Final answer:
To convert an array into a max-heap, arrange the elements such that each parent node is greater than its children. The number of interchanges depends on comparing and swapping elements to maintain the max-heap property. The exact number of required interchanges varies, starting with the largest number being swapped to the root.
Step-by-step explanation:
The student's question involves determining the number of interchanges required to convert a given array into a max-heap. In a max-heap, for every node i other than the root, the value of i is less than or equal to the value of its parent. To convert the given array into a max-heap, one needs to arrange the numbers such that each parent node is greater than its children.
The given array is (70, 23, 60, 19, 13, 16, 1, 4, 8, 12, 7, 10, 85). To convert it into a max-heap, we have to make sure that the largest number is at the root of the heap. The largest number in the array is 85, which is currently at the last position of the array. The array representation of the max-heap for the first three levels is as follows (assuming the index starts at 1):
- Index 1: Max value (root of the heap)
- Index 2 and 3: Children of the root
- Index 4 to 7: Granchildren of the root
To have 85 at the root, we need to swap 85 with the current root, which is 70. This is one interchange. After this interchange, we need to confirm that the max-heap property holds for the rest of the tree by comparing the new root with its children and making any additional necessary swaps to maintain the max-heap characteristic. The number of interchanges can vary, and it's calculated by comparing and swapping to maintain the heap property.
Steps to Convert to Max-Heap
- Swap 85 with 70 in the array to move 85 to the root position.
- Compare 85 with its children (23 and 60) - No swap needed since 85 is larger.
- Continue down the heap, swapping where necessary to maintain the max-heap property.
After following these steps, you'll have the array ordered as a max-heap.