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 OCaml (4.02.3) (judge verdict: Wrong Answer)

module Union_find : sig
  type t
  val make : int -> t
  val find : int -> t -> int
  val size : int -> t -> int
  val same : int -> int -> t -> bool
  val unite : int -> int -> t -> unit
end = struct
  type t = int array
  let make n = Array.make n (-1)
  let rec find x uf =
    if uf.(x) < 0 then x
    else (( let x' = find uf.(x) uf in uf.(x) <- x'; x' ))
  let same x y uf = find x uf = find y uf
  let size x uf = - uf.(find x uf)
  let unite x y uf =
    let x = find x uf and y = find y uf in
    if x <> y then
      if uf.(x) < uf.(y) then ((
        uf.(x) <- uf.(x) + uf.(y);
        uf.(y) <- x
      )) else ((
        uf.(y) <- uf.(x) + uf.(y);
        uf.(x) <- y
      ))
end
module UF = Union_find

module M = Map.Make(struct type t = int let compare = (-) end)
module S = Set.Make(struct type t = int let compare = (-) end)

let () =
  Scanf.scanf "%d" @@ fun n ->
  let uf = UF.make 100000 in
  let xy = Array.init n (fun _ -> Scanf.scanf " %d %d" (fun x y -> x-1, y-1)) in
  Array.sort (fun (a,_) (c,_) -> a-c) xy;
  let _ = Array.fold_left (fun (u, v) (w, x) ->
    if u = w then UF.unite v x uf; (w, x)
  ) (-1, -1) xy in
  let ss = Array.fold_left (fun ss (x, y) ->
    let k = UF.find x uf in
    try M.add k (S.add y (M.find k ss)) ss
    with Not_found -> M.add k (S.singleton y) ss
  ) M.empty xy in
  let t = M.fold (fun k v s -> UF.size k uf * S.cardinal v + s) ss 0 in
  Printf.printf "%d\n" (t - n)