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

Basically, work is done only on non-leaf nodes while building a heap...and the work done is the amount of swapping down to satisfy heap condition...in other words (in worst case) the amount is proportional to the height of the node...all in all the complexity of the problem is proportional to the sum of heights of all the non-leaf nodes..which is (2^h+1 - 1)-h-1=n-h-1= O(n)