Kahn’s algorithm for Topological Sorting

Last Updated : 07 Mar, 2024

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: 5 4 2 3 1 0
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 for Topological Sorting is a method used to order the vertices of a directed graph in a linear order such that for every directed edge from vertex A to vertex B, A comes before B in the order. The algorithm works by repeatedly finding vertices with no incoming edges, removing them from the graph, and updating the incoming edges of the remaining vertices. This process continues until all vertices have been ordered.

Algorithm:

Add all nodes with in-degree 0 to a queue.
While the queue is not empty:
- Remove a node from the queue.
- For each outgoing edge from the removed node, decrement the in-degree of the destination node by 1.
- If the in-degree of a destination node becomes 0, add it to the queue.
If the queue is empty and there are still nodes in the graph, the graph contains a cycle and cannot be topologically sorted.
The nodes in the queue represent the topological ordering of the graph.

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.

Below is the implementation of the above algorithm.

C++

// Including necessary header file
#include <bits/stdc++.h>
using namespace std;
 
// Function to return list containing vertices in
// Topological order.
vector<int> topologicalSort(vector<vector<int> >& adj,
                            int V)
{
    // 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 n = 4;
 
    // Edges
    vector<vector<int> > edges
        = { { 0, 1 }, { 1, 2 }, { 3, 1 }, { 3, 2 } };
 
    // Graph represented as an adjacency list
    vector<vector<int> > adj(n);
 
    // Constructing adjacency list
    for (auto i : edges) {
        adj[i[0]].push_back(i[1]);
    }
 
    // Performing topological sort
    cout << "Topological sorting of the graph: ";
    vector<int> result = topologicalSort(adj, n);
 
    // Displaying result
    for (auto i : result) {
        cout << i << " ";
    }
 
    return 0;
}

Java

import java.util.*;
 
public class TopologicalSort {
 
    // Function to return list containing vertices in
    // Topological order.
    public static List<Integer>
    topologicalSort(List<List<Integer> > adj, int V)
    {
        // Vector to store indegree of each vertex
        int[] indegree = new int[V];
        for (List<Integer> list : adj) {
            for (int vertex : list) {
                indegree[vertex]++;
            }
        }
 
        // Queue to store vertices with indegree 0
        Queue<Integer> q = new LinkedList<>();
        for (int i = 0; i < V; i++) {
            if (indegree[i] == 0) {
                q.add(i);
            }
        }
        List<Integer> result = new ArrayList<>();
        while (!q.isEmpty()) {
            int node = q.poll();
            result.add(node);
            // Decrease indegree of adjacent vertices as the
            // current node is in topological order
            for (int adjacent : adj.get(node)) {
                indegree[adjacent]--;
                // If indegree becomes 0, push it to the
                // queue
                if (indegree[adjacent] == 0)
                    q.add(adjacent);
            }
        }
 
        // Check for cycle
        if (result.size() != V) {
            System.out.println("Graph contains cycle!");
            return new ArrayList<>();
        }
        return result;
    }
 
    public static void main(String[] args)
    {
        int n = 4; // Number of nodes
 
        // Edges
        List<List<Integer> > edges = Arrays.asList(
            Arrays.asList(0, 1), Arrays.asList(1, 2),
            Arrays.asList(3, 1), Arrays.asList(3, 2));
 
        // Graph represented as an adjacency list
        List<List<Integer> > adj = new ArrayList<>();
        for (int i = 0; i < n; i++) {
            adj.add(new ArrayList<>());
        }
        // Constructing adjacency list
        for (List<Integer> edge : edges) {
            adj.get(edge.get(0)).add(edge.get(1));
        }
 
        // Performing topological sort
        System.out.print(
            "Topological sorting of the graph: ");
        List<Integer> result = topologicalSort(adj, n);
 
        // Displaying result
        for (int vertex : result) {
            System.out.print(vertex + " ");
        }
    }
}

C#

using System;
using System.Collections.Generic;
 
class Program {
    // Function to return list containing vertices in
    // Topological order.
    static List<int> TopologicalSort(List<List<int> > adj,
                                     int V)
    {
        // Vector to store indegree of each vertex
        int[] indegree = new int[V];
        foreach(var list in adj)
        {
            foreach(var vertex in list)
            {
                indegree[vertex]++;
            }
        }
 
        // 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);
            }
        }
        List<int> result = new List<int>();
        while (q.Count > 0) {
            int node = q.Dequeue();
            result.Add(node);
            // Decrease indegree of adjacent vertices as the
            // current node is in topological order
            foreach(var adjacent in adj[node])
            {
                indegree[adjacent]--;
                // If indegree becomes 0, push it to the
                // queue
                if (indegree[adjacent] == 0)
                    q.Enqueue(adjacent);
            }
        }
 
