Consolidate duplicate implementations of max subarray (#8849)

* Remove max subarray sum duplicate implementations

* updating DIRECTORY.md

* Rename max_sum_contiguous_subsequence.py

* Fix typo in dynamic_programming/max_subarray_sum.py

* Remove duplicate divide and conquer max subarray

* updating DIRECTORY.md

---------

Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>

dynamic_programming/max_subarray_sum.py

@@ -0,0 +1,60 @@
+"""
+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])
+    19
+    >>> max_subarray_sum([0, 0])
+    0
+    >>> max_subarray_sum([-1.0, 0.0, 1.0])
+    1.0
+    >>> max_subarray_sum([1, 2, 3, 4, -2])
+    10
+    >>> max_subarray_sum([-2, 1, -3, 4, -1, 2, 1, -5, 4])
+    6
+    >>> max_subarray_sum([2, 3, -9, 8, -2])
+    8
+    >>> max_subarray_sum([-2, -3, -1, -4, -6])
+    -1
+    >>> max_subarray_sum([-2, -3, -1, -4, -6], allow_empty_subarrays=True)
+    0
+    >>> max_subarray_sum([])
+    0
+    """
+    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) = }")