boost_1_45_0/libs/graph/doc/prim_minimum_spanning_tree.html - nest-learning-thermostat/5.0/boost - Git at Google

 <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: Prim Minimum Spanning Tree</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><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>

 </BODY>
 </HTML>
	<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: Prim Minimum Spanning Tree</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><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 <class Graph, class PredMap, class P, class T, class R>
	void prim_minimum_spanning_tree(const Graph& g, PredMap p_map,
	const bgl_named_params<P, T, R>& params)

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

	<P>
	This is Prim's algorithm [<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 != Ø</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& 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<null_visitor></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 © 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>