[ceph.git] / ceph / src / boost / libs / graph / doc / dijkstra_shortest_paths_no_color_map.html

<HTML>
<!--
     Copyright (c) Michael Hansen 2009
    
     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: Dijkstra's Shortest Paths (No Color Map)</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>
<TT>dijkstra_shortest_paths_no_color_map</TT>
</H1>

<P>
<PRE>
<i>// named parameter version</i>
template &lt;typename Graph, typename Param, typename Tag, typename Rest&gt;
void dijkstra_shortest_paths_no_color_map
  (const Graph&amp; graph,
   typename graph_traits&lt;Graph&gt;::vertex_descriptor start_vertex,
   const bgl_named_params<Param,Tag,Rest>& 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">DistanceCompare</a>, typename <a href="http://www.sgi.com/tech/stl/BinaryFunction.html">DistanceWeightCombine</a>, 
	  typename DistanceInfinity, typename DistanceZero&gt;
void dijkstra_shortest_paths_no_color_map
  (const Graph&amp; graph,
   typename graph_traits&lt;Graph&gt;::vertex_descriptor start_vertex, 
   PredecessorMap predecessor_map, DistanceMap distance_map, WeightMap weight_map, 
   VertexIndexMap index_map,
   DistanceCompare distance_compare, DistanceWeightCombine distance_weight_combine,
   DistanceInfinity distance_infinity, DistanceZero distance_zero);

<i>// version that does not initialize the property maps</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">DistanceCompare</a>, typename <a href="http://www.sgi.com/tech/stl/BinaryFunction.html">DistanceWeightCombine</a>, 
	  typename DistanceInfinity, typename DistanceZero&gt;
void dijkstra_shortest_paths_no_color_map_no_init
  (const Graph&amp; graph,
   typename graph_traits&lt;Graph&gt;::vertex_descriptor start_vertex, 
   PredecessorMap predecessor_map, DistanceMap distance_map, WeightMap weight_map, 
   VertexIndexMap index_map,
   DistanceCompare distance_compare, DistanceWeightCombine distance_weight_combine,
   DistanceInfinity distance_infinity, DistanceZero distance_zero);
</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>
	<tt>dijkstra_shortest_paths_no_color_map</tt> differs from the original <tt>dijkstra_shortest_paths</tt> algorithm by not using a color map to identify vertices as discovered or undiscovered.  Instead, this is done with the distance map: a vertex <i>u</i> such that <i>distance_compare(distance_map[u], distance_infinity) == false</i> is considered to be undiscovered.  Note that this means that edges with infinite weight will not work correctly in this algorithm.
</P>

<P>
There are two main options for obtaining output from the
<tt>dijkstra_shortest_paths_no_color_map()</tt> function. If you provide a
distance property map through the <tt>distance_map()</tt> parameter
then the shortest distance from the start 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 <a href="#4">[4]</a>.</P>

<P>
Dijkstra's algorithm finds all the shortest paths from the source
vertex to every other vertex by iteratively &quot;growing&quot; 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 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>)
  <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>)
    <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>d[v]</i> was originally infinity) 
          INSERT(<i>Q</i>, <i>v</i>)   
        <b>else</b>
          DECREASE-KEY(<i>Q</i>, <i>v</i>)
      <b>else</b>
      	...
    <b>end for</b>
  <b>end while</b>
  return (<i>d</i>, <i>p</i>)
</pre>
</td>
<td valign="top">
<pre>


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_no_color_map.hpp"><tt>boost/graph/dijkstra_shortest_paths_no_color_map.hpp</tt></a>

<h3>Parameters</h3>

IN: <tt>const Graph&amp; graph</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>
</blockquote>

IN: <tt>vertex_descriptor start_vertex</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>
</blockquote>

<h3>Named Parameters</h3>

IN: <tt>weight_map(WeightMap weight_map)</tt>   
<blockquote>
  The weight or ``length'' of each edge in the graph. The weights
  must all be non-negative and non-infinite <a href="#3">[3]</a>. 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, graph)</tt><br>
</blockquote>

