Parallel Reachability in Almost Linear Work and Square Root Depth

In this paper we provide a parallel algorithm that given any n-node m-edge directed graph and source vertex s computes all vertices reachable from s with Õ(m) work and n^{1/2 + o(1)} depth with high probability in n. This algorithm also computes a set of Õ(n) edges which when added to the graph preserves reachability and ensures that the diameter of the resulting graph is at most n^{1/2 + o(1)}. Our result improves upon the previous best known almost linear work reachability algorithm due to Fineman [1] which had depth Õ(n^2/3). Further, we show how to leverage this algorithm to achieve improved distributed algorithms for single source reachability in the CONGEST model. In particular, we provide a distributed algorithm that given a n-node digraph of undirected hop-diameter D solves the single source reachability problem with Õ(n^1/2 + n^1/3+o(1) D^2/3) rounds of the communication in the congest~model with high probability in n. Our algorithm is nearly optimal whenever D = O(n^1/4-ε) for any constant ε > 0 and is the first nearly optimal algorithm for general graphs whose diameter is Ω(n^δ) for any constant δ.