A Nearly Optimal Oracle for Avoiding Failed Vertices and Edges

Abstract

We present an improved oracle for the distance sensitivity problem. The goal is to preprocess a directed graph G = (V,E) with non-negative edge weights to answer queries of the form: what is the length of the shortest path from x to y that does not go through some failed vertex or edge f. The previous best algorithm produces an oracle of size ~O(n[superscript 2]) that has an O(1) query time, and an ~O(nn[superscript 2]√m) construction time. It was a randomized Monte Carlo algorithm that worked with high probability. Our oracle also has a constant query time and an ~O(n[superscript 2]) space requirement, but it has an improved construction time of ~O(mn), and it is deterministic. Note that O(1) query, O(n[superscript 2]) space, and O(mn) construction time is also the best known bound (up to logarithmic factors) for the simpler problem of finding all pairs shortest paths in a weighted, directed graph. Thus, barring improved solutions to the all pairs shortest path problem, our oracle is optimal up to logarithmic factors.