結果
| 問題 | No.2605 Pickup Parentheses |
| ユーザー |
 SSRS
|
| 提出日時 | 2024-01-12 21:42:10 |
| 言語 | C++17 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 138 ms / 2,000 ms |
| コード長 | 15,911 bytes |
| コンパイル時間 | 2,743 ms |
| コンパイル使用メモリ | 219,060 KB |
| 最終ジャッジ日時 | 2025-02-18 17:57:46 |
|
ジャッジサーバーID （参考情報） |
judge5 / judge1 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 3 |
| other | AC * 68 |
ソースコード
#include <bits/stdc++.h>using namespace std;const long long MOD = 998244353;//https://judge.yosupo.jp/submission/101681template <int mod>struct Mod_Int {int x;Mod_Int() : x(0) {}Mod_Int(long long y) : x(y >= 0 ? y % mod : (mod - (-y) % mod) % mod) {}static int get_mod() { return mod; }Mod_Int &operator+=(const Mod_Int &p) {if ((x += p.x) >= mod) x -= mod;return *this;}Mod_Int &operator-=(const Mod_Int &p) {if ((x += mod - p.x) >= mod) x -= mod;return *this;}Mod_Int &operator*=(const Mod_Int &p) {x = (int)(1LL * x * p.x % mod);return *this;}Mod_Int &operator/=(const Mod_Int &p) {*this *= p.inverse();return *this;}Mod_Int &operator++() { return *this += Mod_Int(1); }Mod_Int operator++(int) {Mod_Int tmp = *this;++*this;return tmp;}Mod_Int &operator--() { return *this -= Mod_Int(1); }Mod_Int operator--(int) {Mod_Int tmp = *this;--*this;return tmp;}Mod_Int operator-() const { return Mod_Int(-x); }Mod_Int operator+(const Mod_Int &p) const { return Mod_Int(*this) += p; }Mod_Int operator-(const Mod_Int &p) const { return Mod_Int(*this) -= p; }Mod_Int operator*(const Mod_Int &p) const { return Mod_Int(*this) *= p; }Mod_Int operator/(const Mod_Int &p) const { return Mod_Int(*this) /= p; }bool operator==(const Mod_Int &p) const { return x == p.x; }bool operator!=(const Mod_Int &p) const { return x != p.x; }Mod_Int inverse() const {assert(*this != Mod_Int(0));return pow(mod - 2);}Mod_Int pow(long long k) const {Mod_Int now = *this, ret = 1;for (; k > 0; k >>= 1, now *= now) {if (k & 1) ret *= now;}return ret;}friend ostream &operator<<(ostream &os, const Mod_Int &p) { return os << p.x; }friend istream &operator>>(istream &is, Mod_Int &p) {long long a;is >> a;p = Mod_Int<mod>(a);return is;}};using mint = Mod_Int<MOD>;template <typename T>struct Number_Theoretic_Transform {static int max_base;static T root;static vector<T> r, ir;Number_Theoretic_Transform() {}static void init() {if (!r.empty()) return;int mod = T::get_mod();int tmp = mod - 1;root = 2;while (root.pow(tmp >> 1) == 1) root++;max_base = 0;while (tmp % 2 == 0) tmp >>= 1, max_base++;r.resize(max_base), ir.resize(max_base);for (int i = 0; i < max_base; i++) {r[i] = -root.pow((mod - 1) >> (i + 2)); // r[i] := 1 の 2^(i+2) 乗根ir[i] = r[i].inverse(); // ir[i] := 1/r[i]}}static void ntt(vector<T> &a) {init();int n = a.size();assert((n & (n - 1)) == 0);assert(n <= (1 << max_base));for (int k = n; k >>= 1;) {T w = 1;for (int s = 0, t = 0; s < n; s += 2 * k) {for (int i = s, j = s + k; i < s + k; i++, j++) {T x = a[i], y = w * a[j];a[i] = x + y, a[j] = x - y;}w *= r[__builtin_ctz(++t)];}}}static void intt(vector<T> &a) {init();int n = a.size();assert((n & (n - 1)) == 0);assert(n <= (1 << max_base));for (int k = 1; k < n; k <<= 1) {T w = 1;for (int s = 0, t = 