AN ALGORITHM OF LOW-COST MULTICAST BASED ON DELAY AND DELAY VARIATION CONSTRAINT

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1 INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 2, Number 1, Pages c 2006 Institute for Scientific Computing and Information AN ALGORITHM OF LOW-COST MULTICAST BASED ON DELAY AND DELAY VARIATION CONSTRAINT DAN-MEI ZHU, YAN-RUI SUN, HONG-XIA GAO, AND LIN LI Abstract. To meet the quality of service requirements of real-time multimedia, multicast communication is becoming a new approach of communication. Research on how to construct an effective multicast routing algorithm is therefore a hot topic. This paper first introduces a definition of crucial points and proposes a method of dividing the interval of delay, by which establishing an algorithm of minimum cost multicast based on delay and delay variation constraints. The new algorithm is shown to be effective by analyzing time complexity and comparison with other algorithms. Key Words. bounded multicast tree, crucial points, dividing the interval of delay. 1. Introduction In multicast communication, messages are concurrently sent to multiple destinations and all members of the same multicast group. With the development of multimedia applications, there has been a lot of interest in providing real-time multimedia services like audio and video over high-performance networks. These services require certain quality of service (QoS) from the networks. Supporting multimedia applications is a major objective of future high speed networks. Meanwhile, multicast services have been used by various continuous media applications. Multicast routing is a central problem in multicast communication. A typical approach to multicast routing requires the transmission of packets along the branches of a tree spanning the source and destination nodes, which is a problem of how to construct multicast a tree. One frequently considered optimization objective is the minimization of the total cost of the tree, which is taken as the sum of the costs on the links of the multicast tree. The minimum cost tree is known as the Steiner tree and finding such a tree is a well-known NP-hard problem [1,2]. Several heuristic multicast routing algorithms have been proposed recently [3,4]. To meet the quality of service requirements of real-time multimedia, it is necessary to satisfy the bounds of not only delay from source node to destination nodes but also delay variation in destination nodes, namely, we should find a Steiner tree accounting for delay and delay variation constraints. Apparently, it is also an NP-hard problem. The traditional algorithm on the above problem mostly neglect some crucial points in looking for Steiner trees [3-6], which often act as a crucial status in constructing Steiner tree. This paper first introduces a definition of crucial points and propose a method of dividing the area of delay, by which establishing an algorithm of minimum cost multicast based on delay and delay variation constraints. Received by the editors January 1, 2004 and, in revised form, March 22, Mathematics Subject Classification. 35R35, 49J40, 60G40. 51

2 52 DAN-MEI ZHU, YAN-RUI SUN, HONG-XIA GAO, AND LIN LI 2. Notation G(V, E): a weighted digraph. V denotes the set of nodes, and E the set of arcs corresponding to the set of communication links which connect the nodes. C(e): a link-cost function C:e R + which assigns a nonnegative weight to each link in the network; D(e): a link-delay function D:e R + which assigns a nonnegative weight to each link in the network; s : source node; M: destination set M = {V 1, V 2,, V n }; T = (V T, E T ) : multicast tree, V T V, E T E; i : source of the i th destination delay tolerance; : source-destination delay tolerance; δ: the maximum inter-destination delay variation in a tree T; P T s, v: the path from source s to destination v in a tree; e P T (s,v) D(e): a total delay from source s to destination v in path of tree. 3. Network model for multicasting Consider a communication network represented by a directed graph G=(V, E). The network is assumed to be full duplex. In other words, the existence of link e = (u, v) E implies the existence of link e = (v, u) E. Each link e E is assigned a cost value C(e), where C(e) R +. A link cost value can be the utilization of the link or a monetary cost, for example. We assume that the messages are concurrently sent to multiple destinations from a source node. A network model of minimum cost multicast tree based on delay and delay variation constraints is the tree that satisfies the following: Minimize Cost(T ) = C(e) e T subject to e P T (s,v i) e P T (s,v i ) D(e) D(e) v i M (1) e P T (s,v j) D(e) δ, v i, v j M (2) This problem is NP-complete. Therefore, efficient heuristics that build low-cost multicast trees with bounded end-to-end delay and delay variation are needed. 4. Heuristics for the constrained Steiner tree We now present an algorithm to construct a tree satisfying constraints (1) and (2) for the given values of the path delay and the interdestination delay variation tolerances. We assume that the source has all the information necessary to construct the multicast tree. Such information may be collected and updated using an existing topology-broadcast algorithm [9]. The algorithm is given as follows. Step 1: Divide the interval of delay. To satisfy constraints (1) and (2), we change constraint (2) into constraint (1). We can let H = {(x ɛ, x + δ ɛ) x (0, ), ɛ (0, δ) and x, ɛ R}, Let ɛ be a certain real number, and r i a random number, let x = r i and i = 1, 2,, n. We can get a certain interval H i = (x ɛ, x + δ ɛ), i = 1, 2,, n.

