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- Copyright (c) Jeremy Siek 2000
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- <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>
- <b>end for</b>
- <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
- the same as the value type of the distance map, and that type must be <a
- href="http://www.boost.org/sgi/stl/LessThanComparable.html">Less Than
- Comparable</a>.<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>adjacency_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, and the value
- type needs to be the same as the value type of the <tt>weight_map</tt>
- argument.<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>
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