Added Kruskal's algorithm

dsa/Algorithms/kruskal's.md

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+---
+id: kruskal's
+title: Kruskal's Graph Algorithm
+sidebar_label: Kruskal's Algorithm
+tags:
+  - dsa
+  - data-structures
+  - graph
+  - graph-traversal
+  - algorithm
+  - javascript
+  - python
+  - c++
+  - java
+  - Minimum Spanning Tree
+  - programming
+  - tutorial
+sidebar_position: 3
+---
+Kruskal's algorithm is a popular method used to find the minimum spanning tree (MST) of a connected, undirected graph. A minimum spanning tree is a subset of the edges in a graph that connects all the vertices together, wihout any cycles, and with minimum possible total edge weight.
+### Key Concepts:
+* Edge Selection: The algorithm picks the smallest weight edge first and uses a greedy approach to ensure the overall minimum weight for the spanning tree.
+* Cycle Detection: To check efficiently if adding a new edge forms a cycle, the algorithm uses union-find (Disjoint set union) function for that.
+### To find MST using Kruskal's algorithm
+Following steps are used to find the MST:
+1. Sort the edges in ascending order of their weight.
+2. Pick the smallest edge. Check if it forms a cycle with the spanning tree formed so far. If the cycle is not formed, include this edge. Else, discard it. 
+3. Repeat step 2 until there are (V-1) edges in the spanning tree. 
+### Implementation:
+![image](https://wat-images.s3.ap-south-1.amazonaws.com/images/ps/minimum-spanning-tree-using-kruskals-algorithm.svg)
+
+Program for the Kruskal's algorithm:
+
+<Tabs>
+  <TabItem value="C++" label="C++">
+    ```Cpp  showLineNumbers
+    //kruskals algorithm 
+#include<iostream>
+#include<vector>
+#define I INT32_MAX
+using namespace std;
+void Union(int u, int v,vector<int>&s)// here u and v are the head of two sets
+{
+    // the one with more child will become the ultimate head
+    if (s[u]<s[v])//s is the array where sets are stored
+    {
+        s[u]=s[u]+s[v];
+        s[v]=u;
+    }
+    else
+    {
+        s[v]=s[u]+s[v];
+        s[u]=v;
+    }
+
+}
+// write a find function to find the parent of any node
+int find(int u,vector<int>&s)// u is the element whose parent we need to find
+{
+    int x=u;
+    int v=0;
+    while(s[x]>0)
+    {
+        x=s[x];
+    }
+
+    return x;
+}
+vector<vector<int>> kruskals_algo(vector<vector<int>>&G,int n,int a)
+{
+    vector<int>set(a,-1);
+    vector<int>included(n,0);
+    vector<vector<int>>sol(2,vector<int>(a-1));
+    int i=0,j,k,min,u,v;
+    while(i<a-1)
+    {
+        min=I;
+        for (j=0;j<n;j++)
+        {
+            if (included[j]==0 && G[2][j]<min)
+            {
+                min=G[2][j];
+                k=j;
+                u=G[0][j];
+                v=G[1][j];
+            }
+        }
+        //cout<<"out"<<endl;
+        int x=find(u,set),y=find(v,set);
+        if (x!=y)
+        {
+            sol[0][i]=u;
+            sol[1][i]=v;
+            Union(x,y,set);
+            i++;
+        }
+        included[k]=1;
+   // cout<<"worked"<<endl;
+    }
+    //cout<<"worked"<<endl;
+    return sol;
+}
+int main()
+{
+    int n,a;
+    cout<<"Enter the number of edges: ";
+    cin>>n;
+    cout<<"Enter the number of vertexes: ";
+    cin>>a;
+    vector<vector<int>>G(3,vector<int>(n));
+    for (int i=0;i<3;i++)
+    {
+        for (int j=0;j<n;j++)
+            cin>>G[i][j];
+    }
+    vector<vector<int>>res=kruskals_algo(G,n,a);
+    cout<<"Kruskals path is: "<<endl;
+    for (int i=0;i<2;i++)
+    {
+        for (int j=0;j<a-1;j++)
+            cout<<res[i][j]<<" ";
+        cout<<endl;
+    }
+    return 0;
+}
+ ```
+</TabItem>
+
+<TabItem value="Python" label="Python">
+```Python showLineNumbers
+import sys
+
+def union(u, v, s):
+    if s[u] < s[v]:
+        s[u] += s[v]
+        s[v] = u
+    else:
+        s[v] += s[u]
+        s[u] = v
+
+def find(u, s):
+    x = u
+    while s[x] > 0:
+        x = s[x]
+    return x
+
+def kruskals_algo(G, n, a):
+    set = [-1] * a
+    included = [0] * n
+    sol = [[0] * (a - 1) for _ in range(2)]
+    i = 0
+    while i < a - 1:
+        min = sys.maxsize
+        for j in range(n):
+            if included[j] == 0 and G[2][j] < min:
+                min = G[2][j]
+                k = j
+                u = G[0][j]
+                v = G[1][j]
+        x = find(u, set)
+        y = find(v, set)
+        if x != y:
+            sol[0][i] = u
+            sol[1][i] = v
+            union(x, y, set)
+            i += 1
+        included[k] = 1
+    return sol
+
+if __name__ == "__main__":
+    n = int(input("Enter the number of edges: "))
+    a = int(input("Enter the number of vertices: "))
+    G = [list(map(int, input().split())) for _ in range(3)]
+    res = kruskals_algo(G, n, a)
+    print("Kruskal's path is: ")
+    for i in range(2):
+        print(" ".join(map(str, res[i])))
+
+```
+</TabItem>
+
+<TabItem value="Java" label="Java">
+``` jsx showLineNumbers
+import java.util.*;
+
+public class KruskalsAlgorithm {
+    static void union(int u, int v, int[] s) {
+        if (s[u] < s[v]) {
+            s[u] += s[v];
+            s[v] = u;
+        } else {
+            s[v] += s[u];
+            s[u] = v;
+        }
+    }
+
+    static int find(int u, int[] s) {
+        int x = u;
+        while (s[x] > 0) {
+            x = s[x];
+        }
+        return x;
+    }
+
+    static int[][] kruskals_algo(int[][] G, int n, int a) {
+        int[] set = new int[a];
+        Arrays.fill(set, -1);
+        int[] included = new int[n];
+        int[][] sol = new int[2][a - 1];
+        int i = 0;
+        while (i < a - 1) {
+            int min = Integer.MAX_VALUE;
+            int u = 0, v = 0, k = 0;
+            for (int j = 0; j < n; j++) {
+                if (included[j] == 0 && G[2][j] < min) {
+                    min = G[2][j];
+                    k = j;
+                    u = G[0][j];
+                    v = G[1][j];
+                }
+            }
+            int x = find(u, set);
+            int y = find(v, set);
+            if (x != y) {
+                sol[0][i] = u;
+                sol[1][i] = v;
+                union(x, y, set);
+                i++;
+            }
+            included[k] = 1;
+        }
+        return sol;
+    }
+
+    public static void main(String[] args) {
+        Scanner scanner = new Scanner(System.in);
+        System.out.print("Enter the number of edges: ");
+        int n = scanner.nextInt();
+        System.out.print("Enter the number of vertices: ");
+        int a = scanner.nextInt();
+        int[][] G = new int[3][n];
+        for (int i = 0; i < 3; i++) {
+            for (int j = 0; j < n; j++) {
+                G[i][j] = scanner.nextInt();
+            }
+        }
+        int[][] res = kruskals_algo(G, n, a);
+        System.out.println("Kruskal's path is: ");
+        for (int i = 0; i < 2; i++) {
+            for (int j = 0; j < a - 1; j++) {
+                System.out.print(res[i][j] + " ");
+            }
+            System.out.println();
+        }
+        scanner.close();
+    }
+}
+```
+</TabItem>
+
+<TabItem value="JavaScript" label="JavaScript">
+``` jsx showLineNumbers
+function union(u, v, s) {
+    if (s[u] < s[v]) {
+        s[u] += s[v];
+        s[v] = u;
+    } else {
+        s[v] += s[u];
+        s[u] = v;
+    }
+}
+
+function find(u, s) {
+    let x = u;
+    while (s[x] > 0) {
+        x = s[x];
+    }
+    return x;
+}
+
+function kruskalsAlgo(G, n, a) {
+    let set = new Array(a).fill(-1);
+    let included = new Array(n).fill(0);
+    let sol = [new Array(a - 1), new Array(a - 1)];
+    let i = 0;
+    while (i < a - 1) {
+        let min = Number.MAX_SAFE_INTEGER;
+        let u, v, k;
+        for (let j = 0; j < n; j++) {
+            if (included[j] === 0 && G[2][j] < min) {
+                min = G[2][j];
+                k = j;
+                u = G[0][j];
+                v = G[1][j];
+            }
+        }
+        let x = find(u, set);
+        let y = find(v, set);
+        if (x !== y) {
+            sol[0][i] = u;
+            sol[1][i] = v;
+            union(x, y, set);
+            i++;
+        }
+        included[k] = 1;
+    }
+    return sol;
+}
+
+function main() {
+    const n = parseInt(prompt("Enter the number of edges: "));
+    const a = parseInt(prompt("Enter the number of vertices: "));
+    let G = [];
+    for (let i = 0; i < 3; i++) {
+        G.push(prompt(`Enter row ${i} of graph:`).split(' ').map(Number));
+    }
+    let res = kruskalsAlgo(G, n, a);
+    console.log("Kruskal's path is: ");
+    for (let i = 0; i < 2; i++) {
+        console.log(res[i].join(' '));
+    }
+}
+
+main();
+```
+</TabItem>
+</Tabs>
+
+Output:<br />
+
+```
+/*input is in the format row one u, row two v and row three weight between u and v*/
+Enter the number of edges: 8
+Enter the number of vertexes: 7
+0 0 0 1 1 3 6 4
+1 3 6 2 4 4 4 5
+2 3 4 3 2 5 6 7
+Kruskals path is: 
+0 1 0 1 0 4 
+1 4 3 2 6 5 
+
+```
+### Time and Space Complexity:
+* Time complexity: The time complexity of kruskal's algorithm depends on two operations i.e. sorting of all edges and union-find operation.Thus, the overall time complexity is: **O(ElogV)**
+* Space complexity: The space complexity of kruskal's algorithm includes storage for edges, union-find data structure and storage for the MST. Thus the overall space complexity is: **O(E+V)**