpam-5569	457	5	2)−	2)−	NUM
ejpam-5569	457	6	2n+	2n+	NUM
ejpam-5569	457	7	2	2	NUM
ejpam-5569	457	8	+	+	CCONJ
ejpam-5569	457	9	(	(	PUNCT
ejpam-5569	457	10	6k	6k	NOUN
ejpam-5569	457	11	−	−	PROPN
ejpam-5569	457	12	3)|	3)|	NUM
ejpam-5569	457	13	≥	≥	NOUN
ejpam-5569	457	14	n	n	CCONJ
ejpam-5569	457	15	,	,	PUNCT
ejpam-5569	457	16	where	where	SCONJ
ejpam-5569	457	17	n	n	NOUN
ejpam-5569	457	18	=	=	SYM
ejpam-5569	457	19	4k	4k	NOUN
ejpam-5569	457	20	,	,	PUNCT
ejpam-5569	457	21	d(x	d(x	PROPN
ejpam-5569	457	22	,	,	PUNCT
ejpam-5569	457	23	y	y	PROPN
ejpam-5569	457	24	)	)	PUNCT
ejpam-5569	457	25	≥	≥	NOUN
ejpam-5569	457	26	1	1	NUM
ejpam-5569	457	27	≥	≥	NOUN
ejpam-5569	457	28	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	457	29	)	)	PUNCT
ejpam-5569	457	30	)	)	PUNCT
ejpam-5569	458	1	+	+	CCONJ
ejpam-5569	458	2	1−	1−	NUM
ejpam-5569	458	3	d(x	d(x	PROPN
ejpam-5569	458	4	,	,	PUNCT
ejpam-5569	458	5	y	y	NOUN
ejpam-5569	458	6	)	)	PUNCT
ejpam-5569	458	7	case	case	NOUN
ejpam-5569	458	8	6	6	NUM
ejpam-5569	458	9	x	x	NOUN
ejpam-5569	458	10	,	,	PUNCT
ejpam-5569	458	11	y	y	PROPN
ejpam-5569	458	12	∈	∈	PROPN
ejpam-5569	458	13	vv	vv	INTJ
ejpam-5569	458	14	.	.	PUNCT
ejpam-5569	459	1	if	if	SCONJ
ejpam-5569	459	2	{	{	PUNCT
ejpam-5569	459	3	x	x	NOUN
ejpam-5569	459	4	,	,	PUNCT
ejpam-5569	459	5	y	y	PROPN
ejpam-5569	459	6	}	}	PUNCT
ejpam-5569	459	7	=	=	SYM
ejpam-5569	459	8	{	{	PUNCT
ejpam-5569	459	9	v1	v1	PROPN
ejpam-5569	459	10	,	,	PUNCT
ejpam-5569	459	11	vn	vn	VERB
ejpam-5569	459	12	}	}	PUNCT
ejpam-5569	459	13	then	then	ADV
ejpam-5569	459	14	|f(x)−	|f(x)−	NOUN
ejpam-5569	459	15	f(y)|	f(y)|	PROPN
ejpam-5569	459	16	=	=	PROPN
ejpam-5569	460	1	|f(v1)−	|f(v1)−	ADJ
ejpam-5569	460	2	f(vn)|	f(vn)|	NOUN
ejpam-5569	460	3	=	=	X
ejpam-5569	461	1	|f(v1)−	|f(v1)−	PROPN
ejpam-5569	461	2	[	[	X
ejpam-5569	461	3	f(v1)−	f(v1)−	PROPN
ejpam-5569	461	4	1]|	1]|	NUM
ejpam-5569	461	5	=	=	SYM
ejpam-5569	461	6	1	1	NUM
ejpam-5569	461	7	≥	≥	NOUN
ejpam-5569	461	8	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	461	9	)	)	PUNCT
ejpam-5569	461	10	)	)	PUNCT
ejpam-5569	462	1	+	+	CCONJ
ejpam-5569	462	2	1−	1−	NUM
ejpam-5569	462	3	d(x	d(x	PROPN
ejpam-5569	462	4	,	,	PUNCT
ejpam-5569	462	5	y	y	NOUN
ejpam-5569	462	6	)	)	PUNCT
ejpam-5569	462	7	,	,	PUNCT
ejpam-5569	462	8	where	where	SCONJ
ejpam-5569	462	9	d(x	d(x	PROPN
ejpam-5569	462	10	,	,	PUNCT
ejpam-5569	462	11	y	y	NOUN
ejpam-5569	462	12	)	)	PUNCT
ejpam-5569	462	13	=	=	VERB
ejpam-5569	462	14	n−	n−	NOUN
ejpam-5569	462	15	1	1	NUM
ejpam-5569	462	16	.	.	PUNCT
ejpam-5569	462	17	case	case	NOUN
ejpam-5569	462	18	7	7	NUM
ejpam-5569	462	19	if	if	SCONJ
ejpam-5569	462	20	{	{	PUNCT
ejpam-5569	462	21	x	x	NOUN
ejpam-5569	462	22	,	,	PUNCT
ejpam-5569	462	23	y	y	PROPN
ejpam-5569	462	24	}	}	PUNCT
ejpam-5569	462	25	=	=	SYM
ejpam-5569	462	26	{	{	PUNCT
ejpam-5569	462	27	v1	v1	NOUN
ejpam-5569	462	28	,	,	PUNCT
ejpam-5569	462	29	vk−1	vk−1	NOUN
ejpam-5569	462	30	}	}	PUNCT
ejpam-5569	462	31	then	then	ADV
ejpam-5569	462	32	|f(x)−	|f(x)−	NOUN
ejpam-5569	462	33	f(y)|	f(y)|	PROPN
ejpam-5569	462	34	=	=	PROPN
ejpam-5569	463	1	|f(v1)−	|f(v1)−	PROPN
ejpam-5569	463	2	f(vk−1)|	f(vk−1)|	PROPN
ejpam-5569	463	3	=	=	PUNCT
ejpam-5569	463	4	|(n+	|(n+	NOUN
ejpam-5569	463	5	2)−	2)−	NUM
ejpam-5569	463	6	[	[	X
ejpam-5569	463	7	(	(	PUNCT
ejpam-5569	463	8	n+	n+	NOUN
ejpam-5569	463	9	2	2	NUM
ejpam-5569	463	10	)	)	PUNCT
ejpam-5569	463	11	+	+	CCONJ
ejpam-5569	463	12	(	(	PUNCT
ejpam-5569	463	13	2n+	2n+	NUM
ejpam-5569	463	14	7)(k	7)(k	NUM
ejpam-5569	463	15	−	−	NOUN
ejpam-5569	463	16	2	2	NUM
ejpam-5569	463	17	)	)	PUNCT
ejpam-5569	463	18	+	+	CCONJ
ejpam-5569	463	19	2(k	2(k	NUM
ejpam-5569	463	20	−	−	NOUN
ejpam-5569	463	21	2)(k	2)(k	NUM
ejpam-5569	464	1	−	−	NOUN
ejpam-5569	464	2	3)]|	3)]|	NOUN
ejpam-5569	464	3	=	=	SYM
ejpam-5569	464	4	|(2n+	|(2n+	X
ejpam-5569	465	1	7)(k	7)(k	NUM
ejpam-5569	465	2	−	−	NOUN
ejpam-5569	465	3	2	2	NUM
ejpam-5569	465	4	)	)	PUNCT
ejpam-5569	465	5	+	+	CCONJ
ejpam-5569	465	6	2(k	2(k	NUM
ejpam-5569	466	1	−	−	NOUN
ejpam-5569	466	2	2)(k	2)(k	NUM
ejpam-5569	466	3	−	−	PROPN
ejpam-5569	466	4	3)|	3)|	NUM
ejpam-5569	466	5	≥	≥	NOUN
ejpam-5569	466	6	n	n	CCONJ
ejpam-5569	466	7	,	,	PUNCT
ejpam-5569	466	8	where	where	SCONJ
ejpam-5569	466	9	n	n	NOUN
ejpam-5569	466	10	=	=	SYM
ejpam-5569	466	11	4k	4k	X
ejpam-5569	466	12	≥	≥	NOUN
ejpam-5569	466	13	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	466	14	)	)	PUNCT
ejpam-5569	466	15	)	)	PUNCT
ejpam-5569	467	1	+	+	CCONJ
ejpam-5569	467	2	1−	1−	NUM
ejpam-5569	467	3	d(x	d(x	PROPN
ejpam-5569	467	4	,	,	PUNCT
ejpam-5569	467	5	y	y	NOUN
ejpam-5569	467	6	)	)	PUNCT
ejpam-5569	467	7	.	.	PUNCT
ejpam-5569	468	1	case	case	NOUN
ejpam-5569	468	2	8	8	NUM
ejpam-5569	468	3	if	if	SCONJ
ejpam-5569	468	4	{	{	PUNCT
ejpam-5569	468	5	x	x	NOUN
ejpam-5569	468	6	,	,	PUNCT
ejpam-5569	468	7	y	y	PROPN
ejpam-5569	468	8	}	}	PUNCT
ejpam-5569	468	9	=	=	SYM
ejpam-5569	468	10	{	{	PUNCT
ejpam-5569	468	11	v1	v1	NOUN
ejpam-5569	468	12	,	,	PUNCT
ejpam-5569	468	13	v2k−1	v2k−1	PROPN
ejpam-5569	468	14	}	}	PUNCT
ejpam-5569	468	15	then	then	ADV
ejpam-5569	468	16	|f(x)−	|f(x)−	NOUN
ejpam-5569	468	17	f(y)|	f(y)|	PROPN
ejpam-5569	468	18	=	=	PROPN
ejpam-5569	469	1	|f(v1)−	|f(v1)−	PROPN
ejpam-5569	469	2	f(v2k−1)|	f(v2k−1)|	PROPN
ejpam-5569	469	3	=	=	PUNCT
ejpam-5569	470	1	|(n+	|(n+	VERB
ejpam-5569	470	2	2)−	2)−	NUM
ejpam-5569	471	1	[	[	X
ejpam-5569	471	2	(	(	PUNCT
ejpam-5569	471	3	n+	n+	NOUN
ejpam-5569	471	4	2	2	NUM
ejpam-5569	471	5	)	)	PUNCT
ejpam-5569	471	6	+	+	CCONJ
ejpam-5569	471	7	(	(	PUNCT
ejpam-5569	471	8	2n+	2n+	NUM
ejpam-5569	471	9	7)(k	7)(k	NUM
ejpam-5569	471	10	−	−	NOUN
ejpam-5569	471	11	2	2	NUM
ejpam-5569	471	12	)	)	PUNCT
ejpam-5569	471	13	+	+	CCONJ
ejpam-5569	471	14	2(k	2(k	NUM
ejpam-5569	472	1	−	−	NOUN
ejpam-5569	472	2	2)(k	2)(k	NUM
ejpam-5569	472	3	−	−	PROPN
ejpam-5569	472	4	3)+	3)+	NUM
ejpam-5569	472	5	+	+	CCONJ
ejpam-5569	472	6	(	(	PUNCT
ejpam-5569	472	7	2n+	2n+	NUM
ejpam-5569	472	8	1	1	NUM
ejpam-5569	472	9	)	)	PUNCT
ejpam-5569	472	10	+	+	CCONJ
ejpam-5569	472	11	(	(	PUNCT
ejpam-5569	472	12	n+	n+	NUM
ejpam-5569	472	13	1)(k	1)(k	NUM
ejpam-5569	472	14	−	−	PROPN
ejpam-5569	472	15	1	1	NUM
ejpam-5569	472	16	)	)	PUNCT
ejpam-5569	472	17	+	+	CCONJ
ejpam-5569	473	1	2(k	2(k	NUM
ejpam-5569	473	2	−	−	NOUN
ejpam-5569	473	3	1)(k	1)(k	NUM
ejpam-5569	473	4	−	−	NOUN
ejpam-5569	473	5	2)]|	2)]|	NOUN
ejpam-5569	473	6	=	=	SYM
ejpam-5569	473	7	|(2n+	|(2n+	X
ejpam-5569	474	1	7)(k	7)(k	NUM
ejpam-5569	474	2	−	−	NOUN
ejpam-5569	474	3	1	1	NUM
ejpam-5569	474	4	)	)	PUNCT
ejpam-5569	474	5	+	+	CCONJ
ejpam-5569	474	6	2(k	2(k	NUM
ejpam-5569	474	7	−	−	NOUN
ejpam-5569	474	8	2)(k	2)(k	NUM
ejpam-5569	474	9	−	−	NOUN
ejpam-5569	474	10	3	3	NUM
ejpam-5569	474	11	)	)	PUNCT
ejpam-5569	474	12	+	+	CCONJ
ejpam-5569	474	13	(	(	PUNCT
ejpam-5569	474	14	2n+	2n+	NUM
ejpam-5569	474	15	1	1	NUM
ejpam-5569	474	16	)	)	PUNCT
ejpam-5569	474	17	+	+	CCONJ
ejpam-5569	474	18	(	(	PUNCT
ejpam-5569	474	19	n+	n+	NUM
ejpam-5569	474	20	1)(k	1)(k	NUM
ejpam-5569	474	21	−	−	PROPN
ejpam-5569	474	22	1)+	1)+	NUM
ejpam-5569	475	1	+	+	CCONJ
ejpam-5569	476	1	2(k	2(k	NUM
ejpam-5569	476	2	−	−	NOUN
ejpam-5569	476	3	1)(k	1)(k	NUM
ejpam-5569	476	4	−	−	PROPN
ejpam-5569	476	5	2)|	2)|	NUM
ejpam-5569	476	6	≥	≥	NOUN
ejpam-5569	476	7	n	n	CCONJ
ejpam-5569	476	8	,	,	PUNCT
ejpam-5569	476	9	where	where	SCONJ
ejpam-5569	476	10	n	n	NOUN
ejpam-5569	476	11	=	=	SYM
ejpam-5569	476	12	4k	4k	NOUN
ejpam-5569	476	13	,	,	PUNCT
ejpam-5569	476	14	d(x	d(x	PROPN
ejpam-5569	476	15	,	,	PUNCT
ejpam-5569	476	16	y	y	PROPN
ejpam-5569	476	17	)	)	PUNCT
ejpam-5569	476	18	≥	≥	NOUN
ejpam-5569	476	19	1	1	NUM
ejpam-5569	476	20	≥	≥	NOUN
ejpam-5569	476	21	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	476	22	)	)	PUNCT
ejpam-5569	476	23	)	)	PUNCT
ejpam-5569	477	1	+	+	CCONJ
ejpam-5569	477	2	1−	1−	NUM
ejpam-5569	477	3	d(x	d(x	PROPN
ejpam-5569	477	4	,	,	PUNCT
ejpam-5569	477	5	y	y	NOUN
ejpam-5569	477	6	)	)	PUNCT
ejpam-5569	477	7	case	case	NOUN
ejpam-5569	477	8	9if	9if	NOUN
ejpam-5569	477	9	{	{	PUNCT
ejpam-5569	477	10	x	x	NOUN
ejpam-5569	477	11	,	,	PUNCT
ejpam-5569	477	12	y	y	PROPN
ejpam-5569	477	13	}	}	PUNCT
ejpam-5569	477	14	=	=	SYM
ejpam-5569	477	15	{	{	PUNCT
ejpam-5569	477	16	v1	v1	NOUN
ejpam-5569	477	17	,	,	PUNCT
ejpam-5569	477	18	v3k−1	v3k−1	NOUN
ejpam-5569	477	19	}	}	PUNCT
ejpam-5569	477	20	then	then	ADV
ejpam-5569	477	21	|f(x)−	|f(x)−	NOUN
ejpam-5569	477	22	f(y)|	f(y)|	PROPN
ejpam-5569	477	23	=	=	PROPN
ejpam-5569	477	24	|f(v1)−	|f(v1)−	PROPN
ejpam-5569	477	25	f(v3k−1)|	f(v3k−1)|	PROPN
ejpam-5569	477	26	n.	n.	NOUN
ejpam-5569	477	27	m.	m.	NOUN
ejpam-5569	478	1	noureldeen	noureldeen	NOUN
ejpam-5569	478	2	et	et	PROPN
ejpam-5569	478	3	al	al	PROPN
ejpam-5569	478	4	.	.	PUNCT
ejpam-5569	478	5	/	/	SYM
ejpam-5569	478	6	eur	eur	PROPN
ejpam-5569	478	7	.	.	PUNCT
ejpam-5569	479	1	j.	j.	PROPN
ejpam-5569	479	2	pure	pure	PROPN
ejpam-5569	479	3	appl	appl	PROPN
ejpam-5569	479	4	.	.	PROPN
ejpam-5569	479	5	math	math	PROPN
ejpam-5569	479	6	,	,	PUNCT
ejpam-5569	479	7	18	18	NUM
ejpam-5569	479	8	(	(	PUNCT
ejpam-5569	479	9	2	2	NUM
ejpam-5569	479	10	)	)	PUNCT
ejpam-5569	479	11	(	(	PUNCT
ejpam-5569	479	12	2025	2025	NUM
ejpam-5569	479	13	)	)	PUNCT
ejpam-5569	479	14	,	,	PUNCT
ejpam-5569	479	15	5569	5569	NUM
ejpam-5569	479	16	16	16	NUM
ejpam-5569	479	17	of	of	ADP
ejpam-5569	479	18	25	25	NUM
ejpam-5569	479	19	=	=	SYM
ejpam-5569	480	1	|f(v1)−	|f(v1)−	PROPN
ejpam-5569	480	2	[	[	X
ejpam-5569	480	3	f(vk	f(vk	X
ejpam-5569	480	4	)	)	PUNCT
ejpam-5569	480	5	+	+	CCONJ
ejpam-5569	480	6	(	(	PUNCT
ejpam-5569	480	7	2k	2k	NUM
ejpam-5569	480	8	−	−	PROPN
ejpam-5569	480	9	1)]|	1)]|	PROPN
ejpam-5569	480	10	=	=	PROPN
ejpam-5569	480	11	|f(v1)−	|f(v1)−	PROPN
ejpam-5569	480	12	[	[	X
ejpam-5569	480	13	f(vk−1	f(vk−1	X
ejpam-5569	480	14	)	)	PUNCT
ejpam-5569	480	15	+	+	CCONJ
ejpam-5569	480	16	(	(	PUNCT
ejpam-5569	480	17	2n+	2n+	NUM
ejpam-5569	480	18	1	1	NUM
ejpam-5569	480	19	)	)	PUNCT
ejpam-5569	480	20	+	+	CCONJ
ejpam-5569	480	21	(	(	PUNCT
ejpam-5569	480	22	2k	2k	NUM
ejpam-5569	480	23	−	−	PROPN
ejpam-5569	480	24	1)]|	1)]|	NUM
ejpam-5569	480	25	=	=	NOUN
ejpam-5569	480	26	|(n+	|(n+	NOUN
ejpam-5569	480	27	2)−	2)−	NUM
ejpam-5569	480	28	[	[	X
ejpam-5569	480	29	(	(	PUNCT
ejpam-5569	480	30	n+	n+	NOUN
ejpam-5569	480	31	2	2	NUM
ejpam-5569	480	32	)	)	PUNCT
ejpam-5569	480	33	+	+	CCONJ
ejpam-5569	480	34	(	(	PUNCT
ejpam-5569	480	35	2n+	2n+	NUM
ejpam-5569	480	36	7)(k	7)(k	NUM
ejpam-5569	480	37	−	−	NOUN
ejpam-5569	480	38	2	2	NUM
ejpam-5569	480	39	)	)	PUNCT
ejpam-5569	480	40	+	+	CCONJ
ejpam-5569	480	41	2(k	2(k	NUM
ejpam-5569	480	42	−	−	NOUN
ejpam-5569	480	43	2)(k	2)(k	NUM
ejpam-5569	480	44	−	−	NOUN
ejpam-5569	480	45	3	3	NUM
ejpam-5569	480	46	)	)	PUNCT
ejpam-5569	480	47	+	+	CCONJ
ejpam-5569	480	48	(	(	PUNCT
ejpam-5569	480	49	2n+	2n+	NUM
ejpam-5569	480	50	1)+	1)+	NUM
ejpam-5569	480	51	+	+	CCONJ
ejpam-5569	480	52	(	(	PUNCT
ejpam-5569	480	53	2k	2k	NOUN
ejpam-5569	480	54	−	−	PROPN
ejpam-5569	480	55	1)]|	1)]|	NUM
ejpam-5569	480	56	=	=	SYM
ejpam-5569	480	57	|(2n+	|(2n+	X
ejpam-5569	481	1	7)(k	7)(k	NUM
ejpam-5569	481	2	−	−	NOUN
ejpam-5569	481	3	2	2	NUM
ejpam-5569	481	4	)	)	PUNCT
ejpam-5569	481	5	+	+	CCONJ
ejpam-5569	481	6	2(k	2(k	NUM
ejpam-5569	482	1	−	−	NOUN
ejpam-5569	482	2	2)(k	2)(k	NUM
ejpam-5569	482	3	−	−	NOUN
ejpam-5569	482	4	3	3	NUM
ejpam-5569	482	5	)	)	PUNCT
ejpam-5569	482	6	+	+	CCONJ
ejpam-5569	482	7	(	(	PUNCT
ejpam-5569	482	8	2n+	2n+	NUM
ejpam-5569	482	9	1	1	NUM
ejpam-5569	482	10	)	)	PUNCT
ejpam-5569	482	11	+	+	CCONJ
ejpam-5569	482	12	(	(	PUNCT
ejpam-5569	482	13	2k	2k	NUM
ejpam-5569	482	14	−	−	PROPN
ejpam-5569	482	15	1)|	1)|	NUM
ejpam-5569	482	16	≥	≥	NOUN
ejpam-5569	482	17	n	n	CCONJ
ejpam-5569	482	18	,	,	PUNCT
ejpam-5569	482	19	where	where	SCONJ
ejpam-5569	482	20	n	n	NOUN
ejpam-5569	482	21	=	=	SYM
ejpam-5569	482	22	4k	4k	NOUN
ejpam-5569	482	23	,	,	PUNCT
ejpam-5569	482	24	d(x	d(x	PROPN
ejpam-5569	482	25	,	,	PUNCT
ejpam-5569	482	26	y	y	PROPN
ejpam-5569	482	27	)	)	PUNCT
ejpam-5569	482	28	≥	≥	NOUN
ejpam-5569	482	29	1	1	NUM
ejpam-5569	482	30	≥	≥	NOUN
ejpam-5569	482	31	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	482	32	)	)	PUNCT
ejpam-5569	482	33	)	)	PUNCT
ejpam-5569	483	1	+	+	CCONJ
ejpam-5569	483	2	1−	1−	NUM
ejpam-5569	483	3	d(x	d(x	PROPN
ejpam-5569	483	4	,	,	PUNCT
ejpam-5569	483	5	y	y	NOUN
ejpam-5569	483	6	)	)	PUNCT
ejpam-5569	483	7	case	case	NOUN
ejpam-5569	483	8	10	10	NUM
ejpam-5569	483	9	x	x	SYM
ejpam-5569	483	10	∈	∈	NOUN
ejpam-5569	483	11	vu	vu	NOUN
ejpam-5569	483	12	and	and	CCONJ
ejpam-5569	483	13	y	y	PROPN
ejpam-5569	483	14	∈	∈	PROPN
ejpam-5569	483	15	vv	vv	AUX
ejpam-5569	483	16	let	let	VERB
ejpam-5569	483	17	x	x	PUNCT
ejpam-5569	483	18	=	=	PUNCT
ejpam-5569	483	19	ui	ui	PROPN
ejpam-5569	483	20	and	and	CCONJ
ejpam-5569	483	21	y	y	PROPN
ejpam-5569	483	22	=	=	PUNCT
ejpam-5569	483	23	vl	vl	PROPN
ejpam-5569	483	24	,	,	PUNCT
ejpam-5569	483	25	where	where	SCONJ
ejpam-5569	483	26	1	1	NUM
ejpam-5569	483	27	≤	≤	PUNCT
ejpam-5569	483	28	i	i	PRON
ejpam-5569	483	29	,	,	PUNCT
ejpam-5569	483	30	l	l	PROPN
ejpam-5569	483	31	≤	≤	X
ejpam-5569	483	32	n	n	CCONJ
ejpam-5569	483	33	in	in	ADP
ejpam-5569	483	34	this	this	DET
ejpam-5569	483	35	case	case	NOUN
ejpam-5569	483	36	we	we	PRON
ejpam-5569	483	37	have	have	VERB
ejpam-5569	483	38	several	several	ADJ
ejpam-5569	483	39	subcases	subcase	NOUN
ejpam-5569	483	40	:	:	PUNCT
ejpam-5569	483	41	subcase	subcase	NOUN
ejpam-5569	483	42	10.1	10.1	NUM
ejpam-5569	483	43	if	if	SCONJ
ejpam-5569	483	44	i	i	PRON
ejpam-5569	483	45	=	=	NOUN
ejpam-5569	483	46	1	1	NUM
ejpam-5569	483	47	,	,	PUNCT
ejpam-5569	483	48	j	j	PROPN
ejpam-5569	484	1	=	=	SYM
ejpam-5569	484	2	k	k	PROPN
ejpam-5569	485	1	−	−	PROPN
ejpam-5569	485	2	1	1	NUM
ejpam-5569	486	1	then	then	ADV
ejpam-5569	486	2	|f(x)−	|f(x)−	NOUN
ejpam-5569	486	3	f(y)|	f(y)|	PROPN
ejpam-5569	486	4	=	=	PROPN
ejpam-5569	486	5	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	487	1	f(vk−1)|	f(vk−1)|	ADJ
ejpam-5569	487	2	=	=	NOUN
ejpam-5569	487	3	|0−	|0−	NOUN
ejpam-5569	487	4	[	[	X
ejpam-5569	487	5	(	(	PUNCT
ejpam-5569	487	6	n+	n+	NOUN
ejpam-5569	487	7	2	2	NUM
ejpam-5569	487	8	)	)	PUNCT
ejpam-5569	487	9	+	+	CCONJ
ejpam-5569	487	10	(	(	PUNCT
ejpam-5569	487	11	2n+	2n+	NUM
ejpam-5569	487	12	7)(k	7)(k	NUM
ejpam-5569	487	13	−	−	NOUN
ejpam-5569	487	14	2	2	NUM
ejpam-5569	487	15	)	)	PUNCT
ejpam-5569	487	16	+	+	CCONJ
ejpam-5569	488	1	2(k	2(k	NUM
ejpam-5569	488	2	−	−	NOUN
ejpam-5569	488	3	2)(k	2)(k	NUM
ejpam-5569	488	4	−	−	NOUN
ejpam-5569	488	5	3)]|	3)]|	ADJ
ejpam-5569	488	6	=	=	SYM
ejpam-5569	488	7	|(n+	|(n+	NOUN
ejpam-5569	488	8	2	2	NUM
ejpam-5569	488	9	)	)	PUNCT
ejpam-5569	488	10	+	+	CCONJ
ejpam-5569	488	11	(	(	PUNCT
ejpam-5569	488	12	2n+	2n+	NUM
ejpam-5569	488	13	7)(k	7)(k	NUM
ejpam-5569	488	14	−	−	NOUN
ejpam-5569	488	15	2	2	NUM
ejpam-5569	488	16	)	)	PUNCT
ejpam-5569	488	17	+	+	CCONJ
ejpam-5569	488	18	2(k	2(k	NUM
ejpam-5569	489	1	−	−	NOUN
ejpam-5569	489	2	2)(k	2)(k	NUM
ejpam-5569	489	3	−	−	PROPN
ejpam-5569	489	4	3)|	3)|	NUM
ejpam-5569	489	5	≥	≥	NOUN
ejpam-5569	489	6	n	n	CCONJ
ejpam-5569	489	7	,	,	PUNCT
ejpam-5569	489	8	where	where	SCONJ
ejpam-5569	489	9	n	n	NOUN
ejpam-5569	489	10	=	=	SYM
ejpam-5569	489	11	4k	4k	NOUN
ejpam-5569	489	12	,	,	PUNCT
ejpam-5569	489	13	d(x	d(x	PROPN
ejpam-5569	489	14	,	,	PUNCT
ejpam-5569	489	15	y	y	PROPN
ejpam-5569	489	16	)	)	PUNCT
ejpam-5569	489	17	≥	≥	NOUN
ejpam-5569	489	18	1	1	NUM
ejpam-5569	489	19	≥	≥	NOUN
ejpam-5569	489	20	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	489	21	)	)	PUNCT
ejpam-5569	489	22	)	)	PUNCT
ejpam-5569	490	1	+	+	CCONJ
ejpam-5569	490	2	1−	1−	NUM
ejpam-5569	490	3	d(x	d(x	PROPN
ejpam-5569	490	4	,	,	PUNCT
ejpam-5569	490	5	y	y	NOUN
ejpam-5569	490	6	)	)	PUNCT
ejpam-5569	490	7	.	.	PUNCT
ejpam-5569	491	1	otherwise	otherwise	ADV
ejpam-5569	491	2	,	,	PUNCT
ejpam-5569	491	3	|f(x)−	|f(x)−	PROPN
ejpam-5569	491	4	f(y)|	f(y)|	PROPN
ejpam-5569	491	5	≥	≥	PROPN
ejpam-5569	491	6	n.	n.	PROPN
ejpam-5569	491	7	subcase	subcase	PROPN
ejpam-5569	491	8	10.2	10.2	NUM
ejpam-5569	491	9	.	.	PUNCT
ejpam-5569	492	1	suppose	suppose	VERB
ejpam-5569	492	2	,	,	PUNCT
ejpam-5569	492	3	1	1	NUM
ejpam-5569	492	4	≤	≤	NUM
ejpam-5569	492	5	i	i	NOUN
ejpam-5569	492	6	≤	≤	NUM
ejpam-5569	493	1	k	k	PROPN
ejpam-5569	493	2	and	and	CCONJ
ejpam-5569	493	3	k	k	PROPN
ejpam-5569	493	4	≤	≤	NUM
ejpam-5569	493	5	l	l	NOUN
ejpam-5569	493	6	≤	≤	NOUN
ejpam-5569	493	7	2k	2k	NOUN
ejpam-5569	493	8	−	−	NUM
ejpam-5569	493	9	1	1	NUM
ejpam-5569	493	10	|f(x)−	|f(x)−	NOUN
ejpam-5569	493	11	f(y)|	f(y)|	PROPN
ejpam-5569	493	12	=	=	NUM
ejpam-5569	493	13	|f(ui)−	|f(ui)−	ADJ
ejpam-5569	493	14	f(vl)|	f(vl)|	NOUN
ejpam-5569	493	15	=	=	SYM
ejpam-5569	493	16	|f(ui)−	|f(ui)−	NOUN
ejpam-5569	493	17	f(vk−1+j)|	f(vk−1+j)|	NUM
ejpam-5569	493	18	,	,	PUNCT
ejpam-5569	493	19	where	where	SCONJ
ejpam-5569	493	20	1	1	NUM
ejpam-5569	493	21	≤	≤	NUM
ejpam-5569	493	22	j	j	PROPN
ejpam-5569	493	23	≤	≤	PROPN
ejpam-5569	494	1	k	k	NOUN
ejpam-5569	494	2	−	−	NOUN
ejpam-5569	494	3	1	1	NUM
ejpam-5569	495	1	=	=	SYM
ejpam-5569	495	2	|(2n+	|(2n+	PROPN
ejpam-5569	495	3	3)(i−	3)(i−	NUM
ejpam-5569	495	4	1	1	NUM
ejpam-5569	495	5	)	)	PUNCT
ejpam-5569	495	6	+	+	CCONJ
ejpam-5569	496	1	2(i−	2(i−	NUM
ejpam-5569	496	2	1)(i−	1)(i−	NUM
ejpam-5569	496	3	2)−	2)−	NUM
ejpam-5569	497	1	[	[	X
ejpam-5569	497	2	f(vk−1	f(vk−1	X
ejpam-5569	497	3	)	)	PUNCT
ejpam-5569	497	4	+	+	CCONJ
ejpam-5569	497	5	(	(	PUNCT
ejpam-5569	497	6	2n+	2n+	NUM
ejpam-5569	497	7	1	1	NUM
ejpam-5569	497	8	)	)	PUNCT
ejpam-5569	497	9	+	+	CCONJ
ejpam-5569	497	10	(	(	PUNCT
ejpam-5569	497	11	n+	n+	ADP
ejpam-5569	497	12	1)(j	1)(j	NUM
ejpam-5569	497	13	−	−	NOUN
ejpam-5569	497	14	1)+	1)+	NUM
ejpam-5569	498	1	+	+	CCONJ
ejpam-5569	498	2	2(j	2(j	NUM
ejpam-5569	498	3	−	−	PROPN
ejpam-5569	498	4	1)(j	1)(j	NUM
ejpam-5569	498	5	−	−	PROPN
ejpam-5569	498	6	2)]|	2)]|	NOUN
ejpam-5569	498	7	=	=	SYM
ejpam-5569	498	8	|(2n+	|(2n+	PROPN
ejpam-5569	499	1	3)(i−	3)(i−	NUM
ejpam-5569	499	2	1	1	NUM
ejpam-5569	499	3	)	)	PUNCT
ejpam-5569	499	4	+	+	CCONJ
ejpam-5569	500	1	2(i−	2(i−	NUM
ejpam-5569	500	2	1)(i−	1)(i−	NUM
ejpam-5569	500	3	2)−	2)−	NUM
ejpam-5569	501	1	[	[	X
ejpam-5569	501	2	(	(	PUNCT
ejpam-5569	501	3	n+	n+	NOUN
ejpam-5569	501	4	2	2	NUM
ejpam-5569	501	5	)	)	PUNCT
ejpam-5569	501	6	+	+	CCONJ
ejpam-5569	501	7	(	(	PUNCT
ejpam-5569	501	8	2n+	2n+	NUM
ejpam-5569	501	9	7)(k	7)(k	NUM
ejpam-5569	501	10	−	−	PROPN
ejpam-5569	501	11	2)+	2)+	NUM
ejpam-5569	501	12	+	+	CCONJ
ejpam-5569	501	13	2(k	2(k	NUM
ejpam-5569	501	14	−	−	NOUN
ejpam-5569	501	15	2)(k	2)(k	NUM
ejpam-5569	501	16	−	−	NOUN
ejpam-5569	501	17	3	3	NUM
ejpam-5569	501	18	)	)	PUNCT
ejpam-5569	501	19	+	+	CCONJ
ejpam-5569	501	20	(	(	PUNCT
ejpam-5569	501	21	2n+	2n+	NUM
ejpam-5569	501	22	1	1	NUM
ejpam-5569	501	23	)	)	PUNCT
ejpam-5569	501	24	+	+	CCONJ
ejpam-5569	501	25	(	(	PUNCT
ejpam-5569	501	26	n+	n+	NOUN
ejpam-5569	501	27	1)(j	1)(j	NUM
ejpam-5569	501	28	−	−	NOUN
ejpam-5569	501	29	1	1	NUM
ejpam-5569	501	30	)	)	PUNCT
ejpam-5569	501	31	+	+	CCONJ
ejpam-5569	501	32	2(j	2(j	NUM
ejpam-5569	501	33	−	−	DET
ejpam-5569	501	34	1)(j	1)(j	NUM
ejpam-5569	501	35	−	−	NOUN
ejpam-5569	501	36	2)]|	2)]|	NOUN
ejpam-5569	501	37	.	.	PUNCT
ejpam-5569	501	38	subcase	subcase	PROPN
ejpam-5569	501	39	10.3	10.3	NUM
ejpam-5569	501	40	.	.	PUNCT
ejpam-5569	502	1	if	if	SCONJ
ejpam-5569	502	2	i	i	PRON
ejpam-5569	502	3	=	=	NOUN
ejpam-5569	502	4	1	1	NUM
ejpam-5569	502	5	,	,	PUNCT
ejpam-5569	502	6	l	l	NOUN
ejpam-5569	502	7	=	=	PUNCT
ejpam-5569	502	8	k	k	X
ejpam-5569	502	9	i.e.	i.e.	X
ejpam-5569	502	10	j	j	X
ejpam-5569	502	11	=	=	SYM
ejpam-5569	502	12	1	1	NUM
ejpam-5569	502	13	then	then	ADV
ejpam-5569	502	14	|f(x)−	|f(x)−	NOUN
ejpam-5569	502	15	f(y)|	f(y)|	PROPN
ejpam-5569	502	16	=	=	NUM
ejpam-5569	502	17	|f(ui)−	|f(ui)−	ADJ
ejpam-5569	502	18	f(vl)|	f(vl)|	NOUN
ejpam-5569	502	19	=	=	SYM
ejpam-5569	502	20	|f(ui)−	|f(ui)−	NOUN
ejpam-5569	502	21	f(vk−1+j)|	f(vk−1+j)|	NUM
ejpam-5569	502	22	,	,	PUNCT
ejpam-5569	502	23	where	where	SCONJ
ejpam-5569	502	24	1	1	NUM
ejpam-5569	502	25	≤	≤	NUM
ejpam-5569	502	26	j	j	PROPN
ejpam-5569	502	27	≤	≤	PROPN
ejpam-5569	503	1	k	k	NOUN
ejpam-5569	503	2	−	−	NOUN
ejpam-5569	504	1	1	1	NUM
ejpam-5569	504	2	=	=	SYM
ejpam-5569	504	3	|0−	|0−	NOUN
ejpam-5569	504	4	[	[	X
ejpam-5569	504	5	f(vk−1	f(vk−1	X
ejpam-5569	504	6	)	)	PUNCT
ejpam-5569	504	7	+	+	CCONJ
ejpam-5569	504	8	(	(	PUNCT
ejpam-5569	504	9	2n+	2n+	NUM
ejpam-5569	504	10	1	1	NUM
ejpam-5569	504	11	)	)	PUNCT
ejpam-5569	504	12	+	+	CCONJ
ejpam-5569	504	13	(	(	PUNCT
ejpam-5569	504	14	n+	n+	NOUN
ejpam-5569	504	15	1)(j	1)(j	NUM
ejpam-5569	504	16	−	−	NOUN
ejpam-5569	504	17	1	1	NUM
ejpam-5569	504	18	)	)	PUNCT
ejpam-5569	504	19	+	+	CCONJ
ejpam-5569	504	20	2(j	2(j	NUM
ejpam-5569	504	21	−	−	PROPN
ejpam-5569	504	22	1)(j	1)(j	NUM
ejpam-5569	504	23	−	−	PROPN
ejpam-5569	504	24	2)]|	2)]|	NOUN
ejpam-5569	504	25	=	=	NOUN
ejpam-5569	504	26	|	|	ADV
ejpam-5569	504	27	−	−	PROPN
ejpam-5569	504	28	(	(	PUNCT
ejpam-5569	504	29	n+	n+	NOUN
ejpam-5569	504	30	2	2	NUM
ejpam-5569	504	31	)	)	PUNCT
ejpam-5569	504	32	+	+	CCONJ
ejpam-5569	504	33	(	(	PUNCT
ejpam-5569	504	34	2n+	2n+	NUM
ejpam-5569	504	35	7)(k	7)(k	NUM
ejpam-5569	504	36	−	−	NOUN
ejpam-5569	504	37	2	2	NUM
ejpam-5569	504	38	)	)	PUNCT
ejpam-5569	504	39	+	+	CCONJ
ejpam-5569	504	40	2(k	2(k	NUM
ejpam-5569	504	41	−	−	NOUN
ejpam-5569	504	42	2)(k	2)(k	NUM
ejpam-5569	504	43	−	−	NOUN
ejpam-5569	504	44	3	3	NUM
ejpam-5569	504	45	)	)	PUNCT
ejpam-5569	504	46	+	+	CCONJ
ejpam-5569	504	47	(	(	PUNCT
ejpam-5569	504	48	2n+	2n+	NUM
ejpam-5569	504	49	1)|	1)|	NUM
ejpam-5569	504	50	=	=	PUNCT
ejpam-5569	504	51	|(n+	|(n+	NOUN
ejpam-5569	504	52	2	2	NUM
ejpam-5569	504	53	)	)	PUNCT
ejpam-5569	504	54	+	+	CCONJ
ejpam-5569	504	55	(	(	PUNCT
ejpam-5569	504	56	2n+	2n+	NUM
ejpam-5569	504	57	7)(k	7)(k	NUM
ejpam-5569	504	58	−	−	NOUN
ejpam-5569	504	59	2	2	NUM
ejpam-5569	504	60	)	)	PUNCT
ejpam-5569	504	61	+	+	CCONJ
ejpam-5569	505	1	2(k	2(k	NUM
ejpam-5569	505	2	−	−	NOUN
ejpam-5569	505	3	2)(k	2)(k	NUM
ejpam-5569	505	4	−	−	NOUN
ejpam-5569	505	5	3	3	NUM
ejpam-5569	505	6	)	)	PUNCT
ejpam-5569	505	7	+	+	CCONJ
ejpam-5569	505	8	(	(	PUNCT
ejpam-5569	505	9	2n+	2n+	NUM
ejpam-5569	505	10	1)|	1)|	NUM
ejpam-5569	505	11	≥	≥	NOUN
ejpam-5569	505	12	n	n	CCONJ
ejpam-5569	505	13	,	,	PUNCT
ejpam-5569	505	14	where	where	SCONJ
ejpam-5569	505	15	n	n	NOUN
ejpam-5569	505	16	=	=	SYM
ejpam-5569	505	17	4k	4k	NOUN
ejpam-5569	505	18	,	,	PUNCT
ejpam-5569	505	19	d(x	d(x	PROPN
ejpam-5569	505	20	,	,	PUNCT
ejpam-5569	505	21	y	y	PROPN
ejpam-5569	505	22	)	)	PUNCT
ejpam-5569	505	23	≥	≥	NOUN
ejpam-5569	505	24	1	1	NUM
ejpam-5569	505	25	≥	≥	NOUN
ejpam-5569	505	26	diam(d2(pn	diam(d2(pn	NOUN
ejpam-5569	505	27	)	)	PUNCT
ejpam-5569	505	28	)	)	PUNCT
ejpam-5569	506	1	+	+	CCONJ
ejpam-5569	506	2	1−	1−	NUM
ejpam-5569	506	3	d(x	d(x	PROPN
ejpam-5569	506	4	,	,	PUNCT
ejpam-5569	506	5	y	y	NOUN
ejpam-5569	506	6	)	)	PUNCT
ejpam-5569	506	7	.	.	PUNCT
ejpam-5569	507	1	subcase	subcase	PROPN
ejpam-5569	507	2	10.4	10.4	NUM
ejpam-5569	507	3	.	.	PUNCT
ejpam-5569	508	1	suppose	suppose	VERB
ejpam-5569	508	2	,	,	PUNCT
ejpam-5569	508	3	1	1	NUM
ejpam-5569	508	4	≤	≤	NUM
ejpam-5569	509	1	i	i	NOUN
ejpam-5569	509	2	≤	≤	ADJ
ejpam-5569	509	3	k	k	NOUN
ejpam-5569	509	4	and	and	CCONJ
ejpam-5569	509	5	2k	2k	NUM
ejpam-5569	509	6	≤	≤	NUM
ejpam-5569	509	7	l	l	NOUN
ejpam-5569	509	8	≤	≤	NOUN
ejpam-5569	510	1	3k	3k	PRON
ejpam-5569	510	2	−	−	NOUN
ejpam-5569	510	3	1	1	NUM
ejpam-5569	510	4	|f(x)−	|f(x)−	NOUN
ejpam-5569	510	5	f(y)|	f(y)|	PROPN
ejpam-5569	510	6	=	=	NUM
ejpam-5569	510	7	|f(ui)−	|f(ui)−	ADJ
ejpam-5569	510	8	f(vl)|	f(vl)|	PROPN
ejpam-5569	510	9	n.	n.	PROPN
ejpam-5569	510	10	m.	m.	PROPN
ejpam-5569	511	1	noureldeen	noureldeen	INTJ
ejpam-5569	511	2	et	et	PROPN
ejpam-5569	511	3	al	al	PROPN
ejpam-5569	511	4	.	.	PUNCT
ejpam-5569	511	5	/	/	SYM
ejpam-5569	511	6	eur	eur	PROPN
ejpam-5569	511	7	.	.	PUNCT
ejpam-5569	512	1	j.	j.	PROPN
ejpam-5569	512	2	pure	pure	PROPN
ejpam-5569	512	3	appl	appl	PROPN
ejpam-5569	512	4	.	.	PROPN
ejpam-5569	512	5	math	math	PROPN
ejpam-5569	512	6	,	,	PUNCT
ejpam-5569	512	7	18	18	NUM
ejpam-5569	512	8	(	(	PUNCT
ejpam-5569	512	9	2	2	NUM
ejpam-5569	512	10	)	)	PUNCT
ejpam-5569	512	11	(	(	PUNCT
ejpam-5569	512	12	2025	2025	NUM
ejpam-5569	512	13	)	)	PUNCT
ejpam-5569	512	14	,	,	PUNCT
ejpam-5569	512	15	5569	5569	NUM
ejpam-5569	512	16	17	17	NUM
ejpam-5569	512	17	of	of	ADP
ejpam-5569	512	18	25	25	NUM
ejpam-5569	512	19	=	=	NOUN
ejpam-5569	512	20	|f(ui)−	|f(ui)−	NOUN
ejpam-5569	512	21	f(vn−(k+j−2))|	f(vn−(k+j−2))|	NOUN
ejpam-5569	512	22	,	,	PUNCT
ejpam-5569	512	23	where	where	SCONJ
ejpam-5569	512	24	1	1	NUM
ejpam-5569	512	25	≤	≤	NUM
ejpam-5569	512	26	j	j	PROPN
ejpam-5569	512	27	≤	≤	PROPN
ejpam-5569	512	28	k.	k.	PROPN
ejpam-5569	512	29	subcase	subcase	PROPN
ejpam-5569	512	30	10.5	10.5	NUM
ejpam-5569	512	31	.	.	PUNCT
ejpam-5569	513	1	if	if	SCONJ
ejpam-5569	513	2	i	i	PRON
ejpam-5569	513	3	=	=	NOUN
ejpam-5569	513	4	1	1	NUM
ejpam-5569	513	5	,	,	PUNCT
ejpam-5569	513	6	l	l	NOUN
ejpam-5569	513	7	=	=	PUNCT
ejpam-5569	513	8	3k	3k	X
ejpam-5569	514	1	−	−	NOUN
ejpam-5569	514	2	1	1	NUM
ejpam-5569	514	3	i.e.	i.e.	X
ejpam-5569	514	4	j	j	X
ejpam-5569	514	5	=	=	SYM
ejpam-5569	514	6	1	1	NUM
ejpam-5569	514	7	then	then	ADV
ejpam-5569	514	8	|f(x)−	|f(x)−	NOUN
ejpam-5569	514	9	f(y)|	f(y)|	PROPN
ejpam-5569	514	10	=	=	PROPN
ejpam-5569	514	11	|f(u1)−	|f(u1)−	PROPN
ejpam-5569	514	12	f(v3k−1)|	f(v3k−1)|	PROPN
ejpam-5569	514	13	=	=	SYM
ejpam-5569	515	1	|f(ui)−	|f(ui)−	ADJ
ejpam-5569	515	2	f(vn−k−1)|	f(vn−k−1)|	NOUN
ejpam-5569	515	3	=	=	PUNCT
ejpam-5569	515	4	|0−	|0−	NOUN
ejpam-5569	515	5	f(vk	f(vk	PROPN
ejpam-5569	515	6	)	)	PUNCT
ejpam-5569	515	7	+	+	NUM
ejpam-5569	515	8	2k	2k	NUM
ejpam-5569	515	9	−	−	NUM
ejpam-5569	515	10	1|	1|	NUM
ejpam-5569	516	1	=	=	SYM
ejpam-5569	516	2	|(n+	|(n+	NOUN
ejpam-5569	516	3	2	2	NUM
ejpam-5569	516	4	)	)	PUNCT
ejpam-5569	517	1	+	+	CCONJ
ejpam-5569	517	2	(	(	PUNCT
ejpam-5569	517	3	2n+	2n+	NUM
ejpam-5569	517	4	7)(k	7)(k	NUM
ejpam-5569	517	5	−	−	NOUN
ejpam-5569	517	6	2	2	NUM
ejpam-5569	517	7	)	)	PUNCT
ejpam-5569	517	8	+	+	CCONJ
ejpam-5569	518	1	2(k	2(k	NUM
ejpam-5569	518	2	−	−	NOUN
ejpam-5569	518	3	2)(k	2)(k	NUM
ejpam-5569	518	4	−	−	NOUN
ejpam-5569	518	5	3	3	NUM
ejpam-5569	518	6	)	)	PUNCT
ejpam-5569	518	7	+	+	NOUN
ejpam-5569	518	8	2k	2k	NUM
ejpam-5569	518	9	−	−	NUM
ejpam-5569	518	10	1|	1|	NUM
ejpam-5569	518	11	≥	≥	NOUN
ejpam-5569	518	12	n	n	CCONJ
ejpam-5569	518	13	,	,	PUNCT
ejpam-5569	518	14	where	where	SCONJ
ejpam-5569	518	15	n	n	NOUN
ejpam-5569	518	16	=	=	SYM
ejpam-5569	518	17	4k	4k	NOUN
ejpam-5569	518	18	,	,	PUNCT
ejpam-5569	518	19	d(x	d(x	PROPN
ejpam-5569	518	20	,	,	PUNCT
ejpam-5569	518	21	y	y	PROPN
ejpam-5569	518	22	)	)	PUNCT
ejpam-5569	518	23	≥	≥	NOUN
ejpam-5569	518	24	1	1	NUM
ejpam-5569	518	25	.	.	PUNCT
ejpam-5569	519	1	otherwise	otherwise	ADV
ejpam-5569	519	2	,	,	PUNCT
ejpam-5569	519	3	|f(x)−f(y)|	|f(x)−f(y)|	PROPN
ejpam-5569	519	4	≥	≥	PROPN
ejpam-5569	519	5	n.	n.	PROPN
ejpam-5569	519	6	similarly	similarly	ADV
ejpam-5569	519	7	,	,	PUNCT
ejpam-5569	519	8	proved	prove	VERB
ejpam-5569	519	9	if	if	SCONJ
ejpam-5569	519	10	k+2	k+2	ADP
ejpam-5569	519	11	≤	≤	NOUN
ejpam-5569	519	12	i	i	PRON
ejpam-5569	519	13	≤	≤	NOUN
ejpam-5569	519	14	2k+1	2k+1	PROPN
ejpam-5569	519	15	and	and	CCONJ
ejpam-5569	519	16	k+1	k+1	X
ejpam-5569	519	17	≤	≤	NUM
ejpam-5569	519	18	l	l	NOUN
ejpam-5569	519	19	≤	≤	NUM
ejpam-5569	519	20	2k+1	2k+1	NOUN
ejpam-5569	519	21	,	,	PUNCT
ejpam-5569	519	22	for	for	ADP
ejpam-5569	519	23	this	this	PRON
ejpam-5569	519	24	,	,	PUNCT
ejpam-5569	519	25	the	the	DET
ejpam-5569	519	26	radio	radio	NOUN
ejpam-5569	519	27	condition	condition	NOUN
ejpam-5569	519	28	is	be	AUX
ejpam-5569	519	29	|f(x)−	|f(x)−	PROPN
ejpam-5569	519	30	f(y)|	f(y)|	PROPN
ejpam-5569	519	31	≥	≥	PRON
ejpam-5569	519	32	n	n	CCONJ
ejpam-5569	519	33	satisfied	satisfied	ADJ
ejpam-5569	519	34	by	by	ADP
ejpam-5569	519	35	every	every	DET
ejpam-5569	519	36	pair	pair	NOUN
ejpam-5569	519	37	of	of	ADP
ejpam-5569	519	38	vertices	vertex	NOUN
ejpam-5569	519	39	.	.	PUNCT
ejpam-5569	520	1	figure	figure	NOUN
ejpam-5569	520	2	4	4	NUM
ejpam-5569	520	3	.	.	NUM
ejpam-5569	520	4	,	,	PUNCT
ejpam-5569	520	5	illustrates	illustrate	VERB
ejpam-5569	520	6	the	the	DET
ejpam-5569	520	7	proof	proof	NOUN
ejpam-5569	520	8	of	of	ADP
ejpam-5569	520	9	this	this	DET
ejpam-5569	520	10	case	case	NOUN
ejpam-5569	520	11	,	,	PUNCT
ejpam-5569	520	12	presents	present	VERB
ejpam-5569	520	13	the	the	DET
ejpam-5569	520	14	labeling	labeling	NOUN
ejpam-5569	520	15	d2(p16	d2(p16	NOUN
ejpam-5569	520	16	)	)	PUNCT
ejpam-5569	520	17	according	accord	VERB
ejpam-5569	520	18	theorem	theorem	ADJ
ejpam-5569	520	19	3	3	NUM
ejpam-5569	520	20	.	.	PUNCT
ejpam-5569	520	21	figure	figure	NOUN
ejpam-5569	520	22	4	4	NUM
ejpam-5569	520	23	:	:	PUNCT
ejpam-5569	520	24	radio	radio	NOUN
ejpam-5569	520	25	labeling	labeling	NOUN
ejpam-5569	520	26	of	of	ADP
ejpam-5569	520	27	d2(p16	d2(p16	NOUN
ejpam-5569	520	28	)	)	PUNCT
ejpam-5569	520	29	following	follow	VERB
ejpam-5569	520	30	theorem	theorem	VERB
ejpam-5569	520	31	3	3	NUM
ejpam-5569	520	32	.	.	PUNCT
ejpam-5569	520	33	theorem	theorem	NOUN
ejpam-5569	520	34	4	4	NUM
ejpam-5569	520	35	.	.	PUNCT
ejpam-5569	521	1	let	let	VERB
ejpam-5569	521	2	n	n	PRON
ejpam-5569	521	3	>	>	X
ejpam-5569	521	4	6	6	NUM
ejpam-5569	521	5	be	be	AUX
ejpam-5569	521	6	a	a	DET
ejpam-5569	521	7	positive	positive	ADJ
ejpam-5569	521	8	integer	integer	NOUN
ejpam-5569	521	9	such	such	ADJ
ejpam-5569	521	10	that	that	SCONJ
ejpam-5569	521	11	n	n	NUM
ejpam-5569	521	12	≡	≡	PROPN
ejpam-5569	521	13	1(mod4	1(mod4	NUM
ejpam-5569	521	14	)	)	PUNCT
ejpam-5569	521	15	.	.	PUNCT
ejpam-5569	522	1	then	then	ADV
ejpam-5569	522	2	rn(d2(pn	rn(d2(pn	PROPN
ejpam-5569	522	3	)	)	PUNCT
ejpam-5569	522	4	)	)	PUNCT
ejpam-5569	523	1	≤	≤	NUM
ejpam-5569	523	2	n2	n2	NOUN
ejpam-5569	523	3	−	−	PROPN
ejpam-5569	523	4	2	2	NUM
ejpam-5569	523	5	.	.	PUNCT
ejpam-5569	523	6	proof	proof	NOUN
ejpam-5569	523	7	.	.	PUNCT
ejpam-5569	524	1	for	for	ADP
ejpam-5569	524	2	any	any	DET
ejpam-5569	524	3	positive	positive	ADJ
ejpam-5569	524	4	integer	integer	NOUN
ejpam-5569	524	5	n	n	PRON
ejpam-5569	524	6	≡	≡	PROPN
ejpam-5569	524	7	1(mod4	1(mod4	NUM
ejpam-5569	524	8	)	)	PUNCT
ejpam-5569	524	9	,	,	PUNCT
ejpam-5569	524	10	such	such	ADJ
ejpam-5569	524	11	that	that	SCONJ
ejpam-5569	524	12	n	n	NOUN
ejpam-5569	524	13	=	=	SYM
ejpam-5569	524	14	4k	4k	NOUN
ejpam-5569	524	15	+	+	NOUN
ejpam-5569	524	16	1	1	NUM
ejpam-5569	524	17	,	,	PUNCT
ejpam-5569	524	18	k	k	X
ejpam-5569	524	19	≥	≥	NUM
ejpam-5569	524	20	2	2	NUM
ejpam-5569	524	21	,	,	PUNCT
ejpam-5569	524	22	let	let	VERB
ejpam-5569	524	23	f	f	PRON
ejpam-5569	524	24	:	:	PUNCT
ejpam-5569	524	25	v	v	PROPN
ejpam-5569	524	26	(	(	PUNCT
ejpam-5569	524	27	d2(pn	d2(pn	PROPN
ejpam-5569	524	28	)	)	PUNCT
ejpam-5569	524	29	)	)	PUNCT
ejpam-5569	524	30	→	→	SYM
ejpam-5569	524	31	n	n	CCONJ
ejpam-5569	524	32	define	define	VERB
ejpam-5569	524	33	as	as	SCONJ
ejpam-5569	524	34	follows	follow	VERB
ejpam-5569	524	35	:	:	PUNCT
ejpam-5569	524	36	f(ui	f(ui	PROPN
ejpam-5569	524	37	)	)	PUNCT
ejpam-5569	525	1	=	=	SYM
ejpam-5569	525	2			PROPN
ejpam-5569	525	3	(	(	PUNCT
ejpam-5569	525	4	2n+	2n+	NUM
ejpam-5569	525	5	3)(i−	3)(i−	NUM
ejpam-5569	525	6	1	1	NUM
ejpam-5569	525	7	)	)	PUNCT
ejpam-5569	525	8	+	+	CCONJ
ejpam-5569	525	9	2(i−	2(i−	NUM
ejpam-5569	525	10	1)(i−	1)(i−	NUM
ejpam-5569	525	11	2	2	NUM
ejpam-5569	525	12	)	)	PUNCT
ejpam-5569	525	13	,	,	PUNCT
ejpam-5569	525	14	if	if	SCONJ
ejpam-5569	525	15	1	1	NUM
ejpam-5569	525	16	≤	≤	NUM
ejpam-5569	525	17	i	i	NOUN
ejpam-5569	525	18	≤	≤	NOUN
ejpam-5569	525	19	k	k	X
ejpam-5569	525	20	f(uk	f(uk	PROPN
ejpam-5569	525	21	)	)	PUNCT
ejpam-5569	525	22	+	+	PUNCT
ejpam-5569	525	23	2n−	2n−	NUM
ejpam-5569	525	24	3	3	NUM
ejpam-5569	525	25	if	if	SCONJ
ejpam-5569	525	26	i	i	PRON
ejpam-5569	525	27	=	=	PUNCT
ejpam-5569	525	28	k	k	PROPN
ejpam-5569	526	1	+	+	CCONJ
ejpam-5569	526	2	1	1	NUM
ejpam-5569	526	3	f(uk)−	f(uk)−	NOUN
ejpam-5569	526	4	(	(	PUNCT
ejpam-5569	526	5	2n−	2n−	NUM
ejpam-5569	526	6	3)−	3)−	PROPN
ejpam-5569	526	7	(	(	PUNCT
ejpam-5569	526	8	3n−	3n−	NUM
ejpam-5569	526	9	8)(j	8)(j	NUM
ejpam-5569	526	10	−	−	NUM
ejpam-5569	526	11	1	1	NUM
ejpam-5569	526	12	)	)	PUNCT
ejpam-5569	526	13	+	+	CCONJ
ejpam-5569	526	14	2(j	2(j	NUM
ejpam-5569	526	15	−	−	PROPN
ejpam-5569	526	16	1)(j	1)(j	NUM
ejpam-5569	526	17	−	−	PROPN
ejpam-5569	526	18	2	2	NUM
ejpam-5569	526	19	)	)	PUNCT
ejpam-5569	526	20	,	,	PUNCT
ejpam-5569	526	21	if	if	SCONJ
ejpam-5569	526	22	k	k	PROPN
ejpam-5569	526	23	+	+	CCONJ
ejpam-5569	526	24	2	2	X
ejpam-5569	526	25	≤	≤	NUM
ejpam-5569	526	26	i	i	NOUN
ejpam-5569	526	27	≤	≤	ADJ
ejpam-5569	526	28	2k	2k	NUM
ejpam-5569	526	29	where	where	SCONJ
ejpam-5569	526	30	1	1	NUM
ejpam-5569	526	31	≤	≤	NUM
ejpam-5569	526	32	j	j	PROPN
ejpam-5569	526	33	≤	≤	PROPN
ejpam-5569	526	34	k	k	NOUN
ejpam-5569	527	1	−	−	PROPN
ejpam-5569	527	2	1	1	NUM
ejpam-5569	527	3	,	,	PUNCT
ejpam-5569	527	4	and	and	CCONJ
ejpam-5569	527	5	i	i	PRON
ejpam-5569	527	6	=	=	PUNCT
ejpam-5569	528	1	k	k	PROPN
ejpam-5569	529	1	+	+	CCONJ
ejpam-5569	529	2	1	1	NUM
ejpam-5569	529	3	+	+	CCONJ
ejpam-5569	529	4	j	j	PROPN
ejpam-5569	529	5	n2	n2	NOUN
ejpam-5569	529	6	−	−	PROPN
ejpam-5569	529	7	2	2	NUM
ejpam-5569	529	8	,	,	PUNCT
ejpam-5569	529	9	if	if	SCONJ
ejpam-5569	529	10	i	i	PRON
ejpam-5569	529	11	=	=	SYM
ejpam-5569	529	12	2k	2k	NUM
ejpam-5569	529	13	+	+	CCONJ
ejpam-5569	529	14	1	1	NUM
ejpam-5569	529	15	f(uk+1+j	f(uk+1+j	NUM
ejpam-5569	529	16	)	)	PUNCT
ejpam-5569	530	1	+	+	CCONJ
ejpam-5569	530	2	(	(	PUNCT
ejpam-5569	530	3	6k	6k	NOUN
ejpam-5569	530	4	−	−	PROPN
ejpam-5569	530	5	2)−	2)−	NUM
ejpam-5569	530	6	2(j	2(j	NUM
ejpam-5569	530	7	−	−	NOUN
ejpam-5569	530	8	1	1	NUM
ejpam-5569	530	9	)	)	PUNCT
ejpam-5569	530	10	,	,	PUNCT
ejpam-5569	530	11	if	if	SCONJ
ejpam-5569	530	12	2k	2k	NUM
ejpam-5569	530	13	+	+	CCONJ
ejpam-5569	530	14	2	2	NUM
ejpam-5569	530	15	≤	≤	NUM
ejpam-5569	530	16	i	i	PRON
ejpam-5569	530	17	≤	≤	NOUN
ejpam-5569	530	18	3k	3k	NUM
ejpam-5569	530	19	,	,	PUNCT
ejpam-5569	530	20	where	where	SCONJ
ejpam-5569	530	21	1	1	NUM
ejpam-5569	530	22	≤	≤	NUM
ejpam-5569	530	23	j	j	PROPN
ejpam-5569	530	24	≤	≤	PROPN
ejpam-5569	531	1	k	k	NOUN
ejpam-5569	532	1	−	−	PROPN
ejpam-5569	532	2	1	1	NUM
ejpam-5569	532	3	,	,	PUNCT
ejpam-5569	532	4	and	and	CCONJ
ejpam-5569	532	5	i	i	PRON
ejpam-5569	532	6	=	=	NOUN
ejpam-5569	533	1	3k	3k	PRON
ejpam-5569	533	2	+	+	CCONJ
ejpam-5569	533	3	1−	1−	NUM
ejpam-5569	533	4	j	j	NOUN
ejpam-5569	533	5	f(uj	f(uj	PROPN
ejpam-5569	533	6	)	)	PUNCT
ejpam-5569	534	1	+	+	CCONJ
ejpam-5569	534	2	2j	2j	NUM
ejpam-5569	534	3	−	−	NOUN
ejpam-5569	534	4	1	1	NUM
ejpam-5569	534	5	,	,	PUNCT
ejpam-5569	534	6	if	if	SCONJ
ejpam-5569	534	7	3k	3k	PRON
ejpam-5569	534	8	+	+	CCONJ
ejpam-5569	534	9	1	1	NUM
ejpam-5569	534	10	≤	≤	NUM
ejpam-5569	534	11	i	i	PRON
ejpam-5569	534	12	≤	≤	NOUN
ejpam-5569	534	13	4k	4k	NUM
ejpam-5569	534	14	+	+	CCONJ
ejpam-5569	534	15	1	1	NUM
ejpam-5569	534	16	=	=	SYM
ejpam-5569	534	17	n	n	X
ejpam-5569	534	18	where	where	SCONJ
ejpam-5569	534	19	1	1	NUM
ejpam-5569	534	20	≤	≤	NUM
ejpam-5569	534	21	j	j	PROPN
ejpam-5569	534	22	≤	≤	PROPN
ejpam-5569	534	23	k	k	PROPN
ejpam-5569	535	1	+	+	CCONJ
ejpam-5569	535	2	1	1	NUM
ejpam-5569	535	3	and	and	CCONJ
ejpam-5569	535	4	i	i	PRON
ejpam-5569	535	5	=	=	PUNCT
ejpam-5569	535	6	4k	4k	NUM
ejpam-5569	535	7	+	+	CCONJ
ejpam-5569	535	8	2−	2−	NUM
ejpam-5569	535	9	j	j	NOUN
ejpam-5569	535	10	and	and	CCONJ
ejpam-5569	535	11	n.	n.	PROPN
ejpam-5569	535	12	m.	m.	NOUN
ejpam-5569	535	13	noureldeen	noureldeen	NOUN
ejpam-5569	535	14	et	et	PROPN
ejpam-5569	536	1	al	al	PROPN
ejpam-5569	536	2	.	.	PUNCT
ejpam-5569	536	3	/	/	SYM
ejpam-5569	536	4	eur	eur	PROPN
ejpam-5569	536	5	.	.	PUNCT
ejpam-5569	537	1	j.	j.	PROPN
ejpam-5569	537	2	pure	pure	PROPN
ejpam-5569	537	3	appl	appl	PROPN
ejpam-5569	537	4	.	.	PROPN
ejpam-5569	537	5	math	math	PROPN
ejpam-5569	537	6	,	,	PUNCT
ejpam-5569	537	7	18	18	NUM
ejpam-5569	537	8	(	(	PUNCT
ejpam-5569	537	9	2	2	NUM
ejpam-5569	537	10	)	)	PUNCT
ejpam-5569	537	11	(	(	PUNCT
ejpam-5569	537	12	2025	2025	NUM
ejpam-5569	537	13	)	)	PUNCT
ejpam-5569	537	14	,	,	PUNCT
ejpam-5569	537	15	5569	5569	NUM
ejpam-5569	537	16	18	18	NUM
ejpam-5569	537	17	of	of	ADP
ejpam-5569	537	18	25	25	NUM
ejpam-5569	537	19	f(vi	f(vi	NOUN
ejpam-5569	537	20	)	)	PUNCT
ejpam-5569	537	21	=	=	SYM
ejpam-5569	537	22			PROPN
ejpam-5569	537	23	(	(	PUNCT
ejpam-5569	537	24	n+	n+	NOUN
ejpam-5569	537	25	3	3	NUM
ejpam-5569	537	26	)	)	PUNCT
ejpam-5569	537	27	+	+	CCONJ
ejpam-5569	537	28	(	(	PUNCT
ejpam-5569	537	29	2n+	2n+	NUM
ejpam-5569	537	30	7)(i−	7)(i−	NUM
ejpam-5569	537	31	1	1	NUM
ejpam-5569	537	32	)	)	PUNCT
ejpam-5569	537	33	+	+	CCONJ
ejpam-5569	538	1	2(i−	2(i−	NUM
ejpam-5569	538	2	1)(i−	1)(i−	NUM
ejpam-5569	538	3	2	2	NUM
ejpam-5569	538	4	)	)	PUNCT
ejpam-5569	538	5	,	,	PUNCT
ejpam-5569	538	6	if	if	SCONJ
ejpam-5569	538	7	1	1	NUM
ejpam-5569	538	8	≤	≤	NUM
ejpam-5569	538	9	i	i	NOUN
ejpam-5569	538	10	≤	≤	NOUN
ejpam-5569	539	1	k	k	PRON
ejpam-5569	539	2	−	−	NUM
ejpam-5569	539	3	1	1	NUM
ejpam-5569	539	4	f(uk−1	f(uk−1	NOUN
ejpam-5569	539	5	)	)	PUNCT
ejpam-5569	539	6	+	+	CCONJ
ejpam-5569	540	1	2n−	2n−	NUM
ejpam-5569	540	2	2	2	NUM
ejpam-5569	540	3	if	if	SCONJ
ejpam-5569	540	4	i	i	PRON
ejpam-5569	540	5	=	=	SYM
ejpam-5569	540	6	k	k	X
ejpam-5569	540	7	f(vk	f(vk	PROPN
ejpam-5569	540	8	)	)	PUNCT
ejpam-5569	540	9	+	+	CCONJ
ejpam-5569	540	10	(	(	PUNCT
ejpam-5569	540	11	2n	2n	NUM
ejpam-5569	540	12	)	)	PUNCT
ejpam-5569	540	13	+	+	CCONJ
ejpam-5569	540	14	(	(	PUNCT
ejpam-5569	540	15	n+	n+	NUM
ejpam-5569	540	16	2)(j	2)(j	NUM
ejpam-5569	540	17	−	−	NOUN
ejpam-5569	540	18	1	1	NUM
ejpam-5569	540	19	)	)	PUNCT
ejpam-5569	540	20	+	+	CCONJ
ejpam-5569	541	1	2(j	2(j	NUM
ejpam-5569	541	2	−	−	PROPN
ejpam-5569	541	3	1)(j	1)(j	NUM
ejpam-5569	541	4	−	−	PROPN
ejpam-5569	541	5	2	2	NUM
ejpam-5569	541	6	)	)	PUNCT
ejpam-5569	541	7	,	,	PUNCT
ejpam-5569	541	8	if	if	SCONJ
ejpam-5569	541	9	k	k	PROPN
ejpam-5569	541	10	+	+	PROPN
ejpam-5569	541	11	1	1	X
ejpam-5569	541	12	≤	≤	NUM
ejpam-5569	541	13	i	i	NOUN
ejpam-5569	541	14	≤	≤	ADJ
ejpam-5569	541	15	2k	2k	NUM
ejpam-5569	541	16	,	,	PUNCT
ejpam-5569	541	17	where	where	SCONJ
ejpam-5569	541	18	1	1	NUM
ejpam-5569	541	19	≤	≤	NUM
ejpam-5569	541	20	j	j	PROPN
ejpam-5569	541	21	≤	≤	PROPN
ejpam-5569	541	22	k	k	PROPN
ejpam-5569	541	23	,	,	PUNCT
ejpam-5569	541	24	and	and	CCONJ
ejpam-5569	541	25	i	i	PRON
ejpam-5569	541	26	=	=	PUNCT
ejpam-5569	541	27	k	k	PROPN
ejpam-5569	542	1	+	+	CCONJ
ejpam-5569	542	2	j	j	PROPN
ejpam-5569	542	3	n2	n2	NOUN
ejpam-5569	542	4	−	−	PROPN
ejpam-5569	542	5	2	2	NUM
ejpam-5569	542	6	,	,	PUNCT
ejpam-5569	542	7	if	if	SCONJ
ejpam-5569	542	8	i	i	PRON
ejpam-5569	542	9	=	=	SYM
ejpam-5569	542	10	2k	2k	NUM
ejpam-5569	542	11	+	+	CCONJ
ejpam-5569	542	12	1	1	NUM
ejpam-5569	542	13	f(vk−1+j	f(vk−1+j	NOUN
ejpam-5569	542	14	)	)	PUNCT
ejpam-5569	542	15	+	+	CCONJ
ejpam-5569	542	16	(	(	PUNCT
ejpam-5569	542	17	2k	2k	NUM
ejpam-5569	542	18	+	+	CCONJ
ejpam-5569	542	19	1	1	NUM
ejpam-5569	542	20	)	)	PUNCT
ejpam-5569	542	21	+	+	CCONJ
ejpam-5569	542	22	2(j	2(j	NUM
ejpam-5569	542	23	−	−	NOUN
ejpam-5569	542	24	1	1	NUM
ejpam-5569	542	25	)	)	PUNCT
ejpam-5569	542	26	,	,	PUNCT
ejpam-5569	542	27	if	if	SCONJ
ejpam-5569	542	28	2k	2k	NUM
ejpam-5569	542	29	+	+	CCONJ
ejpam-5569	542	30	2	2	NUM
ejpam-5569	542	31	≤	≤	NUM
ejpam-5569	542	32	i	i	PRON
ejpam-5569	542	33	≤	≤	NOUN
ejpam-5569	542	34	3k	3k	X
ejpam-5569	543	1	+	+	CCONJ
ejpam-5569	543	2	2	2	NUM
ejpam-5569	543	3	where	where	SCONJ
ejpam-5569	543	4	1	1	NUM
ejpam-5569	543	5	≤	≤	NUM
ejpam-5569	543	6	j	j	PROPN
ejpam-5569	543	7	≤	≤	PROPN
ejpam-5569	543	8	k	k	PROPN
ejpam-5569	544	1	+	+	CCONJ
ejpam-5569	544	2	1	1	NUM
ejpam-5569	544	3	,	,	PUNCT
