Chinese Restaurant (Three-Star Version)

AtCoder

IDabc268_e

Time2000ms

Memory256MB

Difficulty—

Person $0$, Person $1$, $\ldots$, and Person $(N-1)$ are sitting around a turntable in counterclockwise order, evenly spaced. Dish $p_i$ is in front of Person $i$ on the table. You may perform the following operation $0$ or more times: * Rotate the turntable by one $N$\-th of a counterclockwise turn. The dish that was in front of Person $i$ right before the rotation is now in front of Person $(i+1) \bmod N$. When you are finished, Person $i$ gains frustration of $k$, where $k$ is the minimum integer such that Dish $i$ is in front of either Person $(i-k) \bmod N$ or Person $(i+k) \bmod N$. Find the minimum possible sum of frustration of the $N$ people. What is $a \bmod m$? For an integer $a$ and a positive integer $m$, $a \bmod m$ denotes the integer $x$ between $0$ and $(m-1)$ (inclusive) such that $(a-x)$ is a multiple of $m$. (It can be proved that such $x$ is unique.) ## Constraints * $3 \leq N \leq 2 \times 10^5$ * $0 \leq p_i \leq N-1$ * $p_i \neq p_j$ if $i \neq j$. * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $p_0$ $\ldots$ $p_{N-1}$ [samples]

Samples

Input #1

4
1 2 0 3

Output #1

2

The figure below shows the table after one operation.
![image](https://img.atcoder.jp/abc268/70536a7b7fad87d6a49ad00df89a4a30.png)
Here, the sum of their frustration is $2$ because:

*   Person $0$ gains a frustration of $1$ since Dish $0$ is in front of Person $3\ (=(0-1) \bmod 4)$;
*   Person $1$ gains a frustration of $0$ since Dish $1$ is in front of Person $1\ (=(1+0) \bmod 4)$;
*   Person $2$ gains a frustration of $0$ since Dish $2$ is in front of Person $2\ (=(2+0) \bmod 4)$;
*   Person $3$ gains a frustration of $1$ since Dish $3$ is in front of Person $0\ (=(3+1) \bmod 4)$.

We cannot make the sum of their frustration less than $2$, so the answer is $2$.

Input #2

3
0 1 2

Output #2

Input #3

10
3 9 6 1 7 2 8 0 5 4

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Chinese Restaurant (Three-Star Version)",
    "description": {
      "content": "Person $0$, Person $1$, $\\ldots$, and Person $(N-1)$ are sitting around a turntable in counterclockwise order, evenly spaced. Dish $p_i$ is in front of Person $i$ on the table.   You may perform the f",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc268_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Person $0$, Person $1$, $\\ldots$, and Person $(N-1)$ are sitting around a turntable in counterclockwise order, evenly spaced. Dish $p_i$ is in front of Person $i$ on the table.  \nYou may perform the f...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments