# 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(√(n/ logn) ) 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(√(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(√(n) ) for this problem.

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