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Ex - 01? Queries

Score : $600$ points

Problem Statement

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:

$N$ $Q$

$S$

$x_1$ $c_1$

$x_2$ $c_2$

$\vdots$

$x_Q$ $c_Q$

Output

Print $Q$ lines. For each $i = 1, 2, \ldots, Q$, the $i$-th line should contain the answer to the $i$-th query $(x_i, c_i)$ (that is, the number of strings modulo $998244353$ at the step 2. in the statement).

3 3
100
2 1
2 ?
3 ?

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 $10$.

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

746884092
532460539
299568633
541985786
217532539
217532539
217532539
573323772
483176957
236273405

Be sure to print the count modulo $998244353$.