



"""simple docstring"""

from typing import Any







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

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

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

															snake_case_					:		Any 					=					arg_max

    # The final observation
					snake_case_					:		Dict 					=					observations_space[len(SCREAMING_SNAKE_CASE__			) - 1]

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

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

					return result







def 							SCREAMING_SNAKE_CASE__      (		SCREAMING_SNAKE_CASE__ :  Any				,    SCREAMING_SNAKE_CASE__ :  Any				,    SCREAMING_SNAKE_CASE__ :  Any				,    SCREAMING_SNAKE_CASE__ :  Any				,    SCREAMING_SNAKE_CASE__ :  Any				,    ):
					_validate_not_empty(
					    SCREAMING_SNAKE_CASE__				,    SCREAMING_SNAKE_CASE__				,    SCREAMING_SNAKE_CASE__				,    SCREAMING_SNAKE_CASE__				,    SCREAMING_SNAKE_CASE__				,    )
					_validate_lists(SCREAMING_SNAKE_CASE__				,    SCREAMING_SNAKE_CASE__			)
					_validate_dicts(
					    SCREAMING_SNAKE_CASE__				,    SCREAMING_SNAKE_CASE__				,    SCREAMING_SNAKE_CASE__			)







def 							SCREAMING_SNAKE_CASE__      (		SCREAMING_SNAKE_CASE__ :  Any				,    SCREAMING_SNAKE_CASE__ :  Any				,    SCREAMING_SNAKE_CASE__ :  Any				,    SCREAMING_SNAKE_CASE__ :  Any				,    SCREAMING_SNAKE_CASE__ :  Any				,    ):





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







def 							SCREAMING_SNAKE_CASE__      (		SCREAMING_SNAKE_CASE__ :  Any				,    SCREAMING_SNAKE_CASE__ :  Any			):
					_validate_list(SCREAMING_SNAKE_CASE__				,    """observations_space"""			)
					_validate_list(SCREAMING_SNAKE_CASE__				,    """states_space"""			)







def 							SCREAMING_SNAKE_CASE__      (		SCREAMING_SNAKE_CASE__ :  Any				,    SCREAMING_SNAKE_CASE__ :  str			):





					if not isinstance(_object				,    SCREAMING_SNAKE_CASE__			):
										snake_case_					:		Optional[int] 					=					f'{var_name} must be a list'
										raise ValueError(SCREAMING_SNAKE_CASE__			)
					else:
										for x in _object:
															if not isinstance(SCREAMING_SNAKE_CASE__				,    SCREAMING_SNAKE_CASE__			):
																				snake_case_					:		Dict 					=					f'{var_name} must be a list of strings'
																				raise ValueError(SCREAMING_SNAKE_CASE__			)







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







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







def 							SCREAMING_SNAKE_CASE__      (		SCREAMING_SNAKE_CASE__ :  Any				,    SCREAMING_SNAKE_CASE__ :  str				,    SCREAMING_SNAKE_CASE__ :  type				,    SCREAMING_SNAKE_CASE__ :  bool = False			):





					if not isinstance(_object				,    SCREAMING_SNAKE_CASE__			):
										snake_case_					:		Optional[Any] 					=					f'{var_name} must be a dict'
										raise ValueError(SCREAMING_SNAKE_CASE__			)
					if not all(isinstance(SCREAMING_SNAKE_CASE__				,    SCREAMING_SNAKE_CASE__			) for x in _object			):
										snake_case_					:		Optional[Any] 					=					f'{var_name} all keys must be strings'
										raise ValueError(SCREAMING_SNAKE_CASE__			)
					if not all(isinstance(SCREAMING_SNAKE_CASE__				,    SCREAMING_SNAKE_CASE__			) for x in _object.values()			):
										snake_case_					:		Optional[Any] 					=					"""nested dictionary """ if nested else """"""
										snake_case_					:		int 					=					f'{var_name} {nested_text}all values must be {value_type.__name__}'
										raise ValueError(SCREAMING_SNAKE_CASE__			)


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

			testmod()









"""simple docstring"""

from copy import deepcopy







class    __lowercase     :







						"""simple docstring"""







						def __init__(self   ,		lowercase__ = None   ,		lowercase__ = None       ):

											if arr is None and size is not None:
																snake_case_					:		str 					=					size
																snake_case_					:		Optional[Any] 					=					[0] * size
											elif arr is not None:
																self.init(lowercase__       )
											else:
																raise ValueError("""Either arr or size must be specified"""       )







						def 						__UpperCamelCase							(self   ,		lowercase__       ):
											snake_case_					:		Optional[Any] 					=					len(lowercase__       )
											snake_case_					:		int 					=					deepcopy(lowercase__       )
											for i in range(1   ,		self.size       ):
																snake_case_					:		Optional[Any] 					=					self.next_(lowercase__       )
																if j < self.size:
																					self.tree[j] += self.tree[i]







						def 						__UpperCamelCase							(self       ):
											snake_case_					:		Dict 					=					self.tree[:]
											for i in range(self.size - 1   ,		0   ,		-1       ):
																snake_case_					:		Optional[int] 					=					self.next_(lowercase__       )
																if j < self.size:
																					arr[j] -= arr[i]
											return arr







						@staticmethod
						def 						__UpperCamelCase							(lowercase__       ):
											return index + (index & (-index))







						@staticmethod
						def 						__UpperCamelCase							(lowercase__       ):
											return index - (index & (-index))







						def 						__UpperCamelCase							(self   ,		lowercase__   ,		lowercase__       ):
											if index == 0:
																self.tree[0] += value
																return
											while index < self.size:
																self.tree[index] += value
																snake_case_					:		Tuple 					=					self.next_(lowercase__       )







						def 						__UpperCamelCase							(self   ,		lowercase__   ,		lowercase__       ):
											self.add(lowercase__   ,		value - self.get(lowercase__       )       )







						def 						__UpperCamelCase							(self   ,		lowercase__       ):
											if right == 0:
																return 0
											snake_case_					:		List[str] 					=					self.tree[0]
											right -= 1  # make right inclusive
											while right > 0:
																result += self.tree[right]
																snake_case_					:		Optional[int] 					=					self.prev(lowercase__       )
											return result







						def 						__UpperCamelCase							(self   ,		lowercase__   ,		lowercase__       ):
											return self.prefix(lowercase__       ) - self.prefix(lowercase__       )







						def 						__UpperCamelCase							(self   ,		lowercase__       ):
											return self.query(lowercase__   ,		index + 1       )







						def 						__UpperCamelCase							(self   ,		lowercase__       ):
											value -= self.tree[0]
											if value < 0:
																return -1

											snake_case_					:		Tuple 					=					1  # Largest power of 2 <= size
											while j * 2 < self.size:
																j *= 2

											snake_case_					:		Tuple 					=					0

											while j > 0:
																if i + j < self.size and self.tree[i + j] <= value:
																					value -= self.tree[i + j]
																					i += j
																j //= 2
											return i


if __name__ == "__main__":
			import doctest

			doctest.testmod()




