One-way communication complexity of symmetric Boolean functions

¹ Institut für Theoretische Informatik, Universität zu Lübeck, Razeburger Allee 160, 23538 Lübeck, Germany; arpe@tcs.uni-luebeck.de;jakoby@tcs.uni-luebeck.de; liskiewi@tcs.uni-luebeck.de
² Supported by DFG research grant Re 672/3.
³ Part of this work was done while visiting International University Bremen, Germany.
⁴ On leave from Instytut Informatyki, Uniwersytet Wrocławski, Wrocław, Poland.

Received: 23 August 2004
Accepted: 5 February 2005

Abstract

We study deterministic one-way communication complexity of functions with Hankel communication matrices. Some structural properties of such matrices are established and applied to the one-way two-party communication complexity of symmetric Boolean functions. It is shown that the number of required communication bits does not depend on the communication direction, provided that neither direction needs maximum complexity. Moreover, in order to obtain an optimal protocol, it is in any case sufficient to consider only the communication direction from the party with the shorter input to the other party. These facts do not hold for arbitrary Boolean functions in general. Next, gaps between one-way and two-way communication complexity for symmetric Boolean functions are discussed. Finally, we give some generalizations to the case of multiple parties.