## Abstract

The broadcast congested clique model is a message-passing model of distributed computation where n nodes communicate with each other in synchronous rounds. The joint input to the n nodes is an undirected graph G on the same set of nodes, with each node receiving the list of its immediate neighbors in G. In each round each node sends the same message to all other nodes, depending on its own input, on the messages it has received so far, and on a public sequence of random bits. One parameter of this model is an upper bound b on the size of the messages, also known as bandwidth. In this paper we introduce another parameter to the model. We study the situation where the nodes, initially, instead of knowing their immediate neighbors, know their neighborhood up to a fixed radius r. In this new framework we study one of the hardest problems in distributed graph algorithms, this is, the problem of detecting, in G, an induced cycle of length at most k (CYCLE≤k) and the problem of detecting in G an induced cycle of length at least k + 1 (CYCLE>k). For r= 1, we exhibit a deterministic, one-round algorithm for solving CYCLE≤k with b= O(n^{2/k}log n) if k is even, and b= O(n^{2}^{/}^{(}^{k}^{-}^{1}^{)}log n) if k is odd. We also prove, assuming the Erdős Girth Conjecture, that this result is tight for k≥ 4, up to logarithmic factors. More precisely, if each node, instead of being able to see only its immediate neighbors, is allowed to see up to distance r= ⌊ k/ 4 ⌋, and if we also allowed randomization and multiple rounds, then the number of rounds R needed to solve CYCLE≤k must be such that R· b= Ω(n^{2}^{/}^{k}) if k is even, and R· b= Ω(n^{2}^{/}^{(}^{k}^{-}^{1}^{)}) if k is odd. On the other hand, we show that, if each node is allowed to see up to distance r= ⌊ k/ 2 ⌋ + 1, then a polylogarithmic bandwidth is sufficient for solving CYCLE>k with only two rounds. Nevertheless, if nodes were allowed to see up to distance r= ⌊ k/ 3 ⌋, then any one-round algorithm that solves CYCLE>k needs the bandwidth b to be at least Ω(n/ log n).

Original language | English |
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Title of host publication | LATIN 2018 |

Subtitle of host publication | Theoretical Informatics - 13th Latin American Symposium, Proceedings |

Editors | Miguel A. Mosteiro, Michael A. Bender, Martin Farach-Colton |

Publisher | Springer Verlag |

Pages | 134-145 |

Number of pages | 12 |

ISBN (Print) | 9783319774039 |

DOIs | |

State | Published - 2018 |

Externally published | Yes |

Event | 13th International Symposium on Latin American Theoretical Informatics, LATIN 2018 - Buenos Aires, Argentina Duration: 16 Apr 2018 → 19 Apr 2018 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 10807 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 13th International Symposium on Latin American Theoretical Informatics, LATIN 2018 |
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Country/Territory | Argentina |

City | Buenos Aires |

Period | 16/04/18 → 19/04/18 |

## Keywords

- Broadcast congested clique
- Graph degeneracy
- Induced cycles