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<Title>Boost Graph Library: Kruskal Minimum Spanning Tree</Title>
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<H1><A NAME="sec:kruskal">
<img src="figs/python.gif" alt="(Python)"/>
<TT>kruskal_minimum_spanning_tree</TT>
</H1>

<PRE>
template &lt;class Graph, class OutputIterator, class P, class T, class R&gt;
OutputIterator
kruskal_minimum_spanning_tree(Graph&amp; g, OutputIterator tree_edges, 
    const bgl_named_params&lt;P, T, R&gt;&amp; params = <i>all defaults</i>);
</PRE>

<P>
The <tt>kruskal_minimum_spanning_tree()</tt> function find a minimum
spanning tree (MST) in 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.  For more details,
see section <a
href="graph_theory_review.html#sec:minimum-spanning-tree">Minimum
Spanning Tree Problem</a>.  The edges in the MST are output to the
<tt>tree_edges</tt> output iterator.  This function uses Kruskal's
algorithm to compute the MST&nbsp;[<A
HREF="bibliography.html#kruskal56">18</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>].
</p>
<p>
Kruskal's algorithm starts with each vertex in a tree by itself, and
with no edges in the minimum spanning tree <i>T</i>. The algorithm
then examines each edge in the graph in order of increasing edge
weight. If an edge connects two vertices in different trees the
algorithm merges the two trees into a single tree and adds the edge to
<i>T</i>. We use the ``union by rank'' and ``path compression''
heuristics to provide fast implementations of the disjoint set
operations (<tt>MAKE-SET</tt>, <tt>FIND-SET</tt>, and
<tt>UNION-SET</tt>).  The algorithm is as follows:
</p>

<pre>
KRUSKAL-MST(<i>G</i>, <i>w</i>) 
  <i>T := &Oslash;</i> 
  <b>for</b> each vertex <i>u in V</i> 
    MAKE-SET(<i>DS</i>, <i>u</i>) 
  <b>end for</b>
  <b>for</b> each edge <i>(u,v) in E</i> in order of nondecreasing weight 
    <b>if</b> FIND-SET(<i>DS</i>, <i>u</i>) != FIND-SET(<i>DS</i>, <i>v</i>) 
      UNION-SET(<i>DS</i>, <i>u</i>, <i>v</i>) 
      <i>T := T U {(u,v)}</i> 
  <b>end for</b>
  <b>return</b> <i>T</i>
</pre>


<H3>Where Defined</H3>

<P>
<a href="../../../boost/graph/kruskal_min_spanning_tree.hpp"><TT>boost/graph/kruskal_min_spanning_tree.hpp</TT></a>

<P>

<h3>Parameters</h3>

IN: <tt>const Graph&amp; g</tt> 
<blockquote>
  An undirected graph. The graph type must be a model of
  <a href="./VertexListGraph.html">Vertex List Graph</a>
  and <a href="./EdgeListGraph.html">Edge List Graph</a>.<br>

  <b>Python</b>: The parameter is named <tt>graph</tt>.
</blockquote>

IN: <tt>OutputIterator spanning_tree_edges</tt>
<blockquote>
   The edges of the minimum spanning tree are output to this <a
   href="http://www.sgi.com/tech/stl/OutputIterator.html">Output
   Iterator</a>.<br>

   <b>Python</b>: This parameter is not used in Python. Instead, a
   Python <tt>list</tt> containing all of the spanning tree edges is
   returned.
</blockquote>


<h3>Named Parameters</h3>

IN: <tt>weight_map(WeightMap w_map)</tt> 
<blockquote>
The weight or ``length'' of
  each edge in the graph.  The <tt>WeightMap</tt> type must be a model
  of <a href="../../property_map/doc/ReadablePropertyMap.html">Readable
  Property Map</a> and its value type must be <a
  href="http://www.sgi.com/tech/stl/LessThanComparable.html">Less Than
  Comparable</a>. The key type of this map needs to be the graph's
  edge descriptor type.<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>

UTIL: <tt>rank_map(RankMap r_map)</tt> 
<blockquote>
  This is used by the disjoint sets data structure.
  The type <tt>RankMap</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 rank map. The value type of the
  rank map must be an integer type.<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 the integers of size
  <tt>num_vertices(g)</tt> and using the <tt>i_map</tt> for the index
  map.<br>

  <b>Python</b>: Unsupported parameter.
</blockquote>

UTIL: <tt>predecessor_map(PredecessorMap p_map)</tt> 
<blockquote>
  This is used by the disjoint sets data structure, and is <b>not</b>
  used for storing predecessors in the spanning tree.  The predecessors
  of the spanning tree can be obtained from the spanning tree edges
  output. The type <tt>PredecessorMap</tt> must be a model of <a
  href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
  Property Map</a>. The key type value types of the predecessor map
  must be the vertex descriptor type of the graph. <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 vertex descriptors of size
  <tt>num_vertices(g)</tt> and using the <tt>i_map</tt> for the index
  map.<br>

  <b>Python</b>: Unsupported parameter.
</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 only necessary if the default is used
  for the rank or predecessor maps. 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>


<H3>Complexity</H3>

<P>
The time complexity is <i>O(E log E)</i>

<H3>Example</H3>

<P>
The file <a
href="../example/kruskal-example.cpp"><TT>examples/kruskal-example.cpp</TT></a>
contains an example of using Kruskal's algorithm.


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<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>)
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