Kruskal’s Minimum Spanning Tree Algorithm | Greedy Algo-2

What is Minimum Spanning Tree?
Given a connected and undirected graph, a spanning tree of that graph is a subgraph that is a tree and connects all the vertices together. A single graph can have many different spanning trees. A minimum spanning tree (MST) or minimum weight spanning tree for a weighted, connected and undirected graph is a spanning tree with weight less than or equal to the weight of every other spanning tree. The weight of a spanning tree is the sum of weights given to each edge of the spanning tree.

How many edges does a minimum spanning tree has?
A minimum spanning tree has (V – 1) edges where V is the number of vertices in the given graph.

What are the applications of Minimum Spanning Tree?
See this for applications of MST.



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.

The step#2 uses Union-Find algorithm to detect cycle. So we recommend to read following post as a prerequisite.
Union-Find Algorithm | Set 1 (Detect Cycle in a Graph)
Union-Find Algorithm | Set 2 (Union By Rank and Path Compression)

The algorithm is a Greedy Algorithm. The Greedy Choice is to pick the smallest weight edge that does not cause a cycle in the MST constructed so far. Let us understand it with an example: Consider the below input graph.

The graph contains 9 vertices and 14 edges. So, the minimum spanning tree formed will be having (9 – 1) = 8 edges. 

After sorting:
Weight   Src    Dest
1         7      6
2         8      2
2         6      5
4         0      1
4         2      5
6         8      6
7         2      3
7         7      8
8         0      7
8         1      2
9         3      4
10        5      4
11        1      7
14        3      5


Now pick all edges one by one from sorted list of edges
1. Pick edge 7-6: No cycle is formed, include it.

2. Pick edge 8-2: No cycle is formed, include it.

3. Pick edge 6-5: No cycle is formed, include it.

4. Pick edge 0-1: No cycle is formed, include it.



5. Pick edge 2-5: No cycle is formed, include it.

6. Pick edge 8-6: Since including this edge results in cycle, discard it.

7. Pick edge 2-3: No cycle is formed, include it.

8. Pick edge 7-8: Since including this edge results in cycle, discard it.

9. Pick edge 0-7: No cycle is formed, include it.

10. Pick edge 1-2: Since including this edge results in cycle, discard it.

11. Pick edge 3-4: No cycle is formed, include it.

Since the number of edges included equals (V – 1), the algorithm stops here.

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// C++ program for Kruskal's algorithm to find Minimum Spanning Tree 
// of a given connected, undirected and weighted graph 
#include <bits/stdc++.h>
using namespace std;
  
// a structure to represent a weighted edge in graph 
class Edge 
{ 
    public:
    int src, dest, weight; 
}; 
  
// a structure to represent a connected, undirected 
// and weighted graph 
class Graph 
{ 
    public:
    // V-> Number of vertices, E-> Number of edges 
    int V, E; 
  
    // graph is represented as an array of edges. 
    // Since the graph is undirected, the edge 
    // from src to dest is also edge from dest 
    // to src. Both are counted as 1 edge here. 
    Edge* edge; 
}; 
  
// Creates a graph with V vertices and E edges 
Graph* createGraph(int V, int E) 
{ 
    Graph* graph = new Graph; 
    graph->V = V; 
    graph->E = E; 
  
    graph->edge = new Edge[E]; 
  
    return graph; 
} 
  
// A structure to represent a subset for union-find 
class subset 
{ 
    public:
    int parent; 
    int rank; 
}; 
  
// A utility function to find set of an element i 
// (uses path compression technique) 
int find(subset subsets[], int i) 
{ 
    // find root and make root as parent of i 
    // (path compression) 
    if (subsets[i].parent != i) 
        subsets[i].parent = find(subsets, subsets[i].parent); 
  
    return subsets[i].parent; 
} 
  
// A function that does union of two sets of x and y 
// (uses union by rank) 
void Union(subset subsets[], int x, int y) 
{ 
    int xroot = find(subsets, x); 
    int yroot = find(subsets, y); 
  
    // Attach smaller rank tree under root of high 
    // rank tree (Union by Rank) 
    if (subsets[xroot].rank < subsets[yroot].rank) 
        subsets[xroot].parent = yroot; 
    else if (subsets[xroot].rank > subsets[yroot].rank) 
        subsets[yroot].parent = xroot; 
  
    // If ranks are same, then make one as root and 
    // increment its rank by one 
    else
    { 
        subsets[yroot].parent = xroot; 
        subsets[xroot].rank++; 
    } 
} 
  
// Compare two edges according to their weights. 
// Used in qsort() for sorting an array of edges 
int myComp(const void* a, const void* b) 
{ 
    Edge* a1 = (Edge*)a; 
    Edge* b1 = (Edge*)b; 
    return a1->weight > b1->weight; 
} 
  
// The main function to construct MST using Kruskal's algorithm 
void KruskalMST(Graph* graph) 
{ 
    int V = graph->V; 
    Edge result[V]; // Tnis will store the resultant MST 
    int e = 0; // An index variable, used for result[] 
    int i = 0; // An index variable, used for sorted edges 
  
    // Step 1: Sort all the edges in non-decreasing 
    // order of their weight. If we are not allowed to 
    // change the given graph, we can create a copy of 
    // array of edges 
    qsort(graph->edge, graph->E, sizeof(graph->edge[0]), myComp); 
  
    // Allocate memory for creating V ssubsets 
    subset *subsets = new subset[( V * sizeof(subset) )]; 
  
    // Create V subsets with single elements 
    for (int v = 0; v < V; ++v) 
    { 
        subsets[v].parent = v; 
        subsets[v].rank = 0; 
    } 
  
    // Number of edges to be taken is equal to V-1 
    while (e < V - 1 && i < graph->E) 
    { 
        // Step 2: Pick the smallest edge. And increment 
        // the index for next iteration 
        Edge next_edge = graph->edge[i++]; 
  
        int x = find(subsets, next_edge.src); 
        int y = find(subsets, next_edge.dest); 
  
        // If including this edge does't cause cycle, 
        // include it in result and increment the index 
        // of result for next edge 
        if (x != y) 
        { 
            result[e++] = next_edge; 
            Union(subsets, x, y); 
        } 
        // Else discard the next_edge 
    } 
  
    // print the contents of result[] to display the 
    // built MST 
    cout<<"Following are the edges in the constructed MST\n"; 
    for (i = 0; i < e; ++i) 
        cout<<result[i].src<<" -- "<<result[i].dest<<" == "<<result[i].weight<<endl; 
    return; 
} 
  
// Driver code
int main() 
{ 
    /* Let us create following weighted graph 
            10 
        0--------1 
        | \ | 
    6| 5\ |15 
        | \ | 
        2--------3 
            4 */
    int V = 4; // Number of vertices in graph 
    int E = 5; // Number of edges in graph 
    Graph* graph = createGraph(V, E); 
  
  
    // add edge 0-1 
    graph->edge[0].src = 0; 
    graph->edge[0].dest = 1; 
    graph->edge[0].weight = 10; 
  
    // add edge 0-2 
    graph->edge[1].src = 0; 
    graph->edge[1].dest = 2; 
    graph->edge[1].weight = 6; 
  
    // add edge 0-3 
    graph->edge[2].src = 0; 
    graph->edge[2].dest = 3; 
    graph->edge[2].weight = 5; 
  
    // add edge 1-3 
    graph->edge[3].src = 1; 
    graph->edge[3].dest = 3; 
    graph->edge[3].weight = 15; 
  
    // add edge 2-3 
    graph->edge[4].src = 2; 
    graph->edge[4].dest = 3; 
    graph->edge[4].weight = 4; 
  
    KruskalMST(graph); 
  
    return 0; 
} 
  
// This code is contributed by rathbhupendra

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// C program for Kruskal's algorithm to find Minimum Spanning Tree 


// of a given connected, undirected and weighted graph 


#include <stdio.h> 


#include <stdlib.h> 


#include <string.h> 


  


// a structure to represent a weighted edge in graph 


struct Edge 


{ 


    int src, dest, weight; 


}; 


  


// a structure to represent a connected, undirected 


// and weighted graph 


struct Graph 


{ 


    // V-> Number of vertices, E-> Number of edges 


    int V, E; 


  


    // graph is represented as an array of edges.  


