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  ~~ Copyright (c) Jeremy Siek 2000
  ~~
     Distributed under the Boost Software License, Version 1.0.
     (See accompanying file LICENSE_1_0.txt or copy at
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<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 &lt;class IncidenceGraph, class OutputIterator,
            class ColorMap, class DegreeMap, class VertexIndexMap&gt;
  OutputIterator
  king_ordering(const IncidenceGraph&amp; g,
                typename graph_traits&lt;Graph&gt;::vertex_descriptor s,
                OutputIterator inverse_permutation, 
                ColorMap color, DegreeMap degree, VertexIndexMap index_map);

  (2)
  template &lt;class IncidenceGraph, class OutputIterator&gt;
  OutputIterator
  king_ordering(const IncidenceGraph&amp; g, OutputIterator inverse_permutation);

  template &lt;class IncidenceGraph, class OutputItrator, class VertexIndexMap&gt;
  OutputIterator
  king_ordering(const IncidenceGraph&amp; g, OutputIterator inverse_permutation,
                VertexIndexMap index_map);

  template &lt;class VertexListGraph, class OutputIterator, 
            class ColorMap, class DegreeMap, class VertexIndexMap&gt;
  OutputIterator
  king_ordering(const VertexListGraph&amp; G, OutputIterator inverse_permutation, 
                ColorMap color, DegreeMap degree, VertexIndexMap index_map);
                         
  (3)
  template &lt;class IncidenceGraph, class OutputIterator,
            class ColorMap, class DegreeMap, class VertexIndexMap&gt;
  OutputIterator
  king_ordering(const IncidenceGraph&amp; g,
    		std::deque&lt; typename
		graph_traits&lt;Graph&gt;::vertex_descriptor &gt; 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&amp; g</tt> &nbsp;(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> &nbsp(IN) <br>
  The starting vertex.<br>
  <b>Python</b>: Unsupported parameter.

<li> <tt>OutputIterator permutation</tt> &nbsp(OUT) <br> 
  The new vertex ordering. The vertices are written to the <a
  href="http://www.boost.org/sgi/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> &nbsp(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> &nbsp(IN) <br>
  This must map vertices to their degree.<br>
  <b>Python</b>: Unsupported parameter.

</ul>


For version 2:

<ul>

<li> <tt>VertexListGraph&amp; g</tt> &nbsp;(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.boost.org/sgi/stl/OutputIterator.html">
  OutputIterator</a> permutation</tt> &nbsp(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> &nbsp(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> &nbsp(IN) <br>
  This must map vertices to their degree.<br>
  <b>Python</b>: Unsupported parameter.
</ul>


For version 3:

<ul>

<li> <tt>IncidenceGraph&amp; g</tt> &nbsp;(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&lt; typename graph_traits&lt;Graph&gt;::vertex_descriptor &gt; vertex_queue </tt> &nbsp(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> &nbsp(OUT) <br> 
  The new vertex ordering. The vertices are written to the <a
  href="http://www.boost.org/sgi/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> &nbsp(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> &nbsp(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 &copy; 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>

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