# Sub-cubic cost algorithms for the all pairs shortest path problem

## Abstract

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(log^{2}*n*) 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(log^{2}*n*) time with o(n^{3}) subcubic cost. Previously this cost was greater than O(n^{3}). 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(M^{1/3}*n*^{6+1/3}/3(log *n*)^{2/3}(log log *n*)^{1/3}) in the APSD problem.

## Keywords

Short Path Directed Graph Matrix Multiplication Parallel Algorithm Short Path Problem## Preview

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