Question: Suppose we are given a collection A={a_1,...,a_n} of
n positive integers that add up to N. Design an O(nN) algorithm for
determining whether there is a subset B\subset A such that
\sum_{a_i\in B} a_i = \sum_{a_i\in A-B} a_i.

Solution: We actually solve a more general problem.
Assuming that N is the total sum of the numbers in A,
for any given 0<= S <= N, we solve the problem of wether
there is a subset of A, call A', such that the sum
 of numbers in A' is equal to S. Using this procedure,
we can solve the original problem by just letting S=N/2
(assuming that N is even, and returning NO if N is odd).

We use dynamic programming.
For this problem, define table B[i,j] where 0<= i<= n and
0<= j <= N which holds 1 (True) or 0 (false) saying that there is a 
subset of the items in A_i={a_1,a_2,...,a_i} that adds to 
exactly j. Our solution will be B[n,S].

Clearly, B[0,0]=1, and for any i>0: 
B[0,i]=0  and B[i,0]=1 (by picking nothing).

For any i>=1 and j>=1 we have:

B[i,j]= max {B[i-1,j] ,  
	    {B[i-1,j-a_i] (if j>= a_i) 

 The first corresponds to the case that we find a subset with
total j from items a_1,...,a_{i-1} and the second case corresponds
to the case that we find a subset with total j-a_i from items 
a_1,...,a_{i-1} and then add item a_i to it.

The running time is constant to compute every entry of table
B and the size of the table is n*N so the time is O(n*N).