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)

#![allow(unused_imports)]
#![allow(non_snake_case)]
use std::cmp::*;
use std::collections::*;
use std::io::Write;

#[allow(unused_macros)]
macro_rules! debug {
    ($($e:expr),*) => {
        #[cfg(debug_assertions)]
        $({
            let (e, mut err) = (stringify!($e), std::io::stderr());
            writeln!(err, "{} = {:?}", e, $e).unwrap()
        })*
    };
}

struct Solver1D {
    l: [i64; 3],
    r: [i64; 3],
}

const INF: i64 = 100000_00000;

impl Solver1D {
    fn new() -> Solver1D {
        let l = [INF; 3];
        let r = [-INF; 3];
        Solver1D { l: l, r: r }
    }

    fn add(&mut self, coord: i64, idx: usize) {
        self.l[idx] = min(self.l[idx], coord);
        self.r[idx] = max(self.r[idx], coord);
    }

    fn calc_width(&self, t: f64) -> f64 {
        let mut left = std::f64::MAX;
        for i in 0..3 {
            let v = i as f64 - 1.0;
            left = fmin(left, self.l[i] as f64 + v * t);
        }
        let mut right = -std::f64::MAX;
        for i in 0..3 {
            let v = i as f64 - 1.0;
            right = fmax(right, self.r[i] as f64 + v * t);
        }
        right - left
    }

    fn get_events(&self) -> Vec<f64> {
        let mut events = vec![];
        for lr in &[self.l, self.r] {
            for i in 0..3 {
                if lr[i].abs() == INF {
                    continue;
                }
                for j in 0..i {
                    if lr[j].abs() == INF {
                        continue;
                    }
                    let event = -(lr[i] - lr[j]) as f64 / (i as f64 - j as f64);
                    if event > 0.0 {
                        events.push(event);
                    }
                }
            }
        }
        events
    }
}

fn main() {
    let n = read::<usize>();

    let mut lr_solver = Solver1D::new();
    let mut ud_solver = Solver1D::new();

    for _ in 0..n {
        let v = read_vec::<String>();
        let x = i64::from_str_radix(&v[0], 10).unwrap();
        let y = i64::from_str_radix(&v[1], 10).unwrap();
        if v[2] == "R" {
            lr_solver.add(x, 2);
            ud_solver.add(y, 1);
        } else if v[2] == "L" {
            lr_solver.add(x, 0);
            ud_solver.add(y, 1);
        } else if v[2] == "U" {
            lr_solver.add(x, 1);
            ud_solver.add(y, 2);
        } else {
            lr_solver.add(x, 1);
            ud_solver.add(y, 0);
        }
    }
    let mut events = vec![0.0, 1e9];
    events.append(&mut lr_solver.get_events());
    events.append(&mut ud_solver.get_events());

    let mut ans = std::f64::MAX;
    for t in events {
        let temp = lr_solver.calc_width(t) * ud_solver.calc_width(t);
        ans = fmin(ans, temp);
    }
    println!("{0:.10}", ans);
}

fn read<T: std::str::FromStr>() -> T {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    s.trim().parse().ok().unwrap()
}

fn read_vec<T: std::str::FromStr>() -> Vec<T> {
    read::<String>()
        .split_whitespace()
        .map(|e| e.parse().ok().unwrap())
        .collect()
}

#[derive(PartialEq, PartialOrd)]
pub struct Total<T>(pub T);

impl<T: PartialEq> Eq for Total<T> {}

impl<T: PartialOrd> Ord for Total<T> {
    fn cmp(&self, other: &Total<T>) -> std::cmp::Ordering {
        self.0.partial_cmp(&other.0).unwrap()
    }
}

fn fmin(a: f64, b: f64) -> f64 {
    min(Total(a), Total(b)).0
}

fn fmax(a: f64, b: f64) -> f64 {
    max(Total(a), Total(b)).0
}