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17
On the Minimum Diameter Spanning Tree Problem
 Information Processing Letters
, 1997
"... : We point out a relation between the minimum diameter spanning tree of a graph and its absolute 1center. We use this relation to solve the diameter problem and an extension of it efficiently. Keywords: Minimum diameter spanning tree, Absolute 1center. 1 Introduction Let G = (V; E) be an undire ..."
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Cited by 24 (3 self)
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: We point out a relation between the minimum diameter spanning tree of a graph and its absolute 1center. We use this relation to solve the diameter problem and an extension of it efficiently. Keywords: Minimum diameter spanning tree, Absolute 1center. 1 Introduction Let G = (V; E) be an undirected graph, where V is the set of nodes and E is the set of edges. Also let jV j = n and jEj = m. Suppose that each edge e 2 E is associated with a positive weight (length) d e . A spanning tree of G is a connected subgraph T = (V; E T ) without cycles. The diameter of T , D(T ), is defined as the longest of the shortest paths in T among all the pairs of nodes in V . The minimum diameter spanning tree (MDST) problem is to find a spanning tree of G of minimum diameter. Ho, Lee, Chang and Wong [HLCW] consider the case where the graph G is a complete Euclidean graph, induced by a set S of n points in the Euclidean plane. They call this special case the geometric MDST problem. They prove that ...
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...
On the Power of BFS to Determine a Graph's Diameter
 Networks
, 2003
"... this paper, we show that, in some cases, the full power of LBFS is not required and that other variations of Breadth First Search (BFS) suffice. The restricted graph classes that are amenable to this approach all have a small constant upper bound on the maximumsized cycle that may appear as an indu ..."
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Cited by 9 (0 self)
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this paper, we show that, in some cases, the full power of LBFS is not required and that other variations of Breadth First Search (BFS) suffice. The restricted graph classes that are amenable to this approach all have a small constant upper bound on the maximumsized cycle that may appear as an induced subgraph. We show that, on graphs that have no induced cycle of size greater than k, BFS finds an estimate of the diameter that is no worse than diam(G) # #k/2#. 2003 Wiley Periodicals, Inc
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...
Fast Computation of Empirically Tight Bounds for the Diameter of Massive Graphs
"... The diameter of a graph is among its most basic parameters. Since a few years, it moreover became a key issue to compute it for massive graphs in the context of complex network analysis. However, known algorithms, including the ones producing approximate values, have too high a time and/or space com ..."
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Cited by 6 (0 self)
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The diameter of a graph is among its most basic parameters. Since a few years, it moreover became a key issue to compute it for massive graphs in the context of complex network analysis. However, known algorithms, including the ones producing approximate values, have too high a time and/or space complexity to be used in such cases. We propose here a new approach relying on very simple and fast algorithms that compute (upper and lower) bounds for the diameter. We show empirically that, on various realworld cases representative of complex networks studied in the literature, the obtained bounds are very tight (and even equal in some cases). This leads to rigorous and very accurate estimations of the actual diameter in cases which were previously untractable in practice. 1 Context. Throughout the paper, we consider a connected undirected unweighted graph G = (V, E) with n = V  vertices and m = E  edges. We denote by d(u, v) the distance between u and v in G, by ecc(v) = maxu d(v, u) the eccentricity of v in G, and by D = maxu,v d(u, v) = maxvecc(v) the diameter of G.
Combinatorial algorithms for inverse absolute and vertex 1center location problems on trees
, 2009
"... location problems on trees ..."
The inverse 1center location problem on a tree
, 2009
"... Abstract. In an inverse network 1center location problem the parameters of a given network, like edge lengths or vertex weights, have to be modified at minimum total cost such that a prespecified vertex s becomes a 1center of the network. In this paper, we consider the inverse 1center location pr ..."
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Cited by 4 (3 self)
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Abstract. In an inverse network 1center location problem the parameters of a given network, like edge lengths or vertex weights, have to be modified at minimum total cost such that a prespecified vertex s becomes a 1center of the network. In this paper, we consider the inverse 1center location problem on an unweighted tree network T with n + 1 vertices in which we are allowed to increase or reduce the edge lengths within certain bounds. The problem can be solved in O(n 2) time provided that no essential topology change occurs. Dropping this condition, an O(n 2 r) time algorithm solves the problem where r, r < n, is the compressed depth of the tree network T rooted in s. Keywords: optimization network center location, inverse optimization, combinatorial 1 Introduction and problem statement Location problems are fundamental optimization models in operations research and play a considerable role in practice and theory. These problems are concerned with
The Tree Center Problems and the Relationship with the Bottleneck Knapsack Problems
, 1997
"... The tree center problems are to nd a subtree minimizing the maximum distance from any vertex. This paper shows that these problems in a tree network are related to the bottleneck knapsack problems, and presents lineartime algorithms for the tree center problems by using the relation. 1 The Tree Cen ..."
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Cited by 3 (0 self)
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The tree center problems are to nd a subtree minimizing the maximum distance from any vertex. This paper shows that these problems in a tree network are related to the bottleneck knapsack problems, and presents lineartime algorithms for the tree center problems by using the relation. 1 The Tree Center Problems Let T = (V; E) be a tree network with a vertex set V = fv 1 ; ; v n g and an edge set E = fe 2 ; ; e n g; where an edge e i is incident to the vertex v i and on the path connecting v i and v 1 : We denote by N the index set f2; ; ng of edges. For each i 2 N; the edge e i has a positive length l i and a positive weight w i : In this paper, we assume that T is drawn in the Euclidean plane so that each edge e i is a line segment with length l i ; and regard T as a closed and connected subset of points in the Euclidean plane. For two points p; q on an edge e i ; a partial edge connecting p and q is the set of points lying between p and q on e i : The length...
Almost diameter of a householefree graph in linear time via LexBFS
 DISCRETE APPL. MATH
, 1999
"... We show that the vertex visited last by a LexBFS has eccentricity at least diam(G)  2 for householefree graphs, at least diam(G)  1 for householedominofree graphs, and equal to diam(G) for householedominofree and ATfree graphs. To prove these results we use special metric properties o ..."
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Cited by 2 (2 self)
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We show that the vertex visited last by a LexBFS has eccentricity at least diam(G)  2 for householefree graphs, at least diam(G)  1 for householedominofree graphs, and equal to diam(G) for householedominofree and ATfree graphs. To prove these results we use special metric properties of householefree graphs with respect to LexBFS.