'''simple docstring'''




class     a__:


			'''simple docstring'''






			def __init__(       self	,   __lowerCAmelCase):




					"""simple docstring"""


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







			def 							a_	(       self	,   __lowerCAmelCase):




					"""simple docstring"""


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

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







class     a__:


			'''simple docstring'''






			def __init__(       self	,   __lowerCAmelCase=None	,   __lowerCAmelCase=None	,   __lowerCAmelCase=0):




					"""simple docstring"""


					lowerCAmelCase        =  heap_size
					lowerCAmelCase        =  bottom_root
					lowerCAmelCase        =  min_node



			def 							a_	(       self	,   __lowerCAmelCase):




					"""simple docstring"""


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

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

					# Order roots by left_subtree_size
					lowerCAmelCase        =  []
					lowerCAmelCase ,					lowerCAmelCase        =  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))
									lowerCAmelCase        =  i.parent
							else:
									combined_roots_list.append((j, False))
									lowerCAmelCase        =  j.parent
        # Insert links between them
					for i in range(len(__lowerCAmelCase) - 1):
							if combined_roots_list[i][1] != combined_roots_list[i + 1][1]:
									lowerCAmelCase        =  combined_roots_list[i + 1][0]
									lowerCAmelCase        =  combined_roots_list[i][0]
        # Consecutively merge roots with same left_tree_size
					lowerCAmelCase        =  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
									lowerCAmelCase        =  i.left
									lowerCAmelCase        =  i.parent.parent

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

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

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

					# Return the merged heap
					return self



			def 							a_	(       self	,   __lowerCAmelCase):




					"""simple docstring"""


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

							# Update size
							self.size += 1

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

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

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



			def 							a_	(       self):




					"""simple docstring"""


					return self.min_node.val



			def 							a_	(       self):




					"""simple docstring"""


					return self.size == 0



			def 							a_	(       self):




					"""simple docstring"""


					lowerCAmelCase        =  self.min_node.val

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

							# Update bottom root
							lowerCAmelCase        =  None

							# Update min_node
							lowerCAmelCase        =  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
							lowerCAmelCase        =  self.bottom_root.parent
							lowerCAmelCase        =  None

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

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

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

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

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

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

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

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

					return min_value



			def 							a_	(       self):




					"""simple docstring"""


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



			def 							a_	(       self	,   __lowerCAmelCase	,   __lowerCAmelCase	,   __lowerCAmelCase=0):




					"""simple docstring"""


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







			def __str__(       self):




					"""simple docstring"""


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

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


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

						doctest.testmod()





'''simple docstring'''




from __future__ import annotations

__lowercase				   =  [2, 1, 4, 5, 6, 0, 8, 9, 1, 2, 0, 6, 4, 2, 0, 6, 5, 3, 2, 7]






class     a__:


			'''simple docstring'''






			def __init__(       self	,   __lowerCAmelCase):




					"""simple docstring"""


					lowerCAmelCase        =  -1
					lowerCAmelCase        =  -1
					lowerCAmelCase        =  [-1] * length
					lowerCAmelCase        =  None
					lowerCAmelCase        =  None







			def __repr__(       self):




					"""simple docstring"""


					return f"Node(min_value={self.minn} max_value={self.maxx})"







def     snake_case__							(      _A: list[int]			)   ->     Node | None:

		'''simple docstring'''


		lowerCAmelCase        =  Node(len(_A			)			)
		lowerCAmelCase ,					lowerCAmelCase        =  min(_A			), max(_A			)
		# Leaf node case where the node contains only one unique value
		if root.minn == root.maxx:
				return root
		lowerCAmelCase        =  (root.minn + root.maxx) // 2

		lowerCAmelCase        =  []
		lowerCAmelCase        =  []

		for index, num in enumerate(_A			):
				if num <= pivot:
						left_arr.append(_A			)
				else:
						right_arr.append(_A			)
				lowerCAmelCase        =  len(_A			)
		lowerCAmelCase        =  build_tree(_A			)
		lowerCAmelCase        =  build_tree(_A			)
		return root






def     snake_case__							(      _A: Node | None				,       _A: int				,       _A: int			)   ->     int:

		'''simple docstring'''


		if index < 0 or node is None:
				return 0
		# Leaf node cases
		if node.minn == node.maxx:
				return index + 1 if node.minn == num else 0
		lowerCAmelCase        =  (node.minn + node.maxx) // 2
		if num <= pivot:
				# go the left subtree and map index to the left subtree
				return rank_till_index(node.left				,       _A				,       node.map_left[index] - 1			)
		else:
				# go to the right subtree and map index to the right subtree
				return rank_till_index(node.right				,       _A				,       index - node.map_left[index]			)






def     snake_case__							(      _A: Node | None				,       _A: int				,       _A: int				,       _A: int			)   ->     int:

		'''simple docstring'''


		if start > end:
				return 0
		lowerCAmelCase        =  rank_till_index(_A				,       _A				,       _A			)
		lowerCAmelCase        =  rank_till_index(_A				,       _A				,       start - 1			)
		return rank_till_end - rank_before_start






def     snake_case__							(      _A: Node | None				,       _A: int				,       _A: int				,       _A: int			)   ->     int:

		'''simple docstring'''


		if index > (end - start) or start > end or node is None:
				return -1
		# Leaf node case
		if node.minn == node.maxx:
				return node.minn
		# Number of elements in the left subtree in interval [start, end]
		lowerCAmelCase        =  node.map_left[end] - (
		    node.map_left[start - 1] if start else 0
		)
		if num_elements_in_left_tree > index:
				return quantile(
				    node.left				,       _A				,       (node.map_left[start - 1] if start else 0)				,       node.map_left[end] - 1				,       )
		else:
				return quantile(
				    node.right				,       index - num_elements_in_left_tree				,       start - (node.map_left[start - 1] if start else 0)				,       end - node.map_left[end]				,       )






def     snake_case__							(      _A: Node | None				,       _A: int				,       _A: int				,       _A: int				,       _A: int			)   ->     int:

		'''simple docstring'''


		if (
		    start > end
		    or node is None
		    or start_num > end_num
		    or node.minn > end_num
		    or node.maxx < start_num
		):
				return 0
		if start_num <= node.minn and node.maxx <= end_num:
				return end - start + 1
		lowerCAmelCase        =  range_counting(
		    node.left				,       (node.map_left[start - 1] if start else 0)				,       node.map_left[end] - 1				,       _A				,       _A				,       )
		lowerCAmelCase        =  range_counting(
		    node.right				,       start - (node.map_left[start - 1] if start else 0)				,       end - node.map_left[end]				,       _A				,       _A				,       )
		return left + right


if __name__ == "__main__":
						import doctest

						doctest.testmod()




