結果
| 問題 |
No.3273 Exactly One Match
|
| コンテスト | |
| ユーザー |
 akakimidori
|
| 提出日時 | 2025-09-13 10:57:17 |
| 言語 | Rust (1.83.0 + proconio) |
| 結果 |
AC
|
| 実行時間 | 1,151 ms / 4,000 ms |
| コード長 | 33,460 bytes |
| コンパイル時間 | 22,105 ms |
| コンパイル使用メモリ | 399,400 KB |
| 実行使用メモリ | 49,856 KB |
| 最終ジャッジ日時 | 2025-09-13 10:57:52 |
| 合計ジャッジ時間 | 28,754 ms |
|
ジャッジサーバーID （参考情報） |
judge3 / judge2 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 3 |
| other | AC * 26 |
コンパイルメッセージ
warning: unused import: `std::io::Write`
--> src/main.rs:9:5
|
9 | use std::io::Write;
| ^^^^^^^^^^^^^^
|
= note: `#[warn(unused_imports)]` on by default
warning: type alias `Map` is never used
--> src/main.rs:11:6
|
11 | type Map<K, V> = BTreeMap<K, V>;
| ^^^
|
= note: `#[warn(dead_code)]` on by default
warning: type alias `Set` is never used
--> src/main.rs:12:6
|
12 | type Set<T> = BTreeSet<T>;
| ^^^
warning: type alias `Deque` is never used
--> src/main.rs:13:6
|
13 | type Deque<T> = VecDeque<T>;
| ^^^^^
warning: method `join` is never used
--> src/main.rs:1014:12
|
1013 | pub trait Join {
| ---- method in this trait
1014 | fn join(self, sep: &str) -> String;
| ^^^^
ソースコード
// functional graph を色塗りして同じ色に向かう辺が一つだけあるようにする
// という感じの問
//
// 長さNのサイクルで色が異なるようなもの
// 長さNのサイクルで1つだけ一緒なもの
//
use std::collections::*;
use std::io::Write;
type Map<K, V> = BTreeMap<K, V>;
type Set<T> = BTreeSet<T>;
type Deque<T> = VecDeque<T>;
type M = ModInt<998244353>;
fn main() {
input! {
n: usize,
k: usize,
}
let pc = Precalc::new(n);
let mut cycle = vec![M::zero(); n + 1];
let mut one = vec![M::zero(); n + 1];
one[1] = M::from(k);
let mut pow = M::one();
for i in 2..=n {
pow *= M::from(k - 1);
cycle[i] = M::from(k) * pow - cycle[i - 1];
if i > 2 {
one[i] = cycle[i - 1] * M::from(i);
}
}
for i in 1..=n {
one[i] *= pc.inv(i);
cycle[i] *= pc.inv(i);
}
let free = cycle.exp(n + 1);
let one = free.convolution(&one);
let mut ans = one[n];
let mut pow = M::one();
for i in (1..n).rev() {
if i + 1 < n {
pow *= M::from(n);
}
ans += free[i] * pow * pc.ifact(n - i) * M::from(n - i) * M::from(i);
pow *= M::from(k - 1);
ans += one[i] * pow * pc.ifact(n - i) * M::from(i);
}
ans *= pc.fact(n);
println!("{}", ans);
}
// ---------- 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 ----------
// ---------- 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 VALID: () = assert!(is_prime(M) && M % 2 == 1 && M < (1 << 30));
const PRIMITIVE_ROOT: u32 = primitive_root(M);
const ORDER: usize = 1 << (M - 1).trailing_zeros();
const fn reduce(x: u64) -> u32 {
let _ = Self::VALID;
let b = (x as u32 * Self::REM) as u64;
let t = x + b * M as u64;
(t >> 32) 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 {
Self(Self::reduce((v % M) 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 {
let mut res = Self::reduce(self.0 as u64);
if res >= M {
res -= M;
}
res
}
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 >= 2 * M {
v -= 2 * 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 += 2 * 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(2 * 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)
