




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

def 					UpperCAmelCase_							(      __UpperCAmelCase						:	List[Any]			,			__UpperCAmelCase						:	List[Any]			,			__UpperCAmelCase						:	List[str]			,			__UpperCAmelCase						:	Union[str, Any]							)					-> Optional[Any]:
      SCREAMING_SNAKE_CASE_							=			[[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_:
                        SCREAMING_SNAKE_CASE_							=			max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]			,			dp[i - 1][w_]							)
                  else:
                        SCREAMING_SNAKE_CASE_							=			dp[i - 1][w_]

      return dp[n][w_], dp

def 					UpperCAmelCase_							(      __UpperCAmelCase						:	int			,			__UpperCAmelCase						:	list			,			__UpperCAmelCase						:	list							)					-> int:
      if not (isinstance(__UpperCAmelCase			,			(list, tuple)							) and isinstance(__UpperCAmelCase			,			(list, tuple)							)):
            raise ValueError(
                'Both the weights and values vectors must be either lists or tuples'							)

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

      SCREAMING_SNAKE_CASE_						, SCREAMING_SNAKE_CASE_							=			knapsack(__UpperCAmelCase			,			__UpperCAmelCase			,			__UpperCAmelCase			,			__UpperCAmelCase							)
      SCREAMING_SNAKE_CASE_							=			set()
      _construct_solution(__UpperCAmelCase			,			__UpperCAmelCase			,			__UpperCAmelCase			,			__UpperCAmelCase			,			__UpperCAmelCase							)

      return optimal_val, example_optional_set

def 					UpperCAmelCase_							(      __UpperCAmelCase						:	list			,			__UpperCAmelCase						:	list			,			__UpperCAmelCase						:	int			,			__UpperCAmelCase						:	int			,			__UpperCAmelCase						:	set							)					-> 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(__UpperCAmelCase			,			__UpperCAmelCase			,			i - 1			,			__UpperCAmelCase			,			__UpperCAmelCase							)
            else:
                  optimal_set.add(__UpperCAmelCase							)
                  _construct_solution(__UpperCAmelCase			,			__UpperCAmelCase			,			i - 1			,			j - wt[i - 1]			,			__UpperCAmelCase							)


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

def 					UpperCAmelCase_							(      __UpperCAmelCase						:	int							)					-> bool:

      if 1 < number < 4:
            # 2 and 3 are primes
            return True
      elif number < 2 or number % 2 == 0 or number % 3 == 0:
            # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
            return False

      # All primes number are in format of 6k +/- 1
      for i in range(5			,			int(sqrt(__UpperCAmelCase							) + 1							)			,			6							):
            if number % i == 0 or number % (i + 2) == 0:
                  return False
      return True

def 					UpperCAmelCase_							(      __UpperCAmelCase						:	int = 1_00_01							)					-> int:
      SCREAMING_SNAKE_CASE_							=			0
      SCREAMING_SNAKE_CASE_							=			1
      while count != nth and number < 3:
            number += 1
            if is_prime(__UpperCAmelCase							):
                  count += 1
      while count != nth:
            number += 2
            if is_prime(__UpperCAmelCase							):
                  count += 1
      return number


if __name__ == "__main__":
   print(f'''{solution() = }''')