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<HTML>
<!--
-- Copyright (c) Jeremy Siek, Lie-Quan Lee, and Andrew Lumsdaine 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: Incremental 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>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>
typedef adjacency_list <vecS, vecS, undirectedS> Graph;
typedef graph_traits<Graph>::vertex_descriptor Vertex;
typedef graph_traits<Graph>::vertices_size_type size_type;
const int N = 6;
Graph G(N);
std::vector<size_type> rank(num_vertices(G));
std::vector<Vertex> parent(num_vertices(G));
typedef size_type* Rank;
typedef Vertex* Parent;
disjoint_sets<Rank, Parent> ds(&rank[0], &parent[0]);
initialize_incremental_components(G, ds);
incremental_components(G, ds);
graph_traits<Graph>::edge_descriptor e;
bool flag;
boost::tie(e,flag) = add_edge(0, 1, G);
ds.union_set(0,1);
boost::tie(e,flag) = add_edge(1, 4, G);
ds.union_set(1,4);
boost::tie(e,flag) = add_edge(4, 0, G);
ds.union_set(4,0);
boost::tie(e,flag) = add_edge(2, 5, G);
ds.union_set(2,5);
cout << "An undirected graph:" << endl;
print_graph(G, get(vertex_index, G));
cout << endl;
graph_traits<Graph>::vertex_iterator i,end;
for (boost::tie(i, end) = vertices(G); i != end; ++i)
cout << "representative[" << *i << "] = " <<
ds.find_set(*i) << endl;;
cout << endl;
typedef component_index<unsigned int> Components;
Components components(&parent[0], &parent[0] + parent.size());
for (Components::size_type c = 0; c < components.size(); ++c) {
cout << "component " << c << " contains: ";
Components::value_type::iterator
j = components[c].begin(),
jend = components[c].end();
for ( ; j != jend; ++j)
cout << *j << " ";
cout << endl;
}
</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 is a class that provides an STL container-like view for the
components of the graph. Each component is a container-like object,
and the <TT>component_index</TT> object provides access to the
component objects via <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.
<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>size_type</tt></td>
<td>The type used for representing the number of components.</td>
</tr>
<tr>
<td><tt>value_type</tt></td>
<td>
The type for a component object. The component type has the following members.
</td>
</tr>
<tr>
<td><tt>value_type::value_type</tt></td>
<td>
The value type of a component object is a vertex ID.
</td>
</tr>
<tr>
<td><tt>value_type::iterator</tt></td>
<td>
This iterator can be used to traverse all of the vertices
in the component. This iterator dereferences to give a vertex ID.
</td>
</tr>
<tr>
<td><tt>value_type::const_iterator</tt></td>
<td>
The const iterator.
</td>
</tr>
<tr>
<td><tt>value_type::iterator value_type::begin() const</tt></td>
<td>
Return an iterator pointing to the first vertex in the component.
</td>
</tr>
<tr>
<td><tt>value_type::iterator value_type::end() const</tt></td>
<td>
Return an iterator pointing past the end of the last vertex in the
component.
</td>
<tr>
<tr>
<td>
<tt>
template <class ComponentsContainer>
component_index(const ComponentsContainer& c)
</tt>
</td>
<td>
Construct the <TT>component_index</TT> using the information
from the components container <TT>c</TT> which was the result
of executing <TT>connected_components_on_edgelist</TT>.
</td>
</tr>
<tr>
<td><tt>value_type operator[](size_type i)</tt></td>
<td>
Returns the <TT>i</TT>th component in the graph.
</td>
</tr>
<tr>
<td><tt>size_type component_index::size()</tt></td>
<td>
Returns the number of components in the graph.
</td>
</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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