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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="SECTION003190000000000000000">Best-Case and Worst-Case Running Times</A></H2>
<P>
In Section&nbsp;<A HREF="page43.html#secmodelavg" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page43.html#secmodelavg"><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 derived the average running
time of Program&nbsp;<A HREF="page42.html#progfindmaxc" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page42.html#progfindmaxc"><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> which finds the largest element of an array.
In order to do this we had to determine the probability that a certain
program statement is executed.
To do this, we made an assumption about
the <em>average</em> input to the program.
<P>
The analysis can be significantly simplified if we simply wish
to determine the <em>worst case</em> running time.
For Program&nbsp;<A HREF="page42.html#progfindmaxc" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page42.html#progfindmaxc"><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>,
the worst-case scenario occurs when line&nbsp;6 is executed in every iteration
of the loop.
We saw that this corresponds to the case in which the input array is
ordered from smallest to largest.
In terms of Equation&nbsp;<A HREF="page43.html#eqnmodelavg" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page43.html#eqnmodelavg"><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>,
this occurs when  <IMG WIDTH=42 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline58429" SRC="img88.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img88.gif"  >.
Thus, the worst-case running time is given by
<P> <IMG WIDTH=500 HEIGHT=154 ALIGN=BOTTOM ALT="eqnarray768" SRC="img125.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img125.gif"  ><P>
<P>
Similarly, the <em>best-case</em> running time occurs when line&nbsp;6
is never executed.
This corresponds to the case in which the input array is ordered from
largest to smallest.
This occurs when  <IMG WIDTH=43 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline58431" SRC="img89.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img89.gif"  > and
best-case running time is
<P> <IMG WIDTH=500 HEIGHT=71 ALIGN=BOTTOM ALT="eqnarray777" SRC="img126.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img126.gif"  ><P>
<P>
In summary we have the following results for the running time of
Program&nbsp;<A HREF="page42.html#progfindmaxc" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page42.html#progfindmaxc"><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>:
<P> <IMG WIDTH=363 HEIGHT=134 ALIGN=BOTTOM ALT="gather784" SRC="img127.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img127.gif"  ><P><HR><A NAME="tex2html2447" HREF="page46.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page46.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/next_motif.gif"></A> <A NAME="tex2html2445" HREF="page35.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page35.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/up_motif.gif"></A> <A NAME="tex2html2439" HREF="page44.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page44.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/previous_motif.gif"></A> <A NAME="tex2html2449" HREF="page9.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page9.html"><IMG WIDTH=65 HEIGHT=24 ALIGN=BOTTOM ALT="contents" SRC="contents_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/contents_motif.gif"></A> <A NAME="tex2html2450" HREF="page620.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page620.html"><IMG WIDTH=43 HEIGHT=24 ALIGN=BOTTOM ALT="index" SRC="index_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/index_motif.gif"></A> <P><ADDRESS>
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