



'''simple docstring'''




from __future__ import annotations


def     lowerCAmelCase_       (     _lowerCamelCase:       list[int] ):
		if not list_of_ints:
				return []

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

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


def     lowerCAmelCase_       (     _lowerCamelCase:       list[int]	,				_lowerCamelCase:       int ):
		if bit_position == 0 or len(_lowerCamelCase ) in [0, 1]:
				return list_of_ints

		__SCREAMING_SNAKE_CASE	:		List[Any]   	=		[]
		__SCREAMING_SNAKE_CASE	:		Dict   	=		[]
		# 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(_lowerCamelCase )
				else:
						# number has a zero at bit bit_position
						zeros.append(_lowerCamelCase )

    # recursively split both lists further
		__SCREAMING_SNAKE_CASE	:		List[Any]   	=		_msd_radix_sort(_lowerCamelCase	,				bit_position - 1 )
		__SCREAMING_SNAKE_CASE	:		str   	=		_msd_radix_sort(_lowerCamelCase	,				bit_position - 1 )

		# recombine lists
		__SCREAMING_SNAKE_CASE	:		str   	=		zeros
		res.extend(_lowerCamelCase )

		return res


def     lowerCAmelCase_       (     _lowerCamelCase:       list[int] ):
		__SCREAMING_SNAKE_CASE	:		Union[str, Any]   	=		len(_lowerCamelCase )
		if not list_of_ints or length == 1:
				return

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

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


def     lowerCAmelCase_       (     _lowerCamelCase:       list[int]	,				_lowerCamelCase:       int	,				_lowerCamelCase:       int	,				_lowerCamelCase:       int ):
		if bit_position == 0 or end_index - begin_index <= 1:
				return

		bit_position -= 1

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

				if changed:
						continue

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

		_msd_radix_sort_inplace(_lowerCamelCase	,				_lowerCamelCase	,				_lowerCamelCase	,				_lowerCamelCase )
		_msd_radix_sort_inplace(_lowerCamelCase	,				_lowerCamelCase	,				_lowerCamelCase	,				_lowerCamelCase )


if __name__ == "__main__":
						import doctest

						doctest.testmod()





'''simple docstring'''




from __future__ import annotations

import random
from collections.abc import Iterable
from typing import Any, Generic, TypeVar

UpperCamelCase__	:     str    =			TypeVar('''T''', bound=bool)






class 						_UpperCamelCase  (							Generic[T]   ):


			'''simple docstring'''

			def __init__(							self					:			str							,					lowerCAmelCase__					:			T						):





					"""simple docstring"""






					__SCREAMING_SNAKE_CASE	:		T   	=		value
					__SCREAMING_SNAKE_CASE	:		RandomizedHeapNode[T] | None   	=		None
					__SCREAMING_SNAKE_CASE	:		RandomizedHeapNode[T] | None   	=		None

			@property
			def UpperCamelCase__ (							self					:			Any						):





					"""simple docstring"""






					return self._value



			@staticmethod
			def UpperCamelCase__ (							lowerCAmelCase__					:			RandomizedHeapNode[T] | None							,					lowerCAmelCase__					:			RandomizedHeapNode[T] | None						):





					"""simple docstring"""






					if not roota:
							return roota

					if not roota:
							return roota

					if roota.value > roota.value:
							__SCREAMING_SNAKE_CASE	,						__SCREAMING_SNAKE_CASE	:		List[str]   	=		roota, roota

					if random.choice([True, False]						):
							__SCREAMING_SNAKE_CASE	,						__SCREAMING_SNAKE_CASE	:		Tuple   	=		roota.right, roota.left

					__SCREAMING_SNAKE_CASE	:		Tuple   	=		RandomizedHeapNode.merge(roota.left							,					lowerCAmelCase__						)

					return roota






class 						_UpperCamelCase  (							Generic[T]   ):


			'''simple docstring'''

			def __init__(							self					:			Optional[int]							,					lowerCAmelCase__					:			Iterable[T] | None = ()						):





					"""simple docstring"""






					__SCREAMING_SNAKE_CASE	:		RandomizedHeapNode[T] | None   	=		None

					if data:
							for item in data:
									self.insert(lowerCAmelCase__						)

			def UpperCamelCase__ (							self					:			Dict							,					lowerCAmelCase__					:			T						):





					"""simple docstring"""






					__SCREAMING_SNAKE_CASE	:		Dict   	=		RandomizedHeapNode.merge(self._root							,					RandomizedHeapNode(lowerCAmelCase__						)						)

			def UpperCamelCase__ (							self					:			Union[str, Any]						):





					"""simple docstring"""






					__SCREAMING_SNAKE_CASE	:		Tuple   	=		self.top()

					if self._root is None:
							return None

					__SCREAMING_SNAKE_CASE	:		Union[str, Any]   	=		RandomizedHeapNode.merge(self._root.left							,					self._root.right						)

					return result

			def UpperCamelCase__ (							self					:			Union[str, Any]						):





					"""simple docstring"""






					if not self._root:
							raise IndexError("""Can't get top element for the empty heap."""						)
					return self._root.value

			def UpperCamelCase__ (							self					:			List[Any]						):





					"""simple docstring"""






					__SCREAMING_SNAKE_CASE	:		List[str]   	=		None

			def UpperCamelCase__ (							self					:			Optional[Any]						):





					"""simple docstring"""






					__SCREAMING_SNAKE_CASE	:		int   	=		[]
					while self:
							result.append(self.pop()						)

					return result



			def __bool__(							self					:			List[Any]						):





					"""simple docstring"""






					return self._root is not None


if __name__ == "__main__":
						import doctest

						doctest.testmod()