Topological sorting is a common problem in computer science that involves arranging the vertices of a directed acyclic graph (DAG) in a linear order such that for every directed edge (u, v), vertex u comes before vertex v in the ordering.
Two important methods to solve are:
- Kahn’s Algorithm
- Depth-First Search (DFS) Algorithm
Let us take the following example as input for both algorithms:
Directed Acyclic Graph
Kahn’s Algorithm:
Kahn’s Algorithm is a topological sorting algorithm that uses a queue-based approach to sort vertices in a DAG. It starts by finding vertices that have no incoming edges and adds them to a queue. It then removes a vertex from the queue and adds it to the sorted list. The algorithm continues this process, removing vertices with no incoming edges until all vertices have been sorted.
Below is the implementation of the Kahn’s approach:
C++
// C++ Implementation#include <bits/stdc++.h>using namespace std;class Solution {public: // Topological sort using bfs is called // Kahn's Algorithm void topo_sort(vector<int> adj[], int n) { // Here concept of indegree used // as it represents number of incoming // edges on a node vector<int> indegree(n, 0); // Code for make indegree for (int i = 0; i < n; i++) { for (auto it : adj[i]) { indegree[it]++; } } vector<int> ans; queue<int> qrr; // Initially insert elements who has // indegree 0 for (int i = 0; i < n; i++) { if (indegree[i] == 0) { qrr.push(i); } } while (!qrr.empty()) { // push those elements in queue which // poses 0 indegree int node = qrr.front(); qrr.pop(); ans.push_back(node); for (auto it : adj[node]) { indegree[it]--; if (indegree[it] == 0) { qrr.push(it); } } } for (int i = 0; i < n; i++) { cout << ans[i] << " "; } }};// Function to add edgevoid addEdge(vector<int> adj[], int u, int v){ adj[u].push_back(v);}// Driver codeint main(){ int n = 6; vector<int> adj[n]; addEdge(adj, 5, 0); addEdge(adj, 5, 2); addEdge(adj, 2, 0); addEdge(adj, 2, 3); addEdge(adj, 3, 0); addEdge(adj, 3, 1); addEdge(adj, 1, 0); addEdge(adj, 4, 0); addEdge(adj, 4, 1); Solution s1; // Function call s1.topo_sort(adj, n); return 0;} |
Java
import java.util.*;class Solution { public void topo_sort(List<Integer>[] adj, int n) { List<Integer> indegree = new ArrayList<>(Collections.nCopies(n, 0)); for (int i = 0; i < n; i++) { for (int it : adj[i]) { indegree.set(it, indegree.get(it) + 1); } } List<Integer> ans = new ArrayList<>(); Queue<Integer> qrr = new LinkedList<>(); for (int i = 0; i < n; i++) { if (indegree.get(i) == 0) { qrr.add(i); } } while (!qrr.isEmpty()) { int node = qrr.poll(); ans.add(node); for (int it : adj[node]) { indegree.set(it, indegree.get(it) - 1); if (indegree.get(it) == 0) { qrr.add(it); } } } for (int i = 0; i < n; i++) { System.out.print(ans.get(i) + " "); } }}class Main { @SuppressWarnings("unchecked") public static void main(String[] args) { int n = 6; List<Integer>[] adj = new ArrayList[n]; for (int i = 0; i < n; i++) { adj[i] = new ArrayList<>(); } addEdge(adj, 5, 0); addEdge(adj, 5, 2); addEdge(adj, 2, 0); addEdge(adj, 2, 3); addEdge(adj, 3, 0); addEdge(adj, 3, 1); addEdge(adj, 1, 0); addEdge(adj, 4, 0); addEdge(adj, 4, 1); Solution s1 = new Solution(); s1.topo_sort(adj, n); } private static void addEdge(List<Integer>[] adj, int u, int v) { adj[u].add(v); }} |
Python3
