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 Python (3.4.3) (judge verdict: Wrong Answer)

## 場合分けが無理すぎる

N = int(input())
xy_R = []
xy_L = []
xy_U = []
xy_D = []
for i in range(N):
  x, y, d = input().split()
  if d == 'R':
    xy_R.append([int(x), int(y)])
  elif d == 'L':
    xy_L.append([int(x), int(y)])
  elif d == 'U':
    xy_U.append([int(x), int(y)])
  else:
    xy_D.append([int(x), int(y)])

inf = float('inf')

## 最大、最小になり得る点だけを取り出して考える
x_max_R = sorted(xy_R)[-1][0] if len(xy_R)>0 else -inf
x_max_L = sorted(xy_L)[-1][0] if len(xy_L)>0 else -inf
x_max_U = sorted(xy_U)[-1][0] if len(xy_U)>0 else -inf
x_max_D = sorted(xy_D)[-1][0] if len(xy_D)>0 else -inf
x_max = [x_max_R, x_max_L, max(x_max_U, x_max_D)]
x_min_R = sorted(xy_R)[0][0] if len(xy_R)>0 else inf
x_min_L = sorted(xy_L)[0][0] if len(xy_L)>0 else inf
x_min_U = sorted(xy_U)[0][0] if len(xy_U)>0 else inf
x_min_D = sorted(xy_D)[0][0] if len(xy_D)>0 else inf
x_min = [x_min_R, x_min_L, min(x_min_U, x_min_D)]
y_max_R = sorted(xy_R)[-1][0] if len(xy_R)>0 else -inf
y_max_L = sorted(xy_L)[-1][0] if len(xy_L)>0 else -inf
y_max_U = sorted(xy_U)[-1][0] if len(xy_U)>0 else -inf
y_max_D = sorted(xy_D)[-1][0] if len(xy_D)>0 else -inf
y_max = [max(y_max_R, y_max_L), y_max_U, y_max_D]
y_min_R = sorted(xy_R)[0][0] if len(xy_R)>0 else inf
y_min_L = sorted(xy_L)[0][0] if len(xy_L)>0 else inf
y_min_U = sorted(xy_U)[0][0] if len(xy_U)>0 else inf
y_min_D = sorted(xy_D)[0][0] if len(xy_D)>0 else inf
y_min = [min(y_min_R, y_min_L), y_min_U, y_min_D]

#print(x_max, x_min, y_max, y_min)

if x_max[0] == max(x_max):
  case_x_max = 0
elif x_max[2] == max(x_max):
  case_x_max = 1