Document Open Access Logo

Correlation in Hard Distributions in Communication Complexity

Authors Ralph Christian Bottesch, Dmitry Gavinsky, Hartmut Klauck

PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2015.544.pdf
  • Filesize: 0.59 MB
  • 29 pages

Document Identifiers

Author Details

Ralph Christian Bottesch
Dmitry Gavinsky
Hartmut Klauck

Cite As Get BibTex

Ralph Christian Bottesch, Dmitry Gavinsky, and Hartmut Klauck. Correlation in Hard Distributions in Communication Complexity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 544-572, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.544

Abstract

We study the effect that the amount of correlation in a bipartite distribution has on the communication complexity of a problem under that distribution. We introduce a new family of complexity measures that interpolates between the two previously studied extreme cases: the (standard) randomised communication complexity and the case of distributional complexity under product distributions.

- We give a tight characterisation of the randomised complexity of Disjointness under distributions with mutual information k, showing that it is Theta(sqrt(n(k+1))) for all 0 <= k <= n. This smoothly interpolates between the lower bounds of Babai, Frankl and Simon for the product distribution case (k=0), and the bound of Razborov for the randomised case. The upper bounds improve and generalise what was known for product distributions, and imply that any tight bound for Disjointness needs Omega(n) bits of mutual information in the corresponding distribution.

- We study the same question in the distributional quantum setting, and show a lower bound of Omega((n(k+1))^{1/4}), and an upper bound (via constructing communication protocols), matching up to a logarithmic factor.

- We show that there are total Boolean functions f_d that have distributional communication complexity O(log(n)) under all distributions of information up to o(n), while the (interactive) distributional complexity maximised over all distributions is Theta(log(d)) for n <= d <= 2^{n/100}. This shows, in particular, that the correlation needed to show that a problem is hard can be much larger than the communication complexity of the problem.

- We show that in the setting of one-way communication under product distributions, the dependence of communication cost on the allowed error epsilon is multiplicative in log(1/epsilon) - the previous upper bounds had the dependence of more than 1/epsilon. This result, for the first time, explains how one-way communication complexity under product distributions is stronger than PAC-learning: both tasks are characterised by the VC-dimension, but have very different error dependence (learning from examples, it costs more to reduce the error).

Subject Classification

Keywords
  • communication complexity; information theory

Metrics

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

References

  1. S. Aaronson and A. Ambainis. Quantum search of spatial regions. Theory of Computing, 1(1):47-79, 2005. Earlier version in FOCS'03. quant-ph/0303041. Google Scholar
  2. Noga Alon, Shay Moran, and Amir Yehudayoff. Sign rank, VC dimension and spectral gaps. Electronic Colloquium on Computational Complexity (ECCC), 21:135, 2014. Google Scholar
  3. A. Ambainis, A. Nayak, A. Ta-Shma, and U. V. Vazirani. Dense quantum coding and quantum finite automata. Journal of the ACM, 49(4):496-511, 2002. Earlier version in STOC'99. Google Scholar
  4. L. Babai, P. Frankl, and J. Simon. Complexity classes in communication complexity theory. In Proceedings of 27th IEEE FOCS, pages 337-347, 1986. Google Scholar
  5. Z. Bar-Yossef, T. S. Jayram, R. Kumar, and D. Sivakumar. An information statistics approach to data stream and communication complexity. In Proceedings of 43rd IEEE FOCS, pages 209-218, 2002. Google Scholar
  6. H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation. In Proceedings of 30th ACM STOC, pages 63-68, 1998. quant-ph/9802040. Google Scholar
  7. B. Chor and O. Goldreich. Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM Journal on Computing, 17(2):230-261, 1988. Earlier version in FOCS'85. Google Scholar
  8. T. M. Cover and J. A. Thomas. Elements of Information Theory. Wiley, 1991. Google Scholar
  9. Ronald de Wolf. Quantum communication and complexity. Theoretical Computer Science, 287(1):337-353, 2002. Google Scholar
  10. Prahladh Harsha, Rahul Jain, David A. McAllester, and Jaikumar Radhakrishnan. The communication complexity of correlation. IEEE Transactions on Information Theory, 56(1):438-449, 2010. Google Scholar
  11. Johan Håstad and Avi Wigderson. The randomized communication complexity of set disjointness. Theory of Computing, 3(1):211-219, 2007. Google Scholar
  12. R. Jain, H. Klauck, and A. Nayak. Direct product theorems for classical communication complexity via subdistribution bounds. In Proc. of 40th ACM STOC, pages 599-608, 2008. Google Scholar
  13. R. Jain, J. Radhakrishnan, and P. Sen. Privacy and interaction in quantum communication complexity and a theorem about the relative entropy of quantum states. In Proceedings of 43rd IEEE FOCS, pages 429-438, 2002. Google Scholar
  14. R. Jain and S. Zhang. New bounds on classical and quantum one-way communication complexity. Theoretical Computer Science, 410(26):2463-2477, 2009. Google Scholar
  15. Rahul Jain, Jaikumar Radhakrishnan, and Pranab Sen. A direct sum theorem in communication complexity via message compression. In ICALP, page 187, 2003. Google Scholar
  16. B. Kalyanasundaram and G. Schnitger. The probabilistic communication complexity of set intersection. SIAM Journal on Discrete Mathematics, 5(4):545-557, 1992. Earlier version in Structures'87. Google Scholar
  17. Michael J. Kearns and Umesh V. Vazirani. An Introduction to Computational Learning Theory. MIT Press, 1994. Google Scholar
  18. H. Klauck. On quantum and probabilistic communication: Las Vegas and one-way protocols. In Proceedings of 32nd ACM STOC, pages 644-651, 2000. Google Scholar
  19. I. Kremer, N. Nisan, and D. Ron. On randomized one-round communication complexity. Computational Complexity, 8(1):21-49, 1999. Earlier version in STOC'95. Correction at http://www.eng.tau.ac.il/~ danar/Public/KNR-fix.ps. Google Scholar
  20. E. Kushilevitz and N. Nisan. Communication Complexity. Cambridge Univ. Press, 1997. Google Scholar
  21. Marco Molinaro, David P. Woodruff, and Grigory Yaroslavtsev. Amplification of one-way information complexity via codes and noise sensitivity. In ICALP, pages 960-972, 2015. Google Scholar
  22. A. Razborov. On the distributional complexity of disjointness. Theoretical Computer Science, 106(2):385-390, 1992. Google Scholar
  23. A. Razborov. Quantum communication complexity of symmetric predicates. Izvestiya of the Russian Academy of Sciences, mathematics, 67(1):159-176, 2003. quant-ph/0204025. Google Scholar
  24. Mert Saglam and Gábor Tardos. On the communication complexity of sparse set disjointness and exists-equal problems. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS, pages 678-687, 2013. Google Scholar
  25. Alexander A. Sherstov. Communication complexity under product and nonproduct distributions. In Proceedings of the 23rd Annual IEEE Conference on Computational Complexity, CCC, pages 64-70, 2008. Google Scholar
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact