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 Common Lisp (SBCL 1.1.14) (judge verdict: Accepted)

;; -*- coding: utf-8 -*-
(eval-when (:compile-toplevel :load-toplevel :execute)
  (sb-int:defconstant-eqx OPT
    #+swank '(optimize (speed 3) (safety 2))
    #-swank '(optimize (speed 3) (safety 0) (debug 0))
    #'equal)
  #+swank (ql:quickload '(:cl-debug-print :fiveam) :silent t)
  #-swank (set-dispatch-macro-character
           ;; enclose the form with VALUES to avoid being captured by LOOP macro
           #\# #\> (lambda (s c p) (declare (ignore c p)) `(values ,(read s nil nil t)))))
#+swank (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)
#-swank (disable-debugger) ; for CS Academy

;; BEGIN_INSERTED_CONTENTS
(macrolet ((def (name fname)
             `(define-modify-macro ,name (new-value) ,fname)))
  (def minf min)
  (def maxf max)
  (def mulf *)
  (def divf /)
  (def iorf logior)
  (def xorf logxor)
  (def andf logand))

(defmacro dbg (&rest forms)
  #+swank
  (if (= (length forms) 1)
      `(format *error-output* "~A => ~A~%" ',(car forms) ,(car forms))
      `(format *error-output* "~A => ~A~%" ',forms `(,,@forms)))
  #-swank (declare (ignore forms)))

(defmacro define-int-types (&rest bits)
  `(progn
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "UINT~A" b)) () '(unsigned-byte ,b))) bits)
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "INT~A" b)) () '(signed-byte ,b))) bits)))
(define-int-types 2 4 7 8 15 16 31 32 62 63 64)

(declaim (inline println))
(defun println (obj &optional (stream *standard-output*))
  (let ((*read-default-float-format* 'double-float))
    (prog1 (princ obj stream) (terpri stream))))

(defconstant +mod+ 1000000007)

;;;
;;; Body
;;;


;; http://www.ambrsoft.com/trigocalc/circle3d.htm
;; FIXME: more sane handling of degeneracy
(declaim (inline calc-circumcenter))
(defun calc-circumcenter (p1 p2 p3)
  "Returns the center of circumcirlce if it exists, otherwise returns NIL."
  (declare (complex p1 p2 p3))
  (let* ((x1 (realpart p1))
         (y1 (imagpart p1))
         (x2 (realpart p2))
         (y2 (imagpart p2))
         (x3 (realpart p3))
         (y3 (imagpart p3))
         (a (+ (* x1 (- y2 y3))
               (- (* y1 (- x2 x3)))
               (* x2 y3)
               (- (* x3 y2))))
         (b (+ (* (+ (* x1 x1) (* y1 y1)) (- y3 y2))
               (* (+ (* x2 x2) (* y2 y2)) (- y1 y3))
               (* (+ (* x3 x3) (* y3 y3)) (- y2 y1))))
         (c (+ (* (+ (* x1 x1) (* y1 y1)) (- x2 x3))
               (* (+ (* x2 x2) (* y2 y2)) (- x3 x1))
               (* (+ (* x3 x3) (* y3 y3)) (- x1 x2)))))
    (handler-bind ((error (lambda (c)
                            (declare (ignore c))
                            (return-from calc-circumcenter nil))))
      (complex (- (/ b (* 2 a)))
               (- (/ c (* 2 a)))))))

(defun mini-disc-with-2-points (points end q1 q2 eps)
  "Returns the smallest circle that contains points, q1 and q2 (contains q1 and
q2 at the perimeter)."
  (declare (complex q1 q2)
           (double-float eps)
           ((integer 0 #.most-positive-fixnum) end))
  (let* ((center (* 0.5d0 (+ q1 q2)))
         (radius (abs (- q1 center))))
    (declare (complex center)
             (double-float radius))
    (dotimes (i end)
      (let ((new-point (aref points i)))
        (declare (complex new-point))
        (when (>= (abs (- new-point center)) (+ radius eps))
          (let ((new-center (calc-circumcenter q1 q2 new-point)))
            (setq center new-center
                  radius (abs (- new-point new-center)))))))
    (values center radius)))

(declaim (inline shuffle!))
(defun shuffle! (vector &optional end)
  "Destructively shuffles VECTOR by Fisher-Yates algorithm."
  (declare (vector vector))
  (loop for i from (- (or end (length vector)) 1) above 0
        for j = (random (+ i 1))
        do (rotatef (aref vector i) (aref vector j)))
  vector)

(defun mini-disc-with-point (points end q eps)
  (declare (complex q)
           (double-float eps))
  (let* ((points (shuffle! (copy-seq points) end))
         (center (* 0.5d0 (+ (aref points 0) q)))
         (radius (abs (- q center))))
    (declare (complex center)
             (double-float radius))
    (loop for i from 1 below end
          for new-point of-type complex = (aref points i)
          when (>= (abs (- new-point center)) (+ radius eps))
          do (setf (values center radius)
                   (mini-disc-with-2-points points i (aref points i) q eps)))
    (values center radius)))

(defun calc-smallest-circle (points eps)
  (assert (>= (length points) 2))
  (let* ((points (shuffle! (copy-seq points)))
         (p0 (aref points 0))
         (p1 (aref points 1))
         (center (* 0.5d0 (+ p0 p1)))
         (radius (abs (- p0 center))))
    (loop for i from 2 below (length points)
          for new-point = (aref points i)
          when (>= (abs (- new-point center)) (+ radius eps))
          do (setf (values center radius)
                   (mini-disc-with-point points i new-point eps)))
    (values center radius)))

(defun main ()
  (let* ((n (read))
         (points (make-array n :element-type '(complex double-float))))
    (declare (uint16 n))
    (dotimes (i n)
      (let ((x (float (read) 1d0))
            (y (float (read) 1d0)))
        (setf (aref points i) (complex x y))))
    (println (nth-value 1 (calc-smallest-circle points 1d-9)))))

#-swank (main)

;;;
;;; Test and benchmark
;;;

#+swank
(defun io-equal (in-string out-string &key (function #'main) (test #'equal))
  "Passes IN-STRING to *STANDARD-INPUT*, executes FUNCTION, and returns true if
the string output to *STANDARD-OUTPUT* is equal to OUT-STRING."
  (labels ((ensure-last-lf (s)
             (if (eql (uiop:last-char s) #\Linefeed)
                 s
                 (uiop:strcat s uiop:+lf+))))
    (funcall test
             (ensure-last-lf out-string)
             (with-output-to-string (out)
               (let ((*standard-output* out))
                 (with-input-from-string (*standard-input* (ensure-last-lf in-string))
                   (funcall function)))))))

#+swank
(defun get-clipbrd ()
  (with-output-to-string (out)
    ;; (run-program "C:/Windows/System32/WindowsPowerShell/v1.0/powershell.exe" '("get-clipboard") :output out)
    (run-program "powershell.exe" '("-Command" "Get-Clipboard") :output out :search t)))

#+swank (defparameter *this-pathname* (uiop:current-lisp-file-pathname))
#+swank (defparameter *dat-pathname* (uiop:merge-pathnames* "test.dat" *this-pathname*))

#+swank
(defun run (&optional thing (out *standard-output*))
  "THING := null | string | symbol | pathname

null: run #'MAIN using the text on clipboard as input.
string: run #'MAIN using the string as input.
symbol: alias of FIVEAM:RUN!.
pathname: run #'MAIN using the text file as input."
  (let ((*standard-output* out))
    (etypecase thing
      (null
       (with-input-from-string (*standard-input* (delete #\Return (get-clipbrd)))
         (main)))
      (string
       (with-input-from-string (*standard-input* (delete #\Return thing))
         (main)))
      (symbol (5am:run! thing))
      (pathname
       (with-open-file (*standard-input* thing)
         (main))))))

#+swank
(defun gen-dat ()
  (uiop:with-output-file (out *dat-pathname* :if-exists :supersede)
    (format out "")))

#+swank
(defun bench (&optional (out (make-broadcast-stream)))
  (time (run *dat-pathname* out)))

;; To run: (5am:run! :sample)
#+swank
(it.bese.fiveam:test :sample
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "2
0 0
1 0
"
    "0.500000000000000000
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "3
0 0
0 1
1 0
"
    "0.707106781186497524
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "10
10 9
5 9
2 0
0 0
2 7
3 3
2 5
10 0
3 7
1 9
"
    "6.726812023536805158
")))