All Pairs Shortest Paths via Matrix Multiplication

Years and Authors of Summarized Original Work

• 2002; Zwick

Problem Definition

The all pairs shortest path (APSP) problem is to compute shortest paths between all pairs of vertices of a directed graph with nonnegative 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), where V = { 1, …, n}, the set of vertices, and E is the set of edges. The cost of edge (i, j) ∈ 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 ≠ 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 the edges in the path. The length of a path is the number of edges in the...