EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7091 ISSN 1307-5543 – ejpam.com Published by New York Business Global Optimizing Communication Network Routing with A* Algorithm: A Comparative Study Against Bellman-Ford and Dijkstra for Enhanced Quality of Service Mohamed Ayari1,∗, Atef Gharbi2, Zeineb Klai3, Abdelhalim Hasnaoui4 , Mahmoud Salaheldin Elsayed3, Elsaid Md. Abdelrahim5 1 Department of Information Technology, Faculty of Computing and Information Technology, Northern Border University, Kingdom of Saudi Arabia 2 Department of Information Systems, Faculty of Computing and Information Technology, Northern Border University, Kingdom of Saudi Arabia 3 Department of Computer Sciences, Faculty of Computing and Information Technology, Northern Border University, Kingdom of Saudi Arabia 4 Mathematics Department, College of Sciences and Arts, Northern Border University, Kingdom of Saudi Arabia 5 Computer Science Department, Science College, Northern Border University (NBU), Arar 73213, Saudi Arabia Abstract. Efficient routing is critical to modern networks, where low latency, QoS guarantees, and energy efficiency must be maintained despite dynamic conditions. This paper applied the A* algorithm to QoS-aware routing and compared it with Bellman–Ford and Dijkstra in 5G/6G slicing, IoT multi-hop, and SDN/backbone settings. We introduced admissible domain-specific heuristics that integrate graph distance with live network indicators (queue delay, link loss, available band- width, and residual energy). Across trace-driven and synthetic topologies, A* achieved lower end- to-end latency and jitter, faster convergence, and reduced routing overhead relative to baselines, while maintaining competitive delivery ratio and energy per delivered bit. When admissibility held, A-Star preserved optimal paths with fewer expansions. With non-admissible but informa- tive heuristics, it delivered favorable latency–overhead trade-offs. The results demonstrate that heuristic-guided routing can meet emerging QoS demands in real-time and resource-constrained networks, providing a scalable and implementation-ready alternative to traditional shortest-path methods. 2020 Mathematics Subject Classifications: 91A10, 91A35, 90C59, 78M50 Key Words and Phrases: A-Star (A*) Algorithm, Heuristic Routing, QoS-Aware Routing, Dijkstra Algorithm, Bellman–Ford Algorithm, 5G/6G Networks ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7091 Email address: Mohamed.ayari@nbu.edu.sa (M. Ayari) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 2 of 29 1. Introduction Next-generation communication systems—from 5G/6G mobile networks and cloud backbones to dense Internet of Things (IoT) deployments—demand routing solutions that deliver low latency, predictable Quality of Service (QoS), and energy efficiency under time- varying traffic and topology changes. In 5G, for example, QoS flows and network slicing impose differentiated performance targets (latency, loss, jitter, and bandwidth reservation) across heterogeneous segments; moving toward 6G further amplifies the requirements for reliability, ultra-low latency, and pervasive intelligence across the control and user planes [1–3]. Software-Defined Networking (SDN) and centralized control provide global visibility to implement QoS-aware routing and traffic engineering, but rapidly finding near-optimal paths as conditions evolve remains a challenge at scale [4–9]. Classic shortest-path algorithms underpin network routing. Bellman–Ford supports negative edge weights and underlies distance-vector protocols, whereas Dijkstra—with heap-based priority queues—underlies link-state protocols under non-negative weights. This motivates our A-Star comparison across both families. However, both expand the search space without considering destination-directed guidance, which can inflate con- vergence time and control overhead in dynamic environments with tight QoS constraints [10, 11]. In contrast, the A* algorithm increases the cost-to-come with a problem-informed heuristic estimate of cost-to-go, prioritizing promising frontiers and reducing unnecessary node expansions while preserving optimality when the heuristic is admissible and consis- tent [12, 13]. These properties make A* an attractive candidate for QoS-aware routing, provided that we can craft admissible or bounded heuristics that encode network condi- tions such as queueing delay, loss probability, residual bandwidth, and energy budgets. Recent work across SDN, IoT, and satellite networks highlights the benefits of heuristic or learning-guided path selection. In SDN, dynamic, traffic-aware controllers and deep reinforcement learning (DRL) agents have been shown to reduce delay, smooth jitter, and balance load while responding to failures and bursts [6–9, 14]. In IoT and wireless sensor networks (WSNs), where energy is at a premium, energy-aware QoS routing remains central; contemporary surveys and designs report improvements in network lifetime and delivery ratio by using multi-constraint metrics (delay, reliability, residual energy) [15– 19]. Even in non-terrestrial networks such as LEO satellite constellations, route selection increasingly couples energy models with QoS requirements [20]. Meanwhile, 6G-oriented studies emphasize that tight QoS guarantees must co-exist with high mobility and extreme heterogeneity, strengthening the case for search strategies that prune the exploration space effectively [2, 3]. Multi-criteria evaluation of uncertain, context-sensitive constraints can be formalized using intuitionistic fuzzy-set frameworks, which aligns with QoS trade-off decisions in heuristic routing [21]. Against this backdrop, we investigate A* for QoS-aware routing in communica- tion networks and compare it with Bellman–Ford and Dijkstra. Our core thesis is that domain-informed heuristics—built from graph-geodesic priors and live signals (e.g., measured queuing delay, short-term loss, available bandwidth, residual energy)—allow A* to (i) reduce search overhead and convergence time, (ii) maintain optimality under M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 3 of 29 admissible heuristics (or provide bounded suboptimality under weighted heuristics), and (iii) realize favorable latency–energy trade-offs in diverse topologies (5G/6G slices, IoT multi-hop meshes, and SDN backbones). We evaluate