I asked this question to explain dynamic programming. I hope question is clear. Okay now, our challenge is to find the start and end of a subarray in the given array so that sum is maximum. i.e., given array

$$A[1..n]$$,

we have to find $i$ and $j$ such that $\sum_{k=i}^j A[k]$ is maximum. (Well, we don't need to output $i$ and $j$ but just the corresponding sum).

Okay, so one important thing is we are looking at "subarray" and not "subsequence" and hence the elements in the required subarray must be continuous. This makes the problem harder (subsequence solution will just be the sum of all positive array numbers).

Now, lets see if the problem has any sub-part which can be solved and which overlaps (if so we can do dynamic programming).

So, let me take an array element - say $A[i]$. There are only 2 choices for it- either it is in the required subsequence or it is not. So what is the condition for inclusion?

- Let us assume the array ends at i for now.
- We include A[i] in our required subarray, if the sum of the subarray ending at i-1 + A[i] produces a sum greater than any other subarray.

Well, the second statement here is the key- it gives our problem sub problems and moreover they are overlapping. So, lets try to apply it. What we need is an array say "sum" to store the max continuous sum for each position of the array ($sum[i]$ gives the maximum sum of the substring ending at $i$) and one more element say MAXS, which stores the max sum seen so far. Now, we can say

sum[i] = max(sum[i-1] + A[i], A[i]);

if (sum[i] > MAXS), MAXS = sum[i];

with sum[1] = A[1] and MAXS = A[1] as the boundary conditions.

Now, trying to code this? - 10 lines? :P