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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="SECTION0017261000000000000000">Space Comparison</A></H3>
<P>
Consider the representation of 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"  >.
In addition to the  <IMG WIDTH=17 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline71781" SRC="img2365.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2365.gif"  > <tt>Vertex</tt> class instances
and the  <IMG WIDTH=16 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline71793" SRC="img2367.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2367.gif"  > <tt>Edge</tt> class instances contained by the graph,
there is the storage required by the adjacency matrix.
In this case, the matrix is a  <IMG WIDTH=56 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline71885" SRC="img2383.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2383.gif"  > matrix
of pointers to <tt>Edge</tt> instances.
Therefore, the amount of storage required by an adjacency matrix
implementation is
<P><A NAME="eqnspaceam">&#160;</A> <IMG WIDTH=500 HEIGHT=41 ALIGN=BOTTOM ALT="equation49979" SRC="img2384.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2384.gif"  ><P>
<P>
On the other hand,
consider the amount of storage required
when we represent the same graph using adjacency lists.
In addition to the vertices and the edges themselves,
there are  <IMG WIDTH=17 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline71781" SRC="img2365.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2365.gif"  > linked lists.
If we use the <tt>LinkedList</tt> class defined in Section&nbsp;<A HREF="page88.html#secfdslinklist" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page88.html#secfdslinklist"><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>,
each such list has a <tt>head</tt> and <tt>tail</tt> pointer.
Altogether there are  <IMG WIDTH=16 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline71793" SRC="img2367.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2367.gif"  > linked lists elements,
each of which consists of a pointer to the next element of the lists
and a pointer to an edge.
Therefore, the total space required is
<P><A NAME="eqnspaceal">&#160;</A> <IMG WIDTH=533 HEIGHT=63 ALIGN=BOTTOM ALT="equation49991" SRC="img2385.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2385.gif"  ><P>
<P>
Notice that the space for the vertices and edges themselves
cancels out when we compare Equation&nbsp;<A HREF="page548.html#eqnspaceam" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page548.html#eqnspaceam"><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> with Equation&nbsp;<A HREF="page548.html#eqnspaceal" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page548.html#eqnspaceal"><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>.
If we assume that all pointers require the same amount of space,
we can conclude that adjacency lists use less space than
adjacency matrices when
<P> <IMG WIDTH=305 HEIGHT=35 ALIGN=BOTTOM ALT="displaymath71877" SRC="img2386.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2386.gif"  ><P>
For example, given a 10 node graph,
the adjacency lists version uses less space
when there are fewer than 45 edges.
As a rough rule of thumb,
we can say that adjacency lists use less space
when the average degree of a node,  <IMG WIDTH=76 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline71891" SRC="img2387.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2387.gif"  >,
satisfies  <IMG WIDTH=64 HEIGHT=31 ALIGN=MIDDLE ALT="tex2html_wrap_inline71893" SRC="img2388.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2388.gif"  >.
<P>
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