結果

問題 No.2688 Cell Proliferation (Hard)
ユーザー akakimidoriakakimidori
提出日時 2024-03-20 22:26:54
言語 Rust
(1.77.0)
結果
AC  
実行時間 1,381 ms / 4,000 ms
コード長 20,789 bytes
コンパイル時間 3,148 ms
コンパイル使用メモリ 207,344 KB
実行使用メモリ 27,668 KB
最終ジャッジ日時 2024-03-20 22:27:20
合計ジャッジ時間 25,543 ms
ジャッジサーバーID
（参考情報） 		judge12 / judge15
このコードへのチャレンジ
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テストケース

テストケース表示
入力 結果 実行時間
実行使用メモリ
testcase_00 AC 1 ms
6,548 KB
testcase_01 AC 1 ms
6,548 KB
testcase_02 AC 1 ms
6,548 KB
testcase_03 AC 100 ms
6,548 KB
testcase_04 AC 1,166 ms
25,820 KB
testcase_05 AC 654 ms
14,720 KB
testcase_06 AC 277 ms
8,176 KB
testcase_07 AC 268 ms
8,128 KB
testcase_08 AC 1,092 ms
25,324 KB
testcase_09 AC 993 ms
24,908 KB
testcase_10 AC 1,289 ms
26,724 KB
testcase_11 AC 1,131 ms
25,468 KB
testcase_12 AC 327 ms
8,632 KB
testcase_13 AC 1,381 ms
27,660 KB
testcase_14 AC 847 ms
24,108 KB
testcase_15 AC 1,380 ms
27,668 KB
testcase_16 AC 825 ms
23,948 KB
testcase_17 AC 735 ms
15,408 KB
testcase_18 AC 1,313 ms
26,948 KB
testcase_19 AC 709 ms
15,448 KB
testcase_20 AC 411 ms
13,168 KB
testcase_21 AC 403 ms
13,160 KB
testcase_22 AC 708 ms
15,144 KB
testcase_23 AC 469 ms
13,480 KB
testcase_24 AC 1,292 ms
26,628 KB
testcase_25 AC 336 ms
8,808 KB
testcase_26 AC 1,333 ms
27,060 KB
testcase_27 AC 720 ms
15,504 KB
testcase_28 AC 863 ms
24,044 KB
権限があれば一括ダウンロードができます
コンパイルメッセージ
warning: unused import: `std::io::Write`
 --> Main.rs:1:5
  |
1 | use std::io::Write;
  |     ^^^^^^^^^^^^^^
  |
  = note: `#[warn(unused_imports)]` on by default

warning: type alias `Map` is never used
 --> Main.rs:4:6
  |
4 | type Map<K, V> = BTreeMap<K, V>;
  |      ^^^
  |
  = note: `#[warn(dead_code)]` on by default

warning: type alias `Set` is never used
 --> Main.rs:5:6
  |
5 | type Set<T> = BTreeSet<T>;
  |      ^^^

warning: type alias `Deque` is never used
 --> Main.rs:6:6
  |
6 | type Deque<T> = VecDeque<T>;
  |      ^^^^^

warning: 4 warnings emitted

ソースコード

diff # 

use std::io::Write;
use std::collections::*;

type Map<K, V> = BTreeMap<K, V>;
type Set<T> = BTreeSet<T>;
type Deque<T> = VecDeque<T>;

