Parallel, Distributed, and Quantum Exact Single-Source Shortest Paths with Negative Edge Weights

TL;DR

Negative-weight SSSP on directed graphs with integer weights $w:E\to\mathbb{Z}$ is the central problem. The paper proposes a framework reducing negative-weight SSSP to $n^{o(1)}$ calls to a virtual-source SSSP routine, enabling parallel, distributed, and quantum variants. A key technical contribution is an efficient low-diameter decomposition (LDD) for directed graphs, extending prior undirected LDD results to directed and enabling the virtual-source reduction. With state-of-the-art SSSP techniques, the framework yields concrete randomized bounds: in parallel, $W_{SSSP}(m,n)n^{o(1)}$ work and $S_{SSSP}(m,n)n^{o(1)}$ span; in CONGEST, $T_{SSSP}(n,D)n^{o(1)}$ rounds; and in the quantum edge-query model, $Q_{SSSP}(m,n)n^{o(1)}$ queries, including concrete instances such as $m^{1+o(1)}$ work and $n^{1/2+o(1)}$ span in parallel, $(n^{2/5}D^{2/5}+ obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak )n^{o(1)}$ rounds in CONGEST, and $m^{1/2}n^{1/2+o(1)}$ (or $n^{1.5+o(1)}$) quantum queries. These results build on Bernstein et al. (FOCS'22) and Chen et al.'s sequential near-linear-time negative-weight SSSP to extend to parallel, distributed, and quantum settings.

Abstract

This paper presents parallel, distributed and quantum algorithms for single-source shortest paths when edges can have negative weights (negative-weight SSSP). We show a framework that reduces negative-weight SSSP in all these setting to $n^{o(1)}$ calls to any SSSP algorithm that works with a virtual source. More specifically, for a graph with $m$ edges, $n$ vertices, undirected hop-diameter $D$, and polynomially bounded integer edge weights, we show randomized algorithms for negative-weight SSSP with (i) $W_{SSSP}(m,n)n^{o(1)}$ work and $S_{SSSP}(m,n)n^{o(1)}$ span, given access to an SSSP algorithm with $W_{SSSP}(m,n)$ work and $S_{SSSP}(m,n)$ span in the parallel model, (ii) $T_{SSSP}(n,D)n^{o(1)}$, given access to an SSSP algorithm that takes $T_{SSSP}(n,D)$ rounds in $\mathsf{CONGEST}$, (iii) $Q_{SSSP}(m,n)n^{o(1)}$ quantum edge queries, given access to a non-negative-weight SSSP algorithm that takes $Q_{SSSP}(m,n)$ queries in the quantum edge query model. This work builds off the recent result of [Bernstein, Nanongkai, Wulff-Nilsen, FOCS'22], which gives a near-linear time algorithm for negative-weight SSSP in the sequential setting. Using current state-of-the-art SSSP algorithms yields randomized algorithms for negative-weight SSSP with (i) $m^{1+o(1)}$ work and $n^{1/2+o(1)}$ span in the parallel model, (ii) $(n^{2/5}D^{2/5} + \sqrt{n} + D)n^{o(1)}$ rounds in $\mathsf{CONGEST}$, (iii) $m^{1/2}n^{1/2+o(1)}$ quantum queries to the adjacency list or $n^{1.5+o(1)}$ quantum queries to the adjacency matrix. Our main technical contribution is an efficient reduction for computing a low-diameter decomposition (LDD) of directed graphs to computations of SSSP with a virtual source. Efficiently computing an LDD has heretofore only been known for undirected graphs in both the parallel and distributed models.