# Kruskal’s Algorithm

In this article I show you the implementation of Kruskal’s Algorithm using C and C++ Programming Languages. This a popular computer science algorithm which is directly based on the generic Minimum Spanning Tree (MST) algorithm. A minimum spanning tree is a subgraph of the graph (a tree) with the minimum sum of edge weights.

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.

## Key Concepts

1. Graph: A collection of vertices (or nodes) and edges connecting pairs of vertices.
2. Weighted Graph: A graph where each edge has a numerical value (weight) associated with it.
3. Minimum Spanning Tree (MST): A spanning tree of a graph whose total weight is minimized.

Kruskal’s algorithm addresses two problems as mentioned below.

• PROBLEM 1. Give a practical method for constructing a spanning subtree of minimum length.
• PROBLEM 2. Give a practical method for constructing an unbranched spanning subtree of minimum length.

Kruskal’s algorithm is most suitable for sparse graphs (low number of edges).  This algorithm is practically used in many fields such as Traveling Salesman Problem, Creating Mazes and Computer Networks etc. It is also helpful in cluster analysis in machine learning as well as geographical mapping.

## Kruskal’s Algorithm Steps

1. Sort all the edges in the graph in non-decreasing order of their weight.
2. Initialize a forest (a set of trees), where each vertex in the graph is a separate tree.
3. Initialize an empty set for the MST.
4. Process each edge in the sorted order:
• Check if the current edge forms a cycle with the spanning tree formed so far.
• If it does not form a cycle, include this edge in the MST.
• If it forms a cycle, discard the edge.
5. Repeat until there are V−1V−1 edges in the MST (where VV is the number of vertices in the graph).

## Pseudo code of the Kruskal’s Algorithm

```Kruskal(graph):
sort all edges in graph by their weight in non-decreasing order
create an empty set for the MST
initialize a Union-Find structure

for each edge (u, v) in sorted order:
if find(u) != find(v):
add edge (u, v) to the MST
union(u, v)
if MST has V-1 edges:
break

return MST

find(x):
if parent[x] != x:
parent[x] = find(parent[x])
return parent[x]

union(x, y):
rootX = find(x)
rootY = find(y)
if rootX != rootY:
if rank[rootX] > rank[rootY]:
parent[rootY] = rootX
else if rank[rootX] < rank[rootY]:
parent[rootX] = rootY
else:
parent[rootY] = rootX
rank[rootX] += 1
```

## Complexity of Kruskal’s Algorithm

Time Complexity: The time complexity Of Kruskal’s Algorithm is: O(ElogE+ElogV) where E is number of edges and V is the number of vertices. Sorting the edges takes 0(ElogE)O(ElogE) and the union-find operations take 0(ElogV)O(ElogV).

Space Complexity: 0(V+E)O(V+E), due to the storage required for the graph representation and the Union-Find data structure.

Explanation:

Kruskal’s algorithm takes o(e log e) time in sorting of the edges. Here e is numbers of edges and v is the number of vertices in the graph. Further, it iterates all edges and runs a subroutine to find the cycles in the graph which is called union-find algorithm. The union-find algorithm requires o(log v) time and is applied after sorting of edges is completed.

So, Overall Kruskal’s algorithm requires o(e log v) time to run.

## C Programming Implementation of Kruskal’s Algorithm

Here is the C program that implements this 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() {
printf("\n\tImplementation of Kruskal's Algorithm\n");
printf("\nEnter the no. of vertices:");
scanf("%d", & 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 of the C Program

Implementation of Kruskal’s Algorithm

Enter the no. of vertices:3

9

8

7

6

5

4

3

2

3

The edges of Minimum Cost Spanning Tree are

1 edge (3,2) =2

2 edge (3,1) =3

Minimum cost = 5

## C++ Programming Implementation of Kruskal’s Algorithm

```#include <iostream>
#include <vector>
#include <algorithm>

using namespace std;

// Define a structure to represent an edge
struct Edge {
int src, dest, weight;
};

// Define a structure to represent a graph
class Graph {
public:
int V, E; // Number of vertices and edges
vector<Edge> edges; // Collection of all edges

Graph(int V, int E);
void addEdge(int src, int dest, int weight);
};

// Constructor
Graph::Graph(int V, int E) {
this->V = V;
this->E = E;
}

// Function to add an edge to the graph
void Graph::addEdge(int src, int dest, int weight) {
edges.push_back({src, dest, weight});
}

// Find set of an element i (uses path compression technique)
int find(vector<int>& parent, int i) {
if (parent[i] != i)
parent[i] = find(parent, parent[i]);
return parent[i];
}

// Do union of two sets (uses union by rank)
void Union(vector<int>& parent, vector<int>& rank, int x, int y) {
int rootX = find(parent, x);
int rootY = find(parent, y);

if (rootX != rootY) {
if (rank[rootX] > rank[rootY])
parent[rootY] = rootX;
else if (rank[rootX] < rank[rootY])
parent[rootX] = rootY;
else {
parent[rootY] = rootX;
rank[rootX]++;
}
}
}

// Compare two edges according to their weights
bool compare(Edge a, Edge b) {
return a.weight < b.weight;
}

// Function to perform Kruskal's algorithm
void KruskalMST(Graph& graph) {
int V = graph.V;
vector<Edge> result; // Store the resultant MST
int e = 0; // An index variable for result[]

// Step 1: Sort all the edges in non-decreasing order of their weight
sort(graph.edges.begin(), graph.edges.end(), compare);

// Allocate memory for creating V subsets
vector<int> parent(V);
vector<int> rank(V, 0);

// Create V single-item sets
for (int v = 0; v < V; ++v)
parent[v] = v;

// Number of edges to be taken is equal to V-1
for (auto edge : graph.edges) {
if (e >= V - 1)
break;

int u = find(parent, edge.src);
int v = find(parent, edge.dest);

// If including this edge does not cause cycle, include it in result
if (u != v) {
result.push_back(edge);
Union(parent, rank, u, v);
e++;
}
}

// Print the contents of result[] to display the built MST
cout << "Following are the edges in the constructed MST:\n";
for (auto edge : result)
cout << edge.src << " -- " << edge.dest << " == " << edge.weight << endl;
}

int main() {
int V = 7; // Number of vertices
int E = 11; // Number of edges
Graph graph(V, E);

// Add edges to the graph