1<HTML> 2 3<!-- Copyright (c) Jeremy Siek 2000 4 5 Distributed under the Boost Software License, Version 1.0. 6 (See accompanying file LICENSE_1_0.txt or copy at 7 http://www.boost.org/LICENSE_1_0.txt) 8 --> 9 10<Head> 11<Title>Boost Graph Library: Cuthill-Mckee Ordering</Title> 12</Head> 13<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b" 14 ALINK="#ff0000"> 15<IMG SRC="../../../boost.png" 16 ALT="C++ Boost" width="277" height="86"> 17 18<BR Clear> 19 20<H1> 21<img src="figs/python.gif" alt="(Python)"/> 22<TT>cuthill_mckee_ordering</TT> 23</H1> 24 25 26<P> 27<DIV ALIGN="LEFT"> 28<TABLE CELLPADDING=3 border> 29<TR><TH ALIGN="LEFT"><B>Graphs:</B></TH> 30<TD ALIGN="LEFT">undirected</TD> 31</TR> 32<TR><TH ALIGN="LEFT"><B>Properties:</B></TH> 33<TD ALIGN="LEFT">color, degree</TD> 34</TR> 35<TR><TH ALIGN="LEFT"><B>Complexity:</B></TH> 36<TD ALIGN="LEFT">time: <i>O(m log(m)|V|)</i> where <i>m = max { degree(v) | v in V }</i> </TD> 37</TR> 38</TABLE> 39</DIV> 40 41 42<pre> 43 (1) 44 template <class IncidenceGraph, class OutputIterator, 45 class ColorMap, class DegreeMap> 46 OutputIterator 47 cuthill_mckee_ordering(const IncidenceGraph& g, 48 typename graph_traits<IncidenceGraph>::vertex_descriptor s, 49 OutputIterator inverse_permutation, 50 ColorMap color, DegreeMap degree) 51 52 (2) 53 template <class VertexListGraph, class OutputIterator> 54 OutputIterator 55 cuthill_mckee_ordering(const VertexListGraph& g, OutputIterator inverse_permutation); 56 57 template <class VertexListGraph, class OutputIterator, class VertexIndexMap> 58 OutputIterator 59 cuthill_mckee_ordering(const VertexListGraph& g, OutputIterator inverse_permutation, 60 VertexIndexMap index_map); 61 62 template <class VertexListGraph, class OutputIterator, 63 class ColorMap, class DegreeMap> 64 OutputIterator 65 cuthill_mckee_ordering(const VertexListGraph& g, OutputIterator inverse_permutation, 66 ColorMap color, DegreeMap degree) 67 68 (3) 69 template <class IncidenceGraph, class OutputIterator, 70 class ColorMap, class DegreeMap> 71 OutputIterator 72 cuthill_mckee_ordering(const IncidenceGraph& g, 73 std::deque< typename 74 graph_traits<IncidenceGraph>::vertex_descriptor > vertex_queue, 75 OutputIterator inverse_permutation, 76 ColorMap color, DegreeMap degree) 77</pre> 78 79The goal of the Cuthill-Mckee (and reverse Cuthill-Mckee) ordering 80algorithm[<A 81HREF="bibliography.html#george81:__sparse_pos_def">14</A>, <A 82HREF="bibliography.html#cuthill69:reducing_bandwith">43</A>, <a 83href="bibliography.html#liu75:anal_cm_rcm">44</a>, <a 84href="bibliography.html#george71:fem">45</a> ] is to reduce the <a 85href="./bandwidth.html">bandwidth</a> of a graph by reordering the 86indices assigned to each vertex. The Cuthill-Mckee ordering algorithm 87works by a local minimization of the i-th bandwidths. The vertices are 88basically assigned a breadth-first search order, except that at each 89step, the adjacent vertices are placed in the queue in order of 90increasing degree. 91 92<p> 93Version 1 of the algorithm lets the user choose the ``starting 94vertex'', version 2 finds a good starting vertex using the 95pseudo-peripheral pair heuristic (among each component), while version 3 96contains the starting nodes for each vertex in the deque. The choice of the 97``starting vertex'' can have a significant effect on the quality of the 98ordering. For versions 2 and 3, <tt>find_starting_vertex</tt> will be called 99for each component in the graph, increasing run time significantly. 100</p> 101 102<p> 103The output of the algorithm are the vertices in the new ordering. 104Depending on what kind of output iterator you use, you can get either 105the Cuthill-Mckee ordering or the reverse Cuthill-McKee ordering. For 106example, if you store the output into a vector using the vector's 107reverse iterator, then you get the reverse Cuthill-McKee ordering. 108</p> 109 110<pre> 111 std::vector<vertex_descriptor> inv_perm(num_vertices(G)); 112 cuthill_mckee_ordering(G, inv_perm.rbegin(), ...); 113</pre> 114 115<p> 116Either way, storing the output into a vector gives you the 117permutation from the new ordering to the old ordering. 