


'''simple docstring'''

def __snake_case					(      UpperCAmelCase_						:						Union[str, Any]					,				UpperCAmelCase_						:						Union[str, Any]					,				UpperCAmelCase_						:						Tuple					,				UpperCAmelCase_						:						Optional[Any]		):
 global f  # a global dp table for knapsack
 if f[i][j] < 0:
  if j < wt[i - 1]:
   lowerCamelCase_      =  mf_knapsack(i - 1					,				_SCREAMING_SNAKE_CASE					,				_SCREAMING_SNAKE_CASE					,				_SCREAMING_SNAKE_CASE		)
  else:
   lowerCamelCase_      =  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]					,				)
  lowerCamelCase_      =  val
 return f[i][j]





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

 return dp[n][w_], dp





def __snake_case					(      UpperCAmelCase_						:						int					,				UpperCAmelCase_						:						Dict					,				UpperCAmelCase_						:						Tuple		):
 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"		)

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

 lowerCamelCase_      =  knapsack(_SCREAMING_SNAKE_CASE					,				_SCREAMING_SNAKE_CASE					,				_SCREAMING_SNAKE_CASE					,				_SCREAMING_SNAKE_CASE		)
 lowerCamelCase_      =  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					(      UpperCAmelCase_						:						List[Any]					,				UpperCAmelCase_						:						List[Any]					,				UpperCAmelCase_						:						int					,				UpperCAmelCase_						:						Any					,				UpperCAmelCase_						:						Tuple		):
 # 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__":
     a_  :				List[Any]						= [3, 2, 4, 4]
     a_  :				str						= [4, 3, 2, 3]
     a_  :				Union[str, Any]						= 4
     a_  :				Union[str, Any]						= 6
     a_  :				List[str]						= [[0] * (w + 1)] + [[0] + [-1] * (w + 1) for _ in range(n + 1)]
     a_	,   a_  :				Union[str, 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_  :				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)




"""simple docstring"""
def 						__a   (	_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )	->Union[str, Any]:
  if height >= 1:
    move_tower(height - 1		,     _SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )
    move_disk(_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )
    move_tower(height - 1		,     _SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )







def 						__a   (	_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )	->Union[str, Any]:
  print('moving disk from'		,     _SCREAMING_SNAKE_CASE		,     'to'		,     _SCREAMING_SNAKE_CASE   )







def 						__a   (	)	->List[str]:
  a__:     Dict										=	int(input('Height of hanoi: '   ).strip()   )
  move_tower(_SCREAMING_SNAKE_CASE		,     'A'		,     'B'		,     'C'   )


if __name__ == "__main__":
 main()


