Consider the process of inserting an element into a Max Heap, where the Max…

Answer: Θ(log log n)

Key idea: The path from the newly inserted leaf up to the root has length Θ(log n). If you perform a binary search on that path to find the correct position for the new element, the number of comparisons is logarithmic in the path length.

• Let h be the path length from the new leaf to the root. For a heap of size n, h = Θ(log n).

• A binary search on a list of h elements requires Θ(log h) comparisons.

• Substituting h = Θ(log n) gives Θ(log (log n)) = Θ(log log n) comparisons.

Note: This count refers to the number of comparisons between the new element and ancestor nodes. If you count element moves or swaps required to place the element after locating the position, you may still perform up to Θ(log n) moves, but the comparison count is Θ(log log n) under a binary-search-on-path strategy.