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)]
use std::cmp::{Ordering};

#[allow(unused_imports)]
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};

#[allow(unused_imports)]
use std::io::{stdin, stdout, BufWriter, Write};

#[allow(unused_imports)] use std::iter::FromIterator; #[allow(unused_macros)]
macro_rules! min {
    ($x: expr) => ($x);
    ($x: expr, $($z: expr),+) => {{
        std::cmp::min($x, min!($($z),*))
    }}
}

#[allow(unused_macros)]
macro_rules! max {
    ($x: expr) => ($x);
    ($x: expr, $($z: expr),+) => {{
        std::cmp::max($x, max!($($z),*))
    }}
}

#[allow(unused_macros)]
macro_rules! get { 
    ([$t: ty]) => { 
        { 
            let mut line = String::new(); 
            stdin().read_line(&mut line).unwrap(); 
            line.split_whitespace().map(|t|t.parse::<$t>().unwrap()).collect::<Vec<_>>()
        }
    };
    ([$t: ty];$n: expr) => {
        (0..$n).map(|_|get!([$t])).collect::<Vec<_>>()
    };
    ($t: ty) => {
        {
            let mut line = String::new();
            stdin().read_line(&mut line).unwrap();
            line.trim().parse::<$t>().unwrap()
        }
    };
    ($($t: ty),*) => {
        { 
            let mut line = String::new();
            stdin().read_line(&mut line).unwrap();
            let mut iter = line.split_whitespace();
            ($(iter.next().unwrap().parse::<$t>().unwrap(),)*)
        }
    };
    ($t: ty; $n: expr) => {
        (0..$n).map(|_|get!($t)).collect::<Vec<_>>()
    };
    ($($t: ty),*; $n: expr) => {
        (0..$n).map(|_|get!($($t),*)).collect::<Vec<_>>()
    };
}

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

const BIG_STACK_SIZE: bool = true;

#[allow(dead_code)]
fn main() {
    use std::thread;
    if BIG_STACK_SIZE {
        thread::Builder::new()
            .stack_size(32 * 1024 * 1024)
            .name("solve".into())
            .spawn(solve)
            .unwrap()
            .join()
            .unwrap();
    } else {
        solve();
    }
}

#[derive(Debug)]
struct Calculator {
    left: Vec<f64>,
    right: Vec<f64>,
}
impl Calculator {
    fn new() -> Self {
        Calculator {
            left: vec![std::f64::INFINITY; 3],
            right: vec![std::f64::NEG_INFINITY;3],
        }
    }

    fn add(&mut self, n: f64, dir: usize) {
        self.left[dir] = self.left[dir].min(n);
        self.right[dir] = self.right[dir].max(n);
    }

    fn get(&self, t: f64) -> f64 {
        let mut mn = std::f64::INFINITY;
        let mut mx = std::f64::NEG_INFINITY;
        for i in 0..3 {
            mn = mn.min(self.left[i]+t*(i as f64));
            mx = mx.max(self.right[i]+t*(i as f64));
        }
        mx - mn
    }

    fn events(&self) -> Vec<f64> {
        let mut events = Vec::new();
        events.push(0.0);
        for i in 0..3 {
            for j in 0..i {
                let diff = i as f64 - j as f64;
                let t = (self.left[j]-self.left[i])/diff;
                if t > 0.0 {
                    events.push(t);
                }
                let t = (self.right[j]-self.right[i])/diff;
                if t > 0.0 {
                    events.push(t);
                }
            }
        }
        events
    }
}


fn solve() {
    let n = get!(usize);
    let xyd = get!(f64, f64, String; n);

    let mut calcx = Calculator::new();
    let mut calcy = Calculator::new();

    for (x,y,d) in xyd {
        match d.chars().next().unwrap() {
            'L' => {
                calcx.add(x, 0);
                calcy.add(y, 1);
            },
            'R' => {
                calcx.add(x, 2);
                calcy.add(y, 1);
            },
            'U' => {
                calcx.add(x, 1);
                calcy.add(y, 2);
            },
            'D' => {
                calcx.add(x, 1);
                calcy.add(y, 0);
            },
            _ => (),
        }
    }

    let mut mn = std::f64::INFINITY;
    for t in calcx.events().into_iter().chain(calcy.events().into_iter()) {
        mn = mn.min(calcx.get(t)*calcy.get(t));
    }
    println!("{}", mn);
}