ceph/src/boost/libs/graph/doc/dijkstra_shortest_paths.html

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   9 <Head>
  10 <Title>Boost Graph Library: Dijkstra's Shortest Paths</Title>
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  17
  18 <H1><A NAME="sec:dijkstra"></A><img src="figs/python.gif" alt="(Python)"/>
  19 <TT>dijkstra_shortest_paths</TT>
  20 </H1>
  21
  22 <P>
  23 <PRE>
  24 <i>// named parameter version</i>
  25 template &lt;typename Graph, typename P, typename T, typename R&gt;
  26 void
  27 dijkstra_shortest_paths(Graph&amp; g,
  28   typename graph_traits&lt;Graph&gt;::vertex_descriptor s,
  29   const bgl_named_params&lt;P, T, R&gt;&amp; params);
  30
  31 <i>// non-named parameter version</i>
  32 template &lt;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">CompareFunction</a>, typename <a href="http://www.sgi.com/tech/stl/BinaryFunction.html">CombineFunction</a>,
  35           typename DistInf, typename DistZero, typename ColorMap = <i>default</i>&gt;
  36 void dijkstra_shortest_paths
  37   (const Graph&amp; g,
  38    typename graph_traits&lt;Graph&gt;::vertex_descriptor s,
  39    PredecessorMap predecessor, DistanceMap distance, WeightMap weight,
  40    VertexIndexMap index_map,
  41    CompareFunction compare, CombineFunction combine, DistInf inf, DistZero zero,
  42    DijkstraVisitor vis, ColorMap color = <i>default</i>)
  43
  44 <i>// version that does not initialize the property maps (except for the default color map)</i>
  45 template &lt;class Graph, class DijkstraVisitor,
  46           class PredecessorMap, class DistanceMap,
  47           class WeightMap, class IndexMap, class Compare, class Combine,
  48           class DistZero, class ColorMap&gt;
  49 void
  50 dijkstra_shortest_paths_no_init
  51   (const Graph&amp; g,
  52    typename graph_traits&lt;Graph&gt;::vertex_descriptor s,
  53    PredecessorMap predecessor, DistanceMap distance, WeightMap weight,
  54    IndexMap index_map,
  55    Compare compare, Combine combine, DistZero zero,
  56    DijkstraVisitor vis, ColorMap color = <i>default</i>);
  57 </PRE>
  58
  59 <P>
  60 This algorithm&nbsp;[<A HREF="bibliography.html#dijkstra59">10</A>,<A
  61 HREF="bibliography.html#clr90">8</A>] solves the single-source
  62 shortest-paths problem on a weighted, directed or undirected graph for
  63 the case where all edge weights are nonnegative.  Use the Bellman-Ford
  64 algorithm for the case when some edge weights are negative.  Use
  65 breadth-first search instead of Dijkstra's algorithm when all edge
  66 weights are equal to one.  For the definition of the shortest-path
  67 problem see Section <A
  68 HREF="graph_theory_review.html#sec:shortest-paths-algorithms">Shortest-Paths
  69 Algorithms</A> for some background to the shortest-path problem.
  70 </P>
  71
  72 <P>
  73 There are two main options for obtaining output from the
  74 <tt>dijkstra_shortest_paths()</tt> function. If you provide a
  75 distance property map through the <tt>distance_map()</tt> parameter
  76 then the shortest distance from the source vertex to every other
  77 vertex in the graph will be recorded in the distance map. Also you can
  78 record the shortest paths tree in a predecessor map: for each vertex
  79 <i>u in V</i>, <i>p[u]</i> will be the predecessor of <i>u</i> in
  80 the shortest paths tree (unless <i>p[u] = u</i>, in which case <i>u</i> is
  81 either the source or a vertex unreachable from the source).  In
  82 addition to these two options, the user can provide their own
  83 custom-made visitor that takes actions during any of the
  84 algorithm's event points.</P>
  85
  86 <P>
  87 Dijkstra's algorithm finds all the shortest paths from the source
  88 vertex to every other vertex by iteratively ``growing'' the set of
  89 vertices <i>S</i> to which it knows the shortest path. At each step of
  90 the algorithm, the next vertex added to <i>S</i> is determined by a
  91 priority queue.  The queue contains the vertices in <i>V - S</i><a
  92 href="#1">[1]</a> prioritized by their distance label, which is the
  93 length of the shortest path seen so far for each vertex. The vertex
  94 <i>u</i> at the top of the priority queue is then added to <i>S</i>,
  95 and each of its out-edges is relaxed: if the distance to <i>u</i> plus
  96 the weight of the out-edge <i>(u,v)</i> is less than the distance
  97 label for <i>v</i> then the estimated distance for vertex <i>v</i> is
  98 reduced.  The algorithm then loops back, processing the next vertex at
  99 the top of the priority queue. The algorithm finishes when the
 100 priority queue is empty.
