

"""simple docstring"""




from typing import Any

def 						A							(  snake_case__		,					snake_case__		,					snake_case__		,					snake_case__		,					snake_case__		,					):




			'''simple docstring'''


			_validation(
			    snake_case__		,					snake_case__		,					snake_case__		,					snake_case__		,					snake_case__		,					)
			# Creates data structures and fill initial step
			SCREAMING_SNAKE_CASE__										=   {}
			SCREAMING_SNAKE_CASE__										=   {}
			for state in states_space:
						SCREAMING_SNAKE_CASE__										=   observations_space[0]
						SCREAMING_SNAKE_CASE__										=   (
						    initial_probabilities[state] * emission_probabilities[state][observation]
						)
						SCREAMING_SNAKE_CASE__										=   None

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

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

									SCREAMING_SNAKE_CASE__										=   arg_max

    # The final observation
			SCREAMING_SNAKE_CASE__										=   observations_space[len(snake_case__     ) - 1]

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

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

			return result

def 						A							(  snake_case__		,					snake_case__		,					snake_case__		,					snake_case__		,					snake_case__		,					):




			'''simple docstring'''


			_validate_not_empty(
			    snake_case__		,					snake_case__		,					snake_case__		,					snake_case__		,					snake_case__		,					)
			_validate_lists(snake_case__		,					snake_case__     )
			_validate_dicts(
			    snake_case__		,					snake_case__		,					snake_case__     )

def 						A							(  snake_case__		,					snake_case__		,					snake_case__		,					snake_case__		,					snake_case__		,					):




			'''simple docstring'''


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

def 						A							(  snake_case__		,					snake_case__     ):




			'''simple docstring'''


			_validate_list(snake_case__		,					"""observations_space"""     )
			_validate_list(snake_case__		,					"""states_space"""     )

def 						A							(  snake_case__		,					snake_case__     ):




			'''simple docstring'''


			if not isinstance(_object		,					snake_case__     ):
						SCREAMING_SNAKE_CASE__										=   f"""{var_name} must be a list"""
						raise ValueError(snake_case__     )
			else:
						for x in _object:
									if not isinstance(snake_case__		,					snake_case__     ):
												SCREAMING_SNAKE_CASE__										=   f"""{var_name} must be a list of strings"""
												raise ValueError(snake_case__     )

def 						A							(  snake_case__		,					snake_case__		,					snake_case__		,					):




			'''simple docstring'''


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

def 						A							(  snake_case__		,					snake_case__     ):




			'''simple docstring'''


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

def 						A							(  snake_case__		,					snake_case__		,					snake_case__		,					snake_case__ = False     ):




			'''simple docstring'''


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


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

						testmod()





"""simple docstring"""




from __future__ import annotations

def 						A							(  snake_case__     ):




			'''simple docstring'''


			for i in range(1		,					len(matrix[0]     )     ):
						matrix[0][i] += matrix[0][i - 1]

			# preprocessing the first column
			for i in range(1		,					len(snake_case__     )     ):
						matrix[i][0] += matrix[i - 1][0]

			# updating the path cost for current position
			for i in range(1		,					len(snake_case__     )     ):
						for j in range(1		,					len(matrix[0]     )     ):
									matrix[i][j] += min(matrix[i - 1][j]		,					matrix[i][j - 1]     )

			return matrix[-1][-1]


if __name__ == "__main__":
						import doctest

						doctest.testmod()


