Uniformly Distributed

AtCoder

IDarc120_b

Time2000ms

Memory256MB

Difficulty—

We have a grid with $H$ rows and $W$ columns. Let $(i, j)$ denote the square at the $i$\-th row and $j$\-th column. You are given $H$ strings $S_1, S_2, S_3, \dots, S_H$ that describe the square, as follows: * if the $j$\-th character of $S_i$ is `R`, $(i, j)$ is painted red; * if the $j$\-th character of $S_i$ is `B`, $(i, j)$ is painted blue; * if the $j$\-th character of $S_i$ is `.`, $(i, j)$ is unpainted. Let $k$ be the number of unpainted squares. Among the $2^k$ ways to paint each of those squares red or blue, how many satisfy the following condition? * The number of red squares visited while getting from $(1, 1)$ to $(H, W)$ by repeatedly moving one square right or down, including $(1, 1)$ and $(H, W)$, is always the same regardless of the path taken. Since the count may be enormous, print it modulo $998244353$. ## Constraints * $2 \le H \le 500$ * $2 \le W \le 500$ * $S_i$ is a string of length $W$ consisting of `R`, `B`, and `.`. ## Input Input is given from Standard Input in the following format: $H$ $W$ $S_1$ $S_2$ $S_3$ $\hspace{3pt} \vdots$ $S_H$ [samples]

Samples

Input #1

2 2
B.
.R

Output #1

2

There are two ways to get from $(1, 1)$ to $(2, 2)$ by repeatedly moving right or down, as follows:

*   $(1, 1) \rightarrow (1, 2) \rightarrow (2, 2)$
*   $(1, 1) \rightarrow (2, 1) \rightarrow (2, 2)$

If we paint $(1, 2)$ and $(2, 1)$ in different colors, the above two paths will contain different numbers of red squares, violating the condition.  
On the other hand, if we paint $(1, 2)$ and $(2, 1)$ in the same color, the two paths will contain the same numbers of red squares, satisfying the condition.  
Thus, there are two ways to satisfy the condition.

Input #2

3 3
R.R
BBR
...

Output #2

0

There may be no way to satisfy the condition.

Input #3

2 2
BB
BB

Output #3

1

There is no unpainted square, and the current grid satisfies the condition, so the answer is $1$.

API Response (JSON)

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    "name": "Uniformly Distributed",
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      "content": "We have a grid with $H$ rows and $W$ columns. Let $(i, j)$ denote the square at the $i$\\-th row and $j$\\-th column.   You are given $H$ strings $S_1, S_2, S_3, \\dots, S_H$ that describe the square, as",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
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    "sync_url": null,
    "sign": "arc120_b"
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  "statements": [
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      "statement_type": "Markdown",
      "content": "We have a grid with $H$ rows and $W$ columns. Let $(i, j)$ denote the square at the $i$\\-th row and $j$\\-th column.  \nYou are given $H$ strings $S_1, S_2, S_3, \\dots, S_H$ that describe the square, as...",
      "is_translate": false,
      "language": "English"
    }
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}

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