#904204 (Rust) No.2413 Multiple of 99

提出ソース
結果

問題	No.2413 Multiple of 99
ユーザー	koba-e964
提出日時	2023-08-15 22:51:46
言語	Rust (1.77.0 + proconio)
結果	AC
実行時間	5,111 ms / 8,000 ms
コード長	16,704 bytes
コンパイル時間	21,871 ms
コンパイル使用メモリ	378,512 KB
実行使用メモリ	92,044 KB
最終ジャッジ日時	2024-11-24 08:03:01
合計ジャッジ時間	143,303 ms
ジャッジサーバーID （参考情報）	judge5 / judge4
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	5,051 ms 92,040 KB
testcase_01	AC	4,917 ms 92,036 KB
testcase_02	AC	5,020 ms 92,040 KB
testcase_03	AC	4,968 ms 92,024 KB
testcase_04	AC	5,087 ms 92,036 KB
testcase_05	AC	5,105 ms 91,912 KB
testcase_06	AC	5,093 ms 91,912 KB
testcase_07	AC	5,090 ms 92,040 KB
testcase_08	AC	5,111 ms 92,016 KB
testcase_09	AC	5,010 ms 92,040 KB
testcase_10	AC	5,083 ms 92,040 KB
testcase_11	AC	5,096 ms 92,036 KB
testcase_12	AC	5,101 ms 92,040 KB
testcase_13	AC	5,093 ms 92,036 KB
testcase_14	AC	5,093 ms 92,040 KB
testcase_15	AC	5,056 ms 92,040 KB
testcase_16	AC	5,077 ms 92,036 KB
testcase_17	AC	5,049 ms 92,040 KB
testcase_18	AC	5,042 ms 92,040 KB
testcase_19	AC	5,041 ms 92,040 KB
testcase_20	AC	5,076 ms 91,916 KB
testcase_21	AC	5,060 ms 92,036 KB
testcase_22	AC	4,996 ms 92,044 KB
testcase_23	AC	5,043 ms 92,036 KB
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ソースコード

raw source code
#[allow(unused_imports)]
use std::cmp::*;
#[allow(unused_imports)]
use std::collections::*;
use std::io::Read;

#[allow(dead_code)]
fn getline() -> String {
    let mut ret = String::new();
    std::io::stdin().read_line(&mut ret).ok().unwrap();
    ret
}

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;
        }
    }
}

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 = 998_244_353;
define_mod!(P, MOD);
type MInt = mod_int::ModInt<P>;

// 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;
            }
        }
    }
}

// Computes f^{-1} mod x^{f.len()}.
// Reference: https://codeforces.com/blog/entry/56422
// Complexity: O(n log n)
// Verified by: https://judge.yosupo.jp/submission/3219
// Depends on: MInt.rs, fft.rs
fn fps_inv<P: mod_int::Mod + PartialEq>(
    f: &[mod_int::ModInt<P>],
    gen: mod_int::ModInt<P>
) -> Vec<mod_int::ModInt<P>> {
    let n = f.len();
    assert!(n.is_power_of_two());
    assert_eq!(f[0], 1.into());
    let mut sz = 1;
    let mut r = vec![mod_int::ModInt::new(0); n];
    let mut tmp_f = vec![mod_int::ModInt::new(0); n];
    let mut tmp_r = vec![mod_int::ModInt::new(0); n];
    r[0] = 1.into();
    // Adopts the technique used in https://judge.yosupo.jp/submission/3153
    while sz < n {
        let zeta = gen.pow((P::m() - 1) / sz as i64 / 2);
        tmp_f[..2 * sz].copy_from_slice(&f[..2 * sz]);
        tmp_r[..2 * sz].copy_from_slice(&r[..2 * sz]);
        fft::fft(&mut tmp_r[..2 * sz], zeta, 1.into());
        fft::fft(&mut tmp_f[..2 * sz], zeta, 1.into());
        let fac = mod_int::ModInt::new(2 * sz as i64).inv().pow(2);
        for i in 0..2 * sz {
            tmp_f[i] *= tmp_r[i];
        }
        fft::inv_fft(&mut tmp_f[..2 * sz], zeta.inv(), 1.into());
        for v in &mut tmp_f[..sz] {
            *v = 0.into();
        }
        fft::fft(&mut tmp_f[..2 * sz], zeta, 1.into());
        for i in 0..2 * sz {
            tmp_f[i] = -tmp_f[i] * tmp_r[i] * fac;
        }
        fft::inv_fft(&mut tmp_f[..2 * sz], zeta.inv(), 1.into());
        r[sz..2 * sz].copy_from_slice(&tmp_f[sz..2 * sz]);
        sz *= 2;
    }
    r
}

