結果
| 問題 |
No.3182 recurrence relation’s intersection sum
|
| ユーザー |
 koba-e964
|
| 提出日時 | 2025-06-13 22:58:58 |
| 言語 | Rust (1.83.0 + proconio) |
| 結果 |
AC
|
| 実行時間 | 617 ms / 2,000 ms |
| コード長 | 6,415 bytes |
| コンパイル時間 | 12,899 ms |
| コンパイル使用メモリ | 400,364 KB |
| 実行使用メモリ | 7,844 KB |
| 最終ジャッジ日時 | 2025-06-13 22:59:22 |
| 合計ジャッジ時間 | 23,627 ms |
|
ジャッジサーバーID （参考情報） |
judge2 / judge5 |
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| ファイルパターン | 結果 |
|---|---|
| sample | AC * 3 |
| other | AC * 40 |
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)]
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>;
// 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)
}
// Depends on MInt.rs
// Verified by: https://atcoder.jp/contests/abc199/submissions/22259436
fn squmul(a: &[Vec<MInt>], b: &[Vec<MInt>]) -> Vec<Vec<MInt>> {
let n = a.len();
let mut ret = vec![vec![MInt::new(0); n]; n];
for i in 0..n {
for j in 0..n {
for k in 0..n {
ret[i][k] += a[i][j] * b[j][k];
}
}
}
ret
}
fn squpow(a: &[Vec<MInt>], mut e: i64) -> Vec<Vec<MInt>> {
let n = a.len();
let mut sum = vec![vec![MInt::new(0); n]; n];
for i in 0..n { sum[i][i] = 1.into(); }
let mut cur = a.to_vec();
while e > 0 {
if e % 2 == 1 {
sum = squmul(&sum, &cur);
}
cur = squmul(&cur, &cur);
e /= 2;
}
sum
}
fn f(k: usize, n: i64) -> MInt {
let (fac, invfac) = fact_init(k + 1);
let mut mat = vec![vec![MInt::new(0); k + 4]; k + 4];
let kk = MInt::new(k as i64);
for i in 0..k + 1 {
for j in 0..i + 1 {
mat[j][i] = fac[i] * invfac[j] * invfac[i - j];
}
}
mat[k + 1][k + 1] = kk;
mat[k + 3][k + 2] = MInt::new(1);
mat[k][k + 3] = MInt::new(1);
mat[k + 1][k + 3] = MInt::new(1);
mat[k + 2][k + 3] = -kk;
mat[k + 3][k + 3] = kk + 1;
let pw = squpow(&mat, n);
let mut ret = MInt::new(0);
ret += pw[0][k + 2];
ret += pw[k + 1][k + 2];
ret += pw[k + 3][k + 2];
ret
}
// https://yukicoder.me/problems/no/3182 (3.5)
// b_n = \sum_{i=0}^{k - 1} a_i とする。
// b_0 = 0, b_1 = 1
// b_{n+2} - b_{n+1} = K (b_{n+1} - b_n) + n^K + K^n
// b_{n+2} = (K + 1) b_{n+1} - K b_n + n^K + K^n
// これは行列累乗で解ける。 i^0, i^1, ..., i^K, K^i, b_i, b_{i+1} からなるベクトルに行列を掛けるようにする。
fn main() {
let ints = getline()
.trim()
.split_whitespace()
.map(|x| x.parse::<i64>().unwrap())
.collect::<Vec<_>>();
let k = ints[0] as usize;
let l = ints[1];
let r = ints[2];
println!("{}", f(k, r + 1) - f(k, l))
}