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Trading Information Complexity for Error

Authors Yuval Dagan, Yuval Filmus, Hamed Hatami, Yaqiao Li



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Yuval Dagan
Yuval Filmus
Hamed Hatami
Yaqiao Li

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Yuval Dagan, Yuval Filmus, Hamed Hatami, and Yaqiao Li. Trading Information Complexity for Error. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 16:1-16:59, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.CCC.2017.16

Abstract

We consider the standard two-party communication model. The central problem studied in this article is how much can one save in information complexity by allowing a certain error. * For arbitrary functions, we obtain lower bounds and upper bounds indicating a gain that is of order Omega(h(epsilon)) and O(h(sqrt{epsilon})). Here h denotes the binary entropy function. * We analyze the case of the two-bit AND function in detail to show that for this function the gain is Theta(h(epsilon)). This answers a question of Braverman et al. [Braverman, STOC 2013]. * We obtain sharp bounds for the set disjointness function of order n. For the case of the distributional error, we introduce a new protocol that achieves a gain of Theta(sqrt{h(epsilon)}) provided that n is sufficiently large. We apply these results to answer another of question of Braverman et al. regarding the randomized communication complexity of the set disjointness function. * Answering a question of Braverman [Braverman, STOC 2012], we apply our analysis of the set disjointness function to establish a gap between the two different notions of the prior-free information cost. In light of [Braverman, STOC 2012], this implies that amortized randomized communication complexity is not necessarily equal to the amortized distributional communication complexity with respect to the hardest distribution. As a consequence, we show that the epsilon-error randomized communication complexity of the set disjointness function of order n is n[C_{DISJ} - Theta(h(epsilon))] + o(n), where C_{DISJ} ~ 0.4827$ is the constant found by Braverman et al. [Braverman, STOC 2012].
Keywords
  • communication complexity
  • information complexity

