Shortest-Path Reconstruction Algorithms

Abstract

The shortest path problem in graphs is well known in computer science and operations research. This paper studies the problem of computing shortest paths between pairs of vertices in an n-vertex graph, given only the all pairs shortest paths distance matrix. This computation is called a reconstruction, since the algorithm has no access to explicit information about edges in the original graph. We present the following results: A shortest path between a single pair of vertices can be reconstructed in O(n log n) time. Shortest paths between all pairs of vertices can be reconstructed in O(n3) time. A shortest path between a single pair of vertices can be reconstructed in O((log n)2) time on an exclusive read, exclusive write parallel random access machine (EREW PRAM) with n/log n processors. Shortest path between all pairs of vertices can be reconstructed in O(n2) time on an EREW PRAM with n processors. The significance of our results is in showing that the all pairs shortest paths distance matrix provides highly useful information on the original graph, which can be extracted very simply and efficiently. It is also hoped that this work would lead to new information theoretic insights on data structures in general.