The Strict Strong Coloring Based Graph Distribution Algorithm

State space explosion is a fundamental obstacle in formal verification of concurrent systems. As a solution for this problem, this paper deals with distributed state space. The authors’ solution is to introduce the coloring concept and dominance relation in graphs for finding the good distribution of given graphs. This basic solution is improved in two steps: the initialization and optimization step. The authors also report on a thorough experimental study to evaluate the performance of this new algorithm which depends strongly on the size, nature of the graphs, and the chosen number of workers. In addition, the quality of this algorithm is illustrated by comparison with the hash function (MD5) based algorithm. To the best of the authors’ knowledge, it is the first time when coloring concept is used to solve this problem. DOI: 10.4018/jamc.2013010104 International Journal of Applied Metaheuristic Computing, 4(1), 50-66, January-March 2013 51 Copyright © 2013, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. Several factors should be taken into account to have a good distribution. The most important of them are the workload balancing of the workers (i.e. no unemployed or overloaded workers) and the minimization of the distribution cost (i.e. edges to be cut). Distributing the state space among several workstations (workers) which communicate through a message in a network is the subject of this paper. A new heuristic based on coloring concept and dominance relation in graphs is presented to find a good distribution within a reasonable time. Many papers present several approaches for solving this NP-hard problem (Bixby et al., 1993) and other distributed applications on large graphs (Bouneb & Saïdouni, 2009; Orzan et al., 2005; Stanton & Kliot, 2011). Authors in Saad et al. (2009) have presented deeply the different solutions proposed up to 2009. All these solutions are based on a partition function which assigns each state to a fixed worker. These approaches differ mainly by the nature of this function. In Stanton and Kliot (2011), authors have used ten heuristics such that each one of them gives an algorithm for selecting the index of the partition where a state is assigned. However, coloring approach has not yet used for distributing graphs. Graph coloring has been studied profoundly in literature (Dharwadker, 2006; Klotz, 2002). As a variant, graph strict strong coloring has been defined in Haddad and Kheddouci (2009) where an exact algorithm for trees has been proposed. This coloring approach has been extended for general graphs (Bouzenada et al., 2012) which is named GGSSCA (for Generalized Graph Strict Strong Coloring Algorithm). In this paper, the proposed approach is the first heuristic algorithm, for resolving the state space distribution, based on GGSSCA (Bouzenada et al., 2012). Particularly, we use its first step which insures the dominance property to make the initial distribution. After that, another process will be started to build the final and optimal distribution. The goal of the proposed approach is, given a graph G as input and a number W, to find the better distribution of G which respects the balancing constraint (i.e. holds good load balancing), and minimizes the communication costs (i.e. edges between states on different parts). The remainder of this paper is organized as follows: Section 2 gives some preliminary notations and definitions. In section 3, we describe the relation between graph distribution and graph strict strong coloring. In section 4, the proposed algorithm for graph distribution is presented with its explanation. Section 5 discusses implementation issues and presents various experimental results. In this section, to improve formal distribution verification process, it will be shown that the proposed algorithm is better than the hash function (MD5) based algorithm. Conclusion and future work are given in the last section.