
from __future__ import annotations

import math
import random
from typing import Any







class 				lowercase		:




							def __init__(				self):
												lowercase							=						[]
												lowercase							=						0
												lowercase							=						0




							def  A__      (				self):
												return self.head == self.tail




							def  A__      (				self   ,A__):
												self.data.append(A__)
												lowercase							=						self.tail + 1




							def  A__      (				self):
												lowercase							=						self.data[self.head]
												lowercase							=						self.head + 1
												return ret




							def  A__      (				self):
												return self.tail - self.head





							def  A__      (				self):
												print(self.data)
												print('''**************''')
												print(self.data[self.head : self.tail])







class 				lowercase		:




							def __init__(				self   ,A__):
												lowercase							=						data
												lowercase							=						None
												lowercase							=						None
												lowercase							=						1




							def  A__      (				self):
												return self.data




							def  A__      (				self):
												return self.left




							def  A__      (				self):
												return self.right




							def  A__      (				self):
												return self.height




							def  A__      (				self   ,A__):
												lowercase							=						data




							def  A__      (				self   ,A__):
												lowercase							=						node




							def  A__      (				self   ,A__):
												lowercase							=						node





							def  A__      (				self   ,A__):
												lowercase							=						height



def        UpperCamelCase    (							lowerCAmelCase__		):
					'''simple docstring'''
					if node is None:
										return 0
					return node.get_height()



def        UpperCamelCase    (							lowerCAmelCase__ ,			lowerCAmelCase__		):
					'''simple docstring'''
					if a > b:
										return a
					return b



def        UpperCamelCase    (							lowerCAmelCase__		):
					'''simple docstring'''
					print('''left rotation node:''' ,			node.get_data()		)
					lowercase							=						node.get_left()
					assert ret is not None
					node.set_left(ret.get_right()		)
					ret.set_right(lowerCAmelCase__		)
					lowercase							=						my_max(get_height(node.get_right()		) ,			get_height(node.get_left()		)		) + 1
					node.set_height(lowerCAmelCase__		)
					lowercase							=						my_max(get_height(ret.get_right()		) ,			get_height(ret.get_left()		)		) + 1
					ret.set_height(lowerCAmelCase__		)
					return ret



def        UpperCamelCase    (							lowerCAmelCase__		):
					'''simple docstring'''
					print('''right rotation node:''' ,			node.get_data()		)
					lowercase							=						node.get_right()
					assert ret is not None
					node.set_right(ret.get_left()		)
					ret.set_left(lowerCAmelCase__		)
					lowercase							=						my_max(get_height(node.get_right()		) ,			get_height(node.get_left()		)		) + 1
					node.set_height(lowerCAmelCase__		)
					lowercase							=						my_max(get_height(ret.get_right()		) ,			get_height(ret.get_left()		)		) + 1
					ret.set_height(lowerCAmelCase__		)
					return ret



def        UpperCamelCase    (							lowerCAmelCase__		):
					'''simple docstring'''
					lowercase							=						node.get_left()
					assert left_child is not None
					node.set_left(left_rotation(lowerCAmelCase__		)		)
					return right_rotation(lowerCAmelCase__		)



def        UpperCamelCase    (							lowerCAmelCase__		):
					'''simple docstring'''
					lowercase							=						node.get_right()
					assert right_child is not None
					node.set_right(right_rotation(lowerCAmelCase__		)		)
					return left_rotation(lowerCAmelCase__		)



def        UpperCamelCase    (							lowerCAmelCase__ ,			lowerCAmelCase__		):
					'''simple docstring'''
					if node is None:
										return MyNode(lowerCAmelCase__		)
					if data < node.get_data():
										node.set_left(insert_node(node.get_left() ,			lowerCAmelCase__		)		)
										if (
										    get_height(node.get_left()		) - get_height(node.get_right()		) == 2
										):  # an unbalance detected
															lowercase							=						node.get_left()
															assert left_child is not None
															if (
															    data < left_child.get_data()
															):  # new node is the left child of the left child
																				lowercase							=						right_rotation(lowerCAmelCase__		)
															else:
																				lowercase							=						lr_rotation(lowerCAmelCase__		)
					else:
										node.set_right(insert_node(node.get_right() ,			lowerCAmelCase__		)		)
										if get_height(node.get_right()		) - get_height(node.get_left()		) == 2:
															lowercase							=						node.get_right()
															assert right_child is not None
															if data < right_child.get_data():
																				lowercase							=						rl_rotation(lowerCAmelCase__		)
															else:
																				lowercase							=						left_rotation(lowerCAmelCase__		)
					lowercase							=						my_max(get_height(node.get_right()		) ,			get_height(node.get_left()		)		) + 1
					node.set_height(lowerCAmelCase__		)
					return node



