<HTML>
<!--
     Copyright (c) Jeremy Siek 2000
    
     Distributed under the Boost Software License, Version 1.0.
     (See accompanying file LICENSE_1_0.txt or copy at
     http://www.boost.org/LICENSE_1_0.txt)
  -->
<Head>
<Title>Boost Graph Library: Strongly Connected Components</Title>
<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b" 
        ALINK="#ff0000"> 
<IMG SRC="../../../boost.png" 
     ALT="C++ Boost" width="277" height="86"> 

<BR Clear>


<H1>
<A NAME="sec:connected-components"></A><A NAME="sec:strongly-connected-components"></A>
<img src="figs/python.gif" alt="(Python)"/>
<TT>strong_components</TT>
</H1>

<PRE>
<i>// named parameter version</i>
template &lt;class Graph, class ComponentMap, class P, class T, class R&gt;
typename property_traits&lt;ComponentMap&gt;::value_type
strong_components(Graph& g, ComponentMap comp,
    const bgl_named_params&lt;P, T, R&gt;&amp; params = <i>all defaults</i>)

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

<P>
The <TT>strong_components()</TT> functions compute the strongly
connected components of a directed graph using Tarjan's algorithm
based on DFS&nbsp;[<A
HREF="bibliography.html#tarjan72:dfs_and_linear_algo">41</A>].
</p>

<P>
The output of the algorithm is recorded in the component property
map <TT>comp</TT>, which will contain numbers giving the component ID
assigned to each vertex. The number of components is the return value
of the function.
</p>

<H3>Where Defined</H3>

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

<P>

<H3>Definitions</H3>

<P>
A <a name="def:strongly-connected-component"><b><I>strongly connected
component</I></b></a> of a directed graph <i>G=(V,E)</i> is a maximal
set of vertices <i>U</i> which is in <i>V</i> such that for every pair
of vertices <i>u</i> and <i>v</i> in <i>U</i>, we have both a path
from <i>u</i> to <i>v</i> and path from <i>v</i> to <i>u</i>. That is
to say that <i>u</i> and <i>v</i> are reachable from each other.

<P>

<h3>Parameters</h3>

IN: <tt>const Graph&amp; g</tt>
<blockquote>
A directed graph. The graph type must be a model of <a
href="VertexListGraph.html">Vertex List Graph</a> and <a
href="IncidenceGraph.html">Incidence Graph</a>.<br>

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

OUT: <tt>ComponentMap c</tt>
<blockquote>
The algorithm computes how many connected components are in the graph,
and assigning each component an integer label. The algorithm then
records which component each vertex in the graph belongs to by
recording the component number in the component property map. The
<tt>ComponentMap</tt> type must be a model of <a
href="../../property_map/doc/WritablePropertyMap.html">Writable Property
Map</a>. The value type shouch be an integer type, preferably the same
as the <tt>vertices_size_type</tt> of the graph. The key type must be
the graph's vertex descriptor type.<br>

  <b>Python</b>: Must be an <tt>vertex_int_map</tt> for the graph.<br>
  <b>Python default</b>: <tt>graph.get_vertex_int_map("component")</tt>
</blockquote>


<h3>Named Parameters</h3>

UTIL: <tt>root_map(RootMap r_map)</tt>
<blockquote>
  This is used by the algorithm to record the candidate root vertex for
  each vertex. By the end of the algorithm, there is a single root vertex
  for each component and <tt>get(r_map, v)</tt> returns the root
  vertex for whichever component vertex <tt>v</tt> is a member.
  The <TT>RootMap</TT> must be a <a
  href="../../property_map/doc/ReadWritePropertyMap.html">
  Read/Write Property Map</a>, where the key type and the value type are
  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>

UTIL: <tt>discover_time_map(TimeMap t_map)</tt>
<blockquote>
  This is used by the algorithm to keep track of the DFS ordering
  of the vertices. The <TT>TimeMap</TT> must be a model
  of <a href="../../property_map/doc/ReadWritePropertyMap.html"> Read/Write
  Property Map</a> and its value type must be an integer type. The key
  type 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 integers with 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>color_map(ColorMap c_map)</tt>
<blockquote>
  This is used by the algorithm to keep track of its progress through
  the graph. The type <tt>ColorMap</tt> must be a model of <a
  href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
  Property Map</a> and its key type must be the graph's vertex
  descriptor type and the value type of the color map must model
  <a href="./ColorValue.html">ColorValue</a>.<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 <tt>default_color_type</tt> 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 parameter is only necessary when a
  default is used for one of the other named parameters. 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 for the strongly connected components algorithm is
<i>O(V + E)</i>.

<P>

<h3>See Also</h3>

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

<H3>Example</H3>

<P>
See <a
href="../example/strong_components.cpp"><tt>examples/strong_components.cpp</tt></a>.

<br>
<HR>
<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>)
</TD></TR></TABLE>

</BODY>
</HTML>