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<title>Parallel BGL Dijkstra's Single-Source Shortest Paths</title>
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<div class="document" id="logo-dijkstra-s-single-source-shortest-paths">
<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> Dijkstra's Single-Source Shortest Paths</h1>

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<pre class="literal-block">
// named parameter version
template &lt;typename Graph, typename P, typename T, typename R&gt;
void
dijkstra_shortest_paths(Graph&amp; g,
  typename graph_traits&lt;Graph&gt;::vertex_descriptor s,
  const bgl_named_params&lt;P, T, R&gt;&amp; params);

// non-named parameter version
template &lt;typename Graph, typename DijkstraVisitor,
          typename PredecessorMap, typename DistanceMap,
          typename WeightMap, typename VertexIndexMap, typename CompareFunction, typename CombineFunction,
          typename DistInf, typename DistZero&gt;
void dijkstra_shortest_paths
  (const Graph&amp; g,
   typename graph_traits&lt;Graph&gt;::vertex_descriptor s,
   PredecessorMap predecessor, DistanceMap distance, WeightMap weight,
   VertexIndexMap index_map,
   CompareFunction compare, CombineFunction combine, DistInf inf, DistZero zero,
   DijkstraVisitor vis);
</pre>
<p>The <tt class="docutils literal"><span class="pre">dijkstra_shortest_paths()</span></tt> function solves the single-source
shortest paths problem on a weighted, undirected or directed
distributed graph. There are two implementations of distributed
Dijkstra's algorithm, which offer different performance
tradeoffs. Both are accessible via the <tt class="docutils literal"><span class="pre">dijkstra_shortest_paths()</span></tt>
function (for compatibility with sequential BGL programs). The
distributed Dijkstra algorithms are very similar to their sequential
counterparts. Only the differences are highlighted here; please refer
to the <a class="reference external" href="http://www.boost.org/libs/graph/doc/dijkstra_shortest_paths.html">sequential Dijkstra implementation</a> for additional
details. The best-performing implementation for most cases is the
<a class="reference internal" href="#delta-stepping-algorithm">Delta-Stepping algorithm</a>; however, one can also employ the more
conservative <a class="reference internal" href="#crauser-et-al-s-algorithm">Crauser et al.'s algorithm</a> or the simlistic  <a class="reference internal" href="#eager-dijkstra-s-algorithm">Eager
Dijkstra's algorithm</a>.</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="id10">Where Defined</a></li>
<li><a class="reference internal" href="#parameters" id="id11">Parameters</a></li>
<li><a class="reference internal" href="#visitor-event-points" id="id12">Visitor Event Points</a></li>
<li><a class="reference internal" href="#crauser-et-al-s-algorithm" id="id13">Crauser et al.'s algorithm</a><ul>
<li><a class="reference internal" href="#id2" id="id14">Where Defined</a></li>
<li><a class="reference internal" href="#complexity" id="id15">Complexity</a></li>
<li><a class="reference internal" href="#performance" id="id16">Performance</a></li>
</ul>
</li>
<li><a class="reference internal" href="#eager-dijkstra-s-algorithm" id="id17">Eager Dijkstra's algorithm</a><ul>
<li><a class="reference internal" href="#id5" id="id18">Where Defined</a></li>
<li><a class="reference internal" href="#id6" id="id19">Complexity</a></li>
<li><a class="reference internal" href="#id7" id="id20">Performance</a></li>
</ul>
</li>
<li><a class="reference internal" href="#delta-stepping-algorithm" id="id21">Delta-Stepping algorithm</a><ul>
