101305: [AtCoder]ABC130 F - Minimum Bounding Box

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