




"""simple docstring"""






from __future__ import annotations







def 	_snake_case     (							lowercase__						):
				_lowerCamelCase	:     int       =       len(lowercase__						)
				# We need to create solution object to save path.
				_lowerCamelCase	:     Tuple       =       [[0 for _ in range(lowercase__						)] for _ in range(lowercase__						)]
				_lowerCamelCase	:     Optional[Any]       =       run_maze(lowercase__				,       0				,       0				,       lowercase__						)
				if solved:
								print('\n'.join(str(lowercase__						) for row in solutions						)						)
				else:
								print('No solution exists!'						)
				return solved







def 	_snake_case     (							lowercase__				,       lowercase__				,       lowercase__				,       lowercase__						):
				_lowerCamelCase	:     Optional[int]       =       len(lowercase__						)
				# Final check point.
				if i == j == (size - 1):
								_lowerCamelCase	:     Optional[Any]       =       1
								return True

				_lowerCamelCase	:     List[str]       =       (not i < 0) and (not j < 0)  # Check lower bounds
				_lowerCamelCase	:     List[str]       =       (i < size) and (j < size)  # Check upper bounds

				if lower_flag and upper_flag:
								# check for already visited and block points.
								_lowerCamelCase	:     List[Any]       =       (not solutions[i][j]) and (not maze[i][j])
								if block_flag:
												# check visited
												_lowerCamelCase	:     Union[str, Any]       =       1

												# check for directions
												if (
												    run_maze(lowercase__				,       i + 1				,       lowercase__				,       lowercase__						)
												    or run_maze(lowercase__				,       lowercase__				,       j + 1				,       lowercase__						)
												    or run_maze(lowercase__				,       i - 1				,       lowercase__				,       lowercase__						)
												    or run_maze(lowercase__				,       lowercase__				,       j - 1				,       lowercase__						)
												):
																return True

												_lowerCamelCase	:     Dict       =       0
												return False
				return False


if __name__ == "__main__":
			import doctest

			doctest.testmod()





"""simple docstring"""






from typing import Any







def 	_snake_case     (							lowercase__				,       lowercase__				,       lowercase__				,       lowercase__				,       lowercase__				,       ):
				_validation(
				    lowercase__				,       lowercase__				,       lowercase__				,       lowercase__				,       lowercase__				,       )
				# Creates data structures and fill initial step
				_lowerCamelCase	:     dict       =       {}
				_lowerCamelCase	:     dict       =       {}
				for state in states_space:
								_lowerCamelCase	:     Optional[Any]       =       observations_space[0]
								_lowerCamelCase	:     int       =       (
								    initial_probabilities[state] * emission_probabilities[state][observation]
								)
								_lowerCamelCase	:     List[Any]       =       None

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

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

												_lowerCamelCase	:     Union[str, Any]       =       arg_max

    # The final observation
				_lowerCamelCase	:     Tuple       =       observations_space[len(lowercase__						) - 1]

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

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

				return result







def 	_snake_case     (							lowercase__				,       lowercase__				,       lowercase__				,       lowercase__				,       lowercase__				,       ):
				_validate_not_empty(
				    lowercase__				,       lowercase__				,       lowercase__				,       lowercase__				,       lowercase__				,       )
				_validate_lists(lowercase__				,       lowercase__						)
				_validate_dicts(
				    lowercase__				,       lowercase__				,       lowercase__						)







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







def 	_snake_case     (							lowercase__				,       lowercase__						):
				_validate_list(lowercase__				,       'observations_space'						)
				_validate_list(lowercase__				,       'states_space'						)







def 	_snake_case     (							lowercase__				,       lowercase__						):
				if not isinstance(_object				,       lowercase__						):
								_lowerCamelCase	:     Tuple       =       f'''{var_name} must be a list'''
								raise ValueError(lowercase__						)
				else:
								for x in _object:
												if not isinstance(lowercase__				,       lowercase__						):
																_lowerCamelCase	:     int       =       f'''{var_name} must be a list of strings'''
																raise ValueError(lowercase__						)







def 	_snake_case     (							lowercase__				,       lowercase__				,       lowercase__				,       ):
				_validate_dict(lowercase__				,       'initial_probabilities'				,       lowercase__						)
				_validate_nested_dict(lowercase__				,       'transition_probabilities'						)
				_validate_nested_dict(lowercase__				,       'emission_probabilities'						)







def 	_snake_case     (							lowercase__				,       lowercase__						):
				_validate_dict(_object				,       lowercase__				,       lowercase__						)
				for x in _object.values():
								_validate_dict(lowercase__				,       lowercase__				,       lowercase__				,       lowercase__						)







def 	_snake_case     (							lowercase__				,       lowercase__				,       lowercase__				,       lowercase__ = False						):
				if not isinstance(_object				,       lowercase__						):
								_lowerCamelCase	:     Tuple       =       f'''{var_name} must be a dict'''
								raise ValueError(lowercase__						)
				if not all(isinstance(lowercase__				,       lowercase__						) for x in _object						):
								_lowerCamelCase	:     Optional[int]       =       f'''{var_name} all keys must be strings'''
								raise ValueError(lowercase__						)
				if not all(isinstance(lowercase__				,       lowercase__						) for x in _object.values()						):
								_lowerCamelCase	:     Optional[int]       =       'nested dictionary ' if nested else ''
								_lowerCamelCase	:     Optional[Any]       =       f'''{var_name} {nested_text}all values must be {value_type.__name__}'''
								raise ValueError(lowercase__						)


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

			testmod()