


from typing import Any
def 				UpperCAmelCase_					(							__snake_case     ,						__snake_case     ,						__snake_case     ,						__snake_case     ,						__snake_case     ,						)     ->			list:







 """simple docstring"""


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

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

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

   _lowercase							      =arg_max

    # The final observation
 _lowercase							      =observations_space[len(__snake_case						) - 1]

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

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

 return result
def 				UpperCAmelCase_					(							__snake_case     ,						__snake_case     ,						__snake_case     ,						__snake_case     ,						__snake_case     ,						)     ->			None:







 """simple docstring"""


 _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 				UpperCAmelCase_					(							__snake_case     ,						__snake_case     ,						__snake_case     ,						__snake_case     ,						__snake_case     ,						)     ->			None:







 """simple docstring"""


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







 """simple docstring"""


 _validate_list(__snake_case     ,						'''observations_space'''						)
 _validate_list(__snake_case     ,						'''states_space'''						)
def 				UpperCAmelCase_					(							__snake_case     ,						__snake_case						)     ->			None:







 """simple docstring"""


 if not isinstance(_object     ,						__snake_case						):
  _lowercase							      =F"{var_name} must be a list"
  raise ValueError(__snake_case						)
 else:
  for x in _object:
   if not isinstance(__snake_case     ,						__snake_case						):
    _lowercase							      =F"{var_name} must be a list of strings"
    raise ValueError(__snake_case						)
def 				UpperCAmelCase_					(							__snake_case     ,						__snake_case     ,						__snake_case     ,						)     ->			None:







 """simple docstring"""


 _validate_dict(__snake_case     ,						'''initial_probabilities'''     ,						__snake_case						)
 _validate_nested_dict(__snake_case     ,						'''transition_probabilities'''						)
 _validate_nested_dict(__snake_case     ,						'''emission_probabilities'''						)
def 				UpperCAmelCase_					(							__snake_case     ,						__snake_case						)     ->			None:







 """simple docstring"""


 _validate_dict(_object     ,						__snake_case     ,						__snake_case						)
 for x in _object.values():
  _validate_dict(__snake_case     ,						__snake_case     ,						__snake_case     ,						__snake_case						)
def 				UpperCAmelCase_					(							__snake_case     ,						__snake_case     ,						__snake_case     ,						__snake_case = False						)     ->			None:







 """simple docstring"""


 if not isinstance(_object     ,						__snake_case						):
  _lowercase							      =F"{var_name} must be a dict"
  raise ValueError(__snake_case						)
 if not all(isinstance(__snake_case     ,						__snake_case						) for x in _object						):
  _lowercase							      =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()						):
  _lowercase							      ='''nested dictionary ''' if nested else ''''''
  _lowercase							      =F"{var_name} {nested_text}all values must be {value_type.__name__}"
  raise ValueError(__snake_case						)


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

  testmod()





import heapq as hq
import math
from collections.abc import Iterator







class       lowerCamelCase__   :





     def __init__(self		,     UpperCAmelCase       )     ->   Any:
      _lowercase							      =str(id_       )
      _lowercase							      =None
      _lowercase							      =None
      _lowercase							      =[]
      _lowercase							      ={}  # {vertex:distance}





     def __lt__(self		,     UpperCAmelCase       )     ->   List[str]:
      return self.key < other.key





     def __repr__(self       )     ->   str:
      return self.id





     def      __A					(self		,     UpperCAmelCase       )     ->   Dict:
      self.neighbors.append(UpperCAmelCase       )





     def      __A					(self		,     UpperCAmelCase		,     UpperCAmelCase       )     ->   Optional[Any]:
      _lowercase							      =weight



def 				UpperCAmelCase_					(							__snake_case     ,						__snake_case     ,						__snake_case     ,						__snake_case						)     ->			List[str]:







 """simple docstring"""


 graph[a - 1].add_neighbor(graph[b - 1]						)
 graph[b - 1].add_neighbor(graph[a - 1]						)
 # add the edges:
 graph[a - 1].add_edge(graph[b - 1]     ,						__snake_case						)
 graph[b - 1].add_edge(graph[a - 1]     ,						__snake_case						)
def 				UpperCAmelCase_					(							__snake_case     ,						__snake_case						)     ->			list:







 """simple docstring"""


 _lowercase							      =[]
 for u in graph:
  _lowercase							      =math.inf
  _lowercase							      =None
 _lowercase							      =0
 _lowercase							      =graph[:]
 while q:
  _lowercase							      =min(__snake_case						)
  q.remove(__snake_case						)
  for v in u.neighbors:
   if (v in q) and (u.edges[v.id] < v.key):
    _lowercase							      =u
    _lowercase							      =u.edges[v.id]
 for i in range(1     ,						len(__snake_case						)						):
  a.append((int(graph[i].id						) + 1, int(graph[i].pi.id						) + 1)						)
 return a
def 				UpperCAmelCase_					(							__snake_case     ,						__snake_case						)     ->			Iterator[tuple]:







 """simple docstring"""


 for u in graph:
  _lowercase							      =math.inf
  _lowercase							      =None
 _lowercase							      =0

 _lowercase							      =list(__snake_case						)
 hq.heapify(__snake_case						)

 while h:
  _lowercase							      =hq.heappop(__snake_case						)
  for v in u.neighbors:
   if (v in h) and (u.edges[v.id] < v.key):
    _lowercase							      =u
    _lowercase							      =u.edges[v.id]
    hq.heapify(__snake_case						)

 for i in range(1     ,						len(__snake_case						)						):
  yield (int(graph[i].id						) + 1, int(graph[i].pi.id						) + 1)
def 				UpperCAmelCase_					(							)     ->			None:







 """simple docstring"""




if __name__ == "__main__":
  import doctest

  doctest.testmod()

