SIAM Journal on Computing, Volume 49, Issue 5, Page STOC18-500-STOC18-539, January 2020.
One of the simplest problems on directed graphs is that of identifying the set of vertices reachable from a designated source vertex. This problem can be solved easily sequentially by performing a graph search, but efficient parallel algorithms have eluded researchers for decades. For sparse high-diameter graphs in particular, there is no known work-efficient parallel algorithm with nontrivial parallelism. This amounts to one of the most fundamental open questions in parallel graph algorithms: Is there a parallel algorithm for digraph reachability with nearly linear work? This article shows that the answer is yes, presenting a randomized parallel algorithm for digraph reachability and related problems with expected work $\tilde{O}(m)$ and span $\tilde{O}(n^{2/3})$, and hence parallelism $\tilde{\Omega}(m/n^{2/3}) = \tilde{\Omega}(n^{1/3})$, on any graph with $n$ vertices and $m$ arcs. This is the first parallel algorithm having both nearly linear work and strongly sublinear span, i.e., span $\tilde{O}(n^{1-\epsilon})$ for any constant $\epsilon>0$. The algorithm can be extended to produce a directed spanning tree, determine whether the graph is acyclic, topologically sort the strongly connected components of the graph, or produce a directed ear decomposition, all with work $\tilde{O}(m)$ and span $\tilde{O}(n^{2/3})$. The main technical contribution is an efficient Monte Carlo algorithm that, through the addition of $\tilde{O}(n)$ shortcuts, reduces the diameter of the graph to $\tilde{O}(n^{2/3})$ with high probability. While both sequential and parallel algorithms are known with those combinatorial properties, even the sequential algorithms are not efficient, having sequential runtime $\Omega(mn^{\Omega(1)})$. This article presents a surprisingly simple sequential algorithm that achieves the stated diameter reduction and runs in $\tilde{O}(m)$ time. Parallelizing that algorithm yields the main result, but doing so involves overcoming several other challenges.

中文翻译：

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图可及性几乎有效的并行算法

SIAM计算杂志，第49卷，第5期，第STOC18-500-STOC18-539页，2020年1月。
有向图上最简单的问题之一是识别从指定源顶点可到达的一组顶点。通过执行图搜索可以轻松地顺序解决此问题，但是数十年来，高效的并行算法一直困扰着研究人员。特别是对于稀疏的大直径图，没有已知的具有非平凡并行度的高效工作并行算法。这等于平行图算法中最基本的开放性问题之一：是否有一种并行的算法可实现具有几乎线性工作的有向图？本文表明答案是肯定的，提出了一种有向图可达性和预期工作$ \ tilde {O}（m）$和跨度$ \ tilde {O}（n ^ {2/3}）相关问题的随机并行算法。 $，因此并行度$ \ tilde {\ Omega}（m / n ^ {2/3}）= \ tilde {\ Omega}（n ^ {1/3}）$，在具有$ n $个顶点和$ m $弧的任何图形上。这是第一个具有几乎线性功和强次线性跨度的并行算法，即，对于任何常数$ \ epsilon> 0 $，跨度$ \ tilde {O}（n ^ {1- \ epsilon}）$。可以扩展该算法以生成有向生成树，确定图是否为非循环图，对图的强连接组件进行拓扑排序或生成有向耳分解，所有工作都用\\ tilde {O}（m）$和跨度$ \ tilde {O}（n ^ {2/3}）$。主要技术贡献是有效的蒙特卡洛算法，该算法通过添加$ \ tilde {O}（n）$快捷键，将图形的直径减小为$ \ tilde {O}（n ^ {2/3}） $的可能性很高。虽然顺序算法和并行算法都具有这些组合属性，但即使顺序算法也不高效，具有顺序运行时$ \ Omega（mn ^ {\ Omega（1）}）$。本文介绍了一种令人惊讶的简单序列算法，该算法可实现指定的直径减小并以$ \ tilde {O}（m）$的时间运行。并行执行该算法可以得出主要结果，但这样做需要克服其他几个挑战。