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 <Head>
 <Title>Boost Graph Library: Directed Acyclic Graph Shortest Paths</Title>
 <BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
         ALINK="#ff0000">
 <IMG SRC="../../../boost.png"
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 <BR Clear>

 <H1><A NAME="sec:dag_shortest_paths"></A>
 <img src="figs/python.gif" alt="(Python)"/>
 <TT>dag_shortest_paths</TT>
 </H1>


 <P>
 <PRE>
 <i>// named paramter version</i>
 template &lt;class VertexListGraph, class Param, class Tag, class Rest&gt;
 void dag_shortest_paths(const VertexListGraph&amp; g,
    typename graph_traits&lt;VertexListGraph&gt;::vertex_descriptor s,
    const bgl_named_params&lt;Param,Tag,Rest&gt;&amp; params)

 <i>// non-named parameter version</i>
 template &lt;class VertexListGraph, class DijkstraVisitor,
 	  class DistanceMap, class WeightMap, class ColorMap,
 	  class PredecessorMap,
 	  class Compare, class Combine,
 	  class DistInf, class DistZero&gt;
 void dag_shortest_paths(const VertexListGraph&amp; g,
    typename graph_traits&lt;VertexListGraph&gt;::vertex_descriptor s,
    DistanceMap distance, WeightMap weight, ColorMap color,
    PredecessorMap pred, DijkstraVisitor vis,
    Compare compare, Combine combine, DistInf inf, DistZero zero)
 </PRE>

 <P>
 This algorithm&nbsp;[<A HREF="bibliography.html#clr90">8</A>] solves
 the single-source shortest-paths problem on a weighted, directed
 acyclic graph (DAG). This algorithm is more efficient for DAG's
 than either the Dijkstra or Bellman-Ford algorithm.
 Use breadth-first search instead of this algorithm
 when all edge weights are equal to one.  For the definition of the
 shortest-path problem see Section <A
 HREF="graph_theory_review.html#sec:shortest-paths-algorithms">Shortest-Paths
 Algorithms</A> for some background to the shortest-path problem.
 </P>

 <P>
 There are two main options for obtaining output from the
 <tt>dag_shortest_paths()</tt> function. If you provide a
 distance property map through the <tt>distance_map()</tt> parameter
 then the shortest distance from the source vertex to every other
 vertex in the graph will be recorded in the distance map. Also you can
 record the shortest paths tree in a predecessor map: for each vertex
 <i>u in V</i>, <i>p[u]</i> will be the predecessor of <i>u</i> in
 the shortest paths tree (unless <i>p[u] = u</i>, in which case <i>u</i> is
 either the source or a vertex unreachable from the source).  In
 addition to these two options, the user can provide there own
 custom-made visitor that can takes actions during any of the
 algorithm's event points.</P>

 <h3>Where Defined</h3>

 <a href="../../../boost/graph/dag_shortest_paths.hpp"><tt>boost/graph/dag_shortest_paths.hpp</tt></a>

 <h3>Parameters</h3>

 IN: <tt>const VertexListGraph&amp; g</tt>
 <blockquote>
   The graph object on which the algorithm will be applied.
   The type <tt>VertexListGraph</tt> must be a model of \concept{VertexListGraph}.<br>

   <b>Python</b>: The parameter is named <tt>graph</tt>.
 </blockquote>

 IN: <tt>vertex_descriptor s</tt>
 <blockquote>
   The source vertex. All distance will be calculated from this vertex,
   and the shortest paths tree will be rooted at this vertex.<br>

   <b>Python</b>: The parameter is named <tt>root_vertex</tt>.
 </blockquote>

 <h3>Named Parameters</h3>

 IN: <tt>weight_map(WeightMap w_map)</tt>
 <blockquote>
   The weight or ``length'' of each edge in the graph.
   The type <tt>WeightMap</tt> must be a model of
   <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The edge descriptor type of
   the graph needs to be usable as the key type for the weight
   map. The value type for the map must be
   <i>Addable</i> with the value type of the distance map.<br>
   <b>Default:</b>  <tt>get(edge_weight, g)</tt><br>
   <b>Python</b>: Must be an <tt>edge_double_map</tt> for the graph.<br>
   <b>Python default</b>: <tt>graph.get_edge_double_map("weight")</tt>

