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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="SECTION0017111000000000000000">Adjacency Matrices</A></H3>
<P>
Consider a directed 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> vertices,
<IMG WIDTH=137 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline71661" SRC="img2346.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2346.gif" >.
The simplest graph representation scheme
uses an <IMG WIDTH=38 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline68887" SRC="img1918.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1918.gif" > matrix <I>A</I> of zeroes and ones given by
<P> <IMG WIDTH=332 HEIGHT=48 ALIGN=BOTTOM ALT="displaymath71647" SRC="img2347.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2347.gif" ><P>
I.e., the <IMG WIDTH=43 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline61089" SRC="img639.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img639.gif" > element of the matrix,
is a one only if <IMG WIDTH=51 HEIGHT=17 ALIGN=MIDDLE ALT="tex2html_wrap_inline71669" SRC="img2348.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2348.gif" > is an edge in <I>G</I>.
The matrix <I>A</I> is called an
<em>adjacency matrix</em><A NAME=49354> </A><A NAME=49355> </A>.
<P>
For example, the adjacency matrix for graph <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline71449" SRC="img2298.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2298.gif" > in Figure <A HREF="page525.html#figgraph1" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page525.html#figgraph1"><IMG ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A> is
<P> <IMG WIDTH=323 HEIGHT=76 ALIGN=BOTTOM ALT="displaymath71648" SRC="img2349.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2349.gif" ><P>
Clearly, the number of ones in the adjacency matrix is equal
to the number of edges in the graph.
<P>
One advantage of using an adjacency matrix is that it is easy to
determine the sets of edges emanating from and incident on a given vertex.
E.g., consider vertex <IMG WIDTH=11 HEIGHT=15 ALIGN=MIDDLE ALT="tex2html_wrap_inline71521" SRC="img2317.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2317.gif" >.
Each one in the <IMG WIDTH=17 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline58387" SRC="img77.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img77.gif" > row corresponds to an edge
that emanates from vertex <IMG WIDTH=11 HEIGHT=15 ALIGN=MIDDLE ALT="tex2html_wrap_inline71521" SRC="img2317.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2317.gif" >.
Conversely, each one in the <IMG WIDTH=17 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline58387" SRC="img77.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img77.gif" > column
corresponds to an edge incident on vertex <IMG WIDTH=11 HEIGHT=15 ALIGN=MIDDLE ALT="tex2html_wrap_inline71521" SRC="img2317.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2317.gif" >.
<P>
We can also use adjacency matrices to represent undirected graphs.
I.e., we represent an undirected 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> vertices,
using an <IMG WIDTH=38 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline68887" SRC="img1918.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1918.gif" > matrix <I>A</I> of zeroes and ones given by
<P> <IMG WIDTH=334 HEIGHT=48 ALIGN=BOTTOM ALT="displaymath71649" SRC="img2350.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2350.gif" ><P>
Since the two sets <IMG WIDTH=47 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline71695" SRC="img2351.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2351.gif" > and <IMG WIDTH=47 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline71697" SRC="img2352.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2352.gif" > are equivalent,
matrix <I>A</I> is symmetric about the diagonal.
I.e., <IMG WIDTH=73 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline71701" SRC="img2353.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2353.gif" >.
Furthermore, all of the entries on the diagonal are zero.
I.e., <IMG WIDTH=54 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline71703" SRC="img2354.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2354.gif" > for <IMG WIDTH=64 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline69739" SRC="img2095.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2095.gif" >.
<P>
For example, the adjacency matrix for graph <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline71603" SRC="img2334.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2334.gif" > in Figure <A HREF="page529.html#figgraph3" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page529.html#figgraph3"><IMG ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A> is
<P> <IMG WIDTH=318 HEIGHT=76 ALIGN=BOTTOM ALT="displaymath71650" SRC="img2355.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2355.gif" ><P>
In this case, there are twice as many ones in the adjacency
matrix as there are edges in the undirected graph.
<P>
A simple variation allows us to use an adjacency matrix
to represent an edge-labeled graph.
For example, given numeric edge labels,
we can represent a graph (directed or undirected)
using an <IMG WIDTH=38 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline68887" SRC="img1918.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1918.gif" > matrix <I>A</I> in which the <IMG WIDTH=25 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline68907" SRC="img1924.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1924.gif" >
is the numeric label associated with edge <IMG WIDTH=45 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline71715" SRC="img2356.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2356.gif" >
in the case of a directed graph,
and edge <IMG WIDTH=47 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline71695" SRC="img2351.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2351.gif" >,
in an undirected graph.
<P>
For example, the adjacency matrix for the graph <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline71633" SRC="img2343.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2343.gif" > in Figure <A HREF="page531.html#figgraph4" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page531.html#figgraph4"><IMG ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A> is
<P> <IMG WIDTH=334 HEIGHT=76 ALIGN=BOTTOM ALT="displaymath71651" SRC="img2357.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2357.gif" ><P>
In this case, the array entries corresponding to non-existent edges
have all been set to <IMG WIDTH=14 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline69187" SRC="img1984.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1984.gif" >.
Here <IMG WIDTH=14 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline69187" SRC="img1984.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1984.gif" > serves as a kind of <em>sentinel</em><A NAME=49376> </A>.
The value to use for the sentinel depends on the application.
For example, if the edges represent routes between geographic locations,
then a route of length <IMG WIDTH=14 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline69187" SRC="img1984.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1984.gif" > is much like one that does not exist.
<P>
Since the adjacency matrix has <IMG WIDTH=24 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline71641" SRC="img2344.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2344.gif" > entries,
the amount of spaced needed to represent the edges of
a graph is <IMG WIDTH=50 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline71729" SRC="img2358.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2358.gif" >,
<em>regardless of the actual number of edges</em> in the graph.
If the graph contains relatively few edges,
e.g., if <IMG WIDTH=68 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline71731" SRC="img2359.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2359.gif" >,
then most of the elements of the adjacency matrix will be zero (or <IMG WIDTH=14 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline69187" SRC="img1984.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1984.gif" >).
A matrix in which most of the elements are zero (or <IMG WIDTH=14 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline69187" SRC="img1984.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1984.gif" >)
is a <em>sparse matrix</em><A NAME=49379> </A><A NAME=49380> </A>.
<P>
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