結果
| 問題 |
No.1938 Lagrange Sum
|
| コンテスト | |
| ユーザー |
 koba-e964
|
| 提出日時 | 2023-07-28 11:34:47 |
| 言語 | Rust (1.83.0 + proconio) |
| 結果 |
AC
|
| 実行時間 | 162 ms / 3,000 ms |
| コード長 | 16,780 bytes |
| コンパイル時間 | 14,647 ms |
| コンパイル使用メモリ | 383,400 KB |
| 実行使用メモリ | 8,372 KB |
| 最終ジャッジ日時 | 2024-10-06 00:26:29 |
| 合計ジャッジ時間 | 16,561 ms |
|
ジャッジサーバーID （参考情報） |
judge5 / judge4 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 3 |
| other | AC * 25 |
ソースコード
// 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),* )) => { ($(read_value!($next, $t)),*) };
($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>;
// 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 = P> {
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
}
}
// Computes f^{-1} mod x^{f.len()}.
// Reference: https://codeforces.com/blog/entry/56422
// Complexity: O(n log n)
// Verified by: https://judge.yosupo.jp/submission/3219
// Depends on: MInt.rs, fft.rs
fn fps_inv<P: mod_int::Mod + PartialEq>(
f: &[mod_int::ModInt<P>],
gen: mod_int::ModInt<P>
) -> Vec<mod_int::ModInt<P>> {
let n = f.len();
assert!(n.is_power_of_two());
assert_eq!(f[0], 1.into());
let mut sz = 1;
let mut r = vec![mod_int::ModInt::new(0); n];
let mut tmp_f = vec![mod_int::ModInt::new(0); n];
let mut tmp_r = vec![mod_int::ModInt::new(0); n];
r[0] = 1.into();
// Adopts the technique used in https://judge.yosupo.jp/submission/3153
while sz < n {
let zeta = gen.pow((P::m() - 1) / sz as i64 / 2);
tmp_f[..2 * sz].copy_from_slice(&f[..2 * sz]);
tmp_r[..2 * sz].copy_from_slice(&r[..2 * sz]);
fft::fft(&mut tmp_r[..2 * sz], zeta, 1.into());
fft::fft(&mut tmp_f[..2 * sz], zeta, 1.into());
let fac = mod_int::ModInt::new(2 * sz as i64).inv().pow(2);
for i in 0..2 * sz {
tmp_f[i] *= tmp_r[i];
}
fft::inv_fft(&mut tmp_f[..2 * sz], zeta.inv(), 1.into());
for v in &mut tmp_f[..sz] {
*v = 0.into();
}
fft::fft(&mut tmp_f[..2 * sz], zeta, 1.into());
for i in 0..2 * sz {
tmp_f[i] = -tmp_f[i] * tmp_r[i] * fac;
}
fft::inv_fft(&mut tmp_f[..2 * sz], zeta.inv(), 1.into());
r[sz..2 * sz].copy_from_slice(&tmp_f[sz..2 * sz]);
sz *= 2;
}
r
}
type M = MInt;
// Copied and modified from https://judge.yosupo.jp/submission/133199.
// Originally by sansen.
fn middle_product(c: &[M], a: &[M]) -> Vec<M> {
assert!(c.len() >= a.len());
if a.len() <= (1 << 5) {
return c
.windows(a.len())
.map(|c| {
c.iter()
.zip(a.iter())
.fold(MInt::new(0), |s, a| s + *a.0 * *a.1)
})
.collect();
}
let size = c.len().next_power_of_two();
let mut x = Vec::from(c);
x.resize(size, MInt::new(0));
let mut y = Vec::from(a);
y.reverse();
y.resize(size, MInt::new(0));
let zeta = MInt::new(3).pow((MOD - 1) / size as i64);
fft::fft(&mut x, zeta, 1.into());
fft::fft(&mut y, zeta, 1.into());
let factor = MInt::new(size as i64).inv();
for i in 0..size {
x[i] *= y[i] * factor;
}
fft::inv_fft(&mut x, zeta.inv(), 1.into());
(a.len()..=c.len()).map(|z| x[z - 1]).collect()
}
fn multipoint_evaluation(ops: &FPSOps, c: &[MInt], p: &[MInt]) -> Vec<M> {
if p.is_empty() {
return vec![];
}
let n = c.len();
let m = p.len();
let mut prod = vec![vec![]; 2 * m];
for (prod, p) in prod[m..].iter_mut().zip(p.iter()) {
*prod = vec![MInt::new(1), -*p];
}
for i in (1..m).rev() {
prod[i] = ops.mul(prod[2 * i].clone(), prod[2 * i + 1].clone());
}
let mut prod1 = prod[1].clone();
let mut sz = 1;
while sz < n { sz *= 2; }
prod1.resize(sz, 0.into());
let mut inv = fps_inv(&prod1, 3.into());
inv.truncate(n);
let mut c = c.to_vec();
c.resize(n + m - 1, MInt::new(0));
let mut dp = vec![vec![]; 2 * m];
dp[1] = middle_product(&c, &inv);
for i in 1..m {
dp[2 * i] = middle_product(&dp[i], &prod[2 * i + 1]);
dp[2 * i + 1] = middle_product(&dp[i], &prod[2 * i]);
}
dp[m..].iter().map(|dp| dp[0]).collect()
}
// End of copy-pasted part.
