Kahn's algorithm for Topological Sorting (original) (raw)

Last Updated : 05 Apr, 2025

Given a **Directed Acyclic Graph having **V vertices and **E edges, your task is to find any Topological Sorted order of the graph.

**Topological Sorted order: It is a linear ordering of vertices such that for every directed edge **u -> v, where vertex **u comes before **v in the ordering.

**Example:

**Input: V=6 , E = [[2,3], [3,1], [4,0], [4,1], [5,0], [5,2]]

**Output: 4 5 2 0 3 1
**Explanation: In the above output, each dependent vertex is printed after the vertices it depends upon.

**Input: V=5 , E=[[0,1], [1,2[, [3,2], [3,4]]

**Output: 0 3 4 1 2
**Explanation: In the above output, each dependent vertex is printed after the vertices it depends upon.

Kahn's Algorithm for Topological Sorting:

**Kahn's Algorithm works by repeatedly finding vertices with no incoming edges, removing them from the graph, and updating the incoming edges of the vertices connected from the removed removed edges. This process continues until all vertices have been ordered.

**Algorithm:

**How to find the in-degree of each node?

To find the in-degree of each node by initially calculating the number of incoming edges to each node. Iterate through all the edges in the graph and **increment the **in-degree of the **destination **node for each edge. This way, you can determine the in-degree of each node before starting the sorting process.

**Working of the above algorithm:

Below is the implementation of the above algorithm.

C++ `

// Including necessary header file #include <bits/stdc++.h> using namespace std;

// We mainly take input graph as a set of edges. This function is // mainly a utility function to convert the edges to an adjacency // list vector<vector> constructadj(int V,vector<vector> &edges){

// Graph represented as an adjacency list
vector<vector<int> > adj(V);

// Constructing adjacency list
for (auto i : edges) {
    adj[i[0]].push_back(i[1]);
}

return adj;

}

// Function to return list containing vertices in // Topological order. vector topologicalSort(int V, vector<vector >& edges) { vector<vector> adj = constructadj(V,edges);

// Vector to store indegree of each vertex
vector<int> indegree(V);
for (int i = 0; i < V; i++) {
    for (auto it : adj[i]) {
        indegree[it]++;
    }
}
// Queue to store vertices with indegree 0
queue<int> q;
for (int i = 0; i < V; i++) {
    if (indegree[i] == 0) {
        q.push(i);
    }
}
vector<int> result;
while (!q.empty()) {
    int node = q.front();
    q.pop();
    result.push_back(node);
    
    // Decrease indegree of adjacent vertices as the
    // current node is in topological order
    for (auto it : adj[node]) {
        indegree[it]--;

        // If indegree becomes 0, push it to the queue
        if (indegree[it] == 0)
            q.push(it);
    }
}

// Check for cycle
if (result.size() != V) {
    cout << "Graph contains cycle!" << endl;
    return {};
}

return result;

}

int main() {

// Number of nodes
int V = 6;

// Edges
vector<vector<int> > edges
    = {{0, 1}, {1, 2}, {2, 3},
       {4, 5}, {5, 1}, {5, 2}};

vector<int> result = topologicalSort(V, edges);

// Displaying result
for (auto i : result) {
    cout << i << " ";
}

return 0;

}

Java

import java.util.*;

public class GfG {

// We mainly take input graph as a set of edges. This function is
// mainly a utility function to convert the edges to an adjacency
// list
static List<Integer>[] constructadj(int V, int[][] edges) {
    List<Integer>[] adj = new ArrayList[V];

    for (int i = 0; i < V; i++) {
        adj[i] = new ArrayList<>();
    }

    for (int[] edge : edges) {
        adj[edge[0]].add(edge[1]);
    }

    return adj;
}

// Function to return list containing vertices in Topological order
static int[] topologicalSort(int V, int[][] edges) {
    List<Integer>[] adj = constructadj(V, edges);
    int[] indegree = new int[V];

    // Calculate indegree of each vertex
    for (int i = 0; i < V; i++) {
        for (int neighbor : adj[i]) {
            indegree[neighbor]++;
        }
    }

    // Queue to store vertices with indegree 0
    Queue<Integer> q = new LinkedList<>();
    for (int i = 0; i < V; i++) {
        if (indegree[i] == 0) {
            q.offer(i);
        }
    }

    int[] result = new int[V];
    int index = 0;

    while (!q.isEmpty()) {
        int node = q.poll();
        result[index++] = node;

        for (int neighbor : adj[node]) {
            indegree[neighbor]--;
            if (indegree[neighbor] == 0) {
                q.offer(neighbor);
            }
        }
    }

    // Check for cycle
    if (index != V) {
        System.out.println("Graph contains a cycle!");
        return new int[0];
    }

    return result;
}

public static void main(String[] args) {
    int V = 6;
    int[][] edges = {{0, 1}, {1, 2}, {2, 3}, {4, 5}, {5, 1}, {5, 2}};
    
    int[] result = topologicalSort(V, edges);

    if (result.length > 0) {
        for (int i : result) {
            System.out.print(i + " ");
        }
    }
}

}

Python

from collections import deque

We mainly take input graph as a set of edges. This function is

mainly a utility function to convert the edges to an adjacency

list

def constructadj(V, edges): adj = [[] for _ in range(V)] for u, v in edges: adj[u].append(v) return adj

Function to return list containing vertices in Topological order

def topologicalSort(V, edges): adj = constructadj(V, edges) indegree = [0] * V

# Calculate indegree of each vertex
for u in range(V):
    for v in adj[u]:
        indegree[v] += 1

# Queue to store vertices with indegree 0
q = deque([i for i in range(V) if indegree[i] == 0])

result = []
while q:
    node = q.popleft()
    result.append(node)

    for neighbor in adj[node]:
        indegree[neighbor] -= 1
        if indegree[neighbor] == 0:
            q.append(neighbor)

# Check for cycle
if len(result) != V:
    print("Graph contains cycle!")
    return []

return result

if name == "main": V = 6 edges = [[0, 1], [1, 2], [2, 3], [4, 5], [5, 1], [5, 2]]

result = topologicalSort(V, edges)
if result:
    print("Topological Order:", result)

C#

using System; using System.Collections.Generic;

class GfG {

// We mainly take input graph as a set of edges. This function is
// mainly a utility function to convert the edges to an adjacency
// list
static List<int>[] ConstructAdj(int V, int[,] edges)
{
    List<int>[] adj = new List<int>[V];

    for (int i = 0; i < V; i++)
    {
        adj[i] = new List<int>();
    }

    for (int i = 0; i < edges.GetLength(0); i++)
    {
        adj[edges[i, 0]].Add(edges[i, 1]);
    }

    return adj;
}

// Function to return list containing vertices in Topological order
static int[] TopologicalSort(int V, int[,] edges)
{
    List<int>[] adj = ConstructAdj(V, edges);
    int[] indegree = new int[V];

    // Calculate indegree of each vertex
    for (int i = 0; i < V; i++)
    {
        foreach (var neighbor in adj[i])
        {
            indegree[neighbor]++;
        }
    }

    // Queue to store vertices with indegree 0
    Queue<int> q = new Queue<int>();
    for (int i = 0; i < V; i++)
    {
        if (indegree[i] == 0)
        {
            q.Enqueue(i);
        }
    }

    int[] result = new int[V];
    int index = 0;

    while (q.Count > 0)
    {
        int node = q.Dequeue();
        result[index++] = node;

        foreach (var neighbor in adj[node])
        {
            indegree[neighbor]--;
            if (indegree[neighbor] == 0)
            {
                q.Enqueue(neighbor);
            }
        }
    }

    // Check for cycle
    if (index != V)
    {
        Console.WriteLine("Graph contains a cycle!");
        return new int[0];
    }

    return result;
}

static void Main()
{
    int V = 6;
    int[,] edges = {{0, 1}, {1, 2}, {2, 3}, {4, 5}, {5, 1}, {5, 2}};

    int[] result = TopologicalSort(V, edges);

    if (result.Length > 0)
    {
        Console.WriteLine("Topological Order:");
        Console.WriteLine(string.Join(" ", result));
    }
}

}

JavaScript

// We mainly take input graph as a set of edges. This function is // mainly a utility function to convert the edges to an adjacency // list function constructadj(V, edges) { let adj = Array.from({ length: V }, () => []); for (let [u, v] of edges) { adj[u].push(v); } return adj; }

// Function to return list containing vertices in Topological order function topologicalSort(V, edges) { let adj = constructadj(V, edges); let indegree = Array(V).fill(0);

// Calculate indegree of each vertex
for (let u = 0; u < V; u++) {
    for (let v of adj[u]) {
        indegree[v]++;
    }
}

// Queue to store vertices with indegree 0
let queue = [];
for (let i = 0; i < V; i++) {
    if (indegree[i] === 0) queue.push(i);
}

let result = [];
while (queue.length > 0) {
    let node = queue.shift();
    result.push(node);

    for (let neighbor of adj[node]) {
        indegree[neighbor]--;
        if (indegree[neighbor] === 0) {
            queue.push(neighbor);
        }
    }
}

// Check for cycle
if (result.length !== V) {
    console.log("Graph contains cycle!");
    return [];
}

return result;

}

// Test the function let V = 6; let edges = [[0, 1], [1, 2], [2, 3], [4, 5], [5, 1], [5, 2]];

let result = topologicalSort(V, edges); if (result.length > 0) { console.log("Topological Order:", result.join(" ")); }

`

**Time Complexity: O(V+E). The outer for loop will be executed V number of times and the inner for loop will be executed E number of times.
**Auxiliary Space: O(V). The queue needs to store all the vertices of the graph.

We do not count the adjacency list in auxiliary space as it is necessary for representing the input graph.

**Applications of Kahn's algorithm for Topological Sort: