Adjacent Difference

AtCoder

IDagc066_a

Time2000ms

Memory256MB

Difficulty—

There is a grid with $N$ rows and $N$ columns. Let $(i, j)$ denote the cell at the $i$\-th row from the top and the $j$\-th column from the left. Each cell contains an integer. Initially, cell $(i,j)$ contains the integer $A_{i,j}$. You can repeatedly perform the following operation: * Choose integers $i$ and $j$ such that $1\leq i, j\leq N$ and an integer $x$, and add $x$ to $A_{i,j}$. The cost of this operation is $|x|$. You are given a positive integer $d$. Your goal is to satisfy the following condition: * The difference between the integers written in any two cells that are vertically or horizontally adjacent is at least $d$. More formally, the following two conditions are satisfied: * $|A_{i,j}-A_{i+1,j}|\geq d$ for all integers $i$ and $j$ such that $1\leq i\leq N-1$ and $1\leq j\leq N$. * $|A_{i,j}-A_{i,j+1}|\geq d$ for all integers $i$ and $j$ such that $1\leq i\leq N$ and $1\leq j\leq N-1$. Achieve this goal with a total cost of $\frac12 dN^2$ or less. ## Constraints * $2\leq N\leq 500$ * $1\leq d\leq 1000$ * $-1000\leq A_{i,j}\leq 1000$ ## Input The input is given from Standard Input in the following format: $N$ $d$ $A_{1,1}$ $\cdots$ $A_{1,N}$ $\vdots$ $A_{N,1}$ $\cdots$ $A_{N,N}$ [samples]

Samples

Input #1

Output #1

\-2 8 3
3 -9 -4
-2 8 3

Perform the operation four times as follows to yield the grid shown in the sample output:

*   Perform the operation with $(i,j,x)=(1,2,7)$.
*   Perform the operation with $(i,j,x)=(2,2,-5)$.
*   Perform the operation with $(i,j,x)=(3,1,-2)$.
*   Perform the operation with $(i,j,x)=(3,2,7)$.

The total cost is $7+5+2+7=21$, which is not greater than $\frac{1}{2}dN^2=\frac{45}{2}$.

Input #2

Output #2

API Response (JSON)

{
  "problem": {
    "name": "Adjacent Difference",
    "description": {
      "content": "There is a grid with $N$ rows and $N$ columns. Let $(i, j)$ denote the cell at the $i$\\-th row from the top and the $j$\\-th column from the left. Each cell contains an integer. Initially, cell $(i,j)$",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc066_a"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a grid with $N$ rows and $N$ columns. Let $(i, j)$ denote the cell at the $i$\\-th row from the top and the $j$\\-th column from the left.\nEach cell contains an integer. Initially, cell $(i,j)$...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments