


'''simple docstring'''







def _UpperCamelCase       (     __A      ,			__A      ,			__A      ,			__A    )							->		Optional[int]:


      '''simple docstring'''



      global f  # a global dp table for knapsack
      if f[i][j] < 0:
            if j < wt[i - 1]:
                  UpperCamelCase__										=				mf_knapsack(i - 1      ,			__SCREAMING_SNAKE_CASE      ,			__SCREAMING_SNAKE_CASE      ,			__SCREAMING_SNAKE_CASE    )
            else:
                  UpperCamelCase__										=				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]      ,			)
            UpperCamelCase__										=				val
      return f[i][j]







def _UpperCamelCase       (     __A      ,			__A      ,			__A      ,			__A    )							->		Optional[Any]:


      '''simple docstring'''



      UpperCamelCase__										=				[[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_:
                        UpperCamelCase__										=				max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]      ,			dp[i - 1][w_]    )
                  else:
                        UpperCamelCase__										=				dp[i - 1][w_]

      return dp[n][w_], dp







def _UpperCamelCase       (     __A      ,			__A      ,			__A    )							->		Dict:


      '''simple docstring'''



      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"    )

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

      UpperCamelCase__					,	UpperCamelCase__										=				knapsack(__SCREAMING_SNAKE_CASE      ,			__SCREAMING_SNAKE_CASE      ,			__SCREAMING_SNAKE_CASE      ,			__SCREAMING_SNAKE_CASE    )
      UpperCamelCase__										=				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 _UpperCamelCase       (     __A      ,			__A      ,			__A      ,			__A      ,			__A    )							->		Optional[int]:


      '''simple docstring'''



      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__":
  a__  :					Tuple        =       [3, 2, 4, 4]
  a__  :					int        =       [4, 3, 2, 3]
  a__  :					Dict        =       4
  a__  :					int        =       6
  a__  :					Dict        =       [[0] * (w + 1)] + [[0] + [-1] * (w + 1) for _ in range(n + 1)]
  a__ , a__  :					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
  a__ , a__  :					List[str]        =       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)







def 							UpperCAmelCase_		(							__SCREAMING_SNAKE_CASE							):
	if not isinstance(__SCREAMING_SNAKE_CASE ,							__SCREAMING_SNAKE_CASE							):
		raise TypeError('Input value must be an \'int\' type'							)

	lowercase       =    0
	while number:
		position += 1
		number >>= 1

	return position


if __name__ == "__main__":
	import doctest

	doctest.testmod()

