An expected-case sub-cubic solution to the all-pairs shortest path problem in R

TL;DR

This paper presents an expected-case sub-cubic algorithm for the all-pairs shortest path problem in real-valued graphs, outperforming classical methods like Floyd-Warshall under certain data distribution assumptions.

Contribution

It introduces a novel expected-case sub-cubic solution for all-pairs shortest paths in R, extending previous integer-based results to real-valued graphs.

Findings

Expected time complexity is O(V^2.5 * log V) under certain data conditions.

Algorithm outperforms Floyd-Warshall in practical dense graph scenarios.

Performance depends on data correlation properties, with potential for better or worse results.

Abstract

It has been shown by Alon et al. that the so-called 'all-pairs shortest-path' problem can be solved in O((MV)^2.688 * log^3(V)) for graphs with V vertices, with integer distances bounded by M. We solve the more general problem for graphs in R (assuming no negative cycles), with expected-case running time O(V^2.5 * log(V)). While our result appears to violate the Omega(V^3) requirement of "Funny Matrix Multiplication" (due to Kerr), we find that it has a sub-cubic expected time solution subject to reasonable conditions on the data distribution. The expected time solution arises when certain sub-problems are uncorrelated, though we can do better/worse than the expected-case under positive/negative correlation (respectively). Whether we observe positive/negative correlation depends on the statistics of the graph in question. In practice, our algorithm is significantly faster than Floyd-Warshall, even for dense graphs.