2026-08-11T00:06:57Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:9434 2024-03-06T10:44:23Z ddc:004 open_access

Optimal Deterministic Extractors for Generalized Santha-Vazirani Sources Beigi, Salman Bogdanov, Andrej Etesami, Omid Guo, Siyao feasibility of randomness extraction extractor lower bounds martingales 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 in 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^{-Theta(1)}; (3) extractable with error 2^{-Omega(n)}. We provide essentially randomness-optimal extraction algorithms for extractable sources. Our algorithm for category (2) sources extracts one bit with error epsilon from n = poly(1/epsilon) samples in time linear in n. Our algorithm for category (3) sources extracts m bits with error epsilon from n = O(m + log 1/epsilon) samples in time min{O(m2^m * n),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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Salman Beigi and Andrej Bogdanov and Omid Etesami and Siyao Guo 2018 Is Part Of LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.APPROX-RANDOM.2018.30 urn:nbn:de:0030-drops-94349 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.30 eng https://creativecommons.org/licenses/by/3.0/legalcode