





'''simple docstring'''


from __future__ import annotations

# Define the Gravitational Constant G and the function
lowerCAmelCase      :  int      							=6.6_7_4_3E-1_1  # unit of G : m^3 * kg^-1 * s^-2
def 							UpperCAmelCase_						(		__lowerCamelCase      :      float			,__lowerCamelCase      :      float			,__lowerCamelCase      :      float			,__lowerCamelCase      :      float				):
   lowercase_    :Dict					    =				mass_a * mass_a

   if (force, mass_a, mass_a, distance).count(0				) != 1:
      raise ValueError("One and only one argument must be 0"				)
   if force < 0:
      raise ValueError("Gravitational force can not be negative"				)
   if distance < 0:
      raise ValueError("Distance can not be negative"				)
   if mass_a < 0 or mass_a < 0:
      raise ValueError("Mass can not be negative"				)
   if force == 0:
      lowercase_    :Optional[int]					    =				GRAVITATIONAL_CONSTANT * product_of_mass / (distance**2)
      return {"force": force}
   elif mass_a == 0:
      lowercase_    :Any					    =				(force) * (distance**2) / (GRAVITATIONAL_CONSTANT * mass_a)
      return {"mass_1": mass_a}
   elif mass_a == 0:
      lowercase_    :Any					    =				(force) * (distance**2) / (GRAVITATIONAL_CONSTANT * mass_a)
      return {"mass_2": mass_a}
   elif distance == 0:
      lowercase_    :Any					    =				(GRAVITATIONAL_CONSTANT * product_of_mass / (force)) ** 0.5
      return {"distance": distance}
   raise ValueError("One and only one argument must be 0"				)


# Run doctest
if __name__ == "__main__":
 import doctest

 doctest.testmod()









'''simple docstring'''


class    a_     :





     def __init__(				self	:      str			,   lowercase	:      List[Any]		):





        """simple docstring"""




        lowercase_    :List[str]					    =				val
        # Number of nodes in left subtree
        lowercase_    :Optional[int]					    =				0
        lowercase_    :List[str]					    =				None
        lowercase_    :List[Any]					    =				None
        lowercase_    :Optional[Any]					    =				None







     def 							lowercase__   (				self	:      Union[str, Any]			,   lowercase	:      Dict		):





        """simple docstring"""




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

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







class    a_     :





     def __init__(				self	:      Optional[Any]			,   lowercase	:      Any=None			,   lowercase	:      Optional[Any]=None			,   lowercase	:      int=0		):





        """simple docstring"""




        lowercase_    :Optional[int]					    =				heap_size
        lowercase_    :List[str]					    =				bottom_root
        lowercase_    :Dict					    =				min_node





     def 							lowercase__   (				self	:      List[str]			,   lowercase	:      List[Any]		):





        """simple docstring"""




        if other.size == 0:
           return None
        if self.size == 0:
           lowercase_    :Dict					    =				other.size
           lowercase_    :int					    =				other.bottom_root
           lowercase_    :Tuple					    =				other.min_node
           return None
        # Update size
        lowercase_    :Optional[int]					    =				self.size + other.size

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

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

              # Merging trees
              lowercase_    :Any					    =				i.merge_trees(i.parent		)

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

        # Update other
        lowercase_    :Any					    =				self.size
        lowercase_    :Optional[Any]					    =				self.bottom_root
        lowercase_    :int					    =				self.min_node

        # Return the merged heap
        return self





     def 							lowercase__   (				self	:      int			,   lowercase	:      List[Any]		):





        """simple docstring"""




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

           # Update size
           self.size += 1

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

              # Merge
              lowercase_    :Union[str, Any]					    =				self.bottom_root.merge_trees(self.bottom_root.parent		)

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





     def 							lowercase__   (				self	:      str		):





        """simple docstring"""




        return self.min_node.val





     def 							lowercase__   (				self	:      Optional[int]		):





        """simple docstring"""




        return self.size == 0





     def 							lowercase__   (				self	:      str		):





        """simple docstring"""




        lowercase_    :str					    =				self.min_node.val

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

           # Update bottom root
           lowercase_    :int					    =				None

           # Update min_node
           lowercase_    :Tuple					    =				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_    :Optional[Any]					    =				self.bottom_root.parent
           lowercase_    :Tuple					    =				None

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

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

        # Update size
        lowercase_    :Dict					    =				self.size - 1 - size_of_new

        # Neighbour nodes
        lowercase_    :Optional[Any]					    =				self.min_node.left
        lowercase_    :str					    =				self.min_node.parent

        # Initialize new bottom_root and min_node
        lowercase_    :int					    =				previous_node or next_node
        lowercase_    :Any					    =				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_    :str					    =				previous_node
           lowercase_    :List[Any]					    =				previous_node
           while self.bottom_root.left:
              lowercase_    :List[str]					    =				self.bottom_root.left
              if self.bottom_root.val < self.min_node.val:
                 lowercase_    :List[Any]					    =				self.bottom_root
        if next_node:
           lowercase_    :List[Any]					    =				previous_node

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

        return min_value





     def 							lowercase__   (				self	:      Dict		):





        """simple docstring"""




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





     def 							lowercase__   (				self	:      Union[str, Any]			,   lowercase	:      List[Any]			,   lowercase	:      List[str]			,   lowercase	:      Dict=0		):





        """simple docstring"""




        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	:      Optional[int]		):





        """simple docstring"""




        if self.is_empty():
           return ""
        lowercase_    :int					    =				self.pre_order()

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


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

 doctest.testmod()


