TSDLMRA: an efficient multicast routing algorithm based on Tabu search

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1 Journal of Network and Computer Applications 27 (2004) TSDLMRA: an efficient multicast routing algorithm based on Tabu search Heng Wang*, Jie Fang, Hua Wang, Ya-Min Sun Department of Computer Science and Technology, Nanjing University of Science and Technology, 608 Staff Room, Nanjing , China Received 17 June 2003; received in revised form 24 September 2003; accepted 2 October 2003 Abstract As a NP-Complete problem, multicast routing with delay constraint is a research difficulty in routing problem. Tabu Search is artificial intelligence algorithm, which is an extension of local search algorithm and has simple realization and well properties. In this paper, an efficient algorithm based on Tabu Search for Delay-Constrained Low-Cost Multicast Routing is proposed to solve delay-constrained multicast routing problem. This problem is known to be NP-Complete. The proposed heuristic algorithm makes use of the characteristics of flexible memory function and tabu rule in TS algorithm, generates neighborhood structure base on paths-switching operations, and finds multicast tree satisfying constraint. A large number of simulations demonstrate that the algorithm performs excellent performance of cost, rapid convergence and stable performance of delay. q 2003 Elsevier Ltd. All rights reserved. Keywords: Multicast routing; Tabu search; Delay constrained; Constrained Steiner tree; NP-Complete 1. Introduction Distributed real-time applications, such as audio- and video-conferencing, collaborative environments and distributed interactive simulation, by and large, involve a source sending messages to a selected set of destinations with varying Quality-of-Service (QoS) delivery constraints. This requires the underlying network to provide multicasting and QoS capabilities to efficiently support these applications. At the routing level, these * Corresponding author. Tel.: þ ; fax: þ addresses: puchengew@yahoo.com.cn (H. Wang), fangjie80@yahoo.com.cn (J. Fang), wanghua_nust@yahoo.com (H. Wang), sunym@mail.njust.edu.cn (Y.M. Sun) /$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi: /j.jnca

2 78 H. Wang et al. / Journal of Network and Computer Applications 27 (2004) requirements are translated into the problem of determining a multicast tree, usually rooted at the source node and spanning the set of receiver (destination) nodes. The QoS constraints typically impose a restriction on the acceptable multicast trees. With the advent of the real-time interactive applications, minimizing delay of the multicast tree is also an important objective along with minimizing cost. Therefore, both cost optimization and delay optimization goals are important for the multicast routing tree construction. The problem of minim izing tree cost under the constrained that all path delays are within a user-specified delay bound is referred to as the delay constraint multicast routing problem. This problem is formalized into constrained Steiner tree problem, which is known to be NP-Complete. We briefly discuss some recently proposed delay-constrained Steiner tree heuristics. KPP (Kompella et al., 1993) is the first heuristic algorithm for delay-constrained Steiner tree problem. KPP extends KMB (Kou et al., 1981), an unconstrained Steiner tree heuristic, to compute delay-bounded paths assuming the delay bound and link delays are all integers. Then KPP converts the original network graph into a closure graph connecting multicast group members. KPP uses Prim s algorithm (Cormen et al., 2001) to obtain a minimum spanning tree of the closure graph. Finally, KPP replaces the edges in the minimum spanning tree with paths in the original graph. The time complexity of KPP is OðDlVl 3 Þ: Zhu et al. (1995) proposed another delay constrained Steiner tree heuristic algorithm, called BSMA. The algorithm starts by computing a least delay tree (LDT) rooted at a given source and spanning all the group members. It then iteratively replaces superedges in the tree with cheaper superedges not in the tree while not violating the delay constraint, until the total cost of the tree cannot be further reduced. Cheaper superedges are located by using a Kth shortest path algorithm. The time complexity for BSMA is OðKlVl 3 log lvlþ; where K may be very large in case of large, densely connected networks, and it may be difficult to achieve acceptable running times (Salama et al., 1997). As an extension of local neighborhood search, Tabu Search (TS) has been successful in a variety of problem settings like scheduling, transportation, layout and circuit design, and graphs (Wang, 2001). Procedures based on TS explores the search space by moving from a solution to its best neighbor (the one with the best objective value among all examined candidates), even if this results in a deterioration of the objective function value, in order to increase the likelihood of escaping from a poor local optimum. In this paper, we propose an efficient algorithm based on TS for generating a low-cost multicast tree subject to delay constraints. We call this algorithm Tabu Search for Delay- Constrained Low-Cost Multicast Routing (TSDLMRA). The algorithm starts with an initial shortest path tree constructed by using Zhang and Mouftah s DDSP algorithm (Zhang and Mouftah, 2002). Then algorithm constructs a backup-paths-set for each destination using Kth shortest path algorithm, and generates neighborhood structure by paths-switching operation. For those trees that violate delay constraint, we assign an extra penalty by increasing their cost, so that final solutions are feasible. Simulation results show our algorithm has features of well performance of cost, fast convergence and stable delay. TSDLMRA belongs to source-based routing algorithm, because it assumes that sufficient global information is available to the source. The rest of the paper is organized as follows. In Section 2, delay-constrained low-cost multicast routing problem is formulated. Section 3 introduces briefly TS method.

3 H. Wang et al. / Journal of Network and Computer Applications 27 (2004) The proposed algorithm is described in Section 4. The complexity analysis of TSDLMRA is introduced in Section 5. Simulation results and comparison with other reported heuristics are presented in Section 6. Section 7 concludes the paper. 2. The delay-constrained multicast routing problem The communication network is modeled as a undirected graph G ¼ðV; EÞ; where V denotes the set of nodes and E is the set of edges (links) representing physical connectivity between nodes. Any link e [ E has a cost cðeþ : E! R þ and a delay dðeþ : E! R þ associated with it. The function cð Þ represents the utilization of the link and the function dð Þ represents the delay that the packet experiences on link including queuing, transmission, and propagation delay. Let s [ V be a source node and M # V 2 {s} be the set of destination nodes, called the multicast group. A multicast tree TðT # GÞ is a tree rooted at s and spanning all members of M: Assume that pðu; vþ represents the path from u to v, then there exist following relationships in multicast tree T : delayðpðs; mþþ ¼ X e[pðs;mþ dðeþ; m [ M ð1þ costðtþ ¼ X cðeþ e[t ð2þ delayðtþ ¼max ðdelayðpðs; mþþþ ð3þ m[m Given these definitions, we can formally present the delay-constrained multicast routing problem as follows: Definition 1. Delay-constrained low-cost multicast routing problem. Given a network G ¼ðV; EÞ; a source node s; destination node set M; a link delay function dð Þ; a link cost function cð Þ; and a positive delay constraint D; the objective of the delay-constrained lowcost multicast routing problem is to construct a multicast tree TðV T ; E T Þ such that the delay constraint is satisfied, i.e. delayðtþ # D ð4þ and that the tree cost costðtþ is minimized. 3. Tabu search method TS was introduced by Glover as a general iterative metaheuristic for solving combinatorial optimization problems (Wang, 2001). TS is conceptually simple and elegant. It is a form of local neighborhood search. Each solution x [ V has an associated set of neighbors NðxÞ; where V is the set of feasible solutions. A solution x 0 [ NðxÞ can be reached from x by an operation called a move to x 0 : TS moves from a solution to its best

