% Template LaTeX source file for homework problem solutions.
% Alan T. Sherman (9/9/98)

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\newcommand{\myZ}{\hbox{Z}^+}
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\newcommand{\myalgsheader}[0]
{\myheader{UMBC, Department of Computer Science and Electrical Engineering}
{CMSC-441 Algorithms}{Fall 1998}{Section 0101}}

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{\myrunningalgshead{Solution to HW {#1}, Problem {#2}}}

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{\large {\bf Solution to HW {#1}, Problem {#2}}}\\
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\myhwtitle{1}{4 (3.2-2)}{Ben Bitdiddle}

\bigskip

%%%%% Begin Solution %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent
We must find an asymptotic upper bound on the summation
\myzrfunction{S} defined by

% Here's an example of how to do display equations.
%
\begin{equation} \label{a}
S(n) = \sum_{k=0}^{\lfloor \lg n \rfloor} 
\left\lceil {{n} \over {2^k}} \right\rceil,
\end{equation}

\noindent whenever $n \in {\myZ}$.  

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As a partial check on our work,  in Table~\ref{tvalues}
we computationally verify Equation~\ref{a}
for $n=1,2$.

% Here's an example of how to do tables.
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\begin{table}[h] \label{tvalues}
\begin{center}
\begin{tabular}{|l|l|l|l|}
$n$ & $2n-2$ & $S(n)$ & $2n+\lg n + 1$ \\ \hline
$1$ & $2(1)-2=0$ &
$\lceil 1/2^0 \rceil = 1$ & $2(1) + \lg 1 + 1= 3 $\\
$2$ & $2(2)-2 = 2$ &
$\lceil 2/2^0\rceil + \lceil 2/2^1 \rceil = 2 + 1 = 3$ &
$2(2) + \lg 2 + 1= 6$ \\ \hline
\end{tabular}.
\end{center}

\caption{Values of $S(n)$ for $n = 1, 2$.}
\end{table}

\vfill
\noindent
{\it Acknowledgment:  I thank Mary Lou
for suggesting to use the method of splitting and bounding.}

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