[ceph.git] / ceph / src / boost / libs / graph_parallel / doc / html / non_distributed_betweenness_centrality.html

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<h1 class="title"><a class="reference external" href="http://www.osl.iu.edu/research/pbgl"><img align="middle" alt="Parallel BGL" class="align-middle" src="pbgl-logo.png" /></a> Non-Distributed Betweenness Centrality</h1>

<!-- Copyright (C) 2004-2009 The Trustees of Indiana University.
Use, modification and distribution is subject to 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) -->
<pre class="literal-block">
// named parameter versions
template&lt;typename ProcessGroup, typename Graph, typename Param, typename Tag, typename Rest&gt;
void
non_distributed_brandes_betweenness_centrality(const ProcessGroup&amp; pg, const Graph&amp; g,
                                               const bgl_named_params&lt;Param,Tag,Rest&gt;&amp; params);

template&lt;typename ProcessGroup, typename Graph, typename CentralityMap,
         typename Buffer&gt;
void
non_distributed_brandes_betweenness_centrality(const ProcessGroup&amp; pg, const Graph&amp; g,
                                               CentralityMap centrality, Buffer sources);

template&lt;typename ProcessGroup, typename Graph, typename CentralityMap,
         typename EdgeCentralityMap, typename Buffer&gt;
void
non_distributed_brandes_betweenness_centrality(const ProcessGroup&amp; pg, const Graph&amp; g,
                                               CentralityMap centrality,
                                               EdgeCentralityMap edge_centrality_map,
                                               Buffer sources);

// non-named parameter versions
template&lt;typename ProcessGroup, typename Graph, typename CentralityMap,
         typename EdgeCentralityMap, typename IncomingMap, typename DistanceMap,
         typename DependencyMap, typename PathCountMap, typename VertexIndexMap,
         typename Buffer&gt;
void
non_distributed_brandes_betweenness_centrality(const ProcessGroup&amp; pg,
                                               const Graph&amp; g,
                                               CentralityMap centrality,
                                               EdgeCentralityMap edge_centrality_map,
                                               IncomingMap incoming,
                                               DistanceMap distance,
                                               DependencyMap dependency,
                                               PathCountMap path_count,
                                               VertexIndexMap vertex_index,
                                               Buffer sources);

template&lt;typename ProcessGroup, typename Graph, typename CentralityMap,
         typename EdgeCentralityMap, typename IncomingMap, typename DistanceMap,
         typename DependencyMap, typename PathCountMap, typename VertexIndexMap,
         typename WeightMap, typename Buffer&gt;
void
non_distributed_brandes_betweenness_centrality(const ProcessGroup&amp; pg,
                                               const Graph&amp; g,
                                               CentralityMap centrality,
                                               EdgeCentralityMap edge_centrality_map,
                                               IncomingMap incoming,
                                               DistanceMap distance,
                                               DependencyMap dependency,
                                               PathCountMap path_count,
                                               VertexIndexMap vertex_index,
                                               WeightMap weight_map,
                                               Buffer sources);

// helper functions
template&lt;typename Graph, typename CentralityMap&gt;
typename property_traits&lt;CentralityMap&gt;::value_type
central_point_dominance(const Graph&amp; g, CentralityMap centrality);
</pre>
