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

Authors Yong Gu, Hanlin Ren



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Yong Gu
  • Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China
Hanlin Ren
  • Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China

Acknowledgements

We would like to thank Ran Duan and Tianyi Zhang for helpful discussions during the initial stage of this research. We are grateful to anonymous reviewers for helpful comments. We would also like to thank an anonymous reviewer for suggesting the title of Section 5, and another anonymous reviewer for pointing out a subtle issue regarding the invertibility of polynomial matrices (and fixing the issue).

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Yong Gu and Hanlin Ren. Constructing a Distance Sensitivity Oracle in O(n^2.5794 M) Time. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 76:1-76:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ICALP.2021.76

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Shortest paths
  • Theory of computation → Design and analysis of algorithms
Keywords
  • graph theory
  • shortest paths
  • distance sensitivity oracles

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