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 means 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:

👁 Untitled-Diagram-(1)

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 an O(V³) solution for this here. The solution was based on Floyd Warshall Algorithm. In this post, DFS solution is discussed. So for dense graph, it would become O(V³) and for sparse graph, it would become O(V²).

Below are the abstract steps of the algorithm. 

• Create a matrix tc[V][V] that would finally have transitive closure of the given graph. Initialize all entries of tc[][] as 0.
• Call DFS for every node of the 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 the 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.

Implementation:

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

Time Complexity : O(V^3) where V is the number of vertexes . For dense graph, it would become O(V³) and for sparse graph, it would become O(V²).
Auxiliary Space: O(V^2) where V is number of vertices.