IN: <tt>index_map(VertexIndexMap index_map)</tt> 
<blockquote>
  This maps each vertex to an integer in the range <tt>[0,
    num_vertices(graph))</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, graph)</tt>.
    Note: if you use this default, make sure your graph has
    an internal <tt>vertex_index</tt> property. For example,
    <tt>adjacency_list</tt> with <tt>VertexList=listS</tt> does
    not have an internal <tt>vertex_index</tt> property.
   <br>
</blockquote>

OUT: <tt>predecessor_map(PredecessorMap predecessor_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> 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 distance_map)</tt> 
<blockquote>
  The shortest path weight from the source vertex <tt>start_vertex</tt> to each
  vertex in the graph <tt>graph</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>distance_weight_combine</tt>
  function object and the <tt>distance_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>distance_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(graph)</tt> and using the <tt>index_map</tt> for the index
  map.<br>
</blockquote>

IN: <tt>distance_compare(CompareFunction distance_compare)</tt> 
<blockquote>
  This function is use to compare distances to determine which vertex
  is closer to the source vertex.  The <tt>DistanceCompareFunction</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>
</blockquote>

IN: <tt>distance_combine(CombineFunction distance_weight_combine)</tt> 
<blockquote>
  This function is used to combine distances to compute the distance
  of a path. The <tt>DistanceWeightCombineFunction</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>boost::closed_plus&lt;D&gt;</tt> with
   <tt>D=typename property_traits&lt;DistanceMap&gt;::value_type</tt><br>
</blockquote>

IN: <tt>distance_inf(D distance_infinity)</tt> 
<blockquote>
  The <tt>distance_infinity</tt> object must be the greatest value of any <tt>D</tt> object.
  That is, <tt>distance_compare(d, distance_infinity) == true</tt> for any <tt>d != distance_infinity</tt>.
  The type <tt>D</tt> is the value type of the <tt>DistanceMap</tt>.  All edges
  are assumed to have weight less than (by <tt>distance_compare</tt>) this
  value.<br>
  <b>Default:</b> <tt>std::numeric_limits&lt;D&gt;::max()</tt><br>
</blockquote>

IN: <tt>distance_zero(D distance_zero)</tt> 
<blockquote>
  The <tt>distance_zero</tt> value must be the identity element for the
  <a href="./Monoid.html">Monoid</a> formed by the distance values
  and the <tt>distance_weight_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>
</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>
</blockquote>


<H3>Complexity</H3>

<P>
The time complexity is <i>O(V log 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 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> has not yet been discovered (i.e. its distance was infinity before relaxation was attempted on the edge).  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-no-color-map-example.cpp">
<TT>example/dijkstra-no-color-map-example.cpp</TT></a> for an example of using Dijkstra's algorithm.

<H3>See also</H3> <a href="dijkstra_shortest_paths.html">dijkstra_shortest_paths</a> for a version of dijkstra's shortest path that uses a color map.

<H3>Notes</H3>

<p>Based on the documentation for <a href="dijkstra_shortest_paths.html">dijkstra_shortest_paths</a>.

<p><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.
  
<p><a name="3">[3]</a> 
  The algorithm will not work correctly if any of the edge weights are equal to infinity since the infinite distance value is used to determine if a vertex has been discovered.
  
<p><a name="4">[4]</a> 
  Calls to the visitor events occur in the same order as <tt>dijkstra_shortest_paths</tt> (i.e. <i>discover_vertex(u)</i> will always be called after <i>examine_vertex(u)</i> for an undiscovered vertex <i>u</i>).  However, the vertices of the graph given to <i>dijkstra_shortest_paths_no_color_map</i> will <b>not</b> necessarily be visited in the same order as <i>dijkstra_shortest_paths</i>.

