結果
| 問題 | No.718 行列のできるフィボナッチ数列道場 (1) |
| ユーザー |
 へのく
|
| 提出日時 | 2021-06-24 18:38:07 |
| 言語 | Rust (1.77.0 + proconio) |
| 結果 |
AC
|
| 実行時間 | 2 ms / 2,000 ms |
| コード長 | 43,975 bytes |
| コンパイル時間 | 15,954 ms |
| コンパイル使用メモリ | 378,680 KB |
| 実行使用メモリ | 5,376 KB |
| 最終ジャッジ日時 | 2024-06-25 07:31:55 |
| 合計ジャッジ時間 | 17,367 ms |
|
ジャッジサーバーID （参考情報） |
judge2 / judge5 |
（要ログイン）
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
5,248 KB |
| testcase_01 | AC | 1 ms
5,248 KB |
| testcase_02 | AC | 1 ms
5,248 KB |
| testcase_03 | AC | 2 ms
5,376 KB |
| testcase_04 | AC | 1 ms
5,376 KB |
| testcase_05 | AC | 1 ms
5,376 KB |
| testcase_06 | AC | 1 ms
5,376 KB |
| testcase_07 | AC | 1 ms
5,376 KB |
| testcase_08 | AC | 1 ms
5,376 KB |
| testcase_09 | AC | 1 ms
5,376 KB |
| testcase_10 | AC | 1 ms
5,376 KB |
| testcase_11 | AC | 1 ms
5,376 KB |
| testcase_12 | AC | 1 ms
5,376 KB |
| testcase_13 | AC | 1 ms
5,376 KB |
| testcase_14 | AC | 1 ms
5,376 KB |
| testcase_15 | AC | 1 ms
5,376 KB |
| testcase_16 | AC | 1 ms
5,376 KB |
| testcase_17 | AC | 1 ms
5,376 KB |
| testcase_18 | AC | 1 ms
5,376 KB |
| testcase_19 | AC | 1 ms
5,376 KB |
| testcase_20 | AC | 1 ms
5,376 KB |
| testcase_21 | AC | 1 ms
5,376 KB |
| testcase_22 | AC | 1 ms
5,376 KB |
ソースコード
#![allow(non_snake_case)]
use crate::{
fps::{conv_anymod::ConvAnyMod, nth_term::nth_term},
scanner::Scanner,
};
modint!();
fn main() {
let mut scan = Scanner::new();
let n = scan.long();
let ret = nth_term::<_, ConvAnyMod>(n - 1, &list![1, 2, 6, 15, 40, 104].map(Z::new));
println!("{}", ret);
}
pub mod independent {
pub mod integer {
use std::fmt::Display;
use std::ops::*;
pub trait Int:
Add<Output = Self>
+ Sub<Output = Self>
+ Mul<Output = Self>
+ Div<Output = Self>
+ Rem<Output = Self>
+ AddAssign
+ SubAssign
+ MulAssign
+ DivAssign
+ RemAssign
+ std::hash::Hash
+ PartialEq
+ Eq
+ PartialOrd
+ Ord
+ Copy
+ Display
{
fn to_u8(&self) -> u8;
fn to_u16(&self) -> u16;
fn to_u32(&self) -> u32;
fn to_u64(&self) -> u64;
fn to_u128(&self) -> u128;
fn to_i8(&self) -> i8;
fn to_i16(&self) -> i16;
fn to_i32(&self) -> i32;
fn to_i64(&self) -> i64;
fn to_i128(&self) -> i128;
fn to_usize(&self) -> usize;
fn to_isize(&self) -> isize;
fn from_u8(x: u8) -> Self;
fn from_u16(x: u16) -> Self;
fn from_u32(x: u32) -> Self;
fn from_u64(x: u64) -> Self;
fn from_u128(x: u128) -> Self;
fn from_i8(x: i8) -> Self;
fn from_i16(x: i16) -> Self;
fn from_i32(x: i32) -> Self;
fn from_i64(x: i64) -> Self;
fn from_i128(x: i128) -> Self;
fn from_usize(x: usize) -> Self;
fn from_isize(x: isize) -> Self;
fn zero() -> Self;
fn one() -> Self;
fn next(&self) -> Self {
*self + Self::one()
}
fn powint(&self, n: i64) -> Self;
}
#[macro_export]
macro_rules! impl_integer_functions {
($to_op:expr, $from_op:expr, $pow:expr) => {
impl_integer_functions!(
$to_op, $from_op, $pow,
to_u8, from_u8, u8,
to_u16, from_u16, u16,
to_u32, from_u32, u32,
to_u64, from_u64, u64,
to_u128, from_u128, u128,
to_i8, from_i8, i8,
to_i16, from_i16, i16,
to_i32, from_i32, i32,
to_i64, from_i64, i64,
to_i128, from_i128, i128,
to_usize, from_usize, usize,
to_isize, from_isize, isize
);
};
($to_op:expr, $from_op:expr, $pow:expr, $($tofn:ident, $fromfn:ident, $tpe:ident),*) => {
$(
fn $tofn(&self) -> $tpe {
$to_op(self) as $tpe
}
fn $fromfn(x: $tpe) -> Self {
$from_op(x)
}
)*
fn zero() -> Self {$from_op(0)}
fn one() -> Self {$from_op(1)}
fn powint(&self, n: i64) -> Self {$pow(self, n)}
};
}
macro_rules! impl_integer {
($($tpe:ident),*) => {
$(
impl Int for $tpe {
impl_integer_functions!(
|s: &Self| *s, |x| x as $tpe, |s: &Self, n: i64| s.pow(n as u32)
);
}
)*
};
}
impl_integer!(u8, u16, u32, u64, u128, i8, i16, i32, i64, i128, usize, isize);
}
}
