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.Strict as M
import qualified Data.Set as S
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 uf0 0
  where
    uf0 = fromDistinctAscList [1..n]

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

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

built xysx xysy = kruskal n edge
  where
    n = length xysx
    edge = sortOn snd (edgex ++ edgey)
    edgex = zipWith mkedge xysx (tail xysx)
    edgey = zipWith mkedge xysy (tail xysy)

mkedge ((xi,yi),i) ((xj,yj),j) = ((i,j), min (abs (xj-xi)) (abs (yj-yi)))

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

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

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

merge :: UnionFind -> Point -> Point -> UnionFind
merge uf p1 p2
  | r1 /= r2 && rank uf r1 > rank uf r2 = UnionFind (M.insert p2 r1 (M.insert r2 r1 (tree uf))) (M.insert r1 (r1 + r2) (rk uf)) 
  | r1 /= r2 = UnionFind (M.insert p1 r2 (M.insert r1 r2 (tree uf))) (M.insert r2 (r1 + r2) (rk uf)) 
  | otherwise = uf
  where
    r1 = untilFix (rep uf) p1
    r2 = untilFix (rep uf) p2

same :: UnionFind -> Point -> Point -> Bool
same uf p1 p2 = untilFix (rep uf) p1 == untilFix (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]