Knapsack with continuous (non distinct) constraint

Question

I watched Dynamic Programming - Kapsack Problem (YouTube). However, I am solving a slightly different problem where the constraint is the budget, price, in double, not integer. So I am wondering how can I modify that? Double is "continuous" unlike integer where I can have 1,2,3 .... I don't suppose I do 0.0, 0.1, 0.2 ...?

UPDATE 1

I thought of converting double to int by multiply by 100. Money is only 2 decimal places. But that will mean the range of values will be very large?

UPDATE 2

The problem I need to solve is:

Items have a price (double) & satisfaction (integer) value. I have a budget as a constraint and I need to maximize satisfaction value.

In the youtube video, the author created two 2d array like int[numItems][possibleCapacity(weight)]. Here, I can't as budget is a double not integer

Answer 1

If you want to use floating point numbers with arbitrary precision (i.e., don't have a fixed number of decimals), and these are not fractions, dynamic programming won't work.

The basis of dynamic programming is to store previous results of a calculation for specific inputs. Therefore, if you used floating point numbers with arbitrary precision, you would need practically infinite memory for each of the possible floating point numbers and, of course, do infinite calculations, something that is impossible and non-optimal.

However, if these numbers have a fixed precision (as with the money, which only have two decimal numbers), you can convert these into integers by multiplying them (as you've mentioned), and then solve the knapsack problem as usual.