#744022 (Rust) No.1866 Unfair Tournament

提出ソース
結果

問題	No.1866 Unfair Tournament
ユーザー	koba-e964
提出日時	2022-03-06 23:39:10
言語	Rust (1.77.0 + proconio)
結果	AC
実行時間	591 ms / 3,000 ms
コード長	11,683 bytes
コンパイル時間	16,219 ms
コンパイル使用メモリ	379,016 KB
実行使用メモリ	19,260 KB
最終ジャッジ日時	2024-07-21 10:09:27
合計ジャッジ時間	21,240 ms
ジャッジサーバーID （参考情報）	judge4 / judge2
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	1 ms 5,376 KB
testcase_03	AC	1 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	5 ms 5,376 KB
testcase_08	AC	1 ms 5,376 KB
testcase_09	AC	9 ms 5,376 KB
testcase_10	AC	3 ms 5,376 KB
testcase_11	AC	2 ms 5,376 KB
testcase_12	AC	588 ms 19,260 KB
testcase_13	AC	587 ms 19,260 KB
testcase_14	AC	589 ms 19,260 KB
testcase_15	AC	591 ms 19,260 KB
testcase_16	AC	591 ms 19,260 KB
testcase_17	AC	589 ms 19,260 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;
        }
    }
}

#[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 = 998_244_353;
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;
            }
        }
    }
}

// Depends on: fft.rs, MInt.rs
// Primitive root defaults to 3 (for 998244353); for other moduli change the value of it.
fn conv(a: Vec<MInt>, b: Vec<MInt>) -> Vec<MInt> {
    let n = a.len() - 1;
    let m = b.len() - 1;
    let mut p = 1;
    while p <= n + m { p *= 2; }
    let mut f = vec![MInt::new(0); p];
    let mut g = vec![MInt::new(0); p];
    for i in 0..n + 1 { f[i] = a[i]; }
    for i in 0..m + 1 { g[i] = b[i]; }
    let fac = MInt::new(p as i64).inv();
    let zeta = MInt::new(3).pow((MOD - 1) / p as i64);
    fft::fft(&mut f, zeta, 1.into());
    fft::fft(&mut g, zeta, 1.into());
    for i in 0..p { f[i] *= g[i] * fac; }
    fft::inv_fft(&mut f, zeta.inv(), 1.into());
    f[..n + m + 1].to_vec()
}

// https://yukicoder.me/problems/no/1866 (4)
// 選手 i にとって興味があるのは、自分より若い番号、老いた番号の選手と何回当たるかという情報だけである。
// p = A/B として、前者が x 回、後者が N - x 回であるような順列に対して、優勝確率は p^(N-x)(1-p)^x である。
// 欲しい情報は、2^k 人のグループの中で、自分より若い番号の人数が x であるときの自分より若い番号の優勝確率である。
// これを dp[k][x] と置くと、dp[k] は dp[k - 1] から畳み込みで求めることができる。
// 具体的には C(2^{k-1}, x) C(2^{k-1}, y) C(2^k, x+y)^{-1} *
// (dp[k-1][x] * dp[k-1][y] + p * dp[k-1][x] * (1 - dp[k-1][y]) + p * (1 - dp[k-1][x]) * dp[k-1][y]) -> dp[k][x+y] という遷移がある。
// これは畳み込みで計算できる。
fn solve() {
    let n: usize = get();
    let a: i64 = get();
    let b: i64 = get();
    let (fac, invfac) = fact_init(1 << n);
    let p = MInt::new(b).inv() * a;
    let mut dp = vec![vec![]; n];
    dp[0] = vec![MInt::new(0), MInt::new(1)];
    for i in 1..n {
        let mut t = dp[i - 1].clone();
        let mut mt = t.clone();
        for v in &mut mt {
            *v = -*v + 1;
        }
        for j in 0..t.len() {
            let tmp = fac[1 << (i - 1)] * invfac[j] * invfac[(1 << (i - 1)) - j];
            t[j] *= tmp;
            mt[j] *= tmp;
        }
        let mut u = conv(t.clone(), t.clone());
        let w = conv(t, mt);
        for i in 0..u.len() {
            u[i] += w[i] * p * 2;
        }
        dp[i] = u;
        for k in 0..(1 << i) + 1 {
            dp[i][k] *= invfac[1 << i] * fac[k] * fac[(1 << i) - k];
        }
        // eprintln!("{} => {:?}", i, dp[i]);
    }
    for j in 0..n {
        for i in 0..(1 << j) + 1 {
            dp[j][i] = (-p + 1) * dp[j][i] + p * (-dp[j][i] + 1);
            dp[j][i] *= fac[1 << j] * invfac[i] * invfac[(1 << j) - i];
        }
    }
    let mut ans = vec![MInt::new(1)];
    for j in 0..n {
        ans = conv(ans, dp[j].clone());
    }
    let mut tot = MInt::new(0);
    for i in 0..1 << n {
        println!("{}", ans[i] * invfac[(1 << n) - 1] * fac[i] * fac[(1 << n) - 1 - i]);
        tot += ans[i] * invfac[(1 << n) - 1] * fac[i] * fac[(1 << n) - 1 - i];
    }
    eprintln!("tot = {:?}", tot);
}

fn main() {
    // In order to avoid potential stack overflow, spawn a new thread.
    let stack_size = 104_857_600; // 100 MB
    let thd = std::thread::Builder::new().stack_size(stack_size);
    thd.spawn(|| solve()).unwrap().join().unwrap();
}