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| <Head> |
| <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 <class Graph, class OutputIterator, class P, class T, class R> |
| OutputIterator |
| kruskal_minimum_spanning_tree(Graph& g, OutputIterator tree_edges, |
| const bgl_named_params<P, T, R>& 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 [<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 := Ø</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& 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. |
| |
| |
| <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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