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%%%  EDIT THE FOLLOWING TO UPDATE DATE AND LECTURE NUMBER

\lecturehead{CS G399}{Algorithmic Power Tools I}{Fall 2007}{27
November 2007}{Shahzad Rajput}

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\noindent {\bf Lecture Outline:}

\begin{itemize}
\item
The Sparsest Cut Problem
\item
Previous Approximation Algorithms
\item
Arora-Rao-Vazirani Algorithm - $O(\sqrt{\log n})$ Approximation for the Sparsest Cut Problem
\item
Some Observations on High Dimension Geometry
\end{itemize}

In this lecture, we have introduced the $O(\sqrt{\log n})$
approximation algorithm for the sparsest cut problem.  Good references
for the material covered in this lecture include a seminal paper on
graph partioning by Sanjeev Arora, Satish Rao and Umesh
Vazirani~\cite{ARVFULL} and lecture notes from Avner Megan's class at
University of Toronto.

\section{The Sparsest Cut Problem}
Recall the Sparsest Cut Problem. For a graph $G = (V,E)$ and non-empty
set $S \subset V$ of the vertices, $(S,\overline{S})$ denotes the set
of edges of $G$ with exactly one end vertex in $S$. An edge set of
this form is called an $\emph{edge cut}$, or cut. The $\emph{density}$
of an edge cut $(S,\overline{S})$ is the ratio between the number of
edges that are present in the cut, and the maximum number of edges
that are possible between $S$ and $\overline{S}: d(S,\overline{S})$ =
$|(S,\overline{S})|/(|S||\overline{S}|)$.  An edge cut with minimum
density is called a sparsest cut of the graph.
\section{Previous Approximation Algorithms}
\begin{itemize}
\item
Eigenvalue approaches (Cheeger'70, Alon'85, Alon-Milman'85)\\
Only yield  factor n approximation.
\item
$O(log n)$-approximation via LP (multicommodity flows) - (Leighton-Rao'88)
\item
Embeddings of finite metric spaces into $\ell_{1}$ - (Linial, London, Rabinovich'94, AR'94)\\
Geometric approach; more general result (but still $O(log n)$
approximation); we studied this last lecture for the uniform case.
\end{itemize}
\section{Arora-Rao-Vazirani algorithm - $O(\sqrt{\log n})$ approximation for the sparsest cut problem}
The key idea underlying algorithms for graph partitioning is to spread
out the vertices in some abstract space while not stretching the edges
too much. Finding a good graph partition is then accomplished by
partitioning this abstract space.~\cite{ARVFULL}

The approach of the ARV algorithm is based on the geometry of metric
spaces. In particular, this approach maps the vertices to points in a
high-dimensional space such that the average square distance between
vertices is a fixed constant, but the average squared distance between
the endpoints of edges is minimized. Furthermore, we insist that these
squared distances form a metric. This embedding, which we refer to as
an $\ell_{2}^2$ representation, can be computed in polynomial time
using semi-definite programming. It is shown in ~\cite{ARVFULL} that
this $\ell_{2}^2$ representation embeds into $\ell_{1}$ with
$O(\sqrt{\log n})$ distortion.

\begin{Definition}
\label{def:uniform.spasestcut}
Given a graph $G=(V,E)$, the goal in the UNIFORM SPARSEST CUT problem
is to determine the cut $(S,\overline{S})$ (where
$|S|\leq|\overline{S}|$ without loss of generality) that
minimizes
\[
\frac{|E(S,\overline{S})|}{|S||\overline{S}|}
\]
\end{Definition}

From previous lecture, we know that
\begin{eqnarray*}
& & \min_{S} \frac{|E(S,\overline{S})|}{|S||\overline{S}|}\\
& = & \min_{l_{1}} \frac{\sum_{i,j\epsilon E} d_{l_{1}}(i,j)}{\sum_{i,j} d_{l_{1}}(i,j)}\\
& = & \min_{v_{i}\epsilon \{-1,+1\}} \frac{\sum_{i,j\epsilon E} |v_{i}-v_{j}|^2}{\sum_{i,j} |v_{i}-v_{j}|^2}\\
\end{eqnarray*}
Here 
\begin{eqnarray*}
& & \sum_{i,j\epsilon E} |v_{i}-v_{j}|^2 = 4E(S,\overline{S})\\
& & \sum_{i,j} |v_{i}-v_{j}|^2 = 4|S||\overline{S}|
\end{eqnarray*}

Relaxing $v_{i}$ to arbitrary vector ($v_{1},v_{2},\cdots,v_{n} \in
\Re^n$), we obtain that the sparsest cut value is at least
\begin{eqnarray*}
\min_{v_{i}\epsilon \Re^m} \frac{\sum_{i,j\epsilon E} \parallel v_{i}-v_{j} \parallel^2 _{2}}{\sum_{i,j} \parallel v_{i}-v_{j} \parallel^2 _{2}}
\end{eqnarray*}

This can be written as the following SDP:
\begin{eqnarray*}
& & \min_{v_{i}\epsilon \Re^m} \sum_{i,j\epsilon E} \parallel v_{i}-v_{j}
\parallel^2 _{2}\\
& s.t. & \sum_{i,j} \parallel v_{i}-v_{j} \parallel^2 _{2}  =  n^2
\end{eqnarray*}

$n^2$ is used for simplicity, as this would make the average distance equal to 1. We can use value other than $n^2$ and then scale the distance accordingly.

