ceph/src/boost/libs/graph/doc/stoer_wagner_min_cut.html

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  10 <title>Boost Graph Library: Stoer&ndash;Wagner Min-Cut</title>
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  13 <img src="../../../boost.png"  alt="C++ Boost">
  14
  15 <h1><a name="sec:stoer_wagner"><tt>stoer_wagner_min_cut</tt></a></h1>
  16 <table border="0" cellspacing="0" style="float: right">
  17 <caption align="bottom">A min-cut of a weighted graph<br>having min-cut weight 4</caption>
  18 <tr><td style="border: #666 1px solid"><img src="stoer_wagner_imgs/stoer_wagner-example-min-cut.gif" width="376"></td></tr>
  19 </table>
  20 <pre>
  21 template &lt;class UndirectedGraph, class WeightMap, class P, class T, class R&gt;
  22 weight_type
  23 stoer_wagner_min_cut(const UndirectedGraph&amp; g, WeightMap weights,
  24     const bgl_named_params&lt;P, T, R&gt;&amp; params = <i>all defaults</i>);
  25 </pre>
  26
  27 <p>The <tt>stoer_wagner_min_cut</tt> function determines a min-cut and the min-cut weight of a connected, undirected graph.
  28
  29 <p>A <em>cut</em> of a graph <i>G</i> is a partition of the vertices into two, non-empty sets. The <em>weight</em> of such a partition is the number of edges between the two sets if <i>G</i> is unweighted, or the sum of the weights of all edges between the two sets if <i>G</i> is weighted. A <em>min-cut</em> is a cut having the least weight.
  30
  31 <p>Sometimes a graph has multiple min-cuts, but all have the same weight. The <tt>stoer_wagner_min_cut</tt> function determines exactly one of the min-cuts as well as its weight.
  32
  33 <h3>Where Defined</h3>
  34 <p><a href="../../../boost/graph/stoer_wagner_min_cut.hpp"><tt>boost/graph/stoer_wagner_min_cut.hpp</tt></a>
  35
  36 <h3>Parameters</h3>
  37
  38 <p>IN: <tt>const UndirectedGraph&amp; g</tt>
  39 <blockquote>
  40   A connected, undirected graph. The graph type must be a model of
  41   <a href="./VertexListGraph.html">Vertex List Graph</a>
  42   and <a href="./IncidenceGraph.html">Incidence Graph</a>.
  43 </blockquote>
  44
  45 <p>IN: <tt>WeightMap weights</tt>
  46 <blockquote>
  47   The weight or length of each edge in the graph. The <tt>WeightMap</tt> type must be a model
  48   of <a href="../../property_map/doc/ReadablePropertyMap.html">Readable
  49   Property Map</a> and its value type must be <a class="external" href="http://www.sgi.com/tech/stl/LessThanComparable.html">Less Than
  50   Comparable</a> and summable. The key type of this map needs to be the graph's
  51   edge descriptor type.
  52 </blockquote>
  53
  54 <h3>Named Parameters</h3>
  55
  56 <p>OUT: <tt>parity_map(ParityMap parities)</tt>
  57 <blockquote>
  58   The algorithm computes a min-cut, which divides the set of vertices into two,
  59   non-empty sets. The <tt>stoer_wagner_min_cut</tt> function records which of
  60   the two sets that each vertex belongs to by setting the parity to <tt>true</tt>
  61   (representing one set) or <tt>false</tt> (for the other). <tt>ParityMap</tt>
  62   must be a model of a <a href="../../property_map/doc/WritablePropertyMap.html">Writable
  63   Property Map</a> and its value type should be a bool type. The
  64   key type must be the graph's vertex descriptor type.<br>
  65   <b>Default:</b> <tt>boost::dummy_property_map</tt>
  66 </blockquote>
  67
  68 <h4>Expert Parameters</h4>
  69
  70 <p>IN: <tt>vertex_index_map(VertexIndexMap vertexIndices)</tt>
  71 <blockquote>
  72   This maps each vertex to an integer in the range [0, <tt>num_vertices(g)</tt>). This
  73   is only necessary if the default is used for the assignment, index-in-heap, or distance maps.
  74   <tt>VertexIndexMap</tt> must be a model of <a
  75   href="../../property_map/doc/ReadablePropertyMap.html">Readable Property
  76   Map</a>. The value type of the map must be an integer type. The
  77   key type must be the graph's vertex descriptor type.<br>
  78   <b>Default:</b> <tt>get(boost::vertex_index, g)</tt>
  79 </blockquote>
  80
  81 <p>UTIL: <tt>assignment_map(AssignmentMap assignments)</tt>
  82 <blockquote>
  83   <tt>AssignmentMap</tt> must be a model of <a
  84   href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write Property
  85   Map</a>. The key and value types must be the graph's vertex descriptor type.<br>
  86   <b>Default:</b> A <tt>boost::iterator_property_map</tt> using a <tt>std::vector</tt>
  87   of <tt>num_vertices(g)</tt> vertex descriptors and <tt>vertexIndices</tt> for
  88   the index map.
