Problem:
Score : 600 points

Problem Statement

Given are N points (x_i, y_i) in a two-dimensional plane.

Find the minimum radius of a circle such that all the points are inside or on it.

Constraints

2 \leq N \leq 50

0 \leq x_i \leq 1000

0 \leq y_i \leq 1000

The given N points are all different.

The values in input are all integers.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1
:
x_N y_N

Output

Print the minimum radius of a circle such that all the N points are inside or on it.

Your output will be considered correct if the absolute or relative error from our answer is at most 10^{-6}.

Sample Input 1

2
0 0
1 0

Sample Output 1

0.500000000000000000

Both points are contained in the circle centered at (0.5,0) with a radius of 0.5.

Sample Input 2

3
0 0
0 1
1 0

Sample Output 2

0.707106781186497524

Sample Input 3

10
10 9
5 9
2 0
0 0
2 7
3 3
2 5
10 0
3 7
1 9

Sample Output 3

6.726812023536805158

If the absolute or relative error from our answer is at most 10^{-6}, the output will be considered correct.

Solution in PyPy3 (7.3.0) (judge verdict: Accepted)

from copy import copy
from itertools import combinations
from random import random, sample, seed


class circle:
    def __init__(self, pts):
        assert (0 <= len(pts) and len(pts) <= 3)
        self.__pts = copy(pts)
        if len(pts) == 0:
            self.__cx, self.__cy, self.__r = float('inf'), float('inf'), 0
        elif len(pts) == 1:
            self.__cx, self.__cy, self.__r = *pts[0], 0
        elif len(pts) == 2:
            self.__cx, self.__cy, self.__r = self.__c2()
        else:
            self.__cx, self.__cy, self.__r = self.__c3()

    def __c2(self):
        assert (len(self.__pts) == 2)
        (x0, y0), (x1, y1) = self.__pts
        return ((x0 + x1) / 2, (y0 + y1) / 2,
                0.5 * ((x0 - x1) * (x0 - x1) + (y0 - y1) * (y0 - y1)) ** 0.5)

    def __c3(self):
        assert (len(self.__pts) == 3)
        return self.__o3() if self.__obtuse() else self.__a3()

    def __o3(self):
        p, q = max(((xi - xj) ** 2 + (yi - yj) ** 2, ((xi, yi), (xj, yj)))
                   for (xi, yi), (xj, yj) in combinations(self.__pts, 2))[1]
        self.__pts = [p, q]
        return self.__c2()

    def __obtuse(self):
        """ Determine whether given tripe forms obtuse, cf.,
                https://math.stackexchange.com/questions/3200316/
        """
        (x0, y0), (x1, y1), (x2, y2) = self.__pts
        a = (x0 - x1) ** 2 + (y0 - y1) ** 2
        b = (x1 - x2) ** 2 + (y1 - y2) ** 2
        c = (x2 - x0) ** 2 + (y2 - y0) ** 2
        return a > b + c or b > c + a or c > a + b

    def __a3(self):
        """ Equation of circle when three points on the circle are given.
            https://www.geeksforgeeks.org/equation-of-circle-when-three-points-on-the-circle-are-given/
        """
        (x0, y0), (x1, y1), (x2, y2) = self.__pts

        x01, x02 = x0 - x1, x0 - x2
        y01, y02 = y0 - y1, y0 - y2

        y20, y10 = y2 - y0, y1 - y0
        x20, x10 = x2 - x0, x1 - x0

        sx02, sy02 = x0 * x0 - x2 * x2, y0 * y0 - y2 * y2
        sx10, sy10 = x1 * x1 - x0 * x0, y1 * y1 - y0 * y0

        f = (sx02 * x01 + sy02 * x01 + sx10 * x02 + sy10 * x02) / (2 * (y20 * x01 - y10 * x02))
        g = (sx02 * y01 + sy02 * y01 + sx10 * y02 + sy10 * y02) / (2 * (x20 * y01 - x10 * y02))
        c = -x0 * x0 - y0 * y0 - 2 * g * x0 - 2 * f * y0

        return (-g, -f, (f * f + g * g - c) ** 0.5)

    def draw(self, c='g', ax=None):
        if ax is None:
            ax = plt.gca()
        c = plt.Circle((self.__cx, self.__cy), self.__r, fill=False, color=c)
        ax.add_artist(c)

    def contains(self, x, y):
        return ((x - self.__cx) ** 2 + (y - self.__cy) ** 2) ** 0.5 <= self.__r + 1e-9

    def pts(self):
        return set(self.__pts)

    @property
    def center(self):
        return (self.__cx, self.__cy)

    @property
    def radius(self):
        return self.__r


def welzl(p, r=set()):
    """ Find the smallest enclosing circle for given a set of points in plane.
        from https://en.wikipedia.org/wiki/Smallest-circle_problem

        input:  Finite sets P and R of points in the plane |R| <= 3.
        output: Minimal disk enclosing P with R on the boundary.
    """
    if len(p) == 0 or len(r) == 3:
        return circle(list(r))
    q = sample(p, 1)[0]  # randomly and uniformly
    d = welzl(set(p) - {q}, r)
    if d.contains(*q):
        return d
    return welzl(set(p) - {q}, r | {q})


def msw(pts):
    """ Matousek, Sharir, Welzl's Algorithm for the minimum-circle problem """
    if len(pts) <= 3:
        return circle(pts)
    return _msw(set(pts), set(sample(pts, 3)))


def _msw(p, r):
    """ Find the circle in recursively
        input:  Finite sets P and R of points in the plane |R| = 3.
        output: Minimal disk enclosing P U R.
        from wikipedia.
    """
    if len(p) == 0:
        return circle(list(r))
    q = sample(p, 1)[0]
    d = _msw(p - {q}, r)
    if d.contains(*q):
        return d
    x = (r | {q}) - welzl(r | {q}).pts()
    return _msw((p - {q}) | x, (r | {q}) - x)


def gen(n=20, s=1):
    seed(s)
    return [(random(), random()) for i in range(n)]


if __name__ == '__main__':
    N = int(input())
    pts = []
    for _ in range(N):
        x, y = map(int, input().split())
        pts.append((x,y))

    print(msw(pts).radius)