




"""simple docstring"""






from __future__ import annotations

from random import random
from typing import Generic, TypeVar

lowercase__			=		TypeVar("""KT""")
lowercase__			=		TypeVar("""VT""")






class lowerCAmelCase__      (	Generic[KT, VT]		):






			'''simple docstring'''



			def __init__(    self    , lowercase = "root"    , lowercase = None    ):
							_lowerCamelCase	:     List[Any]       =       key
							_lowerCamelCase	:     Union[str, Any]       =       value
							_lowerCamelCase	:     list[Node[KT, VT]]       =       []
			def __repr__(    self    ):
							return F'''Node({self.key}: {self.value})'''






			@property
			def    A_       (    self    ):
							return len(self.forward    )






class lowerCAmelCase__      (	Generic[KT, VT]		):






			'''simple docstring'''



			def __init__(    self    , lowercase = 0.5    , lowercase = 16    ):
							_lowerCamelCase	:     Node[KT, VT]       =       Node[KT, VT]()
							_lowerCamelCase	:     Optional[Any]       =       0
							_lowerCamelCase	:     Tuple       =       p
							_lowerCamelCase	:     Any       =       max_level
			def __str__(    self    ):
							_lowerCamelCase	:     List[Any]       =       list(self    )

							if len(lowercase    ) == 0:
											return F'''SkipList(level={self.level})'''

							_lowerCamelCase	:     Dict       =       max((len(str(lowercase    )    ) for item in items)    , default=4    )
							_lowerCamelCase	:     Optional[int]       =       max(lowercase    , 4    ) + 4

							_lowerCamelCase	:     Union[str, Any]       =       self.head
							_lowerCamelCase	:     int       =       []

							_lowerCamelCase	:     Optional[Any]       =       node.forward.copy()
							lines.append(F'''[{node.key}]'''.ljust(lowercase    , '-'    ) + '* ' * len(lowercase    )    )
							lines.append(' ' * label_size + '| ' * len(lowercase    )    )

							while len(node.forward    ) != 0:
											_lowerCamelCase	:     List[Any]       =       node.forward[0]

											lines.append(
											    F'''[{node.key}]'''.ljust(lowercase    , '-'    )
											    + ' '.join(str(n.key    ) if n.key == node.key else '|' for n in forwards    )    )
											lines.append(' ' * label_size + '| ' * len(lowercase    )    )
											_lowerCamelCase	:     List[str]       =       node.forward

							lines.append('None'.ljust(lowercase    ) + '* ' * len(lowercase    )    )
							return F'''SkipList(level={self.level})\n''' + "\n".join(lowercase    )
			def __iter__(    self    ):
							_lowerCamelCase	:     List[Any]       =       self.head

							while len(node.forward    ) != 0:
											yield node.forward[0].key
											_lowerCamelCase	:     Union[str, Any]       =       node.forward[0]
			def    A_       (    self    ):
							_lowerCamelCase	:     Optional[int]       =       1
							while random() < self.p and level < self.max_level:
											level += 1

							return level
			def    A_       (    self    , lowercase    ):
							_lowerCamelCase	:     List[Any]       =       []

							_lowerCamelCase	:     Dict       =       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:
															_lowerCamelCase	:     Any       =       node.forward[i]
											# Each leftmost node (relative to searched node) will potentially have to
											# be updated.
											update_vector.append(lowercase    )

							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    A_       (    self    , lowercase    ):
							_lowerCamelCase,					_lowerCamelCase	:     Union[str, Any]       =       self._locate_node(lowercase    )

							if node is not None:
											for i, update_node in enumerate(lowercase    ):
															# Remove or replace all references to removed node.
															if update_node.level > i and update_node.forward[i].key == key:
																			if node.level > i:
																							_lowerCamelCase	:     Tuple       =       node.forward[i]
																			else:
																							_lowerCamelCase	:     Dict       =       update_node.forward[:i]
			def    A_       (    self    , lowercase    , lowercase    ):
							_lowerCamelCase,					_lowerCamelCase	:     List[Any]       =       self._locate_node(lowercase    )
							if node is not None:
											_lowerCamelCase	:     int       =       value
							else:
											_lowerCamelCase	:     Tuple       =       self.random_level()

											if level > self.level:
															# After level increase we have to add additional nodes to head.
															for _ in range(self.level - 1    , lowercase    ):
																			update_vector.append(self.head    )
															_lowerCamelCase	:     List[Any]       =       level

											_lowerCamelCase	:     List[Any]       =       Node(lowercase    , lowercase    )

											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(lowercase    )
															else:
																			_lowerCamelCase	:     Any       =       new_node






			def    A_       (    self    , lowercase    ):
							_lowerCamelCase,					_lowerCamelCase	:     Dict       =       self._locate_node(lowercase    )

							if node is not None:
											return node.value

							return None







def 	_snake_case     (							):
				_lowerCamelCase	:     Union[str, Any]       =       SkipList()
				skip_list.insert('Key1'				,       3						)
				skip_list.insert('Key2'				,       12						)
				skip_list.insert('Key3'				,       41						)
				skip_list.insert('Key4'				,       -19						)

				_lowerCamelCase	:     List[str]       =       skip_list.head
				_lowerCamelCase	:     int       =       {}
				while node.level != 0:
								_lowerCamelCase	:     Any       =       node.forward[0]
								_lowerCamelCase	:     Any       =       node.value

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







def 	_snake_case     (							):
				_lowerCamelCase	:     Optional[Any]       =       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						)

				_lowerCamelCase	:     Optional[int]       =       skip_list.head
				_lowerCamelCase	:     Dict       =       {}
				while node.level != 0:
								_lowerCamelCase	:     List[Any]       =       node.forward[0]
								_lowerCamelCase	:     Dict       =       node.value

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







def 	_snake_case     (							):
				_lowerCamelCase	:     Optional[int]       =       SkipList()
				assert skip_list.find('Some key'						) is None







def 	_snake_case     (							):
				_lowerCamelCase	:     Optional[Any]       =       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 	_snake_case     (							):
				_lowerCamelCase	:     Dict       =       SkipList()
				skip_list.delete('Some key'						)

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







def 	_snake_case     (							):
				_lowerCamelCase	:     Any       =       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 	_snake_case     (							):
				_lowerCamelCase	:     List[Any]       =       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 	_snake_case     (							):
				_lowerCamelCase	:     Optional[int]       =       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(lowercase__						):
								yield node.key
								for forward_node in node.forward:
												yield from traverse_keys(lowercase__						)

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







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

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







def 	_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 	_snake_case     (							):
				_lowerCamelCase	:     int       =       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(lowercase__						)


if __name__ == "__main__":
			import doctest

			doctest.testmod()
			main()





"""simple docstring"""






def 	_snake_case     (							lowercase__ = 1				,       lowercase__ = 1000						):
				_lowerCamelCase	:     Optional[int]       =       1
				_lowerCamelCase	:     List[Any]       =       0

				for divide_by_number in range(lowercase__				,       digit + 1						):
								_lowerCamelCase	:     list[int]       =       []
								_lowerCamelCase	:     Dict       =       numerator
								for _ in range(1				,       digit + 1						):
												if now_divide in has_been_divided:
																if longest_list_length < len(lowercase__						):
																				_lowerCamelCase	:     Any       =       len(lowercase__						)
																				_lowerCamelCase	:     Any       =       divide_by_number
												else:
																has_been_divided.append(lowercase__						)
																_lowerCamelCase	:     int       =       now_divide * 10 % divide_by_number

				return the_digit


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

			doctest.testmod()