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159 lines
5.6 KiB
HTML
<HTML>
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<!--
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-- Copyright (c) Lie-Quan Lee and Jeremy Siek, 2001
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--
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-- Permission to use, copy, modify, distribute and sell this software
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-- and its documentation for any purpose is hereby granted without fee,
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-- provided that the above copyright notice appears in all copies and
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-- that both that copyright notice and this permission notice appear
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-- in supporting documentation. Silicon Graphics makes no
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-- representations about the suitability of this software for any
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-- purpose. It is provided "as is" without express or implied warranty.
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-->
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<Head>
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<Title>Boost Graph Library: Minimum Degree Ordering</Title>
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<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
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ALINK="#ff0000">
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<IMG SRC="../../../c++boost.gif"
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ALT="C++ Boost" width="277" height="86">
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<BR Clear>
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<H1><A NAME="sec:mmd">
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<TT>minimum_degree_ordering</TT>
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</H1>
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<P>
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<DIV ALIGN="LEFT">
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<TABLE CELLPADDING=3 border>
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<TR><TH ALIGN="LEFT"><B>Graphs:</B></TH>
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<TD ALIGN="LEFT">directed</TD>
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</TR>
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<TR><TH ALIGN="LEFT"><B>Properties:</B></TH>
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<TD ALIGN="LEFT">out degree</TD>
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</TR>
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<!--
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<TR><TH ALIGN="LEFT"><B>Complexity:</B></TH>
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<TD ALIGN="LEFT">time: <i>O(log(m)|E|)</i> where <i>m = max { degree(v) | v in V }</i> </TD>
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</TR>
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-->
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</TABLE>
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</DIV>
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<pre>
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template <class AdjacencyGraph, class OutDegreeMap,
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class InversePermutationMap,
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class PermutationMap, class SuperNodeSizeMap, class VertexIndexMap>
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void minimum_degree_ordering
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(AdjacencyGraph& G,
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OutDegreeMap outdegree,
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InversePermutationMap inverse_perm,
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PermutationMap perm,
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SuperNodeSizeMap supernode_size, int delta, VertexIndexMap id)
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</pre>
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The minimum degree ordering algorithm [<A
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HREF="bibliography.html#LIU:MMD">21</A>, <A
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href="bibliography.html#George:evolution">35</a>] is a fill-in
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reduction reordering algorithm. It chooses the vertex with minimum
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degree in the graph at each step of simulating Gaussian elimination
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process. This implementation provided a number of enhancements
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including mass elimination, incomplete degree update, multiple
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elimination, and external degree. See [<A
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href="bibliography.html#George:evolution">35</a>] for a historical
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survey of the minimum degree algorithm.
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<p>
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The graph <TT>G</TT> corresponding to a symmetric matrix <TT>A</TT>
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should be directed one instead of a undirected graph in this
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implementation. Therefore, nonzero entry <TT>A(i, j)</TT> corresponds
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two directed edges (<TT>e(i,j)</TT> and <TT>e(j,i)</TT>) in <TT>G</TT>.
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<p>
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The output of the algorithm are the vertices in the new ordering.
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<pre>
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inverse_perm[new_index[u]] == old_index[u]
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</pre>
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<p> and the permutation from the old index to the new index.
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<pre>
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perm[old_index[u]] == new_index[u]
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</pre>
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<p>The following equations are always held.
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<pre>
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for (size_type i = 0; i != inverse_perm.size(); ++i)
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perm[inverse_perm[i]] == i;
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</pre>
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<h3>Parameters</h3>
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<ul>
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<li> <tt>AdjacencyGraph& G</tt> (IN) <br>
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A directed graph. The graph's type must be a model of <a
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href="./AdjacencyGraph.html">Adjacency Graph</a>,
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<a
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href="./VertexListGraph.html">Vertex List Graph</a>,
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and <a href="./MutableIncidenceGraph.html">Mutable Incidence Graph</a>.
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In addition, the functions <tt>num_vertices()</tt> and
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<TT>out_degree()<TT> are required.
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<li> <tt>OutDegreeMap outdegree</tt>  (WORK) <br>
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This is used internally to store the out degree of vertices. This
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must be a <a href="../../property_map/LvaluePropertyMap.html">
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LvaluePropertyMap</a> with key type the same as the vertex
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descriptor type of the graph, and with a value type that is an
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integer type.
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<li> <tt>InversePermutationMap inverse_perm</tt>  (OUT) <br>
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The new vertex ordering, given as the mapping from the
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new indices to the old indices (an inverse permutation).
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This must be an <a href="../../property_map/LvaluePropertyMap.html">
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LvaluePropertyMap</a> with a value type and key type a signed integer.
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<li> <tt>PermutationMap perm</tt>  (OUT) <br>
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The new vertex ordering, given as the mapping from the
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old indices to the new indices (a permutation).
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This must be an <a href="../../property_map/LvaluePropertyMap.html">
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LvaluePropertyMap</a> with a value type and key type a signed integer.
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<li> <tt>SuperNodeSizeMap supernode_size</tt>  (WORK/OUT) <br>
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This is used internally to record the size of supernodes and is also
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useful information to have. This is a <a
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href="../../property_map/LvaluePropertyMap.html">
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LvaluePropertyMap</a> with an unsigned integer value type and key
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type of vertex descriptor.
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<li> <tt>int delta</tt>  (IN) <br>
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Multiple elimination control variable. If it is larger than or equal
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to zero then multiple elimination is enabled. The value of
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<tt>delta</tt> specifies the difference between the minimum degree
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and the degree of vertices that are to be eliminated.
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<li> <tt>VertexIndexMap id</tt>  (IN) <br>
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Used internally to map vertices to their indices. This must be a <a
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href="../../property_map/ReadablePropertyMap.html"> Readable
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Property Map</a> with key type the same as the vertex descriptor of
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the graph and a value type that is some unsigned integer type.
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</ul>
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<h3>Example</h3>
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See <a
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href="../example/minimum_degree_ordering.cpp"><tt>example/minimum_degree_ordering.cpp</tt></a>.
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<br>
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<HR>
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<TABLE>
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<TR valign=top>
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<TD nowrap>Copyright © 2001</TD><TD>
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<A HREF="../../../people/liequan_lee.htm">Lie-Quan Lee</A>, Univ.of Notre Dame (<A HREF="mailto:llee1@lsc.nd.edu">llee1@lsc.nd.edu</A>) <br>
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<A HREF="../../../people/jeremy_siek.htm">Jeremy Siek</A>, Univ.of Notre Dame (<A HREF="mailto:jsiek@lsc.nd.edu">jsiek@lsc.nd.edu</A>)
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</TD></TR></TABLE>
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</BODY>
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</HTML>
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