Document Open Access Logo

Equality Is Far Weaker Than Constant-Cost Communication

Authors Mika Göös, Nathaniel Harms , Artur Riazanov

PDF

File

Thumbnail PDF
PDF
LIPIcs.APPROX-RANDOM.2025.58.pdf
  • Filesize: 0.87 MB
  • 14 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
Keywords
  • Equality oracle
  • constant-cost communication
  • gamma-2 norm
  • spectral norm

Metrics

Abstract

We exhibit an n-bit communication problem with a constant-cost randomized protocol but which requires n^Ω(1) deterministic (or even non-deterministic) queries to an Equality oracle. Therefore, even constant-cost randomized protocols cannot be efficiently "derandomized" using Equality oracles. This improves on several recent results and answers a question from the survey of Hatami and Hatami (SIGACT News 2024). It also gives a significantly simpler and quantitatively superior proof of the main result of Fang, Göös, Harms, and Hatami (STOC 2025), that constant-cost communication does not reduce to the k-Hamming Distance hierarchy.

Cite As Get BibTex

Mika Göös, Nathaniel Harms, and Artur Riazanov. Equality Is Far Weaker Than Constant-Cost Communication. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 58:1-58:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.58

Author Details

Mika Göös
  • EPFL, Lausanne, Switzerland
Nathaniel Harms
  • EPFL, Lausanne, Switzerland
  • University of British Columbia, Vancouver, Canada
Artur Riazanov
  • EPFL, Lausanne, Switzerland

Funding

The project is supported by Swiss State Secretariat for Education, Research, and Innovation (SERI) under contract number MB22.00026.

Acknowledgements

Thanks to Kaave Hosseini for suggesting Corollary 18 and to Tsun-Ming Cheung for comments on the presentation of this article. Thanks to an anonymous reviewer for pointing out a simpler proof of Claim 13.

References

  1. Mark Braverman and Omri Weinstein. A discrepancy lower bound for information complexity. Algorithmica, 76:846-864, 2016. URL: https://doi.org/10.1007/s00453-015-0093-8.
  2. Arkadev Chattopadhyay, Shachar Lovett, and Marc Vinyals. Equality alone does not simulate randomness. In Proceedings of the Computational Complexity Conference (CCC), pages 14-1. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.CCC.2019.14.
  3. Tsun-Ming Cheung, Hamed Hatami, Kaave Hosseini, Aleksandar Nikolov, Toniann Pitassi, and Morgan Shirley. A lower bound on the trace norm of boolean matrices and its applications. In Proceedings of the Innovations in Theoretical Computer Science Conference (ITCS), pages 37-1. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. URL: https://doi.org/10.4230/LIPIcs.ITCS.2025.37.
  4. Tsun-Ming Cheung, Hamed Hatami, Kaave Hosseini, and Morgan Shirley. Separation of the factorization norm and randomized communication complexity. In Proceedings of the Computational Complexity Conference (CCC), pages 1-1. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.CCC.2023.1.
  5. Tsun Ming Cheung, Hamed Hatami, Rosie Zhao, and Itai Zilberstein. Boolean functions with small approximate spectral norm. Discrete Analysis, 2024. URL: https://doi.org/10.19086/da.122971.
  6. Louis Esperet, Nathaniel Harms, and Andrey Kupavskii. Sketching distances in monotone graph classes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM),, pages 18-1. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.18.
  7. Yuting Fang, Mika Göös, Nathaniel Harms, and Pooya Hatami. Constant-cost communication does not reduce to k-Hamming Distance. In Proceedings of the ACM SIGACT Symposium on Theory of Computing (STOC), 2025. URL: https://doi.org/10.48550/arXiv.2407.20204.
  8. Yuting Fang, Lianna Hambardzumyan, Nathaniel Harms, and Pooya Hatami. No complete problem for constant-cost randomized communication. In Proceedings of the ACM SIGACT Symposium on Theory of Computing (STOC), 2024. URL: https://doi.org/10.48550/arXiv.2404.00812.
  9. Lianna Hambardzumyan, Hamed Hatami, and Pooya Hatami. Dimension-free bounds and structural results in communication complexity. Israel Journal of Mathematics, 253(2):555-616, 2023. URL: https://doi.org/10.1007/s11856-022-2365-8.
  10. Nathaniel Harms. Universal communication, universal graphs, and graph labeling. In Proceedings of the Innovations in Theoretical Computer Science Conference (ITCS), volume 151, page 33. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.33.
  11. Nathaniel Harms and Artur Riazanov. Better boosting of communication oracles, or not. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), 2024. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2024.25.
  12. Nathaniel Harms, Sebastian Wild, and Viktor Zamaraev. Randomized communication and implicit graph representations. In Proceedings of the ACM SIGACT Symposium on Theory of Computing (STOC), 2022. URL: https://doi.org/10.1145/3519935.3519978.
  13. Nathaniel Harms and Viktor Zamaraev. Randomized communication and implicit representations for matrices and graphs of small sign-rank. In Proceedings of the ACMSIAM Symposium on Discrete Algorithms (SODA), pages 1810-1833. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.72.
  14. Hamed Hatami and Pooya Hatami. Guest column: Structure in communication complexity and constant-cost complexity classes. ACM SIGACT News, 55(1):67-93, 2024. URL: https://doi.org/10.1145/3654780.3654788.
  15. Eyal Kushilevitz and Noam Nisan. Communication complexity. Cambridge University Press, 1996. Google Scholar
  16. Noam Nisan. The communication complexity of threshold gates. Combinatorics, Paul Erdos is Eighty, 1(301-315):6, 1993. Google Scholar
  17. Toniann Pitassi, Morgan Shirley, and Adi Shraibman. The strength of equality oracles in communication. In Proceedings of the Innovations in Theoretical Computer Science Conference (ITCS), pages 89-1. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ITCS.2023.89.
  18. Anup Rao and Amir Yehudayoff. Communication Complexity: and Applications. Cambridge University Press, 2020. Google Scholar
  19. Tom Sanders. Boolean functions with small spectral norm, revisited. Mathematical Proceedings of the Cambridge Philosophical Society, 167(2):335-344, 2019. URL: https://doi.org/10.1017/S030500411800035X.
  20. Alexander Sherstov and Andrey Storozhenko. The communication complexity of approximating matrix rank. In Proceedings of the IEEE Symposium on Foundations of Computer Science (FOCS), pages 433-462. IEEE, 2024. URL: https://doi.org/10.1109/FOCS61266.2024.00035.
  21. Srikanth Srinivasan and Amir Yehudayoff. The discrepancy of greater-than. arXiv preprint, 2023. URL: https://doi.org/10.48550/arXiv.2309.08703.
  22. István Tomon. Factorization norms and Zarankiewicz problems. arXiv preprint, 2025. URL: https://doi.org/10.48550/arXiv.2502.18429.
  23. Emanuele Viola. The communication complexity of addition. Combinatorica, 35(6):703-747, 2015. URL: https://doi.org/10.1007/s00493-014-3078-3.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact