結果
| 問題 |
No.2605 Pickup Parentheses
|
| コンテスト | |
| ユーザー |
 Misuki
|
| 提出日時 | 2024-01-12 22:35:19 |
| 言語 | C++23 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 318 ms / 2,000 ms |
| コード長 | 13,438 bytes |
| コンパイル時間 | 2,425 ms |
| コンパイル使用メモリ | 213,936 KB |
| 実行使用メモリ | 15,504 KB |
| 最終ジャッジ日時 | 2024-09-30 06:27:23 |
| 合計ジャッジ時間 | 8,062 ms |
|
ジャッジサーバーID （参考情報） |
judge4 / 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
#pragma GCC optimize("O2")
#include <algorithm>
#include <array>
#include <bit>
#include <bitset>
#include <cassert>
#include <cctype>
#include <cfenv>
#include <cfloat>
#include <chrono>
#include <cinttypes>
#include <climits>
#include <cmath>
#include <compare>
#include <complex>
#include <concepts>
#include <cstdarg>
#include <cstddef>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <deque>
#include <fstream>
#include <functional>
#include <initializer_list>
#include <iomanip>
#include <ios>
#include <iostream>
#include <istream>
#include <iterator>
#include <limits>
#include <list>
#include <map>
#include <memory>
#include <new>
#include <numbers>
#include <numeric>
#include <ostream>
#include <queue>
#include <random>
#include <ranges>
#include <set>
#include <span>
#include <sstream>
#include <stack>
#include <streambuf>
#include <string>
#include <tuple>
#include <type_traits>
#include <variant>
//#define int ll
#define INT128_MAX (__int128)(((unsigned __int128) 1 << ((sizeof(__int128) * __CHAR_BIT__) - 1)) - 1)
#define INT128_MIN (-INT128_MAX - 1)
#ifdef DEBUG
#define dbg(x) cout << (#x) << " = " << x << '\n'
#else
#define dbg(x)
#endif
namespace R = std::ranges;
namespace V = std::views;
using namespace std;
using ll = long long;
using ull = unsigned long long;
using ldb = long double;
//#define double ldb
template<class T, size_t N>
ostream& operator<<(ostream& os, const array<T, N> &arr) {
for(const T &X : arr)
os << X << ' ';
return os;
}
template<class T>
ostream& operator<<(ostream& os, const vector<T> &vec) {
for(const T &X : vec)
os << X << ' ';
return os;
}
template<class T>
ostream& operator<<(ostream& os, const set<T> &s) {
for(const T &x : s)
os << x << ' ';
return os;
}
/**
* template name: MontgomeryModInt
* author: Misuki
* reference: https://github.com/NyaanNyaan/library/blob/master/modint/montgomery-modint.hpp#L10
* last update: 2023/11/30
* note: mod should be a prime less than 2^30.
*/
template<uint32_t mod>
struct MontgomeryModInt {
using mint = MontgomeryModInt;
using i32 = int32_t;
using u32 = uint32_t;
using u64 = uint64_t;
static constexpr u32 get_r() {
u32 res = 1, base = mod;
for(i32 i = 0; i < 31; i++)
res *= base, base *= base;
return -res;
}
static constexpr u32 get_mod() {
return mod;
}
static constexpr u32 n2 = -u64(mod) % mod; //2^64 % mod
static constexpr u32 r = get_r(); //-P^{-1} % 2^32
u32 a;
static u32 reduce(const u64 &b) {
return (b + u64(u32(b) * r) * mod) >> 32;
}
static u32 transform(const u64 &b) {
return reduce(u64(b) * n2);
}
MontgomeryModInt() : a(0) {}
MontgomeryModInt(const int64_t &b)
: a(transform(b % mod + mod)) {}
mint pow(u64 k) const {
mint res(1), base(*this);
while(k) {
if (k & 1)
res *= base;
base *= base, k >>= 1;
}
return res;
}
mint inverse() const { return (*this).pow(mod - 2); }
u32 get() const {
u32 res = reduce(a);
return res >= mod ? res - mod : res;
}
mint& operator+=(const mint &b) {
if (i32(a += b.a - 2 * mod) < 0) a += 2 * mod;
return *this;
}
mint& operator-=(const mint &b) {
if (i32(a -= b.a) < 0) a += 2 * mod;
return *this;
}
mint& operator*=(const mint &b) {
a = reduce(u64(a) * b.a);
return *this;
}
mint& operator/=(const mint &b) {
a = reduce(u64(a) * b.inverse().a);
return *this;
}
mint operator-() { return mint() - mint(*this); }
bool operator==(mint b) const {
return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a);
}
bool operator!=(mint b) const {
return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a);
}
friend mint operator+(mint a, mint b) { return a += b; }
friend mint operator-(mint a, mint b) { return a -= b; }
friend mint operator*(mint a, mint b) { return a *= b; }
friend mint operator/(mint a, mint b) { return a /= b; }
friend ostream& operator<<(ostream& os, const mint& b) {
return os << b.get();
}
friend istream& operator>>(istream& is, mint& b) {
int64_t val;
is >> val;
b = mint(val);
return is;
}
};
using mint = MontgomeryModInt<998244353>;
/**
* template name: comb
* author: Misuki
* last update: 2023/01/22
* note: remember to call init() before using it.
