Fault-Tolerant Approximate Distance Oracles with a Source Set

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 $G-f$. The work of Bil{\`{o}} et al. (Algorithmica 2022) on multiple-edge fault-tolerant approximate shortest path trees implies a compact oracle for the above problem with a stretch of at most 3 and with query answering time $O(\log^2 n)$. We show a very simple construction of an $S\times V$ approximate distance oracle with $O(1)$ query answering time; its size is $\widetilde{O}(|S|n + n^{3/2})$ and multiplicative stretch is at most 5. A single-edge fault-tolerant $ST$-distance oracle from the work of Bil{\`{o}} et al. (STACS 2018) plays a key role in our construction. We also give a construction of a fault-tolerant $S \times V$ approximate distance oracle of size $\widetilde{O}(|S|n + n^{4/3})$ with multiplicative stretch at most 13 and as before, with $O(1)$ query answering time.