Transitive Closure of a Graph using DFS

Difficulty Level : Medium
Last Updated : 28 Jan, 2021

Given a directed graph, find out if a vertex v is reachable from another vertex u for all vertex pairs (u, v) in the given graph. Here reachable mean that there is a path from vertex u to v. The reach-ability matrix is called transitive closure of a graph.

For example, consider below graph

Transitive closure of above graphs is 
     1 1 1 1 
     1 1 1 1 
     1 1 1 1 
     0 0 0 1

We have discussed a O(V³) solution for this here. The solution was based on Floyd Warshall Algorithm. In this post a O(V²) algorithm for the same is discussed.
Below are abstract steps of algorithm.

Create a matrix tc[V][V] that would finally have transitive closure of given graph. Initialize all entries of tc[][] as 0.
Call DFS for every node of graph to mark reachable vertices in tc[][]. In recursive calls to DFS, we don’t call DFS for an adjacent vertex if it is already marked as reachable in tc[][].

Below is implementation of the above idea. The code uses adjacency list representation of input graph and builds a matrix tc[V][V] such that tc[u][v] would be true if v is reachable from u.

C++

// C++ program to print transitive closure of a graph 
#include<bits/stdc++.h> 
using namespace std; 
   
class Graph 
{ 
    int V; // No. of vertices 
    bool **tc; // To store transitive closure 
    list<int> *adj; // array of adjacency lists 
    void DFSUtil(int u, int v); 
public: 
    Graph(int V); // Constructor 
   
    // function to add an edge to graph 
    void addEdge(int v, int w) { adj[v].push_back(w); } 
   
    // prints transitive closure matrix 
    void transitiveClosure(); 
}; 
   
Graph::Graph(int V) 
{ 
    this->V = V; 
    adj = new list<int>[V]; 
   
    tc = new bool* [V]; 
    for (int i = 0; i < V; i++) 
    { 
        tc[i] = new bool[V]; 
        memset(tc[i], false, V*sizeof(bool)); 
    } 
} 
   
// A recursive DFS traversal function that finds 
// all reachable vertices for s. 
void Graph::DFSUtil(int s, int v) 
{ 
    // Mark reachability from s to t as true. 
    tc[s][v] = true; 
   
    // Find all the vertices reachable through v 
    list<int>::iterator i; 
    for (i = adj[v].begin(); i != adj[v].end(); ++i) 
        if (tc[s][*i] == false) 
            DFSUtil(s, *i); 
} 
   
// The function to find transitive closure. It uses 
// recursive DFSUtil() 
void Graph::transitiveClosure() 
{ 
    // Call the recursive helper function to print DFS 
    // traversal starting from all vertices one by one 
    for (int i = 0; i < V; i++) 
        DFSUtil(i, i); // Every vertex is reachable from self. 
   
    for (int i=0; i<V; i++) 
    { 
        for (int j=0; j<V; j++) 
            cout << tc[i][j] << " "; 
        cout << endl; 
    } 
} 
   
// Driver code 
int main() 
{ 
   
    // Create a graph given in the above diagram 
    Graph g(4); 
    g.addEdge(0, 1); 
    g.addEdge(0, 2); 
    g.addEdge(1, 2); 
    g.addEdge(2, 0); 
    g.addEdge(2, 3); 
    g.addEdge(3, 3);   
    cout << "Transitive closure matrix is \n"; 
    g.transitiveClosure(); 
    return 0; 
}

Java

// JAVA program to print transitive
// closure of a graph.
 
import java.util.ArrayList;
import java.util.Arrays;
 
// A directed graph using 
// adjacency list representation
public class Graph {
 
        // No. of vertices in graph
    private int vertices; 
 
        // adjacency list
    private ArrayList<Integer>[] adjList;
 
        // To store transitive closure
    private int[][] tc;
 
    // Constructor
    public Graph(int vertices) {
 
             // initialise vertex count
             this.vertices = vertices; 
             this.tc = new int[this.vertices][this.vertices];
 
             // initialise adjacency list
             initAdjList(); 
    }
 
    // utility method to initialise adjacency list
    @SuppressWarnings("unchecked")
    private void initAdjList() {
 
        adjList = new ArrayList[vertices];
        for (int i = 0; i < vertices; i++) {
            adjList[i] = new ArrayList<>();
        }
    }
 
    // add edge from u to v
    public void addEdge(int u, int v) {
                  
      // Add v to u's list.
        adjList[u].add(v); 
    }
 
    // The function to find transitive
    // closure. It uses
    // recursive DFSUtil()
    public void transitiveClosure() {
 
        // Call the recursive helper
        // function to print DFS
        // traversal starting from all
        // vertices one by one
        for (int i = 0; i < vertices; i++) {
            dfsUtil(i, i);
        }
 
        for (int i = 0; i < vertices; i++) {
          System.out.println(Arrays.toString(tc[i]));
        }
    }
 
    // A recursive DFS traversal
    // function that finds
    // all reachable vertices for s
    private void dfsUtil(int s, int v) {
 
        // Mark reachability from 
        // s to v as true.
        tc[s][v] = 1;
         
        // Find all the vertices reachable
        // through v
        for (int adj : adjList[v]) {            
            if (tc[s][adj]==0) {
                dfsUtil(s, adj);
            }
        }
    }
     
    // Driver Code
    public static void main(String[] args) {
 
        // Create a graph given
        // in the above diagram
        Graph g = new Graph(4);
 
        g.addEdge(0, 1);
        g.addEdge(0, 2);
        g.addEdge(1, 2);
        g.addEdge(2, 0);
        g.addEdge(2, 3);
        g.addEdge(3, 3);
        System.out.println("Transitive closure " +
                "matrix is");
 
        g.transitiveClosure();
 
    }
}
 
 
// This code is contributed
// by Himanshu Shekhar

Python

# Python program to print transitive closure of a graph
from collections import defaultdict
 
# This class represents a directed graph using adjacency
# list representation
class Graph:
 
    def __init__(self,vertices):
        # No. of vertices
        self.V= vertices
 
        # default dictionary to store graph
        self.graph= defaultdict(list)
 
        # To store transitive closure
        self.tc = [[0 for j in range(self.V)] for i in range(self.V)]
 
    # function to add an edge to graph
    def addEdge(self,u,v):
        self.graph[u].append(v)
 
    # A recursive DFS traversal function that finds
    # all reachable vertices for s
    def DFSUtil(self,s,v):
 
        # Mark reachability from s to v as true.
        self.tc[s][v] = 1
 
        # Find all the vertices reachable through v
        for i in self.graph[v]:
            if self.tc[s][i]==0:
                self.DFSUtil(s,i)
 
    # The function to find transitive closure. It uses
    # recursive DFSUtil()
    def transitiveClosure(self):
 
        # Call the recursive helper function to print DFS
        # traversal starting from all vertices one by one
        for i in range(self.V):
            self.DFSUtil(i, i)
        print self.tc
 
# Create a graph given in the above diagram
g = Graph(4)
g.addEdge(0, 1)
g.addEdge(0, 2)
g.addEdge(1, 2)
g.addEdge(2, 0)
g.addEdge(2, 3)
g.addEdge(3, 3)
 
print "Transitive closure matrix is"
g.transitiveClosure();
 
# This code is contributed by Neelam Yadav

C#

// C# program to print transitive
// closure of a graph.
using System;
using System.Collections.Generic;
 
// A directed graph using 
// adjacency list representation
public class Graph
{
 
  // No. of vertices in graph
  private int vertices; 
 
  // adjacency list
  private List<int>[] adjList;
 
  // To store transitive closure
  private int[,] tc;
 
  // Constructor
  public Graph(int vertices) 
  {
 
    // initialise vertex count
    this.vertices = vertices; 
    this.tc = new int[this.vertices, this.vertices];
 
    // initialise adjacency list
    initAdjList(); 
  }
 
  // utility method to initialise adjacency list
  private void initAdjList() 
  {
 
    adjList = new List<int>[vertices];
    for (int i = 0; i < vertices; i++) 
    {
      adjList[i] = new List<int>();
    }
  }
 
  // add edge from u to v
  public void addEdge(int u, int v)
  {
 
    // Add v to u's list.
    adjList[u].Add(v); 
  }
 
  // The function to find transitive
  // closure. It uses
  // recursive DFSUtil()
  public void transitiveClosure() {
 
    // Call the recursive helper
    // function to print DFS
    // traversal starting from all
    // vertices one by one
    for (int i = 0; i < vertices; i++) {
      dfsUtil(i, i);
    }
 
    for (int i = 0; i < vertices; i++) {
      for(int j = 0; j < vertices; j++)
        Console.Write(tc[i, j] + " ");
      Console.WriteLine();
    }
  }
 
  // A recursive DFS traversal
  // function that finds
  // all reachable vertices for s
  private void dfsUtil(int s, int v) {
 
    // Mark reachability from 
    // s to v as true.
    tc[s, v] = 1;
 
    // Find all the vertices reachable
    // through v
    foreach (int adj in adjList[v])
    {            
      if (tc[s, adj] == 0) {
        dfsUtil(s, adj);
      }
    }
  }
 
  // Driver Code
  public static void Main(String[] args) {
 
    // Create a graph given
    // in the above diagram
    Graph g = new Graph(4);
    g.addEdge(0, 1);
    g.addEdge(0, 2);
    g.addEdge(1, 2);
    g.addEdge(2, 0);
    g.addEdge(2, 3);
    g.addEdge(3, 3);
    Console.WriteLine("Transitive closure " +
                      "matrix is");
    g.transitiveClosure();
  }
}
 
// This code is contributed by Rajput-Ji

Output:

Transitive closure matrix is 
1 1 1 1 
1 1 1 1 
1 1 1 1 
0 0 0 1

References:
http://www.cs.princeton.edu/courses/archive/spr03/cs226/lectures/digraph.4up.pdf
This article is contributed by Aditya Goel. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

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