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 $\tilde O(n^{2.5-r}+n^{\omega(r)})$ time, for any $r\in[0,1]$. This is $O(n^{2.032})$ time, using known bounds for rectangular matrix multiplication~$n^{\omega(r)}$~[Le Gall, Urrutia, SODA 2018]. Our result improves on the $\tilde{O}(n^{2.25})$ bound of [Roddity, STOC 2023], and on the $\tilde{O}(m\sqrt n+n^2)$ bound of [Baswana, Kavitha, SICOMP 2010] for graphs with $m\geq n^{1.532}$ edges. For weighted graphs, we obtain $(2+\epsilon)$-approximate APSP in $\tilde O(n^{3-r}+n^{\omega(r)})$ time, for any $r\in [0,1]$. This is $O(n^{2.214})$ time using known bounds for $\omega(r)$. 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 $\tilde O(mn^{2/3})$ 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 $(1+\epsilon,k)$-approximate APSP, for $k=2,4,6,8$. 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.

翻译：本文重新审视了无向图中经典的近似全源最短路径（APSP）问题。对于无权图，我们提供了一种算法，可在 $\tilde O(n^{2.5-r}+n^{\omega(r)})$ 时间内实现 $2$-近似APSP，其中 $r\in[0,1]$。利用已知的矩形矩阵乘法上界 $n^{\omega(r)}$ [Le Gall, Urrutia, SODA 2018]，该算法时间复杂度为 $O(n^{2.032})$。我们的结果改进了 [Roddity, STOC 2023] 的 $\tilde{O}(n^{2.25})$ 上界，以及 [Baswana, Kavitha, SICOMP 2010] 针对 $m\geq n^{1.532}$ 边图的 $\tilde{O}(m\sqrt n+n^2)$ 上界。对于加权图，我们实现了 $(2+\epsilon)$-近似APSP，时间复杂度为 $\tilde O(n^{3-r}+n^{\omega(r)})$，其中 $r\in[0,1]$。利用 $\omega(r)$ 的已知上界，该时间复杂度为 $O(n^{2.214})$，改进了 [Kavitha, Algorithmica 2012] 的当前最优上界 $O(n^{2.25})$。我们的技术进一步在加权图的广泛密度范围内带来了改进的上界。特别是，对于稀疏图，我们构建了一个距离预言机，预处理时间为 $\tilde O(mn^{2/3})$，支持常数时间内的 $2$-近似查询。对于稀疏图，该算法的预处理时间匹配条件性下界 [Patrascu, Roditty, Thorup, FOCS 2012; Abboud, Bringmann, Fischer, STOC 2023]。据我们所知，这是首个在稀疏图中具有次二次预处理时间的2-近似距离预言机。我们还在无权图的近加性范围内获得了新的上界。对于 $k=2,4,6,8$，我们给出了更快的 $(1+\epsilon,k)$-近似APSP算法。这些结果通过将快速矩形矩阵乘法融入多种组合算法中实现，这些算法精心平衡了在保留特定距离信息的稀疏图层次上的距离计算。