3 LOW-COST MULTICAST BASED ON DELAY VARIATION CONSTRAINT 53 If H i does not belong to [0, ], we discard it. Initialize n = 1, then go to Step 2. Step 2: Find a lowest cost source-destination path satisfying delay in H 1 with a k-shortest path algorithm. If some does not satisfy the delay given in H 1, then replace this path with the second lowest path, and so on. We can gain an effective path set P 1 = {p 1, p 2,, p m }. Step 3: Find crucial points. Counting the frequency of the relay nodes (not source nor destination nodes) arising on effective paths. Taking out high frequency nodes whose degrees are more than 3 in networks. We define these nodes as crucial points. Step 4: Construct closure graph G. The closure graph G is a complete graph with source, destination nodes and crucial points as its set of nodes. Its set of edges consists of the lowest cost path satisfying constraints(1) and(2). Step 5: Constructing a multicast constrained tree. In the closure graph G, we use the Prim algorithm to construct a multicast constrained tree. Firstly, initialize a multicast tree with the source node; secondly, we use a greedy approach to add edges to a subtree of the constrained spanning tree until all the destination nodes are covered and then record the total cost of this tree. Finally, expand the edges of the constrained spanning tree into the constrained lowest paths they present, and remove any loops that may be caused by this expansion. In this process, assume v is in the tree constructed so far, and we consider whether to include some edges adjacent to v. Adding edges subject to two selection functions which are given in Ref. [3] and revising the selection function. Those two selection functions are: P C (v, w) f CD (v, w) = if P D (s, v) H H i (P (v) + P D (v, w)) i ; f C (v, w) = P C (v, w). In the above functions, P C (v, w),p D (v, w) respectively denote the cost of link and the delay of link from node v to w; P (v) denotes the delay of link from source s to node v in tree; H i, H i respectively denote the maximum and minimum convergence points in H i. Step 6: When n = k, compare the total cost of all trees, choose a tree which has a lowest cost. If there are more than one trees of lowest cost, we then get a satisfied solution through increasing the value of k. 5. Performance analysis of the new heuristic algorithm 5.1. Analysis of algorithm rationality Theorem 1: The new heuristic algorithm can find a solution if a solution exists. Proof: If the heuristic is able to find a feasible solution covering all the k nodes, then we know two facts. Firstly, the algorithm produces a tree that spans all the k nodes. Secondly, no constraints conditions has been violated. Thus, there must be a path from s to each destination with delay and delay variation constraint, indicating that a solution exists. Step 4 and Step 5 in the new heuristic algorithm can ensure that this algorithm satisfies the first condition because we use the Prim algorithm in the closure graph whose nodes are consisted of source, destination nodes and crucial points. Meantime, Step 1 ensures this algorithm satisfies the second condition. Therefore, the new heuristic algorithm finds a feasible solution if a solution exists.