}
}
impl<const M: u32> From<u64> for ModInt<{ M }> {
fn from(val: u64) -> ModInt<{ M }> {
ModInt::new((val % M as u64) as u32)
}
}
impl<const M: u32> Into<usize> for ModInt<{ M }> {
fn into(self) -> usize {
self.get() as usize
}
}
// ---------- end modint ----------
// ---------- begin precalc ----------
pub struct Precalc<T> {
fact: Vec<T>,
ifact: Vec<T>,
inv: Vec<T>,
}
impl<T> Precalc<T>
where
T: Copy + Field,
{
pub fn new(size: usize) -> Self {
let mut fact = vec![T::one(); size + 1];
let mut ifact = vec![T::one(); size + 1];
let mut inv = vec![T::one(); size + 1];
let mut mul = T::one();
for i in 2..=size {
mul = mul + T::one();
fact[i] = fact[i - 1] * mul;
}
ifact[size] = T::one() / fact[size];
for i in (2..=size).rev() {
inv[i] = ifact[i] * fact[i - 1];
ifact[i - 1] = ifact[i] * mul;
mul = mul - T::one();
}
Self { fact, ifact, inv }
}
pub fn fact(&self, n: usize) -> T {
self.fact[n]
}
pub fn ifact(&self, n: usize) -> T {
self.ifact[n]
}
pub fn inv(&self, n: usize) -> T {
assert!(0 < n);
self.inv[n]
}
pub fn perm(&self, n: usize, k: usize) -> T {
if k > n {
return T::zero();
}
self.fact[n] * self.ifact[n - k]
}
pub fn binom(&self, n: usize, k: usize) -> T {
if n < k {
return T::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
}
}
pub trait NTT {
fn ntt(&mut self);
fn intt(&mut self);
fn transform(&mut self, len: usize);
fn inverse_transform(&mut self, len: usize);
fn dot_product_ntt(&mut self, rhs: &Self, len: usize);
}
impl<const M: u32> NTT for [ModInt<{ M }>] {
fn ntt(&mut self) {
self.transform(1);
}
fn intt(&mut self) {
self.inverse_transform(1);
}
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 dot_product_ntt(&mut self, rhs: &Self, len: usize) {
let mut buf = [ModInt::zero(); 20];
let buf = &mut buf[..(2 * len - 1)];
let pre = &NTTPrecalcHelper::<{ M }>::A;
let mut now = ModInt::one();
for (i, (f, g)) in self
.chunks_exact_mut(2 * len)
.zip(rhs.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];
}
}
}
// transform でlen=1を指定すればNTTになる
pub trait ArrayConvolution {
type Item;
fn convolution(&self, rhs: &[Self::Item]) -> Vec<Self::Item>;
fn middle_product(&self, a: &[Self::Item]) -> Vec<Self::Item>;
}
pub fn convolution_modulo<const MOD: u32, const A: u32>(
a: &[ModInt<MOD>],
b: &[ModInt<MOD>],
) -> Vec<ModInt<A>> {
let a = a
.iter()
.map(|a| ModInt::<A>::new(a.get()))
.collect::<Vec<_>>();
let b = b
.iter()
.map(|a| ModInt::<A>::new(a.get()))
.collect::<Vec<_>>();
a.convolution(&b)
}
pub fn middle_product_modulo<const MOD: u32, const A: u32>(
a: &[ModInt<MOD>],
b: &[ModInt<MOD>],
) -> Vec<ModInt<A>> {
let a = a
.iter()
.map(|a| ModInt::<A>::new(a.get()))
.collect::<Vec<_>>();
let b = b
.iter()
.map(|a| ModInt::<A>::new(a.get()))
.collect::<Vec<_>>();
a.middle_product(&b)
}
impl<const M: u32> ArrayConvolution for [ModInt<{ M }>] {
type Item = ModInt<{ M }>;
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;
}
if ModInt::<{ M }>::ORDER < k {
const A: u32 = 167772161;
const B: u32 = 469762049;
const C: u32 = 754974721;
assert!(ModInt::<A>::ORDER >= k);
