Open In App
Related Articles

Topological Sorting

Improve
Improve
Improve
Like Article
Like
Save Article
Save
Report issue
Report

Topological sorting for Directed Acyclic Graph (DAG) is a linear ordering of vertices such that for every directed edge u-v, vertex u comes before v in the ordering.

Note: Topological Sorting for a graph is not possible if the graph is not a DAG.

Example:

Input: Graph :

example

Example

Output: 5 4 2 3 1 0
Explanation: The first vertex in topological sorting is always a vertex with an in-degree of 0 (a vertex with no incoming edges).  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. Another topological sorting of the following graph is “4 5 2 3 1 0”.

Recommended Practice

Topological Sorting vs Depth First Traversal (DFS)

In DFS, we print a vertex and then recursively call DFS for its adjacent vertices. In topological sorting, we need to print a vertex before its adjacent vertices. 

For example, In the above given graph, the vertex ‘5’ should be printed before vertex ‘0’, but unlike DFS, the vertex ‘4’ should also be printed before vertex ‘0’. So Topological sorting is different from DFS. For example, a DFS of the shown graph is “5 2 3 1 0 4”, but it is not a topological sorting.

Algorithm for Topological Sorting:

Prerequisite: DFS

We can modify DFS to find the Topological Sorting of a graph. In DFS

  • We start from a vertex, we first print it, and then 
  • Recursively call DFS for its adjacent vertices. 

In topological sorting:

  • We use a temporary stack. 
  • We don’t print the vertex immediately, 
  • We first recursively call topological sorting for all its adjacent vertices, then push it to a stack. 
  • Finally, print the contents of the stack. 

Note: A vertex is pushed to stack only when all of its adjacent vertices (and their adjacent vertices and so on) are already in the stack

Approach:

  • Create a stack to store the nodes.
  • Initialize visited array of size N to keep the record of visited nodes.
  • Run a loop from 0 till N :
  • if the node is not marked True in visited array then call the recursive function for topological sort and perform the following steps:
    • Mark the current node as True in the visited array.
    • Run a loop on all the nodes which has a directed edge to the current node
    • if the node is not marked True in the visited array:
      • Recursively call the topological sort function on the node
    • Push the current node in the stack.
  • Print all the elements in the stack.

Illustration Topological Sorting Algorithm:

Below image is an illustration of the above approach:

Topological-sorting

Overall workflow of topological sorting

Step1:

  • We start DFS from node 0 because it has zero incoming Nodes
  • We push node 0 in the stack and move to next node having minimum number of adjacent nodes i.e. node 1.

file

Step 2:

  • In this step , because there is no adjacent of this node so push the node 1 in the stack and move to next node.

file

Step 3:

  • In this step , We choose node 2 because it has minimum number of adjacent nodes after 0 and 1 .
  • We call DFS for node 2 and push all the nodes which comes in traversal from node 2 in reverse order.
  • So push 3 then push 2 .

file

Step 4:

  • We now call DFS for node 4
  • Because 0 and 1 already present in the stack so we just push node 4 in the stack and return.

file

Step 5:

  • In this step because all the adjacent nodes of 5 is already in the stack we push node 5 in the stack and return.

file

Step 6: This is the final step of the Topological sorting in which we pop all the element from the stack and print it in that order .

Below is the implementation of the above approach:

C++

// A C++ program to print topological
// sorting of a DAG
#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;
 
    // A function used by topologicalSort
    void topologicalSortUtil(int v, bool visited[],
                             stack<int>& Stack);
 
public:
    // Constructor
    Graph(int V);
 
    // function to add an edge to graph
    void addEdge(int v, int w);
 
    // 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 v, int w)
{
    // Add w to v’s list.
    adj[v].push_back(w);
}
 
// A recursive function used by topologicalSort
void Graph::topologicalSortUtil(int v, bool visited[],
                                stack<int>& Stack)
{
    // Mark the current node as visited.
    visited[v] = true;
 
    // Recur for all the vertices
    // adjacent to this vertex
    list<int>::iterator i;
    for (i = adj[v].begin(); i != adj[v].end(); ++i)
        if (!visited[*i])
            topologicalSortUtil(*i, visited, Stack);
 
    // Push current vertex to stack
    // which stores result
    Stack.push(v);
}
 
// The function to do Topological Sort.
// It uses recursive topologicalSortUtil()
void Graph::topologicalSort()
{
    stack<int> Stack;
 
