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<b>Data Structures and Algorithms
with Object-Oriented Design Patterns in C++</b><br>
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<H2><A NAME="SECTION004210000000000000000">A Simple Example</A></H2>
<P>
Consider the function <IMG WIDTH=163 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59933" SRC="img410.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img410.gif" >
which is shown in Figure <A HREF="page67.html#figomega" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page67.html#figomega"><IMG ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A>.
Clearly, <I>f</I>(<I>n</I>) is non-negative for all integers <IMG WIDTH=38 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59063" SRC="img241.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img241.gif" >.
We wish to show that <IMG WIDTH=92 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59939" SRC="img411.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img411.gif" >.
According to Definition <A HREF="page66.html#defnomega" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page66.html#defnomega"><IMG ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A>,
in order to show this we need to find an integer <IMG WIDTH=16 HEIGHT=15 ALIGN=MIDDLE ALT="tex2html_wrap_inline59043" SRC="img238.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img238.gif" > and a constant <I>c</I><I>></I>0
such that for all integers <IMG WIDTH=46 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline59075" SRC="img242.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img242.gif" >, <IMG WIDTH=74 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59947" SRC="img412.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img412.gif" >.
<P>
As with big oh, it does not matter what the particular constants are--as long as they exist!
E.g., suppose we choose <I>c</I>=1.
Then
<P> <IMG WIDTH=500 HEIGHT=65 ALIGN=BOTTOM ALT="eqnarray1662" SRC="img413.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img413.gif" ><P>
Since <IMG WIDTH=84 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59951" SRC="img414.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img414.gif" > for all values of <IMG WIDTH=38 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59063" SRC="img241.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img241.gif" >,
we conclude that <IMG WIDTH=45 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline59955" SRC="img415.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img415.gif" >.
<P>
So, we have that for <I>c</I>=1 and <IMG WIDTH=45 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline59955" SRC="img415.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img415.gif" >, <IMG WIDTH=74 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59947" SRC="img412.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img412.gif" >
for all integers <IMG WIDTH=46 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline59075" SRC="img242.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img242.gif" >.
Hence, <IMG WIDTH=92 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59939" SRC="img411.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img411.gif" >.
Figure <A HREF="page67.html#figomega" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page67.html#figomega"><IMG ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A> clearly shows
that the function <IMG WIDTH=68 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59115" SRC="img249.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img249.gif" > is less than
the function <I>f</I>(<I>n</I>)=5<I>n</I>-64<I>n</I>+256 for all values of <IMG WIDTH=38 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59063" SRC="img241.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img241.gif" >.
Of course, there are many other values of <I>c</I> and <IMG WIDTH=16 HEIGHT=15 ALIGN=MIDDLE ALT="tex2html_wrap_inline59043" SRC="img238.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img238.gif" > that will do.
For example, <I>c</I>=2 and <IMG WIDTH=53 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline59103" SRC="img248.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img248.gif" >.
<P>
<P><A NAME="1719"> </A><A NAME="figomega"> </A> <IMG WIDTH=575 HEIGHT=322 ALIGN=BOTTOM ALT="figure1665" SRC="img417.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img417.gif" ><BR>
<STRONG>Figure:</STRONG> Showing that <IMG WIDTH=225 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline60011" SRC="img416.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img416.gif" >.<BR>
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