






"""simple docstring"""
def 							snake_case (  A__      ,A__      ,A__      ,A__		):
			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      ,A__      ,A__      ,A__		)
						else:
									UpperCAmelCase_	:					str     	=   max(
									    mf_knapsack(i - 1      ,A__      ,A__      ,A__		)      ,mf_knapsack(i - 1      ,A__      ,A__      ,j - wt[i - 1]		) + val[i - 1]      ,)
						UpperCAmelCase_	:					int     	=   val
			return f[i][j]

def 							snake_case (  A__      ,A__      ,A__      ,A__		):
			UpperCAmelCase_	:					Optional[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_	:					Optional[int]     	=   dp[i - 1][w_]

			return dp[n][w_], dp

def 							snake_case (  A__      ,A__      ,A__		):
			if not (isinstance(A__      ,(list, tuple)		) and isinstance(A__      ,(list, tuple)		)):
						raise ValueError(
						    "Both the weights and values vectors must be either lists or tuples"		)

			UpperCAmelCase_	:					str     	=   len(A__		)
			if num_items != len(A__		):
						UpperCAmelCase_	:					Tuple     	=   (
						    "The number of weights must be the same as the number of values.\n"
						    F"""But got {num_items} weights and {len(A__		)} values"""
						)
						raise ValueError(A__		)
			for i in range(A__		):
						if not isinstance(wt[i]      ,A__		):
									UpperCAmelCase_	:					str     	=   (
									    "All weights must be integers but got weight of "
									    F"""type {type(wt[i]		)} at index {i}"""
									)
									raise TypeError(A__		)

			UpperCAmelCase_	,      UpperCAmelCase_	:					Tuple     	=   knapsack(A__      ,A__      ,A__      ,A__		)
			UpperCAmelCase_	:					set     	=   set()
			_construct_solution(A__      ,A__      ,A__      ,A__      ,A__		)

			return optimal_val, example_optional_set

def 							snake_case (  A__      ,A__      ,A__      ,A__      ,A__		):
			# 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(A__      ,A__      ,i - 1      ,A__      ,A__		)
						else:
									optimal_set.add(A__		)
									_construct_solution(A__      ,A__      ,i - 1      ,j - wt[i - 1]      ,A__		)


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)








_lowercase    	=		[0, 2, 4, 6, 8]
_lowercase    	=		[1, 3, 5, 7, 9]



def 		lowerCAmelCase__     (       UpperCamelCase_					:							int			,							UpperCamelCase_					:							int			,							UpperCamelCase_					:							list[int]			,							UpperCamelCase_					:							int						)->	int:
    if remaining_length == 0:
        if digits[0] == 0 or digits[-1] == 0:
            return 0

        for i in range(length // 2 - 1			,							-1			,							-1						):
            remainder += digits[i] + digits[length - i - 1]

            if remainder % 2 == 0:
                return 0

            remainder //= 1_0

        return 1

    if remaining_length == 1:
        if remainder % 2 == 0:
            return 0

        A__					=       0
        for digit in range(1_0						):
            A__					=       digit
            result += reversible_numbers(
                0			,							(remainder + 2 * digit) // 1_0			,							UpperCamelCase_			,							UpperCamelCase_						)
        return result

    A__					=       0
    for digita in range(1_0						):
        A__					=       digita

        if (remainder + digita) % 2 == 0:
            A__					=       ODD_DIGITS
        else:
            A__					=       EVEN_DIGITS

        for digita in other_parity_digits:
            A__					=       digita
            result += reversible_numbers(
                remaining_length - 2			,							(remainder + digita + digita) // 1_0			,							UpperCamelCase_			,							UpperCamelCase_			,							)
    return result



def 		lowerCAmelCase__     (       UpperCamelCase_					:							int = 9						)->	int:
    A__					=       0
    for length in range(1			,							max_power + 1						):
        result += reversible_numbers(UpperCamelCase_			,							0			,							[0] * length			,							UpperCamelCase_						)
    return result


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