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
// Edgesconst edges = [[0, 1], [1, 2], [3, 1], [3, 2]];// Graph represented as an adjacency listconst adj = Array.from({ length: n }, () => []);// Constructing adjacency listfor (const edge of edges) {
adj[edge[0]].push(edge[1]);
}// Performing topological sortconsole.log("Topological sorting of the graph: ");
const result = topologicalSort(adj, n);// Displaying resultfor (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.