P. Winter, J. M. Smith, Path-distance heuristics for the Steiner problem in undirected networks, Algorithmica (1992) 309--327.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is a Steiner tree produced by A and c : c(T ) is the cost of a minimum cost Steiner tree T for the same instance. Note that recently Zelikovsky [21] announced an 11 6 approximation algorithm. However, we do not study his heuristic. Also very recent combined (2 approximation) heuristics from [5, 14, 20] remain undiscussed here. There are several experimental studies [12, 14, 16, 20] comparing various heuristics. One of the best heuristics is that developed by Rayward Smith [11, 12] In Section 3 we present and analyse a modification which usually takes a less number of iterations than the ....

....Steiner tree T for the same instance. Note that recently Zelikovsky [21] announced an 11 6 approximation algorithm. However, we do not study his heuristic. Also very recent combined (2 approximation) heuristics from [5, 14, 20] remain undiscussed here. There are several experimental studies [12, 14, 16, 20] comparing various heuristics. One of the best heuristics is that developed by Rayward Smith [11, 12] In Section 3 we present and analyse a modification which usually takes a less number of iterations than the original Rayward Smith heuristic does. Note that before actually solving the Steiner ....

Winter P. and Smith MacGregor J., Path-distance heuristics for the Steiner problem in undirected networks, Algorithmica (to appear).

....here, may need to be incorporated in the criteria 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 ....

P. Winter and J. MacGregor Smith, "Path-distance heuristics for the Steiner problem in undirected networks," Algorithmica, vol 7, no. 2--3, pp. 309--327, 1992.

....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 ....

.... chose the following four heuristics as candidates for distributed implementation: the shortest path heuristic (SPH) a variant of SPH known as SPH Z, the Kruskal based shortest path heuristic (K SPH) and the Average distance heuristic (ADH) Each heuristic s unconstrained version is described in [22] and each heuristic s degree constrained version is described in [1] A brief summary of each degree constrained heuristic follows. DCSP SPH Heuristic SPH, whose unconstrained version was introduced in [23] initializes the multicast tree to an arbitrary multicast member. It then grows the tree ....

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M. Smith and P. Winter. "Path-distance heuristics for the Steiner problem in undirected networks," Algorithmica, vol. 7, no. 2-3, pp. 309--327, 1992. 27

....for the online, low cost, delay bounded multicast problem. A conclusion of this work is presented in the last section. 2 RELATED WORK Many different algorithms were proposed for the offline multicasting problem. Most of these proposed algorithms aimed at reducing the cost of the multicast tree [22, 1]. Few other algorithms were proposed that bounds the delay while reducing the cost. Based on their design objectives, the multicast algorithms proposed in the literature can be viewed as members of one of three possible classes. The first class includes algorithms which are designed to ....

....thereby allowing a tradeoff between delay and cost. Thus, the objective is to produce a tree that has minimal cost among all possible trees that bound end toend traffic delay between all source destination pairs. Many heuristics were developed for the low cost unbounded delay multicast problem [22, 1]. However, there are few attempts to develop low cost boundeddelay multicast heuristics. In the following, we review off line, low cost, delay bounded, multicast heuristics. A simple approach to solving this problem is to use a tree that is composed of the least delay paths (LDPs) from the source ....

Pawel Winter and J. MacGregor Smith. Pathdistance heuristics for the steiner problem in undirected networks. Algorithmica, 7:309--327, 1992.

.... 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 ....

M. Smith and P. Winter, "Path-distance heuristics for the steiner problem in undirected networks.," Algorithmica 7(2-3), pp. 309--327, 1992.

....the proposed solution methods are exact algorithms, heuristic procedures, approximation algorithms, polynomial algorithms for special instances, polyhedral approaches, and many more. Excellent surveys are given in Winter [1987] Maculan [1987] Hwang and Richards [1992] and Hwang, Richards, and Winter [1992]. To solve the Steiner tree problem to optimality, Aneja [1980] proposes a row generation algorithm based on an undirected formulation, Dreyfus and Wagner [1971] and Lawler [1976] use dynamic programming techniques, Beasley [1984, 1989] presents a Lagrangean relaxation approach, Wong [1984] ....

