An \emph{$\alpha$-approximate vertex fault-tolerant distance sensitivity oracle} (\emph{$\alpha$-VSDO}) for a weighted input graph $G=(V, E, w)$ and a source vertex $s \in V$ is the data structure answering an $\alpha$-approximate distance from $s$ to $t$ in $G-x$ for any given query $(x, t) \in V \times V$. It is a data structure version of the so-called single-source replacement path problem (SSRP). In this paper, we present a new \emph{nearly linear-time} algorithm of constructing a $(1 + \epsilon)$-VSDO for any directed input graph with polynomially bounded integer edge weights. More precisely, the presented oracle attains $\tilde{O}(m \log (nW)/ \epsilon + n \log^2 (nW)/\epsilon^2)$ construction time, $\tilde{O}(n \log (nW) / \epsilon)$ size, and $\tilde{O}(1/\epsilon)$ query time, where $n$ is the number of vertices, $m$ is the number of edges, and $W$ is the maximum edge weight. These bounds are all optimal up to polylogarithmic factors. To the best of our knowledge, this is the first non-trivial algorithm for SSRP/VSDO beating $\tilde{O}(mn)$ computation time for directed graphs with general edge weight functions, and also the first nearly linear-time construction breaking approximation factor 3. Such a construction has been unknown even for undirected and unweighted graphs. In addition, our result implies that the known conditional lower bounds for the exact SSRP computation does not apply to the case of approximation.

翻译：对于带权输入图 $G=(V, E, w)$ 和源顶点 $s \in V$，一个 \emph{$\alpha$近似顶点容错距离敏感性预言机}（\emph{$\alpha$-VSDO}）是一种数据结构，它能够针对任意给定的查询 $(x, t) \in V \times V$，返回从 $s$ 到 $t$ 在 $G-x$ 中的 $\alpha$ 近似距离。该结构是所谓的单源替代路径问题（SSRP）的数据结构版本。本文提出了一种新的 \emph{近线性时间} 算法，用于为任何具有多项式有界整数边权重的有向输入图构建 $(1 + \epsilon)$-VSDO。更精确地说，所构建的预言机实现了 $\tilde{O}(m \log (nW)/ \epsilon + n \log^2 (nW)/\epsilon^2)$ 的构建时间、$\tilde{O}(n \log (nW) / \epsilon)$ 的空间占用以及 $\tilde{O}(1/\epsilon)$ 的查询时间，其中 $n$ 是顶点数，$m$ 是边数，$W$ 是最大边权重。这些界限在忽略多对数因子的意义下均是最优的。据我们所知，这是首个针对具有一般边权重函数的有向图，其计算时间优于 $\tilde{O}(mn)$ 的非平凡 SSRP/VSDO 算法，同时也是首个突破近似因子 3 的近线性时间构建方法。即使对于无向且无权重的图，此类构建方法此前也属未知。此外，我们的结果表明，已知的关于精确 SSRP 计算的条件性下界并不适用于近似计算的情形。