




"""simple docstring"""





class      a__				:


			def __init__(      self      :    Any):


						"""simple docstring"""




						__UpperCAmelCase	: Union[str, Any]		=					{}


			def       a_     (      self      :    Optional[Any]):


						"""simple docstring"""




						print(self.vertex)
						for i in self.vertex:
									print(UpperCamelCase_      ,  " -> "      ,  " -> ".join([str(UpperCamelCase_) for j in self.vertex[i]]))


			def       a_     (      self      :    Tuple      ,  UpperCamelCase_      :    int      ,  UpperCamelCase_      :    int):


						"""simple docstring"""




						if from_vertex in self.vertex:
									self.vertex[from_vertex].append(UpperCamelCase_)
						else:
									# else make a new vertex
									__UpperCAmelCase	: Tuple		=					[to_vertex]


			def       a_     (      self      :    str):


						"""simple docstring"""




						__UpperCAmelCase	: int		=					[False] * len(self.vertex)

						# call the recursive helper function
						for i in range(len(self.vertex)):
									if not visited[i]:
												self.dfs_recursive(UpperCamelCase_      ,  UpperCamelCase_)


			def       a_     (      self      :    Tuple      ,  UpperCamelCase_      :    int      ,  UpperCamelCase_      :    list):


						"""simple docstring"""




						__UpperCAmelCase	: List[str]		=					True

						print(UpperCamelCase_      ,  end=" ")

						# Recur for all the vertices that are adjacent to this node
						for i in self.vertex:
									if not visited[i]:
												self.dfs_recursive(UpperCamelCase_      ,  UpperCamelCase_)


if __name__ == "__main__":
					A   =       Graph()
					g.add_edge(0, 1)
					g.add_edge(0, 2)
					g.add_edge(1, 2)
					g.add_edge(2, 0)
					g.add_edge(2, 3)
					g.add_edge(3, 3)

					g.print_graph()
					print("""DFS:""")
					g.dfs()

					# OUTPUT:
					# 0  ->  1 -> 2
					# 1  ->  2
					# 2  ->  0 -> 3
					# 3  ->  3
					# DFS:
					#  0 1 2 3








"""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)