pub mod fps {
pub mod convolution {
use crate::{
arraylist::List,
independent::integer::Int,
modint,
modulo::{ButterflyCache, Modulus, StaticModInt},
nums::*,
};
fn prepare<M: Modulus>() -> ButterflyCache<M> {
let g = StaticModInt::<M>::raw(primitive_root(M::M as i32) as u32);
let mut es = [StaticModInt::<M>::raw(0); 30];
let mut ies = [StaticModInt::<M>::raw(0); 30];
let cnt2 = (M::M - 1).trailing_zeros() as usize;
let mut e = g.pow((M::M - 1) >> cnt2);
let mut ie = e.inv();
for i in (2..=cnt2).rev() {
es[i - 2] = e;
ies[i - 2] = ie;
e *= e;
ie *= ie;
}
let sum_e = es
.iter()
.scan(StaticModInt::new(1), |acc, e| {
*acc *= *e;
Some(*acc)
})
.collect();
let sum_ie = ies
.iter()
.scan(StaticModInt::new(1), |acc, ie| {
*acc *= *ie;
Some(*acc)
})
.collect();
ButterflyCache { sum_e, sum_ie }
}
fn butterfly<M: Modulus>(a: &mut [StaticModInt<M>]) {
let n = a.len();
let h = ceil_pow2(n as u32);
M::butterfly_cache().with(|cache| {
let mut cache = cache.borrow_mut();
let ButterflyCache { sum_e, .. } = cache.get_or_insert_with(prepare);
for ph in 1..=h {
let w = 1 << (ph - 1);
let p = 1 << (h - ph);
let mut now = StaticModInt::<M>::new(1);
for s in 0..w {
let offset = s << (h - ph + 1);
for i in 0..p {
let l = a[i + offset];
let r = a[i + offset + p] * now;
a[i + offset] = l + r;
a[i + offset + p] = l - r;
}
now *= sum_e[(!s).trailing_zeros() as usize];
}
}
});
}
fn butterfly_inv<M: Modulus>(a: &mut [StaticModInt<M>]) {
let n = a.len();
let h = ceil_pow2(n as u32);
M::butterfly_cache().with(|cache| {
let mut cache = cache.borrow_mut();
let ButterflyCache { sum_ie, .. } = cache.get_or_insert_with(prepare);
for ph in (1..=h).rev() {
let w = 1 << (ph - 1);
let p = 1 << (h - ph);
let mut inow = StaticModInt::<M>::new(1);
for s in 0..w {
let offset = s << (h - ph + 1);
for i in 0..p {
let l = a[i + offset];
let r = a[i + offset + p];
a[i + offset] = l + r;
a[i + offset + p] = StaticModInt::new(M::M + l.val - r.val) * inow;
}
inow *= sum_ie[(!s).trailing_zeros() as usize];
}
}
})
}
pub fn convolution_naive<T: Int>(a: &[T], b: &[T]) -> List<T> {
if a.is_empty() || b.is_empty() {
return vec![].into();
}
let (n, m) = (a.len(), b.len());
let (n, m, a, b) = if n < m { (m, n, b, a) } else { (n, m, a, b) };
let mut ans = vec![T::zero(); n + m - 1];
for i in 0..n {
for j in 0..m {
ans[i + j] += a[i] * b[j];
}
}
ans.into()
}
pub fn convolution_ntt<M: Modulus>(
a: &[StaticModInt<M>],
b: &[StaticModInt<M>],
) -> List<StaticModInt<M>> {
if a.is_empty() || b.is_empty() {
return vec![].into();
}
let (n, m) = (a.len(), b.len());
if n.min(m) <= 60 {
return convolution_naive(a, b);
}
let (mut a, mut b) = (a.to_owned(), b.to_owned());
let z = 1 << ceil_pow2((n + m - 1) as _);
a.resize(z, StaticModInt::raw(0));
butterfly(&mut a);
b.resize(z, StaticModInt::raw(0));
butterfly(&mut b);
for (a, b) in a.iter_mut().zip(&b) {
*a *= *b;
}
butterfly_inv(&mut a);
a.resize(n + m - 1, StaticModInt::raw(0));
let iz = StaticModInt::new(z).inv();
for a in &mut a {
*a *= iz;
}
a.into()
}
fn convolution_raw_mod<T: Int, M: Modulus>(a: &[T], b: &[T]) -> List<StaticModInt<M>> {
let a = a
.iter()
.cloned()
.map(StaticModInt::<M>::new)
.collect::<Vec<_>>();
let b = b
.iter()
.cloned()
.map(StaticModInt::<M>::new)
.collect::<Vec<_>>();
convolution_ntt::<M>(&a, &b)
}
pub fn convolution_anymod<M: Modulus>(
a: &[StaticModInt<M>],
b: &[StaticModInt<M>],
) -> List<StaticModInt<M>> {
const M1: u32 = 998244353;
const M2: u32 = 1012924417;
const M3: u32 = 924844033;
modint!(M1);
modint!(M2);
modint!(M3);
let a = a.iter().map(StaticModInt::to_u32).collect::<Vec<_>>();
let b = b.iter().map(StaticModInt::to_u32).collect::<Vec<_>>();
let v1 = convolution_raw_mod::<_, M1>(&a, &b);
