Topological Sort

Topological sorting of a Directed Acyclic Graph (DAG)

Criteria for topological sorting : Vertex with no incoming edges is accessed first followed by the vertices on the outgoing paths.
Example 1 : In the below graph vertex 1 has no incoming edges and can be reached before 3, 6, 0, 5, 2 and 4.
Example 2 : In the below graph vertex 0 has no incoming edges and can be reached before 5, 2, 1, 3, 6 and 4.


Algorithm : Topological Sort (Graph G, Source_Vertex S)
1.    Mark the source vertex S as visited.
2.    For every vertex V adjacent to the source vertex S.
3.        If the vertex V is not visited
4.              Topological Sort (Graph G, Source_Vertex V)
5.    Push the source vertex S into the stack STK.

Printing the topological sorting order
1.    While the stack STK is not empty.
2.        Print the vertex V at the stack (STK) top.
3.        Pop the vertex V at the stack (STK) top.


Example of topological sorting in a graph Topological_Search_Graph_Image Data structure used for storing graph: Adjacency list
Data structure used for DFS: Stack

Time complexity of topological sort : O(V+E) for an adjacency list implementation of a graph. ‘V’ is the number of vertices and ‘E’ is the number of edges in a graph.


Python

Python : Topological Sorting implementation Python 3.7 using dataclass


Java

Java : Topological Sorting implementation Java 8


C++ : Topological Sorting implementation C++14

#include<iostream>
#include<list>
#include<vector>
#include<stack>

using namespace std;

class Graph{

    private:
        int nodes;
        list<int> *adjlist;
        vector<bool> visited;
        stack<int> stack_topological_order;

    public:
        Graph () {
        }

        Graph (int nodes) {
            adjlist = new list<int> [nodes];
            visited.resize(nodes, false);
            this->nodes = nodes;
        }

        ~Graph(){ 
            delete [] adjlist;
        }

        void AddEdge (int src, int dst) {
            adjlist[src].push_back(dst);
        }
    
        void TopologicalSort (int src) {
            visited[src] = true;
            for (auto& itr : adjlist[src]) {
                if (!visited[itr]) {
                    TopologicalSort (itr);
                }   
            }   
            /* Only after all the outgoing edges are visited push the 
               source node in the stack */
            stack_topological_order.push(src);
        }  

        void Traverse () {
            for(int i=0; i<nodes; i++) {
                if (!visited[i]) {
                   TopologicalSort (i);
                }   
            }  
            while (!stack_topological_order.empty()) {
                cout << stack_topological_order.top() << " ";
                stack_topological_order.pop();
            }
        }
};
   
int main()
{
    Graph g(7);
    // Store the edges of the directed graph in adjacency list.
    g.AddEdge(0,2);
    g.AddEdge(0,5);
    g.AddEdge(1,3);
    g.AddEdge(1,6);
    g.AddEdge(2,4);
    g.AddEdge(3,5);
    g.AddEdge(5,2);
    g.AddEdge(5,4);
    g.AddEdge(6,2);

    cout << "Topological Sorting Order: "; 
    g.Traverse();

    return 0;
}

Output showing topological sorting of nodes in a graph

Topological Sorting Order: 1 6 3 0 5 2 4

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