#938423 (Rust) No.2578 Jewelry Store

• 問題: No.2578 Jewelry Store
• ユーザー: koba-e964
• 提出日時: 2023-12-15 11:43:36
• 言語: Rust (1.77.0)
• 結果: AC
• 実行時間: 1,015 ms / 3,500 ms
• コード長: 10,629 bytes
• コンパイル時間: 1,653 ms
• コンパイル使用メモリ: 207,216 KB
• 実行使用メモリ: 9,948 KB
• 最終ジャッジ日時: 2023-12-15 11:43:48
• 合計ジャッジ時間: 10,724 ms
• ジャッジサーバーID （参考情報）: judge15 / judge11

入力	結果	実行時間; 実行使用メモリ
testcase_00	AC	1 ms; 6,676 KB
testcase_01	AC	0 ms; 6,676 KB
testcase_02	AC	9 ms; 6,676 KB
testcase_03	AC	6 ms; 6,676 KB
testcase_04	AC	23 ms; 6,676 KB
testcase_05	AC	4 ms; 6,676 KB
testcase_06	AC	9 ms; 6,676 KB
testcase_07	AC	23 ms; 6,676 KB
testcase_08	AC	6 ms; 6,676 KB
testcase_09	AC	23 ms; 6,676 KB
testcase_10	AC	44 ms; 6,676 KB
testcase_11	AC	4 ms; 6,676 KB
testcase_12	AC	5 ms; 6,676 KB
testcase_13	AC	5 ms; 6,676 KB
testcase_14	AC	4 ms; 6,676 KB
testcase_15	AC	4 ms; 6,676 KB
testcase_16	AC	45 ms; 6,676 KB
testcase_17	AC	23 ms; 6,676 KB
testcase_18	AC	4 ms; 6,676 KB
testcase_19	AC	13 ms; 6,676 KB
testcase_20	AC	23 ms; 6,676 KB
testcase_21	AC	23 ms; 6,676 KB
testcase_22	AC	8 ms; 6,676 KB
testcase_23	AC	12 ms; 6,676 KB
testcase_24	AC	6 ms; 6,676 KB
testcase_25	AC	8 ms; 6,676 KB
testcase_26	AC	7 ms; 6,676 KB
testcase_27	AC	142 ms; 6,676 KB
testcase_28	AC	198 ms; 6,676 KB
testcase_29	AC	197 ms; 6,676 KB
testcase_30	AC	246 ms; 6,676 KB
testcase_31	AC	141 ms; 6,676 KB
testcase_32	AC	100 ms; 6,676 KB
testcase_33	AC	117 ms; 6,676 KB
testcase_34	AC	157 ms; 6,676 KB
testcase_35	AC	39 ms; 6,676 KB
testcase_36	AC	132 ms; 7,300 KB
testcase_37	AC	128 ms; 6,676 KB
testcase_38	AC	137 ms; 6,676 KB
testcase_39	AC	139 ms; 6,676 KB
testcase_40	AC	129 ms; 6,676 KB
testcase_41	AC	129 ms; 6,676 KB
testcase_42	AC	127 ms; 6,676 KB
testcase_43	AC	136 ms; 6,676 KB
testcase_44	AC	123 ms; 6,676 KB
testcase_45	AC	136 ms; 6,676 KB
testcase_46	AC	137 ms; 6,676 KB
testcase_47	AC	444 ms; 6,676 KB
testcase_48	AC	174 ms; 9,948 KB
testcase_49	AC	1,015 ms; 6,676 KB
testcase_50	AC	1,012 ms; 6,676 KB
testcase_51	AC	124 ms; 6,676 KB
testcase_52	AC	219 ms; 6,676 KB
testcase_53	AC	56 ms; 6,676 KB

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;
        }
    }
}

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)]
        pub 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>;

// https://judge.yosupo.jp/submission/5155
mod pollard_rho {
    /// binary gcd
    pub fn gcd(mut x: i64, mut y: i64) -> i64 {
        if y == 0 { return x; }
        if x == 0 { return y; }
        let k = (x | y).trailing_zeros();
        y >>= k;
        x >>= x.trailing_zeros();
        while y != 0 {
            y >>= y.trailing_zeros();
            if x > y { let t = x; x = y; y = t; }
            y -= x;
        }
        x << k
    }

    fn add_mod(x: i64, y: i64, n: i64) -> i64 {
        let z = x + y;
        if z >= n { z - n } else { z }
    }

    fn mul_mod(x: i64, mut y: i64, n: i64) -> i64 {
        assert!(x >= 0);
        assert!(x < n);
        let mut sum = 0;
        let mut cur = x;
        while y > 0 {
            if (y & 1) == 1 { sum = add_mod(sum, cur, n); }
            cur = add_mod(cur, cur, n);
            y >>= 1;
        }
        sum
    }

    fn mod_pow(x: i64, mut e: i64, n: i64) -> i64 {
        let mut prod = if n == 1 { 0 } else { 1 };
        let mut cur = x % n;
        while e > 0 {
            if (e & 1) == 1 { prod = mul_mod(prod, cur, n); }
            e >>= 1;
            if e > 0 { cur = mul_mod(cur, cur, n); }
        }
        prod
    }

