


'''simple docstring'''
from typing import Any





def 					a__       (					lowerCAmelCase__  ,					lowerCAmelCase__  ,					lowerCAmelCase__  ,					lowerCAmelCase__  ,					lowerCAmelCase__  ,					)    ->	list:
    _validation(
        lowerCAmelCase__  ,					lowerCAmelCase__  ,					lowerCAmelCase__  ,					lowerCAmelCase__  ,					lowerCAmelCase__  ,					)
    # Creates data structures and fill initial step
    UpperCAmelCase__			:		dict													=				{}
    UpperCAmelCase__			:		dict													=				{}
    for state in states_space:
        UpperCAmelCase__			:		List[Any]													=				observations_space[0]
        UpperCAmelCase__			:		List[str]													=				(
            initial_probabilities[state] * emission_probabilities[state][observation]
        )
        UpperCAmelCase__			:		int													=				None

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

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

            UpperCAmelCase__			:		str													=				arg_max

    # The final observation
    UpperCAmelCase__			:		Tuple													=				observations_space[len(lowerCAmelCase__						) - 1]

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

    # Process pointers backwards
    UpperCAmelCase__			:		Dict													=				last_state
    UpperCAmelCase__			:		str													=				[]
    for o in range(len(lowerCAmelCase__						) - 1  ,					-1  ,					-1						):
        result.append(lowerCAmelCase__						)
        UpperCAmelCase__			:		str													=				pointers[previous, observations_space[o]]
    result.reverse()

    return result





def 					a__       (					lowerCAmelCase__  ,					lowerCAmelCase__  ,					lowerCAmelCase__  ,					lowerCAmelCase__  ,					lowerCAmelCase__  ,					)    ->	None:
    _validate_not_empty(
        lowerCAmelCase__  ,					lowerCAmelCase__  ,					lowerCAmelCase__  ,					lowerCAmelCase__  ,					lowerCAmelCase__  ,					)
    _validate_lists(lowerCAmelCase__  ,					lowerCAmelCase__						)
    _validate_dicts(
        lowerCAmelCase__  ,					lowerCAmelCase__  ,					lowerCAmelCase__						)





def 					a__       (					lowerCAmelCase__  ,					lowerCAmelCase__  ,					lowerCAmelCase__  ,					lowerCAmelCase__  ,					lowerCAmelCase__  ,					)    ->	None:
    if not all(
        [
            observations_space,
            states_space,
            initial_probabilities,
            transition_probabilities,
            emission_probabilities,
        ]						):
        raise ValueError('''There\'s an empty parameter'''						)





def 					a__       (					lowerCAmelCase__  ,					lowerCAmelCase__						)    ->	None:
    _validate_list(lowerCAmelCase__  ,					'''observations_space'''						)
    _validate_list(lowerCAmelCase__  ,					'''states_space'''						)





def 					a__       (					lowerCAmelCase__  ,					lowerCAmelCase__						)    ->	None:
    if not isinstance(_object  ,					lowerCAmelCase__						):
        UpperCAmelCase__			:		Optional[int]													=				F"""{var_name} must be a list"""
        raise ValueError(lowerCAmelCase__						)
    else:
        for x in _object:
            if not isinstance(lowerCAmelCase__  ,					lowerCAmelCase__						):
                UpperCAmelCase__			:		Optional[int]													=				F"""{var_name} must be a list of strings"""
                raise ValueError(lowerCAmelCase__						)





def 					a__       (					lowerCAmelCase__  ,					lowerCAmelCase__  ,					lowerCAmelCase__  ,					)    ->	None:
    _validate_dict(lowerCAmelCase__  ,					'''initial_probabilities'''  ,					lowerCAmelCase__						)
    _validate_nested_dict(lowerCAmelCase__  ,					'''transition_probabilities'''						)
    _validate_nested_dict(lowerCAmelCase__  ,					'''emission_probabilities'''						)





def 					a__       (					lowerCAmelCase__  ,					lowerCAmelCase__						)    ->	None:
    _validate_dict(_object  ,					lowerCAmelCase__  ,					lowerCAmelCase__						)
    for x in _object.values():
        _validate_dict(lowerCAmelCase__  ,					lowerCAmelCase__  ,					lowerCAmelCase__  ,					lowerCAmelCase__						)





def 					a__       (					lowerCAmelCase__  ,					lowerCAmelCase__  ,					lowerCAmelCase__  ,					lowerCAmelCase__ = False						)    ->	None:
    if not isinstance(_object  ,					lowerCAmelCase__						):
        UpperCAmelCase__			:		Any													=				F"""{var_name} must be a dict"""
        raise ValueError(lowerCAmelCase__						)
    if not all(isinstance(lowerCAmelCase__  ,					lowerCAmelCase__						) for x in _object						):
        UpperCAmelCase__			:		Optional[Any]													=				F"""{var_name} all keys must be strings"""
        raise ValueError(lowerCAmelCase__						)
    if not all(isinstance(lowerCAmelCase__  ,					lowerCAmelCase__						) for x in _object.values()						):
        UpperCAmelCase__			:		List[Any]													=				'''nested dictionary ''' if nested else ''''''
        UpperCAmelCase__			:		Any													=				F"""{var_name} {nested_text}all values must be {value_type.__name__}"""
        raise ValueError(lowerCAmelCase__						)


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

  testmod()




'''simple docstring'''
# Algorithm for the pigeonhole sorting







def 					a__       (					lowerCAmelCase__						)    ->	Optional[Any]:
    UpperCAmelCase__			:		Any													=				min(lowerCAmelCase__						)  # min() finds the minimum value
    UpperCAmelCase__			:		Optional[int]													=				max(lowerCAmelCase__						)  # max() finds the maximum value

    UpperCAmelCase__			:		int													=				max_val - min_val + 1  # size is difference of max and min values plus one

    # list of pigeonholes of size equal to the variable size
    UpperCAmelCase__			:		Any													=				[0] * size

    # Populate the pigeonholes.
    for x in a:
        assert isinstance(lowerCAmelCase__  ,					lowerCAmelCase__						), "integers only please"
        holes[x - min_val] += 1

    # Putting the elements back into the array in an order.
    UpperCAmelCase__			:		Optional[int]													=				0
    for count in range(lowerCAmelCase__						):
        while holes[count] > 0:
            holes[count] -= 1
            UpperCAmelCase__			:		Dict													=				count + min_val
            i += 1





def 					a__       (					)    ->	Union[str, Any]:
    UpperCAmelCase__			:		List[str]													=				[8, 3, 2, 7, 4, 6, 8]
    pigeonhole_sort(lowerCAmelCase__						)
    print('''Sorted order is:'''  ,					''' '''.join(lowerCAmelCase__						)						)


if __name__ == "__main__":
  main()
