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 Scala (2.11.7) (judge verdict: Wrong Answer)

import java.io._
import java.util.StringTokenizer
import scala.util.control.Breaks.{breakable,break}
import scala.collection.mutable.{Map, Stack, Queue, PriorityQueue, Set}

case class Pos(x: Long, y: Long, dir: String) {

  // calc (x, y)
  def calc(t: Double): (Double, Double) = {
    dir match {
      case "R" => (x+t, y)
      case "L" => (x-t, y)
      case "U" => (x, y-t)
      case "D" => (x, y+t)
    }
  }
}

object Main extends App {
  val in = new InputReader(System.in)
  val pw = new PrintWriter(System.out)

  val N = in.next().toInt
  val p = new Array[Pos](N+1)

  for (i <- 1 to N) {
    val x, y = in.next().toLong
    val d = in.next()

    p(i) = Pos(x, y, d)
  }


  //
  def check(n: Double): Double = {
    import Math.{max, min}

    var xmax = -2e8
    var ymax = -2e8
    var xmin = 2e8
    var ymin = 2e8

    for (i <- 1 to N) {
      val (x, y) = p(i).calc(n)

      xmax = max(xmax, x)
      ymax = max(ymax, y)
      xmin = min(xmin, x)
      ymin = min(ymin, y)
    }

    (xmax - xmin) * (ymax - ymin)
  }

  // 下に凸のグラフの最小値
  def ternarySearch(small: Double, big: Double): Double = {
    val END = 1e-3

    def func(small: Double, big: Double): Double = {
      val vol = (big - small) / 3
      val midl = small + vol
      val midr = small + vol * 2
      if (big - small < END) big
      else {
        if (check(midl) >= check(midr))
          func(midl, big)
        else
          func(small, midr)
      }
    }
    func(small, big)
  }

  def timeProc(time: Double): Double = (time * 10).round / 10.0

  var ans =  Double.MaxValue

  for (i <- 1 to 1000) {
    val a = 1e8 / i
    val time = ternarySearch(a * (i-1), a * i)
    ans = Math.min(ans, check(timeProc(time)))
  }

  println(ans)
}

object Sub {
  def print2DimArray[A](mat: Array[Array[A]]): Unit = {
    mat.foreach(lin => println(lin.deep))
  }
}

class InputReader(stream: InputStream) {
  private[this] val reader = new BufferedReader(new InputStreamReader(stream))
  private[this] var tokenizer: StringTokenizer = new StringTokenizer(reader.readLine())
  def next(): String = {
    while (!tokenizer.hasMoreTokens()) {
      tokenizer = new StringTokenizer(reader.readLine())
    }
    tokenizer.nextToken()
  }
}