Median Replace

AtCoder

IDagc022_e

Time2000ms

Memory256MB

Difficulty—

Taichi thinks a binary string $X$ of odd length $N$ is **beautiful** if it is possible to apply the following operation $\frac{N-1}{2}$ times so that the only character of the resulting string is `1` : * Choose three **consecutive** bits of $X$ and replace them by their median. For example, we can turn `00110` into `010` by applying the operation to the middle three bits. Taichi has a string $S$ consisting of characters `0`, `1` and `?`. Taichi wants to know the number of ways to replace the question marks with `1` or `0` so that the resulting string is beautiful, modulo $10^{9} + 7$. ## Constraints * $1 \leq |S| \leq 300000$ * $|S|$ is odd. * All characters of $S$ are either `0`, `1` or `?`. ## Input Input is given from Standard Input in the following format: $S$ [samples]

Samples

Input #1

1??00

Output #1

2

There are $4$ ways to replace the question marks with `0` or `1` :

*   `11100` : This string is beautiful because we can first perform the operation on the last $3$ bits to get `110` and then on the only $3$ bits to get `1`.
    
*   `11000` : This string is beautiful because we can first perform the operation on the last $3$ bits to get `110` and then on the only $3$ bits to get `1`.
    
*   `10100` : This string is not beautiful because there is no sequence of operations such that the final string is `1`.
    
*   `10000` : This string is not beautiful because there is no sequence of operations such that the final string is `1`.
    

Thus, there are $2$ ways to form a beautiful string.

Input #2

Output #2

1

In this case, `1` is the only beautiful string.

Input #3

?0101???10???00?1???????????????0????????????1????0

Output #3

402589311

Remember to output your answer modulo $10^{9} + 7$.

API Response (JSON)

{
  "problem": {
    "name": "Median Replace",
    "description": {
      "content": "Taichi thinks a binary string $X$ of odd length $N$ is **beautiful** if it is possible to apply the following operation $\\frac{N-1}{2}$ times so that the only character of the resulting string is `1` ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc022_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Taichi thinks a binary string $X$ of odd length $N$ is **beautiful** if it is possible to apply the following operation $\\frac{N-1}{2}$ times so that the only character of the resulting string is `1` ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments