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A Min-Entropy Approach to Multi-Party Communication Lower Bounds

Authors Mi-Ying (Miryam) Huang , Xinyu Mao , Shuo Wang , Guangxu Yang , Jiapeng Zhang

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  • Theory of computation → Communication complexity
Keywords
  • communication complexity
  • lifting theorems
  • set intersection
  • chained index

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Abstract

Information complexity is one of the most powerful techniques to prove information-theoretical lower bounds, in which Shannon entropy plays a central role. Though Shannon entropy has some convenient properties, such as the chain rule, it still has inherent limitations. One of the most notable barriers is the square-root loss, which appears in the square-root gap between entropy gaps and statistical distances, e.g., Pinsker’s inequality. To bypass this barrier, we introduce a new method based on min-entropy analysis. Building on this new method, we prove the following results.  
- An Ω(N^{∑_i α_i - max_i {α_i}}/k) randomized communication lower bound of the k-party set-intersection problem where the i-th party holds a random set of size ≈ N^{1-α_i}. 
- A tight Ω(n/k) randomized lower bound of the k-party Tree Pointer Jumping problems, improving an Ω(n/k²) lower bound by Chakrabarti, Cormode, and McGregor (STOC 08). 
- An Ω(n/k+√n) lower bound of the Chained Index problem, improving an Ω(n/k²) lower bound by Cormode, Dark, and Konrad (ICALP 19).  Since these problems served as hard problems for numerous applications in streaming lower bounds and cryptography, our new lower bounds directly improve these streaming lower bounds and cryptography lower bounds. 
On the technical side, min-entropy does not have nice properties such as the chain rule. To address this issue, we enhance the structure-vs-pseudorandomness decomposition used by Göös, Pitassi, and Watson (FOCS 17) and Yang and Zhang (STOC 24); both papers used this decomposition to prove communication lower bounds. In this paper, we give a new breath to this method in the multi-party setting, presenting a new toolkit for proving multi-party communication lower bounds.

Cite As Get BibTex

Mi-Ying (Miryam) Huang, Xinyu Mao, Shuo Wang, Guangxu Yang, and Jiapeng Zhang. A Min-Entropy Approach to Multi-Party Communication Lower Bounds. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 33:1-33:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CCC.2025.33

Author Details

Mi-Ying (Miryam) Huang
  • Thomas Lord Department of Computer Science, University of Southern California, Los Angeles, CA, USA
Xinyu Mao
  • Thomas Lord Department of Computer Science, University of Southern California, Los Angeles, CA, USA
Shuo Wang
  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA
Guangxu Yang
  • Thomas Lord Department of Computer Science, University of Southern California, Los Angeles, CA, USA
Jiapeng Zhang
  • Thomas Lord Department of Computer Science, University of Southern California, Los Angeles, CA, USA

Funding

  • Huang, Mi-Ying (Miryam): Supported by NSF CAREER award 2141536.
  • Mao, Xinyu: Supported by NSF CAREER award 2141536.
  • Wang, Shuo: Supported by NSF CCF award 2227876.
  • Yang, Guangxu: Supported by NSF CAREER award 2141536.
  • Zhang, Jiapeng: Supported by NSF CAREER award 2141536.

Acknowledgements

We thank anonymous reviewers for their helpful comments.

