Document Open Access Logo

Nearly Optimal Separations Between Communication (or Query) Complexity and Partitions

Authors Andris Ambainis, Martins Kokainis, Robin Kothari



PDF
Thumbnail PDF

File

LIPIcs.CCC.2016.4.pdf
  • Filesize: 0.52 MB
  • 14 pages

Document Identifiers

Author Details

Andris Ambainis
Martins Kokainis
Robin Kothari

Cite AsGet BibTex

Andris Ambainis, Martins Kokainis, and Robin Kothari. Nearly Optimal Separations Between Communication (or Query) Complexity and Partitions. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 4:1-4:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.CCC.2016.4

Abstract

We show a nearly quadratic separation between deterministic communication complexity and the logarithm of the partition number, which is essentially optimal. This improves upon a recent power 1.5 separation of Göös, Pitassi, and Watson (FOCS 2015). In query complexity, we establish a nearly quadratic separation between deterministic (and even randomized) query complexity and subcube partition complexity, which is also essentially optimal. We also establish a nearly power 1.5 separation between quantum query complexity and subcube partition complexity, the first superlinear separation between the two measures. Lastly, we show a quadratic separation between quantum query complexity and one-sided subcube partition complexity. Our query complexity separations use the recent cheat sheet framework of Aaronson, Ben-David, and Kothari. Our query functions are built up in stages by alternating function composition with the cheat sheet construction. The communication complexity separation follows from "lifting" the query separation to communication complexity.
Keywords
  • Query Complexity
  • Communication Complexity
  • Subcube Partition Complexity
  • Partition Bound

