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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="SECTION0016440000000000000000">Average Running Time</A></H2>
<P>
To determine the average running time for the quicksort algorithm,
we shall assume that each element of the sequence has an equal chance
of being selected for the pivot.
Therefore, if <I>i</I> is the number of elements in a sequence of length <I>n</I>
less than the pivot,
then <I>i</I> is uniformly distributed in the interval [0,<I>n</I>-1].
Consequently, the average value of <IMG WIDTH=139 HEIGHT=32 ALIGN=MIDDLE ALT="tex2html_wrap_inline70219" SRC="img2168.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2168.gif" >.
Similarly, the average the value of
<IMG WIDTH=253 HEIGHT=32 ALIGN=MIDDLE ALT="tex2html_wrap_inline70221" SRC="img2169.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2169.gif" >.
To determine the average running time,
we rewrite Equation <A HREF="page496.html#eqnsortingii" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page496.html#eqnsortingii"><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> thus:
<P><A NAME="eqnsortingiii"> </A> <IMG WIDTH=506 HEIGHT=102 ALIGN=BOTTOM ALT="eqnarray37899" SRC="img2170.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2170.gif" ><P>
<P>
To solve this recurrence we consider the case <I>n</I><I>></I>2
and then multiply Equation <A HREF="page499.html#eqnsortingiii" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page499.html#eqnsortingiii"><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> by <I>n</I> to get
<P> <IMG WIDTH=500 HEIGHT=49 ALIGN=BOTTOM ALT="equation37919" SRC="img2171.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2171.gif" ><P>
Since this equation is valid for any <I>n</I><I>></I>2,
by substituting <I>n</I>-1 for <I>n</I> we can also write
<P><A NAME="eqnsortingiv"> </A> <IMG WIDTH=500 HEIGHT=49 ALIGN=BOTTOM ALT="equation37923" SRC="img2172.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2172.gif" ><P>
which is valid for <I>n</I><I>></I>3.
Subtracting Equation <A HREF="page499.html#eqnsortingiv" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page499.html#eqnsortingiv"><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> from Equation <A HREF="page499.html#eqnsortingiii" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page499.html#eqnsortingiii"><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> gives
<P> <IMG WIDTH=413 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath70207" SRC="img2173.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2173.gif" ><P>
which can be rewritten as
<P><A NAME="eqnsortingv"> </A> <IMG WIDTH=500 HEIGHT=38 ALIGN=BOTTOM ALT="equation37930" SRC="img2174.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2174.gif" ><P>
<P>
Equation <A HREF="page499.html#eqnsortingv" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page499.html#eqnsortingv"><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> can be solved by telescoping like this:
<P><A NAME="eqnsortingvii"> </A><A NAME="eqnsortingvi"> </A> <IMG WIDTH=508 HEIGHT=274 ALIGN=BOTTOM ALT="eqnarray37942" SRC="img2175.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2175.gif" ><P>
Adding together Equation <A HREF="page499.html#eqnsortingvi" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page499.html#eqnsortingvi"><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> through Equation <A HREF="page499.html#eqnsortingvii" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page499.html#eqnsortingvii"><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> gives
<P> <IMG WIDTH=500 HEIGHT=143 ALIGN=BOTTOM ALT="eqnarray37988" SRC="img2176.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2176.gif" ><P>
where <IMG WIDTH=38 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline70235" SRC="img2177.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2177.gif" > is the <IMG WIDTH=60 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline70237" SRC="img2178.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2178.gif" >
<em>harmonic number</em><A NAME=38016> </A>.
Finally, multiplying through by <I>n</I>+1 gives
<P> <IMG WIDTH=358 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath70208" SRC="img2179.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2179.gif" ><P>
<P>
In Section <A HREF="page44.html#secmodelharmonic" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page44.html#secmodelharmonic"><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> it is shown that
<IMG WIDTH=96 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline64680" SRC="img1268.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1268.gif" >,
where <IMG WIDTH=93 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline64682" SRC="img1269.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1269.gif" >
is called <em>Euler's constant</em><A NAME=38020> </A>.
Thus, we get that the average running time of quicksort is
<P> <IMG WIDTH=500 HEIGHT=41 ALIGN=BOTTOM ALT="eqnarray38021" SRC="img2180.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2180.gif" ><P>
Table <A HREF="page499.html#tblsortingexchange" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page499.html#tblsortingexchange"><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> summarizes the asymptotic running times
for the quicksort routine and compares it to those of bubble sort.
Notice that the best-case and average case running times
for the quicksort algorithm have the same asymptotic bound!
<P>
<P><A NAME="38025"> </A>
<P>
<A NAME="tblsortingexchange"> </A>
<DIV ALIGN=CENTER><P ALIGN=CENTER><TABLE COLS=4 BORDER FRAME=HSIDES RULES=GROUPS>
<COL ALIGN=LEFT><COL ALIGN=CENTER><COL ALIGN=CENTER><COL ALIGN=CENTER>
<TBODY>
<TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP>
</TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP COLSPAN=3> running time</TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP>
<P>
algorithm </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> best case </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> average case </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> worst case </TD></TR>
</TBODY><TBODY>
<TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP>bubble sort </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=40 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59179" SRC="img261.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img261.gif" > </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=40 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59179" SRC="img261.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img261.gif" > </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=40 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59179" SRC="img261.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img261.gif" > </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP>
quicksort (random pivot selection)
</TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=68 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59897" SRC="img405.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img405.gif" > </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=68 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59897" SRC="img405.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img405.gif" > </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=40 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59179" SRC="img261.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img261.gif" > </TD></TR>
</TBODY>
<CAPTION ALIGN=BOTTOM><STRONG>Table:</STRONG> Running Times for Exchange Sorting</CAPTION></TABLE>
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