

'''simple docstring'''







from __future__ import annotations






def 					_SCREAMING_SNAKE_CASE		(						snake_case_	):
	if not list_of_ints:
		return []

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

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






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

	_lowercase							=     []
	_lowercase							=     []
	# 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							=     _msd_radix_sort(snake_case_							,       bit_position - 1	)
	_lowercase							=     _msd_radix_sort(snake_case_							,       bit_position - 1	)

	# recombine lists
	_lowercase							=     zeros
	res.extend(snake_case_	)

	return res






def 					_SCREAMING_SNAKE_CASE		(						snake_case_	):
	_lowercase							=     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							=     max(len(bin(snake_case_	)[2:]	) for x in list_of_ints	)
	_msd_radix_sort_inplace(snake_case_							,       snake_case_							,       0							,       snake_case_	)






def 					_SCREAMING_SNAKE_CASE		(						snake_case_							,       snake_case_							,       snake_case_							,       snake_case_	):
	if bit_position == 0 or end_index - begin_index <= 1:
		return

	bit_position -= 1

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

		if changed:
			continue

		_lowercase		,		_lowercase							=     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'''







class 					__a   :

						def __init__(		self	:						str    ,			lowercase__	:						List[str])						->List[str]:

							"""simple docstring"""



							_lowercase							=     val
							# Number of nodes in left subtree
							_lowercase							=     0
							_lowercase							=     None
							_lowercase							=     None
							_lowercase							=     None


						def 	_UpperCAmelCase							(		self	:						List[Any]    ,			lowercase__	:						Union[str, Any])						->str:

							"""simple docstring"""



							assert self.left_tree_size == other.left_tree_size, "Unequal Sizes of Blocks"

							if self.val < other.val:
								_lowercase							=     self.right
								_lowercase							=     None
								if self.right:
									_lowercase							=     other
								_lowercase							=     other
								_lowercase							=     self.left_tree_size * 2 + 1
								return self
							else:
								_lowercase							=     other.right
								_lowercase							=     None
								if other.right:
									_lowercase							=     self
								_lowercase							=     self
								_lowercase							=     other.left_tree_size * 2 + 1
								return other






class 					__a   :

						def __init__(		self	:						Optional[Any]    ,			lowercase__	:						List[Any]=None    ,			lowercase__	:						Any=None    ,			lowercase__	:						Any=0)						->Optional[int]:

							"""simple docstring"""



							_lowercase							=     heap_size
							_lowercase							=     bottom_root
							_lowercase							=     min_node

						def 	_UpperCAmelCase							(		self	:						Dict    ,			lowercase__	:						Union[str, Any])						->Any:

							"""simple docstring"""



							if other.size == 0:
								return None
							if self.size == 0:
								_lowercase							=     other.size
								_lowercase							=     other.bottom_root
								_lowercase							=     other.min_node
								return None
							# Update size
							_lowercase							=     self.size + other.size

							# Update min.node
							if self.min_node.val > other.min_node.val:
								_lowercase							=     other.min_node
							# Merge

							# Order roots by left_subtree_size
							_lowercase							=     []
							_lowercase		,		_lowercase							=     self.bottom_root, other.bottom_root
							while i or j:
								if i and ((not j) or i.left_tree_size < j.left_tree_size):
									combined_roots_list.append((i, True))
									_lowercase							=     i.parent
								else:
									combined_roots_list.append((j, False))
									_lowercase							=     j.parent
        # Insert links between them
							for i in range(len(lowercase__) - 1):
								if combined_roots_list[i][1] != combined_roots_list[i + 1][1]:
									_lowercase							=     combined_roots_list[i + 1][0]
									_lowercase							=     combined_roots_list[i][0]
        # Consecutively merge roots with same left_tree_size
							_lowercase							=     combined_roots_list[0][0]
							while i.parent:
								if (
								    (i.left_tree_size == i.parent.left_tree_size) and (not i.parent.parent)
								) or (
								    i.left_tree_size == i.parent.left_tree_size
								    and i.left_tree_size != i.parent.parent.left_tree_size
								):
									# Neighbouring Nodes
									_lowercase							=     i.left
									_lowercase							=     i.parent.parent

									# Merging trees
									_lowercase							=     i.merge_trees(i.parent)

									# Updating links
									_lowercase							=     previous_node
									_lowercase							=     next_node
									if previous_node:
										_lowercase							=     i
									if next_node:
										_lowercase							=     i
								else:
									_lowercase							=     i.parent
        # Updating self.bottom_root
							while i.left:
								_lowercase							=     i.left
							_lowercase							=     i

							# Update other
							_lowercase							=     self.size
							_lowercase							=     self.bottom_root
							_lowercase							=     self.min_node

							# Return the merged heap
							return self

						def 	_UpperCAmelCase							(		self	:						Optional[Any]    ,			lowercase__	:						List[Any])						->List[str]:

							"""simple docstring"""



							if self.size == 0:
								_lowercase							=     Node(lowercase__)
								_lowercase							=     1
								_lowercase							=     self.bottom_root
							else:
								# Create new node
								_lowercase							=     Node(lowercase__)