 101 </P>
 102 <P>
 103 The algorithm uses color markers (white, gray, and black) to keep
 104 track of which set each vertex is in. Vertices colored black are in
 105 <i>S</i>. Vertices colored white or gray are in <i>V-S</i>. White vertices have
 106 not yet been discovered and gray vertices are in the priority queue.
 107 By default, the algorithm will allocate an array to store a color
 108 marker for each vertex in the graph. You can provide your own storage
 109 and access for colors with the <tt>color_map()</tt> parameter.
 110 </P>
 111 <p>
 112 The following is the pseudo-code for Dijkstra's single-source shortest
 113 paths algorithm. <i>w</i> is the edge weight, <i>d</i> is the distance label,
 114 and <i>p</i> is the predecessor of each vertex which is used to encode
 115 the shortest paths tree. <i>Q</i> is a priority queue that supports the
 116 DECREASE-KEY operation.  The visitor event points for the algorithm are
 117 indicated by the labels on the right.
 118 </p>
 119
 120 <table>
 121 <tr>
 122 <td valign="top">
 123 <pre>
 124 DIJKSTRA(<i>G</i>, <i>s</i>, <i>w</i>)
 125   <b>for</b> each vertex <i>u in V</i> <b>(This loop is not run in dijkstra_shortest_paths_no_init)</b>
 126     <i>d[u] := infinity</i>
 127     <i>p[u] := u</i>
 128     <i>color[u] :=</i> WHITE
 129   <b>end for</b>
 130   <i>color[s] := </i>GRAY
 131   <i>d[s] := 0</i>
 132   INSERT(<i>Q</i>, <i>s</i>)
 133   <b>while</b> (<i>Q != &Oslash;</i>)
 134     <i>u :=</i> EXTRACT-MIN(<i>Q</i>)
 135     <i>S := S U { u }</i>
 136     <b>for</b> each vertex <i>v in Adj[u]</i>
 137       <b>if</b> (<i>w(u,v) + d[u] < d[v]</i>)
 138         <i>d[v] := w(u,v) + d[u]</i>
 139         <i>p[v] := u</i>
 140         <b>if</b> (<i>color[v] =</i> WHITE)
 141           <i>color[v] :=</i> GRAY
 142           INSERT(<i>Q</i>, <i>v</i>)
 143         <b>else if</b> (<i>color[v] =</i> GRAY)
 144           DECREASE-KEY(<i>Q</i>, <i>v</i>)
 145       <b>else</b>
 146         <i>...</i>
 147     <b>end for</b>
 148     <i>color[u] :=</i> BLACK
 149   <b>end while</b>
 150   return (<i>d</i>, <i>p</i>)
 151 </pre>
 152 </td>
 153 <td valign="top">
 154 <pre>
 155
 156 initialize vertex <i>u</i>
 157
 158
 159
 160
 161
 162
 163 discover vertex <i>s</i>
 164
 165 examine vertex <i>u</i>
 166
 167 examine edge <i>(u,v)</i>
 168
 169 edge <i>(u,v)</i> relaxed
 170
 171
 172
 173 discover vertex <i>v</i>
 174
 175
 176
 177 edge <i>(u,v)</i> not relaxed
 178
 179 finish vertex <i>u</i>
 180 </pre>
 181 </td>
 182 </tr>
 183 </table>
 184
 185 <h3>Where Defined</h3>
 186
 187 <a href="../../../boost/graph/dijkstra_shortest_paths.hpp"><tt>boost/graph/dijkstra_shortest_paths.hpp</tt></a>
 188
 189 <h3>Parameters</h3>
 190
 191 IN: <tt>const Graph&amp; g</tt>
 192 <blockquote>
 193   The graph object on which the algorithm will be applied.
 194   The type <tt>Graph</tt> must be a model of
 195   <a href="./VertexListGraph.html">Vertex List Graph</a>
 196   and <a href="./IncidenceGraph.html">Incidence Graph</a>.<br>
 197
 198   <b>Python</b>: The parameter is named <tt>graph</tt>.
