






'''simple docstring'''




def       __UpperCAmelCase						(  lowerCamelCase_     ,      lowerCamelCase_     ,      lowerCamelCase_     ,      lowerCamelCase_)							->     Optional[int]:
		global f  # a global dp table for knapsack
		if f[i][j] < 0:
				if j < wt[i - 1]:
						UpperCamelCase__				:					Tuple					=					mf_knapsack(i - 1     ,      lowerCamelCase_     ,      lowerCamelCase_     ,      lowerCamelCase_)
				else:
						UpperCamelCase__				:					Union[str, Any]					=					max(
						    mf_knapsack(i - 1     ,      lowerCamelCase_     ,      lowerCamelCase_     ,      lowerCamelCase_)     ,      mf_knapsack(i - 1     ,      lowerCamelCase_     ,      lowerCamelCase_     ,      j - wt[i - 1]) + val[i - 1]     ,      )
				UpperCamelCase__				:					List[str]					=					val
		return f[i][j]







def       __UpperCAmelCase						(  lowerCamelCase_     ,      lowerCamelCase_     ,      lowerCamelCase_     ,      lowerCamelCase_)							->     List[str]:
		UpperCamelCase__				:					Any					=					[[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_:
								UpperCamelCase__				:					Tuple					=					max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]     ,      dp[i - 1][w_])
						else:
								UpperCamelCase__				:					str					=					dp[i - 1][w_]

		return dp[n][w_], dp







def       __UpperCAmelCase						(  lowerCamelCase_     ,      lowerCamelCase_     ,      lowerCamelCase_)							->     Any:
		if not (isinstance(lowerCamelCase_     ,      (list, tuple)) and isinstance(lowerCamelCase_     ,      (list, tuple))):
				raise ValueError(
				    'Both the weights and values vectors must be either lists or tuples')

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

		UpperCamelCase__,	UpperCamelCase__				:					List[Any]					=					knapsack(lowerCamelCase_     ,      lowerCamelCase_     ,      lowerCamelCase_     ,      lowerCamelCase_)
		UpperCamelCase__				:					set					=					set()
		_construct_solution(lowerCamelCase_     ,      lowerCamelCase_     ,      lowerCamelCase_     ,      lowerCamelCase_     ,      lowerCamelCase_)

		return optimal_val, example_optional_set







def       __UpperCAmelCase						(  lowerCamelCase_     ,      lowerCamelCase_     ,      lowerCamelCase_     ,      lowerCamelCase_     ,      lowerCamelCase_)							->     Union[str, Any]:
		# for the current item i at a maximum weight j to be part of an optimal subset,
		# the optimal value at (i, j) must be greater than the optimal value at (i-1, j).
		# where i - 1 means considering only the previous items at the given maximum weight
		if i > 0 and j > 0:
				if dp[i - 1][j] == dp[i][j]:
						_construct_solution(lowerCamelCase_     ,      lowerCamelCase_     ,      i - 1     ,      lowerCamelCase_     ,      lowerCamelCase_)
				else:
						optimal_set.add(lowerCamelCase_)
						_construct_solution(lowerCamelCase_     ,      lowerCamelCase_     ,      i - 1     ,      j - wt[i - 1]     ,      lowerCamelCase_)


if __name__ == "__main__":
		lowerCAmelCase__          =						[3, 2, 4, 4]
		lowerCAmelCase__          =						[4, 3, 2, 3]
		lowerCAmelCase__          =						4
		lowerCAmelCase__          =						6
		lowerCAmelCase__          =						[[0] * (w + 1)] + [[0] + [-1] * (w + 1) for _ in range(n + 1)]
		lowerCAmelCase__       ,							lowerCAmelCase__          =						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
		lowerCAmelCase__       ,							lowerCAmelCase__          =						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)












'''simple docstring'''




def       __UpperCAmelCase						(  lowerCamelCase_     ,      lowerCamelCase_)							->     list[str]:
		return [sentence[i : i + ngram_size] for i in range(len(lowerCamelCase_) - ngram_size + 1)]


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

		testmod()




