

from __future__ import annotations

from random import random
from typing import Generic, TypeVar

lowerCAmelCase_			      =      TypeVar('''KT''')
lowerCAmelCase_			      =      TypeVar('''VT''')





class       snake_case_  (						Generic[KT, VT]   ):







			'''simple docstring'''

			def __init__(	self					:							Optional[Any]				,  _UpperCamelCase					:							KT | str = "root"				,  _UpperCamelCase					:							VT | None = None				) ->Dict:
									snake_case_  =      key
									snake_case_  =      value
									snake_case_  =      []







			def __repr__(	self					:							str				) ->str:
									return f'''Node({self.key}: {self.value})'''







			@property
			def 							snake_case__(	self					:							Dict				) ->int:
									return len(self.forward				)






class       snake_case_  (						Generic[KT, VT]   ):







			'''simple docstring'''

			def __init__(	self					:							int				,  _UpperCamelCase					:							float = 0.5				,  _UpperCamelCase					:							int = 1_6				) ->Union[str, Any]:
									snake_case_  =      Node[KT, VT]()
									snake_case_  =      0
									snake_case_  =      p
									snake_case_  =      max_level







			def __str__(	self					:							List[str]				) ->str:
									snake_case_  =      list(self				)

									if len(_UpperCamelCase				) == 0:
															return f'''SkipList(level={self.level})'''

									snake_case_  =      max((len(str(_UpperCamelCase				)				) for item in items)				,  default=4				)
									snake_case_  =      max(_UpperCamelCase				,  4				) + 4

									snake_case_  =      self.head
									snake_case_  =      []

									snake_case_  =      node.forward.copy()
									lines.append(f'''[{node.key}]'''.ljust(_UpperCamelCase				,  '''-'''				) + '''* ''' * len(_UpperCamelCase				)				)
									lines.append(''' ''' * label_size + '''| ''' * len(_UpperCamelCase				)				)

									while len(node.forward				) != 0:
															snake_case_  =      node.forward[0]

															lines.append(
															    f'''[{node.key}]'''.ljust(_UpperCamelCase				,  '''-'''				)
															    + ''' '''.join(str(n.key				) if n.key == node.key else '''|''' for n in forwards				)				)
															lines.append(''' ''' * label_size + '''| ''' * len(_UpperCamelCase				)				)
															snake_case_  =      node.forward

									lines.append('''None'''.ljust(_UpperCamelCase				) + '''* ''' * len(_UpperCamelCase				)				)
									return f'''SkipList(level={self.level})\n''' + "\n".join(_UpperCamelCase				)







			def __iter__(	self					:							Optional[Any]				) ->List[Any]:
									snake_case_  =      self.head

									while len(node.forward				) != 0:
															yield node.forward[0].key
															snake_case_  =      node.forward[0]







			def 							snake_case__(	self					:							str				) ->int:
									snake_case_  =      1
									while random() < self.p and level < self.max_level:
															level += 1

									return level







			def 							snake_case__(	self					:							str				,  _UpperCamelCase					:							Union[str, Any]				) ->tuple[Node[KT, VT] | None, list[Node[KT, VT]]]:
									snake_case_  =      []

									snake_case_  =      self.head

									for i in reversed(range(self.level				)				):
															# i < node.level - When node level is lesser than `i` decrement `i`.
															# node.forward[i].key < key - Jumping to node with key value higher
															#                             or equal to searched key would result
															#                             in skipping searched key.
															while i < node.level and node.forward[i].key < key:
																					snake_case_  =      node.forward[i]
															# Each leftmost node (relative to searched node) will potentially have to
															# be updated.
															update_vector.append(_UpperCamelCase				)

									update_vector.reverse()  # Note that we were inserting values in reverse order.

									# len(node.forward) != 0 - If current node doesn't contain any further
									#                          references then searched key is not present.
									# node.forward[0].key == key - Next node key should be equal to search key
									#                              if key is present.
									if len(node.forward				) != 0 and node.forward[0].key == key:
															return node.forward[0], update_vector
									else:
															return None, update_vector







			def 							snake_case__(	self					:							List[Any]				,  _UpperCamelCase					:							KT				) ->Any:
									snake_case_,       snake_case_  =      self._locate_node(_UpperCamelCase				)

									if node is not None:
															for i, update_node in enumerate(_UpperCamelCase				):
																					# Remove or replace all references to removed node.
																					if update_node.level > i and update_node.forward[i].key == key:
																											if node.level > i:
																																	snake_case_  =      node.forward[i]
																											else:
																																	snake_case_  =      update_node.forward[:i]







			def 							snake_case__(	self					:							str				,  _UpperCamelCase					:							KT				,  _UpperCamelCase					:							VT				) ->Optional[Any]:
									snake_case_,       snake_case_  =      self._locate_node(_UpperCamelCase				)
									if node is not None:
															snake_case_  =      value
									else:
															snake_case_  =      self.random_level()

