How to prove the performance, Big Omega ,of building a binary heap using recursive method is Ω(nlog(n))

Question

We can learn the big-O of building a binary heap using recursive method is O(n log n) from wiki "This approach, called Williams’ method after the inventor of binary heaps, is easily seen to run in O(n log n) time: it performs n insertions at O(log n) cost each.[a]" we can also know if we build the heap using other methods. the big O could be better. Where should we begin if we would like to prove that the performance, Big Omega ,of building a binary heap using recursive method is Ω(nlog(n)), which is same as Big-O?

insert method from wiki

To add an element to a heap we must perform an up-heap operation (also known as bubble-up, percolate-up, sift-up, trickle-up, swim-up, heapify-up, or cascade-up), by following this algorithm:

Add the element to the bottom level of the heap. Compare the added element with its parent; if they are in the correct order, stop. If not, swap the element with its parent and return to the previous step

then we have the buildheap as

buildheap((𝑂1,𝐾1),(𝑂𝑛,𝐾𝑛)) 
   if n==1: 
      return [(𝑂1,𝐾1)] 
return insert((𝑂𝑛 ,𝐾𝑛),buildheap((𝑂1,𝐾1),...(𝑂𝑛−1,𝐾𝑛−1))

Answer 1

Suppose that your heap is a max-heap: a parent should be larger than its children. Consider what happens when you call buildheap with the numbers $1,\ldots,n$ in that order. Each time you add a new element to the bottom level of the heap, and the heapify procedure diffuses it all the way to the root. Since element $k$ is put at level $\Theta(\log k)$, inserting the $k$'th element takes time $\Theta(\log k)$. Overall, buildheap takes time $\Theta(\log 1 + \cdots + \log n) = \Theta(\log n!) = \Theta(n\log n)$.