Heap sort

Heap sort is a comparison based sorting technique based on Binary Heap data structure. It is similar to selection sort where we first find the maximum element and place the maximum element at the end. We repeat the same process for remaining element.

A Binary Heap is a Complete Binary Tree where items are stored in a special order such that value in a parent node is greater(or smaller) than the values in its two children nodes. The former is called as max heap and the latter is called min heap. The heap can be represented by binary tree or array.

Since a Binary Heap is a Complete Binary Tree, it can be easily represented as array and array based representation is space efficient. If the parent node is stored at index I, the left child can be calculated by 2 * I + 1 and right child by 2 * I + 2 (assuming the indexing starts at 0).

Heap Sort Algorithm for sorting in increasing order: 1. Build a max heap from the input data. 2. At this point, the largest item is stored at the root of the heap. Replace it with the last item of the heap followed by reducing the size of heap by 1. Finally, heapify the root of tree. 3. Repeat above steps while size of heap is greater than 1.

How to build the heap? Heapify procedure can be applied to a node only if its children nodes are heapified. So the heapification must be performed in the bottom up order.

Lets understand with the help of an example:

Input data: 4, 10, 3, 5, 1
         4(0)
        /   \
     10(1)   3(2)
    /   \
 5(3)    1(4)

The numbers in bracket represent the indices in the array 
representation of data.

Applying heapify procedure to index 1:
         4(0)
        /   \
    10(1)    3(2)
    /   \
5(3)    1(4)

Applying heapify procedure to index 0:
        10(0)
        /  \
     5(1)  3(2)
    /   \
 4(3)    1(4)
The heapify procedure calls itself recursively to build heap
 in top down manner.

Algrithom:

heapify the array;

while the array isn’t empty {

remove and replace the root;

reheap the new root node; }

code:

def heapify(arr, n, i):
   largest = i # Initialize largest as root
   l = 2 * i + 1   # left = 2*i + 1
   r = 2 * i + 2   # right = 2*i + 2

   # See if left child of root exists and is
   # greater than root
   if l < n and arr[i] < arr[l]:
      largest = l

   # See if right child of root exists and is
   # greater than root
   if r < n and arr[largest] < arr[r]:
      largest = r

   # Change root, if needed
   if largest != i:
      arr[i],arr[largest] = arr[largest],arr[i] # swap

      # Heapify the root.
      heapify(arr, n, largest)

# The main function to sort an array of given size
def heapSort(arr):
   n = len(arr)

   # Build a maxheap.
   for i in range(n, -1, -1):
      heapify(arr, n, i)

   # One by one extract elements
   for i in range(n-1, 0, -1):
      arr[i], arr[0] = arr[0], arr[i] # swap
      heapify(arr, i, 0)