249 lines
8.3 KiB
HTML
249 lines
8.3 KiB
HTML
<HTML>
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<!-- Copyright (c) Jeremy Siek 2000
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt)
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-->
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<Head>
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<Title>Boost Graph Library: Cuthill-Mckee Ordering</Title>
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</Head>
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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="../../../boost.png"
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ALT="C++ Boost" width="277" height="86">
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<BR Clear>
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<H1>
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<img src="figs/python.gif" alt="(Python)"/>
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<TT>cuthill_mckee_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">undirected</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">color, degree</TD>
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</TR>
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<TR><TH ALIGN="LEFT"><B>Complexity:</B></TH>
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<TD ALIGN="LEFT">time: <i>O(m log(m)|V|)</i> where <i>m = max { degree(v) | v in V }</i> </TD>
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</TR>
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</TABLE>
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</DIV>
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<pre>
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(1)
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template <class IncidenceGraph, class OutputIterator,
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class ColorMap, class DegreeMap>
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OutputIterator
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cuthill_mckee_ordering(const IncidenceGraph& g,
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typename graph_traits<IncidenceGraph>::vertex_descriptor s,
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OutputIterator inverse_permutation,
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ColorMap color, DegreeMap degree)
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(2)
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template <class VertexListGraph, class OutputIterator>
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OutputIterator
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cuthill_mckee_ordering(const VertexListGraph& g, OutputIterator inverse_permutation);
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template <class VertexListGraph, class OutputIterator, class VertexIndexMap>
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OutputIterator
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cuthill_mckee_ordering(const VertexListGraph& g, OutputIterator inverse_permutation,
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VertexIndexMap index_map);
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template <class VertexListGraph, class OutputIterator,
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class ColorMap, class DegreeMap>
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OutputIterator
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cuthill_mckee_ordering(const VertexListGraph& g, OutputIterator inverse_permutation,
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ColorMap color, DegreeMap degree)
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(3)
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template <class IncidenceGraph, class OutputIterator,
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class ColorMap, class DegreeMap>
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OutputIterator
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cuthill_mckee_ordering(const IncidenceGraph& g,
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std::deque< typename
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graph_traits<IncidenceGraph>::vertex_descriptor > vertex_queue,
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OutputIterator inverse_permutation,
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ColorMap color, DegreeMap degree)
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</pre>
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The goal of the Cuthill-Mckee (and reverse Cuthill-Mckee) ordering
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algorithm[<A
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HREF="bibliography.html#george81:__sparse_pos_def">14</A>, <A
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HREF="bibliography.html#cuthill69:reducing_bandwith">43</A>, <a
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href="bibliography.html#liu75:anal_cm_rcm">44</a>, <a
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href="bibliography.html#george71:fem">45</a> ] is to reduce the <a
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href="./bandwidth.html">bandwidth</a> of a graph by reordering the
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indices assigned to each vertex. The Cuthill-Mckee ordering algorithm
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works by a local minimization of the i-th bandwidths. The vertices are
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basically assigned a breadth-first search order, except that at each
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step, the adjacent vertices are placed in the queue in order of
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increasing degree.
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<p>
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Version 1 of the algorithm lets the user choose the ``starting
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vertex'', version 2 finds a good starting vertex using the
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pseudo-peripheral pair heuristic (among each component), while version 3
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contains the starting nodes for each vertex in the deque. The choice of the
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``starting vertex'' can have a significant effect on the quality of the
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ordering. For versions 2 and 3, <tt>find_starting_vertex</tt> will be called
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for each component in the graph, increasing run time significantly.
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</p>
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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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Depending on what kind of output iterator you use, you can get either
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the Cuthill-Mckee ordering or the reverse Cuthill-McKee ordering. For
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example, if you store the output into a vector using the vector's
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reverse iterator, then you get the reverse Cuthill-McKee ordering.
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</p>
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<pre>
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std::vector<vertex_descriptor> inv_perm(num_vertices(G));
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cuthill_mckee_ordering(G, inv_perm.rbegin(), ...);
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</pre>
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<p>
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Either way, storing the output into a vector gives you the
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permutation from the new ordering to the old ordering.
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</p>
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<pre>
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inv_perm[new_index[u]] == u
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</pre>
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<p>
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Often times, it is the opposite permutation that you want, the
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permutation from the old index to the new index. This can easily be
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computed in the following way.
