


from typing import Any



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

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

            # Update probabilities and pointers dicts
         __lowerCamelCase:				Optional[Any]     =	(
             probabilities[(arg_max, prior_observation)]
             * transition_probabilities[arg_max][state]
             * emission_probabilities[state][observation]
         )

         __lowerCamelCase:				Optional[int]     =	arg_max

    # The final observation
   __lowerCamelCase:				Tuple     =	observations_space[len(snake_case ) - 1]

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

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

   return result



def     __lowerCAmelCase						(							snake_case	:    Any						,      snake_case	:    Any						,      snake_case	:    Any						,      snake_case	:    Any						,      snake_case	:    Any						,      )       ->							None:
   _validate_not_empty(
       snake_case						,      snake_case						,      snake_case						,      snake_case						,      snake_case						,      )
   _validate_lists(snake_case						,      snake_case )
   _validate_dicts(
       snake_case						,      snake_case						,      snake_case )



def     __lowerCAmelCase						(							snake_case	:    Any						,      snake_case	:    Any						,      snake_case	:    Any						,      snake_case	:    Any						,      snake_case	:    Any						,      )       ->							None:
   if not all(
       [
           observations_space,
           states_space,
           initial_probabilities,
           transition_probabilities,
           emission_probabilities,
       ] ):
      raise ValueError("""There's an empty parameter""" )



def     __lowerCAmelCase						(							snake_case	:    Any						,      snake_case	:    Any )       ->							None:
   _validate_list(snake_case						,      """observations_space""" )
   _validate_list(snake_case						,      """states_space""" )



def     __lowerCAmelCase						(							snake_case	:    Any						,      snake_case	:    str )       ->							None:
   if not isinstance(_object						,      snake_case ):
      __lowerCamelCase:				Optional[int]     =	f'{var_name} must be a list'
      raise ValueError(snake_case )
   else:
      for x in _object:
         if not isinstance(snake_case						,      snake_case ):
            __lowerCamelCase:				Optional[Any]     =	f'{var_name} must be a list of strings'
            raise ValueError(snake_case )



def     __lowerCAmelCase						(							snake_case	:    Any						,      snake_case	:    Any						,      snake_case	:    Any						,      )       ->							None:
   _validate_dict(snake_case						,      """initial_probabilities"""						,      snake_case )
   _validate_nested_dict(snake_case						,      """transition_probabilities""" )
   _validate_nested_dict(snake_case						,      """emission_probabilities""" )



def     __lowerCAmelCase						(							snake_case	:    Any						,      snake_case	:    str )       ->							None:
   _validate_dict(_object						,      snake_case						,      snake_case )
   for x in _object.values():
      _validate_dict(snake_case						,      snake_case						,      snake_case						,      snake_case )



def     __lowerCAmelCase						(							snake_case	:    Any						,      snake_case	:    str						,      snake_case	:    type						,      snake_case	:    bool = False )       ->							None:
   if not isinstance(_object						,      snake_case ):
      __lowerCamelCase:				Any     =	f'{var_name} must be a dict'
      raise ValueError(snake_case )
   if not all(isinstance(snake_case						,      snake_case ) for x in _object ):
      __lowerCamelCase:				List[str]     =	f'{var_name} all keys must be strings'
      raise ValueError(snake_case )
   if not all(isinstance(snake_case						,      snake_case ) for x in _object.values() ):
      __lowerCamelCase:				Any     =	"""nested dictionary """ if nested else """"""
      __lowerCamelCase:				List[Any]     =	f'{var_name} {nested_text}all values must be {value_type.__name__}'
      raise ValueError(snake_case )


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

   testmod()







from __future__ import annotations

_A		:				List[str]									=       {
    '''A''': ['''B''', '''C''', '''E'''],
    '''B''': ['''A''', '''D''', '''E'''],
    '''C''': ['''A''', '''F''', '''G'''],
    '''D''': ['''B'''],
    '''E''': ['''A''', '''B''', '''D'''],
    '''F''': ['''C'''],
    '''G''': ['''C'''],
}



class        a					:





     def __init__(    self					:				Any    ,	SCREAMING_SNAKE_CASE_					:				dict[str, list[str]]    ,	SCREAMING_SNAKE_CASE_					:				str							):
        __lowerCamelCase:				Dict     =	graph
        # mapping node to its parent in resulting breadth first tree
        __lowerCamelCase:				dict[str, str | None]     =	{}
        __lowerCamelCase:				str     =	source_vertex





     def  SCREAMING_SNAKE_CASE__       (    self					:				int							):
        __lowerCamelCase:				Optional[int]     =	{self.source_vertex}
        __lowerCamelCase:				Optional[int]     =	None
        __lowerCamelCase:				Optional[int]     =	[self.source_vertex]  # first in first out queue

        while queue:
           __lowerCamelCase:				int     =	queue.pop(0							)
           for adjacent_vertex in self.graph[vertex]:
              if adjacent_vertex not in visited:
                 visited.add(SCREAMING_SNAKE_CASE_							)
                 __lowerCamelCase:				Dict     =	vertex
                 queue.append(SCREAMING_SNAKE_CASE_							)







     def  SCREAMING_SNAKE_CASE__       (    self					:				Optional[Any]    ,	SCREAMING_SNAKE_CASE_					:				str							):
        if target_vertex == self.source_vertex:
           return self.source_vertex

        __lowerCamelCase:				List[str]     =	self.parent.get(SCREAMING_SNAKE_CASE_							)
        if target_vertex_parent is None:
           __lowerCamelCase:				Optional[int]     =	(
               F'No path from vertex: {self.source_vertex} to vertex: {target_vertex}'
           )
           raise ValueError(SCREAMING_SNAKE_CASE_							)

        return self.shortest_path(SCREAMING_SNAKE_CASE_							) + F'->{target_vertex}'


if __name__ == "__main__":
   _A		:				int									=       Graph(graph, '''G''')
   g.breath_first_search()
   print(g.shortest_path('''D'''))
   print(g.shortest_path('''G'''))
   print(g.shortest_path('''Foo'''))


