
'''simple docstring'''
def        A_     (						__SCREAMING_SNAKE_CASE						:  List[Any]			,     __SCREAMING_SNAKE_CASE						:  Optional[Any] )				->       int:





  """simple docstring"""

  _enforce_args(_lowercase			,     _lowercase )
  if n == 0:
    return 0
  __A		:     List[Any]												=		float("""-inf""" )
  for i in range(1			,     n + 1 ):
    __A		:     int												=		max(
        _lowercase			,     prices[i - 1] + naive_cut_rod_recursive(n - i			,     _lowercase ) )

  return max_revue







def        A_     (						__SCREAMING_SNAKE_CASE						:  Optional[Any]			,     __SCREAMING_SNAKE_CASE						:  Optional[int] )				->       Union[str, Any]:





  """simple docstring"""

  _enforce_args(_lowercase			,     _lowercase )
  __A		:     int												=		[float("""-inf""" ) for _ in range(n + 1 )]
  return _top_down_cut_rod_recursive(_lowercase			,     _lowercase			,     _lowercase )







def        A_     (						__SCREAMING_SNAKE_CASE						:  Tuple			,     __SCREAMING_SNAKE_CASE						:  Tuple			,     __SCREAMING_SNAKE_CASE						:  Optional[int] )				->       List[str]:





  """simple docstring"""

  if max_rev[n] >= 0:
    return max_rev[n]
  elif n == 0:
    return 0
  else:
    __A		:     int												=		float("""-inf""" )
    for i in range(1			,     n + 1 ):
      __A		:     List[str]												=		max(
          _lowercase			,     prices[i - 1] + _top_down_cut_rod_recursive(n - i			,     _lowercase			,     _lowercase )			,     )

    __A		:     List[str]												=		max_revenue

  return max_rev[n]







def        A_     (						__SCREAMING_SNAKE_CASE						:  Dict			,     __SCREAMING_SNAKE_CASE						:  str )				->       str:





  """simple docstring"""

  _enforce_args(_lowercase			,     _lowercase )

  # length(max_rev) = n + 1, to accommodate for the revenue obtainable from a rod of
  # length 0.
  __A		:     Any												=		[float("""-inf""" ) for _ in range(n + 1 )]
  __A		:     List[str]												=		0

  for i in range(1			,     n + 1 ):
    __A		:     Tuple												=		max_rev[i]
    for j in range(1			,     i + 1 ):
      __A		:     Optional[int]												=		max(_lowercase			,     prices[j - 1] + max_rev[i - j] )

    __A		:     Union[str, Any]												=		max_revenue_i

  return max_rev[n]







def        A_     (						__SCREAMING_SNAKE_CASE						:  str			,     __SCREAMING_SNAKE_CASE						:  Tuple )				->       Union[str, Any]:





  """simple docstring"""

  if n < 0:
    __A		:     int												=		F"n must be greater than or equal to 0. Got n = {n}"
    raise ValueError(_lowercase )

  if n > len(_lowercase ):
    __A		:     Any												=		(
        'Each integral piece of rod must have a corresponding price. '
        F"Got n = {n} but length of prices = {len(_lowercase )}"
    )
    raise ValueError(_lowercase )







def        A_     (						)				->       List[str]:





  """simple docstring"""

  __A		:     Optional[Any]												=		[6, 10, 12, 15, 20, 23]
  __A		:     Optional[int]												=		len(_lowercase )

  # the best revenue comes from cutting the rod into 6 pieces, each
  # of length 1 resulting in a revenue of 6 * 6 = 36.
  __A		:     int												=		36

  __A		:     Optional[int]												=		top_down_cut_rod(_lowercase			,     _lowercase )
  __A		:     List[Any]												=		bottom_up_cut_rod(_lowercase			,     _lowercase )
  __A		:     List[str]												=		naive_cut_rod_recursive(_lowercase			,     _lowercase )

  assert expected_max_revenue == max_rev_top_down
  assert max_rev_top_down == max_rev_bottom_up
  assert max_rev_bottom_up == max_rev_naive


if __name__ == "__main__":
    main()





'''simple docstring'''
def        A_     (						__SCREAMING_SNAKE_CASE						:  int )				->       None:





  """simple docstring"""

  __A		:     Tuple												=		generate_pascal_triangle(__SCREAMING_SNAKE_CASE )
  for row_idx in range(__SCREAMING_SNAKE_CASE ):
    # 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        A_     (						__SCREAMING_SNAKE_CASE						:  int )				->       list[list[int]]:





  """simple docstring"""

  if not isinstance(__SCREAMING_SNAKE_CASE			,     __SCREAMING_SNAKE_CASE ):
    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""" )

  __A		:     list[list[int]]												=		[]
  for current_row_idx in range(__SCREAMING_SNAKE_CASE ):
    __A		:     Tuple												=		populate_current_row(__SCREAMING_SNAKE_CASE			,     __SCREAMING_SNAKE_CASE )
    triangle.append(__SCREAMING_SNAKE_CASE )
  return triangle







def        A_     (						__SCREAMING_SNAKE_CASE						:  list[list[int]]			,     __SCREAMING_SNAKE_CASE						:  int )				->       list[int]:





  """simple docstring"""

  __A		:     Union[str, Any]												=		[-1] * (current_row_idx + 1)
  # first and last elements of current row are equal to 1
  __A							,					__A		:     Tuple												=		1, 1
  for current_col_idx in range(1			,     __SCREAMING_SNAKE_CASE ):
    calculate_current_element(
        __SCREAMING_SNAKE_CASE			,     __SCREAMING_SNAKE_CASE			,     __SCREAMING_SNAKE_CASE			,     __SCREAMING_SNAKE_CASE )
  return current_row







def        A_     (						__SCREAMING_SNAKE_CASE						:  list[list[int]]			,     __SCREAMING_SNAKE_CASE						:  list[int]			,     __SCREAMING_SNAKE_CASE						:  int			,     __SCREAMING_SNAKE_CASE						:  int			,     )				->       None:





  """simple docstring"""

  __A		:     str												=		triangle[current_row_idx - 1][current_col_idx - 1]
  __A		:     Union[str, Any]												=		triangle[current_row_idx - 1][current_col_idx]
  __A		:     Any												=		above_to_left_elt + above_to_right_elt







def        A_     (						__SCREAMING_SNAKE_CASE						:  int )				->       list[list[int]]:





  """simple docstring"""

  if not isinstance(__SCREAMING_SNAKE_CASE			,     __SCREAMING_SNAKE_CASE ):
    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""" )

  __A		:     list[list[int]]												=		[[1]]

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

  return result







def        A_     (						)				->       None:





  """simple docstring"""

  from collections.abc import Callable
  from timeit import timeit

  def benchmark_a_function(__SCREAMING_SNAKE_CASE						:  Callable			,     __SCREAMING_SNAKE_CASE						:  int ) -> None:
    __A		:     List[Any]												=		F"{func.__name__}({value})"
    __A		:     Dict												=		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(__SCREAMING_SNAKE_CASE			,     __SCREAMING_SNAKE_CASE )
    print()


if __name__ == "__main__":
     import doctest

     doctest.testmod()
     benchmark()



