How can building a heap be O(n) time complexity?

Question

Can someone help explain how can building a heap be O(n) complexity?

Inserting an item into a heap is O(log n), and the insert is repeated n/2 times (t

Answer 1

@bcorso has already demonstrated the proof of the complexity analysis. But for the sake of those still learning complexity analysis, I have this to add:

The basis of your original mistake is due to a misinterpretation of the meaning of the statement, "insertion into a heap takes O(log n) time". Insertion into a heap is indeed O(log n), but you have to recognise that n is the size of the heap during the insertion.

In the context of inserting n objects into a heap, the complexity of the ith insertion is O(log n_i) where n_i is the size of the heap as at insertion i. Only the last insertion has a complexity of O (log n).