0; s < n; s += 2 * k) {for (int i = s, j = s + k; i < s + k; i++, j++) {T x = a[i], y = a[j];a[i] = x + y, a[j] = w * (x - y);}w *= ir[__builtin_ctz(++t)];}}T inv = T(n).inverse();for (auto &e : a) e *= inv;}static vector<T> convolve(vector<T> a, vector<T> b) {if (a.empty() || b.empty()) return {};int k = (int)a.size() + (int)b.size() - 1, n = 1;while (n < k) n <<= 1;a.resize(n), b.resize(n);ntt(a), ntt(b);for (int i = 0; i < n; i++) a[i] *= b[i];intt(a), a.resize(k);return a;}};template <typename T>int Number_Theoretic_Transform<T>::max_base = 0;template <typename T>T Number_Theoretic_Transform<T>::root = T();template <typename T>vector<T> Number_Theoretic_Transform<T>::r = vector<T>();template <typename T>vector<T> Number_Theoretic_Transform<T>::ir = vector<T>();using NTT = Number_Theoretic_Transform<mint>;template <typename T>struct Formal_Power_Series : vector<T> {using NTT_ = Number_Theoretic_Transform<T>;using vector<T>::vector;Formal_Power_Series(const vector<T> &f) : vector<T>(f) {}// f(x) mod x^nFormal_Power_Series pre(int n) const {Formal_Power_Series ret(begin(*this), begin(*this) + min((int)this->size(), n));ret.resize(n, 0);return ret;}// f(1/x)x^{n-1}Formal_Power_Series rev(int n = -1) const {Formal_Power_Series ret = *this;if (n != -1) ret.resize(n, 0);reverse(begin(ret), end(ret));return ret;}void normalize() {while (!this->empty() && this->back() == 0) this->pop_back();}Formal_Power_Series operator-() const {Formal_Power_Series ret = *this;for (int i = 0; i < (int)ret.size(); i++) ret[i] = -ret[i];return ret;}Formal_Power_Series &operator+=(const T &t) {if (this->empty()) this->resize(1, 0);(*this)[0] += t;return *this;}Formal_Power_Series &operator+=(const Formal_Power_Series &g) {if (g.size() > this->size()) this->resize(g.size());for (int i = 0; i < (int)g.size(); i++) (*this)[i] += g[i];this->normalize();return *this;}Formal_Power_Series &operator-=(const T &t) {if (this->empty()) this->resize(1, 0);*this[0] -= t;return *this;}Formal_Power_Series &operator-=(const Formal_Power_Series &g) {if (g.size() > this->size()) this->resize(g.size());for (int i = 0; i < (int)g.size(); i++) (*this)[i] -= g[i];this->normalize();return *this;}Formal_Power_Series &operator*=(const T &t) {for (int i = 0; i < (int)this->size(); i++) (*this)[i] *= t;return *this;}Formal_Power_Series &operator*=(const Formal_Power_Series &g) {if (empty(*this) || empty(g)) {this->clear();return *this;}return *this = NTT_::convolve(*this, g);}Formal_Power_Series &operator/=(const T &t) {assert(t != 0);T inv = t.inverse();return *this *= inv;}// f(x) を g(x) で割った商Formal_Power_Series &operator/=(const Formal_Power_Series &g) {if (g.size() > this->size()) {this->clear();return *this;}int n = this->size(), m = g.size();return *this = (rev() * g.rev().inv(n - m + 1)).pre(n - m + 1).rev();}// f(x) を g(x) で割った余りFormal_Power_Series &operator%=(const Formal_Power_Series &g) { return *this -= (*this / g) * g; }// f(x)/x^kFormal_Power_Series &operator<<=(int k) {Formal_Power_Series ret(k, 0);ret.insert(end(ret), begin(*this), end(*this));return *this = ret;}// f(x)x^kFormal_Power_Series &operator>>=(int k) {Formal_Power_Series ret;ret.insert(end(ret), begin(*this) + k, end(*this));return *this = ret;}Formal_Power_Series operator+(const T &x) const { return Formal_Power_Series(*this) += x; }Formal_Power_Series operator+(const Formal_Power_Series &v) const { return Formal_Power_Series(*this) += v; }Formal_Power_Series operator-(const T &x) const { return Formal_Power_Series(*this) -= x; }Formal_Power_Series operator-(const Formal_Power_Series &v) const { return Formal_Power_Series(*this) -= v; }Formal_Power_Series operator*(const T &x) const { return Formal_Power_Series(*this) *= x; }Formal_Power_Series operator*(const Formal_Power_Series &v) const { return Formal_Power_Series(*this) *= v; }Formal_Power_Series operator/(const T &x) const { return Formal_Power_Series(*this) /= x; }Formal_Power_Series operator/(const Formal_Power_Series &v) const { return Formal_Power_Series(*this) /= v; }Formal_Power_Series operator%(const Formal_Power_Series &v) const { return Formal_Power_Series(*this) %= v; }Formal_Power_Series operator<<(int x) const { return Formal_Power_Series(*this) <<= x; }Formal_Power_Series operator>>(int x) const { return Formal_Power_Series(*this) >>= x; }// f(c)T val(const T &c) const {T ret = 0;for (int i = (int)this->size() - 1; i >= 0; i--) ret *= c, ret += (*this)[i];return ret;}// df/dxFormal_Power_Series derivative() const {if (empty(*this)) return *this;int n = this->size();Formal_Power_Series ret(n - 1);for (int i = 1; i < n; i++) ret[i - 1] = (*this)[i] * i;return ret;}// ∫f(x)dxFormal_Power_Series integral() const {if (empty(*this)) return *this;int n = this->size();vector<T> inv(n + 1, 0);inv[1] = 1;int mod = T::get_mod();for (int i = 2; i <= n; i++) inv[i] = -inv[mod % i] * T(mod / i);Formal_Power_Series ret(n + 1, 0);for (int i = 0; i < n; i++) ret[i + 1] = (*this)[i] * inv[i + 1];return ret;}// 1/f(x) mod x^n (f[0] != 0)Formal_Power_Series inv(int n = -1) const {assert((*this)[0] != 0);if (n == -1) n = this->size();Formal_Power_Series ret(1, (*this)[0].inverse());for (int m = 1; m < n; m <<= 1) {Formal_Power_Series f = pre(2 * m), g = ret;f.resize(2 * m), g.resize(2 * m);NTT_::ntt(f), NTT_::ntt(g);Formal_Power_Series h(2 * m);for (int i = 0; i < 2 * m; i++) h[i] = f[i] * g[i];NTT_::intt(h);for (int i = 0; i < m; i++) h[i] = 0;NTT_::ntt(h);for (int i = 0; i < 2 * m; i++) h[i] *= g[i];NTT_::intt(h);for (int i = 0; i < m; i++) h[i] = 0;ret -= h;}ret.resize(n);return ret;}// log(f(x)) mod x^n (f[0] = 1)Formal_Power_Series log(int n = -1) const {assert((*this)[0] == 1);if (n == -1) n = this->size();Formal_Power_Series ret = (derivative() * inv(n)).pre(n - 1).integral();ret.resize(n);return ret;}// exp(f(x)) mod x^n (f[0] = 0)Formal_Power_Series exp(int n = -1) const {assert((*this)[0] == 0);if (n == -1) n = this->size();vector<T> inv(2 * n + 1, 0);inv[1] = 1;int mod = T::get_mod();for (int i = 2; i <= 2 * n; i++) inv[i] = -inv[mod % i] * T(mod / i);auto inplace_integral = [inv](Formal_Power_Series &f) {if (empty(f)) return;int n = f.size();f.insert(begin(f), 0);for (int i = 1; i <= n; i++) f[i] *= inv[i];};auto inplace_derivative = [](Formal_Power_Series &f) {if (empty(f)) return;int n = f.size();f.erase(begin(f));for (int i = 0; i < n - 1; i++) f[i] *= T(i + 1);};Formal_Power_Series ret{1, this->size() > 1 ? (*this)[1] : 0}, c{1}, z1, z2{1, 1};for (int m = 2; m < n; m *= 2) {auto y = ret;y.resize(2 * m);NTT_::ntt(y);z1 = z2;Formal_Power_Series z(m);for (int i = 0; i < m; i++) z[i] = y[i] * z1[i];NTT_::intt(z);fill(begin(z), begin(z) + m / 2, 0);NTT_::ntt(z);for (int i = 0; i < m; i++) z[i] *= -z1[i];NTT_::intt(z);c.insert(end(c), begin(z) + m / 2, end(z));z2 = c, z2.resize(2 * m);NTT_::ntt(z2);Formal_Power_Series x(begin(*this), begin(*this) + min((int)this->size(), m));inplace_derivative(x);x.resize(m, 0);NTT_::ntt(x);for (int i = 0; i < m; i++) x[i] *= y[i];NTT_::intt(x);x -= ret.derivative(), x.resize(2 * m);for (int i = 0; i < m - 1; i++) x[m + i] = x[i], x[i] = 0;NTT_::ntt(x);for (int i = 0; i < 2 * m; i++) x[i] *= z2[i];NTT_::intt(x);x.pop_back();inplace_integral(x);for (int i = m; i < min((int)this->size(), 2 * m); i++) x[i] += (*this)[i];fill(begin(x), begin(x) + m, 0);NTT_::ntt(x);for (int i = 0; i < 2 * m; i++) x[i] *= y[i];NTT_::intt(x);ret.insert(end(ret), begin(x) + m, end(x));}ret.resize(n);return ret;}// f(x)^k mod x^nFormal_Power_Series pow(long long k, int n = -1) const {if (n == -1) n = this->size();int m = this->size();for (int i = 0; i < m; i++) {if ((*this)[i] == 0) continue;T rev = (*this)[i].inverse();Formal_Power_Series C(*this * rev), D(m - i, 0);for (int j = i; j < m; j++) D[j - i] = C[j];D = (D.log() * k).exp() * ((*this)[i].pow(k));Formal_Power_Series E(n, 0);if (i > 0 && k > n / i) return E;long long S = i * k;for (int j = 0; j + S < n && j < D.size(); j++) E[j + S] = D[j];E.resize(n);return E;}return Formal_Power_Series(n, 0);}// f(x+c)Formal_Power_Series Taylor_shift(T c) const {int n = this->size();vector<T> ifac(n, 1);Formal_Power_Series f(n), g(n);for (int i = 0; i < n; i++) {f[n - 1 - i] = (*this)[i] * ifac[n - 1];if (i < n - 1) ifac[n - 1] *= i + 1;}ifac[n - 1] = ifac[n - 1].inverse();for (int i = n - 1; i > 0; i--) ifac[i - 1] = ifac[i] * i;T pw = 1;for (int i = 0; i < n; i++) {g[i] = pw * ifac[i];pw *= c;}f *= g;Formal_Power_Series b(n);for (int i = 0; i < n; i++) b[i] = f[n - 1 - i] * ifac[i];return b;}};using fps = Formal_Power_Series<mint>;int main(){int N, M;cin >> N >> M;vector<int> L(M), R(M);for (int i = 0; i < M; i++){cin >> L[i] >> R[i];L[i]--;}if (N % 2 == 1){cout << 0 << endl;} else {N /= 2;vector<long long> inv(N + 2);inv[1] = 1;for (int i = 2; i <= N + 1; i++){inv[i] = MOD - inv[MOD % i] * (MOD / i) % MOD;}vector<long long> cat(N + 1);cat[0] = 1;cat[1] = 1;for (int i = 1; i < N; i++){cat[i + 1] = cat[i] * 2 * (2 * i + 1) % MOD * inv[i + 2] % MOD;}vector<int> cnt(N + 1, 0);for (int i = 0; i < M; i++){if ((R[i] - L[i]) % 2 == 0){cnt[(R[i] - L[i]) / 2]++;}}vector<long long> f(N + 1, 0);for (int i = 1; i <= N; i++){long long tmp = 1;for (int j = 1; i * j <= N; j++){tmp *= cat[i];tmp %= MOD;f[i * j] += MOD - tmp * inv[j] % MOD * cnt[i] % MOD;f[i * j] %= MOD;}}fps f2(N + 1);for (int i = 0; i <= N; i++){f2[i] = f[i];}fps g = f2.exp();long long ans = 0;for (int i = 0; i <= N; i++){ans += g[i].x * cat[N - i] % MOD;ans %= MOD;}cout << ans << endl;}}