This paper presents parallel and distributed 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 either 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 $\mathsf{CONGEST}$. 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 $\mathsf{CONGEST}$. 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. The LDD is a crucial step of the algorithm in [Bernstein, Nanongkai, Wulff-Nilsen, FOCS'22], and we think that its applications to other problems in parallel and distributed models are far from being exhausted.

翻译：本文提出了边权可为负值的单源最短路径（含负权边的SSSP）的并行与分布式算法。我们构建了一个框架，可将任意设置下的含负权边SSSP问题归约为对利用虚拟源点运行的任意SSSP算法的$n^{o(1)}$次调用。具体而言，对于具有$m$条边、$n$个顶点、无向跃点直径$D$且边权为多项式有界整数的图，我们给出了含负权边SSSP的随机化算法：（i）在并行模型下，若可调用的SSSP算法工作量为$W_{SSSP}(m,n)$、跨度为$S_{SSSP}(m,n)$，则本文算法的工作量为$W_{SSSP}(m,n)n^{o(1)}$、跨度为$S_{SSSP}(m,n)n^{o(1)}$；（ii）在$\mathsf{CONGEST}$模型中，若可调用的SSSP算法运行$T_{SSSP}(n,D)$轮，则本文算法运行$T_{SSSP}(n,D)n^{o(1)}$轮。本研究基于[Bernstein, Nanongkai, Wulff-Nilsen, FOCS'22]的最新成果，该成果给出了顺序设置下含负权边SSSP的近线性时间算法。利用当前最先进的SSSP算法，可得到含负权边SSSP的随机化算法：（i）在并行模型下，工作量为$m^{1+o(1)}$、跨度为$n^{1/2+o(1)}$；（ii）在$\mathsf{CONGEST}$模型中，运行轮数为$(n^{2/5}D^{2/5} + \sqrt{n} + D)n^{o(1)}$。本文的主要技术贡献在于将计算有向图的低直径分解（LDD）高效归约为利用虚拟源点计算SSSP。此前，高效计算LDD仅在并行和分布式模型的无向图中实现。LDD是[Bernstein, Nanongkai, Wulff-Nilsen, FOCS'22]算法中的关键步骤，我们认为其在并行与分布式模型中的其他问题应用仍有广阔前景。