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 (F1, . . . , Fn) in Fn, where Fi is a sample from some distribution d ∈ D whose choice may depend on F1, . . . , Fi−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/ logn or 2− √ . 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(nm2m), nO(|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 polynomialtime decidable.