Directed Shortest Paths via Approximate Cost Balancing

TL;DR

This paper introduces an $O(nm)$ algorithm for all-pairs shortest paths in directed graphs with nonnegative integer costs, leveraging approximate cost balancing to match undirected graph complexities.

Contribution

It presents a novel approach to shortest path computation in directed graphs using approximate cost balancing, achieving complexity bounds similar to undirected cases.

Findings

Achieves $O(nm)$ complexity for directed all-pairs shortest paths.

Introduces an $O(m\sqrt{n}\log n)$ preprocessing step for cost balancing.

Enables $O(m)$ shortest path queries after preprocessing.

Abstract

We present an $O(nm)$ algorithm for all-pairs shortest paths computations in a directed graph with $n$ nodes, $m$ arcs, and nonnegative integer arc costs. This matches the complexity bound attained by Thorup \cite{Thorup1999} for the all-pairs problems in undirected graphs. The main insight is that shortest paths problems with approximately balanced directed cost functions can be solved similarly to the undirected case. The algorithm finds an approximately balanced reduced cost function in an $O(m\sqrt{n}\log n)$ preprocessing step. Using these reduced costs, every shortest path query can be solved in $O(m)$ time using an adaptation of Thorup's component hierarchy method. The balancing result can also be applied to the $\ell_\infty$-matrix balancing problem.