A15029 | Optimal Partition

You are given an array $a$ consisting of $n$ integers. You should divide $a$ into continuous non-empty subarrays (there are $2^{n-1}$ ways to do that).

Let $s=a_l+a_{l+1}+\ldots+a_r$ . The value of a subarray $a_l, a_{l+1}, \ldots, a_r$ is:

- $(r-l+1)$ if $s>0$ ,
- $0$ if $s=0$ ,
- $-(r-l+1)$ if $s<0$ .

What is the maximum sum of values you can get with a partition?

The input consists of multiple test cases. The first line contains a single integer $t$ ( $1 \le t \le 5 \cdot 10^5$ ) — the number of test cases. The description of the test cases follows.

The first line of each test case contains a single integer $n$ ( $1 \le n \le 5 \cdot 10^5$ ).

The second line of each test case contains $n$ integers $a_1$ , $a_2$ , ..., $a_n$ ( $-10^9 \le a_i \le 10^9$ ).

It is guaranteed that the sum of $n$ over all test cases does not exceed $5 \cdot 10^5$ .

For each test case print a single integer — the maximum sum of values you can get with an optimal parition.