Metrics

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

References

  1. Scott Aaronson. Quantum certificate complexity. SIAM Journal on Computing, 35(4):804-824, 2006. Google Scholar
  2. Scott Aaronson. Lower bounds for local search by quantum arguments. Journal of Computer and System Sciences, 74(3):313-332, 2008. Google Scholar
  3. Scott Aaronson, Shalev Ben-David, and Robin Kothari. Separations in query complexity using cheat sheets. In Proceedings of the 48th ACM Symposium on Theory of Computing (STOC 2016), to appear. URL: http://arxiv.org/abs/arXiv:1511.01937.
  4. Alfred V. Aho, Jeffrey D. Ullman, and Mihalis Yannakakis. On notions of information transfer in VLSI circuits. In Proceedings of the 15th ACM Symposium on Theory of Computing (STOC 1983), pages 133-139, 1983. URL: http://dx.doi.org/10.1145/800061.808742.
  5. Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani. Strengths and weaknesses of quantum computing. SIAM Journal on Computing (special issue on quantum computing), 26:1510-1523, 1997. Google Scholar
  6. Harry Buhrman and Ronald de Wolf. Complexity measures and decision tree complexity: a survey. Theoretical Computer Science, 288(1):21-43, 2002. URL: http://dx.doi.org/10.1016/S0304-3975(01)00144-X.
  7. Justin Gilmer, Michael Saks, and Srikanth Srinivasan. Composition limits and separating examples for some Boolean function complexity measures. In Proceedings of 2013 IEEE Conference on Computational Complexity (CCC 2013), pages 185-196, June 2013. URL: http://dx.doi.org/10.1109/CCC.2013.27.
  8. Mika Göös. Lower bounds for clique vs. independent set. In Proceedings of the 56th IEEE Symposium on Foundations of Computer Science (FOCS 2015), pages 1066-1076, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.69.
  9. Mika Göös, T.S. Jayram, Toniann Pitassi, and Thomas Watson. Randomized communication vs. partition number. Electronic Colloquium on Computational Complexity (ECCC) http://eccc.hpi-web.de/report/2015/169/, 2015.
  10. Mika Göös, Toniann Pitassi, and Thomas Watson. Deterministic communication vs. partition number. In Proceedings of the 56th IEEE Symposium on Foundations of Computer Science (FOCS 2015), pages 1077-1088, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.70.
  11. Lov K. Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the 28th ACM Symposium on Theory of Computing (STOC 1996), pages 212-219, 1996. URL: http://dx.doi.org/10.1145/237814.237866.
  12. Peter Høyer, Troy Lee, and Robert Špalek. Negative weights make adversaries stronger. In Proceedings of the 39th ACM Symposium on Theory of Computing (STOC 2007), pages 526-535, 2007. URL: http://dx.doi.org/10.1145/1250790.1250867.
  13. Rahul Jain and Hartmut Klauck. The partition bound for classical communication complexity and query complexity. In Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity, CCC'10, pages 247-258, 2010. URL: http://dx.doi.org/10.1109/CCC.2010.31.
  14. Rahul Jain, Troy Lee, and Nisheeth K. Vishnoi. A quadratically tight partition bound for classical communication complexity and query complexity. arXiv preprint http://arxiv.org/abs/arXiv:1401.4512, 2014. Google Scholar
  15. Stasys Jukna. Boolean Function Complexity: Advances and Frontiers. Algorithms and Combinatorics. Springer, 2012. Google Scholar
  16. Jedrzej Kaniewski, Troy Lee, and Ronald de Wolf. Query complexity in expectation. In Automata, Languages, and Programming, volume 9134 of Lecture Notes in Computer Science, pages 761-772. Springer Berlin Heidelberg, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47672-7_62.
  17. Robin Kothari, David Racicot-Desloges, and Miklos Santha. Separating Decision Tree Complexity from Subcube Partition Complexity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015), volume 40 of Leibniz International Proceedings in Informatics (LIPIcs), pages 915-930, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.915.
  18. Eyal Kushilevitz, Nathan Linial, and Rafail Ostrovsky. The linear-array conjecture in communication complexity is false. Combinatorica, 19(2):241-254, 1999. URL: http://dx.doi.org/10.1007/s004930050054.
  19. Eyal Kushilevitz and Noam Nisan. Communication Complexity. Cambridge University Press, 2006. Google Scholar
  20. Sophie Laplante and Frédéric Magniez. Lower bounds for randomized and quantum query complexity using Kolmogorov arguments. SIAM Journal on Computing, 38(1):46-62, 2008. Google Scholar
  21. Troy Lee, Rajat Mittal, Ben W. Reichardt, Robert Špalek, and Mario Szegedy. Quantum query complexity of state conversion. In Proceedings of the 52nd IEEE Symposium on Foundations of Computer Science (FOCS 2011), pages 344-353, 2011. http://arxiv.org/abs/arXiv:1011.3020, URL: http://dx.doi.org/10.1109/FOCS.2011.75.
  22. Troy Lee and Adi Shraibman. Lower bounds in communication complexity. Foundations and Trends^ ® in Theoretical Computer Science, 3(4):263-399, 2007. URL: http://dx.doi.org/10.1561/0400000040.
  23. Ashley Montanaro. A composition theorem for decision tree complexity. Chicago Journal of Theoretical Computer Science, 2014(6), July 2014. Google Scholar
  24. Noam Nisan. CREW PRAMs and decision trees. SIAM Journal on Computing, 20(6):999-1007, 1991. URL: http://dx.doi.org/10.1137/0220062.
  25. Noam Nisan and Mario Szegedy. On the degree of Boolean functions as real polynomials. Computational Complexity, 15(4):557-565, 1995. Google Scholar
  26. Ben W. Reichardt. Reflections for quantum query algorithms. In Proceedings of the 22nd ACM-SIAM Symposium on Discrete Algorithms (SODA 2011), SODA'11, pages 560-569, 2011. URL: http://dl.acm.org/citation.cfm?id=2133036.2133080.
  27. Petr Savický. On determinism versus unambiquous nondeterminism for decision trees. ECCC, TR02-009, 2002. Google Scholar
  28. Robert Špalek and Mario Szegedy. All quantum adversary methods are equivalent. Theory of Computing, 2(1):1-18, 2006. Google Scholar
  29. Avishay Tal. Properties and applications of Boolean function composition. In Proceedings of the 4th Conference on Innovations in Theoretical Computer Science, ITCS'13, pages 441-454, 2013. URL: http://dx.doi.org/10.1145/2422436.2422485.
  30. Andrew Chi-Chih Yao. Some complexity questions related to distributive computing. In Proceedings of the 11th ACM Symposium on Theory of Computing, STOC'79, pages 209-213, 1979. URL: http://dx.doi.org/10.1145/800135.804414.
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