Complete Classification of Generalized Santha-Vazirani Sources
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Let F be a finite alphabet and D be a finite set of distributions over F. A Generalized Santha-Vazirani (GSV) source of type (F, D), introduced by Beigi, Etesami and Gohari (ICALP 2015, SICOMP 2017), is a random sequence (F_1, ..., F_n) in F^n, where F_i is a sample from some distribution d ∈D whose choice may depend on F_1, ..., F_i-1. We show that all GSV source types (F, 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/ n or 2^-√(n). We provide essentially randomness-optimal extraction algorithms for extractable sources. Our algorithm for category (2) sources extracts with error ε from n = poly(1/ε) samples in time linear in n. Our algorithm for category (3) sources extracts m bits with error ε from n = O(m + 1/ε) samples in time {O(nm2^m),n^O(F)}. We also give algorithms for classifying a GSV source type (F, D): Membership in category (1) can be decided in NP, while membership in category (3) is polynomial-time decidable.
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