Manhattan Multifocal Ellipse

AtCoder

IDabc366_e

Time2000ms

Memory256MB

Difficulty—

You are given $N$ points $(x_1, y_1), (x_2, y_2), \dots, (x_N, y_N)$ on a two-dimensional plane, and a non-negative integer $D$. Find the number of integer pairs $(x, y)$ such that $\displaystyle \sum_{i=1}^N (|x-x_i|+|y-y_i|) \leq D$. ## Constraints * $1 \leq N \leq 2 \times 10^5$ * $0 \leq D \leq 10^6$ * $-10^6 \leq x_i, y_i \leq 10^6$ * $(x_i, y_i) \neq (x_j, y_j)$ for $i \neq j$. * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $D$ $x_1$ $y_1$ $x_2$ $y_2$ $\vdots$ $x_N$ $y_N$ [samples]

Samples

Input #1

2 3
0 0
1 0

Output #1

8

The following figure visualizes the input and the answer for Sample $1$. The blue points represent the input. The blue and red points, eight in total, satisfy the condition in the statement.
![image](https://img.atcoder.jp/abc366/2b6d85ce3511e14c65dc80e052d62bca.png)

Input #2

2 0
0 0
2 0

Output #2

Input #3

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Manhattan Multifocal Ellipse",
    "description": {
      "content": "You are given $N$ points $(x_1, y_1), (x_2, y_2), \\dots, (x_N, y_N)$ on a two-dimensional plane, and a non-negative integer $D$. Find the number of integer pairs $(x, y)$ such that $\\displaystyle \\sum",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc366_e"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given $N$ points $(x_1, y_1), (x_2, y_2), \\dots, (x_N, y_N)$ on a two-dimensional plane, and a non-negative integer $D$.\nFind the number of integer pairs $(x, y)$ such that $\\displaystyle \\sum...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments