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|
|Title of host publication||10th Innovations in Theoretical Computer Science, ITCS 2019|
|Publisher||Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing|
|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
|Name||Leibniz International Proceedings in Informatics, LIPIcs|
|Conference||10th Innovations in Theoretical Computer Science, ITCS 2019|
|Period||10/01/19 → 12/01/19|
Bibliographical noteFunding Information:
1 Supported in part by ERC grant 339096, High-dimensional combinatorics.
© Nati Linial, Toniann Pitassi, and Adi Shraibman.
- Additive combinatorics
- High dimensional permutations
- Number On the Forehead model