Minimum-Cost QoS Multicast and Unicast Routing in Communication Networks

Transcription

1 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 5, MAY Minimum-Cost QoS Multicast and Unicast Routing in Communication Networks Guoliang Xue, Senior Member, IEEE Abstract In this paper, we study the minimum-cost quality-ofservice multicast and unicast routing problems in communication networks. For the multicast problem, we present an efficient approximation algorithm to find a balance between a minimum-cost multicast tree and a minimum-delay multicast tree, with a provably good performance under the condition that link delay and link cost are identical. For the unicast problem, we present an efficient primal-dual heuristic algorithm to find a path which balances path cost and path delay, together with an error bound. The lack of a provably good performance for the second algorithm is complemented by computational results on randomly generated networks. Our algorithm finds optimal solutions in more than 80% of the cases and finds close to optimal solutions in all other cases, while using much less time. Index Terms Communication networks, minimum-cost delayconstrained end-to-end routing, minimum-cost delay-constrained multicast trees. I. INTRODUCTION ACOMMUNICATION network can be modeled by an edge-weighted undirected graph, where is the set of vertices, is the set of edges, is the cost of edge, and is the delay of edge. Note that this is a somewhat simplied model because in a realistic communication system, delays consist of delays due to propagation, link bandwidth, and queueing at intermediate nodes. This simplied model has been (and still is) widely used in the scientic literature on computer communications and networks because many efficient algorithms for the more complicated model are based on efficient algorithms for this simplied model [26], [29]. Multicasting consists of concurrently sending the same data from a source to a group of destinations in a computer or communication network. Multicast service plays a more and more important role in computer or communication networks supporting multimedia applications [32]. To support a large number of multicast sessions, a network must minimize the sessions resource usage, while meeting their quality-of-service (QoS) requirements [12]. To reduce resource usage, a packet from the source node to several destination nodes may share some communication links in the early stages before forking to the destinations. As a result, the packet is transmitted from the source Paper approved by P. E. Rynes, the Editor for Switching Systems of the IEEE Communications Society. Manuscript received October 10, 2000; revised February 23, 2002 and November 29, This work was supported in part by the Army Research Office (ARO) under Grant DAAD The author is with the Department of Computer Science and Engineering, Arizona State University, Tempe, AZ USA ( xue@asu.edu). Digital Object Identier /TCOMM to the destinations in a tree-like network known as a multicast tree [2]. Algorithms for computing multicast trees are known as multicast algorithms [23]. Unicast refers to the special case of a multicast where there is only one destination. Also known as end-to-end routing [26], unicast is another basic operation in communication networks and is often used as a subproblem in the computation of optimal multicast trees. Dferent optimization goals can be used in multicast/unicast algorithms to define what constitutes a good multicast/unicast routing. One such goal is to guarantee the minimum routing delay from the source to the destination(s), which is important for delay-sensitive multimedia applications, such as real-time teleconferencing. We call this kind of objective performance driven, because it emphasizes the performance of the QoS. Another optimization goal is to minimize the cost of the multicast/unicast routing, which is important in managing network resources efficiently. We call this kind of objective cost driven, because it minimizes the total cost required. Early studies have considered the delay and cost optimization objectives separately. Dijkstra s shortest-path algorithm [9] can be used to compute a delay-preserving multicast tree in polynomial time. It can also be used to compute a minimum-delay unicast routing, as well as a minimum-cost unicast routing, in polynomial time. The computation of a minimum-cost multicast tree [known as a Steiner minimum tree (SMT)], on the other hand, is NP-hard [10], [14]. Nevertheless, a minimum spanning tree-based greedy heuristic proposed by Takahashi and Matsuyama [31] can compute, in polynomial time, a multicast tree whose cost is, at most, twice that of an SMT. The best-known polynomial time approximation algorithm to this problem belongs to Robins