// 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)
}

// Computes exp(f) mod x^{f.len()}.
// Reference: https://arxiv.org/pdf/1301.5804.pdf
// Complexity: O(n log n)
// Depends on: MInt.rs, fact_init.rs, fft.rs
fn fps_exp<P: mod_int::Mod + PartialEq>(
    h: &[mod_int::ModInt<P>],
    gen: mod_int::ModInt<P>,
    fac: &[mod_int::ModInt<P>],
    invfac: &[mod_int::ModInt<P>],
) -> Vec<mod_int::ModInt<P>> {
    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_f = vec![mod_int::ModInt::new(0); n];
    let mut tmp_g = vec![mod_int::ModInt::new(0); n];
    let mut tmp = 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))
        // Complexity: 4 * fft(2 * m) + 2 * fft(m) + 2 * inv_fft(2 * m) + 3 * inv_fft(m)
        // ~= 8.5 * fft(2 * m)
        let zeta2m = gen.pow((P::m() - 1) / m as i64 / 2);
        let zeta = zeta2m * zeta2m;
        // 2.a': g = 2g - fg^2 mod x^m
        let factor2m = mod_int::ModInt::new(m as i64 * 2).inv();
        let factor = factor2m * 2;
        let factor2 = factor * factor;
        // Here we only need FFT(f[..m]), but we use it later at 2.c'
        tmp_f[..2 * m].copy_from_slice(&f[..2 * m]);
        fft::fft(&mut tmp_f[..2 * m], zeta2m, 1.into());
        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());
            for i in 0..m {
                tmp[i] = tmp_f[i] * tmp_g[i];
            }
            fft::inv_fft(&mut tmp[..m], zeta.inv(), 1.into());
            for v in &mut tmp[..m / 2] {
                *v = 0.into();
            }
            fft::fft(&mut tmp[..m], zeta, 1.into());
            for i in 0..m {
                tmp[i] = -tmp[i] * tmp_g[i] * factor2;
            }
            fft::inv_fft(&mut tmp[..m], zeta.inv(), 1.into());
            g[m / 2..m].copy_from_slice(&tmp[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)
        fft::fft(&mut tmp[..m], zeta, 1.into());
        // FFT(f[..2m])[..m] == FFT(f[..m])
        // Note that the result of FFT is bit-reversed.
        for i in 0..m {
            tmp[i] *= tmp_f[i] * factor;
        }
        fft::inv_fft(&mut tmp[..m], zeta.inv(), 1.into());
        // 2.d' s = x(f' - r) mod (x^m - 1)
        for i in (0..m - 1).rev() {
            tmp.swap(i, i + 1);
        }
        for i in 0..m {
            tmp[i] = f[i] * i as i64 - tmp[i];
        }
        // 2.e': t = gs mod x^m
        tmp_g[..2 * m].copy_from_slice(&g[..2 * m]);
        fft::fft(&mut tmp_g[..2 * m], zeta2m, 1.into());
        fft::fft(&mut tmp[..2 * m], zeta2m, 1.into());
        for i in 0..2 * m {
            tmp[i] *= tmp_g[i] * factor2m;
        }
        fft::inv_fft(&mut tmp[..2 * m], zeta2m.inv(), 1.into());
        // 2.f': u = (h mod x^2m - \int tx^(m-1)) / x^m
        for i in 0..m {
            tmp[i] = h[i + m] - tmp[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
        fft::fft(&mut tmp[..2 * m], zeta2m, 1.into());
        for i in 0..2 * m {
            tmp[i] *= tmp_f[i] * factor2m;
        }
        fft::inv_fft(&mut tmp[..2 * m], zeta2m.inv(), 1.into());
        // 2.h': f += vx^m
        f[m..2 * m].copy_from_slice(&tmp[..m]);
        // 2.i': m *= 2
        m *= 2;
    }
    f
}

