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; ` `} ` |

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## 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 ` |

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## 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 ` |

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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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