and Zelikovsky [28], which computes a multicast tree whose cost is about times the cost of an SMT. Bharath-Kumar and Jaffe [2] discussed minimizing both cost and delay in a multicast tree, assuming that the cost and delay functions are identical. Awerbuch et al. [1] proposed a heuristic that starts from a minimum spanning tree and refines the tree to bound the diameter of the tree. This same idea is used in [8] in global routing in the rectilinear plane to bound the radius of the tree, motivated by applications in very large-scale integration (VLSI) circuit design. Khuller et al. [18] published a best-possible algorithm to balance a minimum spanning tree and a shortest-path tree (SPT). This idea has been used by Lin and Xue [19] in the design of performance-driven rectilinear SMTs. Recently, there has been increased interest in efficient algorithms for computing low-cost delay-constrained multicast trees. Parsa et al. [23] proposed an algorithm which refines an SPT to a low-cost delay-constrained multicast tree by iteratively replacing the most expensive super-edge by a less expensive /03$ IEEE

2 818 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 5, MAY 2003 path, without violating the delay constraints. Jia [15] and Jia et al. [17] proposed efficient admission-control methods based on the balance of SMTs and SPTs. Faloutsos et al. proposed a QoS-sensitive multicast Internet protocol known as QoSMIC [12]. Carlberg and Crowcroft proposed a shared-tree approach for multicast [5]. Shields and Garcia-Luna-Aceves proposed a hierarchical multicast routing approach [30]. Makki et al. presented an algorithm which computes a multicast tree whose cost is, at most, twice the cost of the smallest possible multicast tree (without considering delay) [21]. Other related work can be found in [7], [16], [25], [27], and the references therein. Although unicast is a special case of multicast, it is an important problem itself and has received extensive study in recent years [6], [20], [22], [24], [26], [29], [34], [35]. Most known algorithms for the unicast problem find a quickest path between two vertices in a communication network. When the delay and cost of an edge are not identical, computing a minimum-cost delay-constrained path is NP-hard [14]. In this paper, we extend the idea of Khuller et al. [18] for balancing minimum spanning trees and SPTs to compute good approximations to delay-constrained multicast trees. When the cost and delay for every edge is identical (as in the model proposed by Bharath-Kumar and Jaffe [2]), our algorithm has a provably good approximation ratio. Since our algorithm is very similar to the ones proposed in [15], [17], and [23], it also provides some theoretical support to the success of those algorithms. For unicast with independent link cost and link delay, we present an efficient primal-dual algorithm to find a path which balances path cost and path delay, together with an error bound. Computational results show that our algorithm finds optimal solutions in over 80% of the test cases and near-optimal solutions in all other cases. The rest of this paper is organized as follows. In Section II, we formally define the problems being studied, as well as some notations that will be used in later sections. In Section III, we present our approximation algorithm for QoS multicast, together with illustrations and performance analysis. In Section IV, we present our primal-dual algorithm for QoS unicast, together with performance analysis and computational results. We conclude this paper in Section V. II. DEFINITIONS AND NOTATIONS In this section, we formally define the problems that will be studied in later sections. The communication network is modeled using an edge-weighted undirected graph as defined in Section I. Let be a subgraph of. The cost of, denoted by, is the sum of the edge costs over the edges in. Let be a path in. The delay of, denoted by, is the sum of the edge delays over the edges on path. We will assume standard graph-theoretic notations [33] throughout this paper unless specied otherwise. Definition II.1: Given a source node and a set of destination nodes, a multicast tree is a tree subgraph of whose nodes form a superset of and whose leaf nodes form a subset of. Let be a multicast tree. We use to denote the cost of. For any two nodes,, we use to denote the unique path from to in. The cost of is denoted by. The delay of is denoted by. It is clear from the definition that a multicast tree may contain other vertices, in addition to those in. However, none of those additional vertices may be a leaf vertex in the tree, for they are introduced to reduce the network resource usage. Definition II.2: An SMT is a multicast tree whose cost is minimum among all multicast trees with source and