let v2 = convolution_raw_mod::<_, M2>(&a, &b);
let v3 = convolution_raw_mod::<_, M3>(&a, &b);
let (_, f2f1) = inv_gcd(M1 as _, M2 as _);
let (_, f3f1) = inv_gcd(M1 as _, M3 as _);
let (_, f3f2) = inv_gcd(M2 as _, M3 as _);
v1.into_iter()
.zip(v2)
.zip(v3)
.map(|((e1, e2), e3)| {
let x1 = e1;
let x2 = (e2 - x1.val.into()) * f2f1.into();
let x3 = ((e3 - x1.val.into()) * f3f1.into() - x2.val.into()) * f3f2.into();
StaticModInt::<M>::new(x1.val)
+ StaticModInt::<M>::new(x2.val) * M1.into()
+ StaticModInt::<M>::new(x3.val) * M1.into() * M2.into()
})
.collect()
}
}
pub mod formal_power_series {
use std::{fmt::Debug, iter::FromIterator, marker::PhantomData, ops::*};
use crate::{
arraylist::{lst, List},
independent::integer::Int,
list,
modulo::ModInt,
};
pub trait Conv<T> {
fn convolution(a: &[T], b: &[T]) -> List<T>;
}
impl<T: Int, F: Conv<T>> From<&lst<T>> for Fps<T, F> {
fn from(lst: &lst<T>) -> Self {
Self::new(lst.list())
}
}
impl<T: Int, F: Conv<T>> FromIterator<T> for Fps<T, F> {
fn from_iter<U: IntoIterator<Item = T>>(iter: U) -> Self {
Self::new(iter.into_iter().collect())
}
}
#[derive(PartialEq, Eq)]
pub struct Fps<T, F: Conv<T>> {
pub data: List<T>,
phantom: PhantomData<fn() -> F>,
}
impl<T: Int, F: Conv<T>> Clone for Fps<T, F> {
fn clone(&self) -> Self {
Self::new(self.data.clone())
}
}
impl<T: Int + Debug, F: Conv<T>> Debug for Fps<T, F> {
fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
self.data.fmt(f)
}
}
impl<T: Int, F: Conv<T>> Fps<T, F> {
pub fn new(data: List<T>) -> Fps<T, F> {
Fps {
data,
phantom: PhantomData,
}
}
pub fn empty() -> Self {
Self::new(List::new())
}
pub fn lens(&self) -> isize {
self.data.lens()
}
pub fn is_empty(&self) -> bool {
self.data.lens() == 0
}
pub fn resize(&mut self, new_len: isize) {
self.data.resize(new_len, T::zero());
}
pub fn shrink(&mut self) {
while self.data.last() == Some(&T::zero()) {
self.data.pop();
}
}
pub fn pre(&self, size: isize) -> Self {
Self::from(&self.data[..self.lens().min(size)])
}
pub fn rev(&self, deg: impl Into<Option<isize>>) -> Self {
let deg = deg.into();
let mut data = self.data.clone();
if let Some(deg) = deg {
data.resize(deg, T::zero());
}
data.reverse();
Self::new(data)
}
pub fn inv(&self, deg: impl Into<Option<isize>>) -> Self
where
T: ModInt,
{
assert!(self[0] != T::zero());
let n = self.lens();
let deg = deg.into().unwrap_or(n);
let mut ret = Self::new(list![T::one() / self[0]]);
let mut i = 1;
while i < deg {
ret = (&ret + &ret - &ret * &ret * &self.pre(i << 1)).pre(i << 1);
i <<= 1;
}
ret.pre(deg)
}
pub fn div_polynomial(&mut self, rhs: &Self)
where
T: ModInt,
{
if self.lens() < rhs.lens() {
self.data.data.clear();
return;
}
let n = self.lens() - rhs.lens() + 1;
*self = (self.rev(None).pre(n) * rhs.rev(None).inv(n)).pre(n).rev(n);
}
}
macro_rules! impl_ops {
($({$guard:ident, $tpe:ident, $fname:ident, $op:tt, [$($lhs:tt)+], [$($rhs:tt)+]})*) => {
$(impl<T: $guard, F: Conv<T>> $tpe<$($rhs)+> for $($lhs)+ {
type Output = Fps<T, F>;
fn $fname(self, rhs: $($rhs)+) -> Self::Output {
let mut ret: Fps<T, F> = self.clone();
ret $op rhs;
ret
}
})*
};
}
macro_rules! impl_ops_times {
($([$($lhs:tt)+] * [$($rhs:tt)+]),*; $([$($lhs2:tt)+] * [$($rhs2:tt)+]),*) => {
$(
impl_ops!(
{Int, Add, add, +=, [$($lhs)+], [$($rhs)+]}
{Int, Sub, sub, -=, [$($lhs)+], [$($rhs)+]}
{Int, Mul, mul, *=, [$($lhs)+], [$($rhs)+]}
{ModInt, Div, div, /=, [$($lhs)+], [$($rhs)+]}
{ModInt, Rem, rem, %=, [$($lhs)+], [$($rhs)+]}
);
)*
$(
impl_ops!(
{Int, Add, add, +=, [$($lhs2)+], [$($rhs2)+]}
{Int, Sub, sub, -=, [$($lhs2)+], [$($rhs2)+]}
{Int, Mul, mul, *=, [$($lhs2)+], [$($rhs2)+]}
);
)*
};
}
impl_ops_times!([Fps<T, F>] * [&Self], [Fps<T, F>] * [Self], [&Fps<T, F>] * [Self]; [Fps<T, F>] * [T], [&Fps<T, F>] * [T]);
