## 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 hyperedge set of a hypergraph H = (VH, EH) into k = Θ(log^{2} n) disjoint subsets F1 ∪ · · · ∪ F_{k} = EH such that the (sub-)hypergraph (VH, Fi) is bipartite in the sense that there exists a vertex subset U ⊆ V such that |U ∩ e| = 1 for every e ∈ Fi. This procedure turns out to be instrumental in speeding up shortest path constructions under the beeping model.

Original language | English |
---|---|

Title of host publication | 14th Innovations in Theoretical Computer Science Conference, ITCS 2023 |

Editors | Yael Tauman Kalai |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959772631 |

DOIs | |

State | Published - 1 Jan 2023 |

Event | 14th Innovations in Theoretical Computer Science Conference, ITCS 2023 - Cambridge, United States Duration: 10 Jan 2023 → 13 Jan 2023 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
---|---|

Volume | 251 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 14th Innovations in Theoretical Computer Science Conference, ITCS 2023 |
---|---|

Country/Territory | United States |

City | Cambridge |

Period | 10/01/23 → 13/01/23 |

### Bibliographical note

Publisher Copyright:© Fabien Dufoulon, Yuval Emek, and Ran Gelles; licensed under Creative Commons License CC-BY 4.0.

### Funding

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

Funders | Funder number |
---|---|

National Science Foundation | CCF-1540512, IIS-1633720, CCF-1717075 |

Bloom's Syndrome Foundation | 2016419 |

Iowa Science Foundation | 2020277, 1078/17 |

Israel National Cyber Directorate | |

Technion Hiroshi Fujiwara Cyber Security Research Center |

## Keywords

- Beeping Networks
- Hypergraph Bipartite Decomposition
- Shortest Paths