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)

const INF: f64 = 1e15;
const EPS: f64 = 1e-15;
fn main() {
    let s = std::io::stdin();
    let mut sc = Scanner { stdin: s.lock() };

    let n: usize = sc.read();
    let v: Vec<((f64, f64), (f64, f64))> = (0..n)
        .map(|_| {
            let x: f64 = sc.read();
            let y: f64 = sc.read();
            let c = sc.chars()[0];
            let d = match c {
                'R' => (1.0, 0.0),
                'L' => (-1.0, 0.0),
                'U' => (0.0, 1.0),
                'D' => (0.0, -1.0),
                _ => unreachable!(),
            };
            ((x, y), d)
        })
        .collect();
    let x = v
        .iter()
        .map(|&((x, _), (dx, _))| (x, dx))
        .collect::<Vec<_>>();
    let y = v
        .iter()
        .map(|&((_, y), (_, dy))| (y, dy))
        .collect::<Vec<_>>();

    let event_x = get_events(&x);
    let event_y = get_events(&y);
    let mut times = vec![];
    for e in event_x.into_iter() {
        times.push(e);
    }
    for e in event_y.into_iter() {
        times.push(e);
    }

    let mut ans = INF;

    for time in times.into_iter() {
        let dx = get_length(&x, time);
        let dy = get_length(&y, time);
        let s = dx * dy;
        if ans > s {
            ans = s;
        }
    }

    let ans = ((ans * 10.0) as i64) as f64;
    println!("{}", ans / 10.0);
}

fn get_events(x: &Vec<(f64, f64)>) -> Vec<f64> {
    let t0 = 0.0;
    let t1 = binary_search(x, t0, -2.0);
    let t2 = binary_search(x, t1, -1.0);
    let t3 = binary_search(x, t2, 0.0);
    let t4 = binary_search(x, t3, 1.0);
    vec![t0, t1, t2, t3, t4]
}

fn binary_search(x: &Vec<(f64, f64)>, start: f64, dx: f64) -> f64 {
    let start_length = get_length(x, start);
    let mut low = start;
    let mut high = INF;
    let mut p_low = low;
    let mut p_high = high;
    loop {
        let time = (low + high) / 2.0;
        let cur_length = get_length(x, time);
        if eq(cur_length - start_length, dx * (time - start)) {
            low = time;
        } else {
            high = time;
        }

        if low == p_low && high == p_high {
            break;
        }
        p_low = low;
        p_high = high;
    }
    (high + low) / 2.0
}

fn eq(a: f64, b: f64) -> bool {
    (a - b).abs() < 1e-15
}

fn get_length(x: &Vec<(f64, f64)>, time: f64) -> f64 {
    let mut max = -INF;
    let mut min = INF;
    for &(x, dx) in x.iter() {
        let cur_x = x + dx * time;
        if max < cur_x {
            max = cur_x;
        }
        if min > cur_x {
            min = cur_x;
        }
    }
    assert!(max >= min);
    max - min
}

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()
    }
}