





'''simple docstring'''






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




class  UpperCAmelCase     :




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




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




      def __repr__(			self			: Optional[Any]  )				->      Optional[Any]:
            return self.id




      def 				lowercase__    (			self			: Union[str, Any]			,			__snake_case			: Tuple  )				->      Optional[Any]:
            self.neighbors.append(__snake_case  )




      def 				lowercase__    (			self			: Tuple			,			__snake_case			: List[str]			,			__snake_case			: Tuple  )				->      Any:
            _lowerCAmelCase							     =							weight




def    UpperCamelCase__      (		lowerCAmelCase							,	lowerCAmelCase							,	lowerCAmelCase							,	lowerCAmelCase  ):





      """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]							,	lowerCAmelCase  )
      graph[b - 1].add_edge(graph[a - 1]							,	lowerCAmelCase  )
def    UpperCamelCase__      (		lowerCAmelCase							,	lowerCAmelCase  ):





      """simple docstring"""


      _lowerCAmelCase							     =							[]
      for u in graph:
            _lowerCAmelCase							     =							math.inf
            _lowerCAmelCase							     =							None
      _lowerCAmelCase							     =							0
      _lowerCAmelCase							     =							graph[:]
      while q:
            _lowerCAmelCase							     =							min(lowerCAmelCase  )
            q.remove(lowerCAmelCase  )
            for v in u.neighbors:
                  if (v in q) and (u.edges[v.id] < v.key):
                        _lowerCAmelCase							     =							u
                        _lowerCAmelCase							     =							u.edges[v.id]
      for i in range(1							,	len(lowerCAmelCase  )  ):
            a.append((int(graph[i].id  ) + 1, int(graph[i].pi.id  ) + 1)  )
      return a
def    UpperCamelCase__      (		lowerCAmelCase							,	lowerCAmelCase  ):





      """simple docstring"""


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

      _lowerCAmelCase							     =							list(lowerCAmelCase  )
      hq.heapify(lowerCAmelCase  )

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

      for i in range(1							,	len(lowerCAmelCase  )  ):
            yield (int(graph[i].id  ) + 1, int(graph[i].pi.id  ) + 1)
def    UpperCamelCase__      (		):





      """simple docstring"""




if __name__ == "__main__":
      import doctest

      doctest.testmod()













from typing import Any





def      lowerCAmelCase_ (       __a	,    __a	,    __a	,    __a	,    __a	,    )     ->		list:







  """simple docstring"""





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

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

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

      lowerCamelCase__:      int			  =arg_max

    # The final observation
  lowerCamelCase__:      Any			  =observations_space[len(__a		) - 1]

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

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

  return result





def      lowerCAmelCase_ (       __a	,    __a	,    __a	,    __a	,    __a	,    )     ->		None:







  """simple docstring"""





  _validate_not_empty(
      __a	,    __a	,    __a	,    __a	,    __a	,    )
  _validate_lists(__a	,    __a		)
  _validate_dicts(
      __a	,    __a	,    __a		)





def      lowerCAmelCase_ (       __a	,    __a	,    __a	,    __a	,    __a	,    )     ->		None:







  """simple docstring"""





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





def      lowerCAmelCase_ (       __a	,    __a		)     ->		None:







  """simple docstring"""





  _validate_list(__a	,    "observations_space"		)
  _validate_list(__a	,    "states_space"		)





def      lowerCAmelCase_ (       __a	,    __a		)     ->		None:







  """simple docstring"""





  if not isinstance(_object	,    __a		):
    lowerCamelCase__:      Tuple			  =F"""{var_name} must be a list"""
    raise ValueError(__a		)
  else:
    for x in _object:
      if not isinstance(__a	,    __a		):
        lowerCamelCase__:      str			  =F"""{var_name} must be a list of strings"""
        raise ValueError(__a		)





def      lowerCAmelCase_ (       __a	,    __a	,    __a	,    )     ->		None:







  """simple docstring"""





  _validate_dict(__a	,    "initial_probabilities"	,    __a		)
  _validate_nested_dict(__a	,    "transition_probabilities"		)
  _validate_nested_dict(__a	,    "emission_probabilities"		)





def      lowerCAmelCase_ (       __a	,    __a		)     ->		None:







  """simple docstring"""





  _validate_dict(_object	,    __a	,    __a		)
  for x in _object.values():
    _validate_dict(__a	,    __a	,    __a	,    __a		)





def      lowerCAmelCase_ (       __a	,    __a	,    __a	,    __a = False		)     ->		None:







  """simple docstring"""





  if not isinstance(_object	,    __a		):
    lowerCamelCase__:      Optional[int]			  =F"""{var_name} must be a dict"""
    raise ValueError(__a		)
  if not all(isinstance(__a	,    __a		) for x in _object		):
    lowerCamelCase__:      Tuple			  =F"""{var_name} all keys must be strings"""
    raise ValueError(__a		)
  if not all(isinstance(__a	,    __a		) for x in _object.values()		):
    lowerCamelCase__:      Dict			  ="nested dictionary " if nested else ""
    lowerCamelCase__:      List[str]			  =F"""{var_name} {nested_text}all values must be {value_type.__name__}"""
    raise ValueError(__a		)


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

     testmod()

