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| <Head> |
| <Title>Boost Graph Library: Dijkstra's Shortest Paths (No Color Map)</Title> |
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| <BR Clear> |
| |
| <H1><A NAME="sec:dijkstra"></A> |
| <TT>dijkstra_shortest_paths_no_color_map</TT> |
| </H1> |
| |
| <P> |
| <PRE> |
| <i>// named parameter version</i> |
| template <typename Graph, typename Param, typename Tag, typename Rest> |
| void dijkstra_shortest_paths_no_color_map |
| (const Graph& graph, |
| typename graph_traits<Graph>::vertex_descriptor start_vertex, |
| const bgl_named_params<Param,Tag,Rest>& 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.sgi.com/tech/stl/BinaryPredicate.html">DistanceCompare</a>, typename <a href="http://www.sgi.com/tech/stl/BinaryFunction.html">DistanceWeightCombine</a>, |
| typename DistanceInfinity, typename DistanceZero> |
| void dijkstra_shortest_paths_no_color_map |
| (const Graph& graph, |
| typename graph_traits<Graph>::vertex_descriptor start_vertex, |
| PredecessorMap predecessor_map, DistanceMap distance_map, WeightMap weight_map, |
| VertexIndexMap index_map, |
| DistanceCompare distance_compare, DistanceWeightCombine distance_weight_combine, |
| DistanceInfinity distance_infinity, DistanceZero distance_zero); |
| |
| <i>// version that does not initialize the property maps</i> |
| template <typename Graph, typename <a href="DijkstraVisitor.html">DijkstraVisitor</a>, |
| typename PredecessorMap, typename DistanceMap, |
| typename WeightMap, typename VertexIndexMap, typename <a href="http://www.sgi.com/tech/stl/BinaryPredicate.html">DistanceCompare</a>, typename <a href="http://www.sgi.com/tech/stl/BinaryFunction.html">DistanceWeightCombine</a>, |
| typename DistanceInfinity, typename DistanceZero> |
| void dijkstra_shortest_paths_no_color_map_no_init |
| (const Graph& graph, |
| typename graph_traits<Graph>::vertex_descriptor start_vertex, |
| PredecessorMap predecessor_map, DistanceMap distance_map, WeightMap weight_map, |
| VertexIndexMap index_map, |
| DistanceCompare distance_compare, DistanceWeightCombine distance_weight_combine, |
| DistanceInfinity distance_infinity, DistanceZero distance_zero); |
| </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> |
| <tt>dijkstra_shortest_paths_no_color_map</tt> differs from the original <tt>dijkstra_shortest_paths</tt> algorithm by not using a color map to identify vertices as discovered or undiscovered. Instead, this is done with the distance map: a vertex <i>u</i> such that <i>distance_compare(distance_map[u], distance_infinity) == false</i> is considered to be undiscovered. |
| </P> |
| |
| <P> |
| There are two main options for obtaining output from the |
| <tt>dijkstra_shortest_paths_no_color_map()</tt> function. If you provide a |
| distance property map through the <tt>distance_map()</tt> parameter |
| then the shortest distance from the start 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 <a href="#4">[4]</a>.</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 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_color_map_no_init)</b> |
| <i>d[u] := infinity</i> |
| <i>p[u] := u</i> |
| <b>end for</b> |
| <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>) |
| <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> |
| DECREASE-KEY(<i>Q</i>, <i>v</i>) |
| <b>else</b> |
| ... |
| <b>if</b> (<i>d[v]</i> was originally infinity) |
| INSERT(<i>Q</i>, <i>v</i>) |
| <b>end for</b> |
| <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 |
| |
| |
| |
| edge <i>(u,v)</i> not relaxed |
| |
| discover vertex <i>v</i> |
| finish vertex <i>u</i> |
| </pre> |
| </td> |
| </tr> |
| </table> |
| |
| <h3>Where Defined</h3> |
| |
| <a href="../../../boost/graph/dijkstra_shortest_paths_no_color_map.hpp"><tt>boost/graph/dijkstra_shortest_paths_no_color_map.hpp</tt></a> |
| |
| <h3>Parameters</h3> |
| |
| IN: <tt>const Graph& graph</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> |
| </blockquote> |
| |
| IN: <tt>vertex_descriptor start_vertex</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> |
| </blockquote> |
| |
| <h3>Named Parameters</h3> |
| |
| IN: <tt>weight_map(WeightMap weight_map)</tt> |
| <blockquote> |
| The weight or ``length'' of each edge in the graph. The weights |
| must all be non-negative and non-infinite <a href="#3">[3]</a>. 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, graph)</tt><br> |
| </blockquote> |
| |
| IN: <tt>index_map(VertexIndexMap index_map)</tt> |
| <blockquote> |
| This maps each vertex to an integer in the range <tt>[0, |
| num_vertices(graph))</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, graph)</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> |
| </blockquote> |
| |
| OUT: <tt>predecessor_map(PredecessorMap predecessor_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> 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 distance_map)</tt> |
| <blockquote> |
| The shortest path weight from the source vertex <tt>start_vertex</tt> to each |
| vertex in the graph <tt>graph</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>distance_weight_combine</tt> |
| function object and the <tt>distance_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>distance_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(graph)</tt> and using the <tt>index_map</tt> for the index |
| map.<br> |
| </blockquote> |
| |
| IN: <tt>distance_compare(CompareFunction distance_compare)</tt> |
| <blockquote> |
| This function is use to compare distances to determine which vertex |
| is closer to the source vertex. The <tt>DistanceCompareFunction</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> |
| </blockquote> |
| |
| IN: <tt>distance_combine(CombineFunction distance_weight_combine)</tt> |
| <blockquote> |
| This function is used to combine distances to compute the distance |
| of a path. The <tt>DistanceWeightCombineFunction</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>boost::closed_plus<D></tt> with |
| <tt>D=typename property_traits<DistanceMap>::value_type</tt><br> |
| </blockquote> |
| |
| IN: <tt>distance_inf(D distance_infinity)</tt> |
| <blockquote> |
| The <tt>distance_infinity</tt> object must be the greatest value of any <tt>D</tt> object. |
| That is, <tt>distance_compare(d, distance_infinity) == true</tt> for any <tt>d != distance_infinity</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> |
| </blockquote> |
| |
| IN: <tt>distance_zero(D distance_zero)</tt> |
| <blockquote> |
| The <tt>distance_zero</tt> value must be the identity element for the |
| <a href="./Monoid.html">Monoid</a> formed by the distance values |
| and the <tt>distance_weight_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> |
| </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> |
| </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> has not yet been discovered (i.e. its distance was infinity before relaxation was attempted on the edge). 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-no-color-map-example.cpp"> |
| <TT>example/dijkstra-no-color-map-example.cpp</TT></a> for an example of using Dijkstra's algorithm. |
| |
| <H3>See also</H3> <a href="dijkstra_shortest_paths.html">dijkstra_shortest_paths</a> for a version of dijkstra's shortest path that uses a color map. |
| |
| <H3>Notes</H3> |
| |
| <p>Based on the documentation for <a href="dijkstra_shortest_paths.html">dijkstra_shortest_paths</a>. |
| |
| <p><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. |
| |
| <p><a name="3">[3]</a> |
| The algorithm will not work correctly if any of the edge weights are equal to infinity since the infinite distance value is used to determine if a vertex has been discovered. |
| |
| <p><a name="4">[4]</a> |
| Calls to the visitor events occur in the same order as <tt>dijkstra_shortest_paths</tt> (i.e. <i>discover_vertex(u)</i> will always be called after <i>examine_vertex(u)</i> for an undiscovered vertex <i>u</i>). However, the vertices of the graph given to <i>dijkstra_shortest_paths_no_color_map</i> will <b>not</b> necessarily be visited in the same order as <i>dijkstra_shortest_paths</i>. |
| |
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| <TD nowrap>Copyright © 2009</TD><TD> |
| Trustees of Indiana University</TD></TR></TABLE> |
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