fn main() {
    input! {
        a: [u32; 4],
        t: usize,
    }
    let p = M::new(a[0]) * M::new(a[1]).inv();
    let q = M::new(a[2]) * M::new(a[3]).inv();
    let mut trans = vec![M::zero(); t + 1];
    let mut prod = M::one();
    let mut s = M::one();
    for i in 0..trans.len() {
        trans[i] = prod;
        s *= q;
        prod *= s;
    }
    let mut solver = RelaxedMultiplication::new();
    let mut memo = vec![M::zero(); t + 1];
    memo[0] = M::one();
    for i in 1..memo.len() {
        memo[i] = solver.next(memo[i - 1], trans[i - 1]) * p;
    }
    /*
    // dp[i]: 時刻iに生まれた細胞の個数の期待値
    let mut dp = vec![M::zero(); t + 1];
    dp[0] = M::one();
    for i in 0..t {
        let mut v = dp[i];
        let mut del = M::one();
        for j in (i + 1)..dp.len() {
            dp[j] += v * p;
            del *= q;
            v *= del;
        }
    }
    */
    let mut ans = M::zero();
    let mut del = M::one();
    let mut s = M::one();
    for dp in memo.iter().rev() {
        ans += *dp * del;
        s *= q;
        del *= s;
    }
    println!("{}", ans);
}

type M = ModInt<998244353>;

// ---------- begin input macro ----------
// reference: https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8
#[macro_export]
macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        input_inner!{iter, $($r)*}
    };
    ($($r:tt)*) => {
        let s = {
            use std::io::Read;
            let mut s = String::new();
            std::io::stdin().read_to_string(&mut s).unwrap();
            s
        };
        let mut iter = s.split_whitespace();
        input_inner!{iter, $($r)*}
    };
}

#[macro_export]
macro_rules! input_inner {
    ($iter:expr) => {};
    ($iter:expr, ) => {};
    ($iter:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($iter, $t);
        input_inner!{$iter $($r)*}
    };
}

#[macro_export]
macro_rules! read_value {
    ($iter:expr, ( $($t:tt),* )) => {
        ( $(read_value!($iter, $t)),* )
    };
    ($iter:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($iter, $t)).collect::<Vec<_>>()
    };
    ($iter:expr, chars) => {
        read_value!($iter, String).chars().collect::<Vec<char>>()
    };
    ($iter:expr, bytes) => {
        read_value!($iter, String).bytes().collect::<Vec<u8>>()
    };
    ($iter:expr, usize1) => {
        read_value!($iter, usize) - 1
    };
    ($iter:expr, $t:ty) => {
        $iter.next().unwrap().parse::<$t>().expect("Parse error")
    };
}
// ---------- end input macro ----------

use std::ops::*;

// ---------- begin trait ----------
pub trait Zero: Sized + Add<Self, Output = Self> {
    fn zero() -> Self;
    fn is_zero(&self) -> bool;
}

pub trait One: Sized + Mul<Self, Output = Self> {
    fn one() -> Self;
    fn is_one(&self) -> bool;
}

pub trait SemiRing: Zero + One {}

pub trait Ring: SemiRing + Sub<Output = Self> + Neg<Output = Self> {}

pub trait Field: Ring + Div<Output = Self> {}

impl<T> SemiRing for T where T: Zero + One {}

impl<T> Ring for T where T: SemiRing + Sub<Output = Self> + Neg<Output = Self> {}

impl<T> Field for T where T: Ring + Div<Output = Self> {}
// ---------- end trait ----------
// ---------- begin modint ----------
pub const fn pow_mod(mut r: u32, mut n: u32, m: u32) -> u32 {
    let mut t = 1;
    while n > 0 {
        if n & 1 == 1 {
            t = (t as u64 * r as u64 % m as u64) as u32;
        }
        r = (r as u64 * r as u64 % m as u64) as u32;
        n >>= 1;
    }
    t
}

pub const fn primitive_root(p: u32) -> u32 {
    let mut m = p - 1;
    let mut f = [1; 30];
    let mut k = 0;
    let mut d = 2;
    while d * d <= m {
        if m % d == 0 {
            f[k] = d;
            k += 1;
        }
        while m % d == 0 {
            m /= d;
        }
        d += 1;
    }
    if m > 1 {
        f[k] = m;
        k += 1;
    }
    let mut g = 1;
    while g < p {
        let mut ok = true;
        let mut i = 0;
        while i < k {
            ok &= pow_mod(g, (p - 1) / f[i], p) > 1;
            i += 1;
        }
        if ok {
            break;
        }
        g += 1;
    }
    g
}

pub const fn is_prime(n: u32) -> bool {
    if n <= 1 {
        return false;
    }
    let mut d = 2;
    while d * d <= n {
        if n % d == 0 {
            return false;
        }
        d += 1;
    }
    true
}

#[derive(Clone, Copy, PartialEq, Eq)]
pub struct ModInt<const M: u32>(u32);

impl<const M: u32> ModInt<{ M }> {
    const REM: u32 = {
        let mut t = 1u32;
        let mut s = !M + 1;
        let mut n = !0u32 >> 2;
        while n > 0 {
            if n & 1 == 1 {
                t = t.wrapping_mul(s);
            }
            s = s.wrapping_mul(s);
            n >>= 1;
        }
        t
    };
    const INI: u64 = ((1u128 << 64) % M as u128) as u64;
    const IS_PRIME: () = assert!(is_prime(M));
    const PRIMITIVE_ROOT: u32 = primitive_root(M);
    const ORDER: usize = 1 << (M - 1).trailing_zeros();
    const fn reduce(x: u64) -> u32 {
        let _ = Self::IS_PRIME;
        let b = (x as u32 * Self::REM) as u64;
        let t = x + b * M as u64;
        let mut c = (t >> 32) as u32;
        if c >= M {
            c -= M;
        }
        c as u32
    }
    const fn multiply(a: u32, b: u32) -> u32 {
        Self::reduce(a as u64 * b as u64)
    }
    pub const fn new(v: u32) -> Self {
        assert!(v < M);
        Self(Self::reduce(v as u64 * Self::INI))
    }
    pub const fn const_mul(&self, rhs: Self) -> Self {
        Self(Self::multiply(self.0, rhs.0))
    }
    pub const fn pow(&self, mut n: u64) -> Self {
        let mut t = Self::new(1);
        let mut r = *self;
        while n > 0 {
            if n & 1 == 1 {
                t = t.const_mul(r);
            }
            r = r.const_mul(r);
            n >>= 1;
        }
        t
    }
    pub const fn inv(&self) -> Self {
        assert!(self.0 != 0);
        self.pow(M as u64 - 2)
    }
    pub const fn get(&self) -> u32 {
        Self::reduce(self.0 as u64)
    }
    pub const fn zero() -> Self {
        Self::new(0)
    }
    pub const fn one() -> Self {
        Self::new(1)
    }
}

impl<const M: u32> Add for ModInt<{ M }> {
    type Output = Self;
    fn add(self, rhs: Self) -> Self::Output {
        let mut v = self.0 + rhs.0;
        if v >= M {
            v -= M;
        }
        Self(v)
    }
}

impl<const M: u32> Sub for ModInt<{ M }> {
    type Output = Self;
    fn sub(self, rhs: Self) -> Self::Output {
        let mut v = self.0 - rhs.0;
        if self.0 < rhs.0 {
            v += M;
        }
        Self(v)
    }
}

impl<const M: u32> Mul for ModInt<{ M }> {
    type Output = Self;
    fn mul(self, rhs: Self) -> Self::Output {
        self.const_mul(rhs)
    }
}

impl<const M: u32> Div for ModInt<{ M }> {
    type Output = Self;
    fn div(self, rhs: Self) -> Self::Output {
        self * rhs.inv()
    }
}

impl<const M: u32> AddAssign for ModInt<{ M }> {
    fn add_assign(&mut self, rhs: Self) {
        *self = *self + rhs;
    }
}

impl<const M: u32> SubAssign for ModInt<{ M }> {
    fn sub_assign(&mut self, rhs: Self) {
        *self = *self - rhs;
    }
}

impl<const M: u32> MulAssign for ModInt<{ M }> {
    fn mul_assign(&mut self, rhs: Self) {
        *self = *self * rhs;
    }
}

impl<const M: u32> DivAssign for ModInt<{ M }> {
    fn div_assign(&mut self, rhs: Self) {
        *self = *self / rhs;
    }
}

impl<const M: u32> Neg for ModInt<{ M }> {
    type Output = Self;
    fn neg(self) -> Self::Output {
        if self.0 == 0 {
            self
        } else {
            Self(M - self.0)
        }
    }
}

impl<const M: u32> std::fmt::Display for ModInt<{ M }> {
    fn fmt<'a>(&self, f: &mut std::fmt::Formatter<'a>) -> std::fmt::Result {
        write!(f, "{}", self.get())
    }
}

impl<const M: u32> std::fmt::Debug for ModInt<{ M }> {
    fn fmt<'a>(&self, f: &mut std::fmt::Formatter<'a>) -> std::fmt::Result {
        write!(f, "{}", self.get())
    }
}

impl<const M: u32> std::str::FromStr for ModInt<{ M }> {
    type Err = std::num::ParseIntError;
    fn from_str(s: &str) -> Result<Self, Self::Err> {
        let val = s.parse::<u32>()?;
        Ok(ModInt::new(val))
    }
}

impl<const M: u32> From<usize> for ModInt<{ M }> {
    fn from(val: usize) -> ModInt<{ M }> {
        ModInt::new((val % M as usize) as u32)
    }
}
// ---------- end modint ----------
// ---------- begin precalc ----------
pub struct Precalc<const MOD: u32> {
    fact: Vec<ModInt<MOD>>,
    ifact: Vec<ModInt<MOD>>,
    inv: Vec<ModInt<MOD>>,
}

impl<const MOD: u32> Precalc<MOD> {
    pub fn new(size: usize) -> Self {
        let mut fact = vec![ModInt::one(); size + 1];
        let mut ifact = vec![ModInt::one(); size + 1];
        let mut inv = vec![ModInt::one(); size + 1];
        for i in 2..=size {
            fact[i] = fact[i - 1] * ModInt::from(i);
        }
        ifact[size] = fact[size].inv();
        for i in (2..=size).rev() {
            inv[i] = ifact[i] * fact[i - 1];
            ifact[i - 1] = ifact[i] * ModInt::from(i);
        }
        Self { fact, ifact, inv }
    }
    pub fn fact(&self, n: usize) -> ModInt<MOD> {
        self.fact[n]
    }
    pub fn ifact(&self, n: usize) -> ModInt<MOD> {
        self.ifact[n]
    }
    pub fn inv(&self, n: usize) -> ModInt<MOD> {
        assert!(0 < n);
        self.inv[n]
    }
    pub fn perm(&self, n: usize, k: usize) -> ModInt<MOD> {
        if k > n {
            return ModInt::zero();
        }
        self.fact[n] * self.ifact[n - k]
    }
    pub fn binom(&self, n: usize, k: usize) -> ModInt<MOD> {
        if n < k {
            return ModInt::zero();
        }
        self.fact[n] * self.ifact[k] * self.ifact[n - k]
    }
}
// ---------- end precalc ----------

impl<const M: u32> Zero for ModInt<{ M }> {
    fn zero() -> Self {
        Self::zero()
    }
    fn is_zero(&self) -> bool {
        self.0 == 0
    }
}

impl<const M: u32> One for ModInt<{ M }> {
    fn one() -> Self {
        Self::one()
    }
    fn is_one(&self) -> bool {
        self.get() == 1
    }
}

