Algorithm to pick values from array that sum closest to a target value?

Question

I have an array of nearly sorted values 28 elements long. I need to find the set of values that sums to a target value provided to the algorithm (or if exa

Answer 1

This problem is known as the subset sum problem, which is a special case of the Knapsack problem. Wikipedia is a good starting point for some algorithms.

Answer 2

If values are reasonably small, it's a simple dynamic programming (DP). Time complexity would be O(n * target) and memory requirements O(target). If that satisfies you, there're lots of DP tutorials on the web. For example, here the first discussed problem (with couns) is very much like yours (except they allow to use each number more than once):
http://www.topcoder.com/tc?module=Static&d1=tutorials&d2=dynProg

update Yep, as other person noted, it's a simple case of knapsack problem.

Answer 3

As others have noted, this is same as the optimization version of subset sum problem, which is NP-Complete.

Since you mentioned that you are short in memory and can probably work with approximate solutions (based on your current solution), there are polynomial time approximation algorithms for solving the optimization version of subset sum.

For instance, given an e > 0, there is a polynomial time algorithm which uses O((n*logt)/e) space, (t is the target sum, n is the size of the array) which gives you a subset such that the sum z is no less than 1/(1+e) times the optimal.

i.e If the largest subset sum was y, then the algorithm finds a subset sum z such that

z <= y <= (1+e)z

and uses space O((n*logt)/e).

Such an algorithm can be found here: http://www.cs.dartmouth.edu/~ac/Teach/CS105-Winter05/Notes/nanda-scribe-3.pdf

Hope this helps.