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%$Header$ |
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|
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\chapter[Solutions: \cratzeroxrefcomma{}Chapter \ref{crat0}] |
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{Solutions: \cratzeroxrefcomma{}Chapter \ref{crat0}, \cratzerolongtitle{}} |
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|
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\label{cras0} |
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|
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\vworkexercisechapterheader{} |
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\begin{vworkexercisesolution}{\ref{exe:crat0:sexe0:01}} |
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The value of the \texttt{state} variable when |
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evaluating the \emph{if()} clause on the |
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$n+1$'th invocation is |
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|
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\begin{equation} |
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\label{eq:cras0:exe01:01} |
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K_1 - nK_4 + (n+1) K_2 . |
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\end{equation} |
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|
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We require on the $n+1$'th invocation, in order for the |
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test of the \emph{if()} clause to fail (i.e. that the function |
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``\texttt{A()}'' has been run on the first $n$ invocations of |
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the base subroutine but is not run on the $n+1$'th invocation), |
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that: |
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|
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\begin{equation} |
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\label{eq:cras0:exe01:01b} |
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K_1 - nK_4 + (n+1) K_2 < K_3. |
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\end{equation} |
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|
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Solving this inequality for the smallest integral |
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value of $n$ yields: |
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|
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\begin{equation} |
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\label{eq:cras0:exe01:01c} |
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n = \left\lceil |
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\frac{K_1 + K_2 - K_3 + 1}{K_4 - K_2} |
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\right\rceil . |
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\end{equation} |
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|
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It can be verified using an example that |
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(\ref{eq:cras0:exe01:01c}). For example, with |
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$K_1 = 10$, $K_2 = 3$, $K_3 = 7$, and $K_4 = 5$, |
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(\ref{eq:cras0:exe01:01}) predicts that on the first |
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$\lceil 7/2 \rceil = 4$ invocations of the base subroutine |
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the subroutine ``\texttt{A()}'' will be run but on the 5th |
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invocation it will not. Tracing the algorithm with the |
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parameters specified reveals that at the |
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test in the \emph{if()} statement |
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on the first invocation of the |
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subroutine, \texttt{state}=13 (``\texttt{A()}'' executed); |
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on the second invocation of the |
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subroutine, \texttt{state}=11 (``\texttt{A()}'' executed); |
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on the third invocation of the |
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subroutine, \texttt{state}=9 (``\texttt{A()}'' executed); |
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on the fourth invocation of the |
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subroutine, \texttt{state}=7 (``\texttt{A()}'' executed); |
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and on the fifth invocation of the |
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subroutine, \texttt{state}=5 (``\texttt{A()}'' not executed). |
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This is in agreement with |
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(\ref{eq:cras0:exe01:01}). |
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\end{vworkexercisesolution} |
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%\vworkexerciseseparator |
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%\begin{vworkexercisesolution}{\ref{exe:cfry0:sexe0:02}} |
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%Placeholder. |
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%\end{vworkexercisesolution} |
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\vworkexercisechapterfooter |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\vfill |
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\noindent\begin{figure}[!b] |
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\noindent\rule[-0.25in]{\textwidth}{1pt} |
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\begin{tiny} |
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\begin{verbatim} |
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$HeadURL$ |
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$Revision$ |
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$Date$ |
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$Author$ |
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\end{verbatim} |
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\end{tiny} |
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\noindent\rule[0.25in]{\textwidth}{1pt} |
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\end{figure} |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% |
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%End of file C_RAS0.TEX |
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