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 $ .

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。