Add yukicoder/2484.rs

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, $t)).collect::<Vec<_>>()
+    };
+    ($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<M> { pub x: i64, phantom: ::std::marker::PhantomData<M> }
+    impl<M: Mod> ModInt<M> {
+        // x >= 0
+        pub fn new(x: i64) -> Self { ModInt::new_internal(x % M::m()) }
+        fn new_internal(x: i64) -> Self {
+            ModInt { x: x, phantom: ::std::marker::PhantomData }
+        }
+        pub fn pow(self, mut e: i64) -> Self {
+            debug_assert!(e >= 0);
+            let mut sum = ModInt::new_internal(1);
+            let mut cur = self;
+            while e > 0 {
+                if e % 2 != 0 { sum *= cur; }
+                cur *= cur;
+                e /= 2;
+            }
+            sum
+        }
+        #[allow(dead_code)]
+        pub fn inv(self) -> Self { self.pow(M::m() - 2) }
+    }
+    impl<M: Mod> Default for ModInt<M> {
+        fn default() -> Self { Self::new_internal(0) }
+    }
+    impl<M: Mod, T: Into<ModInt<M>>> Add<T> for ModInt<M> {
+        type Output = Self;
+        fn add(self, other: T) -> Self {
+            let other = other.into();
+            let mut sum = self.x + other.x;
+            if sum >= M::m() { sum -= M::m(); }
+            ModInt::new_internal(sum)
+        }
+    }
+    impl<M: Mod, T: Into<ModInt<M>>> Sub<T> for ModInt<M> {
+        type Output = Self;
+        fn sub(self, other: T) -> Self {
+            let other = other.into();
+            let mut sum = self.x - other.x;
+            if sum < 0 { sum += M::m(); }
+            ModInt::new_internal(sum)
+        }
+    }
+    impl<M: Mod, T: Into<ModInt<M>>> Mul<T> for ModInt<M> {
+        type Output = Self;
+        fn mul(self, other: T) -> Self { ModInt::new(self.x * other.into().x % M::m()) }
+    }
+    impl<M: Mod, T: Into<ModInt<M>>> AddAssign<T> for ModInt<M> {
+        fn add_assign(&mut self, other: T) { *self = *self + other; }
+    }
+    impl<M: Mod, T: Into<ModInt<M>>> SubAssign<T> for ModInt<M> {
+        fn sub_assign(&mut self, other: T) { *self = *self - other; }
+    }
+    impl<M: Mod, T: Into<ModInt<M>>> MulAssign<T> for ModInt<M> {
+        fn mul_assign(&mut self, other: T) { *self = *self * other; }
+    }
+    impl<M: Mod> Neg for ModInt<M> {
+        type Output = Self;
+        fn neg(self) -> Self { ModInt::new(0) - self }
+    }
+    impl<M> ::std::fmt::Display for ModInt<M> {
+        fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
+            self.x.fmt(f)
+        }
+    }
+    impl<M: Mod> ::std::fmt::Debug for ModInt<M> {
+        fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
+            let (mut a, mut b, _) = red(self.x, M::m());
+            if b < 0 {
+                a = -a;
+                b = -b;
+            }
+            write!(f, "{}/{}", a, b)
+        }
+    }
+    impl<M: Mod> From<i64> for ModInt<M> {
+        fn from(x: i64) -> Self { Self::new(x) }
+    }
+    // Finds the simplest fraction x/y congruent to r mod p.
+    // The return value (x, y, z) satisfies x = y * r + z * p.
+    fn red(r: i64, p: i64) -> (i64, i64, i64) {
+        if r.abs() <= 10000 {
+            return (r, 1, 0);
+        }
+        let mut nxt_r = p % r;
+        let mut q = p / r;
+        if 2 * nxt_r >= r {
+            nxt_r -= r;
+            q += 1;
+        }
+        if 2 * nxt_r <= -r {
+            nxt_r += r;
+            q -= 1;
+        }
+        let (x, z, y) = red(nxt_r, r);
+        (x, y - q * z, z)
+    }
+} // mod mod_int
+
+macro_rules! define_mod {
+    ($struct_name: ident, $modulo: expr) => {
+        #[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Hash)]
+        pub struct $struct_name {}
+        impl mod_int::Mod for $struct_name { fn m() -> i64 { $modulo } }
+    }
+}
+const MOD: i64 = 998_244_353;
+define_mod!(P, MOD);
+type MInt = mod_int::ModInt<P>;
+
+// Depends on MInt.rs
+fn fact_init(w: usize) -> (Vec<MInt>, Vec<MInt>) {
+    let mut fac = vec![MInt::new(1); w];
+    let mut invfac = vec![0.into(); w];
+    for i in 1..w {
+        fac[i] = fac[i - 1] * i as i64;
+    }
+    invfac[w - 1] = fac[w - 1].inv();
+    for i in (0..w - 1).rev() {
+        invfac[i] = invfac[i + 1] * (i as i64 + 1);
+    }
+    (fac, invfac)
+}
+
+// FFT (in-place, verified as NTT only)
+// R: Ring + Copy
+// Verified by: https://judge.yosupo.jp/submission/53831
+// Adopts the technique used in https://judge.yosupo.jp/submission/3153.
