



'''simple docstring'''






def  snake_case_     (					_lowerCAmelCase  :						int   ,      _lowerCAmelCase  :						int   ,      _lowerCAmelCase  :						List[str]   ,      _lowerCAmelCase  :						str  )      ->   List[str]:
							global f  # a global dp table for knapsack
							if f[i][j] < 0:
														if j < wt[i - 1]:
																					UpperCAmelCase :       str 							=		mf_knapsack(i - 1   ,      _lowerCAmelCase   ,      _lowerCAmelCase   ,      _lowerCAmelCase  )
														else:
																					UpperCAmelCase :       int 							=		max(
																					    mf_knapsack(i - 1   ,      _lowerCAmelCase   ,      _lowerCAmelCase   ,      _lowerCAmelCase  )   ,      mf_knapsack(i - 1   ,      _lowerCAmelCase   ,      _lowerCAmelCase   ,      j - wt[i - 1]  ) + val[i - 1]   ,      )
														UpperCAmelCase :       List[str] 							=		val
							return f[i][j]


def  snake_case_     (					_lowerCAmelCase  :						Optional[Any]   ,      _lowerCAmelCase  :						str   ,      _lowerCAmelCase  :						str   ,      _lowerCAmelCase  :						Dict  )      ->   List[str]:
							UpperCAmelCase :       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 :       int 							=		max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]   ,      dp[i - 1][w_]  )
																					else:
																												UpperCAmelCase :       Union[str, Any] 							=		dp[i - 1][w_]

							return dp[n][w_], dp


def  snake_case_     (					_lowerCAmelCase  :						int   ,      _lowerCAmelCase  :						list   ,      _lowerCAmelCase  :						list  )      ->   Dict:
							if not (isinstance(_lowerCAmelCase   ,      (list, tuple)  ) and isinstance(_lowerCAmelCase   ,      (list, tuple)  )):
														raise ValueError(
														    '''Both the weights and values vectors must be either lists or tuples'''  )

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

							UpperCAmelCase							,		UpperCAmelCase :       Union[str, Any] 							=		knapsack(_lowerCAmelCase   ,      _lowerCAmelCase   ,      _lowerCAmelCase   ,      _lowerCAmelCase  )
							UpperCAmelCase :       set 							=		set()
							_construct_solution(_lowerCAmelCase   ,      _lowerCAmelCase   ,      _lowerCAmelCase   ,      _lowerCAmelCase   ,      _lowerCAmelCase  )

							return optimal_val, example_optional_set


def  snake_case_     (					_lowerCAmelCase  :						list   ,      _lowerCAmelCase  :						list   ,      _lowerCAmelCase  :						int   ,      _lowerCAmelCase  :						int   ,      _lowerCAmelCase  :						set  )      ->   int:
							# 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(_lowerCAmelCase   ,      _lowerCAmelCase   ,      i - 1   ,      _lowerCAmelCase   ,      _lowerCAmelCase  )
														else:
																					optimal_set.add(_lowerCAmelCase  )
																					_construct_solution(_lowerCAmelCase   ,      _lowerCAmelCase   ,      i - 1   ,      j - wt[i - 1]   ,      _lowerCAmelCase  )


if __name__ == "__main__":
						UpperCamelCase__:   Optional[int]								=			[3, 2, 4, 4]
						UpperCamelCase__:   List[str]								=			[4, 3, 2, 3]
						UpperCamelCase__:   List[Any]								=			4
						UpperCamelCase__:   Tuple								=			6
						UpperCamelCase__:   Optional[Any]								=			[[0] * (w + 1)] + [[0] + [-1] * (w + 1) for _ in range(n + 1)]
						UpperCamelCase__	,					UpperCamelCase__:   Union[str, 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
						UpperCamelCase__	,					UpperCamelCase__:   Tuple								=			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)









from __future__ import annotations

from typing import Any

def       lowercase_		(		_A			:    list[Any]						):

      """simple docstring"""




      create_state_space_tree(_A    ,					[]    ,					0						)

def       lowercase_		(		_A			:    list[Any]    ,					_A			:    list[Any]    ,					_A			:    int						):

      """simple docstring"""




      if index == len(_A						):
            print(_A						)
            return

      create_state_space_tree(_A    ,					_A    ,					index + 1						)
      current_subsequence.append(sequence[index]						)
      create_state_space_tree(_A    ,					_A    ,					index + 1						)
      current_subsequence.pop()


if __name__ == "__main__":
    A      :    list[Any]		=					[3, 1, 2, 4]
    generate_all_subsequences(seq)

    seq.clear()
    seq.extend(["A", "B", "C"])
    generate_all_subsequences(seq)


