Kahn’s algorithm for Topological Sorting

Topological sorting for Directed Acyclic Graph (DAG) is a linear ordering of vertices such that for every directed edge uv, vertex u comes before v in the ordering. Topological Sorting for a graph is not possible if the graph is not a DAG.
For example, a topological sorting of the following graph is “5 4 2 3 1 0?. There can be more than one topological sorting for a graph. For example, another topological sorting of the following graph is “4 5 2 0 3 1″. The first vertex in topological sorting is always a vertex with an in-degree of 0 (a vertex with no incoming edges).

graph

Let’s look at a few examples with proper explanation,
Example:

Input:
Output: 5 4 2 3 1 0
Explanation: The topological sorting of a DAG is done in a order such that for every directed edge uv, vertex u comes before v in the ordering. 5 has no incoming edge. 4 has no incoming edge, 2 and 0 have incoming edge from 4 and 5 and 1 is placed at last.
Input:
Output: 0 3 4 1 2
Explanation: 0 and 3 have no incoming edge, 4 and 1 has incoming edge from 0 and 3. 2 is placed at last.

A DFS based solution to find a topological sort has already been discussed.
Solution: In this article, we will see another way to find the linear ordering of vertices in a directed acyclic graph (DAG). The approach is based on the below fact:
A DAG G has at least one vertex with in-degree 0 and one vertex with out-degree 0.
Proof: There’s a simple proof to the above fact that a DAG does not contain a cycle which means that all paths will be of finite length. Now let S be the longest path from u(source) to v(destination). Since S is the longest path there can be no incoming edge to u and no outgoing edge from v, if this situation had occurred then S would not have been the longest path
=> indegree(u) = 0 and outdegree(v) = 0

Intuition:

Topological sorting is a kind of dependencies problem so, we can start with the tasks which do not have any dependencies and can be done straight away or simply if we talk about in the term of node,

We will always try to execute those nodes that have outdegree 0.
Then after execution of all those 0 outdegrees, we will execute which are directly dependent on currently resolved tasks (currently resolved tasks’ outdegrees will become 0 now) and so on will execute all other tasks.

We look closely we are doing these executions are done level-wise or in a Breadth-first search (BFS) manner. Similarly, we can also perform the same task for indegree=0.

Algorithm: Steps involved in finding the topological ordering of a DAG:
Step-1: Compute in-degree (number of incoming edges) for each of the vertex present in the DAG and initialize the count of visited nodes as 0.
Step-2: Pick all the vertices with in-degree as 0 and add them into a queue (Enqueue operation)
Step-3: Remove a vertex from the queue (Dequeue operation) and then.

Increment the count of visited nodes by 1.
Decrease in-degree by 1 for all its neighbouring nodes.
If the in-degree of neighbouring nodes is reduced to zero, then add it to the queue.

Step 4: Repeat Step 3 until the queue is empty.
Step 5: If the count of visited nodes is not equal to the number of nodes in the graph then the topological sort is not possible for the given graph.
How to find the in-degree of each node?
There are 2 ways to calculate in-degree of every vertex:
1. Take an in-degree array which will keep track of
Traverse the array of edges and simply increase the counter of the destination node by 1.

for each node in Nodes
    indegree[node] = 0;
for each edge(src, dest) in Edges
    indegree[dest]++

Time Complexity: O(V+E)

2. Traverse the list for every node and then increment the in-degree of all the nodes connected to it by 1.

    for each node in Nodes
        If (list[node].size()!=0) then
        for each dest in list
            indegree[dest]++;

Time Complexity: The outer for loop will be executed V number of times and the inner for loop will be executed E number of times, Thus overall time complexity is O(V+E).
The overall time complexity of the algorithm is O(V+E)

Below is the implementation of the above algorithm. The implementation uses method 2 discussed above for finding in degrees.