 199 </blockquote>
 200
 201 IN: <tt>vertex_descriptor s</tt>
 202 <blockquote>
 203   The source vertex. All distance will be calculated from this vertex,
 204   and the shortest paths tree will be rooted at this vertex.<br>
 205
 206   <b>Python</b>: The parameter is named <tt>root_vertex</tt>.
 207 </blockquote>
 208
 209 <h3>Named Parameters</h3>
 210
 211 IN: <tt>weight_map(WeightMap w_map)</tt>
 212 <blockquote>
 213   The weight or ``length'' of each edge in the graph. The weights
 214   must all be non-negative, and the algorithm will throw a
 215   <a href="./exception.html#negative_edge"><tt>negative_edge</tt></a>
 216   exception is one of the edges is negative.
 217   The type <tt>WeightMap</tt> must be a model of
 218   <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The edge descriptor type of
 219   the graph needs to be usable as the key type for the weight
 220   map. The value type for this map must be
 221   the same as the value type of the distance map.<br>
 222   <b>Default:</b>  <tt>get(edge_weight, g)</tt><br>
 223
 224   <b>Python</b>: Must be an <tt>edge_double_map</tt> for the graph.<br>
 225   <b>Python default</b>: <tt>graph.get_edge_double_map("weight")</tt>
 226 </blockquote>
 227
 228 IN: <tt>vertex_index_map(VertexIndexMap i_map)</tt>
 229 <blockquote>
 230   This maps each vertex to an integer in the range <tt>[0,
 231     num_vertices(g))</tt>. This is necessary for efficient updates of the
 232   heap data structure&nbsp;[<A
 233   HREF="bibliography.html#driscoll88">61</A>] when an edge is relaxed.
 234   The type
 235   <tt>VertexIndexMap</tt> must be a model of
 236   <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The value type of the map must be an
 237   integer type. The vertex descriptor type of the graph needs to be
 238   usable as the key type of the map.<br>
 239   <b>Default:</b> <tt>get(vertex_index, g)</tt>.
 240     Note: if you use this default, make sure your graph has
 241     an internal <tt>vertex_index</tt> property. For example,
 242     <tt>adjacency_list</tt> with <tt>VertexList=listS</tt> does
 243     not have an internal <tt>vertex_index</tt> property.
 244    <br>
 245
 246   <b>Python</b>: Unsupported parameter.
 247 </blockquote>
 248
 249 OUT: <tt>predecessor_map(PredecessorMap p_map)</tt>
 250 <blockquote>
 251   The predecessor map records the edges in the shortest path tree, the tree computed
 252   by the traversal of the graph. Upon completion of the algorithm, the edges
 253   <i>(p[u],u)</i> for all <i>u in V</i> are in the tree.  The shortest path
 254   from vertex <i>s</i> to each vertex <i>v</i> in the graph consists of the
 255   vertices <i>v</i>, <i>p[v]</i>, <i>p[p[v]]</i>, and so on until <i>s</i> is
 256   reached, in reverse order.  The
 257   tree is not guaranteed to be a minimum spanning tree. If <i>p[u] =
 258   u</i> then <i>u</i> is either the source vertex or a vertex that is
 259   not reachable from the source.  The <tt>PredecessorMap</tt> type
 260   must be a <a
 261   href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
 262   Property Map</a> whose key and value types are the same as the vertex
 263   descriptor type of the graph.<br>
 264   <b>Default:</b> <tt>dummy_property_map</tt><br>
 265
 266   <b>Python</b>: Must be a <tt>vertex_vertex_map</tt> for the graph.<br>
 267 </blockquote>
 268
 269 UTIL/OUT: <tt>distance_map(DistanceMap d_map)</tt>
 270 <blockquote>
 271   The shortest path weight from the source vertex <tt>s</tt> to each
 272   vertex in the graph <tt>g</tt> is recorded in this property map. The
 273   shortest path weight is the sum of the edge weights along the
 274   shortest path.  The type <tt>DistanceMap</tt> must be a model of <a
 275   href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
 276   Property Map</a>. The vertex descriptor type of the graph needs to
 277   be usable as the key type of the distance map.