 </blockquote>

 IN: <tt>vertex_index_map(VertexIndexMap i_map)</tt>
 <blockquote>
   This maps each vertex to an integer in the range <tt>[0,
     num_vertices(g))</tt>. This is necessary for efficient updates of the
   heap data structure when an edge is relaxed.  The type
   <tt>VertexIndexMap</tt> must be a model of
   <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The value type of the map must be an
   integer type. The vertex descriptor type of the graph needs to be
   usable as the key type of the map.<br>
   <b>Default:</b> <tt>get(vertex_index, g)</tt>.
     Note: if you use this default, make sure your graph has
     an internal <tt>vertex_index</tt> property. For example,
     <tt>adjacenty_list</tt> with <tt>VertexList=listS</tt> does
     not have an internal <tt>vertex_index</tt> property.<br>

   <b>Python</b>: Unsupported parameter.
 </blockquote>

 OUT: <tt>predecessor_map(PredecessorMap p_map)</tt>
 <blockquote>
   The predecessor map records the edges in the minimum spanning
   tree. Upon completion of the algorithm, the edges <i>(p[u],u)</i>
   for all <i>u in V</i> are in the minimum spanning tree. If <i>p[u] =
   u</i> then <i>u</i> is either the source vertex or a vertex that is
   not reachable from the source.  The <tt>PredecessorMap</tt> type
   must be a <a
   href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
   Property Map</a> which key and vertex types the same as the vertex
   descriptor type of the graph.<br>
   <b>Default:</b> <tt>dummy_property_map</tt><br>
   <b>Python</b>: Must be a <tt>vertex_vertex_map</tt> for the graph.<br>
 </blockquote>

 UTIL/OUT: <tt>distance_map(DistanceMap d_map)</tt>
 <blockquote>
   The shortest path weight from the source vertex <tt>s</tt> to each
   vertex in the graph <tt>g</tt> is recorded in this property map. The
   shortest path weight is the sum of the edge weights along the
   shortest path.  The type <tt>DistanceMap</tt> must be a model of <a
   href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
   Property Map</a>. The vertex descriptor type of the graph needs to
   be usable as the key type of the distance map.

   The value type of the distance map is the element type of a <a
   href="./Monoid.html">Monoid</tt> formed with the <tt>combine</tt>
   function object and the <tt>zero</tt> object for the identity
   element. Also the distance value type must have a <a
   href="http://www.sgi.com/tech/stl/StrictWeakOrdering.html">
   StrictWeakOrdering</a> provided by the <tt>compare</tt> function
   object.<br>
   <b>Default:</b> <a
   href="../../property_map/doc/iterator_property_map.html">
   <tt>iterator_property_map</tt></a> created from a
   <tt>std::vector</tt> of the <tt>WeightMap</tt>'s value type of size
   <tt>num_vertices(g)</tt> and using the <tt>i_map</tt> for the index
   map.<br>

   <b>Python</b>: Must be a <tt>vertex_double_map</tt> for the graph.
 </blockquote>

 IN: <tt>distance_compare(CompareFunction cmp)</tt>
 <blockquote>
   This function is use to compare distances to determine which vertex
   is closer to the source vertex.  The <tt>CompareFunction</tt> type
   must be a model of <a
   href="http://www.sgi.com/tech/stl/BinaryPredicate.html">Binary
   Predicate</a> and have argument types that match the value type of
   the <tt>DistanceMap</tt> property map.<br>

   <b>Default:</b>
   <tt>std::less&lt;D&gt;</tt> with <tt>D=typename
   property_traits&lt;DistanceMap&gt;::value_type</tt><br>

