A Simple Parallel Algorithm with Near-Linear Work for Negative-Weight Single-Source Shortest Paths

TL;DR

This paper introduces a simple, randomized parallel algorithm for the Single-Source Shortest Paths problem in graphs with negative weights, achieving near-linear work and improved span, advancing the state-of-the-art in parallel graph algorithms.

Contribution

It presents the first parallel algorithm with near-linear work for SSSP in general graphs with negative weights, using a novel bottom-up approach and a parallel black-box reduction.

Findings

Achieves near-linear work  O(m) for SSSP in graphs with negative weights.

Provides a parallel algorithm with span n^{1/2 + o(1)}.

Offers a simple and clean algorithm with minimal overhead.

Abstract

We give the first parallel algorithm with optimal $\tilde{O}(m)$ work for the classical problem of computing Single-Source Shortest Paths in general graphs with negative-weight edges. In graphs without negative edges, Dijkstra's algorithm solves the Single-Source Shortest Paths (SSSP) problem with optimal $\tilde O(m)$ work, but is inherently sequential. A recent breakthrough by Bernstein, Nanongkai, Wulff-Nilsen; FOCS '22 achieves the same for general graphs. Parallel shortest path algorithms are more difficult and have been intensely studied for decades. Only very recently, multiple lines of research culminated in parallel algorithms with optimal work $\tilde O(m)$ for various restricted settings, such as approximate or exact algorithms for directed or undirected graphs without negative edges. For general graphs, the best known algorithm by [shvinkumar, Bernstein, Cao, Grunau, Haeupler, Jiang, Nanongkai, Su; ESA '24 still requires $m^{1+o(1)}$ work. This paper presents a randomized parallel algorithm for SSSP in general graphs with near-linear work $\tilde O(m)$ and state-of-the-art span $n^{1/2 + o(1)}$. We follow a novel bottom-up approach leading to a particularly clean and simple algorithm. Our algorithm can be seen as a \emph{near-optimal parallel black-box reduction} from SSSP in general graphs to graphs without negative edges. In contrast to prior works, the reduction in this paper is both parallel and essentially without overhead, only affecting work and span by polylogarithmic factors.