DAG Shortest Paths

Solves the single-source shortest-paths problem on a weighted, directed acyclic graph (DAG), more efficiently than either Dijkstra or Bellman-Ford.

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

Example

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

struct Node {};
struct Edge { int weight; };

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

int main() {
    Graph g{5};
    add_edge(0, 1, Edge{2}, g);
    add_edge(0, 2, Edge{6}, g);
    add_edge(1, 3, Edge{5}, g);
    add_edge(2, 3, Edge{1}, g);
    add_edge(3, 4, Edge{3}, g);

    std::vector<int> dist(num_vertices(g));
    std::vector<Vertex> pred(num_vertices(g));
    std::vector<default_color_type> color(num_vertices(g));

    auto idx = get(vertex_index, g);
    dag_shortest_paths(g, vertex(0, g),
        make_iterator_property_map(dist.begin(), idx),
        get(&Edge::weight, g),
        make_iterator_property_map(color.begin(), idx),
        make_iterator_property_map(pred.begin(), idx),
        dijkstra_visitor<null_visitor>(),
        std::less<int>(), std::plus<int>(),
        (std::numeric_limits<int>::max)(), 0);

    for (auto v : make_iterator_range(vertices(g)))
        std::cout << "distance to " << v << " = " << dist[v] << "\n";
}
distance to 0 = 0
distance to 1 = 2
distance to 2 = 6
distance to 3 = 7
distance to 4 = 10

(1) Positional version

template <class VertexListGraph, class DijkstraVisitor,
          class DistanceMap, class WeightMap, class ColorMap,
          class PredecessorMap,
          class Compare, class Combine,
          class DistInf, class DistZero>
void dag_shortest_paths(
    const VertexListGraph& g,
    typename graph_traits<VertexListGraph>::vertex_descriptor s,
    DistanceMap distance, WeightMap weight, ColorMap color,
    PredecessorMap pred, DijkstraVisitor vis,
    Compare compare, Combine combine, DistInf inf, DistZero zero);
Direction Parameter Description

IN

const VertexListGraph& g

The graph object on which the algorithm will be applied. The type VertexListGraph must be a model of Vertex List 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.

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 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 the map must be Addable with the value type of the distance map.

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.

OUT

PredecessorMap pred

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.

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 [1].

IN

Compare compare

This function is used to compare distances to determine which vertex is closer to the source vertex. The Compare type must be a model of Binary Predicate and have argument types that match the value type of the DistanceMap property map.

IN

Combine combine

This function is used to combine distances to compute the distance of a path. The Combine 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.

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.

(2) Named parameter version

template <class VertexListGraph, class Param, class Tag, class Rest>
void dag_shortest_paths(
    const VertexListGraph& g,
    typename graph_traits<VertexListGraph>::vertex_descriptor s,
    const bgl_named_params<Param, Tag, Rest>& params);
Direction Parameter Description

IN

const VertexListGraph& g

The graph object on which the algorithm will be applied. The type VertexListGraph must be a model of Vertex List 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.

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)

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 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, adjacency_list with VertexList=listS does not have an internal vertex_index property.

OUT

predecessor_map(PredecessorMap p_map)

Predecessor map (see (1) PredecessorMap pred).
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) Compare compare).
Default: std::less<D> with D=typename property_traits<DistanceMap>::value_type

IN

distance_combine(CombineFunction cmb)

Distance combination function (see (1) Combine 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>

Description

This algorithm [5] solves the single-source shortest-paths problem on a weighted, directed acyclic graph (DAG). This algorithm is more efficient for DAG’s than either the Dijkstra or Bellman-Ford algorithm. Use breadth-first search instead of this 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 dag_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.

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 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.

  • 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/dag_shortest_paths.cpp for an example of using this algorithm.

Notes

[1] 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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