4 54 DAN-MEI ZHU, YAN-RUI SUN, HONG-XIA GAO, AND LIN LI Rationality of introducing crucial points. Constructing a multicast tree is a central problem for solving a multicast routing problem, and finding Steiner tree is the main method for constructing a multicast tree. This paper constructs multicast tree through heuristic seeking a bounded Steiner tree. There are some points that should not be neglected, which is often crucial in constructing a Steiner tree. The existence of these points usually can greatly decrease the total cost of the tree. Rayward -Smith once tried to seek these points in solving the Steiner tree problem and defined them as Steiner points [7]. He tried to find some potential Steiner points through iterations of functions. Generally speaking, Steiner points exist in rally nodes whose degrees are more than three in a Steiner tree [8]. This paper is also based on this idea. Selected crucial points satisfies the necessary condition of Steiner points, meanwhile, counting the frequency of the relay nodes (not source nor destination nodes) which arise on effective paths. Taking out high frequency nodes whose degree more than 3 in networks. We define these points as crucial points. Because crucial points are high frequency nodes on effective paths, these points must have a crucial effect on the optimization of a multicast routing tree. Accordingly, these points are important in constructing a Steiner tree. Therefore, we introduce these points and define them as crucial points Rationality of dividing the interval of delay. There usually is a conflict between constraints (1), (2) [6] in a multicast model. Constraint (1) seeks a shortest source-destination delay path, while as constraint (2) generally gives up a shortest source-destination delay path. Therefore, this paper changes two constraints into one constraint and divides the delay interval, by which make multicast routing satisfying constraints (1),(2). Theorem 2: The new heuristic algorithm finds a feasible solution in finite steps if a solution exists. Proof: The delay of the source-destination path should be 0 i when it satisfies constraint (1). This paper converts constraint( 2) into constraint (1) through restraining the delay interval. Assumed H is an open covering of the interval [0, ], and H = {(x ɛ, x + δ ɛ) x (0, ), ɛ (0, δ) and x, ɛ R}, Based on Heine- Borel-Lebesgue theorem of the finite open covering, we know that there are finite open intervals in H, which constitutes an open covering of the interval [0, ]. Because the solution satisfying constraint (1) can only appear in [0, ], meantime, the solution satisfying constraint (2) can only appear in H. According to the theorem of finite open covering, there must exist finite open intervals in H, which constitutes an open covering of the interval [0, ]. Therefore, we conclude that the solution must be found in finite steps while concurrently satisfying constraints (1),(2) Analyzing complexity of the Heuristics. Consider a communication network with n nodes and the number of multicast member is assumed m. In heuristics, k-shortest path algorithm takes O(n 3 ) time in Step 2; finding crucial points takes O((n m 1) 2 ) time in Step 3; As for the construction of closure graph G in Step 4, if there are h nodes in the closure graph G, it will take Oh 3 time; the Prim algorithm takes O(n 2 ) time in Step5, and expanding the edges of the constrained spanning tree into the constrained lowest paths they present, and removing any loops that may be caused by this expansion will cost O(hn). Summarize above, whole algorithm will take O(n 3 ) time.

5 LOW-COST MULTICAST BASED ON DELAY VARIATION CONSTRAINT 55 Figure 1. a multicast routing illustration 5.3. Comparing with others algorithm. Consider a communication network with n nodes, and suppose the number of multicast member is m. We compare this heuristic algorithm with SPT[5], DDVBM[10], DVMA[6], BMSTA[5]. The complexity of DDVBM is O(n 3 ), DVMA is O(mn 3 ), BMSTA is O(mn 4 ) and complexity of this heuristic algorithm is O(n 3 ). We adopt the illustration in paper [5] to compare with other algorithms as shown if Figure 1. In Figure 1, the source node is s and destination set is M = c, e, f, g; the weight is (C, D), where C denotes the cost of link and D denotes the delay of link. The result of different algorithm shows as Table 1. the maximum the maximum the cost of algorithm interdestination interdestination delay multicast delay max variation δ max tree Cost(T) SPT DDVBM DVMA BMSTA Algorithm in this paper Table 1: Comparison with other algorithms From above comparison, we can see that the proposed algorithm here shows a better performance. Compared with other algorithms, the value of cost if our algorithm is better. Although the cost of multicast tree in our algorithm is a little higher than the algorithm of BMSTA s, the value of max and the time complexity of BMSTA is even more high than this algorithm. Therefore, this algorithm is better. 6. Conclusions This paper has introduced a definition of crucial points and proposed to divide the interval of delay, by which establishing a new algorithm of low-cost multicast based on delay and delay variation constraints. Through performance analysis of the new heuristic algorithm and comparison with other algorithms, it has been shown that this algorithm has better performance.