C++

// A C++ program to print topological
// sorting of a graph using indegrees.
#include <bits/stdc++.h>
using namespace std;
 
// Class to represent a graph
class Graph {
    // No. of vertices'
    int V;
 
    // Pointer to an array containing
    // adjacency listsList
    list<int>* adj;
 
public:
    // Constructor
    Graph(int V);
 
    // Function to add an edge to graph
    void addEdge(int u, int v);
 
    // prints a Topological Sort of
    // the complete graph
    void topologicalSort();
};
 
Graph::Graph(int V)
{
    this->V = V;
    adj = new list<int>[V];
}
 
void Graph::addEdge(int u, int v)
{
    adj[u].push_back(v);
}
 
// The function to do
// Topological Sort.
void Graph::topologicalSort()
{
    // Create a vector to store
    // indegrees of all
    // vertices. Initialize all
    // indegrees as 0.
    vector<int> in_degree(V, 0);
 
    // Traverse adjacency lists
    // to fill indegrees of
    // vertices.  This step
    // takes O(V+E) time
    for (int u = 0; u < V; u++) {
        list<int>::iterator itr;
        for (itr = adj[u].begin();
             itr != adj[u].end(); itr++)
            in_degree[*itr]++;
    }
 
    // Create an queue and enqueue
    // all vertices with indegree 0
    queue<int> q;
    for (int i = 0; i < V; i++)
        if (in_degree[i] == 0)
            q.push(i);
 
    // Initialize count of visited vertices
    int cnt = 0;
 
    // Create a vector to store
    // result (A topological
    // ordering of the vertices)
    vector<int> top_order;
 
    // One by one dequeue vertices
    // from queue and enqueue
    // adjacents if indegree of
    // adjacent becomes 0
    while (!q.empty()) {
        // Extract front of queue
        // (or perform dequeue)
        // and add it to topological order
        int u = q.front();
        q.pop();
        top_order.push_back(u);
 
        // Iterate through all its
        // neighbouring nodes
        // of dequeued node u and
        // decrease their in-degree
        // by 1
        list<int>::iterator itr;
        for (itr = adj[u].begin();
             itr != adj[u].end(); itr++)
 
            // If in-degree becomes zero,
            // add it to queue
            if (--in_degree[*itr] == 0)
                q.push(*itr);
 
        cnt++;
    }
 
    // Check if there was a cycle
    if (cnt != V) {
        cout << "There exists a cycle in the graph\n";
        return;
    }
 
    // Print topological order
    for (int i = 0; i < top_order.size(); i++)
        cout << top_order[i] << " ";
    cout << endl;
}
 
// Driver program to test above functions
int main()
{
    // Create a graph given in the
    // above diagram
    Graph g(6);
    g.addEdge(5, 2);
    g.addEdge(5, 0);
    g.addEdge(4, 0);
    g.addEdge(4, 1);
    g.addEdge(2, 3);
    g.addEdge(3, 1);
 
    cout << "Following is a Topological Sort of\n";
    g.topologicalSort();
 
    return 0;
}

Java

// A Java program to print topological
// sorting of a graph using indegrees
import java.util.*;
 
// Class to represent a graph
class Graph {
    // No. of vertices
    int V;
 
    // An Array of List which contains
    // references to the Adjacency List of
    // each vertex
    List<Integer> adj[];
    // Constructor
    public Graph(int V)
    {
        this.V = V;
        adj = new ArrayList[V];
        for (int i = 0; i < V; i++)
            adj[i] = new ArrayList<Integer>();
    }
 
    // Function to add an edge to graph
    public void addEdge(int u, int v)
    {
        adj[u].add(v);
    }
    // prints a Topological Sort of the
    // complete graph
    public void topologicalSort()
    {
        // Create a array to store
        // indegrees of all
        // vertices. Initialize all
        // indegrees as 0.
        int indegree[] = new int[V];
 
        // Traverse adjacency lists
        // to fill indegrees of
        // vertices. This step takes
        // O(V+E) time
        for (int i = 0; i < V; i++) {
            ArrayList<Integer> temp
                = (ArrayList<Integer>)adj[i];
            for (int node : temp) {
                indegree[node]++;
            }
        }
 
        // Create a queue and enqueue
        // all vertices with indegree 0
        Queue<Integer> q
            = new LinkedList<Integer>();
        for (int i = 0; i < V; i++) {
            if (indegree[i] == 0)
                q.add(i);
        }
 
        // Initialize count of visited vertices
        int cnt = 0;
 
        // Create a vector to store result
        // (A topological ordering of the vertices)
        Vector<Integer> topOrder = new Vector<Integer>();
        while (!q.isEmpty()) {
            // Extract front of queue
            // (or perform dequeue)
            // and add it to topological order
            int u = q.poll();
            topOrder.add(u);
 
            // Iterate through all its
            // neighbouring nodes
            // of dequeued node u and
            // decrease their in-degree
            // by 1
            for (int node : adj[u]) {
                // If in-degree becomes zero,
                // add it to queue
                if (--indegree[node] == 0)
                    q.add(node);
            }
            cnt++;
        }
 
