An Improved Protocol for ExactlyN with More Than 3 Players

Authors Lianna Hambardzumyan, Toniann Pitassi, Suhail Sherif, Morgan Shirley, Adi Shraibman

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Author Details

Lianna Hambardzumyan
  • The Hebrew University of Jerusalem, Israel
Toniann Pitassi
  • Columbia University, New York, NY, USA
Suhail Sherif
  • LASIGE, Faculdade de Ciências, Universidade de Lisboa, Portugal
Morgan Shirley
  • University of Toronto, Canada
Adi Shraibman
  • The Academic College of Tel Aviv-Yaffo, Israel


Most of the work was done while Suhail Sherif was at Vector Institute, Toronto, Canada. We thank Zach Hunter for his helpful comments on an earlier version of the paper.

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Lianna Hambardzumyan, Toniann Pitassi, Suhail Sherif, Morgan Shirley, and Adi Shraibman. An Improved Protocol for ExactlyN with More Than 3 Players. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 58:1-58:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


The ExactlyN problem in the number-on-forehead (NOF) communication setting asks k players, each of whom can see every input but their own, if the k input numbers add up to N. Introduced by Chandra, Furst and Lipton in 1983, ExactlyN is important for its role in understanding the strength of randomness in communication complexity with many players. It is also tightly connected to the field of combinatorics: its k-party NOF communication complexity is related to the size of the largest corner-free subset in [N]^{k-1}. In 2021, Linial and Shraibman gave more efficient protocols for ExactlyN for 3 players. As an immediate consequence, this also gave a new construction of larger corner-free subsets in [N]². Later that year Green gave a further refinement to their argument. These results represent the first improvements to the highest-order term for k = 3 since the famous work of Behrend in 1946. In this paper we give a corresponding improvement to the highest-order term for k > 3, the first since Rankin in 1961. That is, we give a more efficient protocol for ExactlyN as well as larger corner-free sets in higher dimensions. Nearly all previous results in this line of research approached the problem from the combinatorics perspective, implicitly resulting in non-constructive protocols for ExactlyN. Approaching the problem from the communication complexity point of view and constructing explicit protocols for ExactlyN was key to the improvements in the k = 3 setting. As a further contribution we provide explicit protocols for ExactlyN for any number of players which serves as a base for our improvement.

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Mathematics of computing → Combinatorics
  • Corner-free sets
  • number-on-forehead communication


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