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<TITLE>Testing for Cycles in a Directed Graph</TITLE>
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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="SECTION0017343000000000000000">Testing for Cycles in a Directed Graph</A></H3>
<P>
The final application of graph traversal
that we consider in this section
is to test a directed graph for cycles.
An easy way to do this is to attempt a topological-order traversal
using the algorithm given in Section <A HREF="page557.html#secgraphstoposort" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page557.html#secgraphstoposort"><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>.
This algorithm only visits all the vertices of a directed graph
if that graph contains no cycles.
<P>
To see why this is so,
consider the directed cyclic graph <IMG WIDTH=24 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline72171" SRC="img2436.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2436.gif" > shown in Figure <A HREF="page563.html#figgraph11" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page563.html#figgraph11"><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>.
The topological traversal algorithm begins
by computing the <em>in-degrees</em> of the vertices.
(The number shown below each vertex in Figure <A HREF="page563.html#figgraph11" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page563.html#figgraph11"><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 the in-degree of that vertex).
<P>
<P><A NAME="51406"> </A><A NAME="figgraph11"> </A> <IMG WIDTH=575 HEIGHT=95 ALIGN=BOTTOM ALT="figure51247" SRC="img2437.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2437.gif" ><BR>
<STRONG>Figure:</STRONG> A Directed Cyclic Graph<BR>
<P>
<P>
At each step of the traversal,
a vertex with in-degree of zero is visited.
After a vertex is visited,
the vertex and all the edges emanating from that vertex
are removed from the graph.
Notice that if we remove vertex <I>a</I> and edge (<I>a</I>,<I>b</I>) from <IMG WIDTH=24 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline72171" SRC="img2436.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2436.gif" >,
all the remaining vertices have in-degrees of one.
The presence of the cycle prevents
the topological-order traversal from completing.
<P>
Therefore, the a simple way to test whether a directed graph is cyclic
is to attempt a topological traversal of its vertices.
If all the vertices are not visited,
the graph must be cyclic.
<P>
Program <A HREF="page563.html#proggraph6c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page563.html#proggraph6c"><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> gives the implementation of the <tt>IsCyclic</tt>
member function of the <tt>Digraph</tt> class.
This Boolean-valued accessor returns <tt>true</tt> if the graph is cyclic.
The implementation simply makes use of a <tt>CountingVisitor</tt>
to count the number of vertices visited during a
<tt>TopologicalOrderTraversal</tt> of the graph.
<P>
<P><A NAME="51466"> </A><A NAME="proggraph6c"> </A> <IMG WIDTH=575 HEIGHT=124 ALIGN=BOTTOM ALT="program51416" SRC="img2438.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2438.gif" ><BR>
<STRONG>Program:</STRONG> <tt>Digraph</tt> Class <tt>IsCyclic</tt> Member Function Definition<BR>
<P>
<P>
The worst-case running time of the <tt>IsCyclic</tt> routine
is determined by the time taken by the <tt>TopologicalOrderTraversal</tt>.
Since <IMG WIDTH=120 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline61332" SRC="img715.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img715.gif" >,
the running time of <tt>IsCyclic</tt> 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" > when adjacency matrices are used to represent the graph
and <IMG WIDTH=82 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline71915" SRC="img2391.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2391.gif" > when adjacency lists are used.
<P>
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