The time complexity of computing the transitive closure of a binary relation on a set of n elements is known to be:a)O(n)b)O(nLogn)c)O(n(3/2))d)O(n3)Correct answer is option 'D'. Can you explain this answer?

Question

Answer

Explanation:

The transitive closure of a binary relation on a set of n elements is a matrix that represents the reachability information between pairs of elements in the set.

To compute the transitive closure of a binary relation, we can use the Floyd-Warshall algorithm. This algorithm iteratively considers all pairs of elements and checks if there exists a path between them through any intermediate element. If such a path exists, the algorithm updates the matrix to reflect this reachability information.

The time complexity of the Floyd-Warshall algorithm is O(n^3), where n is the number of elements in the set. Let's break down the complexity of each step of the algorithm:

1. Initialization: Initializing the matrix takes O(n^2) time as we need to set the initial reachability information for all pairs of elements.

2. Iterations: The algorithm iterates n times, considering each element as an intermediate element. For each iteration, it checks all pairs of elements. Since there are n^2 pairs, this step takes O(n^2) time.

3. Update: For each pair of elements, the algorithm checks if there exists a path through the intermediate element. This check takes constant time, O(1).

Overall, the time complexity of the algorithm is O(n^2) for initialization and O(n^2) for each iteration, resulting in a total time complexity of O(n^2 + n^2) = O(n^2).

However, in the worst case, the algorithm may need to iterate n times, resulting in a time complexity of O(n^3).

Therefore, the correct answer is option D: O(n^3).

Similar Computer Science Engineering (CSE) Doubts

The time complexity of computing the transitive closure of a binary relation on a set of n elements is known to be

Consider an unrolled linked list with n elements. This list stores multiple elements in each node. What is the worst case time complexity to find the kthelement if the number of nodes and the number of elements in each node are equal?

There are n unsorted arrays: A1 , A2 ,....., An . Assume that n is odd. Each of A1 , A2 ,....., An contains n distinct elements. There are no common elements between any two arrays. The worst-case time complexity of computing the median of the medians of A1 , A2 ,....., An is

Consider an array which stores a maximum of 100 elements. For case 1, the user inserts 1 element into an array. For case 2, the user inserts 100 elements into an array. What is the time complexity for both the cases?

If a queue is implemented using two stacks. In enqueue operation, all the elements are pushed from the first stack to the second stack. In dequeue operation pop an element from 1st stack.Which of the following is/are true?

Top Courses for Computer Science Engineering (CSE)View all

GATE Computer Science Engineering(CSE) 2025 Mock Test Series

Question Bank for GATE Computer Science Engineering

General Aptitude for GATE

Computer Networks

Programming and Data Structures

Top Courses for Computer Science Engineering (CSE)

Top Courses for Computer Science Engineering (CSE)

Suggested Free Tests

Related Content

Mock Test Series

Subject-wise Courses

Subject-wise Courses

Other Engg Branches

Company