Answer Posted / shishir bajpai
An (N,M,D,K,ε) -extractor is a bipartite graph with N nodes
on the left and M nodes on the right such that each node on
the left has D neighbors (on the right), which has the
added property that for any subset A of the left vertices
of size at least K, the distribution on right vertices
obtained by choosing a random node in A and then following
a random edge to get a node x on the right side is ε-close
to the uniform distribution in terms of total variation
distance.
A disperser is a related graph.
An equivalent way to view an extractor is as a bivariate
function
in the natural way. With this view it turns out that the
extractor property is equivalent to: for any source of
randomness X that gives n bits with min-entropy logK, the
distribution E(X,UD) is ε-close to UM, where UT denotes the
uniform distribution on [T].
Extractors are interesting when they can be constructed
with small K,D,ε relative to N and M is as close to KD (the
total randomness in the input sources) as possible.
Extractor functions were originally researched as a way to
extract randomness from weakly random sources. See
randomness extractor.
Using the probabilistic method it is easy to show that
extractor graphs with really good parameters exist. The
challenge is to find explicit or polynomial time computable
examples of such graphs with good parameters. Algorithms
that compute extractor (and disperser) graphs have found
many applications in computer science.
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