Computing rooted communication reliability in an almost acyclic digraph

Abstract

AbstractThis article considers the reliability problem of computing the probability that the root‐vertex of a directed network can communicate with all other vertices of the network when arcs are subject to random failure. Ball and Provan have observed that when the network contains no directed cycles this probability can be computed in linear time. This paper considers the problem when the network is almost acyclic, that is, when the number of simple directed cycles in the network is small. A new algorithm for computing network reliability is presented. The algorithm is based on expressing the reliability in terms of the derivative of the network reliability with respect to the reliability of an arc. Given a class of networks with a fixed number of simple cycles c, the computation requirements of the new algorithm are O(|E|c+1), where |E| is the number of arcs. Comparison of coded versions of this algorithm and one of Ball's as well as theoretical comparison with one of Buzacott's confirm that this algorithm is to be preferred when the number of directed cycles is small in comparison to the number of vertices.