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%$Header: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_ras0/c_ras0.tex,v 1.4 2003/11/03 02:14:24 dtashley Exp $
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\chapter[Solutions: \cratzeroxrefcomma{}Chapter \ref{crat0}]
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{Solutions: \cratzeroxrefcomma{}Chapter \ref{crat0}, \cratzerolongtitle{}}
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\label{cras0}
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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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\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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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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\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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Solving this inequality for the smallest integral
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value of $n$ yields:
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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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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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\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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$RCSfile: c_ras0.tex,v $
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$Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_ras0/c_ras0.tex,v $
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$Revision: 1.4 $
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$Author: dtashley $
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$Date: 2003/11/03 02:14:24 $
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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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% $Log: c_ras0.tex,v $
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% Revision 1.4 2003/11/03 02:14:24 dtashley
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% All duplicate labels as flagged by LaTeX removed. Additional files added
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% to Microsoft Visual Studio edit context.
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%
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% Revision 1.3 2003/03/08 04:11:19 dtashley
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% Friday evening safety checkin.
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%
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% Revision 1.2 2001/07/01 21:10:59 dtashley
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% Safety check-in after major re-org.
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%
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% Revision 1.1.1.1 2001/07/01 00:14:03 dtashley
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% Initial import.
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%
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%
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%End of file C_RAS0.TEX
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