   <b>Python</b>: Unsupported parameter.
 </blockquote>

 IN: <tt>distance_combine(CombineFunction cmb)</tt>
 <blockquote>
   This function is used to combine distances to compute the distance
   of a path. The <tt>CombineFunction</tt> type must be a model of <a
   href="http://www.sgi.com/tech/stl/BinaryFunction.html">Binary
   Function</a>. The first argument type of the binary function must
   match the value type of the <tt>DistanceMap</tt> property map and
   the second argument type must match the value type of the
   <tt>WeightMap</tt> property map.  The result type must be the same
   type as the distance value type.<br>

   <b>Default:</b> <tt>std::plus&lt;D&gt;</tt> with
    <tt>D=typename property_traits&lt;DistanceMap&gt;::value_type</tt><br>

   <b>Python</b>: Unsupported parameter.
 </blockquote>

 IN: <tt>distance_inf(D inf)</tt>
 <blockquote>
   The <tt>inf</tt> object must be the greatest value of any <tt>D</tt> object.
   That is, <tt>compare(d, inf) == true</tt> for any <tt>d != inf</tt>.
   The type <tt>D</tt> is the value type of the <tt>DistanceMap</tt>.<br>
   <b>Default:</b> <tt>std::numeric_limits&lt;D&gt;::max()</tt><br>

   <b>Python</b>: Unsupported parameter.
 </blockquote>

 IN: <tt>distance_zero(D zero)</tt>
 <blockquote>
   The <tt>zero</tt> value must be the identity element for the
   <a href="./Monoid.html">Monoid</a> formed by the distance values
   and the <tt>combine</tt> function object.
   The type \code{D} is the value type of the \code{DistanceMap}
   <b>Default:</b> <tt>D()</tt><br>

   <b>Python</b>: Unsupported parameter.
 </blockquote>

 UTIL/OUT: <tt>color_map(ColorMap c_map)</tt>
 <blockquote>
   This is used during the execution of the algorithm to mark the
   vertices. The vertices start out white and become gray when they are
   inserted in the queue. They then turn black when they are removed
   from the queue. At the end of the algorithm, vertices reachable from
   the source vertex will have been colored black. All other vertices
   will still be white. The type <tt>ColorMap</tt> must be a model of
   <a href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
   Property Map</a>. A vertex descriptor must be usable as the key type
   of the map, and the value type of the map must be a model of
   <a href="./ColorValue.html">Color Value</a>.<br>
   <b>Default:</b> an <a
   href="../../property_map/doc/iterator_property_map.html">
   <tt>iterator_property_map</tt></a> created from a <tt>std::vector</tt>
   of <tt>default_color_type</tt> of size <tt>num_vertices(g)</tt> and
   using the <tt>i_map</tt> for the index map.<br>

   <b>Python</b>: The color map must be a <tt>vertex_color_map</tt> for
   the graph.

 </blockquote>

 OUT: <tt>visitor(DijkstraVisitor v)</tt>
 <blockquote>
   Use this to specify actions that you would like to happen
   during certain event points within the algorithm.
   The type <tt>DijkstraVisitor</tt> must be a model of the
   <a href="./DijkstraVisitor.html">Dijkstra Visitor</a> concept.
  The visitor object is passed by value <a
   href="#1">[1]</a>.<br>
   <b>Default:</b> <tt>dijkstra_visitor&lt;null_visitor&gt;</tt><br>

   <b>Python</b>: The parameter should be an object that derives from
   the <a
   href="DijkstraVisitor.html#python"><tt>DijkstraVisitor</tt></a> type
   of the graph.
 </blockquote>


 <H3>Complexity</H3>

 <P>
 The time complexity is <i>O(V + E)</i>.

 <h3>Visitor Event Points</h3>

 <ul>
 <li><b><tt>vis.initialize_vertex(u, g)</tt></b>
   is invoked on each vertex in the graph before the start of the
   algorithm.
 <li><b><tt>vis.examine_vertex(u, g)</tt></b>
   is invoked on a vertex as it is added to set <i>S</i>.
   At this point we know that <i>(p[u],u)</i>
   is a shortest-paths tree edge so
   <i>d[u] = delta(s,u) = d[p[u]] + w(p[u],u)</i>. Also, the distances
   of the examined vertices is monotonically increasing
   <i>d[u<sub>1</sub>] <= d[u<sub>2</sub>] <= d[u<sub>n</sub>]</i>.
 <li><b><tt>vis.examine_edge(e, g)</tt></b>
   is invoked on each out-edge of a vertex immediately after it has
   been added to set <i>S</i>.
 <li><b><tt>vis.edge_relaxed(e, g)</tt></b>
   is invoked on edge <i>(u,v)</i> if <i>d[u] + w(u,v) < d[v]</i>.
   The edge <i>(u,v)</i> that participated in the last
   relaxation for vertex <i>v</i> is an edge in the shortest paths tree.
 <li><b><tt>vis.discover_vertex(v, g)</tt></b>
   is invoked on vertex <i>v</i> when the edge
   <i>(u,v)</i> is examined and <i>v</i> is WHITE. Since
   a vertex is colored GRAY when it is discovered,
   each reacable vertex is discovered exactly once.
 <li><b><tt>vis.edge_not_relaxed(e, g)</tt></b>
   is invoked if the edge is not relaxed (see above).
 <li><b><tt>vis.finish_vertex(u, g)</tt></b>
    is invoked on a vertex after all of its out edges have
   been examined.
 </ul>

 <H3>Example</H3>

 <P>
 See <a href="../example/dag_shortest_paths.cpp">
 <TT>example/dag_shortest_paths.cpp</TT></a> for an example of using this
 algorithm.

 <H3>Notes</H3>

 <p><a name="1">[1]</a>
   Since the visitor parameter is passed by value, if your visitor
   contains state then any changes to the state during the algorithm
   will be made to a copy of the visitor object, not the visitor object
   passed in. Therefore you may want the visitor to hold this state by
   pointer or reference.

 <br>
 <HR>
 <TABLE>
 <TR valign=top>
 <TD nowrap>Copyright &copy; 2000-2001</TD><TD>
 <A HREF="http://www.boost.org/people/jeremy_siek.htm">Jeremy Siek</A>, Indiana University (<A HREF="mailto:jsiek@osl.iu.edu">jsiek@osl.iu.edu</A>)
 </TD></TR></TABLE>

 </BODY>
 </HTML>
	<HTML>
	<!--
	Copyright (c) Jeremy Siek 2000

	Distributed under 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)
	-->
	<Head>
	<Title>Boost Graph Library: Directed Acyclic Graph Shortest Paths</Title>
	<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
	ALINK="#ff0000">
	<IMG SRC="../../../boost.png"
	ALT="C++ Boost" width="277" height="86">

	<BR Clear>

	<H1><A NAME="sec:dag_shortest_paths"></A>
	<img src="figs/python.gif" alt="(Python)"/>
	<TT>dag_shortest_paths</TT>
	</H1>


	<P>
	<PRE>
	<i>// named paramter version</i>
	template <class VertexListGraph, class Param, class Tag, class Rest>
	void dag_shortest_paths(const VertexListGraph& g,
	typename graph_traits<VertexListGraph>::vertex_descriptor s,
	const bgl_named_params<Param,Tag,Rest>& params)

	<i>// non-named parameter version</i>
	template <class VertexListGraph, class DijkstraVisitor,
	class DistanceMap, class WeightMap, class ColorMap,
	class PredecessorMap,
	class Compare, class Combine,
	class DistInf, class DistZero>
	void dag_shortest_paths(const VertexListGraph& g,
	typename graph_traits<VertexListGraph>::vertex_descriptor s,
	DistanceMap distance, WeightMap weight, ColorMap color,
	PredecessorMap pred, DijkstraVisitor vis,
	Compare compare, Combine combine, DistInf inf, DistZero zero)
	</PRE>

	<P>
	This algorithm [<A HREF="bibliography.html#clr90">8</A>] solves
	the single-source shortest-paths problem on a weighted, directed
	acyclic graph (DAG). This algorithm is more efficient for DAG's
	than either the Dijkstra or Bellman-Ford algorithm.
	Use breadth-first search instead of this algorithm
	when all edge weights are equal to one. For the definition of the
	shortest-path problem see Section <A
	HREF="graph_theory_review.html#sec:shortest-paths-algorithms">Shortest-Paths
	Algorithms</A> for some background to the shortest-path problem.
	</P>

