Introduction to Communication Complexity

In this essay, we give a basic introduction to the exciting eld of communication complexity. Communication complexity was rst formalized by Yao in 1979 [3]. e context is easy enough to understand: two parties (usually named Alice and Bob) wish to compute the value of some function f of two variables. Alice knows the value of one of the variables, while Bob knows the value of the other. How much must Alice and Bob communicate in order to nd the value of f on their combined input? Since Yao’s seminal work on the subject, the eld of communication complexity has exploded. is growth is driven by the eld’s diverse applications to computational complexity and connections to other branches of discrete math and computer science. Our brief exposition here is inspired by Yao’s original work, but is strongly inžuenced by Kushilevitz and Nisan’s more modern treatise on communication complexity [1]. We focus on a deterministic theory of communication, although we hint at other models in the nal section. is essay is broken into four modules. In the rst, we formalize the notion of a communication protocol and dene the communication complexity of a function. e bulk of the theoretical material resides in the second section, where we give techniques for proving communication lower bounds. In the third section we apply the techniques of the previous section to prove lower bounds for particular functions. In the nal section, we give examples that suggest other models of communication.