Our input is an undirected weighted graph $G = (V,E)$ on $n$ vertices along with a source set $S\subseteq V$. The problem is to preprocess $G$ and build a compact data structure such that upon query $Qu(s,v,f)$ where $(s,v) \in S\times V$ and $f$ is any faulty edge, we can quickly find a good estimate (i.e., within a small multiplicative stretch) of the $s$-$v$ distance in $G-f$. We use a fault-tolerant $ST$-distance oracle from the work of Bil{\`{o}} et al. (STACS 2018) to construct an $S\times V$ approximate distance oracle or {\em sourcewise} approximate distance oracle of size $\widetilde{O}(|S|n + n^{3/2})$ with multiplicative stretch at most 5. We construct another fault-tolerant sourcewise approximate distance oracle of size $\widetilde{O}(|S|n + n^{4/3})$ with multiplicative stretch at most 13. Both the oracles have $O(1)$ query answering time.

翻译：我们的输入是一个具有 n 个顶点的无向加权图 $G = (V,E)$ 以及一个源集 $S\\subseteq V$。该问题旨在对 $G$ 进行预处理并构建一个紧凑的数据结构，使得对于查询 $Qu(s,v,f)$（其中 $(s,v) \\in S\\times V$，$f$ 为任意故障边），我们能够快速找到 $G-f$ 中 $s$-$v$ 距离的良好估计（即在一个较小的乘法伸缩范围内）。我们利用 Bil{\\`{o}} 等人（STACS 2018）工作中的容错 $ST$-距离预言机构建了一个 $S\\times V$ 近似距离预言机或{\\em 源向}近似距离预言机，其大小为 $\\widetilde{O}(|S|n + n^{3/2})$，乘法伸缩至多为 5。我们还构建了另一个容错源向近似距离预言机，其大小为 $\\widetilde{O}(|S|n + n^{4/3})$，乘法伸缩至多为 13。这两个预言机均具有 $O(1)$ 的查询响应时间。