destinations. An SPT is a multicast tree such that is the smallest possible for every node. An SPT is also called a delay-preserving tree (DPT) since it preserves the shortest delay from the source to every destination. An SPT can be computed in time [13]. Computing an SMT is NP-complete [14], although many good approximation algorithms are known [21], [28], [31]. In many real-time applications, we are interested in computing a low-cost multicast tree such that the delays at the destinations are within a given bound. Such multicast trees are called delay-bounded multicast trees or delay-constrained multicast trees [15], [17], [23]. Problem II.1: [DCMT] Let be a given constant. A multicast tree is called -optimal is true for every node. The delay-constrained multicast tree problem asks for a minimum-cost -optimal multicast tree for a given constant. The problem of computing a minimum-cost delay-constrained multicast tree is NP-hard because it contains, as a special case, the shortest weight-constrained path problem, which is known to be NP-hard [14]. Therefore, we are interested in approximate solutions only. We will study this problem in more detail in Section III. Problem II.2: [DCUP] Let be a given positive number. Let be a path connecting two vertices and in the network. is called a -delay-constrained path. is called a minimum-cost -delay-constrained path is minimum among all -delay-constrained paths. The delay-constrained unicast path problem asks for a minimum-cost -delay-constrained path. This problem is known to be NP-complete and is listed as [ND30] shortest weight-constrained path problem in [14]. Therefore, computing an optimal solution for this problem in polynomial time seems unlikely. On the other hand, this problem is very important in computer and communication networks and is an important subproblem for computing close-to-optimal solutions to the minimum-cost delay-constrained multicast tree problem [15], [16], [34], [35]. In Section IV, we will apply combined techniques in discrete optimization and mathematical programming to design a simple but effective primal-dual algorithm for computing approximate solutions to this problem. III. MINIMUM-COST QoS MULTICAST ROUTING An efficient algorithm for approximating a minimum-cost delay-constrained multicast tree is listed in Algorithm 1. It starts

3 XUE: MINIMUM-COST QoS MULTICAST AND UNICAST ROUTING IN COMMUNICATION NETWORKS 819 out with a lower cost tree and eliminates delay violations by adding minimum-delay paths while traversing the tree in depth first search (DFS) order. Algorithm 1 Approximating a minimum-cost -optimal multicast tree FIND-TREE (, ) Input: Edge-weighted graph, source node, and a constant. Output: A low-cost -optimal multicast tree. Step 1: INITIALIZATION: Compute an SPT and a low-cost tree interconnecting the vertices in with a good approximation algorithm. Let and. for each node in, let and be the parent of in. Step 2: RELAXING EDGES (IN DFS ORDER) WHOSE DELAYS VIOLATE THE BOUND: Perform the following when node is visited: do Find the smallest ancestor of on and add in a shortest path to, adjusting the and fields for nodes on this path necessary. for each child of in do visit node ; ; ; ; ; endfor Step 3: RETURN THE APPROXIMATE SOLUTION: return tree null. Whenever we find a node in the tree which violates the delay constraints, we find the smallest ancestor of in the tree and add a shortest path to the tree (we also update the tree by updating the and fields of the nodes on this path). The algorithm eventually changes the copy of to a low-cost -optimal multicast tree. Note that the cost of the tree will usually increase every time we eliminate a delay violation. This idea has been used by most algorithms for minimum-cost delayconstrained multicast trees [15], [17] (which may be viewed as the dual to the approach used by [23]). We will show that the tree produced by Algorithm 1 has a provably good performance under certain conditions. Before proceeding to the performance analysis, we will illustrate Algorithm 1 with an example. Since the best-known approximation algorithm for computing an SMT is very involved, we will use the simple algorithm of Takahashi and Matsuyama (a) (c) Fig. 1. An illustrative example. (a) The network. (b) Low-cost tree T. (c) SPT T. (d) The final tree T. [31] (listed here as Algorithm 2) to compute the approximation to an SMT for this illustration. Note that the cost of the tree produced by this algorithm is, at most, twice the cost of an SMT. This simplication helps with the understanding of Algorithm 1. Algorithm 2 A 2-approximation of an SMT Step 1: Begin with a subtree of consisting of the source node only. ;. Step 2: If, stop. Step 3: Determine a destination node closest to. Add to a shortest path joining it with.. goto