K. Kumar and J. Jaffe, "Routing to multiple destinations in computer networks, " IEEE Transactions on Communications, pp. 31:343--351, 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....An even more general problem in the context of network multicast is the dynamic MST problem [7] where the tree needs to efficiently adapt to dynamic changes in the multicast group. The basic MST problem is known to be NP Complete, and heuristics for the problem have been proposed in the past [8] [9]. However, all the heuristics assume that the exact topology and location of the multicast group members are known, which is not the case in hierarchical networks. In this paper we propose a framework for the adaptation of multicast routing protocols to hierarchical networks. In the context of ....

K. Kumar and J. Jaffe, "Routing to multiple destinations in computer networks, " IEEE Transactions on Communications, pp. 31:343--351, 1983.

....are often used to obtain a near minimum cost multicast tree. Many multicast tree formation algorithms, which construct a source based tree given the full knowledge of network topology and multicast session membership, have been proposed and their performance evaluated in the literature [2] [4], 7] 9] 18] These heuristic algorithms can roughly be classified into two categories. The first one contains algorithms based on the shortest path heuristic (SPH) which minimizes the cost of the path from a multicast source to each of the members, while the second one contains algorithms ....

K. Bharath-Kumar and J. M. Jaffe, "Routing to multiple destinations in computer networks," IEEE Transactions Commun., vol. COM-31, no. 3, pp. 343--351, Mar. 1983.

....is NP complete [12] It remains NP complete even if all edge weights are equal. Several heuristics are known for this problem [22] One of them, suggested in [13, 20] is the minimum cost paths heuristic (MPH) This heuristic has a worst case performance of two times the optimum cost solution [5]. No heuristic with a better worst case performance is known [22] The minimum cost path heuristic (MPH) works as follows [22] Step 1: Choose an arbitrary vertex z from Z.LetZ # = z and T = z . Step 2: Find in Z Z # the vertex z closest to T.Addzto Z # ,andaddtoT the minimum cost path ....

....and 3 (Figure 5(b) The left graphs show absolute numbers while the right ones show these numbers in relative to the group size. These graphs show that for almost all sizes of multicast groups, the multicast tree created 4 Recall that MPH has a worst case performance of two times the optimum [5]. However, simulations show [21] that its actual performance is only 5 worst than the optimum. 17 0 10 20 30 40 50 60 70 80 90 100 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 average group size average tree cost R DGA vs DGA DGA oo improved R DGA R DGA Figure 4: Average Cost of ....

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K. Bharath-Kumar and J. Ja#e. Routing to multiple destinations in computer networks. IEEE Transactions on Communications, 31:343--351, March 1983.

....0 x 0 (a) Figure 5: Example for ASPH algorithm. 4. T,A) is the desired solution. The second algorithm we present for building a CPE based solution to MC VPN is the Active Double Tree Heuristic (ADTH) Many algorithms for solving the Steiner Tree Problem (STP) are known in the literature (i.e. [1], 9] 12] ADTH can use any of these algorithms as a basis for constructing a solution. As an example we use the SPH algorithm [12] Figure 6(a) shows the spanning tree found by SPH in the network of Figure 5(a) Note that starting with a tree implies that some fork nodes may not belong to the ....

K. Bharath-Kumar and J. Ja#e. Routing to multiple destinations in computer networks. IEEE Transactions on Communications, 31:343--351, March 1983.

....6) Given ff 1, a minimum spanning tree, and a shortest path tree, an (ff; 1 2 ff Gamma1 ) LAST can be found by n processors in O(log n) time on a CREW PRAM. 2 Related Work Trees realizing tradeoffs between weight and distance requirements were first studied by Bharath Kumar and Jaffe [4]. The authors weight requirement was the same as ours, but their distance requirement was that the sum of the distances from the root to each vertex should be at most fi times the minimum possible sum. They showed the weaker tradeoff that the desired tree exists if fffi Theta(n) Awerbuch, ....

