11301: [yosupo] Polynomial - Sqrt of Formal Power Series (Sparse)
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Description
You are given a formal power series $f(x) = \sum_{i=0}^{N-1} a_i x^i \in \mathbb{F}\_{998244353}[[x]]$.
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 a square root of $f(x)$.
In other words, find $g(x) = \sum_{i=0}^{N-1} b_i x^i \in \mathbb{F}\_{998244353}[[x]]$ such that
$f(x) \equiv g(x)^2 \pmod{x^N}.$
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 a square root of $f(x)$.
In other words, find $g(x) = \sum_{i=0}^{N-1} b_i x^i \in \mathbb{F}\_{998244353}[[x]]$ such that
$f(x) \equiv g(x)^2 \pmod{x^N}.$
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
If there are no $g(x)$ satisfying the condition, print
-1
and if such $g(x)$ exists, choose any and print
$b_0$ $b_1$ $\cdots$ $b_{N - 1}$
-1
and if such $g(x)$ exists, choose any and print
$b_0$ $b_1$ $\cdots$ $b_{N - 1}$
Constraints
- $1 \leq N \leq 10^6$
- $0 \leq K \leq 10$
- $0 \leq i_0 < \cdots < i_{K-1} \leq N - 1$
- $1 \leq a_{i_k} < 998244353$
- $0 \leq K \leq 10$
- $0 \leq i_0 < \cdots < i_{K-1} \leq N - 1$
- $1 \leq a_{i_k} < 998244353$
Sample 1 Input
4 2
2 9
3 12
Sample 1 Output
0 998244350 998244351 0
Sample 2 Input
4 2
2 10
3 12
Sample 2 Output
-1
Sample 3 Input
5 0
Sample 3 Output
0 0 0 0 0