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







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

       return dp[n][w_], dp







def 						_snake_case(  SCREAMING_SNAKE_CASE__ ,     SCREAMING_SNAKE_CASE__ ,     SCREAMING_SNAKE_CASE__    )			->   Optional[Any]:
       if not (isinstance(SCREAMING_SNAKE_CASE__ ,     (list, tuple)    ) and isinstance(SCREAMING_SNAKE_CASE__ ,     (list, tuple)    )):
              raise ValueError(
                  """Both the weights and values vectors must be either lists or tuples"""    )

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

       lowercase     ,      lowercase							:					Any	     =				knapsack(SCREAMING_SNAKE_CASE__ ,     SCREAMING_SNAKE_CASE__ ,     SCREAMING_SNAKE_CASE__ ,     SCREAMING_SNAKE_CASE__    )
       lowercase							:					set	     =				set()
       _construct_solution(SCREAMING_SNAKE_CASE__ ,     SCREAMING_SNAKE_CASE__ ,     SCREAMING_SNAKE_CASE__ ,     SCREAMING_SNAKE_CASE__ ,     SCREAMING_SNAKE_CASE__    )

       return optimal_val, example_optional_set







def 						_snake_case(  SCREAMING_SNAKE_CASE__ ,     SCREAMING_SNAKE_CASE__ ,     SCREAMING_SNAKE_CASE__ ,     SCREAMING_SNAKE_CASE__ ,     SCREAMING_SNAKE_CASE__    )			->   Optional[int]:
       # 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(SCREAMING_SNAKE_CASE__ ,     SCREAMING_SNAKE_CASE__ ,     i - 1 ,     SCREAMING_SNAKE_CASE__ ,     SCREAMING_SNAKE_CASE__    )
              else:
                     optimal_set.add(SCREAMING_SNAKE_CASE__    )
                     _construct_solution(SCREAMING_SNAKE_CASE__ ,     SCREAMING_SNAKE_CASE__ ,     i - 1 ,     j - wt[i - 1] ,     SCREAMING_SNAKE_CASE__    )


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


class __snake_case		:
    def __init__(  self		,snake_case		,snake_case=None		,snake_case=None				):



           '''simple docstring'''







           lowercase							:					Tuple	     =				data
           lowercase							:					List[Any]	     =				previous
           lowercase							:					List[str]	     =				next_node
    def __str__(  self				):



           '''simple docstring'''







           return f"{self.data}"
    def 				_SCREAMING_SNAKE_CASE	(  self				):



           '''simple docstring'''







           return self.data
    def 				_SCREAMING_SNAKE_CASE	(  self				):



           '''simple docstring'''







           return self.next
    def 				_SCREAMING_SNAKE_CASE	(  self				):



           '''simple docstring'''







           return self.previous






class __snake_case		:
    def __init__(  self		,snake_case				):



           '''simple docstring'''







           lowercase							:					Optional[int]	     =				head
    def __iter__(  self				):



           '''simple docstring'''







           return self
    def 				_SCREAMING_SNAKE_CASE	(  self				):



           '''simple docstring'''







           if not self.current:
                  raise StopIteration
           else:
                  lowercase							:					Union[str, Any]	     =				self.current.get_data()
                  lowercase							:					Optional[Any]	     =				self.current.get_next()
                  return value






class __snake_case		:
    def __init__(  self				):



           '''simple docstring'''







           lowercase							:					str	     =				None  # First node in list
           lowercase							:					str	     =				None  # Last node in list
    def __str__(  self				):



           '''simple docstring'''







           lowercase							:					int	     =				self.head
           lowercase							:					str	     =				[]
           while current is not None:
                  nodes.append(current.get_data()				)
                  lowercase							:					Dict	     =				current.get_next()
           return " ".join(str(snake_case				) for node in nodes				)
    def __contains__(  self		,snake_case				):



           '''simple docstring'''







           lowercase							:					Dict	     =				self.head
           while current:
                  if current.get_data() == value:
                         return True
                  lowercase							:					Any	     =				current.get_next()
           return False
    def __iter__(  self				):



