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<!DOCTYPE html>

<head>

<title>Kruskal Algorithm</title>

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<h3>Page Index</h3>

<ul>

<li><a href="/dsa.html#data-structure-1" title="Data Structures (I)">Data Structures

(I)</a></li>

<li><a href="/dsa.html#data-structure-2" title="Data Structures (II)">Data Structures

(II)</a></li>

<li><a href="/dsa.html#tree-1" title="Tree based DSA (I)">Tree based DSA (I)</a></li>

<li><a href="/dsa.html#tree-2" title="Tree based DSA (II)">Tree based DSA (II)</a></li>

<li><a href="/dsa.html#graph" title="Graph Data Structures and Algorithm">Graph based

DSA</a></li>

<li><a href="/dsa.html#sorting-searching" title="Sorting and Searching">Sorting and

Searching</a></li>

<li><a href="/dsa.html#greedy-algorithm" title="Greedy Algorithms">Greedy Algorithms</a>

</li>

<li><a href="/dsa.html#dynamic-programming" title="Dynamic Programming">Dynamic

Programming</a></li>

<li><a href="/dsa.html#other-algorithms" title="Other Algorithms">Other Algorithms</a>

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</ul>

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<h1>Kruskal's Algorithm</h1>

In this tutorial, you will learn how Kruskal's Algorithmworks. Also, you will find working examples of Kruskal's Algorithm in C, C++, Java and Python.

Kruskal's algorithm is a <a href="/dsa/spanning-tree-and-minimum-spanning-tree.html">minimum spanning tree</a> algorithm that takes a graph as input and finds the subset of the edges of that graph which

<ul><li>form a tree that includes every vertex</li>

<li>has the minimum sum of weights among all the trees that can be formed from the graph</li>

</ul><hr><h2 id="working">How Kruskal's algorithm works</h2>

It falls under a class of algorithms called <a href="/dsa/greedy-algorithm.html" >greedy algorithms</a> that find the local optimum in the hopes of finding a global optimum.

We start from the edges with the lowest weight and keep adding edges until we reach our goal.

The steps for implementing Kruskal's algorithm are as follows:

<ol><li>Sort all the edges from low weight to high</li>

<li>Take the edge with the lowest weight and add it to the spanning tree. If adding the edge created a cycle, then reject this edge.</li>

<li>Keep adding edges until we reach all vertices.</li>

</ol><hr><h2 id="example">Example of Kruskal's algorithm</h2>

<figure><img alt="Start with a weighted graph" src="//cdn.programiz.com/sites/tutorial2program/files/ka-1.png" title="Kruskal's algorithm example " width="730" height="308"><figcaption>Start with a weighted graph</figcaption></figure><figure><img alt="Choose the edge with the least weight, if there are more than 1, choose anyone" src="//cdn.programiz.com/sites/tutorial2program/files/ka-2.png" title="Kruskal's algorithm example " width="730" height="240"><figcaption>Choose the edge with the least weight, if there are more than 1, choose anyone</figcaption></figure><figure><img alt="Choose the next shortest edge and add it" src="//cdn.programiz.com/sites/tutorial2program/files/ka-3.png" title="Kruskal's algorithm example " width="730" height="240"><figcaption>Choose the next shortest edge and add it</figcaption></figure><figure><img alt="Choose the next shortest edge that doesn't create a cycle and add it" src="//cdn.programiz.com/sites/tutorial2program/files/ka-4.png" title="Kruskal's algorithm example " width="730" height="306"><figcaption>Choose the next shortest edge that doesn't create a cycle and add it</figcaption></figure><figure><img alt="Choose the next shortest edge that doesn't create a cycle and add it" src="//cdn.programiz.com/sites/tutorial2program/files/ka-5.png" title="Kruskal's algorithm example " width="730" height="306"><figcaption>Choose the next shortest edge that doesn't create a cycle and add it</figcaption></figure><figure><img alt="Repeat until you have a spanning tree" src="//cdn.programiz.com/sites/tutorial2program/files/ka-6.png" title="Kruskal's algorithm example " width="730" height="306"><figcaption>Repeat until you have a spanning tree</figcaption></figure><hr><h2 id="pseudocode">Kruskal Algorithm Pseudocode</h2>

Any minimum spanning tree algorithm revolves around checking if adding an edge creates a loop or not.

<div class="clearfix"></div>The most common way to find this out is an algorithm called <a href="https://www.cs.duke.edu/courses/cps100e/fall09/notes/UnionFind.pdf" target="_blank">Union FInd</a>. The Union-Find algorithm divides the vertices into clusters and allows us to check if two vertices belong to the same cluster or not and hence decide whether adding an edge creates a cycle.

<pre style="max-height: 600px;"><code class="dsa hljs sql">KRUSKAL(G):

A = ∅

For each vertex v ∈ G.V:

MAKE-SET(v)

For each edge (u, v) ∈ G.E ordered by increasing order by weight(u, v):

if FIND-SET(u) ≠ FIND-SET(v):

A = A ∪ {(u, v)}

UNION(u, v)

return A</code></pre>

<hr><h2 id="code">Python, Java and C/C++ Examples</h2>

<div id="py1" onclick="changepy()" class="tabbed-editor__node tabbed-editor__node--active"><a href="#python-code">Python</a></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="python hljs"># Kruskal's algorithm in Python

class Graph:

def __init__(self, vertices):

self.V = vertices

self.graph = []

def add_edge(self, u, v, w):

self.graph.append([u, v, w])

# Search function

def find(self, parent, i):

if parent[i] == i:

return i

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

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

xroot = self.find(parent, x)

yroot = self.find(parent, y)

if rank[xroot] < rank[yroot]:

parent[xroot] = yroot

elif rank[xroot] > rank[yroot]:

parent[yroot] = xroot

else:

parent[yroot] = xroot

rank[xroot] += 1

# Applying Kruskal algorithm

def kruskal_algo(self):

result = []

i, e = 0, 0

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

parent = []

rank = []

for node in range(self.V):

parent.append(node)

rank.append(0)

while e < self.V - 1:

u, v, w = self.graph[i]

i = i + 1

x = self.find(parent, u)

y = self.find(parent, v)

if x != y:

e = e + 1

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

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

for u, v, weight in result:

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

g = Graph(6)

g.add_edge(0, 1, 4)

g.add_edge(0, 2, 4)

g.add_edge(1, 2, 2)

g.add_edge(1, 0, 4)

g.add_edge(2, 0, 4)

g.add_edge(2, 1, 2)

g.add_edge(2, 3, 3)

g.add_edge(2, 5, 2)

g.add_edge(2, 4, 4)

g.add_edge(3, 2, 3)

g.add_edge(3, 4, 3)

g.add_edge(4, 2, 4)

g.add_edge(4, 3, 3)

g.add_edge(5, 2, 2)

g.add_edge(5, 4, 3)

g.kruskal_algo()</code></pre></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="java hljs">// Kruskal's algorithm in Java

import java.util.*;

class Graph {

class Edge implements Comparable<Edge> {

int src, dest, weight;

public int compareTo(Edge compareEdge) {

return this.weight - compareEdge.weight;

}

};

// Union

class subset {

int parent, rank;

};

int vertices, edges;

Edge edge[];

// Graph creation

Graph(int v, int e) {

vertices = v;

edges = e;

edge = new Edge[edges];

for (int i = 0; i < e; ++i)

edge[i] = new Edge();

}

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

if (subsets[i].parent != i)

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

return subsets[i].parent;

}

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

int xroot = find(subsets, x);

int yroot = find(subsets, y);

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

subsets[xroot].parent = yroot;

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

subsets[yroot].parent = xroot;

else {

subsets[yroot].parent = xroot;

subsets[xroot].rank++;

}

// Applying Krushkal Algorithm

void KruskalAlgo() {

Edge result[] = new Edge[vertices];

int e = 0;

int i = 0;

for (i = 0; i < vertices; ++i)

result[i] = new Edge();

// Sorting the edges

Arrays.sort(edge);

subset subsets[] = new subset[vertices];

for (i = 0; i < vertices; ++i)

subsets[i] = new subset();

for (int v = 0; v < vertices; ++v) {

subsets[v].parent = v;

subsets[v].rank = 0;

}

i = 0;

while (e < vertices - 1) {

Edge next_edge = new Edge();

next_edge = edge[i++];

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

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

if (x != y) {

result[e++] = next_edge;

Union(subsets, x, y);

}

for (i = 0; i < e; ++i)

System.out.println(result[i].src + " - " + result[i].dest + ": " + result[i].weight);

}

public static void main(String[] args) {

int vertices = 6; // Number of vertices

int edges = 8; // Number of edges

Graph G = new Graph(vertices, edges);

G.edge[0].src = 0;

G.edge[0].dest = 1;

G.edge[0].weight = 4;

G.edge[1].src = 0;

G.edge[1].dest = 2;

G.edge[1].weight = 4;

G.edge[2].src = 1;

G.edge[2].dest = 2;

G.edge[2].weight = 2;

G.edge[3].src = 2;

G.edge[3].dest = 3;

G.edge[3].weight = 3;

G.edge[4].src = 2;

G.edge[4].dest = 5;

G.edge[4].weight = 2;

G.edge[5].src = 2;

G.edge[5].dest = 4;

G.edge[5].weight = 4;

G.edge[6].src = 3;

G.edge[6].dest = 4;

G.edge[6].weight = 3;

G.edge[7].src = 5;

G.edge[7].dest = 4;

G.edge[7].weight = 3;

G.KruskalAlgo();

}

}</code></pre></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="c hljs cpp">// Kruskal's algorithm in C

#include <stdio.h>

#define MAX 30

typedef struct edge {

int u, v, w;

} edge;

typedef struct edge_list {

edge data[MAX];

int n;

} edge_list;

edge_list elist;

int Graph[MAX][MAX], n;

edge_list spanlist;

void kruskalAlgo();

int find(int belongs[], int vertexno);

void applyUnion(int belongs[], int c1, int c2);

void sort();

void print();

// Applying Krushkal Algo

void kruskalAlgo() {

int belongs[MAX], i, j, cno1, cno2;

elist.n = 0;

for (i = 1; i < n; i++)

for (j = 0; j < i; j++) {

if (Graph[i][j] != 0) {

elist.data[elist.n].u = i;

elist.data[elist.n].v = j;

elist.data[elist.n].w = Graph[i][j];

elist.n++;

}

sort();

for (i = 0; i < n; i++)

belongs[i] = i;

spanlist.n = 0;

for (i = 0; i < elist.n; i++) {

cno1 = find(belongs, elist.data[i].u);

cno2 = find(belongs, elist.data[i].v);

if (cno1 != cno2) {

spanlist.data[spanlist.n] = elist.data[i];

spanlist.n = spanlist.n + 1;

applyUnion(belongs, cno1, cno2);

}

return (belongs[vertexno]);

}

int i;

for (i = 0; i < n; i++)

if (belongs[i] == c2)

belongs[i] = c1;

}

// Sorting algo

void sort() {

int i, j;

edge temp;

for (i = 1; i < elist.n; i++)

for (j = 0; j < elist.n - 1; j++)

if (elist.data[j].w > elist.data[j + 1].w) {

temp = elist.data[j];

elist.data[j] = elist.data[j + 1];

elist.data[j + 1] = temp;

}

// Printing the result

void print() {

int i, cost = 0;

for (i = 0; i < spanlist.n; i++) {

printf("\n%d - %d : %d", spanlist.data[i].u, spanlist.data[i].v, spanlist.data[i].w);

cost = cost + spanlist.data[i].w;

}

printf("\nSpanning tree cost: %d", cost);

}

int main() {

int i, j, total_cost;

n = 6;

Graph[0][0] = 0;

Graph[0][1] = 4;

Graph[0][2] = 4;

Graph[0][3] = 0;

Graph[0][4] = 0;

Graph[0][5] = 0;

Graph[0][6] = 0;

Graph[1][0] = 4;

Graph[1][1] = 0;

Graph[1][2] = 2;

Graph[1][3] = 0;

Graph[1][4] = 0;

Graph[1][5] = 0;

Graph[1][6] = 0;

Graph[2][0] = 4;

Graph[2][1] = 2;

Graph[2][2] = 0;

Graph[2][3] = 3;

Graph[2][4] = 4;

Graph[2][5] = 0;

Graph[2][6] = 0;

Graph[3][0] = 0;

Graph[3][1] = 0;

Graph[3][2] = 3;

Graph[3][3] = 0;

Graph[3][4] = 3;

Graph[3][5] = 0;

Graph[3][6] = 0;

Graph[4][0] = 0;

Graph[4][1] = 0;

Graph[4][2] = 4;

Graph[4][3] = 3;

Graph[4][4] = 0;

Graph[4][5] = 0;

Graph[4][6] = 0;

Graph[5][0] = 0;

Graph[5][1] = 0;

Graph[5][2] = 2;

