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.IntMap as M
import qualified Data.ByteString.Char8 as B

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 = fromDistinctAscList [1..n]

kruskalDP _ _ acc _ [] = acc
kruskalDP n uf acc x (((a,b),d):abd)
  | x == n = acc
  | same uf a b = kruskalDP n uf acc x abd
  | otherwise = kruskalDP n (merge uf a b) (acc + d) (succ x) abd

main = do
  n <- readLn
  xys <- map ((\[x,y]->(x,y)) . map readInt . B.words) . B.lines <$> B.getContents
  let xyis = zip xys [1..]
  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,_),i) ((xj,_),j) -> ((i,j), xj-xi)) xysx (tail xysx)
    edgey = zipWith (\((_,yi),i) ((_,yj),j) -> ((i,j), yj-yi)) xysy (tail xysy)

data UnionFind = UnionFind {tree :: M.IntMap Point, rk :: M.IntMap Rank}
type Point = Int
type Rank = Int

rep :: UnionFind -> Point -> Point
rep uf = untilFix (tree uf M.!)

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

merge :: UnionFind -> Point -> Point -> UnionFind
merge uf p1 p2
  | rp1 /= rp2 && rk1 > rk2 = UnionFind tree12 rank1
  | rp1 /= rp2 = UnionFind tree21 rank2
  | otherwise = uf
  where
    tree12 = M.insert p2 rp1 (M.insert rp2 rp1 (tree uf))
    tree21 = M.insert p1 rp2 (M.insert rp1 rp2 (tree uf))
    rank1 = M.insert rp1 (rk1 + rk2) (rk uf)
    rank2 = M.insert rp2 (rk1 + rk2) (rk uf)
    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 = rep uf p1 == rep uf p2

fromDistinctAscList :: [Point] -> UnionFind
fromDistinctAscList ps = UnionFind tree0 rk0
  where
    tree0 = M.fromDistinctAscList [(i, i) | i <- ps]
    rk0 = M.fromDistinctAscList [(i, 1) | i <- ps]