*/
const int MAX = 200002;
mint fac[MAX], facInv[MAX], pow2[MAX];
void init() {
fac[0] = 1;
for(int i = 1; i < MAX; i++)
fac[i] = fac[i - 1] * i;
facInv[MAX - 1] = 1 / fac[MAX - 1];
for(int i = MAX - 2; i >= 0; i--)
facInv[i] = facInv[i + 1] * (i + 1);
pow2[0] = 1;
for(int i = 1; i < MAX; i++)
pow2[i] = pow2[i - 1] * 2;
}
mint binom(int a, int b) {
if (b < 0 or a < b)
return 0;
else
return fac[a] * facInv[b] * facInv[a - b];
}
mint C(int k) {
return fac[k] * facInv[k + 1] * binom(2 * k, k);
}
/**
* template name: NTTmint
* reference: https://judge.yosupo.jp/submission/69896
* last update: 2024/01/07
* include: mint
* remark: MOD = 2^K * C + 1, R is a primitive root modulo MOD
* remark: a.size() <= 2^K must be satisfied
* some common modulo: 998244353 = 2^23 * 119 + 1, R = 3
* 469762049 = 2^26 * 7 + 1, R = 3
* 1224736769 = 2^24 * 73 + 1, R = 3
* verify: Library Checker - Convolution
*/
template<int32_t k = 23, int32_t c = 119, int32_t r = 3, class Mint = MontgomeryModInt<998244353>>
struct NTT {
using u32 = uint32_t;
static constexpr u32 mod = (1 << k) * c + 1;
static constexpr u32 get_mod() { return mod; }
static void ntt(vector<Mint> &a, bool inverse) {
static array<Mint, 30> w, w_inv;
if (w[0] == 0) {
Mint root = 2;
while(root.pow((mod - 1) / 2) == 1) root += 1;
for(int i = 0; i < 30; i++)
w[i] = -(root.pow((mod - 1) >> (i + 2))), w_inv[i] = 1 / w[i];
}
int n = ssize(a);
if (not inverse) {
for(int m = n; m >>= 1; ) {
Mint ww = 1;
for(int s = 0, l = 0; s < n; s += 2 * m) {
for(int i = s, j = s + m; i < s + m; i++, j++) {
Mint x = a[i], y = a[j] * ww;
a[i] = x + y, a[j] = x - y;
}
ww *= w[__builtin_ctz(++l)];
}
}
} else {
for(int m = 1; m < n; m *= 2) {
Mint ww = 1;
for(int s = 0, l = 0; s < n; s += 2 * m) {
for(int i = s, j = s + m; i < s + m; i++, j++) {
Mint x = a[i], y = a[j];
a[i] = x + y, a[j] = (x - y) * ww;
}
ww *= w_inv[__builtin_ctz(++l)];
}
}
Mint inv = 1 / Mint(n);
for(Mint &x : a) x *= inv;
}
}
static vector<Mint> conv(vector<Mint> a, vector<Mint> b) {
int sz = ssize(a) + ssize(b) - 1;
int n = bit_ceil((u32)sz);
a.resize(n, 0);
ntt(a, false);
b.resize(n, 0);
ntt(b, false);
for(int i = 0; i < n; i++)
a[i] *= b[i];
ntt(a, true);
a.resize(sz);
return a;
}
};
/**
* template name: FPS
* author: Misuki
* last update: 2024/01/10
* include: NTT/mint
* verify: Library Checker - Inv of Formal Power Series
* Library Checker - Exp of Formal Power Series
* Library Checker - Log of Formal Power Series
* Library Checker - Pow of Formal Power Series
* Library Checker - Convolution
* Library Checker - Division of Polynomials
* Library Checker - Multipoint Evaluation
* Library Checker - Polynomial Interpolation
*/
template<class Mint>
struct FPS : vector<Mint> {
static function<vector<Mint>(vector<Mint>, vector<Mint>)> conv;
FPS(vector<Mint> v) { *this = v; }
using vector<Mint>::vector;
FPS& operator+=(FPS b) {
if (ssize(*this) < ssize(b)) this -> resize(ssize(b), 0);
for(int i = 0; i < ssize(b); i++)
(*this)[i] += b[i];
return *this;
}
FPS& operator-=(FPS b) {
if (ssize(*this) < ssize(b)) this -> resize(ssize(b), 0);
for(int i = 0; i < ssize(b); i++)
(*this)[i] -= b[i];
return *this;
}
FPS& operator*=(FPS b) {
auto c = conv(*this, b);
this -> resize(ssize(*this) + ssize(b) - 1);
copy(c.begin(), c.end(), this -> begin());
return *this;
}
FPS& operator*=(Mint b) {
for(int i = 0; i < ssize(*this); i++)
(*this)[i] *= b;
return *this;
}
FPS& operator/=(Mint b) {
b = Mint(1) / b;
for(int i = 0; i < ssize(*this); i++)
(*this)[i] *= b;
return *this;
}
FPS shrink() {
FPS F = *this;
while(ssize(F) > 1 and F.back() == 0)
F.pop_back();
return F;
}
FPS integral() {
vector<Mint> Inv(ssize(*this) + 1);
Inv[1] = 1;
for(int i = 2; i < ssize(Inv); i++)
Inv[i] = (Mint::get_mod() - Mint::get_mod() / i) * Inv[Mint::get_mod() % i];
FPS Q(ssize(*this) + 1, 0);
for(int i = 0; i < ssize(*this); i++)
Q[i + 1] = (*this)[i] * Inv[i + 1];
return Q;
}
FPS derivative() {
assert(!this -> empty());
FPS Q(ssize(*this) - 1);
for(int i = 1; i < ssize(*this); i++)
Q[i - 1] = (*this)[i] * i;
return Q;
}
Mint eval(Mint x) {
Mint base = 1, res = 0;
for(int i = 0; i < ssize(*this); i++, base *= x)
res += (*this)[i] * base;
return res;
}
FPS inv(int k) { // 1 / FPS (mod x^k)
assert(!this -> empty() and (*this)[0] != 0);
FPS Q(1, 1 / (*this)[0]);
for(int i = 1; (1 << (i - 1)) < k; i++) {
FPS P = (*this);
P.resize(1 << i, 0);
Q = Q * (FPS(1, 2) - P * Q);
Q.resize(1 << i, 0);
}
Q.resize(k);
return Q;
}
array<FPS, 2> div(FPS G) {
FPS F = this -> shrink();
G = G.shrink();
assert(!G.empty());
if (ssize(G) > ssize(F))
return {{{}, F}};
int n = ssize(F) - ssize(G) + 1;
auto FR = F, GR = G;
R::reverse(FR);
R::reverse(GR);
FPS Q = FR * GR.inv(n);
Q.resize(n);
R::reverse(Q);
return {Q, (F - G * Q).shrink()};
}
FPS log(int k) {
assert(!this -> empty() and (*this)[0] == 1);
FPS Q = *this;
Q = (Q.derivative() * Q.inv(k));
Q.resize(k - 1);
return Q.integral();
}
FPS exp(int k) {
assert(!this -> empty() and (*this)[0] == 0);
FPS Q(1, 1);
for(int i = 1; (1 << (i - 1)) < k; i++) {
FPS P = (*this);
P.resize(1 << i, 0);
Q = Q * (FPS(1, 1) + P - Q.log(1 << i));
Q.resize(1 << i, 0);
}
Q.resize(k);
return Q;
}
FPS pow(ll idx, int k) {
