





from typing import Any







def    A				(							UpperCAmelCase  ,  UpperCAmelCase  ,  UpperCAmelCase  ,  UpperCAmelCase  ,  UpperCAmelCase  ,  ):
					_validation(
					    UpperCAmelCase  ,  UpperCAmelCase  ,  UpperCAmelCase  ,  UpperCAmelCase  ,  UpperCAmelCase  ,  )
					# Creates data structures and fill initial step
					_snake_case       :					dict				       =   {}
					_snake_case       :					dict				       =   {}
					for state in states_space:
										_snake_case       :					List[Any]				       =   observations_space[0]
										_snake_case       :					Any				       =   (
										    initial_probabilities[state] * emission_probabilities[state][observation]
										)
										_snake_case       :					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				)				):
										_snake_case       :					int				       =   observations_space[o]
										_snake_case       :					Dict				       =   observations_space[o - 1]
										for state in states_space:
															# Calculates the argmax for probability function
															_snake_case       :					str				       =   ""
															_snake_case       :					Any				       =   -1
															for k_state in states_space:
																				_snake_case       :					List[str]				       =   (
																				    probabilities[(k_state, prior_observation)]
																				    * transition_probabilities[k_state][state]
																				    * emission_probabilities[state][observation]
																				)
																				if probability > max_probability:
																									_snake_case       :					List[str]				       =   probability
																									_snake_case       :					Optional[int]				       =   k_state

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

															_snake_case       :					List[str]				       =   arg_max

    # The final observation
					_snake_case       :					Dict				       =   observations_space[len(UpperCAmelCase				) - 1]

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

					# Process pointers backwards
					_snake_case       :					Any				       =   last_state
					_snake_case       :					Dict				       =   []
					for o in range(len(UpperCAmelCase				) - 1  ,  -1  ,  -1				):
										result.append(UpperCAmelCase				)
										_snake_case       :					Optional[Any]				       =   pointers[previous, observations_space[o]]
					result.reverse()

					return result







def    A				(							UpperCAmelCase  ,  UpperCAmelCase  ,  UpperCAmelCase  ,  UpperCAmelCase  ,  UpperCAmelCase  ,  ):
					_validate_not_empty(
					    UpperCAmelCase  ,  UpperCAmelCase  ,  UpperCAmelCase  ,  UpperCAmelCase  ,  UpperCAmelCase  ,  )
					_validate_lists(UpperCAmelCase  ,  UpperCAmelCase				)
					_validate_dicts(
					    UpperCAmelCase  ,  UpperCAmelCase  ,  UpperCAmelCase				)







def    A				(							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    A				(							UpperCAmelCase  ,  UpperCAmelCase				):
					_validate_list(UpperCAmelCase  ,  "observations_space"				)
					_validate_list(UpperCAmelCase  ,  "states_space"				)







def    A				(							UpperCAmelCase  ,  UpperCAmelCase				):
					if not isinstance(_object  ,  UpperCAmelCase				):
										_snake_case       :					Dict				       =   F"""{var_name} must be a list"""
										raise ValueError(UpperCAmelCase				)
					else:
										for x in _object:
															if not isinstance(UpperCAmelCase  ,  UpperCAmelCase				):
																				_snake_case       :					Dict				       =   F"""{var_name} must be a list of strings"""
																				raise ValueError(UpperCAmelCase				)







def    A				(							UpperCAmelCase  ,  UpperCAmelCase  ,  UpperCAmelCase  ,  ):
					_validate_dict(UpperCAmelCase  ,  "initial_probabilities"  ,  UpperCAmelCase				)
					_validate_nested_dict(UpperCAmelCase  ,  "transition_probabilities"				)
					_validate_nested_dict(UpperCAmelCase  ,  "emission_probabilities"				)







def    A				(							UpperCAmelCase  ,  UpperCAmelCase				):
					_validate_dict(_object  ,  UpperCAmelCase  ,  UpperCAmelCase				)
					for x in _object.values():
										_validate_dict(UpperCAmelCase  ,  UpperCAmelCase  ,  UpperCAmelCase  ,  UpperCAmelCase				)







def    A				(							UpperCAmelCase  ,  UpperCAmelCase  ,  UpperCAmelCase  ,  UpperCAmelCase = False				):
					if not isinstance(_object  ,  UpperCAmelCase				):
										_snake_case       :					List[str]				       =   F"""{var_name} must be a dict"""
										raise ValueError(UpperCAmelCase				)
					if not all(isinstance(UpperCAmelCase  ,  UpperCAmelCase				) for x in _object				):
										_snake_case       :					Tuple				       =   F"""{var_name} all keys must be strings"""
										raise ValueError(UpperCAmelCase				)
					if not all(isinstance(UpperCAmelCase  ,  UpperCAmelCase				) for x in _object.values()				):
										_snake_case       :					Tuple				       =   "nested dictionary " if nested else ""
										_snake_case       :					List[Any]				       =   F"""{var_name} {nested_text}all values must be {value_type.__name__}"""
										raise ValueError(UpperCAmelCase				)


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

							testmod()






from __future__ import annotations

import typing
from collections.abc import Iterable

import numpy as np

__lowerCAmelCase :Union[str, Any]     =    typing.Union[Iterable[float], Iterable[int], np.ndarray]  # noqa: UP007
__lowerCAmelCase :Optional[Any]     =    typing.Union[np.floataa, int, float]  # noqa: UP007







def    A				(							UpperCAmelCase  ,  UpperCAmelCase				):
					return np.sqrt(np.sum((np.asarray(UpperCAmelCase				) - np.asarray(UpperCAmelCase				)) ** 2				)				)







def    A				(							UpperCAmelCase  ,  UpperCAmelCase				):
					return sum((va - va) ** 2 for va, va in zip(UpperCAmelCase  ,  UpperCAmelCase				)				) ** (1 / 2)


if __name__ == "__main__":







							def    A				(							):
												from timeit import timeit

												print("Without Numpy"				)
												print(
												    timeit(
												        "euclidean_distance_no_np([1, 2, 3], [4, 5, 6])"  ,  number=10_000  ,  globals=globals()  ,  )				)
												print("With Numpy"				)
												print(
												    timeit(
												        "euclidean_distance([1, 2, 3], [4, 5, 6])"  ,  number=10_000  ,  globals=globals()  ,  )				)

							benchmark()