assert!(ModInt::<B>::ORDER >= k);
assert!(ModInt::<C>::ORDER >= k);
const P: u32 = pow_mod(A, B - 2, B);
const Q: u32 = pow_mod(A, C - 2, C);
const R: u32 = pow_mod(B, C - 2, C);
const QR: u32 = (Q as u64 * R as u64 % C as u64) as u32;
const W1: u32 = A;
let w2: u32 = (A as u64 * B as u64 % M as u64) as u32;
let x: Vec<ModInt<A>> = convolution_modulo(self, rhs);
let y: Vec<ModInt<B>> = convolution_modulo(self, rhs);
let z: Vec<ModInt<C>> = convolution_modulo(self, rhs);
let mut ans = vec![ModInt::<{ M }>::zero(); x.len()];
for (((ans, x), y), z) in ans.iter_mut().zip(x).zip(y).zip(z) {
let a = x.get();
let b = ((y.get() + B - a) as u64 * P as u64 % B as u64) as u32;
let c = (((z.get() + C - a) as u64 * QR as u64 + (C - b) as u64 * R as u64)
% C as u64) as u32;
*ans = (a as u64 + b as u64 * W1 as u64 + c as u64 * w2 as u64).into();
}
return ans;
}
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);
f.dot_product_ntt(&g, len);
f.inverse_transform(len);
f.truncate(self.len() + rhs.len() - 1);
f
}
fn middle_product(&self, rhs: &[Self::Item]) -> Vec<Self::Item> {
assert!(self.len() >= rhs.len());
if self.len() - rhs.len() <= 32 {
return self
.windows(rhs.len())
.map(|a| {
a.iter()
.zip(rhs.iter())
.fold(ModInt::zero(), |s, p| s + *p.0 * *p.1)
})
.collect();
}
const PARAM: usize = 10;
let size = self.len();
let mut k = 0;
while (size + (1 << k) - 1) >> k > PARAM {
k += 1;
}
if ModInt::<{ M }>::ORDER < k {
const A: u32 = 167772161;
const B: u32 = 469762049;
const C: u32 = 754974721;
assert!(ModInt::<A>::ORDER >= k);
assert!(ModInt::<B>::ORDER >= k);
assert!(ModInt::<C>::ORDER >= k);
const P: u32 = pow_mod(A, B - 2, B);
const Q: u32 = pow_mod(A, C - 2, C);
const R: u32 = pow_mod(B, C - 2, C);
const QR: u32 = (Q as u64 * R as u64 % C as u64) as u32;
const W1: u32 = A;
let w2: u32 = (A as u64 * B as u64 % M as u64) as u32;
let x: Vec<ModInt<A>> = middle_product_modulo(self, rhs);
let y: Vec<ModInt<B>> = middle_product_modulo(self, rhs);
let z: Vec<ModInt<C>> = middle_product_modulo(self, rhs);
let mut ans = vec![ModInt::<{ M }>::zero(); x.len()];
for (((ans, x), y), z) in ans.iter_mut().zip(x).zip(y).zip(z) {
let a = x.get();
let b = ((y.get() + B - a) as u64 * P as u64 % B as u64) as u32;
let c = (((z.get() + C - a) as u64 * QR as u64 + (C - b) as u64 * R as u64)
% C as u64) as u32;
*ans = (a as u64 + b as u64 * W1 as u64 + c as u64 * w2 as u64).into();
}
return ans;
}
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);
g[..rhs.len()].reverse();
f.transform(len);
g.transform(len);
f.dot_product_ntt(&g, len);
f.inverse_transform(len);
(rhs.len()..=self.len()).map(|i| f[i - 1]).collect()
}
}
pub trait PolynomialOperation {
type Item;
fn eval(&self, x: Self::Item) -> Self::Item;
fn derivative(&self) -> Vec<Self::Item>;
fn integral(&self) -> Vec<Self::Item>;
}
impl<T> PolynomialOperation for [T]
where
T: Field + Copy,
{
type Item = T;
fn eval(&self, x: Self::Item) -> Self::Item {
self.iter().rfold(T::zero(), |s, a| s * x + *a)
}
fn derivative(&self) -> Vec<Self::Item> {
if self.len() <= 1 {
return vec![];
}
self[1..]