  89 </blockquote>
  90
  91 <p>UTIL: <tt>max_priority_queue(MaxPriorityQueue&amp; pq)</tt>
  92 <blockquote>
  93   <tt>MaxPriorityQueue</tt> must be a model of <a href="./KeyedUpdatableQueue.html">Keyed Updatable Queue</a>
  94   and a max-<a href="./UpdatableQueue.html#concept%3AUpdatablePriorityQueue">Updatable Priority Queue</a>.
  95   The value type must be the graph's vertex descriptor and the key type must be
  96   the weight type.
  97   <b>Default:</b> A <tt>boost::d_ary_heap_indirect</tt> using a default index-in-heap
  98   and distance map.
  99 </blockquote>
 100
 101 <p>UTIL: <tt>index_in_heap_map(IndexInHeapMap indicesInHeap)</tt>
 102 <blockquote>
 103   This parameter only has an effect when the default max-priority queue is used.<br>
 104   <tt>IndexInHeapMap</tt> must be a model of <a
 105   href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write Property
 106   Map</a>. The key type must be the graph's vertex descriptor type. The value type
 107   must be a size type (<tt>typename&nbsp;std::vector&lt;vertex_descriptor&gt;::size_type</tt>).<br>
 108   <b>Default:</b> A <tt>boost::iterator_property_map</tt> using a <tt>std::vector</tt>
 109   of <tt>num_vertices(g)</tt> size type objects and <tt>vertexIndices</tt> for
 110   the index map.
 111 </blockquote>
 112
 113 <p>UTIL: <tt>distance_map(DistanceMap wAs)</tt>
 114 <blockquote>
 115   This parameter only has an effect when the default max-priority queue is used.<br>
 116   <tt>DistanceMap</tt> must be a model of <a
 117   href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write Property
 118   Map</a>. The key type must be the graph's vertex descriptor type. The value type
 119   must be the weight type (<tt>typename&nbsp;boost::property_traits&lt;WeightMap&gt;::value_type</tt>).<br>
 120   <b>Default:</b> A <tt>boost::iterator_property_map</tt> using a <tt>std::vector</tt>
 121   of <tt>num_vertices(g)</tt> weight type objects and <tt>vertexIndices</tt> for
 122   the index map.
 123 </blockquote>
 124
 125 <h3>Returns</h3>
 126 <p>The weight of the min-cut
 127
 128 <h3>Throws</h3>
 129
 130 <p><tt>bad_graph</tt>
 131 <blockquote>
 132   If <tt>num_vertices(g)</tt> is less than 2
 133 </blockquote>
 134
 135 <p><tt>std::invalid_argument</tt>
 136 <blockquote>
 137   If a max-priority queue is given as an argument and it is not empty
 138 </blockquote>
 139
 140 <h3>Complexity</h3>
 141
 142 <p>The time complexity is <i>O</i>(<i>V</i>&#xb7;<i>E</i> + <i>V</i><sup>2</sup> log <i>V</i>).
 143
 144 <h3>Example</h3>
 145
 146 <p>The file <a href="../example/stoer_wagner.cpp"><tt>examples/stoer_wagner.cpp</tt></a> contains an example of calculating a min-cut of a weighted, undirected graph and its min-cut weight.
 147
 148 <h3>References</h3>
 149 <ul>
 150 <li>Mehlhorn, Kurt and Christian Uhrig (1995). <q><a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.31.614&amp;rep=rep1&amp;type=pdf">The minimum cut algorithm of Stoer and Wagner</a></q>.
 151 <li>Stoer, Mechthild and Frank Wagner (1997). <q><a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.114.6687&amp;rep=rep1&amp;type=pdf">A simple min-cut algorithm</a></q>. <i>Journal of the ACM</i> <b>44</b> (4), 585&ndash;591.
 152 <li>Zwick, Uri (2008). <q><a href="http://www.cs.tau.ac.il/~zwick/grad-algo-08/gmc.pdf">Global minimum cuts</a></q>.
 153 </ul>
 154
 155 <br>
 156 <hr>
 157 <table>
 158 <tr>
 159 <td>Copyright&nbsp;&copy;&nbsp;2010</td>
 160 <td>Daniel Trebbien (<a href="mailto:dtrebbien@gmail.com">dtrebbien@gmail.com</a>)
 161 </td>
 162 </tr>
 163 </table>
 164
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