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<b>Data Structures and Algorithms
with Object-Oriented Design Patterns in C++</b><br>
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<H1><A NAME="SECTION004000000000000000000">Asymptotic Notation</A></H1>
<P>
<A NAME="chapasymptotic"> </A>
<P>
Suppose we are considering two algorithms, <I>A</I> and <I>B</I>,
for solving a given problem.
Furthermore, let us say that we have done a careful analysis
of the running times of each of the algorithms and determined them to be
<IMG WIDTH=40 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59029" SRC="img236.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img236.gif" > and <IMG WIDTH=42 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59031" SRC="img237.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img237.gif" >, respectively,
where <I>n</I> is a measure of the problem size.
Then it should be a fairly simple matter to compare the
two functions <IMG WIDTH=40 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59029" SRC="img236.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img236.gif" > and <IMG WIDTH=42 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59031" SRC="img237.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img237.gif" > to determine which
algorithm is <em>the best</em>!
<P>
But is it really that simple?
What exactly does it mean for one function, say <IMG WIDTH=40 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59029" SRC="img236.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img236.gif" >,
to be <em>better than</em> another function, <IMG WIDTH=42 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59031" SRC="img237.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img237.gif" >?
One possibility arises if we know the problem size <em>a priori</em>.
E.g., suppose the problem size is <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 <IMG WIDTH=119 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59045" SRC="img239.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img239.gif" >.
Then clearly algorithm <I>A</I> is better than algorithm <I>B</I>
for problem size <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" >.
<P>
In the general case, we have no <em>a priori</em> knowledge of the problem size.
However, if it can be shown, say,
that <IMG WIDTH=164 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59053" SRC="img240.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img240.gif" >,
then algorithm <I>A</I> is better than algorithm <I>B</I>
regardless of the problem size.
<P>
Unfortunately, we usually don't know the problem size beforehand,
nor is it true that one of the functions is less than or equal the other
over the entire range of problem sizes.
In this case,
we consider the <em>asymptotic</em> behavior<A NAME=1317> </A>
of the two functions for very large problem sizes.
<P>
<BR> <HR>
<UL>
<LI> <A NAME="tex2html2583" HREF="page57.html#SECTION004100000000000000000" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page57.html#SECTION004100000000000000000">An Asymptotic Upper Bound-Big Oh</A>
<LI> <A NAME="tex2html2584" HREF="page66.html#SECTION004200000000000000000" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page66.html#SECTION004200000000000000000">An Asymptotic Lower Bound-Omega</A>
<LI> <A NAME="tex2html2585" HREF="page69.html#SECTION004300000000000000000" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page69.html#SECTION004300000000000000000">More Notation-Theta and Little Oh</A>
<LI> <A NAME="tex2html2586" HREF="page70.html#SECTION004400000000000000000" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page70.html#SECTION004400000000000000000">Asymptotic Analysis of Algorithms</A>
<LI> <A NAME="tex2html2587" HREF="page77.html#SECTION004500000000000000000" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page77.html#SECTION004500000000000000000">Exercises</A>
<LI> <A NAME="tex2html2588" HREF="page78.html#SECTION004600000000000000000" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page78.html#SECTION004600000000000000000">Projects</A>
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