
"""simple docstring"""







def    __lowerCAmelCase    (_UpperCamelCase	,  _UpperCamelCase	,  _UpperCamelCase	,  _UpperCamelCase		):
	global f  # a global dp table for knapsack
	if f[i][j] < 0:
		if j < wt[i - 1]:
			__lowerCAmelCase      :						int						      =				mf_knapsack(i - 1	,  _UpperCamelCase	,  _UpperCamelCase	,  _UpperCamelCase		)
		else:
			__lowerCAmelCase      :						Any						      =				max(
			    mf_knapsack(i - 1	,  _UpperCamelCase	,  _UpperCamelCase	,  _UpperCamelCase		)	,  mf_knapsack(i - 1	,  _UpperCamelCase	,  _UpperCamelCase	,  j - wt[i - 1]		) + val[i - 1]	,  )
		__lowerCAmelCase      :						int						      =				val
	return f[i][j]
def    __lowerCAmelCase    (_UpperCamelCase	,  _UpperCamelCase	,  _UpperCamelCase	,  _UpperCamelCase		):
	__lowerCAmelCase      :						Tuple						      =				[[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      :						Optional[Any]						      =				max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]	,  dp[i - 1][w_]		)
			else:
				__lowerCAmelCase      :						int						      =				dp[i - 1][w_]

	return dp[n][w_], dp
def    __lowerCAmelCase    (_UpperCamelCase	,  _UpperCamelCase	,  _UpperCamelCase		):
	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'		)

	__lowerCAmelCase      :						Tuple						      =				len(_UpperCamelCase		)
	if num_items != len(_UpperCamelCase		):
		__lowerCAmelCase      :						str						      =				(
		    '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		):
			__lowerCAmelCase      :						List[Any]						      =				(
			    'All weights must be integers but got weight of '
			    F"type {type(wt[i]		)} at index {i}"
			)
			raise TypeError(_UpperCamelCase		)

	__lowerCAmelCase      ,       __lowerCAmelCase      :						Tuple						      =				knapsack(_UpperCamelCase	,  _UpperCamelCase	,  _UpperCamelCase	,  _UpperCamelCase		)
	__lowerCAmelCase      :						set						      =				set()
	_construct_solution(_UpperCamelCase	,  _UpperCamelCase	,  _UpperCamelCase	,  _UpperCamelCase	,  _UpperCamelCase		)

	return optimal_val, example_optional_set
def    __lowerCAmelCase    (_UpperCamelCase	,  _UpperCamelCase	,  _UpperCamelCase	,  _UpperCamelCase	,  _UpperCamelCase		):
	# 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__        =		[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__ ,						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__ ,						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)

"""simple docstring"""







import os
def    __lowerCAmelCase    (_UpperCamelCase = "input.txt"		):

	with open(os.path.join(os.path.dirname(_UpperCamelCase		)	,  _UpperCamelCase		)		) as input_file:
		__lowerCAmelCase      :						Optional[Any]						      =				[
		    [int(_UpperCamelCase		) for element in line.split(','		)]
		    for line in input_file.readlines()
		]

	__lowerCAmelCase      :						List[str]						      =				len(_UpperCamelCase		)
	__lowerCAmelCase      :						Tuple						      =				len(matrix[0]		)

	__lowerCAmelCase      :						int						      =				[[-1 for _ in range(_UpperCamelCase		)] for _ in range(_UpperCamelCase		)]
	for i in range(_UpperCamelCase		):
		__lowerCAmelCase      :						Any						      =				matrix[i][0]

	for j in range(1	,  _UpperCamelCase		):
		for i in range(_UpperCamelCase		):
			__lowerCAmelCase      :						Optional[Any]						      =				minimal_path_sums[i][j - 1] + matrix[i][j]

		for i in range(1	,  _UpperCamelCase		):
			__lowerCAmelCase      :						Optional[Any]						      =				min(
			    minimal_path_sums[i][j]	,  minimal_path_sums[i - 1][j] + matrix[i][j]		)

		for i in range(rows - 2	,  -1	,  -1		):
			__lowerCAmelCase      :						List[str]						      =				min(
			    minimal_path_sums[i][j]	,  minimal_path_sums[i + 1][j] + matrix[i][j]		)

	return min(minimal_path_sums_row[-1] for minimal_path_sums_row in minimal_path_sums		)


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