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 .
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Ablayev, F.: Lower bounds for one-way probabilistic communication complexity and their application to space complexity. Theoretical Computer Science 157(2), 139–159 (1996)
Andoni, A., Jayram, T.S., Patrascu, M.: Non-embeddability and sketching complexity via information geometry (2009)
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)
Bar-Yehuda, R., Chor, B., Kushilevitz, E., Orlitsky, A.: Privacy, additional information, and communication. IEEE Transactions on Information Theory 39(6), 1930–1943 (1993)
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)
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)
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)
Charikar, M., Chen, K., Farach-Colton, M.: Finding frequent items in data streams. Theor. Comput. Sci. 312(1), 3–15 (2004)
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)
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)
Deza, M., Laurent, M.: Geometry of Cuts and Metrics. Springer, Heidelberg (1997)
Flajolet, P., Martin, G.N.: Probabilistic counting algorithms for data base applications. J. Comput. Syst. Sci. 31(2), 182–209 (1985)
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)
Indyk, P.: Stable distributions, pseudorandom generators, embeddings, and data stream computation. J. ACM 53(3), 307–323 (2006)
Indyk, P., Woodruff, D.P.: Optimal approximations of the frequency moments of data streams. In: STOC, pp. 202–208 (2005)
Jayram, T.S., Woodruff, D.: The data stream space complexity of cascaded norms (submitted, 2009)
Monemizadeh, M., Woodruff, D.: l p -sampling with applications (manuscript)
Sloane, N.: The on-line encyclopedia of integer sequences!, http://www.research.att.com/~njas/sequences/A048651
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)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Jayram, T.S. (2009). Hellinger Strikes Back: A Note on the Multi-party Information Complexity of AND. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2009 2009. Lecture Notes in Computer Science, vol 5687. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03685-9_42
Download citation
DOI: https://doi.org/10.1007/978-3-642-03685-9_42
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03684-2
Online ISBN: 978-3-642-03685-9
eBook Packages: Computer ScienceComputer Science (R0)