<?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 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. </div> </body> </html>