from typing import List, Dequefrom collections import dequeclass Solution: def topo_sort(self, adj: List[int], n: int) -> List[int]: # Create a list to store the in-degree of each node indegree = [0] * n # Calculate the in-degree of each node for i in range(n): for j in adj[i]: indegree[j] += 1 # Create a queue to store nodes with in-degree 0 q = deque() for i in range(n): if indegree[i] == 0: q.append(i) # Perform topological sorting using BFS ans = [] while q: node = q.popleft() ans.append(node) for i in adj[node]: indegree[i] -= 1 if indegree[i] == 0: q.append(i) # Return the sorted nodes return ans# Function to add an edgedef addEdge(adj: List[int], u: int, v: int) -> None: adj[u].append(v)# Driver codeif __name__ == "__main__": n = 6 adj = [[] for _ in range(n)] addEdge(adj, 5, 0) addEdge(adj, 5, 2) addEdge(adj, 2, 0) addEdge(adj, 2, 3) addEdge(adj, 3, 0) addEdge(adj, 3, 1) addEdge(adj, 1, 0) addEdge(adj, 4, 0) addEdge(adj, 4, 1) s = Solution() # Function call print(s.topo_sort(adj, n)) |
C#
// C# Implementationusing System;using System.Collections.Generic;class Solution { // Topological sort using bfs is called // Kahn's Algorithm public void TopoSort(List<int>[] adj, int n) { // Here concept of indegree used // as it represents number of incoming // edges on a node int[] indegree = new int[n]; // Code for make indegree for (int i = 0; i < n; i++) { foreach(int it in adj[i]) { indegree[it]++; } } List<int> ans = new List<int>(); Queue<int> qrr = new Queue<int>(); // Initially insert elements who has // indegree 0 for (int i = 0; i < n; i++) { if (indegree[i] == 0) { qrr.Enqueue(i); } } while (qrr.Count > 0) { // push those elements in queue which // poses 0 indegree int node = qrr.Dequeue(); ans.Add(node); foreach(int it in adj[node]) { indegree[it]--; if (indegree[it] == 0) { qrr.Enqueue(it); } } } for (int i = 0; i < n; i++) { Console.Write(ans[i] + " "); } }}// Driver codepublic class GFG { static void Main(string[] args) { int n = 6; List<int>[] adj = new List<int>[ n ]; for (int i = 0; i < n; i++) { adj[i] = new List<int>(); } adj[5].Add(0); adj[5].Add(2); adj[2].Add(0); adj[2].Add(3); adj[3].Add(0); adj[3].Add(1); adj[1].Add(0); adj[4].Add(0); adj[4].Add(1); Solution s1 = new Solution(); // Function call s1.TopoSort(adj, n); }} |
Javascript
// Javadcript code addition class Solution { topoSort(adj, n) { let indegree = new Array(n).fill(0); let ans = []; let qrr = []; // Code for make indegree for (let i = 0; i < n; i++) { for (let it of adj[i]) { indegree[it]++; } } // Initially insert elements who has indegree 0 for (let i = 0; i < n; i++) { if (indegree[i] == 0) { qrr.push(i); } } while (qrr.length > 0) { let node = qrr.shift(); ans.push(node); for (let it of adj[node]) { indegree[it]--; if (indegree[it] == 0) { qrr.push(it); } } } for (let i = 0; i < n; i++) { process.stdout.write(ans[i] + " "); } }}// Function to add edgefunction addEdge(adj, u, v) { adj[u].push(v);}// Driver codelet n = 6;let adj = Array.from({ length: n }, () => []);addEdge(adj, 5, 0);addEdge(adj, 5, 2);addEdge(adj, 2, 0);addEdge(adj, 2, 3);addEdge(adj, 3, 0);addEdge(adj, 3, 1);addEdge(adj, 1, 0);addEdge(adj, 4, 0);addEdge(adj, 4, 1);let s1 = new Solution();// Function calls1.topoSort(adj, n);// The code is contributed by Nidhi goel. |
Output
4 5 2 3 1 0
Time Complexity: O( V + E)
Auxiliary Space: O(V)
DFS Approach:
DFS Approach is a recursive algorithm that performs a depth-first search on the DAG. It starts at a vertex, explores as far as possible along each branch before backtracking, and marks visited vertices. During the DFS traversal, vertices are added to a stack in the order they are visited. Once the DFS traversal is complete, the stack is reversed to obtain the topological ordering.
