11297: [yosupo] Polynomial - Inv of Formal Power Series (Sparse)
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Description
You are given a formal power series $f(x) = \sum_{i=0}^{N-1} a_i x^i \in \mathbb{F}\_{998244353}[[x]]$ with $a_0 \ne 0$.
Only $K$ coefficients of $f$ are non-zero, and only such coefficients $a_{i_k}$ are given from input.
Calculate the first $N$ terms of $\frac{1}{f(x)} = \sum_{i=0}^{\infty} b_i x^i$.
In other words, find $g(x) = \sum_{i=0}^{N-1} b_i x^i \in \mathbb{F}\_{998244353}[[x]]$ such that
$f(x) g(x) \equiv 1 \pmod{x^N}.$
Only $K$ coefficients of $f$ are non-zero, and only such coefficients $a_{i_k}$ are given from input.
Calculate the first $N$ terms of $\frac{1}{f(x)} = \sum_{i=0}^{\infty} b_i x^i$.
In other words, find $g(x) = \sum_{i=0}^{N-1} b_i x^i \in \mathbb{F}\_{998244353}[[x]]$ such that
$f(x) g(x) \equiv 1 \pmod{x^N}.$
Input
$N$ $K$
$i_0$ $a_{i_0}$
$\vdots$
$i_{K-1}$ $a_{i_{K-1}}$
$i_0$ $a_{i_0}$
$\vdots$
$i_{K-1}$ $a_{i_{K-1}}$
Output
$b_0$ $b_1$ $\cdots$ $b_{N - 1}$
Constraints
- $1 \leq N \leq 10^6$
- $1 \leq K \leq 10$
- $0 = i_0 < \cdots < i_{K-1} \leq N - 1$
- $1 \leq a_{i_k} < 998244353$
- $1 \leq K \leq 10$
- $0 = i_0 < \cdots < i_{K-1} \leq N - 1$
- $1 \leq a_{i_k} < 998244353$
Sample 1 Input
5 2
0 1
2 1
Sample 1 Output
1 0 998244352 0 1
Sample 2 Input
10 5
0 5
1 4
2 3
3 2
4 1
Sample 2 Output
598946612 718735934 862483121 635682004 163871793 995241634 275905156 386987871 291888821 422779055