Problem:
Score : 500 points

Problem Statement

There are N towns on a plane. The i-th town is located at the coordinates (x_i,y_i). There may be more than one town at the same coordinates.

You can build a road between two towns at coordinates (a,b) and (c,d) for a cost of min(|a-c|,|b-d|) yen (the currency of Japan). It is not possible to build other types of roads.

Your objective is to build roads so that it will be possible to travel between every pair of towns by traversing roads. At least how much money is necessary to achieve this?

Constraints

2 ≤ N ≤ 10^5

0 ≤ x_i,y_i ≤ 10^9

All input values are integers.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1
x_2 y_2
:
x_N y_N

Output

Print the minimum necessary amount of money in order to build roads so that it will be possible to travel between every pair of towns by traversing roads.

Sample Input 1

3
1 5
3 9
7 8

Sample Output 1

3

Build a road between Towns 1 and 2, and another between Towns 2 and 3. The total cost is 2+1=3 yen.

Sample Input 2

6
8 3
4 9
12 19
18 1
13 5
7 6

Sample Output 2

8

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

(eval-when (:compile-toplevel :load-toplevel :execute)
  (defparameter OPT
    #+swank '(optimize (speed 3) (safety 2))
    #-swank '(optimize (speed 3) (safety 0) (debug 0)))
  #+swank (progn (ql:quickload '(:cl-debug-print :fiveam))
                 (shadow :run)
                 (use-package :fiveam)))
#+swank (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)

;; BEGIN_INSERTED_CONTENTS
(defmacro split-ints-and-bind (vars string &body body)
  (let ((pos1 (gensym "POS"))
	(pos2 (gensym "POS"))
	(str (gensym "STR")))
    (labels ((expand (vars &optional (init-pos1 t))
	       (if (null vars)
		   body
		   `((let* ((,pos1 ,(if init-pos1 0 `(1+ ,pos2)))
			    (,pos2 (position #\space ,str :start ,pos1 :test #'char=))
			    (,(car vars) (parse-integer ,str :start ,pos1 :end ,pos2)))
		       ,@(expand (cdr vars) nil))))))
      `(let ((,str ,string))
         (declare (string ,str))
	 ,@(expand vars)))))


(defmacro buffered-read-line (&optional (buffer-size 30) (in '*standard-input*) (terminate-char #\Space))
  "Note that the returned string will be reused."
  (let ((buffer (gensym))
        (character (gensym))
        (idx (gensym)))
    `(let ((,buffer (load-time-value (make-string ,buffer-size
                                                  :element-type 'base-char))))
       (declare (simple-base-string ,buffer))
       (loop for ,character of-type base-char =
                #-swank (code-char (read-byte ,in nil #\Newline))
                #+swank (read-char ,in nil #\Newline)
             for ,idx from 0
             until (char= ,character #\Newline)
             do (setf (schar ,buffer ,idx) ,character)
             finally (setf (schar ,buffer ,idx) ,terminate-char)
                     (return (values ,buffer ,idx))))))

(deftype non-negative-fixnum () '(integer 0 #.most-positive-fixnum))

(defstruct (union-find
            (:constructor make-union-find
                (size &aux (parents (let ((seq (make-array size :element-type 'non-negative-fixnum)))
                                      (dotimes (i size seq) (setf (aref seq i) i))))
                           (ranks (make-array size :element-type 'non-negative-fixnum
                                                   :initial-element 0)))))
  (parents nil :type (simple-array non-negative-fixnum (*)))
  (ranks nil :type (simple-array non-negative-fixnum (*))))

(declaim (ftype (function * (values non-negative-fixnum &optional)) uf-root))
(defun uf-root (x uf-tree)
  "Returns the root of X."
  (declare (optimize (speed 3))
           (non-negative-fixnum x))
  (let ((parents (union-find-parents uf-tree)))
    (if (= x (aref parents x))
        x
        (setf (aref parents x)
              (uf-root (aref parents x) uf-tree)))))

(declaim (inline uf-unite!))
(defun uf-unite! (x1 x2 uf-tree)
  "Unites X1 and X2 destructively."
  (let ((root1 (uf-root x1 uf-tree))
        (root2 (uf-root x2 uf-tree))
        (parents (union-find-parents uf-tree))
        (ranks (union-find-ranks uf-tree)))
    (cond ((= root1 root2) nil)
          ((< (aref ranks root1) (aref ranks root2))
           (setf (aref parents root1) root2))
          ((= (aref ranks root1) (aref ranks root2))
           (setf (aref parents root2) root1)
           (incf (aref ranks root1)))
          (t (setf (aref parents root2) root1)))))

(declaim (inline uf-connected-p))
(defun uf-connected-p (x1 x2 uf-tree)
  "Checks if X1 and X2 have the same root."
  (= (uf-root x1 uf-tree) (uf-root x2 uf-tree)))

;; (defun bench ()
;;   (let* ((size 5000000)
;;          (tree (make-union-find size))
;;          (seed (seed-random-state 0)))
;;     (dotimes (i 5000000)
;;       (uf-unite! (random size seed) (random size seed) tree))))

(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)

(defmacro println (obj &optional (stream '*standard-output*))
  `(let ((*read-default-float-format* 'double-float))
     (prog1 (princ ,obj ,stream) (terpri ,stream))))

;; Hauptteil

(defun main ()
  (declare #.OPT)
  (let* ((n (read))
         ;; (x y idx)
         (table (make-array n :element-type 'list)))
    (dotimes (i n)
      (split-ints-and-bind (x y) (buffered-read-line)
        (setf (aref table i) (list x y i))))
    (let ((table-x (stable-sort (copy-seq table) #'< :key #'first))
          (table-y (stable-sort table #'< :key #'second))
          edges) ; (cost from to)
      (declare (list edges))
      (loop for j below (- n 1)
            do (destructuring-bind (x1 y1 city1) (aref table-x j)
                 (destructuring-bind (x2 y2 city2) (aref table-x (+ j 1))
                   (declare (uint32 x1 y1 x2 y2 city1 city2))
                   (push (list (min (abs (- x2 x1)) (abs (- y2 y1)))
                               city1 city2)
                         edges)))
               (destructuring-bind (x1 y1 city1) (aref table-y j)
                 (destructuring-bind (x2 y2 city2) (aref table-y (+ j 1))
                   (declare (uint32 x1 y1 x2 y2 city1 city2))
                   (push (list (min (abs (- x2 x1)) (abs (- y2 y1)))
                               city1 city2)
                         edges)))
            finally (setf edges (sort edges #'< :key #'first)))
      (let ((tree (make-union-find n))
            (cost-sum 0))
        (dolist (edge edges)
          (destructuring-bind (cost from to) edge
            (unless (uf-connected-p from to tree)
              (uf-unite! from to tree)
              (incf cost-sum cost))))
        (println cost-sum)))))

#-swank(main)