Minimum Bounding Box

AtCoder

IDabc130_f

Time5000ms

Memory256MB

Difficulty—

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: $N$ $x_1$ $y_1$ $d_1$ $x_2$ $y_2$ $d_2$ $.$ $.$ $.$ $x_N$ $y_N$ $d_N$ [samples]

Samples

Input #1

2
0 3 D
3 0 L

Output #1

0

After three seconds, the two points will meet at the origin. The value in question will be $0$ at that moment.

Input #2

5
-7 -10 U
7 -6 U
-8 7 D
-3 3 D
0 -6 R

Output #2

97.5

The answer may not be an integer.

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

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Minimum Bounding Box",
    "description": {
      "content": "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",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 5000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc130_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "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...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments