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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Diagnostics;
using System.Linq;
using System.Numerics;
using System.Text;
using static System.Math;

class Program
{
    static int n;
    static long xl1, xl2, xs1, xs2, yl1, yl2, ys1, ys2;
    static long xls, xrs, yls, yss;

    static void Main()
    {
        n = int.Parse(Console.ReadLine());
        xl1 = -((long)1e8);
        xl2 = -((long)1e8);
        yl1 = -((long)1e8);
        yl2 = -((long)1e8);
        xs2 = ((long)1e8);
        xs1 = ((long)1e8);
        ys2 = ((long)1e8);
        ys1 = ((long)1e8);
        xls = -(long)1e8;
        yls = -(long)1e8;
        xrs = (long)1e8;
        yss = (long)1e8;
        long x, y;
        char d;
        for (int i = 0; i < n; i++)
        {
            string[] input = Console.ReadLine().Split();
            x = long.Parse(input[0]);
            y = long.Parse(input[1]);
            d = input[2][0];
            if (d != 'R' && d != 'L')
                xl1 = Max(xl1, x);
            if (d != 'R' && d != 'L')
                xs1 = Min(xs1, x);
            if (d != 'U' && d != 'D')
                yl1 = Max(yl1, y);
            if (d != 'U' && d != 'D')
                ys1 = Min(ys1, y);
            if (d == 'R')
            {
                xl2 = Max(xl2, x);
                xrs = Min(xrs, x);
            }
            if (d == 'L')
            {
                xs2 = Min(xs2, x);
                xls = Max(xls, x);
            }
            if (d == 'U')
            {
                yl2 = Max(yl2, y);
                yss = Min(yss, y);
            }
            if (d == 'D')
            {
                ys2 = Min(ys2, y);
                yls = Max(yls, y);
            }
        }

        double low = 0.0, high = 1e8;
        while (high - low > 1e-9)
        {
            double width = high - low, ml = low + width / 3.0, mr = low + 2.0 * width / 3.0;
            if (Abs(f(ml) - f(mr)) < 1e-9)
            {
                Console.WriteLine(f(ml));
                return;
            }
            if (f(ml) <= f(mr))
            {
                high = mr;
            }
            else
            {
                low = ml;
            }
        }
        Console.WriteLine(f((high + low) / 2.0));
    }

    static double xmax(double t)
    {
        return Max(Max((double)xl1, t + xl2), -t + xls);
    }

    static double xmin(double t)
    {
        return Min(t + xrs, Min((double)xs1, -t + xs2));
    }

    static double ymax(double t)
    {
        return Max(-t + yls, Max((double)yl1, t + yl2));
    }

    static double ymin(double t)
    {
        return Min(t + yss, Min((double)ys1, -t + ys2));
    }

    static double f(double t)
    {
        return (xmax(t) - xmin(t)) * (ymax(t) - ymin(t));
    }
}