Beeping Shortest Paths via Hypergraph Bipartite Decomposition

TL;DR

This paper introduces randomized schemes for constructing shortest paths in beeping networks, utilizing hypergraph bipartite decomposition to achieve efficient routing between nodes with high probability.

Contribution

It presents novel hypergraph bipartite decomposition techniques and applies them to develop faster shortest path algorithms in the beeping communication model.

Findings

Single shortest path construction in $O(D \\log\log n + \\\log^3 n)$ rounds.

Multiple shortest paths construction in $O(D \\log^2 n + \\\log^3 n)$ rounds.

Hypergraph bipartite decomposition accelerates shortest path algorithms.

Abstract

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.