The All-Pairs Shortest Paths (APSP) problem is one of the fundamental problems in theoretical computer science. It asks to compute the distance matrix of a given $n$-vertex graph. We revisit the classical problem of maintaining the distance matrix under a fully dynamic setting undergoing vertex insertions and deletions with a fast worst-case running time and efficient space usage. Although an algorithm with amortized update-time $\tilde O(n ^ 2)$ has been known for nearly two decades [Demetrescu and Italiano, STOC 2003], the current best algorithm for worst-case running time with efficient space usage runs is due to [Gutenberg and Wulff-Nilsen, SODA 2020], which improves the space usage of the previous algorithm due to [Abraham, Chechik, and Krinninger, SODA 2017] to $\tilde O(n ^ 2)$ but fails to improve their running time of $\tilde O(n ^ {2 + 2 / 3})$. It has been conjectured that no algorithm in $O(n ^ {2.5 - \epsilon})$ worst-case update time exists. For graphs without negative cycles, we meet this conjectured lower bound by introducing a Monte Carlo algorithm running in randomized $\tilde O(n ^ {2.5})$ time while keeping the $\tilde O(n ^ 2)$ space bound from the previous algorithm. Our breakthrough is made possible by the idea of ``hop-dominant shortest paths,'' which are shortest paths with a constraint on hops (number of vertices) that remain shortest after we relax the constraint by a constant factor.

翻译：全源最短路径（APSP）问题是理论计算机科学中的基本问题之一，要求计算给定 $n$ 顶点图的距离矩阵。本文重新审视在完全动态环境下（支持顶点插入和删除）维护距离矩阵的经典问题，目标是实现快速的最坏情况运行时间和高效的空间使用。尽管一种均摊更新时间为 $\tilde O(n ^ 2)$ 的算法已问世近二十年 [Demetrescu and Italiano, STOC 2003]，但当前在高效空间使用下实现最坏情况运行时间的最佳算法由 [Gutenberg and Wulff-Nilsen, SODA 2020] 提出，该算法将 [Abraham, Chechik, and Krinninger, SODA 2017] 先前算法的空间使用改进为 $\tilde O(n ^ 2)$，但未能改进其 $\tilde O(n ^ {2 + 2 / 3})$ 的运行时间。已有猜想认为不存在最坏情况更新时间为 $O(n ^ {2.5 - \epsilon})$ 的算法。对于无负环图，我们通过引入一种随机化 $\tilde O(n ^ {2.5})$ 时间的蒙特卡罗算法，满足该猜想的下界，同时保持先前算法的 $\tilde O(n ^ 2)$ 空间上界。这一突破得益于“跳主导最短路径”的概念，即满足跳数（顶点数）约束的最短路径，在将约束放宽常数倍后仍保持最短性。