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 $alpha$ of the length of the shortest $x$-$y$ path, in time $tilde O(|E(P)|)$, where $alpha$ 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 $(1+epsilon)$-approximation with total update time $tilde O(mn/epsilon)$; the best current total update time of $n^{2.5+O(epsilon)}$ is achieved by the deterministic algorithm of [Chuzhoy, Saranurak, SODA21], with $2^{O(1/epsilon)}$-multiplicative and $2^{O(log^{3/4}n/epsilon)}$-additive approximation. To the best of our knowledge, for arbitrary non-negative edge weights, the fastest current adaptive-update algorithm has total update time $O(n^{3}log L/epsilon)$, achieving a $(1+epsilon)$-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 $Omega(1/loglog m)leq epsilon< 1$, achieves approximation factor $(log m)^{2^{O(1/epsilon)}}$, with total update time $Oleft (m^{1+O(epsilon)}cdot (log m)^{O(1/epsilon^2)}cdot log Lright )$.
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