....without refilling if you choose among all possible routes. Note that the following relations s(u; v) d(u; v) c [u;v] hold, where d(u; v) denotes the length of a shortest path between u and v. The special distance can be computed by a modified shortest path algorithm (cf. Hwang, Richards, and Winter [1992]) Given the values s(u; v) for all u; v 2 V there is an easy and very effective test for deleting edges. An optimal solution S of a Steiner tree problem ST(G; T; c) cannot contain any edge [u; v] 2 E with s(u; v) c [u;v] The Special Distance Test is published in Duin and Volgenant ....

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Winter, P. and Smith, J. M. (1992). Path-distance heuristics for the steiner problem in undirected networks. Algorithmica, pages 309--327.

....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 ....

....set threshold, a rearrangement is triggered. Although any static Steiner heuristic can be used to perform the rearrangement, we use the Kruskal shortest path heuristic (KSPH) for a number of reasons. First, it is naturally suited to constructing multicast trees by combining fragments of the tree [1, 11, 16]. Second, the algorithm lends itself to distributed, asynchronous implementation [2] Finally, K SPH has been shown to # Given existing tree T i Gamma1 , find new tree T i taking into account request r i case r i of add request for node v: if v 62 T i Gamma1 then # Join node v to T i Gamma1 ....

M. Smith and P. Winter. "Path-distance heuristics for the Steiner problem in undirected networks," Algorithmica, vol. 7, no. 2-3, pp. 309--327, 1992.

....algorithmic complexity 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 heuristics exist for the SPN and have been reviewed extensively elsewhere [5, 12, 16, 20, 19, 26, 27]. Some have been shown through analysis to produce solutions no worse than twice the optimal solution [27] Our empirical evidence indicates that these heuristics find solutions much better than twice the optimal with reasonable speed in most cases. Finding a multicast tree is complicated by the ....

....Likewise, the distance between a node and a tree is the distance of the shortest path between the node and any node in the tree. Finally, the distance between two trees is the distance of the shortest among all paths between any node in one tree and any node in the other tree. Smith and Winter [19] divide Steiner heuristics into a morphological structure similar to the one shown in Figure 1. At the highest level, heuristics are divided between those that are path distance heuristics (PDH) and others. Path distance heuristics rely on distance calculations and iteratively improve an initial ....

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M. Smith and P. Winter. "Path-distance heuristics for the Steiner problem in undirected networks," Algorithmica, vol. 7, no. 2-3, pp. 309--327, 1992.

....developed for the SPG. Exact algorithms can be found in e.g. 2, 3, 5, 8, 13, 23, 26] However, since the SPG is NP complete [19] these algorithms have exponential worst case time complexities. Therefore, a significant research e#ort has been directed towards polynomial time heuristics, cf. e.g. [2, 20, 24, 25, 27, 31]. Simulated annealing has also been applied to SPG [7] The Rectilinear Steiner Problem (RSP) is an important special case of SPG [14] which is still NP complete [11] While at least two genetic algorithms for RSP have been published [15, 17] we are aware of only one previous genetic algorithm ....

....This test suite consists of randomly generated graphs with up to 2,500 vertices and 62,500 edges. The obtained performance is compared to that of the GA by Kapsalis et al. 18] an iterated version of the Shortest Path Heuristic called SPH I, which is one of the very best deterministic heuristics [31], and two recent branch and cut algorithms by Lucena and Beasley [23] and Chopra, Gorres and Rao [5] The experimental results shows the following: The GA presented here clearly outperforms the GA in [18] with respect to solution quality as well as runtime. The solution quality obtained ....

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Pawel Winter, J. MacGregor Smith, "Path-Distance Heuristics for the Steiner Problem in Undirected Networks, " Algorithmica, Vol. 7, pp. 309-327, 1992.

....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 [3, 8, 10, 15, 16, 17, 20]. Some have been shown through analysis to produce solutions no worse than twice the optimal solution [20] Empirical evidence from our previous papers indicate that these heuristics find solutions much better than twice the optimal with reasonable speed in most cases [1] Most of the algorithms ....

.... in [1] we chose the following four heuristics as candidates for distributed implementation: the shortest path heuristic (SPH) a variant of SPH known as SPH Z, the Kruskal based shortest path heuristic (K SPH) and the Average distance heuristic (ADH) Each of these heuristics is described in [15]. A brief summary of heuristics SPH and K SPH follows. SPH Heuristic SPH, introduced in [16] initializes the multicast tree to an arbitrary multicast member. It then grows the tree by successively adding the next closest multicast member to the multicast tree by the shortest path between the ....

