Given a Directed Acyclic Graph (DAG), find Topological Sort of the graph.

**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 3 1 0”.

Please note that the first vertex in topological sorting is always a vertex with in-degree as 0 (a vertex with no incoming edges). For above graph, vertex 4 and 5 have no incoming edges.

We have already discussed a DFS-based algorithm using stack and Kahn’s Algorithm for Topological Sorting. We have also discussed how to print all topological sorts of the DAG here. In this post, another DFS based approach is discussed for finding Topological sort of a graph by introducing **concept of arrival and departure time of a vertex** in DFS.

**What is Arrival Time & Departure Time of Vertices in DFS?**

In DFS, **Arrival Time** is the time at which the vertex was explored for the first time and **Departure Time** is the time at which we have explored all the neighbors of the vertex and we are ready to backtrack.

**How to find Topological Sort of a graph using departure time?**

To find Topological Sort of a graph, we run DFS starting from all unvisited vertices one by one. For any vertex, before exploring any of its neighbors, we note the arrival time of that vertex and after exploring all the neighbors of the vertex, we note its departure time. Please note only departure time is needed to find Topological Sort of a graph, so we can skip arrival time of vertex. Finally, after we have visited all the vertices of the graph, we print the vertices in order of their decreasing departure time which is our desired Topological Order of Vertices.

Below is C++ implementation of above idea –

`// A C++ program to print topological sorting of a DAG ` `#include<bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Graph class represents a directed graph using adjacency ` `// list representation ` `class` `Graph ` `{ ` ` ` `int` `V; ` `// No. of vertices ` ` ` `// Pointer to an array containing adjacency lists ` ` ` `list<` `int` `>* adj; ` `public` `: ` ` ` `Graph(` `int` `); ` `// Constructor ` ` ` `~Graph(); ` `// Destructor ` ` ` ` ` `// function to add an edge to graph ` ` ` `void` `addEdge(` `int` `, ` `int` `); ` ` ` ` ` `// The function to do DFS traversal ` ` ` `void` `DFS(` `int` `, vector<` `bool` `> &, vector<` `int` `> &, ` `int` `&); ` ` ` ` ` `// The function to do Topological Sort. ` ` ` `void` `topologicalSort(); ` `}; ` ` ` `Graph::Graph(` `int` `V) ` `{ ` ` ` `this` `->V = V; ` ` ` `adj = ` `new` `list<` `int` `>[V]; ` `} ` ` ` `Graph::~Graph() ` `{ ` ` ` `delete` `[] adj; ` `} ` ` ` `void` `Graph::addEdge(` `int` `v, ` `int` `w) ` `{ ` ` ` `adj[v].push_back(w); ` `// Add w to v's list. ` `} ` ` ` `// The function to do DFS() and stores departure time ` `// of all vertex ` `void` `Graph::DFS(` `int` `v, vector<` `bool` `> &visited, ` ` ` `vector<` `int` `> &departure, ` `int` `&` `time` `) ` `{ ` ` ` `visited[v] = ` `true` `; ` ` ` `// time++; // arrival time of vertex v ` ` ` ` ` `for` `(` `int` `i : adj[v]) ` ` ` `if` `(!visited[i]) ` ` ` `DFS(i, visited, departure, ` `time` `); ` ` ` ` ` `// set departure time of vertex v ` ` ` `departure[++` `time` `] = v; ` `} ` ` ` `// The function to do Topological Sort. It uses DFS(). ` `void` `Graph::topologicalSort() ` `{ ` ` ` `// vector to store departure time of vertex. ` ` ` `vector<` `int` `> departure(V, -1); ` ` ` ` ` `// Mark all the vertices as not visited ` ` ` `vector<` `bool` `> visited(V, ` `false` `); ` ` ` `int` `time` `= -1; ` ` ` ` ` `// perform DFS on all unvisited vertices ` ` ` `for` `(` `int` `i = 0; i < V; i++) ` ` ` `if` `(!visited[i]) ` ` ` `DFS(i, visited, departure, ` `time` `); ` ` ` ` ` `// Print vertices in topological order ` ` ` `for` `(` `int` `i = V - 1; i >= 0; i--) ` ` ` `cout << departure[i] << ` `" "` `; ` `} ` ` ` `// 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 << ` ```
"Topological Sort of the given graph is
"
``` `; ` ` ` `g.topologicalSort(); ` ` ` ` ` `return` `0; ` `} ` |

Output:

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

**Time Complexity** of above solution is O(V + E).

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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