ejpam-5569	544	4	and	and	CCONJ
ejpam-5569	544	5	i	i	PRON
ejpam-5569	544	6	=	=	NOUN
ejpam-5569	545	1	3k	3k	X
ejpam-5569	546	1	+	+	CCONJ
ejpam-5569	546	2	3−	3−	NUM
ejpam-5569	546	3	j	j	PROPN
ejpam-5569	546	4	f(vj)−	f(vj)−	NOUN
ejpam-5569	546	5	2j	2j	NUM
ejpam-5569	546	6	,	,	PUNCT
ejpam-5569	546	7	if	if	SCONJ
ejpam-5569	546	8	3k	3k	PRON
ejpam-5569	546	9	+	+	CCONJ
ejpam-5569	546	10	3	3	NUM
ejpam-5569	546	11	≤	≤	NUM
ejpam-5569	547	1	i	i	PRON
ejpam-5569	547	2	≤	≤	NOUN
ejpam-5569	547	3	4k	4k	NUM
ejpam-5569	547	4	+	+	CCONJ
ejpam-5569	547	5	1	1	NUM
ejpam-5569	547	6	=	=	SYM
ejpam-5569	547	7	n	n	X
ejpam-5569	547	8	where	where	SCONJ
ejpam-5569	547	9	1	1	NUM
ejpam-5569	547	10	≤	≤	NUM
ejpam-5569	547	11	j	j	PROPN
ejpam-5569	547	12	≤	≤	PROPN
ejpam-5569	548	1	k	k	NOUN
ejpam-5569	549	1	−	−	PROPN
ejpam-5569	549	2	1	1	NUM
ejpam-5569	549	3	,	,	PUNCT
ejpam-5569	549	4	and	and	CCONJ
ejpam-5569	549	5	i	i	PRON
ejpam-5569	549	6	=	=	PUNCT
ejpam-5569	549	7	4k	4k	NUM
ejpam-5569	549	8	+	+	CCONJ
ejpam-5569	549	9	2−	2−	NUM
ejpam-5569	549	10	j	j	NOUN
ejpam-5569	549	11	the	the	DET
ejpam-5569	549	12	proof	proof	NOUN
ejpam-5569	549	13	is	be	AUX
ejpam-5569	549	14	left	leave	VERB
ejpam-5569	549	15	to	to	ADP
ejpam-5569	549	16	the	the	DET
ejpam-5569	549	17	reader	reader	NOUN
ejpam-5569	549	18	.	.	PUNCT
ejpam-5569	550	1	we	we	PRON
ejpam-5569	550	2	omit	omit	VERB
ejpam-5569	550	3	the	the	DET
ejpam-5569	550	4	proof	proof	NOUN
ejpam-5569	550	5	,	,	PUNCT
ejpam-5569	550	6	but	but	CCONJ
ejpam-5569	550	7	figure	figure	VERB
ejpam-5569	550	8	5	5	NUM
ejpam-5569	550	9	,	,	PUNCT
ejpam-5569	550	10	presents	present	VERB
ejpam-5569	550	11	the	the	DET
ejpam-5569	550	12	labeling	labeling	NOUN
ejpam-5569	550	13	d2(p13	d2(p13	NOUN
ejpam-5569	550	14	)	)	PUNCT
ejpam-5569	550	15	illustrates	illustrate	VERB
ejpam-5569	550	16	theorem	theorem	VERB
ejpam-5569	550	17	4	4	NUM
ejpam-5569	550	18	namely	namely	ADV
ejpam-5569	550	19	,	,	PUNCT
ejpam-5569	550	20	n	n	CCONJ
ejpam-5569	550	21	≡	≡	PROPN
ejpam-5569	550	22	1	1	NUM
ejpam-5569	550	23	(	(	PUNCT
ejpam-5569	550	24	mod	mod	NOUN
ejpam-5569	550	25	4	4	NUM
ejpam-5569	550	26	)	)	PUNCT
ejpam-5569	550	27	,	,	PUNCT
ejpam-5569	550	28	if	if	SCONJ
ejpam-5569	550	29	n	n	NOUN
ejpam-5569	550	30	=	=	SYM
ejpam-5569	550	31	4k	4k	NUM
ejpam-5569	550	32	+	+	NOUN
ejpam-5569	550	33	1	1	NUM
ejpam-5569	550	34	,	,	PUNCT
ejpam-5569	550	35	k	k	NOUN
ejpam-5569	550	36	=	=	SYM
ejpam-5569	550	37	3	3	X
ejpam-5569	550	38	.	.	X
ejpam-5569	550	39	figure	figure	NOUN
ejpam-5569	550	40	5	5	NUM
ejpam-5569	550	41	:	:	PUNCT
ejpam-5569	550	42	radio	radio	NOUN
ejpam-5569	550	43	labeling	labeling	NOUN
ejpam-5569	550	44	d2(p13	d2(p13	PROPN
ejpam-5569	550	45	)	)	PUNCT
ejpam-5569	550	46	following	follow	VERB
ejpam-5569	550	47	theorem	theorem	VERB
ejpam-5569	550	48	4	4	NUM
ejpam-5569	550	49	.	.	NOUN
ejpam-5569	550	50	4	4	NUM
ejpam-5569	550	51	.	.	X
ejpam-5569	551	1	integer	integer	PROPN
ejpam-5569	551	2	linear	linear	PROPN
ejpam-5569	551	3	programming	programming	NOUN
ejpam-5569	551	4	model	model	NOUN
ejpam-5569	551	5	(	(	PUNCT
ejpam-5569	551	6	ilpm	ilpm	PROPN
ejpam-5569	551	7	)	)	PUNCT
ejpam-5569	551	8	for	for	ADP
ejpam-5569	551	9	resolving	resolve	VERB
ejpam-5569	551	10	optimization	optimization	NOUN
ejpam-5569	551	11	issues	issue	NOUN
ejpam-5569	551	12	,	,	PUNCT
ejpam-5569	551	13	mathematical	mathematical	ADJ
ejpam-5569	551	14	programming	programming	NOUN
ejpam-5569	551	15	is	be	AUX
ejpam-5569	551	16	a	a	DET
ejpam-5569	551	17	crucial	crucial	ADJ
ejpam-5569	551	18	instrument	instrument	NOUN
ejpam-5569	551	19	.	.	PUNCT
ejpam-5569	552	1	the	the	DET
ejpam-5569	552	2	references	reference	NOUN
ejpam-5569	552	3	[	[	X
ejpam-5569	552	4	18–20	18–20	NUM
ejpam-5569	552	5	]	]	PUNCT
ejpam-5569	552	6	provide	provide	VERB
ejpam-5569	552	7	specific	specific	ADJ
ejpam-5569	552	8	information	information	NOUN
ejpam-5569	552	9	regarding	regard	VERB
ejpam-5569	552	10	the	the	DET
ejpam-5569	552	11	process	process	NOUN
ejpam-5569	552	12	of	of	ADP
ejpam-5569	552	13	conceptualizing	conceptualize	VERB
ejpam-5569	552	14	real	real	ADJ
ejpam-5569	552	15	-	-	PUNCT
ejpam-5569	552	16	life	life	NOUN
ejpam-5569	552	17	problems	problem	NOUN
ejpam-5569	552	18	as	as	ADP
ejpam-5569	552	19	mathematical	mathematical	ADJ
ejpam-5569	552	20	models	model	NOUN
ejpam-5569	552	21	.	.	PUNCT
ejpam-5569	553	1	for	for	ADP
ejpam-5569	553	2	a	a	DET
ejpam-5569	553	3	connected	connected	ADJ
ejpam-5569	553	4	graph	graph	NOUN
ejpam-5569	553	5	g	g	NOUN
ejpam-5569	553	6	of	of	ADP
ejpam-5569	553	7	order	order	NOUN
ejpam-5569	553	8	n.	n.	NOUN
ejpam-5569	553	9	let	let	VERB
ejpam-5569	553	10	v	v	NOUN
ejpam-5569	553	11	(	(	PUNCT
ejpam-5569	553	12	g	g	NOUN
ejpam-5569	553	13	)	)	PUNCT
ejpam-5569	553	14	=	=	SYM
ejpam-5569	553	15	{	{	PUNCT
ejpam-5569	553	16	x1	x1	PROPN
ejpam-5569	553	17	,	,	PUNCT
ejpam-5569	553	18	x2	x2	PROPN
ejpam-5569	553	19	,	,	PUNCT
ejpam-5569	553	20	·	·	PUNCT
ejpam-5569	553	21	·	·	PUNCT
ejpam-5569	553	22	·	·	PUNCT
ejpam-5569	553	23	,	,	PUNCT
ejpam-5569	553	24	xn	xn	PROPN
ejpam-5569	553	25	}	}	PUNCT
ejpam-5569	553	26	.	.	PUNCT
ejpam-5569	554	1	we	we	PRON
ejpam-5569	554	2	construct	construct	VERB
ejpam-5569	554	3	the	the	DET
ejpam-5569	554	4	distance	distance	NOUN
ejpam-5569	554	5	matrix	matrix	NOUN
ejpam-5569	554	6	d	d	NOUN
ejpam-5569	555	1	=	=	PUNCT
ejpam-5569	556	1	[	[	X
ejpam-5569	556	2	dij	dij	X
ejpam-5569	556	3	]	]	X
ejpam-5569	556	4	of	of	ADP
ejpam-5569	556	5	g	g	PROPN
ejpam-5569	556	6	where	where	SCONJ
ejpam-5569	556	7	dij	dij	PROPN
ejpam-5569	556	8	=	=	SYM
ejpam-5569	557	1	d	d	PROPN
ejpam-5569	557	2	(	(	PUNCT
ejpam-5569	557	3	xi	xi	PROPN
ejpam-5569	557	4	,	,	PUNCT
ejpam-5569	557	5	xj	xj	PROPN
ejpam-5569	557	6	)	)	PUNCT
ejpam-5569	557	7	for	for	ADP
ejpam-5569	557	8	1	1	NUM
ejpam-5569	557	9	≤	≤	NOUN
ejpam-5569	558	1	i	i	PRON
ejpam-5569	558	2	,	,	PUNCT
ejpam-5569	558	3	j	j	PROPN
ejpam-5569	558	4	≤	≤	PROPN
ejpam-5569	558	5	n.	n.	NOUN
ejpam-5569	558	6	for	for	ADP
ejpam-5569	558	7	1	1	NUM
ejpam-5569	558	8	≤	≤	NUM
ejpam-5569	558	9	i	i	PRON
ejpam-5569	558	10	≤	≤	PROPN
ejpam-5569	558	11	n	n	CCONJ
ejpam-5569	558	12	,	,	PUNCT
ejpam-5569	558	13	assume	assume	VERB
ejpam-5569	558	14	that	that	SCONJ
ejpam-5569	558	15	yi	yi	PROPN
ejpam-5569	558	16	is	be	AUX
ejpam-5569	558	17	the	the	DET
ejpam-5569	558	18	radio	radio	NOUN
ejpam-5569	558	19	label	label	NOUN
ejpam-5569	558	20	of	of	ADP
ejpam-5569	558	21	the	the	DET
ejpam-5569	558	22	vertex	vertex	NOUN
ejpam-5569	558	23	xi	xi	PROPN
ejpam-5569	558	24	.	.	PUNCT
ejpam-5569	559	1	now	now	ADV
ejpam-5569	559	2	,	,	PUNCT
ejpam-5569	559	3	we	we	PRON
ejpam-5569	559	4	can	can	AUX
ejpam-5569	559	5	use	use	VERB
ejpam-5569	559	6	integer	integer	NOUN
ejpam-5569	559	7	programming	programming	NOUN
ejpam-5569	559	8	to	to	PART
ejpam-5569	559	9	present	present	VERB
ejpam-5569	559	10	the	the	DET
ejpam-5569	559	11	mathematical	mathematical	ADJ
ejpam-5569	559	12	model	model	NOUN
ejpam-5569	559	13	for	for	ADP
ejpam-5569	559	14	the	the	DET
ejpam-5569	559	15	radio	radio	NOUN
ejpam-5569	559	16	labeling	labeling	NOUN
ejpam-5569	559	17	problem	problem	NOUN
ejpam-5569	559	18	.	.	PUNCT
ejpam-5569	560	1	the	the	DET
ejpam-5569	560	2	function	function	NOUN
ejpam-5569	560	3	f	f	PROPN
ejpam-5569	560	4	is	be	AUX
ejpam-5569	560	5	denoted	denote	VERB
ejpam-5569	560	6	as	as	ADP
ejpam-5569	560	7	f	f	PROPN
ejpam-5569	560	8	=	=	PROPN
ejpam-5569	560	9	y1	y1	PROPN
ejpam-5569	560	10	+	+	CCONJ
ejpam-5569	560	11	y2	y2	PROPN
ejpam-5569	560	12	+	+	SYM
ejpam-5569	560	13	·	·	PUNCT
ejpam-5569	560	14	·	·	PUNCT
ejpam-5569	560	15	·	·	PUNCT
ejpam-5569	561	1	+	+	CCONJ
ejpam-5569	561	2	yn	yn	PRON
ejpam-5569	561	3	the	the	DET
ejpam-5569	561	4	claim	claim	NOUN
ejpam-5569	561	5	is	be	AUX
ejpam-5569	561	6	minimizing	minimize	VERB
ejpam-5569	561	7	f	f	PROPN
ejpam-5569	561	8	subject	subject	NOUN
ejpam-5569	561	9	to	to	ADP
ejpam-5569	561	10	the	the	DET
ejpam-5569	561	11	(	(	PUNCT
ejpam-5569	561	12	n	n	CCONJ
ejpam-5569	561	13	2	2	NUM
ejpam-5569	561	14	)	)	PUNCT
ejpam-5569	561	15	constraints	constraint	NOUN
ejpam-5569	561	16	|yi	|yi	PRON
ejpam-5569	561	17	−	−	PROPN
ejpam-5569	561	18	yj	yj	PROPN
ejpam-5569	561	19	|	|	ADV
ejpam-5569	561	20	≥	≥	NUM
ejpam-5569	561	21	n−	n−	PROPN
ejpam-5569	561	22	d	d	X
ejpam-5569	561	23	(	(	PUNCT
ejpam-5569	561	24	xi	xi	PROPN
ejpam-5569	561	25	,	,	PUNCT
ejpam-5569	561	26	xj	xj	PROPN
ejpam-5569	561	27	)	)	PUNCT
ejpam-5569	561	28	for	for	ADP
ejpam-5569	561	29	1	1	NUM
ejpam-5569	561	30	≤	≤	NUM
ejpam-5569	562	1	i	i	PRON
ejpam-5569	562	2	≤	≤	ADJ
ejpam-5569	562	3	n−	n−	PROPN
ejpam-5569	562	4	1	1	NUM
ejpam-5569	562	5	;	;	PUNCT
ejpam-5569	562	6	2	2	NUM
ejpam-5569	562	7	≤	≤	NUM
ejpam-5569	562	8	j	j	PROPN
ejpam-5569	562	9	≤	≤	PROPN
ejpam-5569	562	10	n	n	CCONJ
ejpam-5569	562	11	and	and	CCONJ
ejpam-5569	562	12	i	i	PRON
ejpam-5569	562	13	<	<	X
ejpam-5569	562	14	j	j	PROPN
ejpam-5569	562	15	where	where	SCONJ
ejpam-5569	562	16	y1	y1	NOUN
ejpam-5569	562	17	,	,	PUNCT
ejpam-5569	562	18	y2	y2	PROPN
ejpam-5569	562	19	,	,	PUNCT
ejpam-5569	562	20	·	·	PUNCT
ejpam-5569	562	21	·	·	PUNCT
ejpam-5569	562	22	·	·	PUNCT
ejpam-5569	562	23	,	,	PUNCT
ejpam-5569	562	24	yn	yn	PRON
ejpam-5569	562	25	are	be	AUX
ejpam-5569	562	26	integer	integer	NOUN
ejpam-5569	562	27	numbers	number	NOUN
ejpam-5569	562	28	.	.	PUNCT
ejpam-5569	563	1	therefore	therefore	ADV
ejpam-5569	563	2	,	,	PUNCT
ejpam-5569	563	3	rn	rn	PROPN
ejpam-5569	563	4	(	(	PUNCT
ejpam-5569	563	5	g	g	NOUN
ejpam-5569	563	6	)	)	PUNCT
ejpam-5569	563	7	=	=	PUNCT
ejpam-5569	563	8	max1≤i≤n	max1≤i≤n	NOUN
ejpam-5569	563	9	{	{	PUNCT
ejpam-5569	563	10	yi	yi	PROPN
ejpam-5569	563	11	}	}	PUNCT
ejpam-5569	563	12	.	.	PUNCT
ejpam-5569	564	1	5	5	X
ejpam-5569	564	2	.	.	X
ejpam-5569	564	3	experimental	experimental	ADJ
ejpam-5569	564	4	study	study	PROPN
ejpam-5569	564	5	hereafter	hereafter	ADV
ejpam-5569	564	6	,	,	PUNCT
ejpam-5569	564	7	a	a	DET
ejpam-5569	564	8	set	set	NOUN
ejpam-5569	564	9	of	of	ADP
ejpam-5569	564	10	experiments	experiment	NOUN
ejpam-5569	564	11	have	have	AUX
ejpam-5569	564	12	been	be	AUX
ejpam-5569	564	13	performed	perform	VERB
ejpam-5569	564	14	in	in	ADP
ejpam-5569	564	15	order	order	NOUN
ejpam-5569	564	16	to	to	PART
ejpam-5569	564	17	compare	compare	VERB
ejpam-5569	564	18	the	the	DET
ejpam-5569	564	19	performance	performance	NOUN
ejpam-5569	564	20	of	of	ADP
ejpam-5569	564	21	algorithms	algorithm	NOUN
ejpam-5569	564	22	presented	present	VERB
ejpam-5569	564	23	in	in	ADP
ejpam-5569	564	24	[	[	X
ejpam-5569	564	25	6	6	NUM
ejpam-5569	564	26	]	]	PUNCT
ejpam-5569	564	27	with	with	ADP
ejpam-5569	564	28	the	the	DET
ejpam-5569	564	29	obtained	obtain	VERB
ejpam-5569	564	30	upper	upper	ADJ
ejpam-5569	564	31	bounds	bound	NOUN
ejpam-5569	564	32	by	by	ADP
ejpam-5569	564	33	theorems	theorem	NOUN
ejpam-5569	564	34	1	1	NUM
ejpam-5569	564	35	-	-	SYM
ejpam-5569	564	36	4	4	NUM
ejpam-5569	564	37	.	.	PUNCT
ejpam-5569	565	1	furthermore	furthermore	ADV
ejpam-5569	565	2	,	,	PUNCT
ejpam-5569	565	3	a	a	DET
ejpam-5569	565	4	comparison	comparison	NOUN
ejpam-5569	565	5	is	be	AUX
ejpam-5569	565	6	carried	carry	VERB
ejpam-5569	565	7	out	out	ADP
ejpam-5569	565	8	between	between	ADP
ejpam-5569	565	9	the	the	DET
ejpam-5569	565	10	findings	finding	NOUN
ejpam-5569	565	11	of	of	ADP
ejpam-5569	565	12	those	those	DET
ejpam-5569	565	13	theorems	theorem	NOUN
ejpam-5569	565	14	and	and	CCONJ
ejpam-5569	565	15	the	the	DET
ejpam-5569	565	16	mathematical	mathematical	ADJ
ejpam-5569	565	17	model	model	NOUN
ejpam-5569	565	18	that	that	PRON
ejpam-5569	565	19	was	be	AUX
ejpam-5569	565	20	presented	present	VERB
ejpam-5569	565	21	in	in	ADP
ejpam-5569	565	22	section	section	NOUN
ejpam-5569	565	23	4	4	NUM
ejpam-5569	565	24	.	.	PUNCT
ejpam-5569	566	1	in	in	ADP
ejpam-5569	566	2	our	our	PRON
ejpam-5569	566	3	experiments	experiment	NOUN
ejpam-5569	566	4	we	we	PRON
ejpam-5569	566	5	have	have	AUX
ejpam-5569	566	6	used	use	VERB
ejpam-5569	566	7	a	a	DET
ejpam-5569	566	8	n.	n.	NOUN
ejpam-5569	566	9	m.	m.	NOUN
ejpam-5569	566	10	noureldeen	noureldeen	NOUN
ejpam-5569	566	11	et	et	PROPN
ejpam-5569	566	12	al	al	PROPN
ejpam-5569	566	13	.	.	PUNCT
ejpam-5569	566	14	/	/	SYM
ejpam-5569	566	15	eur	eur	PROPN
ejpam-5569	566	16	.	.	PUNCT
ejpam-5569	567	1	j.	j.	PROPN
ejpam-5569	567	2	pure	pure	PROPN
ejpam-5569	567	3	appl	appl	PROPN
ejpam-5569	567	4	.	.	PROPN
ejpam-5569	567	5	math	math	PROPN
ejpam-5569	567	6	,	,	PUNCT
ejpam-5569	567	7	18	18	NUM
ejpam-5569	567	8	(	(	PUNCT
ejpam-5569	567	9	2	2	NUM
ejpam-5569	567	10	)	)	PUNCT
ejpam-5569	567	11	(	(	PUNCT
ejpam-5569	567	12	2025	2025	NUM
ejpam-5569	567	13	)	)	PUNCT
ejpam-5569	567	14	,	,	PUNCT
ejpam-5569	567	15	5569	5569	NUM
ejpam-5569	567	16	19	19	NUM
ejpam-5569	567	17	of	of	ADP
ejpam-5569	567	18	25	25	NUM
ejpam-5569	567	19	pc	pc	NOUN
ejpam-5569	567	20	with	with	ADP
ejpam-5569	567	21	core	core	NOUN
ejpam-5569	567	22	i7	i7	NOUN
ejpam-5569	567	23	processor	processor	NOUN
ejpam-5569	567	24	with	with	ADP
ejpam-5569	567	25	2.8	2.8	NUM
ejpam-5569	567	26	ghz	ghz	NOUN
ejpam-5569	567	27	cpu	cpu	NOUN
ejpam-5569	567	28	and	and	CCONJ
ejpam-5569	567	29	8	8	NUM
ejpam-5569	567	30	gb	gb	NOUN
ejpam-5569	567	31	of	of	ADP
ejpam-5569	567	32	ram	ram	NOUN
ejpam-5569	567	33	.	.	PUNCT
ejpam-5569	568	1	our	our	PRON
ejpam-5569	568	2	implementations	implementation	NOUN
ejpam-5569	568	3	are	be	AUX
ejpam-5569	568	4	done	do	VERB
ejpam-5569	568	5	with	with	ADP
ejpam-5569	568	6	matlab	matlab	PROPN
ejpam-5569	568	7	r2016a	r2016a	ADP
ejpam-5569	568	8	and	and	CCONJ
ejpam-5569	568	9	ms	ms	PROPN
ejpam-5569	568	10	windows	windows	PROPN
ejpam-5569	568	11	7	7	NUM
ejpam-5569	568	12	professional	professional	ADJ
ejpam-5569	568	13	system	system	NOUN
ejpam-5569	568	14	.	.	PUNCT
ejpam-5569	569	1	according	accord	VERB
ejpam-5569	569	2	to	to	ADP
ejpam-5569	569	3	table	table	NOUN
ejpam-5569	569	4	2	2	NUM
ejpam-5569	569	5	,	,	PUNCT
ejpam-5569	569	6	the	the	DET
ejpam-5569	569	7	results	result	NOUN
ejpam-5569	569	8	presented	present	VERB
ejpam-5569	569	9	in	in	ADP
ejpam-5569	569	10	lemma	lemma	PROPN
ejpam-5569	569	11	4	4	NUM
ejpam-5569	569	12	demonstrate	demonstrate	VERB
ejpam-5569	569	13	best	good	ADJ
ejpam-5569	569	14	performance	performance	NOUN
ejpam-5569	569	15	compared	compare	VERB
ejpam-5569	569	16	to	to	ADP
ejpam-5569	569	17	those	those	PRON
ejpam-5569	569	18	in	in	ADP
ejpam-5569	569	19	[	[	X
ejpam-5569	569	20	6	6	NUM
ejpam-5569	569	21	]	]	PUNCT
ejpam-5569	569	22	for	for	ADP
ejpam-5569	569	23	n	n	NOUN
ejpam-5569	569	24	=	=	SYM
ejpam-5569	569	25	5	5	NUM
ejpam-5569	569	26	.	.	X
ejpam-5569	569	27	for	for	ADP
ejpam-5569	569	28	n	n	NOUN
ejpam-5569	569	29	=	=	SYM
ejpam-5569	569	30	2	2	NUM
ejpam-5569	569	31	,	,	PUNCT
ejpam-5569	569	32	3	3	NUM
ejpam-5569	569	33	,	,	PUNCT
ejpam-5569	569	34	and	and	CCONJ
ejpam-5569	569	35	4	4	NUM
ejpam-5569	569	36	,	,	PUNCT
ejpam-5569	569	37	similar	similar	ADJ
ejpam-5569	569	38	outcomes	outcome	NOUN
ejpam-5569	569	39	are	be	AUX
ejpam-5569	569	40	observed	observe	VERB
ejpam-5569	569	41	.	.	PUNCT
ejpam-5569	570	1	conversely	conversely	ADV
ejpam-5569	570	2	,	,	PUNCT
ejpam-5569	570	3	the	the	DET
ejpam-5569	570	4	outcomes	outcome	NOUN
ejpam-5569	570	5	in	in	ADP
ejpam-5569	570	6	lemma	lemma	PROPN
ejpam-5569	570	7	1	1	NUM
ejpam-5569	570	8	,	,	PUNCT
ejpam-5569	570	9	2	2	NUM
ejpam-5569	570	10	,	,	PUNCT
ejpam-5569	570	11	3	3	NUM
ejpam-5569	570	12	,	,	PUNCT
ejpam-5569	570	13	and	and	CCONJ
ejpam-5569	570	14	4	4	NUM
ejpam-5569	570	15	outperform	outperform	VERB
ejpam-5569	570	16	those	those	PRON
ejpam-5569	570	17	in	in	ADP
ejpam-5569	570	18	[	[	X
ejpam-5569	570	19	7	7	X
ejpam-5569	570	20	]	]	PUNCT
ejpam-5569	570	21	across	across	ADP
ejpam-5569	570	22	all	all	DET
ejpam-5569	570	23	values	value	NOUN
ejpam-5569	570	24	of	of	ADP
ejpam-5569	570	25	n.	n.	NOUN
ejpam-5569	570	26	in	in	ADP
ejpam-5569	570	27	terms	term	NOUN
ejpam-5569	570	28	of	of	ADP
ejpam-5569	570	29	time	time	NOUN
ejpam-5569	570	30	complexity	complexity	NOUN
ejpam-5569	570	31	table	table	NOUN
ejpam-5569	570	32	2	2	NUM
ejpam-5569	570	33	shows	show	VERB
ejpam-5569	570	34	that	that	SCONJ
ejpam-5569	570	35	the	the	DET
ejpam-5569	570	36	computations	computation	NOUN
ejpam-5569	570	37	of	of	ADP
ejpam-5569	570	38	lemmas	lemmas	PROPN
ejpam-5569	570	39	1	1	NUM
ejpam-5569	570	40	,	,	PUNCT
ejpam-5569	570	41	2	2	NUM
ejpam-5569	570	42	,	,	PUNCT
ejpam-5569	570	43	3	3	NUM
ejpam-5569	570	44	,	,	PUNCT
ejpam-5569	570	45	and	and	CCONJ
ejpam-5569	570	46	4	4	NUM
ejpam-5569	570	47	give	give	VERB
ejpam-5569	570	48	superior	superior	ADJ
ejpam-5569	570	49	performance	performance	NOUN
ejpam-5569	570	50	compared	compare	VERB
ejpam-5569	570	51	to	to	ADP
ejpam-5569	570	52	those	those	PRON
ejpam-5569	570	53	in	in	ADP
ejpam-5569	570	54	[	[	X
ejpam-5569	570	55	6	6	NUM
ejpam-5569	570	56	,	,	PUNCT
ejpam-5569	570	57	7	7	NUM
ejpam-5569	570	58	]	]	PUNCT
ejpam-5569	570	59	.	.	PUNCT
ejpam-5569	571	1	table	table	NOUN
ejpam-5569	571	2	2	2	NUM
ejpam-5569	571	3	:	:	PUNCT
ejpam-5569	571	4	comparison	comparison	NOUN
ejpam-5569	571	5	among	among	ADP
ejpam-5569	571	6	our	our	PRON
ejpam-5569	571	7	approach	approach	NOUN
ejpam-5569	571	8	,	,	PUNCT
ejpam-5569	571	9	saha	saha	PROPN
ejpam-5569	572	1	[	[	X
ejpam-5569	572	2	6	6	NUM
ejpam-5569	572	3	]	]	PUNCT
ejpam-5569	572	4	and	and	CCONJ
ejpam-5569	572	5	ilpm	ilpm	NOUN
ejpam-5569	573	1	[	[	X
ejpam-5569	573	2	7	7	X
ejpam-5569	573	3	]	]	PUNCT
ejpam-5569	573	4	for	for	SCONJ
ejpam-5569	573	5	the	the	DET
ejpam-5569	573	6	radio	radio	NOUN
ejpam-5569	573	7	number	number	NOUN
ejpam-5569	573	8	of	of	ADP
ejpam-5569	573	9	shadow	shadow	NOUN
ejpam-5569	573	10	graph	graph	NOUN
ejpam-5569	573	11	with	with	ADP
ejpam-5569	573	12	n	n	NOUN
ejpam-5569	573	13	=	=	SYM
ejpam-5569	573	14	2	2	NUM
ejpam-5569	573	15	,	,	PUNCT
ejpam-5569	573	16	3	3	NUM
ejpam-5569	573	17	,	,	PUNCT
ejpam-5569	573	18	4	4	NUM
ejpam-5569	573	19	,	,	PUNCT
ejpam-5569	573	20	5	5	NUM
ejpam-5569	573	21	.	.	X
ejpam-5569	574	1	n	n	PROPN
ejpam-5569	574	2	2n	2n	NUM
ejpam-5569	574	3	lemma	lemma	PROPN
ejpam-5569	574	4	1,2,3,4	1,2,3,4	NUM
ejpam-5569	574	5	saha	saha	NOUN
ejpam-5569	574	6	[	[	X
ejpam-5569	574	7	6	6	NUM
ejpam-5569	574	8	]	]	X
ejpam-5569	574	9	ilpm	ilpm	NOUN
ejpam-5569	575	1	[	[	X
ejpam-5569	575	2	7	7	NUM
ejpam-5569	575	3	]	]	X
ejpam-5569	575	4	rn	rn	PROPN
ejpam-5569	575	5	cpu	cpu	PROPN
ejpam-5569	575	6	time	time	PROPN
ejpam-5569	575	7	rn	rn	PROPN
ejpam-5569	575	8	cpu	cpu	VERB
ejpam-5569	575	9	time	time	PROPN
ejpam-5569	575	10	rn	rn	PROPN
ejpam-5569	575	11	cpu	cpu	VERB
ejpam-5569	575	12	time	time	NOUN
ejpam-5569	575	13	2	2	NUM
ejpam-5569	575	14	4	4	NUM
ejpam-5569	575	15	4	4	NUM
ejpam-5569	575	16	o(1	o(1	NOUN
ejpam-5569	575	17	)	)	PUNCT
ejpam-5569	575	18	4	4	NUM
ejpam-5569	575	19	0.002396	0.002396	NUM
ejpam-5569	575	20	5	5	NUM
ejpam-5569	575	21	0.003654	0.003654	NUM
ejpam-5569	575	22	3	3	NUM
ejpam-5569	575	23	6	6	NUM
ejpam-5569	575	24	6	6	NUM
ejpam-5569	575	25	o(1	o(1	NOUN
ejpam-5569	575	26	)	)	PUNCT
ejpam-5569	575	27	6	6	NUM
ejpam-5569	575	28	0.004862	0.004862	NUM
ejpam-5569	575	29	9	9	NUM
ejpam-5569	575	30	0.004854	0.004854	NUM
ejpam-5569	575	31	4	4	NUM
ejpam-5569	575	32	8	8	NUM
ejpam-5569	575	33	12	12	NUM
ejpam-5569	575	34	o(1	o(1	NOUN
ejpam-5569	575	35	)	)	PUNCT
ejpam-5569	575	36	12	12	NUM
ejpam-5569	575	37	0.005359	0.005359	NUM
ejpam-5569	575	38	19	19	NUM
ejpam-5569	575	39	0.019494	0.019494	NUM
ejpam-5569	575	40	5	5	NUM
ejpam-5569	575	41	10	10	NUM
ejpam-5569	575	42	22	22	NUM
ejpam-5569	575	43	o(1	o(1	NOUN
ejpam-5569	575	44	)	)	PUNCT
ejpam-5569	575	45	23	23	NUM
ejpam-5569	575	46	0.005539	0.005539	NUM
ejpam-5569	575	47	33	33	NUM
ejpam-5569	575	48	0.034559	0.034559	NUM
ejpam-5569	575	49	table	table	NOUN
ejpam-5569	575	50	3	3	NUM
ejpam-5569	575	51	:	:	PUNCT
ejpam-5569	575	52	comparison	comparison	NOUN
ejpam-5569	575	53	among	among	ADP
ejpam-5569	575	54	theorem	theorem	NOUN
ejpam-5569	575	55	1	1	NUM
