Ex - Rating Estimator

AtCoder

IDabc292_h

Time3000ms

Memory256MB

Difficulty—

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 $c$ 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: $N$ $B$ $Q$ $a_1$ $a_2$ $\dots$ $a_N$ $c_1$ $x_1$ $c_2$ $x_2$ $\vdots$ $c_Q$ $x_Q$ [samples]

Samples

Input #1

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.

API Response (JSON)

{
  "problem": {
    "name": "Ex - Rating Estimator",
    "description": {
      "content": "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 perfor",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc292_h"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You will participate in $N$ contests, numbered $1$ to $N$ in chronological order.  \nA participant in each contest will be given a value called **performance** for that contest. Let $P_i$ be the perfor...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments