added a problem on kadane's algo and its solution. (#8569)

* added kadane's algorithm directory with one problem's solution.

* added type hints

* Rename kaadne_algorithm/max_product_subarray.py to dynamic_programming/max_product_subarray.py

* Update dynamic_programming/max_product_subarray.py

Co-authored-by: Christian Clauss <cclauss@me.com>

* Update max_product_subarray.py

* Update max_product_subarray.py

* Update dynamic_programming/max_product_subarray.py

Co-authored-by: Christian Clauss <cclauss@me.com>

* Update max_product_subarray.py

* Update max_product_subarray.py

* Update max_product_subarray.py

* Update max_product_subarray.py

* Update max_product_subarray.py

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* Update max_product_subarray.py

* [pre-commit.ci] auto fixes from pre-commit.com hooks

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* Update max_product_subarray.py

* [pre-commit.ci] auto fixes from pre-commit.com hooks

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* Update max_product_subarray.py

* Update max_product_subarray.py

* Update dynamic_programming/max_product_subarray.py

Co-authored-by: Christian Clauss <cclauss@me.com>

* [pre-commit.ci] auto fixes from pre-commit.com hooks

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* Update dynamic_programming/max_product_subarray.py

Co-authored-by: Christian Clauss <cclauss@me.com>

* Update max_product_subarray.py

---------

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dynamic_programming/max_product_subarray.py

@@ -0,0 +1,53 @@
+def max_product_subarray(numbers: list[int]) -> int:
+    """
+    Returns the maximum product that can be obtained by multiplying a
+    contiguous subarray of the given integer list `nums`.
+
+    Example:
+    >>> max_product_subarray([2, 3, -2, 4])
+    6
+    >>> max_product_subarray((-2, 0, -1))
+    0
+    >>> max_product_subarray([2, 3, -2, 4, -1])
+    48
+    >>> max_product_subarray([-1])
+    -1
+    >>> max_product_subarray([0])
+    0
+    >>> max_product_subarray([])
+    0
+    >>> max_product_subarray("")
+    0
+    >>> max_product_subarray(None)
+    0
+    >>> max_product_subarray([2, 3, -2, 4.5, -1])
+    Traceback (most recent call last):
+        ...
+    ValueError: numbers must be an iterable of integers
+    >>> max_product_subarray("ABC")
+    Traceback (most recent call last):
+        ...
+    ValueError: numbers must be an iterable of integers
+    """
+    if not numbers:
+        return 0
+
+    if not isinstance(numbers, (list, tuple)) or not all(
+        isinstance(number, int) for number in numbers
+    ):
+        raise ValueError("numbers must be an iterable of integers")
+
+    max_till_now = min_till_now = max_prod = numbers[0]
+
+    for i in range(1, len(numbers)):
+        # update the maximum and minimum subarray products
+        number = numbers[i]
+        if number < 0:
+            max_till_now, min_till_now = min_till_now, max_till_now
+        max_till_now = max(number, max_till_now * number)
+        min_till_now = min(number, min_till_now * number)
+
+        # update the maximum product found till now
+        max_prod = max(max_prod, max_till_now)
+
+    return max_prod