ejpam-5569	575	56	,	,	PUNCT
ejpam-5569	575	57	saha	saha	NOUN
ejpam-5569	576	1	[	[	X
ejpam-5569	576	2	6	6	NUM
ejpam-5569	576	3	]	]	PUNCT
ejpam-5569	576	4	and	and	CCONJ
ejpam-5569	576	5	ilpm	ilpm	NOUN
ejpam-5569	577	1	[	[	X
ejpam-5569	577	2	7	7	X
ejpam-5569	577	3	]	]	PUNCT
ejpam-5569	577	4	for	for	ADP
ejpam-5569	577	5	the	the	DET
ejpam-5569	577	6	radio	radio	NOUN
ejpam-5569	577	7	number	number	NOUN
ejpam-5569	577	8	of	of	ADP
ejpam-5569	577	9	shadow	shadow	NOUN
ejpam-5569	577	10	graph	graph	NOUN
ejpam-5569	577	11	with	with	ADP
ejpam-5569	577	12	n	n	NOUN
ejpam-5569	577	13	=	=	SYM
ejpam-5569	577	14	2(mod	2(mod	NUM
ejpam-5569	577	15	)	)	PUNCT
ejpam-5569	577	16	4	4	NUM
ejpam-5569	577	17	.	.	X
ejpam-5569	578	1	n	n	PROPN
ejpam-5569	578	2	2n	2n	NUM
ejpam-5569	578	3	theorem	theorem	VERB
ejpam-5569	578	4	1	1	NUM
ejpam-5569	578	5	saha	saha	NOUN
ejpam-5569	579	1	[	[	X
ejpam-5569	579	2	6	6	NUM
ejpam-5569	579	3	]	]	X
ejpam-5569	579	4	ilpm	ilpm	NOUN
ejpam-5569	580	1	[	[	X
ejpam-5569	580	2	7	7	NUM
ejpam-5569	580	3	]	]	X
ejpam-5569	580	4	rn	rn	PROPN
ejpam-5569	580	5	cpu	cpu	PROPN
ejpam-5569	580	6	time	time	PROPN
ejpam-5569	580	7	rn	rn	PROPN
ejpam-5569	580	8	cpu	cpu	VERB
ejpam-5569	580	9	time	time	PROPN
ejpam-5569	580	10	rn	rn	PROPN
ejpam-5569	580	11	cpu	cpu	VERB
ejpam-5569	580	12	time	time	NOUN
ejpam-5569	580	13	6	6	NUM
ejpam-5569	580	14	12	12	NUM
ejpam-5569	580	15	33	33	NUM
ejpam-5569	580	16	o(1	o(1	NOUN
ejpam-5569	580	17	)	)	PUNCT
ejpam-5569	580	18	33	33	NUM
ejpam-5569	580	19	0.042993	0.042993	NUM
ejpam-5569	580	20	51	51	NUM
ejpam-5569	580	21	0.036009	0.036009	NUM
ejpam-5569	580	22	10	10	NUM
ejpam-5569	580	23	20	20	NUM
ejpam-5569	580	24	95	95	NUM
ejpam-5569	580	25	o(1	o(1	NOUN
ejpam-5569	580	26	)	)	PUNCT
ejpam-5569	580	27	95	95	NUM
ejpam-5569	580	28	0.043329	0.043329	NUM
ejpam-5569	580	29	163	163	NUM
ejpam-5569	580	30	0.100475	0.100475	NUM
ejpam-5569	580	31	14	14	NUM
ejpam-5569	580	32	28	28	NUM
ejpam-5569	580	33	189	189	NUM
ejpam-5569	580	34	o(1	o(1	NOUN
ejpam-5569	580	35	)	)	PUNCT
ejpam-5569	580	36	189	189	NUM
ejpam-5569	580	37	0.043968	0.043968	NUM
ejpam-5569	580	38	343	343	NUM
ejpam-5569	580	39	0.150742	0.150742	NUM
ejpam-5569	580	40	18	18	NUM
ejpam-5569	580	41	36	36	NUM
ejpam-5569	580	42	315	315	NUM
ejpam-5569	580	43	o(1	o(1	NOUN
ejpam-5569	580	44	)	)	PUNCT
ejpam-5569	580	45	315	315	NUM
ejpam-5569	580	46	0.048544	0.048544	NUM
ejpam-5569	580	47	579	579	NUM
ejpam-5569	580	48	0.151717	0.151717	NUM
ejpam-5569	580	49	22	22	NUM
ejpam-5569	580	50	44	44	NUM
ejpam-5569	580	51	473	473	NUM
ejpam-5569	580	52	o(1	o(1	NOUN
ejpam-5569	580	53	)	)	PUNCT
ejpam-5569	580	54	473	473	NUM
ejpam-5569	580	55	0.054947	0.054947	NUM
ejpam-5569	580	56	883	883	NUM
ejpam-5569	580	57	0.162229	0.162229	NUM
ejpam-5569	580	58	26	26	NUM
ejpam-5569	580	59	52	52	NUM
ejpam-5569	580	60	663	663	NUM
ejpam-5569	580	61	o(1	o(1	NOUN
ejpam-5569	580	62	)	)	PUNCT
ejpam-5569	580	63	663	663	NUM
ejpam-5569	580	64	0.082347	0.082347	NUM
ejpam-5569	580	65	1251	1251	NUM
ejpam-5569	580	66	0.182440	0.182440	NUM
ejpam-5569	580	67	30	30	NUM
ejpam-5569	580	68	60	60	NUM
ejpam-5569	580	69	885	885	NUM
ejpam-5569	580	70	o(1	o(1	NOUN
ejpam-5569	580	71	)	)	PUNCT
ejpam-5569	580	72	885	885	NUM
ejpam-5569	580	73	0.083501	0.083501	NUM
ejpam-5569	580	74	1683	1683	NUM
ejpam-5569	580	75	0.190866	0.190866	NUM
ejpam-5569	580	76	34	34	NUM
ejpam-5569	580	77	68	68	NUM
ejpam-5569	580	78	1139	1139	NUM
ejpam-5569	580	79	o(1	o(1	NOUN
ejpam-5569	580	80	)	)	PUNCT
ejpam-5569	580	81	1139	1139	NUM
ejpam-5569	580	82	0.096042	0.096042	NUM
ejpam-5569	580	83	2179	2179	NUM
ejpam-5569	580	84	0.218329	0.218329	NUM
ejpam-5569	580	85	38	38	NUM
ejpam-5569	580	86	76	76	NUM
ejpam-5569	580	87	1425	1425	NUM
ejpam-5569	580	88	o(1	o(1	NOUN
ejpam-5569	580	89	)	)	PUNCT
ejpam-5569	580	90	1425	1425	NUM
ejpam-5569	580	91	0.096626	0.096626	NUM
ejpam-5569	580	92	2739	2739	NUM
ejpam-5569	580	93	0.222084	0.222084	NUM
ejpam-5569	580	94	42	42	NUM
ejpam-5569	580	95	84	84	NUM
ejpam-5569	580	96	1743	1743	NUM
ejpam-5569	580	97	o(1	o(1	NOUN
ejpam-5569	580	98	)	)	PUNCT
ejpam-5569	580	99	1743	1743	NUM
ejpam-5569	580	100	0.117833	0.117833	NUM
ejpam-5569	580	101	3363	3363	NUM
ejpam-5569	580	102	0.264229	0.264229	NUM
ejpam-5569	580	103	46	46	NUM
ejpam-5569	580	104	92	92	NUM
ejpam-5569	580	105	2093	2093	NUM
ejpam-5569	580	106	o(1	o(1	NOUN
ejpam-5569	580	107	)	)	PUNCT
ejpam-5569	580	108	2093	2093	NUM
ejpam-5569	580	109	0.118883	0.118883	NUM
ejpam-5569	580	110	4051	4051	NUM
ejpam-5569	580	111	0.278491	0.278491	NUM
ejpam-5569	580	112	50	50	NUM
ejpam-5569	580	113	100	100	NUM
ejpam-5569	580	114	2475	2475	NUM
ejpam-5569	580	115	o(1	o(1	NOUN
ejpam-5569	580	116	)	)	PUNCT
ejpam-5569	580	117	2475	2475	NUM
ejpam-5569	580	118	0.121298	0.121298	NUM
ejpam-5569	580	119	4803	4803	NUM
ejpam-5569	580	120	0.301669	0.301669	NUM
ejpam-5569	580	121	54	54	NUM
ejpam-5569	580	122	108	108	NUM
ejpam-5569	580	123	2889	2889	NUM
ejpam-5569	580	124	o(1	o(1	NOUN
ejpam-5569	580	125	)	)	PUNCT
ejpam-5569	580	126	2889	2889	NUM
ejpam-5569	580	127	0.128889	0.128889	NUM
ejpam-5569	580	128	5619	5619	NUM
ejpam-5569	580	129	0.420457	0.420457	NUM
ejpam-5569	580	130	58	58	NUM
ejpam-5569	580	131	116	116	NUM
ejpam-5569	580	132	3335	3335	NUM
ejpam-5569	580	133	o(1	o(1	NOUN
ejpam-5569	580	134	)	)	PUNCT
ejpam-5569	580	135	3335	3335	NUM
ejpam-5569	580	136	0.131909	0.131909	NUM
ejpam-5569	580	137	6499	6499	NUM
ejpam-5569	580	138	0.475030	0.475030	NUM
ejpam-5569	580	139	62	62	NUM
ejpam-5569	580	140	124	124	NUM
ejpam-5569	580	141	3813	3813	NUM
ejpam-5569	580	142	o(1	o(1	NOUN
ejpam-5569	580	143	)	)	PUNCT
ejpam-5569	580	144	3813	3813	NUM
ejpam-5569	580	145	0.148007	0.148007	NUM
ejpam-5569	580	146	7443	7443	NUM
ejpam-5569	580	147	0.477373	0.477373	NUM
ejpam-5569	580	148	64	64	NUM
ejpam-5569	580	149	128	128	NUM
ejpam-5569	580	150	4064	4064	NUM
ejpam-5569	580	151	o(1	o(1	NOUN
ejpam-5569	580	152	)	)	PUNCT
ejpam-5569	580	153	4064	4064	NUM
ejpam-5569	580	154	0.178655	0.178655	NUM
ejpam-5569	580	155	7939	7939	NUM
ejpam-5569	580	156	0.527846	0.527846	NUM
ejpam-5569	580	157	68	68	NUM
ejpam-5569	580	158	136	136	NUM
ejpam-5569	580	159	4590	4590	NUM
ejpam-5569	580	160	o(1	o(1	NOUN
ejpam-5569	580	161	)	)	PUNCT
ejpam-5569	580	162	4590	4590	NUM
ejpam-5569	580	163	0.198579	0.198579	NUM
ejpam-5569	580	164	8979	8979	NUM
ejpam-5569	580	165	0.635992	0.635992	NUM
ejpam-5569	580	166	72	72	NUM
ejpam-5569	580	167	144	144	NUM
ejpam-5569	580	168	5148	5148	NUM
ejpam-5569	580	169	o(1	o(1	NOUN
ejpam-5569	580	170	)	)	PUNCT
ejpam-5569	580	171	5148	5148	NUM
ejpam-5569	580	172	0.20673	0.20673	NUM
ejpam-5569	580	173	10083	10083	NUM
ejpam-5569	580	174	0.720457	0.720457	NUM
ejpam-5569	580	175	76	76	NUM
ejpam-5569	580	176	152	152	NUM
ejpam-5569	580	177	5738	5738	NUM
ejpam-5569	580	178	o(1	o(1	NOUN
ejpam-5569	580	179	)	)	PUNCT
ejpam-5569	580	180	5738	5738	NUM
ejpam-5569	580	181	0.213292	0.213292	NUM
ejpam-5569	580	182	11251	11251	NUM
ejpam-5569	580	183	0.875030	0.875030	NUM
ejpam-5569	580	184	80	80	NUM
ejpam-5569	580	185	160	160	NUM
ejpam-5569	580	186	6360	6360	NUM
ejpam-5569	580	187	o(1	o(1	NOUN
ejpam-5569	580	188	)	)	PUNCT
ejpam-5569	580	189	6360	6360	NUM
ejpam-5569	580	190	0.231493	0.231493	NUM
ejpam-5569	580	191	12483	12483	NUM
ejpam-5569	580	192	0.976116	0.976116	NUM
ejpam-5569	580	193	based	base	VERB
ejpam-5569	580	194	on	on	ADP
ejpam-5569	580	195	the	the	DET
ejpam-5569	580	196	upper	upper	ADJ
ejpam-5569	580	197	bound	bound	NOUN
ejpam-5569	580	198	of	of	ADP
ejpam-5569	580	199	the	the	DET
ejpam-5569	580	200	radio	radio	NOUN
ejpam-5569	580	201	number	number	NOUN
ejpam-5569	580	202	of	of	ADP
ejpam-5569	580	203	shadow	shadow	NOUN
ejpam-5569	580	204	graph	graph	NOUN
ejpam-5569	580	205	with	with	ADP
ejpam-5569	580	206	n	n	NOUN
ejpam-5569	580	207	=	=	SYM
ejpam-5569	580	208	2	2	NUM
ejpam-5569	580	209	(	(	PUNCT
ejpam-5569	580	210	mod	mod	PROPN
ejpam-5569	580	211	)	)	PUNCT
ejpam-5569	580	212	4	4	NUM
ejpam-5569	580	213	,	,	PUNCT
ejpam-5569	580	214	it	it	PRON
ejpam-5569	580	215	can	can	AUX
ejpam-5569	580	216	be	be	AUX
ejpam-5569	580	217	observed	observe	VERB
ejpam-5569	580	218	from	from	ADP
ejpam-5569	580	219	table	table	NOUN
ejpam-5569	580	220	3	3	NUM
ejpam-5569	580	221	and	and	CCONJ
ejpam-5569	580	222	figure	figure	VERB
ejpam-5569	580	223	6	6	NUM
ejpam-5569	580	224	that	that	SCONJ
ejpam-5569	580	225	the	the	DET
ejpam-5569	580	226	outcomes	outcome	NOUN
ejpam-5569	580	227	presented	present	VERB
ejpam-5569	580	228	in	in	ADP
ejpam-5569	580	229	theorem	theorem	ADJ
ejpam-5569	580	230	1	1	NUM
ejpam-5569	580	231	coincide	coincide	NOUN
ejpam-5569	580	232	with	with	ADP
ejpam-5569	580	233	those	those	PRON
ejpam-5569	580	234	in	in	ADP
ejpam-5569	580	235	[	[	X
ejpam-5569	580	236	6	6	NUM
ejpam-5569	580	237	]	]	PUNCT
ejpam-5569	580	238	.	.	PUNCT
ejpam-5569	581	1	conversely	conversely	ADV
ejpam-5569	581	2	,	,	PUNCT
ejpam-5569	581	3	the	the	DET
ejpam-5569	581	4	results	result	NOUN
ejpam-5569	581	5	from	from	ADP
ejpam-5569	581	6	theorem	theorem	ADJ
ejpam-5569	581	7	1	1	NUM
ejpam-5569	581	8	surpass	surpass	NOUN
ejpam-5569	581	9	the	the	DET
ejpam-5569	581	10	findings	finding	NOUN
ejpam-5569	581	11	in	in	ADP
ejpam-5569	581	12	[	[	X
ejpam-5569	581	13	7	7	NUM
ejpam-5569	581	14	]	]	PUNCT
ejpam-5569	581	15	for	for	ADP
ejpam-5569	581	16	all	all	DET
ejpam-5569	581	17	values	value	NOUN
ejpam-5569	581	18	of	of	ADP
ejpam-5569	581	19	n.	n.	NOUN
ejpam-5569	581	20	table	table	NOUN
ejpam-5569	581	21	3	3	NUM
ejpam-5569	581	22	indicates	indicate	VERB
ejpam-5569	581	23	that	that	SCONJ
ejpam-5569	581	24	the	the	DET
ejpam-5569	581	25	results	result	NOUN
ejpam-5569	581	26	from	from	ADP
ejpam-5569	581	27	theorem	theorem	ADJ
ejpam-5569	581	28	1	1	NUM
ejpam-5569	581	29	have	have	VERB
ejpam-5569	581	30	a	a	DET
ejpam-5569	581	31	time	time	NOUN
ejpam-5569	581	32	n.	n.	PROPN
ejpam-5569	581	33	m.	m.	NOUN
ejpam-5569	582	1	noureldeen	noureldeen	NOUN
ejpam-5569	582	2	et	et	PROPN
ejpam-5569	582	3	al	al	PROPN
ejpam-5569	582	4	.	.	PUNCT
ejpam-5569	582	5	/	/	SYM
ejpam-5569	582	6	eur	eur	PROPN
ejpam-5569	582	7	.	.	PUNCT
ejpam-5569	583	1	j.	j.	PROPN
ejpam-5569	583	2	pure	pure	PROPN
ejpam-5569	583	3	appl	appl	PROPN
ejpam-5569	583	4	.	.	PROPN
ejpam-5569	583	5	math	math	PROPN
ejpam-5569	583	6	,	,	PUNCT
ejpam-5569	583	7	18	18	NUM
ejpam-5569	583	8	(	(	PUNCT
ejpam-5569	583	9	2	2	NUM
ejpam-5569	583	10	)	)	PUNCT
ejpam-5569	583	11	(	(	PUNCT
ejpam-5569	583	12	2025	2025	NUM
ejpam-5569	583	13	)	)	PUNCT
ejpam-5569	583	14	,	,	PUNCT
ejpam-5569	583	15	5569	5569	NUM
ejpam-5569	583	16	20	20	NUM
ejpam-5569	583	17	of	of	ADP
ejpam-5569	583	18	25	25	NUM
ejpam-5569	583	19	complexity	complexity	NOUN
ejpam-5569	583	20	of	of	ADP
ejpam-5569	583	21	o(1	o(1	NOUN
ejpam-5569	583	22	)	)	PUNCT
ejpam-5569	583	23	,	,	PUNCT
ejpam-5569	583	24	while	while	SCONJ
ejpam-5569	583	25	those	those	PRON
ejpam-5569	583	26	in	in	ADP
ejpam-5569	583	27	[	[	X
ejpam-5569	583	28	6	6	NUM
ejpam-5569	583	29	]	]	PUNCT
ejpam-5569	583	30	have	have	VERB
ejpam-5569	583	31	a	a	DET
ejpam-5569	583	32	complexity	complexity	NOUN
ejpam-5569	583	33	of	of	ADP
ejpam-5569	583	34	o(n3	o(n3	NOUN
ejpam-5569	583	35	)	)	PUNCT
ejpam-5569	583	36	.	.	PUNCT
ejpam-5569	584	1	furthermore	furthermore	ADV
ejpam-5569	584	2	,	,	PUNCT
ejpam-5569	584	3	the	the	DET
ejpam-5569	584	4	results	result	NOUN
ejpam-5569	584	5	from	from	ADP
ejpam-5569	584	6	theorem	theorem	ADJ
ejpam-5569	584	7	1	1	NUM
ejpam-5569	584	8	require	require	NOUN
ejpam-5569	584	9	less	less	ADJ
ejpam-5569	584	10	time	time	NOUN
ejpam-5569	584	11	compared	compare	VERB
ejpam-5569	584	12	to	to	ADP
ejpam-5569	584	13	the	the	DET
ejpam-5569	584	14	results	result	NOUN
ejpam-5569	584	15	in	in	ADP
ejpam-5569	584	16	[	[	X
ejpam-5569	584	17	7	7	NUM
ejpam-5569	584	18	]	]	PUNCT
ejpam-5569	584	19	.	.	PUNCT
ejpam-5569	585	1	figure	figure	VERB
ejpam-5569	585	2	6	6	NUM
ejpam-5569	585	3	:	:	PUNCT
ejpam-5569	585	4	comparison	comparison	NOUN
ejpam-5569	585	5	among	among	ADP
ejpam-5569	585	6	theorem	theorem	NOUN
ejpam-5569	585	7	1	1	NUM
ejpam-5569	585	8	,	,	PUNCT
ejpam-5569	585	9	saha	saha	NOUN
ejpam-5569	586	1	[	[	X
ejpam-5569	586	2	6	6	NUM
ejpam-5569	586	3	]	]	PUNCT
ejpam-5569	586	4	and	and	CCONJ
ejpam-5569	586	5	ilpm	ilpm	NOUN
ejpam-5569	587	1	[	[	X
ejpam-5569	587	2	7	7	X
ejpam-5569	587	3	]	]	PUNCT
ejpam-5569	587	4	for	for	ADP
ejpam-5569	587	5	the	the	DET
ejpam-5569	587	6	radio	radio	NOUN
ejpam-5569	587	7	number	number	NOUN
ejpam-5569	587	8	of	of	ADP
ejpam-5569	587	9	shadow	shadow	NOUN
ejpam-5569	587	10	graph	graph	NOUN
ejpam-5569	587	11	with	with	ADP
ejpam-5569	587	12	n	n	NOUN
ejpam-5569	587	13	=	=	SYM
ejpam-5569	587	14	2(mod	2(mod	NUM
ejpam-5569	587	15	)	)	PUNCT
ejpam-5569	587	16	4	4	NUM
ejpam-5569	587	17	.	.	NOUN
ejpam-5569	587	18	table	table	NOUN
ejpam-5569	587	19	4	4	NUM
ejpam-5569	587	20	:	:	PUNCT
ejpam-5569	587	21	comparison	comparison	NOUN
ejpam-5569	587	22	among	among	ADP
ejpam-5569	587	23	theorem	theorem	NOUN
ejpam-5569	587	24	2	2	NUM
ejpam-5569	587	25	,	,	PUNCT
ejpam-5569	587	26	saha	saha	NOUN
ejpam-5569	588	1	[	[	X
ejpam-5569	588	2	6	6	NUM
ejpam-5569	588	3	]	]	PUNCT
ejpam-5569	588	4	and	and	CCONJ
ejpam-5569	588	5	ilpm	ilpm	NOUN
ejpam-5569	589	1	[	[	X
ejpam-5569	589	2	7	7	X
ejpam-5569	589	3	]	]	PUNCT
ejpam-5569	589	4	for	for	SCONJ
ejpam-5569	589	5	the	the	DET
ejpam-5569	589	6	radio	radio	NOUN
ejpam-5569	589	7	number	number	NOUN
ejpam-5569	589	8	of	of	ADP
ejpam-5569	589	9	shadow	shadow	NOUN
ejpam-5569	589	10	graph	graph	NOUN
ejpam-5569	589	11	with	with	ADP
ejpam-5569	589	12	n	n	NOUN
ejpam-5569	589	13	=	=	SYM
ejpam-5569	589	14	3(mod)4	3(mod)4	NUM
ejpam-5569	589	15	.	.	PUNCT
ejpam-5569	590	1	n	n	PROPN
ejpam-5569	590	2	2n	2n	NUM
ejpam-5569	590	3	theorem	theorem	VERB
ejpam-5569	590	4	2	2	NUM
ejpam-5569	590	5	saha	saha	NOUN
ejpam-5569	591	1	[	[	X
ejpam-5569	591	2	6	6	NUM
ejpam-5569	591	3	]	]	X
ejpam-5569	591	4	ilpm	ilpm	NOUN
ejpam-5569	592	1	[	[	X
ejpam-5569	592	2	7	7	NUM
ejpam-5569	592	3	]	]	X
ejpam-5569	592	4	rn	rn	PROPN
ejpam-5569	592	5	cpu	cpu	PROPN
ejpam-5569	592	6	time	time	PROPN
ejpam-5569	592	7	rn	rn	PROPN
ejpam-5569	592	8	cpu	cpu	VERB
ejpam-5569	592	9	time	time	PROPN
ejpam-5569	592	10	rn	rn	PROPN
ejpam-5569	592	11	cpu	cpu	VERB
ejpam-5569	592	12	time	time	NOUN
ejpam-5569	592	13	7	7	NUM
ejpam-5569	592	14	14	14	NUM
ejpam-5569	592	15	46	46	NUM
ejpam-5569	592	16	o(1	o(1	NOUN
ejpam-5569	592	17	)	)	PUNCT
ejpam-5569	592	18	46	46	NUM
ejpam-5569	592	19	0.043355	0.043355	NUM
ejpam-5569	592	20	73	73	NUM
ejpam-5569	592	21	0.089586	0.089586	NUM
ejpam-5569	592	22	11	11	NUM
ejpam-5569	592	23	22	22	NUM
ejpam-5569	592	24	118	118	NUM
ejpam-5569	592	25	o(1	o(1	NOUN
ejpam-5569	592	26	)	)	PUNCT
ejpam-5569	592	27	118	118	NUM
ejpam-5569	592	28	0.048763	0.048763	NUM
ejpam-5569	592	29	201	201	NUM
ejpam-5569	592	30	0.090475	0.090475	NUM
ejpam-5569	592	31	15	15	NUM
ejpam-5569	592	32	30	30	NUM
ejpam-5569	592	33	222	222	NUM
ejpam-5569	592	34	o(1	o(1	NOUN
ejpam-5569	592	35	)	)	PUNCT
ejpam-5569	592	36	222	222	NUM
ejpam-5569	592	37	0.050551	0.050551	NUM
ejpam-5569	592	38	393	393	NUM
ejpam-5569	592	39	0.130588	0.130588	NUM
ejpam-5569	592	40	19	19	NUM
ejpam-5569	592	41	38	38	NUM
ejpam-5569	592	42	358	358	NUM
ejpam-5569	592	43	o(1	o(1	NOUN
ejpam-5569	592	44	)	)	PUNCT
ejpam-5569	592	45	358	358	NUM
ejpam-5569	592	46	0.050987	0.050987	NUM
ejpam-5569	592	47	649	649	NUM
ejpam-5569	592	48	0.161792	0.161792	NUM
ejpam-5569	592	49	23	23	NUM
ejpam-5569	592	50	46	46	NUM
ejpam-5569	592	51	526	526	NUM
ejpam-5569	592	52	o(1	o(1	NOUN
ejpam-5569	592	53	)	)	PUNCT
ejpam-5569	592	54	526	526	NUM
ejpam-5569	592	55	0.051813	0.051813	NUM
ejpam-5569	592	56	969	969	NUM
ejpam-5569	592	57	0.220337	0.220337	NUM
ejpam-5569	592	58	27	27	NUM
ejpam-5569	592	59	54	54	NUM
ejpam-5569	592	60	726	726	NUM
ejpam-5569	592	61	o(1	o(1	NOUN
ejpam-5569	592	62	)	)	PUNCT
ejpam-5569	592	63	726	726	NUM
ejpam-5569	592	64	0.081522	0.081522	NUM
ejpam-5569	592	65	1353	1353	NUM
ejpam-5569	592	66	0.305778	0.305778	NUM
ejpam-5569	592	67	31	31	NUM
ejpam-5569	592	68	62	62	NUM
ejpam-5569	592	69	958	958	NUM
ejpam-5569	592	70	o(1	o(1	NOUN
ejpam-5569	592	71	)	)	PUNCT
ejpam-5569	592	72	958	958	NUM
ejpam-5569	592	73	0.086948	0.086948	NUM
ejpam-5569	592	74	1801	1801	NUM
ejpam-5569	592	75	0.207713	0.207713	NUM
ejpam-5569	592	76	35	35	NUM
ejpam-5569	592	77	70	70	NUM
ejpam-5569	592	78	1222	1222	NUM
ejpam-5569	592	79	o(1	o(1	NOUN
ejpam-5569	592	80	)	)	PUNCT
ejpam-5569	592	81	1222	1222	NUM
ejpam-5569	592	82	0.092306	0.092306	NUM
ejpam-5569	592	83	2313	2313	NUM
ejpam-5569	592	84	0.266207	0.266207	NUM
ejpam-5569	592	85	39	39	NUM
ejpam-5569	592	86	78	78	NUM
ejpam-5569	592	87	1518	1518	NUM
ejpam-5569	592	88	o(1	o(1	NOUN
ejpam-5569	592	89	)	)	PUNCT
ejpam-5569	592	90	1518	1518	NUM
ejpam-5569	592	91	0.095106	0.095106	NUM
ejpam-5569	592	92	2889	2889	NUM
ejpam-5569	592	93	0.288857	0.288857	NUM
ejpam-5569	592	94	43	43	NUM
ejpam-5569	592	95	86	86	NUM
ejpam-5569	592	96	1846	1846	NUM
ejpam-5569	592	97	o(1	o(1	NOUN
ejpam-5569	592	98	)	)	PUNCT
ejpam-5569	592	99	1846	1846	NUM
ejpam-5569	592	100	0.102625	0.102625	NUM
ejpam-5569	592	101	3529	3529	NUM
ejpam-5569	592	102	0.368033	0.368033	NUM
ejpam-5569	592	103	47	47	NUM
ejpam-5569	592	104	94	94	NUM
ejpam-5569	592	105	2206	2206	NUM
ejpam-5569	592	106	o(1	o(1	NOUN
ejpam-5569	592	107	)	)	PUNCT
ejpam-5569	592	108	2206	2206	NUM
ejpam-5569	592	109	0.116411	0.116411	NUM
ejpam-5569	592	110	4233	4233	NUM
ejpam-5569	592	111	0.486650	0.486650	NUM
ejpam-5569	592	112	51	51	NUM
ejpam-5569	592	113	102	102	NUM
ejpam-5569	592	114	2598	2598	NUM
ejpam-5569	592	115	o(1	o(1	NOUN
ejpam-5569	592	116	)	)	PUNCT
ejpam-5569	592	117	2598	2598	NUM
ejpam-5569	592	118	0.121707	0.121707	NUM
ejpam-5569	592	119	5001	5001	NUM
ejpam-5569	592	120	0.530145	0.530145	NUM
ejpam-5569	592	121	55	55	NUM
ejpam-5569	592	122	110	110	NUM
ejpam-5569	592	123	3022	3022	NUM
ejpam-5569	592	124	o(1	o(1	NOUN
ejpam-5569	592	125	)	)	PUNCT
ejpam-5569	592	126	3022	3022	NUM
ejpam-5569	592	127	0.124297	0.124297	NUM
ejpam-5569	592	128	5833	5833	NUM
ejpam-5569	592	129	0.544023	0.544023	NUM
ejpam-5569	592	130	59	59	NUM
ejpam-5569	592	131	118	118	NUM
ejpam-5569	592	132	3478	3478	NUM
ejpam-5569	592	133	o(1	o(1	NOUN
ejpam-5569	592	134	)	)	PUNCT
ejpam-5569	592	135	3478	3478	NUM
ejpam-5569	593	1	0.140879	0.140879	NUM
ejpam-5569	593	2	6729	6729	NUM
ejpam-5569	593	3	0.551847	0.551847	NUM
ejpam-5569	593	4	63	63	NUM
ejpam-5569	593	5	126	126	NUM
ejpam-5569	593	6	3966	3966	NUM
ejpam-5569	593	7	o(1	o(1	NOUN
ejpam-5569	593	8	)	)	PUNCT
ejpam-5569	593	9	3966	3966	NUM
ejpam-5569	593	10	0.15269	0.15269	NUM
ejpam-5569	593	11	7689	7689	NUM
ejpam-5569	593	12	0.635992	0.635992	NUM
ejpam-5569	593	13	67	67	NUM
ejpam-5569	593	14	134	134	NUM
ejpam-5569	593	15	4486	4486	NUM
ejpam-5569	593	16	o(1	o(1	NOUN
ejpam-5569	593	17	)	)	PUNCT
ejpam-5569	593	18	4486	4486	NUM
ejpam-5569	593	19	0.164255	0.164255	NUM
ejpam-5569	593	20	8713	8713	NUM
ejpam-5569	593	21	0.720457	0.720457	NUM
ejpam-5569	593	22	71	71	NUM
ejpam-5569	593	23	142	142	NUM
ejpam-5569	593	24	5038	5038	NUM
ejpam-5569	593	25	o(1	o(1	NOUN
ejpam-5569	593	26	)	)	PUNCT
ejpam-5569	593	27	5038	5038	NUM
ejpam-5569	593	28	0.190907	0.190907	NUM
ejpam-5569	593	29	9801	9801	NUM
ejpam-5569	593	30	0.829255	0.829255	NUM
ejpam-5569	593	31	75	75	NUM
ejpam-5569	593	32	150	150	NUM
ejpam-5569	593	33	5622	5622	NUM
ejpam-5569	593	34	o(1	o(1	NOUN
ejpam-5569	593	35	)	)	PUNCT
ejpam-5569	593	36	5622	5622	NUM
ejpam-5569	593	37	0.206634	0.206634	NUM
ejpam-5569	593	38	10953	10953	NUM
ejpam-5569	593	39	0.972498	0.972498	NUM
ejpam-5569	593	40	79	79	NUM
ejpam-5569	593	41	158	158	NUM
ejpam-5569	593	42	6238	6238	NUM
ejpam-5569	593	43	o(1	o(1	NOUN
ejpam-5569	593	44	)	)	PUNCT
ejpam-5569	593	45	6238	6238	NUM
ejpam-5569	593	46	0.224507	0.224507	NUM
ejpam-5569	593	47	12169	12169	NUM
ejpam-5569	593	48	0.998945	0.998945	NUM
