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<TITLE>An Asymptotic Lower Bound-Omega</TITLE>
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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="SECTION004200000000000000000">An Asymptotic Lower Bound-Omega</A></H1>
<P>
The big oh notation introduced in the preceding section
is an asymptotic <em>upper bound</em>.
In this section, we introduce a similar notation
for characterizing the asymptotic behavior of a function,
but in this case it is a <em>lower bound</em>.
<P>
<BLOCKQUOTE> <b>Definition (Omega)</b><A NAME=1655> </A><A NAME=1690> </A>
<A NAME="defnomega"> </A>
Consider a function <I>f</I>(<I>n</I>) which 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 say that ``<I>f</I>(<I>n</I>) is omega <I>g</I>(<I>n</I>),''
which we write <IMG WIDTH=104 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59919" SRC="img408.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img408.gif" >,
if there exists 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=88 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59927" SRC="img409.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img409.gif" >.
</BLOCKQUOTE>
<P>
The definition of omega is almost identical to that of big oh.
The only difference is in the comparison--for big oh it is
<IMG WIDTH=88 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59077" SRC="img243.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img243.gif" >; for omega, it is <IMG WIDTH=88 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59927" SRC="img409.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img409.gif" >.
All of the same conventions and caveats apply to omega as they do to big oh.
<P>
<BR> <HR>
<UL>
<LI> <A NAME="tex2html2715" HREF="page67.html#SECTION004210000000000000000" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page67.html#SECTION004210000000000000000">A Simple Example</A>
<LI> <A NAME="tex2html2716" HREF="page68.html#SECTION004220000000000000000" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page68.html#SECTION004220000000000000000">About Polynomials Again</A>
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