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