ejpam-5569	593	49	83	83	NUM
ejpam-5569	593	50	166	166	NUM
ejpam-5569	593	51	6886	6886	NUM
ejpam-5569	593	52	o(1	o(1	NOUN
ejpam-5569	593	53	)	)	PUNCT
ejpam-5569	593	54	6886	6886	NUM
ejpam-5569	593	55	0.273817	0.273817	NUM
ejpam-5569	593	56	13449	13449	NUM
ejpam-5569	593	57	1.034510	1.034510	NUM
ejpam-5569	593	58	87	87	NUM
ejpam-5569	593	59	174	174	NUM
ejpam-5569	593	60	7566	7566	NUM
ejpam-5569	593	61	o(1	o(1	NOUN
ejpam-5569	593	62	)	)	PUNCT
ejpam-5569	593	63	7566	7566	NUM
ejpam-5569	593	64	0.351691	0.351691	NUM
ejpam-5569	593	65	14793	14793	NUM
ejpam-5569	593	66	1.293168	1.293168	NUM
ejpam-5569	593	67	n.	n.	NOUN
ejpam-5569	593	68	m.	m.	NOUN
ejpam-5569	593	69	noureldeen	noureldeen	NOUN
ejpam-5569	593	70	et	et	PROPN
ejpam-5569	593	71	al	al	PROPN
ejpam-5569	593	72	.	.	PUNCT
ejpam-5569	593	73	/	/	SYM
ejpam-5569	593	74	eur	eur	PROPN
ejpam-5569	593	75	.	.	PUNCT
ejpam-5569	594	1	j.	j.	PROPN
ejpam-5569	594	2	pure	pure	PROPN
ejpam-5569	594	3	appl	appl	PROPN
ejpam-5569	594	4	.	.	PROPN
ejpam-5569	594	5	math	math	PROPN
ejpam-5569	594	6	,	,	PUNCT
ejpam-5569	594	7	18	18	NUM
ejpam-5569	594	8	(	(	PUNCT
ejpam-5569	594	9	2	2	NUM
ejpam-5569	594	10	)	)	PUNCT
ejpam-5569	594	11	(	(	PUNCT
ejpam-5569	594	12	2025	2025	NUM
ejpam-5569	594	13	)	)	PUNCT
ejpam-5569	594	14	,	,	PUNCT
ejpam-5569	594	15	5569	5569	NUM
ejpam-5569	594	16	21	21	NUM
ejpam-5569	594	17	of	of	ADP
ejpam-5569	594	18	25	25	NUM
ejpam-5569	594	19	table	table	NOUN
ejpam-5569	594	20	4	4	NUM
ejpam-5569	594	21	and	and	CCONJ
ejpam-5569	594	22	figure	figure	VERB
ejpam-5569	594	23	7	7	NUM
ejpam-5569	594	24	demonstrate	demonstrate	VERB
ejpam-5569	594	25	that	that	SCONJ
ejpam-5569	594	26	the	the	DET
ejpam-5569	594	27	results	result	NOUN
ejpam-5569	594	28	presented	present	VERB
ejpam-5569	594	29	in	in	ADP
ejpam-5569	594	30	theorem	theorem	ADJ
ejpam-5569	594	31	2	2	NUM
ejpam-5569	594	32	align	align	NOUN
ejpam-5569	594	33	with	with	ADP
ejpam-5569	594	34	the	the	DET
ejpam-5569	594	35	findings	finding	NOUN
ejpam-5569	594	36	in	in	ADP
ejpam-5569	594	37	[	[	X
ejpam-5569	594	38	6	6	NUM
ejpam-5569	594	39	]	]	PUNCT
ejpam-5569	594	40	,	,	PUNCT
ejpam-5569	594	41	considering	consider	VERB
ejpam-5569	594	42	the	the	DET
ejpam-5569	594	43	upper	upper	ADJ
ejpam-5569	594	44	bound	bound	NOUN
ejpam-5569	594	45	of	of	ADP
ejpam-5569	594	46	the	the	DET
ejpam-5569	594	47	radio	radio	NOUN
ejpam-5569	594	48	number	number	NOUN
ejpam-5569	594	49	of	of	ADP
ejpam-5569	594	50	shadow	shadow	NOUN
ejpam-5569	594	51	graph	graph	NOUN
ejpam-5569	594	52	with	with	ADP
ejpam-5569	594	53	n	n	NOUN
ejpam-5569	594	54	=	=	SYM
ejpam-5569	594	55	3(mod	3(mod	NUM
ejpam-5569	594	56	)	)	PUNCT
ejpam-5569	594	57	4	4	NUM
ejpam-5569	594	58	.	.	PUNCT
ejpam-5569	595	1	additionally	additionally	ADV
ejpam-5569	595	2	,	,	PUNCT
ejpam-5569	595	3	it	it	PRON
ejpam-5569	595	4	is	be	AUX
ejpam-5569	595	5	worth	worth	ADJ
ejpam-5569	595	6	noting	note	VERB
ejpam-5569	595	7	that	that	SCONJ
ejpam-5569	595	8	the	the	DET
ejpam-5569	595	9	results	result	NOUN
ejpam-5569	595	10	obtained	obtain	VERB
ejpam-5569	595	11	in	in	ADP
ejpam-5569	595	12	theorem	theorem	ADJ
ejpam-5569	595	13	2	2	NUM
ejpam-5569	595	14	outperform	outperform	VERB
ejpam-5569	595	15	the	the	DET
ejpam-5569	595	16	results	result	NOUN
ejpam-5569	595	17	in	in	ADP
ejpam-5569	595	18	[	[	X
ejpam-5569	595	19	7	7	NUM
ejpam-5569	595	20	]	]	PUNCT
ejpam-5569	595	21	for	for	ADP
ejpam-5569	595	22	all	all	DET
ejpam-5569	595	23	values	value	NOUN
ejpam-5569	595	24	of	of	ADP
ejpam-5569	595	25	n.	n.	NOUN
ejpam-5569	595	26	in	in	ADP
ejpam-5569	595	27	terms	term	NOUN
ejpam-5569	595	28	of	of	ADP
ejpam-5569	595	29	running	run	VERB
ejpam-5569	595	30	time	time	NOUN
ejpam-5569	595	31	,	,	PUNCT
ejpam-5569	595	32	table	table	NOUN
ejpam-5569	595	33	4	4	NUM
ejpam-5569	595	34	indicates	indicate	VERB
ejpam-5569	595	35	that	that	SCONJ
ejpam-5569	595	36	the	the	DET
ejpam-5569	595	37	results	result	NOUN
ejpam-5569	595	38	derived	derive	VERB
ejpam-5569	595	39	from	from	ADP
ejpam-5569	595	40	theorem	theorem	ADJ
ejpam-5569	595	41	2	2	NUM
ejpam-5569	595	42	have	have	VERB
ejpam-5569	595	43	a	a	DET
ejpam-5569	595	44	constant	constant	ADJ
ejpam-5569	595	45	time	time	NOUN
ejpam-5569	595	46	complexity	complexity	NOUN
ejpam-5569	595	47	of	of	ADP
ejpam-5569	595	48	o(1	o(1	NOUN
ejpam-5569	595	49	)	)	PUNCT
ejpam-5569	595	50	,	,	PUNCT
ejpam-5569	595	51	whereas	whereas	SCONJ
ejpam-5569	595	52	the	the	DET
ejpam-5569	595	53	results	result	NOUN
ejpam-5569	595	54	in	in	ADP
ejpam-5569	595	55	[	[	X
ejpam-5569	595	56	6	6	NUM
ejpam-5569	595	57	]	]	PUNCT
ejpam-5569	595	58	have	have	VERB
ejpam-5569	595	59	a	a	DET
ejpam-5569	595	60	time	time	NOUN
ejpam-5569	595	61	complexity	complexity	NOUN
ejpam-5569	595	62	of	of	ADP
ejpam-5569	595	63	o(n3	o(n3	NOUN
ejpam-5569	595	64	)	)	PUNCT
ejpam-5569	595	65	.	.	PUNCT
ejpam-5569	596	1	furthermore	furthermore	ADV
ejpam-5569	596	2	,	,	PUNCT
ejpam-5569	596	3	the	the	DET
ejpam-5569	596	4	results	result	NOUN
ejpam-5569	596	5	from	from	ADP
ejpam-5569	596	6	theorem	theorem	ADJ
ejpam-5569	596	7	2	2	NUM
ejpam-5569	596	8	require	require	NOUN
ejpam-5569	596	9	less	less	ADJ
ejpam-5569	596	10	time	time	NOUN
ejpam-5569	596	11	compared	compare	VERB
ejpam-5569	596	12	to	to	ADP
ejpam-5569	596	13	the	the	DET
ejpam-5569	596	14	results	result	NOUN
ejpam-5569	596	15	in	in	ADP
ejpam-5569	596	16	[	[	X
ejpam-5569	596	17	7	7	NUM
ejpam-5569	596	18	]	]	PUNCT
ejpam-5569	596	19	.	.	PUNCT
ejpam-5569	597	1	figure	figure	VERB
ejpam-5569	597	2	7	7	NUM
ejpam-5569	597	3	:	:	PUNCT
ejpam-5569	597	4	comparison	comparison	NOUN
ejpam-5569	597	5	among	among	ADP
ejpam-5569	597	6	theorem	theorem	NOUN
ejpam-5569	597	7	2	2	NUM
ejpam-5569	597	8	,	,	PUNCT
ejpam-5569	597	9	saha	saha	NOUN
ejpam-5569	598	1	[	[	X
ejpam-5569	598	2	6	6	NUM
ejpam-5569	598	3	]	]	PUNCT
ejpam-5569	598	4	and	and	CCONJ
ejpam-5569	598	5	ilpm	ilpm	NOUN
ejpam-5569	599	1	[	[	X
ejpam-5569	599	2	7	7	X
ejpam-5569	599	3	]	]	PUNCT
ejpam-5569	599	4	for	for	SCONJ
ejpam-5569	599	5	the	the	DET
ejpam-5569	599	6	radio	radio	NOUN
ejpam-5569	599	7	number	number	NOUN
ejpam-5569	599	8	of	of	ADP
ejpam-5569	599	9	shadow	shadow	NOUN
ejpam-5569	599	10	graph	graph	NOUN
ejpam-5569	599	11	with	with	ADP
ejpam-5569	599	12	n	n	NOUN
ejpam-5569	599	13	=	=	SYM
ejpam-5569	599	14	3(mod)4	3(mod)4	NUM
ejpam-5569	599	15	figure	figure	NOUN
ejpam-5569	599	16	8	8	NUM
ejpam-5569	599	17	:	:	PUNCT
ejpam-5569	599	18	comparison	comparison	NOUN
ejpam-5569	599	19	among	among	ADP
ejpam-5569	599	20	theorem	theorem	NOUN
ejpam-5569	599	21	3	3	NUM
ejpam-5569	599	22	,	,	PUNCT
ejpam-5569	599	23	saha	saha	NOUN
ejpam-5569	600	1	[	[	X
ejpam-5569	600	2	6	6	NUM
ejpam-5569	600	3	]	]	PUNCT
ejpam-5569	600	4	and	and	CCONJ
ejpam-5569	600	5	ilpm	ilpm	NOUN
ejpam-5569	601	1	[	[	X
ejpam-5569	601	2	7	7	X
ejpam-5569	601	3	]	]	PUNCT
ejpam-5569	601	4	for	for	SCONJ
ejpam-5569	601	5	the	the	DET
ejpam-5569	601	6	radio	radio	NOUN
ejpam-5569	601	7	number	number	NOUN
ejpam-5569	601	8	of	of	ADP
ejpam-5569	601	9	shadow	shadow	NOUN
ejpam-5569	601	10	graph	graph	NOUN
ejpam-5569	601	11	with	with	ADP
ejpam-5569	601	12	n	n	NOUN
ejpam-5569	601	13	=	=	SYM
ejpam-5569	601	14	0(mod)4	0(mod)4	NUM
ejpam-5569	601	15	.	.	PUNCT
ejpam-5569	601	16	table	table	NOUN
ejpam-5569	601	17	5	5	NUM
ejpam-5569	601	18	and	and	CCONJ
ejpam-5569	601	19	figure	figure	VERB
ejpam-5569	601	20	8	8	NUM
ejpam-5569	601	21	demonstrate	demonstrate	VERB
ejpam-5569	601	22	that	that	SCONJ
ejpam-5569	601	23	the	the	DET
ejpam-5569	601	24	results	result	NOUN
ejpam-5569	601	25	presented	present	VERB
ejpam-5569	601	26	in	in	ADP
ejpam-5569	601	27	theorem	theorem	ADJ
ejpam-5569	601	28	3	3	NUM
ejpam-5569	601	29	are	be	AUX
ejpam-5569	601	30	identical	identical	ADJ
ejpam-5569	601	31	to	to	ADP
ejpam-5569	601	32	the	the	DET
ejpam-5569	601	33	results	result	NOUN
ejpam-5569	601	34	in	in	ADP
ejpam-5569	601	35	[	[	X
ejpam-5569	601	36	6	6	NUM
ejpam-5569	601	37	]	]	PUNCT
ejpam-5569	601	38	,	,	PUNCT
ejpam-5569	601	39	considering	consider	VERB
ejpam-5569	601	40	the	the	DET
ejpam-5569	601	41	upper	upper	ADJ
ejpam-5569	601	42	bound	bound	NOUN
ejpam-5569	601	43	of	of	ADP
ejpam-5569	601	44	the	the	DET
ejpam-5569	601	45	radio	radio	NOUN
ejpam-5569	601	46	number	number	NOUN
ejpam-5569	601	47	of	of	ADP
ejpam-5569	601	48	shadow	shadow	NOUN
ejpam-5569	601	49	graph	graph	NOUN
ejpam-5569	601	50	with	with	ADP
ejpam-5569	601	51	n	n	NOUN
ejpam-5569	601	52	=	=	SYM
ejpam-5569	601	53	0	0	NUM
ejpam-5569	601	54	(	(	PUNCT
ejpam-5569	601	55	mod	mod	PROPN
ejpam-5569	601	56	)	)	PUNCT
ejpam-5569	601	57	4	4	NUM
ejpam-5569	601	58	.	.	PUNCT
ejpam-5569	602	1	additionally	additionally	ADV
ejpam-5569	602	2	,	,	PUNCT
ejpam-5569	602	3	it	it	PRON
ejpam-5569	602	4	is	be	AUX
ejpam-5569	602	5	worth	worth	ADJ
ejpam-5569	602	6	noting	note	VERB
ejpam-5569	602	7	that	that	SCONJ
ejpam-5569	602	8	the	the	DET
ejpam-5569	602	9	results	result	NOUN
ejpam-5569	602	10	obtained	obtain	VERB
ejpam-5569	602	11	in	in	ADP
ejpam-5569	602	12	theorem	theorem	ADJ
ejpam-5569	602	13	3	3	NUM
ejpam-5569	602	14	outperform	outperform	VERB
ejpam-5569	602	15	the	the	DET
ejpam-5569	602	16	results	result	NOUN
ejpam-5569	602	17	in	in	ADP
ejpam-5569	602	18	[	[	X
ejpam-5569	602	19	7	7	NUM
ejpam-5569	602	20	]	]	PUNCT
ejpam-5569	602	21	for	for	ADP
ejpam-5569	602	22	all	all	DET
ejpam-5569	602	23	values	value	NOUN
ejpam-5569	602	24	of	of	ADP
ejpam-5569	602	25	n.	n.	NOUN
ejpam-5569	602	26	in	in	ADP
ejpam-5569	602	27	terms	term	NOUN
ejpam-5569	602	28	of	of	ADP
ejpam-5569	602	29	running	run	VERB
ejpam-5569	602	30	time	time	NOUN
ejpam-5569	602	31	,	,	PUNCT
ejpam-5569	602	32	table	table	NOUN
ejpam-5569	602	33	5	5	NUM
ejpam-5569	602	34	indicates	indicate	VERB
ejpam-5569	602	35	that	that	SCONJ
ejpam-5569	602	36	the	the	DET
ejpam-5569	602	37	results	result	NOUN
ejpam-5569	602	38	derived	derive	VERB
ejpam-5569	602	39	from	from	ADP
ejpam-5569	602	40	theorem	theorem	ADJ
ejpam-5569	602	41	3	3	NUM
ejpam-5569	602	42	have	have	VERB
ejpam-5569	602	43	a	a	DET
ejpam-5569	602	44	constant	constant	ADJ
ejpam-5569	602	45	time	time	NOUN
ejpam-5569	602	46	complexity	complexity	NOUN
ejpam-5569	602	47	of	of	ADP
ejpam-5569	602	48	o(1	o(1	NOUN
ejpam-5569	602	49	)	)	PUNCT
ejpam-5569	602	50	,	,	PUNCT
ejpam-5569	602	51	whereas	whereas	SCONJ
ejpam-5569	602	52	the	the	DET
ejpam-5569	602	53	results	result	NOUN
ejpam-5569	602	54	in	in	ADP
ejpam-5569	602	55	[	[	X
ejpam-5569	602	56	6	6	NUM
ejpam-5569	602	57	]	]	PUNCT
ejpam-5569	602	58	have	have	VERB
ejpam-5569	602	59	a	a	DET
ejpam-5569	602	60	time	time	NOUN
ejpam-5569	602	61	complexity	complexity	NOUN
ejpam-5569	602	62	of	of	ADP
ejpam-5569	602	63	o(n3	o(n3	NOUN
ejpam-5569	602	64	)	)	PUNCT
ejpam-5569	602	65	.	.	PUNCT
ejpam-5569	603	1	furthermore	furthermore	ADV
ejpam-5569	603	2	,	,	PUNCT
ejpam-5569	603	3	the	the	DET
ejpam-5569	603	4	results	result	NOUN
ejpam-5569	603	5	from	from	ADP
ejpam-5569	603	6	theorem	theorem	ADJ
ejpam-5569	603	7	3	3	NUM
ejpam-5569	603	8	require	require	NOUN
ejpam-5569	603	9	less	less	ADJ
ejpam-5569	603	10	time	time	NOUN
ejpam-5569	603	11	compared	compare	VERB
ejpam-5569	603	12	to	to	ADP
ejpam-5569	603	13	the	the	DET
ejpam-5569	603	14	results	result	NOUN
ejpam-5569	603	15	in	in	ADP
ejpam-5569	603	16	[	[	X
ejpam-5569	603	17	7	7	NUM
ejpam-5569	603	18	]	]	PUNCT
ejpam-5569	603	19	.	.	PUNCT
ejpam-5569	604	1	n.	n.	PROPN
ejpam-5569	604	2	m.	m.	PROPN
ejpam-5569	604	3	noureldeen	noureldeen	INTJ
ejpam-5569	604	4	et	et	PROPN
ejpam-5569	604	5	al	al	PROPN
ejpam-5569	604	6	.	.	PUNCT
ejpam-5569	604	7	/	/	SYM
ejpam-5569	604	8	eur	eur	PROPN
ejpam-5569	604	9	.	.	PUNCT
ejpam-5569	605	1	j.	j.	PROPN
ejpam-5569	605	2	pure	pure	PROPN
ejpam-5569	605	3	appl	appl	PROPN
ejpam-5569	605	4	.	.	PROPN
ejpam-5569	605	5	math	math	PROPN
ejpam-5569	605	6	,	,	PUNCT
ejpam-5569	605	7	18	18	NUM
ejpam-5569	605	8	(	(	PUNCT
ejpam-5569	605	9	2	2	NUM
ejpam-5569	605	10	)	)	PUNCT
ejpam-5569	605	11	(	(	PUNCT
ejpam-5569	605	12	2025	2025	NUM
ejpam-5569	605	13	)	)	PUNCT
ejpam-5569	605	14	,	,	PUNCT
ejpam-5569	605	15	5569	5569	NUM
ejpam-5569	605	16	22	22	NUM
ejpam-5569	605	17	of	of	ADP
ejpam-5569	605	18	25	25	NUM
ejpam-5569	605	19	table	table	NOUN
ejpam-5569	605	20	5	5	NUM
ejpam-5569	605	21	:	:	PUNCT
ejpam-5569	605	22	comparison	comparison	NOUN
ejpam-5569	605	23	among	among	ADP
ejpam-5569	605	24	theorem	theorem	NOUN
ejpam-5569	605	25	3	3	NUM
ejpam-5569	605	26	,	,	PUNCT
ejpam-5569	605	27	saha	saha	PROPN
ejpam-5569	605	28	and	and	CCONJ
ejpam-5569	605	29	ilpm	ilpm	PROPN
ejpam-5569	606	1	[	[	X
ejpam-5569	606	2	7	7	X
ejpam-5569	606	3	]	]	PUNCT
ejpam-5569	606	4	for	for	SCONJ
ejpam-5569	606	5	the	the	DET
ejpam-5569	606	6	radio	radio	NOUN
ejpam-5569	606	7	number	number	NOUN
ejpam-5569	606	8	of	of	ADP
ejpam-5569	606	9	shadow	shadow	NOUN
ejpam-5569	606	10	graph	graph	NOUN
ejpam-5569	606	11	with	with	ADP
ejpam-5569	606	12	n	n	NOUN
ejpam-5569	606	13	=	=	SYM
ejpam-5569	606	14	0(mod)4	0(mod)4	NUM
ejpam-5569	606	15	.	.	PUNCT
ejpam-5569	607	1	n	n	PROPN
ejpam-5569	607	2	2n	2n	NUM
ejpam-5569	607	3	theorem	theorem	VERB
ejpam-5569	607	4	3	3	NUM
ejpam-5569	607	5	saha	saha	NOUN
ejpam-5569	608	1	[	[	X
ejpam-5569	608	2	6	6	NUM
ejpam-5569	608	3	]	]	X
ejpam-5569	608	4	ilpm	ilpm	NOUN
ejpam-5569	609	1	[	[	X
ejpam-5569	609	2	7	7	NUM
ejpam-5569	609	3	]	]	X
ejpam-5569	609	4	rn	rn	PROPN
ejpam-5569	609	5	cpu	cpu	PROPN
ejpam-5569	609	6	time	time	NOUN
ejpam-5569	609	7	m	m	PROPN
ejpam-5569	609	8	cpu	cpu	VERB
ejpam-5569	609	9	time	time	NOUN
ejpam-5569	609	10	rn	rn	PROPN
ejpam-5569	609	11	cpu	cpu	VERB
ejpam-5569	609	12	time	time	NOUN
ejpam-5569	609	13	8	8	NUM
ejpam-5569	609	14	16	16	NUM
ejpam-5569	609	15	60	60	NUM
ejpam-5569	609	16	o(1	o(1	NOUN
ejpam-5569	609	17	)	)	PUNCT
ejpam-5569	609	18	60	60	NUM
ejpam-5569	609	19	0.036009	0.036009	NUM
ejpam-5569	609	20	99	99	NUM
ejpam-5569	609	21	0.033053	0.033053	NUM
ejpam-5569	609	22	12	12	NUM
ejpam-5569	609	23	24	24	NUM
ejpam-5569	609	24	138	138	NUM
ejpam-5569	609	25	o(1	o(1	NOUN
ejpam-5569	609	26	)	)	PUNCT
ejpam-5569	609	27	138	138	NUM
ejpam-5569	609	28	0.046854	0.046854	NUM
ejpam-5569	609	29	243	243	NUM
ejpam-5569	609	30	0.089839	0.089839	NUM
ejpam-5569	609	31	16	16	NUM
ejpam-5569	609	32	32	32	NUM
ejpam-5569	609	33	248	248	NUM
ejpam-5569	609	34	o(1	o(1	NOUN
ejpam-5569	609	35	)	)	PUNCT
ejpam-5569	609	36	248	248	NUM
ejpam-5569	609	37	0.052537	0.052537	NUM
ejpam-5569	609	38	451	451	NUM
ejpam-5569	609	39	0.097359	0.097359	NUM
ejpam-5569	609	40	20	20	NUM
ejpam-5569	609	41	40	40	NUM
ejpam-5569	609	42	390	390	NUM
ejpam-5569	609	43	o(1	o(1	NOUN
ejpam-5569	609	44	)	)	PUNCT
ejpam-5569	609	45	390	390	NUM
ejpam-5569	609	46	0.053037	0.053037	NUM
ejpam-5569	609	47	723	723	NUM
ejpam-5569	609	48	0.102315	0.102315	NUM
ejpam-5569	609	49	24	24	NUM
ejpam-5569	609	50	48	48	NUM
ejpam-5569	609	51	564	564	NUM
ejpam-5569	609	52	o(1	o(1	NOUN
ejpam-5569	609	53	)	)	PUNCT
ejpam-5569	609	54	564	564	NUM
ejpam-5569	609	55	0.054454	0.054454	NUM
ejpam-5569	609	56	1059	1059	NUM
ejpam-5569	609	57	0.124432	0.124432	NUM
ejpam-5569	609	58	28	28	NUM
ejpam-5569	609	59	56	56	NUM
ejpam-5569	609	60	770	770	NUM
ejpam-5569	609	61	o(1	o(1	NOUN
ejpam-5569	609	62	)	)	PUNCT
ejpam-5569	609	63	770	770	NUM
ejpam-5569	609	64	0.082983	0.082983	NUM
ejpam-5569	609	65	1459	1459	NUM
ejpam-5569	609	66	0.171284	0.171284	NUM
ejpam-5569	609	67	32	32	NUM
ejpam-5569	609	68	64	64	NUM
ejpam-5569	609	69	1008	1008	NUM
ejpam-5569	609	70	o(1	o(1	NOUN
ejpam-5569	609	71	)	)	PUNCT
ejpam-5569	609	72	1008	1008	NUM
ejpam-5569	609	73	0.086663	0.086663	NUM
ejpam-5569	609	74	1923	1923	NUM
ejpam-5569	609	75	0.185284	0.185284	NUM
ejpam-5569	609	76	36	36	NUM
ejpam-5569	609	77	72	72	NUM
ejpam-5569	609	78	1278	1278	NUM
ejpam-5569	609	79	o(1	o(1	NOUN
ejpam-5569	609	80	)	)	PUNCT
ejpam-5569	609	81	1278	1278	NUM
ejpam-5569	609	82	0.099965	0.099965	NUM
ejpam-5569	609	83	2451	2451	NUM
ejpam-5569	609	84	0.218490	0.218490	NUM
ejpam-5569	609	85	40	40	NUM
ejpam-5569	609	86	80	80	NUM
ejpam-5569	609	87	1580	1580	NUM
ejpam-5569	609	88	o(1	o(1	NOUN
ejpam-5569	609	89	)	)	PUNCT
ejpam-5569	609	90	1580	1580	NUM
ejpam-5569	609	91	0.102013	0.102013	NUM
ejpam-5569	609	92	3043	3043	NUM
ejpam-5569	609	93	0.235134	0.235134	NUM
ejpam-5569	609	94	44	44	NUM
ejpam-5569	609	95	88	88	NUM
ejpam-5569	609	96	1914	1914	NUM
ejpam-5569	609	97	o(1	o(1	NOUN
ejpam-5569	609	98	)	)	PUNCT
ejpam-5569	609	99	1914	1914	NUM
ejpam-5569	609	100	0.123509	0.123509	NUM
ejpam-5569	609	101	3699	3699	NUM
ejpam-5569	609	102	0.254231	0.254231	NUM
ejpam-5569	609	103	48	48	NUM
ejpam-5569	609	104	96	96	NUM
ejpam-5569	609	105	2280	2280	NUM
ejpam-5569	609	106	o(1	o(1	NOUN
ejpam-5569	609	107	)	)	PUNCT
ejpam-5569	609	108	2280	2280	NUM
ejpam-5569	609	109	0.123897	0.123897	NUM
ejpam-5569	609	110	4419	4419	NUM
ejpam-5569	609	111	0.287511	0.287511	NUM
ejpam-5569	609	112	52	52	NUM
ejpam-5569	609	113	104	104	NUM
ejpam-5569	609	114	2678	2678	NUM
ejpam-5569	609	115	o(1	o(1	NOUN
ejpam-5569	609	116	)	)	PUNCT
ejpam-5569	609	117	2678	2678	NUM
ejpam-5569	609	118	0.134616	0.134616	NUM
ejpam-5569	609	119	5203	5203	NUM
ejpam-5569	609	120	0.451393	0.451393	NUM
ejpam-5569	609	121	56	56	NUM
ejpam-5569	609	122	112	112	NUM
ejpam-5569	609	123	3108	3108	NUM
ejpam-5569	609	124	o(1	o(1	NOUN
ejpam-5569	609	125	)	)	PUNCT
ejpam-5569	609	126	3108	3108	NUM
ejpam-5569	609	127	0.141561	0.141561	NUM
ejpam-5569	609	128	6051	6051	NUM
ejpam-5569	609	129	0.466374	0.466374	NUM
ejpam-5569	609	130	60	60	NUM
ejpam-5569	609	131	120	120	NUM
ejpam-5569	609	132	3570	3570	NUM
ejpam-5569	609	133	o(1	o(1	NOUN
ejpam-5569	609	134	)	)	PUNCT
ejpam-5569	609	135	3570	3570	NUM
ejpam-5569	609	136	0.149171	0.149171	NUM
ejpam-5569	609	137	6963	6963	NUM
ejpam-5569	609	138	0.477208	0.477208	NUM
ejpam-5569	609	139	64	64	NUM
ejpam-5569	609	140	128	128	NUM
ejpam-5569	609	141	4064	4064	NUM
ejpam-5569	609	142	o(1	o(1	NOUN
ejpam-5569	609	143	)	)	PUNCT
ejpam-5569	609	144	4064	4064	NUM
ejpam-5569	609	145	0.178655	0.178655	NUM
ejpam-5569	609	146	7939	7939	NUM
ejpam-5569	609	147	0.527846	0.527846	NUM
ejpam-5569	609	148	68	68	NUM
ejpam-5569	609	149	136	136	NUM
ejpam-5569	609	150	4590	4590	NUM
ejpam-5569	609	151	o(1	o(1	NOUN
ejpam-5569	609	152	)	)	PUNCT
ejpam-5569	609	153	4590	4590	NUM
ejpam-5569	609	154	0.198579	0.198579	NUM
ejpam-5569	609	155	8979	8979	NUM
ejpam-5569	609	156	0.635992	0.635992	NUM
ejpam-5569	609	157	72	72	NUM
ejpam-5569	609	158	144	144	NUM
ejpam-5569	609	159	5148	5148	NUM
ejpam-5569	609	160	o(1	o(1	NOUN
ejpam-5569	609	161	)	)	PUNCT
ejpam-5569	609	162	5148	5148	NUM
ejpam-5569	609	163	0.20673	0.20673	NUM
ejpam-5569	609	164	10083	10083	NUM
ejpam-5569	609	165	0.976116	0.976116	NUM
ejpam-5569	609	166	76	76	NUM
ejpam-5569	609	167	152	152	NUM
ejpam-5569	609	168	5738	5738	NUM
ejpam-5569	609	169	o(1	o(1	NOUN
ejpam-5569	609	170	)	)	PUNCT
ejpam-5569	609	171	5738	5738	NUM
ejpam-5569	609	172	0.213292	0.213292	NUM
ejpam-5569	609	173	11251	11251	NUM
ejpam-5569	609	174	0.983264	0.983264	NUM
ejpam-5569	609	175	80	80	NUM
ejpam-5569	609	176	160	160	NUM
ejpam-5569	609	177	6360	6360	NUM
ejpam-5569	609	178	o(1	o(1	NOUN
ejpam-5569	609	179	)	)	PUNCT
ejpam-5569	609	180	6360	6360	NUM
ejpam-5569	609	181	0.231493	0.231493	NUM
ejpam-5569	609	182	12483	12483	NUM
ejpam-5569	609	183	0.998945	0.998945	NUM
ejpam-5569	609	184	84	84	NUM
ejpam-5569	609	185	168	168	NUM
ejpam-5569	609	186	7014	7014	NUM
ejpam-5569	609	187	o(1	o(1	NOUN
ejpam-5569	609	188	)	)	PUNCT
ejpam-5569	609	189	7014	7014	NUM
ejpam-5569	610	1	0.252509	0.252509	NUM
ejpam-5569	610	2	13779	13779	NUM
ejpam-5569	610	3	1.034510	1.034510	NUM
ejpam-5569	610	4	figure	figure	NOUN
ejpam-5569	610	5	9	9	NUM
ejpam-5569	610	6	:	:	PUNCT
ejpam-5569	610	7	comparison	comparison	NOUN
ejpam-5569	610	8	among	among	ADP
ejpam-5569	610	9	theorem	theorem	NOUN
ejpam-5569	610	10	4	4	NUM
ejpam-5569	610	11	,	,	PUNCT
ejpam-5569	610	12	saha	saha	NOUN
