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  3.      Copyright (c) Jeremy Siek 2000
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  9. <Head>
    10. 

  10. <Title>Boost Graph Library: Strongly Connected Components</Title>
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  14.      ALT="C++ Boost" width="277" height="86"> 
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  17. 

  18. 

  19. <H1>
    20. 

  20. <A NAME="sec:connected-components"></A><A NAME="sec:strongly-connected-components"></A>
    21. 

  21. <img src="figs/python.gif" alt="(Python)"/>
    22. 

  22. <TT>strong_components</TT>
    23. 

  23. </H1>
    24. 

  24. 

  25. <PRE>
    26. 

  26. <i>// named parameter version</i>
    27. 

  27. template &lt;class Graph, class ComponentMap, class P, class T, class R&gt;
    28. 

  28. typename property_traits&lt;ComponentMap&gt;::value_type
    29. 

  29. strong_components(Graph& g, ComponentMap comp,
    30. 

  30.     const bgl_named_params&lt;P, T, R&gt;&amp; params = <i>all defaults</i>)
    31. 

  31. 

  32. <i>// there is not a non-named parameter version of this function</i>
    33. 

  33. </PRE>
    34. 

  34. 

  35. <P>
    36. 

  36. The <TT>strong_components()</TT> functions compute the strongly
    37. 

  37. connected components of a directed graph using Tarjan's algorithm
    38. 

  38. based on DFS&nbsp;[<A
    39. 

  39. HREF="bibliography.html#tarjan72:dfs_and_linear_algo">41</A>].
    40. 

  40. </p>
    41. 

  41. 

  42. <P>
    43. 

  43. The output of the algorithm is recorded in the component property
    44. 

  44. map <TT>comp</TT>, which will contain numbers giving the component ID
    45. 

  45. assigned to each vertex. The number of components is the return value
    46. 

  46. of the function.
    47. 

  47. </p>
    48. 

  48. 

  49. <H3>Where Defined</H3>
    50. 

  50. 

  51. <P>
    52. 

  52. <a href="../../../boost/graph/strong_components.hpp"><TT>boost/graph/strong_components.hpp</TT></a>
    53. 

  53. 

  54. <P>
    55. 

  55. 

  56. <H3>Definitions</H3>
    57. 

  57. 

  58. <P>
    59. 

  59. A <a name="def:strongly-connected-component"><b><I>strongly connected
    60. 

  60. component</I></b></a> of a directed graph <i>G=(V,E)</i> is a maximal
    61. 

  61. set of vertices <i>U</i> which is in <i>V</i> such that for every pair
    62. 

  62. of vertices <i>u</i> and <i>v</i> in <i>U</i>, we have both a path
    63. 

  63. from <i>u</i> to <i>v</i> and path from <i>v</i> to <i>u</i>. That is
    64. 

  64. to say that <i>u</i> and <i>v</i> are reachable from each other.
    65. 

  65. 

  66. <P>
    67. 

  67. 

  68. <h3>Parameters</h3>
    69. 

  69. 

  70. IN: <tt>const Graph&amp; g</tt>
    71. 

  71. <blockquote>
    72. 

  72. A directed graph. The graph type must be a model of <a
    73. 

  73. href="VertexListGraph.html">Vertex List Graph</a> and <a
    74. 

  74. href="IncidenceGraph.html">Incidence Graph</a>.<br>
    75. 

  75. 

  76. <b>Python</b>: The parameter is named <tt>graph</tt>.
    77. 

  77. </blockquote>
    78. 

  78. 

  79. OUT: <tt>ComponentMap c</tt>
    80. 

  80. <blockquote>
    81. 

  81. The algorithm computes how many connected components are in the graph,
    82. 

  82. and assigning each component an integer label. The algorithm then
    83. 

  83. records which component each vertex in the graph belongs to by
    84. 

  84. recording the component number in the component property map. The
    85. 

  85. <tt>ComponentMap</tt> type must be a model of <a
    86. 

  86. href="../../property_map/doc/WritablePropertyMap.html">Writable Property
    87. 

  87. Map</a>. The value type shouch be an integer type, preferably the same
    88. 

  88. as the <tt>vertices_size_type</tt> of the graph. The key type must be
    89. 

  89. the graph's vertex descriptor type.<br>
    90. 

    91. 

  91.   <b>Python</b>: Must be an <tt>vertex_int_map</tt> for the graph.<br>
    92. 

  92.   <b>Python default</b>: <tt>graph.get_vertex_int_map("component")</tt>
    93. 

  93. </blockquote>
    94. 

  94. 

  95. 

  96. <h3>Named Parameters</h3>
    97. 

  97. 

  98. UTIL: <tt>root_map(RootMap r_map)</tt>
    99. 

  99. <blockquote>
    100. 

  100.   This is used by the algorithm to record the candidate root vertex for
    101. 

  101.   each vertex. By the end of the algorithm, there is a single root vertex
    102. 

  102.   for each component and <tt>get(r_map, v)</tt> returns the root
    103. 

  103.   vertex for whichever component vertex <tt>v</tt> is a member.
    104. 

  104.   The <TT>RootMap</TT> must be a <a
    105. 

  105.   href="../../property_map/doc/ReadWritePropertyMap.html">
    106. 

