Fewest decrements to make an array a 1-to-k-to-1 pyramid

You are given an array arr of n non-negative integers. One operation picks any index and lowers that element by exactly 1. Elements can never be raised.

Return the minimum number of operations needed to turn arr into a pyramid array:

• it may start and/or end with any number of zeros, and
• the remaining (non-zero) part reads 1, 2, 3, ..., k, ..., 3, 2, 1 for some peak k >= 1: it climbs by exactly 1 to a single peak, then falls by exactly 1 back to 1.

An array of all zeros counts as a pyramid (its non-zero part is empty).

Implement:

def min_ops_to_bitonic(arr: list[int]) -> int: ...

Constraints: 1 <= n <= 10^5, 0 <= arr[i] <= 10^9. Aim for better than O(n²).

Example 1

arr = [2, 5, 4, 6, 2]
output: 10

Lower it to [1, 2, 3, 2, 1]. The array sums to 19, the pyramid to 9, so 10 decrements.

Example 2

arr = [4, 4, 4, 4]
output: 12

No height-3 pyramid fits in 4 slots (it needs 5). The best is a height-2 pyramid such as [1, 2, 1, 0], sum 4, so 16 - 4 = 12.

Reconstructed from the reported question; details may differ from what interviewers ask.