Kruskal’s Algorithm (Simple Implementation for Adjacency Matrix)

Below are the steps for finding MST using Kruskal’s algorithm

1. Sort all the edges in non-decreasing order of their weight. 
2. Pick the smallest edge. Check if it forms a cycle with the spanning tree formed so far. If cycle is not formed, include this edge. Else, discard it. 
3. Repeat step#2 until there are (V-1) edges in the spanning tree.

We have discussed one implementation of Kruskal’s algorithm in previous post. In this post, a simpler implementation for adjacency matrix is discussed. 

Implementation:

C++

// Simple C++ implementation for Kruskal's
// algorithm
#include <bits/stdc++.h>
using namespace std;

#define V 5
int parent[V];

// Find set of vertex i
int find(int i)
{
    while (parent[i] != i)
        i = parent[i];
    return i;
}

// Does union of i and j. It returns
// false if i and j are already in same
// set.
void union1(int i, int j)
{
    int a = find(i);
    int b = find(j);
    parent[a] = b;
}

// Finds MST using Kruskal's algorithm
void kruskalMST(int cost[][V])
{
    int mincost = 0; // Cost of min MST.

    // Initialize sets of disjoint sets.
    for (int i = 0; i < V; i++)
        parent[i] = i;

    // Include minimum weight edges one by one
    int edge_count = 0;
    while (edge_count < V - 1) {
        int min = INT_MAX, a = -1, b = -1;
        for (int i = 0; i < V; i++) {
            for (int j = 0; j < V; j++) {
                if (find(i) != find(j) && cost[i][j] < min) {
                    min = cost[i][j];
                    a = i;
                    b = j;
                }
            }
        }

        union1(a, b);
        printf("Edge %d:(%d, %d) cost:%d \n",
               edge_count++, a, b, min);
        mincost += min;
    }
    printf("\n Minimum cost= %d \n", mincost);
}

// driver program to test above function
int main()
{
    /* Let us create the following graph
          2    3
      (0)--(1)--(2)
       |   / \   |
      6| 8/   \5 |7
       | /     \ |
      (3)-------(4)
            9          */
    int cost[][V] = {
        { INT_MAX, 2, INT_MAX, 6, INT_MAX },
        { 2, INT_MAX, 3, 8, 5 },
        { INT_MAX, 3, INT_MAX, INT_MAX, 7 },
        { 6, 8, INT_MAX, INT_MAX, 9 },
        { INT_MAX, 5, 7, 9, INT_MAX },
    };

    // Print the solution
    kruskalMST(cost);

    return 0;
}

Java

// Simple Java implementation for Kruskal's
// algorithm
import java.util.*;

class GFG
{

static int V = 5;
static int[] parent = new int[V];
static int INF = Integer.MAX_VALUE;

// Find set of vertex i
static int find(int i)
{
    while (parent[i] != i)
        i = parent[i];
    return i;
}

// Does union of i and j. It returns
// false if i and j are already in same
// set.
static void union1(int i, int j)
{
    int a = find(i);
    int b = find(j);
    parent[a] = b;
}

// Finds MST using Kruskal's algorithm
static void kruskalMST(int cost[][])
{
    int mincost = 0; // Cost of min MST.

    // Initialize sets of disjoint sets.
    for (int i = 0; i < V; i++)
        parent[i] = i;

    // Include minimum weight edges one by one
    int edge_count = 0;
    while (edge_count < V - 1)
    {
        int min = INF, a = -1, b = -1;
        for (int i = 0; i < V; i++)
        {
            for (int j = 0; j < V; j++)
            {
                if (find(i) != find(j) && cost[i][j] < min)
                {
                    min = cost[i][j];
                    a = i;
                    b = j;
                }
            }
        }

        union1(a, b);
        System.out.printf("Edge %d:(%d, %d) cost:%d \n",
            edge_count++, a, b, min);
        mincost += min;
    }
    System.out.printf("\n Minimum cost= %d \n", mincost);
}

// Driver code
public static void main(String[] args)
{
/* Let us create the following graph
        2 3
    (0)--(1)--(2)
    | / \ |
    6| 8/ \5 |7
    | /     \ |
    (3)-------(4)
            9         */
    int cost[][] = {
        { INF, 2, INF, 6, INF },
        { 2, INF, 3, 8, 5 },
        { INF, 3, INF, INF, 7 },
        { 6, 8, INF, INF, 9 },
        { INF, 5, 7, 9, INF },
    };

    // Print the solution
    kruskalMST(cost);
    }
}

// This code contributed by Rajput-Ji

Python3

# Python implementation for Kruskal's
# algorithm

# Find set of vertex i
def find(i):
    while parent[i] != i:
        i = parent[i]
    return i

# Does union of i and j. It returns
# false if i and j are already in same
# set.
def union(i, j):
    a = find(i)
    b = find(j)
    parent[a] = b

# Finds MST using Kruskal's algorithm
def kruskalMST(cost):
    mincost = 0 # Cost of min MST

    # Initialize sets of disjoint sets
    for i in range(V):
        parent[i] = i

    # Include minimum weight edges one by one
    edge_count = 0
    while edge_count < V - 1:
        min = INF
        a = -1
        b = -1
        for i in range(V):
            for j in range(V):
                if find(i) != find(j) and cost[i][j] < min:
                    min = cost[i][j]
                    a = i
                    b = j
        union(a, b)
        print('Edge {}:({}, {}) cost:{}'.format(edge_count, a, b, min))
        edge_count += 1
        mincost += min

    print("Minimum cost= {}".format(mincost))