6 56 DAN-MEI ZHU, YAN-RUI SUN, HONG-XIA GAO, AND LIN LI Acknowledgments The author thanks the anonymous authors whose work largely constitutes this sample file. This research was supported by???. References [1] Hakkimi S L. Steiner s problem in graphs and its implications[j], Networks,1971,(1): [2] Garey M R, Graham R L, Johnson D S. The complexity of computing Steiner minimal tree[j]. SIAM J. Of Applied mathematics, 1977, 32(4): [3] Kompella V P, Pasquale J C, Polyzos G C. Multicast Routing for Multimedia Communication [J]. IEEE/ACM Transactions on Networking, 1993, 1(3): [4] Ravikumar C P, Rajneesh Bajpai. Sourcebased delaybounded multicasting in multimedia networks [J]. Computer Communications, 1998, 21(2): [5] Fan Xiu-meiChen Chang-jia.Research on minimum cost multicast tree based on delay and delay variation constraints[j]. Journal of the China Railway 2000, 22(4): [6] Rouskas G N, Baldine I. Multicast Routing with End-to End Delay and Delay Variation Constraints[J]. IEEE JSAC , 15(3): [7] V. J. Rayward - Smith and A. Clare. On finding Steiner vertices. Networks, vol.16, pp , [8] Bernard M. Waxman. Routing of multipoint connections. IEEE JSAC.1988, vol.6, no.9, [9] D. Bertsekas and R. Gallager, Data Networks. Englewood Cliffs, NJ: Prentice Hall,1992. [10] C.P.Low, Y.J.Lee. Distributed multicast routing, with end-to-end delay and delay variation constraints. Computer Communication. 23(2000) Appendix: The Constrained Steiner Tree Algorithm / LetG = (V, E) describe the network topology s=source node M=mluticast group C=crucial points = source destination delay constraint δ =the maximum interdestination delay variation in a tree T / 1. Begin 2. ɛ = ɛ 0, = 0,δ = δ 0 3. for i = 1 to k 4. x = r i / r i = random(0, 1) / 5. H i = (x ɛ, x + δ ɛ) Else 8. Multicast(G(V, E), s, M, C,, δ) 9. if C Ti+1 < C Ti then C T = C Ti+1 / Record the cost of T i as C Ti / 10. end if 11. end if 12. i = i end for 14. Output T and C T 15. End 16. Multicast(G(V, E), s, M, C,, δ) / ConstructM ulticastroutingtree / 17. Begin / Compute the cheapest constrained Steiner Tree in M C {s} / 18. V M C {s} 19. for each v, w V do 20. begin 21. P C [V, W ] cost of cheapest constrained path from v to w

7 LOW-COST MULTICAST BASED ON DELAY VARIATION CONSTRAINT P D [V, W ] path delay along cheapest constrained path from v to w 23. end / Q = set of nodes already visited / / P [v] =path delay from s to v in the tree / / T =spanning tree on the closure graph whose nodes is consisted of V / 24. Q = {s} 25. P [s] = T = / until all nodes in have been spanned / 27. while (Q V ) do 28. begin 29. min = 30. for eachv Q do 31. begin 32. for eachw V \Q do 33. begin / if delay from s to w is within limits, consider edge (v, w) as a candidate and compute the selection function f s (v, w) = f CD (v, w) or f C (v, w) f CD (v, w) = P C (v,w) H i (P (v)+p D (v,w)) if P D (s, v) H i ; f C (v, w) = P C (v, w) / 34. if (f s (v, w) < min) then 35. begin 36. nextedge = (v, w) 37. min = f s (v, w) 38. end if 39. end for 40 end for 41. Q = Q w P [w] = P [v] + P D [v, w] T = T {nextedge} 44. end / while / 45. end / Multicast / 46. Crucial points(h) / find crucial points / 47. Begin 48. h = h for i = 1 to m 50. P j i = Dijkstra(i) original graph / find the first k- shortest paths from s to v i in the 51. Q i = {Pi 1,, P j i,, P i k} / G(V, E), such that the delay from s to v i over these paths belong to H i. / 52. P i = minq i / Pick out a shortest path P j from them, label this path and record the relay nodes of degree3 in this path / 53. If P i = then go to End 54. else 55. record P i 56. end if 57. i = i end for 59. P = {P 1, P 2,, P m } / construct a effective path set /

8 58 DAN-MEI ZHU, YAN-RUI SUN, HONG-XIA GAO, AND LIN LI 60. if Statistics (RN) > h / the statistic frequency of relay nodes of degree 3 in P / 61. C = Statistics(RN) 62. end if College of Science, Northeastern University, Shenyang, , P. R. China zhudanmei@126.com College of Science, Northeastern University, Shenyang, , P. R. China yanruisun@126.com and yanggao0910@126.com and LL @126.com