| 1 |
dashley |
140 |
%$Header$ |
| 2 |
|
|
|
| 3 |
|
|
\chapter[Solutions: \cratzeroxrefcomma{}Chapter \ref{crat0}] |
| 4 |
|
|
{Solutions: \cratzeroxrefcomma{}Chapter \ref{crat0}, \cratzerolongtitle{}} |
| 5 |
|
|
|
| 6 |
|
|
\label{cras0} |
| 7 |
|
|
|
| 8 |
|
|
\vworkexercisechapterheader{} |
| 9 |
|
|
\begin{vworkexercisesolution}{\ref{exe:crat0:sexe0:01}} |
| 10 |
|
|
The value of the \texttt{state} variable when |
| 11 |
|
|
evaluating the \emph{if()} clause on the |
| 12 |
|
|
$n+1$'th invocation is |
| 13 |
|
|
|
| 14 |
|
|
\begin{equation} |
| 15 |
|
|
\label{eq:cras0:exe01:01} |
| 16 |
|
|
K_1 - nK_4 + (n+1) K_2 . |
| 17 |
|
|
\end{equation} |
| 18 |
|
|
|
| 19 |
|
|
We require on the $n+1$'th invocation, in order for the |
| 20 |
|
|
test of the \emph{if()} clause to fail (i.e. that the function |
| 21 |
|
|
``\texttt{A()}'' has been run on the first $n$ invocations of |
| 22 |
|
|
the base subroutine but is not run on the $n+1$'th invocation), |
| 23 |
|
|
that: |
| 24 |
|
|
|
| 25 |
|
|
\begin{equation} |
| 26 |
|
|
\label{eq:cras0:exe01:01b} |
| 27 |
|
|
K_1 - nK_4 + (n+1) K_2 < K_3. |
| 28 |
|
|
\end{equation} |
| 29 |
|
|
|
| 30 |
|
|
Solving this inequality for the smallest integral |
| 31 |
|
|
value of $n$ yields: |
| 32 |
|
|
|
| 33 |
|
|
\begin{equation} |
| 34 |
|
|
\label{eq:cras0:exe01:01c} |
| 35 |
|
|
n = \left\lceil |
| 36 |
|
|
\frac{K_1 + K_2 - K_3 + 1}{K_4 - K_2} |
| 37 |
|
|
\right\rceil . |
| 38 |
|
|
\end{equation} |
| 39 |
|
|
|
| 40 |
|
|
It can be verified using an example that |
| 41 |
|
|
(\ref{eq:cras0:exe01:01c}). For example, with |
| 42 |
|
|
$K_1 = 10$, $K_2 = 3$, $K_3 = 7$, and $K_4 = 5$, |
| 43 |
|
|
(\ref{eq:cras0:exe01:01}) predicts that on the first |
| 44 |
|
|
$\lceil 7/2 \rceil = 4$ invocations of the base subroutine |
| 45 |
|
|
the subroutine ``\texttt{A()}'' will be run but on the 5th |
| 46 |
|
|
invocation it will not. Tracing the algorithm with the |
| 47 |
|
|
parameters specified reveals that at the |
| 48 |
|
|
test in the \emph{if()} statement |
| 49 |
|
|
on the first invocation of the |
| 50 |
|
|
subroutine, \texttt{state}=13 (``\texttt{A()}'' executed); |
| 51 |
|
|
on the second invocation of the |
| 52 |
|
|
subroutine, \texttt{state}=11 (``\texttt{A()}'' executed); |
| 53 |
|
|
on the third invocation of the |
| 54 |
|
|
subroutine, \texttt{state}=9 (``\texttt{A()}'' executed); |
| 55 |
|
|
on the fourth invocation of the |
| 56 |
|
|
subroutine, \texttt{state}=7 (``\texttt{A()}'' executed); |
| 57 |
|
|
and on the fifth invocation of the |
| 58 |
|
|
subroutine, \texttt{state}=5 (``\texttt{A()}'' not executed). |
| 59 |
|
|
This is in agreement with |
| 60 |
|
|
(\ref{eq:cras0:exe01:01}). |
| 61 |
|
|
\end{vworkexercisesolution} |
| 62 |
|
|
%\vworkexerciseseparator |
| 63 |
|
|
%\begin{vworkexercisesolution}{\ref{exe:cfry0:sexe0:02}} |
| 64 |
|
|
%Placeholder. |
| 65 |
|
|
%\end{vworkexercisesolution} |
| 66 |
|
|
\vworkexercisechapterfooter |
| 67 |
|
|
|
| 68 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 69 |
|
|
\vfill |
| 70 |
|
|
\noindent\begin{figure}[!b] |
| 71 |
|
|
\noindent\rule[-0.25in]{\textwidth}{1pt} |
| 72 |
|
|
\begin{tiny} |
| 73 |
|
|
\begin{verbatim} |
| 74 |
|
|
$RCSfile: c_ras0.tex,v $ |
| 75 |
|
|
$Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_ras0/c_ras0.tex,v $ |
| 76 |
|
|
$Revision: 1.4 $ |
| 77 |
|
|
$Author: dtashley $ |
| 78 |
|
|
$Date: 2003/11/03 02:14:24 $ |
| 79 |
|
|
\end{verbatim} |
| 80 |
|
|
\end{tiny} |
| 81 |
|
|
\noindent\rule[0.25in]{\textwidth}{1pt} |
| 82 |
|
|
\end{figure} |
| 83 |
|
|
|
| 84 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 85 |
|
|
% $Log: c_ras0.tex,v $ |
| 86 |
|
|
% Revision 1.4 2003/11/03 02:14:24 dtashley |
| 87 |
|
|
% All duplicate labels as flagged by LaTeX removed. Additional files added |
| 88 |
|
|
% to Microsoft Visual Studio edit context. |
| 89 |
|
|
% |
| 90 |
|
|
% Revision 1.3 2003/03/08 04:11:19 dtashley |
| 91 |
|
|
% Friday evening safety checkin. |
| 92 |
|
|
% |
| 93 |
|
|
% Revision 1.2 2001/07/01 21:10:59 dtashley |
| 94 |
|
|
% Safety check-in after major re-org. |
| 95 |
|
|
% |
| 96 |
|
|
% Revision 1.1.1.1 2001/07/01 00:14:03 dtashley |
| 97 |
|
|
% Initial import. |
| 98 |
|
|
% |
| 99 |
|
|
% |
| 100 |
|
|
%End of file C_RAS0.TEX |
Parent Directory
| 
Revision Log