How to find which elements are in the bag, using Knapsack Algorithm [and not only the bag's value]?

Question

Here I have code which calculates the optimal value using the knapsack algorithm (bin packing NP-hard problem):

int Knapsack::knapsack(std::vector<Item>& items, int W)
{
    size_t n = items.size();
    std::vector<std::vector<int> > dp(W + 1, std::vector<int>(n + 1, 0));
    for (size_t j = 1; j <= n; j++)
    {
        for ( int w = 1; w <= W; w++)
        {
            if (items[j-1].getWeight() <= w)
            {
                dp[w][j] = std::max(dp[w][j-1], dp[w - items[j-1].getWeight()][j-1] + items[j-1].getWeight());
            }
            else
            {
                dp[w][j] = dp[w][j - 1];
            }
        }
    }
    return dp[W][n];
}

I also need the elements included in the pack to be shown. I want to create an array to put the chosen elements. So the question is, in which step can I perform this selection? Is there any other more efficient way to determine which items have been taken?

I want to be able to know the items that give me the optimal solution, and not just the value of the best solution.

Answer 1

Getting the elements you packed from the matrix can be done using the data from the matrix without storing any additional data.

Pseudo code:

line <- W
i <- n
while (i > 0):
  if dp[line][i] - dp[line - weight(i)][i-1] == value(i):
      // the element 'i' is in the knapsack
      i <- i-1 // only in 0-1 knapsack
      line <- line - weight(i)
  else: 
      i <- i-1 

The idea behind it is that you iterate the matrix; if the weight difference is exactly the element's size, it is in the knapsack. If it is not, the item is not in the knapsack, go on without it.