Decremental All-Pairs Shortest Paths in Deterministic Near-Linear Time
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We study the decremental All-Pairs Shortest Paths (APSP) problem in undirected edge-weighted graphs. The input to the problem is an n-vertex m-edge graph G with non-negative edge lengths, that undergoes a sequence of edge deletions. The goal is to support approximate shortest-path queries: given a pair x,y of vertices of G, return a path P connecting x to y, whose length is within factor α of the length of the shortest x-y path, in time Õ(|E(P)|), where α is the approximation factor of the algorithm. APSP is one of the most basic and extensively studied dynamic graph problems. A long line of work culminated in the algorithm of [Chechik, FOCS 2018] with near optimal guarantees for the oblivious-adversary setting. Unfortunately, adaptive-adversary setting is still poorly understood. For unweighted graphs, the algorithm of [Henzinger, Krinninger and Nanongkai, FOCS '13, SICOMP '16] achieves a (1+ϵ)-approximation with total update time Õ(mn/ϵ); the best current total update time of n^2.5+O(ϵ) is achieved by the deterministic algorithm of [Chuzhoy, Saranurak, SODA'21], with 2^O(1/ϵ)-multiplicative and 2^O(log^3/4n/ϵ)-additive approximation. To the best of our knowledge, for arbitrary non-negative edge weights, the fastest current adaptive-update algorithm has total update time O(n^3log L/ϵ), achieving a (1+ϵ)-approximation. Here, L is the ratio of longest to shortest edge lengths. Our main result is a deterministic algorithm for decremental APSP in undirected edge-weighted graphs, that, for any Ω(1/loglog m)≤ϵ< 1, achieves approximation factor (log m)^2^O(1/ϵ), with total update time O (m^1+O(ϵ)· (log m)^O(1/ϵ^2)·log L ).
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