....u in TM 4 do Relax(u; v) 5 DFS(v) 6 Relax(v;u) Add Path(v) Relax edges along path from r to v in T S . 1 if d[v] D T S (r; v) 2 then Add Path(parent T S (v) 3 Relax(parent T S (v) v) Figure 2: Algorithm to compute a LAST. relaxation changed vertex 4 s parent to vertex 5 and updated d[4] to reflect the new shorter path (1; 5) 5; 4) The algorithm then traversed and relaxed edges (4; 6) and (6; 7) bringing vertices 6 and 7 into the tree. When vertex 7 was encountered, its distance estimate (40) exceeded twice the shortest path distance (15) so the edges on the shortest path ....

K. Bharath-Kumar and J. M. Jaffe, Routing to multiple destinations in computer networks, IEEE Transactions on Communications 31 (3), pp. 343--351, (1983).

....for tree selections. Note that the problem of finding minimum cost trees for a certain source and a set of destinations is the Steiner tree problem, which is NP hard. Good surveys of the problem and heuristics can be found in [63] 70] 68] 74] and applications to networks can be found in [6], 48] 10] Because the Steiner tree problem is NP hard, the issue of cost minimization and capacity utilization may be difficult. Finally, we may comment on extending our algorithm to loopback so as to achieve link or node rerouting rather than path rerouting. The placing of traffic onto the ....

K. Bharath-Kumar and J. M. Jaffe, "Routing to multiple destinations in computer networks," IEEE Trans. Commun., vol. COM-31, pp. 343--351, Mar. 1983.

....data. The goal is to construct a tree T connecting the source to R so that the sum of the cost of the tree and the delays seen by the receivers is small. This problem is equivalent to COST DISTANCE if the distance metric are the delays. This problem has been extensively in the networks community [9, 18, 22, 26, 31], and many heuristics have been proposed. This problem also arises in the context of wire routing in programmable gate arrays and VLSI circuits [3] We present the first approximation for this problem. Our approximation ratio is O(log jRj) Note that our scheme works for the following problem as ....

K. Bharath-Kumar and J.M. Jaffe. Routing to multiple destinations in computer networks. IEEE Transactions on Communications, 31(3):343--51, 1983.

.... algorithms based on the shortest path heuristic which minimizes the cost of the path from a (multicast) source to each of the (multicast) group members (or destinations) while the second one contains algorithms based on the minimum Steiner tree, which minimizes the total cost of a multicast tree [2 4,6,8,13]. Although a minimum Steiner tree is more desirable, finding the minimum Steiner tree for a multicast session when only a subset of the nodes in a network are destinations is an NP complete problem [7] and accordingly, heuristics are often used. Supporting multicast in WDM networks requires ....

K. Bharath-Kumar and Jaffe. Routing to multiple destinations in computer networks. IEEE Transactions on Communications, COM-31(3):343--351, March 1983.

....have been shown to approach the ideal solution of small networks. See for example Waxman [26] or Rayward Smith [19] Other theoretical work has concentrated on slightly different aspects. BharathKumar and Jaffe consider multicast trees which trade efficiency with low average delay to recipients [3]. Kadirire introduces the concept of geographic spread [15] which makes it more likely that a node wishing to join an existing multicast group will find a cheap path to the tree. Jiang developed a Steiner tree variation which takes account of link capacities for high bandwidth applications, ....

K. Bharath-Kumar and J.M. Jaffe. Routing to multiple destinations in computer networks. IEEE Trans. on Communications, COM-31(3):343--351, March 1983.

....to minimize 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 [2] and finding such a tree is a well known NP complete problem [3] Heuristics to construct low cost trees have been developed in [4, 5, 6, 7]. While total tree cost as a measure of bandwidth efficiency is certainly an important parameter, it is not sufficient to characterize the quality of the tree as perceived by interactive multimedia and real time applications. Networks sup This work was supported in part by a Faculty Development ....

.... constructs the tree of shortest paths (SPT) from the source to any node in the network; 4) Prim s algorithm [14] which constructs a tree of minimum weight (MST) spanning all nodes in the network; the weight of each link is set to the delay incurred along the link; 5) The tradeoff (TDF) algorithm [5] between the minimumspanning tree heuristic [7] and SPT, considered here because it was conjectured in [10] that it may yield good performance in terms of ffi T . We have studied the average case behavior of the five algorithms by generating random graphs for a wide range of values for the total ....

K. Bharath-Kumar and J. M. Jaffe. Routing to multiple destinations in computer networks. IEEE Transactions on Communications, COM-31(3):343--351, March 1983.

....set up from the source of the multicast to the destinations before data transmission occurs. Determining this optimal multicast tree for the virtual circuit is a difficult problem. Previous authors have established that the multicast tree problem may be modeled as the Steiner problem in networks [3, 4, 5, 12, 30], referred to hereafter as the SPN, and that explicit solutions are prohibitively expensive. For example, two popular explicit algorithms, the spanning tree enumeration algorithm and the dynamic programming algorithm [30] have algorithmic complexities of O(p 2 2 (n Gammap) n 3 ) and O(n3 ....

....complexities of O(p 2 2 (n Gammap) n 3 ) and O(n3 p n 2 2 p n 3 ) respectively, where n is the number of nodes in the graph and p the number of multicast members. A number of good, inexpensive, centralized heuristics exist for the SPN and have been reviewed extensively elsewhere [5, 12, 17, 22, 23, 29, 30]. Some have been shown through analysis to produce solutions no worse than twice the optimal solution. 30] That is, the sum of the edge weights of the heuristic tree is no more than twice the sum of edge weights of an optimal tree. In practice, our empirical evidence indicates that these ....

K. Bharath-Kumar and Jaffe. "Routing to multiple destinations in computer networks," IEEE Transactions on Communications, vol. COM-31, no. 3, pp. 343--351, Mar. 1983.

....[3] 6] and [12] or a tree generated by the Greedy algorithm (see [14] A Steiner tree, or Steiner Minimal Tree (SMT) is the multicast tree that minimizes total path cost. An algorithm to find an exact SMT belongs to the class NP complete, hence heuristic algorithms have been studied (see [2], 13] and [16] However, due to their high algorithm complexity, routing algorithms based on SMT heuristics are not popular in actual use. One of the advantages of the SPT algorithm is that it always guarantees the shortest path from each receiver node to the sender. However, the SPT algorithm ....

K. Bharath-Kumar, and J. Jaffe, "Routing to Multiple Destinations in Computer Networks," IEEE Transactions on Communications, Vol. COM-31, No. 3, pp. 343 - 351, March 1983.

....case performance that can be proportional to the number of nodes in the network. Clearly, this also applies to directed graphs since an undirected graph is a special case of a directed one. In this paper, we focus on the join only case. For the undirected join only problem, Bharath Kumar and Jaffe [BKJ83] and later Imase and Waxman [IW91] proved that the greedy solution is never more expensive than log(M) times the optimal solution. In a more recent work, Awerbuch and Azar [AA95] study the join only problem for multiple interleaving sessions on undirected graphs, and propose a O(log N log M) ....

K. Bharath-Kumar and J.M. Jaffe. Routing to multiple destinations in computer networks. IEEE Trans. on Communications, 31:343--351, 1983.

....do not quit a session. For both problems, we define M to be the number of the join requests plus one 1 . For the undirected join leave problem, Imase and Waxman [IW91] prove a lower bound of M , a bound that we increase to 2 M . For the undirected join only problem, Bharath Kumar and Jaffe [BKJ83], later Imase and Waxman [IW91] proved that the WCI of the on line NPF is tightly bounded by log(M ) In a more recent work, Awerbuch and Azar [AA95] study the join only problem for multiple interleaved sessions on undirected graphs, and propose a (log N log M) competitive algorithm with respect ....

K. Bharath-Kumar and J.M. Jaffe. Routing to multiple destinations in computer networks. IEEE Trans. on Communications, 31:343--351, 1983.

....An even more general problem in the context of network multicast is the dynamic MST problem [7] where the tree needs to efficiently adapt to dynamic changes in the multicast group. The basic MST problem is known to be NP Complete, and heuristics for the problem have been proposed in the past [8] [9]. However, all the heuristics assume that the exact topology and location of the multicast group members are known, which is not the case in hierarchical networks. In this paper we propose a framework for the adaptation of multicast routing protocols to hierarchical networks. In the context of ....

K. Kumar and J. Jaffe, "Routing to multiple destinations in computer networks, " IEEE Transactions on Communications, pp. 31:343--351, 1983.

....and bit rate transparency across both unicast and multicast. Many multicast tree formation algorithms, which construct a source based tree given the full knowledge of network topology and multicast session (or group) membership, have been proposed and their performance evaluated in the literature [2 4,6,8,16]. These algorithms can roughly be classified into two categories. The first one contains algorithms based on the shortest path heuristic which minimizes the cost of the path from a multicast source to each of the members, while the second one contains algorithms based on the minimum Steiner tree, ....

K. Bharath-Kumar and Jaffe. Routing to multiple destinations in computer networks. IEEE Transactions on Communications, COM-31(3):343-- 351, March 1983.

....1.00 Average Waiting time to Join 15.95 Average Number of Lost Packet per Handoff 1.61 V. RELATED WORK AND CONCLUSION Multicast routing has a long history in the context of wireline networks. Those works have typically focused on the optimality of the tree, and the scalability of the algorithm [5], 21] 15] 20] 18] 7] 3] 8] 13] 16] 21] In the context of mobile computing environments, there has been very little work in the area of multicast routing algorithms. While related work has dealt with the issues of supporting multicast communication for mobile hosts [1] two ....

K. Bharath-Kumar, J. M. Jaffe, "Routing to Multiple Destinations in Computer Networks", IEEE Transactions on Communications, p. 343-351, vol. 31, no. 3, March 1983.

.... Determining the optimal multicasting tree (i.e. i.e. the one having the minimal cost) in an arbitrary network is often modeled as the Steiner problem [21 27] While exact solutions to the Steiner problem are very time consuming, a number of inexpensive and effective heuristics already exist [23 26, 28 30]. In this paper, we evaluate the benefit of multicasting over unicasting in WDM networks and the effect of equipping some nodes with the wavelength conversion capability and or the splitting (or copying) capability. The later refers to the number of copies of data that a node can forward to other ....

K. Bharath-Kumar and Jaffe, "Routing to multiple destinations in computer networks.," IEEE Transactions on Communications COM-31, pp. 343--351, Mar. 1983.

....updated without reconstruction. The problem is the broadcast tree is fixed, not dynamic, which does not respond to the change of traffic in the network, and the maintenance of a separate multicast tree is still required for each source group pair. 23 Early in 1983, Bharath Kumar and Jaffe[5] studied several heuristic algorithms for multicast routing and evaluated their performance. These algorithms are Minimum Spanning Tree (MST) Minimum Destination Cost (MINDC) Nearby Heuristic 1 (NH1) Nearby Heuristic 2 (NH2) and Nearest Neighbor Heuristic (NN) A minimum spanning tree for the ....

K. Bharath-Kumar and J. M. Jaffe. Routing to multiple destination in computer networks. IEEE Tran. on Communications, COM-31(3):343--351, March 1983.

....be NP complete [2] Many traditional heuristics for Steiner Tree problem are not suitable for multicasting when the network is large (see [2] Part II, Chapter 4 for coverage of several heuristics) A number of algorithms have been proposed for multicast routing problems. The earlier algorithms [3, 4] do not support group addressing. The optimal multicast routing algorithms [5, 6] are too costly to be practical. The algorithms described in [1] and [7] are mainly low delay algorithms. A low cost algorithm, such as [8] that incrementally updates a delivery tree will lose effectiveness when ....

K. Bharath-Kumar and J. M. Jaffe, "Routing to multiple destinations in computer networks," IEEE Trans. Comm., vol. COM-31, no. 3, pp. 343--351, March 1983.

....trees. In this paper, we consider the construction of a single tree that is used by all nodes. It is important that the cost of this tree be as low as possible. The problem of constructing a minimum cost multicast tree can be modeled as the Steiner tree problem in graphs and is NP complete [Win87, BKJ93, HR92]. Therefore, finding an exact solution for this problem in a large network can be very expensive. Several heuristics have been proposed to provide close to optimal solutions [BKJ93, KPP93, TM80] For example, it has been shown that some heuristics produce trees whose cost is no greater than twice ....

....a minimum cost multicast tree can be modeled as the Steiner tree problem in graphs and is NP complete [Win87, BKJ93, HR92] Therefore, finding an exact solution for this problem in a large network can be very expensive. Several heuristics have been proposed to provide close to optimal solutions [BKJ93, KPP93, TM80]. For example, it has been shown that some heuristics produce trees whose cost is no greater than twice the cost of the optimal tree [Win87, BV95a] Most of these heuristics are centralized or serial in nature (that is, they require that the entire topology information be available at a single ....