<p>The <tt class="docutils literal"><span class="pre">non_distributed_betweenness_centrality()</span></tt> function computes the
betweenness centrality of the vertices and edges in a graph.  The name
is somewhat confusing, the graph <tt class="docutils literal"><span class="pre">g</span></tt> is not a distributed graph,
however the algorithm does run in parallel.  Rather than parallelizing
the individual shortest paths calculations that are required by
betweenness centrality, this variant of the algorithm performs the
shortest paths calculations in parallel with each process in <tt class="docutils literal"><span class="pre">pg</span></tt>
calculating the shortest path from a different set of source vertices
using the BGL's implementation of <a class="reference external" href="http://www.boost.org/libs/graph/doc/dijkstra_shortest_paths.html">Dijkstra shortest paths</a>.  Each
process accumulates into it's local centrality map and once all the
shortest paths calculations are performed a reduction is performed to
combine the centrality from all the processes.</p>
<div class="contents topic" id="contents">
<p class="topic-title first">Contents</p>
<ul class="simple">
<li><a class="reference internal" href="#where-defined" id="id1">Where Defined</a></li>
<li><a class="reference internal" href="#parameters" id="id2">Parameters</a></li>
<li><a class="reference internal" href="#complexity" id="id3">Complexity</a></li>
<li><a class="reference internal" href="#algorithm-description" id="id4">Algorithm Description</a></li>
</ul>
</div>
<div class="section" id="where-defined">
<h1><a class="toc-backref" href="#id1">Where Defined</a></h1>
<p>&lt;<tt class="docutils literal"><span class="pre">boost/graph/distributed/betweenness_centrality.hpp</span></tt>&gt;</p>
</div>
<div class="section" id="parameters">
<h1><a class="toc-backref" href="#id2">Parameters</a></h1>
<dl class="docutils">
<dt>IN: <tt class="docutils literal"><span class="pre">const</span> <span class="pre">ProcessGroup&amp;</span> <span class="pre">pg</span></tt></dt>
<dd>The process group over which the the processes involved in
betweenness centrality communicate.  The process group type must
model the <a class="reference external" href="process_group.html">Process Group</a> concept.</dd>
<dt>IN: <tt class="docutils literal"><span class="pre">const</span> <span class="pre">Graph&amp;</span> <span class="pre">g</span></tt></dt>
<dd>The graph type must be a model of the <a class="reference external" href="http://www.boost.org/libs/graph/doc/IncidenceGraph.html">Incidence Graph</a> concept.</dd>
<dt>IN: <tt class="docutils literal"><span class="pre">CentralityMap</span> <span class="pre">centrality</span></tt></dt>
<dd><p class="first">A centrality map may be supplied to the algorithm, if one is not
supplied a <tt class="docutils literal"><span class="pre">dummy_property_map</span></tt> will be used and no vertex
centrality information will be recorded.  The key type of the
<tt class="docutils literal"><span class="pre">CentralityMap</span></tt> must be the graph's vertex descriptor type.</p>
<p class="last"><strong>Default</strong>: A <tt class="docutils literal"><span class="pre">dummy_property_map</span></tt>.</p>
</dd>
<dt>IN:  <tt class="docutils literal"><span class="pre">EdgeCentralityMap</span> <span class="pre">edge_centrality_map</span></tt></dt>
<dd><p class="first">An edge centrality map may be supplied to the algorithm, if one is
not supplied a <tt class="docutils literal"><span class="pre">dummy_property_map</span></tt> will be used and no edge
centrality information will be recorded.  The key type of the
<tt class="docutils literal"><span class="pre">EdgeCentralityMap</span></tt> must be the graph's vertex descriptor type.</p>
<p class="last"><strong>Default</strong>: A <tt class="docutils literal"><span class="pre">dummy_property_map</span></tt>.</p>
</dd>
<dt>IN:  <tt class="docutils literal"><span class="pre">IncomingMap</span> <span class="pre">incoming</span></tt></dt>