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</p>
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<pre>
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for (size_type i = 0; i != inv_perm.size(); ++i)
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perm[old_index[inv_perm[i]]] = i;
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</pre>
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<h3>Parameters</h3>
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For version 1:
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<ul>
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<li> <tt>IncidenceGraph& g</tt> (IN) <br>
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An undirected graph. The graph's type must be a model of <a
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href="./IncidenceGraph.html">IncidenceGraph</a>.<br>
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<b>Python</b>: The parameter is named <tt>graph</tt>.
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<li> <tt>vertex_descriptor s</tt>  (IN) <br>
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The starting vertex.<br>
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<b>Python</b>: Unsupported parameter.
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<li> <tt>OutputIterator inverse_permutation</tt>  (OUT) <br>
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The new vertex ordering. The vertices are written to the <a
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href="http://www.boost.org/sgi/stl/OutputIterator.html">output
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iterator</a> in their new order.<br>
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<b>Python</b>: This parameter is unused in Python. The new vertex
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ordering is returned as a Python <tt>list</tt>.
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<li> <tt>ColorMap color_map</tt>  (WORK) <br>
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Used internally to keep track of the progress of the algorithm
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(to avoid visiting the same vertex twice).<br>
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<b>Python</b>: Unsupported parameter.
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<li> <tt>DegreeMap degree_map</tt>  (IN) <br>
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This must map vertices to their degree.<br>
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<b>Python</b>: Unsupported parameter.
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</ul>
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For version 2:
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<ul>
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<li> <tt>VertexListGraph& g</tt> (IN) <br>
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An undirected graph. The graph's type must be a model of <a
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href="./VertexListGraph.html">VertexListGraph</a> and <a href="./IncidenceGraph.html">IncidenceGraph</a>.<br>
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<b>Python</b>: The parameter is named <tt>graph</tt>.
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<li> <tt><a href="http://www.boost.org/sgi/stl/OutputIterator.html">
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OutputIterator</a> inverse_permutation</tt>  (OUT) <br>
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The new vertex ordering. The vertices are written to the
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output iterator in their new order.<br>
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<b>Python</b>: This parameter is unused in Python. The new vertex
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ordering is returned as a Python <tt>list</tt>.
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<li> <tt>ColorMap color_map</tt>  (WORK) <br>
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Used internally to keep track of the progress of the algorithm
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(to avoid visiting the same vertex twice).<br>
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<b>Python</b>: Unsupported parameter.
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<li> <tt>DegreeMap degree_map</tt>  (IN) <br>
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This must map vertices to their degree.<br>
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<b>Python</b>: Unsupported parameter.
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</ul>
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For version 3:
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<ul>
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<li> <tt>IncidenceGraph& g</tt> (IN) <br>
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An undirected graph. The graph's type must be a model of <a
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href="./IncidenceGraph.html">IncidenceGraph</a>.<br>
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<b>Python</b>: The parameter is named <tt>graph</tt>.
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<li> <tt> std::deque< typename graph_traits<Graph>::vertex_descriptor > vertex_queue </tt>  (IN) <br>
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The deque containing the starting vertices for each component.<br>
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<b>Python</b>: Unsupported parameter.
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<li> <tt>OutputIterator inverse_permutation</tt>  (OUT) <br>
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The new vertex ordering. The vertices are written to the <a
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href="http://www.boost.org/sgi/stl/OutputIterator.html">output
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iterator</a> in their new order.<br>
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<b>Python</b>: This parameter is unused in Python. The new vertex
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ordering is returned as a Python <tt>list</tt>.
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<li> <tt>ColorMap color_map</tt>  (WORK) <br>
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Used internally to keep track of the progress of the algorithm
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(to avoid visiting the same vertex twice).<br>
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<b>Python</b>: Unsupported parameter.
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<li> <tt>DegreeMap degree_map</tt>  (IN) <br>
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This must map vertices to their degree.<br>
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<b>Python</b>: Unsupported parameter.
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</ul>
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<h3>Example</h3>
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See <a
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href="../example/cuthill_mckee_ordering.cpp"><tt>example/cuthill_mckee_ordering.cpp</tt></a>.
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<h3>See Also</h3>
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<a href="./bandwidth.html">bandwidth</tt></a>,
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and <tt>degree_property_map</tt> in <tt>boost/graph/properties.hpp</tt>.
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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 © 2000-2001</TD><TD>
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<A HREF="http://www.boost.org/people/jeremy_siek.htm">Jeremy Siek</A>, Indiana University (<A HREF="mailto:jsiek@osl.iu.edu">jsiek@osl.iu.edu</A>)
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</TD></TR></TABLE>
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</BODY>
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</HTML>
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