1 . Introduction to communication complexity

The model consists of two parties [Yao79], Alice and Bob, holding inputs x ∈ X and y ∈ Y respectively. They exchange messages in order to compute a function f : X×Y → {0, 1} at (x, y); their goal is to do this with minimal amount of interaction, which is some measure of the communication between the two parties and will usually be the total number of bits exchanged by the parties. Figure 1 illustrates this model. The communication between the two parties is guided by a protocol, which specifies how each message sent depends on the input and the messages received previously. In the definition below, we assume that Alice (who has x) sends the first message.