Reachability and Shortest Paths in the Broadcast CONGEST Model

In this paper we study the time complexity of the single-source reachability problem and the single-source shortest path problem for directed unweighted graphs in the Broadcast CONGEST model. We focus on the case where the diameter $D$ of the underlying network is constant. We show that for the case where $D = 1$ there is, quite surprisingly, a very simple algorithm that solves the reachability problem in $1$(!) round. In contrast, for networks with $D = 2$, we show that any distributed algorithm (possibly randomized) for this problem requires $Ω(\sqrt{n/ \log{n}}\,)$ rounds. Our results therefore completely resolve (up to a small polylogarithmic factor) the complexity of the single-source reachability problem for a wide range of diameters. Furthermore, we show that when $D = 1$, it is even possible to get a $3$-approximation for the all-pairs shortest path problem (for directed unweighted graphs) in just $2$ rounds. We also prove a stronger lower bound of $Ω(\sqrt{n}\,)$ for the single-source shortest path problem for unweighted directed graphs that holds even when the diameter of the underlying network is $2$. As far as we know this is the first lower bound that achieves $Ω(\sqrt{n}\,)$ for this problem.