

'''simple docstring'''





def 				_lowerCAmelCase			(   __magic_name__						:     int						)				->  None:
						lowercase :							int    		=generate_pascal_triangle(__magic_name__						)
						for row_idx in range(__magic_name__						):
												# Print left spaces
												for _ in range(num_rows - row_idx - 1						):
																		print(end=''' '''						)
												# Print row values
												for col_idx in range(row_idx + 1						):
																		if col_idx != row_idx:
																								print(triangle[row_idx][col_idx]   , end=''' '''						)
																		else:
																								print(triangle[row_idx][col_idx]   , end=''''''						)
												print()






def 				_lowerCAmelCase			(   __magic_name__						:     int						)				->  list[list[int]]:

						if not isinstance(__magic_name__   , __magic_name__						):
												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'''						)

						lowercase :							list[list[int]]    		=[]
						for current_row_idx in range(__magic_name__						):
												lowercase :							List[str]    		=populate_current_row(__magic_name__   , __magic_name__						)
												triangle.append(__magic_name__						)
						return triangle






def 				_lowerCAmelCase			(   __magic_name__						:     list[list[int]]   , __magic_name__						:     int						)				->  list[int]:
						lowercase :							int    		=[-1] * (current_row_idx + 1)
						# first and last elements of current row are equal to 1
						lowercase		,	lowercase :							Any    		=1, 1
						for current_col_idx in range(1   , __magic_name__						):
												calculate_current_element(
												    __magic_name__   , __magic_name__   , __magic_name__   , __magic_name__						)
						return current_row






def 				_lowerCAmelCase			(   __magic_name__						:     list[list[int]]   , __magic_name__						:     list[int]   , __magic_name__						:     int   , __magic_name__						:     int   , )				->  None:
						lowercase :							Any    		=triangle[current_row_idx - 1][current_col_idx - 1]
						lowercase :							Union[str, Any]    		=triangle[current_row_idx - 1][current_col_idx]
						lowercase :							int    		=above_to_left_elt + above_to_right_elt






def 				_lowerCAmelCase			(   __magic_name__						:     int						)				->  list[list[int]]:

						if not isinstance(__magic_name__   , __magic_name__						):
												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'''						)

						lowercase :							list[list[int]]    		=[[1]]

						for row_index in range(1   , __magic_name__						):
												lowercase :							List[str]    		=[0] + result[-1] + [0]
												lowercase :							Tuple    		=row_index + 1
												# Calculate the number of distinct elements in a row
												lowercase :							List[Any]    		=sum(divmod(__magic_name__   , 2						)						)
												lowercase :							int    		=[
												    temp_row[i - 1] + temp_row[i] for i in range(1   , distinct_elements + 1						)
												]
												lowercase :							Union[str, Any]    		=row_first_half[: (row_index + 1) // 2]
												row_second_half.reverse()
												lowercase :							str    		=row_first_half + row_second_half
												result.append(__magic_name__						)

						return result






def 				_lowerCAmelCase			(   )				->  None:
						from collections.abc import Callable
						from timeit import timeit

						def benchmark_a_function(__magic_name__						:     Callable   , __magic_name__						:     int						) -> None:
												lowercase :							str    		=f'''{func.__name__}({value})'''
												lowercase :							Optional[Any]    		=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(__magic_name__   , __magic_name__						)
												print()


if __name__ == "__main__":
							import doctest

							doctest.testmod()
							benchmark()











def 		lowerCamelCase__       (						_A	,      _A				):







	'''simple docstring'''





	_validate_point(_A				)
	_validate_point(_A				)
	if len(_A				) != len(_A				):
		raise ValueError("Both points must be in the same n-dimensional space"				)

	return float(sum(abs(a - b				) for a, b in zip(_A	,      _A				)				)				)


def 		lowerCamelCase__       (						_A				):







	'''simple docstring'''





	if point:
		if isinstance(_A	,      _A				):
			for item in point:
				if not isinstance(_A	,      (int, float)				):
					snake_case_						       =	(
					    "Expected a list of numbers as input, found "
					    f"{type(_A				).__name__}"
					)
					raise TypeError(_A				)
		else:
			snake_case_						       =	f"Expected a list of numbers as input, found {type(_A				).__name__}"
			raise TypeError(_A				)
	else:
		raise ValueError("Missing an input"				)


def 		lowerCamelCase__       (						_A	,      _A				):







	'''simple docstring'''





	_validate_point(_A				)
	_validate_point(_A				)
	if len(_A				) != len(_A				):
		raise ValueError("Both points must be in the same n-dimensional space"				)

	return float(sum(abs(x - y				) for x, y in zip(_A	,      _A				)				)				)


if __name__ == "__main__":
		import doctest

		doctest.testmod()
