結果

問題 No.3243 Multiplication 8 1
ユーザー urectanc
提出日時 2025-08-24 13:17:07
言語 Rust
(1.83.0 + proconio)
結果
AC  
実行時間 43 ms / 2,000 ms
コード長 9,086 bytes
コンパイル時間 11,095 ms
コンパイル使用メモリ 403,628 KB
実行使用メモリ 7,716 KB
最終ジャッジ日時 2025-08-24 13:17:22
合計ジャッジ時間 12,216 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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ファイルパターン 結果
sample AC * 1
other AC * 4
権限があれば一括ダウンロードができます

ソースコード

diff # 

use modint::ModInt998244353;
use proconio::input;
use std::io::Write;

type Mint = ModInt998244353;

const ELEMENTS: [i32; 4] = [1, 2, -1, -2];
const STATES: [i32; 7] = [1, -1, 2, -2, 4, -4, -8];
const K: usize = STATES.len();
type Mat = [[Mint; K]; K];

fn main() {
    input! { t: usize }

    let mut writer = std::io::BufWriter::new(std::io::stdout().lock());
    for _ in 0..t {
        input! { n: usize }
        let ans = solve(n);
        writeln!(writer, "{ans}").unwrap();
    }
}

fn solve(n: usize) -> Mint {
    let mut transition = [[Mint::new(0); K]; K];
    for (i, &state) in STATES.iter().enumerate() {
        for &e in &ELEMENTS {
            let nstate = state * e;
            if nstate.abs() > 8 {
                continue;
            }
            let j = STATES.iter().position(|&x| x == nstate).unwrap_or(0);
            transition[i][j] += 1;
        }
    }

    let acc = pow(&transition, n);
    let ans = acc[0][0] - Mint::new(2).pow(n as u64 - 1);
    ans
}

fn mul(a: &Mat, b: &Mat) -> Mat {
    let mut res = [[Mint::new(0); K]; K];
    for i in 0..K {
        for j in 0..K {
            for k in 0..K {
                res[i][j] += a[i][k] * b[k][j];
            }
        }
    }
    res
}

fn pow(a: &Mat, mut k: usize) -> Mat {
    assert!(k > 0);
    k -= 1;
    let mut res = a.clone();
    let mut pow = a.clone();
    while k > 0 {
        if k & 1 == 1 {
            res = mul(&res, &pow);
        }
        pow = mul(&pow, &pow);
        k >>= 1;
    }
    res
}

// f(3) = C(3, 3) * (C(3, 0) + C(3, 2)) = 1 * 4 = 4
// f(4) = C(4, 3) * (C(4, 0) + C(4, 2) + C(4, 4)) = 4 * 8 = 32
// f(5) = C(5, 3) * (C(5, 0) + C(5, 2) + C(5, 4)) = 10 * 16 = 160
// f(6) = C(6, 3) * (C(6, 0) + C(6, 2) + C(6, 4) + C(6, 6)) + 2 * f(3) = 20 * 32 + 2 * 4 = 648

#[allow(dead_code)]
mod modint {
    use std::ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Sub, SubAssign};

    pub type ModInt998244353 = ModInt<998244353>;
    pub type ModInt1000000007 = ModInt<1000000007>;

    type ModIntInner = u64;

    #[derive(Clone, Copy, PartialEq, Eq)]
    pub struct ModInt<const M: ModIntInner> {
        val: ModIntInner,
    }

    impl<const M: ModIntInner> ModInt<M> {
        const IS_PRIME: bool = is_prime(M as u32);

        pub const fn modulus() -> ModIntInner {
            M
        }

        pub const fn new(val: ModIntInner) -> Self {
            assert!(M < (1 << 31));
            Self {
                val: val.rem_euclid(M),
            }
        }

        pub const fn new_unchecked(val: ModIntInner) -> Self {
            Self { val }
        }

        pub const fn val(&self) -> ModIntInner {
            self.val
        }

        pub fn pow(self, mut exp: u64) -> Self {
            let mut result = Self::new(1);
            let mut base = self;
            while exp > 0 {
                if exp & 1 == 1 {
                    result *= base;
                }
                base *= base;
                exp >>= 1;
            }
            result
        }

        pub fn inv(self) -> Self {
            assert!(Self::IS_PRIME);
            self.pow(M as u64 - 2).into()
        }
    }

    impl<const M: ModIntInner> std::fmt::Display for ModInt<M> {
        fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
            write!(f, "{}", self.val)
        }
    }

    impl<const M: ModIntInner> std::fmt::Debug for ModInt<M> {
        fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
            write!(f, "{}", self.val)
        }
    }

    impl<const M: ModIntInner> std::str::FromStr for ModInt<M> {
        type Err = std::num::ParseIntError;
        fn from_str(s: &str) -> Result<Self, Self::Err> {
            let value = s.parse::<ModIntInner>()?;
            Ok(ModInt::new(value))
        }
    }

    impl<const M: ModIntInner> Neg for ModInt<M> {
        type Output = Self;
        fn neg(mut self) -> Self::Output {
            if self.val > 0 {
                self.val = M - self.val;
            }
            self
        }
    }

    impl<const M: ModIntInner, T: Into<ModInt<M>>> AddAssign<T> for ModInt<M> {
        fn add_assign(&mut self, rhs: T) {
            self.val += rhs.into().val;
            if self.val >= M {
                self.val -= M;
            }
        }
    }

