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 main() {
    let s = std::io::stdin();
    let mut sc = Scanner { stdin: s.lock() };
    let n: usize = sc.read();

    let mut right_side = Side::new();
    let mut left_side = Side::new();
    let mut head_side = Side::new();
    let mut bottom_side = Side::new();

    for _ in 0..n {
        let x = sc.read::<f64>();
        let y = sc.read::<f64>();
        let d = sc.chars()[0];
        match d {
            'R' => {
                right_side.increasing = right_side.increasing.max(x);
                left_side.decreasing = left_side.decreasing.max(-x);
                head_side.solid = head_side.solid.max(y);
                bottom_side.solid = bottom_side.solid.max(-y);
            }
            'L' => {
                right_side.decreasing = right_side.decreasing.max(x);
                left_side.increasing = left_side.increasing.max(-x);
                head_side.solid = head_side.solid.max(y);
                bottom_side.solid = bottom_side.solid.max(-y);
            }
            'U' => {
                head_side.increasing = head_side.increasing.max(y);
                bottom_side.decreasing = bottom_side.decreasing.max(-y);
                right_side.solid = right_side.solid.max(x);
                left_side.solid = left_side.solid.max(-x);
            }
            'D' => {
                head_side.decreasing = head_side.decreasing.max(y);
                bottom_side.increasing = bottom_side.increasing.max(-y);
                right_side.solid = right_side.solid.max(x);
                left_side.solid = left_side.solid.max(-x);
            }
            _ => unreachable!(),
        }
    }

    let mut events = vec![0.0];
    head_side.add(&mut events);
    left_side.add(&mut events);
    right_side.add(&mut events);
    bottom_side.add(&mut events);

    let mut ans = std::f64::MAX;
    for t in events.into_iter() {
        let x_max = right_side.get(t);
        let x_min = -left_side.get(t);
        let y_max = head_side.get(t);
        let y_min = -bottom_side.get(t);
        let s = (x_max - x_min) * (y_max - y_min);
        ans = ans.min(s);
    }
    println!("{}", ans);
}

struct Side {
    increasing: f64,
    decreasing: f64,
    solid: f64,
}

impl Side {
    fn new() -> Side {
        Side {
            increasing: std::f64::MIN,
            decreasing: std::f64::MIN,
            solid: std::f64::MIN,
        }
    }

    fn add(&self, events: &mut Vec<f64>) {
        if self.increasing != std::f64::MIN && self.solid != std::f64::MIN {
            let d = self.solid - self.increasing;
            if d > 0.0 {
                events.push(d);
            }
        }
        if self.decreasing != std::f64::MIN && self.solid != std::f64::MIN {
            let d = self.decreasing - self.solid;
            if d > 0.0 {
                events.push(d);
            }
        }
        if self.increasing != std::f64::MIN && self.decreasing != std::f64::MIN {
            let d = (self.decreasing - self.increasing) / 2.0;
            if d > 0.0 {
                events.push(d);
            }
        }
    }

    fn get(&self, t: f64) -> f64 {
        let x1 = self.increasing + t;
        let x2 = self.decreasing - t;
        self.solid.max(x1).max(x2)
    }
}

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()
    }
}