4 80 H. Wang et al. / Journal of Network and Computer Applications 27 (2004) admissible neighbor, even if this causes the objective function to deteriorate. To avoid cycling in the course of the search, the reverses of the last certain number of moves, formed as a tabu list, are forbidden or declared as tabu restricted for certain number of iterations. To prevent a too strict setting the tabu restriction, aspiration criteria are usually introduced to override the tabu restriction and thereby to lead the search to a promising region of the solution space. Intensification and diversification strategies are used to improve the search. The general TS algorithm is given as follows. Step 1. Start with an initial solution x now ; and initialize tabu list H ¼ B; Step 2. If satisfy termination rule, then procedure stop; otherwise, generate candidate solution set Can_Nðx now Þ satisfied tabu restriction in x now s neighbors Nðx now Þ; find a best solution x next in Can_Nðx now Þ; x now U x next ; update tabu list H; repeat Step Delay-constrained low-cost multicast routing algorithm based on tabu search 4.1. Encoding and initial solution In TSDLMRA, a solution is encoded as an array of lml elements, where each element is a path from source s to destination node m [ M in multicast tree T; i.e. x ¼ðp 1 ; p 2 ; ; p k Þ; where k ¼ lml; p i ¼ pðs; m i Þ; m i [ M; 1 # i # k: The initial solution T 0 is a shortest path tree, which is constructed by using DDSP algorithm (Zhang and Mouftah, 2002). DDSP algorithm aims to construct a low-cost SPT by considering link sharing between different destinations, and can be seen as an extension of Dijkstra s shortest path algorithm (Cormen et al., 2001). DDSP algorithm integrates the destination-driven characteristics of Shaikh and Shin s DDMC algorithm (Shaikh and Shin, 1997) and the shortest path characteristics of SPT, so it outperforms SPT in terms of cost performance. A best soulution so far T best is memorized during the whole procedure. If T 0 satisfies the delay constraint, then initial T best is T 0 ; otherwise T best is a LDT rooted at source by using Dijkstra algorithm. If a LDT s delay violates the delay constraint, then algorithm returns. For ease of presentation, we consider an example network as shown in Fig. 1(a), where s ¼ {A} and M ¼ {B; C; D; E}; the numbers in the parentheses along each edge represent the cost and delay for that edge. Given delay constraint D ¼ 7: The initial solution T 0 and the initial best solution T best are shown, respectively, in Fig. 1(b) and (c). T best is a LDT because of its violating delay constraint. The encoding for T 0 and T best are given in Table Evaluation function The evaluation function is used for evaluating state of search, in our algorithm, we take objective function as evaluation function, i.e. f ðxþ ¼costðxÞ:

5 H. Wang et al. / Journal of Network and Computer Applications 27 (2004) Fig. 1. Example network, initial solution T 0 and initial best solution T best :

6 82 H. Wang et al. / Journal of Network and Computer Applications 27 (2004) Table 1 The encoding for T 0 and T best Delay Cost T 0 A; F; D; B A; F; C A; F; D A; F; D; B; E 8 5 T best A; H; E; B A; F; D; C A; F; D A; H; E Backup-paths-set For each destination node m [ M; we compute least-cost paths from s to m by using Kth shortest path algorithm to construct a backup-paths-set for generating neighbors. Let P i be paths set for destination node i; then P i ¼ {p 1 i ; ; p j i ; ; pk i } ð5þ where p j i is the jth path for destination node i: Table 2 shows backup-paths-set from source A to destination nodes in example network Neighborhood structure For generating neighbors, we choose a neighborhood structure based on pathsswitching operations. We randomly select a destination node m [ M; and delete the path from source s to m in current solution T now ; then generate different neighbor solutions by selecting other paths from s to m in backup-paths-set. Each iteration begins by generating neighbor solutions NðT now Þ corresponding to T now ; and the size of neighbor set is dynamic in our algorithm. For T best in Fig. 1(c), we assume that destination node B is selected and path from A to B is deleted, then we select other paths from Table 2 to construct new neighbors. The two neighbors of T best are shown in Fig. 2 and are encoded in Table 3. Among the neighbors, the one with the best cost is selected, and considered as new current solution for the next iteration. The paths-switching fails to consider delay constraint, so it might happen the some of the trees violate delay constraint. In that case, we assign an extra penalty by increasing its cost, so that it is less likely to be accepted in the candidate list as shown in the tree of Fig. 2(b). Obviously, the new multicast tree of Fig. 2(a) is selected for the next iteration and considered as new current solution. Table 2 The backup-paths-set of example network, K ¼ 5 Destination B A; F; D; B A; F; C; B A; H; G; B A; F; C; E; B A; F; C; D; B C A; F; C A; F; D; C A; F; D; B; C A; F; D; B; E; C A; H; E; C D A; F; D A; F; C; D A; F; C; B; D A; F; C; E; B; D A; H; G; B; D E A; F; C; E A; F; D; B; E A; F; C; B; E A; H; E A; F; D; C; E

7 H. Wang et al. / Journal of Network and Computer Applications 27 (2004) Fig. 2. Two neighbors from T best : 4.5. Tabu list and tabu length To avoid local minimum, a tabu list is constructed where forbidden moves are listed. This list contains information that to some extent forbids the search from returning to a previously visited solution. In our algorithm, a multicast tree is considered as an element of tabu list, and the tabu length is set to Aspiration criterion While central to TS, tabus are sometimes too powerful: they may prohibit attractive moves, even when there is no danger of cycling, or they may lead to an overall stagnation of the searching process. It is thus necessary to use algorithmic devices that will allow one to revoke (cancel) tabus. These are called aspiration criteria. The simplest and most commonly used aspiration criterion consists in allowing a move, even if it is tabu, if it results in a solution with an objective value better than that of the current best-known solution (since the new solution has obviously not been previously visited). So, if the cost of a tabu candidate solution is better than T best s, then it is considered as new current solution, and T best is updated at once Termination rule We use a fixed iteration number as stopping criterion, and maximum iteration number is 100. Table 3 The encoding for neighbors of T best Delay Cost (a) A; F; D; B A; F; D; C A; F; D A; H; E 6 10 (b) A; H; G; B A; F; D; C A; F; D A; H; E 8 12 þ penalty

8 84 H. Wang et al. / Journal of Network and Computer Applications 27 (2004) Fig. 3. The pseudo code for TSDLMRA. The pseudo code of our TSDLMRA algorithm is given in Fig. 3. Steps 1 11 show the initialization phase. Steps show the iteration phase. 5. The complexity analysis Theorem 1. The time complexity of TSDLMRA is OðKmn 3 Þ; where m is group size and n is network size.

9 H. Wang et al. / Journal of Network and Computer Applications 27 (2004) Proof. In the initialization phase, the time complexity of generating LDT by using Dijkstra and initial solution by using DDSP are both Oðn 2 Þ: The time complexity of constructing backup-paths-set by using Kth shortest path algorithm is OðKmn 3 Þ: One iteration costs OðKÞ; thus, for Q iterations, the cost becomes OðQKÞ: So the worst time complexity of our algorithm is Oðn 2 þ n 2 þ Kmn 3 þ QKÞ: The term QK is usually much smaller than Kmn 3 ; so the time complexity of TSDLMRA is OðKmn 3 Þ: A 6. Simulation results To evaluate the efficiency of TSDLMRA algorithm, we use the random link generator developed by Salama (1996), which yields networks with an average node degree of 4 6. The positions of the nodes are fixed in a rectangle of size km 2, and the capacity of each link is 155 Mbps. The Euclidean metric is then used to determine the distance between each pair of nodes. Edges are introduced between pairs of nodes u; v with a probability that depends on the distance between them. The edge probability is given by Pðu; vþ ¼ b expð2dðu; vþ=alþ; where dðu; vþ is the distance from node u to v; L is the maximum distance between two nodes. a and b are parameters, and are set to 0.15 and 2.2, respectively. Larger values of b result in graphs with higher edge densities, while small values of a increase the density of short edges relative to longer ones. The link delay function dðeþ is defined as the propagation delay of the link, and queuing and transmission delays are negligible. The propagation speed through the links is taken to be two third the speed of light. The link cost function cðeþ is defined as the current total bandwidth reserved on the link, which is random variable uniformly distributed between 10 and 120 Mbps. In the first set of experiments, TSDLMRA is compared with KPP, BSMA and LD for cost performance. Fig. 4 shows the tree cost for varying network size with the group size ¼ 5; average node degree of network ¼ 4; and D ¼ 40 and 60 ms, respectively. It can be seen from Fig. 4 that our algorithm has a better cost performance than the compared algorithms. Fig. 5 shows the cost performance measures versus group size for a 50-node network and node degree of 4, delay constraint of 40 and 60 ms. As the group size increases, the tree cost of all algorithms grow. Fig. 6 shows the performance of different heuristics for varying delay bound, for a 50-node network and a group size of 5. In general, TSDLMRA has best cost performance among all algorithms. In the second set of experiments, we observe the delay of tree that comes from applying for TSDLMRA and LD algorithm versus varying network size for a group size of 5 nodes, with node degree of 4. We can see from Fig. 7 that the delay of tree generated using by our algorithm always maintains at 24 ms as network size increases whether delay constraint is so stringent or not. In the third set of experiments, TSDLMRA is compared with other two algorithms (Wang and Wang, 2002; Shi et al., 2000) based on genetic algorithms for convergence performance. We call the algorithm in (Wang and Wang, 2002) WGA, and the algorithm in (Shi et al., 2000) SGA. Fig. 8 shows the best solutions for example

