





from __future__ import annotations

from random import random
from typing import Generic, TypeVar

lowercase      :       Any									=  TypeVar("""KT""")
lowercase      :       Any									=  TypeVar("""VT""")





class  A__    (							Generic[KT, VT]	):




   """simple docstring"""



   def __init__(					self					,  lowercase = "root"					,  lowercase = None)			->						Dict:







          '''simple docstring'''

          a__						:	Dict =				key
          a__						:	Dict =				value
          a__						:	list[Node[KT, VT]] =				[]

   def __repr__(					self)			->						str:







          '''simple docstring'''

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

   @property
   def 							__lowercase		(					self)			->						int:







          '''simple docstring'''

          return len(self.forward)





class  A__    (							Generic[KT, VT]	):




   """simple docstring"""



   def __init__(					self					,  lowercase = 0.5					,  lowercase = 16)			->						str:







          '''simple docstring'''

          a__						:	Node[KT, VT] =				Node[KT, VT]()
          a__						:	Tuple =				0
          a__						:	Any =				p
          a__						:	Optional[int] =				max_level

   def __str__(					self)			->						str:







          '''simple docstring'''

          a__						:	List[Any] =				list(self)

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

          a__						:	Any =				max((len(str(lowercase)) for item in items)					,  default=4)
          a__						:	Union[str, Any] =				max(lowercase					,  4) + 4

          a__						:	Any =				self.head
          a__						:	List[Any] =				[]

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

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

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

   def __iter__(					self)			->						Optional[int]:







          '''simple docstring'''

          a__						:	Dict =				self.head

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

   def 							__lowercase		(					self)			->						int:







          '''simple docstring'''

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

          return level

   def 							__lowercase		(					self					,  lowercase)			->						tuple[Node[KT, VT] | None, list[Node[KT, VT]]]:







          '''simple docstring'''

          a__						:	Union[str, Any] =				[]

          a__						:	Tuple =				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__						:	Tuple =				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 							__lowercase		(					self					,  lowercase)			->						Optional[Any]:







          '''simple docstring'''

          a__ ,							a__						:	Dict =				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:
                                      a__						:	int =				node.forward[i]
                               else:
                                      a__						:	Tuple =				update_node.forward[:i]

   def 							__lowercase		(					self					,  lowercase					,  lowercase)			->						List[str]:







          '''simple docstring'''

          a__ ,							a__						:	List[str] =				self._locate_node(lowercase)
          if node is not None:
                 a__						:	Optional[int] =				value
          else:
                 a__						:	Optional[Any] =				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)
                        a__						:	str =				level

                 a__						:	Union[str, 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:
                               a__						:	Any =				new_node

   def 							__lowercase		(					self					,  lowercase)			->						VT | None:







          '''simple docstring'''

          a__ ,							a__						:	Optional[Any] =				self._locate_node(lowercase)

          if node is not None:
                 return node.value

          return None



def A_	(		)  ->		Optional[Any]:
       a__						:	int =				SkipList()
       skip_list.insert('Key1'		,      3				)
       skip_list.insert('Key2'		,      12				)
       skip_list.insert('Key3'		,      41				)
       skip_list.insert('Key4'		,      -19				)

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

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



def A_	(		)  ->		Any:
       a__						:	List[str] =				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__						:	Dict =				skip_list.head
       a__						:	str =				{}
       while node.level != 0:
              a__						:	Dict =				node.forward[0]
              a__						:	Tuple =				node.value

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



def A_	(		)  ->		Tuple:
       a__						:	List[str] =				SkipList()
       assert skip_list.find('Some key'				) is None



def A_	(		)  ->		Optional[int]:
       a__						:	List[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 A_	(		)  ->		Dict:
       a__						:	Any =				SkipList()
       skip_list.delete('Some key'				)

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



def A_	(		)  ->		List[Any]:
       a__						:	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'				)
       skip_list.delete('Key2'				)

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



def A_	(		)  ->		str:
       a__						:	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 A_	(		)  ->		Tuple:
       a__						:	str =				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(A__				):
              yield node.key
              for forward_node in node.forward:
                     yield from traverse_keys(A__				)

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



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

       a__						:	List[Any] =				SkipList()
       for i in range(10				):
              skip_list.insert(A__		,      A__				)
       assert is_sorted(list(A__				)				)
       skip_list.delete(5				)
       skip_list.delete(8				)
       skip_list.delete(2				)
       assert is_sorted(list(A__				)				)
       skip_list.insert(-12		,      -12				)
       skip_list.insert(77		,      77				)
       assert is_sorted(list(A__				)				)



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[Any]:
       a__						:	str =				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(A__				)


if __name__ == "__main__":
    import doctest

    doctest.testmod()
    main()









from math import loga



def A_	(		A__				)  ->		int:
       if a < 0:
              raise ValueError('Input value must be a positive integer'				)
       elif isinstance(A__		,      A__				):
              raise TypeError('Input value must be a \'int\' type'				)
       return 0 if (a == 0) else int(loga(a & -a				)				)


if __name__ == "__main__":
    import doctest

    doctest.testmod()


