結果

問題 No.1080 Strange Squared Score Sum
ユーザー koba-e964
提出日時 2021-11-12 23:06:17
言語 Rust
(1.83.0 + proconio)
結果
AC  
実行時間 1,979 ms / 5,000 ms
コード長 16,337 bytes
コンパイル時間 13,988 ms
コンパイル使用メモリ 378,760 KB
実行使用メモリ 15,420 KB
最終ジャッジ日時 2024-11-25 21:13:54
合計ジャッジ時間 35,511 ms
ジャッジサーバーID
(参考情報)
judge3 / judge1
このコードへのチャレンジ
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ファイルパターン 結果
sample AC * 2
other AC * 20
権限があれば一括ダウンロードができます

ソースコード

diff #
プレゼンテーションモードにする

use std::io::Read;
fn get_word() -> String {
let stdin = std::io::stdin();
let mut stdin=stdin.lock();
let mut u8b: [u8; 1] = [0];
loop {
let mut buf: Vec<u8> = Vec::with_capacity(16);
loop {
let res = stdin.read(&mut u8b);
if res.unwrap_or(0) == 0 || u8b[0] <= b' ' {
break;
} else {
buf.push(u8b[0]);
}
}
if buf.len() >= 1 {
let ret = String::from_utf8(buf).unwrap();
return ret;
}
}
}
#[allow(dead_code)]
fn get<T: std::str::FromStr>() -> T { get_word().parse().ok().unwrap() }
/// Verified by https://atcoder.jp/contests/abc198/submissions/21774342
mod mod_int {
use std::ops::*;
pub trait Mod: Copy { fn m() -> i64; }
#[derive(Copy, Clone, Hash, PartialEq, Eq, PartialOrd, Ord)]
pub struct ModInt<M> { pub x: i64, phantom: ::std::marker::PhantomData<M> }
impl<M: Mod> ModInt<M> {
// x >= 0
pub fn new(x: i64) -> Self { ModInt::new_internal(x % M::m()) }
fn new_internal(x: i64) -> Self {
ModInt { x: x, phantom: ::std::marker::PhantomData }
}
pub fn pow(self, mut e: i64) -> Self {
debug_assert!(e >= 0);
let mut sum = ModInt::new_internal(1);
let mut cur = self;
while e > 0 {
if e % 2 != 0 { sum *= cur; }
cur *= cur;
e /= 2;
}
sum
}
#[allow(dead_code)]
pub fn inv(self) -> Self { self.pow(M::m() - 2) }
}
impl<M: Mod> Default for ModInt<M> {
fn default() -> Self { Self::new_internal(0) }
}
impl<M: Mod, T: Into<ModInt<M>>> Add<T> for ModInt<M> {
type Output = Self;
fn add(self, other: T) -> Self {
let other = other.into();
let mut sum = self.x + other.x;
if sum >= M::m() { sum -= M::m(); }
ModInt::new_internal(sum)
}
}
impl<M: Mod, T: Into<ModInt<M>>> Sub<T> for ModInt<M> {
type Output = Self;
fn sub(self, other: T) -> Self {
let other = other.into();
let mut sum = self.x - other.x;
if sum < 0 { sum += M::m(); }
ModInt::new_internal(sum)
}
}
impl<M: Mod, T: Into<ModInt<M>>> Mul<T> for ModInt<M> {
type Output = Self;
fn mul(self, other: T) -> Self { ModInt::new(self.x * other.into().x % M::m()) }
}
impl<M: Mod, T: Into<ModInt<M>>> AddAssign<T> for ModInt<M> {
fn add_assign(&mut self, other: T) { *self = *self + other; }
}
impl<M: Mod, T: Into<ModInt<M>>> SubAssign<T> for ModInt<M> {
fn sub_assign(&mut self, other: T) { *self = *self - other; }
}
impl<M: Mod, T: Into<ModInt<M>>> MulAssign<T> for ModInt<M> {
fn mul_assign(&mut self, other: T) { *self = *self * other; }
}
impl<M: Mod> Neg for ModInt<M> {
type Output = Self;
fn neg(self) -> Self { ModInt::new(0) - self }
}
impl<M> ::std::fmt::Display for ModInt<M> {
fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
self.x.fmt(f)
}
}
impl<M: Mod> ::std::fmt::Debug for ModInt<M> {
fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
let (mut a, mut b, _) = red(self.x, M::m());
if b < 0 {
a = -a;
b = -b;
}
write!(f, "{}/{}", a, b)
}
}
impl<M: Mod> From<i64> for ModInt<M> {
fn from(x: i64) -> Self { Self::new(x) }
}
// Finds the simplest fraction x/y congruent to r mod p.
// The return value (x, y, z) satisfies x = y * r + z * p.
fn red(r: i64, p: i64) -> (i64, i64, i64) {
if r.abs() <= 10000 {
return (r, 1, 0);
}
let mut nxt_r = p % r;
let mut q = p / r;
if 2 * nxt_r >= r {
nxt_r -= r;
q += 1;
}
if 2 * nxt_r <= -r {
nxt_r += r;
q -= 1;
}
let (x, z, y) = red(nxt_r, r);
(x, y - q * z, z)
}
} // mod mod_int
macro_rules! define_mod {
($struct_name: ident, $modulo: expr) => {
#[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Hash)]
struct $struct_name {}
impl mod_int::Mod for $struct_name { fn m() -> i64 { $modulo } }
}
}
const MOD: i64 = 1_000_000_009;
define_mod!(P, MOD);
type MInt = mod_int::ModInt<P>;
// Depends on MInt.rs
fn fact_init(w: usize) -> (Vec<MInt>, Vec<MInt>) {
let mut fac = vec![MInt::new(1); w];
let mut invfac = vec![0.into(); w];
for i in 1..w {
fac[i] = fac[i - 1] * i as i64;
}
invfac[w - 1] = fac[w - 1].inv();
for i in (0..w - 1).rev() {
invfac[i] = invfac[i + 1] * (i as i64 + 1);
}
(fac, invfac)
}
// FFT (in-place, verified as NTT only)
// R: Ring + Copy
// Verified by: https://judge.yosupo.jp/submission/53831
// Adopts the technique used in https://judge.yosupo.jp/submission/3153.
mod fft {
use std::ops::*;
// n should be a power of 2. zeta is a primitive n-th root of unity.
// one is unity
// Note that the result is bit-reversed.
pub fn fft<R>(f: &mut [R], zeta: R, one: R)
where R: Copy +
Add<Output = R> +
Sub<Output = R> +
Mul<Output = R> {
let n = f.len();
assert!(n.is_power_of_two());
let mut m = n;
let mut base = zeta;
unsafe {
while m > 2 {
m >>= 1;
let mut r = 0;
while r < n {
let mut w = one;
for s in r..r + m {
let &u = f.get_unchecked(s);
let d = *f.get_unchecked(s + m);
*f.get_unchecked_mut(s) = u + d;
*f.get_unchecked_mut(s + m) = w * (u - d);
w = w * base;
}
r += 2 * m;
}
base = base * base;
}
if m > 1 {
// m = 1
let mut r = 0;
while r < n {
let &u = f.get_unchecked(r);
let d = *f.get_unchecked(r + 1);
*f.get_unchecked_mut(r) = u + d;
*f.get_unchecked_mut(r + 1) = u - d;
r += 2;
}
}
}
}
pub fn inv_fft<R>(f: &mut [R], zeta_inv: R, one: R)
where R: Copy +
Add<Output = R> +
Sub<Output = R> +
Mul<Output = R> {
let n = f.len();
assert!(n.is_power_of_two());
let zeta = zeta_inv; // inverse FFT
let mut zetapow = Vec::with_capacity(20);
{
let mut m = 1;
let mut cur = zeta;
while m < n {
zetapow.push(cur);
cur = cur * cur;
m *= 2;
}
}
let mut m = 1;
unsafe {
if m < n {
zetapow.pop();
let mut r = 0;
while r < n {
let &u = f.get_unchecked(r);
let d = *f.get_unchecked(r + 1);
*f.get_unchecked_mut(r) = u + d;
*f.get_unchecked_mut(r + 1) = u - d;
r += 2;
}
m = 2;
}
while m < n {
let base = zetapow.pop().unwrap();
let mut r = 0;
while r < n {
let mut w = one;
for s in r..r + m {
let &u = f.get_unchecked(s);
let d = *f.get_unchecked(s + m) * w;
*f.get_unchecked_mut(s) = u + d;
*f.get_unchecked_mut(s + m) = u - d;
w = w * base;
}
r += 2 * m;
}
m *= 2;
}
}
}
}
mod arbitrary_mod {
use crate::mod_int::{self, ModInt};
use crate::fft;
const MOD1: i64 = 1012924417;
const MOD2: i64 = 1224736769;
const MOD3: i64 = 1007681537;
const G1: i64 = 5;
const G2: i64 = 3;
const G3: i64 = 3;
define_mod!(P1, MOD1);
define_mod!(P2, MOD2);
define_mod!(P3, MOD3);
fn zmod(mut a: i64, b: i64) -> i64 {
a %= b;
if a < 0 { a += b; }
a
}
fn ext_gcd(mut a: i64, mut b: i64) -> (i64, i64, i64) {
let mut x = 0;
let mut y = 1;
let mut u = 1;
let mut v = 0;
while a != 0 {
let q = b / a;
x -= q * u;
std::mem::swap(&mut x, &mut u);
y -= q * v;
std::mem::swap(&mut y, &mut v);
b -= q * a;
std::mem::swap(&mut b, &mut a);
}
(b, x, y)
}
fn invmod(a: i64, b: i64) -> i64 {
let x = ext_gcd(a, b).1;
zmod(x, b)
}
// This function is ported from http://math314.hateblo.jp/entry/2015/05/07/014908
fn garner(mut mr: Vec<(i64, i64)>, mo: i64) -> i64 {
mr.push((mo, 0));
let mut coffs = vec![1; mr.len()];
let mut constants = vec![0; mr.len()];
for i in 0..mr.len() - 1 {
let v = zmod(mr[i].1 - constants[i], mr[i].0) * invmod(coffs[i], mr[i].0) % mr[i].0;
assert!(v >= 0);
for j in i + 1..mr.len() {
constants[j] += coffs[j] * v % mr[j].0;
constants[j] %= mr[j].0;
coffs[j] = coffs[j] * mr[i].0 % mr[j].0;
}
}
constants[mr.len() - 1]
}
// f *= g, g is destroyed
fn convolution_friendly<P: mod_int::Mod>(a: &[i64], b: &[i64], gen: i64) -> Vec<i64> {
let d = a.len();
let mut f = vec![ModInt::<P>::new(0); d];
let mut g = vec![ModInt::<P>::new(0); d];
for i in 0..d {
f[i] = a[i].into();
g[i] = b[i].into();
}
let zeta = ModInt::new(gen).pow((P::m() - 1) / d as i64);
fft::fft(&mut f, zeta, ModInt::new(1));
fft::fft(&mut g, zeta, ModInt::new(1));
for i in 0..d {
f[i] *= g[i];
}
fft::inv_fft(&mut f, zeta.inv(), ModInt::new(1));
let inv = ModInt::new(d as i64).inv();
let mut ans = vec![0; d];
for i in 0..d {
ans[i] = (f[i] * inv).x;
}
ans
}
// Precondition: 0 <= a[i], b[i] < mo
pub fn arbmod_convolution(a: &[i64], b: &[i64], mo: i64, ret: &mut [i64]) {
use crate::mod_int::Mod;
let d = a.len();
assert!(d.is_power_of_two());
assert_eq!(d, b.len());
let x = convolution_friendly::<P1>(&a, &b, G1);
let y = convolution_friendly::<P2>(&a, &b, G2);
let z = convolution_friendly::<P3>(&a, &b, G3);
let mut mr = [(0, 0); 3];
for i in 0..d {
mr[0] = (P1::m(), x[i]);
mr[1] = (P2::m(), y[i]);
mr[2] = (P3::m(), z[i]);
ret[i] = garner(mr.to_vec(), mo);
}
}
pub fn arbmod_convolution_modint<P: mod_int::Mod>(
a: &[ModInt<P>], b: &[ModInt<P>], ret: &mut [ModInt<P>]) {
let mo = P::m();
unsafe {
arbmod_convolution(std::mem::transmute(a), std::mem::transmute(b), mo, std::mem::transmute(ret));
}
}
}
// Computes exp(f) mod x^{f.len()}.
// Reference: https://arxiv.org/pdf/1301.5804.pdf
// Complexity: O(n log n)
fn fps_exp_arb<P: mod_int::Mod + PartialEq>(
h: &[mod_int::ModInt<P>],
fac: &[mod_int::ModInt<P>],
invfac: &[mod_int::ModInt<P>],
) -> Vec<mod_int::ModInt<P>> {
use arbitrary_mod::*;
let n = h.len();
assert!(n.is_power_of_two());
assert_eq!(h[0], 0.into());
let mut m = 1;
let mut f = vec![mod_int::ModInt::new(0); n];
let mut g = vec![mod_int::ModInt::new(0); n];
let mut tmp = vec![mod_int::ModInt::new(0); n];
let mut tmp2 = vec![mod_int::ModInt::new(0); n];
f[0] = 1.into();
g[0] = 1.into();
// Adopts the technique used in https://judge.yosupo.jp/submission/3153
while m < n {
// upheld invariants: f = exp(h) (mod x^m)
// g = exp(-h) (mod x^(m/2))
// 2.a': g = 2g - fg^2 mod x^m
if m > 1 {
// The following can be dropped because the actual
// computation was done in the previous iteration.
