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<Title>Boost Graph Library: Incremental Connected Components</Title>
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<BR Clear>

<H1>Incremental Connected Components</H1>

<P>
This section describes a family of functions and classes that work
together to calculate the connected components of an undirected graph.
The algorithm used here is based on the disjoint-sets (fast
union-find) data structure&nbsp;[<A
HREF="bibliography.html#clr90">8</A>,<A
HREF="bibliography.html#tarjan83:_data_struct_network_algo">27</A>]
which is a good method to use for situations where the graph is
growing (edges are being added) and the connected components
information needs to be updated repeatedly. This method does not cover
the situation where edges are both added and removed from the graph,
hence it is called <b><i>incremental</i></b><a
href="bibliography.html#eppstein97:dynamic_graph">[42]</a> (and not
fully dynamic). The disjoint-sets class is described in Section <A
HREF="../../disjoint_sets/disjoint_sets.html">Disjoint Sets</A>.

<P>
The following five operations are the primary functions that you will
use to calculate and maintain the connected components.  The objects
used here are a graph <TT>g</TT>, a disjoint-sets structure <TT>ds</TT>,
and vertices <TT>u</TT> and <TT>v</TT>.

<P>

<UL>
<LI><TT>initialize_incremental_components(g, ds)</TT> 
<BR>
Basic initialization of the disjoint-sets structure. Each
    vertex in the graph <TT>g</TT> is in its own set.
</LI>
<LI><TT>incremental_components(g, ds)</TT> 
<BR>
The connected components are calculated based on the edges in the graph
    <TT>g</TT> and the information is embedded in <TT>ds</TT>.
</LI>
<LI><TT>ds.find_set(v)</TT> 
<BR>
Extracts the component information for vertex <TT>v</TT> from the
    disjoint-sets.
</LI>
<LI><TT>ds.union_set(u, v)</TT> 
<BR>
Update the disjoint-sets structure when edge <i>(u,v)</i> is added to the graph.
</LI>
</UL>

<P>

<H3>Complexity</H3>

<P>
The time complexity for the whole process is <i>O(V + E
alpha(E,V))</i> where <i>E</i> is the total number of edges in the
graph (by the end of the process) and <i>V</i> is the number of
vertices.  <i>alpha</i> is the inverse of Ackermann's function which
has explosive recursively exponential growth. Therefore its inverse
function grows <I>very</I> slowly. For all practical purposes
<i>alpha(m,n) <= 4</i> which means the time complexity is only
slightly larger than <i>O(V + E)</i>.

<P>

<H3>Example</H3>

<P>
Maintain the connected components of a graph while adding edges using
the disjoint-sets data structure. The full source code for this
example can be found in <a
href="../example/incremental_components.cpp"><TT>examples/incremental_components.cpp</TT></a>.

<P>
<PRE class="code">
using namespace boost;

int main(int argc, char* argv[]) 
{
  typedef adjacency_list <vecS, vecS, undirectedS> Graph;
  typedef graph_traits<Graph>::vertex_descriptor Vertex;
  typedef graph_traits<Graph>::vertices_size_type VertexIndex;

  const int VERTEX_COUNT = 6;
  Graph graph(VERTEX_COUNT);

  std::vector<VertexIndex> rank(num_vertices(graph));
  std::vector<Vertex> parent(num_vertices(graph));

  typedef VertexIndex* Rank;
  typedef Vertex* Parent;

  disjoint_sets<Rank, Parent> ds(&rank[0], &parent[0]);

  initialize_incremental_components(graph, ds);
  incremental_components(graph, ds);

  graph_traits<Graph>::edge_descriptor edge;
  bool flag;

  boost::tie(edge, flag) = add_edge(0, 1, graph);
  ds.union_set(0,1);

  boost::tie(edge, flag) = add_edge(1, 4, graph);
  ds.union_set(1,4);

  boost::tie(edge, flag) = add_edge(4, 0, graph);
  ds.union_set(4,0);

  boost::tie(edge, flag) = add_edge(2, 5, graph);
  ds.union_set(2,5);
    
  std::cout << "An undirected graph:" << std::endl;
  print_graph(graph, get(boost::vertex_index, graph));
  std::cout << std::endl;
    
  BOOST_FOREACH(Vertex current_vertex, vertices(graph)) {
    std::cout << "representative[" << current_vertex << "] = " <<
      ds.find_set(current_vertex) << std::endl;
  }

  std::cout << std::endl;

  typedef component_index<VertexIndex> Components;

  // NOTE: Because we're using vecS for the graph type, we're
  // effectively using identity_property_map for a vertex index map.
  // If we were to use listS instead, the index map would need to be
  // explicity passed to the component_index constructor.
  Components components(parent.begin(), parent.end());

  // Iterate through the component indices
  BOOST_FOREACH(VertexIndex current_index, components) {
    std::cout << "component " << current_index << " contains: ";

    // Iterate through the child vertex indices for [current_index]
    BOOST_FOREACH(VertexIndex child_index,
                  components[current_index]) {
      std::cout << child_index << " ";
    }

    std::cout << std::endl;
  }

  return (0);
}
</PRE>

<P>
<hr>
<p>

<H2><A NAME="sec:initialize-incremental-components"></A>
<TT>initialize_incremental_components</TT>
</H2>

<P>
<DIV ALIGN="left">
<TABLE CELLPADDING=3 border>
<TR><TH ALIGN="LEFT"><B>Graphs:</B></TH>
<TD ALIGN="LEFT">undirected</TD>
</TR>
<TR><TH ALIGN="LEFT"><B>Properties:</B></TH>
<TD ALIGN="LEFT">rank, parent (in disjoint-sets)</TD>
</TR>
<TR><TH ALIGN="LEFT"><B>Complexity:</B></TH>
<TD></TD>
</TR>
</TABLE>
</DIV>

<P>
<PRE>
template &lt;class VertexListGraph, class DisjointSets&gt; 
void initialize_incremental_components(VertexListGraph&amp; G, DisjointSets&amp; ds)
</PRE>

<P>
This prepares the disjoint-sets data structure for the incremental
connected components algorithm by making each vertex in the graph a
member of its own component (or set).

