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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
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<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>
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