結果
問題 | No.1080 Strange Squared Score Sum |
ユーザー |
|
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 |
(要ログイン)
ファイルパターン | 結果 |
---|---|
sample | AC * 2 |
other | AC * 20 |
ソースコード
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/21774342mod 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 >= 0pub 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_intmacro_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.rsfn 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 = 1let 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 FFTlet 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/2015/05/07/014908fn 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 destroyedfn 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] < mopub 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/3153while 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^mif 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^marbmod_convolution_modint(&tmp[..2 * m], &g[..2 * m], &mut tmp2[..2 * m]);// 2.f': u = (h mod x^2m - \int tx^(m-1)) / x^mfor 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^marbmod_convolution_modint(&tmp[..2 * m], &f[..2 * m], &mut tmp2[..2 * m]);// 2.h': f += vx^mf[m..2 * m].copy_from_slice(&tmp2[..m]);// 2.i': m *= 2m *= 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 + …)) の実部 + 虚部が答え。mod (10^9 + 9) なので sqrt(-1) は存在するが、NTT friendly ではない。// Tags: fps, ntt-unfriendly-mod, mod-sqrtfn 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);}}