	<P>
	There are two main options for obtaining output from the
	<tt>dag_shortest_paths()</tt> function. If you provide a
	distance property map through the <tt>distance_map()</tt> parameter
	then the shortest distance from the source vertex to every other
	vertex in the graph will be recorded in the distance map. Also you can
	record the shortest paths tree in a predecessor map: for each vertex
	<i>u in V</i>, <i>p[u]</i> will be the predecessor of <i>u</i> in
	the shortest paths tree (unless <i>p[u] = u</i>, in which case <i>u</i> is
	either the source or a vertex unreachable from the source). In
	addition to these two options, the user can provide there own
	custom-made visitor that can takes actions during any of the
	algorithm's event points.</P>

	<h3>Where Defined</h3>

	<a href="../../../boost/graph/dag_shortest_paths.hpp"><tt>boost/graph/dag_shortest_paths.hpp</tt></a>

	<h3>Parameters</h3>

	IN: <tt>const VertexListGraph& g</tt>
	<blockquote>
	The graph object on which the algorithm will be applied.
	The type <tt>VertexListGraph</tt> must be a model of \concept{VertexListGraph}.<br>

	<b>Python</b>: The parameter is named <tt>graph</tt>.
	</blockquote>

	IN: <tt>vertex_descriptor s</tt>
	<blockquote>
	The source vertex. All distance will be calculated from this vertex,
	and the shortest paths tree will be rooted at this vertex.<br>

	<b>Python</b>: The parameter is named <tt>root_vertex</tt>.
	</blockquote>

	<h3>Named Parameters</h3>

	IN: <tt>weight_map(WeightMap w_map)</tt>
	<blockquote>
	The weight or ``length'' of each edge in the graph.
	The type <tt>WeightMap</tt> must be a model of
	<a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The edge descriptor type of
	the graph needs to be usable as the key type for the weight
	map. The value type for the map must be
	<i>Addable</i> with the value type of the distance map.<br>
	<b>Default:</b> <tt>get(edge_weight, g)</tt><br>
	<b>Python</b>: Must be an <tt>edge_double_map</tt> for the graph.<br>
	<b>Python default</b>: <tt>graph.get_edge_double_map("weight")</tt>

	</blockquote>

	IN: <tt>vertex_index_map(VertexIndexMap i_map)</tt>
	<blockquote>
	This maps each vertex to an integer in the range <tt>[0,
	num_vertices(g))</tt>. This is necessary for efficient updates of the
	heap data structure when an edge is relaxed. The type
	<tt>VertexIndexMap</tt> must be a model of
	<a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The value type of the map must be an
	integer type. The vertex descriptor type of the graph needs to be
	usable as the key type of the map.<br>
	<b>Default:</b> <tt>get(vertex_index, g)</tt>.
	Note: if you use this default, make sure your graph has
	an internal <tt>vertex_index</tt> property. For example,
	<tt>adjacenty_list</tt> with <tt>VertexList=listS</tt> does
	not have an internal <tt>vertex_index</tt> property.<br>

	<b>Python</b>: Unsupported parameter.
	</blockquote>

	OUT: <tt>predecessor_map(PredecessorMap p_map)</tt>
	<blockquote>
	The predecessor map records the edges in the minimum spanning
	tree. Upon completion of the algorithm, the edges <i>(p[u],u)</i>
	for all <i>u in V</i> are in the minimum spanning tree. If <i>p[u] =
	u</i> then <i>u</i> is either the source vertex or a vertex that is
	not reachable from the source. The <tt>PredecessorMap</tt> type
	must be a <a
	href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
	Property Map</a> which key and vertex types the same as the vertex
	descriptor type of the graph.<br>
	<b>Default:</b> <tt>dummy_property_map</tt><br>
	<b>Python</b>: Must be a <tt>vertex_vertex_map</tt> for the graph.<br>
	</blockquote>

	UTIL/OUT: <tt>distance_map(DistanceMap d_map)</tt>
	<blockquote>
	The shortest path weight from the source vertex <tt>s</tt> to each
	vertex in the graph <tt>g</tt> is recorded in this property map. The
	shortest path weight is the sum of the edge weights along the
	shortest path. The type <tt>DistanceMap</tt> must be a model of <a
	href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
	Property Map</a>. The vertex descriptor type of the graph needs to
	be usable as the key type of the distance map.

