algorithm - Finding minimal absolute sum of a subarray

Question

There's an array A containing (positive and negative) integers. Find a (contiguous) subarray whose elements' absolute sum is minimal, e.g.:

A = [2, -4, 6, -3, 9]
|(?4) + 6 + (?3)| = 1 <- minimal absolute sum

I've started by implementing a brute-force algorithm which was O(N^2) or O(N^3), though it produced correct results. But the task specifies:

complexity:
- expected worst-case time complexity is O(N*log(N))
- expected worst-case space complexity is O(N)

After some searching I thought that maybe Kadane's algorithm can be modified to fit this problem but I failed to do it.

My question is - is Kadane's algorithm the right way to go? If not, could you point me in the right direction (or name an algorithm that could help me here)? I don't want a ready-made code, I just need help in finding the right algorithm.

Answer 1

If you compute the partial sums such as

2, 2 +(-4), 2 + (-4) + 6, 2 + (-4) + 6 + (-3)...

Then the sum of any contiguous subarray is the difference of two of the partial sums. So to find the contiguous subarray whose absolute value is minimal, I suggest that you sort the partial sums and then find the two values which are closest together, and use the positions of these two partial sums in the original sequence to find the start and end of the sub-array with smallest absolute value.

The expensive bit here is the sort, so I think this runs in time O(n * log(n)).