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Wed Aug 14 23:10:36 2019 UTC (4 years, 9 months ago) by dashley
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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 dashley 278 $HeadURL$
75     $Revision$
76     $Date$
77     $Author$
78 dashley 140 \end{verbatim}
79     \end{tiny}
80     \noindent\rule[0.25in]{\textwidth}{1pt}
81     \end{figure}
82    
83 dashley 278 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
84 dashley 140 %
85     %End of file C_RAS0.TEX

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