Given a directed graph, find out if a vertex j is reachable from another vertex i for all vertex pairs (i, j) in the given graph. Here reachable mean that there is a path from vertex i to j. 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

The graph is given in the form of adjacency matrix say ‘graph[V][V]’ where graph[i][j] is 1 if there is an edge from vertex i to vertex j or i is equal to j, otherwise graph[i][j] is 0.

Floyd Warshall Algorithm can be used, we can calculate the distance matrix dist[V][V] using Floyd Warshall, if dist[i][j] is infinite, then j is not reachable from i, otherwise j is reachable and value of dist[i][j] will be less than V.

Instead of directly using Floyd Warshall, we can optimize it in terms of space and time, for this particular problem. Following are the optimizations:

**1)** Instead of integer resultant matrix (dist[V][V] in floyd warshall), we can create a boolean reach-ability matrix reach[V][V] (we save space). The value reach[i][j] will be 1 if j is reachable from i, otherwise 0.

**2)** Instead of using arithmetic operations, we can use logical operations. For arithmetic operation ‘+’, logical and ‘&&’ is used, and for min, logical or ‘||’ is used. (We save time by a constant factor. Time complexity is same though)

## C++

// Program for transitive closure using Floyd Warshall Algorithm #include<stdio.h> // Number of vertices in the graph #define V 4 // A function to print the solution matrix void printSolution(int reach[][V]); // Prints transitive closure of graph[][] using Floyd Warshall algorithm void transitiveClosure(int graph[][V]) { /* reach[][] will be the output matrix that will finally have the shortest distances between every pair of vertices */ int reach[V][V], i, j, k; /* Initialize the solution matrix same as input graph matrix. Or we can say the initial values of shortest distances are based on shortest paths considering no intermediate vertex. */ for (i = 0; i < V; i++) for (j = 0; j < V; j++) reach[i][j] = graph[i][j]; /* Add all vertices one by one to the set of intermediate vertices. ---> Before start of a iteration, we have reachability values for all pairs of vertices such that the reachability values consider only the vertices in set {0, 1, 2, .. k-1} as intermediate vertices. ----> After the end of a iteration, vertex no. k is added to the set of intermediate vertices and the set becomes {0, 1, .. k} */ for (k = 0; k < V; k++) { // Pick all vertices as source one by one for (i = 0; i < V; i++) { // Pick all vertices as destination for the // above picked source for (j = 0; j < V; j++) { // If vertex k is on a path from i to j, // then make sure that the value of reach[i][j] is 1 reach[i][j] = reach[i][j] || (reach[i][k] && reach[k][j]); } } } // Print the shortest distance matrix printSolution(reach); } /* A utility function to print solution */ void printSolution(int reach[][V]) { printf ("Following matrix is transitive closure of the given graph\n"); for (int i = 0; i < V; i++) { for (int j = 0; j < V; j++) printf ("%d ", reach[i][j]); printf("\n"); } } // driver program to test above function int main() { /* Let us create the following weighted graph 10 (0)------->(3) | /|\ 5 | | | | 1 \|/ | (1)------->(2) 3 */ int graph[V][V] = { {1, 1, 0, 1}, {0, 1, 1, 0}, {0, 0, 1, 1}, {0, 0, 0, 1} }; // Print the solution transitiveClosure(graph); return 0; }

