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arXiv · 2607.18714

$\tilde{O}$ptimal Algorithm for 2-Approximate All Pair Shortest Paths -- almost

Manoj Gupta·Mrigankashekhar Shandilya
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Abstract

Given an undirected, unweighted graph $G$, we aim to compute a 2-approximation of all-pairs shortest paths (APSP). This problem admits a natural lower bound of $\Omega(n^2)$ since the output size is $\Theta(n^2)$. A central goal in this area is to achieve a running time of $O(n^2)$. Dor, Halperin, and Zwick (FOCS 1996, SICOMP 2001) designed an algorithm with a running time of $\tilde{O}(n^2)$ that guarantees a 2-approximation only for pairs at a distance of at least $O(\log n)$. Recently, Gupta (FOCS 2025) improved this bound, handling all pairs at a distance of at least $O(\log \log n)$. We nearly resolve this problem. We design a randomized algorithm that runs in $\tilde{O}(n^2)$ time and, with high probability, guarantees a 2-approximation for all pairs at distance at least $c$, where $c \ge 0$ is a constant. Unlike the above two results, which were purely combinatorial, our algorithm combines combinatorial techniques with fast matrix multiplication (FMM).

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Manoj Gupta, Mrigankashekhar Shandilya. 2026-07-21. $\tilde{O}$ptimal Algorithm for 2-Approximate All Pair Shortest Paths -- almost. https://arxiv.org/abs/2607.18714

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