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 Scala (2.13.1) (judge verdict: Accepted)

object Main extends App {
  val sc = new FastScanner(System.in)
  val n = sc.nextInt()
  val xy = List.fill(n)((sc.nextInt(), sc.nextInt())).zipWithIndex
  var e = List[(Int, Int, Int)]()
  xy.sortBy(_._1._1).sliding(2).foreach{ case List(((x1, y1), idx1), ((x2, y2), idx2)) =>
    e ::= (idx1, idx2, (x1 - x2).abs.min((y1 - y2).abs))
  }
  xy.sortBy(_._1._2).sliding(2).foreach{ case List(((x1, y1), idx1), ((x2, y2), idx2)) =>
    e ::= (idx1, idx2, (x1 - x2).abs.min((y1 - y2).abs))
  }

  def minimumSpanningTree(n: Int, edges: Seq[(Int, Int, Int)]): Int = {
    class UnionFindTree(n: Int) {
      private val par = Array.tabulate(n)(identity)
      private val rank = Array.fill(n)(1)

      def root(x: Int): Int = {
        var res = x
        var children = List[Int]()
        while (par(res) != res) {
          children ::= res
          res = par(res)
        }

        for (y <- children) {
          par(y) = res
        }
        res
      }

      def unite(x: Int, y: Int): Unit = {
        val xr = root(x)
        val yr = root(y)
        if (xr != yr) {
          if (rank(xr) < rank(yr)) {
            par(xr) = yr
          } else {
            par(yr) = xr
            if (rank(xr) == rank(yr)) {
              rank(xr) += 1
            }
          }
        }
      }
    }

    val uft = new UnionFindTree(n)
    var ans = 0
    for ((x, y, c) <- edges.sortBy(_._3) if uft.root(x) != uft.root(y)) {
      uft.unite(x, y)
      ans += c
    }

    ans
  }

  println(minimumSpanningTree(n, e))
}

class FastScanner(source: java.io.InputStream) {
  val buf = new Array[Byte](1024)
  var now, end = 0

  def isPrintable(b: Byte): Boolean = 33 <= b && b <= 126
  def isNum(b: Byte): Boolean = 48 <= b && b <= 57

  def hasNextByte: Boolean = {
    if (now < end) {
      true
    } else {
      now = 0
      end = source.read(buf)
      end > 0
    }
  }

  def nextByte(): Byte = {
    if (hasNextByte) {
      val b = buf(now)
      now += 1
      b
    } else {
      -1
    }
  }

  def hasNext: Boolean = {
    while (hasNextByte && !isPrintable(buf(now))) {
      now += 1
    }
    hasNextByte
  }

  def next(): String = {
    if (hasNext) {
      val sb = new StringBuilder()
      var b = nextByte()
      while (isPrintable(b)) {
        sb += b.toChar
        b = nextByte()
      }
      sb.toString
    } else {
      throw new NoSuchElementException()
    }
  }

  def nextInt(): Int = {
    nextLong().toInt
  }

  def nextLong(): Long = {
    if (hasNext) {
      var b = nextByte()
      var sign = 1
      var n = 0L
      if (b == '-') {
        sign = -1
        b = nextByte()
      }
      while (isNum(b)) {
        n = n * 10 + (b - 48)
        b = nextByte()
      }
      sign * n
    } else {
      throw new NoSuchElementException()
    }
  }

  def nextDouble(): Double = next().toDouble
}