macro_rules! impl_assign_ops {
($([$($rhs:tt)+]),*) => {
$(
impl<T: Int, F: Conv<T>> AddAssign<$($rhs)+> for Fps<T, F> {
fn add_assign(&mut self, rhs: $($rhs)+) {
if rhs.lens() > self.lens() {self.resize(rhs.lens())};
self.data.iter_mut().zip(rhs.data.iter()).for_each(|(e, r)| *e += *r);
}
}
impl<T: Int, F: Conv<T>> SubAssign<$($rhs)+> for Fps<T, F> {
fn sub_assign(&mut self, rhs: $($rhs)+) {
if rhs.lens() > self.lens() {self.resize(rhs.lens())};
self.data.iter_mut().zip(rhs.data.iter()).for_each(|(e,r)| *e -= *r);
self.shrink();
}
}
impl<T: Int, F: Conv<T>> MulAssign<$($rhs)+> for Fps<T, F> {
fn mul_assign(&mut self, rhs: $($rhs)+) {
self.data = F::convolution(&self.data, &rhs.data);
}
}
impl<T: ModInt, F: Conv<T>> RemAssign<$($rhs)+> for Fps<T, F> {
fn rem_assign(&mut self, rhs: $($rhs)+) {
let mut t = self.clone(); t.div_polynomial(&rhs);
*self -= &(t * rhs);
}
}
impl<T: ModInt, F: Conv<T>> DivAssign<$($rhs)+> for Fps<T, F> {
fn div_assign(&mut self, rhs: $($rhs)+) {
let n = self.lens();
self.data = F::convolution(&self.data, &rhs.inv(n).data);
self.resize(n);
}
}
)*
};
}
impl_assign_ops!([Self], [&Self]);
impl<T: Int, F: Conv<T>> Neg for &Fps<T, F> {
type Output = Fps<T, F>;
fn neg(self) -> Self::Output {
let data = self.data.map(|x| T::zero() - x);
Fps {
data,
phantom: PhantomData,
}
}
}
impl<T: Int, F: Conv<T>> AddAssign<T> for Fps<T, F> {
fn add_assign(&mut self, rhs: T) {
if self.is_empty() {
self.resize(1)
};
self[0] += rhs;
}
}
impl<T: Int, F: Conv<T>> SubAssign<T> for Fps<T, F> {
fn sub_assign(&mut self, rhs: T) {
if self.is_empty() {
self.resize(1);
}
self[0] -= rhs;
self.shrink();
}
}
impl<T: Int, F: Conv<T>> MulAssign<T> for Fps<T, F> {
fn mul_assign(&mut self, rhs: T) {
for e in self.data.iter_mut() {
*e *= rhs;
}
}
}
impl<T: Int, F: Conv<T>> Shr<isize> for &Fps<T, F> {
type Output = Fps<T, F>;
fn shr(self, rhs: isize) -> Self::Output {
if self.lens() <= rhs {
return Fps::<T, F>::empty();
}
Self::Output::from(&self.data[rhs..])
}
}
impl<T: Int, F: Conv<T>> Shl<isize> for &Fps<T, F> {
type Output = Fps<T, F>;
fn shl(self, rhs: isize) -> Self::Output {
let mut data = list![T::zero();rhs];
data.append(&self.data);
Self::Output::new(data)
}
}
impl<T: Int, F: Conv<T>> Index<isize> for Fps<T, F> {
type Output = T;
fn index(&self, index: isize) -> &T {
&self.data[index]
}
}
impl<T: Int, F: Conv<T>> IndexMut<isize> for Fps<T, F> {
fn index_mut(&mut self, index: isize) -> &mut T {
&mut self.data[index]
}
}
#[macro_export]
macro_rules! fps {
([$($args:tt)+] [$($tpe:tt)+]) => {$crate::fps::formal_power_series::Fps::<_, $($tpe)+>::new($crate::list![$($args)+])};
}
}
pub mod berlekamp_massey {
use crate::{arraylist::List, modulo::ModInt};
pub fn berlekamp_massey<T: ModInt>(a: &[T]) -> List<T> {
let n = a.len();
let mut b = Vec::with_capacity(n + 1);
let mut c = Vec::with_capacity(n + 1);
b.push(T::one());
c.push(T::one());
let mut y = T::one();
for ed in 1..=n {
let l = c.len();
let mut m = b.len();
let mut x = T::zero();
for i in 0..l {
x += c[i] * a[ed - l + i];
}
b.push(T::zero());
m += 1;
if x == T::zero() {
continue;
}
let freq = x / y;
if l < m {
let tmp = c.clone();
for _ in 0..m - l {
c.insert(0, T::zero());
}
for i in 0..m {
c[m - 1 - i] -= freq * b[m - 1 - i];
}
b = tmp;
y = x;
} else {
for i in 0..m {
c[l - 1 - i] -= freq * b[m - 1 - i];
}
}
}
c.reverse();
c.into()
}
}
pub mod conv_anymod {
use crate::{
arraylist::List,
fps::{convolution::convolution_anymod, formal_power_series::Conv},
modulo::{Modulus, StaticModInt},
};
#[derive(Debug, PartialEq, Eq, Copy, Clone, Hash, PartialOrd, Ord)]
pub enum ConvAnyMod {}
impl<M: Modulus> Conv<StaticModInt<M>> for ConvAnyMod {