								# Update size
								self.size += 1

								# update min_node
								if val < self.min_node.val:
									_lowercase							=     new_node
								# Put new_node as a bottom_root in heap
								_lowercase							=     new_node
								_lowercase							=     self.bottom_root
								_lowercase							=     new_node

								# Consecutively merge roots with same left_tree_size
								while (
								    self.bottom_root.parent
								    and self.bottom_root.left_tree_size
								    == self.bottom_root.parent.left_tree_size
								):
									# Next node
									_lowercase							=     self.bottom_root.parent.parent

									# Merge
									_lowercase							=     self.bottom_root.merge_trees(self.bottom_root.parent)

									# Update Links
									_lowercase							=     next_node
									_lowercase							=     None
									if next_node:
										_lowercase							=     self.bottom_root

						def 	_UpperCAmelCase							(		self	:						Optional[Any])						->List[str]:

							"""simple docstring"""



							return self.min_node.val

						def 	_UpperCAmelCase							(		self	:						Optional[int])						->Tuple:

							"""simple docstring"""



							return self.size == 0

						def 	_UpperCAmelCase							(		self	:						str)						->List[str]:

							"""simple docstring"""



							_lowercase							=     self.min_node.val

							# Last element in heap corner case
							if self.size == 1:
								# Update size
								_lowercase							=     0

								# Update bottom root
								_lowercase							=     None

								# Update min_node
								_lowercase							=     None

								return min_value
							# No right subtree corner case
							# The structure of the tree implies that this should be the bottom root
							# and there is at least one other root
							if self.min_node.right is None:
								# Update size
								self.size -= 1

								# Update bottom root
								_lowercase							=     self.bottom_root.parent
								_lowercase							=     None

								# Update min_node
								_lowercase							=     self.bottom_root
								_lowercase							=     self.bottom_root.parent
								while i:
									if i.val < self.min_node.val:
										_lowercase							=     i
									_lowercase							=     i.parent
								return min_value
							# General case
							# Find the BinomialHeap of the right subtree of min_node
							_lowercase							=     self.min_node.right
							_lowercase							=     None
							_lowercase							=     bottom_of_new
							_lowercase							=     1

							# Size, min_node and bottom_root
							while bottom_of_new.left:
								_lowercase							=     size_of_new * 2 + 1
								_lowercase							=     bottom_of_new.left
								if bottom_of_new.val < min_of_new.val:
									_lowercase							=     bottom_of_new
        # Corner case of single root on top left path
							if (not self.min_node.left) and (not self.min_node.parent):
								_lowercase							=     size_of_new
								_lowercase							=     bottom_of_new
								_lowercase							=     min_of_new
								# print("Single root, multiple nodes case")
								return min_value
							# Remaining cases
							# Construct heap of right subtree
							_lowercase							=     BinomialHeap(
							    bottom_root=lowercase__    ,			min_node=lowercase__    ,			heap_size=lowercase__)

							# Update size
							_lowercase							=     self.size - 1 - size_of_new

							# Neighbour nodes
							_lowercase							=     self.min_node.left
							_lowercase							=     self.min_node.parent

							# Initialize new bottom_root and min_node
							_lowercase							=     previous_node or next_node
							_lowercase							=     next_node

							# Update links of previous_node and search below for new min_node and
							# bottom_root
							if previous_node:
								_lowercase							=     next_node

								# Update bottom_root and search for min_node below
								_lowercase							=     previous_node
								_lowercase							=     previous_node
								while self.bottom_root.left:
									_lowercase							=     self.bottom_root.left
									if self.bottom_root.val < self.min_node.val:
										_lowercase							=     self.bottom_root
							if next_node:
								_lowercase							=     previous_node

								# Search for new min_node above min_node
								_lowercase							=     next_node
								while i:
									if i.val < self.min_node.val:
										_lowercase							=     i
									_lowercase							=     i.parent
        # Merge heaps
							self.merge_heaps(lowercase__)

							return min_value

						def 	_UpperCAmelCase							(		self	:						List[str])						->str:

							"""simple docstring"""



							_lowercase							=     self.bottom_root
							while top_root.parent:
								_lowercase							=     top_root.parent
							# preorder
							_lowercase							=     []
							self.__traversal(lowercase__    ,			lowercase__)
							return heap_pre_order

						def 	_UpperCAmelCase							(		self	:						Dict    ,			lowercase__	:						Union[str, Any]    ,			lowercase__	:						List[Any]    ,			lowercase__	:						Optional[int]=0)						->int:

							"""simple docstring"""



							if curr_node:
								preorder.append((curr_node.val, level))
								self.__traversal(curr_node.left    ,			lowercase__    ,			level + 1)
								self.__traversal(curr_node.right    ,			lowercase__    ,			level + 1)
							else:
								preorder.append(("""#""", level))


						def __str__(		self	:						Optional[Any])						->str:

							"""simple docstring"""



							if self.is_empty():
								return ""
							_lowercase							=     self.pre_order()

							return "\n".join(("""-""" * level + str(lowercase__)) for value, level in preorder_heap)


# Unit Tests
if __name__ == "__main__":
							import doctest

							doctest.testmod()




