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{\bf CS 174: Combinatorics and Discrete Probability} \hfill Fall 2012 \bigskip \\
\centering{\Large Homework 5} \medskip \\
\centering{Due: Thursday, October 4, 2012 by {\bf 9:30am}}
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\noindent{\it \textbf{Instructions}}: {\it You should upload your homework
solutions on bspace. You are strongly encouraged to type out your solutions
using \LaTeX . You may also want to consider using mathematical mode typing in
some office suite if you are not familiar with \LaTeX . If you must handwrite
your homeworks, please write clearly and legibly. We will not grade homeworks
that are unreadable. You are encouraged to work in groups of 2-4, but you {\bf
must} write solutions on your own. Please review the homework policy carefully
on the class homepage.} \medskip \\
\noindent {\bf Note}: You \emph{must} justify all your answers. In particular, you will get
no credit if you simply write the final answer without any explanation. \medskip
\\
\noindent {\bf Problem 1}. {\it (Exercise 5.9 from MU -- 5 points)} Consider the
probability that every bin receives exactly one ball when $n$ balls are thrown
randomly into $n$ bins.
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\begin{enumerate}
\item[(a)] Give an upper bound on this probability using the Poisson
approximation.
\item[(b)] Determine the \emph{exact} probability of this event.
\end{enumerate} \medskip
\noindent {\bf Problem 2}. {\it (Exercise 5.13 from MU -- 5 points)} Let $Z$ be
a Poisson random variable with mean $\mu$, where $\mu \geq 1$ is an integer.
First, show that $\Pr[Z = \mu + h] \geq \Pr[Z = \mu - h - 1]$ for $0 \leq h \leq
\mu - 1$, and use this to conclude that $\Pr[Z \geq \mu ] \geq 1/2$. \medskip \\
\noindent {\bf Problem 3} {\it (Exercise 5.14 from MU -- 5 points)} Let $Y_1,
\ldots, Y_n$ be Poisson random variables with mean $\mu (= m/n)$. Let $X_1, X_2, \ldots,
X_n$ be the random variables denoting the number of balls in each bin when $m$
balls are thrown in $n$ bins. In class, we showed that for any non-negative
function, $f$,
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\[ \E[f(Y_1, \ldots, Y_n)] \geq \E[f(X_1, \ldots,X_n)] \Pr[ \sum_{i=1}^n Y_i =
m] \]
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When $f$ is monotonically increasing, show that
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\[ \E[f(Y_1, \ldots, Y_n)] \geq \E[f(X_1, \ldots,X_n)] \Pr[ \sum_{i=1}^n Y_i
\geq m] \]
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Use this and problem 2 to conclude that $\E[f(X_1, \ldots, X_n)] \leq 2
\E[f(Y_1, \ldots, Y_n)]$ (see Theorem 5.10). \medskip \\
\noindent {\bf Problem 4} {\it (5 points)} Let $X_1, \ldots, X_n$ be geometric
random variables with mean $2$. Let $X = \sum_{i=1}^n X_i$ and
$\delta > 0$,
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\begin{enumerate}
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\item[(a)] Derive a bound on $\Pr[X \geq (1 + \delta) 2n]$ by applying a
Chernoff bound to a squence of $(1 + \delta)(2n)$ independent coin tosses.
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\item[(b)] Consider the quantity $\E[e^{tX}]$ and derive a Chernoff bound for
$\Pr[X \geq (1 + \delta)(2n)]$ using Markov's inequality for the random variable
$e^{tX}$.
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\item[(c)] Which bound is better?
\end{enumerate} \medskip
\noindent {\bf Problem 5}. {\it (Exercise 4.25 from MU - 10 points)} In this
exercise, we design a randomized algorithm for the following packet routing
problem. We are given a network that is an undirected connected graph, $G$,
where nodes represent processors and the edges between the nodes represent
wires. We are also given a set of $N$ packets to route. For each packet we are
given a source node, a destination node, and the exact route (path in the graph)
that the packet should take from the source to the destination. (We may assume
that there are no loops in the path.) In each time step, at most one packet can
traverse an edge. A packet can wait at any node during any time step, and we
assume unbounded queue sizes at each node.
A \emph{schedule} for a set of packets specifies the timing for the movement of
packets along their respective routes. That is, it specifies which packet should
move and which should wait at each time step. Our goal is to produce a schedule
for the packets that tries to minimize the total time and the maximum queue size
needed to route all the packets to their destinations.
\begin{enumerate}
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\item[(a)] The dilation, $d$, is the maximum distance travelled by any packet. The
congestion, $c$, is the maximum number of packets that must traverse a single
edge during the entire course of the routing. Argue that the time required for
any schedule should be at least $\Omega(c + d)$. (Hint: Show that the time
should be at least $\max\{c, d\}$ which is $\Omega(c + d)$.)
%
\item[(b)] Consider the following \emph{unconstrained} schedule, where many
packets may traverse an edge during a single time step. Assign each packet an
integral delay chosen randomly, independently, and uniformly from the interval
$[1, \lceil \alpha c/\log(Nd)\rceil]$, where $\alpha$ is a \emph{sufficiently
large} constant. A packet that is assigned a delay of $x$ waits in its source
node for $x$ time steps; then it moves on to its final destination through its
specified route without ever stopping. Give an upper bound on the probability
that more than $O(\log(Nd))$ packets use a particular edge $e$ at a particular
time step $t$.
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\item[(c)] Again using the unconstrained schedule of part (b), show that the
probability that more than $O(\log(Nd))$ packets pass through any edge at any
time step is at most $1/(Nd)$ for a sufficiently large $\alpha$.
%
\item[(d)] Use the unconstrained schedule to devise a simple randomized
algorithm that, with high probability, produces a schedule of length $O(c + d
\log(Nd))$ using queues of size $O(\log(Nd))$ and following the constraint that
at most one packet crosses an edge per time step. (By high probability, we mean
$1 - O(1/N)$.)
\end{enumerate}
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