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<HTML><HEAD><TITLE>An Asymptotic Upper Bound-Big Oh</TITLE></HEAD><BODY bgcolor="#FFFFFF"> <a href="../index.html" target="_top"><img src="../icons/usins.gif" alt="Logo" align=right></a><b>Data Structures and Algorithms with Object-Oriented Design Patterns in Python</b><br><A NAME="tex2html1884" HREF="page60.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="../icons/next_motif.gif"></A> <A NAME="tex2html1882" HREF="page58.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="../icons/up_motif.gif"></A> <A NAME="tex2html1876" HREF="page58.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="../icons/previous_motif.gif"></A>  <A NAME="tex2html1886" HREF="page611.html"><IMG WIDTH=43 HEIGHT=24 ALIGN=BOTTOM ALT="index" SRC="../icons/index_motif.gif"></A> <BR><HR><H1><A NAME="SECTION003100000000000000000">An Asymptotic Upper Bound-Big Oh</A></H1><P>In 1892, P. Bachmann<A NAME=1379>&#160;</A>invented a notation for characterizingthe asymptotic behavior of functions.His invention has come to be known as <em>big oh notation</em>:<P><BLOCKQUOTE> <b>Definition (Big Oh)</b><A NAME="defnbigoh">&#160;</A><A NAME=1384>&#160;</A><A NAME=1399>&#160;</A>Consider a function <I>f</I>(<I>n</I>) which is non-negativefor all integers  <IMG WIDTH=39 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline58503" SRC="img238.gif"  >.We say that ``<I>f</I>(<I>n</I>) is big oh <I>g</I>(<I>n</I>),''which we write <I>f</I>(<I>n</I>)=<I>O</I>(<I>g</I>(<I>n</I>)),if there exists an integer  <IMG WIDTH=16 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline58491" SRC="img235.gif"  > and a constant <I>c</I><I>&gt;</I>0such that for all integers  <IMG WIDTH=47 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline58525" SRC="img239.gif"  >, <IMG WIDTH=88 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline58527" SRC="img240.gif"  >.</BLOCKQUOTE><BR> <HR><UL> <LI> <A NAME="tex2html1887" HREF="page60.html#SECTION003110000000000000000">A Simple Example</A><LI> <A NAME="tex2html1888" HREF="page61.html#SECTION003120000000000000000">Big Oh Fallacies and Pitfalls</A><LI> <A NAME="tex2html1889" HREF="page62.html#SECTION003130000000000000000">Properties of Big Oh</A><LI> <A NAME="tex2html1890" HREF="page63.html#SECTION003140000000000000000">About Polynomials</A><LI> <A NAME="tex2html1891" HREF="page64.html#SECTION003150000000000000000">About Logarithms</A><LI> <A NAME="tex2html1892" HREF="page65.html#SECTION003160000000000000000">Tight Big Oh Bounds</A><LI> <A NAME="tex2html1893" HREF="page66.html#SECTION003170000000000000000">More Big Oh Fallacies and Pitfalls</A><LI> <A NAME="tex2html1894" HREF="page67.html#SECTION003180000000000000000">Conventions for Writing Big Oh Expressions</A></UL><HR><A NAME="tex2html1884" HREF="page60.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="../icons/next_motif.gif"></A> <A NAME="tex2html1882" HREF="page58.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="../icons/up_motif.gif"></A> <A NAME="tex2html1876" HREF="page58.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="../icons/previous_motif.gif"></A>  <A NAME="tex2html1886" HREF="page611.html"><IMG WIDTH=43 HEIGHT=24 ALIGN=BOTTOM ALT="index" SRC="../icons/index_motif.gif"></A> <P><ADDRESS><img src="../icons/bruno.gif" alt="Bruno" align=right><a href="../copyright.html">Copyright &#169; 2003</a> by <a href="../signature.html">Bruno R. Preiss, P.Eng.</a>  All rights reserved.</ADDRESS></BODY></HTML>

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