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

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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> Boman et al graph coloring</h1>

 <!-- Copyright (C) 2004-2008 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">
 namespace graph {
   template&lt;typename DistributedGraph, typename ColorMap&gt;
   typename property_traits&lt;ColorMap&gt;::value_type
   boman_et_al_graph_coloring
     (const DistributedGraph&amp; g,
      ColorMap color,
      typename graph_traits&lt;DistributedGraph&gt;::vertices_size_type s = 100);

   template&lt;typename DistributedGraph, typename ColorMap, typename ChooseColor&gt;
   typename property_traits&lt;ColorMap&gt;::value_type
   boman_et_al_graph_coloring
     (const DistributedGraph&amp; g,
      ColorMap color,
      typename graph_traits&lt;DistributedGraph&gt;::vertices_size_type s,
      ChooseColor choose_color);

   template&lt;typename DistributedGraph, typename ColorMap, typename ChooseColor,
            typename VertexOrdering&gt;
   typename property_traits&lt;ColorMap&gt;::value_type
   boman_et_al_graph_coloring
     (const DistributedGraph&amp; g, ColorMap color,
      typename graph_traits&lt;DistributedGraph&gt;::vertices_size_type s,
      ChooseColor choose_color, VertexOrdering ordering);

   template&lt;typename DistributedGraph, typename ColorMap, typename ChooseColor,
            typename VertexOrdering, typename VertexIndexMap&gt;
   typename property_traits&lt;ColorMap&gt;::value_type
   boman_et_al_graph_coloring
     (const DistributedGraph&amp; g,
      ColorMap color,
      typename graph_traits&lt;DistributedGraph&gt;::vertices_size_type s,
      ChooseColor choose_color,
      VertexOrdering ordering, VertexIndexMap vertex_index);
 }
 </pre>
 <p>The <tt class="docutils literal"><span class="pre">boman_et_al_graph_coloring</span></tt> function colors the vertices of an
 undirected, distributed graph such that no two adjacent vertices have
 the same color. All of the vertices of a given color form an
 independent set in the graph. Graph coloring has been used to solve
 various problems, including register allocation in compilers,
 optimization problems, and scheduling problems.</p>
 <img align="right" alt="Vertex coloring example" class="align-right" src="../vertex_coloring.png" style="width: 462px; height: 269px;" />
 <p>The problem of coloring a graph with the fewest possible number of
 colors is NP-complete, so many algorithms (including the one
 implemented here) are heuristic algorithms that try to minimize the
 number of colors but are not guaranteed to provide an optimal
 solution. This algorithm <a class="citation-reference" href="#bbc05" id="id1">[BBC05]</a> is similar to the
 <tt class="docutils literal"><span class="pre">sequential_vertex_coloring</span></tt> algorithm, that iterates through the
 vertices once and selects the lowest-numbered color that the current
 vertex can have. The coloring and the number of colors is therefore
 related to the ordering of the vertices in the sequential case.</p>
 <p>The distributed <tt class="docutils literal"><span class="pre">boman_et_al_graph_coloring</span></tt> algorithm will produce
 different colorings depending on the ordering and distribution of the
 vertices and the number of parallel processes cooperating to perform
 the coloring.</p>
 <p>The algorithm returns the number of colors <tt class="docutils literal"><span class="pre">num_colors</span></tt> used to
 color the graph.</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="#performance" id="id5">Performance</a></li>
 </ul>
 </div>
 <div class="section" id="where-defined">
 <h1><a class="toc-backref" href="#id2">Where Defined</a></h1>
 <p>&lt;<tt class="docutils literal"><span class="pre">boost/graph/distributed/boman_et_al_graph_coloring.hpp</span></tt>&gt;</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">Graph&amp;</span> <span class="pre">g</span></tt></dt>
 <dd>The graph type must be a model of <a class="reference external" href="DistributedVertexListGraph.html">Distributed Vertex List Graph</a> and
 <a class="reference external" href="DistributedEdgeListGraph.html">Distributed Edge List Graph</a>.</dd>
 <dt>UTIL/OUT: <tt class="docutils literal"><span class="pre">ColorMap</span> <span class="pre">color</span></tt></dt>
 <dd>Stores the color of each vertex, which will be a value in the range
 [0, <tt class="docutils literal"><span class="pre">num_colors</span></tt>). The type <tt class="docutils literal"><span class="pre">ColorMap</span></tt> must model the
 <a class="reference external" href="http://www.boost.org/libs/property_map/ReadWritePropertyMap.html">Read/Write Property Map</a> concept and must be a <a class="reference external" href="distributed_property_map.html">distributed
 property map</a>.</dd>
 <dt>IN: <tt class="docutils literal"><span class="pre">vertices_size_type</span> <span class="pre">s</span></tt></dt>
 <dd><p class="first">The number of vertices to color within each superstep. After
 <tt class="docutils literal"><span class="pre">s</span></tt> vertices have been colored, the colors of boundary vertices
 will be sent to their out-of-process neighbors. Smaller values
 communicate more often but may reduce the risk of conflicts,
 whereas larger values do more work in between communication steps
 but may create many conflicts.</p>
