




"""simple docstring"""





def 					_UpperCamelCase    (			UpperCamelCase				,       UpperCamelCase				,       UpperCamelCase				,       UpperCamelCase							)			->			Tuple:


			"""simple docstring"""
			global f  # a global dp table for knapsack
			if f[i][j] < 0:
						if j < wt[i - 1]:
									__UpperCAmelCase	: Optional[int]		=					mf_knapsack(i - 1				,       UpperCamelCase				,       UpperCamelCase				,       UpperCamelCase							)
						else:
									__UpperCAmelCase	: Any		=					max(
									    mf_knapsack(i - 1				,       UpperCamelCase				,       UpperCamelCase				,       UpperCamelCase							)				,       mf_knapsack(i - 1				,       UpperCamelCase				,       UpperCamelCase				,       j - wt[i - 1]							) + val[i - 1]				,       )
						__UpperCAmelCase	: str		=					val
			return f[i][j]

def 					_UpperCamelCase    (			UpperCamelCase				,       UpperCamelCase				,       UpperCamelCase				,       UpperCamelCase							)			->			List[Any]:


			"""simple docstring"""
			__UpperCAmelCase	: List[Any]		=					[[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	: Tuple		=					max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]				,       dp[i - 1][w_]							)
									else:
												__UpperCAmelCase	: int		=					dp[i - 1][w_]

			return dp[n][w_], dp

def 					_UpperCamelCase    (			UpperCamelCase				,       UpperCamelCase				,       UpperCamelCase							)			->			int:


			"""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"							)

			__UpperCAmelCase	: List[Any]		=					len(UpperCamelCase							)
			if num_items != len(UpperCamelCase							):
						__UpperCAmelCase	: int		=					(
						    "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							):
									__UpperCAmelCase	: Tuple		=					(
									    "All weights must be integers but got weight of "
									    f"type {type(wt[i]							)} at index {i}"
									)
									raise TypeError(UpperCamelCase							)

			__UpperCAmelCase	,     __UpperCAmelCase	: Tuple		=					knapsack(UpperCamelCase				,       UpperCamelCase				,       UpperCamelCase				,       UpperCamelCase							)
			__UpperCAmelCase	: set		=					set()
			_construct_solution(UpperCamelCase				,       UpperCamelCase				,       UpperCamelCase				,       UpperCamelCase				,       UpperCamelCase							)

			return optimal_val, example_optional_set

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


			"""simple docstring"""
			# 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__":
					A   =       [3, 2, 4, 4]
					A   =       [4, 3, 2, 3]
					A   =       4
					A   =       6
					A   =       [[0] * (w + 1)] + [[0] + [-1] * (w + 1) for _ in range(n + 1)]
					A ,						A   =       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   =       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 					_UpperCamelCase    (			UpperCamelCase							)			->			int:


			"""simple docstring"""

			if not grid or not grid[0]:
						raise TypeError("The grid does not contain the appropriate information"							)

			for cell_n in range(1				,       len(grid[0]							)							):
						grid[0][cell_n] += grid[0][cell_n - 1]
			__UpperCAmelCase	: int		=					grid[0]

			for row_n in range(1				,       len(UpperCamelCase							)							):
						__UpperCAmelCase	: int		=					grid[row_n]
						__UpperCAmelCase	: str		=					fill_row(UpperCamelCase				,       UpperCamelCase							)
						__UpperCAmelCase	: List[str]		=					grid[row_n]

			return grid[-1][-1]

def 					_UpperCamelCase    (			UpperCamelCase				,       UpperCamelCase							)			->			list:


			"""simple docstring"""
			current_row[0] += row_above[0]
			for cell_n in range(1				,       len(UpperCamelCase							)							):
						current_row[cell_n] += min(current_row[cell_n - 1]				,       row_above[cell_n]							)

			return current_row


if __name__ == "__main__":
					import doctest

					doctest.testmod()