if (idx == 0) {
FPS res(k, 0);
res[0] = 1;
return res;
}
for(int i = 0; i < ssize(*this) and i * idx < k; i++) {
if ((*this)[i] != 0) {
Mint Inv = 1 / (*this)[i];
FPS Q(ssize(*this) - i);
for(int j = i; j < ssize(*this); j++)
Q[j - i] = (*this)[j] * Inv;
Q = (Q.log(k) * idx).exp(k);
FPS Q2(k, 0);
Mint Pow = (*this)[i].pow(idx);
for(int j = 0; j + i * idx < k; j++)
Q2[j + i * idx] = Q[j] * Pow;
return Q2;
}
}
return FPS(k, 0);
}
vector<Mint> multieval(vector<Mint> xs) {
int n = ssize(xs);
vector<FPS> data(2 * n);
for(int i = 0; i < n; i++)
data[n + i] = {-xs[i], 1};
for(int i = n - 1; i > 0; i--)
data[i] = data[i << 1] * data[i << 1 | 1];
data[1] = (this -> div(data[1]))[1];
for(int i = 1; i < n; i++) {
data[i << 1] = data[i].div(data[i << 1])[1];
data[i << 1 | 1] = data[i].div(data[i << 1 | 1])[1];
}
vector<Mint> res(n);
for(int i = 0; i < n; i++)
res[i] = data[n + i][0];
return res;
}
static vector<Mint> interpolate(vector<Mint> xs, vector<Mint> ys) {
assert(ssize(xs) == ssize(ys));
int n = ssize(xs);
vector<FPS> data(2 * n), res(2 * n);
for(int i = 0; i < n; i++)
data[n + i] = {-xs[i], 1};
for(int i = n - 1; i > 0; i--)
data[i] = data[i << 1] * data[i << 1 | 1];
res[1] = data[1].derivative().div(data[1])[1];
for(int i = 1; i < n; i++) {
res[i << 1] = res[i].div(data[i << 1])[1];
res[i << 1 | 1] = res[i].div(data[i << 1 | 1])[1];
}
for(int i = 0; i < n; i++)
res[n + i][0] = ys[i] / res[n + i][0];
for(int i = n - 1; i > 0; i--)
res[i] = res[i << 1] * data[i << 1 | 1] + res[i << 1 | 1] * data[i << 1];
return res[1];
}
friend FPS operator+(FPS a, FPS b) { return a += b; }
friend FPS operator-(FPS a, FPS b) { return a -= b; }
friend FPS operator*(FPS a, FPS b) { return a *= b; }
friend FPS operator*(FPS a, Mint b) { return a *= b; }
friend FPS operator/(FPS a, Mint b) { return a /= b; }
};
NTT ntt;
using fps = FPS<mint>;
template<>
function<vector<mint>(vector<mint>, vector<mint>)> fps::conv = ntt.conv;
signed main() {
ios::sync_with_stdio(false), cin.tie(NULL);
init();
int n, m; cin >> n >> m;
vector<fps> poly;
poly.push_back(fps(1, 1));
while(m--) {
int l, r; cin >> l >> r;
if ((r - l + 1) % 2 == 0) {
poly.push_back(fps(r - l + 2, 0));
poly.back()[0] = 1, poly.back().back() = -C((r - l + 1) / 2);
}
}
if (n & 1) {
cout << 0 << '\n';
return 0;
}
auto calc = [&](int l, int r, auto self) -> fps {
if (l + 1 == r)
return poly[l];
else
return self(l, (l + r) / 2, self) * self((l + r) / 2, r, self);
};
auto f = calc(0, ssize(poly), calc);
mint ans = 0;
for(int i = 0; i < ssize(f); i += 2)
ans += f[i] * C((n - i) / 2);
cout << ans << '\n';
return 0;
}