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 Fortran (gfortran v4.8.4) (judge verdict: Wrong Answer)

module mod_union_find
  implicit none
  type union_find
    integer :: n
    integer, pointer :: par(:), rnk(:), siz(:)
  end type union_find
  private
  public :: union_find
  public :: init_union_find, release_union_find, find, same, unite, size_of
contains
  subroutine init_union_find(uf,n)
    implicit none
    type(union_find), intent(inout) :: uf
    integer, intent(in) :: n
    integer :: i
    allocate(uf%par(-n:n),uf%rnk(-n:n),uf%siz(-n:n))
    uf%rnk = 0
    uf%siz = 1
    do i = -n, n
      uf%par(i) = i
    end do
    return
  end subroutine init_union_find
  subroutine release_union_find(uf)
    implicit none
    type(union_find), intent(inout) :: uf
    if (associated(uf%par)) deallocate(uf%par)
    if (associated(uf%rnk)) deallocate(uf%rnk)
    if (associated(uf%siz)) deallocate(uf%siz)
    return
  end subroutine release_union_find
  recursive function find(uf,i) result(j)
    implicit none
    type(union_find), intent(inout) :: uf
    integer, intent(in) :: i
    integer :: j
    if (uf%par(i).ne.i) then
      uf%par(i) = find(uf,uf%par(i))
    end if
    j = uf%par(i)
    return
  end function find
  function same(uf,i,j) result(y)
    implicit none
    type(union_find), intent(inout) :: uf
    integer, intent(in) :: i, j
    logical :: y
    y = find(uf,i).eq.find(uf,j)
  end function same
  subroutine unite(uf,i,j)
    implicit none
    type(union_find), intent(inout) :: uf
    integer, intent(in) :: i, j
    integer :: x, y
    x = find(uf,i)
    y = find(uf,j)
    if (x.eq.y) return
    if (uf%rnk(x).lt.uf%rnk(y)) then
      uf%par(x) = y
      uf%siz(y) = uf%siz(y)+uf%siz(x)
    else
      uf%par(y) = x
      uf%siz(x) = uf%siz(x)+uf%siz(y)
      if (uf%rnk(x).eq.uf%rnk(y)) uf%rnk(x) = uf%rnk(x)+1
    end if
    return
  end subroutine unite
  function size_of(uf,i) result(s)
    implicit none
    type(union_find), intent(inout) :: uf
    integer, intent(in) :: i
    integer :: s
    s = uf%siz(find(uf,i))
  end function size_of
end module mod_union_find
program must_be_rectangular
  use mod_union_find
  implicit none
  type(union_find) :: uf
  integer :: n, x, y, m(-100000:100001,2), i, j
  integer(8) :: z
  call init_union_find(uf,100000)
  m = 0
  read(*,*) n
  do i = 1, n
    read(*,*) x, y
    call unite(uf,x,-y)
  end do
  m(-100000:100000,1) = uf%par
  m(100001,1) = 1000010000
  do i = -100000, 100000
    m(i,2) = i
  end do
  call merge_sort(m(-100000:100000,1:2),1)
  i = -100000
  z = 0_8
  do while (i.lt.100001)
    x = 0
    y = 0
    do j = i, 100001
      if (m(i,1).eq.m(j,1)) then
        if (m(j,2).gt.0) then
          x = x+1
        else
          y = y+1
        end if
      else
        z = z+int(x,8)*int(y,8)
        i = j
        exit
      end if
    end do
  end do
  write(*,'(i0)') z-int(n,8)
  stop
contains
  subroutine merge_sort(c,k)
    implicit none
    integer, intent(inout) :: c(:,:)
    integer, intent(in) :: k
    integer :: n, m, l, u, i
    n = size(c(:,1))
    m = n
    l = 1
    do while (m.gt.1)
      do i = 1, m/2
        u = min(2*i*l,n)
        call merger(c(2*(i-1)*l+1:(2*i-1)*l,:),c((2*i-1)*l+1:u,:),k)
      end do
      l = 2*l
      m = (m+1)/2
    end do
    return
  end subroutine merge_sort
  subroutine merger(c1,c2,k)
    implicit none
    integer, intent(inout) :: c1(:,:), c2(:,:)
    integer, intent(in) :: k
    integer :: c(size(c1(:,1))+size(c2(:,1)),size(c2(1,:)))
    integer :: i1, i2, n1, n2
    i1 = 1
    i2 = 1
    n1 = size(c1(:,1))
    n2 = size(c2(:,1))
    do while ((i1.le.n1).and.(i2.le.n2))
      if (c1(i1,k).le.c2(i2,k)) then
        c(i1+i2-1,:) = c1(i1,:)
        i1 = i1+1
      else
        c(i1+i2-1,:) = c2(i2,:)
        i2 = i2+1
      end if
    end do
    if (i1.le.n1) then
      c(i1+i2-1:n1+n2,:) = c1(i1:n1,:)
    else if (i2.le.n2) then
      c(i1+i2-1:n1+n2,:) = c2(i2:n2,:)
    end if
    c1 = c(1:n1,:)
    c2 = c(n1+1:n1+n2,:)
    return
  end subroutine merger
end program must_be_rectangular