Complete Classification of Generalized Santha-Vazirani Sources

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/log 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 + log 1/ε) samples in time min{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.