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)

use std::io::prelude::*;

fn input<T: std::str::FromStr>() -> T {
    let stdin = std::io::stdin();
    let token: String = stdin
        .lock()
        .bytes()
        .map(|c| c.unwrap() as char)
        .skip_while(|c| c.is_whitespace())
        .take_while(|c| !c.is_whitespace())
        .collect();
    token.parse().ok().unwrap()
}

fn main() {
    let n: usize = input();
    let xyd: Vec<(i32, i32, String)> = (0..n).map(|_| (input(), input(), input())).collect();

    let mut xmin = vec![i32::max_value(); 3];
    let mut xmax = vec![i32::min_value(); 3];
    let mut ymin = vec![i32::max_value(); 3];
    let mut ymax = vec![i32::min_value(); 3];
    for (x, y, d) in xyd {
        if d == "R" {
            xmin[2] = std::cmp::min(xmin[2], x);
            xmax[2] = std::cmp::max(xmax[2], x);
            ymin[1] = std::cmp::min(ymin[1], y);
            ymax[1] = std::cmp::max(ymax[1], y);
        } else if d == "L" {
            xmin[0] = std::cmp::min(xmin[0], x);
            xmax[0] = std::cmp::max(xmax[0], x);
            ymin[1] = std::cmp::min(ymin[1], y);
            ymax[1] = std::cmp::max(ymax[1], y);
        } else if d == "U" {
            xmin[1] = std::cmp::min(xmin[1], x);
            xmax[1] = std::cmp::max(xmax[1], x);
            ymin[2] = std::cmp::min(ymin[2], y);
            ymax[2] = std::cmp::max(ymax[2], y);
        } else {
            xmin[1] = std::cmp::min(xmin[1], x);
            xmax[1] = std::cmp::max(xmax[1], x);
            ymin[0] = std::cmp::min(ymin[0], y);
            ymax[0] = std::cmp::max(ymax[0], y);
        }
    }

    let mut t = vec![0.0];
    if xmin[0] != i32::max_value() && xmin[1] != i32::max_value() && xmin[0] - xmin[1] > 0 {
        t.push((xmin[0] - xmin[1]) as f64);
    }
    if xmin[0] != i32::max_value() && xmin[2] != i32::max_value() && xmin[0] - xmin[2] > 0 {
        t.push((xmin[0] - xmin[2]) as f64 * 0.5);
    }
    if xmin[1] != i32::max_value() && xmin[2] != i32::max_value() && xmin[1] - xmin[2] > 0 {
        t.push((xmin[1] - xmin[2]) as f64);
    }

    if xmax[0] != i32::min_value() && xmax[1] != i32::min_value() && xmax[0] - xmax[1] > 0 {
        t.push((xmax[0] - xmax[1]) as f64);
    }
    if xmax[0] != i32::min_value() && xmax[2] != i32::min_value() && xmax[0] - xmax[2] > 0 {
        t.push((xmax[0] - xmax[2]) as f64 * 0.5);
    }
    if xmax[1] != i32::min_value() && xmax[2] != i32::min_value() && xmax[1] - xmax[2] > 0 {
        t.push((xmax[1] - xmax[2]) as f64);
    }

    if ymin[0] != i32::max_value() && ymin[1] != i32::max_value() && ymin[0] - ymin[1] > 0 {
        t.push((ymin[0] - ymin[1]) as f64);
    }
    if ymin[0] != i32::max_value() && ymin[2] != i32::max_value() && ymin[0] - ymin[2] > 0 {
        t.push((ymin[0] - ymin[2]) as f64 * 0.5);
    }
    if ymin[1] != i32::max_value() && ymin[2] != i32::max_value() && ymin[1] - ymin[2] > 0 {
        t.push((ymin[1] - ymin[2]) as f64);
    }

    if ymax[0] != i32::min_value() && ymax[1] != i32::min_value() && ymax[0] - ymax[1] > 0 {
        t.push((ymax[0] - ymax[1]) as f64);
    }
    if ymax[0] != i32::min_value() && ymax[2] != i32::min_value() && ymax[0] - ymax[2] > 0 {
        t.push((ymax[0] - ymax[2]) as f64 * 0.5);
    }
    if ymax[1] != i32::min_value() && ymax[2] != i32::min_value() && ymax[1] - ymax[2] > 0 {
        t.push((ymax[1] - ymax[2]) as f64);
    }

    let mut ans = i64::max_value() as f64;
    for t in t {
        let mut xl = i32::max_value() as f64;
        let mut xr = i32::min_value() as f64;
        let mut yl = i32::max_value() as f64;
        let mut yr = i32::min_value() as f64;

        if xmin[0] != i32::max_value() && xmin[0] as f64 - t < xl {
            xl = xmin[0] as f64 - t;
        }
        if xmin[1] != i32::max_value() && (xmin[1] as f64) < xl {
            xl = xmin[1] as f64;
        }
        if xmin[2] != i32::max_value() && xmin[2] as f64 + t < xl {
            xl = xmin[2] as f64 + t;
        }

        if xmax[0] != i32::min_value() && xmax[0] as f64 - t > xr {
            xr = xmax[0] as f64 - t;
        }
        if xmax[1] != i32::min_value() && (xmax[1] as f64) > xr {
            xr = xmax[1] as f64;
        }
        if xmax[2] != i32::min_value() && xmax[2] as f64 + t > xr {
            xr = xmax[2] as f64 + t;
        }

        if ymin[0] != i32::max_value() && ymin[0] as f64 - t < yl {
            yl = ymin[0] as f64 - t;
        }
        if ymin[1] != i32::max_value() && (ymin[1] as f64) < yl {
            yl = ymin[1] as f64;
        }
        if ymin[2] != i32::max_value() && ymin[2] as f64 + t < yl {
            yl = ymin[2] as f64 + t;
        }

        if ymax[0] != i32::min_value() && ymax[0] as f64 - t > yr {
            yr = ymax[0] as f64 - t;
        }
        if ymax[1] != i32::min_value() && (ymax[1] as f64) > yr {
            yr = ymax[1] as f64;
        }
        if ymax[2] != i32::min_value() && ymax[2] as f64 + t > yr {
            yr = ymax[2] as f64 + t;
        }

        let now = (xr - xl) * (yr - yl);
        if now < ans {
            ans = now;
        }
    }

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