// named parameter version 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& params); // non-named parameter version template <typename Graph, typename DijkstraVisitor, typename PredecessorMap, typename DistanceMap, typename WeightMap, typename VertexIndexMap, typename DistanceCompare, typename DistanceWeightCombine, 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); // version that does not initialize the property maps template <typename Graph, typename DijkstraVisitor, typename PredecessorMap, typename DistanceMap, typename WeightMap, typename VertexIndexMap, typename DistanceCompare, typename DistanceWeightCombine, 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);
This algorithm [10,8] 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 Shortest-Paths Algorithms for some background to the shortest-path problem.
dijkstra_shortest_paths_no_color_map differs from the original dijkstra_shortest_paths algorithm by not using a color map to identify vertices as discovered or undiscovered. Instead, this is done with the distance map: a vertex u such that distance_compare(distance_map[u], distance_infinity) == false is considered to be undiscovered. Note that this means that edges with infinite weight will not work correctly in this algorithm.
There are two main options for obtaining output from the dijkstra_shortest_paths_no_color_map() function. If you provide a distance property map through the distance_map() 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 u in V, p[u] will be the predecessor of u in the shortest paths tree (unless p[u] = u, in which case u 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 .
Dijkstra's algorithm finds all the shortest paths from the source vertex to every other vertex by iteratively "growing" the set of vertices S to which it knows the shortest path. At each step of the algorithm, the next vertex added to S is determined by a priority queue. The queue contains the vertices in V - S prioritized by their distance label, which is the length of the shortest path seen so far for each vertex. The vertex u at the top of the priority queue is then added to S, and each of its out-edges is relaxed: if the distance to u plus the weight of the out-edge (u,v) is less than the distance label for v then the estimated distance for vertex v 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.
The following is the pseudo-code for Dijkstra's single-source shortest paths algorithm. w is the edge weight, d is the distance label, and p is the predecessor of each vertex which is used to encode the shortest paths tree. Q 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.
DIJKSTRA(G, s, w) d[s] := 0 INSERT(Q, s) while (Q != Ø) u := EXTRACT-MIN(Q) for each vertex v in Adj[u] if (w(u,v) + d[u] < d[v]) d[v] := w(u,v) + d[u] p[v] := u if (d[v] was originally infinity) INSERT(Q, v) else DECREASE-KEY(Q, v) else ... end for end while return (d, p)
discover vertex s examine vertex u examine edge (u,v) edge (u,v) relaxed discover vertex v edge (u,v) not relaxed finish vertex u
The graph object on which the algorithm will be applied. The type Graph must be a model of Vertex List Graph and Incidence Graph.IN: vertex_descriptor start_vertex
The source vertex. All distance will be calculated from this vertex, and the shortest paths tree will be rooted at this vertex.
The weight or ``length'' of each edge in the graph. The weights must all be non-negative and non-infinite . The algorithm will throw a negative_edge exception is one of the edges is negative. The type WeightMap must be a model of Readable Property Map. 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.IN: index_map(VertexIndexMap index_map)
Default: get(edge_weight, graph)
This maps each vertex to an integer in the range [0, num_vertices(graph)). This is necessary for efficient updates of the heap data structure  when an edge is relaxed. The type VertexIndexMap must be a model of Readable Property Map. 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.OUT: predecessor_map(PredecessorMap predecessor_map)
Default: get(vertex_index, graph). Note: if you use this default, make sure your graph has an internal vertex_index property. For example, adjacenty_list with VertexList=listS does not have an internal vertex_index property.
The predecessor map records the edges in the minimum spanning tree. Upon completion of the algorithm, the edges (p[u],u) for all u in V are in the minimum spanning tree. If p[u] = u then u is either the source vertex or a vertex that is not reachable from the source. The PredecessorMap type must be a Read/Write Property Map whose key and value types are the same as the vertex descriptor type of the graph.UTIL/OUT: distance_map(DistanceMap distance_map)
Python: Must be a vertex_vertex_map for the graph.
The shortest path weight from the source vertex start_vertex to each vertex in the graph graph is recorded in this property map. The shortest path weight is the sum of the edge weights along the shortest path. The type DistanceMap must be a model of Read/Write Property Map. 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 Monoid formed with the distance_weight_combine function object and the distance_zero object for the identity element. Also the distance value type must have a StrictWeakOrdering provided by the distance_compare function object.IN: distance_compare(CompareFunction distance_compare)
Default: iterator_property_map created from a std::vector of the WeightMap's value type of size num_vertices(graph) and using the index_map for the index map.
This function is use to compare distances to determine which vertex is closer to the source vertex. The DistanceCompareFunction type must be a model of Binary Predicate and have argument types that match the value type of the DistanceMap property map.IN: distance_combine(CombineFunction distance_weight_combine)
Default: std::less<D> with D=typename property_traits<DistanceMap>::value_type
This function is used to combine distances to compute the distance of a path. The DistanceWeightCombineFunction type must be a model of Binary Function. The first argument type of the binary function must match the value type of the DistanceMap property map and the second argument type must match the value type of the WeightMap property map. The result type must be the same type as the distance value type.IN: distance_inf(D distance_infinity)
Default: boost::closed_plus<D> with D=typename property_traits<DistanceMap>::value_type
The distance_infinity object must be the greatest value of any D object. That is, distance_compare(d, distance_infinity) == true for any d != distance_infinity. The type D is the value type of the DistanceMap. All edges are assumed to have weight less than (by distance_compare) this value.IN: distance_zero(D distance_zero)
The distance_zero value must be the identity element for the Monoid formed by the distance values and the distance_weight_combine function object. The type D is the value type of the DistanceMap.OUT: visitor(DijkstraVisitor v)
Default: D()with D=typename property_traits<DistanceMap>::value_type
Use this to specify actions that you would like to happen during certain event points within the algorithm. The type DijkstraVisitor must be a model of the Dijkstra Visitor concept. The visitor object is passed by value .
The time complexity is O(V log V + E).
See example/dijkstra-no-color-map-example.cpp for an example of using Dijkstra's algorithm.
Based on the documentation for dijkstra_shortest_paths.
 The algorithm used here saves a little space by not putting all V - S vertices in the priority queue at once, but instead only those vertices in V - S that are discovered and therefore have a distance less than infinity.
 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.
 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.
Calls to the visitor events occur in the same order as dijkstra_shortest_paths (i.e. discover_vertex(u) will always be called after examine_vertex(u) for an undiscovered vertex u). However, the vertices of the graph given to dijkstra_shortest_paths_no_color_map will not necessarily be visited in the same order as dijkstra_shortest_paths.
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