Fooling Pairs in Randomized Communication Complexity

The fooling pairs method is one of the standard methods for proving lower bounds for deterministic two-player communication complexity. We study fooling pairs in the context of randomized communication complexity. We show that every fooling pair induces far away distributions on transcripts of private-coin protocols. We use the above to conclude that the private-coin randomized ε-error communication complexity of a function f with a fooling set S is at least order log log |S| ε . This relationship was earlier known to hold only for constant values of ε.The bound we prove is tight, for example, for the equality and greater-than functions. As an application, we exhibit the following dichotomy: for every boolean function f and integer n, the (1/3)-error public-coin randomized communication complexity of the function ∨n i=1 f(xi, yi) is either at most c or at least n/c, where c > 0 is a universal constant.