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)

#[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(PartialEq, PartialOrd, Copy, Clone)]
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()
    }
}
impl<T: std::str::FromStr> std::str::FromStr for Total<T> {
    type Err = T::Err;
    fn from_str(s: &str) -> Result<Self, Self::Err> {
        let res = T::from_str(s)?;
        Ok(Total(res))
    }
}

impl<T: std::fmt::Display> std::fmt::Debug for Total<T> {
    fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}

impl<T: std::fmt::Display> std::fmt::Display for Total<T> {
    fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}

impl<T: std::ops::Add<Output=T>> std::ops::Add for Total<T> {
    type Output = Self;
    fn add(self, other: Self) -> Self {
        Total(self.0+other.0)
    }
}
impl<T: std::ops::Sub<Output=T>> std::ops::Sub for Total<T> {
    type Output = Self;
    fn sub(self, other: Self) -> Self {
        Total(self.0-other.0)
    }
}
impl<T: std::ops::Mul<Output=T>> std::ops::Mul for Total<T> {
    type Output = Self;
    fn mul(self, other: Self) -> Self {
        Total(self.0*other.0)
    }
}
impl<T: std::ops::Div<Output=T>> std::ops::Div for Total<T> {
    type Output = Self;
    fn div(self, other: Self) -> Self {
        Total(self.0/other.0)
    }
}

#[derive(Debug)]
struct Calculator {
    left: Vec<Total<f64>>,
    right: Vec<Total<f64>>,
    exists: Vec<bool>,
}

impl Calculator {
    fn new() -> Self {
        Calculator {
            left: vec![Total(std::f64::MAX); 3],
            right: vec![Total(std::f64::MIN);3],
            exists: vec![false;3],
        }
    }

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

    fn get(&self, t: Total<f64>) -> Total<f64> {
        let mut mn = Total(std::f64::MAX);
        let mut mx = Total(std::f64::MIN);
        for i in 0..3 {
            if !self.exists[i] {
                continue;
            }
            let dir = Total(i as f64-1.0);
            mn = min!(mn, self.left[i]+t*dir);
            mx = max!(mx, self.right[i]+t*dir);
        }
        mx - mn
    }

    fn events(&self) -> Vec<Total<f64>> {
        let mut events = Vec::new();
        for i in 0..3 {
            for j in (i+1)..3 {
                if !self.exists[i] || !self.exists[j] {
                    continue;
                }
                let diff = Total(i as f64-j as f64);
                events.push((self.left[j]-self.left[i])/diff);
                events.push((self.right[j]-self.right[i])/diff);
            }
        }
        events
    }
}


fn solve() {
    let n = get!(usize);
    let xyd = get!(Total<f64>, Total<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 = Total(std::f64::MAX);
    for t in calcx.events().into_iter().chain(calcy.events().into_iter()) {
        if t >= Total(0.0) {
            mn = min!(mn, calcx.get(t)*calcy.get(t));
        }
    }
    println!("{}", mn);
}