For an input graph $G=(V, E)$ and a source vertex $s \in V$, the \emph{$\alpha$-approximate vertex fault-tolerant distance sensitivity oracle} (\emph{$\alpha$-VSDO}) answers an $\alpha$-approximate distance from $s$ to $t$ in $G-x$ for any query $(x, t)$. 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 the $(1 + \epsilon)$-VSDO for any weighted directed graph of $n$ vertices and $m$ edges with integer weights in range $[1, W]$, and any positive constant $\epsilon \in (0, 1]$. More precisely, the presented oracle attains $\tilde{O}(m / \epsilon + n /\epsilon^2)$ construction time, $\tilde{O}(n/ \epsilon)$ size, and $\tilde{O}(1/\epsilon)$ query time for any polynomially-bounded $W$. To the best of our knowledge, this is the first non-trivial result for SSRP/VSDO beating the trivial $\tilde{O}(mn)$ computation time for directed graphs with polynomially-bounded edge weights. Such a result has been unknown so far even for the setting of $(1 + \epsilon)$-approximation. It also implies that the known barrier of $\Omega(m\sqrt{n})$ time for the exact SSRP by Chechik and Magen~[ICALP2020] does not apply to the case of approximation.

翻译：对于输入图 $G=(V, E)$ 和源顶点 $s \in V$，\emph{$\alpha$-近似顶点容错距离灵敏度预言机}（\emph{$\alpha$-VSDO}）能够针对任意查询 $(x, t)$，回答在 $G-x$ 中从 $s$ 到 $t$ 的 $\alpha$-近似距离。它是所谓单源替换路径问题（SSRP）的数据结构版本。在本文中，我们提出了一种新的\emph{近线性时间}算法，用于构建任意加权有向图（具有 $n$ 个顶点和 $m$ 条边，边权重为 $[1, W]$ 范围内的整数）以及任意正常数 $\epsilon \in (0, 1]$ 的 $(1+\epsilon)$-VSDO。更确切地说，对于任意多项式有界的 $W$，所提出的预言机实现了 $\tilde{O}(m / \epsilon + n /\epsilon^2)$ 的构建时间、$\tilde{O}(n/ \epsilon)$ 的存储大小以及 $\tilde{O}(1/\epsilon)$ 的查询时间。据我们所知，这是首个在边权重多项式有界的有向图上，超越了平凡 $\tilde{O}(mn)$ 计算时间的 SSRP/VSDO 非平凡结果。即使在 $(1+\epsilon)$-近似设置下，此前也未见此类结果。该结果同时意味着，Chechik 和 Magen~[ICALP2020] 提出的精确 SSRP 的 $\Omega(m\sqrt{n})$ 时间已知障碍不适用于近似情形。