Find Any Local Minimum with Iterative Binary Search

Given a nonempty integer array numbers, return the index of any local minimum. An index i is a local minimum when its value is strictly smaller than each neighbor that exists. Treat the missing neighbor beyond either boundary as positive infinity.

Adjacent elements are guaranteed to be different. Your implementation must be iterative and run in O(log n) time.

Function Signature

def find_local_minimum(numbers: list[int]) -> int:
    ...

Constraints

• 1 <= len(numbers) <= 1_000_000
• numbers[i] != numbers[i + 1] for every valid adjacent pair.
• -1_000_000_000 <= numbers[i] <= 1_000_000_000.
• If multiple local minima exist, return any valid index.

Examples

Input: numbers = [9, 7, 3, 5, 8]
Output: 2

Input: numbers = [1, 4, 3]
Output: 0 or 2

Input: numbers = [6]
Output: 0