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Direct Sums for Parity Decision Trees

Authors Tyler Besselman , Mika Göös, Siyao Guo , Gilbert Maystre , Weiqiang Yuan

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ACM Subject Classification
  • Theory of computation → Oracles and decision trees
Keywords
  • direct sum
  • parity decision trees
  • query complexity

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Abstract

Direct sum theorems state that the cost of solving k instances of a problem is at least Ω(k) times the cost of solving a single instance. We prove the first such results in the randomised parity decision tree model. We show that a direct sum theorem holds whenever (1) the lower bound for parity decision trees is proved using the discrepancy method; or (2) the lower bound is proved relative to a product distribution.

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Tyler Besselman, Mika Göös, Siyao Guo, Gilbert Maystre, and Weiqiang Yuan. Direct Sums for Parity Decision Trees. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 16:1-16:38, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CCC.2025.16

Author Details

Tyler Besselman
  • NYU Shanghai, China
Mika Göös
  • EPFL, Lausanne, Switzerland
Siyao Guo
  • NYU Shanghai, China
Gilbert Maystre
  • EPFL, Lausanne, Switzerland
Weiqiang Yuan
  • EPFL, Lausanne, Switzerland

Funding

  • Besselman, Tyler: Supported by the National Natural Science Foundation of China Grant No.62102260, NYTP Grant No.20121201, and NYU Shanghai Boost Fund.
  • Göös, Mika: Supported by the Swiss State Secretariat for Education, Research and Innovation (SERI) under contract number MB22.00026.
  • Guo, Siyao: Supported by the National Natural Science Foundation of China Grant No.62102260, NYTP Grant No.20121201, and NYU Shanghai Boost Fund.
  • Maystre, Gilbert: Supported by the Swiss State Secretariat for Education, Research and Innovation (SERI) under contract number MB22.00026.
  • Yuan, Weiqiang: Supported by the Swiss State Secretariat for Education, Research and Innovation (SERI) under contract number MB22.00026.

Acknowledgements

We thank Farzan Byramji for useful comments on an earlier version of this paper.

References

  1. Yaroslav Alekseev, Yuval Filmus, and Alexander Smal. Lifting Dichotomies. In 39th Computational Complexity Conference (CCC 2024), volume 300 of Leibniz International Proceedings in Informatics (LIPIcs), pages 9:1-9:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.CCC.2024.9.
  2. Yaroslav Alekseev and Dmitry Itsykson. Lifting to bounded-depth and regular resolutions over parities via games. Technical Report TR24-128, ECCC, 2024. URL: https://eccc.weizmann.ac.il/report/2024/128/.
  3. 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.
  4. Boaz Barak, Mark Braverman, Xi Chen, and Anup Rao. How to compress interactive communication. SIAM Journal on Computing, 42(3):1327-1363, 2013. URL: https://doi.org/10.1137/100811969.
  5. Paul Beame and Sajin Koroth. On Disperser/Lifting Properties of the Index and Inner-Product Functions. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023), volume 251 of Leibniz International Proceedings in Informatics (LIPIcs), pages 14:1-14:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ITCS.2023.14.
  6. Shalev Ben-David and Eric Blais. A tight composition theorem for the randomized query complexity of partial functions: Extended abstract. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 240-246, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00031.
  7. Shalev Ben-David and Eric Blais. A new minimax theorem for randomized algorithms. J. ACM, 70(6), 2023. URL: https://doi.org/10.1145/3626514.
  8. Shalev Ben-David, Eric Blais, Mika Göös, and Gilbert Maystre. Randomised Composition and Small-Bias Minimax . In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 624-635. IEEE Computer Society, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00065.
  9. Shalev Ben-David, Mika Göös, Robin Kothari, and Thomas Watson. When Is Amplification Necessary for Composition in Randomized Query Complexity? In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020), volume 176 of Leibniz International Proceedings in Informatics (LIPIcs), pages 28:1-28:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.28.
  10. Shalev Ben-David and Robin Kothari. Randomized query complexity of sabotaged and composed functions. Theory of Computing, 14(5):1-27, 2018. URL: https://doi.org/10.4086/toc.2018.v014a005.
  11. Sreejata Kishor Bhattacharya, Arkadev Chattopadhyay, and Pavel Dvořák. Exponential Separation Between Powers of Regular and General Resolution over Parities. In 39th Computational Complexity Conference (CCC 2024), volume 300 of Leibniz International Proceedings in Informatics (LIPIcs), pages 23:1-23:32. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.CCC.2024.23.
  12. Eric Blais and Joshua Brody. Optimal separation and strong direct sum for randomized query complexity. In Proceedings of the 34th Computational Complexity Conference, CCC '19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.CCC.2019.29.
  13. Guy Blanc, Caleb Koch, Carmen Strassle, and Li-Yang Tan. A Strong Direct Sum Theorem for Distributional Query Complexity. In 39th Computational Complexity Conference (CCC 2024), volume 300 of Leibniz International Proceedings in Informatics (LIPIcs), pages 16:1-16:30. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.CCC.2024.16.
  14. Mark Braverman, Ankit Garg, Denis Pankratov, and Omri Weinstein. Information lower bounds via self-reducibility. Theory of Computing Systems, 59(2):377-396, 2015. URL: https://doi.org/10.1007/s00224-015-9655-z.
  15. Mark Braverman and Anup Rao. Information equals amortized communication. IEEE Transactions on Information Theory, 60(10):6058-6069, 2014. URL: https://doi.org/10.1109/TIT.2014.2347282.
  16. Joshua Brody, Jae Tak Kim, Peem Lerdputtipongporn, and Hariharan Srinivasulu. A strong XOR lemma for randomized query complexity. Theory of Computing, 19(11):1-14, 2023. URL: https://doi.org/10.4086/toc.2023.v019a011.
  17. Farzan Byramji and Russell Impagliazzo. Lifting to randomized parity decision trees. Technical Report TR24-202, ECCC, 2024. URL: https://eccc.weizmann.ac.il/report/2024/202/.
  18. Arkadev Chattopadhyay and Pavel Dvorak. Super-critical trade-offs in resolution over parities via lifting. Technical Report TR24-132, ECCC, 2024. URL: https://eccc.weizmann.ac.il/report/2024/132/.
