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8.0 KiB
HTML
268 lines
8.0 KiB
HTML
<HTML>
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<!--
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-- Copyright (c) Jeremy Siek 2000
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--
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-- Permission to use, copy, modify, distribute and sell this software
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-- and its documentation for any purpose is hereby granted without fee,
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-- provided that the above copyright notice appears in all copies and
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-- that both that copyright notice and this permission notice appear
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-- in supporting documentation. Jeremy Siek makes no
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-- representations about the suitability of this software for any
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-- purpose. It is provided "as is" without express or implied warranty.
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-->
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<Head>
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<Title>Boost Graph Library: Connected Components</Title>
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<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
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ALINK="#ff0000">
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<IMG SRC="../../../c++boost.gif"
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ALT="C++ Boost" width="277" height="86">
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<BR Clear>
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<H1>
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<A NAME="sec:connected-components"></A><A NAME="sec:strongly-connected-components"></A>
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<TT>connected_components</TT>
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</H1>
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<P>
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<DIV ALIGN="left">
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<TABLE CELLPADDING=3 border>
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<TR><TH ALIGN="LEFT"><B>Graphs:</B></TH>
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<TD ALIGN="LEFT">see below</TD>
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</TR>
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<TR><TH ALIGN="LEFT"><B>Properties:</B></TH>
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<TD ALIGN="LEFT">components, color, discover time, finish time</TD>
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</TR>
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<TR><TH ALIGN="LEFT"><B>Complexity:</B></TH>
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<TD ALIGN="LEFT"><i>O(V + E)</i></TD>
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</TR>
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</TABLE>
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</DIV>
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<P>
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<PRE>
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(1)
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template <class VertexListGraph, class Visitor,
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class Components>
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typename property_traits< Components >::value_type
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connected_components(VertexListGraph& G, Components c,
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Visitor v);
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(2)
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template <class VertexListGraph, class Visitor,
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class Components, class Color>
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typename property_traits<Components>::value_type
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connected_components(VertexListGraph& G, Components c,
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Color color, Visitor v);
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(3)
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template <class VertexListGraph, class Visitor,
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class Components, class DiscoverTime,
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class FinishTime, class Color>
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typename property_traits<Components>::value_type
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connected_components(VertexListGraph& G, Components c,
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DiscoverTime d, FinishTime f,
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Color color, Visitor v);
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</PRE>
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<P>
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The <TT>connected_component()</TT> function dispatches to two different
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algorithms depending on whether the graph in question is directed or
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undirected.
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<P>
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<UL>
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<LI>Computes the strongly connected components of a directed graph
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using the DFS/transpose/DFS algorithm [<A
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HREF="bibliography.html#aho83:_data_struct_algo">1</A>,<A
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HREF="bibliography.html#clr90">8</A>].
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<P>
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</LI>
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<LI>Computes the connected components of an undirected graph using
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a DFS-based approach. If the connected-components are to be
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calculated over and over while a graph is changing the disjoint-set
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based approach of function
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<TT>dynamic_connected_components()</TT> is faster. For
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``static'' graphs this DFS-based approach is faster [<A
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HREF="bibliography.html#clr90">8</A>].
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</LI>
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</UL>
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<P>
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The output of the algorithm is recorded in the component property
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map <TT>c</TT>, which will contain numbers giving the component ID
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assigned to each vertex. The number of components is the return value
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of the function.
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<P>
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The algorithm requires the use of several property maps: color,
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discover time, and finish time. There are several versions of this
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algorithm to accommodate whether you wish to use interior or exterior
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property maps.
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<P>
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<H3>Where Defined</H3>
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<P>
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<a href="../../../boost/graph/connected_components.hpp"><TT>boost/graph/connected_components.hpp</TT></a>
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<P>
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<H3>Definitions</H3>
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<P>
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A <I>connected component</I> of an undirected graph is a set of
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vertices that are all reachable from each other. A <I>strongly
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connected component</I> of a directed graph <i>G=(V,E)</i> is a
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maximal set of vertices <i>U</i> which is in <i>V</i> such that for
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every pair of vertices <i>u</i> and <i>v</i> in <i>U</i>, we have both
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a path from <i>u</i> to <i>v</i> and path from <i>v</i> to
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<i>u</i>. That is to say that <i>u</i> and <i>v</i> are reachable from
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each other.
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<P>
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<H3>Requirements on Types</H3>
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<P>
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<UL>
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<LI>The graph type must be a model of <a
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href="./VertexListGraph.html">VertexListGraph</a>.