  106.   Read/Write Property Map</a>, where the key type and the value type are
    107. 

  107.   the vertex descriptor type of the graph.<br>
    108. 

  108. 

  109.   <b>Default:</b> an <a
    110. 

  110.   href="../../property_map/doc/iterator_property_map.html">
    111. 

  111.   <tt>iterator_property_map</tt></a> created from a
    112. 

  112.   <tt>std::vector</tt> of vertex descriptors of size
    113. 

  113.   <tt>num_vertices(g)</tt> and using the <tt>i_map</tt> for the index
    114. 

  114.   map.<br>
    115. 

  115.   <b>Python</b>: Unsupported parameter.
    116. 

  116. </blockquote>
    117. 

  117. 

  118. UTIL: <tt>discover_time_map(TimeMap t_map)</tt>
    119. 

  119. <blockquote>
    120. 

  120.   This is used by the algorithm to keep track of the DFS ordering
    121. 

  121.   of the vertices. The <TT>TimeMap</TT> must be a model
    122. 

  122.   of <a href="../../property_map/doc/ReadWritePropertyMap.html"> Read/Write
    123. 

  123.   Property Map</a> and its value type must be an integer type. The key
    124. 

  124.   type must be the vertex descriptor type of the graph.<br>
    125. 

  125.   <b>Default:</b>an <a
    126. 

  126.   href="../../property_map/doc/iterator_property_map.html">
    127. 

  127.   <tt>iterator_property_map</tt></a> created from a
    128. 

  128.   <tt>std::vector</tt> of integers with size
    129. 

  129.   <tt>num_vertices(g)</tt> and using the <tt>i_map</tt> for the index
    130. 

  130.   map.<br>
    131. 

  131.   <b>Python</b>: Unsupported parameter.
    132. 

  132. </blockquote>
    133. 

  133. 

  134. UTIL: <tt>color_map(ColorMap c_map)</tt>
    135. 

  135. <blockquote>
    136. 

  136.   This is used by the algorithm to keep track of its progress through
    137. 

  137.   the graph. The type <tt>ColorMap</tt> must be a model of <a
    138. 

  138.   href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
    139. 

  139.   Property Map</a> and its key type must be the graph's vertex
    140. 

  140.   descriptor type and the value type of the color map must model
    141. 

  141.   <a href="./ColorValue.html">ColorValue</a>.<br>
    142. 

  142.   <b>Default:</b> an <a
    143. 

  143.   href="../../property_map/doc/iterator_property_map.html">
    144. 

  144.   <tt>iterator_property_map</tt></a> created from a
    145. 

  145.   <tt>std::vector</tt> of <tt>default_color_type</tt> of size
    146. 

  146.   <tt>num_vertices(g)</tt> and using the <tt>i_map</tt> for the index
    147. 

  147.   map.<br>
    148. 

  148.   <b>Python</b>: Unsupported parameter.
    149. 

  149. </blockquote>
    150. 

  150. 

  151. IN: <tt>vertex_index_map(VertexIndexMap i_map)</tt>
    152. 

  152. <blockquote>
    153. 

  153.   This maps each vertex to an integer in the range <tt>[0,
    154. 

  154.   num_vertices(g))</tt>. This parameter is only necessary when a
    155. 

  155.   default is used for one of the other named parameters. The type
    156. 

  156.   <tt>VertexIndexMap</tt> must be a model of <a
    157. 

  157.   href="../../property_map/doc/ReadablePropertyMap.html">Readable Property
    158. 

  158.   Map</a>. The value type of the map must be an integer type. The
    159. 

  159.   vertex descriptor type of the graph needs to be usable as the key
    160. 

  160.   type of the map.<br>
    161. 

  161. 

  162.   <b>Default:</b> <tt>get(vertex_index, g)</tt>
    163. 

  163.     Note: if you use this default, make sure your graph has
    164. 

  164.     an internal <tt>vertex_index</tt> property. For example,
    165. 

  165.     <tt>adjacenty_list</tt> with <tt>VertexList=listS</tt> does
    166. 

  166.     not have an internal <tt>vertex_index</tt> property.
    167. 

  167.    <br>
    168. 

  168. 

  169.   <b>Python</b>: Unsupported parameter.
    170. 

  170. </blockquote>
    171. 

  171. 

  172. 

  173. <H3>Complexity</H3>
    174. 

  174. 

  175. <P>
    176. 

  176. The time complexity for the strongly connected components algorithm is
    177. 

  177. <i>O(V + E)</i>.
    178. 

  178. 

  179. <P>
    180. 

  180. 

  181. <h3>See Also</h3>
    182. 

  182. 

  183. <a href="./connected_components.html"><tt>connected_components()</tt></a>
    184. 

  184. and <a href="./incremental_components.html"><tt>incremental_components()</tt></a>
    185. 

  185. 

  186. <H3>Example</H3>
    187. 

  187. 

  188. <P>
    189. 

  189. See <a
    190. 

  190. href="../example/strong_components.cpp"><tt>examples/strong_components.cpp</tt></a>.
    191. 

  191. 

  192. <br>
    193. 

  193. <HR>
    194. 

  194. <TABLE>
    195. 

  195. <TR valign=top>
    196. 

  196. <TD nowrap>Copyright &copy; 2000-2001</TD><TD>
    197. 

  197. <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>)
    198. 

  198. </TD></TR></TABLE>
    199. 

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