Near Optimal Algorithm for Fault Tolerant Distance Oracle and Single Source Replacement Path problem

In a graph $G$ with a source $s$, we design a distance oracle that can answer the following query: Query$(s,t,e)$ -- find the length of shortest path from a fixed source $s$ to any destination vertex $t$ while avoiding any edge $e$. We design a deterministic algorithm that builds such an oracle in $\tilde{O}(m\sqrt n)$ time. Our oracle uses $\tilde{O}(n\sqrt n)$ space and can answer queries in $\tilde{O}(1)$ time. Our oracle is an improvement of the work of Bilò et al. (ESA 2021) in the preprocessing time, which constructs the first deterministic oracle for this problem in $\tilde{O}(m\sqrt n+n^2)$ time. Using our distance oracle, we also solve the {\em single source replacement path problem} (SSR problem). Chechik and Cohen (SODA 2019) designed a randomized combinatorial algorithm to solve the SSR problem. The running time of their algorithm is $\tilde{O}(m\sqrt n + n^2)$. In this paper, we show that the SSR problem can be solved in $\tilde{O}(m\sqrt n + |\mathcal{R}|)$ time, where $\mathcal{R}$ is the output set of the SSR problem in $G$. Our SSR algorithm is optimal (upto polylogarithmic factor) as there is a conditional lower bound of $Ω(m\sqrt n)$ for any combinatorial algorithm that solves this problem.