 <p class="last"><strong>Default</strong>: 100</p>
 </dd>
 <dt>IN: <tt class="docutils literal"><span class="pre">ChooseColor</span> <span class="pre">choose_color</span></tt></dt>
 <dd><p class="first">A function object that chooses the color for a vertex given the
 colors of its neighbors. The function object will be passed a vector
 of values (<tt class="docutils literal"><span class="pre">marked</span></tt>) and a <tt class="docutils literal"><span class="pre">marked_true</span></tt> value, and should
 return a <tt class="docutils literal"><span class="pre">color</span></tt> value such that <tt class="docutils literal"><span class="pre">color</span> <span class="pre">&gt;=</span> <span class="pre">marked.size()</span></tt> or
 <tt class="docutils literal"><span class="pre">marked[color]</span> <span class="pre">!=</span> <span class="pre">marked_true</span></tt>.</p>
 <p class="last"><strong>Default</strong>:
 <tt class="docutils literal"><span class="pre">boost::graph::distributed::first_fit_color&lt;color_type&gt;()</span></tt>, where
 <tt class="docutils literal"><span class="pre">color_type</span></tt> is the value type of the <tt class="docutils literal"><span class="pre">ColorMap</span></tt> property map.</p>
 </dd>
 <dt>IN: <tt class="docutils literal"><span class="pre">VertexOrdering</span> <span class="pre">ordering</span></tt></dt>
 <dd><p class="first">A binary predicate function object that provides total ordering on
 the vertices in the graph. Whenever a conflict arises, only one of
 the processes involved will recolor the vertex in the next round,
 and this ordering determines which vertex should be considered
 conflicting (its owning process will then handle the
 conflict). Ideally, this predicate should order vertices so that
 conflicting vertices will be spread uniformly across
 processes. However, this predicate <em>must</em> resolve the same way on
 both processors.</p>
 <p class="last"><strong>Default</strong>: <em>unspecified</em></p>
 </dd>
 <dt>IN: <tt class="docutils literal"><span class="pre">VertexIndexMap</span> <span class="pre">index</span></tt></dt>
 <dd><p class="first">A mapping from vertex descriptors to indices in the range <em>[0,
 num_vertices(g))</em>. This must be a <a class="reference external" href="http://www.boost.org/libs/property_map/ReadablePropertyMap.html">Readable Property Map</a> whose
 key type is a vertex descriptor and whose value type is an integral
 type, typically the <tt class="docutils literal"><span class="pre">vertices_size_type</span></tt> of the graph.</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>
 </dl>
 </div>
 <div class="section" id="complexity">
 <h1><a class="toc-backref" href="#id4">Complexity</a></h1>
 <p>The complexity of this algorithm is hard to characterize,
 because it depends greatly on the number of <em>conflicts</em> that occur
 during the algorithm. A conflict occurs when a <em>boundary vertex</em>
 (i.e., a vertex that is adjacent to a vertex stored on a different
 processor) is given the same color is a boundary vertex adjacency to
 it (but on another processor). Conflicting vertices must be assigned
 new colors, requiring additional work and communication. The work
 involved in reassigning a color for a conflicting vertex is <em>O(d)</em>,
 where <em>d</em> is the degree of the vertex and <em>O(1)</em> messages of <em>O(1)</em>
 size are needed to resolve the conflict. Note that the number of
 conflicts grows with (1) the number of processes and (2) the number
 of inter-process edges.</p>
 </div>
 <div class="section" id="performance">
 <h1><a class="toc-backref" href="#id5">Performance</a></h1>
 <p>The performance of this implementation of Bomen et al's graph coloring
 algorithm is illustrated by the following charts. Scaling and
 performance is reasonable for all of the graphs we have tried.</p>
 <img align="left" alt="chart_php_generator_ER_SF_SW_dataset_TimeSparse_cluster_Odin_columns_11.png" class="align-left" src="chart_php_generator_ER_SF_SW_dataset_TimeSparse_cluster_Odin_columns_11.png" />
 <img alt="chart_php_generator_ER_SF_SW_dataset_TimeSparse_cluster_Odin_columns_11_speedup_1.png" src="chart_php_generator_ER_SF_SW_dataset_TimeSparse_cluster_Odin_columns_11_speedup_1.png" />
 <img align="left" alt="chart_php_generator_ER_SF_SW_dataset_TimeDense_cluster_Odin_columns_11.png" class="align-left" src="chart_php_generator_ER_SF_SW_dataset_TimeDense_cluster_Odin_columns_11.png" />
 <img alt="chart_php_generator_ER_SF_SW_dataset_TimeDense_cluster_Odin_columns_11_speedup_1.png" src="chart_php_generator_ER_SF_SW_dataset_TimeDense_cluster_Odin_columns_11_speedup_1.png" />
 <hr class="docutils" />
 <p>Copyright (C) 2005 The Trustees of Indiana University.</p>
 <p>Authors: Douglas Gregor and Andrew Lumsdaine</p>
 <table class="docutils citation" frame="void" id="bbc05" rules="none">
 <colgroup><col class="label" /><col /></colgroup>
 <tbody valign="top">
 <tr><td class="label"><a class="fn-backref" href="#id1">[BBC05]</a></td><td>Erik G. Boman, Doruk Bozdag, Umit Catalyurek, Assefaw
 H. Gebremedhin, and Fredrik Manne. A Scalable Parallel Graph Coloring
 Algorithm for Distributed Memory Computers. [preprint]</td></tr>
 </tbody>
 </table>
 </div>
 </div>
 <div class="footer">
 <hr class="footer" />
 Generated on: 2009-05-31 00:21 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.

