array median transformation minimum steps

Question

Given an array A with n integers. In one turn one can apply the following operation to any consecutive subarray A[l..r] : assign to all A i (l <= i <= r) median of sub

Answer 1

You can double the size of the subarray containing the maximum element in each iteration. After the first iteration, there is a subarray of size 2 containing the maximum. Then apply your operation to a subarray of size 4, containing those 2 elements, giving you a subarray of size 4 containing the maximum. Then apply to a size 8 subarray and so on. You fill the array in log2(N) operations, which is optimal. If N is 30, five operations is enough.

This is optimal in the worst case (i.e. when only one element is the maximum), since it sets the highest possible number of elements in each iteration.

Update 1: I noticed I messed up the 4s and 8s a bit. Corrected.

Update 2: here's an example. Array size 10, start state:

[6 1 5 9 3 2 0 7 4 8]

To get two nines, run op on subarray of size two containing the nine. For instance A[4…5] gets you:

[6 1 5 9 9 2 0 7 4 8]

Now run on size four subarray that contains 4…5, for instance on A[2…5] to get:

[6 9 9 9 9 2 0 7 4 8]

Now on subarray of size 8, for instance A[1…8], get:

[9 9 9 9 9 9 9 9 4 8]

Doubling now would get us 16 nines, but we have only 10 positions, so round of with A[1…10], get:

[9 9 9 9 9 9 9 9 9 9]

Update 3: since this is only optimal in the worst case, it is actually not an answer to the original question, which asks for a way of finding the minimal number of operations for all inputs. I misinterpreted the sentence about brute forcing to be about brute forcing with the median operations, rather than in finding the minimum sequence of operations.