10 86 H. Wang et al. / Journal of Network and Computer Applications 27 (2004) Fig. 4. Tree cost versus network size for group size ¼ 5; node degree ¼ 4: network in Fig. 1(a) when delay constraint is 6 and 7, respectively, and their cost is 7 and 6, respectively. Fig. 9 shows the tree cost for varying iteration number. In Fig. 9(a), WGA and SGA s iteration number is 89 and 107, respectively, for getting best solution when D ¼ 6; and our algorithm converges just within 10 steps. Fig. 9(b) shows that WGA and SGA s iteration number is 220 and 405, respectively, for D ¼ 7; and TSDLMRA converges still within 10 steps. Fig. 5. Tree cost versus group size for network size ¼ 50; node degree ¼ 4:

11 H. Wang et al. / Journal of Network and Computer Applications 27 (2004) Fig. 6. Tree cost versus delay constraint for network size ¼ 50; group size ¼ 5: Finally, we track the cost of the best solution over time. Fig. 10 shows cost of best solution found by TSDLMRA versus iteration number for 90-, 60- and 30-node networks with group size ¼ 5; D ¼ 60 ms and node degree ¼ 4: As is clear, the algorithm converges quickly, which satisfies the real-time requirement of multimedia network. Fig. 7. Tree delay versus network size for group size ¼ 5; node degree ¼ 4:

12 88 H. Wang et al. / Journal of Network and Computer Applications 27 (2004) Fig. 8. The best solution for example network when delay constraint is 6 and 7, respectively. Fig. 9. Tree cost versus iteration number for example network. Fig. 10. Cost of best solution found by TSDLMRA versus iteration number for three networks. Group size ¼ 5; delay constraint D ¼ 60 ms; node degree ¼ 4:

13 7. Conclusion H. Wang et al. / Journal of Network and Computer Applications 27 (2004) In this paper, we presented an efficient algorithm based on TS for obtaining delayconstrained low-cost multicast trees. The proposed TSDLMRA algorithm has following features: (1) the time complexity is low, i.e. OðKmn 3 Þ; (2) it was always able to find a multicast tree if one exists; (3) simulation results show that TSDLMRA performs excellent performance of cost, rapid convergence and stability of delay. References Cormen TH, Leiserson CE, Riverst RL, Stein C. Introduction to algorithms, 2nd ed. Cambridge, MA/New York: MIT Press/McGraw-Hill; Kompella VP, Pasquale JC, Polyzos GC. Multicasting routing for multimedia communication. IEEE/ACM Trans Network 1993;1(3): Kou L, Markowsky G, Berman L. A fast algorithm for Steiner trees in graphs. Acta Informatica 1981;15(2): Salama HF. Multicast routing for real-time communication on high-speed networks. PhD Thesis. North Carolina State University, Department of Electrical and Computer Engineering; Salama HF, Reeves DS, Viniotis Y. Evaluation of multicast routing algorithms for real-time communication on high-speed networks. IEEE J Sel Areas Commun 1997;15(3): Shaikh A, Shin KG. Destination-driven routing for low-cost multicast. IEEE J Sel Areas Commun 1997;15(3): Shi J, Zou L, Dong T-L. The application of genetic algorithm in multicast routing. Acta Electron Sin 2000;28(5): Wang L. Intelligent optimization algorithms with applications. Beijing/Heidelberg: Tsinghua University Press/ Springer; Wang X-H, Wang G-X. A multicast routing approach with delay-constrained minimum-cost based on genetic algorithm. J China Inst Commun 2002;23(3): Zhang B, Mouftah HT. A destination-driven shortest path tree algorithm. Proceedings of the IEEE International Conference on Communications 2002 (ICC 2002), New York, vol. 4. IEEE Communication Society; p Zhu Q, Parsa M, Garcia-Luna-Aceves JJ. A source-based algorithm for delay-constrained mimimum-cost multicasting. Proceedings of IEEE INFOCOM 95, Boston, MA 1995; Heng Wang is currently a PhD candidate at Nanjing University of Science and Technology. He received his BE from Department of Computer Science and Engineering, Nanjing University of Science and Technology, China in His research interests include QoS routing algorithm and multicast routing with QoS constraint.

14 90 H. Wang et al. / Journal of Network and Computer Applications 27 (2004) Jie Fang is currently a Master student at Nanjing University of Science and Technology, her research interests include QoS routing algorithm. Hua Wang is currently a PhD candidate at Nanjing University of Science and Technology, his research interests include MPLS traffic engineering and QoS routing. Ya-Min Sun is professor and PhD supervisor. His current research interests are computer network and communications.