Near Optimal Algorithm for the Directed Single Source Replacement Paths Problem

In the Single Source Replacement Paths (SSRP) problem we are given a graph $G = (V, E)$, and a shortest paths tree $\widehat{K}$ rooted at a node $s$, and the goal is to output for every node $t \in V$ and for every edge $e$ in $\widehat{K}$ the length of the shortest path from $s$ to $t$ avoiding $e$. We present an $\tilde{O}(m\sqrt{n} + n^2)$ time randomized combinatorial algorithm for unweighted directed graphs. Previously such a bound was known in the directed case only for the seemingly easier problem of replacement path where both the source and the target nodes are fixed. Our new upper bound for this problem matches the existing conditional combinatorial lower bounds. Hence, (assuming these conditional lower bounds) our result is essentially optimal and completes the picture of the SSRP problem in the combinatorial setting. Our algorithm extends to the case of small, rational edge weights. We strengthen the existing conditional lower bounds in this case by showing that any $O(mn^{{1/2-\epsilon})$} time (combinatorial or algebraic) algorithm for some fixed $\epsilon >0$ yields a truly subcubic algorithm for the weighted All Pairs Shortest Paths problem (previously such a bound was known only for the combinatorial setting).