


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




def  a   (lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__   ):
   __a					=  [[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_:
            __a					=  max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]		,				dp[i - 1][w_]   )
         else:
            __a					=  dp[i - 1][w_]

   return dp[n][w_], dp




def  a   (lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__   ):
   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"""   )

   __a					=  len(lowerCAmelCase__   )
   if num_items != len(lowerCAmelCase__   ):
      __a					=  (
          """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__   ):
         __a					=  (
             """All weights must be integers but got weight of """
             f'''type {type(wt[i]   )} at index {i}'''
         )
         raise TypeError(lowerCAmelCase__   )

   __a	,    __a					=  knapsack(lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__   )
   __a					=  set()
   _construct_solution(lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__   )

   return optimal_val, example_optional_set




def  a   (lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__   ):
   # 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__":
  SCREAMING_SNAKE_CASE							    =							[3, 2, 4, 4]
  SCREAMING_SNAKE_CASE							    =							[4, 3, 2, 3]
  SCREAMING_SNAKE_CASE							    =							4
  SCREAMING_SNAKE_CASE							    =							6
  SCREAMING_SNAKE_CASE							    =							[[0] * (w + 1)] + [[0] + [-1] * (w + 1) for _ in range(n + 1)]
  SCREAMING_SNAKE_CASE    , SCREAMING_SNAKE_CASE							    =							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
  SCREAMING_SNAKE_CASE    , SCREAMING_SNAKE_CASE							    =							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)







import importlib.metadata
from typing import Union

from packaging.version import Version, parse

from .constants import STR_OPERATION_TO_FUNC


SCREAMING_SNAKE_CASE						:Union[str, Any]			=  parse(importlib.metadata.version('''torch'''))



def       _lowerCAmelCase    (				lowerCAmelCase_				:Union[str, Version]		,    lowerCAmelCase_				:str		,    lowerCAmelCase_				:str				)->Any:
   '''simple docstring'''






   if operation not in STR_OPERATION_TO_FUNC.keys():
      raise ValueError(F'''`operation` must be one of {list(STR_OPERATION_TO_FUNC.keys()				)}, received {operation}'''				)
   snake_case_            =       STR_OPERATION_TO_FUNC[operation]
   if isinstance(lowerCAmelCase_		,    lowerCAmelCase_				):
      snake_case_            =       parse(importlib.metadata.version(lowerCAmelCase_				)				)
   return operation(lowerCAmelCase_		,    parse(lowerCAmelCase_				)				)



def       _lowerCAmelCase    (				lowerCAmelCase_				:str		,    lowerCAmelCase_				:str				)->Any:
   '''simple docstring'''






   return compare_versions(lowerCAmelCase_		,    lowerCAmelCase_		,    lowerCAmelCase_				)




