Consider this algorithm: // PRE: A an array of numbers, initially low = 0, high = size of A - 1 // POST: Returns the position of the pivot, and A is partitioned with elements to the left of the pivot being <= pivot, // // and elements to the right of the pivot being >=pivot int partition (A, low, high): 1. 2. pivot =A[high] i = low-1 for j from low to (high-1) do: if pivot >A[j] then do: i = i + 1 swap A[j] with A[i] swap A[i+1] with the pivot element at A [high] return (i+1) Prove correctenss of partition. Use partition to write a version of Quicksort that sorts a numerical array A in place.

Quicksort is a powerful divide-and-conquer sorting algorithm that can be described in just four lines ofpseudocode. The key to Quicksort is the PARTITION(A, p, r) procedure, which inputs elementsptorof array A,and chooses the final element x = A[r] as the pivot element. The output is an array where all elementsto the left ofxare less thanx, and all elements to the right of x are greater than x. In this question, we will use the Lomuto Partition Method from class and assume that the pivot isalwaysthe last (right-most) element of the input array. Question: Let A be an array withn= 2k−1 elements, where k is some positive integer. Determine a formula (in terms of n) for the minimum possible number of total comparisons required by Quicksort, as well as a formula for the maximum possible number of total comparisons required by Quicksort. Use your formulas to show that the running time of Quicksort is O(nlogn) in the best case and O(n2) in the worst case.

Heapsort has heapified an array to: 96 71 57 36 25 and is about to start the second for loop. What is the array after each loop iteration? i= 4: Ex: 86, 75, 30 i = 3: i = 2: i = 1:

Suppose an array has n elements. This _____ sorts to sort and array works as follows: Find the smallest element and place it in the first position. Then find the smallest of the remaining n-1 elements and place it in the second position. Repeat on n-2 elements, n-3 elements, ..., until the array is sorted.