V.J. Rayward-Smith, The computation of nearly minimal Steiner trees in graphs, Int. J. Math. Educ. Sci. Technol. 14 (1983), 15--23.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... (t) t c i (t) where c i (t) is defined as the difference between the numbers of connected components of (R, t 1 , t i ) and (R, t, t 1 , t i ) Note that in quasi bipartite graphs, algorithm Greedy MSS is identical to Rayward Smith s average distance heuristic [14]. 3 Performance Analysis In general the minimum spanning subhypergraph problem is even harder than the Steiner tree problem. Wolsey [17] showed that the greedy algorithm achieves a performance ratio of H(k 1) if the size of every hyperedge is bounded by k. Here H(k) # k i=1 1 i = Input ....

V.J. Rayward-Smith, The computation of nearly minimal Steiner trees in graphs, Int. J. Math. Educ. Sci. Technol. 14 (1983), 15--23.

....B, VPN, i.e. when the topology is a tree and the customer has not specified any subgraph properties, the searching problem is the same as SMT. This is exactly the problem of determining an optimal multicasting tree in networks, and several heuristic algorithms have been proposed and analyzed [6] [8]. The algorithm most appropriate to the VPN design process is a polynomial time algorithm for finding a directed SMT called selective closest terminal first (SCTF) 9] For a category C, VPN, where the topology is a source based tree and the customer has specified one or more subgraph properties, ....

V. J. Rayward-Smith, "The computation of nearly minimal Steiner trees in graphs," Int. J. Math. Ed. Sci. Technol., vol. 14, no. 1, pp. 15--23, Jan.--Feb. 1983.

....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 problem in practice, some tests may reduce the Received May 13, 1991; revised September 2, 1991. 1980 ....

....complicated) bound than 2 Gamma 2=p was derived in [10] CHR (the revised contraction heuristic) It was suggested and analysed by Plesn ik [10] The values of parameters and ae are the same as for CH. ADH (the minimum average distance heuristic) This heuristic was suggested by Rayward Smith [11, 12]. It can be found also in [10, 19] but for our aims in Section 3 we give its full description here. Step 1: Begin with the collection F of single vertex trees consisting of the p Z vertices. For the sake of simplicity F is called a forest. Step 2: For every vertex v 2 V relabel the trees in the ....

Rayward-Smith V. J., The computation of nearly minimal Steiner trees in graphs, Int. J. Math. Educ. Sci. Technol. 14 (1983), 15--23.

.... that the problem is at least as hard to approximate as the set cover problem, for which a polynomial time approximation algorithm with a factor of (1 Gamma ffl) ln k would imply that NP DTIME[n O(loglogn) 4] The Klein Ravi algorithm [9] is based on an earlier heuristic by Rayward Smith [12] and may be viewed as a generalization of the set cover greedy approach [3] In this scheme, at each step a spider is chosen so as to minimize the ratio of the weight of spider to the number of terminals that it connects. They prove that the process of greedily picking spiders yields a good ....

V. J. Rayward-Smith, "The computation of nearly minimal Steiner trees in graphs", Internat. J. Math. Educ. Sci. Tech., 14: 15--23, (1983).

.... polynomial time heuristics for the Steiner problem [13,3] and it is the purpose of this paper to present three new such heuristics which practically compare favorably to several known ones (including the spanning tree heuristic [1,6,9] the path heuristic [12] and the average distance heuristic [10,11]) and have the same theoretical worst case performance. First, in Section 1 we give a heuristic based on the spanning tree heuristic. In Section 2 we present a heuristic which chooses vertices for a tree according to their sum of all distances to the special vertices. Since a vertex with the ....

Rayward-Smith V. J., The computation of nearly minimal Steiner trees in graphs, Int. J. Math. Educ. Sci. Technol. 14 (1983), 15--23.

....function f . The minimum spanning tree heuristic. The class K consists of all paths in G and f(T ) d(T ) The exact bounds for a 1 were proved in [7, 13] An equality a 2 = 1 follows from the famous fact that the greedy algorithm finds exact MST of a graph. The Rayward Smith s heuristic (RSH) [12]. The class K contains stars and f(T ) d(T ) r Gamma1 , where r is the number of leaves of T . For NSP, exact upper bounds a 1 5=3 and a 1 Delta a 2 2 were proved in [15, 16] and [14] respectively. The generalized greedy heuristic (GGH) 16] The class K consists of trees with k ....

V. J. Rayward-Smith, The computation of nearly minimal Steiner trees in graphs, International J. Math. Ed. Sci. Tech. 14: 15--23, 1983.