ejpam-5569	611	1	[	[	X
ejpam-5569	611	2	6	6	NUM
ejpam-5569	611	3	]	]	PUNCT
ejpam-5569	611	4	and	and	CCONJ
ejpam-5569	611	5	ilpm	ilpm	NOUN
ejpam-5569	612	1	[	[	X
ejpam-5569	612	2	7	7	X
ejpam-5569	612	3	]	]	PUNCT
ejpam-5569	612	4	for	for	SCONJ
ejpam-5569	612	5	the	the	DET
ejpam-5569	612	6	radio	radio	NOUN
ejpam-5569	612	7	number	number	NOUN
ejpam-5569	612	8	of	of	ADP
ejpam-5569	612	9	shadow	shadow	NOUN
ejpam-5569	612	10	graph	graph	NOUN
ejpam-5569	612	11	with	with	ADP
ejpam-5569	612	12	n	n	NOUN
ejpam-5569	612	13	=	=	SYM
ejpam-5569	612	14	1(mod)4	1(mod)4	NUM
ejpam-5569	612	15	.	.	PUNCT
ejpam-5569	612	16	table	table	NOUN
ejpam-5569	612	17	6	6	NUM
ejpam-5569	612	18	and	and	CCONJ
ejpam-5569	612	19	figure	figure	VERB
ejpam-5569	612	20	9	9	NUM
ejpam-5569	612	21	demonstrate	demonstrate	VERB
ejpam-5569	612	22	that	that	SCONJ
ejpam-5569	612	23	the	the	DET
ejpam-5569	612	24	results	result	NOUN
ejpam-5569	612	25	presented	present	VERB
ejpam-5569	612	26	in	in	ADP
ejpam-5569	612	27	theorem	theorem	ADJ
ejpam-5569	612	28	4	4	NUM
ejpam-5569	612	29	align	align	NOUN
ejpam-5569	612	30	with	with	ADP
ejpam-5569	612	31	the	the	DET
ejpam-5569	612	32	results	result	NOUN
ejpam-5569	612	33	in	in	ADP
ejpam-5569	612	34	[	[	X
ejpam-5569	612	35	6	6	NUM
ejpam-5569	612	36	]	]	PUNCT
ejpam-5569	612	37	,	,	PUNCT
ejpam-5569	612	38	considering	consider	VERB
ejpam-5569	612	39	the	the	DET
ejpam-5569	612	40	upper	upper	ADJ
ejpam-5569	612	41	bound	bound	NOUN
ejpam-5569	612	42	of	of	ADP
ejpam-5569	612	43	the	the	DET
ejpam-5569	612	44	radio	radio	NOUN
ejpam-5569	612	45	number	number	NOUN
ejpam-5569	612	46	of	of	ADP
ejpam-5569	612	47	shadow	shadow	NOUN
ejpam-5569	612	48	graph	graph	NOUN
ejpam-5569	612	49	with	with	ADP
ejpam-5569	612	50	n	n	NOUN
ejpam-5569	612	51	=	=	SYM
ejpam-5569	612	52	1	1	NUM
ejpam-5569	612	53	(	(	PUNCT
ejpam-5569	612	54	mod	mod	PROPN
ejpam-5569	612	55	)	)	PUNCT
ejpam-5569	612	56	4	4	NUM
ejpam-5569	612	57	.	.	PUNCT
ejpam-5569	613	1	additionally	additionally	ADV
ejpam-5569	613	2	,	,	PUNCT
ejpam-5569	613	3	it	it	PRON
ejpam-5569	613	4	is	be	AUX
ejpam-5569	613	5	worth	worth	ADJ
ejpam-5569	613	6	noting	note	VERB
ejpam-5569	613	7	that	that	SCONJ
ejpam-5569	613	8	the	the	DET
ejpam-5569	613	9	results	result	NOUN
ejpam-5569	613	10	in	in	ADP
ejpam-5569	613	11	theorem	theorem	ADJ
ejpam-5569	613	12	4	4	NUM
ejpam-5569	613	13	outperform	outperform	VERB
ejpam-5569	613	14	the	the	DET
ejpam-5569	613	15	results	result	NOUN
ejpam-5569	613	16	in	in	ADP
ejpam-5569	613	17	[	[	X
ejpam-5569	613	18	7	7	NUM
ejpam-5569	613	19	]	]	PUNCT
ejpam-5569	613	20	for	for	ADP
ejpam-5569	613	21	all	all	DET
ejpam-5569	613	22	values	value	NOUN
ejpam-5569	613	23	of	of	ADP
ejpam-5569	613	24	n.	n.	NOUN
ejpam-5569	613	25	in	in	ADP
ejpam-5569	613	26	terms	term	NOUN
ejpam-5569	613	27	of	of	ADP
ejpam-5569	613	28	running	run	VERB
ejpam-5569	613	29	time	time	NOUN
ejpam-5569	613	30	,	,	PUNCT
ejpam-5569	613	31	table	table	NOUN
ejpam-5569	613	32	6	6	NUM
ejpam-5569	613	33	indicates	indicate	VERB
ejpam-5569	613	34	that	that	SCONJ
ejpam-5569	613	35	the	the	DET
ejpam-5569	613	36	results	result	NOUN
ejpam-5569	613	37	in	in	ADP
ejpam-5569	613	38	theorem	theorem	NOUN
ejpam-5569	613	39	4	4	NUM
ejpam-5569	613	40	have	have	VERB
ejpam-5569	613	41	a	a	DET
ejpam-5569	613	42	constant	constant	ADJ
ejpam-5569	613	43	time	time	NOUN
ejpam-5569	613	44	complexity	complexity	NOUN
ejpam-5569	613	45	of	of	ADP
ejpam-5569	613	46	o(1	o(1	NOUN
ejpam-5569	613	47	)	)	PUNCT
ejpam-5569	613	48	,	,	PUNCT
ejpam-5569	613	49	whereas	whereas	SCONJ
ejpam-5569	613	50	the	the	DET
ejpam-5569	613	51	results	result	NOUN
ejpam-5569	613	52	in	in	ADP
ejpam-5569	613	53	[	[	X
ejpam-5569	613	54	6	6	NUM
ejpam-5569	613	55	]	]	X
ejpam-5569	613	56	n.	n.	NOUN
ejpam-5569	613	57	m.	m.	NOUN
ejpam-5569	613	58	noureldeen	noureldeen	NOUN
ejpam-5569	613	59	et	et	PROPN
ejpam-5569	613	60	al	al	PROPN
ejpam-5569	613	61	.	.	PUNCT
ejpam-5569	613	62	/	/	SYM
ejpam-5569	613	63	eur	eur	PROPN
ejpam-5569	613	64	.	.	PUNCT
ejpam-5569	614	1	j.	j.	PROPN
ejpam-5569	614	2	pure	pure	PROPN
ejpam-5569	614	3	appl	appl	PROPN
ejpam-5569	614	4	.	.	PROPN
ejpam-5569	614	5	math	math	PROPN
ejpam-5569	614	6	,	,	PUNCT
ejpam-5569	614	7	18	18	NUM
ejpam-5569	614	8	(	(	PUNCT
ejpam-5569	614	9	2	2	NUM
ejpam-5569	614	10	)	)	PUNCT
ejpam-5569	614	11	(	(	PUNCT
ejpam-5569	614	12	2025	2025	NUM
ejpam-5569	614	13	)	)	PUNCT
ejpam-5569	614	14	,	,	PUNCT
ejpam-5569	614	15	5569	5569	NUM
ejpam-5569	614	16	23	23	NUM
ejpam-5569	614	17	of	of	ADP
ejpam-5569	614	18	25	25	NUM
ejpam-5569	614	19	have	have	VERB
ejpam-5569	614	20	a	a	DET
ejpam-5569	614	21	time	time	NOUN
ejpam-5569	614	22	complexity	complexity	NOUN
ejpam-5569	614	23	of	of	ADP
ejpam-5569	614	24	o(n3	o(n3	NOUN
ejpam-5569	614	25	)	)	PUNCT
ejpam-5569	614	26	.	.	PUNCT
ejpam-5569	615	1	furthermore	furthermore	ADV
ejpam-5569	615	2	,	,	PUNCT
ejpam-5569	615	3	the	the	DET
ejpam-5569	615	4	results	result	NOUN
ejpam-5569	615	5	in	in	ADP
ejpam-5569	615	6	theorem	theorem	ADJ
ejpam-5569	615	7	4	4	NUM
ejpam-5569	615	8	require	require	VERB
ejpam-5569	615	9	less	less	ADJ
ejpam-5569	615	10	time	time	NOUN
ejpam-5569	615	11	compared	compare	VERB
ejpam-5569	615	12	to	to	ADP
ejpam-5569	615	13	the	the	DET
ejpam-5569	615	14	results	result	NOUN
ejpam-5569	615	15	in	in	ADP
ejpam-5569	615	16	[	[	X
ejpam-5569	615	17	7	7	NUM
ejpam-5569	615	18	]	]	PUNCT
ejpam-5569	615	19	.	.	PUNCT
ejpam-5569	616	1	table	table	NOUN
ejpam-5569	616	2	6	6	NUM
ejpam-5569	616	3	:	:	PUNCT
ejpam-5569	616	4	comparison	comparison	NOUN
ejpam-5569	616	5	among	among	ADP
ejpam-5569	616	6	theorem	theorem	NOUN
ejpam-5569	616	7	4	4	NUM
ejpam-5569	616	8	,	,	PUNCT
ejpam-5569	616	9	saha	saha	NOUN
ejpam-5569	617	1	[	[	X
ejpam-5569	617	2	6	6	NUM
ejpam-5569	617	3	]	]	PUNCT
ejpam-5569	617	4	and	and	CCONJ
ejpam-5569	617	5	ilpm	ilpm	NOUN
ejpam-5569	618	1	[	[	X
ejpam-5569	618	2	7	7	X
ejpam-5569	618	3	]	]	PUNCT
ejpam-5569	618	4	for	for	SCONJ
ejpam-5569	618	5	the	the	DET
ejpam-5569	618	6	radio	radio	NOUN
ejpam-5569	618	7	number	number	NOUN
ejpam-5569	618	8	of	of	ADP
ejpam-5569	618	9	shadow	shadow	NOUN
ejpam-5569	618	10	graph	graph	NOUN
ejpam-5569	618	11	with	with	ADP
ejpam-5569	618	12	n	n	NOUN
ejpam-5569	618	13	=	=	SYM
ejpam-5569	618	14	1(mod)4	1(mod)4	NUM
ejpam-5569	618	15	.	.	PUNCT
ejpam-5569	619	1	n	n	PROPN
ejpam-5569	619	2	2n	2n	NUM
ejpam-5569	619	3	theorem	theorem	VERB
ejpam-5569	619	4	4	4	NUM
ejpam-5569	619	5	saha	saha	NOUN
ejpam-5569	619	6	[	[	X
ejpam-5569	619	7	6	6	NUM
ejpam-5569	619	8	]	]	X
ejpam-5569	619	9	ilpm	ilpm	NOUN
ejpam-5569	620	1	[	[	X
ejpam-5569	620	2	7	7	NUM
ejpam-5569	620	3	]	]	X
ejpam-5569	620	4	rn	rn	PROPN
ejpam-5569	620	5	cpu	cpu	PROPN
ejpam-5569	620	6	time	time	PROPN
ejpam-5569	620	7	rn	rn	PROPN
ejpam-5569	620	8	cpu	cpu	VERB
ejpam-5569	620	9	time	time	PROPN
ejpam-5569	620	10	rn	rn	PROPN
ejpam-5569	620	11	cpu	cpu	VERB
ejpam-5569	620	12	time	time	NOUN
ejpam-5569	620	13	9	9	NUM
ejpam-5569	620	14	18	18	NUM
ejpam-5569	620	15	79	79	NUM
ejpam-5569	620	16	o(1	o(1	NOUN
ejpam-5569	620	17	)	)	PUNCT
ejpam-5569	620	18	79	79	NUM
ejpam-5569	620	19	0.033053	0.033053	NUM
ejpam-5569	620	20	129	129	NUM
ejpam-5569	620	21	0.091875	0.091875	NUM
ejpam-5569	620	22	13	13	NUM
ejpam-5569	620	23	26	26	NUM
ejpam-5569	620	24	167	167	NUM
ejpam-5569	620	25	o(1	o(1	NOUN
ejpam-5569	620	26	)	)	PUNCT
ejpam-5569	620	27	167	167	NUM
ejpam-5569	620	28	0.046598	0.046598	NUM
ejpam-5569	620	29	289	289	NUM
ejpam-5569	620	30	0.124002	0.124002	NUM
ejpam-5569	620	31	17	17	NUM
ejpam-5569	620	32	34	34	NUM
ejpam-5569	620	33	287	287	NUM
ejpam-5569	620	34	o(1	o(1	NOUN
ejpam-5569	620	35	)	)	PUNCT
ejpam-5569	620	36	287	287	NUM
ejpam-5569	620	37	0.04728	0.04728	NUM
ejpam-5569	620	38	513	513	NUM
ejpam-5569	620	39	0.134112	0.134112	NUM
ejpam-5569	620	40	21	21	NUM
ejpam-5569	620	41	42	42	NUM
ejpam-5569	620	42	439	439	NUM
ejpam-5569	620	43	o(1	o(1	NOUN
ejpam-5569	620	44	)	)	PUNCT
ejpam-5569	620	45	439	439	NUM
ejpam-5569	620	46	0.049498	0.049498	NUM
ejpam-5569	620	47	801	801	NUM
ejpam-5569	620	48	0.158377	0.158377	NUM
ejpam-5569	620	49	25	25	NUM
ejpam-5569	620	50	50	50	NUM
ejpam-5569	620	51	623	623	NUM
ejpam-5569	620	52	o(1	o(1	NOUN
ejpam-5569	620	53	)	)	PUNCT
ejpam-5569	620	54	623	623	NUM
ejpam-5569	620	55	0.052472	0.052472	NUM
ejpam-5569	620	56	1153	1153	NUM
ejpam-5569	620	57	0.205898	0.205898	NUM
ejpam-5569	620	58	29	29	NUM
ejpam-5569	620	59	58	58	NUM
ejpam-5569	620	60	839	839	NUM
ejpam-5569	620	61	o(1	o(1	NOUN
ejpam-5569	620	62	)	)	PUNCT
ejpam-5569	620	63	839	839	NUM
ejpam-5569	620	64	0.08865	0.08865	NUM
ejpam-5569	620	65	1569	1569	NUM
ejpam-5569	620	66	0.218329	0.218329	NUM
ejpam-5569	620	67	33	33	NUM
ejpam-5569	620	68	66	66	NUM
ejpam-5569	620	69	1087	1087	NUM
ejpam-5569	620	70	o(1	o(1	NOUN
ejpam-5569	620	71	)	)	PUNCT
ejpam-5569	620	72	1087	1087	NUM
ejpam-5569	620	73	0.089514	0.089514	NUM
ejpam-5569	620	74	2049	2049	NUM
ejpam-5569	620	75	0.222084	0.222084	NUM
ejpam-5569	620	76	37	37	NUM
ejpam-5569	620	77	74	74	NUM
ejpam-5569	620	78	1367	1367	NUM
ejpam-5569	620	79	o(1	o(1	NOUN
ejpam-5569	620	80	)	)	PUNCT
ejpam-5569	620	81	1367	1367	NUM
ejpam-5569	620	82	0.092204	0.092204	NUM
ejpam-5569	620	83	2593	2593	NUM
ejpam-5569	620	84	0.264229	0.264229	NUM
ejpam-5569	620	85	41	41	NUM
ejpam-5569	620	86	82	82	NUM
ejpam-5569	620	87	1679	1679	NUM
ejpam-5569	620	88	o(1	o(1	NOUN
ejpam-5569	620	89	)	)	PUNCT
ejpam-5569	620	90	1679	1679	NUM
ejpam-5569	620	91	0.100206	0.100206	NUM
ejpam-5569	620	92	3201	3201	NUM
ejpam-5569	620	93	0.278491	0.278491	NUM
ejpam-5569	620	94	45	45	NUM
ejpam-5569	620	95	90	90	NUM
ejpam-5569	620	96	2023	2023	NUM
ejpam-5569	620	97	o(1	o(1	NOUN
ejpam-5569	620	98	)	)	PUNCT
ejpam-5569	620	99	2023	2023	NUM
ejpam-5569	620	100	0.102479	0.102479	NUM
ejpam-5569	620	101	3873	3873	NUM
ejpam-5569	620	102	0.305885	0.305885	NUM
ejpam-5569	620	103	49	49	NUM
ejpam-5569	620	104	98	98	NUM
ejpam-5569	620	105	2399	2399	NUM
ejpam-5569	620	106	o(1	o(1	NOUN
ejpam-5569	620	107	)	)	PUNCT
ejpam-5569	620	108	2399	2399	NUM
ejpam-5569	620	109	0.114788	0.114788	NUM
ejpam-5569	620	110	4609	4609	NUM
ejpam-5569	620	111	0.308508	0.308508	NUM
ejpam-5569	620	112	53	53	NUM
ejpam-5569	620	113	106	106	NUM
ejpam-5569	620	114	2807	2807	NUM
ejpam-5569	620	115	o(1	o(1	NOUN
ejpam-5569	620	116	)	)	PUNCT
ejpam-5569	620	117	2807	2807	NUM
ejpam-5569	620	118	0.133461	0.133461	NUM
ejpam-5569	620	119	5409	5409	NUM
ejpam-5569	620	120	0.451393	0.451393	NUM
ejpam-5569	620	121	57	57	NUM
ejpam-5569	620	122	114	114	NUM
ejpam-5569	620	123	3247	3247	NUM
ejpam-5569	620	124	o(1	o(1	NOUN
ejpam-5569	620	125	)	)	PUNCT
ejpam-5569	620	126	3247	3247	NUM
ejpam-5569	620	127	0.148361	0.148361	NUM
ejpam-5569	620	128	6273	6273	NUM
ejpam-5569	620	129	0.466374	0.466374	NUM
ejpam-5569	620	130	61	61	NUM
ejpam-5569	620	131	122	122	NUM
ejpam-5569	620	132	3719	3719	NUM
ejpam-5569	620	133	o(1	o(1	NOUN
ejpam-5569	620	134	)	)	PUNCT
ejpam-5569	620	135	3719	3719	NUM
ejpam-5569	620	136	0.166031	0.166031	NUM
ejpam-5569	620	137	7201	7201	NUM
ejpam-5569	620	138	0.477208	0.477208	NUM
ejpam-5569	620	139	65	65	NUM
ejpam-5569	620	140	130	130	NUM
ejpam-5569	620	141	4223	4223	NUM
ejpam-5569	620	142	o(1	o(1	NOUN
ejpam-5569	620	143	)	)	PUNCT
ejpam-5569	620	144	4223	4223	NUM
ejpam-5569	620	145	0.16886	0.16886	NUM
ejpam-5569	620	146	8193	8193	NUM
ejpam-5569	620	147	0.527846	0.527846	NUM
ejpam-5569	620	148	69	69	NUM
ejpam-5569	620	149	138	138	NUM
ejpam-5569	620	150	4759	4759	NUM
ejpam-5569	620	151	o(1	o(1	NOUN
ejpam-5569	620	152	)	)	PUNCT
ejpam-5569	620	153	4759	4759	NUM
ejpam-5569	621	1	0.21307	0.21307	NUM
ejpam-5569	621	2	9249	9249	NUM
ejpam-5569	621	3	0.644822	0.644822	NUM
ejpam-5569	621	4	73	73	NUM
ejpam-5569	621	5	146	146	NUM
ejpam-5569	621	6	5327	5327	NUM
ejpam-5569	621	7	o(1	o(1	NOUN
ejpam-5569	621	8	)	)	PUNCT
ejpam-5569	621	9	5327	5327	NUM
ejpam-5569	621	10	0.241707	0.241707	NUM
ejpam-5569	621	11	10369	10369	NUM
ejpam-5569	621	12	0.720457	0.720457	NUM
ejpam-5569	621	13	77	77	NUM
ejpam-5569	621	14	154	154	NUM
ejpam-5569	621	15	5927	5927	NUM
ejpam-5569	621	16	o(1	o(1	NOUN
ejpam-5569	621	17	)	)	PUNCT
ejpam-5569	621	18	5927	5927	NUM
ejpam-5569	621	19	0.260753	0.260753	NUM
ejpam-5569	621	20	11553	11553	NUM
ejpam-5569	621	21	0.829255	0.829255	NUM
ejpam-5569	621	22	81	81	NUM
ejpam-5569	621	23	162	162	NUM
ejpam-5569	621	24	6559	6559	NUM
ejpam-5569	621	25	o(1	o(1	NOUN
ejpam-5569	621	26	)	)	PUNCT
ejpam-5569	621	27	6559	6559	NUM
ejpam-5569	621	28	0.299115	0.299115	NUM
ejpam-5569	621	29	12801	12801	NUM
ejpam-5569	621	30	0.937023	0.937023	NUM
ejpam-5569	621	31	85	85	NUM
ejpam-5569	621	32	170	170	NUM
ejpam-5569	621	33	7223	7223	NUM
ejpam-5569	621	34	o(1	o(1	NOUN
ejpam-5569	621	35	)	)	PUNCT
ejpam-5569	621	36	7223	7223	NUM
ejpam-5569	621	37	0.307929	0.307929	NUM
ejpam-5569	621	38	14113	14113	NUM
ejpam-5569	621	39	1.069794	1.069794	NUM
ejpam-5569	621	40	6	6	NUM
ejpam-5569	621	41	.	.	PUNCT
ejpam-5569	622	1	application	application	NOUN
ejpam-5569	622	2	of	of	ADP
ejpam-5569	622	3	radio	radio	NOUN
ejpam-5569	622	4	labeling	labeling	NOUN
ejpam-5569	622	5	of	of	ADP
ejpam-5569	622	6	graph	graph	NOUN
ejpam-5569	622	7	in	in	ADP
ejpam-5569	622	8	cryptography	cryptography	NOUN
ejpam-5569	622	9	recently	recently	ADV
ejpam-5569	622	10	,	,	PUNCT
ejpam-5569	622	11	information	information	NOUN
ejpam-5569	622	12	technology	technology	NOUN
ejpam-5569	622	13	has	have	AUX
ejpam-5569	622	14	been	be	AUX
ejpam-5569	622	15	widely	widely	ADV
ejpam-5569	622	16	used	use	VERB
ejpam-5569	622	17	in	in	ADP
ejpam-5569	622	18	various	various	ADJ
ejpam-5569	622	19	fields	field	NOUN
ejpam-5569	622	20	of	of	ADP
ejpam-5569	622	21	life	life	NOUN
ejpam-5569	622	22	.	.	PUNCT
ejpam-5569	623	1	hence	hence	ADV
ejpam-5569	623	2	,	,	PUNCT
ejpam-5569	623	3	information	information	NOUN
ejpam-5569	623	4	security	security	NOUN
ejpam-5569	623	5	becomes	become	VERB
ejpam-5569	623	6	a	a	DET
ejpam-5569	623	7	vital	vital	ADJ
ejpam-5569	623	8	issue	issue	NOUN
ejpam-5569	623	9	.	.	PUNCT
ejpam-5569	624	1	cryptography	cryptography	NOUN
ejpam-5569	624	2	is	be	AUX
ejpam-5569	624	3	a	a	DET
ejpam-5569	624	4	significant	significant	ADJ
ejpam-5569	624	5	method	method	NOUN
ejpam-5569	624	6	for	for	ADP
ejpam-5569	624	7	information	information	NOUN
ejpam-5569	624	8	security	security	NOUN
ejpam-5569	624	9	.	.	PUNCT
ejpam-5569	625	1	in	in	ADP
ejpam-5569	625	2	cryptographic	cryptographic	ADJ
ejpam-5569	625	3	algorithms	algorithm	NOUN
ejpam-5569	625	4	,	,	PUNCT
ejpam-5569	625	5	keys	key	NOUN
ejpam-5569	625	6	used	use	VERB
ejpam-5569	625	7	for	for	ADP
ejpam-5569	625	8	encryption	encryption	NOUN
ejpam-5569	625	9	and	and	CCONJ
ejpam-5569	625	10	decryption	decryption	NOUN
ejpam-5569	625	11	are	be	AUX
ejpam-5569	625	12	lengthy	lengthy	ADJ
ejpam-5569	625	13	number	number	NOUN
ejpam-5569	625	14	sequences	sequence	NOUN
ejpam-5569	625	15	which	which	PRON
ejpam-5569	625	16	are	be	AUX
ejpam-5569	625	17	created	create	VERB
ejpam-5569	625	18	using	use	VERB
ejpam-5569	625	19	random	random	ADJ
ejpam-5569	625	20	number	number	NOUN
ejpam-5569	625	21	generators	generator	NOUN
ejpam-5569	625	22	.	.	PUNCT
ejpam-5569	626	1	such	such	ADJ
ejpam-5569	626	2	generators	generator	NOUN
ejpam-5569	626	3	follow	follow	VERB
ejpam-5569	626	4	a	a	DET
ejpam-5569	626	5	uniform	uniform	ADJ
ejpam-5569	626	6	distribution	distribution	NOUN
ejpam-5569	626	7	that	that	PRON
ejpam-5569	626	8	can	can	AUX
ejpam-5569	626	9	be	be	AUX
ejpam-5569	626	10	is	be	AUX
ejpam-5569	626	11	easily	easily	ADV
ejpam-5569	626	12	determined	determine	VERB
ejpam-5569	626	13	by	by	ADP
ejpam-5569	626	14	hackers	hacker	NOUN
ejpam-5569	626	15	,	,	PUNCT
ejpam-5569	626	16	as	as	SCONJ
ejpam-5569	626	17	any	any	DET
ejpam-5569	626	18	number	number	NOUN
ejpam-5569	626	19	has	have	VERB
ejpam-5569	626	20	an	an	DET
ejpam-5569	626	21	equal	equal	ADJ
ejpam-5569	626	22	chance	chance	NOUN
ejpam-5569	626	23	of	of	ADP
ejpam-5569	626	24	being	be	AUX
ejpam-5569	626	25	generated	generate	VERB
ejpam-5569	626	26	.	.	PUNCT
ejpam-5569	627	1	using	use	VERB
ejpam-5569	627	2	radio	radio	NOUN
ejpam-5569	627	3	labeling	labeling	NOUN
ejpam-5569	627	4	of	of	ADP
ejpam-5569	627	5	graphs	graph	NOUN
ejpam-5569	627	6	,	,	PUNCT
ejpam-5569	627	7	keys	key	NOUN
ejpam-5569	627	8	can	can	AUX
ejpam-5569	627	9	be	be	AUX
ejpam-5569	627	10	generated	generate	VERB
ejpam-5569	627	11	in	in	ADP
ejpam-5569	627	12	more	more	ADV
ejpam-5569	627	13	faster	fast	ADJ
ejpam-5569	627	14	and	and	CCONJ
ejpam-5569	627	15	effective	effective	ADJ
ejpam-5569	627	16	manner	manner	NOUN
ejpam-5569	627	17	.	.	PUNCT
ejpam-5569	628	1	the	the	DET
ejpam-5569	628	2	radio	radio	NOUN
ejpam-5569	628	3	labeled	label	VERB
ejpam-5569	628	4	graph	graph	NOUN
ejpam-5569	628	5	is	be	AUX
ejpam-5569	628	6	used	use	VERB
ejpam-5569	628	7	as	as	ADP
ejpam-5569	628	8	the	the	DET
ejpam-5569	628	9	cipher	cipher	ADJ
ejpam-5569	628	10	graph	graph	NOUN
ejpam-5569	628	11	.	.	PUNCT
ejpam-5569	629	1	at	at	ADP
ejpam-5569	629	2	the	the	DET
ejpam-5569	629	3	receiver	receiver	NOUN
ejpam-5569	629	4	end	end	VERB
ejpam-5569	629	5	the	the	DET
ejpam-5569	629	6	encrypted	encrypt	VERB
ejpam-5569	629	7	message	message	NOUN
ejpam-5569	629	8	is	be	AUX
ejpam-5569	629	9	received	receive	VERB
ejpam-5569	629	10	in	in	ADP
ejpam-5569	629	11	the	the	DET
ejpam-5569	629	12	form	form	NOUN
ejpam-5569	629	13	of	of	ADP
ejpam-5569	629	14	edge	edge	NOUN
ejpam-5569	629	15	or	or	CCONJ
ejpam-5569	629	16	vertex	vertex	NOUN
ejpam-5569	629	17	sequence	sequence	NOUN
ejpam-5569	629	18	of	of	ADP
ejpam-5569	629	19	this	this	DET
ejpam-5569	629	20	cipher	cipher	ADJ
ejpam-5569	629	21	graph	graph	NOUN
ejpam-5569	629	22	.	.	PUNCT
ejpam-5569	630	1	consequently	consequently	ADV
ejpam-5569	630	2	,	,	PUNCT
ejpam-5569	630	3	it	it	PRON
ejpam-5569	630	4	is	be	AUX
ejpam-5569	630	5	difficult	difficult	ADJ
ejpam-5569	630	6	for	for	SCONJ
ejpam-5569	630	7	an	an	DET
ejpam-5569	630	8	opponent	opponent	NOUN
ejpam-5569	630	9	to	to	PART
ejpam-5569	630	10	guess	guess	VERB
ejpam-5569	630	11	and	and	CCONJ
ejpam-5569	630	12	hack	hack	VERB
ejpam-5569	630	13	,	,	PUNCT
ejpam-5569	630	14	as	as	SCONJ
ejpam-5569	630	15	the	the	DET
ejpam-5569	630	16	radio	radio	NOUN
ejpam-5569	630	17	labeling	labeling	NOUN
ejpam-5569	630	18	problem	problem	NOUN
ejpam-5569	630	19	is	be	AUX
ejpam-5569	630	20	an	an	DET
ejpam-5569	630	21	np	np	NOUN
ejpam-5569	630	22	-	-	PUNCT
ejpam-5569	630	23	hard	hard	ADJ
ejpam-5569	630	24	problem	problem	NOUN
ejpam-5569	630	25	.	.	PUNCT
ejpam-5569	631	1	the	the	DET
ejpam-5569	631	2	radio	radio	NOUN
ejpam-5569	631	3	number	number	NOUN
ejpam-5569	631	4	of	of	ADP
ejpam-5569	631	5	a	a	DET
ejpam-5569	631	6	graph	graph	NOUN
ejpam-5569	631	7	may	may	AUX
ejpam-5569	631	8	also	also	ADV
ejpam-5569	631	9	be	be	AUX
ejpam-5569	631	10	used	use	VERB
ejpam-5569	631	11	to	to	PART
ejpam-5569	631	12	generate	generate	VERB
ejpam-5569	631	13	invertible	invertible	ADJ
ejpam-5569	631	14	matrices	matrix	NOUN
ejpam-5569	631	15	that	that	PRON
ejpam-5569	631	16	serve	serve	VERB
ejpam-5569	631	17	for	for	ADP
ejpam-5569	631	18	keys	key	NOUN
ejpam-5569	631	19	generation	generation	NOUN
ejpam-5569	631	20	.	.	PUNCT
ejpam-5569	632	1	in	in	ADP
ejpam-5569	632	2	this	this	DET
ejpam-5569	632	3	manner	manner	NOUN
ejpam-5569	632	4	,	,	PUNCT
ejpam-5569	632	5	the	the	DET
ejpam-5569	632	6	radio	radio	NOUN
ejpam-5569	632	7	labeling	labeling	NOUN
ejpam-5569	632	8	considered	consider	VERB
ejpam-5569	632	9	an	an	DET
ejpam-5569	632	10	efficient	efficient	ADJ
ejpam-5569	632	11	tool	tool	NOUN
ejpam-5569	632	12	for	for	ADP
ejpam-5569	632	13	cryptography	cryptography	NOUN
ejpam-5569	632	14	as	as	ADP
ejpam-5569	632	15	the	the	DET
ejpam-5569	632	16	problem	problem	NOUN
ejpam-5569	632	17	of	of	ADP
ejpam-5569	632	18	finding	find	VERB
ejpam-5569	632	19	the	the	DET
ejpam-5569	632	20	radio	radio	NOUN
ejpam-5569	632	21	number	number	NOUN
