Mathematical Methods in Computer Science Lecture 6 : Communication Complexity Lecturer : Boaz Barak

Communication complexity was introduced by Yao in 1979. In this model, there are two parties, Alice and Bob, who have access to strings x and y, respectively, where x, y ∈ {0, 1}. The aim is for them to compute a function f : {0, 1} → {0, 1} with as little communication between them as possible. The minimum k for which there exists a protocol that uses at most k bits of communication to compute f(x, y) for any x and y is called the communication complexity of f . We denote this quantity by CC(f). For a more formal definition we refer the reader to [2]. Trivially, for any f , CC(f) is at most n because one of the parties can send its entire string to the other, who can then compute f . The main problem in communication complexity is to give better upper and lower bounds for CC(f), for specific functions f . Communication complexity has been successful in showing lower bounds in several computational settings, such as data structures and streaming algorithms. In the following section, we will see some simple examples of lower bound proofs and an application that proves that simulating a two-tape Turing machine on a one-tape machine incurs a quadratic slowdown. Section 3 describes other techniques for proving upper and lower bounds on communication complexity. In Section 4 we extend some of these ideas to multi-party communication. We conclude in Section 5.