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 Rust (1.15.1) (judge verdict: Accepted)

use std::io::prelude::*;

fn input<T: std::str::FromStr>() -> T {
    let stdin = std::io::stdin();
    let token: String = stdin
        .lock()
        .bytes()
        .map(|c| c.unwrap() as char)
        .skip_while(|c| c.is_whitespace())
        .take_while(|c| !c.is_whitespace())
        .collect();
    token.parse().ok().unwrap()
}

fn main() {
    let n: usize = input();
    let xyd: Vec<(i32, i32, String)> = (0..n).map(|_| (input(), input(), input())).collect();
    const INF: i32 = std::i32::MAX;

    let mut xmin = vec![INF; 3];
    let mut xmax = vec![-INF; 3];
    let mut ymin = vec![INF; 3];
    let mut ymax = vec![-INF; 3];
    for (x, y, d) in xyd {
        if d == "R" {
            xmin[2] = std::cmp::min(xmin[2], x);
            xmax[2] = std::cmp::max(xmax[2], x);
            ymin[1] = std::cmp::min(ymin[1], y);
            ymax[1] = std::cmp::max(ymax[1], y);
        } else if d == "L" {
            xmin[0] = std::cmp::min(xmin[0], x);
            xmax[0] = std::cmp::max(xmax[0], x);
            ymin[1] = std::cmp::min(ymin[1], y);
            ymax[1] = std::cmp::max(ymax[1], y);
        } else if d == "U" {
            xmin[1] = std::cmp::min(xmin[1], x);
            xmax[1] = std::cmp::max(xmax[1], x);
            ymin[2] = std::cmp::min(ymin[2], y);
            ymax[2] = std::cmp::max(ymax[2], y);
        } else {
            xmin[1] = std::cmp::min(xmin[1], x);
            xmax[1] = std::cmp::max(xmax[1], x);
            ymin[0] = std::cmp::min(ymin[0], y);
            ymax[0] = std::cmp::max(ymax[0], y);
        }
    }

    let mut t = vec![0.0];
    for i in 0..3 {
        for j in i..3 {
            if xmin[i] != INF && xmin[j] != INF && xmin[i] - xmin[j] > 0 {
                t.push((xmin[i] - xmin[j]) as f64 / (j - i) as f64);
            }
            if xmax[i] != -INF && xmax[j] != -INF && xmax[i] - xmax[j] > 0 {
                t.push((xmax[i] - xmax[j]) as f64 / (j - i) as f64);
            }
            if ymin[i] != INF && ymin[j] != INF && ymin[i] - ymin[j] > 0 {
                t.push((ymin[i] - ymin[j]) as f64 / (j - i) as f64);
            }
            if ymax[i] != -INF && ymax[j] != -INF && ymax[i] - ymax[j] > 0 {
                t.push((ymax[i] - ymax[j]) as f64 / (j - i) as f64);
            }
        }
    }

    let mut ans = std::f64::MAX;
    for t in t {
        let mut xl = INF as f64;
        let mut xr = -INF as f64;
        let mut yl = INF as f64;
        let mut yr = -INF as f64;

        for i in 0..3 {
            if xmin[i] != INF {
                xl = f64::min(xl, xmin[i] as f64 + (i as i32 - 1) as f64 * t);
            }
            if xmax[i] != -INF {
                xr = f64::max(xr, xmax[i] as f64 + (i as i32 - 1) as f64 * t);
            }
            if ymin[i] != INF {
                yl = f64::min(yl, ymin[i] as f64 + (i as i32 - 1) as f64 * t);
            }
            if ymax[i] != -INF {
                yr = f64::max(yr, ymax[i] as f64 + (i as i32 - 1) as f64 * t);
            }
        }

        ans = f64::min(ans, (xr - xl) * (yr - yl));
    }

    println!("{}", ans);
}