boost_1_45_0/libs/graph_parallel/doc/html/non_distributed_betweenness_centrality.html - nest-learning-thermostat/5.0/boost - Git at Google

 <?xml version="1.0" encoding="utf-8" ?>
 <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
 <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en">
 <head>
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="Docutils 0.6: http://docutils.sourceforge.net/" />
 <title>Parallel BGL Non-Distributed Betweenness Centrality</title>
 <link rel="stylesheet" href="../../../../rst.css" type="text/css" />
 </head>
 <body>
 <div class="document" id="logo-non-distributed-betweenness-centrality">
 <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>
 <div class="footer">
 <hr class="footer" />
 Generated on: 2009-05-31 00:22 UTC.
 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.

 </div>
 </body>
 </html>
	<?xml version="1.0" encoding="utf-8" ?>
	<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
	<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en">
	<head>
	<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
	<meta name="generator" content="Docutils 0.6: http://docutils.sourceforge.net/" />
	<title>Parallel BGL Non-Distributed Betweenness Centrality</title>
	<link rel="stylesheet" href="../../../../rst.css" type="text/css" />
	</head>
	<body>
	<div class="document" id="logo-non-distributed-betweenness-centrality">
	<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<typename ProcessGroup, typename Graph, typename Param, typename Tag, typename Rest>
	void
	non_distributed_brandes_betweenness_centrality(const ProcessGroup& pg, const Graph& g,
	const bgl_named_params<Param,Tag,Rest>& params);

	template<typename ProcessGroup, typename Graph, typename CentralityMap,
	typename Buffer>
	void
	non_distributed_brandes_betweenness_centrality(const ProcessGroup& pg, const Graph& g,
	CentralityMap centrality, Buffer sources);

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

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

	template<typename ProcessGroup, typename Graph, typename CentralityMap,
	typename EdgeCentralityMap, typename IncomingMap, typename DistanceMap,
	typename DependencyMap, typename PathCountMap, typename VertexIndexMap,
	typename WeightMap, typename Buffer>
	void
	non_distributed_brandes_betweenness_centrality(const ProcessGroup& pg,
	const Graph& 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<typename Graph, typename CentralityMap>
	typename property_traits<CentralityMap>::value_type
	central_point_dominance(const Graph& 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><<tt class="docutils literal"><span class="pre">boost/graph/distributed/betweenness_centrality.hpp</span></tt>></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&</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&</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>
	<div class="footer">
	<hr class="footer" />
	Generated on: 2009-05-31 00:22 UTC.
	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.

	</div>
	</body>
	</html>