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).
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Manuscript received August 24, 2008
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Sherstov, A.A. Communication Complexity Under Product and Nonproduct Distributions. comput. complex. 19, 135–150 (2010). https://doi.org/10.1007/s00037-009-0285-1
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DOI: https://doi.org/10.1007/s00037-009-0285-1
Keywords.
- Randomized and distributional communication complexity
- product and nonproduct distributions
- Yao’s minimax principle