codeforces#P1667B. Optimal Partition
Optimal Partition
Description
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 $s<0$.
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.
Input
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 $5 \cdot 10^5$.
Output
For each test case print a single integer — the maximum sum of values you can get with an optimal parition.
Samples
5
3
1 2 -3
4
0 -2 3 -4
5
-1 -2 3 -1 -1
6
-1 2 -3 4 -5 6
7
1 -1 -1 1 -1 -1 1
1
2
1
6
-1
Note
Test case $1$: one optimal partition is $[1, 2]$, $[-3]$. $1+2>0$ so the value of $[1, 2]$ is $2$. $-3<0$, so the value of $[-3]$ is $-1$. $2+(-1)=1$.
Test case $2$: the optimal partition is $[0, -2, 3]$, $[-4]$, and the sum of values is $3+(-1)=2$.