Prim’s Minimum Spanning Tree Algorithm

There is a connected graph G(V, E) and the weight or cost for every edge is given. Prim’s Algorithm will find the minimum spanning tree from the graph G. 

It is a growing tree approach. This algorithm needs a seed value to start the tree. The seed vertex is grown to form the whole tree.

The problem will be solved using two sets. One set holds the nodes that are already selected, and another set holds the item those are not considered yet. From the seed vertex, it takes adjacent vertices, based on minimum edge cost, thus it grows the tree by taking nodes one by one.

The time complexity of this problem is O(V^2). Here V is the number of vertices.

Input and Output

Input:
The adjacency list:

Output:
(0)---(1|1)  (0)---(2|3)  (0)---(3|4)
(1)---(0|1)  (1)---(4|2)
(2)---(0|3)
(3)---(0|4)
(4)---(1|2)  (4)---(5|2)
(5)---(4|2)  (5)---(6|3)
(6)---(5|3)

Algorithm

prims(g: Graph, t: tree, start)

Input − The graph g, A blank tree and the seed vertex named ‘start’

Output − The Tree after adding edges.

Begin
   define two sets as usedVert, unusedVert
   usedVert[0] := start and unusedVert[0] := φ

   for all vertices except start do
      usedVert[i] := φ
      unusedVert[i] := i    //add all vertices in unused list
   done

   while number of vertices in usedVert ≠ V do    //V is number of total nodes
      min := ∞
      for all vertices of usedVert array do
         for all vertices of the graph do
            if min > cost[i,j] AND i ≠ j then
               min := cost[i,j]
               ed := edge between i and j, and cost of ed := min
         done
      done

      unusedVert[destination of ed] := φ
      add edge ed into the tree t
      add source of ed into usedVert
   done
End

Example

#include<iostream>
#define V 7
#define INF 999
using namespace std;

//Cost matrix of the graph
int costMat[V][V] = {
   {0, 1, 3, 4, INF, 5, INF},
   {1, 0, INF, 7, 2, INF, INF},
   {3, INF, 0, INF, 8, INF, INF},
   {4, 7, INF, 0, INF, INF, INF},
   {INF, 2, 8, INF, 0, 2, 4},
   {5, INF, INF, INF, 2, 0, 3},
   {INF, INF, INF, INF, 4, 3, 0}
};

typedef struct {
   int u, v, cost;
}edge;

class Tree {
   int n;
   edge edges[V-1];    //as a tree has vertex-1 edges
   public:
      Tree() {
         n = 0;
      }

      void addEdge(edge e) {
         edges[n] = e;    //add edge e into the tree
         n++;
      }

      void printEdges() {    //print edge, cost and total cost
         int tCost = 0;

         for(int i = 0; i<n; i++) {
            cout << "Edge: " << char(edges[i].u+'A') << "--" << char(edges[i].v+'A');
            cout << " And Cost: " << edges[i].cost << endl;
            tCost += edges[i].cost;
         }
         cout << "Total Cost: " << tCost << endl;
      }
      friend void prims(Tree &tre, int start);
};

void prims(Tree &tr, int start) {
   int usedVert[V], unusedVert[V];
   int i, j, min, p;
   edge ed;

   //initialize
   usedVert[0] = start; p = 1;
   unusedVert[0] = -1;    //-1 indicates the place is empty

   for(i = 1; i<V; i++) {
      usedVert[i] = -1;    //all places except first is empty
      unusedVert[i] = i;   //fill with vertices
   }

   tr.n = 0;
   //get edges and add to tree
   while(p != V) {     //p is number of vertices in usedVert array
      min = INF;
      for(i = 0; i<p; i++) {
         for(j = 0; j<V; j++) {
            if(unusedVert[j] != -1) {
               if(min > costMat[i][j] && costMat[i][j] != 0) {
                  //find the edge with minimum cost
                  //such that u is considered and v is not considered yet
                  min = costMat[i][j];
                  ed.u = i; ed.v = j; ed.cost = min;
               }
            }
         }
      }
      unusedVert[ed.v] = -1;     //delete v from unusedVertex
      tr.addEdge(ed);
      usedVert[p] = ed.u; p++;   //add u to usedVertex
   }
}

main() {
   Tree tr;
   prims(tr, 0);    //starting node 0
   tr.printEdges();
}

Output

(0)---(1|1)  (0)---(2|3)  (0)---(3|4)
(1)---(0|1)  (1)---(4|2)
(2)---(0|3)
(3)---(0|4)
(4)---(1|2)  (4)---(5|2)
(5)---(4|2)  (5)---(6|3)
(6)---(5|3)