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</LI>
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<LI><TT>DiscoverTime</TT> and <TT>FinishTime</TT> must be models of <a
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href="../../property_map/WritablePropertyMap.html">WritablePropertyMap</a>
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and their value type must be an integer type. Vertex descriptors from
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the graph should be usable as the key type for these maps.
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</LI>
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<LI>The <TT>Color</TT> map must be a <a
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href="../../property_map/ReadWritePropertyMap.html">ReadWritePropertyMap</a>
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and the graph's vertex descriptor type should be usable as the
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map's key type. The value type of the map must be a
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model of <I>ColorValue</I>.
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</LI>
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<LI>The <TT>Components</TT> type must be a model of <a
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href="../../property_map/ReadWritePropertyMap.html">ReadWritePropertyMap</a>. The
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value type of the <TT>Components</TT> property map should be
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an integer type, preferably the same as the <TT>size_type</TT> of
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the graph. The key type should be the graph's vertex descriptor
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type.
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</LI>
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</UL>
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<P>
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<H3>Complexity</H3>
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<P>
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The time complexity for the strongly connected components algorithm is
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<i>O(V + E)</i>. The time complexity for the connected components
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algorithm is also <i>O(V + E)</i>.
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<P>
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<H3>Example</H3>
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<P>
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Calculating the connected components of an undirected graph. The
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complete source is in file <a
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href="../example/connected_components.cpp"><tt>examples/connected_components.cpp</tt></a>.
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<P>
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<PRE>
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typedef discover_time_property< finish_time_property
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< color_property<> > > VertexProperty;
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typedef adjacency_list <vecS, vecS, undirectedS, VertexProperty> Graph;
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typedef graph_traits<Graph>::vertex_descriptor Vertex;
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const int N = 6;
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Graph G(N);
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add_edge(0, 1, G);
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add_edge(1, 4, G);
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add_edge(4, 0, G);
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add_edge(2, 5, G);
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std::vector<int> c(num_vertices(G));
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int num = connected_components(G, c.begin(),
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get_color_map(G), null_visitor());
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cout << endl;
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std::vector<int>::iterator i;
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cout << "Total number of components: " << num << endl;
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for (i = c.begin(); i != c.end(); ++i)
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cout << "Vertex " << i - c.begin()
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<< " is in component " << *i << endl;
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cout << endl;
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</PRE>
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The output is:
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<PRE>
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Total number of components: 3
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Vertex 0 is in component 1
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Vertex 1 is in component 1
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Vertex 2 is in component 2
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Vertex 3 is in component 3
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Vertex 4 is in component 1
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Vertex 5 is in component 2
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</PRE>
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<P>
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Calculating the strongly connected components of a directed graph.
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<PRE>
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typedef discover_time_property< finish_time_property
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< color_property<> > > VertexProperty;
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typedef adjacency_list< vecS, vecS, directedS, VertexProperty > Graph;
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const int N = 6;
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Graph G(N);
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add_edge(0, 1, G);
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add_edge(1, 1, G);
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add_edge(1, 3, G);
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add_edge(1, 4, G);
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add_edge(4, 3, G);
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add_edge(3, 4, G);
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add_edge(3, 0, G);
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add_edge(5, 2, G);
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typedef graph_traits<Graph>::vertex_descriptor Vertex;
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std::vector<int> c(N);
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int num = connected_components(G, c.begin(),
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get_color_map(G), null_visitor());
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cout << endl;
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cout << "Total number of components: " << num << endl;
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std::vector<int>::iterator i;
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for (i = c.begin(); i != c.end(); ++i)
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cout << "Vertex " << i - c.begin()
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<< " is in component " << *i << endl;
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}
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</PRE>
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The output is:
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<PRE>
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Total number of components: 3
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Vertex 0 is in component 3
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Vertex 1 is in component 3
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Vertex 2 is in component 2
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Vertex 3 is in component 3
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Vertex 4 is in component 3
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Vertex 5 is in component 1
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</PRE>
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<P>
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<br>
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<HR>
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<TABLE>
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<TR valign=top>
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<TD nowrap>Copyright © 2000</TD><TD>
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<A HREF="../../../people/jeremy_siek.htm">Jeremy Siek</A>, Univ.of Notre Dame (<A HREF="mailto:jsiek@lsc.nd.edu">jsiek@lsc.nd.edu</A>)
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</TD></TR></TABLE>
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</BODY>
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</HTML>
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