The Pattern Matrix Method

In this paper we give a self-contained proof of Sherstov’s pattern matrix method. This theorem combines two main ideas—an extension of the discrepancy method known as the generalized discrepancy method and the notion of dual polynomials—with the techniques of matrix norms, Fourier analysis, and approximate degree. All together, this results in lower bounds on the randomized communication complexity of a certain class of communication problems, known as pattern matrices. These results extend to lower bounds for quantum communication complexity with almost no addition work, though we do not prove these results here. Sherstov’s pattern matrix method is interesting not only because of the elegance of its proof, but because the class of pattern matrices appears as subproblems of several natural problems, allowing us to give lower bounds on these communication problems as well. We give a short proof of Razborov’s lower bound on the communication complexity of symmetric functions using the pattern matrix method.