Fast 2-Approximate All-Pairs Shortest Paths

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(n^2.032) time, using known bounds for rectangular matrix multiplication nω (r) [Le Gall, Urrutia, SODA 2018]. Our result improves on the Õ (n^2.25) bound of [Roditty, STOC 2023], and on the Õ(m√n + n²) bound of [Baswana, Kavitha, SICOMP 2010] for graphs with m ≥ n^1.532 edges. For weighted graphs, we obtain (2 + ε)-approximate APSP in (equation presented) time, for any r ∈ [0, 1]. This is O(n^2.214) time using known bounds for ω(r). It improves on the state of the art bound of O(n^2.25) 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.