Understanding Heaps and the Problem

A heap is a specialized tree-based data structure that satisfies the heap property. In a max-heap, for any given node C, if P is a parent node of C, then the value of P is greater than or equal to the value of C. The highest element is always at the root of the heap. A heap is a complete binary tree, meaning all levels except possibly the last are completely filled, and nodes at the last level are as far left as possible.

We are given an array A = <4, 1, 3, 2, 16, 9, 10, 14, 8, 7>. We need to perform two tasks:

1. Build a max-heap from this array.
2. Determine the depth of the resulting max-heap (with the root at level 0).
3. Identify the value of the right child of the root in the resulting max-heap.

Initial Array and Depth

The given array A has 10 elements. When an array is used to represent a complete binary tree, the elements are typically placed level by level from left to right. Using 0-based indexing for the array A = [4, 1, 3, 2, 16, 9, 10, 14, 8, 7]:

The tree structure formed by placing these elements in a complete binary tree form has nodes at levels 0, 1, 2, and 3. The maximum level number is 3.

The depth of a tree is the number of edges on the longest path from the root to a leaf. If the root is at level 0, this is equivalent to the maximum level number.

For a complete binary tree with \( n \) nodes, the depth (with root at level 0) is \( \lfloor \log_2 n \rfloor \). Here, \( n = 10 \).

\( \log_2 10 \approx 3.32 \)

Depth \( = \lfloor \log_2 10 \rfloor = \lfloor 3.32 \rfloor = 3 \).

So, the initial depth of the tree represented by array A is 3. Building a heap from this array involves rearranging elements while maintaining the complete binary tree structure, so the depth of the resulting heap will remain the same.

Building the Max-Heap (BUILD-MAX-HEAP)

To build a max-heap from an array, we use the BUILD-MAX-HEAP procedure. This procedure works by running the MAX-HEAPIFY procedure on all non-leaf nodes in the tree, starting from the last non-leaf node and working up to the root.

Using 0-based indexing for array A = [4, 1, 3, 2, 16, 9, 10, 14, 8, 7], the size of the heap is \( n=10 \). The indices of non-leaf nodes range from \( \lfloor n/2 \rfloor - 1 \) down to 0.

\( \lfloor 10/2 \rfloor - 1 = 5 - 1 = 4 \).

So we apply MAX-HEAPIFY to indices 4, 3, 2, 1, and 0.

Let's trace the MAX-HEAPIFY calls (values change as swaps occur):

• MAX-HEAPIFY(A, 4): A[4]=16. Children A[9]=7. 16 is max. No change. Array: [4, 1, 3, 2, 16, 9, 10, 14, 8, 7]
• MAX-HEAPIFY(A, 3): A[3]=2. Children A[7]=14, A[8]=8. Max(2, 14, 8) = 14. Swap A[3] <-> A[7]. Array: [4, 1, 3, 14, 16, 9, 10, 2, 8, 7]. Heapify on index 7 (leaf), stop.
• MAX-HEAPIFY(A, 2): A[2]=3. Children A[5]=9, A[6]=10. Max(3, 9, 10) = 10. Swap A[2] <-> A[6]. Array: [4, 1, 10, 14, 16, 9, 3, 2, 8, 7]. Heapify on index 6 (leaf), stop.
• MAX-HEAPIFY(A, 1): A[1]=1. Children A[3]=14, A[4]=16. Max(1, 14, 16) = 16. Swap A[1] <-> A[4]. Array: [4, 16, 10, 14, 1, 9, 3, 2, 8, 7]. Now call MAX-HEAPIFY(A, 4): A[4]=1. Children A[9]=7. Max(1, 7) = 7. Swap A[4] <-> A[9]. Array: [4, 16, 10, 14, 7, 9, 3, 2, 8, 1]. Heapify on index 9 (leaf), stop.
• MAX-HEAPIFY(A, 0): A[0]=4. Children A[1]=16, A[2]=10. Max(4, 16, 10) = 16. Swap A[0] <-> A[1]. Array: [16, 4, 10, 14, 7, 9, 3, 2, 8, 1]. Now call MAX-HEAPIFY(A, 1): A[1]=4. Children A[3]=14, A[4]=7. Max(4, 14, 7) = 14. Swap A[1] <-> A[3]. Array: [16, 14, 10, 4, 7, 9, 3, 2, 8, 1]. Now call MAX-HEAPIFY(A, 3): A[3]=4. Children A[7]=2, A[8]=8. Max(4, 2, 8) = 8. Swap A[3] <-> A[8]. Array: [16, 14, 10, 8, 7, 9, 3, 2, 4, 1]. Heapify on index 8 (leaf), stop.

Final Max-Heap Structure

After performing BUILD-MAX-HEAP, the array becomes:

[16, 14, 10, 8, 7, 9, 3, 2, 4, 1]

This array represents the max-heap. Let's visualize its tree structure (using 0-based indexing):

Level 0: 16 (Index 0 - Root)

Level 1: 14 (Index 1 - Left Child of Root), 10 (Index 2 - Right Child of Root)

Level 2: 8 (Index 3), 7 (Index 4), 9 (Index 5), 3 (Index 6)

Level 3: 2 (Index 7), 4 (Index 8), 1 (Index 9)

The maximum level number is still 3. Thus, the depth of the max-heap is 3.

Right Child of the Root

In the final max-heap array [16, 14, 10, 8, 7, 9, 3, 2, 4, 1], the root is at index 0, which is the element 16.

In a 0-based indexed array representation of a complete binary tree:

• The left child of a node at index \( i \) is at index \( 2i + 1 \).
• The right child of a node at index \( i \) is at index \( 2i + 2 \).

For the root node at index \( i=0 \):

• Left child is at index \( 2*0 + 1 = 1 \). The value is A[1] = 14.
• Right child is at index \( 2*0 + 2 = 2 \). The value is A[2] = 10.

Therefore, the right child of the root (16) in the max-heap is 10.

Conclusion

After building the max-heap from the array A, the depth of the heap is 3, and the right child of the root is 10.

The required values are 3 and 10.

Revision Table: Max-Heap Concepts

Additional Information: Heap Properties and Applications

Heaps are fundamental data structures used in various algorithms and applications.

• Heap Sort: Heaps are the basis for the heap sort algorithm, an efficient sorting algorithm with a time complexity of \( O(n \log n) \).
• Priority Queues: Heaps are commonly used to implement priority queues. A max-heap is used for a max-priority queue (extracting the maximum element quickly), and a min-heap is used for a min-priority queue (extracting the minimum element quickly).
• Heap Properties: The two main properties are the shape property (it's a complete binary tree) and the heap property (max-heap or min-heap property). These properties allow for efficient operations like insertion, deletion, and finding min/max.
• Array Representation: The complete binary tree nature of a heap allows it to be efficiently stored in an array. The parent/child relationships can be calculated directly from the array indices.