References

  1. Ryan Alweiss, Shachar Lovett, Kewen Wu, and Jiapeng Zhang. Improved bounds for the sunflower lemma. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 624-630, 2020. URL: https://doi.org/10.1145/3357713.3384234.
  2. Alexandr Andoni, Andrew McGregor, Krzysztof Onak, and Rina Panigrahy. Better bounds for frequency moments in random-order streams. arXiv preprint, 2008. URL: https://arxiv.org/abs/0808.2222.
  3. Sepehr Assadi, Yu Chen, and Sanjeev Khanna. Polynomial pass lower bounds for graph streaming algorithms. In Proceedings of the 51st Annual ACM SIGACT Symposium on theory of computing, pages 265-276, 2019. URL: https://doi.org/10.1145/3313276.3316361.
  4. Laszlo Babai, Peter Frankl, and Janos Simon. Complexity classes in communication complexity theory. In 27th Annual Symposium on Foundations of Computer Science (sfcs 1986), pages 337-347, 1986. URL: https://doi.org/10.1109/SFCS.1986.15.
  5. Ziv Bar-Yossef, Thathachar S Jayram, Ravi Kumar, and D Sivakumar. An information statistics approach to data stream and communication complexity. Journal of Computer and System Sciences, 68(4):702-732, 2004. URL: https://doi.org/10.1016/J.JCSS.2003.11.006.
  6. Boaz Barak, Mark Braverman, Xi Chen, and Anup Rao. How to compress interactive communication. In Proceedings of the forty-second ACM symposium on Theory of computing, pages 67-76, 2010. URL: https://doi.org/10.1145/1806689.1806701.
  7. Balthazar Bauer, Pooya Farshim, and Sogol Mazaheri. Combiners for backdoored random oracles. In Advances in Cryptology-CRYPTO 2018: 38th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 19-23, 2018, Proceedings, Part II 38, pages 272-302. Springer, 2018. URL: https://doi.org/10.1007/978-3-319-96881-0_10.
  8. Paul Beame and Michael Whitmeyer. Multiparty communication complexity of collision finding, 2024. URL: https://doi.org/10.48550/arXiv.2411.07400.
  9. Anup Bhattacharya, Sourav Chakraborty, Arijit Ghosh, Gopinath Mishra, and Manaswi Paraashar. Disjointness through the lens of vapnik–chervonenkis dimension: Sparsity and beyond. Comput. Complex., 31(2), December 2022. URL: https://doi.org/10.1007/s00037-022-00225-6.
  10. Sujoy Bhore, Fabian Klute, and Jelle J Oostveen. On streaming algorithms for geometric independent set and clique. In International Workshop on Approximation and Online Algorithms, pages 211-224. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-18367-6_11.
  11. Mark Braverman, Ankit Garg, Denis Pankratov, and Omri Weinstein. From information to exact communication. In Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, STOC '13, pages 151-160, New York, NY, USA, 2013. Association for Computing Machinery. URL: https://doi.org/10.1145/2488608.2488628.
  12. Mark Braverman, Sumegha Garg, Qian Li, Shuo Wang, David P Woodruff, and Jiapeng Zhang. A new information complexity measure for multi-pass streaming with applications. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, pages 1781-1792, 2024. URL: https://doi.org/10.1145/3618260.3649672.
  13. Mark Braverman, Sumegha Garg, and David P Woodruff. The coin problem with applications to data streams. In 2020 ieee 61st annual symposium on foundations of computer science (focs), pages 318-329. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00038.
  14. Joshua Brody, Amit Chakrabarti, Ranganath Kondapally, David P. Woodruff, and Grigory Yaroslavtsev. Beyond set disjointness: The communication complexity of finding the intersection. In Proceedings of the 2014 ACM Symposium on Principles of Distributed Computing, PODC '14, pages 106-113, New York, NY, USA, 2014. Association for Computing Machinery. URL: https://doi.org/10.1145/2611462.2611501.
  15. Amit Chakrabarti. Lower bounds for multi-player pointer jumping. In Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07), pages 33-45. IEEE, 2007. URL: https://doi.org/10.1109/CCC.2007.14.
  16. Amit Chakrabarti, Graham Cormode, and Andrew McGregor. Robust lower bounds for communication and stream computation. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 641-650, 2008. URL: https://doi.org/10.1145/1374376.1374470.
  17. Amit Chakrabarti, Yaoyun Shi, Anthony Wirth, and Andrew Yao. Informational complexity and the direct sum problem for simultaneous message complexity. In Proceedings 42nd IEEE Symposium on Foundations of Computer Science, pages 270-278. IEEE, 2001. Google Scholar
  18. Amit Chakrabarti and Anthony Wirth. Incidence geometries and the pass complexity of semi-streaming set cover. In Proceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete algorithms, pages 1365-1373. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.CH94.
  19. Arkadev Chattopadhyay, Yuval Filmus, Sajin Koroth, Or Meir, and Toniann Pitassi. Query-To-Communication Lifting for BPP Using Inner Product. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), volume 132 of Leibniz International Proceedings in Informatics (LIPIcs), pages 35:1-35:15, Dagstuhl, Germany, 2019. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.35.
  20. Sandro Coretti, Yevgeniy Dodis, Siyao Guo, and John Steinberger. Random oracles and non-uniformity. In Annual International Conference on the Theory and Applications of Cryptographic Techniques, pages 227-258. Springer, 2018. URL: https://doi.org/10.1007/978-3-319-78381-9_9.
  21. Graham Cormode, Jacques Dark, and Christian Konrad. Independent sets in vertex-arrival streams. arXiv preprint, 2018. URL: https://arxiv.org/abs/1807.08331.
  22. Jacques Dark, Adithya Diddapur, and Christian Konrad. Interval selection in data streams: Weighted intervals and the insertion-deletion setting. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023), pages 24:1-24:17. Schloss-Dagstuhl-Leibniz Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2023.24.
