Heap Sort Algorithm

Question

Explore the efficiency of Heap Sort Algorithm, a popular sorting technique, as we delve into its implementation and intricacies. Learn how to optimize your code with this powerful sorting method, ensuring faster and more organized data processing. Master the fundamentals of Heap Sort and enhance your algorithmic skills today.

Answer 1

Heap Sort Algorithm

What is a Heap?

A heap is a specialized tree-based data structure that satisfies the heap property. The heap property depends on whether it's a max-heap or a min-heap:

• Max-Heap Property: For every node i other than the root, the value of i is less than or equal to the value of its parent.

• Min-Heap Property: For every node i other than the root, the value of i is greater than or equal to the value of its parent.

In the context of heap sort, we typically use a max-heap.

What is Heap Sort?

Heap Sort is a comparison-based sorting algorithm that uses a binary heap data structure to build a partially ordered tree and efficiently find the maximum (for max-heap) element repeatedly. The algorithm proceeds by building the heap, extracting the maximum element (root), and then restoring the heap property.

Working of Heap Sort Algorithm

1. Build Max-Heap:

   • Convert the input array into a max-heap. This is done by starting from the last non-leaf node and repeatedly heapifying downward.

2. Extract Maximum:

   • Swap the root (maximum element) with the last element in the heap and reduce the size of the heap.

3. Heapify:

   • Heapify the root to maintain the max-heap property.

4. Repeat:

   • Repeat steps 2 and 3 until the heap size is 1.

Let's delve into the mechanics of the Heapsort Algorithm.

Heapsort entails two key phases in its element sorting process:

1. First and foremost, a heap is fashioned by adjusting the array elements.
2. Subsequently, the root element of the heap is systematically removed, shifting it to the array's end. The remaining elements maintain the heap structure.

To illustrate the intricacies of heapsort, consider an unsorted array. Through a detailed example, we can gain a clearer understanding of the algorithm's operations and how it efficiently transforms an unordered array into a sorted one.

First, we have to construct a heap from the given array and convert it into max heap.

After converting the given heap into max heap, the array elements are -

Next, we have to delete the root element (89) from the max heap. To delete this node, we have to swap it with the last node, i.e. (11). After deleting the root element, we again have to heapify it to convert it into max heap.

After swapping the array element 89 with 11, and converting the heap into max-heap, the elements of array are -

In the next step, again, we have to delete the root element (81) from the max heap. To delete this node, we have to swap it with the last node, i.e. (54). After deleting the root element, we again have to heapify it to convert it into max heap.

After swapping the array element 81 with 54 and converting the heap into max-heap, the elements of array are -

In the next step, we have to delete the root element (76) from the max heap again. To delete this node, we have to swap it with the last node, i.e. (9). After deleting the root element, we again have to heapify it to convert it into max heap.

After swapping the array element 76 with 9 and converting the heap into max-heap, the elements of array are -

In the next step, again we have to delete the root element (54) from the max heap. To delete this node, we have to swap it with the last node, i.e. (14). After deleting the root element, we again have to heapify it to convert it into max heap.

After swapping the array element 54 with 14 and converting the heap into max-heap, the elements of array are -

In the next step, again we have to delete the root element (22) from the max heap. To delete this node, we have to swap it with the last node, i.e. (11). After deleting the root element, we again have to heapify it to convert it into max heap.

After swapping the array element 22 with 11 and converting the heap into max-heap, the elements of array are -

In the next step, again we have to delete the root element (14) from the max heap. To delete this node, we have to swap it with the last node, i.e. (9). After deleting the root element, we again have to heapify it to convert it into max heap.

After swapping the array element 14 with 9 and converting the heap into max-heap, the elements of array are -

In the next step, again we have to delete the root element (11) from the max heap. To delete this node, we have to swap it with the last node, i.e. (9). After deleting the root element, we again have to heapify it to convert it into max heap.

After swapping the array element 11 with 9, the elements of array are -

Now, heap has only one element left. After deleting it, heap will be empty.

After completion of sorting, the array elements are -

Now, the array is completely sorted.

Heap Sort Complexity

• Time Complexity:

  • Build Max-Heap: O(n)
  • Extract Maximum (n times): O(n log n)
  • Overall Time Complexity: O(n log n)

• Space Complexity:

  • O(1) - Heap sort is an in-place sorting algorithm.

Implementation of Heap Sort

Below is a simple implementation of the Heap Sort algorithm in Python:

def heapify(arr, n, i):
    largest = i  # Initialize the largest as the root
    left_child = 2 * i + 1
    right_child = 2 * i + 2

    # Check if left child exists and is greater than the root
    if left_child < n and arr[left_child] > arr[largest]:
        largest = left_child

    # Check if right child exists and is greater than the root
    if right_child < n and arr[right_child] > arr[largest]:
        largest = right_child

    # Swap the root if needed
    if largest != i:
        arr[i], arr[largest] = arr[largest], arr[i]
        # Heapify the affected sub-tree
        heapify(arr, n, largest)

def heap_sort(arr):
    n = len(arr)

    # Build a max heap
    for i in range(n // 2 - 1, -1, -1):
        heapify(arr, n, i)

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

# Example Usage
arr = [12, 11, 13, 5, 6, 7]
heap_sort(arr)
print("Sorted array:", arr)
 

In this example, heapify is a function to heapify a subtree rooted at index i, and heap_sort uses this function to perform the sorting.

This implementation assumes that the input is a list of comparable elements. The heapify function is a crucial part of the algorithm as it ensures that the max-heap property is maintained during the extraction of the maximum element and heapification processes.

Answer 2

FAQs on Heap Sort Algorithm

Q: What is Heap Sort?

A: Heap Sort is a comparison-based sorting algorithm that uses a binary heap data structure to build a max-heap or min-heap. The heap is a specialized tree-based data structure that satisfies the heap property. In the case of max-heaps, for every node 'i' other than the root, the value of 'i' is less than or equal to the value of its parent. In the case of min-heaps, it is the opposite. The basic idea of heap sort is to build a heap, then repeatedly extract the maximum (for max-heap) or minimum (for min-heap) element from the heap and swap it with the last element in the array.

Q: What is the time complexity of Heap Sort?

A: Heap Sort has a time complexity of O(n log n) for the worst, average, and best cases, where 'n' is the number of elements in the array.

Q: What is the space complexity of Heap Sort?

A: Heap Sort has a space complexity of O(1) as it sorts the array in-place and does not require additional space proportional to the input size.

Q: How does Heap Sort work?

A:

1. Build Heap: Build a max-heap (for ascending order) or min-heap (for descending order) from the array.
2. Heapify: Repeatedly remove the root of the heap (which is the maximum or minimum element) and swap it with the last element in the array. After each removal, heapify the remaining heap.

Example Code in Python:

def heapify(arr, n, i):
    largest = i  # Initialize largest as root
    left_child = 2 * i + 1
    right_child = 2 * i + 2

    # Check if left child exists and is greater than root
    if left_child < n and arr[left_child] > arr[largest]:
        largest = left_child

    # Check if right child exists and is greater than root
    if right_child < n and arr[right_child] > arr[largest]:
        largest = right_child

    # Swap the root if needed
    if largest != i:
        arr[i], arr[largest] = arr[largest], arr[i]
        # Recursively heapify the affected sub-tree
        heapify(arr, n, largest)

def heap_sort(arr):
    n = len(arr)

    # Build a max heap
    for i in range(n // 2 - 1, -1, -1):
        heapify(arr, n, i)

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

# Example usage
arr = [12, 11, 13, 5, 6, 7]
heap_sort(arr)
print("Sorted array:", arr)
 

This example demonstrates a max-heap and ascending order. For a min-heap and descending order, adjustments can be made accordingly.

Important Interview Questions and Answers on Heap Sort Algorithm

Q: What is Heap Sort?

Heap Sort is a comparison-based sorting algorithm that uses a binary heap data structure to build a max-heap or min-heap. It then repeatedly extracts the maximum or minimum element from the heap, and the remaining elements are re-heapified.

Q: How does Heap Sort work?

Heap Sort works by first building a heap from the input array. It then repeatedly extracts the maximum (for max-heap) element from the heap, swapping it with the last element of the heap, and reducing the size of the heap. This process is repeated until the entire array is sorted.

Q: What is a binary heap?

A binary heap is a complete binary tree where the value of each node is greater than or equal to (for max-heap) or less than or equal to (for min-heap) the values of its children.

Q: Explain the steps involved in Heap Sort.

1. Build a max-heap from the input array.
2. Swap the root (maximum element) with the last element of the heap.
3. Reduce the size of the heap by one.
4. Heapify the root of the heap.
5. Repeat steps 2-4 until the heap is empty.

Q: What is the time complexity of Heap Sort?

The time complexity of Heap Sort is O(n log n) for the worst, average, and best cases.

Q: Can Heap Sort be implemented in-place?

Yes, Heap Sort can be implemented in-place, meaning that it doesn't require additional memory beyond the input array.

Q: Provide a Python implementation of Heap Sort.

Here is the code.

def heapify(arr, n, i):
    largest = i
    left_child = 2 * i + 1
    right_child = 2 * i + 2

    if left_child < n and arr[left_child] > arr[largest]:
        largest = left_child

    if right_child < n and arr[right_child] > arr[largest]:
        largest = right_child

    if largest != i:
        arr[i], arr[largest] = arr[largest], arr[i]
        heapify(arr, n, largest)

def heap_sort(arr):
    n = len(arr)

    # Build max-heap
    for i in range(n // 2 - 1, -1, -1):
        heapify(arr, n, i)

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

# Example usage:
arr = [12, 11, 13, 5, 6, 7]
heap_sort(arr)
print("Sorted array:", arr)