






"""simple docstring"""


def 			snake_case  (A_      :list[list[int]]      ):







  '''simple docstring'''

  if len(A_      ) == 0 or not all(len(A_      ) == 2 for pair in coordinates      ):
    raise ValueError('The program cannot work out a fitting polynomial.'      )

  if len({tuple(A_      ) for pair in coordinates}      ) != len(A_      ):
    raise ValueError('The program cannot work out a fitting polynomial.'      )

  a     :			Optional[Any]				=   {x for x, _ in coordinates}
  if len(A_      ) == 1:
    return f'''x={coordinates[0][0]}'''

  if len(A_      ) != len(A_      ):
    raise ValueError('The program cannot work out a fitting polynomial.'      )

  a     :			List[Any]				=   len(A_      )

  a     :			Any				=   0
  a     :			list[list[float]]				=   []
  # put the x and x to the power values in a matrix
  while count_of_line < x:
    a     :			Optional[int]				=   0
    a     :			Dict				=   coordinates[count_of_line][0]
    a     :			list[float]				=   []
    while count_in_line < x:
      count_line.append(a ** (x - (count_in_line + 1))      )
      count_in_line += 1
    matrix.append(A_      )
    count_of_line += 1

  a     :			Optional[Any]				=   0
  # put the y values into a vector
  a     :			list[float]				=   []
  while count_of_line < x:
    vector.append(coordinates[count_of_line][1]      )
    count_of_line += 1

  a     :			List[str]				=   0

  while count < x:
    a     :			Tuple				=   0
    while zahlen < x:
      if count == zahlen:
        zahlen += 1
      if zahlen == x:
        break
      a     :			Dict				=   matrix[zahlen][count] / matrix[count][count]
      for counting_columns, item in enumerate(matrix[count]      ):
        # manipulating all the values in the matrix
        matrix[zahlen][counting_columns] -= item * bruch
      # manipulating the values in the vector
      vector[zahlen] -= vector[count] * bruch
      zahlen += 1
    count += 1

  a     :			Dict				=   0
  # make solutions
  a     :			list[str]				=   []
  while count < x:
    solution.append(str(vector[count] / matrix[count][count]      )      )
    count += 1

  a     :			Any				=   0
  a     :			Any				=   'f(x)='

  while count < x:
    a     :			list[str]				=   solution[count].split('E'      )
    if len(A_      ) > 1:
      a     :			str				=   f'''{remove_e[0]}*10^{remove_e[1]}'''
    solved += f'''x^{x - (count + 1)}*{solution[count]}'''
    if count + 1 != x:
      solved += "+"
    count += 1

  return solved


if __name__ == "__main__":
   print(points_to_polynomial([]))
   print(points_to_polynomial([[]]))
   print(points_to_polynomial([[1, 0], [2, 0], [3, 0]]))
   print(points_to_polynomial([[1, 1], [2, 1], [3, 1]]))
   print(points_to_polynomial([[1, 3], [2, 3], [3, 3]]))
   print(points_to_polynomial([[1, 1], [2, 2], [3, 3]]))
   print(points_to_polynomial([[1, 1], [2, 4], [3, 9]]))
   print(points_to_polynomial([[1, 3], [2, 6], [3, 11]]))
   print(points_to_polynomial([[1, -3], [2, -6], [3, -11]]))
   print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))








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 						UpperCamelCase__     (	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 						UpperCamelCase__     (	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 						UpperCamelCase__     (	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 						UpperCamelCase__     (	self					):
       return self.min_node.val



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



 def 						UpperCamelCase__     (	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 						UpperCamelCase__     (	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 						UpperCamelCase__     (	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()