 278
 279   The value type of the distance map is the element type of a <a
 280   href="./Monoid.html">Monoid</a> formed with the <tt>combine</tt>
 281   function object and the <tt>zero</tt> object for the identity
 282   element. Also the distance value type must have a <a
 283   href="http://www.sgi.com/tech/stl/StrictWeakOrdering.html">
 284   StrictWeakOrdering</a> provided by the <tt>compare</tt> function
 285   object.<br>
 286   <b>Default:</b> <a
 287   href="../../property_map/doc/iterator_property_map.html">
 288   <tt>iterator_property_map</tt></a> created from a
 289   <tt>std::vector</tt> of the <tt>WeightMap</tt>'s value type of size
 290   <tt>num_vertices(g)</tt> and using the <tt>i_map</tt> for the index
 291   map.<br>
 292
 293   <b>Python</b>: Must be a <tt>vertex_double_map</tt> for the graph.<br>
 294 </blockquote>
 295
 296 IN: <tt>distance_compare(CompareFunction cmp)</tt>
 297 <blockquote>
 298   This function is use to compare distances to determine which vertex
 299   is closer to the source vertex.  The <tt>CompareFunction</tt> type
 300   must be a model of <a
 301   href="http://www.sgi.com/tech/stl/BinaryPredicate.html">Binary
 302   Predicate</a> and have argument types that match the value type of
 303   the <tt>DistanceMap</tt> property map.<br>
 304
 305   <b>Default:</b>
 306   <tt>std::less&lt;D&gt;</tt> with <tt>D=typename
 307   property_traits&lt;DistanceMap&gt;::value_type</tt><br>
 308
 309   <b>Python</b>: Unsupported parameter.
 310 </blockquote>
 311
 312 IN: <tt>distance_combine(CombineFunction cmb)</tt>
 313 <blockquote>
 314   This function is used to combine distances to compute the distance
 315   of a path. The <tt>CombineFunction</tt> type must be a model of <a
 316   href="http://www.sgi.com/tech/stl/BinaryFunction.html">Binary
 317   Function</a>. The first argument type of the binary function must
 318   match the value type of the <tt>DistanceMap</tt> property map and
 319   the second argument type must match the value type of the
 320   <tt>WeightMap</tt> property map.  The result type must be the same
 321   type as the distance value type.<br>
 322
 323   <b>Default:</b> <tt>closed_plus&lt;D&gt;</tt> with
 324    <tt>D=typename property_traits&lt;DistanceMap&gt;::value_type</tt><br>
 325
 326   <b>Python</b>: Unsupported parameter.
 327 </blockquote>
 328
 329 IN: <tt>distance_inf(D inf)</tt>
 330 <blockquote>
 331   The <tt>inf</tt> object must be the greatest value of any <tt>D</tt> object.
 332   That is, <tt>compare(d, inf) == true</tt> for any <tt>d != inf</tt>.
 333   The type <tt>D</tt> is the value type of the <tt>DistanceMap</tt>.  Edges
 334   are assumed to have a weight less than (using <tt>distance_compare</tt> for
 335   comparison) this value.<br>
 336   <b>Default:</b> <tt>std::numeric_limits&lt;D&gt;::max()</tt><br>
 337
 338   <b>Python</b>: Unsupported parameter.
 339 </blockquote>
 340
 341 IN: <tt>distance_zero(D zero)</tt>
 342 <blockquote>
 343   The <tt>zero</tt> value must be the identity element for the
 344   <a href="./Monoid.html">Monoid</a> formed by the distance values
 345   and the <tt>combine</tt> function object.
 346   The type <tt>D</tt> is the value type of the <tt>DistanceMap</tt>.<br>
 347   <b>Default:</b> <tt>D()</tt>with
 348    <tt>D=typename property_traits&lt;DistanceMap&gt;::value_type</tt><br>
 349
 350   <b>Python</b>: Unsupported parameter.
 351 </blockquote>
 352
 353 UTIL/OUT: <tt>color_map(ColorMap c_map)</tt>
 354 <blockquote>
 355   This is used during the execution of the algorithm to mark the
 356   vertices. The vertices start out white and become gray when they are
 357   inserted in the queue. They then turn black when they are removed
 358   from the queue. At the end of the algorithm, vertices reachable from
 359   the source vertex will have been colored black. All other vertices
 360   will still be white. The type <tt>ColorMap</tt> must be a model of
 361   <a href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
 362   Property Map</a>. A vertex descriptor must be usable as the key type
 363   of the map, and the value type of the map must be a model of
 364   <a href="./ColorValue.html">Color Value</a>.<br>
 365   <b>Default:</b> an <a
 366   href="../../property_map/doc/iterator_property_map.html">
 367   <tt>iterator_property_map</tt></a> created from a <tt>std::vector</tt>
 368   of <tt>default_color_type</tt> of size <tt>num_vertices(g)</tt> and
 369   using the <tt>i_map</tt> for the index map.<br>
 370
 371   <b>Python</b>: The color map must be a <tt>vertex_color_map</tt> for
 372   the graph.
