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| <div class="document" id="logo-connected-components"> |
| <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> Connected Components</h1> |
| |
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| <pre class="literal-block"> |
| template<typename Graph, typename ComponentMap> |
| inline typename property_traits<ComponentMap>::value_type |
| strong_components( const Graph& g, ComponentMap c); |
| |
| namespace graph { |
| template<typename Graph, typename VertexComponentMap> |
| void |
| fleischer_hendrickson_pinar_strong_components(const Graph& g, VertexComponentMap r); |
| |
| template<typename Graph, typename ReverseGraph, |
| typename ComponentMap, typename IsoMapFR, typename IsoMapRF> |
| inline typename property_traits<ComponentMap>::value_type |
| fleischer_hendrickson_pinar_strong_components(const Graph& g, |
| ComponentMap c, |
| const ReverseGraph& gr, |
| IsoMapFR fr, IsoMapRF rf); |
| } |
| </pre> |
| <p>The <tt class="docutils literal"><span class="pre">strong_components()</span></tt> function computes the strongly connected |
| components of a directed graph. The distributed strong components |
| algorithm uses the <a class="reference external" href="http://www.boost.org/libs/graph/doc/strong_components.html">sequential strong components</a> algorithm to |
| identify components local to a processor. The distributed portion of |
| the algorithm is built on the <a class="reference external" href="breadth_first_search.html">distributed breadth first search</a> |
| algorithm and is based on the work of Fleischer, Hendrickson, and |
| Pinar <a class="citation-reference" href="#fhp00" id="id1">[FHP00]</a>. The interface is a superset of the interface to the |
| BGL <a class="reference external" href="http://www.boost.org/libs/graph/doc/strong_components.html">sequential strong components</a> algorithm. The number of |
| strongly-connected components in the graph is returned to all |
| processes.</p> |
| <p>The distributed strong components algorithm works on both <tt class="docutils literal"><span class="pre">directed</span></tt> |
| and <tt class="docutils literal"><span class="pre">bidirectional</span></tt> graphs. In the bidirectional case, a reverse |
| graph adapter is used to produce the required reverse graph. In |
| the directed case, a separate graph is constructed in which all the |
| edges are reversed.</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="id2">Where Defined</a></li> |
| <li><a class="reference internal" href="#parameters" id="id3">Parameters</a></li> |
| <li><a class="reference internal" href="#complexity" id="id4">Complexity</a></li> |
| <li><a class="reference internal" href="#algorithm-description" id="id5">Algorithm Description</a></li> |
| <li><a class="reference internal" href="#bibliography" id="id6">Bibliography</a></li> |
| </ul> |
| </div> |
| <div class="section" id="where-defined"> |
| <h1><a class="toc-backref" href="#id2">Where Defined</a></h1> |
| <p><<tt class="docutils literal"><span class="pre">boost/graph/distributed/strong_components.hpp</span></tt>></p> |
| <p>also accessible from</p> |
| <p><<tt class="docutils literal"><span class="pre">boost/graph/strong_components.hpp</span></tt>></p> |
| </div> |
| <div class="section" id="parameters"> |
| <h1><a class="toc-backref" href="#id3">Parameters</a></h1> |
| <dl class="docutils"> |
| <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 <a class="reference external" href="DistributedGraph.html">Distributed Graph</a>. The graph |
| type must also model the <a class="reference external" href="http://www.boost.org/libs/graph/doc/IncidenceGraph.html">Incidence Graph</a> and be directed.</dd> |
| <dt>OUT: <tt class="docutils literal"><span class="pre">ComponentMap</span> <span class="pre">c</span></tt></dt> |
| <dd>The algorithm computes how many strongly connected components are in the |
| graph, and assigns each component an integer label. The algorithm |
| then records to which component each vertex in the graph belongs by |
| recording the component number in the component property map. The |
| <tt class="docutils literal"><span class="pre">ComponentMap</span></tt> type must be a <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a>. The |
| value type must be the <tt class="docutils literal"><span class="pre">vertices_size_type</span></tt> of the graph. The key |
| type must be the graph's vertex descriptor type.</dd> |
| <dt>UTIL: <tt class="docutils literal"><span class="pre">VertexComponentMap</span> <span class="pre">r</span></tt></dt> |
| <dd>The algorithm computes a mapping from each vertex to the |
| representative of the strong component, stored in this property map. |
| The <tt class="docutils literal"><span class="pre">VertexComponentMap</span></tt> type must be a <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a>. |
| The value and key types must be the vertex descriptor of the graph.</dd> |
| <dt>IN: <tt class="docutils literal"><span class="pre">const</span> <span class="pre">ReverseGraph&</span> <span class="pre">gr</span></tt></dt> |
| <dd><p class="first">The reverse (or transpose) graph of <tt class="docutils literal"><span class="pre">g</span></tt>, such that for each |
| directed edge <em>(u, v)</em> in <tt class="docutils literal"><span class="pre">g</span></tt> there exists a directed edge |
| <em>(fr(v), fr(u))</em> in <tt class="docutils literal"><span class="pre">gr</span></tt> and for each edge <em>(v', u')</em> in <em>gr</em> |
| there exists an edge <em>(rf(u'), rf(v'))</em> in <tt class="docutils literal"><span class="pre">g</span></tt>. The functions |
| <em>fr</em> and <em>rf</em> map from vertices in the graph to the reverse graph |
| and vice-verse, and are represented as property map arguments. The |
| concept requirements on this graph type are equivalent to those on |
| the <tt class="docutils literal"><span class="pre">Graph</span></tt> type, but the types need not be the same.</p> |
| <p class="last"><strong>Default</strong>: Either a <tt class="docutils literal"><span class="pre">reverse_graph</span></tt> adaptor over the original |
