# Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors

@article{Reingold2000EntropyWT,
title={Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors},
author={Omer Reingold and Salil P. Vadhan and Avi Wigderson},
journal={Proceedings 41st Annual Symposium on Foundations of Computer Science},
year={2000},
pages={3-13}
}
• Published 12 November 2000
• Computer Science, Mathematics
• Proceedings 41st Annual Symposium on Foundations of Computer Science
The main contribution 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…
396 Citations

## Topics from this paper

A combinatorial construction of almost-ramanujan graphs using the zig-zag product
• Computer Science, Mathematics
STOC
• 2008
A generalization of the zig-zag product that combines a large graph and several small graphs is proposed that gives a fully-explicit combinatorial construction of D-regular graphs having spectral gap 1-D-1/2 + o(1).
Connectedness and Isomorphism Properties of the Zig-Zag Product of Graphs
• Mathematics, Computer Science
J. Graph Theory
• 2016
The connectedness and the isomorphism problems for zig-zag products of two graphs are investigated and an example coming from the theory of Schreier graphs associated with the action of self-similar groups is considered: the graph products are completely determined and their spectral analysis is developed.
Semi-direct product in groups and Zig-zag product in graphs: Connections and applications
We consider the standard semi-direct product of finite groups . We show that with certain choices of generators for these three groups, the Cayley graph of is (essentially) the zigzag product of the
On Construction of Almost-Ramanujan Graphs
• He Sun, Hong Zhu
• Mathematics, Computer Science
COCOA
• 2009
A generalized theorem of zig-zag product of one "big" graph and several "small" graphs with the same size will be formalized and a family of fully explicitly almost-Ramanujan graphs with locally invertible function waived is presented.
Semi-direct product in groups and zig-zag product in graphs: connections and applications
• Computer Science
Proceedings 2001 IEEE International Conference on Cluster Computing
• 2001
The standard semi-direct product A/spl times/B of finite groups A, B is considered and it is shown that with certain choices of generators for these three groups, the Cayley graph is (essentially) the zigzag product of the Cayleys of A and B.
A Mathematical Introduction to Spectral Expansion
In this introduction to spectral expander graphs, we broadly cover different aspects of the theory, taking care to formally define and explain the various concepts involved. First, we present classic
Generalized zig-zag products of regular digraphs and bounds on their spectral expansions
• Mathematics
• 2007
We introduce a generalization of the zig-zag product of regular digraphs (directed graphs), which allows us to construct regular digraphs with m ore flexible choices of the degrees. In our
Randomness conductors and constant-degree lossless expanders
• Mathematics, Computer Science
STOC '02
• 2002
The introduction and initial study of randomness conductors, a notion which generalizes extractors, expanders, condensers and other similar objects, is introduced and it is shown that the flexibility afforded by the conductor definition leads to interesting combinations of these objects, and to better constructions such as those above.
Randomness Conductors and Constant-Degree Lossless Expanders [Extended Abstract]
• 2009
The main concrete result of this paper is the first explicit construction of constant degree lossless expanders. In these graphs, the expansion factor is almost as large as possible: (1− ǫ)D, where D
Zig-zag and replacement product graphs and LDPC codes
• Computer Science, Mathematics
• 2008
This paper investigates the use of zig-zag and replacement product graphs for the construction of codes on graphs and introduces a modification of the zg-zag product, which can operate on two unbalanced biregular bipartite graphs.

## References

SHOWING 1-10 OF 122 REFERENCES
ENTROPY WAVES, THE ZIG-ZAG GRAPH PRODUCT, AND NEW CONSTANT-DEGREE
• Mathematics
• 2002
The main contribution of this work is a new type of graph product, which we call the {\it zig-zag product}. Taking a product of a large graph with a small graph, the resulting graph inherits
Semi-direct product in groups and zig-zag product in graphs: connections and applications
• Computer Science
Proceedings 2001 IEEE International Conference on Cluster Computing
• 2001
The standard semi-direct product A/spl times/B of finite groups A, B is considered and it is shown that with certain choices of generators for these three groups, the Cayley graph is (essentially) the zigzag product of the Cayleys of A and B.
Eigenvalues, geometric expanders, sorting in rounds, and ramsey theory
• N. Alon
• Mathematics, Computer Science
Comb.
• 1986
It is shown that the points versus hyperplanes incidence graphs of finite geometries form highly (nonlinear) expanding graphs with essentially the smallest possible number of edges.
On the second eigenvalue of random regular graphs
• Computer Science, Mathematics
28th Annual Symposium on Foundations of Computer Science (sfcs 1987)
• 1987
It is shown that the second eigenvalue of d-regular graphs, λ2, is concentrated in an interval of width O(√d) around its mean, and that its mean is O(d3/4).
Ramanujan Graphs
In the last two decades, the theory of Ramanujan graphs has gained prominence primarily for two reasons. First, from a practical viewpoint, these graphs resolve an extremal problem in communication
Expanders That Beat the Eigenvalue Bound: Explicit Construction and Applications
• Mathematics, Computer Science
Comb.
• 1999
These results are based on an improvement to the extractor construction of Nisan & Zuckerman: the algorithm extracts an asymptotically optimal number of random bits from a defective random source using a small additional number of truly random bits.
Expanders obtained from affine transformations
• Mathematics, Computer Science
Comb.
• 1987
The problem of estimating the coefficientδ of a bipartite graph is reduced to that of estimatingThe second largest eigenvalue of a matrix related to the graph, and some general results on estimating the eigenvalues of the matrix by using the discrete Fourier transform are obtained.
Random Cayley Graphs and Expanders
• Mathematics, Computer Science
Random Struct. Algorithms
• 1994
It is shown that for every group G of order n, and for a set S of c(δ) log n random elements in the group, the expected value of the second largest eigenvalue of the normalized adjacency matrix of the Cayley graph X(G, S) is at most (1 - δ).
Explicit Constructions of Linear-Sized Superconcentrators
• Computer Science
J. Comput. Syst. Sci.
• 1981
Recursive construction for 3-regular expanders
• M. Ajtai
• Mathematics, Computer Science
28th Annual Symposium on Foundations of Computer Science (sfcs 1987)
• 1987
An algorithm which in n3(log n)3 time constructs a 3- regular expander graph on n vertices so that the total number of cycles of length s decreases (for some fixed absolute constant c) and when the graph is an expander the proof is completely elementary.