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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2018.30
Optimal Deterministic Extractors for Generalized Santha-Vazirani Sources

Authors: Salman Beigi, Andrej Bogdanov, Omid Etesami, and Siyao Guo

Published in: LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

Abstract
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.
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Salman Beigi, Andrej Bogdanov, Omid Etesami, and Siyao Guo. Optimal Deterministic Extractors for Generalized Santha-Vazirani Sources. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{beigi_et_al:LIPIcs.APPROX-RANDOM.2018.30,
  author =	{Beigi, Salman and Bogdanov, Andrej and Etesami, Omid and Guo, Siyao},
  title =	{{Optimal Deterministic Extractors for Generalized Santha-Vazirani Sources}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{30:1--30:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-085-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{116},
  editor =	{Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.30},
  URN =		{urn:nbn:de:0030-drops-94349},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.30},
  annote =	{Keywords: feasibility of randomness extraction, extractor lower bounds, martingales}
}
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