# Driver code
# Let us create the following graph
#         2 3
#     (0)--(1)--(2)
#     | / \ |
#     6| 8/ \5 |7
#     | /     \ |
#     (3)-------(4)
#         9

V = 5
parent = [i for i in range(V)]
INF = float('inf')
cost = [[INF, 2, INF, 6, INF],
        [2, INF, 3, 8, 5],
        [INF, 3, INF, INF, 7],
        [6, 8, INF, INF, 9],
        [INF, 5, 7, 9, INF]]

# Print the solution
kruskalMST(cost)

# This code is contributed by ng24_7

C#

// Simple C# implementation for Kruskal's
// algorithm
using System;

class GFG
{

static int V = 5;
static int[] parent = new int[V];
static int INF = int.MaxValue;

// Find set of vertex i
static int find(int i)
{
    while (parent[i] != i)
        i = parent[i];
    return i;
}

// Does union of i and j. It returns
// false if i and j are already in same
// set.
static void union1(int i, int j)
{
    int a = find(i);
    int b = find(j);
    parent[a] = b;
}

// Finds MST using Kruskal's algorithm
static void kruskalMST(int [,]cost)
{
    int mincost = 0; // Cost of min MST.

    // Initialize sets of disjoint sets.
    for (int i = 0; i < V; i++)
        parent[i] = i;

    // Include minimum weight edges one by one
    int edge_count = 0;
    while (edge_count < V - 1)
    {
        int min = INF, a = -1, b = -1;
        for (int i = 0; i < V; i++)
        {
            for (int j = 0; j < V; j++)
            {
                if (find(i) != find(j) && cost[i, j] < min)
                {
                    min = cost[i, j];
                    a = i;
                    b = j;
                }
            }
        }

        union1(a, b);
        Console.Write("Edge {0}:({1}, {2}) cost:{3} \n",
            edge_count++, a, b, min);
        mincost += min;
    }
    Console.Write("\n Minimum cost= {0} \n", mincost);
}

// Driver code
public static void Main(String[] args)
{
/* Let us create the following graph
        2 3
    (0)--(1)--(2)
    | / \ |
    6| 8/ \5 |7
    | /     \ |
    (3)-------(4)
            9         */
    int [,]cost = {
        { INF, 2, INF, 6, INF },
        { 2, INF, 3, 8, 5 },
        { INF, 3, INF, INF, 7 },
        { 6, 8, INF, INF, 9 },
        { INF, 5, 7, 9, INF },
    };

    // Print the solution
    kruskalMST(cost);
}
}

/* This code contributed by PrinciRaj1992 */

Javascript

<script>


// Simple Javascript implementation for Kruskal's
// algorithm


var V = 5;
var parent = Array(V).fill(0);
var INF = 1000000000;

// Find set of vertex i
function find(i)
{
    while (parent[i] != i)
        i = parent[i];
    return i;
}

// Does union of i and j. It returns
// false if i and j are already in same
// set.
function union1(i, j)
{
    var a = find(i);
    var b = find(j);
    parent[a] = b;
}

// Finds MST using Kruskal's algorithm
function kruskalMST(cost)
{
    var mincost = 0; // Cost of min MST.

    // Initialize sets of disjoint sets.
    for (var i = 0; i < V; i++)
        parent[i] = i;

    // Include minimum weight edges one by one
    var edge_count = 0;
    while (edge_count < V - 1)
    {
        var min = INF, a = -1, b = -1;
        for (var i = 0; i < V; i++)
        {
            for (var j = 0; j < V; j++)
            {
                if (find(i) != find(j) && cost[i][j] < min)
                {
                    min = cost[i][j];
                    a = i;
                    b = j;
                }
            }
        }

        union1(a, b);
        document.write(`Edge ${edge_count++}:(${a},
        ${b}) cost:${min} <br>`);
        mincost += min;
    }
    document.write(`<br> Minimum cost= ${mincost} <br>`);
}

// Driver code

/* Let us create the following graph
        2 3
    (0)--(1)--(2)
    | / \ |
    6| 8/ \5 |7
    | /     \ |
    (3)-------(4)
            9         */
var cost = [
    [INF, 2, INF, 6, INF],
    [2, INF, 3, 8, 5],
    [INF, 3, INF, INF, 7],
    [6, 8, INF, INF, 9],
    [INF, 5, 7, 9, INF]];
// Print the solution
kruskalMST(cost);


</script>

Output

Edge 0:(0, 1) cost:2 
Edge 1:(1, 2) cost:3 
Edge 2:(1, 4) cost:5 
Edge 3:(0, 3) cost:6 

 Minimum cost= 16 

Time Complexity:
The time complexity of Kruskal’s algorithm using the Union-Find algorithm for finding the cycle and sorting the edges is O(E log E + E log V), where E is the number of edges and V is the number of vertices in the graph. 

Space Complexity:
The space complexity of the program is O(V), where V is the number of vertices in the graph.

Note that the above solution is not efficient. The idea is to provide a simple implementation for adjacency matrix representations. Please see below for efficient implementations. 

Kruskal’s Minimum Spanning Tree Algorithm | Greedy Algo-2
Kruskal’s Minimum Spanning Tree using STL in C++