Graph Transformation Approach for the Shortest Path Search and Length Calculation

Abstract

Abstract We consider a graph with labels of edges. A label means the length of an edge. We present a method to compute the length of the shortest path between two vertices using graph transformations. We introduce graph transformation rules which preserve the length of paths. Reducing to a simple graph which contains two vertices, we finally calculate the length of the shortest path of those two vertices. There were several algorithms for computing network reliabilities using graph transformations. We use the same framework as those algorithms for applying the graph transformation rules, but our transformation rules do not calculate the network reliabilities but calculate the length of the shortest path. 1. Introduction Graph transformation (graph reduction or graph rewriting) techniques are useful for developing an algorithm to compute several values of graph properties in the area such as circuits and networks. Politof and Satyanarayana (1986a) introduced a polynomial-time algorithm for the computation of network reliability for some restricted class of graphs, and Satyanarayana and Wood (1985) studied O (I E I)-time algorithms for kterminal reliability for series and parallel graphs. Politof and Satyanarayana (1986b) studied allterminal reliability for IFCF (inner for cycle free) graphs and they also presented O(IV D)-time algorithms for PCF (planar cube free) graphs. In these studies graph transformation methods were introduced to compute network reliabilities. Network graphs were classified into some class, e.g. IFCF graphs and PCF graphs, to give efficient algorithms using graph transformation methods. Okada and Hayashi (1991) showed the theoretical results about critical pair lemma for the graph transformation rules introduced by Politof and Satyanarayana (1986a; 1986b) . The notion of critical pair was firstly considered in the area such as studies of term rewriting systems. In Mizoguchi and Kawahara (1995) and Mizoguchi (1999) the concept of critical pair lemma was generalized to a graph rewriting system, and a complete graph rewriting system was introduced to computes network reliabilities such as Okada and Hayashi (1991) . In this paper, we introduce graph transformation rules to search the shortest path and compute its length for given graphs and construct an O (I E D-time algorithms for series and parallel graphs. Application of those rules yields to the shortest path and its * Department of Informatics , Kyushu University, 33 Fukuoka 812-8581, Japan. masa@i.kyushuu.ac.jp t Faculty of Mathematics , Kyushu University, 33 Fukuoka 812-8581, Japan. ym@math.kyushuu.ac.jp