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Sample spaces with small bias on neighborhoods and error-correcting communication protocols

  • Michael Saks
  • Shiyu Zhou
Randomness
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1269)

Abstract

We give a deterministic algorithm which, on input an integer n, collection f of subsets of {1,2,⋯, n} and ∃ (0,1) runs in time polynomial in nf‖/∈ and produces a ±1-matrix M with n columns and m=O(log ‖f‖/∈2) rows with the following property: for any subset Ff, the fraction of 1's in the n-vector obtained by coordinate-wise multiplication of the column vectors indexed by F is between (1−∈)/2 and (1+)/2. In the case that f is the set of all subsets of size at most k, k constant, this gives a polynomial time construction for a k-wise -biased sample space of size O(log n/∈2), as compared to the best previous constructions of [NN90] and [AGHP91] which were, respectively, O(log n/∈4) and O((log n)2/2). The number of rows in the construction matches the upper bound given by the probabilistic existence argument. Such constructions are of interest for derandomizing algorithms. As an application, we present a family of essentially optimal deterministic communication protocols for the problem of checking the consistency of two files.

Keywords

Sample Space Explicit Construction Deterministic Algorithm Support Size Efficient Construction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Michael Saks
    • 1
  • Shiyu Zhou
    • 2
  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA
  2. 2.Bell LaboratoriesMurray HillUSA

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