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 <Head>
 <Title>Boost Graph Library: Dijkstra's Shortest Paths</Title>
 <BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
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 <BR Clear>

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

 <P>
 <PRE>
 <i>// named parameter version</i>
 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);

 <i>// non-named parameter version</i>
 template &lt;typename Graph, typename <a href="DijkstraVisitor.html">DijkstraVisitor</a>,
 	  typename PredecessorMap, typename DistanceMap,
 	  typename WeightMap, typename VertexIndexMap, typename <a href="http://www.sgi.com/tech/stl/BinaryPredicate.html">CompareFunction</a>, typename <a href="http://www.sgi.com/tech/stl/BinaryFunction.html">CombineFunction</a>,
 	  typename DistInf, typename DistZero, typename ColorMap = <i>default</i>&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, ColorMap color = <i>default</i>)

 <i>// version that does not initialize the property maps (except for the default color map)</i>
 template &lt;class Graph, class DijkstraVisitor,
           class PredecessorMap, class DistanceMap,
           class WeightMap, class IndexMap, class Compare, class Combine,
           class DistZero, class ColorMap&gt;
 void
 dijkstra_shortest_paths_no_init
   (const Graph&amp; g,
    typename graph_traits&lt;Graph&gt;::vertex_descriptor s,
    PredecessorMap predecessor, DistanceMap distance, WeightMap weight,
    IndexMap index_map,
    Compare compare, Combine combine, DistZero zero,
    DijkstraVisitor vis, ColorMap color = <i>default</i>);
 </PRE>

 <P>
 This algorithm&nbsp;[<A HREF="bibliography.html#dijkstra59">10</A>,<A
 HREF="bibliography.html#clr90">8</A>] solves the single-source
 shortest-paths problem on a weighted, directed or undirected graph for
 the case where all edge weights are nonnegative.  Use the Bellman-Ford
 algorithm for the case when some edge weights are negative.  Use
 breadth-first search instead of Dijkstra's 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>dijkstra_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 their own
 custom-made visitor that takes actions during any of the
 algorithm's event points.</P>

 <P>
 Dijkstra's algorithm finds all the shortest paths from the source
 vertex to every other vertex by iteratively ``growing'' the set of
 vertices <i>S</i> to which it knows the shortest path. At each step of
 the algorithm, the next vertex added to <i>S</i> is determined by a
 priority queue.  The queue contains the vertices in <i>V - S</i><a
 href="#1">[1]</a> prioritized by their distance label, which is the
 length of the shortest path seen so far for each vertex. The vertex
 <i>u</i> at the top of the priority queue is then added to <i>S</i>,
 and each of its out-edges is relaxed: if the distance to <i>u</i> plus
 the weight of the out-edge <i>(u,v)</i> is less than the distance
 label for <i>v</i> then the estimated distance for vertex <i>v</i> is
 reduced.  The algorithm then loops back, processing the next vertex at
 the top of the priority queue. The algorithm finishes when the
 priority queue is empty.
 </P>
 <P>
 The algorithm uses color markers (white, gray, and black) to keep
 track of which set each vertex is in. Vertices colored black are in
 <i>S</i>. Vertices colored white or gray are in <i>V-S</i>. White vertices have
 not yet been discovered and gray vertices are in the priority queue.
 By default, the algorithm will allocate an array to store a color
 marker for each vertex in the graph. You can provide your own storage
 and access for colors with the <tt>color_map()</tt> parameter.
 </P>
 <p>
 The following is the pseudo-code for Dijkstra's single-source shortest
 paths algorithm. <i>w</i> is the edge weight, <i>d</i> is the distance label,
 and <i>p</i> is the predecessor of each vertex which is used to encode
 the shortest paths tree. <i>Q</i> is a priority queue that supports the
 DECREASE-KEY operation.  The visitor event points for the algorithm are
 indicated by the labels on the right.
 </p>