latency, jitter, packet delivery ra- tio, path stretch, convergence time, routing/control overhead, and energy per delivered bit using trace-driven and synthetic workloads under dynamic link conditions and failures. We also situate A* within learning-augmented controllers (e.g., DRL) by using A* as a deterministic inner solver with learned cost-to-go surrogates [3, 7–9]. The paper makes three contributions: (1) a unified QoS-aware heuristic design for A*, integrating readily measurable network signals while maintaining admissibility where required; (2) a head-to-head comparison of A*, Dijkstra, and Bellman–Ford across het- erogeneous topologies and QoS regimes, including energy-sensitive IoT and slice-aware SDN/5G scenarios; and (3) an empirical analysis of convergence, overhead, and QoS com- pliance showing when A* yields the largest gains relative to baselines [10, 14, 15, 17, 20]. Our study targets scenarios where controllers (or distributed agents) can estimate or bound link-level metrics at sub-second timescales and where latency, jitter, or loss SLOs must be met despite churn. This includes URLLC-like flows in 5G/6G, bandwidth and energy-constrained IoT telemetry, and SDN-operated edge/core fabrics [1, 2, 4, 22]. The remainder of this paper is structured as follows: Section 2 reviews background and related work on QoS routing and path-search algorithms; Section 3 formalizes the QoS-aware routing problem and metrics; Section 4 details A*, Dijkstra, and Bellman– Ford implementations and our heuristic design; Section 5 describes datasets, topologies, and experimental methodology; Section 6 reports results and analysis; Section 7 dis- cusses implications, limitations, and extensions (including weighted/learned heuristics); and Section 8 concludes. 2. Background and Related Work Ensuring end-to-end performance in modern networks means meeting diverse Quality of Service (QoS) goals—low latency, bounded jitter, packet delivery guarantees, and band- width reservations—under time-varying demand and failures. The advent of 5G/6G intro- duces explicit mechanisms (e.g., QoS flows and network slicing) that expose these goals as first-class constraints for control-plane decisions, while disaggregation via software-defined networking (SDN) centralizes path computation and admission control [1, 2, 4]. In such settings, routing must rapidly find and maintain paths that optimize multi-constraint ob- jectives while scaling to large fabrics. For algebraic generalization beyond crisp graphs, fuzzy Γ-semimodules over Γ-semirings offer a perspective on relational path structures that could support extensions of our model [23]. M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 4 of 29 2.1. Quality of Service-Aware Routing: Concepts and Challenges Classical shortest-path routing assumes a single additive metric (e.g., hop count or static link cost). QoS-aware routing is more complex: (i) it may combine additive (de- lay), multiplicative (loss), and concave (bottleneck bandwidth) metrics; (ii) it must react on sub-second timescales to congestion, failures, and mobility; and (iii) it often targets flow- or slice-specific Service Level Objectives (SLOs) [1, 4]. Central controllers (SDN) can, in principle, compute globally optimal paths with updated telemetry, but repeated recomputation over large topologies can inflate control overhead and reaction time [4]. 2.2. Canonical Algorithms for Shortest Paths Bellman–Ford and Dijkstra remain the canonical baselines. Bellman–Ford, a label- correcting method, supports negative edge weights and detects negative cycles at the cost of O(V ·E) time [11, 24]. Dijkstra’s label-setting method achieves O((V +E) log V ) with heaps, but assumes non-negative weights [11, 25]. Both expand search frontiers with- out destination-directed guidance, which can slow convergence when telemetry frequently shifts and when multiple QoS constraints narrow the feasible region. 2.3. Heuristic Search and A* A* augments the cost-to-come g(n) with a problem-informed heuristic h(n) estimating the cost-to-go, ranking candidates by f(n) = g(n)+h(n) [13]. For admissible and consistent heuristics, A* preserves optimality while expanding far fewer nodes than Dijkstra; weighted A* trades bounded suboptimality for additional pruning. Systematic reviews confirm A*’s efficiency benefits hinge on heuristic quality [12]. In networking, domain-informed h(·) can incorporate geodesic distance plus live signals (queueing delay, loss, residual bandwidth, residual energy), turning A* into a QoS-aware search that prunes unactionable branches early. 2.4. SDN: Centralized Control and Learning-Augmented Routing With SDN, the controller holds a global view and can embed telemetry into routing objectives. Recent work demonstrates that heuristic and learning-augmented controllers reduce delay and packet loss while balancing load [4, 6, 8, 26, 27]. Reinforcement-learning (RL) agents coupled with graph neural networks (GNNs) accelerate convergence and gen- eralize across topologies; hybrid DRL+GNN methods directly emit routing policies that adapt to nonstationary traffic [28, 29]. These trends support our thesis: when combined with informed heuristics, A* can act as a fast inner solver, while learning refines h(·) or prioritizes path candidates. 2.5. Quality of Service and Energy in IoT/WSN and Beyond In IoT and Wireless Sensor Networks (WSNs), routing must be energy-aware while honoring latency and reliability constraints. Recent studies propose QoS-aware and energy- M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 5 of 29 efficient schemes (e.g., clustering with trust and link-quality estimation; multipath, fuzzy, and metaheuristic designs) that extend lifetime and improve delivery [19, 30–32]. These domains are natural fits for A* with energy-informed heuristics (residual energy, link reliability), where pruning reduces radio activity and control traffic. 2.6. Non-Terrestrial and Multi-Domain Networks Low Earth Orbit (LEO) constellations and multi-domain backbones bring additional constraints (satellite energy budgets, dynamic link geometry), where routing couples QoS with energy models [20]. Here, heuristic guidance—e.g., orbital geometry and predicted contact windows—can concentrate search on feasible corridors, again aligning with A*’s strengths. 2.7. Positioning This Work Prior art demonstrates: (i) scalable SDN controllers benefit from search-space reduc- tion and topology-aware priors [4, 6], (ii) DRL/GNN policies can embed network state for robust routing [26–29], and (iii) energy-aware QoS routing is pivotal in IoT/WSN and satellite networks [16, 20, 30–32]. However, the literature lacks a unified evaluation of A* with QoS-informed heuristics against Bellman–Ford and Dijkstra across heterogeneous topologies under realistic telemetry. This paper fills that gap by (1) proposing admissible and weighted heuristics that combine graph distance with live link metrics; (2) bench- marking A*, Dijkstra, and Bellman–Ford under 5G/6G slices, IoT multi-hop meshes, and SDN backbones; and (3) analyzing latency, jitter, delivery ratio, path stretch, convergence time, routing overhead, and energy per delivered bit. 2.8. Takeaways A* offers a principled way to leverage problem structure in routing. When heuristics encode current network conditions and constraints, A* can reduce expansions and con- trol overhead while preserving optimality or offering bounded trade-offs—a compelling advantage in time-sensitive and energy-constrained environments [4, 12, 13, 16, 26]. 3. Problem Formulation and Quality of Service Metrics This section formalizes the Quality of Service (QoS)–aware routing problem on a di- rected graph, defines the feasibility constraints used throughout the paper, and specifies the scalarized objective and per-edge cost adopted by all algorithms. It then details our adaptation of A-Star for QoS-aware routing, including the heuristic design, indicator nor- malization, weight selection, and the evaluation setup. We provide clear notation and conservative heuristic components to ensure compatibility with the shortest-path solvers evaluated later. M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 6 of 29 3.1. Network Model and Path Metrics We represent the network as a directed graphG = (V,E) with |V | = n and |E| = m. At decision time τ , each directed edge e = (u, v) ∈ E carries the following telemetry: latency de(τ), queueing term qe(τ), loss probability λe(τ), available bandwidth be(τ), and energy per delivered bit εe(τ). For a simple path P = ⟨e1, . . . , ek⟩ from source s to destination t, we use the additive and bottleneck metrics D(P ) = ∑ e∈P de, Q(P ) = ∑ e∈P qe, (1) J(P ) = 1 k ∑ e∈P (qe − q̄)2 , q̄ = 1 k ∑ e∈P qe, (2) We refer to J(P ) as a jitter proxy (queue-variance proxy): while it captures per-link queue variability along path P , it is only an approximation of true end-to-end inter-arrival jitter. Λ(P ) = 1− ∏ e∈P (1− λe) , L̂(P ) = ∑ e∈P (− ln(1− λe)) , (3) B(P ) = min e∈P be, E(P ) = ∑ e∈P εe. (4) Here L̂(P ) is an additive surrogate for end-to-end loss that enables standard graph search. The metrics in (1)–(4) capture latency, jitter proxy J(P ), reliability, bandwidth, and energy in forms amenable to shortest-path formulations while preserving the operational meaning of QoS targets. 3.2. Quality of Service Feasibility and Scalarized Objective A flow f = (s, t, D̄, J̄ , Λ̄, B̄, Ē) is feasible on path P when the service-level objectives (SLOs) are met: D(P ) ≤ D̄, J(P ) ≤ J̄ , Λ(P ) ≤ Λ̄, B(P ) ≥ B̄, E(P ) ≤ Ē. (5) To compare alternative feasible routes with a single score, we use a nonnegative scalariza- tion with weights αD, αL, αJ , αE , αB and normalizers σD, σL, σJ , σE , σB: C(P ) = αD D(P ) σD + αL L̂(P ) σL + αJ J(P ) σJ + αE E(P ) σE + αB ϕ(B(P )) , (6) where the shortfall penalty for bandwidth is ϕ(B) =  0, B ≥ B̄,( B̄ −B σB )2 , B < B̄. (7) M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 7 of 29 The single-flow problem is then P ∗ ∈ arg min P s.t. (5) C(P ). (8) When temporary violations are permitted for comparison purposes, we also report the relaxed objective L(P ) = C(P )+µD max{D(P )−D̄, 0}+µJ max{J(P )−J̄ , 0}+µL max{L̂(P )+ln(1−Λ̄), 0}+µE max{E(P )−Ē, 0}+µB max{B̄−B(P ), 0}, (9) with nonnegative multipliers µ·. The scalarization in (6) preserves feasibility via (5) while aligning the optimizer with delay/loss-sensitive operation; the relaxed score L(P ) provides a consistent way to rank paths in constraint-edge cases. 3.3. Per-Edge Additive Cost for Graph Search To enable single-source shortest-path solvers, we define a nonnegative additive edge weight we = αD de σD + αL − ln(1− λe) σL + αJ (qe − q̂)2 σJ + αE εe σE + αB φ(be), (10) with a running estimate q̂ of queueing level and φ(be) =  0, be ≥ B̄,( B̄ − be σB )2 , be < B̄. (11) For any path P that satisfies B(P ) ≥ B̄, one has C(P ) ≤ ∑ e∈P we, which justifies the use of we within shortest-path algorithms. The additive structure in (10) upper-bounds the path score for bandwidth-feasible routes, allowing direct use of label-setting or heuristic search without violating the intended QoS trade-offs. 3.4. A* Heuristic and Weighted Variant Let g(v) denote the cost-to-come to node v, and let h(v) be a conservative lower bound on the cost-to-go from v to t. A* ranks nodes using f(v) = g(v) + h(v), g(s) = 0. (12) We decompose h(v) into admissible component bounds: h(v) = αD d(v, t) σD + αL ℓ(v, t) σL + αJ ȷ(v, t) σJ + αE ε(v, t) σE + αB β(v, t), (13) M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 8 of 29 where d is a latency lower bound, ℓ is an additive loss-surrogate bound, ȷ bounds the queue-variance proxy, ε bounds energy, and β is zero when a B̄-feasible corridor exists and positive otherwise. The weighted variant uses fω(v) = g(v) + ω h(v), ω ≥ 1, (14) to trade additional pruning for a controlled increase in path cost with respect to the scalarized objective. When the components in (13) are admissible (and preferably consistent), A* preserves optimality and expands fewer nodes than unguided search; the weighted form (??) further reduces expansions with a predictable, bounded stretch in score. 3.5. Normalization and Weighting of Real-Time Indicators We form the heuristic as a weighted combination of normalized link indicators: h(n) = α ℓ̃(n) + β λ̃(n) + γ b̃(n) + δ ẽ(n), where ℓ̃ is normalized queueing delay, λ̃ is normalized link loss, b̃ is normalized inverse bandwidth cost, and ẽ is normalized inverse residual energy cost. We use min–max nor- malization per metric m∈{ℓ, λ, b, e} over all candidate edges Et at time t: m̃(e) = m(e)−mine′∈Etm(e′) maxe′∈Etm(e′)−mine′∈Etm(e′) + ε , with ε = 10−9 to avoid division by zero. We constrain α, β, γ, δ ≥ 0 and α+β+γ+δ = 1. Weight selection. Unless otherwise stated, we select (α, β, γ, δ) by grid search on a held- out scenario set using end-to-end latency as the primary objective and control-message overhead as a tiebreaker. Energy sensitivity is calibrated by increasing δ until path-stretch remains within 1% of the δ=0 baseline. This balances QoS and energy, consistent with energy-driven routing principles in dynamic networks [33]. For completeness, we report the chosen weights with 95% CIs in Table ??. 