Rectangle size bounds and threshold covers in communication complexity

H. Klauck
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引用次数: 78

Abstract

We investigate the power of the most important lower bound technique in randomized communication complexity, which is based on an evaluation of the maximal size of approximately monochromatic rectangles, with respect to arbitrary distributions on the inputs. While it is known that the 0-error version of this bound is polynomially tight for deterministic communication, nothing in this direction is known for constant error and randomized communication complexity. We first study a one-sided version of this bound and obtain that its value lies between the MA- and AM- complexities of the considered function. Hence the lower bound actually works for a (communication) complexity class between MA/spl cap/co - MA and AM/spl cap/co - AM, and allows to show that the MA-complexity of the disjointness problem is /spl Omega/(/spl radic/n). Following this we consider the conjecture that the lower bound method is polynomially tight for randomized communication complexity. First we disprove a distributional version of this conjecture. Then we give a combinatorial characterization of the value of the lower bound method, in which the optimization over all distributions is absent. This characterization is done by what we call a bounded error uniform threshold cover, and reduces showing tightness of the bound to the construction of an efficient protocol for a specific communication problem. We then study relaxations of bounded error uniform threshold covers, namely approximate majority covers and majority covers, and exhibit exponential separations between them. Each of these covers captures a lower bound method previously used for randomized communication complexity.
通信复杂度中的矩形大小边界和阈值覆盖
我们研究了随机通信复杂度中最重要的下界技术的力量,该技术基于对输入上任意分布的近似单色矩形的最大大小的评估。虽然已知该边界的0错误版本对于确定性通信是多项式紧的,但对于恒定错误和随机通信复杂性,这个方向上没有任何已知。我们首先研究了这个界的单侧版本,并得到它的值介于所考虑的函数的MA-和AM-复杂度之间。因此,下界实际上适用于MA/spl cap/co - MA和AM/spl cap/co - AM之间的(通信)复杂性类,并允许显示不连接性问题的MA-复杂性为/spl ω /(/spl根号/n)。在此基础上,我们提出了对于随机通信复杂度,下界方法是多项式紧的猜想。首先,我们反驳这个猜想的一个分布版本。然后,我们给出了下界方法值的组合表征,其中不存在对所有分布的优化。这种表征是通过我们所说的有界误差均匀阈值覆盖来完成的,并减少了对特定通信问题的有效协议构建的绑定的紧密性。然后,我们研究了有界误差均匀阈值覆盖的松弛性,即近似多数覆盖和多数覆盖,并展示了它们之间的指数分离。这些覆盖中的每一个都捕获了以前用于随机通信复杂性的下界方法。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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