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| <Head> |
| <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 <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> |
| |
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