On the parallel computation of the biconnected and strongly connected co-components of graphs

Abstract

In this paper, we consider the problems of co-biconnectivity and strong co-connectivity, i.e., computing the biconnected components and the strongly connected components of the complement of a given graph. We describe simple sequential algorithms for these problems, which work on the input graph and not on its complement, and which for a graph on n vertices and m edges both run in optimal O ( n + m ) time. Our algorithms are not data structure-based and they employ neither breadth-first-search nor depth-first-search. Unlike previous linear co-biconnectivity and strong co-connectivity sequential algorithms, both algorithms admit efficient parallelization. The co-biconnectivity algorithm can be parallelized resulting in an optimal parallel algorithm that runs in O ( log 2 n ) time using O ( ( n + m ) / log 2 n ) processors. The strong co-connectivity algorithm can also be parallelized to yield an O ( log 2 n ) -time and O ( m 1.188 / log n ) -processor solution. As a byproduct, we obtain a simple optimal O ( log n ) -time parallel co-connectivity algorithm. Our results show that, in a parallel process environment, the problems of computing the biconnected components and the strongly connected components can be solved with better time-processor complexity on the complement of a graph rather than on the graph itself.