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_macros)]
macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        let mut next = || { iter.next().unwrap() };
        input_inner!{next, $($r)*}
    };
    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        let mut next = move || -> String{
            bytes
                .by_ref()
                .map(|r|r.unwrap() as char)
                .skip_while(|c|c.is_whitespace())
                .take_while(|c|!c.is_whitespace())
                .collect()
        };
        input_inner!{next, $($r)*}
    };
}

#[allow(unused_macros)]
macro_rules! input_inner {
    ($next:expr) => {};
    ($next:expr, ) => {};

    ($next:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };

    ($next:expr, mut $var:ident : $t:tt $($r:tt)*) => {
        let mut $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
}

#[allow(unused_macros)]
macro_rules! read_value {
    ($next:expr, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };

    ($next:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    };

    ($next:expr, chars) => {
        read_value!($next, String).chars().collect::<Vec<char>>()
    };

    ($next:expr, bytes) => {
        read_value!($next, String).into_bytes()
    };

    ($next:expr, usize1) => {
        read_value!($next, usize) - 1
    };

    ($next:expr, $t:ty) => {
        $next().parse::<$t>().expect("Parse error")
    };
}

#[allow(unused_imports)]
use std::cmp::{max, min};
#[allow(unused_imports)]
use std::collections::{BinaryHeap, HashMap};

#[allow(dead_code)]
type Graph = Vec<Vec<usize>>;

fn golden_section<F>(mut a: f64, mut b: f64, f: F) -> f64
where
    F: Fn(f64) -> f64,
{
    let r = 2.0 / (3.0 + 5.0f64.sqrt());
    let mut c = a + r * (b - a);
    let mut d = b - r * (b - a);
    let mut fc = f(c);
    let mut fd = f(d);
    for _ in 0..100 {
        // <: maximum, >: minimum
        if fc > fd {
            a = c;
            c = d;
            d = b - r * (b - a);
            fc = fd;
            fd = f(d);
        } else {
            b = d;
            d = c;
            c = a + r * (b - a);
            fd = fc;
            fc = f(c);
        }
    }
    c
}

fn main() {
    input! {
        n: usize,
        points: [(f64, f64, chars); n],
    }

    let points = points
        .into_iter()
        .map(|(x, y, chars)| (x, y, chars[0]))
        .collect::<Vec<_>>();

    let xf = |t: f64| {
        let mut maxv = 0.0;
        let mut minv = 1e30;
        for &(x, _, c) in &points {
            let v = match c {
                'R' => x + t,
                'L' => x - t,
                _ => x,
            };
            maxv = if v > maxv { v } else { maxv };
            minv = if v < minv { v } else { minv };
        }
        maxv - minv
    };

    let yf = |t: f64| {
        let mut maxv = 0.0;
        let mut minv = 1e30;
        for &(_, y, c) in &points {
            let v = match c {
                'U' => y + t,
                'D' => y - t,
                _ => y,
            };
            maxv = if v > maxv { v } else { maxv };
            minv = if v < minv { v } else { minv };
        }
        maxv - minv
    };

    let xt = golden_section(0.0, 1e30, &xf);
    let yt = golden_section(0.0, 1e10, &yf);

    let mut xpoints = vec![0.0];
    let mut ypoints = vec![0.0];

    if xt > 0.1 {
        let mut ok = xt;
        let mut ng = 0.0;
        let xv = xf(xt);
        for _ in 0..30 {
            let mid = (ok + ng) / 2.0;
            if (xf(mid) - xv).abs() < 0.1 {
                ok = mid;
            } else {
                ng = mid;
            }
        }
        xpoints.push(ok);

        let mut ok = xt;
        let mut ng = 1e30;
        let xv = xf(xt);
        for _ in 0..30 {
            let mid = (ok + ng) / 2.0;
            if (xf(mid) - xv).abs() < 0.1 {
                ok = mid;
            } else {
                ng = mid;
            }
        }
        xpoints.push(ok);
    }

    if yt > 1e-5 {
        let mut ok = yt;
        let mut ng = 0.0;
        let yv = yf(yt);
        for _ in 0..50 {
            let mid = (ok + ng) / 2.0;
            if (yf(mid) - yv).abs() < 1e-5 {
                ok = mid;
            } else {
                ng = mid;
            }
        }
        ypoints.push(ok);

        let mut ok = yt;
        let mut ng = 1e30;
        let yv = yf(yt);
        for _ in 0..50 {
            let mid = (ok + ng) / 2.0;
            if (yf(mid) - yv).abs() < 1e-5 {
                ok = mid;
            } else {
                ng = mid;
            }
        }
        ypoints.push(ok);
    }

    let mut ans = 1e30;
    for &xt in &xpoints {
        for &yt in &ypoints {
            let v = xf(xt) * yf(yt);
            if v < ans {
                ans = v;
            }
        }
    }

    println!("{}", ans);
}