<li><a class="reference internal" href="#id9" id="id22">Where Defined</a></li>
</ul>
</li>
<li><a class="reference internal" href="#example" id="id23">Example</a></li>
<li><a class="reference internal" href="#bibliography" id="id24">Bibliography</a></li>
</ul>
</div>
<div class="section" id="where-defined">
<h1><a class="toc-backref" href="#id10">Where Defined</a></h1>
<p>&lt;<tt class="docutils literal"><span class="pre">boost/graph/dijkstra_shortest_paths.hpp</span></tt>&gt;</p>
</div>
<div class="section" id="parameters">
<h1><a class="toc-backref" href="#id11">Parameters</a></h1>
<p>All parameters of the <a class="reference external" href="http://www.boost.org/libs/graph/doc/dijkstra_shortest_paths.html">sequential Dijkstra implementation</a> are
supported and have essentially the same meaning. The distributed
Dijkstra implementations introduce a new parameter that allows one to
select <a class="reference internal" href="#eager-dijkstra-s-algorithm">Eager Dijkstra's algorithm</a> and control the amount of work it
performs. Only differences and new parameters are documented here.</p>
<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="DistributedGraph.html">Distributed Graph</a>.</dd>
<dt>IN: <tt class="docutils literal"><span class="pre">vertex_descriptor</span> <span class="pre">s</span></tt></dt>
<dd>The start vertex must be the same in every process.</dd>
<dt>OUT: <tt class="docutils literal"><span class="pre">predecessor_map(PredecessorMap</span> <span class="pre">p_map)</span></tt></dt>
<dd><p class="first">The predecessor map must be a <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a> or
<tt class="docutils literal"><span class="pre">dummy_property_map</span></tt>, although only the local portions of the map
will be written.</p>
<p class="last"><strong>Default:</strong> <tt class="docutils literal"><span class="pre">dummy_property_map</span></tt></p>
</dd>
<dt>UTIL/OUT: <tt class="docutils literal"><span class="pre">distance_map(DistanceMap</span> <span class="pre">d_map)</span></tt></dt>
<dd>The distance map must be either a <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a> or
<tt class="docutils literal"><span class="pre">dummy_property_map</span></tt>. It will be given the <tt class="docutils literal"><span class="pre">vertex_distance</span></tt>
role.</dd>
<dt>IN: <tt class="docutils literal"><span class="pre">visitor(DijkstraVisitor</span> <span class="pre">vis)</span></tt></dt>
<dd>The visitor must be a distributed Dijkstra visitor. The suble differences
between sequential and distributed Dijkstra visitors are discussed in the
section <a class="reference internal" href="#visitor-event-points">Visitor Event Points</a>.</dd>
<dt>UTIL/OUT: <tt class="docutils literal"><span class="pre">color_map(ColorMap</span> <span class="pre">color)</span></tt></dt>
<dd>The color map must be a <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a> with the same
process group as the graph <tt class="docutils literal"><span class="pre">g</span></tt> whose colors must monotonically
darken (white -&gt; gray -&gt; black). The default value is a distributed
<tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> created from a <tt class="docutils literal"><span class="pre">std::vector</span></tt> of
<tt class="docutils literal"><span class="pre">default_color_type</span></tt>.</dd>
</dl>
<p>IN: <tt class="docutils literal"><span class="pre">lookahead(distance_type</span> <span class="pre">look)</span></tt></p>
<blockquote>
<p>When this parameter is supplied, the implementation will use the
<a class="reference internal" href="#eager-dijkstra-s-algorithm">Eager Dijkstra's algorithm</a> with the given lookahead value.
