In the planar, dynamic All-Pairs Shortest Paths (APSP) problem, a planar, weighted digraph $G$ undergoes a sequence of edge weight updates and the goal is to maintain a data structure on $G$, that can quickly answer distance queries between any two vertices $x,y \in V(G)$. The currently best algorithms for this problem require $\tilde{O}(n^{2/3})$ worst-case update and query time, while conditional lower bounds show that either update or query time $n^{0.5-δ}$ is needed for any constant $δ> 0$. In this article, we present the first algorithm with near-optimal $\tilde{O}(\sqrt{n})$ worst-case update and query time for the offline setting, where the update sequence is given initially. This result is obtained by giving the first offline dynamic algorithm for maintaining dense distance graphs (DDGs) faster than recomputing from scratch after each update. Further, we also present an \emph{online} algorithm for the incremental APSP problem with $\tilde{O}(\sqrt{n})$ worst-case update/ query time. This allows us to reduce the online dynamic APSP problem to the online decremental APSP problem, which constitutes partial progress even for the online version of this notorious problem.

翻译：在平面动态全源最短路径(APSP)问题中，一个平面加权有向图$G$经历一系列边权重更新，目标是在$G$上维护一个数据结构，能够快速回答任意两个顶点$x,y \in V(G)$之间的距离查询。当前解决该问题的最优算法需要$\tilde{O}(n^{2/3})$的最坏情况更新和查询时间，而条件下界表明，对于任意常数$δ> 0$，需要$n^{0.5-δ}$的更新或查询时间。本文提出了第一个在离线设置下具有近乎最优$\tilde{O}(\sqrt{n})$最坏情况更新和查询时间的算法，其中更新序列在初始时给定。这一结果通过首次给出一种比每次更新后从头重新计算更快的离线动态维护稠密距离图(DDG)算法而获得。此外，我们还提出了一种增量APSP问题的在线算法，其最坏情况下更新/查询时间为$\tilde{O}(\sqrt{n})$。这使我们能够将在线动态APSP问题归约到在线递减APSP问题，即使对于这个棘手问题的在线版本，也构成了部分进展。