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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>

 </BODY>
 </HTML>
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	<Head>
	<Title>Boost Graph Library: Incremental Connected Components</Title>
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	<IMG SRC="../../../boost.png"
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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 [<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 <class VertexListGraph, class DisjointSets>
	void initialize_incremental_components(VertexListGraph& G, DisjointSets& 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 <class EdgeListGraph, class DisjointSets>
	void incremental_components(EdgeListGraph& g, DisjointSets& 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 <class Vertex, class DisjointSet>
	bool same_component(Vertex u, Vertex v, DisjointSet& 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<Index>
	</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 -> 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<component_iterator, component_iterator> 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 © 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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