from __future__ import annotations





def 		__lowerCamelCase							(UpperCAmelCase__       :							str ,	UpperCAmelCase__       :							list[str] | None = None ,	UpperCAmelCase__       :							dict[str, float] | None = None ,	UpperCAmelCase__       :							bool = False ,	) ->    Optional[int]:
					SCREAMING_SNAKE_CASE           =    cipher_alphabet or [chr(UpperCAmelCase__					) 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)
										SCREAMING_SNAKE_CASE           =    {
										    "a": 0.0_8497,
										    "b": 0.0_1492,
										    "c": 0.0_2202,
										    "d": 0.0_4253,
										    "e": 0.1_1162,
										    "f": 0.0_2228,
										    "g": 0.0_2015,
										    "h": 0.0_6094,
										    "i": 0.0_7546,
										    "j": 0.0_0153,
										    "k": 0.0_1292,
										    "l": 0.0_4025,
										    "m": 0.0_2406,
										    "n": 0.0_6749,
										    "o": 0.0_7507,
										    "p": 0.0_1929,
										    "q": 0.0_0095,
										    "r": 0.0_7587,
										    "s": 0.0_6327,
										    "t": 0.0_9356,
										    "u": 0.0_2758,
										    "v": 0.0_0978,
										    "w": 0.0_2560,
										    "x": 0.0_0150,
										    "y": 0.0_1994,
										    "z": 0.0_0077,
										}
					else:
										# Custom frequencies dictionary
										SCREAMING_SNAKE_CASE           =    frequencies_dict

					if not case_sensitive:
										SCREAMING_SNAKE_CASE           =    ciphertext.lower()

					# Chi squared statistic values
					SCREAMING_SNAKE_CASE           =    {}

					# cycle through all of the shifts
					for shift in range(len(UpperCAmelCase__					)					):
										SCREAMING_SNAKE_CASE           =    ""

										# decrypt the message with the shift
										for letter in ciphertext:
															try:
																				# Try to index the letter in the alphabet
																				SCREAMING_SNAKE_CASE           =    (alphabet_letters.index(letter.lower()					) - shift) % len(
																				    UpperCAmelCase__					)
																				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

										SCREAMING_SNAKE_CASE           =    0.0

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

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

																									# Complete the chi squared statistic formula
																									SCREAMING_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
																									SCREAMING_SNAKE_CASE           =    decrypted_with_shift.count(UpperCAmelCase__					)

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

																									# Complete the chi squared statistic formula
																									SCREAMING_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
										SCREAMING_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__       :							int					) -> tuple[float, str]:
										return chi_squared_statistic_values[key]

					SCREAMING_SNAKE_CASE           =    min(
					    UpperCAmelCase__ ,	key=UpperCAmelCase__ ,	)

					# Get all the data from the most likely cipher (key, decoded message)
					(
					    (
					    SCREAMING_SNAKE_CASE				
					)     ,				(
					    SCREAMING_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,
					)





# Author: OMKAR PATHAK, Nwachukwu Chidiebere

# Use a Python dictionary to construct the graph.
from __future__ import annotations

from pprint import pformat
from typing import Generic, TypeVar

_lowerCamelCase   :				Union[str, Any]     			=						TypeVar('''T''')


class   lowercase      (       Generic[T]       ):




							def __init__(  self				:   int		,							_UpperCamelCase				:   bool = True )   ->			None:



												'''simple docstring'''

												SCREAMING_SNAKE_CASE           =    {}  # dictionary of lists
												SCREAMING_SNAKE_CASE           =    directed




							def 		__snake_case(  self				:   int		,							_UpperCamelCase				:   T		,							_UpperCamelCase				:   T )   ->			GraphAdjacencyList[T]:



												'''simple docstring'''

												if not self.directed:  # For undirected graphs
																	# if both source vertex and destination vertex are both present in the
																	# adjacency list, add destination vertex to source vertex list of adjacent
																	# vertices and add source vertex to destination vertex list of adjacent
																	# vertices.
																	if source_vertex in self.adj_list and destination_vertex in self.adj_list:
																						self.adj_list[source_vertex].append(_UpperCamelCase )
																						self.adj_list[destination_vertex].append(_UpperCamelCase )
																	# if only source vertex is present in adjacency list, add destination vertex
																	# to source vertex list of adjacent vertices, then create a new vertex with
																	# destination vertex as key and assign a list containing the source vertex
																	# as it's first adjacent vertex.
																	elif source_vertex in self.adj_list:
																						self.adj_list[source_vertex].append(_UpperCamelCase )
																						SCREAMING_SNAKE_CASE           =    [source_vertex]
																	# if only destination vertex is present in adjacency list, add source vertex
																	# to destination vertex list of adjacent vertices, then create a new vertex
																	# with source vertex as key and assign a list containing the source vertex
																	# as it's first adjacent vertex.
																	elif destination_vertex in self.adj_list:
																						self.adj_list[destination_vertex].append(_UpperCamelCase )
																						SCREAMING_SNAKE_CASE           =    [destination_vertex]
																	# if both source vertex and destination vertex are not present in adjacency
																	# list, create a new vertex with source vertex as key and assign a list
																	# containing the destination vertex as it's first adjacent vertex also
																	# create a new vertex with destination vertex as key and assign a list
																	# containing the source vertex as it's first adjacent vertex.
																	else:
																						SCREAMING_SNAKE_CASE           =    [destination_vertex]
																						SCREAMING_SNAKE_CASE           =    [source_vertex]
												else:  # For directed graphs
																	# if both source vertex and destination vertex are present in adjacency
																	# list, add destination vertex to source vertex list of adjacent vertices.
																	if source_vertex in self.adj_list and destination_vertex in self.adj_list:
																						self.adj_list[source_vertex].append(_UpperCamelCase )
																	# if only source vertex is present in adjacency list, add destination
																	# vertex to source vertex list of adjacent vertices and create a new vertex
																	# with destination vertex as key, which has no adjacent vertex
																	elif source_vertex in self.adj_list:
																						self.adj_list[source_vertex].append(_UpperCamelCase )
																						SCREAMING_SNAKE_CASE           =    []
																	# if only destination vertex is present in adjacency list, create a new
																	# vertex with source vertex as key and assign a list containing destination
																	# vertex as first adjacent vertex
																	elif destination_vertex in self.adj_list:
																						SCREAMING_SNAKE_CASE           =    [destination_vertex]
																	# if both source vertex and destination vertex are not present in adjacency
																	# list, create a new vertex with source vertex as key and a list containing
																	# destination vertex as it's first adjacent vertex. Then create a new vertex
																	# with destination vertex as key, which has no adjacent vertex
																	else:
																						SCREAMING_SNAKE_CASE           =    [destination_vertex]
																						SCREAMING_SNAKE_CASE           =    []

												return self





							def __repr__(  self				:   Union[str, Any] )   ->			str:



												'''simple docstring'''

												return pformat(self.adj_list )