<dd><p class="first">The incoming map contains the incoming edges to a vertex that are
part of shortest paths to that vertex.  Its key type must be the
graph's vertex descriptor type and its value type must be the
graph's edge descriptor type.</p>
<dl class="last docutils">
<dt><strong>Default</strong>: An <tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> created from a</dt>
<dd><tt class="docutils literal"><span class="pre">std::vector</span></tt> of <tt class="docutils literal"><span class="pre">std::vector</span></tt> of the graph's edge descriptor
type.</dd>
</dl>
</dd>
<dt>IN:  <tt class="docutils literal"><span class="pre">DistanceMap</span> <span class="pre">distance</span></tt></dt>
<dd><p class="first">The distance map records the distance to vertices during the
shortest paths portion of the algorithm.  Its key type must be the
graph's vertex descriptor type.</p>
<dl class="last docutils">
<dt><strong>Default</strong>: An <tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> created from a</dt>
<dd><tt class="docutils literal"><span class="pre">std::vector</span></tt> of the value type of the <tt class="docutils literal"><span class="pre">CentralityMap</span></tt>.</dd>
</dl>
</dd>
<dt>IN: <tt class="docutils literal"><span class="pre">DependencyMap</span> <span class="pre">dependency</span></tt></dt>
<dd><p class="first">The dependency map records the dependency of each vertex during the
centrality calculation portion of the algorithm.  Its key type must
be the graph's vertex descriptor type.</p>
<dl class="last docutils">
<dt><strong>Default</strong>: An <tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> created from a</dt>
<dd><tt class="docutils literal"><span class="pre">std::vector</span></tt> of the value type of the <tt class="docutils literal"><span class="pre">CentralityMap</span></tt>.</dd>
</dl>
</dd>
<dt>IN:  <tt class="docutils literal"><span class="pre">PathCountMap</span> <span class="pre">path_count</span></tt></dt>
<dd><p class="first">The path count map records the number of shortest paths each vertex
is on during the centrality calculation portion of the algorithm.
Its key type must be the graph's vertex descriptor type.</p>
<dl class="last docutils">
<dt><strong>Default</strong>: An <tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> created from a</dt>
<dd><tt class="docutils literal"><span class="pre">std::vector</span></tt> of the graph's degree size type.</dd>
</dl>
</dd>
<dt>IN:  <tt class="docutils literal"><span class="pre">VertexIndexMap</span> <span class="pre">vertex_index</span></tt></dt>
<dd><p class="first">A model of <a class="reference external" href="http://www.boost.org/libs/property_map/ReadablePropertyMap.html">Readable Property Map</a> whose key type is the vertex
descriptor type of the graph <tt class="docutils literal"><span class="pre">g</span></tt> and whose value type is an
integral type. The property map should map from vertices to their
(local) indices in the range <em>[0, num_vertices(g))</em>.</p>
<p class="last"><strong>Default</strong>: <tt class="docutils literal"><span class="pre">get(vertex_index,</span> <span class="pre">g)</span></tt></p>
</dd>
<dt>IN:  <tt class="docutils literal"><span class="pre">WeightMap</span> <span class="pre">weight_map</span></tt></dt>
<dd>A model of <a class="reference external" href="http://www.boost.org/libs/property_map/ReadablePropertyMap.html">Readable Property Map</a> whose key type is the edge
descriptor type of the graph <tt class="docutils literal"><span class="pre">g</span></tt>.  If not supplied the betweenness
centrality calculation will be unweighted.</dd>
<dt>IN: <tt class="docutils literal"><span class="pre">Buffer</span> <span class="pre">sources</span></tt></dt>
<dd><p class="first">A model of <a class="reference external" href="http://www.boost.org/libs/graph/doc/Buffer.html">Buffer</a> containing the starting vertices for the
algorithm.  If <tt class="docutils literal"><span class="pre">sources</span></tt> is empty a complete betweenness
centrality calculation using all vertices in <tt class="docutils literal"><span class="pre">g</span></tt> will be
performed.  The value type of the Buffer must be the graph's vertex
descriptor type.</p>
<p class="last"><strong>Default</strong>: An empty <tt class="docutils literal"><span class="pre">boost::queue</span></tt> of int.</p>
</dd>
</dl>
</div>
<div class="section" id="complexity">
<h1><a class="toc-backref" href="#id3">Complexity</a></h1>
<p>Each of the shortest paths calculations is <em>O(V log V)</em> and each of
them may be run in parallel (one per process in <tt class="docutils literal"><span class="pre">pg</span></tt>).  The
reduction phase to calculate the total centrality at the end of the
shortest paths phase is <em>O(V log V)</em>.</p>
</div>
<div class="section" id="algorithm-description">
<h1><a class="toc-backref" href="#id4">Algorithm Description</a></h1>
<p>The algorithm uses a non-distributed (sequential) graph, as well as
non-distributed property maps.  Each process contains a separate copy
of the sequential graph <tt class="docutils literal"><span class="pre">g</span></tt>.  In order for the algorithm to be
correct, these copies of <tt class="docutils literal"><span class="pre">g</span></tt> must be identical.  The <tt class="docutils literal"><span class="pre">sources</span></tt>
buffer contains the vertices to use as the source of single source
shortest paths calculations when approximating betweenness centrality
using a subset of the vertices in the graph.  If <tt class="docutils literal"><span class="pre">sources</span></tt> is empty
then a complete betweenness centrality calculation is performed using
all vertices.</p>
<p>In the sequential phase of the algorithm each process computes
shortest paths from a subset of the vertices in the graph using the
Brandes shortest paths methods from the BGL's betweenness centrality
implementation.  In the parallel phase of the algorithm a reduction is
performed to sum the values computed by each process for all vertices
in the graph.</p>
<p>Either weighted or unweighted betweenness centrality is calculated
depending on whether a <tt class="docutils literal"><span class="pre">WeightMap</span></tt> is passed.</p>
<hr class="docutils" />
<p>Copyright (C) 2009 The Trustees of Indiana University.</p>
<p>Authors: Nick Edmonds and Andrew Lumsdaine</p>
</div>
</div>
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	4	<head>
	5	<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
	6	<meta name="generator" content="Docutils 0.6: http://docutils.sourceforge.net/" />
	7	<title>Parallel BGL Non-Distributed Betweenness Centrality</title>
	8	<link rel="stylesheet" href="../../../../rst.css" type="text/css" />
	9	</head>
	10	<body>
	11	<div class="document" id="logo-non-distributed-betweenness-centrality">
	12	<h1 class="title"><a class="reference external" href="http://www.osl.iu.edu/research/pbgl"><img align="middle" alt="Parallel BGL" class="align-middle" src="pbgl-logo.png" /></a> Non-Distributed Betweenness Centrality</h1>
	13
	14	<!-- Copyright (C) 2004-2009 The Trustees of Indiana University.
	15	Use, modification and distribution is subject to the Boost Software
	16	License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
	17	http://www.boost.org/LICENSE_1_0.txt) -->
	18	<pre class="literal-block">
	19	// named parameter versions
	20	template<typename ProcessGroup, typename Graph, typename Param, typename Tag, typename Rest>
	21	void
	22	non_distributed_brandes_betweenness_centrality(const ProcessGroup& pg, const Graph& g,
	23	const bgl_named_params<Param,Tag,Rest>& params);
	24
	25	template<typename ProcessGroup, typename Graph, typename CentralityMap,
	26	typename Buffer>
	27	void
	28	non_distributed_brandes_betweenness_centrality(const ProcessGroup& pg, const Graph& g,
	29	CentralityMap centrality, Buffer sources);
	30
	31	template<typename ProcessGroup, typename Graph, typename CentralityMap,
	32	typename EdgeCentralityMap, typename Buffer>
	33	void
