A14947 | Covering Circle

Sam started playing with round buckets in the sandbox, while also scattering pebbles. His mom decided to buy him a new bucket, so she needs to solve the following task.

You are given $n$ distinct points with integer coordinates $A_1, A_2, \ldots, A_n$ . All points were generated from the square $[-10^8, 10^8] \times [-10^8, 10^8]$ uniformly and independently.

You are given positive integers $k$ , $l$ , such that $k \leq l \leq n$ . You want to select a subsegment $A_i, A_{i+1}, \ldots, A_{i+l-1}$ of the points array (for some $1 \leq i \leq n + 1 - l$ ), and some circle on the plane, containing $\geq k$ points of the selected subsegment (inside or on the border).

What is the smallest possible radius of that circle?

Each test contains multiple test cases. The first line contains a single integer $t$ ( $1 \leq t \leq 10^4$ ) — the number of test cases. Descriptions of test cases follow.

The first line of each test case contains three integers $n$ , $l$ , $k$ ( $2 \leq k \leq l \leq n \leq 50\,000$ , $k \leq 20$ ).

Each of the next $n$ lines contains two integers $x_i$ , $y_i$ ( $-10^8 \leq x_i, y_i \leq 10^8$ ) — the coordinates of the point $A_i$ . It is guaranteed that all points are distinct and were generated independently from uniform distribution on $[-10^8, 10^8] \times [-10^8, 10^8]$ .

It is guaranteed that the sum of $n$ for all test cases does not exceed $50\,000$ .

In the first test, points were not generated from the uniform distribution on $[-10^8, 10^8] \times [-10^8, 10^8]$ for simplicity. It is the only such test and your solution must pass it.

Hacks are disabled in this problem.

For each test case print a single real number — the answer to the problem.

Your answer will be considered correct if its absolute or relative error does not exceed $10^{-9}$ . Formally let your answer be $a$ , jury answer be $b$ . Your answer will be considered correct if $\frac{|a - b|}{\max{(1, |b|)}} \le 10^{-9}$ .