To efficiently implement the heap sort algorithm, you can follow these steps:

  1. Build a max-heap: Start by converting the array into a max-heap. This can be done by calling the heapify() procedure on the array. The heapify procedure ensures that each node in the heap satisfies the property that its value is greater than or equal to the values of its children.

  2. Sort the array: After building the max-heap, the largest element will be at the root of the heap. Swap this element with the last element in the heap and reduce the heap size by one. Then, call the heapify procedure on the root element to ensure that the new root satisfies the heap property. Repeat this process until the heap is empty. The elements that are swapped out in each iteration will be in sorted order.

  Below is the implementation in Python:

  python

  def heapify(arr, n, i):

   largest = i

   left = 2 * i + 1

   right = 2 * i + 2

   if left < n and arr[i] < arr[left]:

   largest = left

   if right < n and arr[largest] < arr[right]:

   largest = right

   if largest != i:

   arr[i], arr[largest] = arr[largest], arr[i]

   heapify(arr, n, largest)

  def heapSort(arr):

   n = len(arr)

   # Build max-heap

   for i in range(n // 2 - 1, -1, -1):

   heapify(arr, n, i)

   # Sort the array

   for i in range(n - 1, 0, -1):

   arr[i], arr[0] = arr[0], arr[i]

   heapify(arr, i, 0)

  The time complexity of heap sort is O(n log n), where n is the number of elements in the array. This makes it an efficient sorting algorithm for large data sets.