	The value type of the distance map is the element type of a <a
	href="./Monoid.html">Monoid</tt> formed with the <tt>combine</tt>
	function object and the <tt>zero</tt> object for the identity
	element. Also the distance value type must have a <a
	href="http://www.sgi.com/tech/stl/StrictWeakOrdering.html">
	StrictWeakOrdering</a> provided by the <tt>compare</tt> function
	object.<br>
	<b>Default:</b> <a
	href="../../property_map/doc/iterator_property_map.html">
	<tt>iterator_property_map</tt></a> created from a
	<tt>std::vector</tt> of the <tt>WeightMap</tt>'s value type of size
	<tt>num_vertices(g)</tt> and using the <tt>i_map</tt> for the index
	map.<br>

	<b>Python</b>: Must be a <tt>vertex_double_map</tt> for the graph.
	</blockquote>

	IN: <tt>distance_compare(CompareFunction cmp)</tt>
	<blockquote>
	This function is use to compare distances to determine which vertex
	is closer to the source vertex. The <tt>CompareFunction</tt> type
	must be a model of <a
	href="http://www.sgi.com/tech/stl/BinaryPredicate.html">Binary
	Predicate</a> and have argument types that match the value type of
	the <tt>DistanceMap</tt> property map.<br>

	<b>Default:</b>
	<tt>std::less<D></tt> with <tt>D=typename
	property_traits<DistanceMap>::value_type</tt><br>

	<b>Python</b>: Unsupported parameter.
	</blockquote>

	IN: <tt>distance_combine(CombineFunction cmb)</tt>
	<blockquote>
	This function is used to combine distances to compute the distance
	of a path. The <tt>CombineFunction</tt> type must be a model of <a
	href="http://www.sgi.com/tech/stl/BinaryFunction.html">Binary
	Function</a>. The first argument type of the binary function must
	match the value type of the <tt>DistanceMap</tt> property map and
	the second argument type must match the value type of the
	<tt>WeightMap</tt> property map. The result type must be the same
	type as the distance value type.<br>

	<b>Default:</b> <tt>std::plus<D></tt> with
	<tt>D=typename property_traits<DistanceMap>::value_type</tt><br>

	<b>Python</b>: Unsupported parameter.
	</blockquote>

	IN: <tt>distance_inf(D inf)</tt>
	<blockquote>
	The <tt>inf</tt> object must be the greatest value of any <tt>D</tt> object.
	That is, <tt>compare(d, inf) == true</tt> for any <tt>d != inf</tt>.
	The type <tt>D</tt> is the value type of the <tt>DistanceMap</tt>.<br>
	<b>Default:</b> <tt>std::numeric_limits<D>::max()</tt><br>

	<b>Python</b>: Unsupported parameter.
	</blockquote>

	IN: <tt>distance_zero(D zero)</tt>
	<blockquote>
	The <tt>zero</tt> value must be the identity element for the
	<a href="./Monoid.html">Monoid</a> formed by the distance values
	and the <tt>combine</tt> function object.
	The type \code{D} is the value type of the \code{DistanceMap}
	<b>Default:</b> <tt>D()</tt><br>

	<b>Python</b>: Unsupported parameter.
	</blockquote>

	UTIL/OUT: <tt>color_map(ColorMap c_map)</tt>
	<blockquote>
	This is used during the execution of the algorithm to mark the
	vertices. The vertices start out white and become gray when they are
	inserted in the queue. They then turn black when they are removed
	from the queue. At the end of the algorithm, vertices reachable from
	the source vertex will have been colored black. All other vertices
	will still be white. The type <tt>ColorMap</tt> must be a model of
	<a href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
	Property Map</a>. A vertex descriptor must be usable as the key type
	of the map, and the value type of the map must be a model of
	<a href="./ColorValue.html">Color Value</a>.<br>
	<b>Default:</b> an <a
	href="../../property_map/doc/iterator_property_map.html">
	<tt>iterator_property_map</tt></a> created from a <tt>std::vector</tt>
	of <tt>default_color_type</tt> of size <tt>num_vertices(g)</tt> and
	using the <tt>i_map</tt> for the index map.<br>

	<b>Python</b>: The color map must be a <tt>vertex_color_map</tt> for
	the graph.

