Composition Theorems in Communication Complexity

A well-studied class of functions in communication complexity are composed functions of the form (f ◦ g)(x, y) = f(g(x, y), . . . , g(x, y)). This is a rich family of functions which encompasses many of the important examples in the literature. It is thus of great interest to understand what properties of f and g affect the communication complexity of (f ◦ g), and in what way. Recently, Sherstov [She09] and independently Shi-Zhu [SZ09b] developed conditions on the inner function g which imply that the quantum communication complexity of f ◦g is at least the approximate polynomial degree of f . We generalize both of these frameworks. We show that the pattern matrix framework of Sherstov works whenever the inner function g is strongly balanced—we say that g : X×Y → {−1,+1} is strongly balanced if all rows and columns in the matrix Mg = [g(x, y)]x,y sum to zero. This result strictly generalizes the pattern matrix framework of Sherstov [She09], which has been a very useful idea in a variety of settings [She08b,RS08,Cha07,LS09a,CA08,BHN09]. Shi-Zhu require that the inner function g has small spectral discrepancy, a somewhat awkward condition to verify. We relax this to the usual notion of discrepancy. We also enhance the framework of composed functions studied so far by considering functions F (x, y) = f(g(x, y)), where the range of g is a group G. When G is Abelian, the analogue of the strongly balanced condition becomes a simple group invariance property of g. We are able to formulate a general lower bound on F whenever g satisfies this property.