

"""simple docstring"""





class        SCREAMING_SNAKE_CASE      :






					"""simple docstring"""



					def __init__( self    :       Tuple     ,lowercase_    :       Tuple	):
						lowerCAmelCase__			:		Union[str, Any]						       =			val
						# Number of nodes in left subtree
						lowerCAmelCase__			:		int						       =			0
						lowerCAmelCase__			:		str						       =			None
						lowerCAmelCase__			:		Tuple						       =			None
						lowerCAmelCase__			:		int						       =			None







					def  __lowerCAmelCase				( self    :       Optional[Any]     ,lowercase_    :       Union[str, Any]	):
						assert self.left_tree_size == other.left_tree_size, "Unequal Sizes of Blocks"

						if self.val < other.val:
							lowerCAmelCase__			:		Optional[int]						       =			self.right
							lowerCAmelCase__			:		List[str]						       =			None
							if self.right:
								lowerCAmelCase__			:		Optional[Any]						       =			other
							lowerCAmelCase__			:		Union[str, Any]						       =			other
							lowerCAmelCase__			:		List[Any]						       =			self.left_tree_size * 2 + 1
							return self
						else:
							lowerCAmelCase__			:		str						       =			other.right
							lowerCAmelCase__			:		Union[str, Any]						       =			None
							if other.right:
								lowerCAmelCase__			:		List[Any]						       =			self
							lowerCAmelCase__			:		List[Any]						       =			self
							lowerCAmelCase__			:		Tuple						       =			other.left_tree_size * 2 + 1
							return other







class        SCREAMING_SNAKE_CASE      :






					"""simple docstring"""



					def __init__( self    :       List[str]     ,lowercase_    :       int=None     ,lowercase_    :       Optional[Any]=None     ,lowercase_    :       Tuple=0	):
						lowerCAmelCase__			:		List[str]						       =			heap_size
						lowerCAmelCase__			:		Any						       =			bottom_root
						lowerCAmelCase__			:		Union[str, Any]						       =			min_node







					def  __lowerCAmelCase				( self    :       Union[str, Any]     ,lowercase_    :       Dict	):

						# Empty heaps corner cases
						if other.size == 0:
							return None
						if self.size == 0:
							lowerCAmelCase__			:		Tuple						       =			other.size
							lowerCAmelCase__			:		List[str]						       =			other.bottom_root
							lowerCAmelCase__			:		Union[str, Any]						       =			other.min_node
							return None
						# Update size
						lowerCAmelCase__			:		Union[str, Any]						       =			self.size + other.size

						# Update min.node
						if self.min_node.val > other.min_node.val:
							lowerCAmelCase__			:		List[Any]						       =			other.min_node
						# Merge

						# Order roots by left_subtree_size
						lowerCAmelCase__			:		int						       =			[]
						lowerCAmelCase__			:		int						       =			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__			:		Union[str, Any]						       =			i.parent
							else:
								combined_roots_list.append((j, False)	)
								lowerCAmelCase__			:		str						       =			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]:
								lowerCAmelCase__			:		Union[str, Any]						       =			combined_roots_list[i + 1][0]
								lowerCAmelCase__			:		Dict						       =			combined_roots_list[i][0]
        # Consecutively merge roots with same left_tree_size
						lowerCAmelCase__			:		Any						       =			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__			:		List[Any]						       =			i.left
								lowerCAmelCase__			:		Dict						       =			i.parent.parent

								# Merging trees
								lowerCAmelCase__			:		List[Any]						       =			i.merge_trees(i.parent	)

								# Updating links
								lowerCAmelCase__			:		List[str]						       =			previous_node
								lowerCAmelCase__			:		Tuple						       =			next_node
								if previous_node:
									lowerCAmelCase__			:		List[str]						       =			i
								if next_node:
									lowerCAmelCase__			:		List[str]						       =			i
							else:
								lowerCAmelCase__			:		Optional[Any]						       =			i.parent
        # Updating self.bottom_root
						while i.left:
							lowerCAmelCase__			:		List[str]						       =			i.left
						lowerCAmelCase__			:		int						       =			i

						# Update other
						lowerCAmelCase__			:		List[Any]						       =			self.size
						lowerCAmelCase__			:		int						       =			self.bottom_root
						lowerCAmelCase__			:		Tuple						       =			self.min_node

						# Return the merged heap
						return self







					def  __lowerCAmelCase				( self    :       Union[str, Any]     ,lowercase_    :       Dict	):
						if self.size == 0:
							lowerCAmelCase__			:		Union[str, Any]						       =			Node(lowercase_	)
							lowerCAmelCase__			:		List[Any]						       =			1
							lowerCAmelCase__			:		Optional[Any]						       =			self.bottom_root
						else:
							# Create new node
							lowerCAmelCase__			:		Any						       =			Node(lowercase_	)

							# Update size
							self.size += 1

							# update min_node
							if val < self.min_node.val:
								lowerCAmelCase__			:		List[Any]						       =			new_node
							# Put new_node as a bottom_root in heap
							lowerCAmelCase__			:		List[str]						       =			new_node
							lowerCAmelCase__			:		int						       =			self.bottom_root
							lowerCAmelCase__			:		List[Any]						       =			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__			:		List[str]						       =			self.bottom_root.parent.parent

								# Merge
								lowerCAmelCase__			:		Optional[int]						       =			self.bottom_root.merge_trees(self.bottom_root.parent	)

								# Update Links
								lowerCAmelCase__			:		str						       =			next_node
								lowerCAmelCase__			:		List[str]						       =			None
								if next_node:
									lowerCAmelCase__			:		str						       =			self.bottom_root







					def  __lowerCAmelCase				( self    :       Optional[int]	):
						return self.min_node.val







					def  __lowerCAmelCase				( self    :       Optional[Any]	):
						return self.size == 0







					def  __lowerCAmelCase				( self    :       Tuple	):
						lowerCAmelCase__			:		Union[str, Any]						       =			self.min_node.val

						# Last element in heap corner case
						if self.size == 1:
							# Update size
							lowerCAmelCase__			:		List[Any]						       =			0

							# Update bottom root
							lowerCAmelCase__			:		Optional[Any]						       =			None

							# Update min_node
							lowerCAmelCase__			:		Dict						       =			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__			:		Tuple						       =			self.bottom_root.parent
							lowerCAmelCase__			:		int						       =			None

							# Update min_node
							lowerCAmelCase__			:		List[str]						       =			self.bottom_root
							lowerCAmelCase__			:		Optional[Any]						       =			self.bottom_root.parent
							while i:
								if i.val < self.min_node.val:
									lowerCAmelCase__			:		List[Any]						       =			i
								lowerCAmelCase__			:		Dict						       =			i.parent
							return min_value
						# General case
						# Find the BinomialHeap of the right subtree of min_node
						lowerCAmelCase__			:		Optional[Any]						       =			self.min_node.right
						lowerCAmelCase__			:		List[str]						       =			None
						lowerCAmelCase__			:		Union[str, Any]						       =			bottom_of_new
						lowerCAmelCase__			:		Any						       =			1

						# Size, min_node and bottom_root
						while bottom_of_new.left:
							lowerCAmelCase__			:		Union[str, Any]						       =			size_of_new * 2 + 1
							lowerCAmelCase__			:		Union[str, Any]						       =			bottom_of_new.left
							if bottom_of_new.val < min_of_new.val:
								lowerCAmelCase__			:		Optional[int]						       =			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__			:		int						       =			size_of_new
							lowerCAmelCase__			:		Union[str, Any]						       =			bottom_of_new
							lowerCAmelCase__			:		Tuple						       =			min_of_new
							# print("Single root, multiple nodes case")
							return min_value
						# Remaining cases
						# Construct heap of right subtree
						lowerCAmelCase__			:		Any						       =			BinomialHeap(
						    bottom_root=lowercase_     ,min_node=lowercase_     ,heap_size=lowercase_	)

						# Update size
						lowerCAmelCase__			:		List[str]						       =			self.size - 1 - size_of_new

						# Neighbour nodes
						lowerCAmelCase__			:		List[Any]						       =			self.min_node.left
						lowerCAmelCase__			:		Optional[int]						       =			self.min_node.parent

						# Initialize new bottom_root and min_node
						lowerCAmelCase__			:		int						       =			previous_node or next_node
						lowerCAmelCase__			:		Optional[Any]						       =			next_node

						# Update links of previous_node and search below for new min_node and
						# bottom_root
						if previous_node:
							lowerCAmelCase__			:		Optional[Any]						       =			next_node

							# Update bottom_root and search for min_node below
							lowerCAmelCase__			:		Optional[Any]						       =			previous_node
							lowerCAmelCase__			:		int						       =			previous_node
							while self.bottom_root.left:
								lowerCAmelCase__			:		Union[str, Any]						       =			self.bottom_root.left
								if self.bottom_root.val < self.min_node.val:
									lowerCAmelCase__			:		str						       =			self.bottom_root
						if next_node:
							lowerCAmelCase__			:		str						       =			previous_node

							# Search for new min_node above min_node
							lowerCAmelCase__			:		Optional[int]						       =			next_node
							while i:
								if i.val < self.min_node.val:
									lowerCAmelCase__			:		Union[str, Any]						       =			i
								lowerCAmelCase__			:		int						       =			i.parent
        # Merge heaps
						self.merge_heaps(lowercase_	)

						return min_value







					def  __lowerCAmelCase				( self    :       int	):
						lowerCAmelCase__			:		Optional[int]						       =			self.bottom_root
						while top_root.parent:
							lowerCAmelCase__			:		List[Any]						       =			top_root.parent
						# preorder
						lowerCAmelCase__			:		Union[str, Any]						       =			[]
						self.__traversal(lowercase_     ,lowercase_	)
						return heap_pre_order







					def  __lowerCAmelCase				( self    :       Union[str, Any]     ,lowercase_    :       Optional[Any]     ,lowercase_    :       str     ,lowercase_    :       Union[str, Any]=0	):
						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    :       str	):
						if self.is_empty():
							return ""
						lowerCAmelCase__			:		str						       =			self.pre_order()

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


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

					doctest.testmod()







"""simple docstring"""





def 						__SCREAMING_SNAKE_CASE						(   A_				):
	lowerCAmelCase__			:		List[Any]						       =			53_81
	for x in s:
		lowerCAmelCase__			:		Tuple						       =			((hash_value << 5) + hash_value) + ord(A_				)
	return hash_value & 0Xf_f_f_f_f_f_f_f




