Sub-Cubic Cost Algorithms for the All Pairs Shortest Path Problem

In this paper we give three sub-cubic cost algorithms for the all pairs shortest distance (APSD) and path (APSP) problems. The first is a parallel algorithm that solves the APSD problem for a directed graph with unit edge costs in O(log2n) time with \(O({{n^\mu } \mathord{\left/{\vphantom {{n^\mu } {\sqrt {\log n} }}} \right.\kern-\nulldelimiterspace} {\sqrt {\log n} }})\) processors where Μ=2.688 on an EREW-PRAM. The second parallel algorithm solves the APSP, and consequently APSD, problem for a directed graph with non-negative general costs (real numbers) in O(log2n) time with o(n3) subcubic cost. Previously this cost was greater than O(n3). Finally we improve with respect to M the complexity O((Mn)μ) of a sequential algorithm for a graph with edge costs up to M into O(M1/3n6+1/3/3(log n)2/3(log log n)1/3) in the APSD problem.