






"""simple docstring"""





from __future__ import annotations






def     A(  snake_case_		):
       """simple docstring"""





       if not list_of_ints:
              return []

       if min(snake_case_		) < 0:
              raise ValueError("All numbers must be positive"		)

       lowercase__:    List[str]        =      max(len(bin(snake_case_		)[2:]		) for x in list_of_ints		)
       return _msd_radix_sort(snake_case_						,       snake_case_		)






def     A(  snake_case_						,       snake_case_		):
       """simple docstring"""





       if bit_position == 0 or len(snake_case_		) in [0, 1]:
              return list_of_ints

       lowercase__:    Optional[int]        =      []
       lowercase__:    Optional[int]        =      []
       # Split numbers based on bit at bit_position from the right
       for number in list_of_ints:
              if (number >> (bit_position - 1)) & 1:
                     # number has a one at bit bit_position
                     ones.append(snake_case_		)
              else:
                     # number has a zero at bit bit_position
                     zeros.append(snake_case_		)

    # recursively split both lists further
       lowercase__:    List[Any]        =      _msd_radix_sort(snake_case_						,       bit_position - 1		)
       lowercase__:    int        =      _msd_radix_sort(snake_case_						,       bit_position - 1		)

       # recombine lists
       lowercase__:    Union[str, Any]        =      zeros
       res.extend(snake_case_		)

       return res






def     A(  snake_case_		):
       """simple docstring"""





       lowercase__:    Dict        =      len(snake_case_		)
       if not list_of_ints or length == 1:
              return

       if min(snake_case_		) < 0:
              raise ValueError("All numbers must be positive"		)

       lowercase__:    int        =      max(len(bin(snake_case_		)[2:]		) for x in list_of_ints		)
       _msd_radix_sort_inplace(snake_case_						,       snake_case_						,       0						,       snake_case_		)






def     A(  snake_case_						,       snake_case_						,       snake_case_						,       snake_case_		):
       """simple docstring"""





       if bit_position == 0 or end_index - begin_index <= 1:
              return

       bit_position -= 1

       lowercase__:    str        =      begin_index
       lowercase__:    Union[str, Any]        =      end_index - 1
       while i <= j:
              lowercase__:    Optional[Any]        =      False
              if not (list_of_ints[i] >> bit_position) & 1:
                     # found zero at the beginning
                     i += 1
                     lowercase__:    int        =      True
              if (list_of_ints[j] >> bit_position) & 1:
                     # found one at the end
                     j -= 1
                     lowercase__:    Any        =      True

              if changed:
                     continue

              lowercase__  ,     lowercase__:    Optional[Any]        =      list_of_ints[j], list_of_ints[i]
              j -= 1
              if j != i:
                     i += 1

       _msd_radix_sort_inplace(snake_case_						,       snake_case_						,       snake_case_						,       snake_case_		)
       _msd_radix_sort_inplace(snake_case_						,       snake_case_						,       snake_case_						,       snake_case_		)


if __name__ == "__main__":
  import doctest

  doctest.testmod()














"""simple docstring"""





def     A(  snake_case_		):
       """simple docstring"""





       if not isinstance(snake_case_						,       snake_case_		):
              raise ValueError("Input series is not valid, valid series - [1, 2/3, 2]"		)
       if len(snake_case_		) == 0:
              raise ValueError("Input list must be a non empty list"		)
       if len(snake_case_		) == 1 and series[0] != 0:
              return True
       lowercase__:    str        =      []
       lowercase__:    Optional[int]        =      len(snake_case_		)
       for i in range(0						,       snake_case_		):
              if series[i] == 0:
                     raise ValueError("Input series cannot have 0 as an element"		)
              rec_series.append(1 / series[i]		)
       lowercase__:    Optional[int]        =      rec_series[1] - rec_series[0]
       for index in range(2						,       snake_case_		):
              if rec_series[index] - rec_series[index - 1] != common_diff:
                     return False
       return True






def     A(  snake_case_		):
       """simple docstring"""





       if not isinstance(snake_case_						,       snake_case_		):
              raise ValueError("Input series is not valid, valid series - [2, 4, 6]"		)
       if len(snake_case_		) == 0:
              raise ValueError("Input list must be a non empty list"		)
       lowercase__:    Optional[int]        =      0
       for val in series:
              answer += 1 / val
       return len(snake_case_		) / answer


if __name__ == "__main__":
  import doctest

  doctest.testmod()






