Pseudorandom generators for combinatorial shapes
Pseudorandom generators for combinatorial shapes
复制标题
组合形状的伪随机生成器
DOI：
10.1145/1993636.1993671
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
David Zuckerman
中科院分区：
文献类型：
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作者：
Parikshit Gopalan;Raghu Meka;Omer Reingold;David Zuckerman
我们为组合形状构造了伪和生成器，该组合形状</i> </i> sationary概括了组合矩形，ε偏置空间，0/1个半空间和0/1模块化/sup> - > {0,1}是一个（m，n） - combinatorial形状，如果存在设置a <sub> 1 </sub>，...，a <sub> n </sub>⊆[m ]和对称函数h：{0,1} <sup> n </sup>> - > {0,1}，这样f（x <sub> 1 </sub>，...，x <sub> n </sub>）= h（1 <sub> a <sub> 1 </sub> </sub>（x <sub > 1 </sub>），...，1 <sub> a <sub> n </sub> </sub>（x <sub> n </sub>））。 M + log N + log <Sup> 2 </sup>（1/ε）获取错误ε。 ，这意味着任何子集的重量的分布是ε关闭统计距离的适当二项式分布。经典中心极限定理的强大变体显示统计距离的收敛性，而不是通常的kolmogorov距离。
We construct pseudorandom generators for <i>combinatorial shapes</i>, which substantially generalize combinatorial rectangles, ε-biased spaces, 0/1 halfspaces, and 0/1 modular sums. A function f:[m]<sup>n</sup> -> {0,1} is an (m,n)-combinatorial shape if there exist sets A<sub>1</sub>,...,A<sub>n</sub> ⊆ [m] and a symmetric function h:{0,1}<sup>n</sup> -> {0,1} such that f(x<sub>1</sub>,...,x<sub>n</sub>) = h(1<sub>A<sub>1</sub></sub>(x<sub>1</sub>),...,1<sub>A<sub>n</sub></sub>(x<sub>n</sub>)). Our generator uses seed length O(log m + log n + log<sup>2</sup>(1/ε)) to get error ε. When m = 2, this gives the first generator of seed length O(log n) which fools all weight-based tests, meaning that the distribution of the weight of any subset is ε-close to the appropriate binomial distribution in statistical distance. For our proof we give a simple lemma which allows us to convert closeness in Kolmogorov (cdf) distance to closeness in statistical distance. As a corollary of our technique, we give an alternative proof of a powerful variant of the classical central limit theorem showing convergence in statistical distance, instead of the usual Kolmogorov distance.