Abstract.
We give a deterministic algorithm which, on input an integer n , collection \cal F of subsets of {1,2,\ldots,n} , and ɛ∈ (0,1) , runs in time polynomial in n| \cal F |/ɛ and produces a \pm 1 -matrix M with n columns and m=O(log | \cal F |/ɛ 2 ) rows with the following property: for any subset F ∈ \cal F , the fraction of 1's in the n -vector obtained by coordinatewise multiplication of the column vectors indexed by F is between (1-ɛ)/2 and (1+ɛ)/2 . In the case that \cal F is the set of all subsets of size at most k , k constant, this gives a polynomial time construction for a k -wise ɛ -biased sample space of size O(log n/ɛ 2 ) , compared with the best previous constructions in [NN] and [AGHP] which were, respectively, O(log n/ɛ 4 ) and O(log 2 n/ɛ 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.
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Received October 30, 1997; revised September 17, 1999, and April 17, 2000.
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Saks, M., Zhou, S. Sample Spaces with Small Bias on Neighborhoods and Error-Correcting Communication Protocols. Algorithmica 30, 418–431 (2001). https://doi.org/10.1007/s00453-001-0020-z
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DOI: https://doi.org/10.1007/s00453-001-0020-z