> Prove optimality of quicksort algorithm

Quicksort is a sorting algorithm that works by partitioning an array around a pivot element, then recursively sorting the subarrays on either side of the pivot. It is a popular algorithm due to its simplicity, efficiency, and relatively small memory footprint.

To prove the optimality of quicksort, we need to show that it is both correct and efficient.

Correctness:
We can prove the correctness of quicksort using mathematical induction. Suppose that quicksort correctly sorts an array of size n. We can then prove that it will correctly sort an array of size n+1.

The partitioning step of quicksort involves choosing a pivot element and partitioning the array into two subarrays, one with elements less than the pivot and one with elements greater than the pivot. Because the pivot element is in its final sorted position, all elements in the left subarray are less than or equal to the pivot, and all elements in the right subarray are greater than or equal to the pivot.

We can use induction to show that quicksort correctly sorts an array of size n+1. Suppose we have an array of size n+1. We choose a pivot element and partition the array into two subarrays. By the induction hypothesis, quicksort correctly sorts the left subarray and the right subarray. Because the pivot is in its final sorted position, the left subarray contains only elements less than or equal to the pivot, and the right subarray contains only elements greater than or equal to the pivot. Therefore, the entire array is sorted correctly.

Efficiency:
To prove the efficiency of quicksort, we can analyze its time complexity. In the best case, the pivot element divides the array into two subarrays of equal size, and each subarray is recursively sorted. The time complexity in this case is O(n log n).

In the worst case, the pivot element is either the smallest or largest element in the array, and the partitioning step produces one subarray of size n-1 and one subarray of size 0. The time complexity in this case is O(n^2).

However, the average case time complexity of quicksort is O(n log n). This is because the probability of choosing a bad pivot element decreases as the size of the array increases. The expected number of comparisons required to sort an array of size n is proportional to n log n.

In conclusion, quicksort is a correct and efficient sorting algorithm, with an average time complexity of O(n log n). Therefore, it is considered an optimal algorithm for sorting in most practical situations.