On the Communication Complexity of High-Dimensional Permutations

Authors Nati Linial, Toniann Pitassi, Adi Shraibman

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Author Details

Nati Linial
  • Hebrew University of Jerusalem, Jerusalem, Israel
Toniann Pitassi
  • University of Toronto, Toronto, Canada and IAS, Princeton, U.S.A.
Adi Shraibman
  • The Academic College of Tel-Aviv-Yaffo, Tel-Aviv, Israel

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Nati Linial, Toniann Pitassi, and Adi Shraibman. On the Communication Complexity of High-Dimensional Permutations. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 54:1-54:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • High dimensional permutations
  • Number On the Forehead model
  • Additive combinatorics


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