



def        lowercase_							(A : Optional[int]      ):  # noqa: E741
       snake_case__							:	Dict   	=	len(A      )
       snake_case__							:	str   	=	0
       snake_case__							:	Any   	=	[0] * n
       snake_case__							:	Optional[Any]   	=	[False] * n
       snake_case__							:	List[Any]   	=	[False] * n

       def dfs(A : Dict		,					A : Optional[Any]		,					A : List[Any]		,					A : Union[str, Any]      ):
              if parent == root:
                     out_edge_count += 1
              snake_case__							:	Any   	=	True
              snake_case__							:	List[str]   	=	at

              for to in l[at]:
                     if to == parent:
                            pass
                     elif not visited[to]:
                            snake_case__							:	Union[str, Any]   	=	dfs(A		,					A		,					A		,					A      )
                            snake_case__							:	Optional[Any]   	=	min(low[at]		,					low[to]      )

                            # AP found via bridge
                            if at < low[to]:
                                   snake_case__							:	Any   	=	True
                            # AP found via cycle
                            if at == low[to]:
                                   snake_case__							:	List[Any]   	=	True
                     else:
                            snake_case__							:	Tuple   	=	min(low[at]		,					A      )
              return out_edge_count

       for i in range(A      ):
              if not visited[i]:
                     snake_case__							:	Dict   	=	0
                     snake_case__							:	List[str]   	=	dfs(A		,					A		,					-1		,					A      )
                     snake_case__							:	Dict   	=	out_edge_count > 1

       for x in range(len(A      )      ):
              if is_art[x] is True:
                     print(A      )


# Adjacency list of graph
a_   :Any        =      {
    0: [1, 2],
    1: [0, 2],
    2: [0, 1, 3, 5],
    3: [2, 4],
    4: [3],
    5: [2, 6, 8],
    6: [5, 7],
    7: [6, 8],
    8: [5, 7],
}





compute_ap(data)











def        lowercase_							(A : Union[str, Any]		,					A : List[str]		,					A : int		,					A : Optional[int]      ):
       global f  # a global dp table for knapsack
       if f[i][j] < 0:
              if j < wt[i - 1]:
                     snake_case__							:	Union[str, Any]   	=	mf_knapsack(i - 1		,					A		,					A		,					A      )
              else:
                     snake_case__							:	Any   	=	max(
                         mf_knapsack(i - 1		,					A		,					A		,					A      )		,					mf_knapsack(i - 1		,					A		,					A		,					j - wt[i - 1]      ) + val[i - 1]		,					)
              snake_case__							:	Optional[int]   	=	val
       return f[i][j]
def        lowercase_							(A : Optional[int]		,					A : Union[str, Any]		,					A : str		,					A : Dict      ):
       snake_case__							:	int   	=	[[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_:
                            snake_case__							:	Union[str, Any]   	=	max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]		,					dp[i - 1][w_]      )
                     else:
                            snake_case__							:	str   	=	dp[i - 1][w_]

       return dp[n][w_], dp
def        lowercase_							(A : int		,					A : list		,					A : list      ):
       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'      )

       snake_case__							:	Dict   	=	len(A      )
       if num_items != len(A      ):
              snake_case__							:	str   	=	(
                  '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      ):
                     snake_case__							:	Optional[int]   	=	(
                         'All weights must be integers but got weight of '
                         F'''type {type(wt[i]      )} at index {i}'''
                     )
                     raise TypeError(A      )

       snake_case__	,				snake_case__							:	Optional[int]   	=	knapsack(A		,					A		,					A		,					A      )
       snake_case__							:	set   	=	set()
       _construct_solution(A		,					A		,					A		,					A		,					A      )

       return optimal_val, example_optional_set
def        lowercase_							(A : list		,					A : list		,					A : int		,					A : int		,					A : set      ):
       # 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__":
 a_   :Any        =      [3, 2, 4, 4]
 a_   :List[Any]        =      [4, 3, 2, 3]
 a_   :Union[str, Any]        =      4
 a_   :List[str]        =      6
 a_   :Union[str, Any]        =      [[0] * (w + 1)] + [[0] + [-1] * (w + 1) for _ in range(n + 1)]
 a_  ,						a_   :List[Any]        =      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_   :Any        =      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)






