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 as M
import qualified Data.ByteString.Char8 as B

snub = S.toList . S.fromList

readInt = maybe undefined fst . B.readInt

untilFix f x = let x1 = f x in if x1 == x then x else untilFix f x1

kruskal n = kruskalDP n uf0 0 0
  where
    uf0 = UnionFind (M.singleton 0 0) M.empty

kruskalDP _ _ acc _ [] = acc
kruskalDP n uf acc x (((a,b),d):abd)
  | x == n - 1 = acc
  | same uf a b = kruskalDP n uf acc x abd
  | otherwise = kruskalDP n (merge uf a b) (acc + d) (x + 1) 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 n = length xyis
  let xysx = sortOn fst xyis
  let xysy = sortOn (snd.fst) xyis
  print (built n xysx xysy)

built n xysx xysy = kruskal n edge
  where
    edge = sortOn snd (edgex ++ edgey)
    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}
type Point = Int
type Rank = Int

initial :: UnionFind
initial = UnionFind (M.singleton 0 0) M.empty

(!?) :: UnionFind -> Point -> Point
(!?) uf p = M.findWithDefault 0 p (tree uf)

rep :: UnionFind -> Point -> Point
rep uf = untilFix (uf !?)

rank :: UnionFind -> Point -> Rank
rank uf = (rk uf M.!)

isolate :: UnionFind -> Point -> Bool
isolate uf p = rep uf p == 0

merge :: UnionFind -> Point -> Point -> UnionFind
merge uf p1 p2
  | i1 && i2 = UnionFind (M.insert p2 p1 (M.insert p1 p1 (tree uf))) (M.insert p1 2 (rk uf))
  | i1 = UnionFind (M.insert p1 rp2 (tree uf)) (M.insert rp2 (rk2 + 1) (rk uf))
  | i2 = UnionFind (M.insert p2 rp1 (tree uf)) (M.insert rp1 (rk1 + 1) (rk uf))
  | rp1 /= rp2 && rk1 > rk2 = UnionFind (M.insert p2 rp1 (M.insert rp2 rp1 (tree uf))) (M.insert rp1 (rk1 + rk2) (rk uf))
  | rp1 /= rp2 = UnionFind (M.insert p1 rp2 (M.insert rp1 rp2 (tree uf))) (M.insert rp2 (rk1 + rk2) (rk uf))
  | otherwise = uf
  where
    i1 = isolate uf p1
    i2 = isolate uf p2
    rk1 = rank uf rp1
    rk2 = rank uf rp2
    rp1 = rep uf p1
    rp2 = rep uf p2

same :: UnionFind -> Point -> Point -> Bool
same uf p1 p2
  | p1 == p2 = True
  | otherwise = not (isolate uf p1) && (rep uf p1 == rep uf p2)