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 Haskell (GHC 7.10.3) (judge verdict: Time Limit Exceeded)

import Data.List
import qualified Data.Set as S
import qualified Data.IntMap.Strict as M
import qualified Data.ByteString.Char8 as B

type Vertex = Int
type Weight = Int
type Edge = ((Vertex, Vertex), Weight)

snub = S.toList . S.fromList

readInt = maybe undefined fst . B.readInt

kruskal :: [Edge] -> Int
kruskal = fst . _kruskal initial 0 . sortOn snd

_kruskal :: UnionFind -> Int -> [Edge] -> (Int, UnionFind)
_kruskal uf acc [] = (acc, uf)
_kruskal uf acc (((a,b),d):abd)
  | same uf a b = _kruskal uf acc abd
  | otherwise = _kruskal (merge uf a b) (acc + d) abd

main = do
  _ <- getLine
  xys <- map ((\[x,y] -> (x+1, y+1)) . map readInt . B.words) . B.lines <$> B.getContents
  let xyis = zip (snub xys) [1 ..]
  let xysx = xyis
  let xysy = sortOn (snd . fst) xyis
  print (built xysx xysy)

built xysx xysy = kruskal (edgex ++ edgey)
  where
    edgex = zipWith (\((xi,yi),i) ((xj,yj),j) -> ((i,j), min (abs (xj-xi)) (abs (yj-yi)))) xysx (tail xysx)
    edgey = zipWith (\((xi,yi),i) ((xj,yj),j) -> ((i,j), min (abs (yj-yi)) (abs (xj-xi)))) xysy (tail xysy)

data UnionFind = UnionFind {
  tree :: M.IntMap Point,
  rk :: M.IntMap Rank
} deriving (Eq, Show)

type Point = Int -- greater than 0
type Rank = Int

initial :: UnionFind
initial = UnionFind M.empty M.empty

rep :: UnionFind -> Point -> Point
rep uf p
  | M.notMember p (tree uf) = p
  | tree uf M.! p == p = p
  | otherwise = rep uf (tree uf M.! p)

rank :: UnionFind -> Point -> Rank
rank uf p = M.findWithDefault 1 p (rk uf)

merge :: UnionFind -> Point -> Point -> UnionFind
merge uf p q
  | rp1 == rp2 = uf
  | otherwise = UnionFind (M.insert rp2 rp1 (M.insert p2 rp1 (tree uf))) (M.insert rp1 (rk1 + rk2) (rk uf))
  where
    p1 = ordBy uf p q
    p2 = if p1 == p then q else p
    rp1 = rep uf p1
    rp2 = rep uf p2
    rk1 = rank uf rp1
    rk2 = rank uf rp2

same :: UnionFind -> Point -> Point -> Bool
same uf p1 p2 = rep uf p1 == rep uf p2

ordBy :: UnionFind -> Point -> Point -> Point
ordBy uf x y
  | rx > ry = x
  | rx < ry = y
  | rep uf x < rep uf y = x
  | otherwise = y
  where
    rx = rank uf x
    ry = rank uf y