Ex - 01? Queries

AtCoder

IDabc246_h

Time6000ms

Memory256MB

Difficulty—

You are given a string $S$ of length $N$ consisting of `0`, `1`, and `?`. You are also given $Q$ queries $(x_1, c_1), (x_2, c_2), \ldots, (x_Q, c_Q)$. For each $i = 1, 2, \ldots, Q$, $x_i$ is an integer satisfying $1 \leq x_i \leq N$ and $c_i$ is one of the characters `0` , `1`, and `?`. For $i = 1, 2, \ldots, Q$ in this order, do the following process for the query $(x_i, c_i)$. 1. First, change the $x_i$\-th character from the beginning of $S$ to $c_i$. 2. Then, print the number of non-empty strings, modulo $998244353$, that can be obtained as a (not necessarily contiguous) subsequence of $S$ after replacing each occurrence of `?` in $S$ with `0` or `1` independently. ## Constraints * $1 \leq N, Q \leq 10^5$ * $N$ and $Q$ are integers. * $S$ is a string of length $N$ consisting of `0`, `1`, and `?`. * $1 \leq x_i \leq N$ * $c_i$ is one of the characters `0` , `1`, and `?`. ## Input Input is given from Standard Input in the following format: $N$ $Q$ $S$ $x_1$ $c_1$ $x_2$ $c_2$ $\vdots$ $x_Q$ $c_Q$ [samples]

Samples

Input #1

Output #1

5
7
10

*   The $1$\-st query starts by changing $S$ to `110`. Five strings can be obtained as a subsequence of $S = $ `110`: `0`, `1`, `10`, `11`, `110`. Thus, the $1$\-st query should be answered by $5$.
    
*   The $2$\-nd query starts by changing $S$ to `1?0`. Two strings can be obtained by the `?` in $S = $ `1?0`: `100` and `110`. Seven strings can be obtained as a subsequence of one of these strings: `0`, `1`, `00`, `10`, `11`, `100`, `110`. Thus, the $2$\-nd query should be answered by $7$.
    
*   The $3$\-rd query starts by changing $S$ to `1??`. Four strings can be obtained by the `?`'s in $S = $ `1??`: `100`, `101`, `110`, `111`. Ten strings can be obtained as a subsequence of one of these strings: `0`, `1`, `00`, `01`, `10`, `11`, `100`, `101`, `110`, `111`. Thus, the $3$\-rd query should be answered by $10$.

Input #2

40 10
011?0??001??10?0??0?0?1?11?1?00?11??0?01
5 0
2 ?
30 ?
7 1
11 1
3 1
25 1
40 0
12 1
18 1

Output #2

746884092
532460539
299568633
541985786
217532539
217532539
217532539
573323772
483176957
236273405

Be sure to print the count modulo $998244353$.

API Response (JSON)

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    "name": "Ex - 01? Queries",
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      "statement_type": "Markdown",
      "content": "You are given a string $S$ of length $N$ consisting of `0`, `1`, and `?`.\nYou are also given $Q$ queries $(x_1, c_1), (x_2, c_2), \\ldots, (x_Q, c_Q)$.  \nFor each $i = 1, 2, \\ldots, Q$, $x_i$ is an int...",
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