From 87bd8421a45096392d563a2b303f020c4417e31a Mon Sep 17 00:00:00 2001 From: koba-e964 <3303362+koba-e964@users.noreply.github.com> Date: Sat, 30 Sep 2023 23:26:02 +0900 Subject: [PATCH] Add yukicoder/2484.rs --- yukicoder/2484.rs | 367 ++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 367 insertions(+) create mode 100644 yukicoder/2484.rs diff --git a/yukicoder/2484.rs b/yukicoder/2484.rs new file mode 100644 index 00000000..65ac40ed --- /dev/null +++ b/yukicoder/2484.rs @@ -0,0 +1,367 @@ +use std::cmp::*; +// https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8 +macro_rules! input { + ($($r:tt)*) => { + let stdin = std::io::stdin(); + let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock())); + let mut next = move || -> String{ + bytes.by_ref().map(|r|r.unwrap() as char) + .skip_while(|c|c.is_whitespace()) + .take_while(|c|!c.is_whitespace()) + .collect() + }; + input_inner!{next, $($r)*} + }; +} + +macro_rules! input_inner { + ($next:expr) => {}; + ($next:expr,) => {}; + ($next:expr, $var:ident : $t:tt $($r:tt)*) => { + let $var = read_value!($next, $t); + input_inner!{$next $($r)*} + }; +} + +macro_rules! read_value { + ($next:expr, [ $t:tt ; $len:expr ]) => { + (0..$len).map(|_| read_value!($next, $t)).collect::>() + }; + ($next:expr, $t:ty) => ($next().parse::<$t>().expect("Parse error")); +} + +/// 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 { pub x: i64, phantom: ::std::marker::PhantomData } + impl ModInt { + // 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 Default for ModInt { + fn default() -> Self { Self::new_internal(0) } + } + impl>> Add for ModInt { + 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>> Sub for ModInt { + 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>> Mul for ModInt { + type Output = Self; + fn mul(self, other: T) -> Self { ModInt::new(self.x * other.into().x % M::m()) } + } + impl>> AddAssign for ModInt { + fn add_assign(&mut self, other: T) { *self = *self + other; } + } + impl>> SubAssign for ModInt { + fn sub_assign(&mut self, other: T) { *self = *self - other; } + } + impl>> MulAssign for ModInt { + fn mul_assign(&mut self, other: T) { *self = *self * other; } + } + impl Neg for ModInt { + type Output = Self; + fn neg(self) -> Self { ModInt::new(0) - self } + } + impl ::std::fmt::Display for ModInt { + fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result { + self.x.fmt(f) + } + } + impl ::std::fmt::Debug for ModInt { + 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 From for ModInt { + 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

; + +// Depends on MInt.rs +fn fact_init(w: usize) -> (Vec, Vec) { + 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(f: &mut [R], zeta: R, one: R) + where R: Copy + + Add + + Sub + + Mul { + 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(f: &mut [R], zeta_inv: R, one: R) + where R: Copy + + Add + + Sub + + Mul { + 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; + } + } + } +} + +// Depends on: fft.rs, MInt.rs +// Verified by: ABC269-Ex (https://atcoder.jp/contests/abc269/submissions/39116328) +pub struct FPSOps { + gen: mod_int::ModInt, +} + +impl FPSOps { + pub fn new(gen: mod_int::ModInt) -> Self { + FPSOps { gen: gen } + } +} + +impl FPSOps { + pub fn add(&self, mut a: Vec>, mut b: Vec>) -> Vec> { + if a.len() < b.len() { + std::mem::swap(&mut a, &mut b); + } + for i in 0..b.len() { + a[i] += b[i]; + } + a + } + pub fn mul(&self, a: Vec>, b: Vec>) -> Vec> { + type MInt = mod_int::ModInt; + let n = a.len() - 1; + let m = b.len() - 1; + let mut p = 1; + while p <= n + m { p *= 2; } + let mut f = vec![MInt::new(0); p]; + let mut g = vec![MInt::new(0); p]; + for i in 0..n + 1 { f[i] = a[i]; } + for i in 0..m + 1 { g[i] = b[i]; } + let fac = MInt::new(p as i64).inv(); + let zeta = self.gen.pow((M::m() - 1) / p as i64); + fft::fft(&mut f, zeta, 1.into()); + fft::fft(&mut g, zeta, 1.into()); + for i in 0..p { f[i] *= g[i] * fac; } + fft::inv_fft(&mut f, zeta.inv(), 1.into()); + f.truncate(n + m + 1); + f + } +} + +// https://yukicoder.me/problems/no/2484 (3.5) +// https://yukicoder.me/problems/no/2485 (3.5) +// 操作をそれぞれ S, T_{k+1}, U_{k+1}, V と名付ける。T_k U_{k+1} と SV は同じ結果をもたらすが、それ以外はほとんど自由度がなく 1 通りに定まる。 +// c[i] = B[i+1]-B[i] (i >= 1), c[0] = B[1] とすると、各操作は以下のようになる: +// S: なにもしない +// T_{k+1}: 0 番目を +1, k+1 番目を -1 する (0 <= k <= N-2) +// U_{k+1}: k 番目を +1 する (1 <= k <= N-1) +// V: 0 番目を +1 する +// 最終的にすべてゼロの配列を操作によって c と等しくできればよい。 +// c[0] + sum_{i>=1} min(c[i], 0) < 0 であれば不可能なので 0 通り。 +// そうでないとき、T_? と U_? の必須回数、および追加で必要な V の回数は簡単に求められる。 +// V を何個 T_1 U_2, T_2 U_3, T_3 U_4 にするかを全探索すれば、それぞれに対して組み合わせの和を計算すれば良い。 +// これは畳み込みでできる。 +fn main() { + input! { + n: usize, m: usize, + b: [i64; n], + } + let (fac, invfac) = fact_init(m + 1); + let mut c = vec![0; n]; + c[0] = b[0]; + for i in 1..n { + c[i] = b[i] - b[i - 1]; + } + let mut negsum = 0; + let mut possum = 0; + for i in 1..n { + negsum += min(0, c[i]); + possum += max(0, c[i]); + } + if negsum + c[0] < 0 { + println!("0"); + return; + } + // Rules out e.g. m m-1 m + if possum + c[0] > m as i64 { + println!("0"); + return; + } + let rest = m - (possum + c[0]) as usize; + let t = (c[0] + negsum) as usize; + let mut prod = vec![MInt::new(1)]; + let fps = FPSOps::new(MInt::new(3)); + for i in 1..n { + let mut a = vec![MInt::new(0); m + 1]; + let x = c[i].abs() as usize; + for j in 0..m + 1 { + if 2 * j + x <= m { + a[2 * j + x] += invfac[j + x] * invfac[j]; + } + } + prod = fps.mul(prod, a); + prod.truncate(m + 1); + } + let mut a = vec![MInt::new(0); m + 1]; + for j in 0..min(t, rest) + 1 { + if t - j + rest - j <= m { + a[t - j + rest - j] += invfac[t - j] * invfac[rest - j]; + } + } + prod = fps.mul(prod, a); + prod.truncate(m + 1); + println!("{}", prod[m] * fac[m]); +}