| graph (if the graph type is bidirectional, i.e., models the |
| <a class="reference external" href="http://www.boost.org/libs/graph/doc/BidirectionalGraph.html">Bidirectional Graph</a> concept) or a <a class="reference external" href="distributed_adjacency_list.html">distributed adjacency list</a> |
| constructed from the input graph.</p> |
| </dd> |
| <dt>IN: <tt class="docutils literal"><span class="pre">IsoMapFR</span> <span class="pre">fr</span></tt></dt> |
| <dd><p class="first">A property map that maps from vertices in the input graph <tt class="docutils literal"><span class="pre">g</span></tt> to |
| vertices in the reversed graph <tt class="docutils literal"><span class="pre">gr</span></tt>. The type <tt class="docutils literal"><span class="pre">IsoMapFR</span></tt> must |
| model the <a class="reference external" href="http://www.boost.org/libs/property_map/ReadablePropertyMap.html">Readable Property Map</a> concept and have the graph's |
| vertex descriptor as its key type and the reverse graph's vertex |
| descriptor as its value type.</p> |
| <p class="last"><strong>Default</strong>: An identity property map (if the graph type is |
| bidirectional) or a distributed <tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> (if the |
| graph type is directed).</p> |
| </dd> |
| <dt>IN: <tt class="docutils literal"><span class="pre">IsoMapRF</span> <span class="pre">rf</span></tt></dt> |
| <dd><p class="first">A property map that maps from vertices in the reversed graph <tt class="docutils literal"><span class="pre">gr</span></tt> |
| to vertices in the input graph <tt class="docutils literal"><span class="pre">g</span></tt>. The type <tt class="docutils literal"><span class="pre">IsoMapRF</span></tt> must |
| model the <a class="reference external" href="http://www.boost.org/libs/property_map/ReadablePropertyMap.html">Readable Property Map</a> concept and have the reverse |
| graph's vertex descriptor as its key type and the graph's vertex |
| descriptor as its value type.</p> |
| <p class="last"><strong>Default</strong>: An identity property map (if the graph type is |
| bidirectional) or a distributed <tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> (if the |
| graph type is directed).</p> |
| </dd> |
| </dl> |
| </div> |
| <div class="section" id="complexity"> |
| <h1><a class="toc-backref" href="#id4">Complexity</a></h1> |
| <p>The local phase of the algorithm is <em>O(V + E)</em>. The parallel phase of |
| the algorithm requires at most <em>O(V)</em> supersteps each containing two |
| breadth first searches which are <em>O(V + E)</em> each.</p> |
| </div> |
| <div class="section" id="algorithm-description"> |
| <h1><a class="toc-backref" href="#id5">Algorithm Description</a></h1> |
| <p>Prior to executing the sequential phase of the algorithm, each process |
| identifies any completely local strong components which it labels and |
| removes from the vertex set considered in the parallel phase of the |
| algorithm.</p> |
| <p>The parallel phase of the distributed strong components algorithm |
| consists of series of supersteps. Each superstep starts with one |
| or more vertex sets which are guaranteed to completely contain |
| any remaining strong components. A <a class="reference external" href="breadth_first_search.html">distributed breadth first search</a> |
| is performed starting from the first vertex in each vertex set. |
| All of these breadth first searches are performed in parallel by having |
| each processor call <tt class="docutils literal"><span class="pre">breadth_first_search()</span></tt> with a different starting |
| vertex, and if necessary inserting additional vertices into the |
| <tt class="docutils literal"><span class="pre">distributed</span> <span class="pre">queue</span></tt> used for breadth first search before invoking |
| the algorithm. A second <a class="reference external" href="breadth_first_search.html">distributed breadth first search</a> is |
| performed on the reverse graph in the same fashion. For each initial |
| vertex set, the successor set (the vertices reached in the forward |
| breadth first search), and the predecessor set (the vertices reached |
| in the backward breadth first search) is computed. The intersection |
| of the predecessor and successor sets form a strongly connected |
| component which is labeled as such. The remaining vertices in the |
| initial vertex set are partitioned into three subsets each guaranteed |
| to completely contain any remaining strong components. These three sets |
| are the vertices in the predecessor set not contained in the identified |
| strongly connected component, the vertices in the successor set not |
| in the strongly connected component, and the remaing vertices in the |
| initial vertex set not in the predecessor or successor sets. Once |
| new vertex sets are identified, the algorithm begins a new superstep. |
| The algorithm halts when no vertices remain.</p> |
| <p>To boost performance in sparse graphs when identifying small components, |
| when less than a given portion of the initial number of vertices |
| remain in active vertex sets, a filtered graph adapter is used |
| to limit the graph seen by the breadth first search algorithm to the |
| active vertices.</p> |
| </div> |
| <div class="section" id="bibliography"> |
| <h1><a class="toc-backref" href="#id6">Bibliography</a></h1> |
| <table class="docutils citation" frame="void" id="fhp00" rules="none"> |
| <colgroup><col class="label" /><col /></colgroup> |
| <tbody valign="top"> |
| <tr><td class="label"><a class="fn-backref" href="#id1">[FHP00]</a></td><td>Lisa Fleischer, Bruce Hendrickson, and Ali Pinar. On |
| Identifying Strongly Connected Components in Parallel. In Parallel and |
| Distributed Processing (IPDPS), volume 1800 of Lecture Notes in |
| Computer Science, pages 505--511, 2000. Springer.</td></tr> |
| </tbody> |
| </table> |
| <hr class="docutils" /> |
| <p>Copyright (C) 2004, 2005 The Trustees of Indiana University.</p> |
| <p>Authors: Nick Edmonds, Douglas Gregor, and Andrew Lumsdaine</p> |
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