Constructing a Distance Sensitivity Oracle in $O(n^{2.5794}M)$ Time

Abstract

We continue the study of distance sensitivity oracles (DSOs). Given a directed graph G with n vertices and edge weights in {1, 2, . . . , M }, we want to build a data structure such that given any source vertex u, any target vertex v, and any failure f (which is either a vertex or an edge), it outputs the length of the shortest path from u to v not going through f . Our main result is a DSO with preprocessing time O(n 2.5794 M ) and constant query time. Previously, the best preprocessing time of DSOs for directed graphs is O(n 2.7233 M ), and even in the easier case of undirected graphs, the best preprocessing time is O(n 2.6865 M ) [Ren, ESA 2020]. One drawback of our DSOs, though, is that it only supports distance queries but not path queries.Our main technical ingredient is an algorithm that computes the inverse of a degree-d polynomial matrix (i.e. a matrix whose entries are degree-d univariate polynomials) modulo x r . The algorithm is adapted from [Zhou, Labahn and Storjohann, Journal of Complexity, 2015], and we replace some of its intermediate steps with faster rectangular matrix multiplication algorithms.We also show how to compute unique shortest paths in a directed graph with edge weights in {1, 2, . . . , M }, in O(n 2.5286 M ) time. This algorithm is crucial in the preprocessing algorithm of our DSO. Our solution improves the O(n 2.6865 M ) time bound in [Ren, ESA 2020], and matches the current best time bound for computing all-pairs shortest paths.