# CS294 Lecture 16: Zig-Zag Graph Product

In which we give an explicit construction of expander graphs of polylogarithmic degree, state the properties of the zig-zag product of graphs, and provide an explicit construction of a family of constant-degree expanders using the zig-zag product and the polylogarithmic-degree construction.

A family of expanders is a family of graphs ${G_n = (V_n,E_n)}$, ${|V_n|=n}$, such that each graph is ${d_n}$-regular, and the edge-expansion of each graph is at least ${h}$, for an absolute constant ${h}$ independent of ${n}$. Ideally, we would like to have such a construction for each ${n}$, although it is usually enough for most applications that, for some constant ${c}$ and every ${k}$, there is an ${n}$ for which the construction applies in the interval ${\{ k, k+1, \ldots, ck \}}$, or even the interval ${\{ k, \ldots, ck^c\}}$. We would also like the degree ${d_n}$ to be slowly growing in ${n}$ and, ideally, to be bounded above by an explicit constant. Today we will see a simple construction in which ${d_n = O(\log^2 n)}$ and a more complicated one in which ${d_n = O(1)}$.

# CS294 Lecture 15: Abelian Cayley graphs

In which we show how to find the eigenvalues and eigenvectors of Cayley graphs of Abelian groups, we find tight examples for various results that we proved in earlier lectures, and, along the way, we develop the general theory of harmonic analysis which includes the Fourier transform of periodic functions of a real variable, the discrete Fourier transform of periodic functions of an integer variable, and the Walsh transform of Boolean functions.

Earlier, we prove the Cheeger inequalities

$\displaystyle \frac{\lambda_2}{2} \leq \phi(G) \leq \sqrt{2 \lambda_2}$

and the fact that Fiedler’s algorithm, when given an eigenvector of ${\lambda_2}$, finds a cut ${(S,V-S)}$ such that ${\phi(S,V-S) \leq 2\sqrt{\phi(G)}}$. We will show that all such results are tight, up to constants, by proving that

• The dimension-${d}$ hypercube ${H_d}$ has ${\lambda_2 = 1- \frac 2d}$ and ${h(H_d) = \frac 1d}$, giving an infinite family of graphs for which ${\frac{\lambda_2}{2} = \phi(G)}$, showing that the first Cheeger inequality is exactly tight.
• The ${n}$-cycle ${C_n}$ has ${\lambda_2 = O(n^{-2})}$, and ${\phi(C_n) = \frac 2n}$, giving an infinite family of graphs for which ${\phi(G) = \Omega(\sqrt{\lambda_2})}$, showing that the second Cheeger inequality is tight up to a constant.
• There is an eigenvector of the 2nd eigenvalue of the hypercube ${H_d}$, such that Fiedler’s algorithm, given such a vector, outputs a cut ${(S,V-S)}$ of expansion ${\phi(S,V-S) = \Omega(1/\sqrt{d})}$, showing that the analysis of the Fiedler’s algorithm is tight up to a constant.

In this lecture we will develop some theoretical machinery to find the eigenvalues and eigenvectors of Cayley graphs of finite Abelian groups, a class of graphs that includes the cycle and the hypercube, among several other interesting examples. This theory will also be useful later, as a starting point to talk about constructions of expanders.

For readers familiar with the Fourier analysis of Boolean functions, or the discrete Fourier analysis of functions ${f: {\mathbb Z}/N{\mathbb Z} \rightarrow {\mathbb C}}$, or the standard Fourier analysis of periodic real functions, this theory will give a more general, and hopefully interesting, way to look at what they already know.

# CS 294 Lecture 14: ARV Analysis, Part 3

In which we complete the analysis of the ARV rounding algorithm

We are finally going to complete the analysis of the Arora-Rao-Vazirani rounding algorithm, which rounds a Semidefinite Programming solution of a relaxation of sparsest cut into an actual cut, with an approximation ratio ${O(\sqrt {\log |V|})}$.

In previous lectures, we reduced the analysis of the algorithm to the following claim.

# CS294 Lecture 13: ARV Analysis, cont’d

In which we continue the analysis of the ARV rounding algorithm

We are continuing the analysis of the Arora-Rao-Vazirani rounding algorithm, which rounds a Semidefinite Programming solution of a relaxation of sparsest cut into an actual cut, with an approximation ratio ${O(\sqrt {\log |V|})}$.

In previous lectures, we reduced the analysis of the algorithm to the following claim.

