The distance sensitivity oracle (DSO) problem asks us to preprocess a given graph $G=(V,E)$ in order to answer queries of the form $d(x,y,e)$, which denotes the shortest path distance in $G$ from vertex $x$ to vertex $y$ when edge $e$ is removed. This is an important problem for network communication, and it has been extensively studied in the sequential settingand recently in the distributed CONGEST model. However, no prior DSO results tailored to the parallel setting were known. We present the first PRAM algorithms to construct DSOs in directed weighted graphs, that can answer a query in $O(1)$ time with a single processor after preprocessing. We also present the first work-optimal PRAM algorithms for other graph problems that belong to the sequential $\tilde{O}(mn)$ fine-grained complexity class: Replacement Paths, Second Simple Shortest Path, All Pairs Second Simple Shortest Paths and Minimum Weight Cycle.

翻译：距离敏感度预言机（DSO）问题要求对给定图 $G=(V,E)$ 进行预处理，以便回答形如 $d(x,y,e)$ 的查询，该查询表示在边 $e$ 被移除后，图 $G$ 中从顶点 $x$ 到顶点 $y$ 的最短路径距离。这是网络通信中的一个重要问题，已在顺序计算模型中得到了广泛研究，并最近在分布式CONGEST模型中进行了探索。然而，此前尚无专门针对并行计算环境定制的DSO结果。我们提出了首个在有向加权图中构建DSO的PRAM算法，该算法在预处理后能以单个处理器在 $O(1)$ 时间内回答查询。我们还提出了首个针对其他图问题的工作最优PRAM算法，这些问题属于顺序计算中 $\tilde{O}(mn)$ 细粒度复杂度类：替换路径、次简单最短路径、所有顶点对次简单最短路径以及最小权重环。