S. C. Narula and C. A. Ho, "Degree-constrained minimum spanning trees," Computers and Operations Research, vol. 7, pp. 239--249, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the triangle inequality. In this case, a minimum spanning tree on n points may have degree up to n 1. While polynomial time heuristics exist for finding k MSTs in the plane [11; 31] they are less e#ective on the general problem. One simple but e#ective heuristic is due to Narula and Ho [22]: They modified Prim s algorithm so that at each step it includes the cheapest eligible edge one connecting a vertex currently in the (partial) spanning tree with one not yet connected that does not violate the degree constraint. We refer to this heuristic as k Prim. 3. GENETIC CODINGS OF ....

S. C. Narula and C. A. Ho. Degree-constrained minimum spanning trees. Computers and Operations Research, 7:239--249, 1980.

....no more than five. Although Papadimitriou and Vazirani [20] proved that finding a d MST in the Euclidean plane is when d = 3, and conjectured that it remains NP hard when d = 4, Euclidean problems are relatively simple to solve. Exact branch and bound techniques as described by Narula and Ho [18] and Krishnamoorthy et al. 13] can find optimal solutions even for large problem instances including several hundred vertices in reasonable computing times. Furthermore, there exist effective polynomial time heuristics for finding d MSTs in the plane [7, 25] In the more general case, the costs ....

....inequality. In this case, a minimum spanning tree may have degree up to V 1. Finding a d MST in a graph with such a high degree MST is usually a hard task. Exact approaches are too time consuming, and many existing heuristics are either not applicable or less efficient. Narula and Ho [18] proposed a simple but relatively effective heuristic: They modified Prim s algorithm so that at each step it includes the cheapest eligible edge one connecting a vertex currently in the (partial) spanning tree with one not yet connected that does not violate the degree constraint. We refer to ....

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S. C. Narula and C. A. Ho. Degree-constrained minimum spanning trees. Computers and Operations Research, vol. 7, pp. 239--249, 1980.

....depends on communication demands between each pair of nodes and on the tree s structure. In another version of this problem, edge costs are time dependent [5] Often, only spanning trees that satisfy particular constraints are feasible. Examples of such constraints include the degree constraint [6], which requires that no node in a spanning tree have more than d 2 incident edges; leaf constraints [7] which specify or bound the number of leaves a spanning tree may have; the diameter constraint [8] which bounds the longest path in a spanning tree between any two nodes; and capacity ....

.... minimum spanning tree of degree no more than five [55] Finding a d MST in the plane is NP hard when d = 3 and is conjectured to remain so when d = 4 [56] Branch and bound techniques can find exact solutions for problem instances of several hundred points in reasonable computing times [6], 18] and several authors have described e#ective polynomial time approximation schemes and heuristics [57] 58] In general, however, edge costs need not satisfy the triangle inequality. An unconstrained minimum spanning tree may have degree up to n 1, and when a graph has a highdegree ....

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S. C. Narula and C. A. Ho, "Degree-constrained minimum spanning trees," Computers and Operations Research, vol. 7, pp. 239--249, 1980.

....not always be the case in future SAN design problems. Another confounding feature of the SAN design problem is the presence of degree constraints on nodes. Degree constraints appear only in special cases of the network design problem such as the degree constrained minimum spanning tree problem [12, 11, 25], known to be NP hard [15] The many features of the SAN design problem have been addressed individually or in small subsets in the work mentioned above. The first to address all of its features in a common framework was [26] in which an algorithm called Merge was presented. Merge found ....

S.C. Narula and C.A. Ho, Degree-constrained minimum spanning tree, Computers and Operations Research 7 (1980), 239--49.

.... special case of only one vertex with a degree constraint has been examined [5, 6, 9] a polynomial time algorithm for the case of a fixed number of nodes with a constrained degree was given by Brezovec et al. 2] Computational results for some heuristics for the general problem are presented in [14, 19, 21]. Papadimitriou and Vazirani [15] raised the problem of finding the complexity of computing a minimum weight degree 4 spanning tree of points in the plane. Some geometric aspects are considered in [10, 13, 17] In this paper, we consider modifying a given spanning tree T , to meet the degree ....

S. C. Narula and C. A. Ho. Degree-constrained minimum spanning tree. Comput. Ops. Res. 7 (1980), pp. 239--249.

....best technique previously published for solving the d MST, either using heuristic or evolutionary approaches. The basis of this encoding is a spanning tree construction algorithm which we call the Randomised Primal Method (RPM) based on the well known Prim s algorithm [6] and an extension [4] which we call d Prim s . We describe a novel encoding for spanning trees, which involves using the RPM to interpret lists of potential edges to include in the growing tree. We also describe a random graph generator which produces particularly challenging d MST problems. On these and other ....

....(the best in terms of speed and memory usage for finding the MST [1] which seems to perform very well on a range of benchmark d MST problems. We describe an EA approach which outperforms this. It is based on Prim s well known greedy algorithm for spanning tree construction, in its extended form [4] which deals with degree constrained versions of the problem. We call this extended form d Prims . Interestingly, we also find that d Prim s alone is superior to Boldon et al. s dual simplex approach, but not to the EA. Our goal was to achieve fruitful hybridisation of a spanning tree ....

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Narula, S.C., Ho, C.A.: Degree-Constrained Minimum Spanning Tree. Computers and Operations Research 7(4), (1980), 39--49

....25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 Figure 3: A randomly generated 45 node network. To attend to the requirements above, a random network generator, based on a suboptimal degreeconstrained minimum spanning tree construction algorithm due to Narula and Ho [12], was designed. Due to space restrictions we cannot give details here but a fuller specification is contained in [8] The networks used in the experiments here have links of two bandwidths. Figure 3 shows one of the 45 node networks constructed using the generator. The larger circles represent ....

