"""simple docstring"""
from __future__ import annotations





def       lowerCAmelCase__			(	_UpperCamelCase				: str       ,	_UpperCamelCase				: Tuple = None       ,	_UpperCamelCase				: Optional[int] = None       ,	_UpperCamelCase				: Optional[Any] = False       ,	)    -> Optional[int]:





						"""simple docstring"""






						snake_case	    =	cipher_alphabet or [chr(A__    ) for i in range(9_7       ,	1_2_3    )]

						# If the argument is None or the user provided an empty dictionary
						if not frequencies_dict:
												# Frequencies of letters in the english language (how much they show up)
												snake_case	    =	{
												    'a': 0.0_84_97,
												    'b': 0.0_14_92,
												    'c': 0.0_22_02,
												    'd': 0.0_42_53,
												    'e': 0.1_11_62,
												    'f': 0.0_22_28,
												    'g': 0.0_20_15,
												    'h': 0.0_60_94,
												    'i': 0.0_75_46,
												    'j': 0.0_01_53,
												    'k': 0.0_12_92,
												    'l': 0.0_40_25,
												    'm': 0.0_24_06,
												    'n': 0.0_67_49,
												    'o': 0.0_75_07,
												    'p': 0.0_19_29,
												    'q': 0.0_00_95,
												    'r': 0.0_75_87,
												    's': 0.0_63_27,
												    't': 0.0_93_56,
												    'u': 0.0_27_58,
												    'v': 0.0_09_78,
												    'w': 0.0_25_60,
												    'x': 0.0_01_50,
												    'y': 0.0_19_94,
												    'z': 0.0_00_77,
												}
						else:
												# Custom frequencies dictionary
												snake_case	    =	frequencies_dict

						if not case_sensitive:
												snake_case	    =	ciphertext.lower()

						# Chi squared statistic values
						snake_case	    =	{}

						# cycle through all of the shifts
						for shift in range(len(A__    )    ):
												snake_case	    =	''

												# decrypt the message with the shift
												for letter in ciphertext:
																		try:
																								# Try to index the letter in the alphabet
																								snake_case	    =	(alphabet_letters.index(letter.lower()    ) - shift) % len(
																								    A__    )
																								decrypted_with_shift += (
																								    alphabet_letters[new_key].upper()
																								    if case_sensitive and letter.isupper()
																								    else alphabet_letters[new_key]
																								)
																		except ValueError:
																								# Append the character if it isn't in the alphabet
																								decrypted_with_shift += letter

												snake_case	    =	0.0

												# Loop through each letter in the decoded message with the shift
												for letter in decrypted_with_shift:
																		if case_sensitive:
																								snake_case	    =	letter.lower()
																								if letter in frequencies:
																														# Get the amount of times the letter occurs in the message
																														snake_case	    =	decrypted_with_shift.lower().count(A__    )

																														# Get the excepcted amount of times the letter should appear based
																														# on letter frequencies
																														snake_case	    =	frequencies[letter] * occurrences

																														# Complete the chi squared statistic formula
																														snake_case	    =	((occurrences - expected) ** 2) / expected

																														# Add the margin of error to the total chi squared statistic
																														chi_squared_statistic += chi_letter_value
																		else:
																								if letter.lower() in frequencies:
																														# Get the amount of times the letter occurs in the message
																														snake_case	    =	decrypted_with_shift.count(A__    )

																														# Get the excepcted amount of times the letter should appear based
																														# on letter frequencies
																														snake_case	    =	frequencies[letter] * occurrences

																														# Complete the chi squared statistic formula
																														snake_case	    =	((occurrences - expected) ** 2) / expected

																														# Add the margin of error to the total chi squared statistic
																														chi_squared_statistic += chi_letter_value

																														   # Add the data to the chi_squared_statistic_values dictionary
												snake_case	    =	(
												    chi_squared_statistic,
												    decrypted_with_shift,
												)

						# Get the most likely cipher by finding the cipher with the smallest chi squared
						# statistic
						def chi_squared_statistic_values_sorting_key(_UpperCamelCase				: Optional[Any]    ) -> tuple[float, str]:
												return chi_squared_statistic_values[key]

						snake_case	    =	min(
						    A__       ,	key=A__       ,	)

						# Get all the data from the most likely cipher (key, decoded message)
						(
						    (
						    snake_case	
						)    ,(
						    snake_case	
						)    ,	
						)	    =	chi_squared_statistic_values[most_likely_cipher]

						# Return the data on the most likely shift
						return (
						    most_likely_cipher,
						    most_likely_cipher_chi_squared_value,
						    decoded_most_likely_cipher,
						)