Step 2. We will use the simple network in Fig. 1(a) in our illustration. The source node and the destination nodes,,, and are denoted with filled circles, and the edge labels represent the delays (as well as costs) of the edges. We assume that. Applying Algorithm 2, the edges,,, and are added to tree in that order. This tree is illustrated in Fig. 1(b). The SPT is illustrated in Fig. 1(c). During the execution of Algorithm 1, node is visited first. Since, no path is added at this time. The algorithm proceeds to visit node. Since, the path is added, which brings the delay at node to 102. Note that the edge (, ) is (effectively) deleted from the tree, due to the fact that the parent of node is changed from to. The algorithm visits the nodes and, in that order. Since the new delay at node is 202, which does not violate the delay constraints, no path to is added. For a similar reason, no path to is added. The final tree is illustrated in Fig. 1(d). The multicast tree in Fig. 1(d) is not a delay-preserving tree. However, it is a 1.5-optimal delay-constrained multicast tree. We note that the costs of the three multicast trees are,, and. This shows that the tree consumes less network resources than while still satisfying the delay constraints. In the rest of this section, we will prove that the multicast tree computed by Algorithm 1 has a provably good performance under certain conditions. (b) (d)

4 820 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 5, MAY 2003 Lemma III.1: Let be the multicast tree produced by Algorithm 1. For any node,wehave. In other words, is an -optimal multicast tree. Proof: If, a shortest path from to is added into the tree, where is the smallest ancestor of in the current tree. Therefore, the new value of is. During the execution of the algorithm, is never increased. Theorem III.1: Let be the multicast tree produced by Algorithm 1. If the cost and delay of every link in the network are the same, the cost of is, at most, times the cost of. Proof: Let and let be the nodes that caused the addition of a shortest path during the traversal, in the order they were encountered. When the shortest path from to was added, the net cost of the added edges was, at most,. Also, the edges on the path to consisting of the shortest path to followed by the path in from to had been relaxed in order, so that. The shortest path from to was added because. Therefore, we have Summing over Therefore yields the following: (III.1) (III.2) (III.3) Since DFS traverses each edge exactly twice, the sum on the right-hand side is, at most, twice the cost of. Therefore, we have (III.4) Therefore, the cost of the tree computed by Algorithm 1 is, at most,. This proves the theorem. Note that the performance ratio of our algorithm depends on the quality of. If the approximation algorithm of [28] is used to compute, is an approximation to SMT. As a result, the cost of the tree computed by Algorithm 1 is, at most, times the cost of an SMT. We note that the above analysis is very conservative. At the cost of slightly increased computing time, the practical performance of Algorithm 1 can be further improved. For example, when a shortest path to is to be added, we can reduce the cost increase by reusing some of the edges which are both in the current tree and in the shortest-path network from to [35]. The dference between the algorithm of [18] and our algorithm is that the former deals with minimum spanning trees and one-to-all SPTs, while we are dealing with SMTs and delay-preserving multicast trees. However, the ideas used here originate from [18]. Although relatively restrictive, the assumption that for holds for the models used in [1], [2], [8], and [18], as well as in the case where cost is measured by the number of links in the tree and delay is measured by the number of links along the path. If the cost and delay on an edge are independent, Algorithm 1 can still be used to compute a low-cost -optimal multicast tree, except that the good performance guarantee proved in Theorem III.1 may no longer be true. We would like to point out that whenever a minimum-cost path between and is to be computed, we can always compute a minimum-delay minimum-cost path between and. Similarly, whenever a minimum-delay path between and is to be computed, we can always compute a minimum-cost minimum-delay path between and. Our algorithm is very similar to the ones proposed in [15], [17], and [23], because all these algorithms are seeking a kind of balance between an SPT and an SMT. The theoretical performance guarantee of our algorithm fits well with the successful simulation results reported in [15], [17], and [23]. IV. MINIMUM-COST QoS UNICAST ROUTING In this section, we will present an efficient primal-dual heuristic algorithm for the DCUP problem. Our