S. C. Narula and C. A. Ho. Degree-Constrained Minimum Spanning Tree. Computers and Operations Research, 7:39--49, 1980.

....made available at the ectelnet 2 WWW site, or can be obtained from the authors. 4. 3 Experiments: Offline Routing To generate test problems for offline routing we use a random network generator which is based on a suboptimal degree constrained minimum spanning tree construction algorithm due to Narula and Ho (1980). We use two edge bandwidths, a backbone spanning tree of edges which have a bandwidth of 64 units, through which traffic from distal nodes can be routed, and a set of local edges of bandwidth 16 units. The algorithm begins by generating B points in a two dimensional space: these are the nodes in ....

Narula, S.C. and Ho, C.A. (1980) Degree-Constrained Minimum Spanning Tree. Computers and Operations Research 7(4):39--49.

....In fact, for a given rational number R 1, finding a spanning tree of maximum degree at most d, and of total weight at most R times that of the optimal solution, is NP hard [2] Several different approaches for solving the d MST problem have been taken in previous research. Narula and Ho [3] investigated three methods, one of which (a branch and bound algorithm) is guaranteed to converge to a globally optimal solution. In their methods good upper bounds are generated by a modification to Prim s algorithm [4] which we refer to as d Prim s in this paper. Savelsbergh and Volgenant [5] ....

.... multi start hillclimbing, simulated annealing [10] and a genetic algorithm [11] The quality of solutions found by these techniques are compared with each other, and also with those achieved by the dual simplex algorithm of [8] and the upper bound generated by Narula and Ho s d Prim s method [3]. These comparisons are made on a variety of randomly generated d MST problems of from 50 to 250 nodes. Some of the random graphs used for testing have been created to be particularly difficult and misleading in order to emphasise differences in algorithm performance. The procedures by which they ....

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S.C. Narula and C.A. Ho, "Degree-Constrained Minimum Spanning Tree," Computers and Operations Research, vol. 7, no. 4, pp. 39--49, 1980.

....tree of G in terms of the edge weights w, such that the number of edges incident to each vertex i 2 V is less than or equal to b i . The decision version of the DCMST problem is NP complete [6] as far as it reduces to the shortest Hamiltonian path problem if b i = 2 for all i 2 V . Narula and Ho [13] point out an application of this problem in the design of electrical circuits, in which some terminal nodes should be connected using a minimum amount of wire, with no more than a given number of wires incident to each terminal. Savelsbergh and Volgenant [14] mention two other applications: ....

....the design of a road system in which at most four roads are allowed to meet at any crossing; second, in communication networks, where a degree constraint limits vulnerability in case of drop out of a crossing. Gavish [7] reports an application in the design of computer networks. Narula and Ho [13] propose primal and dual heuristic procedures and a branchand bound algorithm. Savelsbergh and Volgenant [14] present a branch and bound algorithm which makes use of an edge elimination procedure based on edge exchange considerations. Edge exchanges are also used in a branch and bound algorithm ....

S.C. Narula and C.A. Ho, "Degree-constrained minimum spanning tree", Computers and Operations Research 7 (1980), 239--249.

....by at least two paths from every other node, so that if a node or edge fails there is a good chance that all traffic can still be accommodated. We use a random network generator which is based on a suboptimal degree constrained minimum spanning tree construction algorithm due to Narula and Ho [19]. We use two edge bandwidths, a backbone spanning tree of edges which have a bandwidth of 64 units, through which traffic from distal nodes can be routed, and a set of local edges of bandwidth 27 units. The algorithm begins by generating B points in a two dimensional space: these are the nodes ....

Narula, S.C., Ho, C.A. (1980) Degree-Constrained Minimum Spanning Tree. Computers and Operations Research 7(4), pp. 39--49.

.... special case of only one vertex with a degree constraint has been examined [5, 6, 9] A polynomial time algorithm for the case of a fixed number of nodes with a constrained degree was given by Brezovec et al. 2] Computational results for some heuristics for the general problem are presented in [14, 19, 21]. Papadimitriou and Vazirani [15] raised the problem of finding the complexity of computing a minimum weight degree 4 spanning tree of points in the plane. Some geometric aspects are considered in [10, 13, 17] In this paper, we consider modifying a given spanning tree T , to meet the degree ....

S. C. Narula and C. A. Ho. Degree-constrained minimum spanning tree. Comput. Ops. Res. 7 (1980), pp. 239-249.

.... special case of only one vertex with a degree constraint has been examined [5, 6, 9] a polynomial time algorithm for the case of a fixed number of nodes with a constrained degree was given by Brezovec et al. 2] Computational results for some heuristics for the general problem are presented in [14, 19, 21]. Papadimitriou and Vazirani [15] raised the problem of finding the complexity of computing a minimum weight degree 4 spanning tree of points in the plane. Some geometric aspects are considered in [10, 13, 17] In this paper, we consider modifying a given spanning tree T , to meet the degree ....

S. C. Narula and C. A. Ho. Degree-constrained minimum spanning tree. Comput. Ops. Res. 7 (1980), pp. 239--249.

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S. C. Narula and C. A. Ho, "Degree-constrained minimum spanning trees," Computers and Operations Research, vol. 7, pp. 239--249, 1980.

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S.C. Narula and C.A. Ho. Degree-constrained minimum spanning tree. Computers and Operations Research, 7(4):39-49, 1980. 312

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