Question

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?

a)O(1)
b)O(d) but not O(1)
c)O(2ᵈ) but not O(d)
d)O(d2ᵈ) but not O(2ᵈ)

Answer 1

The time complexity of restructuring a binary heap after deleting an element at index I is O(d), as the operation involves sifting the element up or down the heap, which is bounded by the heap's depth. Therefore, the correct answer option is D)

Step-by-step explanation:

When deleting an element from any index I in a binary heap implemented as an array, the time complexity to restructure the heap is primarily concerned with the sift-down or sift-up operations required to maintain the heap's properties. In the worst case, the element to be deleted or replaced at the root might need to be moved down to the last level of the heap.

Therefore, the number of levels, or heap depth d, limits the number of operations needed to restore the heap property. Each step of the sift-down operation might require comparison and possible swapping with child elements. Given the depth of the heap is d, the worst-case scenario for re-establishing the heap property is proportional to the depth of the tree, which leads to a time complexity of O(d) but not O (1).

The options O(2ᵈ) and O(d2ᵈ) represent exponential time complexity and are not suitable for this operation as binary heaps perform better than this.