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<b>Data Structures and Algorithms
with Object-Oriented Design Patterns in C++</b><br>
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<H2><A NAME="SECTION0017330000000000000000">Topological Sort</A></H2>
<A NAME="secgraphstoposort"> </A>
<P>
A topological sort is an ordering of the vertices of a
<em>directed acyclic graph</em>
given by the following definition:
<P>
<BLOCKQUOTE> <b>Definition (Topological Sort)</b>
Consider a directed acyclic 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" >.
A <em>topological sort</em><A NAME=50560> </A><A NAME=50561> </A>
of the vertices of <I>G</I>
is a sequence <IMG WIDTH=145 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline72047" SRC="img2407.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2407.gif" >
in which each element of <IMG WIDTH=10 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline71357" SRC="img2283.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2283.gif" > appears exactly once.
For every pair of distinct vertices <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" > and <IMG WIDTH=13 HEIGHT=16 ALIGN=MIDDLE ALT="tex2html_wrap_inline72053" SRC="img2408.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2408.gif" > in the sequence <I>S</I>,
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>,
i.e., <IMG WIDTH=75 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline72061" SRC="img2409.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2409.gif" >,
then <I>i</I><I><</I><I>j</I>.
</BLOCKQUOTE>
<P>
Informally, a topological sort is a list of the vertices of a DAG
in which all the successors of any given vertex
appear in the sequence after that vertex.
Consider the directed acyclic graph <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline72065" SRC="img2410.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2410.gif" > shown in Figure <A HREF="page557.html#figgraph8" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page557.html#figgraph8"><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 sequence <IMG WIDTH=172 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline72067" SRC="img2411.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2411.gif" > is a topological sort
of the vertices of <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline72065" SRC="img2410.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2410.gif" >.
To see that this is so,
consider the set of vertices:
<P> <IMG WIDTH=415 HEIGHT=36 ALIGN=BOTTOM ALT="displaymath72041" SRC="img2412.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2412.gif" ><P>
The vertices in each edge are in alphabetical order,
and so is the sequence <I>S</I>.
<P>
<P><A NAME="50792"> </A><A NAME="figgraph8"> </A> <IMG WIDTH=575 HEIGHT=238 ALIGN=BOTTOM ALT="figure50567" SRC="img2413.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2413.gif" ><BR>
<STRONG>Figure:</STRONG> A Directed Acyclic Graph<BR>
<P>
<P>
It should also be evident from Figure <A HREF="page557.html#figgraph8" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page557.html#figgraph8"><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> that
a topological sort is not unique.
For example, the following are also valid topological sorts
of the graph <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline72065" SRC="img2410.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2410.gif" >:
<P> <IMG WIDTH=500 HEIGHT=95 ALIGN=BOTTOM ALT="eqnarray50796" SRC="img2414.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2414.gif" ><P>
<P>
One way to find a topological sort
is to consider the <em>in-degrees</em><A NAME=50799> </A> of the vertices.
(The number above a vertex in Figure <A HREF="page557.html#figgraph8" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page557.html#figgraph8"><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).
Clearly the first vertex in a topological sort must have in-degree zero and
every DAG must contain at least one vertex with in-degree zero.
A simple algorithm to create the sort goes like this:
<P>
Repeat the following steps until the graph is empty:
<OL><LI> Select a vertex that has in-degree zero.<LI> Add the vertex to the sort.<LI> Delete the vertex and all the edges emanating from it
from the graph.
</OL><BR> <HR>
<UL>
<LI> <A NAME="tex2html8801" HREF="page558.html#SECTION0017331000000000000000" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page558.html#SECTION0017331000000000000000">Implementation</A>
<LI> <A NAME="tex2html8802" HREF="page559.html#SECTION0017332000000000000000" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page559.html#SECTION0017332000000000000000">Running Time Analysis</A>
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