        // Check if there was a cycle
        if (cnt != V) {
            System.out.println(
                "There exists a cycle in the graph");
            return;
        }
 
        // Print topological order
        for (int i : topOrder) {
            System.out.print(i + " ");
        }
    }
}
// Driver program to test above functions
class Main {
    public static void main(String args[])
    {
        // Create a graph given in the above diagram
        Graph g = new Graph(6);
        g.addEdge(5, 2);
        g.addEdge(5, 0);
        g.addEdge(4, 0);
        g.addEdge(4, 1);
        g.addEdge(2, 3);
        g.addEdge(3, 1);
        System.out.println(
            "Following is a Topological Sort");
        g.topologicalSort();
    }
}

Python3

# A Python program to print topological sorting of a graph
# using indegrees
from collections import defaultdict
 
# Class to represent a graph
class Graph:
    def __init__(self, vertices):
        self.graph = defaultdict(list) # dictionary containing adjacency List
        self.V = vertices # No. of vertices
 
    # function to add an edge to graph
    def addEdge(self, u, v):
        self.graph[u].append(v)
 
 
    # The function to do Topological Sort.
    def topologicalSort(self):
         
        # Create a vector to store indegrees of all
        # vertices. Initialize all indegrees as 0.
        in_degree = [0]*(self.V)
         
        # Traverse adjacency lists to fill indegrees of
           # vertices.  This step takes O(V + E) time
        for i in self.graph:
            for j in self.graph[i]:
                in_degree[j] += 1
 
        # Create an queue and enqueue all vertices with
        # indegree 0
        queue = []
        for i in range(self.V):
            if in_degree[i] == 0:
                queue.append(i)
 
        # Initialize count of visited vertices
        cnt = 0
 
        # Create a vector to store result (A topological
        # ordering of the vertices)
        top_order = []
 
        # One by one dequeue vertices from queue and enqueue
        # adjacents if indegree of adjacent becomes 0
        while queue:
 
            # Extract front of queue (or perform dequeue)
            # and add it to topological order
            u = queue.pop(0)
            top_order.append(u)
 
            # Iterate through all neighbouring nodes
            # of dequeued node u and decrease their in-degree
            # by 1
            for i in self.graph[u]:
                in_degree[i] -= 1
                # If in-degree becomes zero, add it to queue
                if in_degree[i] == 0:
                    queue.append(i)
 
            cnt += 1
 
        # Check if there was a cycle
        if cnt != self.V:
            print ("There exists a cycle in the graph")
        else :
            # Print topological order
            print (top_order)
 
 
g = Graph(6)
g.addEdge(5, 2);
g.addEdge(5, 0);
g.addEdge(4, 0);
g.addEdge(4, 1);
g.addEdge(2, 3);
g.addEdge(3, 1);
 
print ("Following is a Topological Sort of the given graph")
g.topologicalSort()
 
# This code is contributed by Neelam Yadav

C#

// C# program to print topological
// sorting of a graph using indegrees.
 
using System;
using System.Collections.Generic;
 
// Class to represent a graph
public class Graph {
  // No. of vertices'
  private int V;
 
  // Pointer to an array containing
  // adjacency listsList
  private List<int>[] adj;
 
  // Constructor
  public Graph(int V)
  {
    this.V = V;
    adj = new List<int>[ V ];
    for (int i = 0; i < V; i++)
      adj[i] = new List<int>();
  }
 
  // Function to Add an edge to graph
  public void AddEdge(int u, int v) { adj[u].Add(v); }
 
  // The function to do Topological Sort.
  public void TopologicalSort()
  {
     
    // Create a list to store
    // indegrees of all
    // vertices. Initialize all
    // indegrees as 0.
    int[] in_degree = new int[V];
 
    // Traverse adjacency lists
    // to fill indegrees of
    // vertices. This step
    // takes O(V+E) time
    for (int u = 0; u < V; u++) {
      foreach(int itr in adj[u]) in_degree[itr]++;
    }
 
    // Create an queue and enqueue
    // all vertices with indegree 0
    Queue<int> q = new Queue<int>();
    for (int i = 0; i < V; i++)
      if (in_degree[i] == 0)
        q.Enqueue(i);
 
    // Initialize count of visited vertices
    int cnt = 0;
 
    // Create a vector to store
    // result (A topological
    // ordering of the vertices)
    List<int> top_order = new List<int>();
 
