add : trapped water program under dynamic programming (#10027)

* to add the trapped water program

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* to make changes for error : B006

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* to make changes in doctest

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* Update dynamic_programming/trapped_water.py

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

* Update dynamic_programming/trapped_water.py

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

* to make changes in parameters

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* to make changes in parameters

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* Update dynamic_programming/trapped_water.py

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

* to make changes in parameters

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* for negative heights

* Update dynamic_programming/trapped_water.py

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

* to remove falsy

* Final edits

* tuple[int, ...]

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Christian Clauss <cclauss@me.com>

dynamic_programming/trapped_water.py

@@ -0,0 +1,60 @@
+"""
+Given an array of non-negative integers representing an elevation map where the width
+of each bar is 1, this program calculates how much rainwater can be trapped.
+
+Example - height = (0, 1, 0, 2, 1, 0, 1, 3, 2, 1, 2, 1)
+Output: 6
+This problem can be solved using the concept of "DYNAMIC PROGRAMMING".
+
+We calculate the maximum height of bars on the left and right of every bar in array.
+Then iterate over the width of structure and at each index.
+The amount of water that will be stored is equal to minimum of maximum height of bars
+on both sides minus height of bar at current position.
+"""
+
+
+def trapped_rainwater(heights: tuple[int, ...]) -> int:
+    """
+    The trapped_rainwater function calculates the total amount of rainwater that can be
+    trapped given an array of bar heights.
+    It uses a dynamic programming approach, determining the maximum height of bars on
+    both sides for each bar, and then computing the trapped water above each bar.
+    The function returns the total trapped water.
+
+    >>> trapped_rainwater((0, 1, 0, 2, 1, 0, 1, 3, 2, 1, 2, 1))
+    6
+    >>> trapped_rainwater((7, 1, 5, 3, 6, 4))
+    9
+    >>> trapped_rainwater((7, 1, 5, 3, 6, -1))
+    Traceback (most recent call last):
+        ...
+    ValueError: No height can be negative
+    """
+    if not heights:
+        return 0
+    if any(h < 0 for h in heights):
+        raise ValueError("No height can be negative")
+    length = len(heights)
+
+    left_max = [0] * length
+    left_max[0] = heights[0]
+    for i, height in enumerate(heights[1:], start=1):
+        left_max[i] = max(height, left_max[i - 1])
+
+    right_max = [0] * length
+    right_max[-1] = heights[-1]
+    for i in range(length - 2, -1, -1):
+        right_max[i] = max(heights[i], right_max[i + 1])
+
+    return sum(
+        min(left, right) - height
+        for left, right, height in zip(left_max, right_max, heights)
+    )
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+    print(f"{trapped_rainwater((0, 1, 0, 2, 1, 0, 1, 3, 2, 1, 2, 1)) = }")
+    print(f"{trapped_rainwater((7, 1, 5, 3, 6, 4)) = }")