7. Distributional Communication Complexity

In this section, we introduce a new communication cost called the distributional communication complexity, which will become useful while obtaining lower bounds on randomized communication complexity. Here we consider a probability distribution μ over the set of inputs X × Y. We are interested in the best (least communication) deterministic protocol P that has an error at most ε when the input is drawn randomly from the distribution μ. In other words, the deterministic protocol P errs on at most ε fraction of the inputs, when the inputs are weighted according to the distribution μ. I.e.,