





'''simple docstring'''



import sys



def  lowercase__(		_UpperCamelCase  : Any ,   _UpperCamelCase  : Optional[int]				)->   List[str]:






					"""simple docstring"""

					if not numbers:
										return 0
					if target == 0 and target in numbers:
										return 0
					if not isinstance(UpperCamelCase_ ,   (list, tuple)				) or not all(
					    isinstance(UpperCamelCase_ ,   UpperCamelCase_				) for number in numbers				):
										raise ValueError("numbers must be an iterable of integers"				)

					_UpperCamelCase											=   _UpperCamelCase											=   _UpperCamelCase											=   0
					_UpperCamelCase											=   sys.maxsize

					while right < len(UpperCamelCase_				):
										curr_sum += numbers[right]
										while curr_sum >= target and left <= right:
															_UpperCamelCase											=   min(UpperCamelCase_ ,   right - left + 1				)
															curr_sum -= numbers[left]
															left += 1
										right += 1

					return 0 if min_len == sys.maxsize else min_len









'''simple docstring'''



from __future__ import annotations



def  lowercase__(		_UpperCamelCase  : list[int]				)->   list[int]:






					"""simple docstring"""

					if not list_of_ints:
										return []

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

					_UpperCamelCase											=   max(len(bin(_UpperCamelCase				)[2:]				) for x in list_of_ints				)
					return _msd_radix_sort(_UpperCamelCase ,   _UpperCamelCase				)



def  lowercase__(		_UpperCamelCase  : list[int] ,   _UpperCamelCase  : int				)->   list[int]:






					"""simple docstring"""

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

					_UpperCamelCase											=   []
					_UpperCamelCase											=   []
					# 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(_UpperCamelCase				)
										else:
															# number has a zero at bit bit_position
															zeros.append(_UpperCamelCase				)

    # recursively split both lists further
					_UpperCamelCase											=   _msd_radix_sort(_UpperCamelCase ,   bit_position - 1				)
					_UpperCamelCase											=   _msd_radix_sort(_UpperCamelCase ,   bit_position - 1				)

					# recombine lists
					_UpperCamelCase											=   zeros
					res.extend(_UpperCamelCase				)

					return res



def  lowercase__(		_UpperCamelCase  : list[int]				)->   Union[str, Any]:






					"""simple docstring"""

					_UpperCamelCase											=   len(_UpperCamelCase				)
					if not list_of_ints or length == 1:
										return

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

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



def  lowercase__(		_UpperCamelCase  : list[int] ,   _UpperCamelCase  : int ,   _UpperCamelCase  : int ,   _UpperCamelCase  : int				)->   Tuple:






					"""simple docstring"""

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

					bit_position -= 1

					_UpperCamelCase											=   begin_index
					_UpperCamelCase											=   end_index - 1
					while i <= j:
										_UpperCamelCase											=   False
										if not (list_of_ints[i] >> bit_position) & 1:
															# found zero at the beginning
															i += 1
															_UpperCamelCase											=   True
										if (list_of_ints[j] >> bit_position) & 1:
															# found one at the end
															j -= 1
															_UpperCamelCase											=   True

										if changed:
															continue

										_UpperCamelCase				,	_UpperCamelCase											=   list_of_ints[j], list_of_ints[i]
										j -= 1
										if j != i:
															i += 1

					_msd_radix_sort_inplace(_UpperCamelCase ,   _UpperCamelCase ,   _UpperCamelCase ,   _UpperCamelCase				)
					_msd_radix_sort_inplace(_UpperCamelCase ,   _UpperCamelCase ,   _UpperCamelCase ,   _UpperCamelCase				)


if __name__ == "__main__":
			import doctest

			doctest.testmod()



