Fully-Dynamic All-Pairs Shortest Paths: Likely Optimal Worst-Case Update Time
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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 amortized algorithm with Õ(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 Õ(n ^ 2) but fails to improve their running time of Õ(n ^ 2 + 2 / 3). It has been conjectured that no algorithm in O(n ^ 2.5 - ϵ) 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 Õ(n ^ 2.5) time while keeping the Õ(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.
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