<P>

<H3>Where Defined</H3>

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

<p>
<hr>
<P>

<H2><A NAME="sec:incremental-components"></A>
<TT>incremental_components</TT>
</H2>

<P>
<DIV ALIGN="left">
<TABLE CELLPADDING=3 border>
<TR><TH ALIGN="LEFT"><B>Graphs:</B></TH>
<TD ALIGN="LEFT">undirected</TD>
</TR>
<TR><TH ALIGN="LEFT"><B>Properties:</B></TH>
<TD ALIGN="LEFT">rank, parent (in disjoint-sets)</TD>
</TR>
<TR><TH ALIGN="LEFT"><B>Complexity:</B></TH>
<TD ALIGN="LEFT"><i>O(E)</i></TD>
</TR>
</TABLE>
</DIV>

<p>
<PRE>
template &lt;class EdgeListGraph, class DisjointSets&gt;
void incremental_components(EdgeListGraph&amp; g, DisjointSets&amp; ds)
</PRE>

<P>
This function calculates the connected components of the graph,
embedding the results in the disjoint-sets data structure.

<P>

<H3>Where Defined</H3>

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

<P>

<H3>Requirements on Types</H3>

<P>

<UL>
<LI>The graph type must be a model of <a href="./EdgeListGraph.html">EdgeListGraph</a>.
</LI>
</UL>

<P>
<hr>
<p>

<H2><A NAME="sec:same-component">
<TT>same_component</TT></A>
</H2>

<P>
<DIV ALIGN="left">
<TABLE CELLPADDING=3 border>
<TR><TH ALIGN="LEFT"><B>Properties:</B></TH>
<TD ALIGN="LEFT">rank, parent (in disjoint-sets)</TD>
</TR>
<TR><TH ALIGN="LEFT"><B>Complexity:</B></TH>
<TD ALIGN="LEFT"><i>O(alpha(E,V))</i></TD>
</TR>
</TABLE>
</DIV>

<P>
<PRE>
template &lt;class Vertex, class DisjointSet&gt;
bool same_component(Vertex u, Vertex v, DisjointSet&amp; ds)
</PRE>

<P>
This function determines whether <TT>u</TT> and <TT>v</TT> are in the same
component.

<P>

<H3>Where Defined</H3>

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

<P>

<H3>Requirements on Types</H3>

<P>

<UL>
<LI><TT>Vertex</TT> must be compatible with the rank and parent
    property maps of the <TT>DisjointSets</TT> data structure.
</LI>
</UL>

<P>
<hr>
<p>

<H2><A NAME="sec:component-index"></A>
<TT>component_index</TT>
</H2>

<p>
<PRE>
component_index&lt;Index&gt;
</PRE>

<P>
The <tt>component_index</tt> class provides an STL
container-like view for the components of the graph. Each component is
a container-like object, and access is provided via
the <TT>operator[]</TT>.  A <TT>component_index</TT> object is
initialized with the parents property in the disjoint-sets calculated
from the <TT>incremental_components()</TT> function.  Optionally, a
vertex -&gt; index property map is passed in
(<tt>identity_property_map</tt> is used by default).

<P>

<H3>Where Defined</H3>

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

<P>

<H3>Members</H3>

<P>
 
<table border>

<tr>
<th>Member</th> <th>Description</th>
</tr>

<tr>
<td><tt>value_type/size_type</tt></td>
<td>
The type for a component index (same as <tt>Index</tt>).
</td>
</tr>

<tr>
<td><tt>size_type size()</tt></td>
<td>
Returns the number of components in the graph.
</td>
</tr>


<tr>
<td><tt>iterator/const_iterator</tt></td>
<td>
Iterators used to traverse the available component indices [0 to <tt>size()</tt>).
</td>
</tr>

<tr>
<td><tt>iterator begin() const</tt></td>
<td>
Returns an iterator at the start of the component indices (0).
</td>
</tr>

<tr>
<td><tt>iterator end() const</tt></td>
<td>
Returns an iterator past the end of the component indices (<tt>size()</tt>).
</td>
</tr>
 
<tr>
<td><tt>std::pair&lt;component_iterator, component_iterator&gt; operator[size_type index] const</tt></td>
<td>
Returns a pair of iterators for the component at <tt>index</tt> where <tt>index</tt> is in [0, <tt>size()</tt>).
</td>
</tr>

</table> 

<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>)<br>
<A HREF="http://www.boost.org/people/liequan_lee.htm">Lie-Quan Lee</A>, Indiana University (<A HREF="mailto:llee@cs.indiana.edu">llee@cs.indiana.edu</A>)<br>
<A HREF="http://www.osl.iu.edu/~lums">Andrew Lumsdaine</A>,
Indiana University (<A
HREF="mailto:lums@osl.iu.edu">lums@osl.iu.edu</A>)
</TD></TR></TABLE>

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