327 lines
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327 lines
13 KiB
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<!--
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Copyright (c) Jeremy Siek 2000
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt)
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-->
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<Head>
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<Title>Boost Graph Library: Directed Acyclic Graph Shortest Paths</Title>
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<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
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ALINK="#ff0000">
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<IMG SRC="../../../boost.png"
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ALT="C++ Boost" width="277" height="86">
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<BR Clear>
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<H1><A NAME="sec:dag_shortest_paths"></A>
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<img src="figs/python.gif" alt="(Python)"/>
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<TT>dag_shortest_paths</TT>
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</H1>
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<P>
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<PRE>
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<i>// named paramter version</i>
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template <class VertexListGraph, class Param, class Tag, class Rest>
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void dag_shortest_paths(const VertexListGraph& g,
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typename graph_traits<VertexListGraph>::vertex_descriptor s,
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const bgl_named_params<Param,Tag,Rest>& params)
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<i>// non-named parameter version</i>
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template <class VertexListGraph, class DijkstraVisitor,
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class DistanceMap, class WeightMap, class ColorMap,
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class PredecessorMap,
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class Compare, class Combine,
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class DistInf, class DistZero>
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void dag_shortest_paths(const VertexListGraph& g,
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typename graph_traits<VertexListGraph>::vertex_descriptor s,
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DistanceMap distance, WeightMap weight, ColorMap color,
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PredecessorMap pred, DijkstraVisitor vis,
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Compare compare, Combine combine, DistInf inf, DistZero zero)
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</PRE>
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<P>
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This algorithm [<A HREF="bibliography.html#clr90">8</A>] solves
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the single-source shortest-paths problem on a weighted, directed
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acyclic graph (DAG). This algorithm is more efficient for DAG's
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than either the Dijkstra or Bellman-Ford algorithm.
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Use breadth-first search instead of this algorithm
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when all edge weights are equal to one. For the definition of the
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shortest-path problem see Section <A
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HREF="graph_theory_review.html#sec:shortest-paths-algorithms">Shortest-Paths
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Algorithms</A> for some background to the shortest-path problem.
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</P>
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<P>
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There are two main options for obtaining output from the
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<tt>dag_shortest_paths()</tt> function. If you provide a
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distance property map through the <tt>distance_map()</tt> parameter
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then the shortest distance from the source vertex to every other
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vertex in the graph will be recorded in the distance map. Also you can
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record the shortest paths tree in a predecessor map: for each vertex
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<i>u in V</i>, <i>p[u]</i> will be the predecessor of <i>u</i> in
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the shortest paths tree (unless <i>p[u] = u</i>, in which case <i>u</i> is
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either the source or a vertex unreachable from the source). In
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addition to these two options, the user can provide there own
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custom-made visitor that can takes actions during any of the
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algorithm's event points.</P>
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<h3>Where Defined</h3>
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<a href="../../../boost/graph/dag_shortest_paths.hpp"><tt>boost/graph/dag_shortest_paths.hpp</tt></a>
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<h3>Parameters</h3>
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IN: <tt>const VertexListGraph& g</tt>
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<blockquote>
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The graph object on which the algorithm will be applied.
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The type <tt>VertexListGraph</tt> must be a model of \concept{VertexListGraph}.<br>
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<b>Python</b>: The parameter is named <tt>graph</tt>.
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</blockquote>
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IN: <tt>vertex_descriptor s</tt>
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<blockquote>
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The source vertex. All distance will be calculated from this vertex,
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and the shortest paths tree will be rooted at this vertex.<br>
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<b>Python</b>: The parameter is named <tt>root_vertex</tt>.
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</blockquote>
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<h3>Named Parameters</h3>
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IN: <tt>weight_map(WeightMap w_map)</tt>
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<blockquote>
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The weight or ``length'' of each edge in the graph.
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The type <tt>WeightMap</tt> must be a model of
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<a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The edge descriptor type of
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the graph needs to be usable as the key type for the weight
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map. The value type for the map must be
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<i>Addable</i> with the value type of the distance map.<br>
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<b>Default:</b> <tt>get(edge_weight, g)</tt><br>
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<b>Python</b>: Must be an <tt>edge_double_map</tt> for the graph.<br>
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<b>Python default</b>: <tt>graph.get_edge_double_map("weight")</tt>
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</blockquote>
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IN: <tt>vertex_index_map(VertexIndexMap i_map)</tt>
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<blockquote>
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This maps each vertex to an integer in the range <tt>[0,
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num_vertices(g))</tt>. This is necessary for efficient updates of the
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heap data structure when an edge is relaxed. The type
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<tt>VertexIndexMap</tt> must be a model of
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<a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The value type of the map must be an
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integer type. The vertex descriptor type of the graph needs to be
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usable as the key type of the map.<br>
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<b>Default:</b> <tt>get(vertex_index, g)</tt>.
