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      Question

      Java MinHeap class has a heapifyDown method.

      public class MinHeap {     private ArrayList heap;     public MinHeap() {         heap = new ArrayList();     }     // ... other methods like insert, getMin ...     private void heapifyDown(int i) {         int left = 2 * i + 1;         int right = 2 * i + 2;         int smallest = i;         if (left < heap.size() && heap.get(left) < heap.get(smallest)) {             smallest = left;         }         if (right < heap.size() && heap.get(right) < heap.get(smallest)) {             smallest = right;         }         if (smallest != i) {             // Swap heap.get(i) and heap.get(smallest)             int temp = heap.get(i);             heap.set(i, heap.get(smallest));             heap.set(smallest, temp);             // Bug: Missing recursive call         }     } } If heapifyDown is called on a node i where its children are not in heap order, but its grandchild is also out of order, what will be the consequence of the missing recursive call?
      A The heap will become a max-heap instead of a min-heap. Correct Answer Incorrect Answer
      B Only the immediate children will be correctly placed, but the heap property might still be violated further down the tree. Correct Answer Incorrect Answer
      C The method will cause an IndexOutOfBoundsException. Correct Answer Incorrect Answer
      D The smallest variable will always remain i. Correct Answer Incorrect Answer
      E The heap will always remain sorted. Correct Answer Incorrect Answer

      Solution

      • Dry Run: o Consider a min-heap structure where i is the root, left is its left child, right is its right child. o Let heap[i] = 10, heap[left] = 5, heap[right] = 12. o Let heap[2*left + 1] (a grandchild of i) be 2. o Call heapifyDown(i):  smallest is initially i.  heap.get(left) (5) is less than heap.get(smallest) (10), so smallest becomes left.  heap.get(right) (12) is not less than heap.get(smallest) (5).  smallest is left (which is not i), so the swap occurs. heap[i] becomes 5, heap[left] becomes 10.  The method then terminates.  Now, heap[left] is 10, but its child heap[2*left + 1] is 2. The heap property is violated at index left (10 is not less than or equal to 2). The heapifyDown operation is incomplete. • Why Correct Answer (B): Only the immediate children will be correctly placed, but the heap property might still be violated further down the tree. o The element at i will be correctly placed relative to its direct children (left and right). However, the element that was swapped down to smallest might still be larger than its children, meaning the heap property is not fully restored throughout the subtree.

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