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80 lines
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<!Converted with LaTeX2HTML 0.6.5 (Tue Nov 15 1994) by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds >
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<TITLE>28.1.5.1. Topological Sorting</TITLE>
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<meta name="description" value=" Topological Sorting">
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<b>Common Lisp the Language, 2nd Edition</b>
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<BR> <HR><A NAME=tex2html5096 HREF="node276.html"><IMG ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME=tex2html5094 HREF="node274.html"><IMG ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME=tex2html5088 HREF="node274.html"><IMG ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <A NAME=tex2html5098 HREF="node1.html"><IMG ALIGN=BOTTOM ALT="contents" SRC="icons/contents_motif.gif"></A> <A NAME=tex2html5099 HREF="index.html"><IMG ALIGN=BOTTOM ALT="index" SRC="icons/index_motif.gif"></A> <BR>
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<B> Next:</B> <A NAME=tex2html5097 HREF="node276.html"> Examples</A>
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<B>Up:</B> <A NAME=tex2html5095 HREF="node274.html"> Determining the Class </A>
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<B> Previous:</B> <A NAME=tex2html5089 HREF="node274.html"> Determining the Class </A>
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<HR> <P>
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<H3><A NAME=SECTION003215100000000000000>28.1.5.1. Topological Sorting</A></H3>
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<P>
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<img align=bottom alt="change_begin" src="gif/change_begin.gif"><br>
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Topological sorting proceeds by finding a class <i>C</i> in <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43435.gif"> such
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that no other class precedes that element according to the elements
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in <i>R</i>. The class <i>C</i> is placed first in the result.
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Remove <i>C</i> from <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43435.gif">, and remove all pairs of the form
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<b>(C, D)</b>,
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<IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43419.gif">, from <i>R</i>. Repeat the process, adding
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classes with no predecessors to the end of the result. Stop when no
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element can be found that has no predecessor.
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<P>
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If <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43435.gif"> is not empty and the process has stopped, the set <i>R</i> is
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inconsistent. If every class in the finite set of classes is preceded
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by another, then <i>R</i> contains a loop. That is, there is a chain of
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classes <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43423.gif">
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such that <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43425.gif"> precedes
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<IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43347.gif">, 1 <= <i>i</i> < <i>n</i>,
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and <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43341.gif"> precedes <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43387.gif">.
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<P>
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Sometimes there are several classes from <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43435.gif"> with no
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predecessors. In this case select the one that has a direct
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subclass rightmost in the class precedence list computed so far.
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<P>
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If there is no
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such candidate class, <i>R</i> does not generate a partial ordering - the
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<IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43499.gif">, <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43407.gif">, are inconsistent.
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<P>
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In more precise terms, let <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43441.gif">,
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<i>m</i> >= 2, be
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the classes from <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43435.gif"> with no predecessors. Let <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43447.gif">, <i>n</i> >= 1, be the class precedence list
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constructed so far. <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43387.gif"> is the most specific class, and <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43453.gif"> is the least specific. Let 1 <= <i>j</i> <= <i>n</i> be the largest number
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such that there exists an <i>i</i> where 1 <= <i>i</i> <= <i>m</i> and
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<IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43463.gif">
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is a direct superclass of <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43461.gif">;
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<IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43463.gif"> is placed next.
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<P>
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The effect of this rule for selecting from a set of classes with no
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predecessors is that classes in a simple superclass chain are
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adjacent in the class precedence list and that classes in each
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relatively separated subgraph are adjacent in the class
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precedence list. For example, let <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43475.gif"> and <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43487.gif"> be subgraphs
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whose only element in common is the class <i>J</i>. Suppose
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that no superclass of <i>J</i> appears in either <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43475.gif"> or <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43487.gif">.
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Let <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43387.gif"> be the bottom of <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43475.gif">; and let <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43591.gif"> be the
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bottom of <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43487.gif">. Suppose <i>C</i> is a class whose direct superclasses
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are <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43387.gif"> and <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43591.gif"> in that order; then the class precedence
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list for <i>C</i> will start with <i>C</i> and will be followed by all classes
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in <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43475.gif"> except <i>J</i>. All the classes of <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap43487.gif"> will be next.
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The class <i>J</i> and its superclasses will appear last.
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<br><img align=bottom alt="change_end" src="gif/change_end.gif">
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<P>
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<BR> <HR><A NAME=tex2html5096 HREF="node276.html"><IMG ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME=tex2html5094 HREF="node274.html"><IMG ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME=tex2html5088 HREF="node274.html"><IMG ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <A NAME=tex2html5098 HREF="node1.html"><IMG ALIGN=BOTTOM ALT="contents" SRC="icons/contents_motif.gif"></A> <A NAME=tex2html5099 HREF="index.html"><IMG ALIGN=BOTTOM ALT="index" SRC="icons/index_motif.gif"></A> <BR>
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<B> Next:</B> <A NAME=tex2html5097 HREF="node276.html"> Examples</A>
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<B>Up:</B> <A NAME=tex2html5095 HREF="node274.html"> Determining the Class </A>
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<B> Previous:</B> <A NAME=tex2html5089 HREF="node274.html"> Determining the Class </A>
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