結果
| 問題 |
No.2904 Distinct Multisets in a Way
|
| コンテスト | |
| ユーザー |
 Today03
|
| 提出日時 | 2024-09-28 03:52:37 |
| 言語 | C++23 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 508 ms / 2,000 ms |
| コード長 | 8,316 bytes |
| コンパイル時間 | 5,328 ms |
| コンパイル使用メモリ | 296,456 KB |
| 実行使用メモリ | 24,312 KB |
| 最終ジャッジ日時 | 2024-09-28 03:52:50 |
| 合計ジャッジ時間 | 13,151 ms |
|
ジャッジサーバーID （参考情報） |
judge4 / judge2 |
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| ファイルパターン | 結果 |
|---|---|
| sample | AC * 2 |
| other | AC * 42 |
ソースコード
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
const int INF = 1e9 + 10;
const ll INFL = 4e18;
/*
https://x.com/hamamu_kyopro/status/1839680787920322683
https://x.com/eco4kb/status/1839666741406011512
f(x)=(1+x)(1+x^2)(1+x^4)…(1+x^(2^(N-1)))として、
f(x)^2=g(x)h(x)(ただし、係数和が両方ともN)と因数分解する方法に帰着される。
*/
/* from: https://potato167.github.io/po167_library */
#include <atcoder/convolution>
#include <atcoder/modint>
template <typename T>
vector<T> fpsMul(const vector<T> &a, const vector<T> &b) {
return atcoder::convolution(a, b);
}
template <typename T>
vector<T> fpsAdd(const vector<T> &a, const vector<T> &b) {
vector<T> res(max(a.size(), b.size()));
for (int i = 0; i < res.size(); i++) {
if (i < a.size()) res[i] += a[i];
if (i < b.size()) res[i] += b[i];
}
return res;
}
template <typename T>
vector<T> fpsSub(const vector<T> &a, const vector<T> &b) {
vector<T> res(max(a.size(), b.size()));
for (int i = 0; i < res.size(); i++) {
if (i < a.size()) res[i] += a[i];
if (i < b.size()) res[i] -= b[i];
}
return res;
}
template <typename T>
vector<T> fpsInv(vector<T> f, int len = -1) {
if (len == -1) len = f.size();
assert(f[0] != 0);
vector<T> g = {1 / f[0]};
int s = 1;
while (s < len) {
// g = 2g_s - f(g_s)^2 (mod x ^ (2 * s))
// g = g - (fg - 1)g
// (fg - 1) = 0 (mod x ^ (s))
vector<T> n_g(s * 2, 0);
vector<T> f_s(s * 2, 0);
g.resize(s * 2);
for (int i = 0; i < s * 2; i++) {
if (int(f.size()) > i) f_s[i] = f[i];
n_g[i] = g[i];
}
atcoder::internal::butterfly(g);
atcoder::internal::butterfly(f_s);
for (int i = 0; i < s * 2; i++) f_s[i] *= g[i];
atcoder::internal::butterfly_inv(f_s);
T iz = 1 / (T)(s * 2);
for (int i = s; i < s * 2; i++) f_s[i] *= iz;
for (int i = 0; i < s; i++) f_s[i] = 0;
atcoder::internal::butterfly(f_s);
for (int i = 0; i < s * 2; i++) f_s[i] *= g[i];
atcoder::internal::butterfly_inv(f_s);
for (int i = s; i < s * 2; i++) n_g[i] -= f_s[i] * iz;
swap(n_g, g);
s *= 2;
}
g.resize(len);
return g;
}
template <typename T>
vector<T> fpsCyclicConvolution(vector<T> f, vector<T> g) {
atcoder::internal::butterfly(f);
atcoder::internal::butterfly(g);
for (int i = 0; i < (int)f.size(); i++) f[i] *= g[i];
atcoder::internal::butterfly_inv(f);
T iz = (T)(1) / (T)(f.size());
for (int i = 0; i < (int)f.size(); i++) f[i] *= iz;
return f;
}
template <typename T>
vector<T> fpsIntegral(vector<T> f) {
if (f.empty()) return f;
vector<T> num_inv((int)f.size() + 1);
num_inv[0] = 1;
num_inv[1] = 1;
auto m = T::mod();
for (int i = 2; i <= (int)f.size(); i++) num_inv[i] = (0 - num_inv[m % i]) * (T)(m / i);
f.reserve((int)f.size() + 1);
f.push_back(0);
for (int i = (int)f.size() - 1; i > 0; i--) f[i] = f[i - 1] * num_inv[i];
f[0] = 0;
return f;
}
template <typename T>
vector<T> fpsDifferential(vector<T> f) {
if (f.empty()) return f;
for (int i = 0; i < (int)f.size() - 1; i++) f[i] = f[i + 1] * (T)(i + 1);
f.pop_back();
return f;
}
template <typename T>
vector<T> fpsExp(vector<T> f, int len = -1) {
if (len == -1) len = f.size();
if (len == 0) return {};
if (len == 1) return {T(1)};
assert(!f.empty() && f[0] == 0);
int s = 1;
// simple
vector<T> g = {T(1)};
while (s < len) {
// g' / g
// A * B
vector<T> A = g, B = g;
A = fpsDifferential(A);
B = fpsInv(B, 2 * s);
A.resize(2 * s);
A = fpsCyclicConvolution(A, B);
A.pop_back();
A = fpsIntegral(A);
for (int i = 0; i < s; i++) A[i] = 0;
for (int i = s; i < s * 2; i++) A[i] = (i < (int)f.size() ? f[i] : 0) - A[i];
// g_hat = g (1 - g + f)
// g += B = g * A
g.resize(2 * s);
B = fpsCyclicConvolution(A, g);
for (int i = s; i < s * 2; i++) g[i] = B[i];
s *= 2;
}
g.resize(len);
return g;
}
template <typename T>
vector<T> fpsLog(vector<T> f, int len = -1) {
if (len == -1) len = f.size();
if (len == 0) return {};
if (len == 1) return {T(0)};
assert(!f.empty() && f[0] == 1);
vector<T> res = atcoder::convolution(fpsDifferential(f), fpsInv(f, len));
res.resize(len - 1);
return fpsIntegral(res);
}
template <class T>
vector<T> fpsPow(vector<T> f, long long M, int len = -1) {
if (len == -1) len = f.size();
vector<T> res(len, 0);
if (M == 0) {
res[0] = 1;
return res;
}
for (int i = 0; i < (int)f.size(); i++) {
if (f[i] == 0) continue;
if (i > (len - 1) / M) break;
vector<T> g((int)f.size() - i);
T v = (T)(1) / (T)(f[i]);
for (int j = i; j < (int)f.size(); j++) g[j - i] = f[j] * v;
long long zero = i * M;
if (i) len -= i * M;
g = fpsLog(g, len);
for (T &x : g) x *= M;
g = fpsExp(g, len);
v = (T)(1) / v;
T c = 1;
while (M) {
if (M & 1) c = c * v;
v = v * v;
M >>= 1;
}
for (int i = 0; i < len; i++) res[i + zero] = g[i] * c;
return res;
}
return res;
}
// in : DFT(v) (len(v) = z)
// out : DFT(v) (len(v) = 2 * z)
template <typename T>
void fpsExtend(vector<T> &v) {
int z = v.size();
T e = (T(atcoder::internal::primitive_root_constexpr(T::mod()))).pow(T::mod() / (2 * z));
auto cp = v;
atcoder::internal::butterfly_inv(cp);
T tmp = (T)(1) / (T)(z);
for (int i = 0; i < z; i++) {
cp[i] *= tmp;
tmp *= e;
}
atcoder::internal::butterfly(cp);
for (int i = 0; i < z; i++) v.push_back(cp[i]);
}
// s.t |v| = 2 ^ s (no assert)
template <typename T>
void fpsPickEvenOdd(vector<T> &v, int odd) {
int z = v.size() / 2;
T half = (T)(1) / (T)(2);
if (odd == 0) {
for (int i = 0; i < z; i++) v[i] = (v[i * 2] + v[i * 2 + 1]) * half;
v.resize(z);
} else {
T e = (T(atcoder::internal::primitive_root_constexpr(T::mod()))).pow(T::mod() / (2 * z));
T ie = T(1) / e;
vector<T> es = {half};
while ((int)es.size() != z) {
vector<T> n_es((int)es.size() * 2);
for (int i = 0; i < (int)es.size(); i++) {
n_es[i * 2] = (es[i]);
n_es[i * 2 + 1] = (es[i] * ie);
}
ie *= ie;
swap(n_es, es);
}
for (int i = 0; i < z; i++) v[i] = (v[i * 2] - v[i * 2 + 1]) * es[i];
v.resize(z);
}
}
// return [x^k] P(x) / Q(x)
template <typename T>
T bostanMori(long long k, vector<T> P, vector<T> Q) {
assert(!Q.empty() && Q[0] != 0);
int z = 1;
while (z < (int)max(P.size(), Q.size())) z *= 2;
P.resize(z * 2, 0);
Q.resize(z * 2, 0);
atcoder::internal::butterfly(P);
atcoder::internal::butterfly(Q);
// fast
while (k) {
// Q(-x)
vector<T> Q_n(z * 2);
for (int i = 0; i < z; i++) {
Q_n[i * 2] = Q[i * 2 + 1];
Q_n[i * 2 + 1] = Q[i * 2];
}
for (int i = 0; i < z * 2; i++) {
P[i] *= Q_n[i];
Q[i] *= Q_n[i];
}
fpsPickEvenOdd(P, k & 1);
fpsPickEvenOdd(Q, 0);
k /= 2;
if (k == 0) break;
fpsExtend(P);
fpsExtend(Q);
}
T SP = 0, SQ = 0;
for (int i = 0; i < z; i++) SP += P[i], SQ += Q[i];
return SP / SQ;
}
template <typename T>
// 0 = a[i] * c[0] + a[i - 1] * c[1] + a[i - 2] * c[2] + ... + a[i - d] * c[d]
// a.size() + 1 == c.size()
// c[0] = - 1 ?
// return a[k]
T kthLinear(long long k, vector<T> a, vector<T> c) {
int d = a.size();
assert(d + 1 == int(c.size()));
vector<T> P = atcoder::convolution(a, c);
P.resize(d);
return bostanMori(k, P, c);
}
#include <atcoder/modint>
using mint = atcoder::modint998244353;
int main() {
int N;
cin >> N;
vector<mint> F(3);
F[0] = F[1] = F[2] = 1;
F = fpsPow(F, N, N + 1);
mint ans = F[N];
ans--;
ans /= 2;
cout << ans.val() << endl;
}