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### Paper:

TR05-025 | 20th February 2005 00:00

#### Analyzing Linear Mergers

TR05-025
Authors: Zeev Dvir, Ran Raz
Publication: 22nd February 2005 12:30
Keywords:

Abstract:

Mergers are functions that transform k (possibly dependent)
random sources into a single random source, in a way that ensures
that if one of the input sources has min-entropy rate $\delta$
then the output has min-entropy rate close to $\delta$. Mergers
have proven to be a very useful tool in explicit constructions of
extractors and condensers, and are also interesting
objects in their own right. In this work we give a refined
analysis of the merger constructed by \cite{Raz05} (based on
\cite{LRVW03}). Our analysis uses min-entropy instead of Shannon's
entropy to derive tighter results than the ones obtained in
\cite{Raz05}.

We show that for every r it is possible to construct a merger
that takes as input k strings of length n bits each, and
outputs a string of length n/r bits, such that if one of the
input sources has min-entropy b, the output will be close to
having min-entropy b/(r+1). This merger uses a constant number
of additional uniform bits when k and r are constants. One
advantage of our analysis is that b (the min-entropy of the
'good' source ) can be as small as a constant, while in the
analysis given in \cite{Raz05}, b is required to be linear in
n.

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