           '''simple docstring'''







           return LinkedListIterator(self.head				)
    def 				_SCREAMING_SNAKE_CASE	(  self				):



           '''simple docstring'''







           if self.head:
                  return self.head.get_data()
           return None
    def 				_SCREAMING_SNAKE_CASE	(  self				):



           '''simple docstring'''







           if self.tail:
                  return self.tail.get_data()
           return None
    def 				_SCREAMING_SNAKE_CASE	(  self		,snake_case				):



           '''simple docstring'''







           if self.head is None:
                  lowercase							:					Any	     =				node
                  lowercase							:					Dict	     =				node
           else:
                  self.insert_before_node(self.head		,snake_case				)
    def 				_SCREAMING_SNAKE_CASE	(  self		,snake_case				):



           '''simple docstring'''







           if self.head is None:
                  self.set_head(snake_case				)
           else:
                  self.insert_after_node(self.tail		,snake_case				)
    def 				_SCREAMING_SNAKE_CASE	(  self		,snake_case				):



           '''simple docstring'''







           lowercase							:					Dict	     =				Node(snake_case				)
           if self.head is None:
                  self.set_head(snake_case				)
           else:
                  self.set_tail(snake_case				)
    def 				_SCREAMING_SNAKE_CASE	(  self		,snake_case		,snake_case				):



           '''simple docstring'''







           lowercase							:					Optional[int]	     =				node
           lowercase							:					Optional[int]	     =				node.previous

           if node.get_previous() is None:
                  lowercase							:					Optional[int]	     =				node_to_insert
           else:
                  lowercase							:					Optional[int]	     =				node_to_insert

           lowercase							:					List[Any]	     =				node_to_insert
    def 				_SCREAMING_SNAKE_CASE	(  self		,snake_case		,snake_case				):



           '''simple docstring'''







           lowercase							:					Dict	     =				node
           lowercase							:					List[str]	     =				node.next

           if node.get_next() is None:
                  lowercase							:					Union[str, Any]	     =				node_to_insert
           else:
                  lowercase							:					List[str]	     =				node_to_insert

           lowercase							:					Dict	     =				node_to_insert
    def 				_SCREAMING_SNAKE_CASE	(  self		,snake_case		,snake_case				):



           '''simple docstring'''







           lowercase							:					List[Any]	     =				1
           lowercase							:					List[str]	     =				Node(snake_case				)
           lowercase							:					Any	     =				self.head
           while node:
                  if current_position == position:
                         self.insert_before_node(snake_case		,snake_case				)
                         return
                  current_position += 1
                  lowercase							:					List[Any]	     =				node.next
           self.insert_after_node(self.tail		,snake_case				)
    def 				_SCREAMING_SNAKE_CASE	(  self		,snake_case				):



           '''simple docstring'''







           lowercase							:					Optional[int]	     =				self.head
           while node:
                  if node.get_data() == item:
                         return node
                  lowercase							:					Any	     =				node.get_next()
           raise Exception("""Node not found"""				)
    def 				_SCREAMING_SNAKE_CASE	(  self		,snake_case				):



           '''simple docstring'''







           if (node := self.get_node(snake_case				)) is not None:
                  if node == self.head:
                         lowercase							:					Optional[Any]	     =				self.head.get_next()

                  if node == self.tail:
                         lowercase							:					List[str]	     =				self.tail.get_previous()

                  self.remove_node_pointers(snake_case				)
    @staticmethod
    def 				_SCREAMING_SNAKE_CASE	(  snake_case				):



           '''simple docstring'''







           if node.get_next():
                  lowercase							:					Optional[int]	     =				node.previous

           if node.get_previous():
                  lowercase							:					Union[str, Any]	     =				node.next

           lowercase							:					Union[str, Any]	     =				None
           lowercase							:					Optional[Any]	     =				None
    def 				_SCREAMING_SNAKE_CASE	(  self				):



           '''simple docstring'''







           return self.head is None







def 						_snake_case(  )			->   None:
       pass


if __name__ == "__main__":
  import doctest

  doctest.testmod()
