## Abstract

The goal of this paper is to understand the complexity of symmetry breaking problems, specifically maximal independent set (MIS) and the closely related β-ruling set problem, in two computational models suited for large-scale graph processing, namely the k-machine model and the graph streaming model. We present a number of results. For MIS in the k-machine model, we improve the Õ(m/k^{2} + ∆/k)-round upper bound of Klauck et al. (SODA 2015) by presenting an Õ(m/k^{2})-round algorithm. We also present an Ω(n/k^{2}) round lower bound for MIS, the first lower bound for a symmetry breaking problem in the k-machine model. For β-ruling sets, we use hierarchical sampling to obtain more efficient algorithms in the k-machine model and also in the graph streaming model. More specifically, we obtain a k-machine algorithm that runs in Õ(βn∆^{1/β}/k^{2}) rounds and, by using a similar hierarchical sampling technique, we obtain one-pass algorithms for both insertion-only and insertion-deletion streams that use O(β · n^{1+1}/^{2β}−^{1}) space. The latter result establishes a clear separation between MIS, which is known to require Ω(n^{2}) space (Cormode et al., ICALP 2019), and β-ruling sets, even for β = 2. Finally, we present an even faster 2-ruling set algorithm in the k-machine model, one that runs in Õ(n/k^{2}−ε + k^{1}−ε) rounds for any ε, 0 ≤ ε ≤ 1. For a wide range of values of k this round complexity simplifies to Õ(n/k^{2}) rounds, which we conjecture is optimal. Our results use a variety of techniques. For our upper bounds, we prove and use simulation theorems for beeping algorithms, hierarchical sampling, and L0-sampling, whereas for our lower bounds we use information-theoretic arguments and reductions to 2-party communication complexity problems.

Original language | English (US) |
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Title of host publication | 33rd International Symposium on Distributed Computing, DISC 2019 |

Editors | Jukka Suomela |

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

ISBN (Electronic) | 9783959771269 |

DOIs | |

State | Published - Oct 2019 |

Externally published | Yes |

Event | 33rd International Symposium on Distributed Computing, DISC 2019 - Budapest, Hungary Duration: Oct 14 2019 → Oct 18 2019 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 146 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 33rd International Symposium on Distributed Computing, DISC 2019 |
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Country/Territory | Hungary |

City | Budapest |

Period | 10/14/19 → 10/18/19 |

## Keywords

- Communication complexity
- Information theory
- K-machine model
- Maximal independent set
- Ruling set
- Streaming algorithms

## ASJC Scopus subject areas

- Software