293 lines
9.7 KiB
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293 lines
9.7 KiB
HTML
<HTML>
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<!--
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Copyright (c) Jeremy Siek 2000
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt)
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-->
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<Head>
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<Title>Boost Graph Library: Prim Minimum Spanning Tree</Title>
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<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
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ALINK="#ff0000">
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<IMG SRC="../../../boost.png"
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ALT="C++ Boost" width="277" height="86">
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<BR Clear>
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<H1><A NAME="sec:prim"></A>
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<img src="figs/python.gif" alt="(Python)"/>
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<TT>prim_minimum_spanning_tree</TT>
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</H1>
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<P>
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<PRE>
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<i>// named parameter version</i>
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template <class Graph, class PredMap, class P, class T, class R>
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void prim_minimum_spanning_tree(const Graph& g, PredMap p_map,
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const bgl_named_params<P, T, R>& params)
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<i>// non-named parameter version</i>
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template <class Graph, class DijkstraVisitor,
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class PredecessorMap, class DistanceMap,
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class WeightMap, class IndexMap>
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void prim_minimum_spanning_tree(const Graph& g,
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typename graph_traits<Graph>::vertex_descriptor s,
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PredecessorMap predecessor, DistanceMap distance, WeightMap weight,
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IndexMap index_map, DijkstraVisitor vis)
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</PRE>
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<P>
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This is Prim's algorithm [<A
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HREF="bibliography.html#prim57:_short">25</A>,<A
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HREF="bibliography.html#clr90">8</A>,<A
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HREF="bibliography.html#tarjan83:_data_struct_network_algo">27</A>,<A
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HREF="bibliography.html#graham85">15</A>] for solving the minimum
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spanning tree problem for an undirected graph with weighted edges. A
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MST is a set of edges that connects all the vertices in the graph
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where the total weight of the edges in the tree is minimized. See
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Section <A
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HREF="graph_theory_review.html#sec:minimum-spanning-tree">Minimum
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Spanning Tree Problem</A> for more details. The implementation is
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simply a call to <a
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href="./dijkstra_shortest_paths.html"><TT>dijkstra_shortest_paths()</TT></a>
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with the appropriate choice of comparison and combine functors.
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The pseudo-code for Prim's algorithm is listed below.
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The algorithm as implemented in Boost.Graph does not produce correct results on
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graphs with parallel edges.
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</p>
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<table>
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<tr>
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<td valign="top">
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<pre>
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PRIM-MST(<i>G</i>, <i>s</i>, <i>w</i>)
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<b>for</b> each vertex <i>u</i> <i>in</i> <i>V[G]</i>
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<i>color[u] :=</i> WHITE
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<i>d[u] :=</i> <i>infinity</i>
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<b>end for</b>
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<i>color[s] :=</i> GRAY
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<i>d[s] := 0</i>
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ENQUEUE(<i>PQ</i>, <i>s</i>)
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<i>p[s] := s</i>
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<b>while</b> (<i>PQ != Ø</i>)
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<i>u :=</i> DEQUEUE(<i>PQ</i>)
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<b>for</b> each <i>v in Adj[u]</i>
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<b>if</b> (<i>w(u,v) < d[v]</i>)
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<i>d[v] := w(u,v)</i>
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<i>p[v] := u</i>
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<b>if</b> (<i>color[v] = </i> WHITE)
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ENQUEUE(<i>PQ</i>, <i>v</i>)
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<i>color[v] :=</i> GRAY
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<b>else if</b> (<i>color[v] = </i> GRAY)
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UPDATE(<i>PQ</i>, <i>v</i>)
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<b>else</b>
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do nothing
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<b>end for</b>
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<i>color[u] :=</i> BLACK
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<b>end while</b>
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<b>return</b> (<i>p</i>, <i>d</i>)
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</pre>
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</td>
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<td valign="top">
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<pre>
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initialize vertex <i>u</i>
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start vertex <i>s</i>
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discover vertex <i>s</i>
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examine vertex <i>u</i>
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examining edge <i>(u,v)</i>
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edge <i>(u,v)</i> relaxed
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discover vertex <i>v</i>
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edge <i>(u,v)</i> not relaxed
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finish <i>u</i>
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</pre>
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</tr>
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</table>
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<H3>Where Defined</H3>
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<P>
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<a href="../../../boost/graph/prim_minimum_spanning_tree.hpp"><TT>boost/graph/prim_minimum_spanning_tree.hpp</TT></a>
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<P>
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<h3>Parameters</h3>
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IN: <tt>const Graph& g</tt>
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<blockquote>
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An undirected graph. The type <tt>Graph</tt> must be a
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model of <a href="./VertexListGraph.html">Vertex List Graph</a>
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and <a href="./IncidenceGraph.html">Incidence Graph</a>. It should not
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contain parallel edges.<br>
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<b>Python</b>: The parameter is named <tt>graph</tt>.
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</blockquote>
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OUT: <tt>PredecessorMap p_map</tt>
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<blockquote>
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The predecessor map records the edges in the minimum spanning
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tree. Upon completion of the algorithm, the edges
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<i>(p[u],u)</i> for all <i>u in V</i> are in the minimum spanning
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tree. If <i>p[u] = u</i> then <i>u</i> is either the root of the
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tree or is a vertex that is not reachable from the root.
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The <tt>PredecessorMap</tt> type must be a <a
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href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
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Property Map</a>
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with key and vertex types the same as the vertex descriptor type
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of the graph.<br>
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<b>Python</b>: Must be a <tt>vertex_vertex_map</tt> for the graph.<br>
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</blockquote>
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<h3>Named Parameters</h3>
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IN: <tt>root_vertex(vertex_descriptor r)</tt>
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<blockquote>
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The vertex that will be the root of the minimum spanning tree.
