广义Santha-Vazirani源的完全分类
Complete Classification of Generalized Santha-Vazirani Sources
摘要：
设$\mathcal{F}$是一个有限字母表，$\mathcal{D}$是$\mathcal{F}$上的一个有限分布集.由Beigi、Etesami和Gohari（ICALP 2015，SICOMP 2017）引入的$(\mathcal{F}, \mathcal{D})$类型的广义Santha-Vazirani（GSV）源是$\mathcal{F}^n$中的随机序列$(F_1, \dots, F_n)$，其中$F_i$是来自某些分布$d \in \mathcal{D}$的样本，其选择可能取决于$F_1, \dots, F_{i-1}$。我们表明，所有的GSV源类型$(\mathcal{F}, \mathcal{D})$分为三类之一：（1）不可提取;（2）可提取错误$n^{-Θ(1)}$ ;（3）可提取错误$2^{-Ω(n)}$。这排除了其他错误率，如$1/\log n$或$2^{-\sqrt{n}}$。我们提供了基本上随机最佳提取算法的可提取源。我们的算法为类别（2）源提取与错误$\varepsilon$从$n = \mathrm{poly}(1/\varepsilon)$样本在时间上线性$n$。我们的用于类别（3）源的算法在时间$\min\{O(nm2^m),n^{O(\lvert\mathcal{F}\rvert)}\}$上从$n = O(m + \log 1/\varepsilon)$样本中提取具有错误$\varepsilon$的$m$比特。我们还给出了GSV源类型$(\mathcal{F}, \mathcal{D})$的分类算法：在$\mathrm{NP}$中可以判定类别（1）中的成员，而在类别（3）中的成员是多项式时间可判定的。
英文摘要：
Let $\mathcal{F}$ be a finite alphabet and $\mathcal{D}$ be a finite set of distributions over $\mathcal{F}$. A Generalized Santha-Vazirani (GSV) source of type $(\mathcal{F}, \mathcal{D})$, introduced by Beigi, Etesami and Gohari (ICALP 2015, SICOMP 2017), is a random sequence $(F_1, \dots, F_n)$ in $\mathcal{F}^n$, where $F_i$ is a sample from some distribution $d \in \mathcal{D}$ whose choice may depend on $F_1, \dots, F_{i-1}$. We show that all GSV source types $(\mathcal{F}, \mathcal{D})$ fall into one of three categories: (1) non-extractable; (2) extractable with error $n^{-Θ(1)}$; (3) extractable with error $2^{-Ω(n)}$. This rules out other error rates like $1/\log n$ or $2^{-\sqrt{n}}$. We provide essentially randomness-optimal extraction algorithms for extractable sources. Our algorithm for category (2) sources extracts with error $\varepsilon$ from $n = \mathrm{poly}(1/\varepsilon)$ samples in time linear in $n$. Our algorithm for category (3) sources extracts $m$ bits with error $\varepsilon$ from $n = O(m + \log 1/\varepsilon)$ samples in time $\min\{O(nm2^m),n^{O(\lvert\mathcal{F}\rvert)}\}$. We also give algorithms for classifying a GSV source type $(\mathcal{F}, \mathcal{D})$: Membership in category (1) can be decided in $\mathrm{NP}$, while membership in category (3) is polynomial-time decidable.
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