Document Open Access Logo

An Improved Protocol for ExactlyN with More Than 3 Players

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

PDF
Thumbnail PDF

File

LIPIcs.ITCS.2024.58.pdf
  • Filesize: 0.75 MB
  • 23 pages

Document Identifiers

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

Funding

  • Hambardzumyan, Lianna: Research partially supported by ISF grant 921/22.
  • Pitassi, Toniann: Supported by NSF AF:Medium 2212136.
  • Sherif, Suhail: Funded by the European Union (ERC, HOFGA, 101041696). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. Also supported by FCT through the LASIGE Research Unit, ref. UIDB/00408/2020 and ref. UIDP/00408/2020.
  • Shirley, Morgan: Supported by an NSERC grant.

Acknowledgements

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.

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.ITCS.2024.58

Abstract

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
Keywords
  • Corner-free sets
  • number-on-forehead communication

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Miklós Ajtai and Endre Szemerédi. Sets of lattice points that form no squares. Studia Scientiarum Mathematicarum Hungarica, 9:9-11, 1974. Google Scholar
  2. Ryan Alweiss, Shachar Lovett, Kewen Wu, and Jiapeng Zhang. Improved bounds for the sunflower lemma. Annals of Mathematics, 194(3):795-815, November 2021. URL: https://doi.org/10.4007/annals.2021.194.3.5.
  3. Paul Beame, Matei David, Toniann Pitassi, and Philipp Woelfel. Separating deterministic from randomized multiparty communication complexity. Theory of Computing, 6(9):201-225, November 2010. URL: https://doi.org/10.4086/toc.2010.v006a009.
  4. Felix A. Behrend. On sets of integers which contain no three terms in arithmetical progression. Proceedings of the National Academy of Sciences of the United States of America, 32(12):331-332, December 1946. URL: https://doi.org/10.1073/pnas.32.12.331.
  5. Richard Beigel and Jun Tarui. On ACC. computational complexity, 4(4):350-366, December 1994. URL: https://doi.org/10.1007/BF01263423.
  6. Ashok K. Chandra, Merrick L. Furst, and Richard J. Lipton. Multi-party protocols. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, STOC '83, pages 94-99, New York, NY, USA, December 1983. Association for Computing Machinery. URL: https://doi.org/10.1145/800061.808737.
  7. Ernie Croot, Vsevolod F. Lev, and Péter Pál Pach. Progression-free sets in ℤ₄ⁿ are exponentially small. Annals of Mathematics, 185(1):331-337, 2017. URL: https://doi.org/10.4007/annals.2017.185.1.7.
  8. Zeev Dvir. On the size of Kakeya sets in finite fields. Journal of the American Mathematical Society, 22(4):1093-1097, 2009. URL: https://doi.org/10.1090/S0894-0347-08-00607-3.
  9. Michael Elkin. An improved construction of progression-free sets. Israel Journal of Mathematics, 184(1):93, July 2011. URL: https://doi.org/10.1007/s11856-011-0061-1.
  10. Jordan S. Ellenberg and Dion Gijswijt. On large subsets of 𝔽_qⁿ with no three-term arithmetic progression. Annals of Mathematics, 185(1):339-343, 2017. URL: https://doi.org/10.4007/annals.2017.185.1.8.
  11. W. Timothy Gowers. A new proof of Szemerédi’s theorem. Geometric & Functional Analysis GAFA, 11(3):465-588, August 2001. URL: https://doi.org/10.1007/s00039-001-0332-9.
  12. W. Timothy Gowers. Hypergraph regularity and the multidimensional Szemerédi theorem. Annals of Mathematics, 166(3):897-946, 2007. URL: https://doi.org/10.4007/annals.2007.166.897.
  13. Ben Green. Lower bounds for corner-free sets. New Zealand Journal of Mathematics, 51:1-2, July 2021. URL: https://doi.org/10.53733/86.
  14. Ben Green and Terence Tao. New bounds for Szemerédi’s theorem, III: A polylogarithmic bound for r₄(N). Mathematika, 63(3):944-1040, 2017. URL: https://doi.org/10.1112/S0025579317000316.
