# Transitive closure of a graph

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 the 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 the 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:

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

Below is the implementation of the above approach:

## 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"` `);` ` ` `printf` `(` `"closure of the given graph\n"` `);` ` ` `for` `(` `int` `i = 0; i < V; i++)` ` ` `{` ` ` `for` `(` `int` `j = 0; j < V; j++)` ` ` `{` ` ` `/* because "i==j means same vertex"` ` ` `and we can reach same vertex` ` ` `from same vertex. So, we print 1....` ` ` `and we have not considered this in` ` ` `Floyd Warshall Algo. so we need to` ` ` `make this true by ourself` ` ` `while printing transitive closure.*/` ` ` `if` `(i == j)` ` ` `printf` `(` `"1 "` `);` ` ` `else` ` ` `printf` `(` `"%d "` `, reach[i][j]);` ` ` `}` ` ` `printf` `(` `"\n"` `);` ` ` `}` `}` `// Driver Code` `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++) {` ` ` `if` `( i == j)` ` ` `System.out.print(` `"1 "` `);` ` ` `else` ` ` `System.out.print(reach[i][j]+` `" "` `);` ` ` `}` ` ` `System.out.println();` ` ` `}` ` ` `}` ` ` `// Driver Code` ` ` `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):` ` ` `if` `(i ` `=` `=` `j):` ` ` `print` `"%7d\t"` `%` `(` `1` `),` ` ` `else` `:` ` ` `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` |

## C#

`// C# Program for transitive closure` `// using Floyd Warshall Algorithm` `using` `System;` `class` `GFG` `{` ` ` `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)` ` ` `{` ` ` `Console.WriteLine(` `"Following matrix is transitive"` `+` ` ` `" closure of the given graph"` `);` ` ` `for` `(` `int` `i = 0; i < V; i++)` ` ` `{` ` ` `for` `(` `int` `j = 0; j < V; j++){` ` ` `if` `(i == j)` ` ` `Console.Write(` `"1 "` `);` ` ` `else` ` ` `Console.Write(reach[i, j] + ` `" "` `);` ` ` `}` ` ` `Console.WriteLine();` ` ` `}` ` ` `}` ` ` `// Driver Code` ` ` `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` ` ` `GFG g = ` `new` `GFG();` ` ` `g.transitiveClosure(graph);` ` ` `}` `}` `// This code is contributed by 29AjayKumar` |

## Javascript

`<script>` ` ` `// JavaScript Program for transitive closure` ` ` `// using Floyd Warshall Algorithm` ` ` `var` `V = 4; ` `// Number of vertices in a graph` ` ` `// Prints transitive closure of graph[,]` ` ` `// using Floyd Warshall algorithm` ` ` `function` `transitiveClosure(graph) {` ` ` `/* reach[,] will be the output matrix that` ` ` `will finally have the shortest distances` ` ` `between every pair of vertices */` ` ` `var` `reach = Array.from(Array(V), () => ` `new` `Array(V));` ` ` `var` `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 */` ` ` `function` `printSolution(reach) {` ` ` `document.write(` ` ` `"Following matrix is transitive"` `+ ` `" closure of the given graph <br>"` ` ` `);` ` ` `for` `(` `var` `i = 0; i < V; i++) {` ` ` `for` `(` `var` `j = 0; j < V; j++) {` ` ` `if` `(i == j) document.write(` `"1 "` `);` ` ` `else` `document.write(reach[i][j] + ` `" "` `);` ` ` `}` ` ` `document.write(` `"<br>"` `);` ` ` `}` ` ` `}` ` ` `// Driver Code` ` ` `/* 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 */` ` ` `var` `graph = [` ` ` `[1, 1, 0, 1],` ` ` `[0, 1, 1, 0],` ` ` `[0, 0, 1, 1],` ` ` `[0, 0, 0, 1],` ` ` `];` ` ` `// Print the solution` ` ` `transitiveClosure(graph);` ` ` ` ` `// This code is contributed by rdtank.` ` ` `</script>` |

**Output**

Following matrix is transitiveclosure 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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