## Abstract

We continue the study of the communication complexity of gap cycle counting problems. These problems have been introduced by Verbin and Yu [SODA 2011] and have found numerous applications in proving streaming lower bounds. In the noisy gap cycle counting problem (NGC), there is a small integer k ≥ 1 and an n-vertex graph consisted of vertex-disjoint union of either k-cycles or 2k-cycles, plus O(n/k) disjoint paths of length k-1 in both cases ("noise"). The edges of this graph are partitioned between Alice and Bob whose goal is to decide which case the graph belongs to with minimal communication from Alice to Bob. We study the robust communication complexity of NGC-à la Chakrabarti, Cormode, and McGregor [STOC 2008]-namely, when edges are partitioned randomly between the players. This is in contrast to all prior work on gap cycle counting problems in adversarial partitions. While NGC can be solved trivially with zero communication when k < logn, we prove that when k is a constant factor larger than logn, the robust (one-way) communication complexity of NGC is ω(n) bits. As a corollary of this result, we can prove several new graph streaming lower bounds for random order streams. In particular, we show that any streaming algorithm that for every > 0 estimates the number of connected components of a graph presented in a random order stream to within an · n additive factor requires 2ω(1/") space, settling a conjecture of Peng and Sohler [SODA 2018]. We further discuss new implications of our lower bounds to other problems such as estimating size of maximum matchings and independent sets on planar graphs, random walks, as well as to stochastic streams.

Original language | English (US) |
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Title of host publication | STOC 2023 - Proceedings of the 55th Annual ACM Symposium on Theory of Computing |

Editors | Barna Saha, Rocco A. Servedio |

Publisher | Association for Computing Machinery |

Pages | 183-195 |

Number of pages | 13 |

ISBN (Electronic) | 9781450399135 |

DOIs | |

State | Published - Jun 2 2023 |

Event | 55th Annual ACM Symposium on Theory of Computing, STOC 2023 - Orlando, United States Duration: Jun 20 2023 → Jun 23 2023 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Conference

Conference | 55th Annual ACM Symposium on Theory of Computing, STOC 2023 |
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Country/Territory | United States |

City | Orlando |

Period | 6/20/23 → 6/23/23 |

## All Science Journal Classification (ASJC) codes

- Software

## Keywords

- communication complexity
- connected components
- graph streaming
- random order streams