Approximate Single Source Dual Fault Tolerant Distance Oracle

We are given an undirected weighted graph $G$ with $n$ vertices and $m$ edges, edge weights in $[1,W]$, and a designated source vertex $s$. We design a single source dual fault tolerant distance oracle for $G$. Given a destination vertex $t$ and a set $F$ of at most two faulty edges, the oracle returns a $(1+O(\epsilon ))$-approximation of the weight of the shortest path from the source $s$ to $t$ avoiding $F$. Our oracle uses $\tilde{O}(n\sqrt{n})$ space and has $\tilde{O}(1)$ query time. Prior to our result, single source single fault tolerant oracles were known to return a $(1+\epsilon )$ approximation of the weight of the shortest path using $\tilde{O}(n)$ space and $O(1)$ query time. However, extending these approaches to multiple faults remained an open problem. Indeed, all $(1+\epsilon )$-approximate distance oracles that handle multiple faults require $\mathrm{\Omega }(n^2)$ space. We break this bound by presenting the first dual fault tolerant distance oracle with $o(n^2)$ space.