Extractors and lower bounds for locally samplable sources

  • Authors:

  • Anindya De;Thomas Watson

  • Affiliations:

  • Computer Science Division, University of California, Berkeley, CA;Computer Science Division, University of California, Berkeley, CA

  • Venue:
  • APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
  • Year:
  • 2011

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Abstract

We consider the problem of extracting randomness from sources that are efficiently samplable, in the sense that each output bit of the sampler only depends on some small number d of the random input bits. As our main result, we construct a deterministic extractor that, given any d-local source with min-entropy k on n bits, extracts Ω(k2/nd) bits that are 2-nΩ(1)-close to uniform, provided d ≤ o(log n) and k ≥ n2/3+γ (for arbitrarily small constants γ 0). Using our result, we also improve a result of Viola (FOCS 2010), who proved a 1/2-O(1/ log n) statistical distance lower bound for o(log n)-local samplers trying to sample inputoutput pairs of an explicit boolean function, assuming the samplers use at most n+n1-δ random bits for some constant δ 0. Using a different function, we simultaneously improve the lower bound to 1/2 - 2-nΩ(1) and eliminate the restriction on the number of random bits.