Optimal Partition

4000ms

262144K

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 $$$s<0$$$.

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

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.

Sample Input:

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

Sample Output:

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 $$$3+(-1)=2$$$.

Informação

Provedor Codeforces

Código CF1667B