Transitive Closure of a Graph using DFS

2.8

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
transitiveclosure
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.

  1. Create a matrix tc[V][V] that would finally have transitive closure of given graph. Initialize all entries of tc[][] as 0.
  2. 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;
}

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

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