Queen on Grid

AtCoder

IDabc183_e

Time2000ms

Memory256MB

Difficulty—

We have a grid with $H$ horizontal rows and $W$ vertical columns of squares. Square $(i,j)$, which is at the $i$\-th row from the top and $j$\-th column from the left, is _wall_ if $S_{ij}$ is `#` and _road_ if $S_{ij}$ is `.`. There is a queen, the chess piece, at Square $(1, 1)$. In one move, it can move any number of squares to the right, downwards, or diagonally to the lower right to a road square without jumping over wall squares. In how many ways can the queen travel from Square $(1, 1)$ to Square $(H, W)$? Find the count modulo $(10^9+7)$. Here, two ways to travel are considered different if and only if there exists $i$ such that the position of the queen after the $i$\-th move is different in those two ways. ## Constraints * $2 \leq H,W \leq 2000$ * $S_{ij}$ is `#` or `.`. * $S_{11}$ and $S_{HW}$ are `.`. ## Input Input is given from Standard Input in the following format: $H$ $W$ $S_{11}\ldots S_{1W}$ $\vdots$ $S_{H1}\ldots S_{HW}$ [samples]

Samples

Input #1

3 3
...
.#.
...

Output #1

10

There are $10$ ways to travel, as follows:

*   $(1,1)\to (1,2)\to (1,3)\to (2,3)\to (3,3)$
*   $(1,1)\to (1,2)\to (1,3)\to (3,3)$
*   $(1,1)\to (1,2)\to (2,3)\to (3,3)$
*   $(1,1)\to (1,3)\to (2,3)\to (3,3)$
*   $(1,1)\to (1,3)\to (3,3)$
*   $(1,1)\to (2,1)\to (3,1)\to (3,2)\to (3,3)$
*   $(1,1)\to (2,1)\to (3,1)\to (3,3)$
*   $(1,1)\to (2,1)\to (3,2)\to (3,3)$
*   $(1,1)\to (3,1)\to (3,2)\to (3,3)$
*   $(1,1)\to (3,1)\to (3,3)$

Input #2

4 4
...#
....
..#.
....

Output #2

84

From $(1,1)$, the queen can move to $(1,2)$, $(1,3)$, $(2,1)$, $(2,2)$, $(3,1)$, or $(4,1)$.
One possible path to $(4,4)$ is $(1,1)\to (3,1)\to (3,2)\to (4,3)\to (4,4)$.

Input #3

8 10
..........
..........
..........
..........
..........
..........
..........
..........

Output #3

13701937

Find the count modulo $(10^9+7)$.

API Response (JSON)

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  "problem": {
    "name": "Queen on Grid",
    "description": {
      "content": "We have a grid with $H$ horizontal rows and $W$ vertical columns of squares. Square $(i,j)$, which is at the $i$\\-th row from the top and $j$\\-th column from the left, is _wall_ if $S_{ij}$ is `#` and",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc183_e"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have a grid with $H$ horizontal rows and $W$ vertical columns of squares. Square $(i,j)$, which is at the $i$\\-th row from the top and $j$\\-th column from the left, is _wall_ if $S_{ij}$ is `#` and...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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