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：全对最短路径（APSP）问题是理论计算机科学中的基本问题之一，它要求计算给定$n$顶点图的距离矩阵。我们重新审视在全动态设置下维护距离矩阵的经典问题，该设置涉及顶点插入和删除，需要快速的最坏情况运行时间和高效的空间使用。尽管具有$\tilde O(n ^ 2)$摊销更新时间的算法在近二十年前就已为人所知[Demetrescu and Italiano, STOC 2003]，但目前具有高效空间使用的最坏情况运行时间最佳算法源于[Gutenberg and Wulff-Nilsen, SODA 2020]，该算法将先前[Abraham, Chechik, and Krinninger, SODA 2017]算法的空间使用改进至$\tilde O(n ^ 2)$，但未能改进其$\tilde O(n ^ {2 + 2 / 3})$的运行时间。据推测，不存在具有$O(n ^ {2.5 - \epsilon})$最坏情况更新时间的算法。对于无负环的图，我们通过引入一种在随机化$\tilde O(n ^ {2.5})$时间内运行的蒙特卡洛算法，同时保持先前算法的$\tilde O(n ^ 2)$空间界限，达到了这一推测下界。我们的突破得益于“跳数主导最短路径”的思想，即那些在跳数（顶点数量）上受限的最短路径，当我们将约束放宽一个常数因子后，它们仍然保持为最短路径。