    // Mark all the vertices as not visited
    bool* visited = new bool[V];
    for (int i = 0; i < V; i++)
        visited[i] = false;
 
    // Call the recursive helper function
    // to store Topological
    // Sort starting from all
    // vertices one by one
    for (int i = 0; i < V; i++)
        if (visited[i] == false)
            topologicalSortUtil(i, visited, Stack);
 
    // Print contents of stack
    while (Stack.empty() == false) {
        cout << Stack.top() << " ";
        Stack.pop();
    }
 
    delete[] visited;
}
 
// Driver Code
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 the given "
            "graph \n";
 
    // Function Call
    g.topologicalSort();
 
    return 0;
}

                    

Java

// A Java program to print topological
// sorting of a DAG
import java.io.*;
import java.util.*;
 
// This class represents a directed graph
// using adjacency list representation
class Graph {
    // No. of vertices
    private int V;
 
    // Adjacency List as ArrayList of ArrayList's
    private ArrayList<ArrayList<Integer> > adj;
 
    // Constructor
    Graph(int v)
    {
        V = v;
        adj = new ArrayList<ArrayList<Integer> >(v);
        for (int i = 0; i < v; ++i)
            adj.add(new ArrayList<Integer>());
    }
 
    // Function to add an edge into the graph
    void addEdge(int v, int w) { adj.get(v).add(w); }
 
    // A recursive function used by topologicalSort
    void topologicalSortUtil(int v, boolean visited[],
                             Stack<Integer> stack)
    {
        // Mark the current node as visited.
        visited[v] = true;
        Integer i;
 
        // Recur for all the vertices adjacent
        // to thisvertex
        Iterator<Integer> it = adj.get(v).iterator();
        while (it.hasNext()) {
            i = it.next();
            if (!visited[i])
                topologicalSortUtil(i, visited, stack);
        }
 
        // Push current vertex to stack
        // which stores result
        stack.push(new Integer(v));
    }
 
    // The function to do Topological Sort.
    // It uses recursive topologicalSortUtil()
    void topologicalSort()
    {
        Stack<Integer> stack = new Stack<Integer>();
 
        // Mark all the vertices as not visited
        boolean visited[] = new boolean[V];
        for (int i = 0; i < V; i++)
            visited[i] = false;
 
        // Call the recursive helper
        // function to store
        // Topological Sort starting
        // from all vertices one by one
        for (int i = 0; i < V; i++)
            if (visited[i] == false)
                topologicalSortUtil(i, visited, stack);
 
        // Print contents of stack
        while (stack.empty() == false)
            System.out.print(stack.pop() + " ");
    }
 
    // Driver code
    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 of the given graph");
        // Function Call
        g.topologicalSort();
    }
}
// This code is contributed by Aakash Hasija

                    

Python3

# Python program to print topological sorting of a DAG
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)
 
    # A recursive function used by topologicalSort
    def topologicalSortUtil(self, v, visited, stack):
 
        # Mark the current node as visited.
        visited[v] = True
 
        # Recur for all the vertices adjacent to this vertex
        for i in self.graph[v]:
            if visited[i] == False:
                self.topologicalSortUtil(i, visited, stack)
 
        # Push current vertex to stack which stores result
        stack.append(v)
 
    # The function to do Topological Sort. It uses recursive
    # topologicalSortUtil()
    def topologicalSort(self):
        # Mark all the vertices as not visited
        visited = [False]*self.V
        stack = []
 
        # Call the recursive helper function to store Topological
        # Sort starting from all vertices one by one
        for i in range(self.V):
            if visited[i] == False:
                self.topologicalSortUtil(i, visited, stack)
 
        # Print contents of the stack
        print(stack[::-1])  # return list in reverse order
 
 
# Driver Code
if __name__ == '__main__':
    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")
 
    # Function Call
    g.topologicalSort()
 
# This code is contributed by Neelam Yadav

                    

C#

using System;
using System.Collections.Generic;
 
// Class to represent a graph
class Graph
{
    // No. of vertices
    private int V;
 
    // Array containing adjacency lists
    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 v, int w)
    {
        // Add w to v's list
        adj[v].Add(w);
    }
 
    // A recursive function used by TopologicalSort
    private void TopologicalSortUtil(int v, bool[] visited, Stack<int> stack)
    {
        // Mark the current node as visited
        visited[v] = true;
 
        // Recur for all the vertices adjacent to this vertex
        foreach (var i in adj[v])
        {
            if (!visited[i])
            {
                TopologicalSortUtil(i, visited, stack);
            }
        }
 
        // Push current vertex to stack which stores result
        stack.Push(v);
    }
 
    // The function to do Topological Sort. It uses recursive TopologicalSortUtil
    public void TopologicalSort()
    {
        Stack<int> stack = new Stack<int>();
 
        // Mark all the vertices as not visited
        bool[] visited = new bool[V];
        for (int i = 0; i < V; i++)
        {
            visited[i] = false;
        }
 
        // Call the recursive helper function to store Topological Sort starting from all vertices one by one
        for (int i = 0; i < V; i++)
        {
            if (!visited[i])
            {
                TopologicalSortUtil(i, visited, stack);
            }
        }
 
        // Print contents of stack
        Console.Write("Following is a Topological Sort of the given graph: ");
        while (stack.Count > 0)
        {
            Console.Write(stack.Pop() + " ");
        }
        Console.WriteLine(); // Add a new line for better formatting
    }
}
 
// Driver Code
class Program
{
    static void Main()
    {
        // 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("");
 
        // Function Call
        g.TopologicalSort();
    }
}

                    

Javascript

// Javascript for the above approach
 
    // This class represents a directed graph
    // using adjacency list representation
    class Graph{
 
        // Constructor
        constructor(v)
        {
            // Number of vertices
            this.V = v
 
            // Adjacency List as ArrayList of ArrayList's
            this.adj = new Array(this.V)
            for (let i = 0 ; i < this.V ; i+=1){
                this.adj[i] = new Array()
            }
        }
 
        // Function to add an edge into the graph
        addEdge(v, w){
            this.adj[v].push(w)
        }
 
        // A recursive function used by topologicalSort
        topologicalSortUtil(v, visited, stack)
        {
            // Mark the current node as visited.
            visited[v] = true;
            let i = 0;
 
            // Recur for all the vertices adjacent
            // to thisvertex
            for(i = 0 ; i < this.adj[v].length ; i++){
                if(!visited[this.adj[v][i]]){
                    this.topologicalSortUtil(this.adj[v][i], visited, stack)
                }
            }
 
            // Push current vertex to stack
            // which stores result
            stack.push(v);
        }
 
        // The function to do Topological Sort.
        // It uses recursive topologicalSortUtil()
        topologicalSort()
        {
            let stack = new Array()
 
            // Mark all the vertices as not visited
            let visited = new Array(this.V);
            for (let i = 0 ; i < this.V ; i++){
                visited[i] = false;
            }
 
            // Call the recursive helper
            // function to store
            // Topological Sort starting
            // from all vertices one by one
            for (let i = 0 ; i < this.V ; i++){
                if (visited[i] == false){
                    this.topologicalSortUtil(i, visited, stack);
                }
            }
 
            // Print contents of stack
            while (stack.length != 0){
                console.log(stack.pop() + " ")
            }
        }
    }
 
    // Driver Code
    var 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.log("Following is a Topological sort of the given graph")
     
    // Function Call
    g.topologicalSort()
     
    // This code is contributed by subhamgoyal2014.

                    

Output
Following is a Topological Sort of the given graph 
5 4 2 3 1 0 

Time Complexity: O(V+E). The above algorithm is simply DFS with an extra stack. So time complexity is the same as DFS
Auxiliary space: O(V). The extra space is needed for the stack

Note: Here, we can also use a vector instead of the stack. If the vector is used then print the elements in reverse order to get the topological sorting.

Applications of Topological Sorting:

  • Topological Sorting is mainly used for scheduling jobs from the given dependencies among jobs. 
  • In computer science, applications of this type arise in:
    • Instruction scheduling
    • Ordering of formula cell evaluation when recomputing formula values in spreadsheets
    • Logic synthesis
    • Determining the order of compilation tasks to perform in make files
    • Data serialization
    • Resolving symbol dependencies in linkers

Related Articles: 



Last Updated : 29 Nov, 2023
Like Article
Save Article
Previous
Next
Share your thoughts in the comments
Similar Reads