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	<title>Parallel BGL Boman et al graph coloring</title>
	<link rel="stylesheet" href="../../../../rst.css" type="text/css" />
	</head>
	<body>
	<div class="document" id="logo-boman-et-al-graph-coloring">
	<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> Boman et al graph coloring</h1>

	<!-- Copyright (C) 2004-2008 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">
	namespace graph {
	template<typename DistributedGraph, typename ColorMap>
	typename property_traits<ColorMap>::value_type
	boman_et_al_graph_coloring
	(const DistributedGraph& g,
	ColorMap color,
	typename graph_traits<DistributedGraph>::vertices_size_type s = 100);

	template<typename DistributedGraph, typename ColorMap, typename ChooseColor>
	typename property_traits<ColorMap>::value_type
	boman_et_al_graph_coloring
	(const DistributedGraph& g,
	ColorMap color,
	typename graph_traits<DistributedGraph>::vertices_size_type s,
	ChooseColor choose_color);

	template<typename DistributedGraph, typename ColorMap, typename ChooseColor,
	typename VertexOrdering>
	typename property_traits<ColorMap>::value_type
	boman_et_al_graph_coloring
	(const DistributedGraph& g, ColorMap color,
	typename graph_traits<DistributedGraph>::vertices_size_type s,
	ChooseColor choose_color, VertexOrdering ordering);

