Communication Complexity of Set-Disjointness for All Probabilities

Authors Mika Göös, Thomas Watson



PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2014.721.pdf
  • Filesize: 0.71 MB
  • 16 pages

Document Identifiers

Author Details

Mika Göös
Thomas Watson

Cite AsGet BibTex

Mika Göös and Thomas Watson. Communication Complexity of Set-Disjointness for All Probabilities. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 721-736, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.721

Abstract

We study set-disjointness in a generalized model of randomized two-party communication where the probability of acceptance must be at least alpha(n) on yes-inputs and at most beta(n) on no-inputs, for some functions alpha(n)>beta(n). Our main result is a complete characterization of the private-coin communication complexity of set-disjointness for all functions alpha and beta, and a near-complete characterization for public-coin protocols. In particular, we obtain a simple proof of a theorem of Braverman and Moitra (STOC 2013), who studied the case where alpha=1/2+epsilon(n) and beta=1/2-epsilon(n). The following contributions play a crucial role in our characterization and are interesting in their own right. (1) We introduce two communication analogues of the classical complexity class that captures small bounded-error computations: we define a "restricted" class SBP (which lies between MA and AM) and an "unrestricted" class USBP. The distinction between them is analogous to the distinction between the well-known communication classes PP and UPP. (2) We show that the SBP communication complexity is precisely captured by the classical corruption lower bound method. This sharpens a theorem of Klauck (CCC 2003). (3) We use information complexity arguments to prove a linear lower bound on the USBP complexity of set-disjointness.
Keywords
  • Communication Complexity
  • Set-Disjointness
  • All Probabilities

