結果
| 問題 |
No.3045 反復重み付き累積和
|
| ユーザー |
 koba-e964
|
| 提出日時 | 2025-08-10 23:41:50 |
| 言語 | Rust (1.83.0 + proconio) |
| 結果 |
AC
|
| 実行時間 | 1,533 ms / 5,000 ms |
| コード長 | 13,029 bytes |
| コンパイル時間 | 12,372 ms |
| コンパイル使用メモリ | 397,076 KB |
| 実行使用メモリ | 7,716 KB |
| 最終ジャッジ日時 | 2025-08-10 23:42:38 |
| 合計ジャッジ時間 | 42,175 ms |
|
ジャッジサーバーID （参考情報） |
judge5 / judge1 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 3 |
| other | AC * 41 |
コンパイルメッセージ
warning: unused variable: `n`
--> src/main.rs:347:10
|
347 | let [n, q] = ints[..] else { panic!() };
| ^ help: if this is intentional, prefix it with an underscore: `_n`
|
= note: `#[warn(unused_variables)]` on by default
ソースコード
fn getline() -> String {
let mut ret = String::new();
std::io::stdin().read_line(&mut ret).ok().unwrap();
ret
}
/// 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;
}
}
}
}
// 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>],
r#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 = r#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
}
// https://yukicoder.me/problems/no/3045 (4)
// 1 k x は (1-kT)^{-x} を掛ける操作、2 x は [T^x] の係数を求める操作だと思うことができる。
// 1 k x で掛けられたものの ln を管理しておいて、2 x ではそれの exp を O(N log N) で計算すればよい。
fn main() {
const W: usize = 1 << 11;
let (fac, invfac) = fact_init(W);
let ints = getline().trim().split_whitespace()
.map(|s| s.parse::<usize>().unwrap()).collect::<Vec<_>>();
let [n, q] = ints[..] else { panic!() };
let a = getline().trim().split_whitespace()
.map(|s| s.parse::<i64>().unwrap()).collect::<Vec<_>>();
let mut acc_ln = vec![MInt::new(0); W];
for _ in 0..q {
let ints = getline().trim().split_whitespace()
.map(|s| s.parse::<i64>().unwrap()).collect::<Vec<_>>();
if ints[0] == 1 {
let [k, v] = ints[1..] else { panic!() };
let mut ln = vec![MInt::new(0); W];
for i in 1..W {
ln[i] = MInt::new(k).pow(i as i64) * MInt::new(i as i64).inv();
}
let v = MInt::new(v);
for i in 1..W {
acc_ln[i] += ln[i] * v;
}
} else {
let x = ints[1] as usize - 1;
let exp = fps_exp(&acc_ln, MInt::new(3), &fac, &invfac);
let mut ans = MInt::new(0);
for i in 0..=x {
ans += exp[i] * MInt::new(a[x - i]);
}
println!("{ans}");
}
}
}