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: Wrong Answer)

using System;
using System.Collections.Generic;
class Program
{
	static void Main(string[] args)
	{
		int N = int.Parse(Console.ReadLine());
		int[,]A = new int[N,2];
		string[]S = new string[N];
		for(var i=0;i<N;i++){
			string[] str = Console.ReadLine().Split();
			A[i,0] = int.Parse(str[0]);
			A[i,1] = int.Parse(str[1]);
			S[i] = str[2];
		}
		double uymax = -999999999;
		double uymin = 999999999;
		double dymax = -999999999;
		double dymin = 999999999;
		double oymax = -999999999;
		double oymin = 999999999;
		double rxmax = -999999999;
		double rxmin = 999999999;
		double lxmax = -999999999;
		double lxmin = 999999999;
		double oxmax = -999999999;
		double oxmin = 999999999;
		for(var i=0;i<N;i++){
			if(S[i]=="U"){
				uymax = Math.Max(uymax,A[i,1]);
				uymin = Math.Min(uymin,A[i,1]);
			} else if(S[i]=="D"){
				dymax = Math.Max(dymax,A[i,1]);
				dymin = Math.Min(dymin,A[i,1]);
			} else {
				oymax = Math.Max(oymax,A[i,1]);
				oymin = Math.Min(oymin,A[i,1]);
			}
			if(S[i]=="R"){
				rxmax = Math.Max(rxmax,A[i,0]);
				rxmin = Math.Min(rxmin,A[i,0]);
			} else if(S[i]=="L"){
				lxmax = Math.Max(lxmax,A[i,0]);
				lxmin = Math.Min(lxmin,A[i,0]);
			} else {
				oxmax = Math.Max(oxmax,A[i,0]);
				oxmin = Math.Min(oxmin,A[i,0]);
			}
		}
		var list = new List<double>();
		list.Add(0);
		if(dymax>oymax){
			list.Add(dymax-oymax);
		}
		if(oymax>uymax){
			list.Add(oymax-uymax);
		}
		if(dymax>uymax){
			list.Add((dymax-uymax)/2);
		}
		if(uymin>oymin){
			list.Add(uymin-oymin);
		}
		if(oymin>dymin){
			list.Add(oymin-dymin);
		}
		if(uymin>dymin){
			list.Add((uymin-dymin)/2);
		}
		if(lxmax>oxmax){
			list.Add(lxmax-oxmax);
		}
		if(oxmax>rxmax){
			list.Add(oxmax-rxmax);
		}
		if(lxmax>rxmax){
			list.Add((lxmax-rxmax)/2);
		}
		if(rxmin>oxmin){
			list.Add(rxmin-oxmin);
		}
		if(oxmin>lxmin){
			list.Add(oxmin-lxmin);
		}
		if(rxmin>lxmin){
			list.Add((rxmin-lxmin)/2);
		}
		double ans = 999999999999999999;
		for(var i=0;i<list.Count;i++){
			double t = list[i];
			if(t>=300000000){
				continue;
			}
			double xmax = -999999999;
			double xmin = 999999999;
			double ymax = -999999999;
			double ymin = 999999999;
			for(var j=0;j<N;j++){
				double x = A[j,0];
				double y = A[j,1];
				if(S[j]=="U"){
					y += t;
				} else if(S[j]=="D"){
					y -= t;
				} else if(S[j]=="R"){
					x += t;
				} else {
					x -= t;
				}
				xmax = Math.Max(xmax,x);
				xmin = Math.Min(xmin,x);
				ymax = Math.Max(ymax,y);
				ymin = Math.Min(ymin,y);
			}
			ans = Math.Min(ans,(xmax-xmin)*(ymax-ymin));
			Console.Error.WriteLine(t);
		}
		Console.WriteLine(ans);
	}
}