'''simple docstring'''






from __future__ import annotations

from random import random
from typing import Generic, TypeVar

lowerCAmelCase__											=	TypeVar('''KT''')
lowerCAmelCase__											=	TypeVar('''VT''')






class 			lowercase_	(Generic[KT, VT]      ):




  """simple docstring"""




  def __init__(      self   :     Optional[Any]					,lowercase__   :     KT | str = "root"					,lowercase__   :     VT | None = None			):
   __lowercase							  = key
   __lowercase							  = value
   __lowercase							  = []






  def __repr__(      self   :     Tuple			):
   return F"Node({self.key}: {self.value})"






  @property
  def 					SCREAMING_SNAKE_CASE      (      self   :     int			):
   return len(self.forward			)






class 			lowercase_	(Generic[KT, VT]      ):




  """simple docstring"""




  def __init__(      self   :     int					,lowercase__   :     float = 0.5					,lowercase__   :     int = 1_6			):
   __lowercase							  = Node[KT, VT]()
   __lowercase							  = 0
   __lowercase							  = p
   __lowercase							  = max_level






  def __str__(      self   :     List[str]			):
   __lowercase							  = list(self			)

   if len(lowercase__			) == 0:
    return F"SkipList(level={self.level})"

   __lowercase							  = max((len(str(lowercase__			)			) for item in items)					,default=4			)
   __lowercase							  = max(lowercase__					,4			) + 4

   __lowercase							  = self.head
   __lowercase							  = []

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

   while len(node.forward			) != 0:
    __lowercase							  = 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__			)			)
    __lowercase							  = node.forward

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






  def __iter__(      self   :     List[str]			):
   __lowercase							  = self.head

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






  def 					SCREAMING_SNAKE_CASE      (      self   :     Optional[Any]			):
   __lowercase							  = 1
   while random() < self.p and level < self.max_level:
    level += 1

   return level






  def 					SCREAMING_SNAKE_CASE      (      self   :     List[str]					,lowercase__   :     str			):
   __lowercase							  = []

   __lowercase							  = 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:
     __lowercase							  = 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 					SCREAMING_SNAKE_CASE      (      self   :     List[str]					,lowercase__   :     KT			):
   __lowercase  ,	__lowercase							  = 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:
       __lowercase							  = node.forward[i]
      else:
       __lowercase							  = update_node.forward[:i]






  def 					SCREAMING_SNAKE_CASE      (      self   :     List[str]					,lowercase__   :     KT					,lowercase__   :     VT			):
   __lowercase  ,	__lowercase							  = self._locate_node(lowercase__			)
   if node is not None:
    __lowercase							  = value
   else:
    __lowercase							  = 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			)
     __lowercase							  = level

    __lowercase							  = 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:
      __lowercase							  = new_node






  def 					SCREAMING_SNAKE_CASE      (      self   :     List[Any]					,lowercase__   :     VT			):
   __lowercase  ,	__lowercase							  = self._locate_node(lowercase__			)

   if node is not None:
    return node.value

   return None


def 		_A     (    ):






 """simple docstring"""



 __lowercase							  = SkipList()
 skip_list.insert('''Key1'''     ,				3      )
 skip_list.insert('''Key2'''     ,				12      )
 skip_list.insert('''Key3'''     ,				41      )
 skip_list.insert('''Key4'''     ,				-19      )

 __lowercase							  = skip_list.head
 __lowercase							  = {}
 while node.level != 0:
  __lowercase							  = node.forward[0]
  __lowercase							  = 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     (    ):






 """simple docstring"""



 __lowercase							  = 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      )

 __lowercase							  = skip_list.head
 __lowercase							  = {}
 while node.level != 0:
  __lowercase							  = node.forward[0]
  __lowercase							  = 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     (    ):






 """simple docstring"""



 __lowercase							  = SkipList()
 assert skip_list.find('''Some key'''      ) is None


def 		_A     (    ):






 """simple docstring"""



 __lowercase							  = 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     (    ):






 """simple docstring"""



 __lowercase							  = SkipList()
 skip_list.delete('''Some key'''      )

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


def 		_A     (    ):






 """simple docstring"""



 __lowercase							  = 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     (    ):






 """simple docstring"""



 __lowercase							  = 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     (    ):






 """simple docstring"""



 __lowercase							  = 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     (    ):






 """simple docstring"""



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

 __lowercase							  = 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     (    ):






 """simple docstring"""



 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     (    ):






 """simple docstring"""



 __lowercase							  = 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()

