  Constructing a shortest path between two network nodes is a fundamental task
in distributed computing. This work develops schemes for the construction of
shortest paths in randomized beeping networks between a predetermined source
node and an arbitrary set of destination nodes. Our first scheme constructs a
(single) shortest path to an arbitrary destination in $O (D \log\log n + \log^3
n)$ rounds with high probability. Our second scheme constructs multiple
shortest paths, one per each destination, in $O (D \log^2 n + \log^3 n)$ rounds
with high probability.
  Our schemes are based on a reduction of the above shortest path construction
tasks to a decomposition of hypergraphs into bipartite hypergraphs: We develop
a beeping procedure that partitions the (polynomially-large) hyperedge set of a
hypergraph $H = (V_H, E_H)$ into $k = \Theta (\log^2 n)$ disjoint subsets $F_1
\cup \cdots \cup F_k = E_H$ such that the (sub-)hypergraph $(V_H, F_i)$ is
bipartite in the sense that there exists a vertex subset $U \subseteq V$ such
that $|U \cap e| = 1$ for every $e \in F_i$. This procedure turns out to be
instrumental in speeding up shortest path constructions under the beeping
model.