															if level > self.level:
																					# After level increase we have to add additional nodes to head.
																					for _ in range(self.level - 1				,  _UpperCamelCase				):
																											update_vector.append(self.head				)
																					snake_case_  =      level

															snake_case_  =      Node(_UpperCamelCase				,  _UpperCamelCase				)

															for i, update_node in enumerate(update_vector[:level]				):
																					# Change references to pass through new node.
																					if update_node.level > i:
																											new_node.forward.append(update_node.forward[i]				)

																					if update_node.level < i + 1:
																											update_node.forward.append(_UpperCamelCase				)
																					else:
																											snake_case_  =      new_node







			def 							snake_case__(	self					:							Union[str, Any]				,  _UpperCamelCase					:							VT				) ->VT | None:
									snake_case_,       snake_case_  =      self._locate_node(_UpperCamelCase				)

									if node is not None:
															return node.value

									return None






def     __SCREAMING_SNAKE_CASE   ():
						snake_case_  =      SkipList()
						skip_list.insert('''Key1'''    ,						3			)
						skip_list.insert('''Key2'''    ,						12			)
						skip_list.insert('''Key3'''    ,						41			)
						skip_list.insert('''Key4'''    ,						-19			)

						snake_case_  =      skip_list.head
						snake_case_  =      {}
						while node.level != 0:
												snake_case_  =      node.forward[0]
												snake_case_  =      node.value

						assert len(SCREAMING_SNAKE_CASE__			) == 4
						assert all_values["Key1"] == 3
						assert all_values["Key2"] == 12
						assert all_values["Key3"] == 41
						assert all_values["Key4"] == -19


def     __SCREAMING_SNAKE_CASE   ():
						snake_case_  =      SkipList()
						skip_list.insert('''Key1'''    ,						10			)
						skip_list.insert('''Key1'''    ,						12			)

						skip_list.insert('''Key5'''    ,						7			)
						skip_list.insert('''Key7'''    ,						10			)
						skip_list.insert('''Key10'''    ,						5			)

						skip_list.insert('''Key7'''    ,						7			)
						skip_list.insert('''Key5'''    ,						5			)
						skip_list.insert('''Key10'''    ,						10			)

						snake_case_  =      skip_list.head
						snake_case_  =      {}
						while node.level != 0:
												snake_case_  =      node.forward[0]
												snake_case_  =      node.value

						if len(SCREAMING_SNAKE_CASE__			) != 4:
												print()
						assert len(SCREAMING_SNAKE_CASE__			) == 4
						assert all_values["Key1"] == 12
						assert all_values["Key7"] == 7
						assert all_values["Key5"] == 5
						assert all_values["Key10"] == 10


def     __SCREAMING_SNAKE_CASE   ():
						snake_case_  =      SkipList()
						assert skip_list.find('''Some key'''			) is None


def     __SCREAMING_SNAKE_CASE   ():
						snake_case_  =      SkipList()

						skip_list.insert('''Key2'''    ,						20			)
						assert skip_list.find('''Key2'''			) == 20

						skip_list.insert('''Some Key'''    ,						10			)
						skip_list.insert('''Key2'''    ,						8			)
						skip_list.insert('''V'''    ,						13			)

						assert skip_list.find('''Y'''			) is None
						assert skip_list.find('''Key2'''			) == 8
						assert skip_list.find('''Some Key'''			) == 10
						assert skip_list.find('''V'''			) == 13


def     __SCREAMING_SNAKE_CASE   ():
						snake_case_  =      SkipList()
						skip_list.delete('''Some key'''			)

						assert len(skip_list.head.forward			) == 0


def     __SCREAMING_SNAKE_CASE   ():
						snake_case_  =      SkipList()

						skip_list.insert('''Key1'''    ,						12			)
						skip_list.insert('''V'''    ,						13			)
						skip_list.insert('''X'''    ,						14			)
						skip_list.insert('''Key2'''    ,						15			)

						skip_list.delete('''V'''			)
						skip_list.delete('''Key2'''			)

						assert skip_list.find('''V'''			) is None
						assert skip_list.find('''Key2'''			) is None


def     __SCREAMING_SNAKE_CASE   ():
						snake_case_  =      SkipList()

						skip_list.insert('''Key1'''    ,						12			)
						skip_list.insert('''V'''    ,						13			)
						skip_list.insert('''X'''    ,						14			)
						skip_list.insert('''Key2'''    ,						15			)

						skip_list.delete('''V'''			)
						assert skip_list.find('''V'''			) is None
						assert skip_list.find('''X'''			) == 14
						assert skip_list.find('''Key1'''			) == 12
						assert skip_list.find('''Key2'''			) == 15

						skip_list.delete('''X'''			)
						assert skip_list.find('''V'''			) is None
						assert skip_list.find('''X'''			) is None
						assert skip_list.find('''Key1'''			) == 12
						assert skip_list.find('''Key2'''			) == 15

						skip_list.delete('''Key1'''			)
						assert skip_list.find('''V'''			) is None
						assert skip_list.find('''X'''			) is None
						assert skip_list.find('''Key1'''			) is None
						assert skip_list.find('''Key2'''			) == 15

