




"""simple docstring"""





class  __lowercase    :


				'''simple docstring'''


				def __init__(				self				:       int ,SCREAMING_SNAKE_CASE_				:       Optional[int] ):
										snake_case__    :   Optional[Any]							     =   val
										# Number of nodes in left subtree
										snake_case__    :   Any							     =   0
										snake_case__    :   str							     =   None
										snake_case__    :   Optional[Any]							     =   None
										snake_case__    :   int							     =   None





				def  _lowerCamelCase   (				self				:       str ,SCREAMING_SNAKE_CASE_				:       Any ):
										assert self.left_tree_size == other.left_tree_size, "Unequal Sizes of Blocks"

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







class  __lowercase    :


				'''simple docstring'''


				def __init__(				self				:       Dict ,SCREAMING_SNAKE_CASE_				:       str=None ,SCREAMING_SNAKE_CASE_				:       List[str]=None ,SCREAMING_SNAKE_CASE_				:       List[Any]=0 ):
										snake_case__    :   Union[str, Any]							     =   heap_size
										snake_case__    :   List[str]							     =   bottom_root
										snake_case__    :   str							     =   min_node

				def  _lowerCamelCase   (				self				:       Any ,SCREAMING_SNAKE_CASE_				:       Any ):

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

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

										# Order roots by left_subtree_size
										snake_case__    :   List[Any]							     =   []
										snake_case__     ,		snake_case__    :   List[str]							     =   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) )
																						snake_case__    :   Dict							     =   i.parent
																else:
																						combined_roots_list.append((j, False) )
																						snake_case__    :   List[Any]							     =   j.parent
        # Insert links between them
										for i in range(len(SCREAMING_SNAKE_CASE_ ) - 1 ):
																if combined_roots_list[i][1] != combined_roots_list[i + 1][1]:
																						snake_case__    :   Optional[int]							     =   combined_roots_list[i + 1][0]
																						snake_case__    :   Any							     =   combined_roots_list[i][0]
        # Consecutively merge roots with same left_tree_size
										snake_case__    :   str							     =   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
																						snake_case__    :   Dict							     =   i.left
																						snake_case__    :   Union[str, Any]							     =   i.parent.parent

																						# Merging trees
																						snake_case__    :   int							     =   i.merge_trees(i.parent )

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

										# Update other
										snake_case__    :   Any							     =   self.size
										snake_case__    :   List[str]							     =   self.bottom_root
										snake_case__    :   str							     =   self.min_node

										# Return the merged heap
										return self

				def  _lowerCamelCase   (				self				:       int ,SCREAMING_SNAKE_CASE_				:       List[str] ):
										if self.size == 0:
																snake_case__    :   int							     =   Node(SCREAMING_SNAKE_CASE_ )
																snake_case__    :   Tuple							     =   1
																snake_case__    :   str							     =   self.bottom_root
										else:
																# Create new node
																snake_case__    :   Optional[int]							     =   Node(SCREAMING_SNAKE_CASE_ )

																# Update size
																self.size += 1

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

																						# Merge
																						snake_case__    :   Optional[Any]							     =   self.bottom_root.merge_trees(self.bottom_root.parent )

																						# Update Links
																						snake_case__    :   Any							     =   next_node
																						snake_case__    :   Dict							     =   None
																						if next_node:
																												snake_case__    :   Tuple							     =   self.bottom_root

				def  _lowerCamelCase   (				self				:       Optional[Any] ):
										return self.min_node.val

				def  _lowerCamelCase   (				self				:       Dict ):
										return self.size == 0

				def  _lowerCamelCase   (				self				:       Any ):
										snake_case__    :   Optional[int]							     =   self.min_node.val

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

																# Update bottom root
																snake_case__    :   Tuple							     =   None

																# Update min_node
																snake_case__    :   List[Any]							     =   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
																snake_case__    :   List[Any]							     =   self.bottom_root.parent
																snake_case__    :   Union[str, Any]							     =   None

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

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

										# Update size
										snake_case__    :   Tuple							     =   self.size - 1 - size_of_new

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

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

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

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

																# Search for new min_node above min_node
																snake_case__    :   Optional[Any]							     =   next_node
																while i:
																						if i.val < self.min_node.val:
																												snake_case__    :   Optional[Any]							     =   i
																						snake_case__    :   Any							     =   i.parent
        # Merge heaps
										self.merge_heaps(SCREAMING_SNAKE_CASE_ )

										return min_value

				def  _lowerCamelCase   (				self				:       Optional[int] ):
										snake_case__    :   List[Any]							     =   self.bottom_root
										while top_root.parent:
																snake_case__    :   str							     =   top_root.parent
										# preorder
										snake_case__    :   Tuple							     =   []
										self.__traversal(SCREAMING_SNAKE_CASE_ ,SCREAMING_SNAKE_CASE_ )
										return heap_pre_order

				def  _lowerCamelCase   (				self				:       int ,SCREAMING_SNAKE_CASE_				:       Optional[int] ,SCREAMING_SNAKE_CASE_				:       Dict ,SCREAMING_SNAKE_CASE_				:       Dict=0 ):
										if curr_node:
																preorder.append((curr_node.val, level) )
																self.__traversal(curr_node.left ,SCREAMING_SNAKE_CASE_ ,level + 1 )
																self.__traversal(curr_node.right ,SCREAMING_SNAKE_CASE_ ,level + 1 )
										else:
																preorder.append(('''#''', level) )





				def __str__(				self				:       int ):
										if self.is_empty():
																return ""
										snake_case__    :   List[Any]							     =   self.pre_order()

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


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

							doctest.testmod()








"""simple docstring"""





def 						lowerCAmelCase							(	lowerCamelCase						):
						'''simple docstring'''


						return int(lowerCamelCase						) if x - int(lowerCamelCase						) <= 0 else int(lowerCamelCase						) + 1


if __name__ == "__main__":
							import doctest

							doctest.testmod()


