Rooks

AtCoder

IDagc047_f

Time1250ms

Memory256MB

Difficulty—

You are given positions $(X_i, Y_i)$ of $N$ enemy rooks on an infinite chessboard. No two rooks attack each other (at most one rook per row or column). You're going to replace one rook with a king and then move the king repeatedly to beat as many rooks as possible. You can't enter a cell that is being attacked by a rook. Additionally, you **can't move diagonally to an empty cell** (but you can beat a rook diagonally). (So this king moves like a superpawn that beats diagonally in 4 directions and moves horizontally/vertically in 4 directions.) For each rook, consider replacing it with a king, and find the minimum possible number of moves needed to beat the maximum possible number of rooks. ## Constraints * $2 \leq N \leq 200\,000$ * $1 \leq X_i, Y_i \leq 10^6$ * $X_i \neq X_j$ * $Y_i \neq Y_j$ * All values in input are integers. ## Input Input is given from Standard Input in the following format. $N$ $X_1$ $Y_1$ $X_2$ $Y_2$ $\vdots$ $X_N$ $Y_N$ [samples]

Samples

Input #1

Output #1

5
0
7
5
0
0

See the drawing below. If we replace rook 3 with a king, we can beat at most two other rooks. The red path is one of optimal sequences of moves: beat rook 1, then keep going down and right until you can beat rook 4. There are 7 steps and that's the third number in the output.
![image](https://img.atcoder.jp/agc047/rooks_path_small3.png)
_x-coordinate increases from left to right, while y increases bottom to top._
Starting from rook 2, 5 or 6, we can't beat any other rook. The optimal number of moves is 0.

Input #2

Output #2

Input #3

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Rooks",
    "description": {
      "content": "You are given positions $(X_i, Y_i)$ of $N$ enemy rooks on an infinite chessboard. No two rooks attack each other (at most one rook per row or column). You're going to replace one rook with a king and",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 1250,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc047_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given positions $(X_i, Y_i)$ of $N$ enemy rooks on an infinite chessboard. No two rooks attack each other (at most one rook per row or column).\nYou're going to replace one rook with a king and...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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