Modified 'pascal_triangle.py' program (TheAlgorithms#7901)

* Added pascals_triangle.py program to maths directory

* Deleted 'pascals_triangle.py' because of duplication. Added a optimized function to generate pascal's triangle to 'pascal_triangle.py' program. Added some aadditional doctests to the existing function. Added some type check functionality to the existing function.

* Modified type check hints in 'generate_pascal_triangle_optimized' function'

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* Modified 'pascal_triangle' prgram

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* Update pascal_triangle.py

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

other/pascal_triangle.py

@@ -1,13 +1,10 @@
 """
-This implementation demonstrates how to generate the
-elements of a Pascal's triangle. The element having
-a row index of r and column index of c can be derived
-as follows:
+This implementation demonstrates how to generate the elements of a Pascal's triangle.
+The element havingva row index of r and column index of c can be derivedvas follows:
 triangle[r][c] = triangle[r-1][c-1]+triangle[r-1][c]
-What is Pascal's triangle?
-- It is a triangular array containing binomial coefficients.
-Refer to (https://en.wikipedia.org/wiki/Pascal%27s_triangle)
-for more info about this triangle.
+
+A Pascal's triangle is a triangular array containing binomial coefficients.
+https://en.wikipedia.org/wiki/Pascal%27s_triangle
 """
 
 
@@ -38,6 +35,8 @@ def print_pascal_triangle(num_rows: int) -> None:
 def generate_pascal_triangle(num_rows: int) -> list[list[int]]:
     """
     Create Pascal's triangle for different number of rows
+    >>> generate_pascal_triangle(0)
+    []
     >>> generate_pascal_triangle(1)
     [[1]]
     >>> generate_pascal_triangle(2)
@@ -48,7 +47,26 @@ def generate_pascal_triangle(num_rows: int) -> list[list[int]]:
     [[1], [1, 1], [1, 2, 1], [1, 3, 3, 1]]
     >>> generate_pascal_triangle(5)
     [[1], [1, 1], [1, 2, 1], [1, 3, 3, 1], [1, 4, 6, 4, 1]]
+    >>> generate_pascal_triangle(-5)
+    Traceback (most recent call last):
+        ...
+    ValueError: The input value of 'num_rows' should be greater than or equal to 0
+    >>> generate_pascal_triangle(7.89)
+    Traceback (most recent call last):
+        ...
+    TypeError: The input value of 'num_rows' should be 'int'
     """
+
+    if not isinstance(num_rows, int):
+        raise TypeError("The input value of 'num_rows' should be 'int'")
+
+    if num_rows == 0:
+        return []
+    elif num_rows < 0:
+        raise ValueError(
+            "The input value of 'num_rows' should be greater than or equal to 0"
+        )
+
     triangle: list[list[int]] = []
     for current_row_idx in range(num_rows):
         current_row = populate_current_row(triangle, current_row_idx)
@@ -90,7 +108,82 @@ def calculate_current_element(
     current_row[current_col_idx] = above_to_left_elt + above_to_right_elt
 
 
+def generate_pascal_triangle_optimized(num_rows: int) -> list[list[int]]:
+    """
+    This function returns a matrix representing the corresponding pascal's triangle
+    according to the given input of number of rows of Pascal's triangle to be generated.
+    It reduces the operations done to generate a row by half
+    by eliminating redundant calculations.
+
+    :param num_rows: Integer specifying the number of rows in the Pascal's triangle
+    :return: 2-D List (matrix) representing the Pascal's triangle
+
+    Return the Pascal's triangle of given rows
+    >>> generate_pascal_triangle_optimized(3)
+    [[1], [1, 1], [1, 2, 1]]
+    >>> generate_pascal_triangle_optimized(1)
+    [[1]]
+    >>> generate_pascal_triangle_optimized(0)
+    []
+    >>> generate_pascal_triangle_optimized(-5)
+    Traceback (most recent call last):
+        ...
+    ValueError: The input value of 'num_rows' should be greater than or equal to 0
+    >>> generate_pascal_triangle_optimized(7.89)
+    Traceback (most recent call last):
+        ...
+    TypeError: The input value of 'num_rows' should be 'int'
+    """
+
+    if not isinstance(num_rows, int):
+        raise TypeError("The input value of 'num_rows' should be 'int'")
+
+    if num_rows == 0:
+        return []
+    elif num_rows < 0:
+        raise ValueError(
+            "The input value of 'num_rows' should be greater than or equal to 0"
+        )
+
+    result: list[list[int]] = [[1]]
+
+    for row_index in range(1, num_rows):
+        temp_row = [0] + result[-1] + [0]
+        row_length = row_index + 1
+        # Calculate the number of distinct elements in a row
+        distinct_elements = sum(divmod(row_length, 2))
+        row_first_half = [
+            temp_row[i - 1] + temp_row[i] for i in range(1, distinct_elements + 1)
+        ]
+        row_second_half = row_first_half[: (row_index + 1) // 2]
+        row_second_half.reverse()
+        row = row_first_half + row_second_half
+        result.append(row)
+
+    return result
+
+
+def benchmark() -> None:
+    """
+    Benchmark multiple functions, with three different length int values.
+    """
+    from collections.abc import Callable
+    from timeit import timeit
+
+    def benchmark_a_function(func: Callable, value: int) -> None:
+        call = f"{func.__name__}({value})"
+        timing = timeit(f"__main__.{call}", setup="import __main__")
+        # print(f"{call:38} = {func(value)} -- {timing:.4f} seconds")
+        print(f"{call:38} -- {timing:.4f} seconds")
+
+    for value in range(15):  # (1, 7, 14):
+        for func in (generate_pascal_triangle, generate_pascal_triangle_optimized):
+            benchmark_a_function(func, value)
+        print()
+
+
 if __name__ == "__main__":
     import doctest
 
     doctest.testmod()
+    benchmark()