Good Grid

AtCoder

IDabc099_d

Time2000ms

Memory256MB

Difficulty—

There is a grid with $N$ rows and $N$ columns of squares. Let $(i,j)$ be the square at the $i$\-th row from the top and the $j$\-th column from the left. These squares have to be painted in one of the $C$ colors from Color $1$ to Color $C$. Initially, $(i,j)$ is painted in Color $c_{i,j}$. We say the grid is a _good_ grid when the following condition is met for all $i,j,x,y$ satisfying $1 \leq i,j,x,y \leq N$: * If $(i+j) \% 3=(x+y) \% 3$, the color of $(i,j)$ and the color of $(x,y)$ are the same. * If $(i+j) \% 3 \neq (x+y) \% 3$, the color of $(i,j)$ and the color of $(x,y)$ are different. Here, $X \% Y$ represents $X$ modulo $Y$. We will repaint zero or more squares so that the grid will be a good grid. For a square, the _wrongness_ when the color of the square is $X$ before repainting and $Y$ after repainting, is $D_{X,Y}$. Find the minimum possible sum of the wrongness of all the squares. ## Constraints * $1 \leq N \leq 500$ * $3 \leq C \leq 30$ * $1 \leq D_{i,j} \leq 1000 (i \neq j),D_{i,j}=0 (i=j)$ * $1 \leq c_{i,j} \leq C$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $C$ $D_{1,1}$ $...$ $D_{1,C}$ $:$ $D_{C,1}$ $...$ $D_{C,C}$ $c_{1,1}$ $...$ $c_{1,N}$ $:$ $c_{N,1}$ $...$ $c_{N,N}$ [samples]

Samples

Input #1

Output #1

3

*   Repaint $(1,1)$ to Color $2$. The wrongness of $(1,1)$ becomes $D_{1,2}=1$.
*   Repaint $(1,2)$ to Color $3$. The wrongness of $(1,2)$ becomes $D_{2,3}=1$.
*   Repaint $(2,2)$ to Color $1$. The wrongness of $(2,2)$ becomes $D_{3,1}=1$.

In this case, the sum of the wrongness of all the squares is $3$.
Note that $D_{i,j} \neq D_{j,i}$ is possible.

Input #2

Output #2

API Response (JSON)

{
  "problem": {
    "name": "Good Grid",
    "description": {
      "content": "There is a grid with $N$ rows and $N$ columns of squares. Let $(i,j)$ be the square at the $i$\\-th row from the top and the $j$\\-th column from the left. These squares have to be painted in one of the",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc099_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a grid with $N$ rows and $N$ columns of squares. Let $(i,j)$ be the square at the $i$\\-th row from the top and the $j$\\-th column from the left.\nThese squares have to be painted in one of the...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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