102927: [Atcoder]ABC292 Ex - Rating Estimator

Problem Statement

You will participate in $N$ contests, numbered $1$ to $N$ in chronological order.
A participant in each contest will be given a value called performance for that contest. Let $P_i$ be the performance for contest $i$.
Additionally, you have a value called rating, which changes according to the performances in contests. The initial rating is $0$, and the rating after contest $n$ is $\frac{1}{n} \left(\sum_{i=1}^n P_i \right)$.
However, once your rating is $B$ or higher, later contests will not affect your rating.

Before the contests, you have decided to estimate your performance in each contest. Let $a_i$ be the initial estimate of your performance in contest $i$. Process $Q$ queries in the order they are given.

In each query, you are given two integers $c$ and $x$. First, change the estimate of your performance in contest $i$ to $x$. (This change is persistent.) Then, assuming that you get the estimated performances in all $N$ contests, print your final rating after the contests.

Constraints

• $1 \leq N \leq 5 \times 10^5$
• $1 \leq B \leq 10^9$
• $1 \leq Q \leq 10^5$
• $0 \leq a_i \leq 10^9$
• $1 \leq c \leq N$
• $0 \leq x \leq 10^9$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format, where $c_i$ and $x_i$ are the $c$ and $x$ for the $i$-th query:

$N$ $B$ $Q$
$a_1$ $a_2$ $\dots$ $a_N$
$c_1$ $x_1$
$c_2$ $x_2$
$\vdots$
$c_Q$ $x_Q$

Output

Print $Q$ lines. The $i$-th line should contain the answer to the $i$-th query.
Your output is considered correct if the absolute or relative error from the true answer is at most $10^{-9}$.

Sample Input 1

5 6 7
5 1 9 3 8
4 9
2 10
1 0
3 0
3 30
5 100
1 100

Sample Output 1

6.000000000000000
7.500000000000000
6.333333333333333
5.400000000000000
13.333333333333334
13.333333333333334
100.000000000000000

Initially, $(a_1, a_2, a_3, a_4, a_5) = (5, 1, 9, 3, 8)$.
The first query changes $a_4$ to $9$, making $(a_1, a_2, a_3, a_4, a_5) = (5, 1, 9, 9, 8)$.
Here, assuming that your performance in contest $i$ is $a_i$, your rating will change as follows.

• Initially, your rating is $0$.
• After contest $1$, your rating will be $5 / 1 = 5$.
• After contest $2$, your rating will be $(5 + 1) / 2 = 3$.
• After contest $3$, your rating will be $(5 + 1 + 9) / 3 = 5$.
• After contest $4$, your rating will be $(5 + 1 + 9 + 9) / 4 = 6$.
• Your rating will no longer change, because your rating after contest $4$ is not less than $B$.

Thus, your final rating after the contests is $6$, which should be printed in the first line.

问题描述

您将参加编号为1到N的N个比赛，按时间顺序编号。
每场比赛的参赛者将获得该场比赛的称为“表现”的值。设P_i为第i场比赛的表现。
此外，您有一个称为“评级”的值，它会根据比赛中的表现而变化。初始评级为0，比赛n后的评级为$\frac{1}{n} \left(\sum_{i=1}^n P_i \right)$。
但是，一旦您的评级达到或高于B，后续比赛将不会影响您的评级。

在比赛开始前，您已经决定对每场比赛的表现进行预估。设$a_i$为第i场比赛的初始表现预估。按照给定的顺序处理Q个查询。

在每个查询中，您将获得两个整数c和x。首先，将您在比赛c的表现预估更改为x。（此更改是持久的。）然后，假设您在所有N场比赛中都获得了预估的表现，打印比赛结束后您的最终评级。

约束条件

• $1 \leq N \leq 5 \times 10^5$
• $1 \leq B \leq 10^9$
• $1 \leq Q \leq 10^5$
• $0 \leq a_i \leq 10^9$
• $1 \leq c \leq N$
• $0 \leq x \leq 10^9$
• 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，其中$c_i$和$x_i$是第i个查询的$c$和$x$：

$N$ $B$ $Q$
$a_1$ $a_2$ $\dots$ $a_N$
$c_1$ $x_1$
$c_2$ $x_2$
$\vdots$
$c_Q$ $x_Q$

输出

打印Q行。第i行应包含第i个查询的答案。
如果您的输出与真实答案的绝对或相对误差不超过$10^{-9}$，则认为您的输出是正确的。

样例输入1

5 6 7
5 1 9 3 8
4 9
2 10
1 0
3 0
3 30
5 100
1 100

样例输出1

6.000000000000000
7.500000000000000
6.333333333333333
5.400000000000000
13.333333333333334
13.333333333333334
100.000000000000000

最初，$(a_1, a_2, a_3, a_4, a_5) = (5, 1, 9, 3, 8)$。
第一个查询将$a_4$更改为9，使$(a_1, a_2, a_3, a_4, a_5) = (5, 1, 9, 9, 8)$。
假设您在比赛i的表现为$a_i$，您的评级将按以下方式更改。

• 最初，您的评级为0。
• 比赛1后，您的评级将为$5 / 1 = 5$。
• 比赛2后，您的评级将为$(5 + 1) / 2 = 3$。
• 比赛3后，您的评级将为$(5 + 1 + 9) / 3 = 5$。
• 比赛4后，您的评级将为$(5 + 1 + 9 + 9) / 4 = 6$。
• 您的评级将不再更改，因为比赛4后的评级不小于B。

因此，比赛结束后您的最终评级为6，应在第一行打印。