The general communication problem may be described in the following terms: A system must perform some task that depends on information distributed among the different parts of the system (called processors, parties, or players). The players thus need to communicate with each other in order to perform the task. Yao's model of communication complexity, which is the subject of this chapter, is the simplest scenario in which such a situation occurs. Yao's model makes the following simplifying assumptions:

• There are only two parts in the system.

• Each part of the system gets a fixed part of the input information.

• The only resource we care about is communication.

• The task is the computation of some prespecified function of the input.

These assumptions help us concentrate on the core issue of communication. Despite its apparent simplicity, this is a very rich model that exhibits a nice structure and in which issues such as randomization and nondeterminism, among others, can be studied. We can also translate our understanding of this model to many other scenarios in which communication is a key issue.

The Model

Let X, Y, Z be arbitrary finite sets and let f: X × Y → Z be an arbitrary function. There are two players, Alice and Bob, who wish to evaluate f(x, y), for some inputs x ∈ X and y ∈ Y. The difficulty is that Alice only knows x and Bob only knows y. Thus, to evaluate the function, they will need to communicate with each other.