Problem:
Score : 600 points

Problem Statement

There are N dots in a two-dimensional plane. The coordinates of the i-th dot are (x_i, y_i).

We will repeat the following operation as long as possible:

Choose four integers a, b, c, d (a \neq c, b \neq d) such that there are dots at exactly three of the positions (a, b), (a, d), (c, b) and (c, d), and add a dot at the remaining position.

We can prove that we can only do this operation a finite number of times. Find the maximum number of times we can do the operation.

Constraints

1 \leq N \leq 10^5

1 \leq x_i, y_i \leq 10^5

If i \neq j, x_i \neq x_j or y_i \neq y_j.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1
:
x_N y_N

Output

Print the maximum number of times we can do the operation.

Sample Input 1

3
1 1
5 1
5 5

Sample Output 1

1

By choosing a = 1, b = 1, c = 5, d = 5, we can add a dot at (1, 5). We cannot do the operation any more, so the maximum number of operations is 1.

Sample Input 2

2
10 10
20 20

Sample Output 2

0

There are only two dots, so we cannot do the operation at all.

Sample Input 3

9
1 1
2 1
3 1
4 1
5 1
1 2
1 3
1 4
1 5

Sample Output 3

16

We can do the operation for all choices of the form a = 1, b = 1, c = i, d = j (2 \leq i,j \leq 5), and no more. Thus, the maximum number of operations is 16.

Solution in Ruby (2.3.3) (judge verdict: Wrong Answer)

class UnionFind
  attr :parent, :rank
  def initialize
    @parent = Hash.new
    @rank = Hash.new
  end
  def make(x)
    @parent[x] = x
    @rank[x] = 0
  end
  def union(a, b)
    x, y = find(a), find(b)
    x, y = y, x if @rank[x] < @rank[y]
    @parent[y] = x
    @rank[x] += 1 if @rank[x] == @rank[y]
  end
  def find(x)
    @parent[x] == x ? x : @parent[x] = find(@parent[x])
  end
  def size
    @parent.keys.map{|k| find(k)}.sort.uniq.size
  end
end

require 'set'

P = gets.to_i.times.map{|i| gets.split.map &:to_i}
N = 10**9

uf = UnionFind.new
P.size.times do |i|
  n, x, y = N+i, P[i][0], -P[i][1]
  uf.make(n)
  uf.make(x)
  uf.make(y)
end
P.size.times do |i|
  n, x, y = N+i, P[i][0], -P[i][1]
  uf.union(n, x)
  uf.union(n, y)
  uf.union(x, y)
end

X = Hash.new{|h, k| h[k] = Set.new}
Y = Hash.new{|h, k| h[k] = Set.new}
G = Set.new
P.size.times do |i|
  g, x, y = uf.find(N+i), P[i][0], -P[i][1]
  G << g
  X[g] << x
  Y[g] << y
end
p G.map{|g| (X[g].size-1) * (Y[g].size-1)}.inject(:+)