// tmp_g[..m].copy_from_slice(&g[..m]);
// fft::fft(&mut tmp_g[..m], zeta, 1.into());
arbmod_convolution_modint(&f[..m], &g[..m], &mut tmp[..m]);
for v in &mut tmp[..m / 2] {
*v = 0.into();
}
for v in &mut tmp[m / 2..m] {
*v = -*v;
}
arbmod_convolution_modint(&tmp[..m], &g[..m], &mut tmp2[..m]);
g[m / 2..m].copy_from_slice(&tmp2[m / 2..m]);
}
// 2.b': q = h' mod x^(m-1)
for i in 0..m - 1 {
tmp[i] = h[i + 1] * (i + 1) as i64;
}
tmp[m - 1] = 0.into();
// 2.c': r = fq (mod x^m - 1)
arbmod_convolution_modint(&tmp[..m], &f[..m], &mut tmp2[..m]);
// 2.d' s = x(f' - r) mod (x^m - 1)
for i in (0..m - 1).rev() {
tmp2.swap(i, i + 1);
}
for i in 0..m {
tmp[i] = f[i] * i as i64 - tmp2[i];
}
// 2.e': t = gs mod x^m
arbmod_convolution_modint(&tmp[..2 * m], &g[..2 * m], &mut tmp2[..2 * m]);
// 2.f': u = (h mod x^2m - \int tx^(m-1)) / x^m
for i in 0..m {
tmp[i] = h[i + m] - tmp2[i] * fac[i + m - 1] * invfac[i + m];
}
for v in &mut tmp[m..2 * m] {
*v = 0.into();
}
// 2.g': v = fu mod x^m
arbmod_convolution_modint(&tmp[..2 * m], &f[..2 * m], &mut tmp2[..2 * m]);
// 2.h': f += vx^m
f[m..2 * m].copy_from_slice(&tmp2[..m]);
// 2.i': m *= 2
m *= 2;
}
f
}
fn powmod(x: i64, mut e: i64, m: i64) -> i64 {
let mut sum = 1;
let mut cur = x % m;
while e > 0 {
if e % 2 != 0 {
sum = sum * cur % m;
}
cur = cur * cur % m;
e /= 2;
}
sum
}
/**
* Calculates x s.t. x^2 = a (mod p)
* p is prime
* Verified by: CF #395 Div1-C
* (http://codeforces.com/contest/763/submission/24380573)
*/
fn modsqrt(mut a: i64, p: i64) -> Option<i64> {
a %= p;
if a == 0 {
return Some(0);
}
if p == 2 {
return Some(a);
}
if powmod(a, (p - 1) / 2, p) != 1 {
return None;
}
let mut b = 1;
while powmod(b, (p - 1) / 2, p) == 1 {
b += 1;
}
let mut e = 0;
let mut m = p - 1;
while m % 2 == 0 {
m /= 2;
e += 1;
}
let mut x = powmod(a, (m - 1) / 2, p);
let mut y = a * (x * x % p) % p;
x = x * a % p;
let mut z = powmod(b, m, p);
while y != 1 {
let mut j = 0;
let mut t = y;
while t != 1 {
j += 1;
t = t * t % p;
}
assert!(j < e);
z = powmod(z, 1 << (e - j - 1), p);
x = x * z % p;
z = z * z % p;
y = y * z % p;
e = j;
}
Some(x)
}
// https://yukicoder.me/problems/no/1080 (5)
// N![x^K]exp(sqrt(-1) * (4x + 9x^2 + 16x^3 + …)) + mod (10^9 + 9) sqrt(-1) NTT friendly
// Tags: fps, ntt-unfriendly-mod, mod-sqrt
fn main() {
let im = modsqrt(MOD - 1, MOD).unwrap();
eprintln!("{}", im);
let n: usize = get();
let mut p = 1;
while p <= n {
p *= 2;
}
let (fac, invfac) = fact_init(p + 1);
let mut f = vec![MInt::new(0); p];
for i in 1..p {
f[i] = MInt::new(im) * (i as i64 + 1) * (i as i64 + 1);
}
let pos = fps_exp_arb(&f, &fac, &invfac);
for i in 1..p {
f[i] = MInt::new(MOD - im) * (i as i64 + 1) * (i as i64 + 1);
}
let neg = fps_exp_arb(&f, &fac, &invfac);
let inv2 = MInt::new(2).inv();
for i in 1..n + 1 {
let re = pos[i] + neg[i];
let im = -(pos[i] - neg[i]) * MInt::new(im);
println!("{}", (re + im) * fac[n] * inv2);
}
}
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