  19. Arkadev Chattopadhyay, Nikhil Mande, Swagato Sanyal, and Suhail Sherif. Lifting to Parity Decision Trees via Stifling. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023), volume 251 of Leibniz International Proceedings in Informatics (LIPIcs), pages 33:1-33:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ITCS.2023.33.
  20. 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.
  21. Andrew Drucker. Improved direct product theorems for randomized query complexity. Comput. Complex., 21(2):197-244, 2012. URL: https://doi.org/10.1007/s00037-012-0043-7.
  22. Klim Efremenko, Michal Garlík, and Dmitry Itsykson. Lower bounds for regular resolution over parities. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, volume 41 of STOC ’24, pages 640-651. ACM, 2024. URL: https://doi.org/10.1145/3618260.3649652.
  23. Tomás Feder, Eyal Kushilevitz, Moni Naor, and Noam Nisan. Amortized communication complexity. SIAM Journal on Computing, 24(4):736-750, 1995. URL: https://doi.org/10.1137/S0097539792235864.
  24. U. Feige, D. Peleg, P. Raghavan, and E. Upfal. Computing with unreliable information. In Proceedings of the Twenty-Second Annual ACM Symposium on Theory of Computing, STOC '90, pages 128-137. Association for Computing Machinery, 1990. URL: https://doi.org/10.1145/100216.100230.
  25. Yuval Filmus, Edward Hirsch, Artur Riazanov, Alexander Smal, and Marc Vinyals. Proving Unsatisfiability with Hitting Formulas. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024), volume 287 of Leibniz International Proceedings in Informatics (LIPIcs), pages 48:1-48:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.ITCS.2024.48.
  26. Anat Ganor, Gillat Kol, and Ran Raz. Exponential separation of information and communication for boolean functions. J. ACM, 63(5), 2016. URL: https://doi.org/10.1145/2907939.
  27. Uma Girish, Makrand Sinha, Avishay Tal, and Kewen Wu. Fourier growth of communication protocols for XOR functions. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 721-732, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00047.
  28. Uma Girish, Avishay Tal, and Kewen Wu. Fourier growth of parity decision trees. In Proceedings of the 36th Computational Complexity Conference, CCC '21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.CCC.2021.39.
  29. Mika Göös and Gilbert Maystre. A majority lemma for randomised query complexity. In Proceedings of the 36th Computational Complexity Conference, CCC '21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.CCC.2021.18.
  30. Nathaniel Harms and Artur Riazanov. Better Boosting of Communication Oracles, or Not. In Siddharth Barman and Sławomir Lasota, editors, 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024), volume 323 of Leibniz International Proceedings in Informatics (LIPIcs). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2024.25.
  31. Hamed Hatami, Kaave Hosseini, and Shachar Lovett. Structure of protocols for XOR functions. SIAM Journal on Computing, 47(1):208-217, 2018. URL: https://doi.org/10.1137/17M1136869.
  32. Hamed Hatami, Kaave Hosseini, Shachar Lovett, and Anthony Ostuni. Refuting Approaches to the Log-Rank Conjecture for XOR Functions. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024), volume 297 of Leibniz International Proceedings in Informatics (LIPIcs), pages 82:1-82:11. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.ICALP.2024.82.
  33. Dmitry Itsykson and Dmitry Sokolov. Resolution over linear equations modulo two. Annals of Pure and Applied Logic, 171(1):102722, 2020. URL: https://doi.org/10.1016/j.apal.2019.102722.
  34. Siddharth Iyer and Anup Rao. An XOR lemma for deterministic communication complexity, 2024. URL: https://doi.org/10.48550/arXiv.2407.01802.
  35. Siddharth Iyer and Anup Rao. XOR lemmas for communication via marginal information. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, pages 652-658. Association for Computing Machinery, 2024. URL: https://doi.org/10.1145/3618260.3649726.
  36. Rahul Jain, Hartmut Klauck, and Miklos Santha. Optimal direct sum results for deterministic and randomized decision tree complexity. Information Processing Letters, 110(20):893-897, 2010. URL: https://doi.org/10.1016/j.ipl.2010.07.020.
  37. Rahul Jain, Jaikumar Radhakrishnan, and Pranab Sen. A direct sum theorem in communication complexity via message compression. In Jos C. M. Baeten, Jan Karel Lenstra, Joachim Parrow, and Gerhard J. Woeginger, editors, Automata, Languages and Programming, pages 300-315. Springer Berlin Heidelberg, 2003. URL: https://doi.org/10.1007/3-540-45061-0_26.
