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arXiv · 1112.2000

A discrepancy lower bound for information complexity

Mark Braverman·Omri Weinstein
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Abstract

This paper provides the first general technique for proving information lower bounds on two-party unbounded-rounds communication problems. We show that the discrepancy lower bound, which applies to randomized communication complexity, also applies to information complexity. More precisely, if the discrepancy of a two-party function $f$ with respect to a distribution $μ$ is $Disc_μf$, then any two party randomized protocol computing $f$ must reveal at least $Ω(\log (1/Disc_μf))$ bits of information to the participants. As a corollary, we obtain that any two-party protocol for computing a random function on $\{0,1\}^n\times\{0,1\}^n$ must reveal $Ω(n)$ bits of information to the participants. In addition, we prove that the discrepancy of the Greater-Than function is $Ω(1/\sqrt{n})$, which provides an alternative proof to the recent proof of Viola \cite{Viola11} of the $Ω(\log n)$ lower bound on the communication complexity of this well-studied function and, combined with our main result, proves the tight $Ω(\log n)$ lower bound on its information complexity. The proof of our main result develops a new simulation procedure that may be of an independent interest. In a very recent breakthrough work of Kerenidis et al. \cite{kerenidis2012lower}, this simulation procedure was the main building block for proving that almost all known lower bound techniques for communication complexity (and not just discrepancy) apply to information complexity.

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BibTeXRIS

Mark Braverman, Omri Weinstein. 2012-06-12. A discrepancy lower bound for information complexity. https://arxiv.org/abs/1112.2000

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