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 read_stdin() -> String {
    use std::io::Read;
    let mut input = String::new();
    std::io::stdin().read_to_string(&mut input).unwrap();
    input
}

fn scan<'a, I: Iterator<Item = &'a str>, T: std::str::FromStr>(tokens: &mut I) -> T {
    tokens.next().unwrap().parse().ok().unwrap()
}

const INFTIY: f64 = 1e10;

fn main() {
    let input = read_stdin();
    let tokens = &mut input.split_whitespace();
    let n: usize = scan(tokens);
    let mut min_x = PLF {
        segments: vec![Segment {
            // l: 0.0,
            r: INFTIY,
            a: 0.0,
            b: INFTIY,
        }],
    };
    let mut max_x = PLF {
        segments: vec![Segment {
            // l: 0.0,
            r: INFTIY,
            a: 0.0,
            b: -INFTIY,
        }],
    };
    let mut min_y = PLF {
        segments: vec![Segment {
            // l: 0.0,
            r: INFTIY,
            a: 0.0,
            b: INFTIY,
        }],
    };
    let mut max_y = PLF {
        segments: vec![Segment {
            // l: 0.0,
            r: INFTIY,
            a: 0.0,
            b: -INFTIY,
        }],
    };
    for _ in 0..n {
        let x: f64 = scan(tokens);
        let y: f64 = scan(tokens);
        let d: String = scan(tokens);
        let (dx, dy) = match d.chars().next().unwrap() {
            'R' => (1.0, 0.0),
            'L' => (-1.0, 0.0),
            'U' => (0.0, 1.0),
            'D' => (0.0, -1.0),
            _ => unreachable!(),
        };
        min_x = min_x.min(&PLF {
            segments: vec![Segment {
                // l: 0.0,
                r: INFTIY,
                a: dx,
                b: x,
            }],
        });
        max_x = max_x.max(&PLF {
            segments: vec![Segment {
                // l: 0.0,
                r: INFTIY,
                a: dx,
                b: x,
            }],
        });
        min_y = min_y.min(&PLF {
            segments: vec![Segment {
                // l: 0.0,
                r: INFTIY,
                a: dy,
                b: y,
            }],
        });
        max_y = max_y.max(&PLF {
            segments: vec![Segment {
                // l: 0.0,
                r: INFTIY,
                a: dy,
                b: y,
            }],
        });
    }
    // println!("min_x: {:?}", min_x);
    // println!("max_x: {:?}", max_x);
    // println!("min_y: {:?}", min_y);
    // println!("max_y: {:?}", max_y);
    let ans = max_x.sub(&min_x).mul_min(&max_y.sub(&min_y));
    println!("{}", ans);
}

#[derive(Debug)]
struct PLF {
    segments: Vec<Segment>,
}

#[derive(Debug)]
struct Segment {
    // l: f64,
    r: f64,
    a: f64,
    b: f64,
}

impl PLF {
    fn min(&self, other: &PLF) -> PLF {
        let mut segments = Vec::new();
        let mut l = 0.0;
        let mut i = 0;
        let mut j = 0;
        while l < INFTIY {
            let s = &self.segments[i];
            let t = &other.segments[j];
            let r = s.r.min(t.r);
            if s.a * l + s.b <= t.a * l + t.b {
                if s.a * r + s.b <= t.a * r + t.b {
                    segments.push(Segment {
                        // l: l,
                        r: r,
                        a: s.a,
                        b: s.b,
                    });
                } else {
                    let m = -(s.b - t.b) / (s.a - t.a);
                    segments.push(Segment {
                        // l: l,
                        r: m,
                        a: s.a,
                        b: s.b,
                    });
                    segments.push(Segment {
                        // l: m,
                        r: r,
                        a: t.a,
                        b: t.b,
                    })
                }
            } else {
                if s.a * r + s.b < t.a * r + t.b {
                    let m = -(s.b - t.b) / (s.a - t.a);
                    segments.push(Segment {
                        // l: l,
                        r: m,
                        a: t.a,
                        b: t.b,
                    });
                    segments.push(Segment {
                        // l: m,
                        r: r,
                        a: s.a,
                        b: s.b,
                    })
                } else {
                    segments.push(Segment {
                        // l: l,
                        r: r,
                        a: t.a,
                        b: t.b,
                    })
                }
            }
            if s.r == r {
                i += 1;
            }
            if t.r == r {
                j += 1;
            }
            l = r;
        }
        PLF { segments: segments }
    }
    fn max(&self, other: &PLF) -> PLF {
        let mut segments = Vec::new();
        let mut l = 0.0;
        let mut i = 0;
        let mut j = 0;
        while l < INFTIY {
            let s = &self.segments[i];
            let t = &other.segments[j];
            let r = s.r.min(t.r);
            if s.a * l + s.b >= t.a * l + t.b {
                if s.a * r + s.b >= t.a * r + t.b {
                    segments.push(Segment {
                        // l: l,
                        r: r,
                        a: s.a,
                        b: s.b,
                    });
                } else {
                    let m = -(s.b - t.b) / (s.a - t.a);
                    segments.push(Segment {
                        // l: l,
                        r: m,
                        a: s.a,
                        b: s.b,
                    });
                    segments.push(Segment {
                        // l: m,
                        r: r,
                        a: t.a,
                        b: t.b,
                    })
                }
            } else {
                if s.a * r + s.b > t.a * r + t.b {
                    let m = -(s.b - t.b) / (s.a - t.a);
                    segments.push(Segment {
                        // l: l,
                        r: m,
                        a: t.a,
                        b: t.b,
                    });
                    segments.push(Segment {
                        // l: m,
                        r: r,
                        a: s.a,
                        b: s.b,
                    })
                } else {
                    segments.push(Segment {
                        // l: l,
                        r: r,
                        a: t.a,
                        b: t.b,
                    })
                }
            }
            if s.r == r {
                i += 1;
            }
            if t.r == r {
                j += 1;
            }
            l = r;
        }
        PLF { segments: segments }
    }
    fn sub(&self, other: &PLF) -> PLF {
        let mut segments = Vec::new();
        let mut l = 0.0;
        let mut i = 0;
        let mut j = 0;
        while l < INFTIY {
            let s = &self.segments[i];
            let t = &other.segments[j];
            let r = s.r.min(t.r);
            segments.push(Segment {
                // l: l,
                r: r,
                a: s.a - t.a,
                b: s.b - t.b,
            });
            if s.r == r {
                i += 1;
            }
            if t.r == r {
                j += 1;
            }
            l = r;
        }
        PLF { segments: segments }
    }
    fn mul_min(&self, other: &PLF) -> f64 {
        let mut l = 0.0;
        let mut i = 0;
        let mut j = 0;
        let mut min = self.segments[i].b * other.segments[i].b;
        while l < INFTIY {
            let s = &self.segments[i];
            let t = &other.segments[j];
            let r = s.r.min(t.r);
            min = min.min((s.a * r + s.b) * (t.a * r + t.b));
            if s.a * t.a != 0.0 {
                let m = -(s.a * t.b + s.b * t.a) / (s.a * t.a);
                if l < m && m < r {
                    min = min.min((s.a * m + s.b) * (t.a * m + t.b));
                }
            }
            if s.r == r {
                i += 1;
            }
            if t.r == r {
                j += 1;
            }
            l = r;
        }
        min
    }
}