Graph[5][3] = 0;

Graph[5][4] = 3;

Graph[5][5] = 0;

Graph[5][6] = 0;

kruskalAlgo();

print();

}</code></pre></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="cpp hljs">// Kruskal's algorithm in C++

#include <algorithm>

#include <iostream>

#include <vector>

using namespace std;

#define edge pair<int, int>

class Graph {

private:

vector<pair<int, edge> > G; // graph

vector<pair<int, edge> > T; // mst

int *parent;

int V; // number of vertices/nodes in graph

public:

Graph(int V);

void AddWeightedEdge(int u, int v, int w);

int find_set(int i);

void union_set(int u, int v);

void kruskal();

void print();

};

Graph::Graph(int V) {

parent = new int[V];

//i 0 1 2 3 4 5

//parent[i] 0 1 2 3 4 5

for (int i = 0; i < V; i++)

parent[i] = i;

G.clear();

T.clear();

}

void Graph::AddWeightedEdge(int u, int v, int w) {

G.push_back(make_pair(w, edge(u, v)));

}

int Graph::find_set(int i) {

// If i is the parent of itself

if (i == parent[i])

return i;

else

// Else if i is not the parent of itself

// Then i is not the representative of his set,

// so we recursively call Find on its parent

return find_set(parent[i]);

}

void Graph::union_set(int u, int v) {

parent[u] = parent[v];

}

void Graph::kruskal() {

int i, uRep, vRep;

sort(G.begin(), G.end()); // increasing weight

for (i = 0; i < G.size(); i++) {

uRep = find_set(G[i].second.first);

vRep = find_set(G[i].second.second);

if (uRep != vRep) {

T.push_back(G[i]); // add to tree

union_set(uRep, vRep);

}

void Graph::print() {

cout << "Edge :"

<< " Weight" << endl;

for (int i = 0; i < T.size(); i++) {

cout << T[i].second.first << " - " << T[i].second.second << " : "

<< T[i].first;

cout << endl;

}

int main() {

Graph g(6);

g.AddWeightedEdge(0, 1, 4);

g.AddWeightedEdge(0, 2, 4);

g.AddWeightedEdge(1, 2, 2);

g.AddWeightedEdge(1, 0, 4);

g.AddWeightedEdge(2, 0, 4);

g.AddWeightedEdge(2, 1, 2);

g.AddWeightedEdge(2, 3, 3);

g.AddWeightedEdge(2, 5, 2);

g.AddWeightedEdge(2, 4, 4);

g.AddWeightedEdge(3, 2, 3);

g.AddWeightedEdge(3, 4, 3);

g.AddWeightedEdge(4, 2, 4);

g.AddWeightedEdge(4, 3, 3);

g.AddWeightedEdge(5, 2, 2);

g.AddWeightedEdge(5, 4, 3);

g.kruskal();

g.print();

return 0;

}</code></pre></div>

</div>

<hr><h2 id="vs">Kruskal's vs Prim's Algorithm</h2>

<a href="/dsa/prim-algorithm.html">Prim's algorithm</a> is another popular minimum spanning tree algorithm that uses a different logic to find the MST of a graph. Instead of starting from an edge, Prim's algorithm starts from a vertex and keeps adding lowest-weight edges which aren't in the tree, until all vertices have been covered.

<hr><h2 id="complexity">Kruskal's Algorithm Complexity</h2>

The time complexity Of Kruskal's Algorithm is: <var>O(E log E)</var>.

<hr><h2 id="application">Kruskal's Algorithm Applications</h2>

<ul><li>In order to layout electrical wiring</li>

<li>In computer network (LAN connection)</li>

</ul></div>

</div>

<div class="tutorial-toc"><div class="tutorial-toc__inner"><h3 class="tutorial-toc__title">Table of Contents

<li><a href="#working">How Kruskal's algorithm works</a></li>

<li><a href="#example">Example of Kruskal's algorithm</a></li>

<li><a href="#pseudocode">Kruskal Algorithm Pseudocode</a></li>

<li><a href="#code">Python, Java and C/C++ Examples</a></li>

<li><a href="#vs">Kruskal's vs Prim's Algorithm</a></li>

<li><a href="#complexity">Kruskal's Algorithm Complexity</a></li>

<li><a href="#application">Kruskal's Algorithm Applications</a></li>

</ul></div></div></div> </div>

<div>

<ul>