Define transitive closure. Write Warshall's algorithm to compute transitive closure. Mention itstime efficiency

Transitive Closure: In graph theory, the transitive closure of a graph is a graph which contains an edge (i, j) if there is a path from i to j in the original graph. In other words, if a node j is reachable from node i, then the edge (i, j) is in the transitive closure of the graph.

Warshall's Algorithm: Warshall's algorithm is a method to compute the transitive closure of a graph. The algorithm is named after Stephen Warshall, who first described it in 1962. Here is a simple implementation of Warshall's algorithm:

1. Start with an adjacency matrix A of the graph, where A[i][j] is 1 if there is an edge from i to j, and 0 otherwise.
2. For each pair of vertices i and j, check for every other vertex k if there is a path from i to k and a path from k to j. If such a vertex k exists, then there is a path from i to j, so set A[i][j] to 1.
3. Repeat step 2 for all pairs of vertices until no more updates are made to the matrix A.

Here is a pseudo-code representation of the algorithm:

function Warshall(A)
  let n = number of vertices in A
  for k from 1 to n
    for i from 1 to n
      for j from 1 to n
        A[i][j] = A[i][j] or (A[i][k] and A[k][j])
  return A

Time Efficiency: The time complexity of Warshall's algorithm is O(n^3), where n is the number of vertices in the graph. This is because the algorithm contains three nested loops, each of which can iterate up to n times.