Breaking the Bellman-Ford Shortest-Path Bound

TL;DR

This paper introduces a novel shortest-path algorithm that surpasses the long-standing $O(n imes m)$ time bound of Bellman-Ford, using graph transformation and optimized Dijkstra calls.

Contribution

It presents a new algorithm that breaks the classic Bellman-Ford bound by transforming the graph and leveraging efficient Dijkstra's algorithm implementations.

Findings

Achieves faster shortest-path computation than Bellman-Ford for certain graph classes.

Runs in $O(\sqrt{n} imes m + n imes \sqrt{m imes ext{log} n})$ time with Fibonacci heaps.

Uses graph transformation to maintain shortest-path trees after weight adjustments.

Abstract

In this paper we give a single-source shortest-path algorithm that breaks, after over 65 years, the $O(n \cdot m)$ bound for the running time of the Bellman-Ford-Moore algorithm, where $n$ is the number of vertices and $m$ is the number of arcs of the graph. Our algorithm converts the input graph to a graph with nonnegative weights by performing at most $\min(2 \cdot \sqrt{n},2 \cdot \sqrt{m/\log n})$ calls to a modified version of Dijkstra's algorithm, such that the shortest-path trees are the same for the new graph as those for the original. When Dijkstra's algorithm is implemented using Fibonacci heaps, the running time of our algorithm is therefore $O(\sqrt{n} \cdot m + n \cdot \sqrt{m \log n})$.