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 \emph{fully-dynamic} setting undergoing vertex insertions and deletions with a fast \emph{worst-case} running time and efficient space usage. Although amortized algorithm with $\tilde O(n ^ 2)$ update-time 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 improvement is made possible using a novel multi-layer approach that exploits the gaps between hops (number of vertices traversed) of paths.

翻译：全源最短路径（APSP）问题是理论计算机科学中的基础问题之一，要求计算给定 $n$ 顶点图的距离矩阵。本文重新研究了在顶点插入与删除操作下的\emph{全动态}环境中维护距离矩阵的经典问题，目标是在最坏情形下实现快速运行时间与高效空间使用。尽管近二十年前已存在更新时间为 $\tilde O(n ^ 2)$ 的摊还算法 [Demetrescu and Italiano, STOC 2003]，当前兼顾最坏情形运行时间与高效空间使用的最佳算法由 [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优化至 $\tilde O(n ^ 2)$，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形更新时间为 $O(n ^ {2.5 - \epsilon})$ 的算法。对于不含负环的图，我们通过引入一种随机化蒙特卡洛算法满足该猜想的下界，该算法运行时间为 $\tilde O(n ^ {2.5})$，同时保持先前算法的 $\tilde O(n ^ 2)$ 空间界。这一改进得益于一种新颖的多层方法，该方法利用了路径跳数（遍历顶点数）之间的间隙。