						skip_list.delete('''Key2'''			)
						assert skip_list.find('''V'''			) is None
						assert skip_list.find('''X'''			) is None
						assert skip_list.find('''Key1'''			) is None
						assert skip_list.find('''Key2'''			) is None


def     __SCREAMING_SNAKE_CASE   ():
						snake_case_  =      SkipList()

						skip_list.insert('''Key1'''    ,						12			)
						skip_list.insert('''V'''    ,						13			)
						skip_list.insert('''X'''    ,						142			)
						skip_list.insert('''Key2'''    ,						15			)

						skip_list.delete('''X'''			)

						def traverse_keys(SCREAMING_SNAKE_CASE__			):
												yield node.key
												for forward_node in node.forward:
																		yield from traverse_keys(SCREAMING_SNAKE_CASE__			)

						assert len(set(traverse_keys(skip_list.head			)			)			) == 4


def     __SCREAMING_SNAKE_CASE   ():
						def is_sorted(SCREAMING_SNAKE_CASE__			):
												return all(next_item >= item for item, next_item in zip(SCREAMING_SNAKE_CASE__    ,						lst[1:]			)			)

						snake_case_  =      SkipList()
						for i in range(10			):
												skip_list.insert(SCREAMING_SNAKE_CASE__    ,						SCREAMING_SNAKE_CASE__			)
						assert is_sorted(list(SCREAMING_SNAKE_CASE__			)			)
						skip_list.delete(5			)
						skip_list.delete(8			)
						skip_list.delete(2			)
						assert is_sorted(list(SCREAMING_SNAKE_CASE__			)			)
						skip_list.insert(-12    ,						-12			)
						skip_list.insert(77    ,						77			)
						assert is_sorted(list(SCREAMING_SNAKE_CASE__			)			)


def     __SCREAMING_SNAKE_CASE   ():
						for _ in range(100			):
												# Repeat test 100 times due to the probabilistic nature of skip list
												# random values == random bugs
												test_insert()
												test_insert_overrides_existing_value()

												test_searching_empty_list_returns_none()
												test_search()

												test_deleting_item_from_empty_list_do_nothing()
												test_deleted_items_are_not_founded_by_find_method()
												test_delete_removes_only_given_key()
												test_delete_doesnt_leave_dead_nodes()

												test_iter_always_yields_sorted_values()


def     __SCREAMING_SNAKE_CASE   ():
						snake_case_  =      SkipList()
						skip_list.insert(2    ,						'''2'''			)
						skip_list.insert(4    ,						'''4'''			)
						skip_list.insert(6    ,						'''4'''			)
						skip_list.insert(4    ,						'''5'''			)
						skip_list.insert(8    ,						'''4'''			)
						skip_list.insert(9    ,						'''4'''			)

						skip_list.delete(4			)

						print(SCREAMING_SNAKE_CASE__			)


if __name__ == "__main__":
					import doctest

					doctest.testmod()
					main()




import numpy as np







def     UpperCAmelCase_		(					snake_case__    ,			snake_case__    ,			snake_case__ = 1E-12    ,			snake_case__ = 100    ,			)	->     tuple[float, np.ndarray]:



				"""simple docstring"""





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

				lowerCAmelCase__		  =  False
				lowerCAmelCase__		  =  0
				lowerCAmelCase__		  =  0
				lowerCAmelCase__		  =  1E12

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

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

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

								lowerCAmelCase__		  =  lambda_

				if is_complex:
								lowerCAmelCase__		  =  np.real(lambda_					)

				return lambda_, vector







def     UpperCAmelCase_		(					)	->     None:



				"""simple docstring"""





				lowerCAmelCase__		  =  np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]]					)
				lowerCAmelCase__		  =  np.array([41, 4, 20]					)
				lowerCAmelCase__		  =  real_input_matrix.astype(np.complexaaa					)
				lowerCAmelCase__		  =  np.triu(1j * complex_input_matrix    ,			1					)
				complex_input_matrix += imag_matrix
				complex_input_matrix += -1 * imag_matrix.T
				lowerCAmelCase__		  =  np.array([41, 4, 20]					).astype(np.complexaaa					)

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

								# Our implementation.
								lowerCAmelCase__  ,   lowerCAmelCase__		  =  power_iteration(snake_case__    ,			snake_case__					)

								# Numpy implementation.

								# Get eigenvalues and eigenvectors using built-in numpy
								# eigh (eigh used for symmetric or hermetian matrices).
								lowerCAmelCase__  ,   lowerCAmelCase__		  =  np.linalg.eigh(snake_case__					)
								# Last eigenvalue is the maximum one.
								lowerCAmelCase__		  =  eigen_values[-1]
								# Last column in this matrix is eigenvector corresponding to largest eigenvalue.
								lowerCAmelCase__		  =  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(snake_case__					) - np.abs(snake_case__					)					) <= 1E-6


if __name__ == "__main__":
	import doctest

	doctest.testmod()
	test_power_iteration()
