Time Complexity of building a heap

Consider the following algorithm for building a Heap of an input array A. 

BUILD-HEAP(A) 

    heapsize := size(A); 

    for i := floor(heapsize/2) downto 1 

        do HEAPIFY(A, i); 

    end for 

END

A quick look over the above algorithm suggests that the running time is  since each call to Heapify costs and Build-Heap makes such calls. 

This upper bound, though correct, is not asymptotically tight. 

We can derive a tighter bound by observing that the running time of Heapify depends on the height of the tree ‘h’ (which is equal to lg(n), where n is a number of nodes) and the heights of most sub-trees are small. The height ’h’ increases as we move upwards along the tree. Line-3 of Build-Heap runs a loop from the index of the last internal node (heapsize/2) with height=1, to the index of root(1) with height = lg(n). Hence, Heapify takes a different time for each node, which is:

For finding the Time Complexity of building a heap, we must know the number of nodes having height h. For this we use the fact that, A heap of size n has at most nodes with height h. 

a  to derive the time complexity, we express the total cost of Build-Heap as-

Step 2 uses the properties of the Big-Oh notation to ignore the ceiling function and the constant 2(). Similarly in Step three, the upper limit of the summation can be increased to infinity since we are using Big-Oh notation. Sum of infinite G.P. (x < 1)

On differentiating both sides and multiplying by x, we get

Putting the result obtained in (3) back in our derivation (1), we get

Hence Proved that the Time complexity for Building a Binary Heap is .