	34	non_distributed_brandes_betweenness_centrality(const ProcessGroup& pg, const Graph& g,
	35	CentralityMap centrality,
	36	EdgeCentralityMap edge_centrality_map,
	37	Buffer sources);
	38
	39	// non-named parameter versions
	40	template<typename ProcessGroup, typename Graph, typename CentralityMap,
	41	typename EdgeCentralityMap, typename IncomingMap, typename DistanceMap,
	42	typename DependencyMap, typename PathCountMap, typename VertexIndexMap,
	43	typename Buffer>
	44	void
	45	non_distributed_brandes_betweenness_centrality(const ProcessGroup& pg,
	46	const Graph& g,
	47	CentralityMap centrality,
	48	EdgeCentralityMap edge_centrality_map,
	49	IncomingMap incoming,
	50	DistanceMap distance,
	51	DependencyMap dependency,
	52	PathCountMap path_count,
	53	VertexIndexMap vertex_index,
	54	Buffer sources);
	55
	56	template<typename ProcessGroup, typename Graph, typename CentralityMap,
	57	typename EdgeCentralityMap, typename IncomingMap, typename DistanceMap,
	58	typename DependencyMap, typename PathCountMap, typename VertexIndexMap,
	59	typename WeightMap, typename Buffer>
	60	void
	61	non_distributed_brandes_betweenness_centrality(const ProcessGroup& pg,
	62	const Graph& g,
	63	CentralityMap centrality,
	64	EdgeCentralityMap edge_centrality_map,
65	IncomingMap incoming,
66	DistanceMap distance,
67	DependencyMap dependency,
68	PathCountMap path_count,
69	VertexIndexMap vertex_index,
70	WeightMap weight_map,
71	Buffer sources);
72
73	// helper functions
74	template<typename Graph, typename CentralityMap>
75	typename property_traits<CentralityMap>::value_type
76	central_point_dominance(const Graph& g, CentralityMap centrality);
77	</pre>
78	<p>The <tt class="docutils literal"><span class="pre">non_distributed_betweenness_centrality()</span></tt> function computes the
79	betweenness centrality of the vertices and edges in a graph. The name
80	is somewhat confusing, the graph <tt class="docutils literal"><span class="pre">g</span></tt> is not a distributed graph,
81	however the algorithm does run in parallel. Rather than parallelizing
82	the individual shortest paths calculations that are required by
83	betweenness centrality, this variant of the algorithm performs the
84	shortest paths calculations in parallel with each process in <tt class="docutils literal"><span class="pre">pg</span></tt>
85	calculating the shortest path from a different set of source vertices
86	using the BGL's implementation of <a class="reference external" href="http://www.boost.org/libs/graph/doc/dijkstra_shortest_paths.html">Dijkstra shortest paths</a>. Each
87	process accumulates into it's local centrality map and once all the
88	shortest paths calculations are performed a reduction is performed to
89	combine the centrality from all the processes.</p>
90	<div class="contents topic" id="contents">
91	<p class="topic-title first">Contents</p>
92	<ul class="simple">
93	<li><a class="reference internal" href="#where-defined" id="id1">Where Defined</a></li>
94	<li><a class="reference internal" href="#parameters" id="id2">Parameters</a></li>
95	<li><a class="reference internal" href="#complexity" id="id3">Complexity</a></li>
96	<li><a class="reference internal" href="#algorithm-description" id="id4">Algorithm Description</a></li>
97	</ul>
98	</div>
99	<div class="section" id="where-defined">
100	<h1><a class="toc-backref" href="#id1">Where Defined</a></h1>
101	<p><<tt class="docutils literal"><span class="pre">boost/graph/distributed/betweenness_centrality.hpp</span></tt>></p>
102	</div>
103	<div class="section" id="parameters">
104	<h1><a class="toc-backref" href="#id2">Parameters</a></h1>
105	<dl class="docutils">
106	<dt>IN: <tt class="docutils literal"><span class="pre">const</span> <span class="pre">ProcessGroup&</span> <span class="pre">pg</span></tt></dt>
107	<dd>The process group over which the the processes involved in
108	betweenness centrality communicate. The process group type must
109	model the <a class="reference external" href="process_group.html">Process Group</a> concept.</dd>
110	<dt>IN: <tt class="docutils literal"><span class="pre">const</span> <span class="pre">Graph&</span> <span class="pre">g</span></tt></dt>
111	<dd>The graph type must be a model of the <a class="reference external" href="http://www.boost.org/libs/graph/doc/IncidenceGraph.html">Incidence Graph</a> concept.</dd>
112	<dt>IN: <tt class="docutils literal"><span class="pre">CentralityMap</span> <span class="pre">centrality</span></tt></dt>
113	<dd><p class="first">A centrality map may be supplied to the algorithm, if one is not
114	supplied a <tt class="docutils literal"><span class="pre">dummy_property_map</span></tt> will be used and no vertex
115	centrality information will be recorded. The key type of the
116	<tt class="docutils literal"><span class="pre">CentralityMap</span></tt> must be the graph's vertex descriptor type.</p>
117	<p class="last"><strong>Default</strong>: A <tt class="docutils literal"><span class="pre">dummy_property_map</span></tt>.</p>
118	</dd>
119	<dt>IN: <tt class="docutils literal"><span class="pre">EdgeCentralityMap</span> <span class="pre">edge_centrality_map</span></tt></dt>
120	<dd><p class="first">An edge centrality map may be supplied to the algorithm, if one is
121	not supplied a <tt class="docutils literal"><span class="pre">dummy_property_map</span></tt> will be used and no edge
122	centrality information will be recorded. The key type of the
123	<tt class="docutils literal"><span class="pre">EdgeCentralityMap</span></tt> must be the graph's vertex descriptor type.</p>
124	<p class="last"><strong>Default</strong>: A <tt class="docutils literal"><span class="pre">dummy_property_map</span></tt>.</p>
125	</dd>
126	<dt>IN: <tt class="docutils literal"><span class="pre">IncomingMap</span> <span class="pre">incoming</span></tt></dt>
127	<dd><p class="first">The incoming map contains the incoming edges to a vertex that are
128	part of shortest paths to that vertex. Its key type must be the
129	graph's vertex descriptor type and its value type must be the
130	graph's edge descriptor type.</p>
131	<dl class="last docutils">
132	<dt><strong>Default</strong>: An <tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> created from a</dt>
133	<dd><tt class="docutils literal"><span class="pre">std::vector</span></tt> of <tt class="docutils literal"><span class="pre">std::vector</span></tt> of the graph's edge descriptor
134	type.</dd>
135	</dl>
136	</dd>
137	<dt>IN: <tt class="docutils literal"><span class="pre">DistanceMap</span> <span class="pre">distance</span></tt></dt>
138	<dd><p class="first">The distance map records the distance to vertices during the
139	shortest paths portion of the algorithm. Its key type must be the
140	graph's vertex descriptor type.</p>
141	<dl class="last docutils">
142	<dt><strong>Default</strong>: An <tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> created from a</dt>
143	<dd><tt class="docutils literal"><span class="pre">std::vector</span></tt> of the value type of the <tt class="docutils literal"><span class="pre">CentralityMap</span></tt>.</dd>
144	</dl>
145	</dd>
146	<dt>IN: <tt class="docutils literal"><span class="pre">DependencyMap</span> <span class="pre">dependency</span></tt></dt>
147	<dd><p class="first">The dependency map records the dependency of each vertex during the
148	centrality calculation portion of the algorithm. Its key type must
149	be the graph's vertex descriptor type.</p>
150	<dl class="last docutils">
151	<dt><strong>Default</strong>: An <tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> created from a</dt>
152	<dd><tt class="docutils literal"><span class="pre">std::vector</span></tt> of the value type of the <tt class="docutils literal"><span class="pre">CentralityMap</span></tt>.</dd>
153	</dl>
154	</dd>
155	<dt>IN: <tt class="docutils literal"><span class="pre">PathCountMap</span> <span class="pre">path_count</span></tt></dt>
156	<dd><p class="first">The path count map records the number of shortest paths each vertex
157	is on during the centrality calculation portion of the algorithm.
