Optimal Separation and Strong Direct Sum for Randomized Query Complexity

Authors Eric Blais, Joshua Brody

Thumbnail PDF


  • Filesize: 0.52 MB
  • 17 pages

Document Identifiers

Author Details

Eric Blais
  • University of Waterloo, ON, Canada
Joshua Brody
  • Swarthmore College, PA, USA


The first author thanks Alexander Belov and Shalev Ben-David for enlightening discussions and helpful suggestions. The second author thanks Peter Winkler for insightful discussions. Both authors wish to thank the anonymous referees for valuable feedback and for the reference to [Sherstov, 2018].

Cite AsGet BibTex

Eric Blais and Joshua Brody. Optimal Separation and Strong Direct Sum for Randomized Query Complexity. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We establish two results regarding the query complexity of bounded-error randomized algorithms. Bounded-error separation theorem. There exists a total function f : {0,1}^n -> {0,1} whose epsilon-error randomized query complexity satisfies overline{R}_epsilon(f) = Omega(R(f) * log 1/epsilon). Strong direct sum theorem. For every function f and every k >= 2, the randomized query complexity of computing k instances of f simultaneously satisfies overline{R}_epsilon(f^k) = Theta(k * overline{R}_{epsilon/k}(f)). As a consequence of our two main results, we obtain an optimal superlinear direct-sum-type theorem for randomized query complexity: there exists a function f for which R(f^k) = Theta(k log k * R(f)). This answers an open question of Drucker (2012). Combining this result with the query-to-communication complexity lifting theorem of Göös, Pitassi, and Watson (2017), this also shows that there is a total function whose public-coin randomized communication complexity satisfies R^{cc}(f^k) = Theta(k log k * R^{cc}(f)), answering a question of Feder, Kushilevitz, Naor, and Nisan (1995).

Subject Classification

ACM Subject Classification
  • Theory of computation → Probabilistic computation
  • Theory of computation → Oracles and decision trees
  • Decision trees
  • query complexity
  • communication complexity


