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RandomTree Diameter and the DiameterConstrained MST
 MST,” Congressus Numerantium
, 2000
"... A minimum spanning tree (MST) with a small diameter is required in numerous practical situations. It is needed, for example, in distributed mutual exclusion algorithms in order to minimize the number of messages communicated among processors per critical section. Understanding the behavior of tre ..."
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Cited by 9 (1 self)
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A minimum spanning tree (MST) with a small diameter is required in numerous practical situations. It is needed, for example, in distributed mutual exclusion algorithms in order to minimize the number of messages communicated among processors per critical section. Understanding the behavior of tree diameter is useful, for example, in determining an upper bound on the expected number of links between two arbitrary documents on the World Wide Web. The DiameterConstrained MST (DCMST) problem can be stated as follows: given an undirected, edgeweighted graph G with n nodes and a positive integer k, find a spanning tree with the smallest weight among all spanning trees of G which contain no path with more than k edges. This problem is known to be NPcomplete, for all values of k; 4 k #n  2). In this paper, we investigate the behavior of the diameter of MST in randomlyweighted complete graphs (in ErdsRnyi sense) and explore heuristics for the DCMST problem. For the case when the diameter bound k is smallindependent of n, we present a onetimetreeconstruction (OTTC) algorithm. It constructs a DCMST in a modified greedy fashion, employing a heuristic for selecting an edge to be added to the tree at each stage of the tree construction. This algorithm is fast and easily parallelizable. We also present a second algorithm that outperforms OTTC for larger values of k. It starts by generating an unconstrained MST and iteratively refines it by replacing edges, one by one, in the middle of long paths in the spanning tree until there is no path left with more than k edges. As expected, the performance of this heuristic is determined by the diameter of the unconstrained MST in the given graph. We discuss convergence, relative merits, and implementation of t...
Computing A DiameterConstrained Minimum Spanning Tree
, 2001
"... In numerous practical applications, it is necessary to find the smallest possible tree with a bounded diameter. A diameterconstrained minimum spanning tree (DCMST) of a given undirected, edgeweighted graph, G, is the smallestweight spanning tree of all spanning trees of G which contain no path wi ..."
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Cited by 8 (0 self)
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In numerous practical applications, it is necessary to find the smallest possible tree with a bounded diameter. A diameterconstrained minimum spanning tree (DCMST) of a given undirected, edgeweighted graph, G, is the smallestweight spanning tree of all spanning trees of G which contain no path with more than k edges, where k is a given positive integer. The problem of finding a DCMST is NPcomplete for all values of k; 4 k (n  2), except when all edgeweights are identical. A DCMST is essential for the efficiency of various distributed mutual exclusion algorithms, where it can minimize the number of messages communicated among processors per critical section. It is also useful in linear lightwave networks, where it can minimize interference in the network by limiting the traffic in the network lines. Another practical application requiring a DCMST arises in data compression, where some algorithms compress a file utilizing a tree datastructure, and decompress a path in the tree to access a record. A DCMST helps such algorithms to be fast without sacrificing a lot of storage space. We present a survey of the literature on the DCMST problem, study the expected diameter of a random labeled tree, and present five new polynomialtime algorithms for an approximate DCMST. One of our new algorithms constructs an approximate DCMST in a modified greedy fashion, employing a heuristic for selecting an edge to be added to iii the tree in each stage of the construction. Three other new algorithms start with an unconstrained minimum spanning tree, and iteratively refine it into an approximate DCMST. We also present an algorithm designed for the special case when the diameter is required to be no more than 4. Such a diameter4 tree is also used for evaluating the quality of o...