fn convolution(a: &[StaticModInt<M>], b: &[StaticModInt<M>]) -> List<StaticModInt<M>> {
convolution_anymod(a, b)
}
}
#[macro_export]
macro_rules! fps_anymod {
($($args:tt)+) => {$crate::fps!([$($args)+] [$crate::fps::conv_anymod::ConvAnyMod])};
}
}
pub mod nth_term {
use crate::{
arraylist::lst,
fps::{
berlekamp_massey::berlekamp_massey,
bostan_mori::bostan_mori,
formal_power_series::{Conv, Fps},
},
modulo::ModInt,
};
pub fn nth_term<T: ModInt, F: Conv<T>>(n: i64, a: &lst<T>) -> T {
let q = Fps::<_, F>::new(berlekamp_massey(&a));
if n < a.lens() as i64 {
return a[n as isize];
}
let mut p = Fps::from(a).pre(q.lens() - 1) * &q;
p.resize(q.lens() - 1);
bostan_mori(n, p, q)
}
}
pub mod bostan_mori {
use crate::{
fps::formal_power_series::{Conv, Fps},
modulo::ModInt,
};
pub fn bostan_mori<T: ModInt, F: Conv<T>>(
mut n: i64,
mut p: Fps<T, F>,
mut q: Fps<T, F>,
) -> T {
assert!(p.lens() < q.lens());
p.resize(q.lens() - 1);
while n > 0 {
let mq = (0..q.lens())
.map(|i| if i & 1 == 0 { q[i] } else { T::zero() - q[i] })
.collect::<Fps<T, F>>();
let s = &p * &mq;
let t = &q * &mq;
if n & 1 == 1 {
for i in (1..s.lens()).step_by(2) {
p[i >> 1] = s[i];
}
for i in (0..t.lens()).step_by(2) {
q[i >> 1] = t[i];
}
} else {
for i in (0..s.lens()).step_by(2) {
p[i >> 1] = s[i];
}
for i in (0..t.lens()).step_by(2) {
q[i >> 1] = t[i];
}
}
n >>= 1;
}
p[0]
}
}
}
pub mod modulo {
use crate::nums::inv_gcd;
use crate::{impl_integer_functions, independent::integer::Int};
use std::cell::RefCell;
use std::fmt::Debug;
use std::marker::PhantomData;
use std::ops::*;
use std::sync::atomic::{self, AtomicU32};
use std::thread::LocalKey;
pub trait Modulus: 'static + PartialEq + Eq + Copy + Clone + std::hash::Hash + Ord {
const M: u32;
const HINT_M_IS_PRIME: bool;
fn butterfly_cache() -> &'static LocalKey<RefCell<Option<ButterflyCache<Self>>>>;
}
pub trait DynamicModulus:
'static + PartialEq + Eq + Copy + Clone + std::hash::Hash + Ord
{
fn state() -> &'static AtomicU32 {
static M: AtomicU32 = AtomicU32::new(1_000_000_007);
&M
}
fn update(m: u32) {
Self::state().store(m, atomic::Ordering::SeqCst)
}
fn umod() -> u32 {
Self::state().load(atomic::Ordering::SeqCst)
}
}
#[derive(PartialEq, Eq, Copy, Clone, Hash, PartialOrd, Ord)]
pub enum DefaultId {}
impl DynamicModulus for DefaultId {}
macro_rules! impl_from_for_modint {
($name:ident, $guard: ident, $($tpe:ident),*) => {
$(
impl<T: $guard> From<$tpe> for $name<T> {
fn from(n: $tpe) -> Self {
Self::new(n)
}
}
)*
};
}
macro_rules! impl_assign {
($name:ident, $guard:ident, $($t1:ty, $t2:ty, $fa:ident, $f:ident),*) => {
$(
impl<T: $guard> $t1 for $name<T> {
type Output = Self;
#[inline]
fn $f(self, other: Self) -> Self {
<Self as ModInt>::$f(self, other)
}
}
impl<T: $guard> $t2 for $name<T> {
#[inline]
fn $fa(&mut self, other: Self) {
*self = <Self as ModInt>::$f(*self, other);
}
}
)*
};
}
macro_rules! impl_modint_structs {
($name:ident, $guard:ident) => {
#[derive(PartialEq, Eq, Copy, Clone, Hash, PartialOrd, Ord)]
#[repr(transparent)]
pub struct $name<T> {
pub val: u32,
phantom: PhantomData<fn() -> T>,
}
impl<T> Debug for $name<T> {
fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
self.val.fmt(f)
}
}
impl<T: $guard> $name<T> {
#[inline]
pub fn new<U: Int>(a: U) -> Self {
<Self as ModInt>::new(a)
}
#[inline]
pub fn inv(self) -> Self {
<Self as ModInt>::inv(self)
}
#[inline]
pub fn raw(val: u32) -> Self {
Self {
val,
phantom: PhantomData,
}
}
#[inline]
pub fn pow<U: Int>(self, x: U) -> Self {
<Self as ModInt>::pow(self, x)
}
#[inline]
pub fn zero() -> Self {
<Self as Int>::zero()
}
#[inline]
pub fn one() -> Self {
<Self as Int>::one()
}
}
impl<T> std::fmt::Display for $name<T> {
fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
write!(f, "{}", self.val)
}
}
impl_from_for_modint!(
$name, $guard, u8, u16, u32, u64, u128, i8, i16, i32, i64, i128, usize, isize
);
impl_assign!(
$name, $guard, Add, AddAssign, add_assign, add, Sub, SubAssign, sub_assign, sub,
Mul, MulAssign, mul_assign, mul, Div, DivAssign, div_assign, div, Rem, RemAssign,
rem_assign, rem
);
impl<T: $guard> Int for $name<T> {
impl_integer_functions!(|s: &Self| s.val, |x| Self::new(x), |s: &Self, n: i64| s
.pow(n));
}
};
}
impl_modint_structs!(StaticModInt, Modulus);
impl_modint_structs!(DynamicModInt, DynamicModulus);
#[macro_export]
macro_rules! modint {
() => {$crate::modint!(1000000007, true);};
($m:literal) => {$crate::modint!($m, true);};
($m:literal, $is_prime:literal) => {
$crate::modint!($m, ModValue, $is_prime);
#[allow(dead_code)]
type Z = $crate::modulo::StaticModInt<ModValue>;
};
($name:ident) => {
$crate::modint!($name, $name, true);
};
($value:expr, $name:ident, $is_prime:literal) => {
#[derive(Debug, PartialEq, Eq, Copy, Clone, Hash, PartialOrd, Ord)]
pub enum $name {}
impl $crate::modulo::Modulus for $name {
const M: u32 = $value as _;
const HINT_M_IS_PRIME: bool = $is_prime;
fn butterfly_cache() -> &'static ::std::thread::LocalKey<::std::cell::RefCell<::std::option::Option<$crate::modulo::ButterflyCache<Self>>>> {
thread_local! {
static BUTTERFLY_CACHE: ::std::cell::RefCell<::std::option::Option<$crate::modulo::ButterflyCache<$name>>> = ::std::default::Default::default();
}
&BUTTERFLY_CACHE
}
}
};
}
pub type D = DynamicModInt<DefaultId>;
pub trait ModInt: Int {
fn new<U: Int>(val: U) -> Self {
let x = val.to_i128();
Self::raw(x.rem_euclid(Self::modulus() as i128) as _)
}
fn inv(self) -> Self {
if Self::mod_is_prime() {
Self::pow(self, Self::modulus() - 2)
} else {
let (g, x) = inv_gcd(Self::val(self) as _, Self::modulus() as _);
if g != 1 {
panic!("the multiplicative inverse does not exist");
} else {
Self::new(x)
}
}
}
fn raw(val: u32) -> Self;
fn val(self) -> u32;
fn modulus() -> u32;
fn mod_is_prime() -> bool;
fn add(self, other: Self) -> Self {
let mut ret = self.val() + other.val();
if ret >= Self::modulus() {
ret -= Self::modulus();
}
Self::raw(ret)
}
fn sub(self, other: Self) -> Self {
let mut ret = self.val().wrapping_sub(other.val());
if ret >= Self::modulus() {
ret = ret.wrapping_add(Self::modulus());
}
Self::raw(ret)
}
fn mul(self, other: Self) -> Self {
Self::raw(
(u64::from(self.val()) * u64::from(other.val()) % u64::from(Self::modulus())) as _,
)
}
fn div(self, other: Self) -> Self {
self * other.inv()
}
fn rem(self, other: Self) -> Self {
Self::raw(self.val() % other.val())
}
fn pow<U: Int>(self, x: U) -> Self {
let mut n = x.to_i64();
let mut a = self;
let mut res = Self::raw(1);
while n > 0 {
if n & 1 == 1 {
res *= a;
}
a = a * a;
n >>= 1;
}
res
}
}
impl<M: Modulus> ModInt for StaticModInt<M> {
fn raw(val: u32) -> Self {
Self::raw(val)
}
fn val(self) -> u32 {
self.val
}
fn modulus() -> u32 {
M::M
}
fn mod_is_prime() -> bool {
M::HINT_M_IS_PRIME
}
}
impl<M: DynamicModulus> ModInt for DynamicModInt<M> {
fn raw(val: u32) -> Self {
Self::raw(val)
}
fn val(self) -> u32 {
self.val
}
fn modulus() -> u32 {
M::umod()
}
fn mod_is_prime() -> bool {
false
}
}
pub struct ButterflyCache<T> {
pub sum_e: Vec<StaticModInt<T>>,
pub sum_ie: Vec<StaticModInt<T>>,
}
}
pub mod arraylist {
use std::ops::*;
use std::slice::Iter;
use std::fmt::Formatter;
use std::iter::FromIterator;
#[derive(Clone, PartialEq, Eq, PartialOrd, Ord)]
pub struct List<T> {
pub data: Vec<T>,
}
impl<T> List<T> {
#[inline]
pub fn new() -> List<T> {
List { data: vec![] }
}
#[inline]
pub fn init(init: T, n: isize) -> List<T>
where
