



"""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)






'''simple docstring'''

def        UpperCAmelCase_ (  lowerCamelCase_			):







 """simple docstring"""


 lowerCAmelCase__     :   str     		=	len(lowerCamelCase_			)
 lowerCAmelCase__     :   Optional[Any]     		=	len(matrix[0]			)
 lowerCAmelCase__     :   Any     		=	min(lowerCamelCase_      ,			lowerCamelCase_			)

 for row in range(lowerCamelCase_			):
  # Check if diagonal element is not zero
  if matrix[row][row] != 0:
   # Eliminate all the elements below the diagonal
   for col in range(row + 1      ,			lowerCamelCase_			):
    lowerCAmelCase__     :   Tuple     		=	matrix[col][row] / matrix[row][row]
    for i in range(lowerCamelCase_      ,			lowerCamelCase_			):
     matrix[col][i] -= multiplier * matrix[row][i]
  else:
   # Find a non-zero diagonal element to swap rows
   lowerCAmelCase__     :   Dict     		=	True
   for i in range(row + 1      ,			lowerCamelCase_			):
    if matrix[i][row] != 0:
     lowerCAmelCase__		,     lowerCAmelCase__     :   Optional[Any]     		=	matrix[i], matrix[row]
     lowerCAmelCase__     :   Optional[Any]     		=	False
     break
   if reduce:
    rank -= 1
    for i in range(lowerCamelCase_			):
     lowerCAmelCase__     :   Union[str, Any]     		=	matrix[i][rank]

            # Reduce the row pointer by one to stay on the same row
   row -= 1

 return rank


if __name__ == "__main__":
     import doctest

     doctest.testmod()





