Composition Theorems in Communication Complexity

Abstract

A well-studied class of functions in communication complexity are composed functions of the form (f ∘ g ⁿ) (x,y) = f(g(x ¹, y ¹), ..., g(x ⁿ,y ⁿ)). 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 ⁿ), 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 ⁿ 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 M _g  = [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.