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computational complexity

, Volume 19, Issue 1, pp 135–150 | Cite as

Communication Complexity Under Product and Nonproduct Distributions

  • Alexander A. Sherstov
Article

Abstract.

We solve an open problem in communication complexity posed by Kushilevitz and Nisan (1997). Let R(f) and \(D^\mu_\in (f)\) denote the randomized and μ-distributional communication complexities of f, respectively (∈ a small constant). Yao’s well-known minimax principle states that \(R_{\in}(f) = max_\mu \{D^\mu_\in(f)\}\). Kushilevitz and Nisan (1997) ask whether this equality is approximately preserved if the maximum is taken over product distributions only, rather than all distributions μ. We give a strong negative answer to this question. Specifically, we prove the existence of a function \(f : \{0, 1\}^n \times \{0, 1\}^n \rightarrow \{0, 1\}\) for which maxμ product \(\{D^\mu_\in (f)\} = \Theta(1) \,{\textrm but}\, R_{\in} (f) = \Theta(n)\). We also obtain an exponential separation between the statistical query dimension and signrank, solving a problem previously posed by the author (2007).

Keywords.

Randomized and distributional communication complexity product and nonproduct distributions Yao’s minimax principle 

Subject classification.

03D15 68Q17 

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Copyright information

© Birkhäuser / Springer Basel 2010

Authors and Affiliations

  1. 1.Department of Computer SciencesThe University of Texas at AustinAustinUSA

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