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Abstract

The \({\textsc{And}}\) problem on t bits is a promise decision problem where either at most one bit of the input is set to 1 (No instance) or all t bits are set to 1 (\({\textsc{Yes}}\) instance). In this note, I will give a new proof of an Ω(1/t) lower bound on the information complexity of \({\textsc{And}}\) in the number-in-hand model of communication. This was recently established by Gronemeier, STACS 2009. The proof exploits the information geometry of communication protocols via Hellinger distance in a novel manner and avoids the analytic approach inherent in previous work. As previously known, this bound implies an Ω(n/t) lower bound on the communication complexity of multiparty disjointness and consequently a Ω(n 1 − 2/k ) space lower bound on estimating the k-th frequency moment F k .

Keywords

Communication Protocol Communication Complexity Information Complexity Information Geometry Hellinger Distance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [Abl96]
    Ablayev, F.: Lower bounds for one-way probabilistic communication complexity and their application to space complexity. Theoretical Computer Science 157(2), 139–159 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [AJP09]
    Andoni, A., Jayram, T.S., Patrascu, M.: Non-embeddability and sketching complexity via information geometry (2009)Google Scholar
  3. [AMS99]
    Alon, N., Matias, Y., Szegedy, M.: The space complexity of approximating the frequency moments. Journal of Computer and System Sciences 58(1), 137–147 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [BCKO93]
    Bar-Yehuda, R., Chor, B., Kushilevitz, E., Orlitsky, A.: Privacy, additional information, and communication. IEEE Transactions on Information Theory 39(6), 1930–1943 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [BGKS06]
    Bhuvanagiri, L., Ganguly, S., Kesh, D., Saha, C.: Simpler algorithm for estimating frequency moments of data streams. In: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2006, Miami, Florida, USA, January 22-26, 2006, pp. 708–713. ACM Press, New York (2006)CrossRefGoogle Scholar
  6. [BJK+02]
    Bar-Yossef, Z., Jayram, T.S., Kumar, R., Sivakumar, D., Trevisan, L.: Counting distinct elements in a data stream. In: Rolim, J.D.P., Vadhan, S.P. (eds.) RANDOM 2002. LNCS, vol. 2483, pp. 1–10. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. [BJKS04]
    Bar-Yossef, Z., Jayram, T.S., Kumar, R., Sivakumar, D.: An information statistics approach to data stream and communication complexity. J. Comput. Syst. Sci. 68(4), 702–732 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [CCFC04]
    Charikar, M., Chen, K., Farach-Colton, M.: Finding frequent items in data streams. Theor. Comput. Sci. 312(1), 3–15 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [CKS03]
    Chakrabarti, A., Khot, S., Sun, X.: Near-optimal lower bounds on the multiparty communication complexity of set-disjointness. In: Proceedings of the 18th Annual IEEE Conference on Computational Complexity, pp. 107–117 (2003)Google Scholar
  10. [CSWY01]
    Chakrabarti, A., Shi, Y., Wirth, A., Yao, A.C.-C.: Informational complexity and the direct sum problem for simultaneous message complexity. In: Proceedings of the 42nd IEEE Annual Symposium on Foundations of Computer Science (FOCS), pp. 270–278 (2001)Google Scholar
  11. [DL97]
    Deza, M., Laurent, M.: Geometry of Cuts and Metrics. Springer, Heidelberg (1997)zbMATHGoogle Scholar
  12. [FM85]
    Flajolet, P., Martin, G.N.: Probabilistic counting algorithms for data base applications. J. Comput. Syst. Sci. 31(2), 182–209 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [Gro09]
    Gronemeier, A.: Asymptotically optimal lower bounds on the nih-multi-party information complexity of the and-function and disjointness. In: Albers, S., Marion, J.-Y. (eds.) STACS, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany Internationales Begegnungs- und Forschungszentrum fuer Informatik (IBFI), Schloss Dagstuhl, Germany. Dagstuhl Seminar Proceedings, vol. 09001, pp. 505–516 (2009)Google Scholar
  14. [Ind06]
    Indyk, P.: Stable distributions, pseudorandom generators, embeddings, and data stream computation. J. ACM 53(3), 307–323 (2006)MathSciNetCrossRefGoogle Scholar
  15. [IW05]
    Indyk, P., Woodruff, D.P.: Optimal approximations of the frequency moments of data streams. In: STOC, pp. 202–208 (2005)Google Scholar
  16. [JW09]
    Jayram, T.S., Woodruff, D.: The data stream space complexity of cascaded norms (submitted, 2009)Google Scholar
  17. [MW]
    Monemizadeh, M., Woodruff, D.: l p-sampling with applications (manuscript)Google Scholar
  18. [Slo]
    Sloane, N.: The on-line encyclopedia of integer sequences!, http://www.research.att.com/~njas/sequences/A048651
  19. [SS02]
    Saks, M., Sun, X.: Space lower bounds for distance approximation in the data stream model. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC), pp. 360–369 (2002)Google Scholar
  20. [Yao79]
    Yao, A.C.-C.: Some complexity questions related to distributive computing. In: Proceedings of the 11th ACM Symposium on Theory of Computing (STOC), pp. 209–213 (1979)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • T. S. Jayram
    • 1
  1. 1.IBM Almaden Research CenterSan JoseUSA

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