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<HTML><HEAD><TITLE>Sparse vs. Dense Graphs</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="tex2html7260" HREF="page531.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="../icons/next_motif.gif"></A> <A NAME="tex2html7258" HREF="page528.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="../icons/up_motif.gif"></A> <A NAME="tex2html7252" HREF="page529.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="../icons/previous_motif.gif"></A>  <A NAME="tex2html7262" HREF="page611.html"><IMG WIDTH=43 HEIGHT=24 ALIGN=BOTTOM ALT="index" SRC="../icons/index_motif.gif"></A> <BR><HR><H3><A NAME="SECTION0016112000000000000000">Sparse vs. Dense Graphs</A></H3><P>Informally, a graph with relatively few edges is <em>sparse</em>,and a graph with many edges is <em>dense</em>.The following definition defines precisely what we meanwhen we say that a graph ``has relatively few edges'':<P><BLOCKQUOTE> <b>Definition (Sparse Graph)</b>    A <em>sparse graph</em><A NAME=49154>&#160;</A><A NAME=49155>&#160;</A>    is a graph  <IMG WIDTH=73 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline70549" SRC="img2166.gif"  >    in which  <IMG WIDTH=82 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline70933" SRC="img2244.gif"  >.</BLOCKQUOTE><P>For example, consider a graph  <IMG WIDTH=73 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline70549" SRC="img2166.gif"  > with <I>n</I> nodes.Suppose that the out-degree of each vertex in <I>G</I> issome fixed constant <I>k</I>.Graph <I>G</I> is a <em>sparse graph</em> because <IMG WIDTH=133 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline70945" SRC="img2245.gif"  >.<P>A graph that is not sparse is said to be <em>dense</em>:<P><BLOCKQUOTE> <b>Definition (Dense Graph)</b>    A <em>dense graph</em><A NAME=49162>&#160;</A><A NAME=49163>&#160;</A>    is a graph  <IMG WIDTH=73 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline70549" SRC="img2166.gif"  >    in which  <IMG WIDTH=89 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline70949" SRC="img2246.gif"  >.</BLOCKQUOTE><P>For example, consider a graph  <IMG WIDTH=73 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline70549" SRC="img2166.gif"  > with <I>n</I> nodes.Suppose that the out-degree of each vertex in <I>G</I> issome fraction <I>f</I> of <I>n</I>,  <IMG WIDTH=67 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline70961" SRC="img2247.gif"  >.For example, if <I>n</I>=16 and <I>f</I>=0.25,the out-degree of each node is 4.Graph <I>G</I> is a <em>dense graph</em> because <IMG WIDTH=147 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline70971" SRC="img2248.gif"  >.<P><HR><A NAME="tex2html7260" HREF="page531.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="../icons/next_motif.gif"></A> <A NAME="tex2html7258" HREF="page528.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="../icons/up_motif.gif"></A> <A NAME="tex2html7252" HREF="page529.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="../icons/previous_motif.gif"></A>  <A NAME="tex2html7262" 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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