## Abstract

The essential subgraph H of a weighted graph or digraph*G* contains an edge (*v, w*) if that edge is uniquely the least-cost path between its vertices. Let s denote the number of edges of*H.* This paper presents an algorithm for solving all-pairs shortest paths on*G* that requires O(*ns*+n^{2} log*n*) worst-case running time. In general the time is equivalent to that of solving*n* single-source problems using only edges in*H*. For general models of random graphs and digraphs*G, s*=0(*n* log*n*) almost surely. The subgraph*H* is optimal in the sense that it is the smallest subgraph sufficient for solving shortest-path problems in*G*. Lower bounds on the largest-cost edge of*H* and on the diameter of*H* and*G* are obtained for general randomly weighted graphs. Experimental results produce some new conjectures about essential subgraphs and distances in graphs with uniform edge costs.

## Key words

All-pairs shortest path Single-source shortest path Random graph Random weighted graph Graph diameter Minimum spanning tree Essential arcs## Preview

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