Sharing Information in All Pairs Shortest Path Algorithms

We show two improvements on time complexities of the all pairs shortest path (APSP) problem for directed graphs that satisfy certain properties. The idea for speed-up is information sharing by n single source shortest path (SSSP) problems that are solved for APSP. We consider two parameters, in addition to the numbers of vertices, n, and edges, m. First we improve the time complexity of O(mn+n √ log c) to O(mn + nc) for the APSP problem with the integer edge costs bounded by c. When c ≤ O(n√logn), this complexity is better than the previous one. Next we consider a nearly acyclic graph. We measure the degree of acyclicity by the size, r, of a given set of feedback vertices. If r is small, the given graph can be considered to be nearly acyclic. We improve the existing time complexity of O(mn + r) for the all pairs shortest path problem to O(mn + rn log n) by some kind of information sharing. This complexity is better than the previous one for all values of r under a reasonable assumption of m ≥ n.