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     Copyright (c) Lie-Quan Lee and Jeremy Siek, 2001
    
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<Head>
<Title>Boost Graph Library: Minimum Degree Ordering</Title>
<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b" 
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<H1><A NAME="sec:mmd">
<img src="figs/python.gif" alt="(Python)"/>
<TT>minimum_degree_ordering</TT>
</H1>


<pre>
  template &lt;class AdjacencyGraph, class OutDegreeMap,
           class InversePermutationMap, 
           class PermutationMap, class SuperNodeSizeMap, class VertexIndexMap&gt;
  void minimum_degree_ordering
    (AdjacencyGraph& G,
     OutDegreeMap outdegree,
     InversePermutationMap inverse_perm,
     PermutationMap perm, 
     SuperNodeSizeMap supernode_size, int delta, VertexIndexMap id) 
</pre>

<p>The minimum degree ordering algorithm&nbsp;[<A
HREF="bibliography.html#LIU:MMD">21</A>, <A
href="bibliography.html#George:evolution">35</a>] is a fill-in
reduction matrix reordering algorithm.  When solving a system of
equations such as <i>A x = b</i> using a sparse version of Cholesky
factorization (which is a variant of Gaussian elimination for
symmetric matrices), often times elements in the matrix that were once
zero become non-zero due to the elimination process. This is what is
referred to as &quot;fill-in&quot;, and fill-in is bad because it
makes the matrix less sparse and therefore consume more time and space
in later stages of the elimintation and in computations that use the
resulting factorization. Now it turns out that reordering the rows of
a matrix can have a dramatic affect on the amount of fill-in that
occurs. So instead of solving <i>A x = b</i>, we instead solve the
reordered (but equivalent) system <i>(P A P<sup>T</sup>)(P x) = P
b</i>. Finding the optimal ordering is an NP-complete problem (e.i.,
it can not be solved in a reasonable amount of time) so instead we
just find an ordering that is &quot;good enough&quot; using
heuristics. The minimum degree ordering algorithm is one such
approach.

<p>
A symmetric matrix <TT>A</TT> is typically represented with an
undirected graph, however for this function we require the input to be
a directed graph.  Each nonzero matrix entry <TT>A(i, j)</TT> must be
represented by two directed edges (<TT>e(i,j)</TT> and
<TT>e(j,i)</TT>) in <TT>G</TT>.

<p>
The output of the algorithm are the vertices in the new ordering.
<pre>
  inverse_perm[new_index[u]] == old_index[u]
</pre>
<p> and the permutation from the old index to the new index. 
<pre>
  perm[old_index[u]] == new_index[u]
</pre>
<p>The following equations are always held.
<pre>
  for (size_type i = 0; i != inverse_perm.size(); ++i)
    perm[inverse_perm[i]] == i;
</pre>

<h3>Parameters</h3>

<ul>

<li> <tt>AdjacencyGraph&amp; G</tt> &nbsp;(IN) <br> 
  A directed graph. The graph's type must be a model of <a
  href="./AdjacencyGraph.html">Adjacency Graph</a>,
  <a
  href="./VertexListGraph.html">Vertex List Graph</a>,
  <a href="./IncidenceGraph.html">Incidence Graph</a>,
  and <a href="./MutableGraph.html">Mutable Graph</a>.
  In addition, the functions <tt>num_vertices()</tt> and
  <TT>out_degree()</TT> are required.

<li> <tt>OutDegreeMap outdegree</tt> &nbsp(WORK) <br>
  This is used internally to store the out degree of vertices.  This
  must be a <a href="../../property_map/doc/LvaluePropertyMap.html">
  LvaluePropertyMap</a> with key type the same as the vertex
  descriptor type of the graph, and with a value type that is an
  integer type.

<li> <tt>InversePermutationMap inverse_perm</tt> &nbsp(OUT) <br> 
  The new vertex ordering, given as the mapping from the
  new indices to the old indices (an inverse permutation).
  This must be an <a href="../../property_map/doc/LvaluePropertyMap.html">
  LvaluePropertyMap</a> with a value type and key type a signed integer. 

<li> <tt>PermutationMap perm</tt> &nbsp(OUT) <br> 
  The new vertex ordering, given as the mapping from the
  old indices to the new indices (a permutation).
  This must be an <a href="../../property_map/doc/LvaluePropertyMap.html">
  LvaluePropertyMap</a> with a value type and key type a signed integer. 

<li> <tt>SuperNodeSizeMap supernode_size</tt> &nbsp(WORK/OUT) <br> 
  This is used internally to record the size of supernodes and is also
  useful information to have. This is a <a
  href="../../property_map/doc/LvaluePropertyMap.html">
  LvaluePropertyMap</a> with an unsigned integer value type and key
  type of vertex descriptor.

<li> <tt>int delta</tt> &nbsp(IN) <br> 
  Multiple elimination control variable. If it is larger than or equal
  to zero then multiple elimination is enabled. The value of
  <tt>delta</tt> specifies the difference between the minimum degree
  and the degree of vertices that are to be eliminated.
  
<li> <tt>VertexIndexMap id</tt> &nbsp(IN) <br> 
  Used internally to map vertices to their indices. This must be a <a
  href="../../property_map/doc/ReadablePropertyMap.html"> Readable
  Property Map</a> with key type the same as the vertex descriptor of
  the graph and a value type that is some unsigned integer type.

</ul>


<h3>Example</h3>

See <a
href="../example/minimum_degree_ordering.cpp"><tt>example/minimum_degree_ordering.cpp</tt></a>.


<h3>Implementation Notes</h3>

<p>UNDER CONSTRUCTION

<p>It chooses the vertex with minimum degree in the graph at each step of
simulating Gaussian elimination process.

<p>This implementation provided a number of enhancements
including mass elimination, incomplete degree update, multiple
elimination, and external degree. See&nbsp;[<A
href="bibliography.html#George:evolution">35</a>] for a historical
survey of the minimum degree algorithm.

<p>
An <b>elimination graph</b> <i>G<sub>v</sub></i> is obtained from the
graph <i>G</i> by removing vertex <i>v</i> and all of its incident
edges and by then adding edges to make all of the vertices adjacent to
<i>v</i> into a clique (that is, add an edge between each pair of
adjacent vertices if an edge doesn't already exist). 

Suppose that graph <i>G</i> is the graph representing the nonzero
structure of a matrix <i>A</i>. Then performing a step of Guassian
elimination on row <i>i</i> of matrix <i>A</i> is equivalent to


<br>
<HR>
<TABLE>
<TR valign=top>
<TD nowrap>Copyright &copy; 2001</TD><TD>
<A HREF="http://www.boost.org/people/liequan_lee.htm">Lie-Quan Lee</A>, Indiana University (<A HREF="mailto:llee@cs.indiana.edu">llee@cs.indiana.edu</A>) <br>
<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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