 <table>
 <tr>
 <td valign="top">
 <pre>
 DIJKSTRA(<i>G</i>, <i>s</i>, <i>w</i>)
   <b>for</b> each vertex <i>u in V</i> <b>(This loop is not run in dijkstra_shortest_paths_no_init)</b>
     <i>d[u] := infinity</i>
     <i>p[u] := u</i>
     <i>color[u] :=</i> WHITE
   <b>end for</b>
   <i>color[s] := </i>GRAY
   <i>d[s] := 0</i>
   INSERT(<i>Q</i>, <i>s</i>)
   <b>while</b> (<i>Q != &Oslash;</i>)
     <i>u :=</i> EXTRACT-MIN(<i>Q</i>)
     <i>S := S U { u }</i>
     <b>for</b> each vertex <i>v in Adj[u]</i>
       <b>if</b> (<i>w(u,v) + d[u] < d[v]</i>)
         <i>d[v] := w(u,v) + d[u]</i>
         <i>p[v] := u</i>
         <b>if</b> (<i>color[v] =</i> WHITE)
           <i>color[v] :=</i> GRAY
           INSERT(<i>Q</i>, <i>v</i>)
         <b>else if</b> (<i>color[v] =</i> GRAY)
           DECREASE-KEY(<i>Q</i>, <i>v</i>)
       <b>else</b>
         <i>...</i>
     <b>end for</b>
     <i>color[u] :=</i> BLACK
   <b>end while</b>
   return (<i>d</i>, <i>p</i>)
 </pre>
 </td>
 <td valign="top">
 <pre>

 initialize vertex <i>u</i>


 discover vertex <i>s</i>

 examine vertex <i>u</i>

 examine edge <i>(u,v)</i>

 edge <i>(u,v)</i> relaxed


 discover vertex <i>v</i>


 edge <i>(u,v)</i> not relaxed

 finish vertex <i>u</i>
 </pre>
 </td>
 </tr>
 </table>

 <h3>Where Defined</h3>

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

 <h3>Parameters</h3>

 IN: <tt>const Graph&amp; g</tt>
 <blockquote>
   The graph object on which the algorithm will be applied.
   The type <tt>Graph</tt> must be a model of
   <a href="./VertexListGraph.html">Vertex List Graph</a>
   and <a href="./IncidenceGraph.html">Incidence Graph</a>.<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 weights
   must all be non-negative, and the algorithm will throw a
   <a href="./exception.html#negative_edge"><tt>negative_edge</tt></a>
   exception is one of the edges is negative.
   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 this map must be
   the same as 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&nbsp;[<A
   HREF="bibliography.html#driscoll88">61</A>] 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 shortest path tree, the tree computed
   by the traversal of the graph. Upon completion of the algorithm, the edges
   <i>(p[u],u)</i> for all <i>u in V</i> are in the tree.  The shortest path
   from vertex <i>s</i> to each vertex <i>v</i> in the graph consists of the
   vertices <i>v</i>, <i>p[v]</i>, <i>p[p[v]]</i>, and so on until <i>s</i> is
   reached, in reverse order.  The
   tree is not guaranteed to be a 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> whose key and value types are 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</a> 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.<br>
 </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>closed_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 <tt>D</tt> is the value type of the <tt>DistanceMap</tt>.<br>
   <b>Default:</b> <tt>D()</tt>with
    <tt>D=typename property_traits&lt;DistanceMap&gt;::value_type</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="#2">[2]</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 log V)</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 removed from the priority queue
   and 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. This
   is also when the vertex is inserted into the priority queue.
 <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/dijkstra-example.cpp">
 <TT>example/dijkstra-example.cpp</TT></a> for an example of using Dijkstra's
 algorithm.

 <H3>See also</H3> <a href="dijkstra_shortest_paths_no_color_map.html">dijkstra_shortest_paths_no_color_map</a> for a version of dijkstra's shortest path that does not use a color map.

 <H3>Notes</H3>

 <a name="1">[1]</a>
 The algorithm used here saves a little space by not putting all <i>V -
 S</i> vertices in the priority queue at once, but instead only those
 vertices in <i>V - S</i> that are discovered and therefore have a
 distance less than infinity.

