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)

fn read_stdin() -> String {
    use std::io::Read;
    let mut input = String::new();
    std::io::stdin().read_to_string(&mut input).unwrap();
    input
}

fn scan<'a, I: Iterator<Item = &'a str>, T: std::str::FromStr>(tokens: &mut I) -> T {
    tokens.next().unwrap().parse().ok().unwrap()
}

const INFTIY: f64 = 1e10;

const R: usize = 0;
const L: usize = 1;
const U: usize = 2;
const D: usize = 3;

const DX: [f64; 4] = [1.0, -1.0, 0.0, 0.0];
const DY: [f64; 4] = [0.0, 0.0, 1.0, -1.0];

fn main() {
    let input = read_stdin();
    let tokens = &mut input.split_whitespace();
    let n: usize = scan(tokens);
    let mut x_mins = [INFTIY; 4];
    let mut x_maxs = [-INFTIY; 4];
    let mut y_mins = [INFTIY; 4];
    let mut y_maxs = [-INFTIY; 4];
    for _ in 0..n {
        let x: f64 = scan(tokens);
        let y: f64 = scan(tokens);
        let d: String = scan(tokens);
        let d: usize = match d.chars().next().unwrap() {
            'L' => L,
            'R' => R,
            'U' => U,
            'D' => D,
            _ => unreachable!(),
        };
        x_mins[d] = x_mins[d].min(x);
        x_maxs[d] = x_maxs[d].max(x);
        y_mins[d] = y_mins[d].min(y);
        y_maxs[d] = y_maxs[d].max(y);
    }
    let mut min = 4.0 * INFTIY * INFTIY;
    {
        let mut check = |t: f64| {
            if t >= 0.0 {
                let mut x_min = INFTIY;
                let mut x_max = -INFTIY;
                let mut y_min = INFTIY;
                let mut y_max = -INFTIY;
                for d in 0..4 {
                    x_min = x_min.min(x_mins[d] + t * DX[d]);
                    x_max = x_max.max(x_maxs[d] + t * DX[d]);
                    y_min = y_min.min(y_mins[d] + t * DY[d]);
                    y_max = y_max.max(y_maxs[d] + t * DY[d]);
                }
                min = min.min((x_max - x_min) * (y_max - y_min));
            }
        };
        check(0.0);
        for d0 in 0..4 {
            for d1 in d0 + 1..4 {
                if DX[d0] != DX[d1] {
                    check(-(x_mins[d0] - x_mins[d1]) / (DX[d0] - DX[d1]));
                }
                if DX[d0] != DX[d1] {
                    check(-(x_maxs[d0] - x_maxs[d1]) / (DX[d0] - DX[d1]));
                }
                if DY[d0] != DY[d1] {
                    check(-(y_mins[d0] - y_mins[d1]) / (DY[d0] - DY[d1]));
                }
                if DY[d0] != DY[d1] {
                    check(-(y_maxs[d0] - y_maxs[d1]) / (DY[d0] - DY[d1]));
                }
            }
        }
    }
    println!("{}", min);
}