Below is the implementation of the DFS approach:
C++
#include <bits/stdc++.h>using namespace std;class Solution { // Topo sort only exists in DAGs i.e. // Direct Acyclic graph void dfs(vector<int> adj[], vector<int>& vis, int node, int n, stack<int>& stck) { vis[node] = 1; for (auto it : adj[node]) { if (!vis[it]) { dfs(adj, vis, it, n, stck); } } stck.push(node); }public: // During the traversal u must // be visited before v stack<int> topo_sort(vector<int> adj[], int n) { vector<int> vis(n, 0); // using stack ADT stack<int> stck; for (int i = 0; i < n; i++) { if (!vis[i]) { dfs(adj, vis, i, n, stck); } } return stck; }};void addEdge(vector<int> adj[], int u, int v){ adj[u].push_back(v);}// Drivers codeint main(){ int n = 6; vector<int> adj[n]; addEdge(adj, 5, 0); addEdge(adj, 5, 2); addEdge(adj, 2, 0); addEdge(adj, 2, 3); addEdge(adj, 3, 0); addEdge(adj, 3, 1); addEdge(adj, 1, 0); addEdge(adj, 4, 0); addEdge(adj, 4, 1); Solution s1; stack<int> ans = s1.topo_sort(adj, n); for (int i = 0; i < n; i++) { int n = ans.top(); ans.pop(); cout << n << " "; } return 0;} |
Java
// Java Implementationimport java.util.ArrayList;import java.util.List;import java.util.Stack;class Solution { // Topo sort only exists in DAGs i.e. // Direct Acyclic Graph void dfs(List<Integer>[] adj, List<Integer> vis, int node, int n, Stack<Integer> stck) { vis.set(node, 1); for (int it : adj[node]) { if (vis.get(it) == 0) { dfs(adj, vis, it, n, stck); } } stck.push(node); } // During the traversal u must // be visited before v Stack<Integer> topo_sort(List<Integer>[] adj, int n) { List<Integer> vis = new ArrayList<>(n); for (int i = 0; i < n; i++) { vis.add(0); } // using stack ADT Stack<Integer> stck = new Stack<>(); for (int i = 0; i < n; i++) { if (vis.get(i) == 0) { dfs(adj, vis, i, n, stck); } } return stck; }}public class Main { static void addEdge(List<Integer>[] adj, int u, int v) { adj[u].add(v); } public static void main(String[] args) { int n = 6; List<Integer>[] adj = new ArrayList[n]; for (int i = 0; i < n; i++) { adj[i] = new ArrayList<>(); } addEdge(adj, 5, 0); addEdge(adj, 5, 2); addEdge(adj, 2, 0); addEdge(adj, 2, 3); addEdge(adj, 3, 0); addEdge(adj, 3, 1); addEdge(adj, 1, 0); addEdge(adj, 4, 0); addEdge(adj, 4, 1); Solution solution = new Solution(); Stack<Integer> ans = solution.topo_sort(adj, n); while (!ans.isEmpty()) { int node = ans.pop(); System.out.print(node + " "); } }}// This code is contributed by Utkarsh Kumar |
Python3
# Python implementation of the above approachclass Solution: # Topo sort only exists in DAGs i.e. # Direct Acyclic graph def dfs(self, adj, vis, node, n, stck): vis[node] = 1 for it in adj[node]: if not vis[it]: self.dfs(adj, vis, it, n, stck) stck.append(node) # During the traversal u must # be visited before v def topo_sort(self, adj, n): vis = [0] * n # using stack ADT stck = [] for i in range(n): if not vis[i]: self.dfs(adj, vis, i, n, stck) return stckdef addEdge(adj, u, v): adj[u].append(v)# Drivers coden = 6adj = [[] for _ in range(n)]addEdge(adj, 5, 0)addEdge(adj, 5, 2)addEdge(adj, 2, 0)addEdge(adj, 2, 3)addEdge(adj, 3, 0)addEdge(adj, 3, 1)addEdge(adj, 1, 0)addEdge(adj, 4, 0)addEdge(adj, 4, 1)s1 = Solution()ans = s1.topo_sort(adj, n)for i in range(n): num = ans.pop() print(num, end=" ")# This code is contributed by Tapesh(tapeshdua420) |
C#
using System;using System.Collections.Generic;class Solution{ // Topo sort only exists in DAGs i.e. // Direct Acyclic Graph void DFS(List<int>[] adj, List<int> vis, int node, int n, Stack<int> stck) { vis[node] = 1; foreach (int it in adj[node]) { if (vis[it] == 0) { DFS(adj, vis, it, n, stck); } } stck.Push(node); } // During the traversal, u must // be visited before v public Stack<int> TopoSort(List<int>[] adj, int n) { List<int> vis = new List<int>(n); for (int i = 0; i < n; i++) { vis.Add(0); } // using stack ADT Stack<int> stck = new Stack<int>(); for (int i = 0; i < n; i++) { if (vis[i] == 0) { DFS(adj, vis, i, n, stck); } } return stck; }}class Program{ static void AddEdge(List<int>[] adj, int u, int v) { adj[u].Add(v); } static void Main(string[] args) { int n = 6; List<int>[] adj = new List<int>[n]; for (int i = 0; i < n; i++) { adj[i] = new List<int>(); } AddEdge(adj, 5, 0); AddEdge(adj, 5, 2); AddEdge(adj, 2, 0); AddEdge(adj, 2, 3); AddEdge(adj, 3, 0); AddEdge(adj, 3, 1); AddEdge(adj, 1, 0); AddEdge(adj, 4, 0); AddEdge(adj, 4, 1); Solution solution = new Solution(); Stack<int> ans = solution.TopoSort(adj, n); while (ans.Count > 0) { int node = ans.Pop(); Console.Write(node + " "); } }} |
Javascript
class Solution { // Topo sort only exists in DAGs i.e. // Direct Acyclic Graph dfs(adj, vis, node, n, stck) { vis[node] = 1; for (let it of adj[node]) { if (!vis[it]) { this.dfs(adj, vis, it, n, stck); } } stck.push(node); } // During the traversal, u must // be visited before v topo_sort(adj, n) { const vis = new Array(n).fill(0); // Using stack data structure const stck = []; for (let i = 0; i < n; i++) { if (!vis[i]) { this.dfs(adj, vis, i, n, stck); } } return stck.reverse(); }}function addEdge(adj, u, v) { adj[u].push(v);}const n = 6;const adj = Array.from({ length: n }, () => []);addEdge(adj, 5, 0);addEdge(adj, 5, 2);addEdge(adj, 2, 0);addEdge(adj, 2, 3);addEdge(adj, 3, 0);addEdge(adj, 3, 1);addEdge(adj, 1, 0);addEdge(adj, 4, 0);addEdge(adj, 4, 1);const solution = new Solution();const ans = solution.topo_sort(adj, n);console.log(ans.join(" ")); |
Output
5 4 2 3 1 0
Time Complexity: O( V + E)
Auxiliary Space: O(V)
Comparative Analysis :
- Both Kahn’s Algorithm and DFS Approach have their strengths and weaknesses. Kahn’s Algorithm is easy to implement and guarantees a topological sort, but it can be slower for large graphs. On the other hand, DFS Approach is more efficient for larger graphs, but it requires more memory and can be more complex to implement.
- Kahn’s Algorithm is a good choice when dealing with smaller graphs, where simplicity and correctness are more important than performance. It is also useful in situations where you need to detect cycles in the graph. If a cycle is detected during the sorting process, the algorithm will terminate early and report that the graph is not a DAG.
- DFS Approach, on the other hand, is a better choice when dealing with larger graphs where performance is more important than simplicity. It is also useful in situations where you need to find a topological sort quickly, and memory is not a constraint.
Advantages of Kahn’s Algorithm over the DFS Approach:
- Guaranteed to find a topological ordering if one exists.
- Simple and easy to implement.
- Works well for small and medium-sized DAGs.
Disadvantages of Kahn’s Algorithm compared to the DFS Approach:
- Can be slower than DFS Approach for large DAGs.
- Requires more memory than DFS Approach.
- Requires a queue data structure, which may not be available in some programming languages.
Advantages of the DFS Approach over Kahn’s Algorithm:
- Can be faster for larger DAGs.
- Requires less memory than Kahn’s Algorithm.
- Does not require a queue data structure.
Disadvantages of the DFS Approach compared to Kahn’s Algorithm:
- May not find a topological ordering if the graph has cycles.
- More complex to implement than Kahn’s Algorithm.
- Can be less efficient for small and medium-sized DAGs.
Conclusion:
In conclusion, both Kahn’s Algorithm and DFS Approach are effective algorithms for topological sorting in DAGs. The choice between the two depends on the size of the graph, the available memory, and the required performance. In general, Kahn’s Algorithm is simpler and more reliable, but DFS Approach can be more efficient for larger graphs. Both algorithms have their strengths and weaknesses, and the best approach depends on the specific requirements of the problem at hand.
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