Lookahead permits distributed Dijkstra's algorithm to speculatively
process vertices whose shortest distance from the source may not
have been found yet. When the distance found is the shortest
distance, parallelism is improved and the algorithm may terminate
more quickly. However, if the distance is not the shortest distance,
the vertex will need to be reprocessed later, resulting in more
work.</p>
<p>The type <tt class="docutils literal"><span class="pre">distance_type</span></tt> is the value type of the <tt class="docutils literal"><span class="pre">DistanceMap</span></tt>
property map. It is a nonnegative value specifying how far ahead
Dijkstra's algorithm may process values.</p>
<p><strong>Default:</strong> no value (lookahead is not employed; uses <a class="reference internal" href="#crauser-et-al-s-algorithm">Crauser et
al.'s algorithm</a>).</p>
</blockquote>
</div>
<div class="section" id="visitor-event-points">
<h1><a class="toc-backref" href="#id12">Visitor Event Points</a></h1>
<p>The <a class="reference external" href="http://www.boost.org/libs/graph/doc/DijkstraVisitor.html">Dijkstra Visitor</a> concept defines 7 event points that will be
triggered by the <a class="reference external" href="http://www.boost.org/libs/graph/doc/dijkstra_shortest_paths.html">sequential Dijkstra implementation</a>. The distributed
Dijkstra retains these event points, but the sequence of events
triggered and the process in which each event occurs will change
depending on the distribution of the graph, lookahead, and edge
weights.</p>
<dl class="docutils">
<dt><tt class="docutils literal"><span class="pre">initialize_vertex(s,</span> <span class="pre">g)</span></tt></dt>
<dd>This will be invoked by every process for each local vertex.</dd>
<dt><tt class="docutils literal"><span class="pre">discover_vertex(u,</span> <span class="pre">g)</span></tt></dt>
<dd>This will be invoked each type a process discovers a new vertex
<tt class="docutils literal"><span class="pre">u</span></tt>. Due to incomplete information in distributed property maps,
this event may be triggered many times for the same vertex <tt class="docutils literal"><span class="pre">u</span></tt>.</dd>
<dt><tt class="docutils literal"><span class="pre">examine_vertex(u,</span> <span class="pre">g)</span></tt></dt>
<dd>This will be invoked by the process owning the vertex <tt class="docutils literal"><span class="pre">u</span></tt>. This
event may be invoked multiple times for the same vertex when the
graph contains negative edges or lookahead is employed.</dd>
<dt><tt class="docutils literal"><span class="pre">examine_edge(e,</span> <span class="pre">g)</span></tt></dt>
<dd>This will be invoked by the process owning the source vertex of
<tt class="docutils literal"><span class="pre">e</span></tt>. As with <tt class="docutils literal"><span class="pre">examine_vertex</span></tt>, this event may be invoked
multiple times for the same edge.</dd>
<dt><tt class="docutils literal"><span class="pre">edge_relaxed(e,</span> <span class="pre">g)</span></tt></dt>
<dd>Similar to <tt class="docutils literal"><span class="pre">examine_edge</span></tt>, this will be invoked by the process
owning the source vertex and may be invoked multiple times (even
without lookahead or negative edges).</dd>
<dt><tt class="docutils literal"><span class="pre">edge_not_relaxed(e,</span> <span class="pre">g)</span></tt></dt>
<dd>Similar to <tt class="docutils literal"><span class="pre">edge_relaxed</span></tt>. Some <tt class="docutils literal"><span class="pre">edge_not_relaxed</span></tt> events that
would be triggered by sequential Dijkstra's will become
<tt class="docutils literal"><span class="pre">edge_relaxed</span></tt> events in distributed Dijkstra's algorithm.</dd>
<dt><tt class="docutils literal"><span class="pre">finish_vertex(e,</span> <span class="pre">g)</span></tt></dt>
<dd>See documentation for <tt class="docutils literal"><span class="pre">examine_vertex</span></tt>. Note that a &quot;finished&quot;
vertex is not necessarily finished if lookahead is permitted or
negative edges exist in the graph.</dd>
</dl>
</div>
<div class="section" id="crauser-et-al-s-algorithm">
<h1><a class="toc-backref" href="#id13">Crauser et al.'s algorithm</a></h1>
<pre class="literal-block">
namespace graph {
  template&lt;typename DistributedGraph, typename DijkstraVisitor,
           typename PredecessorMap, typename DistanceMap, typename WeightMap,
           typename IndexMap, typename ColorMap, typename Compare,
           typename Combine, typename DistInf, typename DistZero&gt;
  void
  crauser_et_al_shortest_paths
    (const DistributedGraph&amp; g,
     typename graph_traits&lt;DistributedGraph&gt;::vertex_descriptor s,
     PredecessorMap predecessor, DistanceMap distance, WeightMap weight,
     IndexMap index_map, ColorMap color_map,
     Compare compare, Combine combine, DistInf inf, DistZero zero,
     DijkstraVisitor vis);

  template&lt;typename DistributedGraph, typename DijkstraVisitor,
           typename PredecessorMap, typename DistanceMap, typename WeightMap&gt;
  void
  crauser_et_al_shortest_paths
    (const DistributedGraph&amp; g,
     typename graph_traits&lt;DistributedGraph&gt;::vertex_descriptor s,
     PredecessorMap predecessor, DistanceMap distance, WeightMap weight);

  template&lt;typename DistributedGraph, typename DijkstraVisitor,
           typename PredecessorMap, typename DistanceMap&gt;
  void
  crauser_et_al_shortest_paths
    (const DistributedGraph&amp; g,
     typename graph_traits&lt;DistributedGraph&gt;::vertex_descriptor s,
     PredecessorMap predecessor, DistanceMap distance);
}
</pre>
<p>The formulation of Dijkstra's algorithm by Crauser, Mehlhorn, Meyer,
and Sanders <a class="citation-reference" href="#cmms98a" id="id1">[CMMS98a]</a> improves the scalability of parallel Dijkstra's
algorithm by increasing the number of vertices that can be processed
in a given superstep. This algorithm adapts well to various graph
types, and is a simple algorithm to use, requiring no additional user
input to achieve reasonable performance. The disadvantage of this
algorithm is that the implementation is required to manage three
priority queues, which creates a large amount of work at each node.</p>
<p>This algorithm is used by default in distributed
<tt class="docutils literal"><span class="pre">dijkstra_shortest_paths()</span></tt>.</p>
<div class="section" id="id2">
<h2><a class="toc-backref" href="#id14">Where Defined</a></h2>
<p>&lt;<tt class="docutils literal"><span class="pre">boost/graph/distributed/crauser_et_al_shortest_paths.hpp</span></tt>&gt;</p>
</div>
<div class="section" id="complexity">
<h2><a class="toc-backref" href="#id15">Complexity</a></h2>
<p>This algorithm performs <em>O(V log V)</em> work in <em>d + 1</em> BSP supersteps,
where <em>d</em> is at most <em>O(V)</em> but is generally much smaller. On directed
Erdos-Renyi graphs with edge weights in [0, 1), the expected number of
supersteps <em>d</em> is <em>O(n^(1/3))</em> with high probability.</p>
</div>
<div class="section" id="performance">
<h2><a class="toc-backref" href="#id16">Performance</a></h2>
<p>The following charts illustrate the performance of the Parallel BGL implementation of Crauser et al.'s
algorithm on graphs with edge weights uniformly selected from the
range <em>[0, 1)</em>.</p>
<img align="left" alt="chart_php_cluster_Odin_generator_ER_SF_SW_dataset_TimeSparse_columns_4.png" class="align-left" src="chart_php_cluster_Odin_generator_ER_SF_SW_dataset_TimeSparse_columns_4.png" />
<img alt="chart_php_cluster_Odin_generator_ER_SF_SW_dataset_TimeSparse_columns_4_speedup_1.png" src="chart_php_cluster_Odin_generator_ER_SF_SW_dataset_TimeSparse_columns_4_speedup_1.png" />
<img align="left" alt="chart_php_cluster_Odin_generator_ER_SF_SW_dataset_TimeDense_columns_4.png" class="align-left" src="chart_php_cluster_Odin_generator_ER_SF_SW_dataset_TimeDense_columns_4.png" />
<img alt="chart_php_cluster_Odin_generator_ER_SF_SW_dataset_TimeDense_columns_4_speedup_1.png" src="chart_php_cluster_Odin_generator_ER_SF_SW_dataset_TimeDense_columns_4_speedup_1.png" />
</div>
</div>
<div class="section" id="eager-dijkstra-s-algorithm">
<h1><a class="toc-backref" href="#id17">Eager Dijkstra's algorithm</a></h1>
<pre class="literal-block">
namespace graph {
  template&lt;typename DistributedGraph, typename DijkstraVisitor,
           typename PredecessorMap, typename DistanceMap, typename WeightMap,