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


  1. Scott Aaronson, Shalev Ben-David, and Robin Kothari. Separations in query complexity using cheat sheets. In Proceedings 48th Annual ACM Symposium on Theory of Computing, pages 863-876, 2016. Google Scholar
  2. Andris Ambainis, Kaspars Balodis, Aleksandrs Belovs, Troy Lee, Miklos Santha, and Juris Smotrovs. Separations in query complexity based on pointer functions. Journal of the ACM, 64(5):32, 2017. Google Scholar
  3. Andris Ambainis, Martins Kokainis, and Robin Kothari. Nearly optimal separations between communication (or query) complexity and partitions. In Proceedings 31st Annual Conference on Computational Complexity, page 4, 2016. Google Scholar
  4. Anurag Anshu, Aleksandrs Belovs, Shalev Ben-David, Mika Göös, Rahul Jain, Robin Kothari, Troy Lee, and Miklos Santha. Separations in communication complexity using cheat sheets and information complexity. In Proceedings 57th Annual IEEE Symposium on Foundations of Computer Science, pages 555-564, 2016. Google Scholar
  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. Google Scholar
  6. 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.
  7. Yosi Ben-Asher and Ilan Newman. Decision Trees with AND, OR Queries. In Proceedings 10th Annual Structure in Complexity Theory Conference, pages 74-81, 1995. URL: https://doi.org/10.1109/SCT.1995.514729.
  8. Shalev Ben-David and Robin Kothari. Randomized Query Complexity of Sabotaged and Composed Functions. Theory of Computing, 14(1):1-27, 2018. URL: https://doi.org/10.4086/toc.2018.v014a005.
  9. Mark Braverman and Anup Rao. Information Equals Amortized Communication. IEEE Transactions on Information Theory, 60(10):6058-6069, 2014. Google Scholar
  10. Harry Buhrman, Ilan Newman, Hein Röhrig, and Ronald de Wolf. Robust Polynomials and Quantum Algorithms. Theory Comput. Syst., 40(4):379-395, 2007. URL: https://doi.org/10.1007/s00224-006-1313-z.
  11. Amit Chakrabarti, Yaoyun Shi, Anthony Wirth, and Andrew Chi-Chih Yao. Informational Complexity and the Direct Sum Problem for Simultaneous Message Complexity. In Proceedings 42nd Annual IEEE Symposium on Foundations of Computer Science, pages 270-278, 2001. URL: https://doi.org/10.1109/SFCS.2001.959901.
  12. Benny Chor, Oded Goldreich, Johan Håstad, Joel Friedman, Steven Rudich, and Roman Smolensky. The Bit Extraction Problem or t-Resilient Functions. In Proceedings 26th Annual IEEE Symposium on Foundations of Computer Science, pages 396-407, 1985. URL: https://doi.org/10.1109/SFCS.1985.55.
  13. Andrew Drucker. Improved direct product theorems for randomized query complexity. Computational Complexity, 21(2):197-244, 2012. Google Scholar
  14. 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.
  15. Anat Ganor, Gillat Kol, and Ran Raz. Exponential separation of information and communication. In Proceedings 55th Annual IEEE Symposium on Foundations of Computer Science, pages 176-185, 2014. Google Scholar
  16. Mika Göös, Toniann Pitassi, and Thomas Watson. Deterministic communication vs. partition number. In Proceedings 56th Annual IEEE Symposium on Foundations of Computer Science, pages 1077-1088, 2015. Google Scholar
  17. Mika Göös, Toniann Pitassi, and Thomas Watson. Query-to-Communication Lifting for BPP. In Proceedings 58th Annual IEEE Symposium on Foundations of Computer Science, 2017. Google Scholar
  18. Russell Impagliazzo, Ran Raz, and Avi Wigderson. A Direct Product Theorem. In Proceedings 9th Annual Structure in Complexity Theory Conference, pages 88-96, 1994. URL: https://doi.org/10.1109/SCT.1994.315814.
  19. Rahul Jain, Hartmut Klauck, and Miklos Santha. Optimal direct sum results for deterministic and randomized decision tree complexity. Inf. Process. Lett., 110(20):893-897, 2010. URL: https://doi.org/10.1016/j.ipl.2010.07.020.
  20. Mauricio Karchmer, Ran Raz, and Avi Wigderson. Super-logarithmic depth lower bounds via the direct sum in communication complexity. Computational Complexity, 5(3):191-204, 1995. Google Scholar
  21. Marco Molinaro, David P. Woodruff, and Grigory Yaroslavtsev. Beating the Direct Sum Theorem in Communication Complexity with Implications for Sketching. In Proceedings 24th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1738-1756, 2013. URL: https://doi.org/10.1137/1.9781611973105.125.
  22. Marco Molinaro, David P Woodruff, and Grigory Yaroslavtsev. Amplification of one-way information complexity via codes and noise sensitivity. In Proceedings 42nd Annual International Colloquium on Automata, Languages, and Programming, pages 960-972. Springer, 2015. Google Scholar
  23. Sagnik Mukhopadhyay and Swagato Sanyal. Towards Better Separation between Deterministic and Randomized Query Complexity. In Proceedings 35th Annual Foundations of Software Technology and Theoretical Computer Science, pages 206-220, 2015. Google Scholar
  24. Noam Nisan, Steven Rudich, and Michael E. Saks. Products and Help Bits in Decision Trees. SIAM Journal on Computing, 28(3):1035-1050, 1999. URL: https://doi.org/10.1137/S0097539795282444.
  25. Ronen Shaltiel. Towards proving strong direct product theorems. Computational Complexity, 12(1-2):1-22, 2003. Google Scholar
  26. Alexander Sherstov. The Power of Asymmetry in Constant-Depth Circuits. SIAM Journal on Computing, 47(6):2362-2434, 2018. URL: https://doi.org/10.1137/16M1064477.