查找所有总和等于零的不间断子序列

Question

Say that we have an array of N integers and want to find all subsequences of consecutive elements which have the sum of the equal to zero.

Example:

N = 9
array = [1, -2, 4, 5, -7, -4, 8, 3, -7]

Should output:

1 4 

4 7

5 8

1 8

as each of the above are the start and end index of the subsequences with sum equal to zero.

I came to the conclusion that there are N * (N + 1) / 2 such possible subsequences which would conclude in O(N^2) complexity if using a naive algorithm that exhaustively searches over all the possible solutions.

I would like to know if there is any way to achieve this in O(N) complexity or something less than O(N^2)?

I am aware of the Subset sum problem which is a NP-complete problem but mine seems to be a bit easier as it only requires subsequences of consecutive elements.

Thank you in advance.

Answer 1

It is worse than you think.

There are potentially Θ(n²) uninterrupted subsequences that add up to zero, as in the following example:

0 0 0 0 0 0 0 0 0

(Here, every subsequence adds up to zero.)

Therefore, any algorithm that prints out the start and end index of all the required subsequences will necessarily have o(n²) worst-case complexity. Printing out their elements will require Θ(n³) time.