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 Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

import java.util.ArrayList;
import java.util.Arrays;
import java.util.Comparator;
import java.util.List;
import java.util.Scanner;

public class Main {
  private static int INF = 1_000_000_000;
  public static void main(String[] args) {
    Scanner scanner = new Scanner(System.in);

    int N = scanner.nextInt();
    Point[] xPoints = new Point[N];
    Point[] yPoints = new Point[N];
    for (int i = 0; i < N; i++) {
      int x = scanner.nextInt();
      int y = scanner.nextInt();
      char d = scanner.next().charAt(0);
      xPoints[i] = new Point(x, d == 'R' ? Dir.POS : d == 'L' ? Dir.NEG : Dir.ZERO);
      yPoints[i] = new Point(y, d == 'U' ? Dir.POS : d == 'D' ? Dir.NEG : Dir.ZERO);
    }

    List<Double> intersects = new ArrayList<>();
    intersects.add(0.d);
    Arrays.sort(xPoints, Comparator.comparingInt(p -> p.pos));
    PolyLine left = polyline(xPoints);
    intersects.addAll(left.intersects());
    Arrays.sort(xPoints, Comparator.comparingInt(p -> -p.pos));
    PolyLine right = polyline(xPoints);
    intersects.addAll(right.intersects());
    Arrays.sort(yPoints, Comparator.comparingInt(p -> p.pos));
    PolyLine bottom = polyline(yPoints);
    intersects.addAll(bottom.intersects());
    Arrays.sort(yPoints, Comparator.comparingInt(p -> -p.pos));
    PolyLine top = polyline(yPoints);
    intersects.addAll(top.intersects());

    double min = (double) INF * INF;
    for (double i : intersects) {
      if (Double.compare(i, 0.d) >= 0) min = Math.min(min, (right.maxAt(i) - left.minAt(i)) * (top.maxAt(i) - bottom.minAt(i)));
    }
    System.out.println(min);
  }

  private static PolyLine polyline(Point[] points) {
    PolyLine res = new PolyLine();
    boolean pos = false, zero = false, neg = false;
    for (Point p : points) {
      if (!pos && p.direction == Dir.POS) {
        res.lines.add(new Line(1, p.pos));
        pos = true;
      }
      if (!zero && p.direction == Dir.ZERO) {
        res.lines.add(new Line(0, p.pos));
        zero = true;
      }
      if (!neg && p.direction == Dir.NEG) {
        res.lines.add(new Line(-1, p.pos));
        neg = true;
      }
    }
    return res;
  }

  private static class Point {
    final int pos;
    final Dir direction;

    private Point(int pos, Dir direction) {
      this.pos = pos;
      this.direction = direction;
    }
  }

  private enum Dir {
    POS, NEG, ZERO
  }

  private static class PolyLine {
    List<Line> lines = new ArrayList<>();

    public double maxAt(double x) {
      double max = 0;
      for (Line l : lines) {
        max = Math.max(max, l.valueAt(x));
      }
      return max;
    }

    public double minAt(double x) {
      double min = INF;
      for (Line l : lines) {
        min = Math.min(min, l.valueAt(x));
      }
      return min;
    }

    public List<Double> intersects() {
      List<Double> res = new ArrayList<>();
      for (int i = 0; i < lines.size(); i++) {
        for (int j = i + 1; j < lines.size(); j++) {
          Double x = lines.get(i).intersectWith(lines.get(j));
          if (x != null) res.add(x);
        }
      }
      return res;
    }
  }

  private static class Line {
    private final int slope;
    private final int intercept;

    private Line(int slope, int intercept) {
      this.slope = slope;
      this.intercept = intercept;
    }

    public double valueAt(double x) {
      return intercept + slope * x;
    }

    public Double intersectWith(Line line) {
      if (slope == line.slope) return null;
      return (double) (intercept - line.intercept) / (line.slope - slope);
    }
  }
}