

def 			lowerCamelCase__			(       _a				,      _a):
	return numa ^ numa < 0


if __name__ == "__main__":
			import doctest

			doctest.testmod()

def 		UpperCamelCase						(   lowercase_			,			lowercase_			,			lowercase_			,			lowercase_ )		->						List[str]:


  '''simple docstring'''






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

def 		UpperCamelCase						(   lowercase_			,			lowercase_			,			lowercase_			,			lowercase_ )		->						str:


  '''simple docstring'''






  lowercase__	: 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_:
        lowercase__	: List[Any]     		=   max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]			,			dp[i - 1][w_] )
      else:
        lowercase__	: Tuple     		=   dp[i - 1][w_]

  return dp[n][w_], dp

def 		UpperCamelCase						(   lowercase_			,			lowercase_			,			lowercase_ )		->						Optional[Any]:


  '''simple docstring'''






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

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

  lowercase__		,     lowercase__	: Tuple     		=   knapsack(lowercase_			,			lowercase_			,			lowercase_			,			lowercase_ )
  lowercase__	: set     		=   set()
  _construct_solution(lowercase_			,			lowercase_			,			lowercase_			,			lowercase_			,			lowercase_ )

  return optimal_val, example_optional_set

def 		UpperCamelCase						(   lowercase_			,			lowercase_			,			lowercase_			,			lowercase_			,			lowercase_ )		->						Any:


  '''simple docstring'''






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


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