// ---------- begin array op ----------

struct NTTPrecalc<const M: u32> {
    sum_e: [ModInt<{ M }>; 30],
    sum_ie: [ModInt<{ M }>; 30],
}

impl<const M: u32> NTTPrecalc<{ M }> {
    const fn new() -> Self {
        let cnt2 = (M - 1).trailing_zeros() as usize;
        let root = ModInt::new(ModInt::<{ M }>::PRIMITIVE_ROOT);
        let zeta = root.pow((M - 1) as u64 >> cnt2);
        let mut es = [ModInt::zero(); 30];
        let mut ies = [ModInt::zero(); 30];
        let mut sum_e = [ModInt::zero(); 30];
        let mut sum_ie = [ModInt::zero(); 30];
        let mut e = zeta;
        let mut ie = e.inv();
        let mut i = cnt2;
        while i >= 2 {
            es[i - 2] = e;
            ies[i - 2] = ie;
            e = e.const_mul(e);
            ie = ie.const_mul(ie);
            i -= 1;
        }
        let mut now = ModInt::one();
        let mut inow = ModInt::one();
        let mut i = 0;
        while i < cnt2 - 1 {
            sum_e[i] = es[i].const_mul(now);
            sum_ie[i] = ies[i].const_mul(inow);
            now = ies[i].const_mul(now);
            inow = es[i].const_mul(inow);
            i += 1;
        }
        Self { sum_e, sum_ie }
    }
}

struct NTTPrecalcHelper<const MOD: u32>;
impl<const MOD: u32> NTTPrecalcHelper<MOD> {
    const A: NTTPrecalc<MOD> = NTTPrecalc::new();
}

pub trait ArrayAdd {
    type Item;
    fn add(&self, rhs: &[Self::Item]) -> Vec<Self::Item>;
}

impl<T> ArrayAdd for [T]
where
    T: Zero + Copy,
{
    type Item = T;
    fn add(&self, rhs: &[Self::Item]) -> Vec<Self::Item> {
        let mut c = vec![T::zero(); self.len().max(rhs.len())];
        c[..self.len()].copy_from_slice(self);
        c.add_assign(rhs);
        c
    }
}

pub trait ArrayAddAssign {
    type Item;
    fn add_assign(&mut self, rhs: &[Self::Item]);
}

impl<T> ArrayAddAssign for [T]
where
    T: Add<Output = T> + Copy,
{
    type Item = T;
    fn add_assign(&mut self, rhs: &[Self::Item]) {
        assert!(self.len() >= rhs.len());
        self.iter_mut().zip(rhs).for_each(|(x, a)| *x = *x + *a);
    }
}

impl<T> ArrayAddAssign for Vec<T>
where
    T: Zero + Add<Output = T> + Copy,
{
    type Item = T;
    fn add_assign(&mut self, rhs: &[Self::Item]) {
        if self.len() < rhs.len() {
            self.resize(rhs.len(), T::zero());
        }
        self.as_mut_slice().add_assign(rhs);
    }
}

pub trait ArraySub {
    type Item;
    fn sub(&self, rhs: &[Self::Item]) -> Vec<Self::Item>;
}

impl<T> ArraySub for [T]
where
    T: Zero + Sub<Output = T> + Copy,
{
    type Item = T;
    fn sub(&self, rhs: &[Self::Item]) -> Vec<Self::Item> {
        let mut c = vec![T::zero(); self.len().max(rhs.len())];
        c[..self.len()].copy_from_slice(self);
        c.sub_assign(rhs);
        c
    }
}

pub trait ArraySubAssign {
    type Item;
    fn sub_assign(&mut self, rhs: &[Self::Item]);
}

impl<T> ArraySubAssign for [T]
where
    T: Sub<Output = T> + Copy,
{
    type Item = T;
    fn sub_assign(&mut self, rhs: &[Self::Item]) {
        assert!(self.len() >= rhs.len());
        self.iter_mut().zip(rhs).for_each(|(x, a)| *x = *x - *a);
    }
}

impl<T> ArraySubAssign for Vec<T>
where
    T: Zero + Sub<Output = T> + Copy,
{
    type Item = T;
    fn sub_assign(&mut self, rhs: &[Self::Item]) {
        if self.len() < rhs.len() {
            self.resize(rhs.len(), T::zero());
        }
        self.as_mut_slice().sub_assign(rhs);
    }
}