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Note: if you use this default, make sure your graph has
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an internal <tt>vertex_index</tt> property. For example,
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<tt>adjacency_list</tt> with <tt>VertexList=listS</tt> does
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not have an internal <tt>vertex_index</tt> property.<br>
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<b>Python</b>: Unsupported parameter.
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</blockquote>
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OUT: <tt>predecessor_map(PredecessorMap p_map)</tt>
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<blockquote>
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The predecessor map records the edges in the minimum spanning
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tree. Upon completion of the algorithm, the edges <i>(p[u],u)</i>
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for all <i>u in V</i> are in the minimum spanning tree. If <i>p[u] =
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u</i> then <i>u</i> is either the source vertex or a vertex that is
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not reachable from the source. The <tt>PredecessorMap</tt> type
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must be a <a
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href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
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Property Map</a> which key and vertex types the same as the vertex
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descriptor type of the graph.<br>
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<b>Default:</b> <tt>dummy_property_map</tt><br>
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<b>Python</b>: Must be a <tt>vertex_vertex_map</tt> for the graph.<br>
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</blockquote>
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UTIL/OUT: <tt>distance_map(DistanceMap d_map)</tt>
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<blockquote>
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The shortest path weight from the source vertex <tt>s</tt> to each
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vertex in the graph <tt>g</tt> is recorded in this property map. The
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shortest path weight is the sum of the edge weights along the
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shortest path. The type <tt>DistanceMap</tt> must be a model of <a
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href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
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Property Map</a>. The vertex descriptor type of the graph needs to
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be usable as the key type of the distance map.
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The value type of the distance map is the element type of a <a
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href="./Monoid.html">Monoid</tt> formed with the <tt>combine</tt>
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function object and the <tt>zero</tt> object for the identity
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element. Also the distance value type must have a <a
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href="http://www.boost.org/sgi/stl/StrictWeakOrdering.html">
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StrictWeakOrdering</a> provided by the <tt>compare</tt> function
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object.<br>
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<b>Default:</b> <a
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href="../../property_map/doc/iterator_property_map.html">
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<tt>iterator_property_map</tt></a> created from a
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<tt>std::vector</tt> of the <tt>WeightMap</tt>'s value type of size
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<tt>num_vertices(g)</tt> and using the <tt>i_map</tt> for the index
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map.<br>
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<b>Python</b>: Must be a <tt>vertex_double_map</tt> for the graph.
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</blockquote>
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IN: <tt>distance_compare(CompareFunction cmp)</tt>
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<blockquote>
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This function is use to compare distances to determine which vertex
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is closer to the source vertex. The <tt>CompareFunction</tt> type
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must be a model of <a
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href="http://www.boost.org/sgi/stl/BinaryPredicate.html">Binary
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Predicate</a> and have argument types that match the value type of
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the <tt>DistanceMap</tt> property map.<br>
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<b>Default:</b>
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<tt>std::less<D></tt> with <tt>D=typename
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property_traits<DistanceMap>::value_type</tt><br>
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<b>Python</b>: Unsupported parameter.
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</blockquote>
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IN: <tt>distance_combine(CombineFunction cmb)</tt>
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<blockquote>
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This function is used to combine distances to compute the distance
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of a path. The <tt>CombineFunction</tt> type must be a model of <a
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href="http://www.boost.org/sgi/stl/BinaryFunction.html">Binary
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Function</a>. The first argument type of the binary function must
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match the value type of the <tt>DistanceMap</tt> property map and
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the second argument type must match the value type of the
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<tt>WeightMap</tt> property map. The result type must be the same
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type as the distance value type.<br>
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<b>Default:</b> <tt>std::plus<D></tt> with
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<tt>D=typename property_traits<DistanceMap>::value_type</tt><br>
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<b>Python</b>: Unsupported parameter.
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</blockquote>
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IN: <tt>distance_inf(D inf)</tt>
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<blockquote>
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The <tt>inf</tt> object must be the greatest value of any <tt>D</tt> object.
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That is, <tt>compare(d, inf) == true</tt> for any <tt>d != inf</tt>.
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The type <tt>D</tt> is the value type of the <tt>DistanceMap</tt>.<br>
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<b>Default:</b> <tt>std::numeric_limits<D>::max()</tt><br>
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<b>Python</b>: Unsupported parameter.
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</blockquote>
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IN: <tt>distance_zero(D zero)</tt>
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<blockquote>
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The <tt>zero</tt> value must be the identity element for the
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<a href="./Monoid.html">Monoid</a> formed by the distance values
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and the <tt>combine</tt> function object.
