You have an array of n elements. Suppose you implementquicksortby always choosing the central element of the array as the pivot. Then the tightest upper bound for the worst case performance isa)O(n2)b)O(nLogn)c)Theta(nLogn)d)O(n3)Correct answer is option 'A'. Can you explain this answer?

Question

Answer

Explanation:

Quicksort is an efficient sorting algorithm that follows the divide-and-conquer strategy. It works by selecting a pivot element from the array 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 recursively sorted.

Choosing the central element as the pivot:
In the given scenario, we are always choosing the central element of the array as the pivot. This means that the array is divided into two sub-arrays of approximately equal size.

Worst case scenario:
The worst-case performance of quicksort occurs when the chosen pivot element is always the smallest or largest element in the array. In this case, one sub-array will have n-1 elements and the other sub-array will have 0 elements. This leads to highly unbalanced partitions and results in a time complexity of O(n^2).

Explanation:
When the pivot is always chosen as the central element, there is a high probability that the pivot will not be the smallest or largest element in the array. This reduces the chance of ending up with highly unbalanced partitions. However, there is still a possibility that the pivot can be the smallest or largest element in some cases, leading to the worst-case time complexity of O(n^2).

Example:
Let's consider an array of 8 elements: [1, 2, 3, 4, 5, 6, 7, 8]. If we choose the central element (5) as the pivot, the resulting partitions would be [1, 2, 3, 4] and [6, 7, 8]. Now, we need to recursively sort these two partitions. If the pivot is chosen as the central element again, we end up with [1, 2, 3, 4] and [6, 7, 8] as the partitions. This process continues until we reach sub-arrays of size 1, which are already sorted.

Conclusion:
In the worst case scenario, the quicksort algorithm with the central element as the pivot can result in highly unbalanced partitions, leading to a time complexity of O(n^2). Therefore, the tightest upper bound for the worst case performance is O(n^2).

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