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: Accepted)

import scala.annotation.tailrec
import scala.collection.mutable
import scala.io.StdIn._
object Main extends App {
  val MAX = 200000000.0
  val MIN = -MAX
  val n = readInt()
  val coordinates = mutable.Map[Char, List[Coordinate]]().withDefaultValue(Nil)
  for (_ ← 0 until n) {
    val Array(x, y, d) = readLine.trim.split(' ')
    coordinates(d.head) ::= Coordinate(x.toDouble, y.toDouble)
  }
  val constMinX = List('U', 'D').flatMap(coordinates).minByOption(_.x).map(_.x).getOrElse(MAX)
  val constMaxX = List('U', 'D').flatMap(coordinates).maxByOption(_.x).map(_.x).getOrElse(MIN)
  val constMinY = List('L', 'R').flatMap(coordinates).minByOption(_.y).map(_.y).getOrElse(MAX)
  val constMaxY = List('L', 'R').flatMap(coordinates).maxByOption(_.y).map(_.y).getOrElse(MIN)
  val minUp = coordinates('U').minByOption(_.y).map(_.y).getOrElse(MAX)
  val maxUp = coordinates('U').maxByOption(_.y).map(_.y).getOrElse(MIN)
  val minDown = coordinates('D').minByOption(_.y).map(_.y).getOrElse(MAX)
  val maxDown = coordinates('D').maxByOption(_.y).map(_.y).getOrElse(MIN)
  val minRight = coordinates('R').minByOption(_.x).map(_.x).getOrElse(MAX)
  val maxRight = coordinates('R').maxByOption(_.x).map(_.x).getOrElse(MIN)
  val minLeft = coordinates('L').minByOption(_.x).map(_.x).getOrElse(MAX)
  val maxLeft = coordinates('L').maxByOption(_.x).map(_.x).getOrElse(MIN)

  val changeTime = List(0, constMinY - minUp, minDown - constMinY, constMinX - minRight, minLeft - constMinX, constMaxY - maxUp, maxDown - constMaxY, constMaxX - maxRight, maxLeft - constMaxX, (minDown - minUp) / 2, (maxDown - maxUp) / 2, (minLeft - minRight) / 2, (maxLeft - maxRight) / 2).filter(_ >= 0)

  println(
    "%.15f".format(changeTime.map{t ⇒
      val minX = List(constMinX, minRight + t, minLeft - t).min
      val maxX = List(constMaxX, maxRight + t, maxLeft - t).max
      val minY = List(constMinY, minUp + t, minDown - t).min
      val maxY = List(constMaxY, maxUp + t, maxDown - t).max
      (maxX - minX) * (maxY - minY)
    }.min)
  )

  implicit class Scope[T](val receiver: T) extends AnyVal {
    def let[A](@inline func: T ⇒ A): A = func(receiver)
    def tap(@inline func: T ⇒ Unit): T = {
      func(receiver)
      receiver
    }
  }
  implicit class Wrap[T](val receiver: List[T]) extends AnyVal {
    def minByOption[A: Ordering](@inline keySelector: T ⇒ A): Option[T] = {
      receiver match {
        case Nil ⇒ None
        case _ ⇒ Some(receiver.minBy(keySelector))
      }
    }
    def maxByOption[A: Ordering](@inline keySelector: T ⇒ A): Option[T] = {
      receiver match {
        case Nil ⇒ None
        case _ ⇒ Some(receiver.maxBy(keySelector))
      }
    }
  }
}
case class Coordinate(x: Double, y: Double)