algorithm either finds an optimal solution or finds two paths and such that and that the cost of the optimal solution is guaranteed to be in the interval. Definition IV.1: Let be a real-valued parameter. We use to denote a network with the same sets of vertices and edges as network and an edge-weighting function where for every edge. We call the length of in. The parameter is introduced to balance the two edge-weighting functions and. The percentage of the contribution of ( ) is monotonically increasing (decreasing) with. In particular, and. For any pair of vertices, and a given value of, the shortest path connecting and with respect to the edge-weighting function can be computed in time [13]. Using the techniques developed by Xue [34], we can always select, among all shortest paths with respect to, a minimum-cost path with respect to or a minimum-delay path with respect to.we intend to find the smallest possible interval such that the delay of the shortest path in is no more than and that the cost of the optimal path is between the cost of the shortest path in and the cost of the shortest path in. The values and (as well as the corresponding paths) can be found efficiently using bisection. In the rest of this section, we will use the length (cost, delay) of a path to mean the sum of the edge weights with respect to the weighting function (, ) over the edges of the path, where the value of is understood from the context. The length (cost, delay) of a path is denoted by (, ). As a result, we may talk about shortest (minimum cost, minimum delay) paths in. If a path does not exist, the values of,, and are taken as. We present

5 XUE: MINIMUM-COST QoS MULTICAST AND UNICAST ROUTING IN COMMUNICATION NETWORKS 821 our primal-dual algorithm for approximating a minimum-cost -delay constrained path as Algorithm 3. Algorithm 3 Computing a low-cost -delay-constrained path in. Step 1: Compute a minimum-cost shortest path in ; does not exist or stop: there is no -delay-constrained path; Step 2: Compute a minimum-delay shortest path in ; stop: is a minimum-cost -delayconstrained path; else ; ; Step 3: If stop: is a low-cost -delayconstrained path; else in an odd iteration; in an even iteration; Step 4: Compute a minimum-cost shortest path in ; ; ; goto Step 3; else ; ; goto Step 3; We will now establish the correctness of Algorithm 3. Lemma IV.1 below shows that the cost (delay) of any shortest path in is no less than (no greater than) the cost (delay) of any shortest path in provided that. Since all edge-weighting functions are nonnegative, we may restrict our attention to simple paths, without loss of generality. In the rest of this paper, any path is assumed to be simple unless otherwise stated. Lemma IV.1: Let be a shortest path in and a shortest path in where. Then and Proof: Since is a shortest path in, and is a path in, we must have (IV.1) Inequalities (IV.2) and (IV.3), together with the assumption that, imply that (IV.4) This is in contradiction with the assumption that is a shortest path in. This contradiction proves that. Using a similar argument, we can prove that. Theorem IV.1: Let, be two vertices and be a given constant. Let be a minimum-cost shortest path in and a minimum-delay shortest path in. 1) There is a -delay-constrained path in and only. 2) If, is a minimum-cost -delay-constrained path in. 3) Assume that there exists a minimum-cost -delay-constrained path, but. Then Algorithm 3 stops with two numbers, together with two paths and such that,, and that the cost of the optimal path is between and. Proof: The first two assertions follow directly from the definition of. Now let us prove the third assertion. Let and be two paths. If and, wesay and are equivalent, denoted as. If and but, we say dominates, or equivalently, is dominated by, denoted as. Now assume that is a minimum-cost -delay-constrained path. It is clear that is also a minimum-cost -delayconstrained path or. Let be the set of all paths minus all the paths which are dominated by another path and minus all but one path from each set of equivalent paths. It is clear that must contain at least one minimum-cost -delay-constrained path. Assume that and let the paths in be ordered as such that Then we must also have Let be a minimum-cost -delay-constrained path. Since, we know that.if, we claim that is a minimum-cost -delay-constrained path. This is true because is a minimum-delay path and has the smallest cost among all minimum-delay paths. Therefore, we only need to prove the third assertion of the lemma in the case. Since no path in dominates another path in,wehave Assume to the contrary that (note that (IV.2). Then (IV.2) implies ). This, in turn, implies that (IV.3) and In addition, any minimum-cost shortest path in must be equivalent to a path in for any.

