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Dijkstra’s Shortest Paths

Solves the single-source shortest-paths problem on a weighted, directed or undirected graph for the case where all edge weights are nonnegative.

Complexity: O(V log V + E)
Defined in: <boost/graph/dijkstra_shortest_paths.hpp>

Example

#include <boost/graph/adjacency_list.hpp>
#include <boost/graph/dijkstra_shortest_paths.hpp>
#include <boost/graph/visitors.hpp>
#include <iostream>
#include <limits>
#include <vector>

struct City {};
struct Road { int cost; };

using namespace boost;
using Graph = adjacency_list<vecS, vecS, directedS, City, Road>;
using Vertex = graph_traits<Graph>::vertex_descriptor;

int main() {
    Graph g(4);
    add_edge(0, 1, Road{1}, g);
    add_edge(1, 2, Road{2}, g);
    add_edge(0, 2, Road{10}, g);
    add_edge(2, 3, Road{1}, g);

    // Storage: you control allocation, lifetime, and container type
    std::vector<Vertex> storage_pred(num_vertices(g));
    std::vector<int>    storage_dist(num_vertices(g));

    // Property maps: lightweight views into the storage
    auto index_map       = get(vertex_index, g);
    auto costs_map       = get(&Road::cost, g);
    auto predecessor_map = make_iterator_property_map(storage_pred.begin(), index_map);
    auto distance_map    = make_iterator_property_map(storage_dist.begin(), index_map);

    dijkstra_shortest_paths(g, vertex(0, g),
        predecessor_map, distance_map,
        costs_map, index_map,
        std::less<int>(), std::plus<int>(),
        std::numeric_limits<int>::max(), 0,
        dijkstra_visitor<null_visitor>());

    for (auto v : make_iterator_range(vertices(g)))
        std::cout << "distance to " << v << " = " << storage_dist[v] << "\n";
}
distance to 0 = 0
distance to 1 = 1
distance to 2 = 3
distance to 3 = 4

(1) Positional 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);
Direction Parameter Description

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.

IN

vertex_descriptor s

The source vertex. All distances will be calculated from this vertex, and the shortest paths tree will be rooted at this vertex.

OUT

PredecessorMap predecessor

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.

UTIL/OUT

DistanceMap distance

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.

IN

WeightMap weight

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 if 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

VertexIndexMap index_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 [55] 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.

IN

CompareFunction compare

This function is used 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.

IN

CombineFunction combine

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.

IN

DistInf 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. Edges are assumed to have a weight less than (using distance_compare for comparison) this value.

IN

DistZero 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.

IN

DijkstraVisitor vis

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].

UTIL/OUT

ColorMap color

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.

(2) No-init version

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);

Same parameters as (1), but does not initialize the property maps (except for the default color map). Use this when you want to control initialization yourself.

(3) 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);
Direction Parameter Description

IN

const Graph& g

The graph object. Must model Vertex List Graph and Incidence Graph.

IN

vertex_descriptor s

The source vertex. All distances will be calculated from this vertex.

IN

params

Named parameters passed via bgl_named_params. The following are accepted:
Direction Named Parameter Description / Default

IN

weight_map(WeightMap w_map)

Edge weights (see (1) WeightMap weight).
Default: get(edge_weight, g)

IN

vertex_index_map(VertexIndexMap i_map)

Vertex-to-integer mapping (see (1) VertexIndexMap index_map).
Default: get(vertex_index, g). Note: if you use this default, make sure your graph has an internal vertex_index property. For example, adjacency_list with VertexList=listS does not have an internal vertex_index property.

OUT

predecessor_map(PredecessorMap p_map)

Predecessor map (see (1) PredecessorMap predecessor).
Default: dummy_property_map

UTIL/OUT

distance_map(DistanceMap d_map)

Distance map (see (1) DistanceMap distance).
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.

IN

distance_compare(CompareFunction cmp)

Distance comparison function (see (1) CompareFunction compare).
Default: std::less<D> with D=typename property_traits<DistanceMap>::value_type

IN

distance_combine(CombineFunction cmb)

Distance combination function (see (1) CombineFunction combine).
Default: std::plus<D> with D=typename property_traits<DistanceMap>::value_type

IN

distance_inf(D inf)

Infinity value (see (1) DistInf inf).
Default: std::numeric_limits<D>::max()

IN

distance_zero(D zero)

Identity element (see (1) DistZero zero).
Default: D() with D=typename property_traits<DistanceMap>::value_type

UTIL/OUT

color_map(ColorMap c_map)

Color map (see (1) ColorMap color).
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.

OUT

visitor(DijkstraVisitor v)

Visitor (see (1) DijkstraVisitor vis).
Default: dijkstra_visitor<null_visitor>
DijkstraVisitor type of the graph.

Description

Dijkstra’s algorithm

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.

Pseudo-Code

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
      (not run in _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
.
.

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 reachable 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.

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.

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