Abstract
We show a new connection between the information complexity of one-way communication problems under product distributions and a relaxed notion of list-decodable codes. As a consequence, we obtain a characterization of the information complexity of one-way problems under product distributions for any error rate based on covering numbers. This generalizes the characterization via VC dimension for constant error rates given by Kremer, Nisan, and Ron (CCC, 1999). It also provides an exponential improvement in the error rate, yielding tight bounds for a number of problems. In addition, our framework gives a new technique for analyzing the complexity of composition (e.g., XOR and OR) of one-way communication problems, connecting the difficulty of these problems to the noise sensitivity of the composing function. Using this connection, we strengthen the lower bounds obtained by Molinaro, Woodruff and Yaroslavtsev (SODA, 2013) for several problems in the distributed and streaming models, obtaining optimal lower bounds for finding the approximate closest pair of a set of points and the approximate largest entry in a matrix product. Finally, to illustrate the utility and simplicity of our framework, we show how it unifies proofs of existing \(1\)-way lower bounds for sparse set disjointness, the indexing problem, the greater than function under product distributions, and the gap-Hamming problem under the uniform distribution.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ablayev, F.M.: Lower bounds for one-way probabilistic communication complexity and their application to space complexity. Theor. Comput. Sci. 157(2), 139–159 (1996)
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)
Beals, R., Buhrman, H., Cleve, R., Mosca, M., de Wolf, R.: Quantum lower bounds by polynomials. J. ACM 48(4), 778–797 (2001)
Buhrman, H., GarcÃa-Soriano, D., Matsliah, A., de Wolf, R.: The non-adaptive query complexity of testing k-parities. Chicago J. Theor. Comput. Sci. (2013)
Cover, T.M., Thomas, J.A.: Elements of information theory (2. ed.). Wiley (2006)
Dasgupta, Anirban, Kumar, Ravi, Sivakumar, D.: Sparse and lopsided set disjointness via information theory. In: Gupta, Anupam, Jansen, Klaus, Rolim, José, Servedio, Rocco (eds.) APPROX 2012 and RANDOM 2012. LNCS, vol. 7408, pp. 517–528. Springer, Heidelberg (2012)
Dudley, R.M.: Central limit theorems for empirical measures. The Annals of Probability 6(6), 899–929 (1978)
Haussler, D.: Decision theoretic generalizations of the PAC model for neural net and other learning applications. Inform. Comput. 100(1), 78–150 (1992)
Jayram, T.S., Woodruff, D.P.: Optimal bounds for johnson-lindenstrauss transforms and streaming problems with sub-constant error. In: SODA (2011)
Kremer, I., Nisan, N., Ron, D.: On randomized one-round communication complexity. Computational Complexity, pp. 21–49 (1999)
Lee, T., Shraibman, A.: Lower bounds in communication complexity. Foundations and Trends in Theoretical Computer Science 3(4), 263–399 (2009)
Lee, T., Zhang, S.: Composition theorem in communication complexity. In: ICALP (2010)
Molinaro, M., Woodruff, D.P., Yaroslavtsev, G.: Beating the direct sum theorem in communication complexity with implications for sketching. In: SODA (2013)
Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, New York (1995)
Muthukrishnan, S.: Data streams: algorithms and applications. Found. Trends Theor. Comput. Sci. 1(2), 117–236 (2005)
Newman, I., Szegedy, M.: Public vs. private coin flips in one round communication games (extended abstract). In: STOC (1996)
Nisan, N., Szegedy, M.: On the degree of boolean functions as real polynomials. Computational Complexity 4, 301–313 (1994)
O’Donnell, R.: Analysis of Boolean Functions. Cambridge University Press (2014)
Papadimitriou, C.H., Sipser, M.: Communication complexity. J. Comput. Syst. Sci. 28(2), 260–269 (1984)
Saglam, M., Tardos, G.: On the communication complexity of sparse set disjointness and exists-equal problems. In: FOCS (2013)
Sherstov, A.: The pattern matrix method. SIAM J. Comput. 40(6), 1969–2000 (2011)
Woodruff, D.P.: The average-case complexity of counting distinct elements. In: ICDT (2009)
Yao, A.C.: Lower bounds by probabilistic arguments (extended abstract). In: FOCS (1983)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Molinaro, M., Woodruff, D.P., Yaroslavtsev, G. (2015). Amplification of One-Way Information Complexity via Codes and Noise Sensitivity. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47672-7_78
Download citation
DOI: https://doi.org/10.1007/978-3-662-47672-7_78
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-47671-0
Online ISBN: 978-3-662-47672-7
eBook Packages: Computer ScienceComputer Science (R0)