📄 page534.html

📁 wqeqwvrw rkjqhwrjwq jkhrjqwhrwq jkhrwq
💻 HTML
字号:
<HTML>
<HEAD>
<TITLE>Sparse vs. Dense Graphs</TITLE>
</HEAD>
<BODY bgcolor="#FFFFFF">
 <img src="cover75.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cover75.gif" alt="Logo" align=right>
<b>Data Structures and Algorithms 
with Object-Oriented Design Patterns in C++</b><br>
<A NAME="tex2html8514" HREF="page535.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page535.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/next_motif.gif"></A> <A NAME="tex2html8512" HREF="page532.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page532.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/up_motif.gif"></A> <A NAME="tex2html8506" HREF="page533.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page533.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/previous_motif.gif"></A> <A NAME="tex2html8516" HREF="page9.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page9.html"><IMG WIDTH=65 HEIGHT=24 ALIGN=BOTTOM ALT="contents" SRC="contents_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/contents_motif.gif"></A> <A NAME="tex2html8517" HREF="page620.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page620.html"><IMG WIDTH=43 HEIGHT=24 ALIGN=BOTTOM ALT="index" SRC="index_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/index_motif.gif"></A> <BR><HR>
<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>&#160;</A><A NAME=49388>&#160;</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>&#160;</A><A NAME=49396>&#160;</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>
<HR><A NAME="tex2html8514" HREF="page535.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page535.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/next_motif.gif"></A> <A NAME="tex2html8512" HREF="page532.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page532.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/up_motif.gif"></A> <A NAME="tex2html8506" HREF="page533.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page533.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/previous_motif.gif"></A> <A NAME="tex2html8516" HREF="page9.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page9.html"><IMG WIDTH=65 HEIGHT=24 ALIGN=BOTTOM ALT="contents" SRC="contents_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/contents_motif.gif"></A> <A NAME="tex2html8517" HREF="page620.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page620.html"><IMG WIDTH=43 HEIGHT=24 ALIGN=BOTTOM ALT="index" SRC="index_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/index_motif.gif"></A> <P><ADDRESS>
<img src="bruno.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/bruno.gif" alt="Bruno" align=right>
<a href="javascript:if(confirm('http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/copyright.html  \n\nThis file was not retrieved by Teleport Pro, because it is addressed on a domain or path outside the boundaries set for its Starting Address.  \n\nDo you want to open it from the server?'))window.location='http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/copyright.html'" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/copyright.html">Copyright &#169; 1997</a> by <a href="javascript:if(confirm('http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/signature.html  \n\nThis file was not retrieved by Teleport Pro, because it is addressed on a domain or path outside the boundaries set for its Starting Address.  \n\nDo you want to open it from the server?'))window.location='http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/signature.html'" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/signature.html">Bruno R. Preiss, P.Eng.</a>  All rights reserved.

</ADDRESS>
</BODY>
</HTML>
⌨️ 快捷键说明

复制代码 Ctrl + C
搜索代码 Ctrl + F
全屏模式 F11
切换主题 Ctrl + Shift + D
显示快捷键 ?
增大字号 Ctrl + =
减小字号 Ctrl + -