
def        A__					(						lowerCamelCase     ,    lowerCamelCase     ,    lowerCamelCase     ,    lowerCamelCase						)  ->	int:
				global f  # a global dp table for knapsack
				if f[i][j] < 0:
								if j < wt[i - 1]:
												UpperCamelCase_:							int	=						mf_knapsack(i - 1     ,    _lowerCamelCase     ,    _lowerCamelCase     ,    _lowerCamelCase						)
								else:
												UpperCamelCase_:							Any	=						max(
												    mf_knapsack(i - 1     ,    _lowerCamelCase     ,    _lowerCamelCase     ,    _lowerCamelCase						)     ,    mf_knapsack(i - 1     ,    _lowerCamelCase     ,    _lowerCamelCase     ,    j - wt[i - 1]						) + val[i - 1]     ,    )
								UpperCamelCase_:							List[Any]	=						val
				return f[i][j]







def        A__					(						lowerCamelCase     ,    lowerCamelCase     ,    lowerCamelCase     ,    lowerCamelCase						)  ->	Optional[Any]:
				UpperCamelCase_:							str	=						[[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_:							Union[str, Any]	=						max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]     ,    dp[i - 1][w_]						)
												else:
																UpperCamelCase_:							Any	=						dp[i - 1][w_]

				return dp[n][w_], dp







def        A__					(						lowerCamelCase     ,    lowerCamelCase     ,    lowerCamelCase						)  ->	List[str]:
				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_:							List[str]	=						len(_lowerCamelCase						)
				if num_items != len(_lowerCamelCase						):
								UpperCamelCase_:							List[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_:							Optional[Any]	=						knapsack(_lowerCamelCase     ,    _lowerCamelCase     ,    _lowerCamelCase     ,    _lowerCamelCase						)
				UpperCamelCase_:							set	=						set()
				_construct_solution(_lowerCamelCase     ,    _lowerCamelCase     ,    _lowerCamelCase     ,    _lowerCamelCase     ,    _lowerCamelCase						)

				return optimal_val, example_optional_set







def        A__					(						lowerCamelCase     ,    lowerCamelCase     ,    lowerCamelCase     ,    lowerCamelCase     ,    lowerCamelCase						)  ->	Dict:
				# 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_						: List[str]         =  [3, 2, 4, 4]
				lowerCamelCase_						: List[Any]         =  [4, 3, 2, 3]
				lowerCamelCase_						: Tuple         =  4
				lowerCamelCase_						: Optional[int]         =  6
				lowerCamelCase_						: Optional[int]         =  [[0] * (w + 1)] + [[0] + [-1] * (w + 1) for _ in range(n + 1)]
				lowerCamelCase_   ,       lowerCamelCase_						: Tuple         =  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_						: str         =  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)








from typing import TYPE_CHECKING

from ...utils import (
    OptionalDependencyNotAvailable,
    _LazyModule,
    is_flax_available,
    is_tf_available,
    is_tokenizers_available,
    is_torch_available,
)


lowerCamelCase_						: str         =  {
    """configuration_roformer""": ["""ROFORMER_PRETRAINED_CONFIG_ARCHIVE_MAP""", """RoFormerConfig""", """RoFormerOnnxConfig"""],
    """tokenization_roformer""": ["""RoFormerTokenizer"""],
}

try:
							if not is_tokenizers_available():
														raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
							pass
else:
							lowerCamelCase_						: Union[str, Any]         =  ["""RoFormerTokenizerFast"""]

try:
							if not is_torch_available():
														raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
							pass
else:
							lowerCamelCase_						: Any         =  [
							    """ROFORMER_PRETRAINED_MODEL_ARCHIVE_LIST""",
							    """RoFormerForCausalLM""",
							    """RoFormerForMaskedLM""",
							    """RoFormerForMultipleChoice""",
							    """RoFormerForQuestionAnswering""",
							    """RoFormerForSequenceClassification""",
							    """RoFormerForTokenClassification""",
							    """RoFormerLayer""",
							    """RoFormerModel""",
							    """RoFormerPreTrainedModel""",
							    """load_tf_weights_in_roformer""",
							]