.iter()
.scan(T::one(), |s, a| {
let res = *a * *s;
*s = *s + T::one();
Some(res)
})
.collect()
}
fn integral(&self) -> Vec<Self::Item> {
if self.is_empty() {
return vec![];
}
let mut inv = vec![T::one(); self.len() + 1];
let mut val = T::zero();
for i in 1..inv.len() {
val = val + T::one();
inv[i] = val * inv[i - 1];
}
let mut iprod = T::one() / inv[self.len()];
for i in (1..inv.len()).rev() {
inv[i] = iprod * inv[i - 1] * self[i - 1];
iprod = iprod * val;
val = val - T::one();
}
inv[0] = T::zero();
inv
}
}
pub trait FPSOperation {
type Item;
fn inverse(&self, n: usize) -> Vec<Self::Item>;
fn log(&self, n: usize) -> Vec<Self::Item>;
fn exp(&self, n: usize) -> Vec<Self::Item>;
}
impl<T> FPSOperation for [T]
where
T: Field + Copy,
[T]: ArrayConvolution<Item = T>,
{
type Item = T;
fn inverse(&self, n: usize) -> Vec<Self::Item> {
assert!(self.len() > 0 && !self[0].is_zero());
if n == 0 {
return vec![];
}
let mut g = Vec::with_capacity(n);
g.push(T::one() / self[0]);
while g.len() < n {
let size = g.len();
let up = (2 * size).min(n);
let gg = g.convolution(&g);
let mut h = gg.convolution(&self[..up.min(self.len())]);
h.resize(up, T::zero());
g.extend(h[size..up].iter().map(|v| -*v));
}
g
}
fn log(&self, n: usize) -> Vec<Self::Item> {
assert!(self.len() > 0 && self[0].is_one());
if n == 0 {
return vec![];
}
let mut res = self.derivative().convolution(&self.inverse(n));
res.truncate(n - 1);
res.integral()
}
fn exp(&self, n: usize) -> Vec<Self::Item> {
assert!(self.len() > 0 && self[0].is_zero());
if n == 0 {
return vec![];
}
let mut g = Vec::with_capacity(n);
g.push(T::one());
while g.len() < n {
let size = g.len();
let up = (2 * size).min(n);
let lg = g.log(up);
let rhs = self[..up.min(self.len())].sub(&lg);
let mut h = g.convolution(&rhs);
h.resize(up, T::zero());
g.extend(h[size..up].iter().cloned());
}
g
}
}
// ---------- end array op ----------
// ---------- begin trait ----------
use std::ops::*;
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 Group: Zero + Sub<Output = Self> + Neg<Output = Self> {}
pub trait SemiRing: Zero + One {}
pub trait Ring: SemiRing + Group {}
pub trait Field: Ring + Div<Output = Self> {}
impl<T> Group for T where T: Zero + Sub<Output = Self> + Neg<Output = Self> {}
impl<T> SemiRing for T where T: Zero + One {}
impl<T> Ring for T where T: SemiRing + Group {}
impl<T> Field for T where T: Ring + Div<Output = Self> {}
pub fn zero<T: Zero>() -> T {
T::zero()
}
pub fn one<T: One>() -> T {
T::one()
}
pub fn pow<T: One + Clone>(mut r: T, mut n: usize) -> T {
let mut t = one();
while n > 0 {
if n & 1 == 1 {
t = t * r.clone();
}
r = r.clone() * r;
n >>= 1;
}
t
}
pub fn pow_sum<T: SemiRing + Clone>(r: T, n: usize) -> T {
if n == 0 {
T::zero()
} else if n & 1 == 1 {
T::one() + r.clone() * pow_sum(r, n - 1)
} else {
let a = T::one() + r.clone();
let b = r.clone() * r;
a * pow_sum(b, n / 2)
}
}
// ---------- end trait ----------
mod util {
pub trait Join {
fn join(self, sep: &str) -> String;
}
impl<T, I> Join for I
where
I: Iterator<Item = T>,
T: std::fmt::Display,
{
fn join(self, sep: &str) -> String {
let mut s = String::new();
use std::fmt::*;
for (i, v) in self.enumerate() {
if i > 0 {
write!(&mut s, "{}", sep).ok();
}
write!(&mut s, "{}", v).ok();
}
s
}
}
}
// ---------- taylor shift ----------
// f(x) とcを受け取って f(x+c) を返す
pub trait TaylorShift {
type Item;
fn taylor_shift(&self, c: Self::Item) -> Vec<Self::Item>;
}
impl<T> TaylorShift for [T]
where
T: Copy + Field,
[T]: ArrayConvolution<Item = T>,
{
type Item = T;
fn taylor_shift(&self, c: Self::Item) -> Vec<Self::Item> {
if self.is_empty() || c.is_zero() {
return Vec::from(self);
}
let mut fact = vec![T::one(); self.len()];
