103556: [Atcoder]ABC355 G - Baseball

Problem Statement

You are given a sequence $P=(P_1,P_2,\dots,P_N)$ of length $N$. Takahashi and Aoki will play a game using the sequence $P$.

First, Takahashi will choose $K$ distinct integers $x_1,x_2,\dots,x_K$ from $1,2,\dots,N$.

Next, Aoki will choose an integer $y$ from $1,2,\dots,N$ with a probability proportional to $P_y$. That is, the probability that integer $y$ is chosen is $\dfrac{P_y}{\sum_{y'=1}^N P_{y'}}$. Then, Aoki's score will be $\displaystyle \min_{i=1,2,\dots,K} |x_i-y|$.

Takahashi wants to minimize the expected value of Aoki's score. Find the expected value of Aoki's score when Takahashi chooses $x_1,x_2,\dots,x_K$ so that this value is minimized, multiplied by $\sum_{y'=1}^N P_{y'}$. It is guaranteed that the value to be printed is an integer.

Constraints

• $1 \leq N \leq 5 \times 10^4$
• $1 \leq K \leq N$
• $0 \leq P_i \leq 10^5$
• $1 \leq \sum_{y'=1}^N P_{y'} \leq 10^5$
• All input values are integers.

Input

The input is given from Standard Input in the following format:

$N$ $K$
$P_1$ $P_2$ $\dots$ $P_N$

Output

Print the answer.

Sample Input 1

5 2
1 1 1 1 1

Sample Output 1

3

The probabilities of Aoki choosing $1,2,\dots,N$ are all equal: $\frac{1}{5}$.

If Takahashi chooses $x_1=2$ and $x_2=4$, then the expected value of Aoki's score will be $1 \times \frac{1}{5} + 0 \times \frac{1}{5} + 1 \times \frac{1}{5} + 0 \times \frac{1}{5} + 1 \times \frac{1}{5} = \frac{3}{5}$.

If Takahashi chooses $x_1=2$ and $x_2=3$, then the expected value of Aoki's score will be $1 \times \frac{1}{5} + 0 \times \frac{1}{5} + 0 \times \frac{1}{5} + 1 \times \frac{1}{5} + 2 \times \frac{1}{5} = \frac{4}{5}$.

No matter how Takahashi chooses, the expected value of Aoki's score cannot be smaller than $\frac{3}{5}$. Therefore, the minimum value is $\frac{3}{5}$, so print this value multiplied by $5$, which is $3$.

Sample Input 2

5 1
0 0 1 0 0

Sample Output 2

0

Sample Input 3

1 1
100

Sample Output 3

0

Sample Input 4

20 7
4262 9522 2426 3823 7364 964 2743 2423 1955 5274 3684 847 363 35 278 3220 203 2904 6304 1928

Sample Output 4

22809

问题描述

给定一个长度为 $N$ 的序列 $P=(P_1,P_2,\dots,P_N)$。Takahashi 和 Aoki 将使用序列 $P$ 进行游戏。

首先，Takahashi 从 $1,2,\dots,N$ 中选择 $K$ 个不同的整数 $x_1,x_2,\dots,x_K$。

接着，Aoki 将以 $P_y$ 与 $\sum_{y'=1}^N P_{y'}$ 的比例选择一个整数 $y$。即，选择 $y$ 的概率为 $\dfrac{P_y}{\sum_{y'=1}^N P_{y'}}$。然后，Aoki 的分数将是 $\displaystyle \min_{i=1,2,\dots,K} |x_i-y|$。

Takahashi 想要**最小化** Aoki 分数的期望值。找出当 Takahashi 选择 $x_1,x_2,\dots,x_K$ 以使该值最小化时，Aoki 分数的期望值乘以 $\sum_{y'=1}^N P_{y'}$。保证要打印的值是一个整数。

限制条件

• $1 \leq N \leq 5 \times 10^4$
• $1 \leq K \leq N$
• $0 \leq P_i \leq 10^5$
• $1 \leq \sum_{y'=1}^N P_{y'} \leq 10^5$
• 所有输入值都是整数。

输入

输入从标准输入以以下格式给出：

$N$ $K$
$P_1$ $P_2$ $\dots$ $P_N$

输出

打印答案。

样例输入 1

5 2
1 1 1 1 1

样例输出 1

3

Aoki 选择 $1,2,\dots,N$ 的概率都相等：$\frac{1}{5}$。

如果 Takahashi 选择 $x_1=2$ 和 $x_2=4$，那么 Aoki 分数的期望值将是 $1 \times \frac{1}{5} + 0 \times \frac{1}{5} + 1 \times \frac{1}{5} + 0 \times \frac{1}{5} + 1 \times \frac{1}{5} = \frac{3}{5}$。

如果 Takahashi 选择 $x_1=2$ 和 $x_2=3$，那么 Aoki 分数的期望值将是 $1 \times \frac{1}{5} + 0 \times \frac{1}{5} + 0 \times \frac{1}{5} + 1 \times \frac{1}{5} + 2 \times \frac{1}{5} = \frac{4}{5}$。

无论 Takahashi 如何选择，Aoki 分数的期望值都无法小于 $\frac{3}{5}$。因此，最小值是 $\frac{3}{5}$，所以打印这个值乘以 $5$，即 $3$。

样例输入 2

5 1
0 0 1 0 0

样例输出 2

0

样例输入 3

1 1
100

样例输出 3

0

样例输入 4

20 7
4262 9522 2426 3823 7364 964 2743 2423 1955 5274 3684 847 363 35 278 3220 203 2904 6304 1928

样例输出 4

22809