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% macros useful for handouts and homeworks.
\newcommand{\setcourse}[1]{\newcommand{\course}{#1\xspace}}
\newcommand{\setsem}[1]{\newcommand{\sem}{#1\xspace}}
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\newcommand{\solution}[1]{\textbf{Solution:} #1}
\setcourse{CS 4104}
\setsem{Fall 2009}
\sethwnum{5}
\newcommand{\cut}[1]{\ensuremath{\mathnormal{}{cut}(#1)}}
\newcommand{\calg}{\ensuremath{{\cal G}}\xspace}
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\chead{\course (\sem): Homework \hwnum}
\begin{document}
\title{\vspace{-0.5in}\textbf{Homework \hwnum}}
\author{\course (\sem)}
\date{Assigned on Monday, October 5, 2009. \\Hardcopy due at the
beginning of class on Monday, October 12, 2009.}
\maketitle
\paragraph{Instructions:}
\begin{itemize}
\item You are not allowed to consult any sources other than your
textbook, the slides on the course web page, your own class notes, the
TA, and the instructor. In particular, do not use a search engine.
\item Do not forget to typeset your solutions. \emph{Every mathematical
expression must be typeset as a mathematical expression, e.g., the
square of $n$ must appear as $n^2$ and not as ``n\^{}2''.} Students
using \LaTeX\ or LyX can use the corresponding versions of the
homework problems to start entering their solutions.
\item Describe your algorithms as clearly as possible. The style used in
the book is fine, as long as your description is not
ambiguous. Explain your algorithm in words. A step-wise description is
fine. \emph{However, if you submit detailed pseudo-code without an
explanation, we will not grade your solutions.}
\item Do not make any assumptions not stated in the problem. If you do
make any assumptions, state them clearly, and explain why the
assumption does not decrease the generality of your solution.
\item Do not describe your algorithms only for a specific example you
may have worked out.
\item You must also provide a clear proof that your solution is correct
(or a counter-example, where applicable). Type out all the statements you
need to complete your proof. \emph{You must convince us that you can
write out the complete proof. You will lose points if you work out
some details of the proof in your head but do not type them out in
your solution.}
\item Analyse your algorithm and state the running time. You will only
get partial credit if your analysis is not tight, i.e., if the bound
you prove for your algorithm is not the best upper bound possible.
\end{itemize}
\begin{description}
\item[Problem 1] (15 points) Solve the recurrence $T(n) = T( \lfloor
\sqrt{n} \rfloor) +
1$. In words, the $T(n)$ is the running time of an algorithm that
creates one sub-problem of the size equal to the square root of $n$,
solves this sub-problem recursively, and spends one more unit of time.
You can assume that $T(2) = 1$ and that $n \geq 2$.
% \solution{
% }
\item[Problem 2] (40 points) You are given three algorithms to solve the
same problem of size $n$. Analyse each algorithm in $O()$ notation.
Provide a clear proof of the analysis. Which algorithm would you
choose and why? In other words, write down which algorithm is
asymptotically the fastest and provide a proof why this algorithm is
asymptotically the fastest of all three. If you can directly apply a
formula we discussed in class, feel free to do so. For some sub-problems,
you will have to come up with proofs from scratch, although your
proofs will follow the general outlines we have used in
class. Remember to prove your solution by induction, even if you use
the ``unrolling'' method to guess a solution.
\begin{enumerate}[(i)]
\item Algorithm A solves the problem by dividing it into five
sub-problems of half the size, recursively solving each sub-problem,
and then combining the solutions in linear time.
\item Algorithm B solves the problem by dividing it into two
sub-problems of size $n - 1$, recursively solving each subproblem, and
then combining the solutions in constant time.
\item Algorithm C solves the problem by dividing it into three
sub-problems of size $n/3$, recursively solving each sub-problem,
and then combining the solutions in $O(n^2)$ time.
\end{enumerate}
% \solution{
% }
\item[Problem 3] (45 points) Solve exercise~3 in Chapter~5
(pages~246--247) of your textbook. Let us call the equivalence class
with more than $n/2$ cards the \emph{important} class. It is enough for
your algorithm to return a card that belongs to this class, if it
exists, or no card at all. Note that the problem specifies that the
only operation you can perform on a pair of cards is to decide if they
are equivalent. You cannot perform any other operation, e.g., compare
them in order to sort them. Your proof of correctness must clearly
address why your algorithm will find a set of important class, if it
exists. For instance, if you divide the cards into two sets of
$n/2$ cards each and find the important class in each subset,
why should the important class(es) found by the recursive calls have
any relation to the important class for all $n$ cards?
% \solution{
% }
\end{description}
\end{document}