    // One by one dequeue vertices
    // from queue and enqueue
    // adjacents if indegree of
    // adjacent becomes 0
    while (q.Count > 0)
    {
       
      // Extract front of queue
      // (or perform dequeue)
      // and Add it to topological order
      int u = q.Dequeue();
      top_order.Add(u);
 
      // Iterate through all its
      // neighbouring nodes
      // of dequeued node u and
      // decrease their in-degree
      // by 1
      foreach(int itr in adj[u])
      {
 
        // If in-degree becomes zero,
        // Add it to queue
        if (--in_degree[itr] == 0)
          q.Enqueue(itr);
      }
 
      cnt++;
    }
 
    // Check if there was a cycle
    if (cnt != V) {
      Console.WriteLine(
        "There exists a cycle in the graph");
      return;
    }
 
    // Print topological order
    for (int i = 0; i < top_order.Count; i++)
      Console.Write(top_order[i] + " ");
    Console.WriteLine();
  }
}
 
// Driver program to test above functions
public class GFG {
  static void Main(string[] args)
 
  {
 
    // Create a graph given in the
    // above diagram
    Graph g = new Graph(6);
    g.AddEdge(5, 2);
    g.AddEdge(5, 0);
    g.AddEdge(4, 0);
    g.AddEdge(4, 1);
    g.AddEdge(2, 3);
    g.AddEdge(3, 1);
 
    Console.WriteLine(
      "Following is a Topological Sort of");
    g.TopologicalSort();
  }
}
 
// This code is contributed by cavi4762.

Javascript

<script>
// A Javascript program to print topological
// sorting of a graph using indegrees
 
// No. of vertices
let V;
 
// An Array of List which contains
    // references to the Adjacency List of
    // each vertex
let adj;
 
function Graph(v)
{
    V = v;
    adj = new Array(V);
    for (let i = 0; i < V; i++)
            adj[i] = [];
}
 
// Function to add an edge to graph
function addEdge(u, v)
{
    adj[u].push(v);
}
 
// prints a Topological Sort of the
    // complete graph
function topologicalSort()
{
 
    // Create a array to store
        // indegrees of all
        // vertices. Initialize all
        // indegrees as 0.
        let indegree = new Array(V);
         for(let i = 0; i < V; i++)
            indegree[i] = 0;
             
        // Traverse adjacency lists
        // to fill indegrees of
        // vertices. This step takes
        // O(V+E) time
        for (let i = 0; i < V; i++) {
            let temp
                = adj[i];
            for (let node = 0; node < temp.length; node++) {
                indegree[temp[node]]++;
            }
        }
  
        // Create a queue and enqueue
        // all vertices with indegree 0
        let q = [];
        for (let i = 0; i < V; i++) {
            if (indegree[i] == 0)
                q.push(i);
        }
  
        // Initialize count of visited vertices
        let cnt = 0;
  
        // Create a vector to store result
        // (A topological ordering of the vertices)
        let topOrder = [];
        while (q.length!=0)
        {
         
            // Extract front of queue
            // (or perform dequeue)
            // and add it to topological order
            let u = q.shift();
            topOrder.push(u);
  
            // Iterate through all its
            // neighbouring nodes
            // of dequeued node u and
            // decrease their in-degree
            // by 1
            for (let node = 0; node < adj[u].length; node++)
            {
             
                // If in-degree becomes zero,
                // add it to queue
                if (--indegree[adj[u][node]] == 0)
                    q.push(adj[u][node]);
            }
            cnt++;
        }
  
        // Check if there was a cycle
        if (cnt != V) {
            document.write(
                "There exists a cycle in the graph");
            return;
        }
  
        // Print topological order
        for (let i = 0; i < topOrder.length; i++)
        {
            document.write(topOrder[i] + " ");
        }
     
}
 
// Driver program to test above functions
// Create a graph given in the above diagram
Graph(6);
addEdge(5, 2);
addEdge(5, 0);
addEdge(4, 0);
addEdge(4, 1);
addEdge(2, 3);
addEdge(3, 1);
document.write(
"Following is a Topological Sort<br>");
topologicalSort();
 
 
// This code is contributed by avanitrachhadiya2155
</script>

Output

Following is a Topological Sort of
4 5 2 0 3 1

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:

1.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.

2.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.

3.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.

4.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.

5.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.

This article is contributed by Chirag Agarwal. Please write comments if you find anything incorrect, or if you want to share more information about the topic discussed above

Last Updated : 06 Jun, 2023