Fault-Tolerant Approximate Distance Oracles with a Source Set

Abstract: 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.