



"""simple docstring"""





def   a__      ( lowerCAmelCase__      ,     lowerCAmelCase__      ,     lowerCAmelCase__      ,     lowerCAmelCase__ ):
      global f  # a global dp table for knapsack
      if f[i][j] < 0:
            if j < wt[i - 1]:
                  UpperCAmelCase_     					=				mf_knapsack(i - 1      ,     lowerCAmelCase__      ,     lowerCAmelCase__      ,     lowerCAmelCase__ )
            else:
                  UpperCAmelCase_     					=				max(
                      mf_knapsack(i - 1      ,     lowerCAmelCase__      ,     lowerCAmelCase__      ,     lowerCAmelCase__ )      ,     mf_knapsack(i - 1      ,     lowerCAmelCase__      ,     lowerCAmelCase__      ,     j - wt[i - 1] ) + val[i - 1]      ,     )
            UpperCAmelCase_     					=				val
      return f[i][j]
def   a__      ( lowerCAmelCase__      ,     lowerCAmelCase__      ,     lowerCAmelCase__      ,     lowerCAmelCase__ ):
      UpperCAmelCase_     					=				[[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_     					=				max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]      ,     dp[i - 1][w_] )
                  else:
                        UpperCAmelCase_     					=				dp[i - 1][w_]

      return dp[n][w_], dp
def   a__      ( lowerCAmelCase__      ,     lowerCAmelCase__      ,     lowerCAmelCase__ ):
      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_     					=				len(lowerCAmelCase__ )
      if num_items != len(lowerCAmelCase__ ):
            UpperCAmelCase_     					=				(
                "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_     					=				(
                      "All weights must be integers but got weight of "
                      f"""type {type(wt[i] )} at index {i}"""
                  )
                  raise TypeError(lowerCAmelCase__ )

      UpperCAmelCase_					,       UpperCAmelCase_     					=				knapsack(lowerCAmelCase__      ,     lowerCAmelCase__      ,     lowerCAmelCase__      ,     lowerCAmelCase__ )
      UpperCAmelCase_     					=				set()
      _construct_solution(lowerCAmelCase__      ,     lowerCAmelCase__      ,     lowerCAmelCase__      ,     lowerCAmelCase__      ,     lowerCAmelCase__ )

      return optimal_val, example_optional_set
def   a__      ( lowerCAmelCase__      ,     lowerCAmelCase__      ,     lowerCAmelCase__      ,     lowerCAmelCase__      ,     lowerCAmelCase__ ):
      # 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__":
  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)







from __future__ import annotations

import math




def 			lowerCAmelCase(      SCREAMING_SNAKE_CASE_			,						SCREAMING_SNAKE_CASE_			,						SCREAMING_SNAKE_CASE_			,						SCREAMING_SNAKE_CASE_			,						SCREAMING_SNAKE_CASE_			)->						int:



    """simple docstring"""






    if depth < 0:
        raise ValueError("Depth cannot be less than 0"			)

    if len(SCREAMING_SNAKE_CASE_			) == 0:
        raise ValueError("Scores cannot be empty"			)

    if depth == height:
        return scores[node_index]

    if is_max:
        return max(
            minimax(depth + 1			,						node_index * 2			,						SCREAMING_SNAKE_CASE_			,						SCREAMING_SNAKE_CASE_			,						SCREAMING_SNAKE_CASE_			)			,						minimax(depth + 1			,						node_index * 2 + 1			,						SCREAMING_SNAKE_CASE_			,						SCREAMING_SNAKE_CASE_			,						SCREAMING_SNAKE_CASE_			)			,						)

    return min(
        minimax(depth + 1			,						node_index * 2			,						SCREAMING_SNAKE_CASE_			,						SCREAMING_SNAKE_CASE_			,						SCREAMING_SNAKE_CASE_			)			,						minimax(depth + 1			,						node_index * 2 + 1			,						SCREAMING_SNAKE_CASE_			,						SCREAMING_SNAKE_CASE_			,						SCREAMING_SNAKE_CASE_			)			,						)




def 			lowerCAmelCase(      )->						None:



    """simple docstring"""






    UpperCamelCase_			  =						[9_0, 2_3, 6, 3_3, 2_1, 6_5, 1_2_3, 3_4_4_2_3]
    UpperCamelCase_			  =						math.log(len(SCREAMING_SNAKE_CASE_			)			,						2			)
    print("Optimal value : "			,						end=""			)
    print(minimax(0			,						0			,						SCREAMING_SNAKE_CASE_			,						SCREAMING_SNAKE_CASE_			,						SCREAMING_SNAKE_CASE_			)			)


if __name__ == "__main__":
 import doctest

 doctest.testmod()
 main()
