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 $\epsilon$, achieves approximation factor $(\log\log n)^{2^{O(1/\epsilon^3)}}$, and has amortized update time $O(n^{\epsilon}\log L)$ per operation, where $L$ is the ratio of longest to shortest edge length. Query time for distance-query is $O(2^{O(1/\epsilon)}\cdot \log n\cdot \log\log L)$, and query time for shortest-path query is $O(|E(P)|+2^{O(1/\epsilon)}\cdot \log n\cdot \log\log L)$, 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 $\Theta(m)$ amortized update time and better than $\Theta(n)$ 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.

翻译：我们研究无向边加权图中的全动态所有点对最短路径（APSP）问题。给定一个具有非负边长的 $n$ 顶点图 $G$，该图经历在线序列的边插入和删除操作，目标是支持近似距离查询和最短路径查询。我们为此问题提供了一种确定性算法，对于给定精度参数 $\epsilon$，该算法实现了近似因子 $(\log\log n)^{2^{O(1/\epsilon^3)}}$，并且每次操作的均摊更新时间为 $O(n^{\epsilon}\log L)$，其中 $L$ 是最长边与最短边长度的比值。距离查询的查询时间为 $O(2^{O(1/\epsilon)}\cdot \log n\cdot \log\log L)$，最短路径查询的查询时间为 $O(|E(P)|+2^{O(1/\epsilon)}\cdot \log n\cdot \log\log L)$，其中 $P$ 是算法返回的路径。据我们所知，即使允许任意 $o(n)$ 近似因子，此前没有已知的自适应更新算法能够实现优于 $\Theta(m)$ 的均摊更新时间和优于 $\Theta(n)$ 的查询时间。我们还注意到，我们的保证比在自适应对抗设置下递减图 APSP 的最佳当前保证更强。