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<Title>Boost Graph Library: Directed Acyclic Graph Shortest Paths</Title>
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<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>

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