




'''simple docstring'''







class 					lowerCamelCase_							:




					'''simple docstring'''






					def __init__(   self,  __UpperCamelCase			):
							_snake_case     	=	val
							# Number of nodes in left subtree
							_snake_case     	=	0
							_snake_case     	=	None
							_snake_case     	=	None
							_snake_case     	=	None






					def     __SCREAMING_SNAKE_CASE     (   self,  __UpperCamelCase			):
							assert self.left_tree_size == other.left_tree_size, "Unequal Sizes of Blocks"

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





class 					lowerCamelCase_							:




					'''simple docstring'''






					def __init__(   self,  __UpperCamelCase=None,  __UpperCamelCase=None,  __UpperCamelCase=0			):
							_snake_case     	=	heap_size
							_snake_case     	=	bottom_root
							_snake_case     	=	min_node






					def     __SCREAMING_SNAKE_CASE     (   self,  __UpperCamelCase			):

							# Empty heaps corner cases
							if other.size == 0:
									return None
							if self.size == 0:
									_snake_case     	=	other.size
									_snake_case     	=	other.bottom_root
									_snake_case     	=	other.min_node
									return None
							# Update size
							_snake_case     	=	self.size + other.size

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

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

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

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

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

							# Return the merged heap
							return self






					def     __SCREAMING_SNAKE_CASE     (   self,  __UpperCamelCase			):
							if self.size == 0:
									_snake_case     	=	Node(__UpperCamelCase			)
									_snake_case     	=	1
									_snake_case     	=	self.bottom_root
							else:
									# Create new node
									_snake_case     	=	Node(__UpperCamelCase			)

									# Update size
									self.size += 1

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

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

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






					def     __SCREAMING_SNAKE_CASE     (   self			):
							return self.min_node.val






					def     __SCREAMING_SNAKE_CASE     (   self			):
							return self.size == 0






					def     __SCREAMING_SNAKE_CASE     (   self			):
							_snake_case     	=	self.min_node.val

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

									# Update bottom root
									_snake_case     	=	None

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

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

							# Size, min_node and bottom_root
							while bottom_of_new.left:
									_snake_case     	=	size_of_new * 2 + 1
									_snake_case     	=	bottom_of_new.left
									if bottom_of_new.val < min_of_new.val:
											_snake_case     	=	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     	=	size_of_new
									_snake_case     	=	bottom_of_new
									_snake_case     	=	min_of_new
									# print("Single root, multiple nodes case")
									return min_value
							# Remaining cases
							# Construct heap of right subtree
							_snake_case     	=	BinomialHeap(
							    bottom_root=__UpperCamelCase,  min_node=__UpperCamelCase,  heap_size=__UpperCamelCase			)

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

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

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

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

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

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

							return min_value






					def     __SCREAMING_SNAKE_CASE     (   self			):
							_snake_case     	=	self.bottom_root
							while top_root.parent:
									_snake_case     	=	top_root.parent
							# preorder
							_snake_case     	=	[]
							self.__traversal(__UpperCamelCase,  __UpperCamelCase			)
							return heap_pre_order






					def     __SCREAMING_SNAKE_CASE     (   self,  __UpperCamelCase,  __UpperCamelCase,  __UpperCamelCase=0			):
							if curr_node:
									preorder.append((curr_node.val, level)			)
									self.__traversal(curr_node.left,  __UpperCamelCase,  level + 1			)
									self.__traversal(curr_node.right,  __UpperCamelCase,  level + 1			)
							else:
									preorder.append(("#", level)			)






					def __str__(   self			):
							if self.is_empty():
									return ""
							_snake_case     	=	self.pre_order()

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


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

						doctest.testmod()











'''simple docstring'''







import math





class 					lowerCamelCase_							:




					'''simple docstring'''






					def __init__(   self,  __UpperCamelCase,  __UpperCamelCase,  __UpperCamelCase			):
							_snake_case     	=	x
							_snake_case     	=	y
							_snake_case     	=	z






					def __repr__(   self			):
							return f'''Point({self.x}, {self.y}, {self.z})'''






def 		_lowercase		(     lowercase_						,							lowercase_  ):



		'''simple docstring'''







		return math.sqrt(abs((b.x - a.x) ** 2 + (b.y - a.y) ** 2 + (b.z - a.z) ** 2  )  )


if __name__ == "__main__":
						import doctest

						doctest.testmod()





