During the last decades, a host of efficient algorithms have been developed for solving the minimum spanning tree problem in deterministic graphs, where the weight associated with the graph edges is assumed to be fixed. Though it is clear that the edge weight varies with time in realistic applications and such an assumption is wrong, finding the minimum spanning tree of a stochastic graph has not received the attention it merits. This is due to the fact that the minimum spanning tree problem becomes incredibly hard to solve when the edge weight is assumed to be a random variable. This becomes more difficult if we assume that the probability distribution function of the edge weight is unknown. In this paper, we propose a learning automata-based heuristic algorithm to solve the minimum spanning tree problem in stochastic graphs wherein the probability distribution function of the edge weight is unknown. The proposed algorithm taking advantage of learning automata determines the edges that must be sampled at each stage. As the presented algorithm proceeds, the sampling process is concentrated on the edges that constitute the spanning tree with the minimum expected weight. The proposed learning automata-based sampling method decreases the number of samples that need to be taken from the graph by reducing the rate of unnecessary samples. Experimental results show the superiority of the proposed algorithm over the well-known existing methods both in terms of the number of samples and the running time of algorithm.

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A learning automata-based heuristic algorithm for solving the minimum spanning tree problem in stochastic graphs

Due to the hardness of solving the minimum spanning tree (MST) problem in stochastic environments, the stochastic MST (SMST) problem has not received the attention it merits, specifically when the probability distribution function (PDF) of the edge weight is not a priori known. In this paper, we first propose a learning automata-based sampling algorithm (Algorithm 1) to solve the MST problem in stochastic graphs where the PDF of the edge weight is assumed to be unknown. At each stage of the proposed algorithm, a set of learning automata is randomly activated and determines the graph edges that must be sampled in that stage. As the proposed algorithm proceeds, the sampling process focuses on the spanning tree with the minimum expected weight. Therefore, the proposed sampling method is capable of decreasing the rate of unnecessary samplings and shortening the time required for finding the SMST. The convergence of this algorithm is theoretically proved and it is shown that by a proper choice of the learning rate the spanning tree with the minimum expected weight can be found with a probability close enough to unity. Numerical results show that Algorithm 1 outperforms the standard sampling method. Selecting a proper learning rate is the most challenging issue in learning automata theory by which a good trade off can be achieved between the cost and efficiency of algorithm. To improve the efficiency (i.e., the convergence speed and convergence rate) of Algorithm 1, we also propose four methods to adjust the learning rate in Algorithm 1 and the resultant algorithms are called as Algorithm 2 through Algorithm 5. In these algorithms, the probabilistic distribution parameters of the edge weight are taken into consideration for adjusting the learning rate. Simulation experiments show the superiority of Algorithm 5 over the others. To show the efficiency of Algorithm 5, its results are compared with those of the multiple edge sensitivity method (MESM). The obtained results show that Algorithm 5 performs better than MESM both in terms of the running time and sampling rate.

Learning automata-based algorithms for solving stochastic minimum spanning tree problem

The minimum spanning tree problem with fuzzy parameters is studied and an exact algorithm is proposed to solve it. However, as this problem conveys the need of large number of comparisons, a genetic algorithm with special characteristics is proposed to try to avoid the complexity issue. These algorithms use possibility theory, searching for the solution set

An Evolutionary Approach To Solve Minimum Spanning Tree Problem With Fuzzy Parameters

This work shows the procedure to optimize the problem of minimum spanning tree applied to percolation by links problem. The aim is improving the structural quality in a complex network by means of a non-directed graph in a square lattice. Moreover, it defines the size of initial population, the number of edges and the running time for each instance of the proposed problem. Experimental results show that proposed algorithm find good quality solutions efficiently for instances of 10, 100, 200,300,400 and 500 vertices. Tackling an instance of 500 vertices for percolation links problem is an important contribution of this research. This work shows the procedure to optimize the problem of minimum spanning tree applied to percolation by links problem. The aim is improving the structural quality in a complex network by means of a non-directed graph in a square lattice. Moreover, it defines the size of initial population, the number of edges and the running time for each instance of the proposed problem. Experimental results show that proposed algorithm find good quality solutions efficiently for instances of 10, 100, 200,300,400 and 500 vertices. Tackling an instance of 500 vertices for percolation links problem is an important contribution of this research.

Percolationby links appliedto the minimum spanning tree problem

This paper considers the problem of minimum spanning trees in uncertain networks in which the edge weights are random variables. We propose the concept of expected minimum spanning tree and formulate the model according to expected value. In order to solve the model, a hybrid intelligent algorithm combined genetic algorithm and stochastic simulation is given, and the Prufer encoding schemes represented spanning trees are adopted. The algorithm has been proved to be useful for solving practical problems by a numerical example.

Huan Qi

Papers

A Model and Algorithm for Minimum Spanning Tree Problems in Uncertain Networks