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 Ruby (2.3.3) (judge verdict: Time Limit Exceeded)

class UnionFind
  def initialize
    @parent = Hash.new { |hash, key| hash[key] = key }
    @rank = Hash.new { |hash, key| hash[key] = 0 }
  end

  def root(x)
    @parent[x] == x ? x : @parent[x] = root(@parent[x])
  end

  def same?(x, y)
    root(x) == root(y)
  end

  def unite(x, y)
    x, y = root(x), root(y)
    return if x == y
    x, y = y, x if @rank[x] < @rank[y]
    @rank[x] += 1 if @rank[x] == @rank[y]
    @parent[y] = x
  end

  def inspect
    @parent
  end
end

class UndirectedGraph
  class Edge
    include Comparable

    attr_accessor :vertex, :cost

    def initialize(i, j, cost)
      @vertex = [i, j]
      @cost = cost
    end

    def to(i)
      if @vertex[0] == i
        vertex[1]
      elsif @vertex[1] == i
        vertex[0]
      else
        nil
      end
    end

    def <=>(other)
      cost <=> other.cost
    end
  end

  def initialize
    @edges = []
  end

  def connect(i, j, cost)
    @edges << Edge.new(i, j, cost)
  end

  def kluskal
    @edges.sort!
    uf = UnionFind.new
    res = 0
    @edges.each do |e|
      u, v = e.vertex
      unless uf.same?(u, v)
        uf.unite(u, v)
        res += e.cost
      end
    end
    res
  end
end

g = UndirectedGraph.new
n = gets.to_i
ps = []
n.times { ps << gets.split.map(&:to_i) }
qs = ps.dup
ps.sort_by! { |x| x[0] }
qs.sort_by! { |x| x[1] }
for i in 1...n do
  g.connect(ps[i - 1], ps[i], ps[i][0] - ps[i - 1][0])
  g.connect(qs[i - 1], qs[i], qs[i][1] - qs[i - 1][1])
end
p g.kluskal