## Abstract

We study local symmetry breaking problems in the Congest model, focusing on ruling set problems, which generalize the fundamental Maximal Independent Set (MIS) problem. The time (round) complexity of MIS (and ruling sets) have attracted much attention in the Local model. Indeed, recent results (Barenboim et al., FOCS 2012, Ghaffari SODA 2016) for the MIS problem have tried to break the long-standing O(log n)-round "barrier" achieved by Luby's algorithm, but these yield o(log n)-round complexity only when the maximum degree Δ is somewhat small relative to n. More importantly, these results apply only in the Local model. In fact, the best known time bound in the Congest model is still O(log n) (via Luby's algorithm) even for moderately small Δ (i.e., for Δ =Ω(log n) and Δ = o(n)). Furthermore, message complexity has been largely ignored in the context of local symmetry breaking. Luby's algorithm takes O(m) messages on m-edge graphs and this is the best known bound with respect to messages. Our work is motivated by the following central question: can we break the Θ(log n) time complexity barrier and the Θ(m) message complexity barrier in the Congest model for MIS or closelyrelated symmetry breaking problems? This paper presents progress towards this question for the distributed ruling set problem in the Congest model. A β-ruling set is an independent set such that every node in the graph is at most β hops from a node in the independent set. We present the following results: Time Complexity: We show that we can break the O(log n) "barrier" for 2- and 3-ruling sets. We compute 3-ruling sets in O (log n/loglogn) rounds with high probability (whp). More generally we show that 2-ruling sets can be computed in O (logΔ(log n)^{1/2+ϵ} + log n/loglog n) rounds for any ϵ > 0, which is o(log n) for a wide range of Δ values (e.g., Δ = 2^{(log n)1/2-ϵ}). These are the first 2- and 3-ruling set algorithms to improve over the O(log n)-round complexity of Luby's algorithm in the Congest model. Message Complexity: We show an Ω(n^{2}) lower bound on the message complexity of computing an MIS (i.e., 1-ruling set) which holds also for randomized algorithms and present a contrast to this by showing a randomized algorithm for 2-ruling sets that, whp, uses only O(n log^{2} n) messages and runs in O(Delta; log n) rounds. This is the first message-efficient algorithm known for ruling sets, which has message complexity nearly linear in n (which is optimal up to a polylogarithmic factor).

Original language | English (US) |
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Title of host publication | 31st International Symposium on Distributed Computing, DISC 2017 |

Editors | Andrea W. Richa |

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

ISBN (Electronic) | 9783959770538 |

DOIs | |

State | Published - Oct 1 2017 |

Externally published | Yes |

Event | 31st International Symposium on Distributed Computing, DISC 2017 - Vienna, Austria Duration: Oct 16 2017 → Oct 20 2017 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 91 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 31st International Symposium on Distributed Computing, DISC 2017 |
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Country/Territory | Austria |

City | Vienna |

Period | 10/16/17 → 10/20/17 |

## Keywords

- Congest model
- Local model
- Maximal independent set
- Message complexity
- Round complexity
- Ruling sets
- Symmetry breaking

## ASJC Scopus subject areas

- Software