#![allow(unused_parens)] #![allow(unused_imports)] #![allow(non_upper_case_globals)] #![allow(non_snake_case)] #![allow(unused_mut)] #![allow(unused_variables)] #![allow(dead_code)] type Vec2 = Vec>; type Vec3 = Vec>>; #[allow(unused_macros)] macro_rules! invec { ( $ t : ty ) => {{ let mut s = String::new(); match std::io::stdin().read_line(&mut s) { Ok(0) => Vec::<$t>::new(), Ok(n) => s .trim() .split_whitespace() .map(|s| s.parse::<$t>().unwrap()) .collect::>(), Err(_) => Vec::<$t>::new(), } }}; } #[allow(unused_macros)] macro_rules! get { ($t:ty) => { { let mut line: String = String::new(); std::io::stdin().read_line(&mut line).unwrap(); line.trim().parse::<$t>().unwrap() } }; ($($t:ty),*) => { { let mut line: String = String::new(); std::io::stdin().read_line(&mut line).unwrap(); let mut iter = line.split_whitespace(); ( $(iter.next().unwrap().parse::<$t>().unwrap(),)* ) } }; ($t:ty; $n:expr) => { (0..$n).map(|_| get!($t) ).collect::>() }; ($($t:ty),*; $n:expr) => { (0..$n).map(|_| get!($($t),*) ).collect::>() }; ($t:ty ;;) => { { let mut line: String = String::new(); std::io::stdin().read_line(&mut line).unwrap(); line.split_whitespace() .map(|t| t.parse::<$t>().unwrap()) .collect::>() } }; ($t:ty ;; $n:expr) => { (0..$n).map(|_| get!($t ;;)).collect::>() }; } #[allow(unused_macros)] macro_rules! input { (source = $s:expr, $($r:tt)*) => { let mut iter = $s.split_whitespace(); input_inner!{iter, $($r)*} }; ($($r:tt)*) => { let mut s = { use std::io::Read; let mut s = String::new(); std::io::stdin().read_to_string(&mut s).unwrap(); s }; let mut iter = s.split_whitespace(); input_inner!{iter, $($r)*} }; } macro_rules! input_inner { ($iter:expr) => {}; ($iter:expr, ) => {}; ($iter:expr, $var:ident : $t:tt $($r:tt)*) => { let $var = read_value!($iter, $t); input_inner!{$iter $($r)*} }; ($iter:expr, mut $var:ident : $t:tt $($r:tt)*) => { let mut $var = read_value!($iter, $t); input_inner!{$iter $($r)*} }; } #[allow(unused_macros)] macro_rules! read_value { ($iter:expr, ( $($t:tt),* )) => { ( $(read_value!($iter, $t)),* ) }; ($iter:expr, [ $t:tt ; $len:expr ]) => { (0..$len).map(|_| read_value!($iter, $t)).collect::>() }; ($next:expr, [$t:tt]) => { { let len = read_value!($next, usize); (0..len).map(|_| read_value!($next, $t)).collect::>() } }; ($iter:expr, chars) => { read_value!($iter, String).chars().collect::>() }; ($iter:expr, usize1) => { read_value!($iter, usize) - 1 }; ($iter:expr, $t:ty) => { $iter.next().unwrap().parse::<$t>().expect("Parse error") }; } #[allow(unused_macros)] #[cfg(debug_assertions)] macro_rules! mydbg { //($arg:expr) => (dbg!($arg)) //($arg:expr) => (println!("{:?}",$arg)); ($($a:expr),*) => { eprintln!(concat!($(stringify!($a), " = {:?}, "),*), $($a),*); } } #[cfg(not(debug_assertions))] macro_rules! mydbg { ($($arg:expr),*) => {}; } macro_rules! echo { ($($a:expr),*) => { $(println!("{}",$a))* } } use std::cmp::*; use std::collections::*; use std::ops::{Add, Div, Mul, Rem, Sub}; trait SafeRangeContain { fn safe_contains(&self, x: i64) -> bool; } impl SafeRangeContain for std::ops::Range { fn safe_contains(&self, x: i64) -> bool { if x < 0 { return false; } return self.contains(&(x as usize)); } } #[allow(dead_code)] static INF_I64: i64 = i64::max_value() / 2; #[allow(dead_code)] static INF_I32: i32 = i32::max_value() / 2; #[allow(dead_code)] static INF_USIZE: usize = usize::max_value() / 2; #[allow(dead_code)] static M_O_D: usize = 1000000007; #[allow(dead_code)] static PAI: f64 = 3.1415926535897932; trait IteratorExt: Iterator { fn toVec(self) -> Vec; } impl IteratorExt for T { fn toVec(self) -> Vec { self.collect() } } trait CharExt { fn toNum(&self) -> usize; fn toAlphabetIndex(&self) -> usize; fn toNumIndex(&self) -> usize; } impl CharExt for char { fn toNum(&self) -> usize { return *self as usize; } fn toAlphabetIndex(&self) -> usize { return self.toNum() - 'a' as usize; } fn toNumIndex(&self) -> usize { return self.toNum() - '0' as usize; } } trait VectorExt { fn joinToString(&self, s: &str) -> String; } impl VectorExt for Vec { fn joinToString(&self, s: &str) -> String { return self .iter() .map(|x| x.to_string()) .collect::>() .join(s); } } trait StringExt { fn get_reverse(&self) -> String; } impl StringExt for String { fn get_reverse(&self) -> String { self.chars().rev().collect::() } } trait UsizeExt { fn pow(&self, n: usize) -> usize; } impl UsizeExt for usize { fn pow(&self, n: usize) -> usize { return ((*self as u64).pow(n as u32)) as usize; } } //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 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 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::>() .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_rs::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_rs::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>`. //! - 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; pub type ModInt998244353 = StaticModInt; pub type ModInt = DynamicModInt; /// Represents where _m_ is a constant value. /// /// Corresponds to `atcoder::static_modint` in the original ACL. /// /// # Example /// /// ``` /// use ac_library_rs::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 { val: u32, phantom: PhantomData M>, } impl StaticModInt { /// Returns the modulus, which is [`::VALUE`]. /// /// Corresponds to `atcoder::static_modint::mod` in the original ACL. /// /// # Example /// /// ``` /// use ac_library_rs::ModInt1000000007 as Mint; /// /// assert_eq!(1_000_000_007, Mint::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(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 { ::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 ModIntBase for StaticModInt { #[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_rs::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>>> { /// thread_local! { /// static BUTTERFLY_CACHE: ::std::cell::RefCell<::std::option::Option>> = ::std::default::Default::default(); /// } /// &BUTTERFLY_CACHE /// } /// } /// )* /// }; /// } /// /// use ac_library_rs::StaticModInt; /// /// modulus!(Mod101(101, true), Mod103(103, true)); /// /// type Z101 = StaticModInt; /// type Z103 = StaticModInt; /// /// 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>>>; } /// 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>>> { thread_local! { static BUTTERFLY_CACHE: RefCell>> = 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>>> { thread_local! { static BUTTERFLY_CACHE: RefCell>> = RefCell::default(); } &BUTTERFLY_CACHE } } /// Cache for butterfly operations. pub struct ButterflyCache { pub(crate) sum_e: Vec>, pub(crate) sum_ie: Vec>, } /// Represents where _m_ is a dynamic value. /// /// Corresponds to `atcoder::dynamic_modint` in the original ACL. /// /// # Example /// /// ``` /// use ac_library_rs::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 { val: u32, phantom: PhantomData I>, } impl DynamicModInt { /// Returns the modulus. /// /// Corresponds to `atcoder::dynamic_modint::mod` in the original ACL. /// /// # Example /// /// ``` /// use ac_library_rs::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_rs::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(val: T) -> Self { ::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 { ::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 ModIntBase for DynamicModInt { #[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 _ceil(2????/m)_. 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 + From + From + From + From + From + From + From + From + From + From + From + Copy + Eq + Hash + fmt::Display + fmt::Debug + Neg + Add + Sub + Mul + Div + 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_rs::modint::ModIntBase; /// /// fn f() { /// 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_rs::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_rs::modint::ModIntBase; /// /// fn f() -> 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_rs::modint::ModIntBase; /// /// fn f(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_rs::modint::ModIntBase; /// /// fn f(x: Z) { /// let _: Z = x.inv(); /// } /// ``` fn inv(self) -> Self; /// Creates a new `Self`. /// /// Takes [any primitive integer]. /// /// # Example /// /// ``` /// use ac_library_rs::modint::ModIntBase; /// /// fn f() { /// let _ = Z::new(1u32); /// let _ = Z::new(1usize); /// let _ = Z::new(-1i64); /// } /// ``` /// /// [any primitive integer]: ../trait.RemEuclidU32.html #[inline] fn new(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_rs::modint::ModIntBase; /// /// fn f() { /// 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` _mod_ `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 { Ok(s.parse::() .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 InternalImplementations for StaticModInt { #[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 InternalImplementations for DynamicModInt { #[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 $self { type Err = Infallible; #[inline] fn from_str(s: &str) -> Result { Self::from_str_impl(s) } } impl<$generic_param: $generic_param_bound, V: RemEuclidU32> From 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 $self { #[inline] fn hash(&self, state: &mut H) { Self::hash_impl(self, state) } } impl<$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 _ for StaticModInt ; impl _ for DynamicModInt; } 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 O, X, O>(f: F, x: X) -> O { f(x) } impl_bin_ops! { for > ~ > -> StaticModInt { { |x| x } ~ { |x| x } } for > ~ <&'_ StaticModInt > -> StaticModInt { { |x| x } ~ { |&x| x } } for <&'_ StaticModInt > ~ > -> StaticModInt { { |&x| x } ~ { |x| x } } for <&'_ StaticModInt > ~ <&'_ StaticModInt > -> StaticModInt { { |&x| x } ~ { |&x| x } } for > ~ > -> DynamicModInt { { |x| x } ~ { |x| x } } for > ~ <&'_ DynamicModInt> -> DynamicModInt { { |x| x } ~ { |&x| x } } for <&'_ DynamicModInt> ~ > -> DynamicModInt { { |&x| x } ~ { |x| x } } for <&'_ DynamicModInt> ~ <&'_ DynamicModInt> -> DynamicModInt { { |&x| x } ~ { |&x| x } } for > ~ -> StaticModInt { { |x| x } ~ { StaticModInt::::new } } for > ~ -> DynamicModInt { { |x| x } ~ { DynamicModInt::::new } } } impl_assign_ops! { for > ~= > { _ ~= { |x| x } } for > ~= <&'_ StaticModInt > { _ ~= { |&x| x } } for > ~= > { _ ~= { |x| x } } for > ~= <&'_ DynamicModInt> { _ ~= { |&x| x } } for > ~= { _ ~= { StaticModInt::::new } } for > ~= { _ ~= { DynamicModInt::::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 for $self { #[inline] fn $method(iter: S) -> Self where S: Iterator, { iter.fold($unit, $op) } } impl<'a, $generic_param: $generic_param_bound> $trait<&'a Self> for $self { #[inline] fn $method(iter: S) -> Self where S: Iterator, { iter.fold($unit, $op) } } impl_folding!($($rest)*); }; } impl_folding! { impl Sum<_> for StaticModInt { fn sum(_) -> _ { _(Self::raw(0), Add::add) } } impl Product<_> for StaticModInt { fn product(_) -> _ { _(Self::raw(1), Mul::mul) } } impl Sum<_> for DynamicModInt { fn sum(_) -> _ { _(Self::raw(0), Add::add) } } impl Product<_> for DynamicModInt { 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::*; use modint::ModInt998244353 as mint; pub struct Combination { fact: Vec, ifact: Vec, } impl Combination { pub fn new(n: i32) -> Self { if n > 3000000 { panic!("error"); } let mut fact = vec![mint::new(0); (n + 1) as usize]; let mut ifact = vec![mint::new(0); (n + 1) as usize]; fact[0] = mint::new(1); for i in 1..n + 1 { fact[i as usize] = fact[(i - 1) as usize] * mint::new(i as i64) } ifact[n as usize] = fact[n as usize].inv(); for i in (1..n + 1).rev() { ifact[(i - 1) as usize] = ifact[i as usize] * mint::new(i as i64) } let a = Combination { fact: fact, ifact: ifact, }; return a; } #[macro_use] pub fn gen(&mut self, n: i32, k: i32) -> mint { if (k < 0 || k > n) { return mint::new(0 as i64); }; return self.fact[n as usize] * self.ifact[k as usize] * self.ifact[(n - k) as usize]; } pub fn P(&mut self, n: i32, k: i32) -> mint { self.fact[n as usize] * self.ifact[(n - k) as usize] } pub fn H(&mut self, n: i32, k: i32) -> mint { self.fact[(n + k - 1) as usize] * self.ifact[k as usize] } } /// nCr nが大きくrが小さいときに使う fn combination(a: usize, b: usize) -> mint { let mut ret = mint::new(1); let mut k = mint::new(a); let mut d = mint::new(1); for i in 1..b { ret *= k; d *= i; k -= 1; } return ret / d; } /// nHr nが大きくrが小さいときに使う fn multichouse(a: usize, sikiri: usize) -> mint { combination(a + sikiri - 1, sikiri) } macro_rules ! max { ( $x : expr ) => ( $x ) ; ( $x : expr , $ ( $xs : expr ) ,+ ) => { { std :: cmp :: max ( $x , max! ( $ ( $xs ) ,+ ) ) } } ; } macro_rules ! min { ( $x : expr ) => ( $x ) ; ( $x : expr , $ ( $xs : expr ) ,+ ) => { { std :: cmp :: min ( $x , min! ( $ ( $xs ) ,+ ) ) } } ; } macro_rules! chmin { ($a:expr, $($b:expr),+) => {{ let d = min!($($b),+); if $a > d { $a = d; true } else { false } }}; } macro_rules! chmax { ($a:expr, $($b:expr),+) => {{ let d = max!($($b),+); if $a < d { $a = d; true } else { false } }}; } macro_rules! chadd { ($a:expr, $b:expr) => {{ $a = $a + $b; }}; } macro_rules! chsub { ($a:expr, $b:expr) => {{ $a = $a - $b; }}; } macro_rules! chmul { ($a:expr, $b:expr) => {{ $a = $a * $b; }}; } fn main() { solve(); } fn solve() { let mut ans = mint::new(0); let N = get!(usize); let B = invec!(i32); let mut a = vec![]; let mut cmb = Combination::new(2 * 100000 + 10); 'outer: for i in 0..N { for j in 0..60 { if i >> j & 1 == 1 { if (N - 1) >> j & 1 == 1 { continue; } else { a.push(0); continue 'outer; } } } a.push(1); } mydbg!(a); let mut dp = vec![vec![mint::new(0); 2]; N + 1]; dp[0][0] = mint::new(1); for i in 0..N { for j in 0..2 { for next in 0..2 { if B[i] != next && B[i] != -1 { continue; } let nj = (if a[i] == 0 { j } else { j ^ next }) as usize; chadd!(dp[i + 1][nj], dp[i][j as usize]); } } } echo!(dp[N][1]); }