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

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   3           Copyright (C) 2012, Michele Caini.
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   7
   8           Two Graphs Common Spanning Trees Algorithm
   9       Based on academic article of Mint, Read and Tarjan
  10      Efficient Algorithm for Common Spanning Tree Problem
  11  Electron. Lett., 28 April 1983, Volume 19, Issue 9, p.346–347
  12 -->
  13 <head>
  14 <title>Boost Graph Library: Two-Graphs Common Spanning Trees</title>
  15 <body bgcolo="#ffffff" link="#0000ee" text="#000000" vlink="#551a8b" alink="#ff0000">
  16 <img src="../../../boost.png" alt="C++ Boost" width="277" height="86">
  17
  18 <br clear>
  19
  20 <h1><tt><a name="sec:two-graphs-common-spanning-trees">Two-Graphs Common Spanning Trees (MRT Algorithm)</a></tt></h1>
  21
  22 <p>
  23 The <b>MRT</b> algorithm, based on an academic article of <b>Mint</b>, <b>Read</b> and
  24 <b>Tarjan</b>, is an efficient algorithm for the common spanning tree problem.<br/>
  25 This kind of algorithm is widely used in electronics, being the basis of the
  26 analysis of electrical circuits. Another area of interest may be that of the
  27 networking.
  28 </p>
  29
  30 <p>
  31 The proposed algorithm receives several arguments and works with callbacks.<br/>
  32 The prototypes are:
  33 </p>
  34
  35 <p>
  36 <i>// Ordered Edges List</i>
  37 <pre>
  38 template &lt; typename Graph, typename Order, typename Func, typename Seq &gt;
  39 void two_graphs_common_spanning_trees
  40   (
  41     const Graph&amp; iG,
  42     Order iG_map,
  43     const Graph&amp; vG,
  44     Order vG_map,
  45     Func func,
  46     Seq inL
  47   )
  48 </pre>
  49
  50 <i>// Unordered Edges List</i>
  51 <pre>
  52 template &lt; typename Graph, typename Func, typename Seq &gt;
  53 void two_graphs_common_spanning_trees
  54   (
  55     const Graph&amp; iG,
  56     const Graph&amp; vG,
  57     Func func,
  58     Seq inL
  59   )
  60 </pre>
  61 </p>
  62
  63 <p>
  64 The problem of common spanning tree is easily described.<br/>
  65 Imagine we have two graphs that are represented as lists of edges. A common
  66 spanning tree is a set of indices that identifies a spanning tree for both
  67 the first and for the second of the two graphs. Despite it is easily accomplished
  68 with edge list representation for graphs, it is intuitively difficult to achieve
  69 with adjacency list representation. This is due to the fact that it is necessary
  70 to represent an edge with an absolute index.
  71 </p>
  72 <p>
  73 Note that the set of common spanning trees of the two graphs is a subset of
  74 the set of spanning trees of the first graph, as well as those of the second
  75 graph.
  76 </p>
  77
  78 <h3>Where Defined</h3>
  79
  80 <p>
  81 <a href="../../../boost/graph/two_graphs_common_spanning_trees.hpp"><TT>boost/graph/two_graphs_common_spanning_trees.hpp</TT></a>
  82
  83 <h3>Parameters</h3>
  84
  85 <tt>const Graph& iG, const Graph& vG</tt>
  86 <blockquote>
  87 These are the graphs to be analyzed.<br/>
  88 They must comply with the concepts VertexAndEdgeListGraphConcept and IncidenceGraphConcept.<br/>
  89 In addition, the directed_category should be of type undirected_tag.
  90 </blockquote>
  91
  92 <tt>Order iG_map, Order vG_map</tt>
  93 <blockquote>
  94 These are lists of references to edges, that define the preferred order for access to the lists of edges.<br/>
  95 They must comply with the concept RandomAccessContainer.
  96 </blockquote>
  97
  98 <tt>Func func</tt>
  99 <blockquote>
 100 This is a callback that is invoked by the algorithm for each common spanning tree found.<br/>
 101 It must comply with the concept UnaryFunction with void as return value, and an object of type <i>typeof(inL)</i> as argument.
 102 </blockquote>
 103
 104 <tt>Seq inL<a href="#1">[1]</a></tt>
 105 <blockquote>
 106 This is the way in which the edges are marked as belonging to the common spanning tree.<br/>
 107 It must comply with the concept Mutable_RandomAccessContainer. In addition, the value_type should be of type bool.
 108 If the i-th edge or <i>inL[i]</i> is true, then it belongs to the common spanning tree, otherwise it does not belong.
 109 </blockquote>
 110
 111 <h3>Example</h3>
 112
 113 <p>
 114 The file
 115 <a href="../example/two_graphs_common_spanning_trees.cpp"><tt>examples/two_graphs_common_spanning_trees.cpp</tt></a>
 116 contains an example of finding common spanning trees of two undirected graphs.
 117 </p>
 118
 119 <h3>Notes</h3>
 120
 121 <p>
 122 <a name="1">[1]</a><br/>
 123   The presence of <i>inL</i> may seem senseless. The algorithm can use a vector of
 124   placeholders internally generated. However, doing so has more flexibility on
 125   the callback function. Moreover, being largely involved in the electronics
 126   world, there are cases where some edges have to be forced in every tree (ie
 127   you want to search all the trees that have the same root): With this
 128   solution, the problem is easily solved.<br/>
 129   Intuitively from the above, <i>inL</i> must be of a size equal to <i>(V-1)</i>, where
 130   <i>V</i> is the number of vertices of the graph.
 131 </p>
 132
 133 <br/>
 134 <hr>
 135 <table>
 136 <tr valign=top>
 137 <td nowrap>Copyright &copy; 2012</td>
 138 <td>Michele Caini, (<a href="mailto:michele.caini@gmail.com">michele.caini@gmail.com</a>)</td>
 139 </tr>
 140 </table>
 141
 142 </body>
 143 </html>