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DOI: 10.4230/LIPIcs.STACS.2013.634
URN: urn:nbn:de:0030-drops-39717
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Watson, Thomas

Advice Lower Bounds for the Dense Model Theorem

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We prove a lower bound on the amount of nonuniform advice needed by black-box reductions for the Dense Model Theorem of Green, Tao, and Ziegler, and of Reingold, Trevisan, Tulsiani, and Vadhan. The latter theorem roughly says that for every distribution D that is delta-dense in a distribution that is epsilon'-indistinguishable from uniform, there exists a "dense model" for D, that is, a distribution that is delta-dense in the uniform distribution and is epsilon-indistinguishable from D. This epsilon-indistinguishability is with respect to an arbitrary small class of functions F. For the natural case where epsilon' >= Omega(epsilon delta) and epsilon >= delta^{O(1)}, our lower bound implies that Omega(sqrt{(1/epsilon)log(1/delta)} log|F|) advice bits are necessary. There is only a polynomial gap between our lower bound and the best upper bound for this case (due to Zhang), which is O((1/epsilon^2)log(1/delta) log|F|). Our lower bound can be viewed as an analog of list size lower bounds for list-decoding of error-correcting codes, but for "dense model decoding" instead. Our proof introduces some new techniques which may be of independent interest, including an analysis of a majority of majorities of p-biased bits. The latter analysis uses an extremely tight lower bound on the tail of the binomial distribution, which we could not find in the literature.

BibTeX - Entry

  author =	{Thomas Watson},
  title =	{{Advice Lower Bounds for the Dense Model Theorem}},
  booktitle =	{30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)},
  pages =	{634--645},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-50-7},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{20},
  editor =	{Natacha Portier and Thomas Wilke},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-39717},
  doi =		{10.4230/LIPIcs.STACS.2013.634},
  annote =	{Keywords: Pseudorandomness, advice lower bounds, dense model theorem}

Keywords: Pseudorandomness, advice lower bounds, dense model theorem
Collection: 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)
Issue Date: 2013
Date of publication: 26.02.2013

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