Randomized quicksort is an extension of quicksort where the pivot is chosen randomly. What is the worst case complexity of sorting n numbers using randomized quicksort?a)O(n)b)(nlogn)c)0(n2)d)0(n!)Correct answer is option 'C'. Can you explain this answer?

Question

Answer

The worst-case complexity of sorting n numbers using randomized quicksort is O(n^2).

Explanation:
Quick sort is a divide-and-conquer algorithm that works by selecting a pivot element and partitioning the other elements into two sub-arrays, according to whether they are less than or greater than the pivot. The sub-arrays are then sorted recursively.

In the worst case scenario, the pivot chosen in each partition step is the smallest or largest element in the array. This can result in highly unbalanced partitions, where one partition contains n-1 elements and the other partition is empty. As a result, the quicksort algorithm only reduces the problem size by 1 in each recursive call.

To understand the worst-case complexity, let's assume that the pivot is always chosen as the smallest or largest element. In this case, the algorithm will make n recursive calls, each reducing the problem size by 1. Therefore, the number of comparisons made by the algorithm can be represented as the sum of the first n-1 positive integers, which is given by the formula (n-1)(n)/2.

Since the comparison-based sorting algorithms have a lower bound of Ω(nlogn) for the worst-case complexity, we can conclude that the worst-case complexity of randomized quicksort is O(n^2), which is worse than the lower bound.

In practice, randomized quicksort performs much better than the worst-case complexity suggests. The random selection of the pivot significantly reduces the probability of encountering the worst-case scenario, resulting in an average-case time complexity of O(nlogn).

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