

def 						_snake_case(    SCREAMING_SNAKE_CASE__			:	str      )							->      str:





    '''simple docstring'''






    A__          =   0

    # if input_string is "aba" than new_input_string become "a|b|a"
    A__          =   ''
    A__          =   ''

    # append each character + "|" in new_string for range(0, length-1)
    for i in input_string[: len(SCREAMING_SNAKE_CASE__      ) - 1]:
        new_input_string += i + "|"
    # append last character
    new_input_string += input_string[-1]

    # we will store the starting and ending of previous furthest ending palindromic
    # substring
    A__      ,						A__          =   0, 0

    # length[i] shows the length of palindromic substring with center i
    A__          =   [1 for i in range(len(SCREAMING_SNAKE_CASE__      )      )]

    # for each character in new_string find corresponding palindromic string
    A__          =   0
    for j in range(len(SCREAMING_SNAKE_CASE__      )      ):
        A__          =   1 if j > r else min(length[l + r - j] // 2    ,			r - j + 1      )
        while (
            j - k >= 0
            and j + k < len(SCREAMING_SNAKE_CASE__      )
            and new_input_string[k + j] == new_input_string[j - k]
        ):
            k += 1

        A__          =   2 * k - 1

        # does this string is ending after the previously explored end (that is r) ?
        # if yes the update the new r to the last index of this
        if j + k - 1 > r:
            A__          =   j - k + 1  # noqa: E741
            A__          =   j + k - 1

        # update max_length and start position
        if max_length < length[j]:
            A__          =   length[j]
            A__          =   j

    # create that string
    A__          =   new_input_string[start - max_length // 2 : start + max_length // 2 + 1]
    for i in s:
        if i != "|":
            output_string += i

    return output_string


if __name__ == "__main__":
     import doctest

     doctest.testmod()




def 						_snake_case(    SCREAMING_SNAKE_CASE__			:	Union[str, Any]    ,			SCREAMING_SNAKE_CASE__			:	Tuple    ,			SCREAMING_SNAKE_CASE__			:	Union[str, Any]    ,			SCREAMING_SNAKE_CASE__			:	Any      )							->      Optional[int]:





    '''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    ,			SCREAMING_SNAKE_CASE__    ,			SCREAMING_SNAKE_CASE__    ,			SCREAMING_SNAKE_CASE__      )
        else:
            A__          =   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]    ,			)
        A__          =   val
    return f[i][j]
def 						_snake_case(    SCREAMING_SNAKE_CASE__			:	Tuple    ,			SCREAMING_SNAKE_CASE__			:	Optional[int]    ,			SCREAMING_SNAKE_CASE__			:	Optional[int]    ,			SCREAMING_SNAKE_CASE__			:	Optional[int]      )							->      Tuple:





    '''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 						_snake_case(    SCREAMING_SNAKE_CASE__			:	int    ,			SCREAMING_SNAKE_CASE__			:	list    ,			SCREAMING_SNAKE_CASE__			:	list      )							->      Union[str, Any]:





    '''simple docstring'''






    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'      )

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

    A__      ,						A__          =   knapsack(SCREAMING_SNAKE_CASE__    ,			SCREAMING_SNAKE_CASE__    ,			SCREAMING_SNAKE_CASE__    ,			SCREAMING_SNAKE_CASE__      )
    A__          =   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(    SCREAMING_SNAKE_CASE__			:	list    ,			SCREAMING_SNAKE_CASE__			:	list    ,			SCREAMING_SNAKE_CASE__			:	int    ,			SCREAMING_SNAKE_CASE__			:	int    ,			SCREAMING_SNAKE_CASE__			:	set      )							->      Optional[int]:





    '''simple docstring'''






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