3.6. Optional Multi-Flow Formulation For a set of flows F with demands rf , the binary variable xe,f ∈ {0, 1} indicates whether flow f uses edge e. Flow conservation at node u is ∑ (u,v)∈E x(u,v),f − ∑ (v,u)∈E x(v,u),f =  1, u = sf , −1, u = tf , 0, otherwise. (15) Capacity feasibility is enforced as∑ f∈F rf xe,f ≤ be for all e ∈ E. (16) M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 9 of 29 In our single-flow experiments, (16) reduces to the bottleneck constraint B(P ) ≥ B̄ used in (5)–(6). The multi-flow form extends the single-route objective to shared-capacity settings; the single-flow case studied empirically is its natural special instance. 3.7. Complexity and Control Overhead Measures To connect the formulation to operational cost, we track asymptotic complexity and control-plane proxies used in the evaluation: Time(Bellman–Ford) = O(nm), Time(Dijkstra) = O((n+m) log n), Time(A*) = O((n+m) log n) + expansions, (17) and Routing overhead = control bytes second , Controller CPU = search CPU time decision interval . (18) Informative lower bounds reduce expansions in (17) and are expected to lower the overhead proxies in (18), which is consistent with the measurements reported in Section 6. 3.8. Notation and Quality of Service Summary Tables For completeness, Tables 1 and 2 summarize symbols and feasibility conditions used throughout. Table 1: Notation used in Section 3. Symbol Meaning G = (V,E) Directed network graph e = (u, v) Directed edge from u to v de, qe, λe, be, εe Edge latency, queue term, loss prob., bandwidth, energy/bit D(P ), J(P ), L̂(P ), B(P ), E(P ) Path delay, jitter proxy, additive loss, bottleneck, energy D̄, J̄ , Λ̄, B̄, Ē QoS targets for delay, jitter, loss, bandwidth, energy we Additive edge weight in (10) g(v), h(v), f(v) Cost-to-come, heuristic, evaluation key α·, σ· Weights and normalizers in (6) Table 2: QoS metrics and feasibility conditions. Metric Definition Feasibility condition Latency D(P ) = ∑ de D(P ) ≤ D̄ Jitter proxy J(P ) in (2) J(P ) ≤ J̄ Loss Λ(P ) in (3) Λ(P ) ≤ Λ̄ Bandwidth B(P ) = min be B(P ) ≥ B̄ Energy E(P ) = ∑ εe E(P ) ≤ Ē M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 10 of 29 4. Methodology and Experimental Setup This section describes the algorithms under test, the QoS-aware cost and heuristic de- sign, the evaluated topologies and traffic/failure models, the parameterization used across experiments, and the evaluation protocol. Each subsection begins with a concise setup paragraph and ends with a short interpretation that anticipates the analyses reported in Section 6. 4.1. Algorithms Under Test We compare three classical shortest-path solvers configured with the same QoS-aware edge cost we from (10): A* (admissible and weighted variants), Dijkstra, and Bellman– Ford. All run on the same directed graph and telemetry. A* additionally uses a domain- informed heuristic h(·) as in (13). Table 3: Baseline characteristics of the compared algorithms. Algorithm Negative Weights Time Complex- ity Space Optimality Notes Bellman–Ford Yes (no neg. cycle) O(nm) O(n) Yes Label-correcting; iterative relax- ations Dijkstra (heap) No O((n+m) log n) O(n+m) Yes Label-setting; unguided explo- ration A* (heap) No (we ≥ 0) O((n+m) log n) + expansions O(n+m) Yes (ad- missible h) Guided by h; weighted A* allows bounded suboptimality Algorithm 1: QoS-Aware A* (single flow). M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 11 of 29 Table 4: Algorithm 1: QoS-Aware A*. Input: G = (V,E), source s, target t; SLOs (D̄, J̄ , Λ̄, B̄, Ē); weights α; scales σ. Output: Feasible path P minimizing C(P ) in (6) or Infeasible. 1. Compute we for all e ∈ E using (10). 2. Initialize open set O = {s} with g(s) = 0, f(s) = h(s); closed set C = ∅. 3. While O ̸= ∅: (a) Extract u = argminv∈O f(v); move u to C. If u = t, return the reconstructed P . (b) For each (u, v) ∈ E with v /∈ C, set g′(v) = g(u) + w(u,v). (c) Optionally prune partial paths that already violate soft QoS budgets in (9). (d) If v /∈ O or g′(v) < g(v), set g(v)← g′, f(v)← g(v)+h(v), parent(v)← u, and add v to O. 4. If the loop ends without reaching t, return Infeasible. Algorithm 2: Dijkstra (QoS-weighted costs). Table 5: Algorithm 2: Dijkstra with QoS-weighted edge costs. Input: G = (V,E), s, t, we from (10). Output: Path minimizing ∑ e∈P we or Infeasible. 1. Set g(s) = 0, g(v) =∞ for v ̸= s; priority queue keyed by g(v). 2. While the queue is nonempty: extract u with smallest g(u). If u = t, stop. 3. For each (u, v) ∈ E: if g(u) + w(u,v) < g(v), update g(v) and parent(v). 4. Reconstruct P and verify (5); if violated, return Infeasible. Algorithm 3: Bellman–Ford (QoS-weighted costs). Table 6: Algorithm 3: Bellman–Ford with QoS-weighted edge costs. Input: G = (V,E), s, t, we from (10). Output: Path minimizing ∑ e∈P we or Infeasible. 1. Initialize g(s) = 0, g(v) =∞ for v ̸= s; parent(v) = nil. 2. For i = 1 to n−1: for each edge (u, v), if g(u)+w(u,v) < g(v), set g(v)← g(u)+w(u,v) and update parent. 3. Optionally detect negative cycles with an extra relaxation pass. 4. Reconstruct P and verify (5); if violated, return Infeasible. A* uses h(·) to shrink the frontier relative to Dijkstra, reducing expansions and decision latency. Bellman–Ford can handle negative weights in general, but this is not required here since we ≥ 0 by design, and its O(nm) time is less suitable for larger or dynamic fabrics. M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 12 of 29 4.2. QoS-Aware Edge Weight and Heuristic Design We employ a nonnegative composite cost that captures latency, loss, jitter proxy, bandwidth shortfall, and energy: we = αD de σD + αL − ln(1− λe) σL + αJ (qe − q̂)2 σJ + αE εe σE + αB ϕ(be), (19) where ϕ(be) = 0 if be ≥ B̄ and ϕ(be) = ((B̄ − be)/σB) 2 otherwise. A* ranks nodes by f(v) = g(v) + h(v) with h(v) = αD d(v, t) σD + αL ℓ(v, t) σL + αJ ȷ(v, t) σJ + αE ε(v, t) σE + αB β(v, t), (20) where each underlined term is a lower bound to the remaining cost-to-go as specified in Section 3. The weighted variant uses fω(v) = g(v) + ω h(v) with ω ≥ 1. Table 7: Heuristic components and their intended lower-bound role. Component Lower-bound idea Practical construction d(v, t) Propagation/latency Geodesic or minimal-latency hop bound ℓ(v, t) Loss surrogate Sum of per-link minima − ln(1− λmin) ȷ(v, t) Jitter proxy Minimal queue variance along admissible corridor ε(v, t) Energy Sum of per-link minimal energy β(v, t) Bandwidth penalty 0 if a B̄-feasible corridor exists; else positive Admissible component bounds keep h(·) conservative and consistent, which preserves optimality for A* while enabling substantial pruning in congested or lossy regions. 