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<Title>Boost Graph Library: Prim Minimum Spanning Tree</Title>
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<H1><A NAME="sec:prim"></A>
<img src="figs/python.gif" alt="(Python)"/>
<TT>prim_minimum_spanning_tree</TT>
</H1>

<P>
<PRE>
<i>// named parameter version</i>
template &lt;class Graph, class PredMap, class P, class T, class R&gt;
void prim_minimum_spanning_tree(const Graph&amp; g, PredMap p_map,
  const bgl_named_params&lt;P, T, R&gt;&amp; params)

<i>// non-named parameter version</i>
template &lt;class Graph, class DijkstraVisitor, 
	  class PredecessorMap, class DistanceMap,
	  class WeightMap, class IndexMap&gt;
void prim_minimum_spanning_tree(const Graph&amp; g,
   typename graph_traits&lt;Graph&gt;::vertex_descriptor s, 
   PredecessorMap predecessor, DistanceMap distance, WeightMap weight, 
   IndexMap index_map, DijkstraVisitor vis)
</PRE>

<P>
This is Prim's algorithm&nbsp;[<A
HREF="bibliography.html#prim57:_short">25</A>,<A
HREF="bibliography.html#clr90">8</A>,<A
HREF="bibliography.html#tarjan83:_data_struct_network_algo">27</A>,<A
HREF="bibliography.html#graham85">15</A>] for solving the minimum
spanning tree problem for an undirected graph with weighted edges. A
MST is a set of edges that connects all the vertices in the graph
where the total weight of the edges in the tree is minimized.  See
Section <A
HREF="graph_theory_review.html#sec:minimum-spanning-tree">Minimum
Spanning Tree Problem</A> for more details. The implementation is
simply a call to <a
href="./dijkstra_shortest_paths.html"><TT>dijkstra_shortest_paths()</TT></a>
with the appropriate choice of comparison and combine functors.
The pseudo-code for Prim's algorithm is listed below.
The algorithm as implemented in Boost.Graph does not produce correct results on
graphs with parallel edges.
</p>

<table>
<tr>
<td valign="top">
<pre>
PRIM-MST(<i>G</i>, <i>s</i>, <i>w</i>)
  <b>for</b> each vertex <i>u</i> <i>in</i> <i>V[G]</i>  
    <i>color[u] :=</i> WHITE
    <i>d[u] :=</i> <i>infinity</i>  
  <i>color[s] :=</i> GRAY
  <i>d[s] := 0</i> 
  ENQUEUE(<i>PQ</i>, <i>s</i>)  
  <i>p[s] := s</i> 
  <b>while</b> (<i>PQ != &Oslash;</i>) 
    <i>u :=</i> DEQUEUE(<i>PQ</i>)
    <b>for</b> each <i>v in Adj[u]</i>  
      <b>if</b> (<i>w(u,v) < d[v]</i>)
        <i>d[v] := w(u,v)</i>
        <i>p[v] := u</i> 
        <b>if</b> (<i>color[v] = </i> WHITE) 
          ENQUEUE(<i>PQ</i>, <i>v</i>) 
          <i>color[v] :=</i> GRAY 
        <b>else if</b> (<i>color[v] = </i> GRAY) 
          UPDATE(<i>PQ</i>, <i>v</i>) 
      <b>else</b> 
        do nothing
    <b>end for</b>
    <i>color[u] :=</i> BLACK
  <b>end while</b>
  <b>return</b> (<i>p</i>, <i>d</i>)
</pre>
</td>
<td valign="top">
<pre>

initialize vertex <i>u</i>


start vertex <i>s</i>
discover vertex <i>s</i> 


examine vertex <i>u</i>
examining edge <i>(u,v)</i> 

edge <i>(u,v)</i> relaxed


discover vertex <i>v</i>


edge <i>(u,v)</i> not relaxed 

finish <i>u</i>
</pre>
</tr>
</table>


<H3>Where Defined</H3>

<P>
<a href="../../../boost/graph/prim_minimum_spanning_tree.hpp"><TT>boost/graph/prim_minimum_spanning_tree.hpp</TT></a>

<P>

<h3>Parameters</h3>

IN: <tt>const Graph&amp; g</tt> 
<blockquote>
  An undirected graph. The type <tt>Graph</tt> must be a
  model of <a href="./VertexListGraph.html">Vertex List Graph</a>
  and <a href="./IncidenceGraph.html">Incidence Graph</a>.  It should not
  contain parallel edges.<br>

  <b>Python</b>: The parameter is named <tt>graph</tt>.
</blockquote>

OUT: <tt>PredecessorMap p_map</tt>
<blockquote>
  The predecessor map records the edges in the minimum spanning
  tree. Upon completion of the algorithm, the edges 
  <i>(p[u],u)</i> for all <i>u in V</i> are in the minimum spanning
  tree. If <i>p[u] = u</i> then <i>u</i> is either the root of the
  tree or is a vertex that is not reachable from the root.
  The <tt>PredecessorMap</tt> type must be a <a
  href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
  Property Map</a>
  with key and vertex types the same as the vertex descriptor type
  of the graph.<br>

  <b>Python</b>: Must be a <tt>vertex_vertex_map</tt> for the graph.<br>
</blockquote>

