| <HTML> |
| <!-- |
| ~~ Copyright (c) Jeremy Siek 2000 |
| ~~ |
| Distributed under the Boost Software License, Version 1.0. |
| (See accompanying file LICENSE_1_0.txt or copy at |
| http://www.boost.org/LICENSE_1_0.txt) |
| --> |
| |
| |
| <Head> |
| <Title>Boost Graph Library: King Ordering</Title> |
| <BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b" |
| ALINK="#ff0000"> |
| <IMG SRC="../../../boost.png" |
| ALT="C++ Boost" width="277" height="86"> |
| |
| <BR Clear> |
| |
| <H1> |
| <img src="figs/python.gif" alt="(Python)"/> |
| <TT>king_ordering</TT> |
| </H1> |
| |
| |
| <P> |
| <DIV ALIGN="LEFT"> |
| <TABLE CELLPADDING=3 border> |
| <TR><TH ALIGN="LEFT"><B>Graphs:</B></TH> |
| <TD ALIGN="LEFT">undirected</TD> |
| </TR> |
| <TR><TH ALIGN="LEFT"><B>Properties:</B></TH> |
| <TD ALIGN="LEFT">color, degree</TD> |
| </TR> |
| <TR><TH ALIGN="LEFT"><B>Complexity:</B></TH> |
| <TD ALIGN="LEFT">time: <i>O(m<sup>2</sup>log(m)|E|)</i> where <i>m = max { degree(v) | v in V }</i> </TD> |
| </TR> |
| </TABLE> |
| </DIV> |
| |
| |
| <pre> |
| (1) |
| template <class IncidenceGraph, class OutputIterator, |
| class ColorMap, class DegreeMap, class VertexIndexMap> |
| OutputIterator |
| king_ordering(const IncidenceGraph& g, |
| typename graph_traits<Graph>::vertex_descriptor s, |
| OutputIterator inverse_permutation, |
| ColorMap color, DegreeMap degree, VertexIndexMap index_map); |
| |
| (2) |
| template <class IncidenceGraph, class OutputIterator> |
| OutputIterator |
| king_ordering(const IncidenceGraph& g, OutputIterator inverse_permutation); |
| |
| template <class IncidenceGraph, class OutputItrator, class VertexIndexMap> |
| OutputIterator |
| king_ordering(const IncidenceGraph& g, OutputIterator inverse_permutation, |
| VertexIndexMap index_map); |
| |
| template <class VertexListGraph, class OutputIterator, |
| class ColorMap, class DegreeMap, class VertexIndexMap> |
| OutputIterator |
| king_ordering(const VertexListGraph& G, OutputIterator inverse_permutation, |
| ColorMap color, DegreeMap degree, VertexIndexMap index_map); |
| |
| (3) |
| template <class IncidenceGraph, class OutputIterator, |
| class ColorMap, class DegreeMap, class VertexIndexMap> |
| OutputIterator |
| king_ordering(const IncidenceGraph& g, |
| std::deque< typename |
| graph_traits<Graph>::vertex_descriptor > vertex_queue, |
| OutputIterator permutation, |
| ColorMap color, DegreeMap degree, VertexIndexMap index_map); |
| </pre> |
| |
| <!-- King, I.P. An automatic reordering scheme for simultaneous equations derived from network analysis. Int. J. Numer. Methods Engrg. 2 (1970), 523-533 --> |
| |
| The goal of the King ordering |
| algorithm [<a href="bibliography.html#king70">62</a>]is to reduce the <a |
| href="./bandwidth.html">bandwidth</a> of a graph by reordering the |
| indices assigned to each vertex. The King ordering algorithm |
| works by a local minimization of the i-th bandwidths. The vertices are |
| basically assigned a breadth-first search order, except that at each |
| step, the adjacent vertices are placed in the queue in order of |
| increasing pseudo-degree, where pseudo-degree is defined as the number of |
| outgoing edges with white endpoints (vertices yet to be examined). |
| |
| <p> |
| Version 1 of the algorithm lets the user choose the ``starting |
| vertex'', version 2 finds a good starting vertex using the |
| pseudo-peripheral pair heuristic (among each component), while version 3 |
| contains the starting nodes for each vertex in the deque. The choice of the ``starting |
| vertex'' can have a significant effect on the quality of the ordering. |
| </p> |
| |
| <p> |
| The output of the algorithm are the vertices in the new ordering. |
| Storing the output into a vector gives you the |
| permutation from the new ordering to the old ordering. |
| |
| <pre> |
| inv_perm[new_index[u]] == u |
| </pre> |
| |
| <p> |
| Often times, it is the opposite permutation that you want, the |
| permutation from the old index to the new index. This can easily be |
| computed in the following way. |
