NDLI: Bounds for Dispersers, Extractors, and Depth-Two Superconcentrators

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Bounds for Dispersers, Extractors, and Depth-Two Superconcentrators

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Radhakrishnan, Jaikumar Ta-Shma, Amnon
Copyright Year	2000
Abstract	We show that the size of the smallest depth-two N-superconcentrator is $$ \Theta(N\log^2 N/\log\log N). $$ Before this work, optimal bounds were known for all depths except two. For the upper bound, we build superconcentrators by putting together a small number of disperser graphs; these disperser graphs are obtained using a probabilistic argument. For obtaining lower bounds, we present two different methods. First, we show that superconcentrators contain several disjoint disperser graphs. When combined with the lower bound for disperser graphs of Kovari, Ss, and Turn, this gives an almost optimal lower bound of $\Omega( N (\log N/\log \log N)^2)$ on the size of N-superconcentrators. The second method, based on the work of Hansel, gives the optimal lower bound.The method of Kovari, Ss, and Turn can be extended to give tight lower bounds for extractors, in terms of both the number of truly random bits needed to extract one additional bit and the unavoidable entropy loss in the system. If the input is an n-bit source with min-entropy k and the output is required to be within a distance of $\epsilon$ from uniform distribution, then to extract even one additional bit, one must invest at least $\log(n-k) + 2\log(1/\epsilon) - O(1)$ truly random bits; to obtain m output bits one must invest at least $m-k+2\log(1/\epsilon)-O(1)$. Thus, there is a loss of $2\log(1/\epsilon)$ bits during the extraction. Interestingly, in the case of dispersers this loss in entropy is only about $\log\log (1/\epsilon)$.
Starting Page	2
Ending Page	24
Page Count	23
File Format	PDF
ISSN	08954801
DOI	10.1137/S0895480197329508
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	1
Volume Number	13
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-08-03
Access Restriction	Subscribed
Subject Keyword	superconcentrators extractors Applications of graph theory dispersers entropy loss Extremal problems
Content Type	Text
Resource Type	Article
Subject	Mathematics

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