  38. Hartmut Klauck, Robert Špalek, and Ronald de Wolf. Quantum and classical strong direct product theorems and optimal time‐space tradeoffs. SIAM Journal on Computing, 36(5):1472-1493, 2007. URL: https://doi.org/10.1137/05063235X.
  39. A. Knop, S. Lovett, S. McGuire, and W. Yuan. Guest column: Models of computation between decision trees and communication. SIGACT News, 52(2):46-70, 2021. URL: https://doi.org/10.1145/3471469.3471479.
  40. Eyal Kushilevitz and Yishay Mansour. Learning decision trees using the fourier spectrum. SIAM Journal on Computing, 22(6):1331-1348, 1993. URL: https://doi.org/10.1137/0222080.
  41. Troy Lee, Adi Shraibman, and Robert Špalek. A direct product theorem for discrepancy. In 2008 23rd Annual IEEE Conference on Computational Complexity, pages 71-80, 2008. URL: https://doi.org/10.1109/CCC.2008.25.
  42. Nati Linial and Adi Shraibman. Learning complexity vs. communication complexity. In 2008 23rd Annual IEEE Conference on Computational Complexity, pages 53-63, 2008. URL: https://doi.org/10.1109/CCC.2008.28.
  43. Nikhil Mande and Swagato Sanyal. On parity decision trees for fourier-sparse boolean functions. ACM Trans. Comput. Theory, 16(2), 2024. URL: https://doi.org/10.1145/3647629.
  44. Noam Nisan. The communication complexity of threshold gates. Proc. of Combinatorics, Paul Erdős is Eighty, 1993. Google Scholar
  45. Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014. URL: https://doi.org/10.1017/CBO9781139814782.
  46. Ryan ODonnell, John Wright, Yu Zhao, Xiaorui Sun, and Li-Yang Tan. A composition theorem for parity kill number. In 2014 IEEE Conference on Computational Complexity (CCC), pages 144-154. IEEE Computer Society, 2014. URL: https://doi.org/10.1109/CCC.2014.22.
  47. Anup Rao and Makrand Sinha. Simplified separation of information and communication. Theory of Computing, 14(20):1-29, 2018. URL: https://doi.org/10.4086/toc.2018.v014a020.
  48. Swagato Sanyal. Fourier sparsity and dimension. Theory of Computing, 15(11):1-13, 2019. URL: https://doi.org/10.4086/toc.2019.v015a011.
  49. Swagato Sanyal. Randomized query composition and product distributions. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS), volume 289 of LIPIcs, pages 56:1-56:19. Schloss Dagstuhl, 2024. URL: https://doi.org/10.4230/LIPIcs.STACS.2024.56.
  50. Petr Savický. On determinism versus unambiquous nondeterminism for decision trees. Technical Report TR02-009, Electronic Colloquium on Computational Complexity (ECCC), 2002. URL: http://eccc.hpi-web.de/report/2002/009/.
  51. Ronen Shaltiel. Towards proving strong direct product theorems. computational complexity, 12(1):1-22, 2003. URL: https://doi.org/10.1007/s00037-003-0175-x.
  52. Alexander Shekhovtsov and Vladimir Podolskii. Randomized lifting to semi-structured communication complexity via linear diversity. In 16th Innovations in Theoretical Computer Science Conference (ITCS), LIPIcs, pages 78:1-78:21. Schloss Dagstuhl, 2025. URL: https://doi.org/10.4230/LIPIcs.ITCS.2025.78.
  53. Amir Shpilka, Avishay Tal, and Ben Volk. On the structure of boolean functions with small spectral norm. computational complexity, 26(1):229-273, 2017. URL: https://doi.org/10.1007/s00037-015-0110-y.
  54. Hing Yin Tsang, Chung Hoi Wong, Ning Xie, and Shengyu Zhang. Fourier sparsity, spectral norm, and the log-rank conjecture. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pages 658-667, 2013. URL: https://doi.org/10.1109/FOCS.2013.76.
  55. Andrew Yao. Probabilistic computations: Toward a unified measure of complexity. In Proceedings of the 18th Annual Symposium on Foundations of Computer Science, SFCS '77, pages 222-227. IEEE Computer Society, 1977. URL: https://doi.org/10.1109/SFCS.1977.24.
  56. Andrew Yao. Lower bounds by probabilistic arguments. In Proceedings of the 24th Annual Symposium on Foundations of Computer Science, SFCS '83, pages 420-428. IEEE Computer Society, 1983. URL: https://doi.org/10.1109/SFCS.1983.30.
  57. Zhiqiang Zhang and Yaoyun Shi. On the parity complexity measures of boolean functions. Theoretical Computer Science, 411(26-28):2612-2618, 2010. URL: https://doi.org/10.1016/j.tcs.2010.03.027.
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