pub trait ArrayDot {
    type Item;
    fn dot(&self, rhs: &[Self::Item]) -> Vec<Self::Item>;
}

impl<T> ArrayDot for [T]
where
    T: Mul<Output = T> + Copy,
{
    type Item = T;
    fn dot(&self, rhs: &[Self::Item]) -> Vec<Self::Item> {
        assert!(self.len() == rhs.len());
        self.iter().zip(rhs).map(|p| *p.0 * *p.1).collect()
    }
}

pub trait ArrayDotAssign {
    type Item;
    fn dot_assign(&mut self, rhs: &[Self::Item]);
}

impl<T> ArrayDotAssign for [T]
where
    T: MulAssign + Copy,
{
    type Item = T;
    fn dot_assign(&mut self, rhs: &[Self::Item]) {
        assert!(self.len() == rhs.len());
        self.iter_mut().zip(rhs).for_each(|(x, a)| *x *= *a);
    }
}

pub trait ArrayMul {
    type Item;
    fn mul(&self, rhs: &[Self::Item]) -> Vec<Self::Item>;
}

impl<T> ArrayMul for [T]
where
    T: Zero + One + Copy,
{
    type Item = T;
    fn mul(&self, rhs: &[Self::Item]) -> Vec<Self::Item> {
        if self.is_empty() || rhs.is_empty() {
            return vec![];
        }
        let mut res = vec![T::zero(); self.len() + rhs.len() - 1];
        for (i, a) in self.iter().enumerate() {
            for (res, b) in res[i..].iter_mut().zip(rhs.iter()) {
                *res = *res + *a * *b;
            }
        }
        res
    }
}

// transform でlen=1を指定すればNTTになる
pub trait ArrayConvolution {
    type Item;
    fn transform(&mut self, len: usize);
    fn inverse_transform(&mut self, len: usize);
    fn convolution(&self, rhs: &[Self::Item]) -> Vec<Self::Item>;
}

impl<const M: u32> ArrayConvolution for [ModInt<{ M }>] {
    type Item = ModInt<{ M }>;
    fn transform(&mut self, len: usize) {
        let f = self;
        let n = f.len();
        let k = (n / len).trailing_zeros() as usize;
        assert!(len << k == n);
        assert!(k <= ModInt::<{ M }>::ORDER);
        let pre = &NTTPrecalcHelper::<{ M }>::A;
        for ph in 1..=k {
            let p = len << (k - ph);
            let mut now = ModInt::one();
            for (i, f) in f.chunks_exact_mut(2 * p).enumerate() {
                let (x, y) = f.split_at_mut(p);
                for (x, y) in x.iter_mut().zip(y.iter_mut()) {
                    let l = *x;
                    let r = *y * now;
                    *x = l + r;
                    *y = l - r;
                }
                now *= pre.sum_e[(!i).trailing_zeros() as usize];
            }
        }
    }
    fn inverse_transform(&mut self, len: usize) {
        let f = self;
        let n = f.len();
        let k = (n / len).trailing_zeros() as usize;
        assert!(len << k == n);
        assert!(k <= ModInt::<{ M }>::ORDER);
        let pre = &NTTPrecalcHelper::<{ M }>::A;
        for ph in (1..=k).rev() {
            let p = len << (k - ph);
            let mut inow = ModInt::one();
            for (i, f) in f.chunks_exact_mut(2 * p).enumerate() {
                let (x, y) = f.split_at_mut(p);
                for (x, y) in x.iter_mut().zip(y.iter_mut()) {
                    let l = *x;
                    let r = *y;
                    *x = l + r;
                    *y = (l - r) * inow;
                }
                inow *= pre.sum_ie[(!i).trailing_zeros() as usize];
            }
        }
        let ik = ModInt::new(2).inv().pow(k as u64);
        for f in f.iter_mut() {
            *f *= ik;
        }
    }
    fn convolution(&self, rhs: &[Self::Item]) -> Vec<Self::Item> {
        if self.len().min(rhs.len()) <= 32 {
            return self.mul(rhs);
        }
        const PARAM: usize = 10;
        let size = self.len() + rhs.len() - 1;
        let mut k = 0;
        while (size + (1 << k) - 1) >> k > PARAM {
            k += 1;
        }
        let len = (size + (1 << k) - 1) >> k;
        let mut f = vec![ModInt::zero(); len << k];
        let mut g = vec![ModInt::zero(); len << k];
        f[..self.len()].copy_from_slice(self);
        g[..rhs.len()].copy_from_slice(rhs);
        f.transform(len);
        g.transform(len);
        let mut buf = [ModInt::zero(); 2 * PARAM - 1];
        let buf = &mut buf[..(2 * len - 1)];
        let pre = &NTTPrecalcHelper::<{ M }>::A;
        let mut now = ModInt::one();
        for (i, (f, g)) in f
            .chunks_exact_mut(2 * len)
            .zip(g.chunks_exact(2 * len))
            .enumerate()
        {
            let mut r = now;
            for (f, g) in f.chunks_exact_mut(len).zip(g.chunks_exact(len)) {
                buf.fill(ModInt::zero());
                for (i, f) in f.iter().enumerate() {
                    for (buf, g) in buf[i..].iter_mut().zip(g.iter()) {
                        *buf = *buf + *f * *g;
                    }
                }
                f.copy_from_slice(&buf[..len]);
                for (f, buf) in f.iter_mut().zip(buf[len..].iter()) {
                    *f = *f + r * *buf;
                }
                r = -r;
            }
            now *= pre.sum_e[(!i).trailing_zeros() as usize];
        }
        f.inverse_transform(len);
        f.truncate(self.len() + rhs.len() - 1);
        f
    }
}
// ---------- end array op ----------

