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# Beeping Shortest Paths via Hypergraph Bipartite Decomposition

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LIPIcs.ITCS.2023.45.pdf
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## Acknowledgements

We would like to thank Mohsen Ghaffari for pointing out that the work of Alon et al. [Alon et al., 1991] immediately gives a lower bound of Ω(log² n) to the HBD task.

## Cite As

Fabien Dufoulon, Yuval Emek, and Ran Gelles. Beeping Shortest Paths via Hypergraph Bipartite Decomposition. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 45:1-45:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ITCS.2023.45

## 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 + log³ n) rounds with high probability. Our second scheme constructs multiple shortest paths, one per each destination, in O(D log² n + log³ 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 hyperedge set of a hypergraph H = (V_H, E_H) into k = Θ (log² n) disjoint subsets F₁ ∪ ⋯ ∪ 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 ⊆ V such that |U ∩ e| = 1 for every e ∈ F_i. This procedure turns out to be instrumental in speeding up shortest path constructions under the beeping model.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Distributed algorithms
##### Keywords
• Beeping Networks
• Shortest Paths
• Hypergraph Bipartite Decomposition

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## References

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