Beeping Shortest Paths via Hypergraph Bipartite Decomposition

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.