K. Bharath-Kumar and J. Jaffe. Routing to multiple destinations in computer networks. IEEE Transactions on Communications, 31(3):343--351, 1993.

.... delay cost, which minimizes the delay as the primary objective, with the cost as a secondary objective. 4 Previous Work in Evaluation of Multicast Routing Algorithms The algorithms referenced above have been studied in the context of multicast routing by a number of researchers. Kumar and Jaffe [10] considered the problem of routing single multicasts in a network with undirected links. They compared a number of minimum delay and minimum cost algorithms when the link cost and delay weights are the same and derived analytical bounds in cost delay under that assumption. Waxman [11] and later ....

K. Bharath-Kumar and J.M. Jaffe, "Routing to Multiple Destinations in Computer Networks," IEEE Trans. on Comm., vol. COM-31, no. 3, March 1983, pp 34351.

....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 [7] and finding such a tree is a well known NP hard problem [4] Heuristics to construct trees of low overall cost have been developed in [2], 6] 9] 15] While total tree cost as a measure of bandwidth efficiency is certainly an important parameter, networks supporting real time traffic will be required to provide certain quality of service guarantees in terms of the end to end delay along the individual paths from the source to ....

.... (SPT) from the source to any node in the network (the tree is pruned so that all leaves are destination nodes) 3) Prim s algorithm [11] which constructs a tree of minimumweight (MST) spanning all nodes in the network (this tree is also pruned as above) and (4) the tradeoff (TDF) algorithm [2] between the minimumspanning tree heuristic for the Steiner tree problem [6] and SPT. We have run the algorithms on randomly generated graphs constructed to resemble real world networks using the method described in [15] The nodes of the graphs were placed in a grid of dimensions 4900 Theta 4900 ....

K. Bharath-Kumar and J. M. Jaffe. Routing to multiple destinations in computer networks. IEEE Transactions on Communications, COM-31(3):343--351, March 1983.

....special significance. This paper presents a new, efficient heuristic for updating the multicast tree for dynamic multicast groups. Previous authors have established that determining an optimal multicast tree for a static multicast group may be modeled as the NP complete Steiner problem in networks [3, 4, 5, 7, 16]. Consequently. its explicit solutions are prohibitively expensive. For example, two popular explicit algorithms, the spanning tree enumeration algorithm and the dynamic programming algorithm [16] have algorithmic complexities of O(p 2 2 (n Gammap) n 3 ) and O(n3 p n 2 2 p n ....

....O(p 2 2 (n Gammap) n 3 ) and O(n3 p n 2 2 p n 3 ) respectively, where n is the number of nodes in the graph and p the number of multicast members. A number of good, inexpensive heuristics exist for the Steiner problem in networks and have been reviewed extensively elsewhere [1, 5, 7, 10, 11, 12, 13, 16]. This paper addresses the problem of modifying an existing multicast tree when new members enter or existing members leave the multicast group. The problem of updating the multicast tree after each addition and deletion is known as the on line multicast problem in networks. This paper focuses on ....

K. Bharath-Kumar and Jaffe. "Routing to multiple destinations in computer networks," IEEE Transactions on Communications, vol. COM-31, no. 3, pp. 343--351, Mar. 1983.

.... variation of a set covering algorithm is presented, and an MST based procedure is used as a shortcut in the dual ascent algorithm given by Wong [48] Another MST based heuristic was proposed by Choukmane [8] and independently by Plesnk [36] Kou, Markowsky, and Berman [28] BharathKumar and Ja#e [6], and Iwainsky et al. 24] A heuristic based on Kruskal s MST algorithm is given by Wu, Widmayer, and Wong [49] Another heuristic approach based on Prim s MST algorithm is given by Takahashi and Matsuyama [42] While all of the above MTS based approaches implicitly choose the Steiner nodes ....

K. Bharath-Kumar and J.M. Ja#e. Routing to multiple destinations in computer networks. IEEE Trans. Commun., COM-31:343-351, 1983.

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