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Entropy Waves, The Zig-Zag Graph Product, and New Constant-Degree Expanders
and Extractors

Omer Reingold, Salil Vadhan, and Avi Wigderson

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Abstract

The main contribution of this work is a new type of graph product, which
we call the **zig-zag product**. Taking a product of a large graph with
a small graph, the resulting graph inherits (roughly) its size from the
large one, its degree from the small one, and its expansion properties
from both! Iteration yields simple explicit constructions of constant-degree
expanders of every size, starting from one constant-size expander.
Crucial to our intuition (and simple analysis) of the properties of
this graph product is the view of expanders as functions which act as "entropy
wave" propagators --- they transform probability distributions in which
entropy is concentrated in one area to distributions where that concentration
is dissipated. In these terms, the graph product affords the constructive
interference of two such waves.

A variant of this product can be applied to extractors, giving the first
explicit extractors whose seed length depends (poly)logarithmically on
only the entropy deficiency of the source (rather than its length) and
that extract almost all the entropy of high min-entropy sources. These
high min-entropy extractors have several interesting applications, including
the first constant-degree explicit expanders which beat the "eigenvalue
bound."

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