<h3>Named Parameters</h3>

IN: <tt>root_vertex(vertex_descriptor r)</tt>
<blockquote>
  The vertex that will be the root of the minimum spanning tree.
  The choice of the root vertex is arbitrary.<br>
  <b>Default:</b> <tt>*vertices(g).first</tt>
</blockquote>

IN: <tt>weight_map(WeightMap w_map)</tt>   
<blockquote>
  The weight or ``length'' of each edge in the graph.
  The type <tt>WeightMap</tt> must be a model of
  <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The edge descriptor type of
  the graph needs to be usable as the key type for the weight
  map. The value type for the map must be
  <i>Addable</i> with the value type of the distance map.<br>
  <b>Default:</b>  <tt>get(edge_weight, g)</tt><br>
  <b>Python</b>: Must be an <tt>edge_double_map</tt> for the graph.<br>
  <b>Python default</b>: <tt>graph.get_edge_double_map("weight")</tt>
</blockquote>

IN: <tt>vertex_index_map(VertexIndexMap i_map)</tt> 
<blockquote>
  This maps each vertex to an integer in the range <tt>[0,
    num_vertices(g))</tt>. This is necessary for efficient updates of the
  heap data structure when an edge is relaxed.  The type
  <tt>VertexIndexMap</tt> must be a model of
  <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The value type of the map must be an
  integer type. The vertex descriptor type of the graph needs to be
  usable as the key type of the map.<br>
  <b>Default:</b> <tt>get(vertex_index, g)</tt>
    Note: if you use this default, make sure your graph has
    an internal <tt>vertex_index</tt> property. For example,
    <tt>adjacenty_list</tt> with <tt>VertexList=listS</tt> does
    not have an internal <tt>vertex_index</tt> property.
   <br>
  <b>Python</b>: Unsupported parameter.
</blockquote>

UTIL/OUT: <tt>distance_map(DistanceMap d_map)</tt> 
<blockquote>
  The weight of the spanning tree edge into each
  vertex in the graph <tt>g</tt> is recorded in this property map, with edges
  directed away from the spanning tree root. 
  The type <tt>DistanceMap</tt> must be a model of <a
  href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
  Property Map</a>. The vertex descriptor type of the
  graph needs to be usable as the key type of the distance map. The
  value type of the distance map must be <a
  href="http://www.sgi.com/tech/stl/LessThanComparable.html">Less Than
  Comparable</a>.<br> 
  <b>Default:</b> <a href="../../property_map/doc/iterator_property_map.html">
  <tt>iterator_property_map</tt></a> created from a
  <tt>std::vector</tt> of the <tt>WeightMap</tt>'s value type of size
  <tt>num_vertices(g)</tt> and using the <tt>i_map</tt> for the index
  map.<br>

  <b>Python</b>: Must be a <tt>vertex_double_map</tt> for the graph.<br>
</blockquote>

UTIL/OUT: <tt>color_map(ColorMap c_map)</tt> 
<blockquote>
  This is used during the execution of the algorithm to mark the
  vertices. The vertices start out white and become gray when they are
  inserted in the queue. They then turn black when they are removed
  from the queue. At the end of the algorithm, vertices reachable from
  the source vertex will have been colored black. All other vertices
  will still be white. The type <tt>ColorMap</tt> must be a model of
  <a href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
  Property Map</a>. A vertex descriptor must be usable as the key type
  of the map, and the value type of the map must be a model of
  <a href="./ColorValue.html">Color Value</a>.<br>
  <b>Default:</b> an <a
  href="../../property_map/doc/iterator_property_map.html">
  <tt>iterator_property_map</tt></a> created from a <tt>std::vector</tt>
  of <tt>default_color_type</tt> of size <tt>num_vertices(g)</tt> and
  using the <tt>i_map</tt> for the index map.<br>

  <b>Python</b>: The color map must be a <tt>vertex_color_map</tt> for
  the graph.
</blockquote>
  
OUT: <tt>visitor(DijkstraVisitor v)</tt>  
<blockquote>
  Use this to specify actions that you would like to happen
  during certain event points within the algorithm.
  The type <tt>DijkstraVisitor</tt> must be a model of the
  <a href="./DijkstraVisitor.html">Dijkstra Visitor</a> concept. 
  The visitor object is passed by value <a
  href="#1">[1]</a>.<br>
  <b>Default:</b> <tt>dijkstra_visitor&lt;null_visitor&gt;</tt><br>

  <b>Python</b>: The parameter should be an object that derives from
  the <a
  href="DijkstraVisitor.html#python"><tt>DijkstraVisitor</tt></a> type
  of the graph.
</blockquote>

<H3>Complexity</H3>

<P>
The time complexity is <i>O(E log V)</i>.

<P>

<H3>Example</H3>

<P>
The file <a
href="../example/prim-example.cpp"><TT>examples/prim-example.cpp</TT></a>
contains an example of using Prim's algorithm.


<h3>Notes</h3>

<p><a name="1">[1]</a> 
  Since the visitor parameter is passed by value, if your visitor
  contains state then any changes to the state during the algorithm
  will be made to a copy of the visitor object, not the visitor object
  passed in. Therefore you may want the visitor to hold this state by
  pointer or reference.

<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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