def        UpperCamelCase    (							lowerCAmelCase__		):
					'''simple docstring'''
					while True:
										lowercase							=						root.get_right()
										if right_child is None:
															break
										lowercase							=						right_child
					return root.get_data()



def        UpperCamelCase    (							lowerCAmelCase__		):
					'''simple docstring'''
					while True:
										lowercase							=						root.get_left()
										if left_child is None:
															break
										lowercase							=						left_child
					return root.get_data()



def        UpperCamelCase    (							lowerCAmelCase__ ,			lowerCAmelCase__		):
					'''simple docstring'''
					lowercase							=						root.get_left()
					lowercase							=						root.get_right()
					if root.get_data() == data:
										if left_child is not None and right_child is not None:
															lowercase							=						get_left_most(lowerCAmelCase__		)
															root.set_data(lowerCAmelCase__		)
															root.set_right(del_node(lowerCAmelCase__ ,			lowerCAmelCase__		)		)
										elif left_child is not None:
															lowercase							=						left_child
										elif right_child is not None:
															lowercase							=						right_child
										else:
															return None
					elif root.get_data() > data:
										if left_child is None:
															print('''No such data'''		)
															return root
										else:
															root.set_left(del_node(lowerCAmelCase__ ,			lowerCAmelCase__		)		)
					else:  # root.get_data() < data
										if right_child is None:
															return root
										else:
															root.set_right(del_node(lowerCAmelCase__ ,			lowerCAmelCase__		)		)

					if get_height(lowerCAmelCase__		) - get_height(lowerCAmelCase__		) == 2:
										assert right_child is not None
										if get_height(right_child.get_right()		) > get_height(right_child.get_left()		):
															lowercase							=						left_rotation(lowerCAmelCase__		)
										else:
															lowercase							=						rl_rotation(lowerCAmelCase__		)
					elif get_height(lowerCAmelCase__		) - get_height(lowerCAmelCase__		) == -2:
										assert left_child is not None
										if get_height(left_child.get_left()		) > get_height(left_child.get_right()		):
															lowercase							=						right_rotation(lowerCAmelCase__		)
										else:
															lowercase							=						lr_rotation(lowerCAmelCase__		)
					lowercase							=						my_max(get_height(root.get_right()		) ,			get_height(root.get_left()		)		) + 1
					root.set_height(lowerCAmelCase__		)
					return root







class 				lowercase		:




							def __init__(				self):
												lowercase							=						None




							def  A__      (				self):
												return get_height(self.root)




							def  A__      (				self   ,A__):
												print('''insert:''' + str(A__))
												lowercase							=						insert_node(self.root   ,A__)




							def  A__      (				self   ,A__):
												print('''delete:''' + str(A__))
												if self.root is None:
																	print('''Tree is empty!''')
																	return
												lowercase							=						del_node(self.root   ,A__)





							def __str__(				self   ,):  # a level traversale, gives a more intuitive look on the tree
												lowercase							=						''''''
												lowercase							=						MyQueue()
												q.push(self.root)
												lowercase							=						self.get_height()
												if layer == 0:
																	return output
												lowercase							=						0
												while not q.is_empty():
																	lowercase							=						q.pop()
																	lowercase							=						''' ''' * int(math.pow(2   ,layer - 1))
																	output += space
																	if node is None:
																						output += "*"
																						q.push(A__)
																						q.push(A__)
																	else:
																						output += str(node.get_data())
																						q.push(node.get_left())
																						q.push(node.get_right())
																	output += space
																	lowercase							=						cnt + 1
																	for i in range(1_0_0):
																						if cnt == math.pow(2   ,A__) - 1:
																											lowercase							=						layer - 1
																											if layer == 0:
																																output += "\n*************************************"
																																return output
																											output += "\n"
																											break
												output += "\n*************************************"
												return output



def        UpperCamelCase    (							):
					'''simple docstring'''
					import doctest

					doctest.testmod()


if __name__ == "__main__":
							_test()
							lowercase__		:Union[str, Any]   =    AVLtree()
							lowercase__		:List[str]   =    list(range(10))
							random.shuffle(lst)
							for i in lst:
														t.insert(i)
														print(str(t))
							random.shuffle(lst)
							for i in lst:
														t.del_node(i)
														print(str(t))






import numpy as np


def 						UpperCAmelCase			(	a_		,   a_		,   a_ = 1E-12		,   a_ = 1_0_0		,   )		->	tuple[float, np.ndarray]:







				"""simple docstring"""



				assert np.shape(a_						)[0] == np.shape(a_						)[1]
				# Ensure proper dimensionality.
				assert np.shape(a_						)[0] == np.shape(a_						)[0]
				# Ensure inputs are either both complex or both real
				assert np.iscomplexobj(a_						) == np.iscomplexobj(a_						)
				__A   			= np.iscomplexobj(a_						)
				if is_complex:
								# Ensure complex input_matrix is Hermitian
								assert np.array_equal(a_		,   input_matrix.conj().T						)

				# Set convergence to False. Will define convergence when we exceed max_iterations
				# or when we have small changes from one iteration to next.

				__A   			= False
				__A   			= 0
				__A   			= 0
				__A   			= 1E12

				while not convergence:
								# Multiple matrix by the vector.
								__A   			= np.dot(a_		,   a_						)
								# Normalize the resulting output vector.
								__A   			= w / np.linalg.norm(a_						)
								# Find rayleigh quotient
								# (faster than usual b/c we know vector is normalized already)
								__A   			= vector.conj().T if is_complex else vector.T
								__A   			= np.dot(a_		,   np.dot(a_		,   a_						)						)

								# Check convergence.
								__A   			= np.abs(lambda_ - lambda_previous						) / lambda_
								iterations += 1

								if error <= error_tol or iterations >= max_iterations:
												__A   			= True

								__A   			= lambda_

				if is_complex:
								__A   			= np.real(lambda_						)

				return lambda_, vector


def 						UpperCAmelCase			(	)		->	None:







				"""simple docstring"""



				__A   			= np.array([[4_1, 4, 2_0], [4, 2_6, 3_0], [2_0, 3_0, 5_0]]						)
				__A   			= np.array([4_1, 4, 2_0]						)
				__A   			= real_input_matrix.astype(np.complexaaa						)
				__A   			= np.triu(1J * complex_input_matrix		,   1						)
				complex_input_matrix += imag_matrix
				complex_input_matrix += -1 * imag_matrix.T
				__A   			= np.array([4_1, 4, 2_0]						).astype(np.complexaaa						)

				for problem_type in ["real", "complex"]:
								if problem_type == "real":
												__A   			= real_input_matrix
												__A   			= real_vector
								elif problem_type == "complex":
												__A   			= complex_input_matrix
												__A   			= complex_vector

								# Our implementation.
								__A			,							__A   			= power_iteration(a_		,   a_						)

								# Numpy implementation.

								# Get eigenvalues and eigenvectors using built-in numpy
								# eigh (eigh used for symmetric or hermetian matrices).
								__A			,							__A   			= np.linalg.eigh(a_						)
								# Last eigenvalue is the maximum one.
								__A   			= eigen_values[-1]
								# Last column in this matrix is eigenvector corresponding to largest eigenvalue.
								__A   			= eigen_vectors[:, -1]

								# Check our implementation and numpy gives close answers.
								assert np.abs(eigen_value - eigen_value_max						) <= 1E-6
								# Take absolute values element wise of each eigenvector.
								# as they are only unique to a minus sign.
								assert np.linalg.norm(np.abs(a_						) - np.abs(a_						)						) <= 1E-6


if __name__ == "__main__":
					import doctest

					doctest.testmod()
					test_power_iteration()
