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

Successive insertions can be described by:

T = O(log(1) + log(2) + .. + log(n)) = O(log(n!))

By starling approximation, n! =~ O(n^(n + O(1))), therefore T =~ O(nlog(n))

Hope this helps, the optimal way O(n) is using the build heap algorithm for a given set (ordering doesn't matter).