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<b>Data Structures and Algorithms
with Object-Oriented Design Patterns in C++</b><br>
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<H3><A NAME="SECTION0017112000000000000000">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 mean
when we say that a graph ``has relatively few edges'':
<P>
<BLOCKQUOTE> <b>Definition (Sparse Graph)</b>
A <em>sparse graph</em><A NAME=49387> </A><A NAME=49388> </A>
is a graph <IMG WIDTH=72 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline71355" SRC="img2282.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2282.gif" >
in which <IMG WIDTH=82 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline71739" SRC="img2360.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2360.gif" >.
</BLOCKQUOTE>
<P>
For example, consider a graph <IMG WIDTH=72 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline71355" SRC="img2282.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2282.gif" > with <I>n</I> nodes.
Suppose that the out-degree of each vertex in <I>G</I> is
some fixed constant <I>k</I>.
Graph <I>G</I> is a <em>sparse graph</em> because
<IMG WIDTH=132 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline71751" SRC="img2361.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2361.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=49395> </A><A NAME=49396> </A>
is a graph <IMG WIDTH=72 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline71355" SRC="img2282.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2282.gif" >
in which <IMG WIDTH=88 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline71755" SRC="img2362.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2362.gif" >.
</BLOCKQUOTE>
<P>
For example, consider a graph <IMG WIDTH=72 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline71355" SRC="img2282.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2282.gif" > with <I>n</I> nodes.
Suppose that the out-degree of each vertex in <I>G</I> is
some fraction <I>f</I> of <I>n</I>, <IMG WIDTH=66 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline71767" SRC="img2363.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2363.gif" >.
E.g., 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=146 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline71777" SRC="img2364.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2364.gif" >.
<P>
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