 373 </blockquote>
 374
 375 OUT: <tt>visitor(DijkstraVisitor v)</tt>
 376 <blockquote>
 377   Use this to specify actions that you would like to happen
 378   during certain event points within the algorithm.
 379   The type <tt>DijkstraVisitor</tt> must be a model of the
 380   <a href="./DijkstraVisitor.html">Dijkstra Visitor</a> concept.
 381  The visitor object is passed by value <a
 382   href="#2">[2]</a>.<br>
 383   <b>Default:</b> <tt>dijkstra_visitor&lt;null_visitor&gt;</tt><br>
 384
 385   <b>Python</b>: The parameter should be an object that derives from
 386   the <a
 387   href="DijkstraVisitor.html#python"><tt>DijkstraVisitor</tt></a> type
 388   of the graph.
 389 </blockquote>
 390
 391
 392 <H3>Complexity</H3>
 393
 394 <P>
 395 The time complexity is <i>O(V log V + E)</i>.
 396
 397
 398 <h3>Visitor Event Points</h3>
 399
 400 <ul>
 401 <li><b><tt>vis.initialize_vertex(u, g)</tt></b>
 402   is invoked on each vertex in the graph before the start of the
 403   algorithm.
 404 <li><b><tt>vis.examine_vertex(u, g)</tt></b>
 405   is invoked on a vertex as it is removed from the priority queue
 406   and added to set <i>S</i>. At this point we know that <i>(p[u],u)</i>
 407   is a shortest-paths tree edge so
 408   <i>d[u] = delta(s,u) = d[p[u]] + w(p[u],u)</i>. Also, the distances
 409   of the examined vertices is monotonically increasing
 410   <i>d[u<sub>1</sub>] <= d[u<sub>2</sub>] <= d[u<sub>n</sub>]</i>.
 411 <li><b><tt>vis.examine_edge(e, g)</tt></b>
 412   is invoked on each out-edge of a vertex immediately after it has
 413   been added to set <i>S</i>.
 414 <li><b><tt>vis.edge_relaxed(e, g)</tt></b>
 415   is invoked on edge <i>(u,v)</i> if <i>d[u] + w(u,v) < d[v]</i>.
 416   The edge <i>(u,v)</i> that participated in the last
 417   relaxation for vertex <i>v</i> is an edge in the shortest paths tree.
 418 <li><b><tt>vis.discover_vertex(v, g)</tt></b>
 419   is invoked on vertex <i>v</i> when the edge
 420   <i>(u,v)</i> is examined and <i>v</i> is WHITE. Since
 421   a vertex is colored GRAY when it is discovered,
 422   each reacable vertex is discovered exactly once. This
 423   is also when the vertex is inserted into the priority queue.
 424 <li><b><tt>vis.edge_not_relaxed(e, g)</tt></b>
 425   is invoked if the edge is not relaxed (see above).
 426 <li><b><tt>vis.finish_vertex(u, g)</tt></b>
 427    is invoked on a vertex after all of its out edges have
 428   been examined.
 429 </ul>
 430
 431 <H3>Example</H3>
 432
 433 <P>
 434 See <a href="../example/dijkstra-example.cpp">
 435 <TT>example/dijkstra-example.cpp</TT></a> for an example of using Dijkstra's
 436 algorithm.
 437
 438 <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.
 439
 440 <H3>Notes</H3>
 441
 442 <a name="1">[1]</a>
 443 The algorithm used here saves a little space by not putting all <i>V -
 444 S</i> vertices in the priority queue at once, but instead only those
 445 vertices in <i>V - S</i> that are discovered and therefore have a
 446 distance less than infinity.
 447
 448 <p><a name="2">[2]</a>
 449   Since the visitor parameter is passed by value, if your visitor
 450   contains state then any changes to the state during the algorithm
 451   will be made to a copy of the visitor object, not the visitor object
 452   passed in. Therefore you may want the visitor to hold this state by
 453   pointer or reference.
 454
 455 <br>
 456 <HR>
 457 <TABLE>
 458 <TR valign=top>
 459 <TD nowrap>Copyright &copy; 2000-2001</TD><TD>
 460 <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>)
 461 </TD></TR></TABLE>
 462
 463 </BODY>
 464 </HTML>