An operator delete(i) for a binary heap data structure is to be designed to delete the item in the $i$-th node. Assume that the heap is implemented in an array and $i$ refers to the $i$-th index of the array. If the heap tree has depth $d$ (number of edges on the path from the root to the farthest leaf), then what is the time complexity to re-fix the heap efficiently after the removal of the element?

To delete an element at index $i$ in a binary heap of depth $d$:

1. Replace the element at index $i$ with the last element in the heap array (heap size decreases by 1).

2. The replacement element may violate the heap property in one of two directions:

   • It may be greater than its parent (for a max-heap) or smaller than its parent (for a min-heap), requiring a heapify-up (shift-up) operation towards the root.

   • Or it may violate the property with respect to its children, requiring a heapify-down (shift-down) operation towards the leaves.

3. The height/depth of the binary heap tree is $d$.

4. In the worst case, heapify-up travels at most $d$ levels up to the root, and heapify-down travels at most $d$ levels down to a leaf.

Thus, re-fixing the heap takes time proportional to the depth of the tree, which is $O(d)$, but not $O(1)$ in the worst case.

Hence, the correct option is (B).