12 . Introduction to communication complexity

The model [Yao79] consists of two parties, Alice and Bob, each holding inputs x ∈ X and y ∈ Y respectively. They wish to compute a function f : X × Y → Z. Each of them have unbounded computational power, unlike as in other models of computation. However their goal is to compute the value of f(x, y) with the least amount of interaction with each other. The measure of communication between the parties will usually be the number of bits exchanged between them. Typically the inputs are boolean, namely X = {0, 1}n and Y = {0, 1}m, and the output is generally 1 bit, namely Z = {0, 1}. We assume the communication will be carried out according to some fixed protocol π, which Alice and Bob have mutually decided beforehand. At each stage, the protocol must decide whether the run terminates. If it does, then it must specify the answer f(x, y), else it must specify which player communicates next. Moreover, this information must be decided by only the bits communicated between them so far in this run of the protocol. Additionally, if it is Alice’s (Bob’s) turn, the protocol must decide what they send depending on the communication so far and Alice’s (Bob’s) input. The following figure describes a protocol.