We study the fully dynamic All-Pairs Shortest Paths (APSP) problem in undirected edge-weighted graphs. Given an $n$-vertex graph $G$ with non-negative edge lengths, that undergoes an online sequence of edge insertions and deletions, the goal is to support approximate distance queries and shortest-path queries. We provide a deterministic algorithm for this problem, that, for a given precision parameter $ε$, achieves approximation factor $(\log\log n)^{2^{O(1/ε^3)}}$, and has amortized update time $O(n^ε\log L)$ per operation, where $L$ is the ratio of longest to shortest edge length. Query time for distance-query is $O(2^{O(1/ε)}\cdot \log n\cdot \log\log L)$, and query time for shortest-path query is $O(|E(P)|+2^{O(1/ε)}\cdot \log n\cdot \log\log L)$, where $P$ is the path that the algorithm returns. To the best of our knowledge, even allowing any $o(n)$-approximation factor, no adaptive-update algorithms with better than $Θ(m)$ amortized update time and better than $Θ(n)$ query time were known prior to this work. We also note that our guarantees are stronger than the best current guarantees for APSP in decremental graphs in the adaptive-adversary setting.

arXiv admin note: text overlap with arXiv:2109.05621