+mod fft {
+    use std::ops::*;
+    // n should be a power of 2. zeta is a primitive n-th root of unity.
+    // one is unity
+    // Note that the result is bit-reversed.
+    pub fn fft<R>(f: &mut [R], zeta: R, one: R)
+        where R: Copy +
+        Add<Output = R> +
+        Sub<Output = R> +
+        Mul<Output = R> {
+        let n = f.len();
+        assert!(n.is_power_of_two());
+        let mut m = n;
+        let mut base = zeta;
+        unsafe {
+            while m > 2 {
+                m >>= 1;
+                let mut r = 0;
+                while r < n {
+                    let mut w = one;
+                    for s in r..r + m {
+                        let &u = f.get_unchecked(s);
+                        let d = *f.get_unchecked(s + m);
+                        *f.get_unchecked_mut(s) = u + d;
+                        *f.get_unchecked_mut(s + m) = w * (u - d);
+                        w = w * base;
+                    }
+                    r += 2 * m;
+                }
+                base = base * base;
+            }
+            if m > 1 {
+                // m = 1
+                let mut r = 0;
+                while r < n {
+                    let &u = f.get_unchecked(r);
+                    let d = *f.get_unchecked(r + 1);
+                    *f.get_unchecked_mut(r) = u + d;
+                    *f.get_unchecked_mut(r + 1) = u - d;
+                    r += 2;
+                }
+            }
+        }
+    }
+    pub fn inv_fft<R>(f: &mut [R], zeta_inv: R, one: R)
+        where R: Copy +
+        Add<Output = R> +
+        Sub<Output = R> +
+        Mul<Output = R> {
+        let n = f.len();
+        assert!(n.is_power_of_two());
+        let zeta = zeta_inv; // inverse FFT
+        let mut zetapow = Vec::with_capacity(20);
+        {
+            let mut m = 1;
+            let mut cur = zeta;
+            while m < n {
+                zetapow.push(cur);
+                cur = cur * cur;
+                m *= 2;
+            }
+        }
+        let mut m = 1;
+        unsafe {
+            if m < n {
+                zetapow.pop();
+                let mut r = 0;
+                while r < n {
+                    let &u = f.get_unchecked(r);
+                    let d = *f.get_unchecked(r + 1);
+                    *f.get_unchecked_mut(r) = u + d;
+                    *f.get_unchecked_mut(r + 1) = u - d;
+                    r += 2;
+                }
+                m = 2;
+            }
+            while m < n {
+                let base = zetapow.pop().unwrap();
+                let mut r = 0;
+                while r < n {
+                    let mut w = one;
+                    for s in r..r + m {
+                        let &u = f.get_unchecked(s);
+                        let d = *f.get_unchecked(s + m) * w;
+                        *f.get_unchecked_mut(s) = u + d;
+                        *f.get_unchecked_mut(s + m) = u - d;
+                        w = w * base;
+                    }
+                    r += 2 * m;
+                }
+                m *= 2;
+            }
+        }
+    }
+}
+
+// Depends on: fft.rs, MInt.rs
+// Verified by: ABC269-Ex (https://atcoder.jp/contests/abc269/submissions/39116328)
+pub struct FPSOps<M: mod_int::Mod> {
+    gen: mod_int::ModInt<M>,
+}
+
+impl<M: mod_int::Mod> FPSOps<M> {
+    pub fn new(gen: mod_int::ModInt<M>) -> Self {
+        FPSOps { gen: gen }
+    }
+}
+
+impl<M: mod_int::Mod> FPSOps<M> {
+    pub fn add(&self, mut a: Vec<mod_int::ModInt<M>>, mut b: Vec<mod_int::ModInt<M>>) -> Vec<mod_int::ModInt<M>> {
+        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<mod_int::ModInt<M>>, b: Vec<mod_int::ModInt<M>>) -> Vec<mod_int::ModInt<M>> {
+        type MInt<M> = mod_int::ModInt<M>;
+        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]);
+}