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The choice of the root vertex is arbitrary.<br>
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<b>Default:</b> <tt>*vertices(g).first</tt>
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</blockquote>
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IN: <tt>weight_map(WeightMap w_map)</tt>
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<blockquote>
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The weight or ``length'' of each edge in the graph.
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The type <tt>WeightMap</tt> must be a model of
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<a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The edge descriptor type of
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the graph needs to be usable as the key type for the weight
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map. The value type for the map must be
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the same as the value type of the distance map, and that type must be <a
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href="http://www.boost.org/sgi/stl/LessThanComparable.html">Less Than
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Comparable</a>.<br>
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<b>Default:</b> <tt>get(edge_weight, g)</tt><br>
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<b>Python</b>: Must be an <tt>edge_double_map</tt> for the graph.<br>
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<b>Python default</b>: <tt>graph.get_edge_double_map("weight")</tt>
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</blockquote>
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IN: <tt>vertex_index_map(VertexIndexMap i_map)</tt>
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<blockquote>
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This maps each vertex to an integer in the range <tt>[0,
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num_vertices(g))</tt>. This is necessary for efficient updates of the
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heap data structure when an edge is relaxed. The type
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<tt>VertexIndexMap</tt> must be a model of
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<a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The value type of the map must be an
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integer type. The vertex descriptor type of the graph needs to be
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usable as the key type of the map.<br>
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<b>Default:</b> <tt>get(vertex_index, g)</tt>
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Note: if you use this default, make sure your graph has
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an internal <tt>vertex_index</tt> property. For example,
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<tt>adjacency_list</tt> with <tt>VertexList=listS</tt> does
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not have an internal <tt>vertex_index</tt> property.
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<br>
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<b>Python</b>: Unsupported parameter.
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</blockquote>
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UTIL/OUT: <tt>distance_map(DistanceMap d_map)</tt>
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<blockquote>
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The weight of the spanning tree edge into each
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vertex in the graph <tt>g</tt> is recorded in this property map, with edges
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directed away from the spanning tree root.
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The type <tt>DistanceMap</tt> must be a model of <a
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href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
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Property Map</a>. The vertex descriptor type of the
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graph needs to be usable as the key type of the distance map, and the value
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type needs to be the same as the value type of the <tt>weight_map</tt>
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argument.<br>
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<b>Default:</b> <a href="../../property_map/doc/iterator_property_map.html">
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<tt>iterator_property_map</tt></a> created from a
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<tt>std::vector</tt> of the <tt>WeightMap</tt>'s value type of size
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<tt>num_vertices(g)</tt> and using the <tt>i_map</tt> for the index
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map.<br>
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<b>Python</b>: Must be a <tt>vertex_double_map</tt> for the graph.<br>
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</blockquote>
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UTIL/OUT: <tt>color_map(ColorMap c_map)</tt>
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<blockquote>
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This is used during the execution of the algorithm to mark the
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vertices. The vertices start out white and become gray when they are
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inserted in the queue. They then turn black when they are removed
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from the queue. At the end of the algorithm, vertices reachable from
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the source vertex will have been colored black. All other vertices
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will still be white. The type <tt>ColorMap</tt> must be a model of
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<a href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
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Property Map</a>. A vertex descriptor must be usable as the key type
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of the map, and the value type of the map must be a model of
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<a href="./ColorValue.html">Color Value</a>.<br>
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<b>Default:</b> an <a
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href="../../property_map/doc/iterator_property_map.html">
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<tt>iterator_property_map</tt></a> created from a <tt>std::vector</tt>
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of <tt>default_color_type</tt> of size <tt>num_vertices(g)</tt> and
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using the <tt>i_map</tt> for the index map.<br>
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<b>Python</b>: The color map must be a <tt>vertex_color_map</tt> for
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the graph.
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</blockquote>
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OUT: <tt>visitor(DijkstraVisitor v)</tt>
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<blockquote>
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Use this to specify actions that you would like to happen
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during certain event points within the algorithm.
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The type <tt>DijkstraVisitor</tt> must be a model of the
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<a href="./DijkstraVisitor.html">Dijkstra Visitor</a> concept.
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The visitor object is passed by value <a
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href="#1">[1]</a>.<br>
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<b>Default:</b> <tt>dijkstra_visitor<null_visitor></tt><br>
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<b>Python</b>: The parameter should be an object that derives from
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the <a
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href="DijkstraVisitor.html#python"><tt>DijkstraVisitor</tt></a> type
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of the graph.
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</blockquote>
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<H3>Complexity</H3>
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<P>
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The time complexity is <i>O(E log V)</i>.
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<P>
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<H3>Example</H3>
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<P>
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The file <a
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href="../example/prim-example.cpp"><TT>examples/prim-example.cpp</TT></a>
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contains an example of using Prim's algorithm.
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<h3>Notes</h3>
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<p><a name="1">[1]</a>
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Since the visitor parameter is passed by value, if your visitor
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contains state then any changes to the state during the algorithm
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will be made to a copy of the visitor object, not the visitor object
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passed in. Therefore you may want the visitor to hold this state by
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pointer or reference.
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<br>
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<HR>
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<TABLE>
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<TR valign=top>
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<TD nowrap>Copyright © 2000-2001</TD><TD>
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<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>)
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</TD></TR></TABLE>
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</BODY>
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</HTML>
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