Efficient Parallel Algorithms for Testingkand Finding Disjoints-tPaths in Graphs

Abstract

An efficient parallel algorithm for testing whether a graph G is k-vertex connected is presented. The algorithm runs in $O(k^2 \log n)$ time and uses $(n + k^2 )kC(n,m)$ processors on a CROW PRAM, where n and m are the number of vertices and edges of G, and $C(n,m)$ is the number of processors required to compute the connected components of G in logarithmic time. For fixed k, the algorithm runs in logarithmic time and uses $nC(n,m)$ processors. To develop our algorithm, an efficient parallel algorithm is designed for the following disjoint s-t paths problem. Given a graph G, and two specified vertices s and t, find k vertex disjoint paths between s and t, if they exist. If no such paths exist, find a set of at most $k-1$ vertices whose removal disconnects s and t. The parallel algorithm for this problem runs in $O(k^2 \log n)$ time and uses $kC(n,m)$ processors. The way to modify the algorithm to find k-edge disjoint paths, if they exist, is shown. This yields an efficient parallel algorithm for testing w...