 <p><a name="2">[2]</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>
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	<Head>
	<Title>Boost Graph Library: Dijkstra's 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:dijkstra"></A><img src="figs/python.gif" alt="(Python)"/>
	<TT>dijkstra_shortest_paths</TT>
	</H1>

	<P>
	<PRE>
	<i>// named parameter version</i>
	template <typename Graph, typename P, typename T, typename R>
	void
	dijkstra_shortest_paths(Graph& g,
	typename graph_traits<Graph>::vertex_descriptor s,
	const bgl_named_params<P, T, R>& params);

	<i>// non-named parameter version</i>
	template <typename Graph, typename <a href="DijkstraVisitor.html">DijkstraVisitor</a>,
	typename PredecessorMap, typename DistanceMap,
	typename WeightMap, typename VertexIndexMap, typename <a href="http://www.sgi.com/tech/stl/BinaryPredicate.html">CompareFunction</a>, typename <a href="http://www.sgi.com/tech/stl/BinaryFunction.html">CombineFunction</a>,
	typename DistInf, typename DistZero, typename ColorMap = <i>default</i>>
	void dijkstra_shortest_paths
	(const Graph& g,
	typename graph_traits<Graph>::vertex_descriptor s,
	PredecessorMap predecessor, DistanceMap distance, WeightMap weight,
	VertexIndexMap index_map,
	CompareFunction compare, CombineFunction combine, DistInf inf, DistZero zero,
	DijkstraVisitor vis, ColorMap color = <i>default</i>)

	<i>// version that does not initialize the property maps (except for the default color map)</i>
	template <class Graph, class DijkstraVisitor,
	class PredecessorMap, class DistanceMap,
	class WeightMap, class IndexMap, class Compare, class Combine,
	class DistZero, class ColorMap>
	void
	dijkstra_shortest_paths_no_init
	(const Graph& g,
	typename graph_traits<Graph>::vertex_descriptor s,
	PredecessorMap predecessor, DistanceMap distance, WeightMap weight,
	IndexMap index_map,
	Compare compare, Combine combine, DistZero zero,
	DijkstraVisitor vis, ColorMap color = <i>default</i>);
	</PRE>

	<P>
	This algorithm [<A HREF="bibliography.html#dijkstra59">10</A>,<A
	HREF="bibliography.html#clr90">8</A>] solves the single-source
	shortest-paths problem on a weighted, directed or undirected graph for
	the case where all edge weights are nonnegative. Use the Bellman-Ford
	algorithm for the case when some edge weights are negative. Use
	breadth-first search instead of Dijkstra's 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>dijkstra_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 their own
	custom-made visitor that takes actions during any of the
	algorithm's event points.</P>

	<P>
	Dijkstra's algorithm finds all the shortest paths from the source
	vertex to every other vertex by iteratively ``growing'' the set of
	vertices <i>S</i> to which it knows the shortest path. At each step of
	the algorithm, the next vertex added to <i>S</i> is determined by a
	priority queue. The queue contains the vertices in <i>V - S</i><a
	href="#1">[1]</a> prioritized by their distance label, which is the
	length of the shortest path seen so far for each vertex. The vertex
	<i>u</i> at the top of the priority queue is then added to <i>S</i>,
	and each of its out-edges is relaxed: if the distance to <i>u</i> plus
	the weight of the out-edge <i>(u,v)</i> is less than the distance
	label for <i>v</i> then the estimated distance for vertex <i>v</i> is
	reduced. The algorithm then loops back, processing the next vertex at
	the top of the priority queue. The algorithm finishes when the
	priority queue is empty.
	</P>
	<P>
	The algorithm uses color markers (white, gray, and black) to keep
	track of which set each vertex is in. Vertices colored black are in
	<i>S</i>. Vertices colored white or gray are in <i>V-S</i>. White vertices have
	not yet been discovered and gray vertices are in the priority queue.
	By default, the algorithm will allocate an array to store a color
	marker for each vertex in the graph. You can provide your own storage
	and access for colors with the <tt>color_map()</tt> parameter.
	</P>
	<p>
	The following is the pseudo-code for Dijkstra's single-source shortest
	paths algorithm. <i>w</i> is the edge weight, <i>d</i> is the distance label,
	and <i>p</i> is the predecessor of each vertex which is used to encode
	the shortest paths tree. <i>Q</i> is a priority queue that supports the
	DECREASE-KEY operation. The visitor event points for the algorithm are
	indicated by the labels on the right.
	</p>