<br>
<HR>
<TABLE>
<TR valign=top>
<TD nowrap>Copyright &copy; 2009</TD><TD>
Trustees of Indiana University</TD></TR></TABLE>

</BODY>
</HTML>
Commit	Line	Data
7c673cae FG	1	<HTML>
	2	<!--
	3	Copyright (c) Michael Hansen 2009
	4
	5	Distributed under the Boost Software License, Version 1.0.
	6	(See accompanying file LICENSE_1_0.txt or copy at
	7	http://www.boost.org/LICENSE_1_0.txt)
	8	-->
	9	<Head>
	10	<Title>Boost Graph Library: Dijkstra's Shortest Paths (No Color Map)</Title>
	11	<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
	12	ALINK="#ff0000">
	13	<IMG SRC="../../../boost.png"
	14	ALT="C++ Boost" width="277" height="86">
	15
	16	<BR Clear>
	17
	18	<H1><A NAME="sec:dijkstra"></A>
	19	<TT>dijkstra_shortest_paths_no_color_map</TT>
	20	</H1>
	21
	22	<P>
	23	<PRE>
	24	<i>// named parameter version</i>
	25	template <typename Graph, typename Param, typename Tag, typename Rest>
	26	void dijkstra_shortest_paths_no_color_map
	27	(const Graph& graph,
	28	typename graph_traits<Graph>::vertex_descriptor start_vertex,
	29	const bgl_named_params<Param,Tag,Rest>& params);
	30
	31	<i>// non-named parameter version</i>
	32	template <typename Graph, typename <a href="DijkstraVisitor.html">DijkstraVisitor</a>,
	33	typename PredecessorMap, typename DistanceMap,
	34	typename WeightMap, typename VertexIndexMap, typename <a href="http://www.sgi.com/tech/stl/BinaryPredicate.html">DistanceCompare</a>, typename <a href="http://www.sgi.com/tech/stl/BinaryFunction.html">DistanceWeightCombine</a>,
	35	typename DistanceInfinity, typename DistanceZero>
	36	void dijkstra_shortest_paths_no_color_map
	37	(const Graph& graph,
	38	typename graph_traits<Graph>::vertex_descriptor start_vertex,
	39	PredecessorMap predecessor_map, DistanceMap distance_map, WeightMap weight_map,
	40	VertexIndexMap index_map,
	41	DistanceCompare distance_compare, DistanceWeightCombine distance_weight_combine,
	42	DistanceInfinity distance_infinity, DistanceZero distance_zero);
	43
	44	<i>// version that does not initialize the property maps</i>
	45	template <typename Graph, typename <a href="DijkstraVisitor.html">DijkstraVisitor</a>,
	46	typename PredecessorMap, typename DistanceMap,
	47	typename WeightMap, typename VertexIndexMap, typename <a href="http://www.sgi.com/tech/stl/BinaryPredicate.html">DistanceCompare</a>, typename <a href="http://www.sgi.com/tech/stl/BinaryFunction.html">DistanceWeightCombine</a>,
	48	typename DistanceInfinity, typename DistanceZero>
	49	void dijkstra_shortest_paths_no_color_map_no_init
	50	(const Graph& graph,
	51	typename graph_traits<Graph>::vertex_descriptor start_vertex,
	52	PredecessorMap predecessor_map, DistanceMap distance_map, WeightMap weight_map,
	53	VertexIndexMap index_map,
	54	DistanceCompare distance_compare, DistanceWeightCombine distance_weight_combine,
	55	DistanceInfinity distance_infinity, DistanceZero distance_zero);
	56	</PRE>
	57
	58	<P>
	59	This algorithm [<A HREF="bibliography.html#dijkstra59">10</A>,<A
	60	HREF="bibliography.html#clr90">8</A>] solves the single-source
	61	shortest-paths problem on a weighted, directed or undirected graph for
	62	the case where all edge weights are nonnegative. Use the Bellman-Ford
	63	algorithm for the case when some edge weights are negative. Use
	64	breadth-first search instead of Dijkstra's algorithm when all edge
65	weights are equal to one. For the definition of the shortest-path
66	problem see Section <A
67	HREF="graph_theory_review.html#sec:shortest-paths-algorithms">Shortest-Paths
68	Algorithms</A> for some background to the shortest-path problem.
69	</P>
70
71	<P>
