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

TL;DR

This paper introduces a near-optimal deterministic distance oracle for fault-tolerant shortest path queries and applies it to solve the single source replacement path problem efficiently, improving preprocessing time and matching lower bounds.

Contribution

The paper presents a deterministic distance oracle with improved preprocessing time and query efficiency, and demonstrates its application to solve the SSR problem optimally up to polylogarithmic factors.

Findings

Deterministic oracle built in  O(m\u221A n) time.

Oracle answers queries in  O(1) time.

SSR problem solved in  O(m n + |\u211D|) time, matching lower bounds.

Abstract

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 Bil\`{o} 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 $\Omega(m\sqrt n)$ for any combinatorial algorithm that solves this problem.