## Abstract

We study the multiparty communication complexity of high dimensional permutations in the Number On the Forehead (NOF) model. This model is due to Chandra, Furst and Lipton (CFL) who also gave a nontrivial protocol for the Exactly-n problem where three players receive integer inputs and need to decide if their inputs sum to a given integer n. There is a considerable body of literature dealing with the same problem, where (N, +) is replaced by some other abelian group. Our work can be viewed as a far-reaching extension of this line of research. We show that the known lower bounds for that group-theoretic problem apply to all high dimensional permutations. We introduce new proof techniques that reveal new and unexpected connections between NOF communication complexity of permutations and a variety of well-known problems in combinatorics. We also give a direct algorithmic protocol for Exactly-n. In contrast, all previous constructions relied on large sets of integers without a 3-term arithmetic progression.

Original language | American English |
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Title of host publication | 10th Innovations in Theoretical Computer Science, ITCS 2019 |

Editors | Avrim Blum |

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

ISBN (Electronic) | 9783959770958 |

DOIs | |

State | Published - 1 Jan 2019 |

Event | 10th Innovations in Theoretical Computer Science, ITCS 2019 - San Diego, United States Duration: 10 Jan 2019 → 12 Jan 2019 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 124 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 10th Innovations in Theoretical Computer Science, ITCS 2019 |
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Country/Territory | United States |

City | San Diego |

Period | 10/01/19 → 12/01/19 |

### Bibliographical note

Funding Information:1 Supported in part by ERC grant 339096, High-dimensional combinatorics.

Publisher Copyright:

© Nati Linial, Toniann Pitassi, and Adi Shraibman.

## Keywords

- Additive combinatorics
- High dimensional permutations
- Number On the Forehead model