....and the typical exact solution can be obtained by some enumeration of these trees. Some obvious enumeration schemes can assist in reducing the actual number of trees examined, but the problem still remains intractable [47] Many approximate solutions to the Steiner tree problem have been devised [48, 49, 39, 50, 51]. The Steiner tree has a nice property in that it is possible to find an approximate solution that lies within a constant factor of the optimal cost [52] All the approximation algorithms cited above produce solutions that come within a factor of two of the optimal cost. An integer programming ....

V. Rayward-Smith, "The computation of nearly minimal Steiner trees in graphs," Intl. J. Math. Educ. Sci. Tech., vol. 14, no. 1, pp. 15--23, 1983. 155

....There are several other heuristics [21] that try to reduce the cost of the tree by increasing the shared portion of the path. The best known heuristics were proposed by Kou, Markowski, and Berman (KMB heuristic) 8] Takahashi and Matsuyama 2 (TM heuristic) 15] and Rayward Smith (RS heuristic) [12]. PIM SM [13] and CBT [2] are examples of routing protocols that use shared tree approach. It has been shown that the cost of the tree generated by these heuristics are at most twice as much as the optimum tree. In contrast, the ratio of the cost of the tree generated by SPH and the cost of the ....

V. Rayward-Smith. The Computation of Nearly Minimal Steiner Trees in Graphs. International Journal of Mathematical Education in Science and Technology, 14(1):15-- 23, January/February 1983.

....case is known in the literature as the Steiner tree problem in networks. This problem was one of the first seven problems shown NP complete by Karp [19] Given the range of its applications, it is not surprising that this problem has been well studied. Many enumeration algorithms, heuristics [33, 44, 18], and approximation algorithms [6, 38, 25, 11, 31, 37, 29, 45] are known for the problem. Polynomial time solutions for restricted classes of graphs are also known (see [42] However, none of the algorithms addresses the more general case in which each client can specify an arbitrary pair of ....

....and worst case analyses, and algorithms for special classes of graphs. Winter [41] and more recently Hwang and Richards [17] surveyed this body of work. Karp [19] showed that the problem is NP complete. Takahashi and Matsuyama [38] Kou, Markowsky and Berman [25] El Arbi [11] RaywardSmith [33], Aneja [2] and Wong [44] are among those who proposed heuristics. Among these, the heuristics that have been analyzed have a worst case performance ratio of 2 Gamma 2=k, where k is the number of terminals that need to be connected (called Z vertices in [41] One algorithm, proposed by Plesnik ....

V. J. Rayward-Smith, "The computation of nearly minimal Steiner trees in graphs," Int. J. Math. Educ. Sci. Tech., vol. 14 (1983), pp. 15-23.

....in this chapter were obtained jointly with Philip Klein, and appeared in [91] 4.1 Problem Definition The Steiner tree problem in networks that we introduced in Section 1.1 is a classic hard problem in combinatorial optimization. Much research has been devoted to heuristics for its solution [33, 68, 112, 134, 135, 163]. Despite a slew of new approximation algorithms for this problem and some of its variants, no approximation algorithm has been given for perhaps the most natural variant: the node weighted Steiner tree problem, in which costs can be assigned to nodes as well as edges. Indeed, Winter s survey ....

....class Deterministic Quasi polynomial time, or DTIME[n polylog n ] 49 of our approximation is within a constant factor of the best possible approximation achievable in polynomial time. The algorithm we propose is only a slight variant of a heuristic proposed by Rayward Smith and Clare in 1986 [74, 135, 134] for the standard edge weighted Steiner tree problem. The key to our analysis is a decomposition lemma for trees; this lemma may be useful in other contexts as well. We also show how to generalize the algorithm and its analysis to handle more general connectivity requirements modeled using the ....

V. J. Rayward-Smith, "The computation of nearly minimal Steiner trees in graphs," Internat, J. Math. Ed. Sci. Tech., 14 (1), pp. 15-23 (1983).

....a constant factor of the best possible unless P NP . Our algorithm generalizes to handle other network design problems. 1 Introduction The Steiner problem in networks is a classic hard problem in combinatorial optimization. Much research has been devoted to heuristics for its solution [4, 7, 14, 16, 17, 22]. Despite a slew of new approximation algorithms for this problem and some of its variants, no approximation algorithm has been given for perhaps the most natural variant: the node weighted Steiner tree problem, in which costs can be assigned to nodes as well as edges. Indeed, Winter s survey [21] ....