  15. Ben Green and Julia Wolf. A note on Elkin’s improvement of Behrend’s construction. In David Chudnovsky and Gregory Chudnovsky, editors, Additive Number Theory: Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson, pages 141-144. Springer, New York, NY, 2010. URL: https://doi.org/10.1007/978-0-387-68361-4_9.
  16. Zach Hunter. Corner-free sets via the torus, October 2022. URL: https://doi.org/10.48550/arXiv.2209.10012.
  17. Zander Kelley, Shachar Lovett, and Raghu Meka. Explicit separations between randomized and deterministic number-on-forehead communication, August 2023. URL: https://arxiv.org/abs/TR23-124.
  18. Zander Kelley and Raghu Meka. Strong bounds for 3-progressions. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, Santa Cruz, CA, USA, November 6-9, 2023, pages 933-973. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00059.
  19. Izabella Łaba and Michael T. Lacey. On sets of integers not containing long arithmetic progressions, August 2001. URL: https://doi.org/10.48550/arXiv.math/0108155.
  20. Nati Linial, Toniann Pitassi, and Adi Shraibman. On the communication complexity of high-dimensional permutations. In Avrim Blum, editor, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019), volume 124 of Leibniz International Proceedings in Informatics (LIPIcs), pages 54:1-54:20, Dagstuhl, Germany, 2019. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.54.
  21. Nati Linial and Adi Shraibman. An improved protocol for the Exactly-N problem. In Valentine Kabanets, editor, 36th Computational Complexity Conference (CCC 2021), volume 200 of Leibniz International Proceedings in Informatics (LIPIcs), pages 2:1-2:8, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2021.2.
  22. Shachar Lovett. Additive combinatorics and its applications in theoretical computer science. Theory Comput., 8:1-55, 2017. URL: https://doi.org/10.4086/toc.gs.2017.008.
  23. Shachar Lovett, Raghu Meka, Ian Mertz, Toniann Pitassi, and Jiapeng Zhang. Lifting with sunflowers. In Mark Braverman, editor, 13th Innovations in Theoretical Computer Science Conference, ITCS 2022, January 31 - February 3, 2022, Berkeley, CA, USA, volume 215 of LIPIcs, pages 104:1-104:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.104.
  24. Kevin O'Bryant. Sets of integers that do not contain long arithmetic progressions. The Electronic Journal of Combinatorics, 18, November 2008. URL: https://doi.org/10.37236/546.
  25. Pavel Pudlák. Boolean complexity and Ramsey theorems. In Jaroslav Nešetřil and Vojtěch Rödl, editors, Mathematics of Ramsey Theory, Algorithms and Combinatorics, pages 246-252. Springer, Berlin, Heidelberg, 1990. URL: https://doi.org/10.1007/978-3-642-72905-8_17.
  26. Robert A. Rankin. Sets of integers containing not more than a given number of terms in arithmetical progression. Proceedings of the Royal Society of Edinburgh Section A: Mathematical and Physical Sciences, 65(4):332-344, 1961. URL: https://doi.org/10.1017/S0080454100017726.
  27. Alexander A. Razborov. Lower bounds on the monotone complexity of some boolean functions. Dokl. Akad. Nauk SSSR, 281:354-357, 1985. Google Scholar
  28. Raphaël Salem and Donald C. Spencer. On sets of integers which contain no three terms in arithmetical progression. Proceedings of the National Academy of Sciences of the United States of America, 28(12):561-563, 1942. URL: https://arxiv.org/abs/87810.
  29. Ilya D. Shkredov. On a generalization of Szemeredi’s theorem. Proceedings of the London Mathematical Society, 93(3):723-760, November 2006. URL: https://doi.org/10.1017/S0024611506015991.
  30. Terence Tao and Van H. Vu. Additive combinatorics, volume 105. Cambridge University Press, 2006. Google Scholar
  31. Andrew Chi-Chih Yao. On ACC and threshold circuits. In 31st Annual Symposium on Foundations of Computer Science, pages 619-627. IEEE, October 1990. URL: https://doi.org/10.1109/FSCS.1990.89583.
  32. Yufei Zhao. Graph Theory and Additive Combinatorics: Exploring Structure and Randomness. Cambridge University Press, 2023. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact