Score : $200$ points

问题描述

在 $xy$ 平面上有 $N$ 名编号为 $1, 2, \dots, N$ 的人。第 $i$ 个人位于坐标 $(X_i, Y_i)$。

这 $N$ 人中有 $K$ 位，分别是第 $A_1, A_2, \dots, A_K$ 号人，他们将获得强度相同的灯光。
当一个位于坐标 $(x, y)$ 的人拥有强度为 $R$ 的灯光时，它会照亮以 $(x, y)$ 为中心、半径为 $R$ 的圆的内部（包括边界）。
找出使每个人至少被一盏灯照亮所需的灯光最小强度。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

There are $N$ people numbered $1, 2, \dots, N$ in the $xy$-plane. Person $i$ is at the coordinates $(X_i, Y_i)$.
$K$ of these people, Persons $A_1, A_2, \dots, A_K$, will receive lights of the same strength.
When a person at coordinates $(x, y)$ has a light of strength $R$, it lights up the interior of a circle of radius $R$ centered at $(x, y)$ (including the boundary).
Find the minimum strength of the lights needed for every person to be lit by at least one light.

Constraints

• All values in input are integers.
• $1 \le K < N \le 1000$
• $1 \le A_1 < A_2 < \dots < A_K \le N$
• $|X_i|,|Y_i| \le 10^5$
• $(X_i,Y_i) \neq (X_j,Y_j)$, if $i \neq j$.

Input

Input is given from Standard Input in the following format:

$N$ $K$

$A_1$ $A_2$ $\dots$ $A_K$

$X_1$ $Y_1$

$X_2$ $Y_2$

$\vdots$

$X_N$ $Y_N$

Output

Print the answer as a real number.
Your output will be considered correct if its absolute or relative error from the judge's output is at most $10^{-5}$.

Sample Input 1

4 2
2 3
0 0
0 1
1 2
2 0

Sample Output 1

2.23606797749978969

This input contains four people. Among them, Persons $2$ and $3$ will have lights.
Every person will be lit by at least one light if $R \ge \sqrt{5} \approx 2.236068$.

Sample Input 2

2 1
2
-100000 -100000
100000 100000

Sample Output 2

282842.712474619009

Sample Input 3

8 3
2 6 8
-17683 17993
93038 47074
58079 -57520
-41515 -89802
-72739 68805
24324 -73073
71049 72103
47863 19268

Sample Output 3

130379.280458974768

update @ 2024/3/10 10:51:11