Algebraic data structures are the main subroutine for maintaining distances in fully dynamic graphs in subquadratic time. However, these dynamic algebraic algorithms generally cannot maintain the shortest paths, especially against adaptive adversaries. We present the first fully dynamic algorithm that maintains the shortest paths against an adaptive adversary in subquadratic update time. This is obtained via a combinatorial reduction that allows reconstructing the shortest paths with only a few distance estimates. Using this reduction, we obtain the following: On weighted directed graphs with real edge weights in $[1,W]$, we can maintain $(1+\epsilon)$ approximate shortest paths in $\tilde{O}(n^{1.816}\epsilon^{-2} \log W)$ update and $\tilde{O}(n^{1.741} \epsilon^{-2} \log W)$ query time. This improves upon the approximate distance data structures from [v.d.Brand, Nanongkai, FOCS'19], which only returned a distance estimate, by matching their complexity and returning an approximate shortest path. On unweighted directed graphs, we can maintain exact shortest paths in $\tilde{O}(n^{1.823})$ update and $\tilde{O}(n^{1.747})$ query time. This improves upon [Bergamaschi, Henzinger, P.Gutenberg, V.Williams, Wein, SODA'21] who could report the path only against oblivious adversaries. We improve both their update and query time while also handling adaptive adversaries. On unweighted undirected graphs, our reduction holds not just against adaptive adversaries but is also deterministic. We maintain a $(1+\epsilon)$-approximate $st$-shortest path in $O(n^{1.529} / \epsilon^2)$ time per update, and $(1+\epsilon)$-approximate single source shortest paths in $O(n^{1.764} / \epsilon^2)$ time per update. Previous deterministic results by [v.d.Brand, Nazari, Forster, FOCS'22] could only maintain distance estimates but no paths.

翻译：代数数据结构是次二次时间内维护完全动态图中距离的主要子程序。然而，这些动态代数算法通常无法维护最短路径，尤其是面对自适应对手时。我们提出了首个在次二次更新时间内维护面向自适应对手的最短路径的完全动态算法。该算法通过一种组合约简实现，仅需少量距离估计即可重构最短路径。利用此约简，我们获得以下结果：在边权为$[1,W]$内实数的加权有向图上，我们可以在$\tilde{O}(n^{1.816}\epsilon^{-2} \log W)$更新时间和$\tilde{O}(n^{1.741} \epsilon^{-2} \log W)$查询时间内维护$(1+\epsilon)$近似最短路径。这改进了[v.d.Brand, Nanongkai, FOCS'19]中仅返回距离估计的近似距离数据结构，在匹配其复杂度的同时返回近似最短路径。在无权有向图上，我们可以在$\tilde{O}(n^{1.823})$更新时间和$\tilde{O}(n^{1.747})$查询时间内维护精确最短路径。这改进了[Bergamaschi, Henzinger, P.Gutenberg, V.Williams, Wein, SODA'21]中仅能针对 oblivious 对手报告路径的结果，我们在提高更新和查询时间的同时处理了自适应对手。在无权无向图上，我们的约简不仅对抗自适应对手，而且具有确定性。我们以每次更新$O(n^{1.529} / \epsilon^2)$的时间维护$(1+\epsilon)$近似$st$-最短路径，并以每次更新$O(n^{1.764} / \epsilon^2)$的时间维护$(1+\epsilon)$近似单源最短路径。此前[v.d.Brand, Nazari, Forster, FOCS'22]的确定性结果仅能维护距离估计而无法获得路径。