	</blockquote>

	OUT: <tt>visitor(DijkstraVisitor v)</tt>
	<blockquote>
	Use this to specify actions that you would like to happen
	during certain event points within the algorithm.
	The type <tt>DijkstraVisitor</tt> must be a model of the
	<a href="./DijkstraVisitor.html">Dijkstra Visitor</a> concept.
	The visitor object is passed by value <a
	href="#1">[1]</a>.<br>
	<b>Default:</b> <tt>dijkstra_visitor<null_visitor></tt><br>

	<b>Python</b>: The parameter should be an object that derives from
	the <a
	href="DijkstraVisitor.html#python"><tt>DijkstraVisitor</tt></a> type
	of the graph.
	</blockquote>


	<H3>Complexity</H3>

	<P>
	The time complexity is <i>O(V + E)</i>.

	<h3>Visitor Event Points</h3>

	<ul>
	<li><b><tt>vis.initialize_vertex(u, g)</tt></b>
	is invoked on each vertex in the graph before the start of the
	algorithm.
	<li><b><tt>vis.examine_vertex(u, g)</tt></b>
	is invoked on a vertex as it is added to set <i>S</i>.
	At this point we know that <i>(p[u],u)</i>
	is a shortest-paths tree edge so
	<i>d[u] = delta(s,u) = d[p[u]] + w(p[u],u)</i>. Also, the distances
	of the examined vertices is monotonically increasing
	<i>d[u<sub>1</sub>] <= d[u<sub>2</sub>] <= d[u<sub>n</sub>]</i>.
	<li><b><tt>vis.examine_edge(e, g)</tt></b>
	is invoked on each out-edge of a vertex immediately after it has
	been added to set <i>S</i>.
	<li><b><tt>vis.edge_relaxed(e, g)</tt></b>
	is invoked on edge <i>(u,v)</i> if <i>d[u] + w(u,v) < d[v]</i>.
	The edge <i>(u,v)</i> that participated in the last
	relaxation for vertex <i>v</i> is an edge in the shortest paths tree.
	<li><b><tt>vis.discover_vertex(v, g)</tt></b>
	is invoked on vertex <i>v</i> when the edge
	<i>(u,v)</i> is examined and <i>v</i> is WHITE. Since
	a vertex is colored GRAY when it is discovered,
	each reacable vertex is discovered exactly once.
	<li><b><tt>vis.edge_not_relaxed(e, g)</tt></b>
	is invoked if the edge is not relaxed (see above).
	<li><b><tt>vis.finish_vertex(u, g)</tt></b>
	is invoked on a vertex after all of its out edges have
	been examined.
	</ul>

	<H3>Example</H3>

	<P>
	See <a href="../example/dag_shortest_paths.cpp">
	<TT>example/dag_shortest_paths.cpp</TT></a> for an example of using this
	algorithm.

	<H3>Notes</H3>

	<p><a name="1">[1]</a>
	Since the visitor parameter is passed by value, if your visitor
	contains state then any changes to the state during the algorithm
	will be made to a copy of the visitor object, not the visitor object
	passed in. Therefore you may want the visitor to hold this state by
	pointer or reference.

	<br>
	<HR>
	<TABLE>
	<TR valign=top>
	<TD nowrap>Copyright © 2000-2001</TD><TD>
	<A HREF="http://www.boost.org/people/jeremy_siek.htm">Jeremy Siek</A>, Indiana University (<A HREF="mailto:jsiek@osl.iu.edu">jsiek@osl.iu.edu</A>)
	</TD></TR></TABLE>

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