Communication Complexity under Product and Nonproduct Distributions

We solve an open problem in communication complexity posed by Kushilevitz and Nisan (1997). Let R
_epsiv
(f) and D
^mu

_epsiv
(f) denote the randomized and mu-distributional communication complexities off, respectively (e a small constant). Yao's well-known minimax principle states that R
_epsiv
(f) = max
_mu
{D
^mu

_epsiv
(f)}. Kushilevitz and Nisan (1997) ask whether this equality is approximately preserved if the maximization is taken over product distributions only, rather than all distributions mu. We give a strong negative answer to this question. Specifically, we prove the existence of a function f : {0,1}
ⁿ
X {0,1}
ⁿ
rarr {0, 1}for which R
_epsiv
(f) = Omega(n) but max
_muproduct
{D
^mu

_epsiv
(f)} = 0(1).