Metrics

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

References

  1. Scott Aaronson and Avi Wigderson. Algebrization: A new barrier in complexity theory. ACM Transactions on Computation Theory, 1(1), 2009. Google Scholar
  2. László Babai, Peter Frankl, and Janos Simon. Complexity classes in communication complexity theory. In Proceedings of the 27th Symposium on Foundations of Computer Science (FOCS), pages 337-347. IEEE, 1986. Google Scholar
  3. László Babai and Shlomo Moran. Arthur-Merlin games: A randomized proof system, and a hierarchy of complexity classes. Journal of Computer and System Sciences, 36(2):254-276, 1988. Google Scholar
  4. Ziv Bar-Yossef, T.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
  5. Elmar Böhler, Christian Glaßer, and Daniel Meister. Error-bounded probabilistic computations between MA and AM. Journal of Computer and System Sciences, 72(6):1043-1076, 2006. Google Scholar
  6. Mark Braverman and Ankur Moitra. An information complexity approach to extended formulations. In Proceedings of the 45th Symposium on Theory of Computing (STOC), pages 161-170. ACM, 2013. Google Scholar
  7. Mark Braverman and Omri Weinstein. A discrepancy lower bound for information complexity. In Proceedings of the 16th International Workshop on Randomization and Computation (RANDOM), pages 459-470. Springer, 2012. Google Scholar
  8. Harry Buhrman, Nikolai Vereshchagin, and Ronald de Wolf. On computation and communication with small bias. In Proceedings of the 22nd Conference on Computational Complexity (CCC), pages 24-32. IEEE, 2007. Google Scholar
  9. Amit Chakrabarti, Graham Cormode, and Andrew McGregor. Annotations in data streams. In Proceedings of the 36th International Colloquium on Automata, Languages, and Programming (ICALP), pages 222-234. Springer, 2009. Google Scholar
  10. Amit Chakrabarti, Graham Cormode, Andrew McGregor, Justin Thaler, and Suresh Venkatasubramanian. On interactivity in Arthur-Merlin communication and stream computation. Technical Report TR13-180, Electronic Colloquium on Computational Complexity (ECCC), 2013. Google Scholar
  11. Amit Chakrabarti and Oded Regev. An optimal lower bound on the communication complexity of Gap-Hamming-Distance. SIAM Journal on Computing, 41(5):1299-1317, 2012. Google Scholar
  12. Amit Chakrabarti, Yaoyun Shi, Anthony Wirth, and Andrew Yao. Informational complexity and the direct sum problem for simultaneous message complexity. In Proceedings of the 42nd Symposium on Foundations of Computer Science (FOCS), pages 270-278. IEEE, 2001. Google Scholar
  13. Arkadev Chattopadhyay and Toniann Pitassi. The story of set disjointness. SIGACT News, 41(3):59-85, 2010. Google Scholar
  14. Thomas Cover and Joy Thomas. Elements of Information Theory. Wiley, 2006. Google Scholar
  15. Scott Diehl. Lower bounds for swapping Arthur and Merlin. In Proceedings of the 11th International Workshop on Randomization and Computation (RANDOM), pages 449-463. Springer, 2007. Google Scholar
  16. Jürgen Forster. A linear lower bound on the unbounded error probabilistic communication complexity. Journal of Computer and System Sciences, 65(4):612-625, 2002. Google Scholar
  17. Anat Ganor, Gillat Kol, and Ran Raz. Exponential separation of information and communication. Technical Report TR14-049, Electronic Colloquium on Computational Complexity (ECCC), 2014. Google Scholar
  18. Shafi Goldwasser and Michael Sipser. Private coins versus public coins in interactive proof systems. In Proceedings of the 18th Symposium on Theory of Computing (STOC), pages 59-68. ACM, 1986. Google Scholar
  19. Tom Gur and Ran Raz. Arthur-Merlin streaming complexity. In Proceedings of the 40th International Colloquium on Automata, Languages, and Programming (ICALP), pages 528-539. Springer, 2013. Google Scholar
  20. Rahul Jain and Hartmut Klauck. The partition bound for classical communication complexity and query complexity. In Proceedings of the 25th Conference on Computational Complexity (CCC), pages 247-258. IEEE, 2010. Google Scholar
  21. Rahul Jain, Troy Lee, and Nisheeth Vishnoi. A quadratically tight partition bound for classical communication complexity and query complexity. Technical report, arXiv, 2014. Google Scholar
  22. Stasys Jukna. Boolean Function Complexity: Advances and Frontiers, volume 27 of Algorithms and Combinatorics. Springer, 2012. Google Scholar
  23. Bala Kalyanasundaram and Georg Schnitger. The probabilistic communication complexity of set intersection. SIAM Journal on Discrete Mathematics, 5(4):545-557, 1992. Google Scholar
  24. Iordanis Kerenidis, Sophie Laplante, Virginie Lerays, Jérémie Roland, and David Xiao. Lower bounds on information complexity via zero-communication protocols and applications. In Proceedings of the 53rd Symposium on Foundations of Computer Science (FOCS), pages 500-509, 2012. Google Scholar
  25. Hartmut Klauck. Rectangle size bounds and threshold covers in communication complexity. In Proceedings of the 18th Conference on Computational Complexity (CCC), pages 118-134. IEEE, 2003. Google Scholar
  26. Hartmut Klauck. Lower bounds for quantum communication complexity. SIAM Journal on Computing, 37(1):20-46, 2007. Google Scholar
  27. Hartmut Klauck. On Arthur Merlin games in communication complexity. In Proceedings of the 26th Conference on Computational Complexity (CCC), pages 189-199. IEEE, 2011. Google Scholar
  28. Eyal Kushilevitz and Noam Nisan. Communication Complexity. Cambridge University Press, 1997. Google Scholar
  29. Ilan Newman. Private vs. common random bits in communication complexity. Information Processing Letters, 39(2):67-71, 1991. Google Scholar
  30. Ramamohan Paturi and Janos Simon. Probabilistic communication complexity. Journal of Computer and System Sciences, 33(1):106-123, 1986. Google Scholar
  31. Alexander Razborov. On the distributional complexity of disjointness. Theoretical Computer Science, 106(2):385-390, 1992. Google Scholar
  32. Miklos Santha. Relativized Arthur-Merlin versus Merlin-Arthur games. Information and Computation, 80(1):44-49, 1989. Google Scholar
  33. Alexander Sherstov. Halfspace matrices. Computational Complexity, 17(2):149-178, 2008. Google Scholar
  34. Alexander Sherstov. The pattern matrix method. SIAM Journal on Computing, 40(6):1969-2000, 2011. Google Scholar
  35. Alexander Sherstov. The unbounded-error communication complexity of symmetric functions. Combinatorica, 31(5):583-614, 2011. Google Scholar
  36. Alexander Sherstov. The communication complexity of Gap Hamming Distance. Theory of Computing, 8(1):197-208, 2012. Google Scholar
  37. Thomas Vidick. A concentration inequality for the overlap of a vector on a large set, with application to the communication complexity of the Gap-Hamming-Distance problem. Chicago Journal of Theoretical Computer Science, 2012(1):1-12, 2012. Google Scholar
  38. Thomas Watson. The complexity of estimating min-entropy. Computational Complexity, 2015. To appear. Preprint: URL: http://eccc.hpi-web.de/report/2012/070/.
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