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Prim Minimum Spanning Tree

Solves the minimum spanning tree problem for an undirected graph with weighted edges using Prim’s algorithm.

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

Example

// Prim Minimum Spanning Tree example
#include <iostream>
#include <vector>
#include <limits>
#include <boost/graph/adjacency_list.hpp>
#include <boost/graph/prim_minimum_spanning_tree.hpp>

struct EdgeWeight { int weight; };

using Graph = boost::adjacency_list<
    boost::vecS, boost::vecS, boost::undirectedS, boost::no_property, EdgeWeight>;
using Vertex = boost::graph_traits<Graph>::vertex_descriptor;

int main() {
    Graph g{5};
    auto add = [&](int u, int v, int w) { boost::add_edge(u, v, EdgeWeight{w}, g); };
    add(0, 1, 2); add(0, 3, 1); add(1, 2, 3);
    add(1, 3, 2); add(2, 4, 4); add(3, 4, 6);

    auto n = boost::num_vertices(g);
    std::vector<Vertex> pred(n);
    std::vector<int> dist(n);
    auto index_map = boost::get(boost::vertex_index, g);
    auto pred_map = boost::make_iterator_property_map(pred.begin(), index_map);
    auto dist_map = boost::make_iterator_property_map(dist.begin(), index_map);
    auto weight_map = boost::get(&EdgeWeight::weight, g);

    boost::prim_minimum_spanning_tree(g, *boost::vertices(g).first,
        pred_map, dist_map, weight_map, index_map,
        boost::default_dijkstra_visitor{});

    std::cout << "Predecessor map (vertex : parent):\n";
    for (Vertex v = 0; v < boost::num_vertices(g); ++v) {
        std::cout << "  " << v << " : " << pred[v] << "\n";
    }
}
Predecessor map (vertex : parent):
  0 : 0
  1 : 0
  2 : 1
  3 : 0
  4 : 2

(1) Positional version

template <typename Graph, typename DijkstraVisitor,
          typename PredecessorMap, typename DistanceMap,
          typename WeightMap, typename IndexMap>
void prim_minimum_spanning_tree(
    const Graph& g,
    typename graph_traits<Graph>::vertex_descriptor s,
    PredecessorMap predecessor, DistanceMap distance, WeightMap weight,
    IndexMap index_map, DijkstraVisitor vis);
Direction Parameter Description

IN

const Graph& g

An undirected graph. The type Graph must be a model of Vertex List Graph and Incidence Graph. It should not contain parallel edges.

IN

vertex_descriptor s

The root vertex for the minimum spanning tree. The choice of the root vertex is arbitrary.

OUT

PredecessorMap predecessor

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 root of the tree or is a vertex that is not reachable from the root. The PredecessorMap type must be a Read/Write Property Map with key and vertex types the same as the vertex descriptor type of the graph.

UTIL/OUT

DistanceMap distance

The weight of the spanning tree edge into each vertex in the graph g is recorded in this property map, with edges directed away from the spanning tree root. 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, and the value type needs to be the same as the value type of the WeightMap argument.

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 the same as the value type of the distance map, and that type must be Less Than Comparable.

IN

IndexMap 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 when an edge is relaxed. The type IndexMap 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

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].

(2) Named parameter version

template <typename Graph, typename PredMap,
          typename P, typename T, typename R>
void prim_minimum_spanning_tree(
    const Graph& g, PredMap p_map,
    const bgl_named_params<P, T, R>& params);
Direction Parameter Description

IN

const Graph& g

An undirected graph. The type Graph must be a model of Vertex List Graph and Incidence Graph. It should not contain parallel edges.

OUT

PredMap p_map

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 root of the tree or is a vertex that is not reachable from the root. The PredMap type must be a Read/Write Property Map with key and vertex types the same as the vertex descriptor type of the graph.

IN

params

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

IN

root_vertex(vertex_descriptor r)

The vertex that will be the root of the minimum spanning tree. The choice of the root vertex is arbitrary.
Default: *vertices(g).first

IN

weight_map(WeightMap w_map)

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 the same as the value type of the distance map, and that type must be Less Than Comparable.
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.

UTIL/OUT

distance_map(DistanceMap d_map)

The weight of the spanning tree edge into each vertex in the graph g is recorded in this property map, with edges directed away from the spanning tree root. 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, and the value type needs to be the same as the value type of the weight_map argument.
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.

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.

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 [1].
Default: dijkstra_visitor<null_visitor>

Description

This is Prim’s algorithm

for solving the minimum spanning tree problem for an undirected graph with weighted edges. A MST is a set of edges that connects all the vertices in the graph where the total weight of the edges in the tree is minimized. See Section Minimum Spanning Tree Problem for more details. The implementation is simply a call to dijkstra_shortest_paths() with the appropriate choice of comparison and combine functors. The pseudo-code for Prim’s algorithm is listed below. The algorithm as implemented in Boost.Graph does not produce correct results on graphs with parallel edges.

Pseudo-Code

PRIM-MST(G, s, w)
  for each vertex u in V[G]
    color[u] := WHITE
    d[u] := infinity
  end for
  color[s] := GRAY
  d[s] := 0
  ENQUEUE(PQ, s)
  p[s] := s
  while (PQ != Ø)
    u := DEQUEUE(PQ)
    for each v in Adj[u]
      if (w(u,v) < d[v])
        d[v] := w(u,v)
        p[v] := u
        if (color[v] = WHITE)
          ENQUEUE(PQ, v)
          color[v] := GRAY
        else if (color[v] = GRAY)
          UPDATE(PQ, v)
      else
        do nothing
    end for
    color[u] := BLACK
  end while
  return (p, d)
 
.
initialize vertex u
.
.
.
start vertex s
discover vertex s
.
.
examine vertex u
examining edge (u,v)
.
edge (u,v) relaxed
.
.
discover vertex v
.
.
.
.
edge (u,v) not relaxed
.
finish 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.

  • vis.examine_edge(e, g) is invoked on each out-edge of a vertex immediately after it has been examined.

  • vis.edge_relaxed(e, g) is invoked on edge (u,v) if w(u,v) < d[v]. The edge (u,v) that participated in the last relaxation for vertex v is an edge in the minimum spanning 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.

Example

See example/prim-example.cpp for an example of using Prim’s 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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