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The type \code{D} is the value type of the \code{DistanceMap}
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<b>Default:</b> <tt>D()</tt><br>
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<b>Python</b>: Unsupported parameter.
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</blockquote>
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UTIL/OUT: <tt>color_map(ColorMap c_map)</tt>
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<blockquote>
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This is used during the execution of the algorithm to mark the
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vertices. The vertices start out white and become gray when they are
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inserted in the queue. They then turn black when they are removed
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from the queue. At the end of the algorithm, vertices reachable from
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the source vertex will have been colored black. All other vertices
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will still be white. The type <tt>ColorMap</tt> must be a model of
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<a href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
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Property Map</a>. A vertex descriptor must be usable as the key type
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of the map, and the value type of the map must be a model of
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<a href="./ColorValue.html">Color Value</a>.<br>
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<b>Default:</b> an <a
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href="../../property_map/doc/iterator_property_map.html">
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<tt>iterator_property_map</tt></a> created from a <tt>std::vector</tt>
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of <tt>default_color_type</tt> of size <tt>num_vertices(g)</tt> and
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using the <tt>i_map</tt> for the index map.<br>
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<b>Python</b>: The color map must be a <tt>vertex_color_map</tt> for
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the graph.
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</blockquote>
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OUT: <tt>visitor(DijkstraVisitor v)</tt>
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<blockquote>
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Use this to specify actions that you would like to happen
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during certain event points within the algorithm.
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The type <tt>DijkstraVisitor</tt> must be a model of the
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<a href="./DijkstraVisitor.html">Dijkstra Visitor</a> concept.
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The visitor object is passed by value <a
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href="#1">[1]</a>.<br>
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<b>Default:</b> <tt>dijkstra_visitor<null_visitor></tt><br>
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<b>Python</b>: The parameter should be an object that derives from
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the <a
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href="DijkstraVisitor.html#python"><tt>DijkstraVisitor</tt></a> type
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of the graph.
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</blockquote>
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<H3>Complexity</H3>
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<P>
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The time complexity is <i>O(V + E)</i>.
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<h3>Visitor Event Points</h3>
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<ul>
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<li><b><tt>vis.initialize_vertex(u, g)</tt></b>
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is invoked on each vertex in the graph before the start of the
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algorithm.
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<li><b><tt>vis.examine_vertex(u, g)</tt></b>
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is invoked on a vertex as it is added to set <i>S</i>.
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At this point we know that <i>(p[u],u)</i>
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is a shortest-paths tree edge so
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<i>d[u] = delta(s,u) = d[p[u]] + w(p[u],u)</i>. Also, the distances
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of the examined vertices is monotonically increasing
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<i>d[u<sub>1</sub>] <= d[u<sub>2</sub>] <= d[u<sub>n</sub>]</i>.
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<li><b><tt>vis.examine_edge(e, g)</tt></b>
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is invoked on each out-edge of a vertex immediately after it has
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been added to set <i>S</i>.
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<li><b><tt>vis.edge_relaxed(e, g)</tt></b>
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is invoked on edge <i>(u,v)</i> if <i>d[u] + w(u,v) < d[v]</i>.
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The edge <i>(u,v)</i> that participated in the last
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relaxation for vertex <i>v</i> is an edge in the shortest paths tree.
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<li><b><tt>vis.discover_vertex(v, g)</tt></b>
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is invoked on vertex <i>v</i> when the edge
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<i>(u,v)</i> is examined and <i>v</i> is WHITE. Since
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a vertex is colored GRAY when it is discovered,
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each reacable vertex is discovered exactly once.
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<li><b><tt>vis.edge_not_relaxed(e, g)</tt></b>
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is invoked if the edge is not relaxed (see above).
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<li><b><tt>vis.finish_vertex(u, g)</tt></b>
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is invoked on a vertex after all of its out edges have
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been examined.
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</ul>
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<H3>Example</H3>
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<P>
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See <a href="../example/dag_shortest_paths.cpp">
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<TT>example/dag_shortest_paths.cpp</TT></a> for an example of using this
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algorithm.
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<H3>Notes</H3>
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<p><a name="1">[1]</a>
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Since the visitor parameter is passed by value, if your visitor
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contains state then any changes to the state during the algorithm
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will be made to a copy of the visitor object, not the visitor object
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passed in. Therefore you may want the visitor to hold this state by
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pointer or reference.
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<br>
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<HR>
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<TABLE>
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<TR valign=top>
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<TD nowrap>Copyright © 2000-2001</TD><TD>
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<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>)
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</TD></TR></TABLE>
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</BODY>
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</HTML>
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