6 822 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 5, MAY 2003 TABLE I 100 VERTICES AND 800 EDGES, EDGE WEIGHTS UNIFORMLY IN [1], [20] Fig. 2. Interval containing an optimal solution. It is clear that Algorithm 3 stops after a finite number of iterations. When the algorithm stops, it is not guaranteed to find an optimal solution. However, the line segment connecting (, ) and (, ) will be an edge of the convex hull of the point set. In addition, there is a minimum-cost -delay-constrained path such that and, as shown in Fig. 2. We have implemented Algorithm 3 in ANSI C and tested it on randomly generated graphs. Given two integers and and an interval, our graph generator generates a connected graph with vertices and edges with edge weights and drawn from a unorm distribution in, with vertex degrees almost the same for all vertices. The graph generator first generates the vertices. It randomly adds new edges to the graph. If the addition of a new edge makes the degree of one of its end vertices too high, the generator discards this edge and randomly generates another edge. This process is continued until edges are added to the graph. For our test purpose, the delay tolerance has been set to two times the minimum delay between and (with and in our case). Optimal solutions were obtained by computing all the paths with delay no more than, and selecting the path with minimum cost. Note that the set of such paths can be computed using the algorithm of Byers and Waterman [3]. In order to compute the optimal solution, we have to generate all of the paths with delay no more than, because the algorithm of Byers and Waterman does not generate the paths in increasing values of the delay, and the algorithm of Eppstein [11] depends on the number of paths rather than the bound. Because of this nature, we will compare the number of shortest paths computed by the algorithms rather than the CPU time used. Table I shows the computational results with 10 randomly generated graphs of 100 vertices and 800 edges, with edge weights drawn from a unorm distribution in [1] and [20]. The first column in the table is the instance number (which is the seed value of our random number generator), a next to the instance number indicates that the primal-dual solution is also an optimal solution. The second column in the table is the delay tolerance (which is two times the minimum delay between the minimum indexed vertex and the maximum indexed vertex. The third column is the number of shortest-path problems solved by the primal-dual algorithm. In the fourth and fth columns are the delay and cost of the approximate solutions. The sixth column gives the error bound of the corresponding solution, which is an upper bound on the cost deviation from optimal. The seventh column gives the number of paths whose delay is TABLE II 100 VERTICES AND 3000 EDGES, EDGE WEIGHTS UNIFORMLY IN [1], [20] within the given bound. In the eighth and ninth columns are the delay and cost of the optimal solutions. These notations are also applicable to Tables II and III. From Table I we observe that our primal-dual algorithm finds optimal solutions in eight of the ten cases tested. The average cost of the paths computed by our algorithm is 22.14, while the average cost of the optimal solutions is Our primal-dual algorithm requires an average of 2.4 path computations, while the -path-based algorithm requires an average of path computations. Table II shows the computational results with 10 randomly generated graphs of 100 vertices and 3000 edges, with edge weights drawn from a unorm distribution in [1], [20]. From Table II, we observe that our primal-dual algorithm finds optimal solutions in seven of the ten cases tested. The average cost of the paths computed by our algorithm is 13.67, while the average cost of the optimal solutions is Our primal-dual algorithm requires an average of 2.4 path computations, while the -path-based algorithm requires an average of path computations. Table III shows the computational results with 10 randomly generated graphs of 100 vertices and 4000 edges, with edge weights drawn from a unorm distribution in [1] and [10]. From Table III, we observe that our primal-dual algorithm finds optimal solutions in all 10 of the cases tested. Our primal-dual algorithm requires an average of 2.4 path computations, while

