Keeping Track of Dynamic Programming Steps

I'm teaching myself basic programming principles, and I'm stuck on a dynamic programming problem. Let's take the infamous Knapsack Problem:

Given a set of items, each with a weight and a value, determine the count of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.

Let's set the weight limit to 10, and let's give two lists: weights = [2,4,7] and values = [8,4,9] (I just made these up). I can write the code to give the maximum value given the constraint--that's not a problem. But what about if I want to know what values I actually ended up using? Not the total value--the individual values. So for this example, the maximum would be the objects with weights 2 and 7, for a total value of 8 + 9 = 17. I don't want my answer to read开发者_运维知识库 "17" though--I want an output of a list like: (8, 9). It might be easy for this problem, but the problem I'm working on uses lists that are much bigger and that have repeat numbers (for example, multiple objects have a value of 8).

Let me know if anyone can think of anything. As always, much love and appreciation to the community.

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Consider each partial solution a Node. Simply add whatever you use into each of these nodes and whichever node becomes the answer at the end will contain the set of items you used.

So each time you find a new solution you just set the list of items to the list of items of the new optimal solution and repeat for each.

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A basic array implementation can help you keep track of what items enabled a new DP state to get it's value. For example, if your DP array is w[] then you can have another array p[]. Every time a state is generated for w[i], you set p[i] to the item you used to get to 'w[i]'. Then to output the list of items used for w[n], output p[n], and then move to the index n-weightOf(p[n]) until you reach 0 to output all the items.