    // Since the graph is undirected, the edge 


    // from src to dest is also edge from dest 


    // to src. Both are counted as 1 edge here. 


    struct Edge* edge; 


}; 


  


// Creates a graph with V vertices and E edges 


struct Graph* createGraph(int V, int E) 


{ 


    struct Graph* graph = new Graph; 


    graph->V = V; 


    graph->E = E; 


  


    graph->edge = new Edge[E]; 


  


    return graph; 


} 


  


// A structure to represent a subset for union-find 


struct subset 


{ 


    int parent; 


    int rank; 


}; 


  


// A utility function to find set of an element i 


// (uses path compression technique) 


int find(struct subset subsets[], int i) 


{ 


    // find root and make root as parent of i  


    // (path compression) 


    if (subsets[i].parent != i) 


        subsets[i].parent = find(subsets, subsets[i].parent); 


  


    return subsets[i].parent; 


} 


  


// A function that does union of two sets of x and y 


// (uses union by rank) 


void Union(struct subset subsets[], int x, int y) 


{ 


    int xroot = find(subsets, x); 


    int yroot = find(subsets, y); 


  


    // Attach smaller rank tree under root of high  


    // rank tree (Union by Rank) 


    if (subsets[xroot].rank < subsets[yroot].rank) 


        subsets[xroot].parent = yroot; 


    else if (subsets[xroot].rank > subsets[yroot].rank) 


        subsets[yroot].parent = xroot; 


  


    // If ranks are same, then make one as root and  


    // increment its rank by one 


    else


    { 


        subsets[yroot].parent = xroot; 


        subsets[xroot].rank++; 


    } 


} 


  


// Compare two edges according to their weights. 


// Used in qsort() for sorting an array of edges 


int myComp(const void* a, const void* b) 


{ 


    struct Edge* a1 = (struct Edge*)a; 


    struct Edge* b1 = (struct Edge*)b; 


    return a1->weight > b1->weight; 


} 


  


// The main function to construct MST using Kruskal's algorithm 


void KruskalMST(struct Graph* graph) 


{ 


    int V = graph->V; 


    struct Edge result[V];  // Tnis will store the resultant MST 


    int e = 0;  // An index variable, used for result[] 


    int i = 0;  // An index variable, used for sorted edges 


  


    // Step 1:  Sort all the edges in non-decreasing  


    // order of their weight. If we are not allowed to  


    // change the given graph, we can create a copy of 


    // array of edges 


    qsort(graph->edge, graph->E, sizeof(graph->edge[0]), myComp); 


  


    // Allocate memory for creating V ssubsets 


    struct subset *subsets = 


        (struct subset*) malloc( V * sizeof(struct subset) ); 


  


    // Create V subsets with single elements 


    for (int v = 0; v < V; ++v) 


    { 


        subsets[v].parent = v; 


        subsets[v].rank = 0; 


    } 


  


    // Number of edges to be taken is equal to V-1 


    while (e < V - 1 && i < graph->E) 


    { 


        // Step 2: Pick the smallest edge. And increment  


        // the index for next iteration 


        struct Edge next_edge = graph->edge[i++]; 


  


        int x = find(subsets, next_edge.src); 


        int y = find(subsets, next_edge.dest); 


  


        // If including this edge does't cause cycle, 


        // include it in result and increment the index  


        // of result for next edge 


        if (x != y) 


        { 


            result[e++] = next_edge; 


            Union(subsets, x, y); 


        } 


        // Else discard the next_edge 


    } 


  


