Fast 2-Approximate All-Pairs Shortest Paths

Michal Dory, Sebastian Forster, Yael Kirkpatrick, Yasamin Nazari, Virginia Vassilevska Williams, Tijn de Vos

Research output: Contribution to conferencePaperpeer-review


In this paper, we revisit the classic approximate All-Pairs Shortest Paths (APSP) problem in undirected graphs. For unweighted graphs, we provide an algorithm for 2-approximate APSP in (equation presented) time, for any r ∈ [0, 1]. This is O(n2.032) time, using known bounds for rectangular matrix multiplication nω (r) [Le Gall, Urrutia, SODA 2018]. Our result improves on the Õ (n2.25) bound of [Roditty, STOC 2023], and on the Õ(m√n + n2) bound of [Baswana, Kavitha, SICOMP 2010] for graphs with m ≥ n1.532 edges. For weighted graphs, we obtain (2 + ε)-approximate APSP in (equation presented) time, for any r ∈ [0, 1]. This is O(n2.214) time using known bounds for ω(r). It improves on the state of the art bound of O(n2.25) by [Kavitha, Algorithmica 2012]. Our techniques further lead to improved bounds in a wide range of density for weighted graphs. In particular, for the sparse regime we construct a distance oracle in Õ (mn2/3) time that supports 2-approximate queries in constant time. For sparse graphs, the preprocessing time of the algorithm matches conditional lower bounds [Patrascu, Roditty, Thorup, FOCS 2012; Abboud, Bringmann, Fischer, STOC 2023]. To the best of our knowledge, this is the first 2-approximate distance oracle that has subquadratic preprocessing time in sparse graphs. We also obtain new bounds in the near additive regime for unweighted graphs. We give faster algorithms for (1 + ε, κ)approximate APSP, for κ = 2, 4, 6, 8. We obtain these results by incorporating fast rectangular matrix multiplications into various combinatorial algorithms that carefully balance out distance computation on layers of sparse graphs preserving certain distance information.

Original languageEnglish
Number of pages30
StatePublished - 2024
Event35th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2024 - Alexandria, United States
Duration: 7 Jan 202410 Jan 2024


Conference35th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2024
Country/TerritoryUnited States

Bibliographical note

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Copyright © 2024 by SIAM.

ASJC Scopus subject areas

  • Software
  • General Mathematics


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