72	<tt>dijkstra_shortest_paths_no_color_map</tt> differs from the original <tt>dijkstra_shortest_paths</tt> algorithm by not using a color map to identify vertices as discovered or undiscovered. Instead, this is done with the distance map: a vertex <i>u</i> such that <i>distance_compare(distance_map[u], distance_infinity) == false</i> is considered to be undiscovered. Note that this means that edges with infinite weight will not work correctly in this algorithm.
73	</P>
74
75	<P>
76	There are two main options for obtaining output from the
77	<tt>dijkstra_shortest_paths_no_color_map()</tt> function. If you provide a
78	distance property map through the <tt>distance_map()</tt> parameter
79	then the shortest distance from the start vertex to every other
80	vertex in the graph will be recorded in the distance map. Also you can
81	record the shortest paths tree in a predecessor map: for each vertex
82	<i>u in V</i>, <i>p[u]</i> will be the predecessor of <i>u</i> in
83	the shortest paths tree (unless <i>p[u] = u</i>, in which case <i>u</i> is
84	either the source or a vertex unreachable from the source). In
85	addition to these two options, the user can provide their own
86	custom-made visitor that takes actions during any of the
87	algorithm's event points <a href="#4">[4]</a>.</P>
88
89	<P>
90	Dijkstra's algorithm finds all the shortest paths from the source
91	vertex to every other vertex by iteratively "growing" the set of
92	vertices <i>S</i> to which it knows the shortest path. At each step of
93	the algorithm, the next vertex added to <i>S</i> is determined by a
94	priority queue. The queue contains the vertices in <i>V - S</i><a
95	href="#1">[1]</a> prioritized by their distance label, which is the
96	length of the shortest path seen so far for each vertex. The vertex
97	<i>u</i> at the top of the priority queue is then added to <i>S</i>,
98	and each of its out-edges is relaxed: if the distance to <i>u</i> plus
99	the weight of the out-edge <i>(u,v)</i> is less than the distance
100	label for <i>v</i> then the estimated distance for vertex <i>v</i> is
101	reduced. The algorithm then loops back, processing the next vertex at
102	the top of the priority queue. The algorithm finishes when the
103	priority queue is empty.
104	</P>
105	<p>
106	The following is the pseudo-code for Dijkstra's single-source shortest
107	paths algorithm. <i>w</i> is the edge weight, <i>d</i> is the distance label,
108	and <i>p</i> is the predecessor of each vertex which is used to encode
109	the shortest paths tree. <i>Q</i> is a priority queue that supports the
110	DECREASE-KEY operation. The visitor event points for the algorithm are
111	indicated by the labels on the right.
112	</p>
113
114	<table>
115	<tr>
116	<td valign="top">
117	<pre>
118	DIJKSTRA(<i>G</i>, <i>s</i>, <i>w</i>)
119	<i>d[s] := 0</i>
120	INSERT(<i>Q</i>, <i>s</i>)
121	<b>while</b> (<i>Q != Ø</i>)
122	<i>u :=</i> EXTRACT-MIN(<i>Q</i>)
123	<b>for</b> each vertex <i>v in Adj[u]</i>
124	<b>if</b> (<i>w(u,v) + d[u] < d[v]</i>)
125	<i>d[v] := w(u,v) + d[u]</i>
126	<i>p[v] := u</i>
127	<b>if</b> (<i>d[v]</i> was originally infinity)
128	INSERT(<i>Q</i>, <i>v</i>)
129	<b>else</b>
130	DECREASE-KEY(<i>Q</i>, <i>v</i>)
131	<b>else</b>
132	...
133	<b>end for</b>
134	<b>end while</b>
135	return (<i>d</i>, <i>p</i>)
136	</pre>
137	</td>
138	<td valign="top">
139	<pre>
140
141
142	discover vertex <i>s</i>
143
144	examine vertex <i>u</i>
145	examine edge <i>(u,v)</i>
146
147	edge <i>(u,v)</i> relaxed
148
149
150	discover vertex <i>v</i>
151
152
153	edge <i>(u,v)</i> not relaxed
154
155	finish vertex <i>u</i>
156	</pre>
157	</td>
158	</tr>
159	</table>
160
161	<h3>Where Defined</h3>
162
163	<a href="../../../boost/graph/dijkstra_shortest_paths_no_color_map.hpp"><tt>boost/graph/dijkstra_shortest_paths_no_color_map.hpp</tt></a>
164
165	<h3>Parameters</h3>
166
167	IN: <tt>const Graph& graph</tt>
168	<blockquote>
169	The graph object on which the algorithm will be applied.
170	The type <tt>Graph</tt> must be a model of
171	<a href="./VertexListGraph.html">Vertex List Graph</a>
172	and <a href="./IncidenceGraph.html">Incidence Graph</a>.<br>
173	</blockquote>
174
175	IN: <tt>vertex_descriptor start_vertex</tt>
176	<blockquote>
177	The source vertex. All distance will be calculated from this vertex,
178	and the shortest paths tree will be rooted at this vertex.<br>
179	</blockquote>
180
181	<h3>Named Parameters</h3>
182
183	IN: <tt>weight_map(WeightMap weight_map)</tt>
184	<blockquote>
185	The weight or ``length'' of each edge in the graph. The weights
186	must all be non-negative and non-infinite <a href="#3">[3]</a>. The algorithm will throw a
187	<a href="./exception.html#negative_edge"><tt>negative_edge</tt></a>
188	exception is one of the edges is negative.
189	The type <tt>WeightMap</tt> must be a model of
190	<a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The edge descriptor type of
191	the graph needs to be usable as the key type for the weight
192	map. The value type for this map must be
193	the same as the value type of the distance map.<br>
194	<b>Default:</b> <tt>get(edge_weight, graph)</tt><br>
195	</blockquote>
196
197	IN: <tt>index_map(VertexIndexMap index_map)</tt>
198	<blockquote>
199	This maps each vertex to an integer in the range <tt>[0,
200	num_vertices(graph))</tt>. This is necessary for efficient updates of the
201	heap data structure [<A
202	HREF="bibliography.html#driscoll88">61</A>] when an edge is relaxed.
203	The type
204	<tt>VertexIndexMap</tt> must be a model of
205	<a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The value type of the map must be an
206	integer type. The vertex descriptor type of the graph needs to be
207	usable as the key type of the map.<br>
208	<b>Default:</b> <tt>get(vertex_index, graph)</tt>.
209	Note: if you use this default, make sure your graph has
210	an internal <tt>vertex_index</tt> property. For example,
211	<tt>adjacency_list</tt> with <tt>VertexList=listS</tt> does
212	not have an internal <tt>vertex_index</tt> property.
213	<br>
214	</blockquote>
215
216	OUT: <tt>predecessor_map(PredecessorMap predecessor_map)</tt>
217	<blockquote>
218	The predecessor map records the edges in the minimum spanning
219	tree. Upon completion of the algorithm, the edges <i>(p[u],u)</i>
220	for all <i>u in V</i> are in the minimum spanning tree. If <i>p[u] =
221	u</i> then <i>u</i> is either the source vertex or a vertex that is
222	not reachable from the source. The <tt>PredecessorMap</tt> type
223	must be a <a
224	href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
225	Property Map</a> whose key and value types are the same as the vertex
226	descriptor type of the graph.<br>
227	<b>Default:</b> <tt>dummy_property_map</tt><br>
228
229	<b>Python</b>: Must be a <tt>vertex_vertex_map</tt> for the graph.<br>
230	</blockquote>
231
232	UTIL/OUT: <tt>distance_map(DistanceMap distance_map)</tt>
233	<blockquote>
234	The shortest path weight from the source vertex <tt>start_vertex</tt> to each
235	vertex in the graph <tt>graph</tt> is recorded in this property map. The
236	shortest path weight is the sum of the edge weights along the
237	shortest path. The type <tt>DistanceMap</tt> must be a model of <a
238	href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
239	Property Map</a>. The vertex descriptor type of the graph needs to