try:
							if not is_tf_available():
														raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
							pass
else:
							lowerCamelCase_						: Dict         =  [
							    """TF_ROFORMER_PRETRAINED_MODEL_ARCHIVE_LIST""",
							    """TFRoFormerForCausalLM""",
							    """TFRoFormerForMaskedLM""",
							    """TFRoFormerForMultipleChoice""",
							    """TFRoFormerForQuestionAnswering""",
							    """TFRoFormerForSequenceClassification""",
							    """TFRoFormerForTokenClassification""",
							    """TFRoFormerLayer""",
							    """TFRoFormerModel""",
							    """TFRoFormerPreTrainedModel""",
							]


try:
							if not is_flax_available():
														raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
							pass
else:
							lowerCamelCase_						: Optional[Any]         =  [
							    """FLAX_ROFORMER_PRETRAINED_MODEL_ARCHIVE_LIST""",
							    """FlaxRoFormerForMaskedLM""",
							    """FlaxRoFormerForMultipleChoice""",
							    """FlaxRoFormerForQuestionAnswering""",
							    """FlaxRoFormerForSequenceClassification""",
							    """FlaxRoFormerForTokenClassification""",
							    """FlaxRoFormerModel""",
							    """FlaxRoFormerPreTrainedModel""",
							]


if TYPE_CHECKING:
							from .configuration_roformer import ROFORMER_PRETRAINED_CONFIG_ARCHIVE_MAP, RoFormerConfig, RoFormerOnnxConfig
							from .tokenization_roformer import RoFormerTokenizer

							try:
														if not is_tokenizers_available():
																					raise OptionalDependencyNotAvailable()
							except OptionalDependencyNotAvailable:
														pass
							else:
														from .tokenization_roformer_fast import RoFormerTokenizerFast

							try:
														if not is_torch_available():
																					raise OptionalDependencyNotAvailable()
							except OptionalDependencyNotAvailable:
														pass
							else:
														from .modeling_roformer import (
														    ROFORMER_PRETRAINED_MODEL_ARCHIVE_LIST,
														    RoFormerForCausalLM,
														    RoFormerForMaskedLM,
														    RoFormerForMultipleChoice,
														    RoFormerForQuestionAnswering,
														    RoFormerForSequenceClassification,
														    RoFormerForTokenClassification,
														    RoFormerLayer,
														    RoFormerModel,
														    RoFormerPreTrainedModel,
														    load_tf_weights_in_roformer,
														)

							try:
														if not is_tf_available():
																					raise OptionalDependencyNotAvailable()
							except OptionalDependencyNotAvailable:
														pass
							else:
														from .modeling_tf_roformer import (
														    TF_ROFORMER_PRETRAINED_MODEL_ARCHIVE_LIST,
														    TFRoFormerForCausalLM,
														    TFRoFormerForMaskedLM,
														    TFRoFormerForMultipleChoice,
														    TFRoFormerForQuestionAnswering,
														    TFRoFormerForSequenceClassification,
														    TFRoFormerForTokenClassification,
														    TFRoFormerLayer,
														    TFRoFormerModel,
														    TFRoFormerPreTrainedModel,
														)

							try:
														if not is_flax_available():
																					raise OptionalDependencyNotAvailable()
							except OptionalDependencyNotAvailable:
														pass
							else:
														from .modeling_flax_roformer import (
														    FLAX_ROFORMER_PRETRAINED_MODEL_ARCHIVE_LIST,
														    FlaxRoFormerForMaskedLM,
														    FlaxRoFormerForMultipleChoice,
														    FlaxRoFormerForQuestionAnswering,
														    FlaxRoFormerForSequenceClassification,
														    FlaxRoFormerForTokenClassification,
														    FlaxRoFormerModel,
														    FlaxRoFormerPreTrainedModel,
														)


else:
							import sys

							lowerCamelCase_						: Tuple         =  _LazyModule(__name__, globals()["""__file__"""], _import_structure, module_spec=__spec__)