let mut val = T::zero();
for i in 1..fact.len() {
val = val + T::one();
fact[i] = fact[i - 1] * val;
}
let mut ifact = vec![T::one(); self.len()];
ifact[self.len() - 1] = T::one() / fact[self.len() - 1];
for i in (1..fact.len()).rev() {
ifact[i - 1] = ifact[i] * val;
val = val - T::one();
}
let mut a = Vec::from(self);
for (a, f) in a.iter_mut().zip(fact.iter()) {
*a = *a * *f;
}
a.reverse();
let mut pow = T::one();
for (f, i) in fact.iter_mut().zip(ifact.iter()) {
*f = *i * pow;
pow = pow * c;
}
a = a.convolution(&fact);
a.truncate(self.len());
a.reverse();
for (a, i) in a.iter_mut().zip(ifact.iter()) {
*a = *a * *i;
}
a
}
}
// ---------- taylor shift ----------
// f(x) を x = a * r^i (0 <= i < m) で評価した値の列を返す
pub fn multipoint_evaluation_on_geometric_sequence<T>(f: &[T], a: T, r: T, m: usize) -> Vec<T>
where
T: Copy + Field,
[T]: ArrayConvolution<Item = T>,
{
let n = f.len();
assert!(n > 0 && m > 0);
if r.is_zero() {
let mut res = vec![f[0]; m];
res[0] = f.iter().rev().fold(T::zero(), |s, f| s * a + *f);
return res;
}
let ir = T::one() / r;
let mut f = Vec::from(f);
let mut dp = (T::one(), ir);
for f in f.iter_mut() {
*f = *f * dp.0;
dp = (dp.0 * dp.1 * a, dp.1 * ir);
}
let mut g = vec![T::zero(); n + m - 1];
let mut dp = (T::one(), r);
for g in g.iter_mut() {
*g = dp.0;
dp = (dp.0 * dp.1, dp.1 * r);
}
f = g.middle_product(&f);
let mut dp = (T::one(), ir);
for f in f.iter_mut() {
*f = *f * dp.0;
dp = (dp.0 * dp.1, dp.1 * ir);
}
f
}
pub fn multipoint_evaluation<T>(c: Vec<T>, p: Vec<T>) -> Vec<T>
where
T: Copy + Field,
[T]: ArrayConvolution<Item = T> + FPSOperation<Item = T>,
{
if p.is_empty() {
return vec![];
}
let n = c.len();
let m = p.len();
let mut prod = vec![vec![]; 2 * m];
for (prod, p) in prod[m..].iter_mut().zip(p.iter()) {
*prod = vec![T::one(), -*p];
}
for i in (1..m).rev() {
prod[i] = prod[2 * i].convolution(&prod[2 * i + 1]);
}
let inv = prod[1].inverse(n);
let mut c = c;
c.resize(n + m - 1, T::zero());
let mut dp = vec![vec![]; 2 * m];
dp[1] = c.middle_product(&inv);
for i in 1..m {
dp[2 * i] = dp[i].middle_product(&prod[2 * i + 1]);
dp[2 * i + 1] = dp[i].middle_product(&prod[2 * i]);
}
dp[m..].iter().map(|dp| dp[0]).collect()
}
pub fn shift_of_sampling_points_of_polynomial<T>(f: &[T], c: T, m: usize) -> Vec<T>
where
T: Copy + Field + From<usize> + Into<usize>,
[T]: ArrayConvolution<Item = T>,
{
if f.is_empty() {
return vec![T::zero(); m];
}
if m == 0 {
return vec![];
}
let cv: usize = c.into();
if cv < f.len() {
let sub = f.len() - cv;
if m <= sub {
return Vec::from(&f[cv..(cv + m)]);
}
let mut ans = Vec::from(&f[cv..]);
ans.extend(shift_of_sampling_points_of_polynomial(
&f,
T::from(f.len()),
m - sub,
));
return ans;
}
for i in 0..m {
if T::from(cv + i).is_zero() {
let up = i;
let mut ans = shift_of_sampling_points_of_polynomial(&f, c, up);
ans.extend(&shift_of_sampling_points_of_polynomial(
&f,
T::zero(),
m - up,
));
return ans;
}
}
let pc = Precalc::<T>::new(f.len());
let n = f.len();
let mut f = Vec::from(f);
for (i, f) in f.iter_mut().enumerate() {
*f = *f * pc.ifact(i) * pc.ifact(n - 1 - i);
if (n - 1 - i) % 2 == 1 {
*f = -*f;
}
}
let mut prod = vec![one(); n + m - 1];
let mut v = T::one();
for i in 0..prod.len() {
v = v * T::from(cv - (n - 1) + i);
prod[i] = v;
}
let mut inv = vec![one(); n + m - 1];
inv[n + m - 2] = T::one() / prod[n + m - 2];
for i in (0..(inv.len() - 1)).rev() {
inv[i] = inv[i + 1] * T::from(cv - (n - 1) + i + 1);
inv[i + 1] = inv[i + 1] * prod[i];
}
f.reverse();
let mut f = inv.middle_product(&f);
let mut val = prod[n - 1];
for (i, f) in f.iter_mut().enumerate() {
*f = *f * val;
val = val * inv[i] * T::from(cv + 1 + i);
}
f
}