// Computes ln f mod x^{f.len()}.
// Reference: https://codeforces.com/blog/entry/56422
// Complexity: O(n log n)
// Verified by: https://judge.yosupo.jp/submission/53708
// Depends on: MInt.rs, fact_init.rs, fft.rs, fps/fps_inv.rs
fn fps_ln<P: mod_int::Mod + PartialEq>(
    f: &[mod_int::ModInt<P>],
    gen: mod_int::ModInt<P>,
    fac: &[mod_int::ModInt<P>],
    invfac: &[mod_int::ModInt<P>],
) -> Vec<mod_int::ModInt<P>> {
    let n = f.len();
    assert!(n.is_power_of_two());
    assert_eq!(f[0], 1.into());
    let mut inv = fps_inv(&f, gen);
    let mut der = vec![mod_int::ModInt::new(0); 2 * n];
    for i in 1..n {
        der[i - 1] = f[i] * i as i64;
    }
    inv.resize(2 * n, 0.into());
    let zeta = gen.pow((P::m() - 1) / n as i64 / 2);
    fft::fft(&mut der, zeta, 1.into());
    fft::fft(&mut inv, zeta, 1.into());
    let invlen = mod_int::ModInt::new(2 * n as i64).inv();
    for i in 0..2 * n {
        der[i] *= inv[i] * invlen;
    }
    fft::inv_fft(&mut der, zeta.inv(), 1.into());
    // integral of f'/f
    let mut ans = vec![mod_int::ModInt::new(0); n];
    for i in 1..n {
        ans[i] = der[i - 1] * invfac[i] * fac[i - 1];
    }
    ans
}

// https://yukicoder.me/problems/no/2413 (4)
// A = 1 + x + ... + x^9, B = 1 + x^10 + ... + x^90 とする。
// 問題の答えは \sum_{0 <= i,j <= 9N, 99|i+10j}(i+j)^K[x^i]A^{ceil(N/2)}[x^j]A^{floor(N/2)} である。
// -> 解説を見た。A^? を計算してから、次数 mod 11 で見て畳み込みをする。
// 計算量は O(11 K log K) where K = 9N である。
// The author read the editorial before implementing this.
fn main() {
    let n: usize = get();
    let k: usize = get();
    const W: usize = 1 << 20;
    let (fac, invfac) = fact_init(W + 1);
    let mut f = vec![MInt::new(0); W];
    for i in 0..10 {
        f[i] += 1;
    }
    let mut g1 = fps_ln(&f, 3.into(), &fac, &invfac);
    let mut g2 = g1.clone();
    for i in 0..W {
        g1[i] *= (n as i64 + 1) / 2;
        g2[i] *= n as i64 / 2;
    }
    let h1 = fps_exp(&g1, 3.into(), &fac, &invfac);
    let h2 = fps_exp(&g2, 3.into(), &fac, &invfac);
    let factor = MInt::new(W as i64).inv();
    let zeta = MInt::new(3).pow((MOD - 1) / W as i64);
    let mut ans = MInt::new(0);
    for rem in 0..11 {
        let mut c1 = vec![MInt::new(0); W];
        let mut c2 = vec![MInt::new(0); W];
        for i in 0..W {
            if i % 11 == rem {
                c1[i] = h1[i];
                c2[i] = h2[i];
            }
        }
        fft::fft(&mut c1, zeta, 1.into());
        fft::fft(&mut c2, zeta, 1.into());
        for i in 0..W {
            c1[i] *= c2[i] * factor;
        }
        fft::inv_fft(&mut c1, zeta.inv(), 1.into());
        for i in 0..W {
            if i % 9 == 0 {
                ans += MInt::new(i as i64).pow(k as i64) * c1[i];
            }
        }
    }
    println!("{}", ans);
}