# `prim_minimum_spanning_tree`

    template <class VertexListGraph, class P, class T, class R>
    void prim_minimum_spanning_tree(VertexListGraph& g,
      typename graph_traits<VertexListGraph>::vertex_descriptor s,
      const bgl_named_params<P, T, R>& params);

This is Prim's algorithm [25,8,27,15] 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.

    PRIM-MST(G, s, w)
      for each vertex u in V[G]
        color[u] := WHITE
        d[u] := infinity
      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

### Where Defined

`boost/graph/prim_minimum_spanning_tree.hpp`

### Parameters

IN: `const VertexListGraph& g`

> An undirected graph. The type `VertexListGraph` must be a model of Vertex List Graph.

OUT: `PredecessorMap 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 `PredecessorMap` type must be a Read/Write Property Map with key and vertex types the same as the vertex descriptor type of the graph.

### Named Parameters

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 Addable with the value type of the distance map.
> 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)`

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 must be Less Than Comparable.
> 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>`

### Complexity

The time complexity is O(E log V).

### Example

The file `examples/prim.cpp` contains 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.

Copyright © 2000 | Jeremy Siek, Univ.of Notre Dame (jsiek@lsc.nd.edu)