Consider The Array A=30,10,15,9,7,50,8,22,5,3. 1) (5 Points) Write A After Calling The Function BUILD-MAX-HEAP(A)

Consider The Array A=30,10,15,9,7,50,8,22,5,3. 1) (5 Points) Write A After Calling The Function BUILD-MAX-HEAP(A)

Introduction to BUILD-MAX-HEAP and Its Significance

Building a max-heap from an unsorted array is a fundamental operation in the realm of data structures and algorithms, especially in the context of heap sort. The process transforms an arbitrary array into a binary heap where each parent node is greater than or equal to its child nodes. This property ensures efficient retrieval of the maximum element, which is pivotal in priority queues, sorting algorithms, and several other computational applications. Understanding how to convert an array into a max-heap using the BUILD-MAX-HEAP function provides insight into the internal mechanics of heaps and their practical utility.

Initial Array and Its Structure

Let's examine the initial array:

• A = [30, 10, 15, 9, 7, 50, 8, 22, 5, 3]

Assuming the array is 1-based indexed (common in algorithm pseudocode), the elements correspond to nodes in a binary tree representation:

Index:  1   2   3   4   5   6   7   8   9   10

Value: 30  10  15   9   7  50   8  22   5    3

This array can be visualized as a binary tree:

• The root is at index 1 (value 30).


• Left child of node at index i: 2i


• Right child of node at index i: 2i + 1

The initial tree structure:

```
30
/ \
10 15
/ \ / \
9 7 50 8
/ \
22 5
/
3
```

In practice, the array is stored in contiguous memory, but conceptual visualization aids understanding the heapification process.

Understanding BUILD-MAX-HEAP Algorithm

The BUILD-MAX-HEAP function works by applying the MAX-HEAPIFY procedure starting from the lowest non-leaf nodes up to the root. The process ensures that each subtree satisfies the max-heap property:

• For each node, compare it with its children.


• If a child's value is greater than the parent's, swap them.


• Recursively ensure subtrees are max-heaps after swaps.

The algorithm proceeds in a bottom-up manner, starting from the last parent node:

• The last parent node is at index ⌊n/2⌋, where n is the number of elements.


• For our array with 10 elements, last parent index is at ⌊10/2⌋=5.

The process involves:

1. Applying MAX-HEAPIFY at index 5


2. Moving upward to index 4


3. Then index 3


4. Then index 2


5. Finally, index 1

This ensures all subtrees become max-heaps, culminating in the entire array satisfying the max-heap property.

Step-by-Step Heapification Process

Let's detail the process at each relevant node:

Heapify at Index 5 (Value 7)

• Children: left at 10 (value 3), right at 11 (none, since array length is 10)
• Compare 7 with 3
• 7 > 3, so no swap needed; subtree rooted at 5 is already a max-heap

Heapify at Index 4 (Value 9)

• Children: 8 (index 8, value 22), 9 (index 9, value 5)
• Compare 9 with children: 22 and 5
• 22 > 9, swap 9 and 22
• Array after swap:

`[30, 10, 15, 22, 7, 50, 8, 9, 5, 3]`

• Now, at index 8 (value 9), check children: index 16 and 17 — out of bounds, so stop

Heapify at Index 3 (Value 15)

• Children: 6 (value 50), 7 (value 8)
• Compare 15 with children: 50 and 8
• 50 > 15, swap 15 and 50
• Array after swap:

`[30, 10, 50, 22, 7, 15, 8, 9, 5, 3]`

• Now, at index 6 (value 15), check children: 12 (value 22), 13 (none)
• 22 > 15, swap 15 and 22
• Array:

`[30, 10, 50, 22, 7, 22, 8, 9, 5, 3]`

• Wait, but note that after swapping, the subtree rooted at index 6 (value 15) now has 22, which is greater than 15. Since 22 is now at index 6, check its children:
• Children: 12 (value 22), 13 (none).
• But after swap, index 6 has 22, and its children are:
• left: 12 (value 22)
• right: 13 (none)
• Since 22 at index 6 is greater than its children, no further swaps needed.

Heapify at Index 2 (Value 10)

• Children: 4 (value 22), 5 (value 7)
• Compare 10 with children: 22 and 7
• 22 > 10, swap 10 and 22
• Array:

`[30, 22, 50, 10, 7, 15, 8, 9, 5, 3]`

• Now, at index 4 (value 10), check children: 8 (value 9), 9 (value 5)
• 10 > 9 and 10 > 5, no further swaps.

Heapify at Index 1 (Value 30)

• Children: 2 (value 22), 3 (value 50)
• Compare 30 with children: 22 and 50
• 50 > 30, swap 30 and 50
• Array:

`[50, 22, 30, 10, 7, 15, 8, 9, 5, 3]`

• Now, at index 3 (value 30), check children: 6 (value 15), 7 (value 8)
• 30 > 15 and 30 > 8, no swaps needed
• At index 2 (value 22), check children: 4 (value 10), 5 (value 7)
• 22 > 10 and 22 > 7, no swaps needed
• At index 1 (value 50), check children: 2 (value 22), 3 (value 30)
• 50 > 22 and 50 > 30, no swaps needed

The heapification process completes here.

Final Max-Heap Array

After applying the BUILD-MAX-HEAP procedure, the array transforms into:

• [50, 22, 30, 10, 7, 15, 8, 9, 5, 3]

This array satisfies the max-heap property:

• For every parent node, the value is greater than or equal to its children.

Visualizing the heap:

```
50
/ \
22 30
/ \ / \
10 7 15 8
/ \
9 5
/
3
```

Implications and Applications of the Final Max-Heap

Transforming the array into a max-heap has significant computational benefits:

• Efficient Extraction: The maximum element (root) can be retrieved in O(1) time.
• Heap Sort: Repeatedly extracting the maximum and rebuilding the heap sorts the array in O(n log n) time.
• Priority Queues: Max-heaps underpin priority queue implementations, allowing for quick priority updates and retrievals.

Understanding the internal steps of BUILD-MAX-HEAP helps in optimizing algorithms and grasping the underlying data structure principles.

Conclusion

The process of converting an arbitrary array into a max-heap using the BUILD-MAX-HEAP procedure involves systematic heapification starting from the last non-leaf node up to the root. For the array A=30,10,15,9,7,50,8,22