






"""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_     : Dict			     =					mf_knapsack(i - 1	,A__	,A__	,A__						)
						else:
									UpperCAmelCase_     : int			     =					max(
									    mf_knapsack(i - 1	,A__	,A__	,A__						)	,mf_knapsack(i - 1	,A__	,A__	,j - wt[i - 1]						) + val[i - 1]	,)
						UpperCAmelCase_     : Union[str, Any]			     =					val
			return f[i][j]

def 				snake_case      (   A__	,A__	,A__	,A__						):
			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_     : Any			     =					max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]	,dp[i - 1][w_]						)
									else:
												UpperCAmelCase_     : Optional[Any]			     =					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_     : Optional[Any]			     =					len(A__						)
			if num_items != len(A__						):
						UpperCAmelCase_     : Any			     =					(
						    "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_     : int			     =					(
									    "All weights must be integers but got weight of "
									    F"""type {type(wt[i]						)} at index {i}"""
									)
									raise TypeError(A__						)

			UpperCAmelCase_     ,	UpperCAmelCase_     : List[str]			     =					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)











"""simple docstring"""
from __future__ import annotations







class 			UpperCamelCase_       :





	def __init__(      self					: Any	,	lowerCAmelCase_					: int						)						->			None:
				UpperCAmelCase_     : Any			     =					data
				UpperCAmelCase_     : Node | None			     =					None
				UpperCAmelCase_     : Node | None			     =					None





def 				snake_case      (   A__						):  # In Order traversal of the tree
			if tree:
						display(tree.left						)
						print(tree.data						)
						display(tree.right						)

def 				snake_case      (   A__						):
			return 1 + max(depth_of_tree(tree.left						)	,depth_of_tree(tree.right						)						) if tree else 0

def 				snake_case      (   A__						):
			if not tree:
						return True
			if tree.left and tree.right:
						return is_full_binary_tree(tree.left						) and is_full_binary_tree(tree.right						)
			else:
						return not tree.left and not tree.right

def 				snake_case      (   ):  # Main function for testing.
			UpperCAmelCase_     : List[str]			     =					Node(1						)
			UpperCAmelCase_     : Any			     =					Node(2						)
			UpperCAmelCase_     : Optional[Any]			     =					Node(3						)
			UpperCAmelCase_     : Union[str, Any]			     =					Node(4						)
			UpperCAmelCase_     : int			     =					Node(5						)
			UpperCAmelCase_     : Optional[int]			     =					Node(6						)
			UpperCAmelCase_     : Any			     =					Node(7						)
			UpperCAmelCase_     : List[str]			     =					Node(8						)
			UpperCAmelCase_     : List[Any]			     =					Node(9						)

			print(is_full_binary_tree(A__						)						)
			print(depth_of_tree(A__						)						)
			print("Tree is: "						)
			display(A__						)


if __name__ == "__main__":
			main()



