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: Wrong Answer)

use std::cmp;

const INF: i64 = 1e13 as i64;

fn main() {
    let s = std::io::stdin();
    let mut sc = Scanner { stdin: s.lock() };
    let n: usize = sc.read();
    let xyd: Vec<_> = (0..n)
        .map(|_| (sc.read::<i64>(), sc.read::<i64>(), sc.chars()[0]))
        .collect();

    let pxd: Vec<_> = xyd.iter().map(|&(x, _, d)| (x, d)).collect();
    let pyd: Vec<_> = xyd.iter().map(|&(_, y, d)| (y, d)).collect();
    let nxd: Vec<_> = xyd.iter().map(|&(x, _, d)| (-x, d)).collect();
    let nyd: Vec<_> = xyd.iter().map(|&(_, y, d)| (-y, d)).collect();

    let mut times = vec![0.0];
    add_times(&mut times, extract(pxd, 'R', 'L'));
    add_times(&mut times, extract(nxd, 'L', 'R'));
    add_times(&mut times, extract(pyd, 'U', 'D'));
    add_times(&mut times, extract(nyd, 'D', 'U'));
    times.sort_by(|a, b| a.partial_cmp(&b).unwrap());
    times = times
        .into_iter()
        .filter(|&t| t >= 0.0)
        .filter(|&t| t < INF as f64)
        .collect();

    let mut ans = INF as f64;
    for time in times.into_iter() {
        let mut max_x = -INF as f64;
        let mut max_y = -INF as f64;
        let mut min_x = INF as f64;
        let mut min_y = INF as f64;
        for &(x, y, c) in xyd.iter() {
            let mut x = x as f64;
            let mut y = y as f64;
            match c {
                'R' => x += time,
                'L' => x -= time,
                'U' => y += time,
                'D' => y -= time,
                _ => unreachable!(),
            }
            max_x = max(max_x, x);
            max_y = max(max_y, y);
            min_x = min(min_x, x);
            min_y = min(min_y, y);
        }

        ans = min(ans, (max_x - min_x) * (max_y - min_y));
    }

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

fn max<T: PartialOrd>(a: T, b: T) -> T {
    if a > b {
        a
    } else {
        b
    }
}
fn min<T: PartialOrd>(a: T, b: T) -> T {
    if a < b {
        a
    } else {
        b
    }
}

fn add_times(times: &mut Vec<f64>, m: (i64, i64, i64)) {
    let (max, dir, inv) = m;
    let (max, dir, inv) = (max as f64, dir as f64, inv as f64);
    times.push(max - dir);
    times.push(inv - max);
    times.push((inv - dir) / 2.0);
}

fn extract(v: Vec<(i64, char)>, dir: char, inv: char) -> (i64, i64, i64) {
    let mut max = -INF;
    let mut dir_max = -INF;
    let mut inv_max = -INF;
    for (x, d) in v.into_iter() {
        if d == dir {
            dir_max = cmp::max(dir_max, x);
        }
        if d == inv {
            inv_max = cmp::max(inv_max, x);
        }
        max = cmp::max(max, x);
    }
    (max, dir_max, inv_max)
}

pub struct Scanner<R> {
    stdin: R,
}

impl<R: std::io::Read> Scanner<R> {
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .stdin
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n')
            .take_while(|&b| b != b' ' && b != b'\n')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }
    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}