        // Check for cycle
        if (result.Count != V) {
            Console.WriteLine("Graph contains cycle!");
            return new List<int>();
        }
        return result;
    }
 
    static void Main(string[] args)
    {
        int n = 4; // Number of nodes
 
        // Edges
        List<List<int> > edges = new List<List<int> >{
            new List<int>{ 0, 1 }, new List<int>{ 1, 2 },
            new List<int>{ 3, 1 }, new List<int>{ 3, 2 }
        };
 
        // Graph represented as an adjacency list
        List<List<int> > adj = new List<List<int> >();
        for (int i = 0; i < n; i++) {
            adj.Add(new List<int>());
        }
        // Constructing adjacency list
        foreach(var edge in edges)
        {
            adj[edge[0]].Add(edge[1]);
        }
 
        // Performing topological sort
        Console.Write("Topological sorting of the graph: ");
        List<int> result = TopologicalSort(adj, n);
 
        // Displaying result
        foreach(var vertex in result)
        {
            Console.Write(vertex + " ");
        }
    }
}

Javascript

// Function to return list containing vertices in Topological order.
function topologicalSort(adj, V) {
    // Vector to store indegree of each vertex
    const indegree = new Array(V).fill(0);
    for (let i = 0; i < V; i++) {
        for (const vertex of adj[i]) {
            indegree[vertex]++;
        }
    }
 
    // Queue to store vertices with indegree 0
    const q = [];
    for (let i = 0; i < V; i++) {
        if (indegree[i] === 0) {
            q.push(i);
        }
    }
    const result = [];
    while (q.length > 0) {
        const node = q.shift();
        result.push(node);
        // Decrease indegree of adjacent vertices as the current node is in topological order
        for (const adjacent of adj[node]) {
            indegree[adjacent]--;
            // If indegree becomes 0, push it to the queue
            if (indegree[adjacent] === 0) q.push(adjacent);
        }
    }
     
    // Check for cycle
    if (result.length !== V) {
        console.log("Graph contains cycle!");
        return [];
    }
    return result;
}
 
const n = 4; // Number of nodes
 
// Edges
const edges = [[0, 1], [1, 2], [3, 1], [3, 2]];
 
// Graph represented as an adjacency list
const adj = Array.from({ length: n }, () => []);
 
// Constructing adjacency list
for (const edge of edges) {
    adj[edge[0]].push(edge[1]);
}
 
// Performing topological sort
console.log("Topological sorting of the graph: ");
const result = topologicalSort(adj, n);
 
// Displaying result
for (const vertex of result) {
    console.log(vertex + " ");
}

Python3

from collections import deque
 
# Function to return list containing vertices in Topological order.
 
 
def topological_sort(adj, V):
    # Vector to store indegree of each vertex
    indegree = [0] * V
    for i in range(V):
        for vertex in adj[i]:
            indegree[vertex] += 1
 
    # Queue to store vertices with indegree 0
    q = deque()
    for i in range(V):
        if indegree[i] == 0:
            q.append(i)
    result = []
    while q:
        node = q.popleft()
        result.append(node)
        # Decrease indegree of adjacent vertices as the current node is in topological order
        for adjacent in adj[node]:
            indegree[adjacent] -= 1
            # If indegree becomes 0, push it to the queue
            if indegree[adjacent] == 0:
                q.append(adjacent)
 
    # Check for cycle
    if len(result) != V:
        print("Graph contains cycle!")
        return []
    return result
 
 
if __name__ == "__main__":
    n = 4  # Number of nodes
 
    # Edges
    edges = [[0, 1], [1, 2], [3, 1], [3, 2]]
 
    # Graph represented as an adjacency list
    adj = [[] for _ in range(n)]
 
    # Constructing adjacency list
    for edge in edges:
        adj[edge[0]].append(edge[1])
 
    # Performing topological sort
    print("Topological sorting of the graph:", end=" ")
    result = topological_sort(adj, n)
 
    # Displaying result
    for vertex in result:
        print(vertex, end=" ")

Output

Topological sorting of the graph: 0 3 1 2

Complexity Analysis:

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. So the space required is O(V)

Application of Kahn’s algorithm for Topological Sort:

Course sequencing: Courses at universities frequently have prerequisites for other courses. The courses can be scheduled using Kahn’s algorithm so that the prerequisites are taken before the courses that call for them.
Management of software dependencies: When developing software, libraries and modules frequently rely on other libraries and modules. The dependencies can be installed in the proper order by using Kahn’s approach.
Scheduling tasks: In project management, activities frequently depend on one another. The tasks can be scheduled using Kahn’s method so that the dependent tasks are finished before the tasks that depend on them.
Data processing: In data processing pipelines, the outcomes of some processes may be dependent. The stages can be carried out in the right order by using Kahn’s algorithm.
Circuit design: In the creation of an electronic circuit, some components may be dependent on the output of others. The components can be joined in the right order by using Kahn’s technique.

Chirag Agarwal

Improve

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BFS and DFS on Graph

Cycle in a Graph

Shortest Paths in Graph

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Some must do problems on Graph