


from typing import Any




def  a   (lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				):
   _validation(
       lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				)
   # Creates data structures and fill initial step
   __a					=  {}
   __a					=  {}
   for state in states_space:
      __a					=  observations_space[0]
      __a					=  (
          initial_probabilities[state] * emission_probabilities[state][observation]
      )
      __a					=  None

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

            # Update probabilities and pointers dicts
         __a					=  (
             probabilities[(arg_max, prior_observation)]
             * transition_probabilities[arg_max][state]
             * emission_probabilities[state][observation]
         )

         __a					=  arg_max

    # The final observation
   __a					=  observations_space[len(lowerCAmelCase__   ) - 1]

   # argmax for given final observation
   __a					=  """"""
   __a					=  -1
   for k_state in states_space:
      __a					=  probabilities[(k_state, final_observation)]
      if probability > max_probability:
         __a					=  probability
         __a					=  k_state
   __a					=  arg_max

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

   return result




def  a   (lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				):
   _validate_not_empty(
       lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				)
   _validate_lists(lowerCAmelCase__		,				lowerCAmelCase__   )
   _validate_dicts(
       lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__   )




def  a   (lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				):
   if not all(
       [
           observations_space,
           states_space,
           initial_probabilities,
           transition_probabilities,
           emission_probabilities,
       ]   ):
      raise ValueError("""There's an empty parameter"""   )




def  a   (lowerCAmelCase__		,				lowerCAmelCase__   ):
   _validate_list(lowerCAmelCase__		,				"""observations_space"""   )
   _validate_list(lowerCAmelCase__		,				"""states_space"""   )




def  a   (lowerCAmelCase__		,				lowerCAmelCase__   ):
   if not isinstance(_object		,				lowerCAmelCase__   ):
      __a					=  f'''{var_name} must be a list'''
      raise ValueError(lowerCAmelCase__   )
   else:
      for x in _object:
         if not isinstance(lowerCAmelCase__		,				lowerCAmelCase__   ):
            __a					=  f'''{var_name} must be a list of strings'''
            raise ValueError(lowerCAmelCase__   )




def  a   (lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				):
   _validate_dict(lowerCAmelCase__		,				"""initial_probabilities"""		,				lowerCAmelCase__   )
   _validate_nested_dict(lowerCAmelCase__		,				"""transition_probabilities"""   )
   _validate_nested_dict(lowerCAmelCase__		,				"""emission_probabilities"""   )




def  a   (lowerCAmelCase__		,				lowerCAmelCase__   ):
   _validate_dict(_object		,				lowerCAmelCase__		,				lowerCAmelCase__   )
   for x in _object.values():
      _validate_dict(lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__   )




def  a   (lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__ = False   ):
   if not isinstance(_object		,				lowerCAmelCase__   ):
      __a					=  f'''{var_name} must be a dict'''
      raise ValueError(lowerCAmelCase__   )
   if not all(isinstance(lowerCAmelCase__		,				lowerCAmelCase__   ) for x in _object   ):
      __a					=  f'''{var_name} all keys must be strings'''
      raise ValueError(lowerCAmelCase__   )
   if not all(isinstance(lowerCAmelCase__		,				lowerCAmelCase__   ) for x in _object.values()   ):
      __a					=  """nested dictionary """ if nested else """"""
      __a					=  f'''{var_name} {nested_text}all values must be {value_type.__name__}'''
      raise ValueError(lowerCAmelCase__   )


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

  testmod()







class     __UpperCAmelCase							:


     """simple docstring"""


     def __init__(	self  ,       __A    ):
        __a					=  set_counts
        __a					=  max(__A    )
        __a					=  len(__A    )
        __a					=  [1] * num_sets
        __a					=  list(range(__A    )    )


     def 	snake_case_  (	self  ,       __A  ,       __A    ):
        __a					=  self.get_parent(__A    )
        __a					=  self.get_parent(__A    )

        if src_parent == dst_parent:
           return False

        if self.ranks[dst_parent] >= self.ranks[src_parent]:
           self.set_counts[dst_parent] += self.set_counts[src_parent]
           __a					=  0
           __a					=  dst_parent
           if self.ranks[dst_parent] == self.ranks[src_parent]:
              self.ranks[dst_parent] += 1
           __a					=  self.set_counts[dst_parent]
        else:
           self.set_counts[src_parent] += self.set_counts[dst_parent]
           __a					=  0
           __a					=  src_parent
           __a					=  self.set_counts[src_parent]

        __a					=  max(self.max_set  ,       __A    )
        return True


     def 	snake_case_  (	self  ,       __A    ):
        if self.parents[disj_set] == disj_set:
           return disj_set
        __a					=  self.get_parent(self.parents[disj_set]    )
        return self.parents[disj_set]



