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\begin{document}

\begin{titlepage}
\title{CS 591-06, Lecture 9}

\author {Jared Saia \\
{\red University of New Mexico}}

\date{}
\maketitle

\end{titlepage}

\begin{SLIDE}[Class Outline]
\emph{A reasonable probability is the only certainty.} - E.W. Howe

\begin{itemize}
\item Review 
\item Expander Graph Proof
\item In Class Exercise
\item Intro to Chernoff Bounds
\end{itemize}
\end{SLIDE}

\begin{SLIDE}[Union Bound]
{\bf Union Bound Fact}: Let $\Gamma $ be any event space, $P$ be any
probability function and $\gamma _{1}, \gamma _{2}, \dots \gamma _{n}$
be any events in $\Gamma $.  Then:

\[
P (\bigcup _{i=1}^{n} \gamma _{i}) \leq \sum _{i=1}^{n} P (\gamma _{i})
\]

Example:\\

Let $\gamma_{1}$ be the event that coin 1 is heads, $\gamma _{2}$ be the
event that coin 2 is heads.  Then:

\[
P (\gamma _{1} \bigcup \gamma _{2}) \leq P (\gamma _{1}) + P (\gamma _{2})
\]
\end{SLIDE}

\begin{SLIDE}[Expander Graphs]

\begin{itemize}
\item Assume we've got a biparitite graph $G= (L,R)$, where each
vertex in $L$ has $C \log n$ edges to random nodes in $R$.  (The edges
are chosen uniformly and independently at random),

\item Consider some fixed sets $L' \subset L, R' \subset R$ such that
$|L'| = n/2$ and $|R'|=\epsilon n/\log n$, for some fixed $\epsilon >0$

\item {\bf Question:} What is the probability that there is no edge
between the set $L'$ and the set $R'$?

\end{itemize}

\end{SLIDE}

\begin{SLIDE}[Expander Picture]
\ifig{fixed-set}
\end{SLIDE}


\begin{SLIDE}[Probability of Failure]
\begin{itemize}
\item Q: What is probability that a single edge from $L'$ does not land
in $R'$?

\item Q: What is the probability that \emph{no} edge from $L'$ lands
in $R'$?  

\item Note: This probability is the same no matter which fixed set $L'$ and
$R'$ are chosen such that $L'$ and $R'$ are of the appropriate size.

\end{itemize}
\end{SLIDE}



\begin{SLIDE}[Probability of Failure]
\begin{itemize}
\item Q: What is probability that a single edge from $L'$ does not land
in $R'$?

\item A: $1 - \epsilon / \log n$

\item Q: What is the probability that \emph{no} edge from $L'$ lands
in $R'$?  

\item A: Since there are $n/2*C\log n$ independent edges, it is:
$$(1 - (\epsilon / \log n))^{n/2*C\log n} \leq e^{- \frac{\epsilon C n}{2}}.$$

\item This probability is the same no matter which fixed set $L'$ and
$R'$ are chosen such that $L'$ and $R'$ are of the appropriate size.

\end{itemize}
\end{SLIDE}

\begin{SLIDE}[Union Bound]

What is the probability that any of the many, many possible
sets of appropriate size fail?

Q: How many sets of size $n/2$ are there in $L$?


Q: How many sets of size $\epsilon n/\log n$ are there?

Q: What is an upper bound on the probability of \emph{any failure}?

\end{SLIDE}


\begin{SLIDE}[Union Bound]

What is the probability that any of the many, many possible
sets of appropriate size fail?

Q: How many sets of size $n/2$ are there in $L$? A:
$n \choose n/2$

Q: How many sets of size $\epsilon n/\log n$ are there? A: $n \choose
\epsilon n/\log n$ 

Q: What is an upper bound on the probability of \emph{any failure}? A: 
\[
{n \choose n/2} * {n \choose {\epsilon n/\log n}} * (e^{\frac{\epsilon
C n}{2}})
\]

\end{SLIDE}

\begin{SLIDE}[Some Useful Bounds]

\begin{itemize}
\item Stirlings Formula: $n! = \sqrt{2\pi n} \left( \frac{n}{e} \right)^{n} \left( 1+\frac{1}{12n} + O (\frac{1}{n^{2}})\right)$ 

\item $n! \geq  \left( \frac{n}{e} \right)^{n}$

\item ${n \choose {k}} \leq \frac{n^{k}}{k!}$

\item ${n \choose {k}} \leq \left(\frac{ne}{k} \right)^{k}$

\end{itemize}
\end{SLIDE}


\begin{SLIDE}[An upper bound]
Q: What is an upper bound on the probability of \emph{any failure}? A: 

\begin{eqnarray}
& & {n \choose n/2} * {n \choose {\epsilon n/\log n}} * (e^{\frac{-\epsilon Cn}{2}}) \\
& \leq & \left(\frac{ne}{n/2} \right)^{n/2} * \left(\frac{ne}{\epsilon
n/\log n} \right)^{\epsilon n/\log n}*(e^{\frac{-\epsilon Cn}{2}})  \\
& \leq  & \left(2e \right)^{n/2} * \left(2e \right)^{n/2}*(e^{\frac{-\epsilon Cn}{2}})  \\
& = & \left(2e \right)^{n}*(e^{\frac{-\epsilon Cn}{2}})\\
& \leq & e^{2n-\frac{\epsilon Cn}{2}}\\
& \leq & e^{n (2-\frac{\epsilon C}{2})}
\end{eqnarray}

\end{SLIDE}

\begin{SLIDE}[An Upper Bound]

\begin{itemize}
\item For $C > 4/\epsilon $, this quantity can be made
\emph{exponentially} small in the size of $n$ 

\item This implies that for $C$ chosen as a function of $\epsilon$
only, we can be sure that a random bipartite graph with $C \log n$
edges from each vertex on the left side has ``expansion'' properties
with high probability.

\item The probability of failure gets smaller and smaller as $n$ grows large

\end{itemize}

\end{SLIDE}


\begin{SLIDE}[In Class Exercise]
\begin{itemize}
\item Consider a simple random general graph $G$ on $n$ vertices where
each of the $n \choose 2$ possible edges is included with probability
$p$.

\item Q: What is an upperbound on the probability that the graph
contains an independent set of size $n/2$?

\begin{itemize}
\item Q1: What is the probability that some fixed set of $n/2$ vertices is
an independent set? (i.e. has no edges)
\item Q2: What is an upperbound on the probability that the graph
contains \emph{some} independent set of size $n/2$ vertices?
\end{itemize}

\end{itemize}
\end{SLIDE}

\begin{SLIDE}[Chernoff Bounds]
\begin{itemize}
\item Let $X=\avec{X}$ where the $X_{i}$ are random variables which
are: independent and all take on values between $0$ and $1$.
\end{itemize}
then for any $\lambda > 0$,
\begin{itemize}
\item $P (|X-E (X)| > \lambda ) \leq 2e^{\frac{-\lambda ^{2}}{2m}}$
\end{itemize}

\end{SLIDE}

\begin{SLIDE}[Chernoff Example]

\begin{itemize}
\item  Let $X = \avec{X}$, where each $X_{i}$ is $1$ if the $i$-th flip of a
coin flip is heads tails otherwise.  Let the coin turn up heads with
probability $1/2$.

\item Q: What is the probability that the number of heads is $(3/4)m$?

\end{itemize}

\end{SLIDE}




\end{document} 

