All Pairs Shortest Paths via Matrix Multiplication

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(n ³). The problem here is to achieve a sub-cubic time for a graph with small integer costs.

A directed graph is given by ${ G=(V,E) }$, where ${ V = \{1,\ldots,n\} }$, the set of vertices, and E is the set of edges. The cost of edge ${ (i,j)\in E }$ is denoted by d _ij . The (n, n)-matrix D is one whose (i, j) element is d _ij . It is assumed for simplicity that ${ d_{ij} > 0 }$ and ${ d_{ii}=0 }$ for all ${ i \neq j }$. If there is no edge from i to j, let ${ d_{ij}=\infty }$. The cost, or distance, of a path is the sum of costs of...