  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 $\Omega(\sqrt{n/ \log{n}}\,)$ 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 $\Omega(\sqrt{n}\,)$ 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 $\Omega(\sqrt{n}\,)$ for this
problem.
