



def 	lowerCAmelCase__    (				_SCREAMING_SNAKE_CASE							:      int					,  _SCREAMING_SNAKE_CASE							:      str					,  _SCREAMING_SNAKE_CASE							:      Optional[Any]					,  _SCREAMING_SNAKE_CASE							:      List[Any]     ):
				"""simple docstring"""







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

def 	lowerCAmelCase__    (				_SCREAMING_SNAKE_CASE							:      List[Any]					,  _SCREAMING_SNAKE_CASE							:      Optional[int]					,  _SCREAMING_SNAKE_CASE							:      Optional[int]					,  _SCREAMING_SNAKE_CASE							:      Union[str, Any]     ):
				"""simple docstring"""







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

				return dp[n][w_], dp

def 	lowerCAmelCase__    (				_SCREAMING_SNAKE_CASE							:      Tuple					,  _SCREAMING_SNAKE_CASE							:      str					,  _SCREAMING_SNAKE_CASE							:      Any     ):
				"""simple docstring"""







				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"""     )

				__a	  =		len(UpperCAmelCase__     )
				if num_items != len(UpperCAmelCase__     ):
								__a	  =		(
								    """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__     ):
												__a	  =		(
												    """All weights must be integers but got weight of """
												    f"type {type(wt[i]     )} at index {i}"
												)
												raise TypeError(UpperCAmelCase__     )

				__a  ,       __a	  =		knapsack(UpperCAmelCase__					,  UpperCAmelCase__					,  UpperCAmelCase__					,  UpperCAmelCase__     )
				__a	  =		set()
				_construct_solution(UpperCAmelCase__					,  UpperCAmelCase__					,  UpperCAmelCase__					,  UpperCAmelCase__					,  UpperCAmelCase__     )

				return optimal_val, example_optional_set

def 	lowerCAmelCase__    (				_SCREAMING_SNAKE_CASE							:      Tuple					,  _SCREAMING_SNAKE_CASE							:      List[str]					,  _SCREAMING_SNAKE_CASE							:      Optional[Any]					,  _SCREAMING_SNAKE_CASE							:      Optional[int]					,  _SCREAMING_SNAKE_CASE							:      Tuple     ):
				"""simple docstring"""







				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__      =			[3, 2, 4, 4]
						lowerCamelCase__      =			[4, 3, 2, 3]
						lowerCamelCase__      =			4
						lowerCamelCase__      =			6
						lowerCamelCase__      =			[[0] * (w + 1)] + [[0] + [-1] * (w + 1) for _ in range(n + 1)]
						lowerCamelCase__      =			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__      =			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)





a_    :  Dict						  =		8.314462  # Unit - J mol-1 K-1

def      __lowercase(						UpperCAmelCase__ , UpperCAmelCase__ , UpperCAmelCase__			):


      """simple docstring"""



      if moles < 0 or kelvin < 0 or volume < 0:
            raise ValueError("Invalid inputs. Enter positive value."			)
      return moles * kelvin * UNIVERSAL_GAS_CONSTANT / volume

def      __lowercase(						UpperCAmelCase__ , UpperCAmelCase__ , UpperCAmelCase__			):


      """simple docstring"""



      if moles < 0 or kelvin < 0 or pressure < 0:
            raise ValueError("Invalid inputs. Enter positive value."			)
      return moles * kelvin * UNIVERSAL_GAS_CONSTANT / pressure


if __name__ == "__main__":
     from doctest import testmod

     testmod()