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

I really like explanation by Jeremy west.... another approach which is really easy for understanding is given here http://courses.washington.edu/css343/zander/NotesProbs/heapcomplexity

since, buildheap depends using depends on heapify and shiftdown approach is used which depends upon sum of the heights of all nodes. So, to find the sum of height of nodes which is given by S = summation from i = 0 to i = h of (2^i*(h-i)), where h = logn is height of the tree solving s, we get s = 2^(h+1) - 1 - (h+1) since, n = 2^(h+1) - 1 s = n - h - 1 = n- logn - 1 s = O(n), and so complexity of buildheap is O(n).