    impl<const M: ModIntInner, T: Into<ModInt<M>>> SubAssign<T> for ModInt<M> {
        fn sub_assign(&mut self, rhs: T) {
            self.val = self.val.wrapping_sub(rhs.into().val);
            if self.val > M {
                self.val = self.val.wrapping_add(M);
            }
        }
    }

    impl<const M: ModIntInner, T: Into<ModInt<M>>> MulAssign<T> for ModInt<M> {
        fn mul_assign(&mut self, rhs: T) {
            self.val = self.val * rhs.into().val % M;
        }
    }

    impl<const M: ModIntInner, T: Into<ModInt<M>>> DivAssign<T> for ModInt<M> {
        fn div_assign(&mut self, rhs: T) {
            *self *= rhs.into().inv();
        }
    }

    macro_rules! impl_binnary_operators {
    ($({ $op: ident, $op_assign: ident, $fn: ident, $fn_assign: ident}),*) => {$(
        impl<const M: ModIntInner, T: Into<ModInt<M>>> $op<T> for ModInt<M> {
            type Output = ModInt<M>;
            fn $fn(mut self, rhs: T) -> ModInt<M> {
                self.$fn_assign(rhs.into());
                self
            }
        }

        impl<const M: ModIntInner> $op<&ModInt<M>> for ModInt<M> {
            type Output = ModInt<M>;
            fn $fn(self, rhs: &ModInt<M>) -> ModInt<M> {
                self.$fn(*rhs)
            }
        }

        impl<const M: ModIntInner, T: Into<ModInt<M>>> $op<T> for &ModInt<M> {
            type Output = ModInt<M>;
            fn $fn(self, rhs: T) -> ModInt<M> {
                (*self).$fn(rhs.into())
            }
        }

        impl<const M: ModIntInner> $op<&ModInt<M>> for &ModInt<M> {
            type Output = ModInt<M>;
            fn $fn(self, rhs: &ModInt<M>) -> ModInt<M> {
                (*self).$fn(*rhs)
            }
        }

        impl<const M: ModIntInner> $op_assign<&ModInt<M>> for ModInt<M> {
            fn $fn_assign(&mut self, rhs: &ModInt<M>) {
                *self = self.$fn(*rhs);
            }
        }
    )*};
}

    impl_binnary_operators!(
        {Add, AddAssign, add, add_assign},
        {Sub, SubAssign, sub, sub_assign},
        {Mul, MulAssign, mul, mul_assign},
        {Div, DivAssign, div, div_assign}
    );

    impl<const M: ModIntInner> std::iter::Sum for ModInt<M> {
        fn sum<I: Iterator<Item = Self>>(iter: I) -> Self {
            iter.fold(Self::new(0), Add::add)
        }
    }

    impl<'a, const M: ModIntInner> std::iter::Sum<&'a Self> for ModInt<M> {
        fn sum<I: Iterator<Item = &'a Self>>(iter: I) -> Self {
            iter.fold(Self::new(0), Add::add)
        }
    }

    impl<const M: ModIntInner> std::iter::Product for ModInt<M> {
        fn product<I: Iterator<Item = Self>>(iter: I) -> Self {
            iter.fold(Self::new(1), Mul::mul)
        }
    }

    impl<'a, const M: ModIntInner> std::iter::Product<&'a Self> for ModInt<M> {
        fn product<I: Iterator<Item = &'a Self>>(iter: I) -> Self {
            iter.fold(Self::new(1), Mul::mul)
        }
    }

    macro_rules! impl_rem_euclid_signed {
    ($($ty:tt),*) => {
        $(
            impl<const M: ModIntInner> From<$ty> for ModInt<M> {
                fn from(value: $ty) -> ModInt<M> {
                    Self::new_unchecked((value as i64).rem_euclid(M as i64) as ModIntInner)
                }
            }
        )*
    };
}
    impl_rem_euclid_signed!(i8, i16, i32, i64, isize);

    macro_rules! impl_rem_euclid_unsigned {
    ($($ty:tt),*) => {
        $(
            impl<const M: ModIntInner> From<$ty> for ModInt<M> {
                fn from(value: $ty) -> ModInt<M> {
                    Self::new_unchecked((value as u64).rem_euclid(M as u64) as ModIntInner)
                }
            }
        )*
    };
}
    impl_rem_euclid_unsigned!(u8, u16, u32, u64, usize);

    const fn is_prime(n: u32) -> bool {
        const fn miller_rabin(n: u32, witness: u32) -> bool {
            let (n, witness) = (n as u64, witness as u64);
            let mut d = n >> (n - 1).trailing_zeros();
            let mut y = {
                let (mut res, mut pow, mut e) = (1, witness, d);
                while e > 0 {
                    if e & 1 == 1 {
                        res = res * pow % n;
                    }
                    pow = pow * pow % n;
                    e >>= 1;
                }
                res
            };
            while d != n - 1 && y != 1 && y != n - 1 {
                y = y * y % n;
                d <<= 1;
            }
            y == n - 1 || d & 1 == 1
        }

        const WITNESS: [u32; 3] = [2, 7, 61];

        if n == 1 || n % 2 == 0 {
            return n == 2;
        }

        let mut i = 0;
        while i < WITNESS.len() {
            if !miller_rabin(n, WITNESS[i]) {
                return false;
            }
            i += 1;
        }

        true
    }
}
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