    pub fn is_prime(n: i64) -> bool {
        if n <= 1 { return false; }
        let small = [2, 3, 5, 7, 11, 13];
        if small.iter().any(|&u| u == n) { return true; }
        if small.iter().any(|&u| n % u == 0) { return false; }
        let mut d = n - 1;
        let e = d.trailing_zeros();
        d >>= e;
        // https://miller-rabin.appspot.com/
        let a = [2, 325, 9375, 28178, 450775, 9780504, 1795265022];
        a.iter().all(|&a| {
            if a % n == 0 { return true; }
            let mut x = mod_pow(a, d, n);
            if x == 1 { return true; }
            for _ in 0..e {
                if x == n - 1 {
                    return true;
                }
                x = mul_mod(x, x, n);
                if x == 1 { return false; }
            }
            x == 1
        })
    }

    fn pollard_rho(n: i64, c: &mut i64) -> i64 {
        // An improvement with Brent's cycle detection algorithm is performed.
        // https://maths-people.anu.edu.au/~brent/pub/pub051.html
        if n % 2 == 0 { return 2; }
        loop {
            let mut x: i64; // tortoise
            let mut y = 2; // hare
            let mut d = 1;
            let cc = *c;
            let f = |i| add_mod(mul_mod(i, i, n), cc, n);
            let mut r = 1;
            // We don't perform the gcd-once-in-a-while optimization
            // because the plain gcd-every-time algorithm appears to
            // outperform, at least on judge.yosupo.jp :)
            while d == 1 {
                x = y;
                for _ in 0..r {
                    y = f(y);
                    d = gcd((x - y).abs(), n);
                    if d != 1 { break; }
                }
                r *= 2;
            }
            if d == n {
                *c += 1;
                continue;
            }
            return d;
        }
    }

    /// Outputs (p, e) in p's ascending order.
    pub fn factorize(x: i64) -> Vec<(i64, usize)> {
        if x <= 1 { return vec![]; }
        let mut hm = std::collections::HashMap::new();
        let mut pool = vec![x];
        let mut c = 1;
        while let Some(u) = pool.pop() {
            if is_prime(u) {
                *hm.entry(u).or_insert(0) += 1;
                continue;
            }
            let p = pollard_rho(u, &mut c);
            pool.push(p);
            pool.push(u / p);
        }
        let mut v: Vec<_> = hm.into_iter().collect();
        v.sort();
        v
    }
} // mod pollard_rho

// https://yukicoder.me/problems/no/2578 (4)
// Solved with hints
// m <= 10^18 なので m の約数は 10^5 個程度あるので、約数をキーに持つ単純な DP だとうまくいかない。
// g(t) := \sum{lcm | t} \prod W とすると、g(t) = \prod{a_i | t} (W_i + 1) である。(足すものが乗法的であるため分配法則が使える。)
// {m の約数} のように線型束の直積になっている束について、包除原理は (1 -1 0 0 ...) の直積である。
// つまり、 f(m) = \sum_{u|m, u is squarefree} (-1)^{#primes(u)} g(t/u) である。
// これによって、2^(m の素因数の個数) で手間を抑えられるので、高速に計算できる。
// https://gist.github.com/koba-e964/2de3a6480749241f424c4e110a440503 によれば m の素因数の個数は 15 個以下であるため、2^15 * n 程度の手間である。
// -> TLE。高速ゼータ変換を使って (m の素因数の個数)n にする必要があった。
// Tags: product-of-lattices, inclusion-exclusion-principle
fn main() {
    let t: usize = get();
    let m: i64 = get();
    let pe = pollard_rho::factorize(m);
    let l = pe.len();
    for _ in 0..t {
        let n: usize = get();
        let b: i64 = get();
        let c: i64 = get();
        let d: i64 = get();
        let a: Vec<i64> = (0..n).map(|_| get()).collect();
        let mut w = vec![MInt::new(b); n];
        for i in 1..n {
            w[i] = w[i - 1] * c + d;
        }
        let mut dp = vec![MInt::new(1); 1 << l];
        for i in 0..n {
            if m % a[i] != 0 {
                continue;
            }
            let rem = m / a[i];
            let mut bits = 0;
            for j in 0..l {
                if rem % pe[j].0 == 0 {
                    bits |= 1 << j;
                }
            }
            dp[bits] *= w[i] + 1;
        }
        for i in 0..l {
            for bits in 0usize..1 << l {
                if (bits & 1 << i) == 0 {
                    dp[bits] = dp[bits] * dp[bits ^ 1 << i];
                }
            }
        }
        let mut tot = MInt::new(0);
        for bits in 0..1usize << l {
            if bits.count_ones() % 2 == 0 {
                tot += dp[bits];
            } else {
                tot -= dp[bits];
            }
        }
        if m == 1 {
            tot -= 1; // exclude the empty set
        }
        println!("{}", tot);
    }
}