The cut integrality gap is very large, therefore they added \emph{triangle inequality}.

\begin{Definition}
\label{def:l2.representation}
($\ell_{2}^2$ REPRESENTATION) \emph{An $\ell_{2}^2$ representation of a graph is an assignment of a point (vector) to each node, say $v_{i}$ assigned to node i, such that for all i, j, k:}
\begin{eqnarray*}
|v_{i} - v_{j}|^2 + |v_{j} - v_{k}|^2 \geq |v_{i} - v_{k}|^2 & \emph{(triangle inequality)}
\end{eqnarray*}
\emph{An $\ell_{2}^2$ representation of a graph is called a unit-$\ell_{2}^2$ representation if all points lie on the unit sphere (or equivalently, all vectors have unit length.)}
\end{Definition}
Note that an $\ell_2^2$ representation implies that all the angles in
any triangle formed by these vectors are acute angles.

This SDP is then solved and the collection of points returned can then
be embedded in $\ell_{1}$ with average distortion $O(\sqrt{\log
n})$. For embedding $\ell_{2}^2$ into $\ell_{1}$, we rely on the
Master Structure Theorem.

{\noindent \bf The Master Structure Theorem:} Given
$v_{1},v_{2},\cdots,v_{n} \in \Re^m$, such that $\|v_{i}\|^2 \leq 1$
for all $i$, the squared distances form a metric (i.e. all the angles
in any triangle formed by three points are acute)
\[ \frac{1}{n^2}\sum{i,j}\|v_{i}-v_{j}\|^2
= \Omega(1),
\]
there exist $S,T$ such that $|S|$ = $\Omega(n)$, $|T|$ = $\Omega(n)$
and $d(S,T) = \Omega(\frac{1}{\sqrt{\log n}})$.

A tight example of $\Omega(\frac{1}{\sqrt{\log n}})$ is a hypercube
with $n$ points in $\log$ n dimensions.

\section{Some Observations on High Dimensional Geometry}
\begin{itemize}
\item
What is the maximum number of points in $n$-dimensions such that no three points subtend obtuse angle?
\begin{itemize}
\item
2-dimensions $\>\ \>$ 4 points (square)
\item
3-dimensions $\>\ \>$ 8 points (cube)
\item
... $\>\ \>$ $\>$ $\>$ $\>$ $\>$ $\>$ $\>$ $\>$ $\>$ \ ...
\item
n-dimensions $\>\ \>$ $2^n$ points
\end{itemize}
\item
High-Dimensional Spheres in Cubes

In high dimensions, 99$\%$ of the mass of the sphere is concentrated
at its center.

\includegraphics{Highdim_Sphere.jpg}

Here is a puzzle~\cite{HIGHSPH}.  Take a cube (in dimension $n$) with
side length 1.  Consider a sphere of radius $1/2$ around each vertex
of the cube.  Now construct a central sphere around the center of the
cube such that it touches each of the spheres that are centered at the
vertices.  What is the volume of this central sphere?  In particular,
as $n$ increases, does this sphere form a small fraction of the cube?
Or does it contain most of the cube?

Somewhat surprisingly, the central sphere's volume goes to infinity,
as $n$ increases!  To see this, note that the center of the cube is
distance $\sqrt{n}/2$ from each of the cube vertices.  So the radius
of the central sphere is $(\sqrt{n} - 1)/2$, yielding a volume of
infinity, as $n$ tends to infinity.  In fact, for even small values of
$n$ (greater than 9), part of the central sphere bulges outside the
cube.  

\junk{
Cut the cube into orthants, each sub-cube has side length
1.  Let's just look at one of those sub-cubes. The sphere S inscribed
in that sub-cube has diameter 1. When we draw the central sphere, its
center is on a corner of that subcube. Draw the diagonal from that
corner to the opposite corner of the sub-cube. That diagonal has
length $\sqrt{N}$. The part of the diagonal going through the sphere S
has length 1 because that is the diameter of S. Of the part that is
left over, half of it is in the central sphere, and in fact forms the
radius of that central sphere. So the central sphere has radius
$\frac{\sqrt{N}-1}{2}$. If N=9, the radius of the central sphere is 1,
so it is just tangent to the cube. If N$>$9, then part of the central
sphere bulges outside the cube! And, eventually the volume of the
central sphere is actually larger than the cube.~\cite{HIGHSPH}}
\end{itemize}
\begin{thebibliography}{1}
	\bibitem{ARVFULL}
	  Sanjeev Arora, Satish Rao and Umesh Vazirani.
	  \emph{Expander Flows, Geometric Embeddings and Graph Partitioning.}
	  April 2007.
    \bibitem{HIGHSPH}
      http://www.math.hmc.edu/funfacts/ffiles/20007.2.shtml
\end{thebibliography}


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