7 XUE: MINIMUM-COST QoS MULTICAST AND UNICAST ROUTING IN COMMUNICATION NETWORKS 823 TABLE III 100 VERTICES AND 4000 EDGES, EDGE WEIGHTS UNIFORMLY IN [1], [10] networks whose topology model the Internet more closely, such as those discussed in Calvert et al. [4]. REFERENCES the -path-based algorithm requires an average of 223 path computations. Another observation we have is that the quality of the solutions produced by the primal-dual algorithm improves as the number of edges increases. A possible explanation for this behavior is that as the number of edges increases, the number of paths also increases, which leads to the reduction of irregularities. Among the 30 instances solved, our primal-dual algorithm finds the optimal solutions in 25 cases. For the five cases where our primal-dual algorithm fails to find an optimal solution, our solutions are very close to optimal. For all instances, we need to consider a total of 2157 paths to find the optimal solutions using the shortest-paths algorithm, compared to a total 72 shortest paths needed to be computed by the primal-dual algorithm. Note that the primal-dual algorithm finds optimal solutions in an average of 2.4 (never more than four) iterations in all cases, whereas the shortest-path-based algorithm needs to compute an average of paths! The fast computing speed and good solution quality makes our primal-dual algorithm a very attractive tool for computing quality solutions to the delayconstrained shortest-path problem. V. CONCLUSIONS In this paper, we have presented efficient algorithms for approximating the minimum-cost QoS multicast trees and minimum-cost QoS unicast paths in a communication network. For QoS multicast, we have presented an efficient approximation algorithm for computing a low-cost -optimal delay-constrained multicast tree, with a provably good performance when the delay and cost of the edges are identical. This guarantee provides some theoretical support to several similar multicast algorithms proposed in the literature. For the QoS unicast problem, we have presented an efficient primal-dual heuristic algorithm which finds a path balancing cost and delay. Our computational results show that our primal-dual algorithm is very effective on randomly generated test problems, finding optimal solutions in more than 80% of the cases and finding close to optimal solutions in other cases. Future research topics include extending our multicast algorithm to more realistic communication systems and evaluating unicast algorithms on [1] B. Awerbuch, A. Baratz, and D. Peleg, Cost-sensitive analysis of communication protocols, in Proc. ACM PODC, 1990, pp [2] K. 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8 824 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 5, MAY 2003 [26] N. S. V. Rao and S. G. Batsell, On routing algorithms with end-to-end delay guarantee, in Proc. ICCCN, 1998, pp [27] C. P. Ravikumar and R. Bajpai, Source-based delay-bounded multicasting in multimedia networks, Comput. Commun., vol. 21, pp , [28] G. Robins and A. Zelikovsky, Improved Steiner tree approximation in graphs, in Proc. 11th Annu. ACM-SIAM Symp. Discrete Algorithms, 2000, pp [29] J. B. Rosen, S. Z. Sun, and G. L. Xue, Algorithms for the quickest path problem and the enumeration of quickest paths, Comput. Oper. Res., vol. 18, pp , [30] C. Shields and J. J. Garcia-Luna-Aceves, HIP-A protocol for hierarchical multicast routing, Comput. Commun., vol. 23, pp , [31] H. Takahashi and A. Matsuyama, An approximate solution for the Steiner problem in graphs, Math. Japonica, vol. 24, pp , [32] Z. Wang and J. Crowcroft, Quality of service routing for supporting multimedia communications, IEEE J. Select. Areas Commun., vol. 14, pp , Sept [33] D. B. West, Introduction to Graph Theory. Englewood Clfs, NJ: Prentice-Hall, [34] G. L. Xue, End-to-end delay paths: quickest or most reliable?, IEEE Commun. Lett., vol. 2, pp , June [35] G. L. Xue and S. Z. Sun, Optimal multicast trees in communication systems with channel capacities and channel reliabilities, IEEE Trans. Commun., vol. 47, pp , May Guoliang Xue (M 98 SM 99) was born in 1960 in Shandong, China. He received the B.S. degree in mathematics and the M.S. degree in operations research from Qufu Teachers University, Qufu, China. He received the Ph.D. degree in computer science from the University of Minnesota, Minneapolis, in He worked for two years at the Army High Performance Computing Research Center as a Postdoctoral Fellow before joining the University of Vermont, Burlington, as an Assistant Professor of Computer Science, and he was promoted to Associate Professor in Since 2001, he has been an Associate Professor of Computer Science and Engineering at Arizona State University, Tempe. His research interests include design and analysis of algorithms, optimization and computer networks. He has published over 100 technical papers in these areas, including papers in the SIAM Journal on Computing, SIAM Journal on Discrete Mathematics, SIAM Journal on Optimization, IEEE TRANSACTIONS ON COMPUTERS, and IEEE TRANSACTIONS ON COMMUNICATIONS. Dr. Xue received a second prize from the Chinese National Education Committee for his publications in Operations Research. As a graduate student, he received the Doctoral Dissertation Fellowship from the Graduate School of the University of Minnesota. In 1994, he received the Research Initiation Award from the National Science Foundation (NSF). He is the Co-editor of eight journal special issues and conference proceedings. He is the Technical Program Co-chair of IEEE IPCCC 2003 and IEEE HPSR 2004 and is serving as a Technical Program Committee member of many conferences. He has served as a panelist to NSF and a reviewer to NSERC. He is profiled in the Marquis Who s Who in America.