AI 中文总结
针对无向无权图全对最短路径的2近似问题，此前算法有距离限制。本文设计随机算法，运行时间$\tilde{O}(n^2)$，高概率对距离至少为常数$c$的所有对保证2近似，结合组合技术与快速矩阵乘法解决了该问题。
AI 中文摘要
给定一个无向无权图$G$，目标是计算全对最短路径（APSP）的2近似。该问题因输出大小为$\Theta(n^2)$，有$\Omega(n^2)$的自然下界，该领域核心目标是实现$O(n^2)$的运行时间。Dor等人设计了运行时间为$\tilde{O}(n^2)$的算法，但仅对距离至少为$O(\log n)$的对保证2近似，Gupta改进为至少$O(\log \log n)$。本文设计了随机算法，运行时间为$\tilde{O}(n^2)$，以高概率对距离至少为常数$c$的所有对保证2近似，算法结合了组合技术和快速矩阵乘法。
英文摘要
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 $Ω(n^2)$ since the output size is $Θ(n^2)$. 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).