    // print the contents of result[] to display the 


    // built MST 


    printf("Following are the edges in the constructed MST\n"); 


    for (i = 0; i < e; ++i) 


        printf("%d -- %d == %d\n", result[i].src, result[i].dest, 


                                                 result[i].weight); 


    return; 


} 


  


// Driver program to test above functions 


int main() 


{ 


    /* Let us create following weighted graph 


             10 


        0--------1 


        |  \     | 


       6|   5\   |15 


        |      \ | 


        2--------3 


            4       */


    int V = 4;  // Number of vertices in graph 


    int E = 5;  // Number of edges in graph 


    struct Graph* graph = createGraph(V, E); 


  


  


    // add edge 0-1 


    graph->edge[0].src = 0; 


    graph->edge[0].dest = 1; 


    graph->edge[0].weight = 10; 


  


    // add edge 0-2 


    graph->edge[1].src = 0; 


    graph->edge[1].dest = 2; 


    graph->edge[1].weight = 6; 


  


    // add edge 0-3 


    graph->edge[2].src = 0; 


    graph->edge[2].dest = 3; 


    graph->edge[2].weight = 5; 


  


    // add edge 1-3 


    graph->edge[3].src = 1; 


    graph->edge[3].dest = 3; 


    graph->edge[3].weight = 15; 


  


    // add edge 2-3 


    graph->edge[4].src = 2; 


    graph->edge[4].dest = 3; 


    graph->edge[4].weight = 4; 


  


    KruskalMST(graph); 


  


    return 0; 


} 



















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// Java program for Kruskal's algorithm to find Minimum 


// Spanning Tree of a given connected, undirected and  


// weighted graph 


import java.util.*; 


import java.lang.*; 


import java.io.*; 


  


class Graph 


{ 


    // A class to represent a graph edge 


    class Edge implements Comparable<Edge> 


    { 


        int src, dest, weight; 


  


        // Comparator function used for sorting edges  


        // based on their weight 


        public int compareTo(Edge compareEdge) 


        { 


            return this.weight-compareEdge.weight; 


        } 


    }; 


  


    // A class to represent a subset for union-find 


    class subset 


    { 


        int parent, rank; 


    }; 


  


    int V, E;    // V-> no. of vertices & E->no.of edges 


    Edge edge[]; // collection of all edges 


  


    // Creates a graph with V vertices and E edges 


    Graph(int v, int e) 


    { 


        V = v; 


        E = e; 


        edge = new Edge[E]; 


        for (int i=0; i<e; ++i) 


            edge[i] = new Edge(); 


    } 


  


    // A utility function to find set of an element i 


    // (uses path compression technique) 


    int find(subset subsets[], int i) 


    { 


        // find root and make root as parent of i (path compression) 


        if (subsets[i].parent != i) 


            subsets[i].parent = find(subsets, subsets[i].parent); 


  


        return subsets[i].parent; 


    } 


  


    // A function that does union of two sets of x and y 


    // (uses union by rank) 


    void Union(subset subsets[], int x, int y) 


    { 


        int xroot = find(subsets, x); 


        int yroot = find(subsets, y); 


  


        // Attach smaller rank tree under root of high rank tree 


        // (Union by Rank) 


        if (subsets[xroot].rank < subsets[yroot].rank) 


            subsets[xroot].parent = yroot; 


        else if (subsets[xroot].rank > subsets[yroot].rank) 


            subsets[yroot].parent = xroot; 


  


        // If ranks are same, then make one as root and increment 


        // its rank by one 


        else


        { 


            subsets[yroot].parent = xroot; 


            subsets[xroot].rank++; 


        } 


    } 


  


    // The main function to construct MST using Kruskal's algorithm 


    void KruskalMST() 


    { 


        Edge result[] = new Edge[V];  // Tnis will store the resultant MST 


        int e = 0;  // An index variable, used for result[] 


        int i = 0;  // An index variable, used for sorted edges 


        for (i=0; i<V; ++i) 


            result[i] = new Edge(); 


  


        // Step 1:  Sort all the edges in non-decreasing order of their 


        // weight.  If we are not allowed to change the given graph, we 


        // can create a copy of array of edges 


        Arrays.sort(edge); 


  


        // Allocate memory for creating V ssubsets 


        subset subsets[] = new subset[V]; 


        for(i=0; i<V; ++i) 


            subsets[i]=new subset(); 


  


        // Create V subsets with single elements 


        for (int v = 0; v < V; ++v) 


        { 


            subsets[v].parent = v; 


            subsets[v].rank = 0; 


        } 


  


        i = 0;  // Index used to pick next edge 


  


        // Number of edges to be taken is equal to V-1 


        while (e < V - 1) 


        { 


            // Step 2: Pick the smallest edge. And increment  


            // the index for next iteration 


            Edge next_edge = new Edge(); 


            next_edge = edge[i++]; 


  


            int x = find(subsets, next_edge.src); 


            int y = find(subsets, next_edge.dest); 


  


            // If including this edge does't cause cycle, 


            // include it in result and increment the index  


            // of result for next edge 


            if (x != y) 


            { 


                result[e++] = next_edge; 


                Union(subsets, x, y); 


            } 


            // Else discard the next_edge 


        } 


  


        // print the contents of result[] to display 


        // the built MST 


        System.out.println("Following are the edges in " +  


                                     "the constructed MST"); 


        for (i = 0; i < e; ++i) 


            System.out.println(result[i].src+" -- " +  


                   result[i].dest+" == " + result[i].weight); 


    } 


  


    // Driver Program 


    public static void main (String[] args) 


    { 


  


        /* Let us create following weighted graph 


                 10 


            0--------1 


            |  \     | 


           6|   5\   |15 


            |      \ | 


            2--------3 


                4       */


        int V = 4;  // Number of vertices in graph 


        int E = 5;  // Number of edges in graph 


        Graph graph = new Graph(V, E); 


  


        // add edge 0-1 


        graph.edge[0].src = 0; 


        graph.edge[0].dest = 1; 


        graph.edge[0].weight = 10; 


  


        // add edge 0-2 


        graph.edge[1].src = 0; 


        graph.edge[1].dest = 2; 


        graph.edge[1].weight = 6; 


  


        // add edge 0-3 


        graph.edge[2].src = 0; 


        graph.edge[2].dest = 3; 


        graph.edge[2].weight = 5; 


  


        // add edge 1-3 


        graph.edge[3].src = 1; 


        graph.edge[3].dest = 3; 


        graph.edge[3].weight = 15; 


  


        // add edge 2-3 


        graph.edge[4].src = 2; 


        graph.edge[4].dest = 3; 


        graph.edge[4].weight = 4; 


  


        graph.KruskalMST(); 


    } 


} 


//This code is contributed by Aakash Hasija 



















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# Python program for Kruskal's algorithm to find 


# Minimum Spanning Tree of a given connected,  


# undirected and weighted graph 


  


from collections import defaultdict 


  


#Class to represent a graph 


class Graph: 


  


    def __init__(self,vertices): 


        self.V= vertices #No. of vertices 


        self.graph = [] # default dictionary  


                                # to store graph 


          


   


    # function to add an edge to graph 


    def addEdge(self,u,v,w): 


        self.graph.append([u,v,w]) 


  


    # A utility function to find set of an element i 


    # (uses path compression technique) 


    def find(self, parent, i): 


        if parent[i] == i: 


            return i 


        return self.find(parent, parent[i]) 


  


    # A function that does union of two sets of x and y 


    # (uses union by rank) 


    def union(self, parent, rank, x, y): 


        xroot = self.find(parent, x) 


        yroot = self.find(parent, y) 


  


        # Attach smaller rank tree under root of  


        # high rank tree (Union by Rank) 


        if rank[xroot] < rank[yroot]: 


            parent[xroot] = yroot 


        elif rank[xroot] > rank[yroot]: 


            parent[yroot] = xroot 


  


        # If ranks are same, then make one as root  


        # and increment its rank by one 


        else : 


            parent[yroot] = xroot 


            rank[xroot] += 1


  


    # The main function to construct MST using Kruskal's  


        # algorithm 


    def KruskalMST(self): 


  


        result =[] #This will store the resultant MST 


  


        i = 0 # An index variable, used for sorted edges 


        e = 0 # An index variable, used for result[] 


  


            # Step 1:  Sort all the edges in non-decreasing  


                # order of their 


                # weight.  If we are not allowed to change the  


                # given graph, we can create a copy of graph 


        self.graph =  sorted(self.graph,key=lambda item: item[2]) 


  


        parent = [] ; rank = [] 


  


        # Create V subsets with single elements 


        for node in range(self.V): 


            parent.append(node) 


            rank.append(0) 


      


        # Number of edges to be taken is equal to V-1 


        while e < self.V -1 : 


  


            # Step 2: Pick the smallest edge and increment  


                    # the index for next iteration 


            u,v,w =  self.graph[i] 


            i = i + 1


            x = self.find(parent, u) 


            y = self.find(parent ,v) 


  


            # If including this edge does't cause cycle,  


                        # include it in result and increment the index 


                        # of result for next edge 


            if x != y: 


                e = e + 1     


                result.append([u,v,w]) 


                self.union(parent, rank, x, y)             


            # Else discard the edge 


  


        # print the contents of result[] to display the built MST 


        print "Following are the edges in the constructed MST"


        for u,v,weight  in result: 


            #print str(u) + " -- " + str(v) + " == " + str(weight) 


            print ("%d -- %d == %d" % (u,v,weight)) 


  


# Driver code 


g = Graph(4) 


g.addEdge(0, 1, 10) 


g.addEdge(0, 2, 6) 


g.addEdge(0, 3, 5) 


g.addEdge(1, 3, 15) 


g.addEdge(2, 3, 4) 


  


g.KruskalMST() 


  


#This code is contributed by Neelam Yadav 



















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C#














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// C# Code for above approach 


using System; 


  


class Graph 


{ 


      


// A class to represent a graph edge  


class Edge : IComparable<Edge> 


{ 


    public int src, dest, weight; 


  


    // Comparator function used for sorting edges  


    // based on their weight  


    public int CompareTo(Edge compareEdge) 


    { 


        return this.weight - compareEdge.weight; 


    } 


} 


  


// A class to represent a subset for union-find  


public class subset 


{ 


    public int parent, rank; 


}; 


  


int V, E; // V-> no. of vertices & E->no.of edges  


Edge[] edge; // collection of all edges  


  


// Creates a graph with V vertices and E edges  


Graph(int v, int e) 


{ 


    V = v; 


    E = e; 


    edge = new Edge[E]; 


    for (int i = 0; i < e; ++i) 


        edge[i] = new Edge(); 


} 


  


// A utility function to find set of an element i  


// (uses path compression technique)  


int find(subset[] subsets, int i) 


{ 


    // find root and make root as  


    // parent of i (path compression)  


    if (subsets[i].parent != i) 


        subsets[i].parent = find(subsets,  


                                 subsets[i].parent); 


  


    return subsets[i].parent; 


} 


  


// A function that does union of  


// two sets of x and y (uses union by rank)  


void Union(subset[] subsets, int x, int y) 


{ 


    int xroot = find(subsets, x); 


    int yroot = find(subsets, y); 


  


    // Attach smaller rank tree under root of 


    // high rank tree (Union by Rank)  


    if (subsets[xroot].rank < subsets[yroot].rank) 


        subsets[xroot].parent = yroot; 


    else if (subsets[xroot].rank > subsets[yroot].rank) 


        subsets[yroot].parent = xroot; 


  


    // If ranks are same, then make one as root  


    // and increment its rank by one  


    else


    { 


        subsets[yroot].parent = xroot; 


        subsets[xroot].rank++; 


    } 


} 


  


// The main function to construct MST  


// using Kruskal's algorithm  


void KruskalMST() 


{ 


    Edge[] result = new Edge[V]; // This will store the resultant MST  


    int e = 0; // An index variable, used for result[]  


    int i = 0; // An index variable, used for sorted edges  


    for (i = 0; i < V; ++i) 


        result[i] = new Edge(); 


  


    // Step 1: Sort all the edges in non-decreasing  


    // order of their weight. If we are not allowed  


    // to change the given graph, we can create 


    // a copy of array of edges  


    Array.Sort(edge); 


  


    // Allocate memory for creating V ssubsets  


    subset[] subsets = new subset[V]; 


    for (i = 0; i < V; ++i) 


        subsets[i] = new subset(); 


  


    // Create V subsets with single elements  


    for (int v = 0; v < V; ++v) 


    { 


        subsets[v].parent = v; 


        subsets[v].rank = 0; 


    } 


  


    i = 0; // Index used to pick next edge  


  


    // Number of edges to be taken is equal to V-1  


    while (e < V - 1) 


    { 


        // Step 2: Pick the smallest edge. And increment  


        // the index for next iteration  


        Edge next_edge = new Edge(); 


        next_edge = edge[i++]; 


  


        int x = find(subsets, next_edge.src); 


        int y = find(subsets, next_edge.dest); 


  


        // If including this edge does't cause cycle,  


        // include it in result and increment the index  


        // of result for next edge  


        if (x != y) 


        { 


            result[e++] = next_edge; 


            Union(subsets, x, y); 


        } 


        // Else discard the next_edge  


    } 


  


    // print the contents of result[] to display  


    // the built MST  


    Console.WriteLine("Following are the edges in " + 


                              "the constructed MST"); 


    for (i = 0; i < e; ++i) 


            Console.WriteLine(result[i].src + " -- " + 


            result[i].dest + " == " + result[i].weight); 


        Console.ReadLine(); 


} 


  


// Driver Code 


public static void Main(String[] args) 


{ 


  


    /* Let us create following weighted graph  


            10  


        0--------1  


        | \ |  


    6| 5\ |15  


        | \ |  


        2--------3  


            4 */


    int V = 4; // Number of vertices in graph  


    int E = 5; // Number of edges in graph  


    Graph graph = new Graph(V, E); 


  


    // add edge 0-1  


    graph.edge[0].src = 0; 


    graph.edge[0].dest = 1; 


    graph.edge[0].weight = 10; 


  


    // add edge 0-2  


    graph.edge[1].src = 0; 


    graph.edge[1].dest = 2; 


    graph.edge[1].weight = 6; 


  


    // add edge 0-3  


    graph.edge[2].src = 0; 


    graph.edge[2].dest = 3; 


    graph.edge[2].weight = 5; 


  


    // add edge 1-3  


    graph.edge[3].src = 1; 


    graph.edge[3].dest = 3; 


    graph.edge[3].weight = 15; 


  


    // add edge 2-3  


    graph.edge[4].src = 2; 


    graph.edge[4].dest = 3; 


    graph.edge[4].weight = 4; 


  


    graph.KruskalMST(); 


}  


}  


  


// This code is contributed by Aakash Hasija  



















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Following are the edges in the constructed MST
2 -- 3 == 4
0 -- 3 == 5
0 -- 1 == 10


Time Complexity: O(ElogE) or O(ElogV).  Sorting of edges takes O(ELogE) time. After sorting, we iterate through all edges and apply find-union algorithm.  The find and union operations can take atmost O(LogV) time.  So overall complexity is O(ELogE + ELogV) time.  The value of E can be atmost O(V2), so O(LogV) are O(LogE) same. Therefore, overall time complexity is O(ElogE) or O(ElogV)




References:

http://www.ics.uci.edu/~eppstein/161/960206.html

http://en.wikipedia.org/wiki/Minimum_spanning_tree


This article is compiled by Aashish Barnwal and reviewed by GeeksforGeeks team. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.





    

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