



import os
from distutils.util import strtobool
def        lowercase_							(A : Optional[int]		,					A : str      ):
       for e in env_keys:
              snake_case__							:	str   	=	int(os.environ.get(A		,					-1      )      )
              if val >= 0:
                     return val
       return default
def        lowercase_							(A : List[Any]		,					A : Tuple=False      ):
       snake_case__							:	str   	=	os.environ.get(A		,					str(A      )      )
       return strtobool(A      ) == 1  # As its name indicates `strtobool` actually returns an int...





def        lowercase_							(A : int		,					A : int="no"      ):
       snake_case__							:	Optional[int]   	=	os.environ.get(A		,					str(A      )      )
       return value











def        lowercase_							(A : Union[str, Any]		,					A : List[str]		,					A : int		,					A : Optional[int]      ):
       global f  # a global dp table for knapsack
       if f[i][j] < 0:
              if j < wt[i - 1]:
                     snake_case__							:	Union[str, Any]   	=	mf_knapsack(i - 1		,					A		,					A		,					A      )
              else:
                     snake_case__							:	Any   	=	max(
                         mf_knapsack(i - 1		,					A		,					A		,					A      )		,					mf_knapsack(i - 1		,					A		,					A		,					j - wt[i - 1]      ) + val[i - 1]		,					)
              snake_case__							:	Optional[int]   	=	val
       return f[i][j]
def        lowercase_							(A : Optional[int]		,					A : Union[str, Any]		,					A : str		,					A : Dict      ):
       snake_case__							:	int   	=	[[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__							:	Union[str, Any]   	=	max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]		,					dp[i - 1][w_]      )
                     else:
                            snake_case__							:	str   	=	dp[i - 1][w_]

       return dp[n][w_], dp
def        lowercase_							(A : int		,					A : list		,					A : list      ):
       if not (isinstance(A		,					(list, tuple)      ) and isinstance(A		,					(list, tuple)      )):
              raise ValueError(
                  'Both the weights and values vectors must be either lists or tuples'      )

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

       snake_case__	,				snake_case__							:	Optional[int]   	=	knapsack(A		,					A		,					A		,					A      )
       snake_case__							:	set   	=	set()
       _construct_solution(A		,					A		,					A		,					A		,					A      )

       return optimal_val, example_optional_set
def        lowercase_							(A : list		,					A : list		,					A : int		,					A : int		,					A : set      ):
       # 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(A		,					A		,					i - 1		,					A		,					A      )
              else:
                     optimal_set.add(A      )
                     _construct_solution(A		,					A		,					i - 1		,					j - wt[i - 1]		,					A      )


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






