#715973 (Rust) No.1080 Strange Squared Score Sum

提出ソース
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 (1.77.0 + 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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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	956 ms 8,640 KB
testcase_03	AC	1,916 ms 15,292 KB
testcase_04	AC	462 ms 5,440 KB
testcase_05	AC	480 ms 5,312 KB
testcase_06	AC	107 ms 5,248 KB
testcase_07	AC	228 ms 5,248 KB
testcase_08	AC	967 ms 8,640 KB
testcase_09	AC	932 ms 8,640 KB
testcase_10	AC	110 ms 5,248 KB
testcase_11	AC	1,951 ms 15,292 KB
testcase_12	AC	942 ms 8,516 KB
testcase_13	AC	1,933 ms 15,292 KB
testcase_14	AC	953 ms 8,640 KB
testcase_15	AC	1 ms 5,248 KB
testcase_16	AC	1,979 ms 15,292 KB
testcase_17	AC	991 ms 8,640 KB
testcase_18	AC	964 ms 8,768 KB
testcase_19	AC	985 ms 8,636 KB
testcase_20	AC	1,942 ms 15,292 KB
testcase_21	AC	1,920 ms 15,420 KB
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ソースコード

raw source code
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 + …)) 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-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);
    }
}