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Kruskal's Algorithm

Kruskal's algorithm is a greedy algorithm in graph theory that finds a minimum spanning tree for a connected weighted graph.
It finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized.
This algorithm is directly based on the MST( minimum spanning tree) property.



Example

A Simple Weighted Graph Minimum-Cost Spanning Tree



Kruskal's Algorithm

MST-KRUSKAL(G, w)
1.	A ← Ø
2.	for each vertex v  V[G]
3.		do MAKE-SET(v)
4.	sort the edges of E into nondecreasing order by weight w
5.	for each edge (u, v)  E, taken in nondecreasing order by weight
6.		do if FIND-SET(u) ≠ FIND-SET(v)
7.			then A ← A  {(u, v)}
8.				UNION(u, v)
9.	return A



Example


Procedure for finding Minimum Spanning Tree

Step1. Edges are sorted in ascending order by weight.
Edge No.Vertex PairEdge Weight
E1(0,2)1
E2(3,5)2
E3(0,1)3
E4(1,4)3
E5(2,5)4
E6(1,2)5
E7(2,3)5
E8(0,3)6
E9(2,4)6
E10(4,5)6

Step2. Edges are added in sequence.
Graph                                
Add Edge E1
Add Edge E2
Add Edge E3
Add Edge E4
Add Edge E5
Total Cost = 1+2+3+3+4 = 13s


C IMPLEMETATION of Kruskal's Algorithm

#include<stdio.h>
#include<conio.h>
#include<stdlib.h>
int i,j,k,a,b,u,v,n,ne=1;
int min,mincost=0,cost[9][9],parent[9];
int find(int);
int uni(int,int);
void main()
{
	clrscr();
	printf("\n\tImplementation of Kruskal's algorithm\n");
	printf("\nEnter the no. of vertices:");
	scanf("%d",&n);
	printf("\nEnter the cost adjacency matrix:\n");
	for(i=1;i<=n;i++)
	{
		for(j=1;j<=n;j++)
		{
			scanf("%d",&cost[i][j]);
			if(cost[i][j]==0)
				cost[i][j]=999;
		}
	}
	printf("The edges of Minimum Cost Spanning Tree are\n");
	while(ne < n)
	{
		for(i=1,min=999;i<=n;i++)
		{
			for(j=1;j <= n;j++)
			{
				if(cost[i][j] < min)
				{
					min=cost[i][j];
					a=u=i;
					b=v=j;
				}
			}
		}
		u=find(u);
		v=find(v);
		if(uni(u,v))
		{
			printf("%d edge (%d,%d) =%d\n",ne++,a,b,min);
			mincost +=min;
		}
		cost[a][b]=cost[b][a]=999;
	}
	printf("\n\tMinimum cost = %d\n",mincost);
	getch();
}
int find(int i)
{
	while(parent[i])
	i=parent[i];
	return i;
}
int uni(int i,int j)
{
	if(i!=j)
	{
		parent[j]=i;
		return 1;
	}
	return 0;
}

output