## DIMACS TR: 2001-09

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

### Authors: Omer Reingold, Salil Vadhan, and Avi Wigderson

**
ABSTRACT
**

The main contribution
of this work is a new type of graph product, which we
call the \defword{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.''

Paper Available at:
ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/2001/2001-09.ps.gz

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