	<table>
	<tr>
	<td valign="top">
	<pre>
	DIJKSTRA(<i>G</i>, <i>s</i>, <i>w</i>)
	<b>for</b> each vertex <i>u in V</i> <b>(This loop is not run in dijkstra_shortest_paths_no_init)</b>
	<i>d[u] := infinity</i>
	<i>p[u] := u</i>
	<i>color[u] :=</i> WHITE
	<b>end for</b>
	<i>color[s] := </i>GRAY
	<i>d[s] := 0</i>
	INSERT(<i>Q</i>, <i>s</i>)
	<b>while</b> (<i>Q != Ø</i>)
	<i>u :=</i> EXTRACT-MIN(<i>Q</i>)
	<i>S := S U { u }</i>
	<b>for</b> each vertex <i>v in Adj[u]</i>
	<b>if</b> (<i>w(u,v) + d[u] < d[v]</i>)
	<i>d[v] := w(u,v) + d[u]</i>
	<i>p[v] := u</i>
	<b>if</b> (<i>color[v] =</i> WHITE)
	<i>color[v] :=</i> GRAY
	INSERT(<i>Q</i>, <i>v</i>)
	<b>else if</b> (<i>color[v] =</i> GRAY)
	DECREASE-KEY(<i>Q</i>, <i>v</i>)
	<b>else</b>
	<i>...</i>
	<b>end for</b>
	<i>color[u] :=</i> BLACK
	<b>end while</b>
	return (<i>d</i>, <i>p</i>)
	</pre>
	</td>
	<td valign="top">
	<pre>

	initialize vertex <i>u</i>






	discover vertex <i>s</i>

	examine vertex <i>u</i>

	examine edge <i>(u,v)</i>

	edge <i>(u,v)</i> relaxed



	discover vertex <i>v</i>



	edge <i>(u,v)</i> not relaxed

	finish vertex <i>u</i>
	</pre>
	</td>
	</tr>
	</table>

	<h3>Where Defined</h3>

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

	<h3>Parameters</h3>

	IN: <tt>const Graph& g</tt>
	<blockquote>
	The graph object on which the algorithm will be applied.
	The type <tt>Graph</tt> must be a model of
	<a href="./VertexListGraph.html">Vertex List Graph</a>
	and <a href="./IncidenceGraph.html">Incidence Graph</a>.<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 weights
	must all be non-negative, and the algorithm will throw a
	<a href="./exception.html#negative_edge"><tt>negative_edge</tt></a>
	exception is one of the edges is negative.
	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 this map must be
	the same as 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 [<A
	HREF="bibliography.html#driscoll88">61</A>] 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 shortest path tree, the tree computed
	by the traversal of the graph. Upon completion of the algorithm, the edges
	<i>(p[u],u)</i> for all <i>u in V</i> are in the tree. The shortest path
	from vertex <i>s</i> to each vertex <i>v</i> in the graph consists of the
	vertices <i>v</i>, <i>p[v]</i>, <i>p[p[v]]</i>, and so on until <i>s</i> is
	reached, in reverse order. The
	tree is not guaranteed to be a 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> whose key and value types are 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</a> 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.<br>
	</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>closed_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 <tt>D</tt> is the value type of the <tt>DistanceMap</tt>.<br>
	<b>Default:</b> <tt>D()</tt>with
	<tt>D=typename property_traits<DistanceMap>::value_type</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="#2">[2]</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 log V)</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 removed from the priority queue
	and 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. This
	is also when the vertex is inserted into the priority queue.
	<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/dijkstra-example.cpp">
	<TT>example/dijkstra-example.cpp</TT></a> for an example of using Dijkstra's
	algorithm.

	<H3>See also</H3> <a href="dijkstra_shortest_paths_no_color_map.html">dijkstra_shortest_paths_no_color_map</a> for a version of dijkstra's shortest path that does not use a color map.

	<H3>Notes</H3>

	<a name="1">[1]</a>
	The algorithm used here saves a little space by not putting all <i>V -
	S</i> vertices in the priority queue at once, but instead only those
	vertices in <i>V - S</i> that are discovered and therefore have a
	distance less than infinity.

	<p><a name="2">[2]</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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