Nearly work-efficient parallel algorithm for digraph reachability

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 paper shows that the answer is yes. This paper presents a randomized parallel algorithm for digraph reachability and related problems with expected work Õ(m) and span Õ(n2/3), and hence parallelism Ω(m/n2/3) = Ω(n1/3), 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 Õ(n1−є) for any constant є>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 Õ(m) and span Õ(n2/3). The main technical contribution is an efficient Monte Carlo algorithm that, through the addition of Õ(n) shortcuts, reduces the diameter of the graph to Õ(n2/3) 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 Ω(mnΩ(1)). This paper presents a surprisingly simple sequential algorithm that achieves the stated diameter reduction and runs in Õ(m) time. Parallelizing that algorithm yields the main result, but doing so involves overcoming several other challenges.