240	be usable as the key type of the distance map.
241
242	The value type of the distance map is the element type of a <a
243	href="./Monoid.html">Monoid</a> formed with the <tt>distance_weight_combine</tt>
244	function object and the <tt>distance_zero</tt> object for the identity
245	element. Also the distance value type must have a <a
246	href="http://www.sgi.com/tech/stl/StrictWeakOrdering.html">
247	StrictWeakOrdering</a> provided by the <tt>distance_compare</tt> function
248	object.<br>
249	<b>Default:</b> <a
250	href="../../property_map/doc/iterator_property_map.html">
251	<tt>iterator_property_map</tt></a> created from a
252	<tt>std::vector</tt> of the <tt>WeightMap</tt>'s value type of size
253	<tt>num_vertices(graph)</tt> and using the <tt>index_map</tt> for the index
254	map.<br>
255	</blockquote>
256
257	IN: <tt>distance_compare(CompareFunction distance_compare)</tt>
258	<blockquote>
259	This function is use to compare distances to determine which vertex
260	is closer to the source vertex. The <tt>DistanceCompareFunction</tt> type
261	must be a model of <a
262	href="http://www.sgi.com/tech/stl/BinaryPredicate.html">Binary
263	Predicate</a> and have argument types that match the value type of
264	the <tt>DistanceMap</tt> property map.<br>
265
266	<b>Default:</b>
267	<tt>std::less<D></tt> with <tt>D=typename
268	property_traits<DistanceMap>::value_type</tt><br>
269	</blockquote>
270
271	IN: <tt>distance_combine(CombineFunction distance_weight_combine)</tt>
272	<blockquote>
273	This function is used to combine distances to compute the distance
274	of a path. The <tt>DistanceWeightCombineFunction</tt> type must be a model of <a
275	href="http://www.sgi.com/tech/stl/BinaryFunction.html">Binary
276	Function</a>. The first argument type of the binary function must
277	match the value type of the <tt>DistanceMap</tt> property map and
278	the second argument type must match the value type of the
279	<tt>WeightMap</tt> property map. The result type must be the same
280	type as the distance value type.<br>
281
282	<b>Default:</b> <tt>boost::closed_plus<D></tt> with
283	<tt>D=typename property_traits<DistanceMap>::value_type</tt><br>
284	</blockquote>
285
286	IN: <tt>distance_inf(D distance_infinity)</tt>
287	<blockquote>
288	The <tt>distance_infinity</tt> object must be the greatest value of any <tt>D</tt> object.
289	That is, <tt>distance_compare(d, distance_infinity) == true</tt> for any <tt>d != distance_infinity</tt>.
290	The type <tt>D</tt> is the value type of the <tt>DistanceMap</tt>. All edges
291	are assumed to have weight less than (by <tt>distance_compare</tt>) this
292	value.<br>
293	<b>Default:</b> <tt>std::numeric_limits<D>::max()</tt><br>
294	</blockquote>
295
296	IN: <tt>distance_zero(D distance_zero)</tt>
297	<blockquote>
298	The <tt>distance_zero</tt> value must be the identity element for the
299	<a href="./Monoid.html">Monoid</a> formed by the distance values
300	and the <tt>distance_weight_combine</tt> function object.
301	The type <tt>D</tt> is the value type of the <tt>DistanceMap</tt>.<br>
302	<b>Default:</b> <tt>D()</tt>with
303	<tt>D=typename property_traits<DistanceMap>::value_type</tt><br>
304	</blockquote>
305
306	OUT: <tt>visitor(DijkstraVisitor v)</tt>
307	<blockquote>
308	Use this to specify actions that you would like to happen
309	during certain event points within the algorithm.
310	The type <tt>DijkstraVisitor</tt> must be a model of the
311	<a href="./DijkstraVisitor.html">Dijkstra Visitor</a> concept.
312	The visitor object is passed by value <a
313	href="#2">[2]</a>.<br>
314	<b>Default:</b> <tt>dijkstra_visitor<null_visitor></tt><br>
