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Beeping Shortest Paths via Hypergraph Bipartite Decomposition

Authors Fabien Dufoulon , Yuval Emek , Ran Gelles

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Author Details

Fabien Dufoulon
  • Department of Computer Science, University of Houston, TX, USA
Yuval Emek
  • Technion, Haifa, Israel
Ran Gelles
  • Bar-Ilan University, Ramat-Gan, Israel

Funding

  • Dufoulon, Fabien: Supported in part by NSF grants CCF-1717075, CCF-1540512, IIS-1633720, and BSF grant 2016419.
  • Emek, Yuval: Supported in part by the Technion Hiroshi Fujiwara Cyber Security Research Center and the Israel National Cyber Directorate.
  • Gelles, Ran: Supported in part by ISF grant No. 1078/17 and BSF grant No. 2020277.

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 AsGet BibTex

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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