ejpam-5569	632	22	of	of	ADP
ejpam-5569	632	23	a	a	DET
ejpam-5569	632	24	graph	graph	NOUN
ejpam-5569	632	25	is	be	AUX
ejpam-5569	632	26	an	an	DET
ejpam-5569	632	27	np	np	NOUN
ejpam-5569	632	28	-	-	PUNCT
ejpam-5569	632	29	complete	complete	ADJ
ejpam-5569	632	30	problem	problem	NOUN
ejpam-5569	632	31	.	.	PUNCT
ejpam-5569	633	1	we	we	PRON
ejpam-5569	633	2	refer	refer	VERB
ejpam-5569	633	3	the	the	DET
ejpam-5569	633	4	reader	reader	NOUN
ejpam-5569	633	5	to	to	ADP
ejpam-5569	633	6	[	[	X
ejpam-5569	633	7	24–26	24–26	NUM
ejpam-5569	633	8	]	]	PUNCT
ejpam-5569	633	9	for	for	ADP
ejpam-5569	633	10	applications	application	NOUN
ejpam-5569	633	11	of	of	ADP
ejpam-5569	633	12	radio	radio	NOUN
ejpam-5569	633	13	labeling	labeling	NOUN
ejpam-5569	633	14	in	in	ADP
ejpam-5569	633	15	cryptography	cryptography	NOUN
ejpam-5569	633	16	n.	n.	NOUN
ejpam-5569	633	17	m.	m.	PROPN
ejpam-5569	633	18	noureldeen	noureldeen	INTJ
ejpam-5569	633	19	et	et	PROPN
ejpam-5569	633	20	al	al	PROPN
ejpam-5569	633	21	.	.	PUNCT
ejpam-5569	633	22	/	/	SYM
ejpam-5569	633	23	eur	eur	PROPN
ejpam-5569	633	24	.	.	PUNCT
ejpam-5569	634	1	j.	j.	PROPN
ejpam-5569	634	2	pure	pure	PROPN
ejpam-5569	634	3	appl	appl	PROPN
ejpam-5569	634	4	.	.	PROPN
ejpam-5569	634	5	math	math	PROPN
ejpam-5569	634	6	,	,	PUNCT
ejpam-5569	634	7	18	18	NUM
ejpam-5569	634	8	(	(	PUNCT
ejpam-5569	634	9	2	2	NUM
ejpam-5569	634	10	)	)	PUNCT
ejpam-5569	634	11	(	(	PUNCT
ejpam-5569	634	12	2025	2025	NUM
ejpam-5569	634	13	)	)	PUNCT
ejpam-5569	634	14	,	,	PUNCT
ejpam-5569	634	15	5569	5569	NUM
ejpam-5569	634	16	24	24	NUM
ejpam-5569	634	17	of	of	ADP
ejpam-5569	634	18	25	25	NUM
ejpam-5569	634	19	7	7	NUM
ejpam-5569	634	20	.	.	PUNCT
ejpam-5569	635	1	conclusions	conclusion	NOUN
ejpam-5569	635	2	effective	effective	ADJ
ejpam-5569	635	3	allocating	allocating	NOUN
ejpam-5569	635	4	of	of	ADP
ejpam-5569	635	5	frequency	frequency	NOUN
ejpam-5569	635	6	resources	resource	NOUN
ejpam-5569	635	7	in	in	ADP
ejpam-5569	635	8	wireless	wireless	ADJ
ejpam-5569	635	9	networks	network	NOUN
ejpam-5569	635	10	is	be	AUX
ejpam-5569	635	11	a	a	DET
ejpam-5569	635	12	challenging	challenging	ADJ
ejpam-5569	635	13	problem	problem	NOUN
ejpam-5569	635	14	.	.	PUNCT
ejpam-5569	636	1	the	the	DET
ejpam-5569	636	2	claim	claim	NOUN
ejpam-5569	636	3	is	be	AUX
ejpam-5569	636	4	avoiding	avoid	VERB
ejpam-5569	636	5	the	the	DET
ejpam-5569	636	6	interference	interference	NOUN
ejpam-5569	636	7	among	among	ADP
ejpam-5569	636	8	stations	station	NOUN
ejpam-5569	636	9	and	and	CCONJ
ejpam-5569	636	10	minimizing	minimize	VERB
ejpam-5569	636	11	the	the	DET
ejpam-5569	636	12	usage	usage	NOUN
ejpam-5569	636	13	of	of	ADP
ejpam-5569	636	14	the	the	DET
ejpam-5569	636	15	available	available	ADJ
ejpam-5569	636	16	frequencies	frequency	NOUN
ejpam-5569	636	17	.	.	PUNCT
ejpam-5569	637	1	here	here	ADV
ejpam-5569	637	2	we	we	PRON
ejpam-5569	637	3	consider	consider	VERB
ejpam-5569	637	4	the	the	DET
ejpam-5569	637	5	radio	radio	NOUN
ejpam-5569	637	6	graph	graph	NOUN
ejpam-5569	637	7	labeling	labeling	NOUN
ejpam-5569	637	8	as	as	ADP
ejpam-5569	637	9	a	a	DET
ejpam-5569	637	10	graph	graph	NOUN
ejpam-5569	637	11	-	-	PUNCT
ejpam-5569	637	12	theoretical	theoretical	ADJ
ejpam-5569	637	13	approach	approach	NOUN
ejpam-5569	637	14	to	to	ADP
ejpam-5569	637	15	modeling	model	VERB
ejpam-5569	637	16	the	the	DET
ejpam-5569	637	17	problem	problem	NOUN
ejpam-5569	637	18	of	of	ADP
ejpam-5569	637	19	frequency	frequency	NOUN
ejpam-5569	637	20	assignment	assignment	NOUN
ejpam-5569	637	21	.	.	PUNCT
ejpam-5569	638	1	in	in	ADP
ejpam-5569	638	2	such	such	ADJ
ejpam-5569	638	3	modeling	modeling	NOUN
ejpam-5569	638	4	an	an	DET
ejpam-5569	638	5	interference	interference	NOUN
ejpam-5569	638	6	graph	graph	NOUN
ejpam-5569	638	7	g	g	NOUN
ejpam-5569	638	8	is	be	AUX
ejpam-5569	638	9	built	build	VERB
ejpam-5569	638	10	.	.	PUNCT
ejpam-5569	639	1	the	the	DET
ejpam-5569	639	2	vertices	vertex	NOUN
ejpam-5569	639	3	of	of	ADP
ejpam-5569	639	4	the	the	DET
ejpam-5569	639	5	interference	interference	NOUN
ejpam-5569	639	6	graph	graph	NOUN
ejpam-5569	639	7	represent	represent	VERB
ejpam-5569	639	8	the	the	DET
ejpam-5569	639	9	stations	station	NOUN
ejpam-5569	639	10	.	.	PUNCT
ejpam-5569	640	1	two	two	NUM
ejpam-5569	640	2	vertices	vertex	NOUN
ejpam-5569	640	3	are	be	AUX
ejpam-5569	640	4	adjacent	adjacent	ADJ
ejpam-5569	640	5	if	if	SCONJ
ejpam-5569	640	6	their	their	PRON
ejpam-5569	640	7	corresponding	correspond	VERB
ejpam-5569	640	8	stations	station	NOUN
ejpam-5569	640	9	may	may	AUX
ejpam-5569	640	10	interfere	interfere	VERB
ejpam-5569	640	11	.	.	PUNCT
ejpam-5569	641	1	radio	radio	NOUN
ejpam-5569	641	2	labeling	labeling	NOUN
ejpam-5569	641	3	of	of	ADP
ejpam-5569	641	4	the	the	DET
ejpam-5569	641	5	interference	interference	NOUN
ejpam-5569	641	6	graph	graph	NOUN
ejpam-5569	641	7	is	be	AUX
ejpam-5569	641	8	done	do	VERB
ejpam-5569	641	9	to	to	PART
ejpam-5569	641	10	avoid	avoid	VERB
ejpam-5569	641	11	the	the	DET
ejpam-5569	641	12	interference	interference	NOUN
ejpam-5569	641	13	and	and	CCONJ
ejpam-5569	641	14	moreover	moreover	ADV
ejpam-5569	641	15	emphasizing	emphasize	VERB
ejpam-5569	641	16	on	on	ADP
ejpam-5569	641	17	minimization	minimization	NOUN
ejpam-5569	641	18	the	the	DET
ejpam-5569	641	19	using	using	NOUN
ejpam-5569	641	20	of	of	ADP
ejpam-5569	641	21	frequency	frequency	NOUN
ejpam-5569	641	22	resources	resource	NOUN
ejpam-5569	641	23	.	.	PUNCT
ejpam-5569	642	1	then	then	ADV
ejpam-5569	642	2	the	the	DET
ejpam-5569	642	3	minimum	minimum	ADJ
ejpam-5569	642	4	value	value	NOUN
ejpam-5569	642	5	of	of	ADP
ejpam-5569	642	6	the	the	DET
ejpam-5569	642	7	maximum	maximum	ADJ
ejpam-5569	642	8	assigned	assign	VERB
ejpam-5569	642	9	frequency	frequency	NOUN
ejpam-5569	642	10	(	(	PUNCT
ejpam-5569	642	11	radio	radio	NOUN
ejpam-5569	642	12	label	label	NOUN
ejpam-5569	642	13	)	)	PUNCT
ejpam-5569	642	14	among	among	ADP
ejpam-5569	642	15	all	all	DET
ejpam-5569	642	16	radio	radio	NOUN
ejpam-5569	642	17	labeling	labeling	NOUN
ejpam-5569	642	18	of	of	ADP
ejpam-5569	642	19	g	g	PROPN
ejpam-5569	642	20	,	,	PUNCT
ejpam-5569	642	21	called	call	VERB
ejpam-5569	642	22	the	the	DET
ejpam-5569	642	23	radio	radio	NOUN
ejpam-5569	642	24	number	number	NOUN
ejpam-5569	642	25	of	of	ADP
ejpam-5569	642	26	the	the	DET
ejpam-5569	642	27	graph	graph	NOUN
ejpam-5569	642	28	is	be	AUX
ejpam-5569	642	29	obtained	obtain	VERB
ejpam-5569	642	30	,	,	PUNCT
ejpam-5569	642	31	this	this	DET
ejpam-5569	642	32	radio	radio	NOUN
ejpam-5569	642	33	number	number	NOUN
ejpam-5569	642	34	represents	represent	VERB
ejpam-5569	642	35	the	the	DET
ejpam-5569	642	36	highest	high	ADJ
ejpam-5569	642	37	frequency	frequency	NOUN
ejpam-5569	642	38	should	should	AUX
ejpam-5569	642	39	be	be	AUX
ejpam-5569	642	40	used	use	VERB
ejpam-5569	642	41	to	to	PART
ejpam-5569	642	42	avoid	avoid	VERB
ejpam-5569	642	43	interference	interference	NOUN
ejpam-5569	642	44	.	.	PUNCT
ejpam-5569	643	1	in	in	ADP
ejpam-5569	643	2	this	this	DET
ejpam-5569	643	3	paper	paper	NOUN
ejpam-5569	643	4	,	,	PUNCT
ejpam-5569	643	5	the	the	DET
ejpam-5569	643	6	radio	radio	NOUN
ejpam-5569	643	7	labeling	labeling	NOUN
ejpam-5569	643	8	and	and	CCONJ
ejpam-5569	643	9	the	the	DET
ejpam-5569	643	10	upper	upper	ADJ
ejpam-5569	643	11	bound	bind	VERB
ejpam-5569	643	12	of	of	ADP
ejpam-5569	643	13	radio	radio	NOUN
ejpam-5569	643	14	number	number	NOUN
ejpam-5569	643	15	of	of	ADP
ejpam-5569	643	16	networks	network	NOUN
ejpam-5569	643	17	as	as	ADP
ejpam-5569	643	18	shadow	shadow	NOUN
ejpam-5569	643	19	of	of	ADP
ejpam-5569	643	20	path	path	NOUN
ejpam-5569	643	21	graph	graph	NOUN
ejpam-5569	643	22	are	be	AUX
ejpam-5569	643	23	investigated	investigate	VERB
ejpam-5569	643	24	.	.	PUNCT
ejpam-5569	644	1	wherein	wherein	ADJ
ejpam-5569	644	2	,	,	PUNCT
ejpam-5569	644	3	a	a	DET
ejpam-5569	644	4	theoretical	theoretical	ADJ
ejpam-5569	644	5	approach	approach	NOUN
ejpam-5569	644	6	for	for	ADP
ejpam-5569	644	7	finding	find	VERB
ejpam-5569	644	8	such	such	ADJ
ejpam-5569	644	9	number	number	NOUN
ejpam-5569	644	10	is	be	AUX
ejpam-5569	644	11	presented	present	VERB
ejpam-5569	644	12	and	and	CCONJ
ejpam-5569	644	13	mathematical	mathematical	ADJ
ejpam-5569	644	14	model	model	NOUN
ejpam-5569	644	15	for	for	ADP
ejpam-5569	644	16	finding	find	VERB
ejpam-5569	644	17	upper	upper	ADJ
ejpam-5569	644	18	bound	bind	VERB
ejpam-5569	644	19	of	of	ADP
ejpam-5569	644	20	radio	radio	NOUN
ejpam-5569	644	21	number	number	NOUN
ejpam-5569	644	22	of	of	ADP
ejpam-5569	644	23	shadow	shadow	NOUN
ejpam-5569	644	24	of	of	ADP
ejpam-5569	644	25	path	path	NOUN
ejpam-5569	644	26	graph	graph	NOUN
ejpam-5569	644	27	is	be	AUX
ejpam-5569	644	28	proposed	propose	VERB
ejpam-5569	644	29	.	.	PUNCT
ejpam-5569	645	1	in	in	ADP
ejpam-5569	645	2	order	order	NOUN
ejpam-5569	645	3	to	to	PART
ejpam-5569	645	4	validate	validate	VERB
ejpam-5569	645	5	the	the	DET
ejpam-5569	645	6	effectiveness	effectiveness	NOUN
ejpam-5569	645	7	of	of	ADP
ejpam-5569	645	8	our	our	PRON
ejpam-5569	645	9	approach	approach	NOUN
ejpam-5569	645	10	,	,	PUNCT
ejpam-5569	645	11	an	an	DET
ejpam-5569	645	12	experimental	experimental	ADJ
ejpam-5569	645	13	study	study	NOUN
ejpam-5569	645	14	is	be	AUX
ejpam-5569	645	15	conducted	conduct	VERB
ejpam-5569	645	16	,	,	PUNCT
ejpam-5569	645	17	comparing	compare	VERB
ejpam-5569	645	18	our	our	PRON
ejpam-5569	645	19	results	result	NOUN
ejpam-5569	645	20	to	to	ADP
ejpam-5569	645	21	previous	previous	ADJ
ejpam-5569	645	22	findings	finding	NOUN
ejpam-5569	645	23	.	.	PUNCT
ejpam-5569	646	1	the	the	DET
ejpam-5569	646	2	results	result	NOUN
ejpam-5569	646	3	of	of	ADP
ejpam-5569	646	4	the	the	DET
ejpam-5569	646	5	study	study	NOUN
ejpam-5569	646	6	demonstrate	demonstrate	VERB
ejpam-5569	646	7	that	that	SCONJ
ejpam-5569	646	8	our	our	PRON
ejpam-5569	646	9	proposed	propose	VERB
ejpam-5569	646	10	approach	approach	NOUN
ejpam-5569	646	11	surpasses	surpass	VERB
ejpam-5569	646	12	the	the	DET
ejpam-5569	646	13	previous	previous	ADJ
ejpam-5569	646	14	results	result	NOUN
ejpam-5569	646	15	in	in	ADP
ejpam-5569	646	16	terms	term	NOUN
ejpam-5569	646	17	of	of	ADP
ejpam-5569	646	18	both	both	CCONJ
ejpam-5569	646	19	the	the	DET
ejpam-5569	646	20	running	running	ADJ
ejpam-5569	646	21	time	time	NOUN
ejpam-5569	646	22	and	and	CCONJ
ejpam-5569	646	23	the	the	DET
ejpam-5569	646	24	upper	upper	ADJ
ejpam-5569	646	25	bound	bound	NOUN
ejpam-5569	646	26	of	of	ADP
ejpam-5569	646	27	the	the	DET
ejpam-5569	646	28	radio	radio	NOUN
ejpam-5569	646	29	number	number	NOUN
ejpam-5569	646	30	.	.	PUNCT
ejpam-5569	647	1	moreover	moreover	ADV
ejpam-5569	647	2	,	,	PUNCT
ejpam-5569	647	3	we	we	PRON
ejpam-5569	647	4	give	give	VERB
ejpam-5569	647	5	an	an	DET
ejpam-5569	647	6	overview	overview	NOUN
ejpam-5569	647	7	of	of	ADP
ejpam-5569	647	8	the	the	DET
ejpam-5569	647	9	application	application	NOUN
ejpam-5569	647	10	of	of	ADP
ejpam-5569	647	11	radio	radio	NOUN
ejpam-5569	647	12	labeling	labeling	NOUN
ejpam-5569	647	13	in	in	ADP
ejpam-5569	647	14	cryptography	cryptography	NOUN
ejpam-5569	647	15	.	.	PUNCT
ejpam-5569	648	1	for	for	ADP
ejpam-5569	648	2	future	future	ADJ
ejpam-5569	648	3	work	work	NOUN
ejpam-5569	648	4	we	we	PRON
ejpam-5569	648	5	claim	claim	VERB
ejpam-5569	648	6	that	that	SCONJ
ejpam-5569	648	7	the	the	DET
ejpam-5569	648	8	integer	integer	NOUN
ejpam-5569	648	9	linear	linear	PROPN
ejpam-5569	648	10	programming	programming	NOUN
ejpam-5569	648	11	model	model	NOUN
ejpam-5569	648	12	will	will	AUX
ejpam-5569	648	13	be	be	AUX
ejpam-5569	648	14	improved	improve	VERB
ejpam-5569	648	15	to	to	PART
ejpam-5569	648	16	obtain	obtain	VERB
ejpam-5569	648	17	the	the	DET
ejpam-5569	648	18	exact	exact	ADJ
ejpam-5569	648	19	radio	radio	NOUN
ejpam-5569	648	20	number	number	NOUN
ejpam-5569	648	21	.	.	PUNCT
ejpam-5569	649	1	conflicts	conflict	NOUN
ejpam-5569	649	2	of	of	ADP
ejpam-5569	649	3	interest	interest	NOUN
ejpam-5569	649	4	the	the	DET
ejpam-5569	649	5	authors	author	NOUN
ejpam-5569	649	6	declare	declare	VERB
ejpam-5569	649	7	no	no	DET
ejpam-5569	649	8	conflict	conflict	NOUN
ejpam-5569	649	9	of	of	ADP
ejpam-5569	649	10	interest	interest	NOUN
ejpam-5569	649	11	.	.	PUNCT
ejpam-5569	650	1	references	reference	NOUN
ejpam-5569	650	2	[	[	X
ejpam-5569	650	3	1	1	NUM
ejpam-5569	650	4	]	]	PUNCT
ejpam-5569	650	5	w.	w.	PROPN
ejpam-5569	650	6	k.	k.	PROPN
ejpam-5569	650	7	hale	hale	PROPN
ejpam-5569	650	8	.	.	PUNCT
ejpam-5569	651	1	frequency	frequency	PROPN
ejpam-5569	651	2	assignment	assignment	NOUN
ejpam-5569	651	3	:	:	PUNCT
ejpam-5569	651	4	theory	theory	NOUN
ejpam-5569	651	5	and	and	CCONJ
ejpam-5569	651	6	applications	application	NOUN
ejpam-5569	651	7	.	.	PUNCT
ejpam-5569	652	1	proc	proc	NOUN
ejpam-5569	652	2	.	.	PUNCT
ejpam-5569	653	1	ieee	ieee	NOUN
ejpam-5569	653	2	,	,	PUNCT
ejpam-5569	653	3	68(12):1497–1514	68(12):1497–1514	NOUN
ejpam-5569	653	4	,	,	PUNCT
ejpam-5569	653	5	1980	1980	NUM
ejpam-5569	653	6	.	.	PUNCT
ejpam-5569	654	1	[	[	X
ejpam-5569	654	2	2	2	X
ejpam-5569	654	3	]	]	PUNCT
ejpam-5569	654	4	j.	j.	PROPN
ejpam-5569	654	5	a.	a.	PROPN
ejpam-5569	654	6	a.	a.	PROPN
ejpam-5569	654	7	gallian	gallian	PROPN
ejpam-5569	654	8	.	.	PUNCT
ejpam-5569	655	1	dynamic	dynamic	ADJ
ejpam-5569	655	2	survey	survey	NOUN
ejpam-5569	655	3	of	of	ADP
ejpam-5569	655	4	graph	graph	NOUN
ejpam-5569	655	5	labeling	labeling	NOUN
ejpam-5569	655	6	.	.	PUNCT
ejpam-5569	656	1	the	the	DET
ejpam-5569	656	2	electronic	electronic	ADJ
ejpam-5569	656	3	journal	journal	NOUN
ejpam-5569	656	4	of	of	ADP
ejpam-5569	656	5	combinatorics	combinatoric	NOUN
ejpam-5569	656	6	,	,	PUNCT
ejpam-5569	656	7	24	24	NUM
ejpam-5569	656	8	,	,	PUNCT
ejpam-5569	656	9	2021	2021	NUM
ejpam-5569	656	10	.	.	PUNCT
ejpam-5569	657	1	[	[	X
ejpam-5569	657	2	3	3	X
ejpam-5569	657	3	]	]	X
ejpam-5569	657	4	j.	j.	PROPN
ejpam-5569	657	5	griggs	griggs	PROPN
ejpam-5569	657	6	and	and	CCONJ
ejpam-5569	657	7	r.	r.	PROPN
ejpam-5569	657	8	yeh	yeh	PROPN
ejpam-5569	657	9	.	.	PUNCT
ejpam-5569	658	1	labelling	labelling	NOUN
ejpam-5569	658	2	graphs	graph	NOUN
ejpam-5569	658	3	with	with	ADP
ejpam-5569	658	4	a	a	DET
ejpam-5569	658	5	condition	condition	NOUN
ejpam-5569	658	6	at	at	ADP
ejpam-5569	658	7	distance	distance	NOUN
ejpam-5569	658	8	2	2	NUM
ejpam-5569	658	9	.	.	PUNCT
ejpam-5569	658	10	siam	siam	PROPN
ejpam-5569	658	11	journal	journal	PROPN
ejpam-5569	658	12	on	on	ADP
ejpam-5569	658	13	discrete	discrete	ADJ
ejpam-5569	658	14	mathematics	mathematic	NOUN
ejpam-5569	658	15	,	,	PUNCT
ejpam-5569	658	16	5(4):586–595	5(4):586–595	NUM
ejpam-5569	658	17	,	,	PUNCT
ejpam-5569	658	18	1992	1992	NUM
ejpam-5569	658	19	.	.	PUNCT
ejpam-5569	659	1	[	[	X
ejpam-5569	659	2	4	4	X
ejpam-5569	659	3	]	]	X
ejpam-5569	659	4	g.	g.	PROPN
ejpam-5569	659	5	chartrand	chartrand	PROPN
ejpam-5569	659	6	,	,	PUNCT
ejpam-5569	659	7	d.	d.	PROPN
ejpam-5569	659	8	erwin	erwin	PROPN
ejpam-5569	659	9	,	,	PUNCT
ejpam-5569	659	10	p.	p.	PROPN
ejpam-5569	659	11	zhang	zhang	PROPN
ejpam-5569	659	12	,	,	PUNCT
ejpam-5569	659	13	and	and	CCONJ
ejpam-5569	659	14	f.	f.	PROPN
ejpam-5569	659	15	harary	harary	PROPN
ejpam-5569	659	16	.	.	PUNCT
ejpam-5569	660	1	radio	radio	NOUN
ejpam-5569	660	2	labeling	labeling	NOUN
ejpam-5569	660	3	of	of	ADP
ejpam-5569	660	4	graphs	graph	NOUN
ejpam-5569	660	5	.	.	PUNCT
ejpam-5569	661	1	bulletin	bulletin	NOUN
ejpam-5569	661	2	of	of	ADP
ejpam-5569	661	3	the	the	DET
ejpam-5569	661	4	institute	institute	PROPN
ejpam-5569	661	5	of	of	ADP
ejpam-5569	661	6	combinatorics	combinatoric	NOUN
ejpam-5569	661	7	and	and	CCONJ
ejpam-5569	661	8	its	its	PRON
ejpam-5569	661	9	applications	application	NOUN
ejpam-5569	661	10	,	,	PUNCT
ejpam-5569	661	11	33:77–85	33:77–85	NUM
ejpam-5569	661	12	,	,	PUNCT
ejpam-5569	661	13	2001	2001	NUM
ejpam-5569	661	14	.	.	PUNCT
ejpam-5569	662	1	[	[	X
ejpam-5569	662	2	5	5	NUM
ejpam-5569	662	3	]	]	PUNCT
ejpam-5569	662	4	m.	m.	NOUN
ejpam-5569	662	5	kehikech	kehikech	PROPN
ejpam-5569	662	6	,	,	PUNCT
ejpam-5569	662	7	r.	r.	PROPN
ejpam-5569	662	8	khemoufa	khemoufa	PROPN
ejpam-5569	662	9	,	,	PUNCT
ejpam-5569	662	10	and	and	CCONJ
ejpam-5569	662	11	o.	o.	PROPN
ejpam-5569	662	12	togni	togni	PROPN
ejpam-5569	662	13	.	.	PUNCT
ejpam-5569	663	1	linear	linear	ADJ
ejpam-5569	663	2	and	and	CCONJ
ejpam-5569	663	3	cyclic	cyclic	ADJ
ejpam-5569	663	4	radio	radio	NOUN
ejpam-5569	663	5	k	k	NOUN
ejpam-5569	663	6	-	-	PUNCT
ejpam-5569	663	7	labelings	labeling	NOUN
ejpam-5569	663	8	of	of	ADP
ejpam-5569	663	9	trees	tree	NOUN
ejpam-5569	663	10	.	.	PUNCT
ejpam-5569	664	1	discussiones	discussione	NOUN
ejpam-5569	664	2	mathematicae	mathematicae	PROPN
ejpam-5569	664	3	graph	graph	NOUN
ejpam-5569	664	4	theory	theory	NOUN
ejpam-5569	664	5	,	,	PUNCT
ejpam-5569	664	6	27(1):105–123	27(1):105–123	PROPN
ejpam-5569	664	7	,	,	PUNCT
ejpam-5569	664	8	2007	2007	NUM
ejpam-5569	664	9	.	.	PUNCT
ejpam-5569	665	1	[	[	X
ejpam-5569	665	2	6	6	NUM
ejpam-5569	665	3	]	]	PUNCT
ejpam-5569	665	4	l.	l.	PROPN
ejpam-5569	665	5	saha	saha	PROPN
ejpam-5569	665	6	and	and	CCONJ
ejpam-5569	665	7	p.	p.	NOUN
ejpam-5569	665	8	panigrahi	panigrahi	NOUN
ejpam-5569	665	9	.	.	PUNCT
ejpam-5569	666	1	a	a	DET
ejpam-5569	666	2	graph	graph	NOUN
ejpam-5569	666	3	radio	radio	NOUN
ejpam-5569	666	4	k	k	ADJ
ejpam-5569	666	5	-	-	ADJ
ejpam-5569	666	6	coloring	color	VERB
ejpam-5569	666	7	algorithm	algorithm	NOUN
ejpam-5569	666	8	.	.	PUNCT
ejpam-5569	667	1	in	in	ADP
ejpam-5569	667	2	lecture	lecture	NOUN
ejpam-5569	667	3	notes	note	NOUN
ejpam-5569	667	4	in	in	ADP
ejpam-5569	667	5	computer	computer	NOUN
ejpam-5569	667	6	science	science	NOUN
ejpam-5569	667	7	,	,	PUNCT
ejpam-5569	667	8	volume	volume	NOUN
ejpam-5569	667	9	7643	7643	NUM
ejpam-5569	667	10	,	,	PUNCT
ejpam-5569	667	11	pages	page	NOUN
ejpam-5569	667	12	125–129	125–129	NUM
ejpam-5569	667	13	,	,	PUNCT
ejpam-5569	667	14	2012	2012	NUM
ejpam-5569	667	15	.	.	PUNCT
ejpam-5569	668	1	[	[	X
ejpam-5569	668	2	7	7	X
ejpam-5569	668	3	]	]	X
ejpam-5569	668	4	e.	e.	PROPN
ejpam-5569	668	5	m.	m.	PROPN
ejpam-5569	668	6	badr	badr	PROPN
ejpam-5569	668	7	and	and	CCONJ
ejpam-5569	668	8	m.	m.	PROPN
ejpam-5569	668	9	i.	i.	PROPN
ejpam-5569	668	10	moussa	moussa	PROPN
ejpam-5569	668	11	.	.	PUNCT
ejpam-5569	669	1	an	an	DET
ejpam-5569	669	2	upper	upper	ADJ
ejpam-5569	669	3	bound	bind	VERB
ejpam-5569	669	4	of	of	ADP
ejpam-5569	669	5	radio	radio	NOUN
ejpam-5569	669	6	k	k	PROPN
ejpam-5569	669	7	-	-	ADJ
ejpam-5569	669	8	coloring	coloring	ADJ
ejpam-5569	669	9	problem	problem	NOUN
ejpam-5569	669	10	and	and	CCONJ
ejpam-5569	669	11	its	its	PRON
ejpam-5569	669	12	integer	integer	NOUN
ejpam-5569	669	13	linear	linear	PROPN
ejpam-5569	669	14	programming	programming	NOUN
ejpam-5569	669	15	model	model	NOUN
ejpam-5569	669	16	.	.	PUNCT
ejpam-5569	670	1	wireless	wireless	ADJ
ejpam-5569	670	2	networks	network	NOUN
ejpam-5569	670	3	,	,	PUNCT
ejpam-5569	670	4	26(7):4955–4964	26(7):4955–4964	NUM
ejpam-5569	670	5	,	,	PUNCT
ejpam-5569	670	6	2020	2020	NUM
ejpam-5569	670	7	.	.	PUNCT
ejpam-5569	671	1	n.	n.	PROPN
ejpam-5569	671	2	m.	m.	NOUN
ejpam-5569	671	3	noureldeen	noureldeen	INTJ
ejpam-5569	671	4	et	et	PROPN
ejpam-5569	671	5	al	al	PROPN
ejpam-5569	671	6	.	.	PUNCT
ejpam-5569	671	7	/	/	SYM
ejpam-5569	671	8	eur	eur	PROPN
ejpam-5569	671	9	.	.	PUNCT
ejpam-5569	672	1	j.	j.	PROPN
ejpam-5569	672	2	pure	pure	PROPN
ejpam-5569	672	3	appl	appl	PROPN
ejpam-5569	672	4	.	.	PROPN
ejpam-5569	672	5	math	math	PROPN
ejpam-5569	672	6	,	,	PUNCT
ejpam-5569	672	7	18	18	NUM
ejpam-5569	672	8	(	(	PUNCT
ejpam-5569	672	9	2	2	NUM
ejpam-5569	672	10	)	)	PUNCT
ejpam-5569	672	11	(	(	PUNCT
ejpam-5569	672	12	2025	2025	NUM
ejpam-5569	672	13	)	)	PUNCT
ejpam-5569	672	14	,	,	PUNCT
ejpam-5569	672	15	5569	5569	NUM
ejpam-5569	672	16	25	25	NUM
ejpam-5569	672	17	of	of	ADP
ejpam-5569	672	18	25	25	NUM
ejpam-5569	673	1	[	[	SYM
ejpam-5569	673	2	8	8	NUM
ejpam-5569	673	3	]	]	PUNCT
ejpam-5569	673	4	s.	s.	PROPN