....guarantee is logarithmic. Thus assuming P 6 NP , the accuracy of our approximation is within a constant factor of the best possible approximation achievable in polynomial time. The algorithm we propose is only a slight variant of a heuristic proposed by Rayward Smith and Clare in 1986 [8, 17, 16] for the standard edge weighted Steiner tree problem. The key to our analysis is a decomposition lemma for trees; this lemma may be useful in other contexts as well. We also show how to generalize the algorithm and its analysis to handle more general connectivity requirements. Thus we obtain ....

V. J. Rayward-Smith, "The computation of nearly minimal Steiner trees in graphs," Internat, J. Math. Ed. Sci. Tech., 14 (1), pp. 15-23 (1983).

....is specified by some parameter (default is every 5 cutting plane iterations) In 116 out of 334 test examples the first call to the heuristic found the optimal solution and in 88 of the cases the gap ( heuristic solution lower bound lower bound ) was below 5 . We also experimented with the Rayward Smith [1983] heuristic. The results are quite promising, however a main bottleneck is the running time, especially for big problems. The reason is that the heuristic requires all to all node distances and due to memory limitations we must compute these on the fly, so most of the time is spent for ....

Rayward-Smith, V. J. (1983). The computation of nearly minimal steiner trees in graphs. Int. J. Math. Educ. Sci. Technol., 14:15--23.

....Steiner Minimal Tree (SMT) 1, 2] problem. The SMT problem has been well studied from a purely graph theoretical point of view and a good survey can be found in [2, 3] In particular, it has been shown that optimizing SMT is NP complete [4] However, several polynomial time near optimum solutions [5, 6, 7, 8, 9] exist, the most straightforward ones and the ones of most interest here being [5] and [6] Using dissimilar approaches, both guarantee solutions that are at worst twice the optimum. The best known performance guarantee is 11 6 times optimum, and is achieved by a more involved solution given in ....

V.J. Rayward-Smith, "The computation of nearly minimal Steiner trees in graphs", Int. J. Math. Educ. Sci. Technol. 14 (1983) 15-23.

....Steiner tree from trees of LIST . Many famous heuristics can be embedded in this framework considering different definitions of a class K and a criterion function f . The minimum spanning tree heuristic (MSTH) 13] K consists of all paths and f(T ) d(T ) The Rayward Smith s heuristic (RSH) [12]. K contains all stars and f(T ) d(T ) r Gamma1 , where r is the number of leaves of T . The generalized greedy heuristic (GGH) 16] K consists of trees with 3 terminals and f(T ) d(T ) Gamma (M 0 (S) Gamma M 0 (S=T ) The size restricted relative greedy heuristic (SRGH) 17] K = ....

V. J. Rayward--Smith, The computation of nearly minimal Steiner trees in graphs, in: International J. Math. Ed. Sci. Tech. 14 (1983), pp. 15--23.

....(pt mtp) connections, a minimum cost spanning tree can be constructed with the same polynomial complexity. However, for multicast connections, even the construction of unconstrained Steiner Trees is an NP Complete problem. Although the problem of constructing Steiner trees is hardly new (e.g. [Smith83]) its study has been intensified in the last ten years due to its direct applicability to multimedia networks [Frank85] Moreover, multimedia connections are likely to add constraints to feasible trees, such as bandwidth and delay. Research work on the construction of Steiner trees with such ....

V. J. Rayward-Smith, "The Computation of Nearly Minimal Steiner Trees in Graphs", Intl. J. Math. Educ. Sci. Tech., Vol. 14(1), pp. 15-23, 1983.

.... 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 (optional nodes in the Steiner tree) an average distance heuristic that selects those nodes is given by Rayward Smith [39]; refer also to Rayward Smith and Clare [40] and Foulds and Rayward Smith [18] A di#erent heuristic based on neighborhood contraction techniques is provided by Plesnk [36] The reader is referred to Winter [47] Hwang and Richards [22] Hwang, Richards, and Winter [23] and Cieslik [9] for ....

V.J. Rayward-Smith. The computation of nearly minimal Steiner trees in graphs. Int. J. Math. Ed. Sci. Technol., 14:15--23, 1983.

....paths from the source to destination nodes; this provides the optimal solution for delay optimization. Multicast routing algorithms that perform cost optimization have been based on minimum Steiner tree which is known to be NP complete problem [6] Some heuristics for the Steiner tree problem [9, 14, 18] have been developed that take polynomial time and produce near optimum results. In Kou, Markowsky and Berman s (KMB) algorithm [9] a network is abstracted to a complete graph consisting of edges that represent the shortest paths among source node and destination nodes. The KMB algorithm applies ....

V. Rayward-Smith. The computation of nearly minimal steiner trees in graphs. Intl. J. Math. Educ. Sci. Tech., 14(1):15--23, 1983.