# CS294 Lecture 12: ARV Analysis

In which we begin the analysis of the ARV rounding algorithm

We want to prove

Lemma 1 (ARV Main Lemma) Let ${d}$ be a negative-type metric over a set ${V}$ such that the points are contained in a unit ball and have constant average distance, that is,

• there is a vertex ${z}$ such that ${d(v,z)\leq 1}$ for every ${v\in V}$
• ${\sum_{u,v\in V} d(u,v) \geq c\cdot |V|^2}$

Then there are sets ${S,T \subseteq V}$ such that

• ${|S|, |T| \geq \Omega(|V|)}$;
• for every ${u\in S}$ and every ${v\in T}$, ${d(u,v) \geq 1/{O(\sqrt {\log |V|})}}$

where the multiplicative factors hidden in the ${O(\cdot)}$ and ${\Omega(\cdot)}$ notations depend only on ${c}$.

In this lecture, we will show how to reduce the ARV Main Lemma to a statement of the following form: if ${\{ {\bf x}_v \}_{v\in V}}$ is a set of vectors such that the metric ${d(\cdot, \cdot)}$ in the ARV Main Lemma can be written as ${d(u,v) = || {\bf x}_u - {\bf x}_v ||^2}$, and ${{\bf g}}$ is a random Gaussian vectors, and if ${\ell}$ is such that with ${\Omega(1)}$ probability, there are ${\Omega(n)}$ disjoint pairs ${u,v}$ such that ${d(u,v) < \ell}$ and ${| \langle g, {\bf x}_u\rangle - \langle g, {\bf x}_v \rangle | \geq \Omega(1)}$, then ${\ell \geq \Omega(1/\sqrt{\log n})}$. We will then prove such a statement in the next lecture.

# CS294 Lecture 11: ARV

In which we introduce semi-definite programming and a semi-definite programming relaxation of sparsest cut, and we reduce its analysis to a key lemma that we will prove in the next lecture(s)

# CS294 Lecture 10: Bourgain’s Theorem

In which we prove Bourgain’s theorem.

Today we prove the following theorem.

Theorem 1 (Bourgain) Let ${d: V\times V \rightarrow {\mathbb R}}$ be a semimetric defined over a finite set ${V}$. Then there exists a mapping ${F: V \rightarrow {\mathbb R}^m}$ such that, for every two elements ${u,v \in R}$,

$\displaystyle || F(u) - F(v)||_1 \leq d(u,v) \leq ||F(u)-F(v)||_1 \cdot c\cdot \log |V|$

where ${c}$ is an absolute constant. Given ${d}$, the mapping ${F}$ can be found with high probability in randomized polynomial time in ${|V|}$.

Together with the results that we proved in the last lecture, this implies that an optimal solution to the Leighton-Rao relaxation can be rounded to an ${O(\log n)}$-approximate solution to the sparsest cut problem. This was the best known approximation algorithm for sparsest cut for 15 years, until the Arora-Rao-Vazirani algorithm, which will be our next topic.

# CS294 Lecture 9: The Sparsest Cut Problem

In which we introduce the sparsest cut problem and the Leighton-Rao relaxation.

1. The Uniform Sparsest Cut problem, Edge Expansion and ${\lambda_2}$

Let ${G=(V,E)}$ be an undirected graph with ${n:= | V |}$ vertices.

We define the uniform sparsity of a cut ${(S,V-S)}$ as

$\displaystyle {\sf usc}_G(S) := \frac {E(S,V-S)}{ |S| \cdot |V-S| }$

(we will omit the subscript when clear from the context) and the uniform sparsest cut of a graph is

$\displaystyle {\sf usc}(G):= \min_{S} {\sf usc}_G(S)$

In ${d}$-regular graphs, approximating the uniform sparsest cut is equivalent (up to a factor of 2 in the approximation) to approximating the edge expansion, because, for every cut ${(S,V-S)}$, we have

$\displaystyle \phi(S,V-S) = \frac {E(S,V-S)}{d \cdot \min \{ |S|, |V-S| \} }$

and, noting that, for every, ${S}$,

$\displaystyle \frac 1n |S| \cdot |V-S| \leq \min \{ |S|, |V-S| \} \leq \frac 2n |S| \cdot |V-S|$

we have, for every ${S}$,

$\displaystyle \phi (S,V-S) \leq \frac nd \cdot {\sf usc}(S) \leq 2 \phi(S,V-S)$

and so

$\displaystyle \phi(G) \leq \frac nd \cdot {\sf usc}(G) \leq 2 \phi(G)$

It will be instructive to see that, in ${d}$-regular graphs, ${\lambda_2}$ is a relaxation of ${\frac nd {\sf usc}(G)}$, a fact that gives an alternative proof of the easy direction ${\lambda_2 \leq 2 \phi(G)}$ of Cheeger’s inequalities.

# CS294 Lecture 8: Spectral Algorithms Wrap-up

In which we talk about even more generalizations of Cheeger’s inequalities, and we analyze the power method to find approximate eigenvectors, thus having a complete description of a polynomial-time approximation algorithm for sparsest cut