M. Smith and P. Winter. "Path-distance heuristics for the Steiner problem in undirected networks," Algorithmica, vol. 7, no. 2-3, pp. 309--327, 1992.

....and z the number of multicast group members. Steiner tree heuristics, in contrast, are relatively inexpensive and produce very good solutions. For this reason, we focus on Steiner heuristics. A number of good, inexpensive heuristics exist for the SPN and have been reviewed extensively elsewhere [9, 20, 27, 40, 38, 46, 50]. Some have been shown through analysis to produce solutions whose cost is no worse than twice that of an optimal solution [50] However, relatively little effort has been made to develop Steiner heuristics that satisfy the constraints considered here. We define the unconstrained Steiner problem ....

....chapter are used throughout the dissertation, we provide an in depth description of each in this chapter. While modifications to suit specific constraints will be noted in each chapter, the heuristics used remain similar to the unconstrained, centralized versions described below. Smith and Winter [38] divide Steiner heuristics into a morphological structure similar to the one shown in Figure 2.1. At the highest level, heuristics are divided between those that are path distance heuristics (PDH) and others. Path distance heuristics rely on distance calculations and iteratively improve an initial ....

[Article contains additional citation context not shown here]

M. Smith and P. Winter. "Path-distance heuristics for the Steiner problem in undirected networks," Algorithmica, vol. 7, no. 2-3, pp. 309--327, 1992.

....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 heuristics exist for the SPN and have been reviewed extensively elsewhere [4] 8] 12] 16] [15], 22] 23] Some have been shown through analysis to produce solutions no worse than twice the optimal solution [23] Our empirical evidence indicates that these heuristics find solutions much better than twice the optimal with reasonable speed in most cases. Finding a multicast tree is ....

....Likewise, the distance between a node and a tree is the distance of the shortest path between the node and any node in the tree. Finally, the distance between two trees is the distance of the shortest among all paths between any node in one tree and any node in the other tree. Smith and Winter [15] divide Steiner heuristics between those that are path distance heuristics (PDH) and others. Path distance heuristics rely on distance calculations and iteratively improve an initial partial solution using appropriately chosen shortest paths between multicast members until the partial solution ....

[Article contains additional citation context not shown here]

M. Smith and P. Winter. "Path-distance heuristics for the Steiner problem in undirected networks", Algorithmica, vol. 7, no. 2-3, pp. 309--327, 1992.

....22 through the application of the shortest path heuristic SPH, using a new node as the root. This diversification strategy is based on the sensitivity of the shortest path heuristic of Takahashi and Matsuyama [21] with respect to the starting node s 0 , as pointed out by Winter and MacGregor Smith [26]. 7. Computational Results. In this section, we present numerical results obtained with the application of the improved tabu search algorithm described in Section 6. The algorithm was implemented in C, using version 5.5.1 of the SunOS cc compiler with the optimzation flag set to O. All ....

P. Winter and J. MacGregor Smith, "Path-distance heuristics for the Steiner problem in undirected networks", Algorithmica 7 (1992), 309--327.

....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 [2, 6, 10, 13, 14, 15, 16, 19]. Approaches to creating multicast trees that do not use Steiner tree heuristics have been proposed elsewhere. Three notable papers of alternative approaches include DVMRP [7] PIM [8] and core based trees [1] These approaches are appropriate for datagram environments such as the Internet in ....

....set threshold, a rearrangement is triggered. Although any static Steiner heuristic can be used to perform the rearrangement, we use the Kruskal shortest path heuristic (K SPH) for a number of reasons. First, it is naturally suited to constructing multicast trees by combining fragments of the tree [2, 14, 19]. Second, the algorithm lends itself to distributed, asynchronous implementation [3] Finally, K SPH has been shown to be among the best of Steiner heuristics in terms of the cost of the multicast trees produced in our previous evaluations [2, 3] Using K SPH, the rearrangement algorithm proceeds ....

M. Smith and P. Winter. "Path-distance heuristics for the Steiner problem in undirected networks", Algorithmica, vol. 7, no. 2-3, pp. 309--327, 1992.