// ---------- begin relaxed multiplication ----------
pub struct RelaxedMultiplication {
    f: Vec<M>,
    g: Vec<M>,
    h: Vec<M>,
    pos: usize,
}
 
impl RelaxedMultiplication {
    pub fn new() -> Self {
        RelaxedMultiplication {
            f: vec![],
            g: vec![],
            h: vec![],
            pos: 0,
        }
    }
    pub fn next(&mut self, f: M, g: M) -> M {
        self.f.push(f);
        self.g.push(g);
        let a = self.pos + 2;
        let len = 1 << a.trailing_zeros();
        if a == len {
            let c = self.f.convolution(&self.g);
            self.h.extend(c.into_iter().skip(self.pos));
        } else {
            let x = Self::middle_product(&self.f[(self.pos + 1 - len)..], &self.g[..(2 * len - 1)]);
            let y = Self::middle_product(&self.g[(self.pos + 1 - len)..], &self.f[..(2 * len - 1)]);
            if self.pos + x.len() > self.h.len() {
                self.h.resize(self.pos + x.len(), M::zero());
            }
            self.h[self.pos..].add_assign(&x);
            self.h[self.pos..].add_assign(&y);
        }
        self.pos += 1;
        self.h[self.pos - 1]
    }
    fn middle_product(a: &[M], b: &[M]) -> Vec<M> {
        if b.len() - a.len() + 1 <= 32 {
            let mut c = vec![M::zero(); b.len() - a.len() + 1];
            for (i, a) in a.iter().rev().enumerate() {
                for (c, b) in c.iter_mut().zip(b[i..].iter()) {
                    *c += *a * *b;
                }
            }
            return c;
        }
        let size = b.len().next_power_of_two();
        let mut x = vec![M::zero(); size];
        let mut y = vec![M::zero(); size];
        x[..a.len()].copy_from_slice(a);
        y[..b.len()].copy_from_slice(b);
        x.transform(1);
        y.transform(1);
        x.dot_assign(&y);
        x.inverse_transform(1);
        let res = Vec::from(&x[(a.len() - 1)..b.len()]);
        res
    }
}
// ---------- end relaxed multiplication ----------
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