I have array $t$ with size $n \leq 10^6$. It has only two kinds of elements inside: $1$ or $-1$. I need to count how many contiguous subsequences have positive sum.

This pseudocode demonstrates exactly the behavior I want, but is too slow for large $n$:

c = 0
for i = 1 to n do
    for i = j to n do
       if sum(u[i:j]) > 0 then
           c = c + 1
return c

Partial sums can be used to optimize it, but it will still be $O(n^2)$.

Here are some examples if they help:

[-1, 1, -1, -1, -1] $\rightarrow$ 1

[1, -1, 1, 1, -1, 1] $\rightarrow$ 14

[1, 1, 1, 1, 1, 1, 1, 1, 1, 1] $\rightarrow$ 55

Does there exist linear or $n \log n$ solution?

  • $\begingroup$ What did you try? Where did you get stuck? We're happy to help you understand the concepts but just solving practice/contest-style problems for you is unlikely to achieve that. You might find this page helpful in improving your question. $\endgroup$
    – D.W.
    Oct 20, 2017 at 14:46
  • $\begingroup$ @D.W. Actually I think I understand the concepts very well. I just wanted to practice and honestly don't have any idea how to solve this problem I encountered. I tried many solutions, but I got stuck because they had too big time complexity and the best I got was $O(n^2)$ $\endgroup$ Oct 20, 2017 at 18:24

1 Answer 1


Given the array $A_1,\ldots,A_n$, start by computing $B_i = \sum_{j \leq i} A_j$. You want to count the number of pairs $i_1 < i_2$ such that $B_{i_1} < B_{i_2}$, where $0 \leq i_1,i_2 \leq n$. Each such pair corresponds to a a contiguous subsequence $A_{i_1+1},\ldots,A_{i_2}$ whose sum is positive. This is the same as the classical problem of counting the number of inversions in an array (this is what you get if you reverse $B$), which can be solved in $O(n\log n)$.

It is plausible that you can do better in your case since the numbers are $\pm 1$.

  • $\begingroup$ Very similar to what I attempted, but actually works! Thanks! $\endgroup$ Oct 20, 2017 at 18:25
  • $\begingroup$ So, you are storing the presum in B and then finding the no. of inversions(after reversing B). Right? $\endgroup$
    – asn
    Apr 8, 2020 at 23:32

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