4.3. Datasets, Topologies, and Traffic/Failure Models We evaluate on heterogeneous topologies (mesh, scale-free, core/edge) and traffic mixes containing short flows, long-lived transfers, and strict-latency flows. Independent link failures are injected with controlled frequencies. Table 8: Topologies and traffic/failure settings. Topology Nodes Avg. degree Link cap. (Mb/s) Prop. delay (ms) Traffic / Failures Mesh-A 100 4.2 1000 1.0 Mixed (mice/elephants); fail- ures 0–0.2/min Mesh-B 400 5.0 1000 1.2 Mixed with bursts; 0–0.3/min ScaleFree 800 3.5 1000 1.5 Heavy-tailed sizes; 0–0.5/min CoreEdge 1600 3.0 1000 2.0 QoS slices; 0–0.5/min Varying size and structure prevents conclusions from hinging on a single graph family and exposes scaling behavior under realistic bursts and failure dynamics. M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 13 of 29 4.4. Parameterization and Implementation Unless noted, scalarization weights and normalizations are fixed across runs; weighted A* varies ω to study speed–quality trade-offs. All algorithms receive the same telemetry snapshot at decision time. Table 9: Default parameter settings. Parameter Value Rationale αD, αL, αJ , αE , αB 1, 1, 0.5, 0.5, 1 Prioritize delay/loss; moderate jitter/energy σD, σL, σJ , σE , σB data-driven Normalize by median per-metric scale Bandwidth target B̄ 200 Mb/s Slice-style minimum throughput Weighted A* ω {1.0, 1.1, 1.2, 1.5, 2.0} Sensitivity of speed vs. path quality Failure frequency {0, 0.05, . . . , 0.5}/min Robustness sweep The configuration reflects common QoS priorities (delay/loss) while allowing jitter and energy to shape path selection without dominating; ω exposes the search/quality trade-off explicitly. 4.5. Complexity and Resource Footprint We report theoretical time/space costs and the operational measures used in Section 6 (control-plane messages per second and controller CPU time per decision interval). Table 10: Asymptotic complexity and expected operational footprint. Algorithm Time (big-O) Space Operational expectation Bellman–Ford O(nm) O(n) Higher compute; larger control churn under dynamics Dijkstra (heap) O((n+m) log n) O(n+m) Moderate compute; frontier grows with congestion A* (heap) O((n+m) log n) + expansions O(n+m) Fewer expansions; faster re-routing with informative h With informative bounds, A* reduces expansions and decision latency; this typically lowers control-plane overhead during bursts and after failures. 4.6. Heuristic Sensitivity and Scalability Setup. This subsection defines how we quantify (i) the effect of the heuristic weight ω in weighted A* and (ii) algorithmic scalability with topology size, without duplicating any figures from Section 5. The weighted key is fω(v) = g(v) + ω h(v) with ω≥ 1 as in (14); ω = 1 recovers admissible A*. All implementations share data structures and stopping rules to ensure a fair comparison. M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 14 of 29 Sensitivity to ω (weighted A*). For ω ∈ {1.0, 1.1, 1.2, 1.5, 2.0}, we record: • Search effort: the number of node expansions Eω (nodes removed from the priority queue) and, for completeness, the number of priority-queue key updates Qω. We also report normalized values Ẽω = Eω/E1.0 and Q̃ω = Qω/Q1.0 to show relative savings against admissible A*. • Path stretch: the ratio of scalarized costs sω = C(Pω) C(P1.0) , s1.0 = 1, (21) where C(·) is given by (6), Pω is the path returned by weight ω, and P1.0 is the admissible baseline. We also track feasibility with respect to (5) to ensure that improvements in effort do not violate QoS. Per-scenario statistics (means across independent runs) are reported in Section 5. The Ẽω curve visualizes effort reduction as ω increases, while sω quantifies the associated cost stretch. Scalability with topology size. We measure single-query wall-clock runtime Talg(|V |) for each algorithm over |V | ∈ {50, 100, 200, 400, 800, 1600}, with edge count |E| determined by the topology family (Table 8). All runs use identical priority queues and graph layouts. Reported values are the mean across repeated trials at each size. For completeness, we also retain the theoretical costs Bellman–Ford: O(nm), Dijkstra (heap): O((n+m) log n), A*: O((n+m) log n) + expansions, (22) to contextualize empirical Talg(·) trends shown in Section 5. The sensitivity metrics (Ẽω, sω) expose the speed–quality trade-off of weighted A*: increasing ω should reduce expansions markedly while keeping sω near 1 for moderate ω. The scalability protocol isolates implementation-agnostic growth by fixing data structures and inputs; empirical runtimes reported later reflect the expected pruning advantage of A* relative to Dijkstra and the higher time growth of Bellman–Ford on larger graphs. 5. Results and Analysis Evaluation protocol . All methods use the same QoS edge weight we and heuristic h(·) defined in Section 3 (see (10) and (13)). We evaluate the scenarios in Table 8: offered load ρ ∈ [0.1, 0.95], background loss ∈ [0, 0.2], failure frequency {0, 0.05, . . . , 0.5}/min, topology size |V | ∈ {50, 100, 200, 400, 800, 1600}, and weighted A* parameter ω ∈ {1.0,1.1,1.2,1.5,2.0} in (14). Unless noted, each point aggregates R = 10 independent runs with deterministic seeds sr = 2025 + r, and we report means (full per-point statistics appear in the released CSVs). Overhead (messages/s) and convergence time follow the definitions in (18). M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 15 of 29 5.1. Latency under Increasing Offered Load We vary the offered load ρ ∈ [0.1, 0.95] and measure mean end-to-end latency with the QoS-weighted objective driven by (10); A* employs the admissible heuristic in (13). 