....of efficient, quality heuristics remains an active research area. The problem has been addressed both in a multicast communication context [12, 26] and as a purely graph theoretic problem [15, 19] Most of the well known heuristics fall into the class known as path distance heuristics [27] since they approach the problem by iteratively enlarging partial solutions using shortest paths. The three primary path distance approaches are the shortest path heuristic (SPH) 28] average distance heuristic (ADH) 29] and distance network heuristic (DNH) 17] In our study we consider only ....

P. Winter and J. M. Smith, "Path-distance heuristics for the Steiner problem in undirected networks," Algorithmica, vol. 7, pp. 309--327, 1992.

....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, 15, 20, 25, 26, 32, 33]. Some have been shown through analysis to produce solutions no worse than twice the optimal solution. 33] That is to say, 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 ....

.... chose the following four heuristics as candidates for distributed implementation: the shortest path heuristic (SPH) a variant of SPH known as SPH Z, the Kruskal based shortest path heuristic (K SPH) and the Average distance heuristic (ADH) Each heuristic s unconstrained version is described in [25] and each heuristic s degree constrained version is described in [1] A brief summary of each degree constrained heuristic follows. DCSP SPH Heuristic SPH, whose unconstrained version was introduced in [26] initializes the multicast tree to an arbitrary multicast member. It then grows the tree ....

[Article contains additional citation context not shown here]

M. Smith and P. Winter. "Path-distance heuristics for the Steiner problem in undirected networks," Algorithmica, vol. 7, no. 2-3, pp. 309--327, 1992.

....used to produce an exact solution to the undirected SMT problem and the algorithms used to solve the minimum spanning tree problem. Heuristics in this class can be further categorized as Shortest Path Heuristics (SPH) Distance Network Heuristics (DNH) and Average Distance Heuristics (ADH) [17]. Each category is characterized by the metric used in calculating distances and the approach used to enlarge partial solutions by adding shortest paths between appropriately selected vertices. 3.1 Shortest Path Heuristics The TM heuristic, along with several derivatives of this heuristic, are ....

....depicts the proposed example for n = 6. Based on the TM heuristic, the cost of the generated tree is (n Gamma 1) Theta, whereas he cost of the cost of the optimal tree is (n Gamma 1) Theta (1 ffl) An approach to improve the performance of the TM heuristic is to run TM jSj times, iteratively [17]. At the beginning of each iteration, the TM heuristic selects a different node in S to produce a new tree consisting of 1 This bound only applies to graphs with symmetric, undirected edges. For directed graphs, the bound can be worse [10] 2 3 4 5 6 1 2 2 2 2 2 1 e 1 e 1 e 1 e 1 e Figure 1: ....

Pawel Winter and J. MacGregor Smith. Pathdistance heuristics for the steiner problem in undirected networks. Algorithmica, 7:309--327, 1992.

.... 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 [2] 6] 8] 11] [12], 13] 14] 17] 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 ....

....threshold, a rearrangement is triggered. Although any static Steiner heuristic can be used to perform the rearrangement, we use the Kruskal shortest path heuristic (K SPH) for a number of reasons. First, it is naturally suited to constructing multicast trees by combining fragments of the tree [2] [12], 17] Second, the algorithm lends itself to distributed, asynchronous implementation [3] Finally, K SPH has been shown to be among the best of Steiner heuristics in terms of the cost of the multicast trees produced in our previous evaluations [2] 3] Using K SPH, the rearrangement algorithm ....

M. Smith and P. Winter. "Path-distance heuristics for the Steiner problem in undirected networks", Algorithmica, vol. 7, no. 2-3, pp. 309--327, 1992.

....as the Steiner problem in networks [3] 7] 12] referred to hereafter as the SPN, and that finding explicit solutions in large networks is prohibitively expensive. A number of good, inexpensive, centralized heuristics exist for the SPN and have been reviewed extensively elsewhere [3] 7] [9], 10] 11] 12] Most of the algorithms proposed in the literature for SPN are serial in nature. However, a few distributed heuristics exist in the literature [4] 8] Many of these algorithms are based on reducing the SPN to the minimum spanning tree problem, referred to here as the MST, and ....

M. Smith and P. Winter. "Path-distance heuristics for the Steiner problem in undirected networks," Algorithmica, vol. 7, no. 2-3, pp. 309--327, 1992.

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P. Winter, J. M. Smith, Path-distance heuristics for the Steiner problem in undirected networks, Algorithmica (1992) 309--327.

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