200 lines
6.9 KiB
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200 lines
6.9 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: Kruskal 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:kruskal">
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<img src="figs/python.gif" alt="(Python)"/>
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<TT>kruskal_minimum_spanning_tree</TT>
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</H1>
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<PRE>
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template <class Graph, class OutputIterator, class P, class T, class R>
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OutputIterator
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kruskal_minimum_spanning_tree(Graph& g, OutputIterator tree_edges,
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const bgl_named_params<P, T, R>& params = <i>all defaults</i>);
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</PRE>
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<P>
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The <tt>kruskal_minimum_spanning_tree()</tt> function find a minimum
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spanning tree (MST) in an undirected graph with weighted edges. A MST is a
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set of edges that connects all the vertices in the graph where the
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total weight of the edges in the tree is minimized. For more details,
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see section <a
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href="graph_theory_review.html#sec:minimum-spanning-tree">Minimum
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Spanning Tree Problem</a>. The edges in the MST are output to the
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<tt>tree_edges</tt> output iterator. This function uses Kruskal's
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algorithm to compute the MST [<A
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HREF="bibliography.html#kruskal56">18</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>].
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</p>
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<p>
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Kruskal's algorithm starts with each vertex in a tree by itself, and
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with no edges in the minimum spanning tree <i>T</i>. The algorithm
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then examines each edge in the graph in order of increasing edge
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weight. If an edge connects two vertices in different trees the
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algorithm merges the two trees into a single tree and adds the edge to
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<i>T</i>. We use the ``union by rank'' and ``path compression''
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heuristics to provide fast implementations of the disjoint set
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operations (<tt>MAKE-SET</tt>, <tt>FIND-SET</tt>, and
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<tt>UNION-SET</tt>). The algorithm is as follows:
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</p>
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<pre>
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KRUSKAL-MST(<i>G</i>, <i>w</i>)
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<i>T := Ø</i>
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<b>for</b> each vertex <i>u in V</i>
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MAKE-SET(<i>DS</i>, <i>u</i>)
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<b>end for</b>
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<b>for</b> each edge <i>(u,v) in E</i> in order of nondecreasing weight
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<b>if</b> FIND-SET(<i>DS</i>, <i>u</i>) != FIND-SET(<i>DS</i>, <i>v</i>)
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UNION-SET(<i>DS</i>, <i>u</i>, <i>v</i>)
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<i>T := T U {(u,v)}</i>
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<b>end for</b>
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<b>return</b> <i>T</i>
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</pre>
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<H3>Where Defined</H3>
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<P>
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<a href="../../../boost/graph/kruskal_min_spanning_tree.hpp"><TT>boost/graph/kruskal_min_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 graph type must be a model of
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<a href="./VertexListGraph.html">Vertex List Graph</a>
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and <a href="./EdgeListGraph.html">Edge List Graph</a>.<br>
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<b>Python</b>: The parameter is named <tt>graph</tt>.
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</blockquote>
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IN: <tt>OutputIterator spanning_tree_edges</tt>
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<blockquote>
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The edges of the minimum spanning tree are output to this <a
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href="http://www.boost.org/sgi/stl/OutputIterator.html">Output
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Iterator</a>.<br>
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<b>Python</b>: This parameter is not used in Python. Instead, a
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Python <tt>list</tt> containing all of the spanning tree edges is
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returned.
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</blockquote>
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<h3>Named Parameters</h3>
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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
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each edge in the graph. The <tt>WeightMap</tt> type must be a model
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of <a href="../../property_map/doc/ReadablePropertyMap.html">Readable
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Property Map</a> and its value 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>. The key type of this map needs to be the graph's
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edge descriptor type.<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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UTIL: <tt>rank_map(RankMap r_map)</tt>
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<blockquote>
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This is used by the disjoint sets data structure.
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The type <tt>RankMap</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 graph needs to
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be usable as the key type of the rank map. The value type of the
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rank map must be an integer type.<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
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<tt>std::vector</tt> of the integers 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>: Unsupported parameter.
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</blockquote>
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UTIL: <tt>predecessor_map(PredecessorMap p_map)</tt>
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<blockquote>
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This is used by the disjoint sets data structure, and is <b>not</b>
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used for storing predecessors in the spanning tree. The predecessors
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of the spanning tree can be obtained from the spanning tree edges
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output. The type <tt>PredecessorMap</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 key type value types of the predecessor map
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must be the vertex descriptor type of the graph. <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
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<tt>std::vector</tt> of vertex descriptors 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>: Unsupported parameter.
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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 only necessary if the default is used
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for the rank or predecessor maps. The type <tt>VertexIndexMap</tt>
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must be a model of <a
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href="../../property_map/doc/ReadablePropertyMap.html">Readable Property
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Map</a>. The value type of the map must be an integer type. The
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vertex descriptor type of the graph needs to be usable as the key
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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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<H3>Complexity</H3>
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<P>
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The time complexity is <i>O(E log E)</i>
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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/kruskal-example.cpp"><TT>examples/kruskal-example.cpp</TT></a>
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contains an example of using Kruskal's algorithm.
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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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