






'''simple docstring'''







from typing import Any





def 		_UpperCAmelCase   (							__A							:   list						,		__A							:   list						,		__A							:   dict						,		__A							:   dict						,		__A							:   dict						,		):
				_validation(
				    __A						,		__A						,		__A						,		__A						,		__A						,		)
				# Creates data structures and fill initial step
				a_		:  Union[str, Any]											=      {}
				a_		:  Tuple											=      {}
				for state in states_space:
								a_		:  Any											=      observations_space[0]
								a_		:  List[str]											=      (
								    initial_probabilities[state] * emission_probabilities[state][observation]
								)
								a_		:  Union[str, Any]											=      None

				# Fills the data structure with the probabilities of
				# different transitions and pointers to previous states
				for o in range(1						,		len(__A      )      ):
								a_		:  Optional[int]											=      observations_space[o]
								a_		:  Optional[Any]											=      observations_space[o - 1]
								for state in states_space:
												# Calculates the argmax for probability function
												a_		:  str											=      ''''''
												a_		:  int											=      -1
												for k_state in states_space:
																a_		:  int											=      (
																    probabilities[(k_state, prior_observation)]
																    * transition_probabilities[k_state][state]
																    * emission_probabilities[state][observation]
																)
																if probability > max_probability:
																				a_		:  int											=      probability
																				a_		:  Optional[int]											=      k_state

            # Update probabilities and pointers dicts
												a_		:  List[str]											=      (
												    probabilities[(arg_max, prior_observation)]
												    * transition_probabilities[arg_max][state]
												    * emission_probabilities[state][observation]
												)

												a_		:  Dict											=      arg_max

    # The final observation
				a_		:  Tuple											=      observations_space[len(__A      ) - 1]

				# argmax for given final observation
				a_		:  List[str]											=      ''''''
				a_		:  int											=      -1
				for k_state in states_space:
								a_		:  str											=      probabilities[(k_state, final_observation)]
								if probability > max_probability:
												a_		:  Optional[Any]											=      probability
												a_		:  List[str]											=      k_state
				a_		:  List[str]											=      arg_max

				# Process pointers backwards
				a_		:  Union[str, Any]											=      last_state
				a_		:  str											=      []
				for o in range(len(__A      ) - 1						,		-1						,		-1      ):
								result.append(__A      )
								a_		:  Optional[int]											=      pointers[previous, observations_space[o]]
				result.reverse()

				return result





def 		_UpperCAmelCase   (							__A							:   Any						,		__A							:   Any						,		__A							:   Any						,		__A							:   Any						,		__A							:   Any						,		):
				_validate_not_empty(
				    __A						,		__A						,		__A						,		__A						,		__A						,		)
				_validate_lists(__A						,		__A      )
				_validate_dicts(
				    __A						,		__A						,		__A      )





def 		_UpperCAmelCase   (							__A							:   Any						,		__A							:   Any						,		__A							:   Any						,		__A							:   Any						,		__A							:   Any						,		):
				if not all(
				    [
				        observations_space,
				        states_space,
				        initial_probabilities,
				        transition_probabilities,
				        emission_probabilities,
				    ]      ):
								raise ValueError('''There\'s an empty parameter'''      )





def 		_UpperCAmelCase   (							__A							:   Any						,		__A							:   Any      ):
				_validate_list(__A						,		'''observations_space'''      )
				_validate_list(__A						,		'''states_space'''      )





def 		_UpperCAmelCase   (							__A							:   Any						,		__A							:   str      ):
				if not isinstance(_object						,		__A      ):
								a_		:  List[str]											=      f'{var_name} must be a list'
								raise ValueError(__A      )
				else:
								for x in _object:
												if not isinstance(__A						,		__A      ):
																a_		:  Optional[int]											=      f'{var_name} must be a list of strings'
																raise ValueError(__A      )





def 		_UpperCAmelCase   (							__A							:   Any						,		__A							:   Any						,		__A							:   Any						,		):
				_validate_dict(__A						,		'''initial_probabilities'''						,		__A      )
				_validate_nested_dict(__A						,		'''transition_probabilities'''      )
				_validate_nested_dict(__A						,		'''emission_probabilities'''      )





def 		_UpperCAmelCase   (							__A							:   Any						,		__A							:   str      ):
				_validate_dict(_object						,		__A						,		__A      )
				for x in _object.values():
								_validate_dict(__A						,		__A						,		__A						,		__A      )





def 		_UpperCAmelCase   (							__A							:   Any						,		__A							:   str						,		__A							:   type						,		__A							:   bool = False      ):
				if not isinstance(_object						,		__A      ):
								a_		:  Dict											=      f'{var_name} must be a dict'
								raise ValueError(__A      )
				if not all(isinstance(__A						,		__A      ) for x in _object      ):
								a_		:  List[str]											=      f'{var_name} all keys must be strings'
								raise ValueError(__A      )
				if not all(isinstance(__A						,		__A      ) for x in _object.values()      ):
								a_		:  Dict											=      '''nested dictionary ''' if nested else ''''''
								a_		:  int											=      f'{var_name} {nested_text}all values must be {value_type.__name__}'
								raise ValueError(__A      )


if __name__ == "__main__":
		from doctest import testmod

		testmod()












'''simple docstring'''







from __future__ import annotations





def 		_UpperCAmelCase   (							__A							:   tuple[int, int]						,		__A							:   int      ):
				a_				,			a_		:  List[str]											=      position
				a_		:  Optional[int]											=      [
				    (y + 1, x + 2),
				    (y - 1, x + 2),
				    (y + 1, x - 2),
				    (y - 1, x - 2),
				    (y + 2, x + 1),
				    (y + 2, x - 1),
				    (y - 2, x + 1),
				    (y - 2, x - 1),
				]
				a_		:  Any											=      []

				for position in positions:
								a_				,			a_		:  Dict											=      position
								if 0 <= y_test < n and 0 <= x_test < n:
												permissible_positions.append(__A      )

				return permissible_positions





def 		_UpperCAmelCase   (							__A							:   list[list[int]]      ):
				return not any(elem == 0 for row in board for elem in row      )





def 		_UpperCAmelCase   (							__A							:   list[list[int]]						,		__A							:   tuple[int, int]						,		__A							:   int      ):

				if is_complete(__A      ):
								return True

				for position in get_valid_pos(__A						,		len(__A      )      ):
								a_				,			a_		:  Dict											=      position

								if board[y][x] == 0:
												a_		:  Optional[Any]											=      curr + 1
												if open_knight_tour_helper(__A						,		__A						,		curr + 1      ):
																return True
												a_		:  Tuple											=      0

				return False





def 		_UpperCAmelCase   (							__A							:   int      ):
				a_		:  List[str]											=      [[0 for i in range(__A      )] for j in range(__A      )]

				for i in range(__A      ):
								for j in range(__A      ):
												a_		:  Optional[Any]											=      1
												if open_knight_tour_helper(__A						,		(i, j)						,		1      ):
																return board
												a_		:  Union[str, Any]											=      0

				a_		:  Dict											=      f'Open Kight Tour cannot be performed on a board of size {n}'
				raise ValueError(__A      )


if __name__ == "__main__":
		import doctest

		doctest.testmod()




