102405: [AtCoder]ABC240 F - Sum Sum Max

Problem Statement

There are integer sequences $A, B, C$ of length $M$ each.

$C$ is represented by integers $x_1, \dots, x_N, y_1, \dots, y_N$. The first $y_1$ terms of $C$ are $x_1$, the subsequent $y_2$ terms are $x_2$, $\ldots$, the last $y_N$ terms are $x_N$.

$B$ is defined by $B_i = \sum_{k = 1}^i C_k \, (1 \leq i \leq M)$.

$A$ is defined by $A_i = \sum_{k = 1}^i B_k \, (1 \leq i \leq M)$.

Find the maximum value among $A_1, \dots, A_M$.

You will be given $T$ test cases to solve.

Constraints

• $1 \leq T \leq 2 \times 10^5$
• $1 \leq N \leq 2 \times 10^5$
• The sum of $N$ in a single file is at most $2 \times 10^5$.
• $1 \leq M \leq 10^9$
• $|x_i| \leq 4 \, (1 \leq i \leq N)$
• $y_i \gt 0 \, (1 \leq i \leq N)$
• $\sum_{k = 1}^N y_k = M$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$T$
$\mathrm{case}_1$
$\vdots$
$\mathrm{case}_T$

Each case is in the following format:

$N$ $M$
$x_1$ $y_1$
$\vdots$
$x_N$ $y_N$

Output

Print $T$ lines. The $i$-th line $(1 \leq i \leq T)$ should contain the answer to the $i$-th test case.

Sample Input 1

3
3 7
-1 2
2 3
-3 2
10 472
-4 12
1 29
2 77
-1 86
0 51
3 81
3 17
-2 31
-4 65
4 23
1 1000000000
4 1000000000

Sample Output 1

4
53910
2000000002000000000

In the first test case, we have:

• $C = (-1, -1, 2, 2, 2, -3, -3)$
• $B = (-1, -2, 0, 2, 4, 1, -2)$
• $A = (-1, -3, -3, -1, 3, 4, 2)$

Thus, the maximum value among $A_1, \dots, A_M$ is $4$.

问题描述

存在长度为$M$的整数序列$A, B, C$。

$C$由整数$x_1, \dots, x_N, y_1, \dots, y_N$表示。$C$的前$y_1$项为$x_1$，接下来的$y_2$项为$x_2$，$\ldots$，最后的$y_N$项为$x_N$。

$B$定义为$B_i = \sum_{k = 1}^i C_k \, (1 \leq i \leq M)$。

$A$定义为$A_i = \sum_{k = 1}^i B_k \, (1 \leq i \leq M)$。

求$A_1, \dots, A_M$中的最大值。

你需要解决$T$个测试用例。

约束

• $1 \leq T \leq 2 \times 10^5$
• $1 \leq N \leq 2 \times 10^5$
• 单个文件中的$N$的总和最多为$2 \times 10^5$。
• $1 \leq M \leq 10^9$
• $|x_i| \leq 4 \, (1 \leq i \leq N)$
• $y_i \gt 0 \, (1 \leq i \leq N)$
• $\sum_{k = 1}^N y_k = M$
• 输入中的所有值都是整数。

输入

输入从标准输入以以下格式给出：

$T$
$\mathrm{case}_1$
$\vdots$
$\mathrm{case}_T$

每个用例的格式如下：

$N$ $M$
$x_1$ $y_1$
$\vdots$
$x_N$ $y_N$

输出

打印$T$行。第$i$行$(1 \leq i \leq T)$应包含第$i$个测试用例的答案。

样例输入 1

3
3 7
-1 2
2 3
-3 2
10 472
-4 12
1 29
2 77
-1 86
0 51
3 81
3 17
-2 31
-4 65
4 23
1 1000000000
4 1000000000

样例输出 1

4
53910
2000000002000000000

在第一个测试用例中，我们有：

• $C = (-1, -1, 2, 2, 2, -3, -3)$
• $B = (-1, -2, 0, 2, 4, 1, -2)$
• $A = (-1, -3, -3, -1, 3, 4, 2)$

因此，$A_1, \dots, A_M$中的最大值为$4$。