Graphs: | directed and undirected |
---|---|
Properties: | color, distance, weight, vertex index |
Complexity: | O((V + E) log V) |
Where Defined: | boost/graph/dijkstra_shortest_paths.hpp |
(1) template <class VertexListGraph> void dijkstra_shortest_paths(VertexListGraph& g, typename graph_traits<VertexListGraph>::vertex_descriptor s); (2) template <class VertexListGraph, class DistanceMap> void dijkstra_shortest_paths(VertexListGraph& g, typename graph_traits<VertexListGraph>::vertex_descriptor s, DistanceMap d); (3) template <class VertexListGraph, class DistanceMap, class UniformCostVisitor> void dijkstra_shortest_paths(VertexListGraph& g, typename graph_traits<VertexListGraph>::vertex_descriptor s, DistanceMap d, UniformCostVisitor visit); (4) template <class VertexListGraph, class UniformCostVisitor, class DistanceMap, class WeightMap, class ColorMap, class VertexIndexMap> void dijkstra_shortest_paths(VertexListGraph& g, typename graph_traits<VertexListGraph>::vertex_descriptor s, DistanceMap distance, WeightMap weight, ColorMap color, VertexIndexMap id, UniformCostVisitor vis);
This is the modified Dijkstra algorithm [10,8] which solves the single-source shortest-paths problem on a weighted, directed graph for the case where all edge weights are nonnegative. See Section Shortest-Paths Algorithms for some background to the shortest-path problem. The priority queue used inside the algorithm is implemented with a heap for efficiency.
There are four versions of the algorithm to accommodate whether the necessary graph properties will be supplied by the graph object or externally via a argument to this function. The properties needed by the algorithm are distance, weight, color, and vertex index. Version 3 and 4 of the algorithm also include a visitor argument for added extensibility.
The time complexity is O((V + E) log V), or just O(E log V) if all vertices are reachable from the source.
The source code for this example is in examples/dijkstra.cpp.
int main(int , char* []) { using namespace boost; typedef property<edge_weight_t, int> weightp; typedef adjacency_list< listS, vecS, directedS, property<vertex_color_t,default_color_type>, weightp > Graph; typedef graph_traits<Graph>::vertex_descriptor Vertex; typedef std::pair<int,int> E; const int num_nodes = 5; E edges[] = { E(0,2), E(1,1), E(1,3), E(1,4), E(2,1), E(2,3), E(3,4), E(4,0), E(4,1) }; int weights[] = { 1, 2, 1, 2, 7, 3, 1, 1, 1}; Graph G(num_nodes, edges, edges + sizeof(edges)/sizeof(E), weights); std::vector<Vertex> p(num_vertices(G)); std::vector<int> d(num_vertices(G)); Vertex s = *(vertices(G).first); p[s] = s; dijkstra_shortest_paths(G, s, &d[0], make_ucs_visitor(record_predecessors(&p[0], on_edge_relaxed()))); std::cout << "distances from start vertex:" << std::endl; graph_traits<Graph>::vertex_iterator vi, vend; for(tie(vi,vend) = vertices(G); vi != vend; ++vi) std::cout << "distance(" << *vi << ") = " << d[*vi] << std::endl; std::cout << std::endl; std::cout << "shortest paths tree" << std::endl; adjacency_list<> tree(num_nodes); for(tie(vi,vend) = vertices(G); vi != vend; ++vi) if (*vi != p[*vi]) add_edge(p[*vi], *vi, tree); print_graph(tree); return 0; }The output is:
distances from start vertex: distance(0) = 0 distance(1) = 6 distance(2) = 1 distance(3) = 4 distance(4) = 5 shortest paths tree 0 --> 2 1 --> 2 --> 3 3 --> 4 4 --> 1
Copyright © 2000 | Jeremy Siek, Univ.of Notre Dame (jsiek@lsc.nd.edu) |