





'''simple docstring'''




def 			_lowerCAmelCase     (    __snake_case    :					List[Any]     )	->	Union[str, Any]:
     return [
         {
             0: [1, 2],
             1: [0, 2],
             2: [0, 1, 3, 5],
             3: [2, 4],
             4: [3],
             5: [2, 6, 8],
             6: [5, 7],
             7: [6, 8],
             8: [5, 7],
         },
         {
             0: [6],
             1: [9],
             2: [4, 5],
             3: [4],
             4: [2, 3],
             5: [2],
             6: [0, 7],
             7: [6],
             8: [],
             9: [1],
         },
         {
             0: [4],
             1: [6],
             2: [],
             3: [5, 6, 7],
             4: [0, 6],
             5: [3, 8, 9],
             6: [1, 3, 4, 7],
             7: [3, 6, 8, 9],
             8: [5, 7],
             9: [5, 7],
         },
         {
             0: [1, 3],
             1: [0, 2, 4],
             2: [1, 3, 4],
             3: [0, 2, 4],
             4: [1, 2, 3],
         },
     ][index]






def 			_lowerCAmelCase     (    __snake_case    :					Tuple     )	->	int:
     __A		:					int						=			0
     __A		:					List[Any]						=			len(UpperCamelCase__     )  # No of vertices in graph
     __A		:					Optional[int]						=			[0] * n
     __A		:					Tuple						=			[False] * n

     def dfs(__snake_case    :					int					,				__snake_case    :					int					,				__snake_case    :					Optional[Any]					,				__snake_case    :					Any     ):
          __A		:					Any						=			True
          __A		:					int						=			id_
          id_ += 1
          for to in graph[at]:
               if to == parent:
                    pass
               elif not visited[to]:
                    dfs(UpperCamelCase__					,				UpperCamelCase__					,				UpperCamelCase__					,				id_     )
                    __A		:					Any						=			min(low[at]					,				low[to]     )
                    if id_ <= low[to]:
                         bridges.append((at, to) if at < to else (to, at)     )
               else:
                    # This edge is a back edge and cannot be a bridge
                    __A		:					Optional[Any]						=			min(low[at]					,				low[to]     )

     __A		:					list[tuple[int, int]]						=			[]
     for i in range(UpperCamelCase__     ):
          if not visited[i]:
               dfs(UpperCamelCase__					,				-1					,				UpperCamelCase__					,				id_     )
     return bridges


if __name__ == "__main__":
  import doctest

  doctest.testmod()

from typing import Any


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

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

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

                  UpperCamelCase__     : Optional[Any]											=					arg_max

    # The final observation
      UpperCamelCase__     : List[str]											=					observations_space[len(UpperCamelCase__		) - 1]

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

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

      return result


def 				SCREAMING_SNAKE_CASE_   (      UpperCamelCase__			,		UpperCamelCase__			,		UpperCamelCase__			,		UpperCamelCase__			,		UpperCamelCase__			,		):
      _validate_not_empty(
          UpperCamelCase__			,		UpperCamelCase__			,		UpperCamelCase__			,		UpperCamelCase__			,		UpperCamelCase__			,		)
      _validate_lists(UpperCamelCase__			,		UpperCamelCase__		)
      _validate_dicts(
          UpperCamelCase__			,		UpperCamelCase__			,		UpperCamelCase__		)


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


def 				SCREAMING_SNAKE_CASE_   (      UpperCamelCase__			,		UpperCamelCase__		):
      _validate_list(UpperCamelCase__			,		'''observations_space'''		)
      _validate_list(UpperCamelCase__			,		'''states_space'''		)


def 				SCREAMING_SNAKE_CASE_   (      UpperCamelCase__			,		UpperCamelCase__		):
      if not isinstance(_object			,		UpperCamelCase__		):
            UpperCamelCase__     : List[Any]											=					f'''{var_name} must be a list'''
            raise ValueError(UpperCamelCase__		)
      else:
            for x in _object:
                  if not isinstance(UpperCamelCase__			,		UpperCamelCase__		):
                        UpperCamelCase__     : List[Any]											=					f'''{var_name} must be a list of strings'''
                        raise ValueError(UpperCamelCase__		)


def 				SCREAMING_SNAKE_CASE_   (      UpperCamelCase__			,		UpperCamelCase__			,		UpperCamelCase__			,		):
      _validate_dict(UpperCamelCase__			,		'''initial_probabilities'''			,		UpperCamelCase__		)
      _validate_nested_dict(UpperCamelCase__			,		'''transition_probabilities'''		)
      _validate_nested_dict(UpperCamelCase__			,		'''emission_probabilities'''		)


def 				SCREAMING_SNAKE_CASE_   (      UpperCamelCase__			,		UpperCamelCase__		):
      _validate_dict(_object			,		UpperCamelCase__			,		UpperCamelCase__		)
      for x in _object.values():
            _validate_dict(UpperCamelCase__			,		UpperCamelCase__			,		UpperCamelCase__			,		UpperCamelCase__		)


def 				SCREAMING_SNAKE_CASE_   (      UpperCamelCase__			,		UpperCamelCase__			,		UpperCamelCase__			,		UpperCamelCase__ = False		):
      if not isinstance(_object			,		UpperCamelCase__		):
            UpperCamelCase__     : List[str]											=					f'''{var_name} must be a dict'''
            raise ValueError(UpperCamelCase__		)
      if not all(isinstance(UpperCamelCase__			,		UpperCamelCase__		) for x in _object		):
            UpperCamelCase__     : Dict											=					f'''{var_name} all keys must be strings'''
            raise ValueError(UpperCamelCase__		)
      if not all(isinstance(UpperCamelCase__			,		UpperCamelCase__		) for x in _object.values()		):
            UpperCamelCase__     : Optional[Any]											=					'''nested dictionary ''' if nested else ''''''
            UpperCamelCase__     : Optional[Any]											=					f'''{var_name} {nested_text}all values must be {value_type.__name__}'''
            raise ValueError(UpperCamelCase__		)


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

    testmod()


