Optimal Partition
题意翻译
给定一个有 $n$ 个数的数列 $n$,现在把它分成若干段。对于其中一段 $a_l,a_{l+1},\ldots ,a_r$,设 $s=a_l+a_{l+1}+\ldots+a_r$,则定义 $a_l,a_{l+1},\ldots ,a_r$ 一段的价值为:
$ \begin{cases}
(r−l+1) & s>0 \\
0 & s=0 \\
-(r−l+1) & s<0
\end{cases}$
显然,通过某些分段方式,可以使 $a$ 中每段价值和最大,求这个最大价值。
题目描述
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?
输入输出格式
输入格式
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 $ .
输出格式
For each test case print a single integer — the maximum sum of values you can get with an optimal parition.
输入输出样例
输入样例 #1
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
输出样例 #1
1
2
1
6
-1
说明
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 $ .