....D of a graph G = V; E) with edge weights is know as the Steiner Problem for graphs [Ha71] A Steiner tree is thus the optimal shared tree if the sum of the edge weights is to be minimized. The problem has been well studied and is known to be an NP complete problem. Reasonable heuristics exist [TaMa80, KMB81, Ra83], however, for finding an approximate solution in polynomial time. The time complexity of these heuristics is typically O(jV j 3 ) For example, we have effectively used the Takahashi Matsuyama heuristic [TaMa80] which is O(jDjjV j 2 ) to determine approximate Steiner trees [AgRa94] Thus, ....

V. Rayward-Smith, The computation of nearly minimal Steiner trees in graphs, Int. J. Math. Educ. Sci. Tech., vol. 14, pp. 15-23, 1983.

....cost optimization have been based on computing the minimum Steiner tree in a graph. A Steiner tree in a graph must reach a subset of nodes in the graph, called given nodes. The problem of finding the minimum Steiner tree is known to be an NP complete problem [12] and a number of heuristics [15, 19, 22] have been developed to solve this problem in polynomial time and producing near optimum results. In Kou, Markowsky and Berman s (KMB) algorithm [15] a network is abstracted to a complete distance graph consisting of edges that represent the shortest paths between the source node and each ....

V. Rayward-Smith. The Computation of Nearly Minimal Steiner Trees in Graphs. Intl. J. Math. Educ. Sci. Tech., 14(1):15--23, 1983.

.... tree [5] and the problem of finding a Steiner tree is known to be NP complete [7] Furthermore, the problem remains NP complete, even if edges have unit cost [4] Several algorithms that construct low cost multicast routes [1, 6, 13, 14] are based on heuristics for approximate Steiner trees [10, 11, 12]. Empirical observations show that the heuristics produce near optimal trees quickly. The algorithms take polynomial time, ranging from O(n 3 ) to O(n 4 ) Furthermore, they produce solutions that are provably within twice the cost of the optimal solution. The two measures C T and D T are ....

V.J. Rayward-Smith. The computation of nearly minimal Steiner trees in graphs. Intl. J. Math. Educ. Sci. Tech., 14(1):15--23, 1983.

.... that the problem is at least as hard to approximate as the set cover problem, for which a polynomial time approximation algorithm with a factor of (1 Gamma ffl) ln k would imply that NP DTIME[n O(loglogn) 4] The Klein Ravi algorithm [9] is based on an earlier heuristic by Rayward Smith [12] and may be viewed as a generalization of the set cover greedy approach [3] In this scheme, at each step a spider is chosen so as to minimize the ratio of the weight of spider to the number of terminals that it connects. They prove that the process of greedily picking spiders yields a good ....

V. J. Rayward-Smith, "The computation of nearly minimal Steiner trees in graphs", Internat. J. Math. Educ. Sci. Tech., 14: 15--23, (1983).

....connections, a minimum cost spanning tree can be constructed with the same polynomial complexity. However, for truly multicast connections (mtp mtp) we are indeed facing a NP Complete problem, as will be shown shortly. Although the problem of constructing Steiner trees is hardly new (e.g. [Smith83]) its study has intensified in the last ten years due to its direct applicability to multimedia networks [Frank85, Makki95] Moreover, multimedia connections are likely to add some constraints in feasible trees, such as bandwidth and delay. Research work on the construction of Steiner Trees with ....

V. J. Rayward-Smith, "The Computation of Nearly Minimal Steiner Trees in Graphs", Intl. J. Math. Educ. Sci. Tech., Vol. 14(1), pp. 15-23, 1983.

....resources can be efficiently managed. Since the problem of computing minimum cost multicast trees (called Steiner trees) in a network is NP complete [8] most of the proposed algorithms are heuristic ones. Some of the well known Steiner tree heuristics are the KMB heuristic [9] the RS heuristic [10] and the TM heuristic [11] Recently, methods based on neural networks [13] and genetic algorithms [16] have also been proposed for computing low cost multicast trees. As more and more delay sensitive networking applications such as video conferencing, teleteaching and many other distributed ....

V. Rayward-Smith, "The Computation of Nearly Minimal Steiner Trees in Graphs", International Journal of Mathematical Education in Science and Technology, vol. 14, no. 1, pp. 15-23, 1983.

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V. J. Rayward-Smith, The computation of nearly minimal Steiner trees in graphs. International J. Math. Ed. Sci. Tech. 14: 15--23, 1983.

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V. J. Rayward-Smith, "The computation of nearly minimal Steiner trees in graphs", Int. J. Math. Educ. Sci. Tech., vol. 14 (1983), pp. 15-23. 5

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