

from typing import Any





def  __lowerCamelCase     (			UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							):







							'''simple docstring'''






							_validation(
							    UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							)
							# Creates data structures and fill initial step
							snake_case_          =   {}
							snake_case_          =   {}
							for state in states_space:
														snake_case_          =   observations_space[0]
														snake_case_          =   (
														    initial_probabilities[state] * emission_probabilities[state][observation]
														)
														snake_case_          =   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_          =   observations_space[o]
														snake_case_          =   observations_space[o - 1]
														for state in states_space:
																					# Calculates the argmax for probability function
																					snake_case_          =   ''
																					snake_case_          =   -1
																					for k_state in states_space:
																												snake_case_          =   (
																												    probabilities[(k_state, prior_observation)]
																												    * transition_probabilities[k_state][state]
																												    * emission_probabilities[state][observation]
																												)
																												if probability > max_probability:
																																			snake_case_          =   probability
																																			snake_case_          =   k_state

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

																					snake_case_          =   arg_max

    # The final observation
							snake_case_          =   observations_space[len(UpperCamelCase__				) - 1]

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

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

							return result





def  __lowerCamelCase     (			UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							):







							'''simple docstring'''






							_validate_not_empty(
							    UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							)
							_validate_lists(UpperCamelCase__       ,							UpperCamelCase__				)
							_validate_dicts(
							    UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__				)





def  __lowerCamelCase     (			UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							):







							'''simple docstring'''






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





def  __lowerCamelCase     (			UpperCamelCase__       ,							UpperCamelCase__				):







							'''simple docstring'''






							_validate_list(UpperCamelCase__       ,							'observations_space'				)
							_validate_list(UpperCamelCase__       ,							'states_space'				)





def  __lowerCamelCase     (			UpperCamelCase__       ,							UpperCamelCase__				):







							'''simple docstring'''






							if not isinstance(_object       ,							UpperCamelCase__				):
														snake_case_          =   F'''{var_name} must be a list'''
														raise ValueError(UpperCamelCase__				)
							else:
														for x in _object:
																					if not isinstance(UpperCamelCase__       ,							UpperCamelCase__				):
																												snake_case_          =   F'''{var_name} must be a list of strings'''
																												raise ValueError(UpperCamelCase__				)





def  __lowerCamelCase     (			UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							):







							'''simple docstring'''






							_validate_dict(UpperCamelCase__       ,							'initial_probabilities'       ,							UpperCamelCase__				)
							_validate_nested_dict(UpperCamelCase__       ,							'transition_probabilities'				)
							_validate_nested_dict(UpperCamelCase__       ,							'emission_probabilities'				)





def  __lowerCamelCase     (			UpperCamelCase__       ,							UpperCamelCase__				):







							'''simple docstring'''






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





def  __lowerCamelCase     (			UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__ = False				):







							'''simple docstring'''






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


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

					testmod()







def  __lowerCamelCase     (			UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__				):







							'''simple docstring'''






							global f  # a global dp table for knapsack
							if f[i][j] < 0:
														if j < wt[i - 1]:
																					snake_case_          =   mf_knapsack(i - 1       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__				)
														else:
																					snake_case_          =   max(
																					    mf_knapsack(i - 1       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__				)       ,							mf_knapsack(i - 1       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							j - wt[i - 1]				) + val[i - 1]       ,							)
														snake_case_          =   val
							return f[i][j]





def  __lowerCamelCase     (			UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__				):







							'''simple docstring'''






							snake_case_          =   [[0] * (w + 1) for _ in range(n + 1				)]

							for i in range(1       ,							n + 1				):
														for w_ in range(1       ,							w + 1				):
																					if wt[i - 1] <= w_:
																												snake_case_          =   max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]       ,							dp[i - 1][w_]				)
																					else:
																												snake_case_          =   dp[i - 1][w_]

							return dp[n][w_], dp





def  __lowerCamelCase     (			UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__				):







							'''simple docstring'''






							if not (isinstance(UpperCamelCase__       ,							(list, tuple)				) and isinstance(UpperCamelCase__       ,							(list, tuple)				)):
														raise ValueError(
														    'Both the weights and values vectors must be either lists or tuples'				)

							snake_case_          =   len(UpperCamelCase__				)
							if num_items != len(UpperCamelCase__				):
														snake_case_          =   (
														    'The number of weights must be the same as the number of values.\n'
														    F'''But got {num_items} weights and {len(UpperCamelCase__				)} values'''
														)
														raise ValueError(UpperCamelCase__				)
							for i in range(UpperCamelCase__				):
														if not isinstance(wt[i]       ,							UpperCamelCase__				):
																					snake_case_          =   (
																					    'All weights must be integers but got weight of '
																					    F'''type {type(wt[i]				)} at index {i}'''
																					)
																					raise TypeError(UpperCamelCase__				)

							snake_case_    ,   snake_case_          =   knapsack(UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__				)
							snake_case_          =   set()
							_construct_solution(UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__				)

							return optimal_val, example_optional_set





def  __lowerCamelCase     (			UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__				):







							'''simple docstring'''






							if i > 0 and j > 0:
														if dp[i - 1][j] == dp[i][j]:
																					_construct_solution(UpperCamelCase__       ,							UpperCamelCase__       ,							i - 1       ,							UpperCamelCase__       ,							UpperCamelCase__				)
														else:
																					optimal_set.add(UpperCamelCase__				)
																					_construct_solution(UpperCamelCase__       ,							UpperCamelCase__       ,							i - 1       ,							j - wt[i - 1]       ,							UpperCamelCase__				)


if __name__ == "__main__":
					_UpperCAmelCase		:       int						    =	[3, 2, 4, 4]
					_UpperCAmelCase		:       Tuple						    =	[4, 3, 2, 3]
					_UpperCAmelCase		:       Dict						    =	4
					_UpperCAmelCase		:       int						    =	6
					_UpperCAmelCase		:       Union[str, Any]						    =	[[0] * (w + 1)] + [[0] + [-1] * (w + 1) for _ in range(n + 1)]
					_UpperCAmelCase    ,						_UpperCAmelCase		:       Optional[int]						    =	knapsack(w, wt, val, n)
					print(optimal_solution)
					print(mf_knapsack(n, wt, val, w))  # switched the n and w

					# testing the dynamic programming problem with example
					# the optimal subset for the above example are items 3 and 4
					_UpperCAmelCase    ,						_UpperCAmelCase		:       Optional[int]						    =	knapsack_with_example_solution(w, wt, val)
					assert optimal_solution == 8
					assert optimal_subset == {3, 4}
					print("""optimal_value = """, optimal_solution)
					print("""An optimal subset corresponding to the optimal value""", optimal_subset)




