Final answer:
The recurrence formula part 'vi + f[x-wi, i-1]' in the 0-1 knapsack problem represents adding the ith item to the knapsack, where 'vi' is the item's value, 'wi' is its weight, and 'f[x-wi, i-1]' is the optimal value with the remaining capacity.
Step-by-step explanation:
In the context of the 0-1 knapsack problem, the recurrence formula f(x, i) = max(vi + f[x-wi, i-1], f[x, i-1]) is used to determine the optimal subset of items to include in a knapsack without exceeding its capacity. The part of the formula vi + f[x-wi, i-1] represents adding the ith item to the knapsack. Here, vi is the value of the ith item, wi is the weight of the ith item, and f[x-wi, i-1] is the optimal value that can be obtained with the remaining capacity after including the ith item. By choosing the maximum value between including or not including the ith item, the formula ensures that the solution is optimal at each step
The first part vi + f[x-wi , i-1] in the o-1 knapsack recurrence formula f(x,i) = max vi + f[x-wi , i-1] , f[x , i -1] represents the value of adding the ith item to the knapsack. It is the sum of the value of the ith item (vi) and the maximum value that can be obtained by considering the remaining capacity (x-wi) and the previous item (i-1).