






from typing import Any






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

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

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

      __UpperCAmelCase				:      Tuple	 =    arg_max

    # The final observation
  __UpperCAmelCase				:      int	 =    observations_space[len(__lowerCamelCase						) - 1]

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

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

  return result






def 							lowerCamelCase__ (       __lowerCamelCase     :	Any    ,						__lowerCamelCase     :	Any    ,						__lowerCamelCase     :	Any    ,						__lowerCamelCase     :	Any    ,						__lowerCamelCase     :	Any    ,						):
  _validate_not_empty(
      __lowerCamelCase    ,						__lowerCamelCase    ,						__lowerCamelCase    ,						__lowerCamelCase    ,						__lowerCamelCase    ,						)
  _validate_lists(__lowerCamelCase    ,						__lowerCamelCase						)
  _validate_dicts(
      __lowerCamelCase    ,						__lowerCamelCase    ,						__lowerCamelCase						)






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






def 							lowerCamelCase__ (       __lowerCamelCase     :	Any    ,						__lowerCamelCase     :	Any						):
  _validate_list(__lowerCamelCase    ,						"""observations_space"""						)
  _validate_list(__lowerCamelCase    ,						"""states_space"""						)






def 							lowerCamelCase__ (       __lowerCamelCase     :	Any    ,						__lowerCamelCase     :	str						):
  if not isinstance(_object    ,						__lowerCamelCase						):
    __UpperCAmelCase				:      List[str]	 =    f"""{var_name} must be a list"""
    raise ValueError(__lowerCamelCase						)
  else:
    for x in _object:
      if not isinstance(__lowerCamelCase    ,						__lowerCamelCase						):
        __UpperCAmelCase				:      Dict	 =    f"""{var_name} must be a list of strings"""
        raise ValueError(__lowerCamelCase						)






def 							lowerCamelCase__ (       __lowerCamelCase     :	Any    ,						__lowerCamelCase     :	Any    ,						__lowerCamelCase     :	Any    ,						):
  _validate_dict(__lowerCamelCase    ,						"""initial_probabilities"""    ,						__lowerCamelCase						)
  _validate_nested_dict(__lowerCamelCase    ,						"""transition_probabilities"""						)
  _validate_nested_dict(__lowerCamelCase    ,						"""emission_probabilities"""						)






def 							lowerCamelCase__ (       __lowerCamelCase     :	Any    ,						__lowerCamelCase     :	str						):
  _validate_dict(_object    ,						__lowerCamelCase    ,						__lowerCamelCase						)
  for x in _object.values():
    _validate_dict(__lowerCamelCase    ,						__lowerCamelCase    ,						__lowerCamelCase    ,						__lowerCamelCase						)






def 							lowerCamelCase__ (       __lowerCamelCase     :	Any    ,						__lowerCamelCase     :	str    ,						__lowerCamelCase     :	type    ,						__lowerCamelCase     :	bool = False						):
  if not isinstance(_object    ,						__lowerCamelCase						):
    __UpperCAmelCase				:      Any	 =    f"""{var_name} must be a dict"""
    raise ValueError(__lowerCamelCase						)
  if not all(isinstance(__lowerCamelCase    ,						__lowerCamelCase						) for x in _object						):
    __UpperCAmelCase				:      Union[str, Any]	 =    f"""{var_name} all keys must be strings"""
    raise ValueError(__lowerCamelCase						)
  if not all(isinstance(__lowerCamelCase    ,						__lowerCamelCase						) for x in _object.values()						):
    __UpperCAmelCase				:      int	 =    """nested dictionary """ if nested else """"""
    __UpperCAmelCase				:      Optional[Any]	 =    f"""{var_name} {nested_text}all values must be {value_type.__name__}"""
    raise ValueError(__lowerCamelCase						)


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

       testmod()






"""simple docstring"""


def _lowerCamelCase      (						_UpperCamelCase       ,					_UpperCamelCase       ,					_UpperCamelCase       ,					_UpperCamelCase ):

			'''simple docstring'''




			__lowerCAmelCase							       =     [False] * len(_UpperCamelCase )
			__lowerCAmelCase							       =     []
			queue.append(_UpperCamelCase )
			__lowerCAmelCase							       =     True

			while queue:
						__lowerCAmelCase							       =     queue.pop(0 )
						for ind in range(len(graph[u] ) ):
									if visited[ind] is False and graph[u][ind] > 0:
												queue.append(_UpperCamelCase )
												__lowerCAmelCase							       =     True
												__lowerCAmelCase							       =     u

			return visited[t]


def _lowerCamelCase      (						_UpperCamelCase       ,					_UpperCamelCase       ,					_UpperCamelCase ):

			'''simple docstring'''




			__lowerCAmelCase							       =     [-1] * (len(_UpperCamelCase ))
			__lowerCAmelCase							       =     0
			while bfs(_UpperCamelCase       ,					_UpperCamelCase       ,					_UpperCamelCase       ,					_UpperCamelCase ):
						__lowerCAmelCase							       =     float("Inf" )
						__lowerCAmelCase							       =     sink

						while s != source:
									# Find the minimum value in select path
									__lowerCAmelCase							       =     min(_UpperCamelCase       ,					graph[parent[s]][s] )
									__lowerCAmelCase							       =     parent[s]

						max_flow += path_flow
						__lowerCAmelCase							       =     sink

						while v != source:
									__lowerCAmelCase							       =     parent[v]
									graph[u][v] -= path_flow
									graph[v][u] += path_flow
									__lowerCAmelCase							       =     parent[v]
			return max_flow


A	:  Union[str, Any]            =     [
    [0, 1_6, 1_3, 0, 0, 0],
    [0, 0, 1_0, 1_2, 0, 0],
    [0, 4, 0, 0, 1_4, 0],
    [0, 0, 9, 0, 0, 2_0],
    [0, 0, 0, 7, 0, 4],
    [0, 0, 0, 0, 0, 0],
]

A       ,       A	:  Any            =     0, 5


print(ford_fulkerson(graph, source, sink))




