
'''simple docstring'''
# Algorithm for the pigeonhole sorting













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





  """simple docstring"""

  __A		:     Union[str, Any]												=		min(__lowercase )  # min() finds the minimum value
  __A		:     Tuple												=		max(__lowercase )  # max() finds the maximum value

  __A		:     str												=		max_val - min_val + 1  # size is difference of max and min values plus one

  # list of pigeonholes of size equal to the variable size
  __A		:     Tuple												=		[0] * size

  # Populate the pigeonholes.
  for x in a:
    assert isinstance(__lowercase			,     __lowercase ), "integers only please"
    holes[x - min_val] += 1

  # Putting the elements back into the array in an order.
  __A		:     str												=		0
  for count in range(__lowercase ):
    while holes[count] > 0:
      holes[count] -= 1
      __A		:     Tuple												=		count + min_val
      i += 1







def        A_     (						)				->       List[str]:





  """simple docstring"""

  __A		:     Any												=		[8, 3, 2, 7, 4, 6, 8]
  pigeonhole_sort(__lowercase )
  print("""Sorted order is:"""			,     """ """.join(__lowercase ) )


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()



