Problem:
Score : 600 points

Problem Statement

There are N points in a two-dimensional plane. The initial coordinates of the i-th point are (x_i, y_i). Now, each point starts moving at a speed of 1 per second, in a direction parallel to the x- or y- axis. You are given a character d_i that represents the specific direction in which the i-th point moves, as follows:

If d_i = R, the i-th point moves in the positive x direction;

If d_i = L, the i-th point moves in the negative x direction;

If d_i = U, the i-th point moves in the positive y direction;

If d_i = D, the i-th point moves in the negative y direction.

You can stop all the points at some moment of your choice after they start moving (including the moment they start moving).
Then, let x_{max} and x_{min} be the maximum and minimum among the x-coordinates of the N points, respectively. Similarly, let y_{max} and y_{min} be the maximum and minimum among the y-coordinates of the N points, respectively.

Find the minimum possible value of (x_{max} - x_{min}) \times (y_{max} - y_{min}) and print it.

Constraints

1 \leq N \leq 10^5

-10^8 \leq x_i,\ y_i \leq 10^8

x_i and y_i are integers.

d_i is R, L, U, or D.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1 d_1
x_2 y_2 d_2
.
.
.
x_N y_N d_N

Output

Print the minimum possible value of (x_{max} - x_{min}) \times (y_{max} - y_{min}).

The output will be considered correct when its absolute or relative error from the judge's output is at most 10^{-9}.

Sample Input 1

2
0 3 D
3 0 L

Sample Output 1

0

After three seconds, the two points will meet at the origin. The value in question will be 0 at that moment.

Sample Input 2

5
-7 -10 U
7 -6 U
-8 7 D
-3 3 D
0 -6 R

Sample Output 2

97.5

The answer may not be an integer.

Sample Input 3

20
6 -10 R
-4 -9 U
9 6 D
-3 -2 R
0 7 D
4 5 D
10 -10 U
-1 -8 U
10 -6 D
8 -5 U
6 4 D
0 3 D
7 9 R
9 -4 R
3 10 D
1 9 U
1 -6 U
9 -8 R
6 7 D
7 -3 D

Sample Output 3

273

Solution in OCaml (4.02.3) (judge verdict: Wrong Answer)

module Array = ArrayLabels
module List = ListLabels

let inf = 1000000000.
let (>>=) x f = List.map f x |> List.concat

let () =
  Scanf.scanf "%d" @@ fun n ->
  let xyds = Array.init n (fun _ -> Scanf.scanf " %f %f %s" (fun x y d -> x, y, d)) in
  let find_edges axis dir =
    Array.fold_left xyds ~init:(inf, -.inf) ~f:(fun (minv, maxv) (x, y, d) ->
      let v = match axis with `X -> x | `Y -> y in
      if d = dir then (min minv v, max maxv v) else (minv, maxv))
  in
  let dirs = ["R"; "L"; "U"; "D"] in
  let exs = List.map dirs ~f:(fun d ->
    let x1, x2 = find_edges `X d in
    (x1, x2, match d with "R" -> 1. | "L" -> -1. | _ -> 0.))
  and eys = List.map dirs ~f:(fun d ->
    let y1, y2 = find_edges `Y d in
    (y1, y2, match d with "U" -> 1. | "D" -> -1. | _ -> 0.))
  in

  let area_of_time (t : float) =
    let (+), (-), ( * ) = (+.), (-.), ( *.) in
    let xm, xM = List.fold_left exs ~init:(inf, -.inf) ~f:(fun (xm, xM) (x1, x2, mul) ->
      (min xm (x1 + mul * t), max xM (x2 + mul * t)))
    in
    let ym, yM = List.fold_left eys ~init:(inf, -.inf) ~f:(fun (ym, yM) (y1, y2, mul) ->
      (min ym (y1 + mul * t), max yM (y2 + mul * t)))
    in
    (xM - xm) * (yM - ym)
  in

  let txs =
    exs >>= fun (xa1, xa2, da) ->
    exs >>= fun (xb1, xb2, db) ->
      if da = db then [] else
        [xa1; xa2] >>= fun xa ->
        [xb1; xb2] >>= fun xb -> [abs_float @@ (xa -. xb) /. (da -. db)]
  and tys =
    eys >>= fun (ya1, ya2, da) ->
    eys >>= fun (yb1, yb2, db) ->
      if da = db then [] else
        [ya1; ya2] >>= fun ya ->
        [yb1; yb2] >>= fun yb -> [abs_float @@ (ya -. yb) /. (da -. db)]
  in
  List.map area_of_time ([0.] @ txs @ tys) |> List.fold_left ~f:min ~init:inf
  |> Printf.printf "%.11f\n"