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Harmonic analysis, real approximation, and the communication complexity of Boolean functions

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Grolmusz,  Vince
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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MPI-I-93-161.pdf
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Citation

Grolmusz, V.(1993). Harmonic analysis, real approximation, and the communication complexity of Boolean functions (MPI-I-93-161). Saarbrücken: Max-Planck-Institut für Informatik.


Cite as: http://hdl.handle.net/11858/00-001M-0000-0014-B770-0
Abstract
In this paper we prove several fundamental theorems, concerning the multi--party communication complexity of Boolean functions. Let $g$ be a real function which approximates Boolean function $f$ of $n$ variables with error less than $1/5$. Then --- from our Theorem 1 --- there exists a $k=O(\log (n\L_1(g)))$--party protocol which computes $f$ with a communication of $O(\log^3(n\L_1(g)))$ bits, where $\L_1(g)$ denotes the $\L_1$ spectral norm of $g$. We show an upper bound to the symmetric $k$--party communication complexity of Boolean functions in terms of their $\L_1$ norms in our Theorem 3. For $k=2$ it was known that the communication complexity of Boolean functions are closely related with the {\it rank} of their communication matrix [Ya1]. No analogous upper bound was known for the k--party communication complexity of {\it arbitrary} Boolean functions, where $k>2$.