逼近单调布尔函数对O（\sqrt{n}）查询复杂度的影响
Approximating the Influence of a monotone Boolean function in O(\sqrt{n}) query complexity
摘要：
离散函数的{\em总影响}（{\em平均灵敏度）是其基本度量之一。本文研究了单调布尔函数\ifnum \plusminus = 1 $f: \{\pm1\}^n \longrightarrow \{\pm1\}$，\else $f: \bitset^n \to \bitset$，\fi（记为$I[f]$）的全影响的逼近问题.我们提出了一个随机算法，近似的影响，这样的功能，在一个乘法因子的$(1\pm \eps)$通过执行$O(\frac{\sqrt{n}\log n}{I[f]} \poly(1/\eps)) $查询。% \mnote{D: say something about technique?}我们还证明了一个下界%$Ω(\frac{\sqrt{n/\log n}}{I[f]})$ $Ω(\frac{\sqrt{n}}{\log n \cdot I[f]})$的查询复杂性的任何常数因子近似算法为这个问题（适用于$I[f] = Ω(1)$），%和$I[f] = O(\sqrt{n}/\log n)$），因此表明，我们的算法几乎是最佳的$n$的依赖。对于一般的功能，我们给了一个下界的$Ω(\frac{n}{I[f]})$，它匹配的复杂性，一个简单的采样算法。
英文摘要：
The {\em Total Influence} ({\em Average Sensitivity) of a discrete function is one of its fundamental measures. We study the problem of approximating the total influence of a monotone Boolean function \ifnum\plusminus=1 $f: \{\pm1\}^n \longrightarrow \{\pm1\}$, \else $f: \bitset^n \to \bitset$, \fi which we denote by $I[f]$. We present a randomized algorithm that approximates the influence of such functions to within a multiplicative factor of $(1\pm \eps)$ by performing $O(\frac{\sqrt{n}\log n}{I[f]} \poly(1/\eps)) $ queries. % \mnote{D: say something about technique?} We also prove a lower bound of % $Ω(\frac{\sqrt{n/\log n}}{I[f]})$ $Ω(\frac{\sqrt{n}}{\log n \cdot I[f]})$ on the query complexity of any constant-factor approximation algorithm for this problem (which holds for $I[f] = Ω(1)$), % and $I[f] = O(\sqrt{n}/\log n)$), hence showing that our algorithm is almost optimal in terms of its dependence on $n$. For general functions we give a lower bound of $Ω(\frac{n}{I[f]})$, which matches the complexity of a simple sampling algorithm.
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