	template<typename DistributedGraph, typename ColorMap, typename ChooseColor,
	typename VertexOrdering, typename VertexIndexMap>
	typename property_traits<ColorMap>::value_type
	boman_et_al_graph_coloring
	(const DistributedGraph& g,
	ColorMap color,
	typename graph_traits<DistributedGraph>::vertices_size_type s,
	ChooseColor choose_color,
	VertexOrdering ordering, VertexIndexMap vertex_index);
	}
	</pre>
	<p>The <tt class="docutils literal"><span class="pre">boman_et_al_graph_coloring</span></tt> function colors the vertices of an
	undirected, distributed graph such that no two adjacent vertices have
	the same color. All of the vertices of a given color form an
	independent set in the graph. Graph coloring has been used to solve
	various problems, including register allocation in compilers,
	optimization problems, and scheduling problems.</p>
	<img align="right" alt="Vertex coloring example" class="align-right" src="../vertex_coloring.png" style="width: 462px; height: 269px;" />
	<p>The problem of coloring a graph with the fewest possible number of
	colors is NP-complete, so many algorithms (including the one
	implemented here) are heuristic algorithms that try to minimize the
	number of colors but are not guaranteed to provide an optimal
	solution. This algorithm <a class="citation-reference" href="#bbc05" id="id1">[BBC05]</a> is similar to the
	<tt class="docutils literal"><span class="pre">sequential_vertex_coloring</span></tt> algorithm, that iterates through the
	vertices once and selects the lowest-numbered color that the current
	vertex can have. The coloring and the number of colors is therefore
	related to the ordering of the vertices in the sequential case.</p>
	<p>The distributed <tt class="docutils literal"><span class="pre">boman_et_al_graph_coloring</span></tt> algorithm will produce
	different colorings depending on the ordering and distribution of the
	vertices and the number of parallel processes cooperating to perform
	the coloring.</p>
	<p>The algorithm returns the number of colors <tt class="docutils literal"><span class="pre">num_colors</span></tt> used to
	color the graph.</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="#performance" id="id5">Performance</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/boman_et_al_graph_coloring.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">Graph&</span> <span class="pre">g</span></tt></dt>
	<dd>The graph type must be a model of <a class="reference external" href="DistributedVertexListGraph.html">Distributed Vertex List Graph</a> and
	<a class="reference external" href="DistributedEdgeListGraph.html">Distributed Edge List Graph</a>.</dd>
	<dt>UTIL/OUT: <tt class="docutils literal"><span class="pre">ColorMap</span> <span class="pre">color</span></tt></dt>
	<dd>Stores the color of each vertex, which will be a value in the range
	[0, <tt class="docutils literal"><span class="pre">num_colors</span></tt>). The type <tt class="docutils literal"><span class="pre">ColorMap</span></tt> must model the
	<a class="reference external" href="http://www.boost.org/libs/property_map/ReadWritePropertyMap.html">Read/Write Property Map</a> concept and must be a <a class="reference external" href="distributed_property_map.html">distributed
	property map</a>.</dd>
	<dt>IN: <tt class="docutils literal"><span class="pre">vertices_size_type</span> <span class="pre">s</span></tt></dt>
	<dd><p class="first">The number of vertices to color within each superstep. After
	<tt class="docutils literal"><span class="pre">s</span></tt> vertices have been colored, the colors of boundary vertices
	will be sent to their out-of-process neighbors. Smaller values
	communicate more often but may reduce the risk of conflicts,
	whereas larger values do more work in between communication steps
	but may create many conflicts.</p>
	<p class="last"><strong>Default</strong>: 100</p>
	</dd>
	<dt>IN: <tt class="docutils literal"><span class="pre">ChooseColor</span> <span class="pre">choose_color</span></tt></dt>
	<dd><p class="first">A function object that chooses the color for a vertex given the
	colors of its neighbors. The function object will be passed a vector
	of values (<tt class="docutils literal"><span class="pre">marked</span></tt>) and a <tt class="docutils literal"><span class="pre">marked_true</span></tt> value, and should
	return a <tt class="docutils literal"><span class="pre">color</span></tt> value such that <tt class="docutils literal"><span class="pre">color</span> <span class="pre">>=</span> <span class="pre">marked.size()</span></tt> or
	<tt class="docutils literal"><span class="pre">marked[color]</span> <span class="pre">!=</span> <span class="pre">marked_true</span></tt>.</p>
	<p class="last"><strong>Default</strong>:
	<tt class="docutils literal"><span class="pre">boost::graph::distributed::first_fit_color<color_type>()</span></tt>, where
	<tt class="docutils literal"><span class="pre">color_type</span></tt> is the value type of the <tt class="docutils literal"><span class="pre">ColorMap</span></tt> property map.</p>
	</dd>
	<dt>IN: <tt class="docutils literal"><span class="pre">VertexOrdering</span> <span class="pre">ordering</span></tt></dt>
	<dd><p class="first">A binary predicate function object that provides total ordering on
	the vertices in the graph. Whenever a conflict arises, only one of
	the processes involved will recolor the vertex in the next round,
	and this ordering determines which vertex should be considered
	conflicting (its owning process will then handle the
	conflict). Ideally, this predicate should order vertices so that
	conflicting vertices will be spread uniformly across
	processes. However, this predicate <em>must</em> resolve the same way on
	both processors.</p>
	<p class="last"><strong>Default</strong>: <em>unspecified</em></p>
	</dd>
	<dt>IN: <tt class="docutils literal"><span class="pre">VertexIndexMap</span> <span class="pre">index</span></tt></dt>
	<dd><p class="first">A mapping from vertex descriptors to indices in the range <em>[0,
	num_vertices(g))</em>. This must be a <a class="reference external" href="http://www.boost.org/libs/property_map/ReadablePropertyMap.html">Readable Property Map</a> whose
	key type is a vertex descriptor and whose value type is an integral
	type, typically the <tt class="docutils literal"><span class="pre">vertices_size_type</span></tt> of the graph.</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>
	</dl>
	</div>
	<div class="section" id="complexity">
	<h1><a class="toc-backref" href="#id4">Complexity</a></h1>
	<p>The complexity of this algorithm is hard to characterize,
	because it depends greatly on the number of <em>conflicts</em> that occur
	during the algorithm. A conflict occurs when a <em>boundary vertex</em>
	(i.e., a vertex that is adjacent to a vertex stored on a different
	processor) is given the same color is a boundary vertex adjacency to
	it (but on another processor). Conflicting vertices must be assigned
	new colors, requiring additional work and communication. The work
	involved in reassigning a color for a conflicting vertex is <em>O(d)</em>,
	where <em>d</em> is the degree of the vertex and <em>O(1)</em> messages of <em>O(1)</em>
	size are needed to resolve the conflict. Note that the number of
	conflicts grows with (1) the number of processes and (2) the number
	of inter-process edges.</p>
	</div>
	<div class="section" id="performance">
	<h1><a class="toc-backref" href="#id5">Performance</a></h1>
	<p>The performance of this implementation of Bomen et al's graph coloring
	algorithm is illustrated by the following charts. Scaling and
	performance is reasonable for all of the graphs we have tried.</p>
	<img align="left" alt="chart_php_generator_ER_SF_SW_dataset_TimeSparse_cluster_Odin_columns_11.png" class="align-left" src="chart_php_generator_ER_SF_SW_dataset_TimeSparse_cluster_Odin_columns_11.png" />
	<img alt="chart_php_generator_ER_SF_SW_dataset_TimeSparse_cluster_Odin_columns_11_speedup_1.png" src="chart_php_generator_ER_SF_SW_dataset_TimeSparse_cluster_Odin_columns_11_speedup_1.png" />
	<img align="left" alt="chart_php_generator_ER_SF_SW_dataset_TimeDense_cluster_Odin_columns_11.png" class="align-left" src="chart_php_generator_ER_SF_SW_dataset_TimeDense_cluster_Odin_columns_11.png" />
	<img alt="chart_php_generator_ER_SF_SW_dataset_TimeDense_cluster_Odin_columns_11_speedup_1.png" src="chart_php_generator_ER_SF_SW_dataset_TimeDense_cluster_Odin_columns_11_speedup_1.png" />
	<hr class="docutils" />
	<p>Copyright (C) 2005 The Trustees of Indiana University.</p>
	<p>Authors: Douglas Gregor and Andrew Lumsdaine</p>
	<table class="docutils citation" frame="void" id="bbc05" rules="none">
	<colgroup><col class="label" /><col /></colgroup>
	<tbody valign="top">
	<tr><td class="label"><a class="fn-backref" href="#id1">[BBC05]</a></td><td>Erik G. Boman, Doruk Bozdag, Umit Catalyurek, Assefaw
	H. Gebremedhin, and Fredrik Manne. A Scalable Parallel Graph Coloring
	Algorithm for Distributed Memory Computers. [preprint]</td></tr>
	</tbody>
	</table>
	</div>
	</div>
	<div class="footer">
	<hr class="footer" />
	Generated on: 2009-05-31 00:21 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.

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