Sort Permutation

AtCoder

IDarc204_b

Time2000ms

Memory256MB

Difficulty—

You are given positive integers $N, K$ and a permutation $P = (P_{1}, P_{2}, \dots, P_{NK})$ of $(1, 2, \dots, NK)$. Alice repeatedly performs the following operation to sort $P$ in ascending order. * Choose integers $i$ and $j$ $(i\neq j)$ where $1 \leq i, j \leq NK$, and swap $P_{i}$ and $P_{j}$. If $|i - j|$ is a multiple of $N$, Alice gets $1$ point. Alice sorts $P$ in ascending order with the minimum number of operations, and under that condition, she maximizes the sum of points she gets. Output the sum of points Alice gets. ## Constraints * $1\leq N\leq 500$ * $1\leq K\leq 10$ * $(P_{1}, P_{2}, \dots , P_{NK})$ is a permutation of $(1, 2, \dots, NK)$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $K$ $P_{1}$ $P_{2}$ $\dots$ $P_{NK}$ [samples]

Samples

Input #1

3 2
1 6 5 3 2 4

Output #1

2

For example, $P$ can be sorted in ascending order with four operations as follows.

*   Perform the operation with $(i, j) = (2, 5)$. After the operation, $P = (1, 2, 5, 3, 6, 4)$.
*   Perform the operation with $(i, j) = (4, 6)$. After the operation, $P = (1, 2, 5, 4, 6, 3)$.
*   Perform the operation with $(i, j) = (3, 5)$. After the operation, $P = (1, 2, 6, 4, 5, 3)$.
*   Perform the operation with $(i, j) = (3, 6)$. After the operation, $P = (1, 2, 3, 4, 5, 6)$.

Among the above four operations, Alice gets $1$ point from each of the $1$st and $4$th operations, so Alice gets a total of $2$ points.
This sequence of operations is one of the ways to sort $P$ in ascending order with the minimum number of operations and maximize the sum of points under that condition, so output $2$.

Input #2

1 1
1

Output #2

Input #3

4 6
10 24 3 4 8 14 5 2 22 9 21 1 15 6 13 23 18 12 7 17 19 16 20 11

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Sort Permutation",
    "description": {
      "content": "You are given positive integers $N, K$ and a permutation $P = (P_{1}, P_{2}, \\dots, P_{NK})$ of $(1, 2, \\dots, NK)$. Alice repeatedly performs the following operation to sort $P$ in ascending order. ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc204_b"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given positive integers $N, K$ and a permutation $P = (P_{1}, P_{2}, \\dots, P_{NK})$ of $(1, 2, \\dots, NK)$.\nAlice repeatedly performs the following operation to sort $P$ in ascending order.\n\n...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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