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- Copyright (c) Jeremy Siek 2000
-
- Distributed under the Boost Software License, Version 1.0.
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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.boost.org/sgi/stl/BinaryPredicate.html">CompareFunction</a>, typename <a href="http://www.boost.org/sgi/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>adjacency_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.boost.org/sgi/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.boost.org/sgi/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.boost.org/sgi/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>. Edges
- are assumed to have a weight less than (using <tt>distance_compare</tt> for
- comparison) this value.<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 + 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> is WHITE. Since
- a vertex is colored GRAY when it is discovered,
- each reachable 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>
- </BODY>
- </HTML>
|