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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="SECTION0016430000000000000000">Running Time Analysis</A></H2>
<P>
The running time of the recursive <tt>DoSort</tt> routine (Program <A HREF="page495.html#progsorter4c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page495.html#progsorter4c"><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>)
is given by
<P><A NAME="eqnsortingi"> </A> <IMG WIDTH=529 HEIGHT=48 ALIGN=BOTTOM ALT="equation37850" SRC="img2158.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2158.gif" ><P>
where <I>n</I> is the number of elements in sequence to be sorted,
<IMG WIDTH=105 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline70173" SRC="img2159.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2159.gif" > is the running time of the <tt>SelectPivot</tt> function,
and <I>i</I> is the number of elements which end up to the left of the pivot,
<IMG WIDTH=91 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline70177" SRC="img2160.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2160.gif" >.
<P>
The running time of <tt>DoSort</tt> is affected by the <tt>SelectPivot</tt>
routine in two ways:
First, the value of the pivot chosen affects the sizes of the subsequences.
I.e., the pivot determines the value <I>i</I> in Equation <A HREF="page496.html#eqnsortingi" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page496.html#eqnsortingi"><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>.
Second, the running time of the <tt>SelectPivot</tt> routine itself,
<IMG WIDTH=105 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline70173" SRC="img2159.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2159.gif" >,
must be taken into account.
Fortunately, if <IMG WIDTH=161 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline70183" SRC="img2161.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2161.gif" >,
we can ignore its running time because there is already an <I>O</I>(<I>n</I>) term
in the expression.
<P>
In order to solve Equation <A HREF="page496.html#eqnsortingi" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page496.html#eqnsortingi"><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>,
we assume that <IMG WIDTH=161 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline70183" SRC="img2161.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2161.gif" >
and then drop the <IMG WIDTH=28 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline58163" SRC="img1.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1.gif" >s from the recurrence to get
<P><A NAME="eqnsortingii"> </A> <IMG WIDTH=500 HEIGHT=48 ALIGN=BOTTOM ALT="equation37861" SRC="img2162.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2162.gif" ><P>
Clearly the solution depends on the value of <I>i</I>.
<P>
<BR> <HR>
<UL>
<LI> <A NAME="tex2html8052" HREF="page497.html#SECTION0016431000000000000000" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page497.html#SECTION0016431000000000000000">Worst-Case Running Time</A>
<LI> <A NAME="tex2html8053" HREF="page498.html#SECTION0016432000000000000000" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page498.html#SECTION0016432000000000000000">Best-Case Running Time</A>
</UL>
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