Complete Classification of Generalized Santha-Vazirani Sources
Salman Beigi, Andrej Bogdanov, Omid Etesami, Siyao Guo

TL;DR
This paper provides a complete classification of generalized Santha-Vazirani sources into three categories based on their extractability and error rates, along with optimal algorithms for each category.
Contribution
It introduces a full classification of GSV sources into non-extractable, polynomial-error, and exponential-error categories, and offers optimal extraction algorithms for each.
Findings
All GSV sources fall into three categories: non-extractable, polynomial-error, exponential-error.
Provides algorithms for extracting randomness with optimal error bounds.
Decides source category membership efficiently, with NP and polynomial-time algorithms.
Abstract
Let be a finite alphabet and be a finite set of distributions over . A Generalized Santha-Vazirani (GSV) source of type , introduced by Beigi, Etesami and Gohari (ICALP 2015, SICOMP 2017), is a random sequence in , where is a sample from some distribution whose choice may depend on . We show that all GSV source types fall into one of three categories: (1) non-extractable; (2) extractable with error ; (3) extractable with error . This rules out other error rates like or . We provide essentially randomness-optimal extraction algorithms for extractable sources. Our algorithm for category (2) sources extracts with error from $n =…
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Taxonomy
TopicsQuantum chaos and dynamical systems · Nonlinear Waves and Solitons · Advanced Differential Equations and Dynamical Systems