118</p> 119 120<pre> 121 inv_perm[new_index[u]] == u 122</pre> 123 124<p> 125Often times, it is the opposite permutation that you want, the 126permutation from the old index to the new index. This can easily be 127computed in the following way. 128</p> 129 130<pre> 131 for (size_type i = 0; i != inv_perm.size(); ++i) 132 perm[old_index[inv_perm[i]]] = i; 133</pre> 134 135 136 137<h3>Parameters</h3> 138 139For version 1: 140 141<ul> 142 143<li> <tt>IncidenceGraph& g</tt> (IN) <br> 144 An undirected graph. The graph's type must be a model of <a 145 href="./IncidenceGraph.html">IncidenceGraph</a>.<br> 146 <b>Python</b>: The parameter is named <tt>graph</tt>. 147 148<li> <tt>vertex_descriptor s</tt>  (IN) <br> 149 The starting vertex.<br> 150 <b>Python</b>: Unsupported parameter. 151 152<li> <tt>OutputIterator inverse_permutation</tt>  (OUT) <br> 153 The new vertex ordering. The vertices are written to the <a 154 href="http://www.boost.org/sgi/stl/OutputIterator.html">output 155 iterator</a> in their new order.<br> 156 <b>Python</b>: This parameter is unused in Python. The new vertex 157 ordering is returned as a Python <tt>list</tt>. 158 159<li> <tt>ColorMap color_map</tt>  (WORK) <br> 160 Used internally to keep track of the progress of the algorithm 161 (to avoid visiting the same vertex twice).<br> 162 <b>Python</b>: Unsupported parameter. 163 164<li> <tt>DegreeMap degree_map</tt>  (IN) <br> 165 This must map vertices to their degree.<br> 166 <b>Python</b>: Unsupported parameter. 167</ul> 168 169 170For version 2: 171 172<ul> 173 174<li> <tt>VertexListGraph& g</tt> (IN) <br> 175 An undirected graph. The graph's type must be a model of <a 176 href="./VertexListGraph.html">VertexListGraph</a> and <a href="./IncidenceGraph.html">IncidenceGraph</a>.<br> 177 <b>Python</b>: The parameter is named <tt>graph</tt>. 178 179<li> <tt><a href="http://www.boost.org/sgi/stl/OutputIterator.html"> 180 OutputIterator</a> inverse_permutation</tt>  (OUT) <br> 181 The new vertex ordering. The vertices are written to the 182 output iterator in their new order.<br> 183 <b>Python</b>: This parameter is unused in Python. The new vertex 184 ordering is returned as a Python <tt>list</tt>. 185 186<li> <tt>ColorMap color_map</tt>  (WORK) <br> 187 Used internally to keep track of the progress of the algorithm 188 (to avoid visiting the same vertex twice).<br> 189 <b>Python</b>: Unsupported parameter. 190 191<li> <tt>DegreeMap degree_map</tt>  (IN) <br> 192 This must map vertices to their degree.<br> 193 <b>Python</b>: Unsupported parameter. 194</ul> 195 196 197For version 3: 198 199<ul> 200 201<li> <tt>IncidenceGraph& g</tt> (IN) <br> 202 An undirected graph. The graph's type must be a model of <a 203 href="./IncidenceGraph.html">IncidenceGraph</a>.<br> 204 <b>Python</b>: The parameter is named <tt>graph</tt>. 205 206<li> <tt> std::deque< typename graph_traits<Graph>::vertex_descriptor > vertex_queue </tt>  (IN) <br> 207 The deque containing the starting vertices for each component.<br> 208 <b>Python</b>: Unsupported parameter. 209 210<li> <tt>OutputIterator inverse_permutation</tt>  (OUT) <br> 211 The new vertex ordering. The vertices are written to the <a 212 href="http://www.boost.org/sgi/stl/OutputIterator.html">output 213 iterator</a> in their new order.<br> 214 <b>Python</b>: This parameter is unused in Python. The new vertex 215 ordering is returned as a Python <tt>list</tt>. 216 217<li> <tt>ColorMap color_map</tt>  (WORK) <br> 218 Used internally to keep track of the progress of the algorithm 219 (to avoid visiting the same vertex twice).<br> 220 <b>Python</b>: Unsupported parameter. 221 222<li> <tt>DegreeMap degree_map</tt>  (IN) <br> 223 This must map vertices to their degree.<br> 224 <b>Python</b>: Unsupported parameter. 225</ul> 226 227 228 229<h3>Example</h3> 230 231See <a 232href="../example/cuthill_mckee_ordering.cpp"><tt>example/cuthill_mckee_ordering.cpp</tt></a>. 233 234<h3>See Also</h3> 235 236<a href="./bandwidth.html">bandwidth</tt></a>, 237and <tt>degree_property_map</tt> in <tt>boost/graph/properties.hpp</tt>. 238 239<br> 240<HR> 241<TABLE> 242<TR valign=top> 243<TD nowrap>Copyright © 2000-2001</TD><TD> 244<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>) 245</TD></TR></TABLE> 246 247</BODY> 248</HTML> 249
