A13626. Strange Addition

Let $a$ and $b$ be some non-negative integers. Let's define strange addition of $a$ and $b$ as following:

1. write down the numbers one under another and align them by their least significant digit;
2. add them up digit by digit and concatenate the respective sums together.

Assume that both numbers have an infinite number of leading zeros.

For example, let's take a look at a strange addition of numbers $3248$ and $908$ :

You are given a string $c$ , consisting of $n$ digits from $0$ to $9$ . You are also given $m$ updates of form:

- $x~d$ — replace the digit at the $x$ -th position of $c$ with a digit $d$ .

Note that string $c$ might have leading zeros at any point of time.

After each update print the number of pairs $(a, b)$ such that both $a$ and $b$ are non-negative integers and the result of a strange addition of $a$ and $b$ is equal to $c$ .

Note that the numbers of pairs can be quite large, so print them modulo $998244353$ .

The first line contains two integers $n$ and $m$ ( $1 \le n, m \le 5 \cdot 10^5$ ) — the length of the number $c$ and the number of updates.

The second line contains a string $c$ , consisting of exactly $n$ digits from $0$ to $9$ .

Each of the next $m$ lines contains two integers $x$ and $d$ ( $1 \le x \le n$ , $0 \le d \le 9$ ) — the descriptions of updates.

Print $m$ integers — the $i$ -th value should be equal to the number of pairs $(a, b)$ such that both $a$ and $b$ are non-negative integers and the result of a strange addition of $a$ and $b$ is equal to $c$ after $i$ updates are applied.

Note that the numbers of pairs can be quite large, so print them modulo $998244353$ .

After the first update $c$ is equal to $14$ . The pairs that sum up to $14$ are: $(0, 14)$ , $(1, 13)$ , $(2, 12)$ , $(3, 11)$ , $(4, 10)$ , $(5, 9)$ , $(6, 8)$ , $(7, 7)$ , $(8, 6)$ , $(9, 5)$ , $(10, 4)$ , $(11, 3)$ , $(12, 2)$ , $(13, 1)$ , $(14, 0)$ .

After the second update $c$ is equal to $11$ .

After the third update $c$ is equal to $01$ .