158	Its key type must be the graph's vertex descriptor type.</p>
159	<dl class="last docutils">
160	<dt><strong>Default</strong>: An <tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> created from a</dt>
161	<dd><tt class="docutils literal"><span class="pre">std::vector</span></tt> of the graph's degree size type.</dd>
162	</dl>
163	</dd>
164	<dt>IN: <tt class="docutils literal"><span class="pre">VertexIndexMap</span> <span class="pre">vertex_index</span></tt></dt>
165	<dd><p class="first">A model of <a class="reference external" href="http://www.boost.org/libs/property_map/ReadablePropertyMap.html">Readable Property Map</a> whose key type is the vertex
166	descriptor type of the graph <tt class="docutils literal"><span class="pre">g</span></tt> and whose value type is an
167	integral type. The property map should map from vertices to their
168	(local) indices in the range <em>[0, num_vertices(g))</em>.</p>
169	<p class="last"><strong>Default</strong>: <tt class="docutils literal"><span class="pre">get(vertex_index,</span> <span class="pre">g)</span></tt></p>
170	</dd>
171	<dt>IN: <tt class="docutils literal"><span class="pre">WeightMap</span> <span class="pre">weight_map</span></tt></dt>
172	<dd>A model of <a class="reference external" href="http://www.boost.org/libs/property_map/ReadablePropertyMap.html">Readable Property Map</a> whose key type is the edge
173	descriptor type of the graph <tt class="docutils literal"><span class="pre">g</span></tt>. If not supplied the betweenness
174	centrality calculation will be unweighted.</dd>
175	<dt>IN: <tt class="docutils literal"><span class="pre">Buffer</span> <span class="pre">sources</span></tt></dt>
176	<dd><p class="first">A model of <a class="reference external" href="http://www.boost.org/libs/graph/doc/Buffer.html">Buffer</a> containing the starting vertices for the
177	algorithm. If <tt class="docutils literal"><span class="pre">sources</span></tt> is empty a complete betweenness
178	centrality calculation using all vertices in <tt class="docutils literal"><span class="pre">g</span></tt> will be
179	performed. The value type of the Buffer must be the graph's vertex
180	descriptor type.</p>
181	<p class="last"><strong>Default</strong>: An empty <tt class="docutils literal"><span class="pre">boost::queue</span></tt> of int.</p>
182	</dd>
183	</dl>
184	</div>
185	<div class="section" id="complexity">
186	<h1><a class="toc-backref" href="#id3">Complexity</a></h1>
187	<p>Each of the shortest paths calculations is <em>O(V log V)</em> and each of
188	them may be run in parallel (one per process in <tt class="docutils literal"><span class="pre">pg</span></tt>). The
189	reduction phase to calculate the total centrality at the end of the
190	shortest paths phase is <em>O(V log V)</em>.</p>
191	</div>
192	<div class="section" id="algorithm-description">
193	<h1><a class="toc-backref" href="#id4">Algorithm Description</a></h1>
194	<p>The algorithm uses a non-distributed (sequential) graph, as well as
195	non-distributed property maps. Each process contains a separate copy
196	of the sequential graph <tt class="docutils literal"><span class="pre">g</span></tt>. In order for the algorithm to be
197	correct, these copies of <tt class="docutils literal"><span class="pre">g</span></tt> must be identical. The <tt class="docutils literal"><span class="pre">sources</span></tt>
198	buffer contains the vertices to use as the source of single source
199	shortest paths calculations when approximating betweenness centrality
200	using a subset of the vertices in the graph. If <tt class="docutils literal"><span class="pre">sources</span></tt> is empty
201	then a complete betweenness centrality calculation is performed using
202	all vertices.</p>
203	<p>In the sequential phase of the algorithm each process computes
204	shortest paths from a subset of the vertices in the graph using the
205	Brandes shortest paths methods from the BGL's betweenness centrality
206	implementation. In the parallel phase of the algorithm a reduction is
207	performed to sum the values computed by each process for all vertices
208	in the graph.</p>
209	<p>Either weighted or unweighted betweenness centrality is calculated
210	depending on whether a <tt class="docutils literal"><span class="pre">WeightMap</span></tt> is passed.</p>
211	<hr class="docutils" />
212	<p>Copyright (C) 2009 The Trustees of Indiana University.</p>
213	<p>Authors: Nick Edmonds and Andrew Lumsdaine</p>
214	</div>
215	</div>
216	<div class="footer">
217	<hr class="footer" />
218	Generated on: 2009-05-31 00:22 UTC.
219	Generated by <a class="reference external" href="http://docutils.sourceforge.net/">Docutils</a> from <a class="reference external" href="http://docutils.sourceforge.net/rst.html">reStructuredText</a> source.
220
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