fn fps_mul_all(ops: &FPSOps, f: &[Vec<MInt>]) -> Vec<MInt> {
let m = f.len();
let mut seg = vec![vec![]; 2 * m];
for i in 0..m {
seg[i + m] = f[i].to_vec();
}
for i in (1..m).rev() {
seg[i] = ops.mul(
std::mem::replace(&mut seg[2 * i], vec![]),
std::mem::replace(&mut seg[2 * i + 1], vec![]),
);
}
std::mem::replace(&mut seg[1], vec![])
}
fn fps_common_denom(ops: &FPSOps, frac: &[(Vec<MInt>, Vec<MInt>)]) -> (Vec<MInt>, Vec<MInt>) {
let m = frac.len();
let mut seg = vec![(vec![], vec![]); 2 * m];
for i in 0..m {
seg[i + m] = frac[i].clone();
}
for i in (1..m).rev() {
let den = ops.mul(seg[2 * i].1.clone(), seg[2 * i + 1].1.clone());
let mut num = ops.mul(
std::mem::replace(&mut seg[2 * i].1, vec![]),
std::mem::replace(&mut seg[2 * i + 1].0, vec![]),
);
let tmp = ops.mul(
std::mem::replace(&mut seg[2 * i].0, vec![]),
std::mem::replace(&mut seg[2 * i + 1].1, vec![]),
);
num = ops.add(num, tmp);
seg[i] = (num, den);
}
std::mem::replace(&mut seg[1], (vec![], vec![]))
}
// https://37zigen.com/lagrange-interpolation/
fn lagrange_interpolate(ops: &FPSOps, xy: &[(MInt, MInt)]) -> Vec<MInt> {
let n = xy.len();
let mut xs = vec![MInt::new(0); n];
let mut ps = vec![vec![]; n];
for i in 0..n {
xs[i] = xy[i].0;
ps[i] = vec![-xy[i].0, 1.into()];
}
let g = fps_mul_all(ops, &ps);
let mut gdash = vec![MInt::new(0); n];
for i in 0..n {
gdash[i] = g[i + 1] * (i + 1) as i64;
}
let vals = multipoint_evaluation(ops, &gdash, &xs);
let mut fracs = vec![(vec![MInt::new(1)], vec![]); n];
for i in 0..n {
fracs[i].0[0] = vals[i].inv() * xy[i].1;
fracs[i].1 = vec![-xy[i].0, 1.into()];
}
let (num, _) = fps_common_denom(ops, &fracs);
num
}
// https://yukicoder.me/problems/no/1938 (4)
// X がいずれかの x_i と等しい場合、F(X) = (N-1)y_i + F_i(x_i) だから 1 回多項式補間をするだけでよい。
// そうでない場合、G(x) を (x_i, y_i) すべてで補間した N-1 次多項式とすると、
// F(X) = NG(X) - (\sum_i A_i / (X - x_i)) (\prod (X - x_i)) where A_i := (y_i - F_i(x_i)) / \prod_{j != i} (x_i - x_j) である。
// A_i はすべて [x^{N-1}]G(x) に等しいため、これは計算できる。
// 計算量は多項式補間がボトルネックであるため O(N log^2 N) である。
fn main() {
input! {
n: usize, bigx: i64,
xy: [(i64, i64); n],
}
let ops = FPSOps {
gen: 3.into(),
};
let xy: Vec<_> = xy.into_iter().map(|(x, y)| (MInt::new(x), MInt::new(y))).collect();
let mut idx = n;
for i in 0..n {
if MInt::new(bigx) == xy[i].0 {
idx = i;
break;
}
}
if idx < n {
let mut rm = xy.clone();
rm.remove(idx);
let p = lagrange_interpolate(&ops, &rm);
let mut ans = xy[idx].1 * (n - 1) as i64;
let mut cur = MInt::new(1);
for i in 0..p.len() {
ans += cur * p[i];
cur *= xy[idx].0;
}
println!("{}", ans);
return;
}
let g = lagrange_interpolate(&ops, &xy);
let lead = g[n - 1];
let mut fracs = vec![(vec![MInt::new(1)], vec![]); n];
for i in 0..n {
fracs[i].1 = vec![-xy[i].0, 1.into()];
}
let (num, _) = fps_common_denom(&ops, &fracs);
let mut ans = MInt::new(0);
let mut cur = MInt::new(1);
for i in 0..n {
ans += cur * (g[i] * n as i64 - num[i] * lead);
cur *= bigx;
}
println!("{}", ans);
}