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The maximum subarray sum problem is the task of finding the maximum sum that can be
obtained from a contiguous subarray within a given array of numbers. For example, given
the array [-2, 1, -3, 4, -1, 2, 1, -5, 4], the contiguous subarray with the maximum sum
is [4, -1, 2, 1], so the maximum subarray sum is 6.
Kadane's algorithm is a simple dynamic programming algorithm that solves the maximum
subarray sum problem in O(n) time and O(1) space.
Reference: https://en.wikipedia.org/wiki/Maximum_subarray_problem
from collections.abc import Sequence
def max_subarray_sum(
    arr: Sequence[float], allow_empty_subarrays: bool = False
) -> float:
    Solves the maximum subarray sum problem using Kadane's algorithm.
    :param arr: the given array of numbers
    :param allow_empty_subarrays: if True, then the algorithm considers empty subarrays
    >>> max_subarray_sum([2, 8, 9])
    >>> max_subarray_sum([0, 0])
    >>> max_subarray_sum([-1.0, 0.0, 1.0])
    >>> max_subarray_sum([1, 2, 3, 4, -2])
    >>> max_subarray_sum([-2, 1, -3, 4, -1, 2, 1, -5, 4])
    >>> max_subarray_sum([2, 3, -9, 8, -2])
    >>> max_subarray_sum([-2, -3, -1, -4, -6])
    >>> max_subarray_sum([-2, -3, -1, -4, -6], allow_empty_subarrays=True)
    >>> max_subarray_sum([])
    if not arr:
        return 0
    max_sum = 0 if allow_empty_subarrays else float("-inf")
    curr_sum = 0.0
    for num in arr:
        curr_sum = max(0 if allow_empty_subarrays else num, curr_sum + num)
        max_sum = max(max_sum, curr_sum)
    return max_sum
if __name__ == "__main__":
    from doctest import testmod
    testmod()
    nums = [-2, 1, -3, 4, -1, 2, 1, -5, 4]
    print(f"{max_subarray_sum(nums) = }")

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