315	</blockquote>
316
317
318	<H3>Complexity</H3>
319
320	<P>
321	The time complexity is <i>O(V log V + E)</i>.
322
323
324	<h3>Visitor Event Points</h3>
325
326	<ul>
327	<li><b><tt>vis.initialize_vertex(u, g)</tt></b>
328	is invoked on each vertex in the graph before the start of the
329	algorithm.
330	<li><b><tt>vis.examine_vertex(u, g)</tt></b>
331	is invoked on a vertex as it is removed from the priority queue
332	and added to set <i>S</i>. At this point we know that <i>(p[u],u)</i>
333	is a shortest-paths tree edge so
334	<i>d[u] = delta(s,u) = d[p[u]] + w(p[u],u)</i>. Also, the distances
335	of the examined vertices is monotonically increasing
336	<i>d[u<sub>1</sub>] <= d[u<sub>2</sub>] <= d[u<sub>n</sub>]</i>.
337	<li><b><tt>vis.examine_edge(e, g)</tt></b>
338	is invoked on each out-edge of a vertex immediately after it has
339	been added to set <i>S</i>.
340	<li><b><tt>vis.edge_relaxed(e, g)</tt></b>
341	is invoked on edge <i>(u,v)</i> if <i>d[u] + w(u,v) < d[v]</i>.
342	The edge <i>(u,v)</i> that participated in the last
343	relaxation for vertex <i>v</i> is an edge in the shortest paths tree.
344	<li><b><tt>vis.discover_vertex(v, g)</tt></b>
345	is invoked on vertex <i>v</i> when the edge
346	<i>(u,v)</i> is examined and <i>v</i> has not yet been discovered (i.e. its distance was infinity before relaxation was attempted on the edge). This
347	is also when the vertex is inserted into the priority queue.
348	<li><b><tt>vis.edge_not_relaxed(e, g)</tt></b>
349	is invoked if the edge is not relaxed (see above).
350	<li><b><tt>vis.finish_vertex(u, g)</tt></b>
351	is invoked on a vertex after all of its out edges have
352	been examined.
353	</ul>
354
355	<H3>Example</H3>
356
357	<P>
358	See <a href="../example/dijkstra-no-color-map-example.cpp">
359	<TT>example/dijkstra-no-color-map-example.cpp</TT></a> for an example of using Dijkstra's algorithm.
360
361	<H3>See also</H3> <a href="dijkstra_shortest_paths.html">dijkstra_shortest_paths</a> for a version of dijkstra's shortest path that uses a color map.
362
363	<H3>Notes</H3>
364
365	<p>Based on the documentation for <a href="dijkstra_shortest_paths.html">dijkstra_shortest_paths</a>.
366
367	<p><a name="1">[1]</a>
368	The algorithm used here saves a little space by not putting all <i>V -
369	S</i> vertices in the priority queue at once, but instead only those
370	vertices in <i>V - S</i> that are discovered and therefore have a
371	distance less than infinity.
372
373	<p><a name="2">[2]</a>
374	Since the visitor parameter is passed by value, if your visitor
375	contains state then any changes to the state during the algorithm
376	will be made to a copy of the visitor object, not the visitor object
377	passed in. Therefore you may want the visitor to hold this state by
378	pointer or reference.
379
380	<p><a name="3">[3]</a>
381	The algorithm will not work correctly if any of the edge weights are equal to infinity since the infinite distance value is used to determine if a vertex has been discovered.
382
383	<p><a name="4">[4]</a>
384	Calls to the visitor events occur in the same order as <tt>dijkstra_shortest_paths</tt> (i.e. <i>discover_vertex(u)</i> will always be called after <i>examine_vertex(u)</i> for an undiscovered vertex <i>u</i>). However, the vertices of the graph given to <i>dijkstra_shortest_paths_no_color_map</i> will <b>not</b> necessarily be visited in the same order as <i>dijkstra_shortest_paths</i>.
385
386	<br>
387	<HR>
388	<TABLE>
389	<TR valign=top>
390	<TD nowrap>Copyright © 2009</TD><TD>
391	Trustees of Indiana University</TD></TR></TABLE>
392
393	</BODY>
394	</HTML>