           typename IndexMap, typename ColorMap, typename Compare,
           typename Combine, typename DistInf, typename DistZero&gt;
  void
  eager_dijkstra_shortest_paths
    (const DistributedGraph&amp; g,
     typename graph_traits&lt;DistributedGraph&gt;::vertex_descriptor s,
     PredecessorMap predecessor, DistanceMap distance,
     typename property_traits&lt;DistanceMap&gt;::value_type lookahead,
     WeightMap weight, IndexMap index_map, ColorMap color_map,
     Compare compare, Combine combine, DistInf inf, DistZero zero,
     DijkstraVisitor vis);

  template&lt;typename DistributedGraph, typename DijkstraVisitor,
           typename PredecessorMap, typename DistanceMap, typename WeightMap&gt;
  void
  eager_dijkstra_shortest_paths
    (const DistributedGraph&amp; g,
     typename graph_traits&lt;DistributedGraph&gt;::vertex_descriptor s,
     PredecessorMap predecessor, DistanceMap distance,
     typename property_traits&lt;DistanceMap&gt;::value_type lookahead,
     WeightMap weight);

  template&lt;typename DistributedGraph, typename DijkstraVisitor,
           typename PredecessorMap, typename DistanceMap&gt;
  void
  eager_dijkstra_shortest_paths
    (const DistributedGraph&amp; g,
     typename graph_traits&lt;DistributedGraph&gt;::vertex_descriptor s,
     PredecessorMap predecessor, DistanceMap distance,
     typename property_traits&lt;DistanceMap&gt;::value_type lookahead);
}
</pre>
<p>In each superstep, parallel Dijkstra's algorithm typically only
processes nodes whose distances equivalent to the global minimum
distance, because these distances are guaranteed to be correct. This
variation on the algorithm allows the algorithm to process all
vertices whose distances are within some constant value of the
minimum distance. The value is called the &quot;lookahead&quot; value and is
provided by the user as the fifth parameter to the function. Small
values of the lookahead parameter will likely result in limited
parallelization opportunities, whereas large values will expose more
parallelism but may introduce (non-infinite) looping and result in
extra work. The optimal value for the lookahead parameter depends on
the input graph; see <a class="citation-reference" href="#cmms98b" id="id3">[CMMS98b]</a> and <a class="citation-reference" href="#ms98" id="id4">[MS98]</a>.</p>
<p>This algorithm will be used by <tt class="docutils literal"><span class="pre">dijkstra_shortest_paths()</span></tt> when it
is provided with a lookahead value.</p>
<div class="section" id="id5">
<h2><a class="toc-backref" href="#id18">Where Defined</a></h2>
<p>&lt;<tt class="docutils literal"><span class="pre">boost/graph/distributed/eager_dijkstra_shortest_paths.hpp</span></tt>&gt;</p>
</div>
<div class="section" id="id6">
<h2><a class="toc-backref" href="#id19">Complexity</a></h2>
<p>This algorithm performs <em>O(V log V)</em> work in <em>d
+ 1</em> BSP supersteps, where <em>d</em> is at most <em>O(V)</em> but may be smaller
depending on the lookahead value. the algorithm may perform more work
when a large lookahead is provided, because vertices will be
reprocessed.</p>
</div>
<div class="section" id="id7">
<h2><a class="toc-backref" href="#id20">Performance</a></h2>
<p>The performance of the eager Dijkstra's algorithm varies greatly
depending on the lookahead value. The following charts illustrate the
performance of the Parallel BGL on graphs with edge weights uniformly
selected from the range <em>[0, 1)</em> and a constant lookahead of 0.1.</p>
<img align="left" alt="chart_php_cluster_Odin_generator_ER_SF_SW_dataset_TimeSparse_columns_5.png" class="align-left" src="chart_php_cluster_Odin_generator_ER_SF_SW_dataset_TimeSparse_columns_5.png" />
<img alt="chart_php_cluster_Odin_generator_ER_SF_SW_dataset_TimeSparse_columns_5_speedup_1.png" src="chart_php_cluster_Odin_generator_ER_SF_SW_dataset_TimeSparse_columns_5_speedup_1.png" />
<img align="left" alt="chart_php_cluster_Odin_generator_ER_SF_SW_dataset_TimeDense_columns_5.png" class="align-left" src="chart_php_cluster_Odin_generator_ER_SF_SW_dataset_TimeDense_columns_5.png" />
<img alt="chart_php_cluster_Odin_generator_ER_SF_SW_dataset_TimeDense_columns_5_speedup_1.png" src="chart_php_cluster_Odin_generator_ER_SF_SW_dataset_TimeDense_columns_5_speedup_1.png" />
