Transitive Closure of a Graph using DFS

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 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++

// 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 

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

My Personal Notes

C/C++

Java

Python

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