On the Shoshan-Zwick Algorithm for the All-Pairs Shortest Path Problem

TL;DR

This paper identifies errors in the Shoshan-Zwick algorithm for all-pairs shortest paths, revises it for correctness, and discusses implementation details using current matrix multiplication techniques.

Contribution

It corrects the previously flawed Shoshan-Zwick algorithm and provides insights into its practical implementation with existing matrix multiplication methods.

Findings

Original algorithm contained errors affecting correctness.

Revised algorithm ensures accurate shortest path computation.

Implementation discussion leverages current sub-cubic matrix multiplication algorithms.

Abstract

The Shoshan-Zwick algorithm solves the all pairs shortest paths problem in undirected graphs with integer edge costs in the range $\{1, 2, \dots, M\}$. It runs in $\tilde{O}(M\cdot n^{\omega})$ time, where $n$ is the number of vertices, $M$ is the largest integer edge cost, and $\omega < 2.3727$ is the exponent of matrix multiplication. It is the fastest known algorithm for this problem. This paper points out the erroneous behavior of the Shoshan-Zwick algorithm and revises the algorithm to resolve the issues that cause this behavior. Moreover, it discusses implementation aspects of the Shoshan-Zwick algorithm using currently-existing sub-cubic matrix multiplication algorithms.