0.2 0.4 0.6 0.8 Offered load (ρ) 0 100 200 300 400 500 M ea n en d- to -e nd la te nc y (m s) Latency vs Offered Load A* Dijkstra Bellman-Ford Figure 1: Latency vs. offered load. A* consistently achieves the lowest latency, with widening gaps at higher load. As ρ→ 1, queueing becomes dominant; A* prunes congested corridors early via h(·), reducing expansions and avoiding high-delay edges relative to Dijkstra. Bellman–Ford exhibits the highest latency due to broader exploration and slower adaptation. 5.2. Jitter Sensitivity to Load Using the jitter proxy J(P ) in (2), we track variability in queueing delay as ρ increases under the same we and heuristic design. M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 16 of 29 0.2 0.4 0.6 0.8 Offered load (ρ) 0 5 10 15 20 25 30 35 Jit te r ( m s v ar ia nc e pr ox y) Jitter vs Offered Load A* Dijkstra Bellman-Ford Figure 2: Jitter proxy J(P ) (queue-variance proxy) vs. offered load. A* maintains lower variance across the operating range. By prioritizing links with stable queues, A* reduces jitter variability. The gap widens beyond ρ ≈ 0.7, where queue instabilities emerge and heuristic guidance is most beneficial. 5.3. Delay Distribution at High Load For ρ = 0.9, we plot the empirical CDF of end-to-end delay across many flows to assess distributional effects. 0 50 100 150 200 250 End-to-end delay (ms) 0.0 0.2 0.4 0.6 0.8 1.0 CD F Delay Distribution at High Load (ρ=0.9) A* Dijkstra Bellman-Ford Figure 3: Delay CDF at ρ = 0.9. A* shifts the distribution left and tightens the tail. A* improves both median and tail latency, evidencing sharper avoidance of lossy/congested paths. Dijkstra is competitive at the median but suffers heavier tails; Bellman–Ford M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 17 of 29 trails overall. 5.4. Convergence under Failure Dynamics We vary link-failure frequency (events/min) and measure time to converge to a new stable route after a failure, following the operational proxy in (18). 0.0 0.1 0.2 0.3 0.4 0.5 Failure frequency (events/min) 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 Co nv er ge nc e tim e (s ) Convergence vs Failure Frequency A* Dijkstra Bellman-Ford Figure 4: Convergence time vs. failure frequency. A* adapts faster across disturbance rates. Heuristic guidance narrows the re-routing scope after failures, yielding consistently faster convergence for A*. Dijkstra remains moderate; Bellman–Ford is most affected by frequent disturbances. 5.5. Control-Plane Overhead over Time We track controller signaling (messages/s) during two bursty traffic episodes under the common we configuration. M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 18 of 29 0 20 40 60 80 100 120 140 Time (s) 2 4 6 8 10 12 Co nt ro l o ve rh ea d (m sg s/ s) Control-Plane Overhead Over Time A* Dijkstra Bellman-Ford Figure 5: Control-plane overhead as traffic fluctuates. A* reduces signaling compared to Dijkstra and Bellman–Ford, reflecting fewer expansions and more stable decisions during bursts. 5.6. Heuristic Weight Sensitivity (Weighted A*) We vary the weight ω in (14) to quantify the speed–quality trade-off for weighted A*. 1.0 1.2 1.4 1.6 1.8 2.0 Weighted A* parameter (ω) 40000 50000 60000 70000 80000 90000 100000 No de e *p an sio ns (r e at i) e sc a e) Search E*pansions )s He(ristic Weight A* (e*pansions) Dijkstra base ine Figure 6: Search expansions vs. heuristic weight ω. M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 19 of 29 1.0 1.2 1.4 1.6 1.8 2.0 Weighted A* parame er (ω) 1.00 1.02 1.04 1.06 1.08 1.10 Pa h s re c h (D (P )/D (P _s p) ) Pa h Quali y vs Heuris ic Weigh A* pa h s re ch Figure 7: Path stretch vs. heuristic weight ω. Increasing ω substantially decreases expansions (speed gain) with a mild rise in path stretch. In practice, ω ∈ [1.1, 1.5] offers a favorable balance for time-critical flows. 5.7. Energy Efficiency vs. Path Length We plot energy per delivered bit versus hop count across algorithms to assess energy- sensitive routing behavior. 2 4 6 8 10 12 14 Path length (hops) 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 En er gy p er d el iv er ed b it (m J/b it) Energy vs Path Length A* Dijkstra Bellman-Ford Figure 8: Energy per delivered bit vs. path length (hops). A* favors corridors that avoid retransmissions and high-queue edges, lowering energy per bit, especially for longer paths where retransmission and queue penalties accumulate. M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 20 of 29 5.8. Scalability with Topology Size We vary |V | ∈ {50, 100, 200, 400, 800, 1600} and report single-query runtime (ms) under the shared implementation and data structures. 0 200 400 600 800 1000 1200 1400 1600 Topology size (nodes) 0 200 400 600 800 1000 1200 1400 1600 Ru nt im e (m s) Scalability: Runtime vs Topology Size A* Dijkstra Bellman-Ford Figure 9: Runtime vs. topology size. A* scales more gracefully than Dijkstra as the heuristic reduces the effective frontier; Bellman–Ford exhibits quadratic growth, limiting applicability on large fabrics. 5.9. Reliability vs. Background Loss We sweep background loss probability and report packet delivery ratio (PDR= 1−Λ(P ) with Λ(P ) defined in (3)). 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 Background loss probability 0.75 0.80 0.85 0.90 0.95 1.00 Pa ck et d el iv er y ra tio (P DR ) Reliability vs Loss Environment A* Dijkstra Bellman-Ford Figure 10: PDR vs. background loss probability. M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 21 of 29 A* routes away from lossy corridors earlier, sustaining higher PDR across the range; the margin widens as loss intensifies. 