Optimal computation of shortest paths on doubly convex bipartite graphs

An optimal parallel algorithm for computing all-pair shortest paths on doubly convex bipartite graphs is presented here. The input is a (0,1)-matrix with consecutive 1s in each of its rows and columns that represents a doubly convex bipartite graph. Our parallel algorithm runs in O(log n) time with O(n²/log n) processors on an EREW PRAM and is time-and-work-optimal. As a byproduct, we show that the problem can be solved by a sequential algorithm in O(n²) time optimally on any adjacency list or matrix representing a doubly convex bipartite graph. The result in this paper improves a recent work on the problem for bipartite permutation graphs which are properly contained in doubly convex bipartite graphs.