Parallel Algorithms for Maximal Cliques in Circle Graphs and Unrestricted Depth Search

Abstract

We present parallel algorithms on the BSP/CGM model, with p processors, to count and generate all the maximal cliques of a circle graph with n vertices and m edges. To count the number of all the maximal cliques, without actually generating them, our algorithm requires O (log p ) communication rounds with O (nm /p ) local computation time. We also present an algorithm to generate the first maximal clique in O (log p ) communication rounds with O (nm /p ) local computation, and to generate each one of the subsequent maximal cliques this algorithm requires O (log p ) communication rounds with O (m /p ) local computation. The maximal cliques generation algorithm is based on generating all maximal paths in a directed acyclic graph, and we present an algorithm for this problem that uses O log (p) communication rounds with O (m /p ) local computation for each maximal path. We also show that the presented algorithms can be extended to the CREW PRAM model.