Reachable Cells

AtCoder

IDagc028_f

Time4000ms

Memory256MB

Difficulty—

**Problem F and F2 are the same problem, but with different constraints and time limits.** We have a board divided into $N$ horizontal rows and $N$ vertical columns of square cells. The cell at the $i$\-th row from the top and the $j$\-th column from the left is called Cell $(i,j)$. Each cell is either empty or occupied by an obstacle. Also, each empty cell has a digit written on it. If $A_{i,j}=$ `1`, `2`, ..., or `9`, Cell $(i,j)$ is empty and the digit $A_{i,j}$ is written on it. If $A_{i,j}=$ `#`, Cell $(i,j)$ is occupied by an obstacle. Cell $Y$ is _reachable_ from cell $X$ when the following conditions are all met: * Cells $X$ and $Y$ are different. * Cells $X$ and $Y$ are both empty. * One can reach from Cell $X$ to Cell $Y$ by repeatedly moving right or down to an adjacent empty cell. Consider all pairs of cells $(X,Y)$ such that cell $Y$ is reachable from cell $X$. Find the sum of the products of the digits written on cell $X$ and cell $Y$ for all of those pairs. ## Constraints * $1 \leq N \leq 500$ * $A_{i,j}$ is one of the following characters: `1`, `2`, ... `9` and `#`. ## Input Input is given from Standard Input in the following format: $N$ $A_{1,1}A_{1,2}...A_{1,N}$ $A_{2,1}A_{2,2}...A_{2,N}$ $:$ $A_{N,1}A_{N,2}...A_{N,N}$ [samples]

Samples

Input #1

2
11
11

Output #1

5

There are five pairs of cells $(X,Y)$ such that cell $Y$ is reachable from cell $X$, as follows:

*   $X=(1,1)$, $Y=(1,2)$
*   $X=(1,1)$, $Y=(2,1)$
*   $X=(1,1)$, $Y=(2,2)$
*   $X=(1,2)$, $Y=(2,2)$
*   $X=(2,1)$, $Y=(2,2)$

The product of the digits written on cell $X$ and cell $Y$ is $1$ for all of those pairs, so the answer is $5$.

Input #2

Output #2

Input #3

Output #3

Input #4

10
4177143673
7#########
5#1716155#
6#4#####5#
2#3#597#6#
6#9#8#3#5#
5#2#899#9#
1#6#####6#
6#5359657#
5#########

Output #4

API Response (JSON)

{
  "problem": {
    "name": "Reachable Cells",
    "description": {
      "content": "**Problem F and F2 are the same problem, but with different constraints and time limits.** We have a board divided into $N$ horizontal rows and $N$ vertical columns of square cells. The cell at the $i",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 4000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc028_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "**Problem F and F2 are the same problem, but with different constraints and time limits.**\nWe have a board divided into $N$ horizontal rows and $N$ vertical columns of square cells. The cell at the $i...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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