The main contribution of this work is a new fype 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 sizefrom the large one, its degree from the small one, and its expansion propertiesfrom both! Iteration yields simple explicit constructions of constant-degree expanders of every size, startingfrom one constant-size expandel: 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" propagatorsthey 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 integerence of hvo such waves. A variant of this product can be applied to extractors, giving the first explicit extractors whose seed length depends (po1y)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 minentropy extractors have several interesting applications, including the first constant-degree explicit expanders which beat the "eigenvalue bound."
An example of this is the Zig-Zag product REF , where an expander is constructed by repeatedly combining two graphs in a tensor-product manner.
Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors
Proceedings 41st Annual Symposium on Foundations of Computer Science
Proceedings 41st Annual Symposium on Foundations of Computer Science
Mathematics Computer Science