A Discrepancy Lower Bound for Information Complexity
A Discrepancy Lower Bound for Information Complexity
复制标题
信息复杂性的差异下界
DOI：
10.1007/s00453-015-0093-8
复制
发表时间：
2011
期刊：
影响因子：
1.1
通讯作者：
Omri Weinstein
中科院分区：
文献类型：
--
作者：
M. Braverman;Omri Weinstein
本文提供了第一个通用技术，用于在两党无界的沟通问题上提供较低的界限。相对于分布μ\ documentClass [12pt] {minimal} \ usepackage {amsmath} \ usepackage {wasySym} \ usepackage {amsfonts} \ usepackage {amssymb} \ usepackage {amssymb} \ usepackage {amsbsy} \ usepackage {Mathrsfs} \ usepackage { } \ usepackage {amsmath} \ usepackage {wasysym} \ usepackage {amsfonts} \ usepackage {amssymb} \ usepackage {amsbsy {amsbsy} pt} \ begin {document } $$ disc_ \ mu f $$ \ end {document {document}，那么任何两个方随机协议计算f必须至少显示ω（log（1/discμf））\ documentClass [12pt] {minimal} \ usepackage {amsmath} \ usepackage {wasysym} \ wasysym} \ usepackage {amsfonts} \ usepackage {amssymb} \ usepackage {amsbsy} \ usepackage {Mathrsfs} \ usepackage {upgreek} \ setLength mu f））$$ \ end {document}向参与者的信息。作为船体，我们获得了在{0,1} n×{0,1} n \ documentClass [12pt] {minimal} \ usepackage {amsmath} \ usepackage {wasySym} \ wasySym} \ usepackage {amsfonts} \ usepackage {amssymb} \ usepackage {amsbsy} \ usepackage {mathrsfs} \ usepackage {upgreek} \ upgreek} \ setLength \ times \ {0,1 \}^n $$ \ end {document}必须显示ω（n）\ documentClass [12pt] {minimal} \ usepackage {amsmath} \ usepackage {wasysym} \ useym} \ usepackage {amsfonts} \ usepackage {amssymb}长度{ \ orddsidemargin} { - 69pt} \ begin {document} $$ \ omega（n）$$ \ end \ end {document {document} in“ document} novery docement {document}向参与者提供的信息。 1/n）\ documentClass [12pt] {minimal} \ usepackage {amsmath} \ usepackage {asysym} \ usepackage {amsfonts} \ usepackage {amssymb} \ usepackage {amsbsy} } \ begin {document} $$ \ omega（1/\ sqrt {n}）$$ \ end {document {document}，它为最近的中提琴证明提供了替代证明（第24届年度ACM-SIAM-SIAM在离散算法上的会议论文集，Soda 2013，Soda 2013 ， 新的美国路易斯安那州奥尔良，2013年1月6日至8日，第632-651、2013卷，2013年），ω（logn）\ documentClass [12pt] {minimal} \ usepackage {amsmath} \ usepackage {wasysym} {amssymb} \ usepackage {amsbsy} \ usepackage {mathrsfs} \ usepackage {upgreek} \ setLength {\ oddSidemargin} { - 69pt} { - 69pt}绑定了这个经过充分研究的功能的沟通复杂性并且，结合我们的主要结果，证明了紧密的ω（logn）\ documentClass [12pt] {minimal} \ usepackage {amsmath} \ usepackage {wasysym} \ usepackage {wasysym} \ usepackage {amsfonts} {Mathrsfs} \ use-package {upgreek} \ setLength {\ oddSidemargin} { - 69pt} \ begin {document} $$ \ omega（\ log o n）$ n）$ n）$ n）$ \ end \ end document {document {document {document}的信息复杂性。主要结果开发了一个新的模拟过程独立的兴趣。我们的仿真程序是一个基本障碍，证明了几乎所有已知的沟通复杂性（而不仅仅是差异）也适用于信息复杂性。
This paper provides the first general technique for proving information lower bounds on two-party unbounded-rounds communication problems. We show that the discrepancy lower bound, which applies to randomized communication complexity, also applies to information complexity. More precisely, if the discrepancy of a two-party function f with respect to a distribution μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} is Discμf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Disc_\mu f$$\end{document}, then any two party randomized protocol computing f must reveal at least Ω(log(1/Discμf))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega (\log (1/Disc_\mu f))$$\end{document} bits of information to the participants. As a corollary, we obtain that any two-party protocol for computing a random function on {0,1}n×{0,1}n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{0,1\}^n\times \{0,1\}^n$$\end{document} must reveal Ω(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega (n)$$\end{document} bits of information to the participants. In addition, we prove that the discrepancy of the Greater-Than function is Ω(1/n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega (1/\sqrt{n})$$\end{document}, which provides an alternative proof to the recent proof of Viola (Proceedings of the twenty-fourth annual ACM-SIAM symposium on discrete algorithms, SODA 2013, New Orleans, LA, USA, 6–8 Jan 2013, pp 632–651, 2013) of the Ω(logn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega (\log n)$$\end{document} lower bound on the communication complexity of this well-studied function and, combined with our main result, proves the tight Ω(logn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega (\log n)$$\end{document} 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 followup breakthrough work of Kerenidis et al. (53rd annual IEEE symposium on foundations of computer science, FOCS 2012, New Brunswick, NJ, USA, 20–23 Oct 2012, pp 500–509, 2012), our simulation procedure served as a building block towards a proof that almost all known lower bound techniques for communication complexity (and not just discrepancy) apply to information complexity as well.