## Java

// Program for transitive closure using Floyd Warshall Algorithm import java.util.*; import java.lang.*; import java.io.*; class GraphClosure { final static int V = 4; //Number of vertices in a graph // Prints transitive closure of graph[][] using Floyd // Warshall algorithm void transitiveClosure(int graph[][]) { /* reach[][] will be the output matrix that will finally have the shortest distances between every pair of vertices */ int reach[][] = new int[V][V]; int i, j, k; /* Initialize the solution matrix same as input graph matrix. Or we can say the initial values of shortest distances are based on shortest paths considering no intermediate vertex. */ for (i = 0; i < V; i++) for (j = 0; j < V; j++) reach[i][j] = graph[i][j]; /* Add all vertices one by one to the set of intermediate vertices. ---> Before start of a iteration, we have reachability values for all pairs of vertices such that the reachability values consider only the vertices in set {0, 1, 2, .. k-1} as intermediate vertices. ----> After the end of a iteration, vertex no. k is added to the set of intermediate vertices and the set becomes {0, 1, 2, .. k} */ for (k = 0; k < V; k++) { // Pick all vertices as source one by one for (i = 0; i < V; i++) { // Pick all vertices as destination for the // above picked source for (j = 0; j < V; j++) { // If vertex k is on a path from i to j, // then make sure that the value of reach[i][j] is 1 reach[i][j] = (reach[i][j]!=0) || ((reach[i][k]!=0) && (reach[k][j]!=0))?1:0; } } } // Print the shortest distance matrix printSolution(reach); } /* A utility function to print solution */ void printSolution(int reach[][]) { System.out.println("Following matrix is transitive closure"+ " of the given graph"); for (int i = 0; i < V; i++) { for (int j = 0; j < V; j++) System.out.print(reach[i][j]+" "); System.out.println(); } } // Driver program to test above function public static void main (String[] args) { /* Let us create the following weighted graph 10 (0)------->(3) | /|\ 5 | | | | 1 \|/ | (1)------->(2) 3 */ /* Let us create the following weighted graph 10 (0)------->(3) | /|\ 5 | | | | 1 \|/ | (1)------->(2) 3 */ int graph[][] = new int[][]{ {1, 1, 0, 1}, {0, 1, 1, 0}, {0, 0, 1, 1}, {0, 0, 0, 1} }; // Print the solution GraphClosure g = new GraphClosure(); g.transitiveClosure(graph); } } // This code is contributed by Aakash Hasija

## Python

# Python program for transitive closure using Floyd Warshall Algorithm #Complexity : O(V^3) from collections import defaultdict #Class to represent a graph class Graph: def __init__(self, vertices): self.V = vertices # A utility function to print the solution def printSolution(self, reach): print ("Following matrix transitive closure of the given graph ") for i in range(self.V): for j in range(self.V): print "%7d\t" %(reach[i][j]), print "" # Prints transitive closure of graph[][] using Floyd Warshall algorithm def transitiveClosure(self,graph): '''reach[][] will be the output matrix that will finally have reachability values. Initialize the solution matrix same as input graph matrix''' reach =[i[:] for i in graph] '''Add all vertices one by one to the set of intermediate vertices. ---> Before start of a iteration, we have reachability value for all pairs of vertices such that the reachability values consider only the vertices in set {0, 1, 2, .. k-1} as intermediate vertices. ----> After the end of an iteration, vertex no. k is added to the set of intermediate vertices and the set becomes {0, 1, 2, .. k}''' for k in range(self.V): # Pick all vertices as source one by one for i in range(self.V): # Pick all vertices as destination for the # above picked source for j in range(self.V): # If vertex k is on a path from i to j, # then make sure that the value of reach[i][j] is 1 reach[i][j] = reach[i][j] or (reach[i][k] and reach[k][j]) self.printSolution(reach) g= Graph(4) graph = [[1, 1, 0, 1], [0, 1, 1, 0], [0, 0, 1, 1], [0, 0, 0, 1]] #Print the solution g.transitiveClosure(graph) #This code is contributed by Neelam Yadav

Output:

Following matrix is transitive closure of the given graph 1 1 1 1 0 1 1 1 0 0 1 1 0 0 0 1

Time Complexity: O(V^{3}) where V is number of vertices in the given graph.

See below post for a O(V^{2}) solution.

Transitive Closure of a Graph using DFS

**References:**

Introduction to Algorithms by Clifford Stein, Thomas H. Cormen, Charles E. Leiserson, Ronald L.

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