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

// 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)
    {
        // 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;
}

// 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

# 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