</div>
</div>
<div class="section" id="delta-stepping-algorithm">
<h1><a class="toc-backref" href="#id21">Delta-Stepping algorithm</a></h1>
<pre class="literal-block">
namespace boost { namespace graph { namespace distributed {

  template &lt;typename Graph, typename PredecessorMap,
            typename DistanceMap, typename WeightMap&gt;
  void delta_stepping_shortest_paths
    (const Graph&amp; g,
     typename graph_traits&lt;Graph&gt;::vertex_descriptor s,
     PredecessorMap predecessor, DistanceMap distance, WeightMap weight,
     typename property_traits&lt;WeightMap&gt;::value_type delta)


  template &lt;typename Graph, typename PredecessorMap,
            typename DistanceMap, typename WeightMap&gt;
  void delta_stepping_shortest_paths
    (const Graph&amp; g,
     typename graph_traits&lt;Graph&gt;::vertex_descriptor s,
     PredecessorMap predecessor, DistanceMap distance, WeightMap weight)
  }

} } }
</pre>
<p>The delta-stepping algorithm <a class="citation-reference" href="#ms98" id="id8">[MS98]</a> is another variant of the parallel
Dijkstra algorithm. Like the eager Dijkstra algorithm, it employs a
lookahead (<tt class="docutils literal"><span class="pre">delta</span></tt>) value that allows processors to process vertices
before we are guaranteed to find their minimum distances, permitting
more parallelism than a conservative strategy. Delta-stepping also
introduces a multi-level bucket data structure that provides more
relaxed ordering constraints than the priority queues employed by the
other Dijkstra variants, reducing the complexity of insertions,
relaxations, and removals from the central data structure. The
delta-stepping algorithm is the best-performing of the Dijkstra
variants.</p>
<p>The lookahead value <tt class="docutils literal"><span class="pre">delta</span></tt> determines how large each of the
&quot;buckets&quot; within the delta-stepping queue will be, where the ith
bucket contains edges within tentative distances between <tt class="docutils literal"><span class="pre">delta``*i</span>
<span class="pre">and</span> <span class="pre">``delta``*(i+1).</span> <span class="pre">``delta</span></tt> must be a positive value. When omitted,
<tt class="docutils literal"><span class="pre">delta</span></tt> will be set to the maximum edge weight divided by the
maximum degree.</p>
<div class="section" id="id9">
<h2><a class="toc-backref" href="#id22">Where Defined</a></h2>
<p>&lt;<tt class="docutils literal"><span class="pre">boost/graph/distributed/delta_stepping_shortest_paths.hpp</span></tt>&gt;</p>
</div>
</div>
<div class="section" id="example">
<h1><a class="toc-backref" href="#id23">Example</a></h1>
<p>See the separate <a class="reference external" href="dijkstra_example.html">Dijkstra example</a>.</p>
</div>
<div class="section" id="bibliography">
<h1><a class="toc-backref" href="#id24">Bibliography</a></h1>
<table class="docutils citation" frame="void" id="cmms98a" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id1">[CMMS98a]</a></td><td>Andreas Crauser, Kurt Mehlhorn, Ulrich Meyer, and Peter Sanders. A
Parallelization of Dijkstra's Shortest Path Algorithm. In
<em>Mathematical Foundations of Computer Science (MFCS)</em>, volume 1450 of
Lecture Notes in Computer Science, pages 722--731, 1998. Springer.</td></tr>
</tbody>
</table>
<table class="docutils citation" frame="void" id="cmms98b" rules="none">
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<tr><td class="label"><a class="fn-backref" href="#id3">[CMMS98b]</a></td><td>Andreas Crauser, Kurt Mehlhorn, Ulrich Meyer, and Peter
Sanders. Parallelizing Dijkstra's shortest path algorithm. Technical
report, MPI-Informatik, 1998.</td></tr>
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<tr><td class="label">[MS98]</td><td><em>(<a class="fn-backref" href="#id4">1</a>, <a class="fn-backref" href="#id8">2</a>)</em> Ulrich Meyer and Peter Sanders. Delta-stepping: A parallel
shortest path algorithm. In <em>6th ESA</em>, LNCS. Springer, 1998.</td></tr>
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<p>Copyright (C) 2004, 2005, 2006, 2007, 2008 The Trustees of Indiana University.</p>
<p>Authors: Douglas Gregor and Andrew Lumsdaine</p>
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