Question

In the o-1 knapsack recurrence formula f(x,i) = max vi + f[x-wi , i-1] , f[x , i -1] , the first part vi + f[x-wi , i-1] represents?

1) Adding the ith item to the knapsack
2) Not adding the ith item to the knapsack

Answer 1

In the 0-1 knapsack recursion formula, the term vi + f[x-wi, i-1] signifies adding the ith item to the knapsack, while f[x, i-1] signifies not adding it.

Step-by-step explanation:

The first part of the 0-1 knapsack recurrence formula, vi + f[x-wi , i-1], corresponds to the option of adding the ith item to the knapsack. This term represents the value of taking the item (which has a value vi) plus the optimal solution (maximum value) for the remaining capacity (x-wi) using the first i-1 items.

In contrast, the second part, f[x , i-1], represents the option of not adding the ith item to the knapsack and simply taking the optimal solution for the same capacity x with the first i-1 items.