Boost Graph Library: Dijkstra's Shortest Paths

# `dijkstra_shortest_paths`

    // named parameter version
    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);

    // non-named parameter version
    template <typename Graph, typename DijkstraVisitor,
    	  typename PredecessorMap, typename DistanceMap,
    	  typename WeightMap, typename VertexIndexMap, typename CompareFunction, typename CombineFunction,
    	  typename DistInf, typename DistZero, typename ColorMap = default>
    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 = default)

    // version that does not initialize the property maps (except for the default color map)
    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 = default);

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.

There are two main options for obtaining output from the `dijkstra_shortest_paths()` function. If you provide a distance property map through the `distance_map()` 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 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[1] 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 algorithm uses color markers (white, gray, and black) to keep track of which set each vertex is in. Vertices colored black are in S. Vertices colored white or gray are in V-S. 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 `color_map()` parameter.

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)
      for each vertex u in V (This loop is not run in dijkstra_shortest_paths_no_init)
        d[u] := infinity
        p[u] := u
        color[u] := WHITE
      end for
      color[s] := GRAY
      d[s] := 0
      INSERT(Q, s)
      while (Q != Ø)
        u := EXTRACT-MIN(Q)
        S := S U { u }
        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 (color[v] = WHITE)
              color[v] := GRAY
              INSERT(Q, v)
            else if (color[v] = GRAY)
              DECREASE-KEY(Q, v)
          else
            ...
        end for
        color[u] := BLACK
      end while
      return (d, p)
    initialize vertex u

    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

### Where Defined

`boost/graph/dijkstra_shortest_paths.hpp`

### Parameters

IN: `const Graph& g`

> The graph object on which the algorithm will be applied. The type `Graph` must be a model of Vertex List Graph and Incidence Graph.
> Python: The parameter is named `graph`.

IN: `vertex_descriptor s`

> The source vertex. All distance will be calculated from this vertex, and the shortest paths tree will be rooted at this vertex.
> Python: The parameter is named `root_vertex`.

### Named Parameters

IN: `weight_map(WeightMap w_map)`

> The weight or ``length'' of each edge in the graph. The weights must all be non-negative, and 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.
> Default: `get(edge_weight, g)`
> Python: Must be an `edge_double_map` for the graph.
> Python default: `graph.get_edge_double_map("weight")`

IN: `vertex_index_map(VertexIndexMap i_map)`

> This maps each vertex to an integer in the range `[0, num_vertices(g))`. This is necessary for efficient updates of the heap data structure [61] 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.
> Default: `get(vertex_index, g)`. 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.
> Python: Unsupported parameter.

OUT: `predecessor_map(PredecessorMap p_map)`

> 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 (p[u],u) for all u in V are in the tree. The shortest path from vertex s to each vertex v in the graph consists of the vertices v, p[v], p[p[v]], and so on until s is reached, in reverse order. The tree is not guaranteed to be a 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.
> Default: `dummy_property_map`
> Python: Must be a `vertex_vertex_map` for the graph.

UTIL/OUT: `distance_map(DistanceMap d_map)`

> The shortest path weight from the source vertex `s` to each vertex in the graph `g` 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 `combine` function object and the `zero` object for the identity element. Also the distance value type must have a StrictWeakOrdering provided by the `compare` function object.
> Default: `iterator_property_map` created from a `std::vector` of the `WeightMap`'s value type of size `num_vertices(g)` and using the `i_map` for the index map.
> Python: Must be a `vertex_double_map` for the graph.

IN: `distance_compare(CompareFunction cmp)`

> This function is use to compare distances to determine which vertex is closer to the source vertex. The `CompareFunction` type must be a model of Binary Predicate and have argument types that match the value type of the `DistanceMap` property map.
> Default: `std::less<D>` with `D=typename property_traits<DistanceMap>::value_type`
> Python: Unsupported parameter.

IN: `distance_combine(CombineFunction cmb)`

> This function is used to combine distances to compute the distance of a path. The `CombineFunction` 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.
> Default: `closed_plus<D>` with `D=typename property_traits<DistanceMap>::value_type`
> Python: Unsupported parameter.

IN: `distance_inf(D inf)`

> The `inf` object must be the greatest value of any `D` object. That is, `compare(d, inf) == true` for any `d != inf`. The type `D` is the value type of the `DistanceMap`.
> Default: `std::numeric_limits<D>::max()`
> Python: Unsupported parameter.

IN: `distance_zero(D zero)`

> The `zero` value must be the identity element for the Monoid formed by the distance values and the `combine` function object. The type `D` is the value type of the `DistanceMap`.
> Default: `D()`with `D=typename property_traits<DistanceMap>::value_type`
> Python: Unsupported parameter.

UTIL/OUT: `color_map(ColorMap c_map)`

> 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 `ColorMap` must be a model of Read/Write Property Map. 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 Color Value.
> Default: an `iterator_property_map` created from a `std::vector` of `default_color_type` of size `num_vertices(g)` and using the `i_map` for the index map.
> Python: The color map must be a `vertex_color_map` for the graph.

OUT: `visitor(DijkstraVisitor v)`

> 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 [2].
> Default: `dijkstra_visitor<null_visitor>`
> Python: The parameter should be an object that derives from the `DijkstraVisitor` type of the graph.

### Complexity

The time complexity is O(V log V + E).

### Visitor Event Points

- `vis.initialize_vertex(u, g)` is invoked on each vertex in the graph before the start of the algorithm.
- `vis.examine_vertex(u, g)` is invoked on a vertex as it is removed from the priority queue and added to set S. At this point we know that (p[u],u) is a shortest-paths tree edge so d[u] = delta(s,u) = d[p[u]] + w(p[u],u). Also, the distances of the examined vertices is monotonically increasing d[u1] <= d[u2] <= d[un].
- `vis.examine_edge(e, g)` is invoked on each out-edge of a vertex immediately after it has been added to set S.
- `vis.edge_relaxed(e, g)` is invoked on edge (u,v) if d[u] + w(u,v) < d[v]. The edge (u,v) that participated in the last relaxation for vertex v is an edge in the shortest paths tree.
- `vis.discover_vertex(v, g)` is invoked on vertex v when the edge (u,v) is examined and v is WHITE. Since a vertex is colored GRAY when it is discovered, each reacable vertex is discovered exactly once. This is also when the vertex is inserted into the priority queue.
- `vis.edge_not_relaxed(e, g)` is invoked if the edge is not relaxed (see above).
- `vis.finish_vertex(u, g)` is invoked on a vertex after all of its out edges have been examined.

### Example

See `example/dijkstra-example.cpp` for an example of using Dijkstra's algorithm.

### See also

dijkstra_shortest_paths_no_color_map for a version of dijkstra's shortest path that does not use a color map.

### Notes

[1] 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.

[2] 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.

Copyright © 2000-2001 | Jeremy Siek, Indiana University (jsiek@osl.iu.edu)