Random-tree Diameter and the Diameter-constrained MST

Lecture Notes in Computer Science, 2009

Given a connected, weighted, undirected graph G with n vertices and a positive integer bound D, the problem of computing the lowest cost spanning tree from amongst all spanning trees of the graph containing paths with at most D edges is known to be NP-Hard for 4 ≤ D < n-1. This is termed as the Diameter Constrained, or Bounded Diameter Minimum Spanning Tree Problem (BDMST). A well known greedy heuristic for this problem is based on Prim's algorithm, and computes BDMSTs in O(n 4) time. A modified version of this heuristic using a tree-center based approach runs an order faster. A greedy randomized heuristic for the problem runs in O(n 3) time and produces better (lower cost) spanning trees on Euclidean benchmark problem instances when the diameter bound is small. This paper presents two novel heuristics that compute low cost diameter-constrained spanning trees in O(n 3) and O(n 2) time respectively. The performance of these heuristics vis-à-vis the extant heuristics is shown to be better on a wide range of Euclidean benchmark instances used in the literature for the BDMST Problem.

Journal of Mathematical Sciences, 2009

The diameter-constrained minimum spanning tree problem is an NP-hard combinatorial optimization problem that seeks a minimum cost spanning tree with a limit D imposed upon the length of any path in the tree. We begin by presenting four constructive greedy heuristics and, secondly, we present some second-order heuristics, performing some improvements on feasible solutions, hopefully leading to better objective function values. We present a heuristic with an edge exchange mechanism, another that transforms a feasible spanning tree solution into a feasible diameter-constrained spanning tree solution, and finally another with a repetitive mechanism. Computational results show that repetitive heuristics can improve considerably over the results of the greedy constructive heuristics, but using a huge amount of computation time. To obtain computational results, we use instances of the problem corresponding to complete graphs with a number of nodes between 20 and 60 and with the value of D varying between 4 and 9.

Expert Systems, 2020

Given a connected, weighted, undirected graph G = (V, E) and an integer D ≥ 2, the bounded diameter minimum spanning tree (BDMST) problem seeks a spanning tree of minimum cost, whose diameter is no greater than D. The problem is known to be NP-hard, and finds application in various domains such as information retrieval, wireless sensor networks and distributed mutual exclusion. This article presents a two-phase memetic algorithm for the BDMST problem that combines a specialized recombination operator (proposed in this work) with good heuristics in order to more effectively direct the exploration of the search space into regions containing better solutions. A parallel meta-heuristic is also proposed and shown to obtain very good speedupssuper-linear, in several casesvis-à-vis the serial memetic algorithm. To the best of the authors' knowledge, this is the first parallel meta-heuristic proposed for the BDMST problem, and potentially paves the way for handling much larger problems than reported in the literature. Some observations and theorems are presented in order to provide the underlying framework for the proposed algorithms. Further, the proposed memetic algorithm and parallel meta-heuristic are shown, in the course of several computational experiments, to in general obtain superior solution quality with much lesser computational effort in comparison to the best known meta-heuristics in the literature over a wide range of benchmark instances.

Soft Computing, 2007

Given an undirected, connected, weighted graph and a positive integer k, the bounded-diameter minimum spanning tree (BDMST) problem seeks a spanning tree of the graph with smallest weight, among all spanning trees of the graph, which contain no path with more than k edges. In general, this problem is NP-Hard for 4 ≤ k < n − 1, where n is the number of vertices in the graph. This work is an improvement over two existing greedy heuristics, called randomized greedy heuristic (RGH) and centre-based tree construction heuristic (CBTC), and a permutation-coded evolutionary algorithm for the BDMST problem. We have proposed two improvements in RGH/CBTC. The first improvement iteratively tries to modify the bounded-diameter spanning tree obtained by RGH/CBTC so as to reduce its cost, whereas the second improves the speed. We have modified the crossover and mutation operators and the decoder used in permutation-coded evolutionary algorithm so as to improve its performance. Computational results show the effectiveness of our approaches. Our approaches obtained better quality solutions in a much shorter time on all test problem instances considered.

Proceedings of the 2003 ACM symposium on Applied computing - SAC '03, 2003

Given a connected, weighted, undirected graph G and a bound D, the bounded-diameter minimum spanning tree problem seeks a spanning tree on G of lowest weight in which no path between two vertices contains more than D edges. This problem is NP-hard for 4 ≤ D < n − 1, where n is the number of vertices in G. An existing greedy heuristic for the problem, called OTTC, is based on Prim's algorithm. OTTC usually yields poor results on instances in which the triangle inequality approximately holds; it always uses the lowest-weight edges that it can, but such edges do not in general connect the interior nodes of low-weight boundeddiameter trees. A new randomized greedy heuristic builds a bounded-diameter spanning tree from its center vertex or vertices. It chooses each next vertex at random but attaches the vertex with the lowest-weight eligible edge. This algorithm is faster than OTTC and yields substantially better solutions on Euclidean instances. An evolutionary algorithm encodes spanning trees as lists of their edges, augmented with their center vertices. It applies operators that maintain the diameter bound and always generate valid offspring trees. These operators are efficient, so the algorithm scales well to larger problem instances. On 25 Euclidean instances of up to 1 000 vertices, the EA improved substantially on solutions found by the randomized greedy heuristic.

Distributed Computing, 2006

This paper considers the problem of distributively constructing a minimum-weight spanning tree (MST) for graphs of constant diameter in the bounded-messages model, where each message can contain at most bits for some parameter. It is shown that the time required to compute an MST for graphs of diameter or ¿ can be as high as ª´¿ Ô Ò µ and ª´ Ô Ò ¾ Ô µ, respectively. The lower bound holds even if the algorithm is allowed to be randomized. On the other hand, it is shown that Ç´ÐÓ Òµ time units suffice to compute an MST deterministically for graphs with diameter ¾, when Ç´ÐÓ Òµ. These results complement a previously known lower bound of ª´¾ Ô Ò µ for graphs of diameter ª´ÐÓ Òµ.

Informatica (Slovenia), 2015

Given a connected, undirected graph G = (V, E) on n = jV j vertices, an integer bound D&gt;=2 and non-zero edge weights associated with each edge e 2 E, a bounded diameter minimum spanning tree (BDMST) on G is defined as a spanning tree T E on G of minimum edge cost w(T) = P w(e), 8 e 2 T and tree diameter no greater than D. The Euclidean BDMST Problem aims to find the minimum cost BDMST on graphs whose vertices are points in Euclidean space and whose edge weights are the Euclidean distances between the corresponding vertices. The problem of computing BDMSTs is known to be NP-Hard for 4 D n -1, where D the diameter bound. Furthermore, the problem is known to be hard to approximate. Heuristics are extant in the literature which build low cost, diameter-constrained spanning trees in O(n3) time. This paper presents some fast and effective heuristic strategies for the Euclidean BDMST Problem and compares their performance with that of the best known existing heuristics. Two of the propo...

Greedy Algorithms, 2008

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We present a new algorithm, which solves the problem of distributively finding a mini-mum diameter spanning tree of any (non-negatively) real-weighted graph G = (V,E, ω). As an intermediate step, we use a new, fast, linear-time all-pairs shortest paths distributed algo-rithm to find an absolute center of G. The resulting distributed algorithm is asynchronous, it works for named asynchronous arbitrary networks and achieves O(|V |) time complexity and O (|V | |E|) message complexity.