T: Clone,
{
List {
data: vec![init; n as usize],
}
}
#[inline]
pub fn pop(&mut self) -> Option<T> {
self.data.pop()
}
#[inline]
pub fn reverse(&mut self) {
self.data.reverse();
}
#[inline]
pub fn append(&mut self, other: &lst<T>)
where
T: Clone,
{
self.data.append(&mut other.to_vec());
}
#[inline]
pub fn resize(&mut self, new_len: isize, value: T)
where
T: Clone,
{
self.data.resize(new_len as usize, value);
}
}
macro_rules! impl_idx {
($($tpe:ty, $t:ident [$($output:tt)+], $slf:ident, $index:ident, $f:expr),*) => {
$(impl<$t> Index<$tpe> for List<$t> {
type Output = $($output)+;
#[inline]
fn index(&$slf, $index: $tpe) -> &Self::Output {$f}
})*
$(impl<$t> Index<$tpe> for lst<$t> {
type Output = $($output)+;
#[inline]
fn index(&$slf, $index: $tpe) -> &Self::Output {$f}
})*
}
}
macro_rules! impl_idx_mut {
($($tpe:ty, $slf:ident, $index:ident, $f:expr),*) => {
$(impl<T> IndexMut<$tpe> for List<T> {
#[inline]
fn index_mut(&mut $slf, $index: $tpe) -> &mut Self::Output {$f}
})*
$(impl<T> IndexMut<$tpe> for lst<T> {
#[inline]
fn index_mut(&mut $slf, $index: $tpe) -> &mut Self::Output {$f}
})*
};
}
macro_rules! impl_idx_slice {
($($tpe:ty),*) => {
impl_idx!($($tpe, T [lst<T>], self, i, self.as_slice(i)),*);
impl_idx_mut!($($tpe, self, i, self.as_slice_mut(i)),*);
};
}
impl_idx! {
isize, T [T], self, i, self.at(i),
char, T [T], self, i, self.at(i as isize - 'a' as isize)
}
impl_idx_slice! {
Range<isize>, RangeTo<isize>, RangeFrom<isize>, RangeFull, RangeInclusive<isize>, RangeToInclusive<isize>
}
impl_idx_mut! {
isize, self, i, self.at_mut(i),
char, self, i, self.at_mut(i as isize - 'a' as isize)
}
impl<T> FromIterator<T> for List<T> {
#[inline]
fn from_iter<U: IntoIterator<Item = T>>(iter: U) -> Self {
List {
data: iter.into_iter().collect(),
}
}
}
impl<T> IntoIterator for List<T> {
type Item = T;
type IntoIter = std::vec::IntoIter<T>;
#[inline]
fn into_iter(self) -> std::vec::IntoIter<T> {
self.data.into_iter()
}
}
macro_rules! impl_traits {
($($tpe:tt),*) => {
$(
impl<T: std::fmt::Display> std::fmt::Display for $tpe<T> {
fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
write!(f, "{}", self.iter().map(|x| format!("{}", x)).collect::<Vec<_>>().join(" "))
}
}
impl<T: std::fmt::Debug> std::fmt::Debug for $tpe<T> {
fn fmt(&self, f: &mut Formatter<'_>) -> std::fmt::Result {
self.data.fmt(f)
}
}
impl<'a, T> IntoIterator for &'a $tpe<T> {
type Item = &'a T;
type IntoIter = Iter<'a, T>;
#[inline]
fn into_iter(self) -> Iter<'a, T> {
self.iter()
}
}
)*
};
}
impl_traits!(List, lst);
impl<T> From<Vec<T>> for List<T> {
#[inline]
fn from(vec: Vec<T>) -> Self {
List { data: vec }
}
}
impl<T: Clone> From<&[T]> for List<T> {
#[inline]
fn from(slice: &[T]) -> Self {
slice.iter().cloned().collect()
}
}
impl<T> Deref for List<T> {
type Target = lst<T>;
#[inline]
fn deref(&self) -> &lst<T> {
lst::new(&self.data)
}
}
impl<T> DerefMut for List<T> {
#[inline]
fn deref_mut(&mut self) -> &mut lst<T> {
lst::new_mut(&mut self.data)
}
}
#[macro_export]
macro_rules! list {
() => { $crate::arraylist::List::new() };
($($v:expr),+ $(,)?) => { $crate::arraylist::List::from([$($v),+].to_vec()) };
($v:expr; $a:expr) => { $crate::arraylist::List::init($v, $a)};
($v:expr; $a:expr; $($rest:expr);+) => { $crate::arraylist::List::init(list!($v; $($rest);+), $a) };
}
#[allow(non_camel_case_types)]
#[derive(PartialEq, Eq, PartialOrd, Ord)]
#[repr(transparent)]
pub struct lst<T> {
data: [T],
}
impl<T> lst<T> {
#[inline]
pub fn new(slice: &[T]) -> &Self {
unsafe { &*(slice as *const [T] as *const Self) }
}
#[inline]
pub fn new_mut(slice: &mut [T]) -> &mut Self {
unsafe { &mut *(slice as *mut [T] as *mut Self) }
}
#[inline]
pub fn lens(&self) -> isize {
