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Advice Lower Bounds for the Dense Model Theorem

Published:13 January 2015Publication History
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Abstract

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 δ-dense in a distribution that is ε′-indistinguishable from uniform, there exists a “dense model” for D, that is, a distribution that is δ-dense in the uniform distribution and is ε-indistinguishable from D. This ε-indistinguishability is with respect to an arbitrary small class of functions F. For the natural case where ε′ ≥ Ω(εδ) and εδO(1), our lower bound implies that Ω(√(1/ε) log(1/δ)·log|F|) advice bits are necessary for a certain type of reduction that establishes a stronger form of the Dense Model Theorem (and which encompasses all known proofs of the Dense Model Theorem in the literature). 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/ε2)log(1/δ)·log|F|). Our lower bound can be viewed as an analogue of list size lower bounds for list-decoding of error-correcting codes, but for “dense model decoding” instead.

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    • Published in

      cover image ACM Transactions on Computation Theory
      ACM Transactions on Computation Theory  Volume 7, Issue 1
      December 2014
      104 pages
      ISSN:1942-3454
      EISSN:1942-3462
      DOI:10.1145/2692372
      Issue’s Table of Contents

      Copyright © 2015 ACM

      Publisher

      Association for Computing Machinery

      New York, NY, United States

      Publication History

      • Published: 13 January 2015
      • Accepted: 1 October 2014
      • Revised: 1 February 2014
      • Received: 1 December 2012
      Published in toct Volume 7, Issue 1

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