  23. Nachum Dershowitz, Rotem Oshman, and Tal Roth. The communication complexity of multiparty set disjointness under product distributions. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 1194-1207, 2021. URL: https://doi.org/10.1145/3406325.3451106.
  24. Moran Feldman, Ashkan Norouzi-Fard, Ola Svensson, and Rico Zenklusen. The one-way communication complexity of submodular maximization with applications to streaming and robustness. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 1363-1374, 2020. URL: https://doi.org/10.1145/3357713.3384286.
  25. Moran Feldman, Ashkan Norouzi-Fard, Ola Svensson, and Rico Zenklusen. Submodular maximization subject to matroid intersection on the fly. In 30th Annual European Symposium on Algorithms (ESA 2022), pages 52:1-52:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ESA.2022.52.
  26. Dmitry Gavinsky. The communication complexity of the inevitable intersection problem. Chicago Journal of Theoretical Computer Science, 2020(3), August 2020. URL: http://cjtcs.cs.uchicago.edu/articles/2020/3/contents.html.
  27. Satrajit Ghosh and Mark Simkin. The communication complexity of threshold private set intersection. In Alexandra Boldyreva and Daniele Micciancio, editors, Advances in Cryptology - CRYPTO 2019, pages 3-29, Cham, 2019. Springer International Publishing. URL: https://doi.org/10.1007/978-3-030-26951-7_1.
  28. Mika Göös, Tom Gur, Siddhartha Jain, and Jiawei Li. Quantum communication advantage in tfnp, 2024. URL: https://doi.org/10.48550/arXiv.2411.03296.
  29. Mika Göös, Shachar Lovett, Raghu Meka, Thomas Watson, and David Zuckerman. Rectangles are nonnegative juntas. SIAM Journal on Computing, 45(5):1835-1869, 2016. URL: https://doi.org/10.1137/15M103145X.
  30. Mika Göös, Toniann Pitassi, and Thomas Watson. Query-to-communication lifting for bpp. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 132-143, 2017. URL: https://doi.org/10.1109/FOCS.2017.21.
  31. Venkatesan Guruswami and Krzysztof Onak. Superlinear lower bounds for multipass graph processing. Algorithmica, 76:654-683, 2016. URL: https://doi.org/10.1007/S00453-016-0138-7.
  32. Dawei Huang, Seth Pettie, Yixiang Zhang, and Zhijun Zhang. The communication complexity of set intersection and multiple equality testing. SIAM Journal on Computing, 50(2):674-717, 2021. URL: https://doi.org/10.1137/20M1326040.
  33. Mi-Ying(Miryam) Huang, Xinyu Mao, Guangxu Yang, and Jiapeng Zhang. Breaking square-root loss barriers via min-entropy. In In Electronic Colloquium on Computational Complexity (ECCC) (TR24-067), 2024. URL: https://eccc.weizmann.ac.il/report/2024/067/.
  34. Bala Kalyanasundaram and Georg Schintger. The probabilistic communication complexity of set intersection. SIAM Journal on Discrete Mathematics, 5(4):545-557, 1992. URL: https://doi.org/10.1137/0405044.
  35. Shachar Lovett, Raghu Meka, Ian Mertz, Toniann Pitassi, and Jiapeng Zhang. Lifting with sunflowers. Leibniz international proceedings in informatics, 215, 2022. URL: https://doi.org/10.4230/LIPICS.ITCS.2022.104.
  36. Shachar Lovett, Noam Solomon, and Jiapeng Zhang. From dnf compression to sunflower theorems via regularity. In Proceedings of the 34th Computational Complexity Conference, pages 1-14, 2019. URL: https://doi.org/10.4230/LIPICS.CCC.2019.5.
  37. Shachar Lovett and Jiapeng Zhang. Streaming lower bounds and asymmetric set-disjointness. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 871-882. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00056.
  38. Xinyu Mao, Guangxu Yang, and Jiapeng Zhang. Gadgetless lifting beats round elimination: Improved lower bounds for pointer chasing. arXiv preprint, 2024. URL: https://doi.org/10.48550/arXiv.2411.10996.
  39. Noam Nisan and Avi Widgerson. Rounds in communication complexity revisited. In Proceedings of the twenty-third annual ACM symposium on Theory of computing, pages 419-429, 1991. URL: https://doi.org/10.1145/103418.103463.
  40. Rotem Oshman and Tal Roth. The Communication Complexity of Set Intersection Under Product Distributions. In Kousha Etessami, Uriel Feige, and Gabriele Puppis, editors, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023), volume 261 of Leibniz International Proceedings in Informatics (LIPIcs), pages 95:1-95:20, Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.95.
  41. Jeff M. Phillips, Elad Verbin, and Qin Zhang. Lower bounds for number-in-hand multiparty communication complexity, made easy. SIAM Journal on Computing, 45(1):174-196, 2016. URL: https://doi.org/10.1137/15M1007525.
  42. M. Saglam and G. Tardos. On the communication complexity of sparse set disjointness and exists-equal problems. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science (FOCS), pages 678-687, Los Alamitos, CA, USA, October 2013. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS.2013.78.
  43. Janani Sundaresan. Optimal communication complexity of chained index. ITCS, 2025. Google Scholar
  44. Shuo Wang, Guangxu Yang, and Jiapeng Zhang. Communication complexity of set-intersection problems and its applications. In In Electronic Colloquium on Computational Complexity (ECCC) (TR23-164), 2023. Google Scholar
  45. Thomas Watson. Communication Complexity with Small Advantage. In Rocco A. Servedio, editor, 33rd Computational Complexity Conference (CCC 2018), volume 102 of Leibniz International Proceedings in Informatics (LIPIcs), pages 9:1-9:17, Dagstuhl, Germany, 2018. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2018.9.
  46. Guangxu Yang and Jiapeng Zhang. Communication lower bounds for collision problems via density increment arguments. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, pages 630-639, 2024. URL: https://doi.org/10.1145/3618260.3649607.
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