結果

問題 No.1781 LCM
コンテスト
ユーザー akakimidori
提出日時 2021-12-10 08:47:17
言語 Rust
(1.83.0 + proconio)
結果
TLE  
実行時間 -
コード長 11,112 bytes
コンパイル時間 22,322 ms
コンパイル使用メモリ 378,884 KB
実行使用メモリ 384,260 KB
最終ジャッジ日時 2024-07-18 02:28:01
合計ジャッジ時間 29,179 ms
ジャッジサーバーID
（参考情報） 		judge2 / judge5
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 -- * 2
other AC * 21 TLE * 1 -- * 9
権限があれば一括ダウンロードができます
コンパイルメッセージ
warning: type alias `Map` is never used
   --> src/main.rs:305:6
    |
305 | type Map<K, V> = HashMap<K, V>;
    |      ^^^
    |
    = note: `#[warn(dead_code)]` on by default

warning: function `naive` is never used
   --> src/main.rs:395:4
    |
395 | fn naive(n: usize, m: usize) {
    |    ^^^^^

ソースコード

diff # 

// 初期化配列
// 初期値とサイズを与えて適当にやる系
// new(size, zero): zero埋めした長さsizeの配列を返す
// init(&mut self): 初期化
// index(mut) でアクセスしたときその履歴を溜め込む
// それ以外でアクセスすると死ぬので注意
//
// 考えるべきこと
// 1. deref で dataへアクセスできるようにしていいか
//    derefmut はダメ
// 2. 今のままだと二次元配列の初期化とかには対応できない
//    なんか方法を考えたい
// ---------- begin init array ----------
#[derive(Clone)]
pub struct InitArray<T> {
    data: Vec<T>,
    used: Vec<bool>,
    list: Vec<usize>,
    zero: T,
}

impl<T: Copy> InitArray<T> {
    pub fn new(zero: T, size: usize) -> Self {
        InitArray {
            data: vec![zero; size],
            used: vec![false; size],
            list: vec![],
            zero: zero,
        }
    }
    pub fn init(&mut self) {
        for x in self.list.drain(..) {
            self.used[x] = false;
            self.data[x] = self.zero;
        }
    }
    pub fn init_with<F>(&mut self, mut f: F)
    where
        F: FnMut(usize, T),
    {
        for x in self.list.drain(..) {
            self.used[x] = false;
            let v = std::mem::replace(&mut self.data[x], self.zero);
            f(x, v);
        }
    }
}

impl<T> std::ops::Index<usize> for InitArray<T> {
    type Output = T;
    fn index(&self, pos: usize) -> &Self::Output {
        &self.data[pos]
    }
}

impl<T> std::ops::IndexMut<usize> for InitArray<T> {
    fn index_mut(&mut self, pos: usize) -> &mut Self::Output {
        if !self.used[pos] {
            self.used[pos] = true;
            self.list.push(pos);
        }
        &mut self.data[pos]
    }
}
// ---------- end init array ----------
// ---------- begin prime count ----------
// 処理が終わった時
// large[i]: pi(floor(n / i))
// small[i]: pi(i)
// となっている
// O(N^(3/4)/log N)
pub fn prime_count(n: usize) -> (Vec<usize>, Vec<usize>) {
    if n <= 1 {
        return (vec![0, 0], vec![0, 0]);
    }
    let sqrt = (1..).find(|p| p * p > n).unwrap() - 1;
    let mut large = vec![0; sqrt + 1];
    let mut small = vec![0; sqrt + 1];
    for (i, (large, small)) in large.iter_mut().zip(&mut small).enumerate().skip(1) {
        *large = n / i - 1;
        *small = i - 1;
    }
    for p in 2..=sqrt {
        if small[p] == small[p - 1] {
            continue;
        }
        let pi = small[p] - 1;
        let q = p * p;
        for i in 1..=sqrt.min(n / q) {
            large[i] -= *large.get(i * p).unwrap_or_else(|| &small[n / (i * p)]) - pi;
        }
        for i in (q..=sqrt).rev() {
            small[i] -= small[i / p] - pi;
        }
    }
    (small, large)
}
// ---------- end prime count ----------
// ---------- begin enumerate prime ----------
fn enumerate_prime<F>(n: usize, mut f: F)
where
    F: FnMut(usize),
{
    assert!(1 <= n && n <= 5 * 10usize.pow(8));
    let batch = (n as f64).sqrt().ceil() as usize;
    let mut is_prime = vec![true; batch + 1];
    for i in (2..).take_while(|p| p * p <= batch) {
        if is_prime[i] {
            let mut j = i * i;
            while let Some(p) = is_prime.get_mut(j) {
                *p = false;
                j += i;
            }
        }
    }
    let mut prime = vec![];
    for (i, p) in is_prime.iter().enumerate().skip(2) {
        if *p && i <= n {
            f(i);
            prime.push(i);
        }
    }
    let mut l = batch + 1;
    while l <= n {
        let r = std::cmp::min(l + batch, n + 1);
        is_prime.clear();
        is_prime.resize(r - l, true);
        for &p in prime.iter() {
            let mut j = (l + p - 1) / p * p - l;
            while let Some(is_prime) = is_prime.get_mut(j) {
                *is_prime = false;
                j += p;
            }
        }
        for (i, _) in is_prime.iter().enumerate().filter(|p| *p.1) {
            f(i + l);
        }
        l += batch;
    }
}
// ---------- end enumerate prime ----------
use std::marker::*;
use std::ops::*;

pub trait Modulo {
    fn modulo() -> u32;
}

pub struct ConstantModulo<const M: u32>;

impl<const M: u32> Modulo for ConstantModulo<{ M }> {
    fn modulo() -> u32 {
        M
    }
}

pub struct ModInt<T>(u32, PhantomData<T>);

impl<T> Clone for ModInt<T> {
    fn clone(&self) -> Self {
        Self::new_unchecked(self.0)
    }
}

impl<T> Copy for ModInt<T> {}

impl<T: Modulo> Add for ModInt<T> {
    type Output = ModInt<T>;
    fn add(self, rhs: Self) -> Self::Output {
        let mut v = self.0 + rhs.0;
        if v >= T::modulo() {
            v -= T::modulo();
        }
        Self::new_unchecked(v)
    }
}

impl<T: Modulo> AddAssign for ModInt<T> {
    fn add_assign(&mut self, rhs: Self) {
        *self = *self + rhs;
    }
}

impl<T: Modulo> Sub for ModInt<T> {
    type Output = ModInt<T>;
    fn sub(self, rhs: Self) -> Self::Output {
        let mut v = self.0 - rhs.0;
        if self.0 < rhs.0 {
            v += T::modulo();
        }
        Self::new_unchecked(v)
    }
}

impl<T: Modulo> SubAssign for ModInt<T> {
    fn sub_assign(&mut self, rhs: Self) {
        *self = *self - rhs;
    }
}

impl<T: Modulo> Mul for ModInt<T> {
    type Output = ModInt<T>;
    fn mul(self, rhs: Self) -> Self::Output {
        let v = self.0 as u64 * rhs.0 as u64 % T::modulo() as u64;
        Self::new_unchecked(v as u32)
    }
}

impl<T: Modulo> MulAssign for ModInt<T> {
    fn mul_assign(&mut self, rhs: Self) {
        *self = *self * rhs;
    }
}

impl<T: Modulo> Neg for ModInt<T> {
    type Output = ModInt<T>;
    fn neg(self) -> Self::Output {
        if self.is_zero() {
            Self::zero()
        } else {
            Self::new_unchecked(T::modulo() - self.0)
        }
    }
}

impl<T> std::fmt::Display for ModInt<T> {
    fn fmt<'a>(&self, f: &mut std::fmt::Formatter<'a>) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}

impl<T> std::fmt::Debug for ModInt<T> {
    fn fmt<'a>(&self, f: &mut std::fmt::Formatter<'a>) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}

impl<T> Default for ModInt<T> {
    fn default() -> Self {
        Self::zero()
    }
}

impl<T: Modulo> std::str::FromStr for ModInt<T> {
    type Err = std::num::ParseIntError;
    fn from_str(s: &str) -> Result<Self, Self::Err> {
        let val = s.parse::<u32>()?;
        Ok(ModInt::new(val))
    }
}

impl<T: Modulo> From<usize> for ModInt<T> {
    fn from(val: usize) -> ModInt<T> {
        ModInt::new_unchecked((val % T::modulo() as usize) as u32)
    }
}

impl<T: Modulo> From<u64> for ModInt<T> {
    fn from(val: u64) -> ModInt<T> {
        ModInt::new_unchecked((val % T::modulo() as u64) as u32)
    }
}

impl<T> ModInt<T> {
    pub fn new_unchecked(n: u32) -> Self {
        ModInt(n, PhantomData)
    }
    pub fn zero() -> Self {
        ModInt::new_unchecked(0)
    }
    pub fn one() -> Self {
        ModInt::new_unchecked(1)
    }
    pub fn is_zero(&self) -> bool {
        self.0 == 0
    }
}

impl<T: Modulo> ModInt<T> {
    pub fn new(d: u32) -> Self {
        ModInt::new_unchecked(d % T::modulo())
    }
    pub fn pow(&self, mut n: u64) -> Self {
        let mut t = Self::one();
        let mut s = *self;
        while n > 0 {
            if n & 1 == 1 {
                t *= s;
            }
            s *= s;
            n >>= 1;
        }
        t
    }
    pub fn inv(&self) -> Self {
        assert!(!self.is_zero());
        self.pow(T::modulo() as u64 - 2)
    }
}

type M = ModInt<ConstantModulo<998_244_353>>;

use std::collections::*;

type Map<K, V> = HashMap<K, V>;

fn read() -> (usize, usize) {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).unwrap();
    let a = s
        .trim()
        .split_whitespace()
        .flat_map(|s| s.parse())
        .collect::<Vec<_>>();
    (a[0], a[1])
}

fn solve(n: usize, m: usize) {
    let mut pow = vec![M::zero(); 40 + 1];
    for i in 1..=40 {
        pow[i] = M::from(i).pow(n as u64);
    }
    let sqrt = (2..).find(|p| p * p > m).unwrap() - 1;
    let pi = prime_count(m);
    let pi = |x: usize| -> usize {
        assert!(x <= m);
        if x <= sqrt {
            return pi.0[x];
        }
        let q = m / x;
        let r = m / q;
        assert_eq!(x, r);
        pi.1[q]
    };
    let mut prime = vec![];
    enumerate_prime(sqrt + 200, |p| prime.push(p));
    let mut large = InitArray::new(M::zero(), sqrt + 1);
    let mut small = InitArray::new(M::zero(), sqrt + 1);
    let mut next_large = InitArray::new(M::zero(), sqrt + 1);
    let mut next_small = InitArray::new(M::zero(), sqrt + 1);
    large[1] = M::one();
    let mut ans = M::zero();
    for (i, &p) in prime.iter().enumerate() {
        large.init_with(|d, v| {
            let m = m / d;
            if p.pow(2) > m {
                ans += v * M::from(m);
                let mut l = p;
                let mut q = m / l;
                while q > 0 {
                    let r = m / q;
                    let cnt = pi(r) - if l == p { i } else { pi(l - 1) };
                    ans += v * M::from(q * cnt) * (pow[2] - M::one());
                    l = r + 1;
                    q -= 1;
                }
            } else {
                for k in (0usize..).take_while(|k| p.pow(*k as u32) <= m) {
                    let v = v * (pow[k + 1] - pow[k]);
                    let pp = p.pow(k as u32);
                    if pp * d <= sqrt {
                        next_large[d * pp] += v;
                    } else {
                        next_small[m / pp] += v;
                    }
                }
            }
        });
        small.init_with(|m, v| {
            if p.pow(2) > m {
                ans += v * M::from(m);
                let mut l = p;
                let mut q = m / l;
                while q > 0 {
                    let r = m / q;
                    let cnt = pi(r) - if l == p { i } else { pi(l - 1) };
                    ans += v * M::from(q * cnt) * (pow[2] - M::one());
                    l = r + 1;
                    q -= 1;
                }
            } else {
                for k in (0usize..).take_while(|k| p.pow(*k as u32) <= m) {
                    let v = v * (pow[k + 1] - pow[k]);
                    let pp = p.pow(k as u32);
                    next_small[m / pp] += v;
                }
            }
        });
        std::mem::swap(&mut small, &mut next_small);
        std::mem::swap(&mut large, &mut next_large);
    }
    println!("{}", ans);
}

fn naive(n: usize, m: usize) {
    let mut dp = vec![M::one(); m + 1];
    enumerate_prime(m, |p| {
        for i in 1..=(m / p) {
            dp[p * i] = dp[p * i] + dp[i];
        }
    });
    for dp in dp.iter_mut() {
        *dp = dp.pow(n as u64);
    }
    enumerate_prime(m, |p| {
        for i in (1..=(m / p)).rev() {
            dp[p * i] = dp[p * i] - dp[i];
        }
    });
    let mut ans = M::zero();
    for i in 1..=m {
        ans += M::from(m / i) * dp[i];
    }
    println!("{}", ans);
}

fn main() {
    let (n, m) = read();
    solve(n, m);
}
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