| </p> |
| |
| <pre> |
| for (size_type i = 0; i != inv_perm.size(); ++i) |
| perm[old_index[inv_perm[i]]] = i; |
| </pre> |
| |
| |
| |
| <h3>Parameters</h3> |
| |
| For version 1: |
| |
| <ul> |
| |
| <li> <tt>IncidenceGraph& g</tt> (IN) <br> |
| An undirected graph. The graph's type must be a model of <a |
| href="./IncidenceGraph.html">IncidenceGraph</a>.<br> |
| <b>Python</b>: The parameter is named <tt>graph</tt>. |
| |
| <li> <tt>vertex_descriptor s</tt>  (IN) <br> |
| The starting vertex.<br> |
| <b>Python</b>: Unsupported parameter. |
| |
| <li> <tt>OutputIterator permutation</tt>  (OUT) <br> |
| The new vertex ordering. The vertices are written to the <a |
| href="http://www.sgi.com/tech/stl/OutputIterator.html">output |
| iterator</a> in their new order. <br> |
| <b>Python</b>: This parameter is unused in Python. The new vertex |
| ordering is returned as a Python <tt>list</tt>. |
| |
| |
| <li> <tt>ColorMap color_map</tt>  (WORK) <br> |
| Used internally to keep track of the progress of the algorithm |
| (to avoid visiting the same vertex twice).<br> |
| <b>Python</b>: Unsupported parameter. |
| |
| <li> <tt>DegreeMap degree_map</tt>  (IN) <br> |
| This must map vertices to their degree.<br> |
| <b>Python</b>: Unsupported parameter. |
| |
| </ul> |
| |
| |
| For version 2: |
| |
| <ul> |
| |
| <li> <tt>VertexListGraph& g</tt> (IN) <br> |
| An undirected graph. The graph's type must be a model of <a |
| href="./VertexListGraph.html">VertexListGraph</a>.<br> |
| <b>Python</b>: The name of this parameter is <tt>graph</tt>. |
| |
| <li> <tt><a href="http://www.sgi.com/tech/stl/OutputIterator.html"> |
| OutputIterator</a> permutation</tt>  (OUT) <br> |
| The new vertex ordering. The vertices are written to the |
| output iterator in their new order.<br> |
| <b>Python</b>: This parameter is unused in Python. The new vertex |
| ordering is returned as a Python <tt>list</tt>. |
| |
| <li> <tt>ColorMap color_map</tt>  (WORK) <br> |
| Used internally to keep track of the progress of the algorithm |
| (to avoid visiting the same vertex twice).<br> |
| <b>Python</b>: Unsupported parameter. |
| |
| <li> <tt>DegreeMap degree_map</tt>  (IN) <br> |
| This must map vertices to their degree.<br> |
| <b>Python</b>: Unsupported parameter. |
| </ul> |
| |
| |
| For version 3: |
| |
| <ul> |
| |
| <li> <tt>IncidenceGraph& g</tt> (IN) <br> |
| An undirected graph. The graph's type must be a model of <a |
| href="./IncidenceGraph.html">IncidenceGraph</a>.<br> |
| <b>Python</b>: The parameter is named <tt>graph</tt>. |
| |
| <li> <tt> std::deque< typename graph_traits<Graph>::vertex_descriptor > vertex_queue </tt>  (IN) <br> |
| The deque containing the starting vertices for each component.<br> |
| <b>Python</b>: This parameter is unused in Python. The new vertex |
| ordering is returned as a Python <tt>list</tt>. |
| |
| <li> <tt>OutputIterator permutation</tt>  (OUT) <br> |
| The new vertex ordering. The vertices are written to the <a |
| href="http://www.sgi.com/tech/stl/OutputIterator.html">output |
| iterator</a> in their new order.<br> |
| <b>Python</b>: This parameter is unused in Python. The new vertex |
| ordering is returned as a Python <tt>list</tt>. |
| |
| <li> <tt>ColorMap color_map</tt>  (WORK) <br> |
| Used internally to keep track of the progress of the algorithm |
| (to avoid visiting the same vertex twice).<br> |
| <b>Python</b>: Unsupported parameter. |
| |
| <li> <tt>DegreeMap degree_map</tt>  (IN) <br> |
| This must map vertices to their degree.<br> |
| <b>Python</b>: Unsupported parameter. |
| </ul> |
| |
| |
| |
| <h3>Example</h3> |
| |
| See <a |
| href="../example/king_ordering.cpp"><tt>example/king_ordering.cpp</tt></a>. |
| |
| <h3>See Also</h3> |
| |
| <a href="./bandwidth.html">bandwidth</tt></a>, |
| and <tt>degree_property_map</tt> in <tt>boost/graph/properties.hpp</tt>. |
| |
| <br> |
| <HR> |
| <TABLE> |
| <TR valign=top> |
| <TD nowrap>Copyright © 2000-2001</TD><TD> |
| <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>) |
| </TD></TR></TABLE> |
| |
| </BODY> |
| </HTML> |