




from __future__ import annotations

from random import random
from typing import Generic, TypeVar

__UpperCamelCase   :			Any          = TypeVar("KT")
__UpperCamelCase   :			List[Any]          = TypeVar("VT")

class 						__lowerCAmelCase      (      Generic[KT, VT]					):
							def __init__(		self		:Union[str, Any]						, __magic_name__		:KT | str = "root"						, __magic_name__		:VT | None = None							):







								'''simple docstring'''







								a									=				key
								a									=				value
								a									=				[]
							def __repr__(		self		:List[Any]							):







								'''simple docstring'''







								return F'Node({self.key}: {self.value})'





							@property
							def  lowerCamelCase__				(		self		:List[str]							):







								'''simple docstring'''







								return len(self.forward							)







class 						__lowerCAmelCase      (      Generic[KT, VT]					):
							def __init__(		self		:Optional[Any]						, __magic_name__		:float = 0.5						, __magic_name__		:int = 16							):







								'''simple docstring'''







								a									=				Node[KT, VT]()
								a									=				0
								a									=				p
								a									=				max_level
							def __str__(		self		:Tuple							):







								'''simple docstring'''







								a									=				list(self							)

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

								a									=				max((len(str(__magic_name__							)							) for item in items)						, default=4							)
								a									=				max(__magic_name__						, 4							) + 4

								a									=				self.head
								a									=				[]

								a									=				node.forward.copy()
								lines.append(F'[{node.key}]'.ljust(__magic_name__						, """-"""							) + """* """ * len(__magic_name__							)							)
								lines.append(""" """ * label_size + """| """ * len(__magic_name__							)							)

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

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

								lines.append("""None""".ljust(__magic_name__							) + """* """ * len(__magic_name__							)							)
								return F'SkipList(level={self.level})\n' + "\n".join(__magic_name__							)
							def __iter__(		self		:int							):







								'''simple docstring'''







								a									=				self.head

								while len(node.forward							) != 0:
									yield node.forward[0].key
									a									=				node.forward[0]
							def  lowerCamelCase__				(		self		:int							):







								'''simple docstring'''







								a									=				1
								while random() < self.p and level < self.max_level:
									level += 1

								return level
							def  lowerCamelCase__				(		self		:Dict						, __magic_name__		:int							):







								'''simple docstring'''







								a									=				[]

								a									=				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:
										a									=				node.forward[i]
									# Each leftmost node (relative to searched node) will potentially have to
									# be updated.
									update_vector.append(__magic_name__							)

								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  lowerCamelCase__				(		self		:List[str]						, __magic_name__		:KT							):







								'''simple docstring'''







								a							,					a									=				self._locate_node(__magic_name__							)

								if node is not None:
									for i, update_node in enumerate(__magic_name__							):
										# Remove or replace all references to removed node.
										if update_node.level > i and update_node.forward[i].key == key:
											if node.level > i:
												a									=				node.forward[i]
											else:
												a									=				update_node.forward[:i]
							def  lowerCamelCase__				(		self		:List[str]						, __magic_name__		:KT						, __magic_name__		:VT							):







								'''simple docstring'''







								a							,					a									=				self._locate_node(__magic_name__							)
								if node is not None:
									a									=				value
								else:
									a									=				self.random_level()

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

									a									=				Node(__magic_name__						, __magic_name__							)

									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(__magic_name__							)
										else:
											a									=				new_node





							def  lowerCamelCase__				(		self		:Dict						, __magic_name__		:VT							):







								'''simple docstring'''







								a							,					a									=				self._locate_node(__magic_name__							)

								if node is not None:
									return node.value

								return None







def 						__A						(      ) ->			Tuple:
	a									=				SkipList()
	skip_list.insert("""Key1"""				,     3       )
	skip_list.insert("""Key2"""				,     12       )
	skip_list.insert("""Key3"""				,     41       )
	skip_list.insert("""Key4"""				,     -19       )

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

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





def 						__A						(      ) ->			Tuple:
	a									=				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       )

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

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





def 						__A						(      ) ->			Optional[int]:
	a									=				SkipList()
	assert skip_list.find("""Some key"""       ) is None





def 						__A						(      ) ->			Optional[int]:
	a									=				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 						__A						(      ) ->			str:
	a									=				SkipList()
	skip_list.delete("""Some key"""       )

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





def 						__A						(      ) ->			int:
	a									=				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 						__A						(      ) ->			Union[str, Any]:
	a									=				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 						__A						(      ) ->			Dict:
	a									=				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(__lowerCamelCase       ):
		yield node.key
		for forward_node in node.forward:
			yield from traverse_keys(__lowerCamelCase       )

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





def 						__A						(      ) ->			Optional[Any]:
	def is_sorted(__lowerCamelCase       ):
		return all(next_item >= item for item, next_item in zip(__lowerCamelCase				,     lst[1:]       )       )

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





def 						__A						(      ) ->			Optional[Any]:
	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 						__A						(      ) ->			Optional[int]:
	a									=				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(__lowerCamelCase       )


if __name__ == "__main__":
		import doctest

		doctest.testmod()
		main()







from math import factorial, radians





def 						__A						(      __lowerCamelCase				,     __lowerCamelCase = 18				,     __lowerCamelCase = 10       ) ->			float:
	a									=				angle_in_degrees - ((angle_in_degrees // 360.0) * 360.0)

	# Converting from degrees to radians
	a									=				radians(__lowerCamelCase       )

	a									=				angle_in_radians
	a									=				3
	a									=				-1

	for _ in range(__lowerCamelCase       ):
		result += (b * (angle_in_radians**a)) / factorial(__lowerCamelCase       )

		a									=				-b  # One positive term and the next will be negative and so on...
		a += 2  # Increased by 2 for every term.

	return round(__lowerCamelCase				,     __lowerCamelCase       )


if __name__ == "__main__":
		__import__("doctest").testmod()

