結果
| 問題 | No.2527 H and W |
| ユーザー |
 haihamabossu
|
| 提出日時 | 2023-11-03 21:52:09 |
| 言語 | Rust (1.77.0 + proconio) |
| 結果 |
AC
|
| 実行時間 | 60 ms / 2,000 ms |
| コード長 | 55,142 bytes |
| コンパイル時間 | 14,599 ms |
| コンパイル使用メモリ | 390,656 KB |
| 実行使用メモリ | 25,408 KB |
| 最終ジャッジ日時 | 2024-09-25 20:02:26 |
| 合計ジャッジ時間 | 16,355 ms |
|
ジャッジサーバーID （参考情報） |
judge3 / judge1 |
（要ログイン）
テストケース
テストケース表示| 入力 | 結果 | 実行時間 実行使用メモリ |
|---|---|---|
| testcase_00 | AC | 47 ms
25,272 KB |
| testcase_01 | AC | 46 ms
25,224 KB |
| testcase_02 | AC | 59 ms
25,408 KB |
| testcase_03 | AC | 48 ms
25,312 KB |
| testcase_04 | AC | 48 ms
25,280 KB |
| testcase_05 | AC | 50 ms
25,272 KB |
| testcase_06 | AC | 51 ms
25,252 KB |
| testcase_07 | AC | 49 ms
25,344 KB |
| testcase_08 | AC | 60 ms
25,260 KB |
| testcase_09 | AC | 46 ms
25,320 KB |
| testcase_10 | AC | 47 ms
25,352 KB |
| testcase_11 | AC | 49 ms
25,272 KB |
| testcase_12 | AC | 53 ms
25,340 KB |
| testcase_13 | AC | 54 ms
25,404 KB |
| testcase_14 | AC | 57 ms
25,292 KB |
| testcase_15 | AC | 50 ms
25,312 KB |
| testcase_16 | AC | 57 ms
25,296 KB |
| testcase_17 | AC | 58 ms
25,276 KB |
| testcase_18 | AC | 58 ms
25,364 KB |
| testcase_19 | AC | 54 ms
25,268 KB |
| testcase_20 | AC | 55 ms
25,224 KB |
| testcase_21 | AC | 52 ms
25,328 KB |
| testcase_22 | AC | 53 ms
25,240 KB |
| testcase_23 | AC | 52 ms
25,288 KB |
| testcase_24 | AC | 54 ms
25,360 KB |
| testcase_25 | AC | 47 ms
25,248 KB |
コンパイルメッセージ
warning: fields `sum_e` and `sum_ie` are never read
--> src/main.rs:750:20
|
749 | pub struct ButterflyCache<M> {
| -------------- fields in this struct
750 | pub(crate) sum_e: Vec<StaticModInt<M>>,
| ^^^^^
751 | pub(crate) sum_ie: Vec<StaticModInt<M>>,
| ^^^^^^
|
= note: `#[warn(dead_code)]` on by default
ソースコード
//https://github.com/rust-lang-ja/ac-library-rs
pub mod internal_math {
// remove this after dependencies has been added
#![allow(dead_code)]
use std::mem::swap;
/// # Arguments
/// * `m` `1 <= m`
///
/// # Returns
/// x mod m
/* const */
pub(crate) fn safe_mod(mut x: i64, m: i64) -> i64 {
x %= m;
if x < 0 {
x += m;
}
x
}
/// Fast modular by barrett reduction
/// Reference: https://en.wikipedia.org/wiki/Barrett_reduction
/// NOTE: reconsider after Ice Lake
pub(crate) struct Barrett {
pub(crate) _m: u32,
pub(crate) im: u64,
}
impl Barrett {
/// # Arguments
/// * `m` `1 <= m`
/// (Note: `m <= 2^31` should also hold, which is undocumented in the original library.
/// See the [pull reqeust commment](https://github.com/rust-lang-ja/ac-library-rs/pull/3#discussion_r484661007)
/// for more details.)
pub(crate) fn new(m: u32) -> Barrett {
Barrett {
_m: m,
im: (-1i64 as u64 / m as u64).wrapping_add(1),
}
}
/// # Returns
/// `m`
pub(crate) fn umod(&self) -> u32 {
self._m
}
/// # Parameters
/// * `a` `0 <= a < m`
/// * `b` `0 <= b < m`
///
/// # Returns
/// a * b % m
#[allow(clippy::many_single_char_names)]
pub(crate) fn mul(&self, a: u32, b: u32) -> u32 {
mul_mod(a, b, self._m, self.im)
}
}
/// Calculates `a * b % m`.
///
/// * `a` `0 <= a < m`
/// * `b` `0 <= b < m`
/// * `m` `1 <= m <= 2^31`
/// * `im` = ceil(2^64 / `m`)
#[allow(clippy::many_single_char_names)]
pub(crate) fn mul_mod(a: u32, b: u32, m: u32, im: u64) -> u32 {
// [1] m = 1
// a = b = im = 0, so okay
// [2] m >= 2
// im = ceil(2^64 / m)
// -> im * m = 2^64 + r (0 <= r < m)
// let z = a*b = c*m + d (0 <= c, d < m)
// a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im
// c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2
// ((ab * im) >> 64) == c or c + 1
let mut z = a as u64;
z *= b as u64;
let x = (((z as u128) * (im as u128)) >> 64) as u64;
let mut v = z.wrapping_sub(x.wrapping_mul(m as u64)) as u32;
if m <= v {
v = v.wrapping_add(m);
}
v
}
/// # Parameters
/// * `n` `0 <= n`
/// * `m` `1 <= m`
///
/// # Returns
/// `(x ** n) % m`
/* const */
#[allow(clippy::many_single_char_names)]
pub(crate) fn pow_mod(x: i64, mut n: i64, m: i32) -> i64 {
if m == 1 {
return 0;
}
let _m = m as u32;
let mut r: u64 = 1;
let mut y: u64 = safe_mod(x, m as i64) as u64;
while n != 0 {
if (n & 1) > 0 {
r = (r * y) % (_m as u64);
}
y = (y * y) % (_m as u64);
n >>= 1;
}
r as i64
}
/// Reference:
/// M. Forisek and J. Jancina,
/// Fast Primality Testing for Integers That Fit into a Machine Word
///
/// # Parameters
/// * `n` `0 <= n`
/* const */
pub(crate) fn is_prime(n: i32) -> bool {
let n = n as i64;
match n {
_ if n <= 1 => return false,
2 | 7 | 61 => return true,
_ if n % 2 == 0 => return false,
_ => {}
}
let mut d = n - 1;
while d % 2 == 0 {
d /= 2;
}
for &a in &[2, 7, 61] {
let mut t = d;
let mut y = pow_mod(a, t, n as i32);
while t != n - 1 && y != 1 && y != n - 1 {
y = y * y % n;
t <<= 1;
}
if y != n - 1 && t % 2 == 0 {
return false;
}
}
true
}
// omitted
// template <int n> constexpr bool is_prime = is_prime_constexpr(n);
/// # Parameters
/// * `b` `1 <= b`
///
/// # Returns
/// (g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
/* const */
#[allow(clippy::many_single_char_names)]
pub(crate) fn inv_gcd(a: i64, b: i64) -> (i64, i64) {
let a = safe_mod(a, b);
if a == 0 {
return (b, 0);
}
// Contracts:
// [1] s - m0 * a = 0 (mod b)
// [2] t - m1 * a = 0 (mod b)
// [3] s * |m1| + t * |m0| <= b
let mut s = b;
let mut t = a;
let mut m0 = 0;
let mut m1 = 1;
while t != 0 {
let u = s / t;
s -= t * u;
m0 -= m1 * u; // |m1 * u| <= |m1| * s <= b
// [3]:
// (s - t * u) * |m1| + t * |m0 - m1 * u|
// <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)
// = s * |m1| + t * |m0| <= b
swap(&mut s, &mut t);
swap(&mut m0, &mut m1);
}
// by [3]: |m0| <= b/g
// by g != b: |m0| < b/g
if m0 < 0 {
m0 += b / s;
}
(s, m0)
}
/// Compile time (currently not) primitive root
/// @param m must be prime
/// @return primitive root (and minimum in now)
/* const */
pub(crate) fn primitive_root(m: i32) -> i32 {
match m {
2 => return 1,
167_772_161 => return 3,
469_762_049 => return 3,
754_974_721 => return 11,
998_244_353 => return 3,
_ => {}
}
let mut divs = [0; 20];
divs[0] = 2;
let mut cnt = 1;
let mut x = (m - 1) / 2;
while x % 2 == 0 {
x /= 2;
}
for i in (3..std::i32::MAX).step_by(2) {
if i as i64 * i as i64 > x as i64 {
break;
}
if x % i == 0 {
divs[cnt] = i;
cnt += 1;
while x % i == 0 {
x /= i;
}
}
}
if x > 1 {
divs[cnt] = x;
cnt += 1;
}
let mut g = 2;
loop {
if (0..cnt).all(|i| pow_mod(g, ((m - 1) / divs[i]) as i64, m) != 1) {
break g as i32;
}
g += 1;
}
}
// omitted
// template <int m> constexpr int primitive_root = primitive_root_constexpr(m);
#[cfg(test)]
mod tests {
#![allow(clippy::unreadable_literal)]
#![allow(clippy::cognitive_complexity)]
use crate::internal_math::{inv_gcd, is_prime, pow_mod, primitive_root, safe_mod, Barrett};
use std::collections::HashSet;
#[test]
fn test_safe_mod() {
assert_eq!(safe_mod(0, 3), 0);
assert_eq!(safe_mod(1, 3), 1);
assert_eq!(safe_mod(2, 3), 2);
assert_eq!(safe_mod(3, 3), 0);
assert_eq!(safe_mod(4, 3), 1);
assert_eq!(safe_mod(5, 3), 2);
assert_eq!(safe_mod(73, 11), 7);
assert_eq!(safe_mod(2306249155046129918, 6620319213327), 1374210749525);
assert_eq!(safe_mod(-1, 3), 2);
assert_eq!(safe_mod(-2, 3), 1);
assert_eq!(safe_mod(-3, 3), 0);
assert_eq!(safe_mod(-4, 3), 2);
assert_eq!(safe_mod(-5, 3), 1);
assert_eq!(safe_mod(-7170500492396019511, 777567337), 333221848);
}
#[test]
fn test_barrett() {
let b = Barrett::new(7);
assert_eq!(b.umod(), 7);
assert_eq!(b.mul(2, 3), 6);
assert_eq!(b.mul(4, 6), 3);
assert_eq!(b.mul(5, 0), 0);
let b = Barrett::new(998244353);
assert_eq!(b.umod(), 998244353);
assert_eq!(b.mul(2, 3), 6);
assert_eq!(b.mul(3141592, 653589), 919583920);
assert_eq!(b.mul(323846264, 338327950), 568012980);
// make `z - x * self._m as u64` overflow.
// Thanks @koba-e964 (at https://github.com/rust-lang-ja/ac-library-rs/pull/3#discussion_r484932161)
let b = Barrett::new(2147483647);
assert_eq!(b.umod(), 2147483647);
assert_eq!(b.mul(1073741824, 2147483645), 2147483646);
}
#[test]
fn test_pow_mod() {
assert_eq!(pow_mod(0, 0, 1), 0);
assert_eq!(pow_mod(0, 0, 3), 1);
assert_eq!(pow_mod(0, 0, 723), 1);
assert_eq!(pow_mod(0, 0, 998244353), 1);
assert_eq!(pow_mod(0, 0, i32::max_value()), 1);
assert_eq!(pow_mod(0, 1, 1), 0);
assert_eq!(pow_mod(0, 1, 3), 0);
assert_eq!(pow_mod(0, 1, 723), 0);
assert_eq!(pow_mod(0, 1, 998244353), 0);
assert_eq!(pow_mod(0, 1, i32::max_value()), 0);
assert_eq!(pow_mod(0, i64::max_value(), 1), 0);
assert_eq!(pow_mod(0, i64::max_value(), 3), 0);
assert_eq!(pow_mod(0, i64::max_value(), 723), 0);
assert_eq!(pow_mod(0, i64::max_value(), 998244353), 0);
assert_eq!(pow_mod(0, i64::max_value(), i32::max_value()), 0);
assert_eq!(pow_mod(1, 0, 1), 0);
assert_eq!(pow_mod(1, 0, 3), 1);
assert_eq!(pow_mod(1, 0, 723), 1);
assert_eq!(pow_mod(1, 0, 998244353), 1);
assert_eq!(pow_mod(1, 0, i32::max_value()), 1);
assert_eq!(pow_mod(1, 1, 1), 0);
assert_eq!(pow_mod(1, 1, 3), 1);
assert_eq!(pow_mod(1, 1, 723), 1);
assert_eq!(pow_mod(1, 1, 998244353), 1);
assert_eq!(pow_mod(1, 1, i32::max_value()), 1);
assert_eq!(pow_mod(1, i64::max_value(), 1), 0);
assert_eq!(pow_mod(1, i64::max_value(), 3), 1);
assert_eq!(pow_mod(1, i64::max_value(), 723), 1);
assert_eq!(pow_mod(1, i64::max_value(), 998244353), 1);
assert_eq!(pow_mod(1, i64::max_value(), i32::max_value()), 1);
assert_eq!(pow_mod(i64::max_value(), 0, 1), 0);
assert_eq!(pow_mod(i64::max_value(), 0, 3), 1);
assert_eq!(pow_mod(i64::max_value(), 0, 723), 1);
assert_eq!(pow_mod(i64::max_value(), 0, 998244353), 1);
assert_eq!(pow_mod(i64::max_value(), 0, i32::max_value()), 1);
assert_eq!(pow_mod(i64::max_value(), i64::max_value(), 1), 0);
assert_eq!(pow_mod(i64::max_value(), i64::max_value(), 3), 1);
assert_eq!(pow_mod(i64::max_value(), i64::max_value(), 723), 640);
assert_eq!(
pow_mod(i64::max_value(), i64::max_value(), 998244353),
683296792
);
assert_eq!(
pow_mod(i64::max_value(), i64::max_value(), i32::max_value()),
1
);
assert_eq!(pow_mod(2, 3, 1_000_000_007), 8);
assert_eq!(pow_mod(5, 7, 1_000_000_007), 78125);
assert_eq!(pow_mod(123, 456, 1_000_000_007), 565291922);
}
#[test]
fn test_is_prime() {
assert!(!is_prime(0));
assert!(!is_prime(1));
assert!(is_prime(2));
assert!(is_prime(3));
assert!(!is_prime(4));
assert!(is_prime(5));
assert!(!is_prime(6));
assert!(is_prime(7));
assert!(!is_prime(8));
assert!(!is_prime(9));
// assert!(is_prime(57));
assert!(!is_prime(57));
assert!(!is_prime(58));
assert!(is_prime(59));
assert!(!is_prime(60));
assert!(is_prime(61));
assert!(!is_prime(62));
assert!(!is_prime(701928443));
assert!(is_prime(998244353));
assert!(!is_prime(1_000_000_000));
assert!(is_prime(1_000_000_007));
assert!(is_prime(i32::max_value()));
}
#[test]
fn test_is_prime_sieve() {
let n = 1_000_000;
let mut prime = vec![true; n];
prime[0] = false;
prime[1] = false;
for i in 0..n {
assert_eq!(prime[i], is_prime(i as i32));
if prime[i] {
for j in (2 * i..n).step_by(i) {
prime[j] = false;
}
}
}
}
#[test]
fn test_inv_gcd() {
for &(a, b, g) in &[
(0, 1, 1),
(0, 4, 4),
(0, 7, 7),
(2, 3, 1),
(-2, 3, 1),
(4, 6, 2),
(-4, 6, 2),
(13, 23, 1),
(57, 81, 3),
(12345, 67890, 15),
(-3141592 * 6535, 3141592 * 8979, 3141592),
(i64::max_value(), i64::max_value(), i64::max_value()),
(i64::min_value(), i64::max_value(), 1),
] {
let (g_, x) = inv_gcd(a, b);
assert_eq!(g, g_);
let b_ = b as i128;
assert_eq!(((x as i128 * a as i128) % b_ + b_) % b_, g as i128 % b_);
}
}
#[test]
fn test_primitive_root() {
for &p in &[
2,
3,
5,
7,
233,
200003,
998244353,
1_000_000_007,
i32::max_value(),
] {
assert!(is_prime(p));
let g = primitive_root(p);
if p != 2 {
assert_ne!(g, 1);
}
let q = p - 1;
for i in (2..i32::max_value()).take_while(|i| i * i <= q) {
if q % i != 0 {
break;
}
for &r in &[i, q / i] {
assert_ne!(pow_mod(g as i64, r as i64, p), 1);
}
}
assert_eq!(pow_mod(g as i64, q as i64, p), 1);
if p < 1_000_000 {
assert_eq!(
(0..p - 1)
.scan(1, |i, _| {
*i = *i * g % p;
Some(*i)
})
.collect::<HashSet<_>>()
.len() as i32,
p - 1
);
}
}
}
}
}
pub mod modint {
//! Structs that treat the modular arithmetic.
//!
//! For most of the problems, It is sufficient to use [`ModInt1000000007`] or [`ModInt998244353`], which can be used as follows.
//!
//! ```
//! use ac_library::ModInt1000000007 as Mint; // rename to whatever you want
//! use proconio::{input, source::once::OnceSource};
//!
//! input! {
//! from OnceSource::from("1000000006 2\n"),
//! a: Mint,
//! b: Mint,
//! }
//!
//! println!("{}", a + b); // `1`
//! ```
//!
//! If the modulus is not fixed, you can use [`ModInt`] as follows.
//!
//! ```
//! use ac_library::ModInt as Mint; // rename to whatever you want
//! use proconio::{input, source::once::OnceSource};
//!
//! input! {
//! from OnceSource::from("3 3 7\n"),
//! a: u32,
//! b: u32,
//! m: u32,
//! }
//!
//! Mint::set_modulus(m);
//! let a = Mint::new(a);
//! let b = Mint::new(b);
//!
//! println!("{}", a * b); // `2`
//! ```
//!
//! # Major changes from the original ACL
//!
//! - Converted the struct names to PascalCase.
//! - Renamed `mod` → `modulus`.
//! - Moduli are `u32`, not `i32`.
//! - Each `Id` does not have a identifier number. Instead, they explicitly own `&'static LocalKey<RefCell<Barrett>>`.
//! - The type of the argument of `pow` is `u64`, not `i64`.
//! - Modints implement `FromStr` and `Display`. Modints in the original ACL don't have `operator<<` or `operator>>`.
//!
//! [`ModInt1000000007`]: ./type.ModInt1000000007.html
//! [`ModInt998244353`]: ./type.ModInt998244353.html
//! [`ModInt`]: ./type.ModInt.html
use crate::internal_math;
use std::{
cell::RefCell,
convert::{Infallible, TryInto as _},
fmt,
hash::{Hash, Hasher},
iter::{Product, Sum},
marker::PhantomData,
ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Sub, SubAssign},
str::FromStr,
sync::atomic::{self, AtomicU32, AtomicU64},
thread::LocalKey,
};
pub type ModInt1000000007 = StaticModInt<Mod1000000007>;
pub type ModInt998244353 = StaticModInt<Mod998244353>;
pub type ModInt = DynamicModInt<DefaultId>;
/// Represents $\mathbb{Z}/m\mathbb{Z}$ where $m$ is a constant value.
///
/// Corresponds to `atcoder::static_modint` in the original ACL.
///
/// # Example
///
/// ```
/// use ac_library::ModInt1000000007 as Mint;
/// use proconio::{input, source::once::OnceSource};
///
/// input! {
/// from OnceSource::from("1000000006 2\n"),
/// a: Mint,
/// b: Mint,
/// }
///
/// println!("{}", a + b); // `1`
/// ```
#[derive(Copy, Clone, Eq, PartialEq)]
#[repr(transparent)]
pub struct StaticModInt<M> {
val: u32,
phantom: PhantomData<fn() -> M>,
}
impl<M: Modulus> StaticModInt<M> {
/// Returns the modulus, which is [`<M as Modulus>::VALUE`].
///
/// Corresponds to `atcoder::static_modint::mod` in the original ACL.
///
/// # Example
///
/// ```
/// use ac_library::ModInt1000000007 as Mint;
///
/// assert_eq!(1_000_000_007, Mint::modulus());
/// ```
///
/// [`<M as Modulus>::VALUE`]: ../trait.Modulus.html#associatedconstant.VALUE
#[inline(always)]
pub fn modulus() -> u32 {
M::VALUE
}
/// Creates a new `StaticModInt`.
///
/// Takes [any primitive integer].
///
/// Corresponds to the constructor of `atcoder::static_modint` in the original ACL.
///
/// [any primitive integer]: ../trait.RemEuclidU32.html
#[inline]
pub fn new<T: RemEuclidU32>(val: T) -> Self {
Self::raw(val.rem_euclid_u32(M::VALUE))
}
/// Constructs a `StaticModInt` from a `val < Self::modulus()` without checking it.
///
/// Corresponds to `atcoder::static_modint::raw` in the original ACL.
///
/// # Constraints
///
/// - `val` is less than `Self::modulus()`
///
/// See [`ModIntBase::raw`] for more more details.
///
/// [`ModIntBase::raw`]: ./trait.ModIntBase.html#tymethod.raw
#[inline]
pub fn raw(val: u32) -> Self {
Self {
val,
phantom: PhantomData,
}
}
/// Retruns the representative.
///
/// Corresponds to `atcoder::static_modint::val` in the original ACL.
#[inline]
pub fn val(self) -> u32 {
self.val
}
/// Returns `self` to the power of `n`.
///
/// Corresponds to `atcoder::static_modint::pow` in the original ACL.
#[inline]
pub fn pow(self, n: u64) -> Self {
<Self as ModIntBase>::pow(self, n)
}
/// Retruns the multiplicative inverse of `self`.
///
/// Corresponds to `atcoder::static_modint::inv` in the original ACL.
///
/// # Panics
///
/// Panics if the multiplicative inverse does not exist.
#[inline]
pub fn inv(self) -> Self {
if M::HINT_VALUE_IS_PRIME {
if self.val() == 0 {
panic!("attempt to divide by zero");
}
debug_assert!(
internal_math::is_prime(M::VALUE.try_into().unwrap()),
"{} is not a prime number",
M::VALUE,
);
self.pow((M::VALUE - 2).into())
} else {
Self::inv_for_non_prime_modulus(self)
}
}
}
/// These methods are implemented for the struct.
/// You don't need to `use` `ModIntBase` to call methods of `StaticModInt`.
impl<M: Modulus> ModIntBase for StaticModInt<M> {
#[inline(always)]
fn modulus() -> u32 {
Self::modulus()
}
#[inline]
fn raw(val: u32) -> Self {
Self::raw(val)
}
#[inline]
fn val(self) -> u32 {
self.val()
}
#[inline]
fn inv(self) -> Self {
self.inv()
}
}
/// Represents a modulus.
///
/// # Example
///
/// ```
/// macro_rules! modulus {
/// ($($name:ident($value:expr, $is_prime:expr)),*) => {
/// $(
/// #[derive(Copy, Clone, Eq, PartialEq)]
/// enum $name {}
///
/// impl ac_library::modint::Modulus for $name {
/// const VALUE: u32 = $value;
/// const HINT_VALUE_IS_PRIME: bool = $is_prime;
///
/// fn butterfly_cache() -> &'static ::std::thread::LocalKey<::std::cell::RefCell<::std::option::Option<ac_library::modint::ButterflyCache<Self>>>> {
/// thread_local! {
/// static BUTTERFLY_CACHE: ::std::cell::RefCell<::std::option::Option<ac_library::modint::ButterflyCache<$name>>> = ::std::default::Default::default();
/// }
/// &BUTTERFLY_CACHE
/// }
/// }
/// )*
/// };
/// }
///
/// use ac_library::StaticModInt;
///
/// modulus!(Mod101(101, true), Mod103(103, true));
///
/// type Z101 = StaticModInt<Mod101>;
/// type Z103 = StaticModInt<Mod103>;
///
/// assert_eq!(Z101::new(101), Z101::new(0));
/// assert_eq!(Z103::new(103), Z103::new(0));
/// ```
pub trait Modulus: 'static + Copy + Eq {
const VALUE: u32;
const HINT_VALUE_IS_PRIME: bool;
fn butterfly_cache() -> &'static LocalKey<RefCell<Option<ButterflyCache<Self>>>>;
}
/// Represents $1000000007$.
#[derive(Copy, Clone, Ord, PartialOrd, Eq, PartialEq, Hash, Debug)]
pub enum Mod1000000007 {}
impl Modulus for Mod1000000007 {
const VALUE: u32 = 1_000_000_007;
const HINT_VALUE_IS_PRIME: bool = true;
fn butterfly_cache() -> &'static LocalKey<RefCell<Option<ButterflyCache<Self>>>> {
thread_local! {
static BUTTERFLY_CACHE: RefCell<Option<ButterflyCache<Mod1000000007>>> = RefCell::default();
}
&BUTTERFLY_CACHE
}
}
/// Represents $998244353$.
#[derive(Copy, Clone, Ord, PartialOrd, Eq, PartialEq, Hash, Debug)]
pub enum Mod998244353 {}
impl Modulus for Mod998244353 {
const VALUE: u32 = 998_244_353;
const HINT_VALUE_IS_PRIME: bool = true;
fn butterfly_cache() -> &'static LocalKey<RefCell<Option<ButterflyCache<Self>>>> {
thread_local! {
static BUTTERFLY_CACHE: RefCell<Option<ButterflyCache<Mod998244353>>> = RefCell::default();
}
&BUTTERFLY_CACHE
}
}
/// Cache for butterfly operations.
pub struct ButterflyCache<M> {
pub(crate) sum_e: Vec<StaticModInt<M>>,
pub(crate) sum_ie: Vec<StaticModInt<M>>,
}
/// Represents $\mathbb{Z}/m\mathbb{Z}$ where $m$ is a dynamic value.
///
/// Corresponds to `atcoder::dynamic_modint` in the original ACL.
///
/// # Example
///
/// ```
/// use ac_library::ModInt as Mint;
/// use proconio::{input, source::once::OnceSource};
///
/// input! {
/// from OnceSource::from("3 3 7\n"),
/// a: u32,
/// b: u32,
/// m: u32,
/// }
///
/// Mint::set_modulus(m);
/// let a = Mint::new(a);
/// let b = Mint::new(b);
///
/// println!("{}", a * b); // `2`
/// ```
#[derive(Copy, Clone, Eq, PartialEq)]
#[repr(transparent)]
pub struct DynamicModInt<I> {
val: u32,
phantom: PhantomData<fn() -> I>,
}
impl<I: Id> DynamicModInt<I> {
/// Returns the modulus.
///
/// Corresponds to `atcoder::dynamic_modint::mod` in the original ACL.
///
/// # Example
///
/// ```
/// use ac_library::ModInt as Mint;
///
/// assert_eq!(998_244_353, Mint::modulus()); // default modulus
/// ```
#[inline]
pub fn modulus() -> u32 {
I::companion_barrett().umod()
}
/// Sets a modulus.
///
/// Corresponds to `atcoder::dynamic_modint::set_mod` in the original ACL.
///
/// # Constraints
///
/// - This function must be called earlier than any other operation of `Self`.
///
/// # Example
///
/// ```
/// use ac_library::ModInt as Mint;
///
/// Mint::set_modulus(7);
/// assert_eq!(7, Mint::modulus());
/// ```
#[inline]
pub fn set_modulus(modulus: u32) {
if modulus == 0 {
panic!("the modulus must not be 0");
}
I::companion_barrett().update(modulus);
}
/// Creates a new `DynamicModInt`.
///
/// Takes [any primitive integer].
///
/// Corresponds to the constructor of `atcoder::dynamic_modint` in the original ACL.
///
/// [any primitive integer]: ../trait.RemEuclidU32.html
#[inline]
pub fn new<T: RemEuclidU32>(val: T) -> Self {
<Self as ModIntBase>::new(val)
}
/// Constructs a `DynamicModInt` from a `val < Self::modulus()` without checking it.
///
/// Corresponds to `atcoder::dynamic_modint::raw` in the original ACL.
///
/// # Constraints
///
/// - `val` is less than `Self::modulus()`
///
/// See [`ModIntBase::raw`] for more more details.
///
/// [`ModIntBase::raw`]: ./trait.ModIntBase.html#tymethod.raw
#[inline]
pub fn raw(val: u32) -> Self {
Self {
val,
phantom: PhantomData,
}
}
/// Retruns the representative.
///
/// Corresponds to `atcoder::static_modint::val` in the original ACL.
#[inline]
pub fn val(self) -> u32 {
self.val
}
/// Returns `self` to the power of `n`.
///
/// Corresponds to `atcoder::dynamic_modint::pow` in the original ACL.
#[inline]
pub fn pow(self, n: u64) -> Self {
<Self as ModIntBase>::pow(self, n)
}
/// Retruns the multiplicative inverse of `self`.
///
/// Corresponds to `atcoder::dynamic_modint::inv` in the original ACL.
///
/// # Panics
///
/// Panics if the multiplicative inverse does not exist.
#[inline]
pub fn inv(self) -> Self {
Self::inv_for_non_prime_modulus(self)
}
}
/// These methods are implemented for the struct.
/// You don't need to `use` `ModIntBase` to call methods of `DynamicModInt`.
impl<I: Id> ModIntBase for DynamicModInt<I> {
#[inline]
fn modulus() -> u32 {
Self::modulus()
}
#[inline]
fn raw(val: u32) -> Self {
Self::raw(val)
}
#[inline]
fn val(self) -> u32 {
self.val()
}
#[inline]
fn inv(self) -> Self {
self.inv()
}
}
pub trait Id: 'static + Copy + Eq {
fn companion_barrett() -> &'static Barrett;
}
#[derive(Copy, Clone, Ord, PartialOrd, Eq, PartialEq, Hash, Debug)]
pub enum DefaultId {}
impl Id for DefaultId {
fn companion_barrett() -> &'static Barrett {
static BARRETT: Barrett = Barrett::default();
&BARRETT
}
}
/// Pair of $m$ and $\lceil 2^{64}/m \rceil$.
pub struct Barrett {
m: AtomicU32,
im: AtomicU64,
}
impl Barrett {
/// Creates a new `Barrett`.
#[inline]
pub const fn new(m: u32) -> Self {
Self {
m: AtomicU32::new(m),
im: AtomicU64::new((-1i64 as u64 / m as u64).wrapping_add(1)),
}
}
#[inline]
const fn default() -> Self {
Self::new(998_244_353)
}
#[inline]
fn update(&self, m: u32) {
let im = (-1i64 as u64 / m as u64).wrapping_add(1);
self.m.store(m, atomic::Ordering::SeqCst);
self.im.store(im, atomic::Ordering::SeqCst);
}
#[inline]
fn umod(&self) -> u32 {
self.m.load(atomic::Ordering::SeqCst)
}
#[inline]
fn mul(&self, a: u32, b: u32) -> u32 {
let m = self.m.load(atomic::Ordering::SeqCst);
let im = self.im.load(atomic::Ordering::SeqCst);
internal_math::mul_mod(a, b, m, im)
}
}
impl Default for Barrett {
#[inline]
fn default() -> Self {
Self::default()
}
}
/// A trait for [`StaticModInt`] and [`DynamicModInt`].
///
/// Corresponds to `atcoder::internal::modint_base` in the original ACL.
///
/// [`StaticModInt`]: ../struct.StaticModInt.html
/// [`DynamicModInt`]: ../struct.DynamicModInt.html
pub trait ModIntBase:
Default
+ FromStr
+ From<i8>
+ From<i16>
+ From<i32>
+ From<i64>
+ From<i128>
+ From<isize>
+ From<u8>
+ From<u16>
+ From<u32>
+ From<u64>
+ From<u128>
+ From<usize>
+ Copy
+ Eq
+ Hash
+ fmt::Display
+ fmt::Debug
+ Neg<Output = Self>
+ Add<Output = Self>
+ Sub<Output = Self>
+ Mul<Output = Self>
+ Div<Output = Self>
+ AddAssign
+ SubAssign
+ MulAssign
+ DivAssign
{
/// Returns the modulus.
///
/// Corresponds to `atcoder::static_modint::mod` and `atcoder::dynamic_modint::mod` in the original ACL.
///
/// # Example
///
/// ```
/// use ac_library::modint::ModIntBase;
///
/// fn f<Z: ModIntBase>() {
/// let _: u32 = Z::modulus();
/// }
/// ```
fn modulus() -> u32;
/// Constructs a `Self` from a `val < Self::modulus()` without checking it.
///
/// Corresponds to `atcoder::static_modint::raw` and `atcoder::dynamic_modint::raw` in the original ACL.
///
/// # Constraints
///
/// - `val` is less than `Self::modulus()`
///
/// **Note that all operations assume that inner values are smaller than the modulus.**
/// If `val` is greater than or equal to `Self::modulus()`, the behaviors are not defined.
///
/// ```should_panic
/// use ac_library::ModInt1000000007 as Mint;
///
/// let x = Mint::raw(1_000_000_007);
/// let y = x + x;
/// assert_eq!(0, y.val());
/// ```
///
/// ```text
/// thread 'main' panicked at 'assertion failed: `(left == right)`
/// left: `0`,
/// right: `1000000007`', src/modint.rs:8:1
/// note: run with `RUST_BACKTRACE=1` environment variable to display a backtrace
/// ```
///
/// # Example
///
/// ```
/// use ac_library::modint::ModIntBase;
///
/// fn f<Z: ModIntBase>() -> Z {
/// debug_assert!(Z::modulus() >= 100);
///
/// let mut acc = Z::new(0);
/// for i in 0..100 {
/// if i % 3 == 0 {
/// // I know `i` is smaller than the modulus!
/// acc += Z::raw(i);
/// }
/// }
/// acc
/// }
/// ```
fn raw(val: u32) -> Self;
/// Retruns the representative.
///
/// Corresponds to `atcoder::static_modint::val` and `atcoder::dynamic_modint::val` in the original ACL.
///
/// # Example
///
/// ```
/// use ac_library::modint::ModIntBase;
///
/// fn f<Z: ModIntBase>(x: Z) {
/// let _: u32 = x.val();
/// }
/// ```
fn val(self) -> u32;
/// Retruns the multiplicative inverse of `self`.
///
/// Corresponds to `atcoder::static_modint::inv` and `atcoder::dynamic_modint::inv` in the original ACL.
///
/// # Panics
///
/// Panics if the multiplicative inverse does not exist.
///
/// # Example
///
/// ```
/// use ac_library::modint::ModIntBase;
///
/// fn f<Z: ModIntBase>(x: Z) {
/// let _: Z = x.inv();
/// }
/// ```
fn inv(self) -> Self;
/// Creates a new `Self`.
///
/// Takes [any primitive integer].
///
/// # Example
///
/// ```
/// use ac_library::modint::ModIntBase;
///
/// fn f<Z: ModIntBase>() {
/// let _ = Z::new(1u32);
/// let _ = Z::new(1usize);
/// let _ = Z::new(-1i64);
/// }
/// ```
///
/// [any primitive integer]: ../trait.RemEuclidU32.html
#[inline]
fn new<T: RemEuclidU32>(val: T) -> Self {
Self::raw(val.rem_euclid_u32(Self::modulus()))
}
/// Returns `self` to the power of `n`.
///
/// Corresponds to `atcoder::static_modint::pow` and `atcoder::dynamic_modint::pow` in the original ACL.
///
/// # Example
///
/// ```
/// use ac_library::modint::ModIntBase;
///
/// fn f<Z: ModIntBase>() {
/// let _: Z = Z::new(2).pow(3);
/// }
/// ```
#[inline]
fn pow(self, mut n: u64) -> Self {
let mut x = self;
let mut r = Self::raw(1);
while n > 0 {
if n & 1 == 1 {
r *= x;
}
x *= x;
n >>= 1;
}
r
}
}
/// A trait for `{StaticModInt, DynamicModInt, ModIntBase}::new`.
pub trait RemEuclidU32 {
/// Calculates `self` $\bmod$ `modulus` losslessly.
fn rem_euclid_u32(self, modulus: u32) -> u32;
}
macro_rules! impl_rem_euclid_u32_for_small_signed {
($($ty:tt),*) => {
$(
impl RemEuclidU32 for $ty {
#[inline]
fn rem_euclid_u32(self, modulus: u32) -> u32 {
(self as i64).rem_euclid(i64::from(modulus)) as _
}
}
)*
}
}
impl_rem_euclid_u32_for_small_signed!(i8, i16, i32, i64, isize);
impl RemEuclidU32 for i128 {
#[inline]
fn rem_euclid_u32(self, modulus: u32) -> u32 {
self.rem_euclid(i128::from(modulus)) as _
}
}
macro_rules! impl_rem_euclid_u32_for_small_unsigned {
($($ty:tt),*) => {
$(
impl RemEuclidU32 for $ty {
#[inline]
fn rem_euclid_u32(self, modulus: u32) -> u32 {
self as u32 % modulus
}
}
)*
}
}
macro_rules! impl_rem_euclid_u32_for_large_unsigned {
($($ty:tt),*) => {
$(
impl RemEuclidU32 for $ty {
#[inline]
fn rem_euclid_u32(self, modulus: u32) -> u32 {
(self % (modulus as $ty)) as _
}
}
)*
}
}
impl_rem_euclid_u32_for_small_unsigned!(u8, u16, u32);
impl_rem_euclid_u32_for_large_unsigned!(u64, u128);
#[cfg(target_pointer_width = "32")]
impl_rem_euclid_u32_for_small_unsigned!(usize);
#[cfg(target_pointer_width = "64")]
impl_rem_euclid_u32_for_large_unsigned!(usize);
trait InternalImplementations: ModIntBase {
#[inline]
fn inv_for_non_prime_modulus(this: Self) -> Self {
let (gcd, x) = internal_math::inv_gcd(this.val().into(), Self::modulus().into());
if gcd != 1 {
panic!("the multiplicative inverse does not exist");
}
Self::new(x)
}
#[inline]
fn default_impl() -> Self {
Self::raw(0)
}
#[inline]
fn from_str_impl(s: &str) -> Result<Self, Infallible> {
Ok(s.parse::<i64>()
.map(Self::new)
.unwrap_or_else(|_| todo!("parsing as an arbitrary precision integer?")))
}
#[inline]
fn hash_impl(this: &Self, state: &mut impl Hasher) {
this.val().hash(state)
}
#[inline]
fn display_impl(this: &Self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
fmt::Display::fmt(&this.val(), f)
}
#[inline]
fn debug_impl(this: &Self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
fmt::Debug::fmt(&this.val(), f)
}
#[inline]
fn neg_impl(this: Self) -> Self {
Self::sub_impl(Self::raw(0), this)
}
#[inline]
fn add_impl(lhs: Self, rhs: Self) -> Self {
let modulus = Self::modulus();
let mut val = lhs.val() + rhs.val();
if val >= modulus {
val -= modulus;
}
Self::raw(val)
}
#[inline]
fn sub_impl(lhs: Self, rhs: Self) -> Self {
let modulus = Self::modulus();
let mut val = lhs.val().wrapping_sub(rhs.val());
if val >= modulus {
val = val.wrapping_add(modulus)
}
Self::raw(val)
}
fn mul_impl(lhs: Self, rhs: Self) -> Self;
#[inline]
fn div_impl(lhs: Self, rhs: Self) -> Self {
Self::mul_impl(lhs, rhs.inv())
}
}
impl<M: Modulus> InternalImplementations for StaticModInt<M> {
#[inline]
fn mul_impl(lhs: Self, rhs: Self) -> Self {
Self::raw((u64::from(lhs.val()) * u64::from(rhs.val()) % u64::from(M::VALUE)) as u32)
}
}
impl<I: Id> InternalImplementations for DynamicModInt<I> {
#[inline]
fn mul_impl(lhs: Self, rhs: Self) -> Self {
Self::raw(I::companion_barrett().mul(lhs.val, rhs.val))
}
}
macro_rules! impl_basic_traits {
() => {};
(impl <$generic_param:ident : $generic_param_bound:tt> _ for $self:ty; $($rest:tt)*) => {
impl <$generic_param: $generic_param_bound> Default for $self {
#[inline]
fn default() -> Self {
Self::default_impl()
}
}
impl <$generic_param: $generic_param_bound> FromStr for $self {
type Err = Infallible;
#[inline]
fn from_str(s: &str) -> Result<Self, Infallible> {
Self::from_str_impl(s)
}
}
impl<$generic_param: $generic_param_bound, V: RemEuclidU32> From<V> for $self {
#[inline]
fn from(from: V) -> Self {
Self::new(from)
}
}
#[allow(clippy::derive_hash_xor_eq)]
impl<$generic_param: $generic_param_bound> Hash for $self {
#[inline]
fn hash<H: Hasher>(&self, state: &mut H) {
Self::hash_impl(self, state)
}
}
impl<$generic_param: $generic_param_bound> fmt::Display for $self {
#[inline]
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
Self::display_impl(self, f)
}
}
impl<$generic_param: $generic_param_bound> fmt::Debug for $self {
#[inline]
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
Self::debug_impl(self, f)
}
}
impl<$generic_param: $generic_param_bound> Neg for $self {
type Output = $self;
#[inline]
fn neg(self) -> $self {
Self::neg_impl(self)
}
}
impl<$generic_param: $generic_param_bound> Neg for &'_ $self {
type Output = $self;
#[inline]
fn neg(self) -> $self {
<$self>::neg_impl(*self)
}
}
impl_basic_traits!($($rest)*);
};
}
impl_basic_traits! {
impl <M: Modulus> _ for StaticModInt<M> ;
impl <I: Id > _ for DynamicModInt<I>;
}
macro_rules! impl_bin_ops {
() => {};
(for<$($generic_param:ident : $generic_param_bound:tt),*> <$lhs_ty:ty> ~ <$rhs_ty:ty> -> $output:ty { { $lhs_body:expr } ~ { $rhs_body:expr } } $($rest:tt)*) => {
impl <$($generic_param: $generic_param_bound),*> Add<$rhs_ty> for $lhs_ty {
type Output = $output;
#[inline]
fn add(self, rhs: $rhs_ty) -> $output {
<$output>::add_impl(apply($lhs_body, self), apply($rhs_body, rhs))
}
}
impl <$($generic_param: $generic_param_bound),*> Sub<$rhs_ty> for $lhs_ty {
type Output = $output;
#[inline]
fn sub(self, rhs: $rhs_ty) -> $output {
<$output>::sub_impl(apply($lhs_body, self), apply($rhs_body, rhs))
}
}
impl <$($generic_param: $generic_param_bound),*> Mul<$rhs_ty> for $lhs_ty {
type Output = $output;
#[inline]
fn mul(self, rhs: $rhs_ty) -> $output {
<$output>::mul_impl(apply($lhs_body, self), apply($rhs_body, rhs))
}
}
impl <$($generic_param: $generic_param_bound),*> Div<$rhs_ty> for $lhs_ty {
type Output = $output;
#[inline]
fn div(self, rhs: $rhs_ty) -> $output {
<$output>::div_impl(apply($lhs_body, self), apply($rhs_body, rhs))
}
}
impl_bin_ops!($($rest)*);
};
}
macro_rules! impl_assign_ops {
() => {};
(for<$($generic_param:ident : $generic_param_bound:tt),*> <$lhs_ty:ty> ~= <$rhs_ty:ty> { _ ~= { $rhs_body:expr } } $($rest:tt)*) => {
impl <$($generic_param: $generic_param_bound),*> AddAssign<$rhs_ty> for $lhs_ty {
#[inline]
fn add_assign(&mut self, rhs: $rhs_ty) {
*self = *self + apply($rhs_body, rhs);
}
}
impl <$($generic_param: $generic_param_bound),*> SubAssign<$rhs_ty> for $lhs_ty {
#[inline]
fn sub_assign(&mut self, rhs: $rhs_ty) {
*self = *self - apply($rhs_body, rhs);
}
}
impl <$($generic_param: $generic_param_bound),*> MulAssign<$rhs_ty> for $lhs_ty {
#[inline]
fn mul_assign(&mut self, rhs: $rhs_ty) {
*self = *self * apply($rhs_body, rhs);
}
}
impl <$($generic_param: $generic_param_bound),*> DivAssign<$rhs_ty> for $lhs_ty {
#[inline]
fn div_assign(&mut self, rhs: $rhs_ty) {
*self = *self / apply($rhs_body, rhs);
}
}
impl_assign_ops!($($rest)*);
};
}
#[inline]
fn apply<F: FnOnce(X) -> O, X, O>(f: F, x: X) -> O {
f(x)
}
impl_bin_ops! {
for<M: Modulus> <StaticModInt<M> > ~ <StaticModInt<M> > -> StaticModInt<M> { { |x| x } ~ { |x| x } }
for<M: Modulus> <StaticModInt<M> > ~ <&'_ StaticModInt<M> > -> StaticModInt<M> { { |x| x } ~ { |&x| x } }
for<M: Modulus> <&'_ StaticModInt<M> > ~ <StaticModInt<M> > -> StaticModInt<M> { { |&x| x } ~ { |x| x } }
for<M: Modulus> <&'_ StaticModInt<M> > ~ <&'_ StaticModInt<M> > -> StaticModInt<M> { { |&x| x } ~ { |&x| x } }
for<I: Id > <DynamicModInt<I> > ~ <DynamicModInt<I> > -> DynamicModInt<I> { { |x| x } ~ { |x| x } }
for<I: Id > <DynamicModInt<I> > ~ <&'_ DynamicModInt<I>> -> DynamicModInt<I> { { |x| x } ~ { |&x| x } }
for<I: Id > <&'_ DynamicModInt<I>> ~ <DynamicModInt<I> > -> DynamicModInt<I> { { |&x| x } ~ { |x| x } }
for<I: Id > <&'_ DynamicModInt<I>> ~ <&'_ DynamicModInt<I>> -> DynamicModInt<I> { { |&x| x } ~ { |&x| x } }
for<M: Modulus, T: RemEuclidU32> <StaticModInt<M> > ~ <T> -> StaticModInt<M> { { |x| x } ~ { StaticModInt::<M>::new } }
for<I: Id , T: RemEuclidU32> <DynamicModInt<I> > ~ <T> -> DynamicModInt<I> { { |x| x } ~ { DynamicModInt::<I>::new } }
}
impl_assign_ops! {
for<M: Modulus> <StaticModInt<M> > ~= <StaticModInt<M> > { _ ~= { |x| x } }
for<M: Modulus> <StaticModInt<M> > ~= <&'_ StaticModInt<M> > { _ ~= { |&x| x } }
for<I: Id > <DynamicModInt<I>> ~= <DynamicModInt<I> > { _ ~= { |x| x } }
for<I: Id > <DynamicModInt<I>> ~= <&'_ DynamicModInt<I>> { _ ~= { |&x| x } }
for<M: Modulus, T: RemEuclidU32> <StaticModInt<M> > ~= <T> { _ ~= { StaticModInt::<M>::new } }
for<I: Id, T: RemEuclidU32> <DynamicModInt<I>> ~= <T> { _ ~= { DynamicModInt::<I>::new } }
}
macro_rules! impl_folding {
() => {};
(impl<$generic_param:ident : $generic_param_bound:tt> $trait:ident<_> for $self:ty { fn $method:ident(_) -> _ { _($unit:expr, $op:expr) } } $($rest:tt)*) => {
impl<$generic_param: $generic_param_bound> $trait<Self> for $self {
#[inline]
fn $method<S>(iter: S) -> Self
where
S: Iterator<Item = Self>,
{
iter.fold($unit, $op)
}
}
impl<'a, $generic_param: $generic_param_bound> $trait<&'a Self> for $self {
#[inline]
fn $method<S>(iter: S) -> Self
where
S: Iterator<Item = &'a Self>,
{
iter.fold($unit, $op)
}
}
impl_folding!($($rest)*);
};
}
impl_folding! {
impl<M: Modulus> Sum<_> for StaticModInt<M> { fn sum(_) -> _ { _(Self::raw(0), Add::add) } }
impl<M: Modulus> Product<_> for StaticModInt<M> { fn product(_) -> _ { _(Self::raw(1), Mul::mul) } }
impl<I: Id > Sum<_> for DynamicModInt<I> { fn sum(_) -> _ { _(Self::raw(0), Add::add) } }
impl<I: Id > Product<_> for DynamicModInt<I> { fn product(_) -> _ { _(Self::raw(1), Mul::mul) } }
}
#[cfg(test)]
mod tests {
use crate::modint::ModInt1000000007;
#[test]
fn static_modint_new() {
assert_eq!(0, ModInt1000000007::new(0u32).val);
assert_eq!(1, ModInt1000000007::new(1u32).val);
assert_eq!(1, ModInt1000000007::new(1_000_000_008u32).val);
assert_eq!(0, ModInt1000000007::new(0u64).val);
assert_eq!(1, ModInt1000000007::new(1u64).val);
assert_eq!(1, ModInt1000000007::new(1_000_000_008u64).val);
assert_eq!(0, ModInt1000000007::new(0usize).val);
assert_eq!(1, ModInt1000000007::new(1usize).val);
assert_eq!(1, ModInt1000000007::new(1_000_000_008usize).val);
assert_eq!(0, ModInt1000000007::new(0i64).val);
assert_eq!(1, ModInt1000000007::new(1i64).val);
assert_eq!(1, ModInt1000000007::new(1_000_000_008i64).val);
assert_eq!(1_000_000_006, ModInt1000000007::new(-1i64).val);
}
#[test]
fn static_modint_add() {
fn add(lhs: u32, rhs: u32) -> u32 {
(ModInt1000000007::new(lhs) + ModInt1000000007::new(rhs)).val
}
assert_eq!(2, add(1, 1));
assert_eq!(1, add(1_000_000_006, 2));
}
#[test]
fn static_modint_sub() {
fn sub(lhs: u32, rhs: u32) -> u32 {
(ModInt1000000007::new(lhs) - ModInt1000000007::new(rhs)).val
}
assert_eq!(1, sub(2, 1));
assert_eq!(1_000_000_006, sub(0, 1));
}
#[test]
fn static_modint_mul() {
fn mul(lhs: u32, rhs: u32) -> u32 {
(ModInt1000000007::new(lhs) * ModInt1000000007::new(rhs)).val
}
assert_eq!(1, mul(1, 1));
assert_eq!(4, mul(2, 2));
assert_eq!(999_999_937, mul(100_000, 100_000));
}
#[test]
fn static_modint_prime_div() {
fn div(lhs: u32, rhs: u32) -> u32 {
(ModInt1000000007::new(lhs) / ModInt1000000007::new(rhs)).val
}
assert_eq!(0, div(0, 1));
assert_eq!(1, div(1, 1));
assert_eq!(1, div(2, 2));
assert_eq!(23_809_524, div(1, 42));
}
#[test]
fn static_modint_sum() {
fn sum(values: &[i64]) -> ModInt1000000007 {
values.iter().copied().map(ModInt1000000007::new).sum()
}
assert_eq!(ModInt1000000007::new(-3), sum(&[-1, 2, -3, 4, -5]));
}
#[test]
fn static_modint_product() {
fn product(values: &[i64]) -> ModInt1000000007 {
values.iter().copied().map(ModInt1000000007::new).product()
}
assert_eq!(ModInt1000000007::new(-120), product(&[-1, 2, -3, 4, -5]));
}
#[test]
fn static_modint_binop_coercion() {
let f = ModInt1000000007::new;
let a = 10_293_812_usize;
let b = 9_083_240_982_usize;
assert_eq!(f(a) + f(b), f(a) + b);
assert_eq!(f(a) - f(b), f(a) - b);
assert_eq!(f(a) * f(b), f(a) * b);
assert_eq!(f(a) / f(b), f(a) / b);
}
#[test]
fn static_modint_assign_coercion() {
let f = ModInt1000000007::new;
let a = f(10_293_812_usize);
let b = 9_083_240_982_usize;
let expected = (((a + b) * b) - b) / b;
let mut c = a;
c += b;
c *= b;
c -= b;
c /= b;
assert_eq!(expected, c);
}
}
}
use modint::*;
pub mod combinatorics {
pub struct Combinatorics {
modulus: usize,
pub factorial: Vec<usize>,
pub reciprocal_of_factorial: Vec<usize>,
}
impl Combinatorics {
pub fn new(modulus: usize, max_n: usize) -> Self {
let mut inv: Vec<usize> = vec![0; max_n + 10];
inv[1] = 1;
for i in 2..=max_n {
inv[i] = modulus - inv[modulus % i] * (modulus / i) % modulus;
}
let mut factorial: Vec<usize> = vec![0; max_n + 10];
let mut reciprocal_of_factorial: Vec<usize> = vec![0; max_n + 10];
for i in 0..=1 {
factorial[i] = 1;
reciprocal_of_factorial[i] = 1;
}
for i in 2..=max_n {
factorial[i] = factorial[i - 1] * i % modulus;
reciprocal_of_factorial[i] = reciprocal_of_factorial[i - 1] * inv[i] % modulus;
}
Self {
modulus,
factorial,
reciprocal_of_factorial,
}
}
pub fn binomial_coefficient(&self, n: isize, k: isize) -> usize {
if n < k || n < 0 || k < 0 {
return 0;
}
self.binomial_coefficient_usize(n as usize, k as usize)
}
pub fn binomial_coefficient_usize(&self, n: usize, k: usize) -> usize {
if n < k {
return 0;
}
let res = self.reciprocal_of_factorial[k] * self.reciprocal_of_factorial[n - k]
% self.modulus;
let res = self.factorial[n] * res % self.modulus;
res
}
pub fn partial_permutation(&self, n: isize, k: isize) -> usize {
if n < k || n < 0 || k < 0 {
return 0;
}
self.partial_permutation_usize(n as usize, k as usize)
}
pub fn partial_permutation_usize(&self, n: usize, k: usize) -> usize {
if n < k {
return 0;
}
self.factorial[n] * self.reciprocal_of_factorial[n - k] % self.modulus
}
}
}
pub mod scanner {
pub struct Scanner {
buf: Vec<String>,
}
impl Scanner {
pub fn new() -> Self {
Self { buf: vec![] }
}
pub fn new_from(source: &str) -> Self {
let source = String::from(source);
let buf = Self::split(source);
Self { buf }
}
pub fn next<T: std::str::FromStr>(&mut self) -> T {
loop {
if let Some(x) = self.buf.pop() {
return x.parse().ok().expect("");
}
let mut source = String::new();
std::io::stdin().read_line(&mut source).expect("");
self.buf = Self::split(source);
}
}
fn split(source: String) -> Vec<String> {
source
.split_whitespace()
.rev()
.map(String::from)
.collect::<Vec<_>>()
}
}
}
use crate::{combinatorics::Combinatorics, scanner::Scanner};
use crate::{ModInt998244353, RemEuclidU32};
use std::io::Write;
type Mint = ModInt998244353;
fn mint<T: RemEuclidU32>(val: T) -> Mint {
Mint::new(val)
}
fn main() {
let mut scanner = Scanner::new();
let out = std::io::stdout();
let mut out = std::io::BufWriter::new(out.lock());
let t: usize = 1;
for _ in 0..t {
solve(&mut scanner, &mut out);
}
}
fn solve(scanner: &mut Scanner, out: &mut std::io::BufWriter<std::io::StdoutLock>) {
let h = scanner.next::<usize>();
let w = scanner.next::<usize>();
let k = scanner.next::<usize>();
let comb = Combinatorics::new(Mint::modulus() as usize, 1_000_010);
let mut ans = mint(0);
for i in 0..=h {
let rem = (h - i) * w;
if rem < k {
continue;
}
if (rem - k) % (h - i) > 0 {
continue;
}
let j = (rem - k) / (h - i);
ans += comb.binomial_coefficient_usize(h, i) * comb.binomial_coefficient_usize(w, j);
}
writeln!(out, "{}", ans.val()).unwrap();
}