  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, SODA'21], 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 $\Omega(1/\log\log m)\leq
\epsilon< 1$, achieves approximation factor $(\log m)^{2^{O(1/\epsilon)}}$,
with total update time $O\left (m^{1+O(\epsilon)}\cdot (\log
m)^{O(1/\epsilon^2)}\cdot \log L\right )$.
