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.

HeapSort

What is Binary Heap? Let us first define a Complete Binary Tree. A complete binary tree is a binary tree in which every level, except possibly the last, is completely filled, and all nodes are as far left as possible (Source Wikipedia)

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.

Why array based representation for Binary Heap? 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.

# Python program for implementation of heap Sort
 
# To heapify subtree rooted at index i.
# n is size of heap
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)
 
# Driver code to test above
arr = [ 12, 11, 13, 5, 6, 7]
heapSort(arr)
n = len(arr)
print ("Sorted array is")
for i in range(n):
    print ("%d" %arr[i]),
# This code is contributed by Mohit Kumra

Sorted array is
5 6 7 11 12 13

LintCode problems

class Solution:
    """
    @param A: an integer array
    @return: nothing
    """
    def sortIntegers2(self, A):
        # write your code here
        n = len(A)

        # create and build heap
        for i in xrange(n, -1, -1):
            self.heapify(A, n, i)

        # extract elements at the root
        for i in xrange(n - 1, 0, -1):

            # Swap
            A[i], A[0] = A[0], A[i]

            # Re-Heapify
            self.heapify(A, i, 0)


    def heapify(self, A, n, i):
        # Ascending order -> Max Heap
        largest = i
        left    = 2 * i + 1
        right   = 2 * i + 2


        if left < n and A[largest] < A[left]:
            largest = left

        if right < n and A[largest] < A[right]:
            largest = right

        # Swap root with left or right if needed
        if largest != i:

            # Swap
            A[i], A[largest] = A[largest], A[i]

            # Re-Heapify
            self.heapify(A, n, largest)

Notes: Heap sort is an in-place algorithm. Its typical implementation is not stable, but can be made stable (See this)

Time Complexity:

• Time complexity of heapify is O(Logn).

• Time complexity of createAndBuildHeap() is O(n)

• overall time complexity of Heap Sort is O(nLogn).

Applications of HeapSort 1. Sort a nearly sorted (or K sorted) array 2. k largest(or smallest) elements in an array

Heap sort algorithm has limited uses because Quicksort and Mergesort are better in practice. Nevertheless, the Heap data structure itself is enormously used. See Applications of Heap Data Structure

References

HeapSort - GeeksforGeeksGeeksforGeeks