5.10. Numerical Summaries To complement the curves in Figs. 1–10, we compile four compact tables that aggregate the most relevant outcomes at representative operating points: (i) latency and jitter at medium/high load, (ii) convergence time versus failure frequency, (iii) single-query runtime versus topology size, and (iv) weighted-A* sensitivity to ω. Unless noted otherwise, values are means over R = 10 independent runs under the protocol stated at the beginning of this section; units and scenario grid points match those used in the figures. These summaries enable side-by-side comparison of A*, Dijkstra, and Bellman–Ford and provide concise references for discussion in the text. Table 11: Summary at medium/high load (selected points from Figs. 1 and 2). Scenario Metric A* Dijkstra Bellman–Ford Medium load medium load latency ms 40.29 53.26 67.24 medium load jitter proxy 2.483 3.002 4.087 High load high load latency ms 142.39 176.22 227.82 high load jitter proxy 9.915 12.258 14.605 Table 12: Convergence time (s) vs. failure frequency (events/min). Failure freq. (events/min) 0.00 0.05 0.10 0.15 0.20 0.25 A* 0.487 0.579 0.588 0.717 0.811 0.812 Dijkstra 0.697 0.841 0.922 1.058 1.149 1.187 Bellman–Ford 0.987 1.203 1.517 1.658 1.664 1.968 Failure freq. (events/min) 0.30 0.35 0.40 0.45 0.50 A* 1.007 1.053 1.129 1.159 1.162 Dijkstra 1.290 1.432 1.679 1.740 1.830 Bellman–Ford 2.023 2.314 2.340 2.446 2.342 M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 22 of 29 Table 13: Scalability: runtime (ms) vs. topology size (nodes). Nodes 50 100 200 400 800 1600 A* (ms) 6.411 8.322 12.644 22.288 43.575 90.151 Dijkstra (ms) 10.258 13.315 20.230 35.660 69.721 144.241 Bellman–Ford (ms) 11.500 16.000 34.000 106.000 394.000 1546.000 Table 14: Weighted A-Star: expansions (relative to Dijkstra) and path stretch vs. ω. ω Metric 1.0 1.1 1.2 1.5 2.0 A* expansions (relative) 1.000 0.883 0.789 0.590 0.406 Dijkstra expansions (relative) 1.000 1.000 1.000 1.000 1.000 A* path stretch (ratio) 1.000 1.010 1.020 1.050 1.100 Legend: Expansions are normalized by Dijkstra’s expansions at the same ω and scenario. Path stretch is length relative to the shortest path. The tabulated results reinforce the figure-level trends: A* reduces latency and jitter at medium/high load, converges faster after failures, scales more favorably with |V |, and offers a tunable speed–quality trade-off via ω with small cost stretch; Dijkstra remains competitive but less selective under dynamics, while Bellman–Ford is constrained by higher computational overhead. A cross all scenarios, A* satisfies QoS targets with lower overhead and faster conver- gence, while Dijkstra remains competitive but less selective under dynamics. Bellman– Ford is limited primarily by its higher computational cost in large or time-sensitive net- works. Heuristic informativeness vs. computational cost. We observed that more in- formative (tightly admissible or slightly inflated) heuristics reduce node expansions and convergence time but incur extra per-step evaluation due to indicator updates. In our experiments, this cost was outweighed by fewer expansions for moderate inflation factors, yielding net latency gains. Similar reliability–efficiency trade-offs have been reported in mobile ad hoc settings where routing decisions blend timeliness with robustness [34]. 6. Discussion This section synthesizes the empirical findings of Section 5 into practical guidance for deployment, clarifies key trade-offs exposed by the methodology in Section 4, and, in separate subsections, states limitations and outlines extensions and future work. Each subsection opens with a brief setup and closes with an interpretation that distills actionable takeaways. M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 23 of 29 6.1. Practical Implications for Fifth- and Sixth-Generation Networks, Software-Defined Networking, and the Internet of Things / Wire- less Sensor Networks Deployments The Quality of Service–aware edge cost in (10) and heuristic composition in (13) were evaluated under heterogeneous topologies and dynamics (Table 8). Operators of software- defined fabrics, fifth-/sixth-generation transport, and Internet of Things / wireless sensor network backhauls need concrete guidance on parameterization, controller integration, and operational guardrails. Controller integration and telemetry. A central controller can compute routes us- ing the same we and h(·) for all tenants, provided that telemetry for de, qe, λe, be, εe is refreshed at a cadence commensurate with the decision interval. Smoothing short-lived fluctuations (e.g., with an exponential average for qe) stabilizes decisions without masking real congestion. Slicing and service-level objective heterogeneity. Different service classes map to different (D̄, J̄ , Λ̄, B̄, Ē) and weights (α·). Latency-critical slices typically emphasize αD and αL, while energy-constrained Internet of Things flows increase αE . The bandwidth penalty ϕ(·) enforces minimum throughput while remaining compatible with shortest-path search. Failure handling and re-routing. The convergence improvements observed for A* under failures (Fig. 4) suggest restricting re-optimization to affected regions when feasible. Warm-starting the open set with nodes near the failure cut can further reduce expansions. Selecting weighted A*. Weighted A* with ω ∈ [1.1, 1.5] (Fig. 6 and Fig. 7) provides a speed gain with small stretch. In practice, ω can be adapted to the network state: lower during quiet periods; higher during bursts or failure storms to prioritize responsiveness. The formulation aligns with controller-based routing and multi-slice operation. Infor- mative heuristics reduce control-plane work and re-routing time while preserving Quality of Service. A modest ω enables responsive behavior without materially degrading path quality. 6.2. Trade-offs and Ablations The composite cost and heuristic introduce tunable choices that affect latency, relia- bility, energy, and computational effort. We discuss key trade-offs and the main factors that influence performance. Speed vs. path quality. Increasing ω reduces expansions markedly (Fig. 6) at the expense of mild stretch (Fig. 7). For strict-optimality settings, ω = 1 is appropriate; for time-critical control loops, a slightly larger ω accelerates decisions. M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 24 of 29 Loss sensitivity and stability. A higher αL steers traffic away from lossy links, im- proving PDR (Fig. 10) but potentially lengthening routes if loss-free corridors are longer. Excessive sensitivity can oscillate routes when loss jitter is transient; smoothing loss esti- mates mitigates this. Energy vs. latency. Weighting αE improves energy per bit (Fig. 8) but may increase hop count. Balanced settings keep energy competitive without eroding latency targets. Ablation considerations. Removing the bandwidth penalty ϕ(·) increases the risk of bottleneck violations in mixed-traffic regimes; omitting the queue term weakens jitter proxy control (Fig. 2). Heuristic components that are less informative (e.g., loose bounds) still preserve correctness but reduce pruning benefits. The observed gains are robust across sizes and dynamics, but the exact margins depend on workload and telemetry quality. Conservative heuristics and moderate weights provide a safe operating point across diverse conditions. 