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Composition Theorems in Communication Complexity

  • Troy Lee
  • Shengyu Zhang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6198)

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.

Keywords

Boolean Function Communication Complexity Orthogonality Condition Irreducible Character Query Complexity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Troy Lee
    • 1
  • Shengyu Zhang
    • 2
  1. 1.Rutgers University 
  2. 2.The Chinese University of Hong Kong 

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