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 (r, w) = (std::io::stdin(), std::io::stdout());
    let mut sc = IO::new(r.lock(), w.lock());

    let n: usize = sc.read();
    let mut left = Edge::new();
    let mut right = Edge::new();
    let mut head = Edge::new();
    let mut tail = Edge::new();

    for _ in 0..n {
        let x: f64 = sc.read();
        let y: f64 = sc.read();
        let d = sc.read::<String>();
        match d.as_str() {
            "R" => {
                right.increasing_max.set_max(x);
                left.decreasing_max.set_max(-x);
                head.static_max.set_max(y);
                tail.static_max.set_max(-y);
            }
            "L" => {
                right.decreasing_max.set_max(x);
                left.increasing_max.set_max(-x);
                head.static_max.set_max(y);
                tail.static_max.set_max(-y);
            }
            "U" => {
                head.increasing_max.set_max(y);
                tail.decreasing_max.set_max(-y);
                right.static_max.set_max(x);
                left.static_max.set_max(-x);
            }
            "D" => {
                head.decreasing_max.set_max(y);
                tail.increasing_max.set_max(-y);
                right.static_max.set_max(x);
                left.static_max.set_max(-x);
            }
            _ => unreachable!(),
        }
    }

    let mut events = vec![0.0];
    events.extend(left.get_event());
    events.extend(right.get_event());
    events.extend(head.get_event());
    events.extend(tail.get_event());

    let mut min = std::f64::MAX;
    for event in events.into_iter() {
        if event < 0.0 {
            continue;
        }
        let left = left.get_max(event).unwrap();
        let right = right.get_max(event).unwrap();
        let head = head.get_max(event).unwrap();
        let tail = tail.get_max(event).unwrap();

        let h = head + tail;
        let w = left + right;
        min = min.min(h * w);
    }
    println!("{}", min);
}

struct Edge {
    increasing_max: Option<f64>,
    decreasing_max: Option<f64>,
    static_max: Option<f64>,
}

impl Edge {
    fn new() -> Self {
        Edge {
            increasing_max: None,
            decreasing_max: None,
            static_max: None,
        }
    }

    fn get_event(&self) -> Vec<f64> {
        let mut result = vec![];
        if let (Some(i), Some(s)) = (self.increasing_max, self.static_max) {
            result.push(s - i);
        }
        if let (Some(d), Some(s)) = (self.decreasing_max, self.static_max) {
            result.push(d - s);
        }
        if let (Some(d), Some(i)) = (self.decreasing_max, self.increasing_max) {
            result.push((d - i) / 2.0);
        }
        result
    }

    fn get_max(&self, time: f64) -> Option<f64> {
        let mut result = None;
        if let Some(s) = self.static_max {
            result.set_max(s);
        }
        if let Some(s) = self.decreasing_max {
            result.set_max(s - time);
        }
        if let Some(s) = self.increasing_max {
            result.set_max(s + time);
        }
        result
    }
}

trait SetMax<T> {
    fn set_max(&mut self, v: T);
}

impl SetMax<f64> for Option<f64> {
    fn set_max(&mut self, v: f64) {
        match self.clone() {
            Some(p) => {
                if p < v {
                    *self = Some(v)
                }
            }
            None => *self = Some(v),
        }
    }
}

pub struct IO<R, W: std::io::Write>(R, std::io::BufWriter<W>);

impl<R: std::io::Read, W: std::io::Write> IO<R, W> {
    pub fn new(r: R, w: W) -> Self {
        IO(r, std::io::BufWriter::new(w))
    }
    pub fn write<S: ToString>(&mut self, s: S) {
        use std::io::Write;
        self.1.write(s.to_string().as_bytes()).unwrap();
    }
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .0
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n' || b == b'\r' || b == b'\t')
            .take_while(|&b| b != b' ' && b != b'\n' && b != b'\r' && b != b'\t')
            .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()
    }
}