# Encyclopedia of Algorithms

2008 Edition
| Editors: Ming-Yang Kao

# All Pairs Shortest Paths via Matrix Multiplication

2002; Zwick
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30162-4_12

## Keywords and Synonyms

Shortest path problem; Algorithm analysis

## Problem Definition

The all pairs shortest path (APSP) problem is to compute shortest paths between all pairs of vertices of a directed graph with non-negative real numbers as edge costs. Focus is given on shortest distances between vertices, as shortest paths can be obtained with a slight increase of cost. Classically, the APSP problem can be solved in cubic time of O(n3). The problem here is to achieve a sub-cubic time for a graph with small integer costs.

A directed graph is given by $${ G=(V,E) }$$

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