self.data.len() as isize
}
#[inline]
pub fn list(&self) -> List<T>
where
T: Clone,
{
self.cloned().collect()
}
#[inline]
fn at(&self, index: isize) -> &T {
if cfg!(debug_assertions) {
self.data.index(index as usize)
} else {
unsafe { self.data.get_unchecked(index as usize) }
}
}
#[inline]
fn at_mut(&mut self, index: isize) -> &mut T {
if cfg!(debug_assertions) {
self.data.index_mut(index as usize)
} else {
unsafe { self.data.get_unchecked_mut(index as usize) }
}
}
#[inline]
pub fn as_slice(&self, range: impl RangeBounds<isize>) -> &lst<T> {
if cfg!(debug_assertions) {
lst::new(self.data.index(self.rgm(range)))
} else {
unsafe { lst::new(self.data.get_unchecked(self.rgm(range))) }
}
}
#[inline]
pub fn as_slice_mut(&mut self, range: impl RangeBounds<isize>) -> &mut lst<T> {
if cfg!(debug_assertions) {
lst::new_mut(self.data.index_mut(self.rgm(range)))
} else {
let r = self.rgm(range);
unsafe { lst::new_mut(self.data.get_unchecked_mut(r)) }
}
}
#[inline]
pub fn cloned(&self) -> std::iter::Cloned<Iter<T>>
where
T: Clone,
{
self.iter().cloned()
}
#[inline]
pub fn map<B, F>(&self, f: F) -> List<B>
where
T: Clone,
F: FnMut(T) -> B,
{
self.cloned().map(f).collect()
}
#[inline]
fn rgm(&self, r: impl RangeBounds<isize>) -> Range<usize> {
(match r.start_bound() {
Bound::Included(x) => *x as usize,
Bound::Excluded(x) => *x as usize + 1,
_ => 0,
})
.max(0)..(match r.end_bound() {
Bound::Included(x) => *x as usize + 1,
Bound::Excluded(x) => *x as usize,
_ => self.len(),
})
.min(self.len())
}
}
impl lst<isize> {}
impl<T> Deref for lst<T> {
type Target = [T];
#[inline]
fn deref(&self) -> &[T] {
&self.data
}
}
impl<T> DerefMut for lst<T> {
#[inline]
fn deref_mut(&mut self) -> &mut [T] {
&mut self.data
}
}
impl<'a, T> From<&'a [T]> for &'a lst<T> {
#[inline]
fn from(slice: &'a [T]) -> Self {
lst::new(slice)
}
}
}
pub mod nums {
use std::mem::swap;
pub fn ceil_pow2(n: u32) -> u32 {
32 - n.saturating_sub(1).leading_zeros()
}
pub fn inv_gcd(a: i64, b: i64) -> (i64, i64) {
let a = a.rem_euclid(b);
if a == 0 {
return (b, 0);
}
let mut s = b;
let mut t = a;
let mut m0 = 0;
let mut m1 = 1;
while t != 0 {
let u = s / t;
s -= t * u;
m0 -= m1 * u;
swap(&mut s, &mut t);
swap(&mut m0, &mut m1);
}
if m0 < 0 {
m0 += b / s;
}
(s, m0)
}
pub fn primitive_root(m: i32) -> i32 {
match m {
2 => return 1,
167_772_161 => return 3,
469_762_049 => return 3,
754_974_721 => return 11,
998_244_353 => return 3,
_ => {}
}
let mut divs = [0; 20];
divs[0] = 2;
let mut cnt = 1;
let mut x = (m - 1) / 2;
while x % 2 == 0 {
x /= 2;
}
for i in (3..std::i32::MAX).step_by(2) {
if i as i64 * i as i64 > x as i64 {
break;
}
if x % i == 0 {
divs[cnt] = i;
cnt += 1;
while x % i == 0 {
x /= i;
}
}
}
if x > 1 {
divs[cnt] = x;
cnt += 1;
}
let mut g = 2;
loop {
if (0..cnt).all(|i| pow_mod(g as _, ((m - 1) / divs[i]) as _, m as _) != 1) {
break g as i32;
}
g += 1;
}
}
pub fn pow_mod(mut a: i64, mut n: i64, m: i64) -> i64 {
let mut res = 1;
a %= m;
while n > 0 {
if n & 1 == 1 {
res *= a;
res %= m;
}
a = (a * a) % m;
n >>= 1;
}
res
}
}
pub mod scanner {
use std::io::{stdin, BufReader, Bytes, Read, Stdin};
use std::str::FromStr;
pub struct Scanner {
buf: Bytes<BufReader<Stdin>>,
}
impl Scanner {
pub fn new() -> Scanner {
Scanner {
buf: BufReader::new(stdin()).bytes(),
}
}
#[inline]
fn token<T: std::iter::FromIterator<char>>(&mut self) -> T {
self.buf
.by_ref()
.map(|c| c.unwrap() as char)
.skip_while(|c| c.is_whitespace())
.take_while(|c| !c.is_whitespace())
.collect()
}
#[inline]
pub fn read<T: FromStr>(&mut self) -> T {
self.string().parse().ok().unwrap()
}
#[inline]
pub fn string(&mut self) -> String {
self.token()
}
#[inline]
pub fn long(&mut self) -> i64 {
self.read()
}
}
}