




from __future__ import annotations

from queue import Queue







def 		lowercase__ (		__A:						int				,__A:						int					):





					'''simple docstring'''






					a ^= b
					b ^= a
					a ^= b
					return a, b







def 		lowercase__ (		__A:						int				,__A:						list[list[int]]					):





					'''simple docstring'''






					__magic_name__				:						Dict     				=				1
					while (1 << j) < max_node:
										for i in range(1				,max_node + 1					):
															__magic_name__				:						Tuple     				=				parent[j - 1][parent[j - 1][i]]
										j += 1
					return parent







def 		lowercase__ (		__A:						int				,__A:						int				,__A:						list[int]				,__A:						list[list[int]]					):





					'''simple docstring'''






					if level[u] < level[v]:
										__magic_name__				,     __magic_name__				:						List[str]     				=				swap(__A				,__A					)
					# making depth of u same as depth of v
					for i in range(1_8				,-1				,-1					):
										if level[u] - (1 << i) >= level[v]:
															__magic_name__				:						int     				=				parent[i][u]
    # at the same depth if u==v that mean lca is found
					if u == v:
										return u
					# moving both nodes upwards till lca in found
					for i in range(1_8				,-1				,-1					):
										if parent[i][u] not in [0, parent[i][v]]:
															__magic_name__				,     __magic_name__				:						Optional[Any]     				=				parent[i][u], parent[i][v]
    # returning longest common ancestor of u,v
					return parent[0][u]







def 		lowercase__ (		__A:						list[int]				,__A:						list[list[int]]				,__A:						int				,__A:						dict[int, list[int]]				,__A:						int = 1				,):





					'''simple docstring'''






					__magic_name__				:						Dict     				=				0
					__magic_name__				:						Queue[int]     				=				Queue(maxsize=__A					)
					q.put(__A					)
					while q.qsize() != 0:
										__magic_name__				:						List[Any]     				=				q.get()
										for v in graph[u]:
															if level[v] == -1:
																				__magic_name__				:						Optional[Any]     				=				level[u] + 1
																				q.put(__A					)
																				__magic_name__				:						Union[str, Any]     				=				u
					return level, parent







def 		lowercase__ (		):





					'''simple docstring'''






					__magic_name__				:						Optional[Any]     				=				1_3
					# initializing with 0
					__magic_name__				:						Dict     				=				[[0 for _ in range(max_node + 1_0					)] for _ in range(2_0					)]
					# initializing with -1 which means every node is unvisited
					__magic_name__				:						str     				=				[-1 for _ in range(max_node + 1_0					)]
					__magic_name__				:						dict[int, list[int]]     				=				{
					    1: [2, 3, 4],
					    2: [5],
					    3: [6, 7],
					    4: [8],
					    5: [9, 1_0],
					    6: [1_1],
					    7: [],
					    8: [1_2, 1_3],
					    9: [],
					    1_0: [],
					    1_1: [],
					    1_2: [],
					    1_3: [],
					}
					__magic_name__				,     __magic_name__				:						Dict     				=				breadth_first_search(__A				,__A				,__A				,__A				,1					)
					__magic_name__				:						Optional[int]     				=				create_sparse(__A				,__A					)
					print('''LCA of node 1 and 3 is: '''				,lowest_common_ancestor(1				,3				,__A				,__A					)					)
					print('''LCA of node 5 and 6 is: '''				,lowest_common_ancestor(5				,6				,__A				,__A					)					)
					print('''LCA of node 7 and 11 is: '''				,lowest_common_ancestor(7				,1_1				,__A				,__A					)					)
					print('''LCA of node 6 and 7 is: '''				,lowest_common_ancestor(6				,7				,__A				,__A					)					)
					print('''LCA of node 4 and 12 is: '''				,lowest_common_ancestor(4				,1_2				,__A				,__A					)					)
					print('''LCA of node 8 and 8 is: '''				,lowest_common_ancestor(8				,8				,__A				,__A					)					)


if __name__ == "__main__":
	main()









def 		lowercase__ (		__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.'''					)

					__magic_name__				:						Union[str, 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.'''					)

					__magic_name__				:						List[Any]     				=				len(__A					)

					__magic_name__				:						int     				=				0
					__magic_name__				:						list[list[float]]     				=				[]
					# put the x and x to the power values in a matrix
					while count_of_line < x:
										__magic_name__				:						List[Any]     				=				0
										__magic_name__				:						List[str]     				=				coordinates[count_of_line][0]
										__magic_name__				:						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

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

					__magic_name__				:						int     				=				0

					while count < x:
										__magic_name__				:						int     				=				0
										while zahlen < x:
															if count == zahlen:
																				zahlen += 1
															if zahlen == x:
																				break
															__magic_name__				:						Tuple     				=				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

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

					__magic_name__				:						List[Any]     				=				0
					__magic_name__				:						Union[str, Any]     				=				'''f(x)='''

					while count < x:
										__magic_name__				:						list[str]     				=				solution[count].split('''E'''					)
										if len(__A					) > 1:
															__magic_name__				:						Union[str, Any]     				=				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]]))



