### Subset Sum

The subset-sum problem is defined as follows:

You are given a set S of non-negative integers: {A[1], ..., A[n]} and a goal G (also a non-negative integer).

The question is whether, for any subset S' ⊆ S, the sum of the integers in S' is G.

For i=1,2,..,n and h=0,1,...,G, define

- M[i,h] = true if there exists a subset of {A[1], A[2], ..., A[i]} that sums to h.

Claim: M satisfies the following recurrence:

- M[0,0] = true
- M[0,h] = false for h > 0
- M[i,h] = ( M[i-1,h] or M[i-1,h-A[i]] )

This is because any subset S'' of {A[1],A[2],...,A[i]} summing to h either contains A[i] or doesn't.
If the subset S'' doesn't contain A[i], then the subset is also a subset of {A[1],...,A[i-1]} summing to h.
If the subset S'' does contain A[i], then removing A[i] yields a subset of {A[1],...,A[i-1]} that sums to h-A[i].

This gives the following recursive algorithm:

bool subset-sum(Array<int> A, int i, int h) {
if (i == 0) return (h == 0);
else return subset-sum(A, i-1, h) || subset-sum(A, i-1, h-A[i]);
}

As implemented, the running time is exponential (something like 2^{i}).
(What is the recursion tree?)

We can modify it to cache the answers:

bool subset-sum(Array<int> A, int i, int h) {
static Array<Array<bool> > M;

if (M[i].exists(h)) return M[i][h];

if (i == 0) return (h == 0);
else return (M[i][h] = (subset-sum(A, i-1, h) || subset-sum(A, i-1, h-A[i])));
}

Now what does the recursion diagram look like?

Since there are at most n*G distinct subproblems,
and the time spent in each call (not counting the recursion) is O(1),
the total time is O(n*G).

### References