






"""simple docstring"""





class 		_a       :





   '''simple docstring'''






   def __init__(   self			,				UpperCAmelCase_)	-> Tuple:




          '''simple docstring'''




          lowercase__:    str        =      val
          # Number of nodes in left subtree
          lowercase__:    Optional[int]        =      0
          lowercase__:    Optional[Any]        =      None
          lowercase__:    Union[str, Any]        =      None
          lowercase__:    str        =      None





   def __lowercase    (   self			,				UpperCAmelCase_)	-> Any:




          '''simple docstring'''




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

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







class 		_a       :





   '''simple docstring'''






   def __init__(   self			,				UpperCAmelCase_=None			,				UpperCAmelCase_=None			,				UpperCAmelCase_=0)	-> Any:




          '''simple docstring'''




          lowercase__:    int        =      heap_size
          lowercase__:    Dict        =      bottom_root
          lowercase__:    Optional[Any]        =      min_node





   def __lowercase    (   self			,				UpperCAmelCase_)	-> List[Any]:




          '''simple docstring'''




          if other.size == 0:
                 return None
          if self.size == 0:
                 lowercase__:    str        =      other.size
                 lowercase__:    Optional[int]        =      other.bottom_root
                 lowercase__:    int        =      other.min_node
                 return None
          # Update size
          lowercase__:    List[str]        =      self.size + other.size

          # Update min.node
          if self.min_node.val > other.min_node.val:
                 lowercase__:    Optional[int]        =      other.min_node
          # Merge

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

                        # Merging trees
                        lowercase__:    Optional[Any]        =      i.merge_trees(i.parent)

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

          # Update other
          lowercase__:    str        =      self.size
          lowercase__:    Optional[Any]        =      self.bottom_root
          lowercase__:    str        =      self.min_node

          # Return the merged heap
          return self





   def __lowercase    (   self			,				UpperCAmelCase_)	-> Optional[int]:




          '''simple docstring'''




          if self.size == 0:
                 lowercase__:    str        =      Node(UpperCAmelCase_)
                 lowercase__:    Union[str, Any]        =      1
                 lowercase__:    Optional[Any]        =      self.bottom_root
          else:
                 # Create new node
                 lowercase__:    Tuple        =      Node(UpperCAmelCase_)

                 # Update size
                 self.size += 1

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

                        # Merge
                        lowercase__:    Any        =      self.bottom_root.merge_trees(self.bottom_root.parent)

                        # Update Links
                        lowercase__:    str        =      next_node
                        lowercase__:    List[str]        =      None
                        if next_node:
                               lowercase__:    Union[str, Any]        =      self.bottom_root





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




          '''simple docstring'''




          return self.min_node.val





   def __lowercase    (   self)	-> Union[str, Any]:




          '''simple docstring'''




          return self.size == 0





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




          '''simple docstring'''




          lowercase__:    Union[str, Any]        =      self.min_node.val

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

                 # Update bottom root
                 lowercase__:    Any        =      None

                 # Update min_node
                 lowercase__:    int        =      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__:    Dict        =      self.bottom_root.parent
                 lowercase__:    Optional[int]        =      None

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

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

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

          # Neighbour nodes
          lowercase__:    List[Any]        =      self.min_node.left
          lowercase__:    Any        =      self.min_node.parent

          # Initialize new bottom_root and min_node
          lowercase__:    List[str]        =      previous_node or next_node
          lowercase__:    int        =      next_node

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

                 # Update bottom_root and search for min_node below
                 lowercase__:    Dict        =      previous_node
                 lowercase__:    Any        =      previous_node
                 while self.bottom_root.left:
                        lowercase__:    Dict        =      self.bottom_root.left
                        if self.bottom_root.val < self.min_node.val:
                               lowercase__:    str        =      self.bottom_root
          if next_node:
                 lowercase__:    Tuple        =      previous_node

                 # Search for new min_node above min_node
                 lowercase__:    Dict        =      next_node
                 while i:
                        if i.val < self.min_node.val:
                               lowercase__:    List[Any]        =      i
                        lowercase__:    Tuple        =      i.parent
        # Merge heaps
          self.merge_heaps(UpperCAmelCase_)

          return min_value





   def __lowercase    (   self)	-> Optional[Any]:




          '''simple docstring'''




          lowercase__:    Optional[int]        =      self.bottom_root
          while top_root.parent:
                 lowercase__:    Tuple        =      top_root.parent
          # preorder
          lowercase__:    int        =      []
          self.__traversal(UpperCAmelCase_			,				UpperCAmelCase_)
          return heap_pre_order





   def __lowercase    (   self			,				UpperCAmelCase_			,				UpperCAmelCase_			,				UpperCAmelCase_=0)	-> Dict:




          '''simple docstring'''




          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)	-> str:




          '''simple docstring'''




          if self.is_empty():
                 return ""
          lowercase__:    Union[str, Any]        =      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"""





class 		_a       :





   '''simple docstring'''






   def __init__(   self			,				UpperCAmelCase_=None)	-> List[Any]:




          '''simple docstring'''




          lowercase__:    str        =      data
          lowercase__:    Any        =      None





   def __repr__(   self)	-> Optional[Any]:




          '''simple docstring'''




          lowercase__:    Any        =      ""
          lowercase__:    Dict        =      self
          while temp:
                 string_rep += F"""<{temp.data}> ---> """
                 lowercase__:    Tuple        =      temp.next
          string_rep += "<END>"
          return string_rep







def     A(  snake_case_		):
       """simple docstring"""





       if not elements_list:
              raise Exception("The Elements List is empty"		)

       # Set first element as Head
       lowercase__:    Any        =      Node(elements_list[0]		)
       lowercase__:    Optional[int]        =      head
       # Loop through elements from position 1
       for data in elements_list[1:]:
              lowercase__:    Dict        =      Node(snake_case_		)
              lowercase__:    Dict        =      current.next
       return head


UpperCamelCase			=      [1, 3, 5, 32, 44, 12, 43]
print(F"List: {list_data}")
print("""Creating Linked List from List.""")
UpperCamelCase			=      make_linked_list(list_data)
print("""Linked List:""")







print(linked_list)