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References

  1. Ziv Bar-Yossef, T. S. Jayram, Ravi Kumar, and D. Sivakumar. An information statistics approach to data stream and communication complexity. J. Comput. System Sci., 68(4):702-732, 2004. URL: http://dx.doi.org/10.1016/j.jcss.2003.11.006.
  2. Boaz Barak, Mark Braverman, Xi Chen, and Anup Rao. How to compress interactive communication [extended abstract]. In STOC'10 - Proceedings of the 2010 ACM International Symposium on Theory of Computing, pages 67-76. ACM, New York, 2010. Google Scholar
  3. Mark Braverman. Interactive information complexity. In STOC'12 - Proceedings of the 2012 ACM Symposium on Theory of Computing, pages 505-524. ACM, New York, 2012. URL: http://dx.doi.org/10.1145/2213977.2214025.
  4. Mark Braverman, Ankit Garg, Denis Pankratov, and Omri Weinstein. From information to exact communication (extended abstract). In STOC'13 - Proceedings of the 2013 ACM Symposium on Theory of Computing, pages 151-160. ACM, New York, 2013. URL: http://dx.doi.org/10.1145/2488608.2488628.
  5. Mark Braverman, Ankit Garg, Denis Pankratov, and Omri Weinstein. Information lower bounds via self-reducibility. In Andrei A. Bulatov and Arseny M. Shur, editors, Computer Science - Theory and Applications, volume 7913 of Lecture Notes in Computer Science, pages 183-194. Springer Berlin Heidelberg, 2013. URL: http://dx.doi.org/10.1007/978-3-642-38536-0_16.
  6. Mark Braverman and Anup Rao. Information equals amortized communication. IEEE Trans. Inform. Theory, 60(10):6058-6069, 2014. URL: http://dx.doi.org/10.1109/TIT.2014.2347282.
  7. Mark Braverman, Anup Rao, Omri Weinstein, and Amir Yehudayoff. Direct product via round-preserving compression. In Automata, languages, and programming. Part I, volume 7965 of Lecture Notes in Comput. Sci., pages 232-243. Springer, Heidelberg, 2013. URL: http://dx.doi.org/10.1007/978-3-642-39206-1_20.
  8. Mark Braverman, Anup Rao, Omri Weinstein, and Amir Yehudayoff. Direct products in communication complexity. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science - FOCS 2013, pages 746-755. IEEE Computer Soc., Los Alamitos, CA, 2013. URL: http://dx.doi.org/10.1109/FOCS.2013.85.
  9. Mark Braverman and Jon Schneider. Information complexity is computable. CoRR, abs/1502.02971, 2015. URL: http://arxiv.org/abs/1502.02971.
  10. Amit Chakrabarti, Yaoyun Shi, Anthony Wirth, and Andrew Yao. Informational complexity and the direct sum problem for simultaneous message complexity. In 42nd IEEE Symposium on Foundations of Computer Science (Las Vegas, NV, 2001), pages 270-278. IEEE Computer Soc., Los Alamitos, CA, 2001. Google Scholar
  11. Arkadev Chattopadhyay and Toniann Pitassi. The story of set disjointness. ACM SIGACT News, 41(3):59-85, 2010. Google Scholar
  12. Yuval Dagan and Yuval Filmus. Grid protocols. In preparation, 2016. Google Scholar
  13. Tomás Feder, Eyal Kushilevitz, Moni Naor, and Noam Nisan. Amortized communication complexity. SIAM J. Comput., 24(4):736-750, 1995. URL: http://dx.doi.org/10.1137/S0097539792235864.
  14. Yuval Filmus, Hamed Hatami, Yaqiao Li, and Suzin You. Information complexity of the and function in the two-party, and multiparty settings. Submitted, 2017. Google Scholar
  15. Anat Ganor, Gillat Kol, and Ran Raz. Exponential separation of information and communication for Boolean functions [extended abstract]. In STOC'15 - Proceedings of the 2015 ACM Symposium on Theory of Computing, pages 557-566. ACM, New York, 2015. Google Scholar
  16. Prahladh Harsha, Rahul Jain, David McAllester, and Jaikumar Radhakrishnan. The communication complexity of correlation. IEEE Trans. Inform. Theory, 56(1):438-449, 2010. URL: http://dx.doi.org/10.1109/TIT.2009.2034824.
  17. Rahul Jain. New strong direct product results in communication complexity. J. ACM, 62(3):Art. 20, 27, 2015. URL: http://dx.doi.org/10.1145/2699432.
  18. Rahul Jain, Attila Pereszlényi, and Penghui Yao. A direct product theorem for the two-party bounded-round public-coin communication complexity. In 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science - FOCS 2012, pages 167-176. IEEE Computer Soc., Los Alamitos, CA, 2012. Google Scholar
  19. Rahul Jain, Jaikumar Radhakrishnan, and Pranab Sen. A direct sum theorem in communication complexity via message compression. In Automata, languages and programming, volume 2719 of Lecture Notes in Comput. Sci., pages 300-315. Springer, Berlin, 2003. URL: http://dx.doi.org/10.1007/3-540-45061-0_26.
  20. Bala Kalyanasundaram and Georg Schnitger. The probabilistic communication complexity of set intersection. SIAM J. Discrete Math., 5(4):545-557, 1992. URL: http://dx.doi.org/10.1137/0405044.
  21. Hartmut Klauck. A strong direct product theorem for disjointness [extended abstract]. In STOC'10 - Proceedings of the 2010 ACM International Symposium on Theory of Computing, pages 77-86. ACM, New York, 2010. Google Scholar
  22. Eyal Kushilevitz and Noam Nisan. Communication complexity. Cambridge University Press, Cambridge, 1997. Google Scholar
  23. Nan Ma and Prakash Ishwar. Some results on distributed source coding for interactive function computation. IEEE Trans. Inform. Theory, 57(9):6180-6195, 2011. URL: http://dx.doi.org/10.1109/TIT.2011.2161916.
  24. Nan Ma and Prakash Ishwar. The infinite-message limit of two-terminal interactive source coding. IEEE Trans. Inform. Theory, 59(7):4071-4094, 2013. URL: http://dx.doi.org/10.1109/TIT.2013.2251412.
  25. Marco Molinaro, David P. Woodruff, and Grigory Yaroslavtsev. Beating the direct sum theorem in communication complexity with implications for sketching. In Proceedings of the Twenty-fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'13, pages 1738-1756, Philadelphia, PA, USA, 2013. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=2627817.2627942.
  26. A. A. Razborov. On the distributional complexity of disjointness. Theoret. Comput. Sci., 106(2):385-390, 1992. URL: http://dx.doi.org/10.1016/0304-3975(92)90260-M.
  27. Byron Schmuland. On the compacity of the space of probability measures. Mathematics Stack Exchange. URL: https://math.stackexchange.com/q/642888.
  28. C. E. Shannon. A mathematical theory of communication. Bell System Tech. J., 27:379-423, 623-656, 1948. Google Scholar
  29. Alexander A. Sherstov. Communication complexity theory: thirty-five years of set disjointness. In Mathematical foundations of computer science 2014. Part I, volume 8634 of Lecture Notes in Comput. Sci., pages 24-43. Springer, Heidelberg, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44522-8_3.
  30. Andrew Chi-Chih Yao. Some complexity questions related to distributive computing(preliminary report). In Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, STOC'79, pages 209-213, New York, NY, USA, 1979. ACM. URL: http://dx.doi.org/10.1145/800135.804414.
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