Communication Complexity Under Product and Nonproduct Distributions

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 max_{μ 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).