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.

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Example

A Simple Weighted Graph	Minimum-Cost Spanning Tree

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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

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Example

Procedure for finding Minimum Spanning Tree

Step1. Edges are sorted in ascending order by weight.

Edge No.	Vertex Pair	Edge 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

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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

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