6.3. Limitations We summarize the primary limitations and threats to validity of our study to clarify scope and generality. Topology representativeness. While varied, the evaluated graphs may not capture all production fabrics (e.g., extreme scale-free cores or highly clustered edge meshes). Traffic realism. The workload mix approximates common patterns but cannot span every distribution; burstiness and long-range dependence may shift margins. Telemetry noise and delay. Measurement error and staleness can perturb decisions; conservative bounds in h(·) help maintain stability, but rapidly changing states remain challenging. Implementation parity. Identical data structures and termination criteria were used across baselines; different system stacks or controller placements may shift absolute run- times, though relative trends generally persist. These limitations bound external validity. We expect qualitative conclusions to hold under broader conditions, but margins should be recalibrated with in-situ telemetry and topology features. 6.4. Extensions and Future Work We outline opportunities to extend the approach beyond the present scope. M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 25 of 29 State-aware inflation scheduling. Adapt ω to congestion/failure regimes using simple controllers (e.g., hysteresis) to retain stability while improving responsiveness. Adaptive weight learning. Learn (α·) online from slice-level service-level objectives and feedback, with safeguards to preserve admissibility where required. Controller integration. Tighter integration with software-defined controllers (e.g., in- cremental path recomputation and warm starts) can further reduce control-plane load. Generalized heuristics. Investigate multi-criteria formulations that incorporate richer uncertainty models for indicators, while keeping heuristic evaluation cost bounded. Prioritizing adaptive inflation and weight learning offers the highest return with modest engineering complexity; deeper theoretical generalizations can follow as a second phase. 6.5. Guidelines for Choosing Heuristics and Weights We provide a step-by-step recipe to instantiate the cost and heuristic in networks with different Quality of Service priorities and measurement characteristics. Normalization and baselining. Calibrate σ· from recent medians or percentiles to render terms roughly comparable. Revisit normalizers when link speeds or delay baselines change. Weight selection. Start with (αD, αL, αJ , αE , αB) = (1, 1, 0.5, 0.5, 1) and adjust to- ward your service-level objectives: increase αD for latency-critical slices; increase αL for reliability-critical paths; increase αE for battery-constrained flows; keep αB sufficient to guard minimum throughput. Admissible bounds for h(·). Construct d, ℓ, ȷ, ε, β as conservative, efficiently com- putable lower bounds. Prefer bounds that are piecewise-constant or monotone in network state to avoid instability. Maintain consistency (triangle-like) when possible to limit re- expansions. Choosing ω. Begin with ω = 1; increase gradually (e.g., 1.1 or 1.2) if decision latency is a bottleneck. Cap ω where the observed stretch (Fig. 7) remains acceptable for the slice. Operational safeguards. Use short moving averages for qe and λe to reduce jitter- induced route churn. Rate-limit re-optimizations during large failure events, and warm- start A* from the previous solution neighborhood. A small set of principled choices—balanced normalization, conservative bounds, and modest ω—yields predictable Quality of Service and efficient search. These settings trans- fer well across fabrics, requiring only light recalibration as operating conditions evolve. M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7091 26 of 29 7. Conclusion This work studied QoS-aware routing through a composite edge cost and heuristic- guided search, comparing A* (including its weighted variant) against Dijkstra and Bellman– Ford across load, loss, failure dynamics, and scale. Using a single cost model and shared telemetry, A* consistently achieved lower latency and jitter, improved tail delay, faster post-failure convergence, and reduced control-plane signaling. The method scaled well with topology size by shrinking the effective search frontier, while Dijkstra remained com- petitive but less selective under dynamics and Bellman–Ford incurred higher computa- tional cost. Weighted A* exposed a practical speed–quality knob: moderate values of ω provided substantial reductions in expansions with small stretch in path cost. Together with the released figures and data tables, these findings offer a clear and reproducible picture of how heuristic guidance can meet QoS targets with efficient controller behavior. Future work will extend the single-flow focus to joint multi-flow optimization with explicit capacity coupling, admission control, and fairness, and will study uncertainty- aware costs that account for noisy or stale telemetry. A promising direction is to learn or adapt heuristics online from measurements while preserving safety via admissible lower- bound surrogates and incremental consistency checks. Additional experiments on non- terrestrial, mobile, and highly time-varying networks can probe robustness under harsher dynamics, and tighter integration with SDN controllers (e.g., warm starts, locality-aware re-optimization) may further reduce convergence time. Finally, hardware acceleration paths and larger-scale testbeds will help translate these algorithmic gains into deployment guidance for 5G/6G transport and IoT backhauls, while expanding the artifact bundle will support broader replication and secondary analysis. Acknowledgements The author extends his appreciation to the Deanship of Scientific Research at Northern Border University, Arar, KSA for funding this research work through the project number ”NBU-FFR-2025-2443-09”. References [1] 3GPP. System architecture for the 5g system (5gs). 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