ejpam-5569	673	5	gomathi	gomathi	PROPN
ejpam-5569	673	6	and	and	CCONJ
ejpam-5569	673	7	p.	p.	PROPN
ejpam-5569	673	8	venugopal	venugopal	PROPN
ejpam-5569	673	9	.	.	PUNCT
ejpam-5569	674	1	channel	channel	PROPN
ejpam-5569	674	2	assignment	assignment	NOUN
ejpam-5569	674	3	of	of	ADP
ejpam-5569	674	4	triangular	triangular	NOUN
ejpam-5569	674	5	and	and	CCONJ
ejpam-5569	674	6	rhombic	rhombic	ADJ
ejpam-5569	674	7	honeycomb	honeycomb	NOUN
ejpam-5569	674	8	networks	network	NOUN
ejpam-5569	674	9	using	use	VERB
ejpam-5569	674	10	radio	radio	NOUN
ejpam-5569	674	11	labeling	labeling	NOUN
ejpam-5569	674	12	techniques	technique	NOUN
ejpam-5569	674	13	.	.	PUNCT
ejpam-5569	675	1	communications	communication	NOUN
ejpam-5569	675	2	in	in	ADP
ejpam-5569	675	3	mathematics	mathematic	NOUN
ejpam-5569	675	4	and	and	CCONJ
ejpam-5569	675	5	applications	application	NOUN
ejpam-5569	675	6	,	,	PUNCT
ejpam-5569	675	7	12(3):665–676	12(3):665–676	NUM
ejpam-5569	675	8	,	,	PUNCT
ejpam-5569	675	9	2021	2021	NUM
ejpam-5569	675	10	.	.	PUNCT
ejpam-5569	676	1	[	[	X
ejpam-5569	676	2	9	9	NUM
ejpam-5569	676	3	]	]	X
ejpam-5569	676	4	r.	r.	PROPN
ejpam-5569	676	5	mahapatra	mahapatra	PROPN
ejpam-5569	676	6	,	,	PUNCT
ejpam-5569	676	7	s.	s.	PROPN
ejpam-5569	676	8	samanta	samanta	PROPN
ejpam-5569	676	9	,	,	PUNCT
ejpam-5569	676	10	t.	t.	NOUN
ejpam-5569	676	11	allahviranloo	allahviranloo	NOUN
ejpam-5569	676	12	,	,	PUNCT
ejpam-5569	676	13	and	and	CCONJ
ejpam-5569	676	14	m.	m.	NOUN
ejpam-5569	676	15	pal	pal	PROPN
ejpam-5569	676	16	.	.	PUNCT
ejpam-5569	677	1	radio	radio	NOUN
ejpam-5569	677	2	fuzzy	fuzzy	ADJ
ejpam-5569	677	3	graphs	graph	NOUN
ejpam-5569	677	4	and	and	CCONJ
ejpam-5569	677	5	assignment	assignment	NOUN
ejpam-5569	677	6	of	of	ADP
ejpam-5569	677	7	frequency	frequency	NOUN
ejpam-5569	677	8	in	in	ADP
ejpam-5569	677	9	radio	radio	NOUN
ejpam-5569	677	10	stations	station	NOUN
ejpam-5569	677	11	.	.	PUNCT
ejpam-5569	678	1	computational	computational	ADJ
ejpam-5569	678	2	and	and	CCONJ
ejpam-5569	678	3	applied	applied	ADJ
ejpam-5569	678	4	mathematics	mathematic	NOUN
ejpam-5569	678	5	,	,	PUNCT
ejpam-5569	678	6	38:1–20	38:1–20	NUM
ejpam-5569	678	7	,	,	PUNCT
ejpam-5569	678	8	2019	2019	NUM
ejpam-5569	678	9	.	.	PUNCT
ejpam-5569	679	1	[	[	X
ejpam-5569	679	2	10	10	NUM
ejpam-5569	679	3	]	]	PUNCT
ejpam-5569	679	4	x.	x.	NOUN
ejpam-5569	679	5	li	li	PROPN
ejpam-5569	679	6	,	,	PUNCT
ejpam-5569	679	7	v.	v.	PROPN
ejpam-5569	679	8	mak	mak	PROPN
ejpam-5569	679	9	,	,	PUNCT
ejpam-5569	679	10	and	and	CCONJ
ejpam-5569	679	11	s.	s.	PROPN
ejpam-5569	679	12	zhou	zhou	PROPN
ejpam-5569	679	13	.	.	PUNCT
ejpam-5569	680	1	optimal	optimal	ADJ
ejpam-5569	680	2	radio	radio	NOUN
ejpam-5569	680	3	labelings	labeling	NOUN
ejpam-5569	680	4	of	of	ADP
ejpam-5569	680	5	complete	complete	ADJ
ejpam-5569	680	6	m	m	NOUN
ejpam-5569	680	7	-	-	PUNCT
ejpam-5569	680	8	array	array	NOUN
ejpam-5569	680	9	trees	tree	NOUN
ejpam-5569	680	10	.	.	PUNCT
ejpam-5569	681	1	discrete	discrete	ADJ
ejpam-5569	681	2	appl	appl	PROPN
ejpam-5569	681	3	.	.	PUNCT
ejpam-5569	681	4	math	math	PROPN
ejpam-5569	681	5	.	.	PUNCT
ejpam-5569	681	6	,	,	PUNCT
ejpam-5569	681	7	158:507–515	158:507–515	NUM
ejpam-5569	681	8	,	,	PUNCT
ejpam-5569	681	9	2010	2010	NUM
ejpam-5569	681	10	.	.	PUNCT
ejpam-5569	682	1	[	[	X
ejpam-5569	682	2	11	11	NUM
ejpam-5569	682	3	]	]	X
ejpam-5569	682	4	d.	d.	PROPN
ejpam-5569	682	5	d.-f	d.-f	PROPN
ejpam-5569	682	6	.	.	PUNCT
ejpam-5569	683	1	liu	liu	PROPN
ejpam-5569	683	2	.	.	PUNCT
ejpam-5569	683	3	radio	radio	NOUN
ejpam-5569	683	4	number	number	NOUN
ejpam-5569	683	5	for	for	ADP
ejpam-5569	683	6	trees	tree	NOUN
ejpam-5569	683	7	.	.	PUNCT
ejpam-5569	684	1	discrete	discrete	ADJ
ejpam-5569	684	2	math	math	NOUN
ejpam-5569	684	3	.	.	PUNCT
ejpam-5569	684	4	,	,	PUNCT
ejpam-5569	684	5	308:1153–1164	308:1153–1164	NOUN
ejpam-5569	684	6	,	,	PUNCT
ejpam-5569	684	7	2008	2008	NUM
ejpam-5569	684	8	.	.	PUNCT
ejpam-5569	685	1	[	[	X
ejpam-5569	685	2	12	12	NUM
ejpam-5569	685	3	]	]	PUNCT
ejpam-5569	686	1	p.	p.	PROPN
ejpam-5569	686	2	k.	k.	PROPN
ejpam-5569	687	1	niranjan	niranjan	PROPN
ejpam-5569	687	2	and	and	CCONJ
ejpam-5569	687	3	s.	s.	PROPN
ejpam-5569	687	4	r.	r.	PROPN
ejpam-5569	687	5	kola	kola	PROPN
ejpam-5569	687	6	.	.	PUNCT
ejpam-5569	688	1	on	on	ADP
ejpam-5569	688	2	the	the	DET
ejpam-5569	688	3	radio	radio	NOUN
ejpam-5569	688	4	number	number	NOUN
ejpam-5569	688	5	for	for	ADP
ejpam-5569	688	6	corona	corona	NOUN
ejpam-5569	688	7	of	of	ADP
ejpam-5569	688	8	paths	path	NOUN
ejpam-5569	688	9	and	and	CCONJ
ejpam-5569	688	10	cycles	cycle	NOUN
ejpam-5569	688	11	.	.	PUNCT
ejpam-5569	689	1	akce	akce	PROPN
ejpam-5569	689	2	int	int	PROPN
ejpam-5569	689	3	.	.	PUNCT
ejpam-5569	690	1	j.	j.	PROPN
ejpam-5569	690	2	graphs	graphs	PROPN
ejpam-5569	690	3	and	and	CCONJ
ejpam-5569	690	4	combinatorics	combinatoric	NOUN
ejpam-5569	690	5	,	,	PUNCT
ejpam-5569	690	6	17(1):269–275	17(1):269–275	PROPN
ejpam-5569	690	7	,	,	PUNCT
ejpam-5569	690	8	2020	2020	NUM
ejpam-5569	690	9	.	.	PUNCT
ejpam-5569	691	1	[	[	X
ejpam-5569	691	2	13	13	NUM
ejpam-5569	691	3	]	]	PUNCT
ejpam-5569	691	4	d.	d.	PROPN
ejpam-5569	691	5	d.-f	d.-f	PROPN
ejpam-5569	691	6	.	.	PUNCT
ejpam-5569	692	1	liu	liu	PROPN
ejpam-5569	692	2	and	and	CCONJ
ejpam-5569	692	3	m.	m.	PROPN
ejpam-5569	692	4	xie	xie	PROPN
ejpam-5569	692	5	.	.	PUNCT
ejpam-5569	693	1	radio	radio	NOUN
ejpam-5569	693	2	number	number	NOUN
ejpam-5569	693	3	for	for	ADP
ejpam-5569	693	4	square	square	ADJ
ejpam-5569	693	5	paths	path	NOUN
ejpam-5569	693	6	.	.	PUNCT
ejpam-5569	694	1	ars	ars	PROPN
ejpam-5569	694	2	combin	combin	PROPN
ejpam-5569	694	3	.	.	PROPN
ejpam-5569	694	4	,	,	PUNCT
ejpam-5569	694	5	90:307–319	90:307–319	PROPN
ejpam-5569	694	6	,	,	PUNCT
ejpam-5569	694	7	2009	2009	NUM
ejpam-5569	694	8	.	.	PUNCT
ejpam-5569	695	1	[	[	X
ejpam-5569	695	2	14	14	NUM
ejpam-5569	695	3	]	]	X
ejpam-5569	695	4	d.	d.	PROPN
ejpam-5569	695	5	d.-f	d.-f	PROPN
ejpam-5569	695	6	.	.	PUNCT
ejpam-5569	696	1	liu	liu	PROPN
ejpam-5569	696	2	and	and	CCONJ
ejpam-5569	696	3	x.	x.	PROPN
ejpam-5569	696	4	zhu	zhu	PROPN
ejpam-5569	696	5	.	.	PUNCT
ejpam-5569	697	1	multi	multi	ADJ
ejpam-5569	697	2	-	-	ADJ
ejpam-5569	697	3	level	level	ADJ
ejpam-5569	697	4	distance	distance	NOUN
ejpam-5569	697	5	labelings	labeling	NOUN
ejpam-5569	697	6	for	for	ADP
ejpam-5569	697	7	paths	path	NOUN
ejpam-5569	697	8	and	and	CCONJ
ejpam-5569	697	9	cycles	cycle	NOUN
ejpam-5569	697	10	.	.	PUNCT
ejpam-5569	698	1	siam	siam	PROPN
ejpam-5569	698	2	j.	j.	PROPN
ejpam-5569	698	3	discrete	discrete	PROPN
ejpam-5569	698	4	math	math	PROPN
ejpam-5569	698	5	.	.	PUNCT
ejpam-5569	698	6	,	,	PUNCT
ejpam-5569	698	7	19:610–621	19:610–621	NUM
ejpam-5569	698	8	,	,	PUNCT
ejpam-5569	698	9	2005	2005	NUM
ejpam-5569	698	10	.	.	PUNCT
ejpam-5569	699	1	[	[	X
ejpam-5569	699	2	15	15	NUM
ejpam-5569	699	3	]	]	X
ejpam-5569	699	4	h.	h.	PROPN
ejpam-5569	699	5	qi	qi	PROPN
ejpam-5569	699	6	,	,	PUNCT
ejpam-5569	699	7	s.	s.	PROPN
ejpam-5569	699	8	nazeer	nazeer	PROPN
ejpam-5569	699	9	,	,	PUNCT
ejpam-5569	699	10	i.	i.	PROPN
ejpam-5569	699	11	kousar	kousar	PROPN
ejpam-5569	699	12	,	,	PUNCT
ejpam-5569	699	13	m.	m.	NOUN
ejpam-5569	699	14	a.	a.	PROPN
ejpam-5569	699	15	umar	umar	PROPN
ejpam-5569	699	16	,	,	PUNCT
ejpam-5569	699	17	and	and	CCONJ
ejpam-5569	699	18	n.	n.	PROPN
ejpam-5569	699	19	a.	a.	NOUN
ejpam-5569	699	20	shah	shah	PROPN
ejpam-5569	699	21	.	.	PUNCT
ejpam-5569	700	1	radio	radio	NOUN
ejpam-5569	700	2	labeling	labeling	NOUN
ejpam-5569	700	3	for	for	ADP
ejpam-5569	700	4	strong	strong	ADJ
ejpam-5569	700	5	product	product	NOUN
ejpam-5569	700	6	k3	k3	VERB
ejpam-5569	700	7	⊠	⊠	PROPN
ejpam-5569	700	8	pn	pn	PROPN
ejpam-5569	700	9	.	.	PROPN
ejpam-5569	700	10	ieee	ieee	PROPN
ejpam-5569	700	11	access	access	NOUN
ejpam-5569	700	12	,	,	PUNCT
ejpam-5569	700	13	8:109801–109806	8:109801–109806	NUM
ejpam-5569	700	14	,	,	PUNCT
ejpam-5569	700	15	2020	2020	NUM
ejpam-5569	700	16	.	.	PUNCT
ejpam-5569	701	1	[	[	X
ejpam-5569	701	2	16	16	NUM
ejpam-5569	701	3	]	]	X
ejpam-5569	701	4	m.	m.	NOUN
ejpam-5569	701	5	morris	morris	PROPN
ejpam-5569	701	6	-	-	PUNCT
ejpam-5569	701	7	rivera	rivera	PROPN
ejpam-5569	701	8	,	,	PUNCT
ejpam-5569	701	9	m.	m.	NOUN
ejpam-5569	701	10	tomova	tomova	PROPN
ejpam-5569	701	11	,	,	PUNCT
ejpam-5569	701	12	c.	c.	NOUN
ejpam-5569	701	13	wyels	wyel	NOUN
ejpam-5569	701	14	,	,	PUNCT
ejpam-5569	701	15	and	and	CCONJ
ejpam-5569	701	16	y.	y.	PROPN
ejpam-5569	701	17	yeager	yeager	PROPN
ejpam-5569	701	18	.	.	PUNCT
ejpam-5569	702	1	the	the	DET
ejpam-5569	702	2	radio	radio	NOUN
ejpam-5569	702	3	number	number	NOUN
ejpam-5569	702	4	of	of	ADP
ejpam-5569	702	5	cn×cn	cn×cn	PROPN
ejpam-5569	702	6	.	.	PUNCT
ejpam-5569	702	7	ars	ars	PROPN
ejpam-5569	702	8	combin	combin	PROPN
ejpam-5569	702	9	.	.	PROPN
ejpam-5569	702	10	,	,	PUNCT
ejpam-5569	702	11	103:81–96	103:81–96	PROPN
ejpam-5569	702	12	,	,	PUNCT
ejpam-5569	702	13	2012	2012	NUM
ejpam-5569	702	14	.	.	PUNCT
ejpam-5569	703	1	[	[	X
ejpam-5569	703	2	17	17	NUM
ejpam-5569	703	3	]	]	PUNCT
ejpam-5569	703	4	j.	j.	PROPN
ejpam-5569	703	5	p.	p.	PROPN
ejpam-5569	703	6	ortiz	ortiz	PROPN
ejpam-5569	703	7	,	,	PUNCT
ejpam-5569	703	8	p.	p.	PROPN
ejpam-5569	703	9	martinez	martinez	PROPN
ejpam-5569	703	10	,	,	PUNCT
ejpam-5569	703	11	m.	m.	NOUN
ejpam-5569	703	12	tomova	tomova	PROPN
ejpam-5569	703	13	,	,	PUNCT
ejpam-5569	703	14	and	and	CCONJ
ejpam-5569	703	15	c.	c.	NOUN
ejpam-5569	703	16	wyels	wyel	NOUN
ejpam-5569	703	17	.	.	PUNCT
ejpam-5569	704	1	radio	radio	NOUN
ejpam-5569	704	2	numbers	number	NOUN
ejpam-5569	704	3	of	of	ADP
ejpam-5569	704	4	some	some	DET
ejpam-5569	704	5	generalized	generalized	ADJ
ejpam-5569	704	6	prism	prism	NOUN
ejpam-5569	704	7	graphs	graph	NOUN
ejpam-5569	704	8	.	.	PUNCT
ejpam-5569	705	1	discuss	discuss	PROPN
ejpam-5569	705	2	.	.	PUNCT
ejpam-5569	706	1	math	math	NOUN
ejpam-5569	706	2	.	.	PUNCT
ejpam-5569	707	1	graph	graph	NOUN
ejpam-5569	707	2	theory	theory	NOUN
ejpam-5569	707	3	,	,	PUNCT
ejpam-5569	707	4	311:45–62	311:45–62	NUM
ejpam-5569	707	5	,	,	PUNCT
ejpam-5569	707	6	2011	2011	NUM
ejpam-5569	707	7	.	.	PUNCT
ejpam-5569	708	1	[	[	X
ejpam-5569	708	2	18	18	NUM
ejpam-5569	708	3	]	]	PUNCT
ejpam-5569	708	4	v.	v.	CCONJ
ejpam-5569	708	5	s.	s.	PROPN
ejpam-5569	708	6	reddy	reddy	PROPN
ejpam-5569	708	7	and	and	CCONJ
ejpam-5569	708	8	v.	v.	PROPN
ejpam-5569	708	9	k.	k.	PROPN
ejpam-5569	708	10	iyer	iyer	PROPN
ejpam-5569	708	11	.	.	PUNCT
ejpam-5569	709	1	upper	upper	ADJ
ejpam-5569	709	2	bounds	bound	NOUN
ejpam-5569	709	3	on	on	ADP
ejpam-5569	709	4	the	the	DET
ejpam-5569	709	5	radio	radio	NOUN
ejpam-5569	709	6	number	number	NOUN
ejpam-5569	709	7	of	of	ADP
ejpam-5569	709	8	some	some	DET
ejpam-5569	709	9	trees	tree	NOUN
ejpam-5569	709	10	.	.	PUNCT
ejpam-5569	710	1	int	int	NOUN
ejpam-5569	710	2	.	.	PUNCT
ejpam-5569	711	1	j.	j.	PROPN
ejpam-5569	711	2	pure	pure	PROPN
ejpam-5569	711	3	applied	applied	ADJ
ejpam-5569	711	4	math	math	NOUN
ejpam-5569	711	5	.	.	PUNCT
ejpam-5569	711	6	,	,	PUNCT
ejpam-5569	712	1	71(2):207–215	71(2):207–215	NUM
ejpam-5569	712	2	,	,	PUNCT
ejpam-5569	712	3	2011	2011	NUM
ejpam-5569	712	4	.	.	PUNCT
ejpam-5569	713	1	[	[	X
ejpam-5569	713	2	19	19	NUM
ejpam-5569	713	3	]	]	PUNCT
ejpam-5569	713	4	l.	l.	PROPN
ejpam-5569	713	5	saha	saha	PROPN
ejpam-5569	713	6	and	and	CCONJ
ejpam-5569	713	7	p.	p.	NOUN
ejpam-5569	713	8	panigrahi	panigrahi	NOUN
ejpam-5569	713	9	.	.	PUNCT
ejpam-5569	714	1	on	on	ADP
ejpam-5569	714	2	the	the	DET
ejpam-5569	714	3	radio	radio	NOUN
ejpam-5569	714	4	number	number	NOUN
ejpam-5569	714	5	of	of	ADP
ejpam-5569	714	6	toroidal	toroidal	ADJ
ejpam-5569	714	7	grids	grid	NOUN
ejpam-5569	714	8	.	.	PUNCT
ejpam-5569	715	1	australian	australian	ADJ
ejpam-5569	715	2	j.	j.	PROPN
ejpam-5569	715	3	combin	combin	PROPN
ejpam-5569	715	4	.	.	PROPN
ejpam-5569	715	5	,	,	PUNCT
ejpam-5569	715	6	55:273–288	55:273–288	NUM
ejpam-5569	715	7	,	,	PUNCT
ejpam-5569	715	8	2013	2013	NUM
ejpam-5569	715	9	.	.	PUNCT
ejpam-5569	716	1	[	[	X
ejpam-5569	716	2	20	20	NUM
ejpam-5569	716	3	]	]	PUNCT
ejpam-5569	716	4	l.	l.	PROPN
ejpam-5569	716	5	saha	saha	PROPN
ejpam-5569	716	6	and	and	CCONJ
ejpam-5569	716	7	p.	p.	NOUN
ejpam-5569	716	8	panigrahi	panigrahi	NOUN
ejpam-5569	716	9	.	.	PUNCT
ejpam-5569	717	1	a	a	DET
ejpam-5569	717	2	lower	lower	ADV
ejpam-5569	717	3	bound	bind	VERB
ejpam-5569	717	4	for	for	ADP
ejpam-5569	717	5	radio	radio	NOUN
ejpam-5569	717	6	k	k	ADJ
ejpam-5569	717	7	-	-	ADJ
ejpam-5569	717	8	chromatic	chromatic	ADJ
ejpam-5569	717	9	number	number	NOUN
ejpam-5569	717	10	.	.	PUNCT
ejpam-5569	718	1	discrete	discrete	ADJ
ejpam-5569	718	2	appl	appl	PROPN
ejpam-5569	718	3	.	.	PUNCT
ejpam-5569	718	4	math	math	PROPN
ejpam-5569	718	5	.	.	PUNCT
ejpam-5569	718	6	,	,	PUNCT
ejpam-5569	718	7	192:87–100	192:87–100	NUM
ejpam-5569	718	8	,	,	PUNCT
ejpam-5569	718	9	2015	2015	NUM
ejpam-5569	718	10	.	.	PUNCT
ejpam-5569	719	1	[	[	X
ejpam-5569	719	2	21	21	NUM
ejpam-5569	719	3	]	]	PUNCT
ejpam-5569	719	4	p.	p.	PROPN
ejpam-5569	719	5	zhang	zhang	PROPN
ejpam-5569	719	6	.	.	PUNCT
ejpam-5569	720	1	radio	radio	NOUN
ejpam-5569	720	2	labellings	labelling	NOUN
ejpam-5569	720	3	of	of	ADP
ejpam-5569	720	4	cycles	cycle	NOUN
ejpam-5569	720	5	.	.	PUNCT
ejpam-5569	721	1	ars	ars	PROPN
ejpam-5569	721	2	combin	combin	NOUN
ejpam-5569	721	3	,	,	PUNCT
ejpam-5569	721	4	65:21–32	65:21–32	NUM
ejpam-5569	721	5	,	,	PUNCT
ejpam-5569	721	6	2002	2002	NUM
ejpam-5569	721	7	.	.	PUNCT
ejpam-5569	722	1	[	[	X
ejpam-5569	722	2	22	22	NUM
ejpam-5569	722	3	]	]	PUNCT
ejpam-5569	722	4	a.	a.	NOUN
ejpam-5569	722	5	h.	h.	PROPN
ejpam-5569	722	6	alkasasbeh	alkasasbeh	PROPN
ejpam-5569	722	7	,	,	PUNCT
ejpam-5569	722	8	e.	e.	PROPN
ejpam-5569	722	9	badr	badr	PROPN
ejpam-5569	722	10	,	,	PUNCT
ejpam-5569	722	11	h.	h.	PROPN
ejpam-5569	722	12	attiya	attiya	PROPN
ejpam-5569	722	13	,	,	PUNCT
ejpam-5569	722	14	and	and	CCONJ
ejpam-5569	722	15	h.	h.	PROPN
ejpam-5569	722	16	m.	m.	PROPN
ejpam-5569	722	17	shabana	shabana	PROPN
ejpam-5569	722	18	.	.	PUNCT
ejpam-5569	723	1	radio	radio	NOUN
ejpam-5569	723	2	number	number	NOUN
ejpam-5569	723	3	for	for	ADP
ejpam-5569	723	4	friendship	friendship	NOUN
ejpam-5569	723	5	communication	communication	NOUN
ejpam-5569	723	6	networks	network	NOUN
ejpam-5569	723	7	.	.	PUNCT
ejpam-5569	724	1	mathematics	mathematic	NOUN
ejpam-5569	724	2	,	,	PUNCT
ejpam-5569	724	3	11:4232	11:4232	PROPN
ejpam-5569	724	4	,	,	PUNCT
ejpam-5569	724	5	2023	2023	NUM
ejpam-5569	724	6	.	.	PUNCT
ejpam-5569	725	1	[	[	X
ejpam-5569	725	2	23	23	NUM
ejpam-5569	725	3	]	]	X
ejpam-5569	725	4	e.	e.	PROPN
ejpam-5569	725	5	badr	badr	PROPN
ejpam-5569	725	6	,	,	PUNCT
ejpam-5569	725	7	s.	s.	PROPN
ejpam-5569	725	8	nada	nada	PROPN
ejpam-5569	725	9	,	,	PUNCT
ejpam-5569	725	10	m.	m.	NOUN
ejpam-5569	725	11	m.	m.	PROPN
ejpam-5569	725	12	a.	a.	PROPN
ejpam-5569	725	13	al	al	PROPN
ejpam-5569	725	14	-	-	PUNCT
ejpam-5569	725	15	shamiri	shamiri	PROPN
ejpam-5569	725	16	,	,	PUNCT
ejpam-5569	725	17	a.	a.	PROPN
ejpam-5569	725	18	abdel	abdel	PROPN
ejpam-5569	725	19	-	-	PUNCT
ejpam-5569	725	20	hay	hay	PROPN
ejpam-5569	725	21	,	,	PUNCT
ejpam-5569	725	22	and	and	CCONJ
ejpam-5569	725	23	a.	a.	NOUN
ejpam-5569	725	24	elrokh	elrokh	NOUN
ejpam-5569	725	25	.	.	PUNCT
ejpam-5569	726	1	a	a	DET
ejpam-5569	726	2	novel	novel	ADJ
ejpam-5569	726	3	mathematical	mathematical	ADJ
ejpam-5569	726	4	model	model	NOUN
ejpam-5569	726	5	for	for	ADP
ejpam-5569	726	6	radio	radio	NOUN
ejpam-5569	726	7	mean	mean	PROPN
ejpam-5569	726	8	square	square	ADJ
ejpam-5569	726	9	labeling	labeling	NOUN
ejpam-5569	726	10	problem	problem	NOUN
ejpam-5569	726	11	.	.	PUNCT
ejpam-5569	727	1	j.	j.	PROPN
ejpam-5569	727	2	math	math	PROPN
ejpam-5569	727	3	.	.	PUNCT
ejpam-5569	727	4	,	,	PUNCT
ejpam-5569	727	5	page	page	NOUN
ejpam-5569	727	6	3303	3303	NUM
ejpam-5569	727	7	,	,	PUNCT
ejpam-5569	727	8	2022	2022	NUM
ejpam-5569	727	9	.	.	PUNCT
ejpam-5569	728	1	[	[	X
ejpam-5569	728	2	24	24	NUM
ejpam-5569	728	3	]	]	X
ejpam-5569	728	4	n.	n.	PROPN
ejpam-5569	728	5	m.	m.	NOUN
ejpam-5569	728	6	noureldeen	noureldeen	PROPN
ejpam-5569	728	7	,	,	PUNCT
ejpam-5569	728	8	e.	e.	PROPN
ejpam-5569	728	9	badr	badr	PROPN
ejpam-5569	728	10	,	,	PUNCT
ejpam-5569	728	11	i.	i.	PROPN
ejpam-5569	728	12	m.	m.	PROPN
ejpam-5569	728	13	hagag	hagag	PROPN
ejpam-5569	728	14	,	,	PUNCT
ejpam-5569	728	15	and	and	CCONJ
ejpam-5569	728	16	h.	h.	PROPN
ejpam-5569	728	17	shabana	shabana	PROPN
ejpam-5569	728	18	.	.	PUNCT
ejpam-5569	729	1	graph	graph	NOUN
ejpam-5569	729	2	-	-	PUNCT
ejpam-5569	729	3	based	base	VERB
ejpam-5569	729	4	approach	approach	NOUN
ejpam-5569	729	5	to	to	ADP
ejpam-5569	729	6	the	the	DET
ejpam-5569	729	7	radio	radio	NOUN
ejpam-5569	729	8	assignment	assignment	NOUN
ejpam-5569	729	9	problem	problem	NOUN
ejpam-5569	729	10	in	in	ADP
ejpam-5569	729	11	wireless	wireless	ADJ
ejpam-5569	729	12	communication	communication	NOUN
ejpam-5569	729	13	networks	network	NOUN
ejpam-5569	729	14	with	with	ADP
ejpam-5569	729	15	applications	application	NOUN
ejpam-5569	729	16	in	in	ADP
ejpam-5569	729	17	cryptography	cryptography	NOUN
ejpam-5569	729	18	.	.	PUNCT
ejpam-5569	730	1	eur	eur	PROPN
ejpam-5569	730	2	.	.	PUNCT
ejpam-5569	731	1	j.	j.	PROPN
ejpam-5569	731	2	pure	pure	PROPN
ejpam-5569	731	3	appl	appl	PROPN
ejpam-5569	731	4	.	.	PROPN
ejpam-5569	731	5	math	math	PROPN
ejpam-5569	731	6	,	,	PUNCT
ejpam-5569	731	7	18(1):5614	18(1):5614	NUM
ejpam-5569	731	8	,	,	PUNCT
ejpam-5569	731	9	2025	2025	NUM
ejpam-5569	731	10	.	.	PUNCT
ejpam-5569	732	1	[	[	X
ejpam-5569	732	2	25	25	NUM
ejpam-5569	732	3	]	]	PUNCT
ejpam-5569	732	4	m.	m.	NOUN
ejpam-5569	732	5	saraswathi	saraswathi	PROPN
ejpam-5569	732	6	and	and	CCONJ
ejpam-5569	732	7	k.	k.	PROPN
ejpam-5569	732	8	n.	n.	PROPN
ejpam-5569	732	9	meera	meera	PROPN
ejpam-5569	732	10	.	.	PUNCT
ejpam-5569	732	11	bounds	bound	VERB
ejpam-5569	732	12	on	on	ADP
ejpam-5569	732	13	radio	radio	NOUN
ejpam-5569	732	14	mean	mean	NOUN
ejpam-5569	732	15	number	number	NOUN
ejpam-5569	732	16	of	of	ADP
ejpam-5569	732	17	graphs	graph	NOUN
ejpam-5569	732	18	.	.	PUNCT
ejpam-5569	733	1	journal	journal	NOUN
ejpam-5569	733	2	of	of	ADP
ejpam-5569	733	3	intelligent	intelligent	ADJ
ejpam-5569	733	4	&	&	CCONJ
ejpam-5569	733	5	fuzzy	fuzzy	ADJ
ejpam-5569	733	6	systems	system	NOUN
ejpam-5569	733	7	,	,	PUNCT
ejpam-5569	733	8	44(2):1691–1702	44(2):1691–1702	PROPN
ejpam-5569	733	9	,	,	PUNCT
ejpam-5569	733	10	2023	2023	NUM
ejpam-5569	733	11	.	.	PUNCT
ejpam-5569	734	1	[	[	X
ejpam-5569	734	2	26	26	NUM
ejpam-5569	734	3	]	]	PUNCT
ejpam-5569	734	4	m.	m.	NOUN
ejpam-5569	734	5	saraswathi	saraswathi	PROPN
ejpam-5569	734	6	and	and	CCONJ
ejpam-5569	734	7	k.	k.	PROPN
ejpam-5569	734	8	n.	n.	PROPN
ejpam-5569	734	9	meera	meera	PROPN
ejpam-5569	734	10	.	.	PUNCT
ejpam-5569	735	1	radio	radio	NOUN
ejpam-5569	735	2	mean	mean	NOUN
ejpam-5569	735	3	labeled	label	VERB
ejpam-5569	735	4	graphs	graph	NOUN
ejpam-5569	735	5	to	to	PART
ejpam-5569	735	6	generate	generate	VERB
ejpam-5569	735	7	keys	key	NOUN
ejpam-5569	735	8	in	in	ADP
ejpam-5569	735	9	cryptography	cryptography	NOUN
ejpam-5569	735	10	.	.	PUNCT
ejpam-5569	736	1	in	in	ADP
ejpam-5569	736	2	2021	2021	NUM
ejpam-5569	736	3	2nd	2nd	ADJ
ejpam-5569	736	4	international	international	ADJ
ejpam-5569	736	5	conference	conference	NOUN
ejpam-5569	736	6	on	on	ADP
ejpam-5569	736	7	communication	communication	NOUN
ejpam-5569	736	8	,	,	PUNCT
ejpam-5569	736	9	computing	computing	NOUN
ejpam-5569	736	10	and	and	CCONJ
ejpam-5569	736	11	industry	industry	NOUN
ejpam-5569	736	12	4.0	4.0	NUM
ejpam-5569	736	13	(	(	PUNCT
ejpam-5569	736	14	c214	c214	PROPN
ejpam-5569	736	15	)	)	PUNCT
ejpam-5569	736	16	,	,	PUNCT
ejpam-5569	736	17	pages	page	NOUN
ejpam-5569	736	18	1–3	1–3	NUM
ejpam-5569	736	19	,	,	PUNCT
ejpam-5569	736	20	bangalore	bangalore	PROPN
ejpam-5569	736	21	,	,	PUNCT
ejpam-5569	736	22	india	india	PROPN
ejpam-5569	736	23	,	,	PUNCT
ejpam-5569	736	24	2021	2021	NUM
ejpam-5569	736	25	.	.	PUNCT