Bounded Diameter Minimum Spanning Tree Research Articles

Serial and parallel memetic algorithms for the bounded diameter minimum spanning tree problem

AbstractGiven 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 speedups – super‐linear, in several cases – vis‐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.

Stochastic Bounded Diameter Minimum Spanning Tree Problem

In this paper, a learning automata-based algorithm is proposed for approximating a near optimal solution to the bounded diameter minimum spanning tree (BDMST) problem in stochastic graphs. A stochastic graph is a graph in which the weight associated with each edge is a random variable. Stochastic BDMST problem seeks for finding the BDMST in a stochastic graph. To the best of our knowledge, no work has been done on solving the stochastic BDMST problem, where the weight associated with the graph edge is random variable. In this study, we assume that the probability distribution of the edges random weight is unknown a priori. This makes the stochastic BDMST problem incredibly hard-to-solve. To show the efficiency of the proposed algorithm, its results are compared with those of the standard sampling method (SSM). Numerical results show the superiority of the proposed sampling algorithm over the SSM both in terms of the sampling rate and convergence rate.

New Heuristic Approaches for the Bounded-Diameter Minimum Spanning Tree Problem

Given a bound, the bounded-diameter minimum spanning tree (BDMST) problem seeks a spanning tree of minimum total weight with a diameter not exceeding the given diameter bound. Depending on the tightness of the bound, optimal solutions have different structures. For loose bounds, optimal solutions are similar to the much easier minimum spanning tree problem, and greedy heuristics perform best. In contrast, these approaches fail for tight diameter bounds. This paper investigates how the structure of good solutions, and in particular their backbones, change depending on the diameter bound. Two new heuristics are then designed to overcome the shortcomings of existing approaches; required parameters are investigated; and the paper presents performance results for Euclidean BDMST instances.

A learning automata based approach to the bounded-diameter minimum spanning tree problem

The bounded diameter minimum spanning tree (BDMST) problem aims at finding the minimum weight spanning tree subject to a predefined diameter constraint. Due to the NP-hardness of the BDMST problem, several heuristic and meta-heuristic approaches have been proposed to find a near optimal solution in a reasonable time. In this article, a distributed algorithm is proposed for solving the BDMST problem based on learning automata. Generally, the proposed algorithm consists of two main phases. In the former phase, the proposed algorithm forms the action-set of learning automata and paths from the root to every other node. The latter phase constructs different spanning trees until it finds the optimum one. To show the performance of the proposed algorithm, it is compared with one of the well-known methods. Experimental results confirm the superiority of the proposed algorithm both in terms of the computational complexity and the weight of the spanning tree.

A stable virtual backbone for wireless MANETS

Due to the highly dynamicity and absence of a fixed infrastructure in wireless mobile ad hoc networks (MANET), formation of a stable virtual backbone through which all the network hosts are connected is of great importance. In this paper, a learning automata-based distributed algorithm is proposed for constructing the most stable virtual backbone of the MANET. To do so, the backbone formation problem is first modeled by the stochastic version of the bounded diameter minimum spanning tree (BDMST) problem. Then, the network backbone is constructed by solving the stochastic BDMST problem for the network topology graph. Several simulation experiments are conducted to investigate the efficiency of the proposed backbone formation protocol. The obtained results are compared with those of the best existing methods. Numerical results show the superiority of the proposed method over the others in terms of backbone lifetime, end-to-end delay, backbone size, packet delivery ratio, and control message overhead.

Improved heuristics for the bounded-diameter minimum spanning tree problem

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.