結果
| 問題 |
No.3138 Minimum Bracket Subsequence
|
| ユーザー |
👑  potato167
|
| 提出日時 | 2025-04-27 20:01:08 |
| 言語 | C++17 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
TLE
|
| 実行時間 | - |
| コード長 | 9,430 bytes |
| コンパイル時間 | 4,010 ms |
| コンパイル使用メモリ | 239,332 KB |
| 実行使用メモリ | 34,168 KB |
| 最終ジャッジ日時 | 2025-04-27 20:01:17 |
| 合計ジャッジ時間 | 8,040 ms |
|
ジャッジサーバーID （参考情報） |
judge4 / judge3 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 2 |
| other | AC * 16 TLE * 1 -- * 19 |
ソースコード
#line 1 "e.cpp"
#include <bits/stdc++.h>
using namespace std;
using ll=long long;
const ll ILL=2167167167167167167;
const int INF=2100000000;
#define rep(i,a,b) for (int i=(int)(a);i<(int)(b);i++)
#define all(p) p.begin(),p.end()
template<class T> using _pq = priority_queue<T, vector<T>, greater<T>>;
template<class T> int LB(vector<T> &v,T a){return lower_bound(v.begin(),v.end(),a)-v.begin();}
template<class T> int UB(vector<T> &v,T a){return upper_bound(v.begin(),v.end(),a)-v.begin();}
template<class T> bool chmin(T &a,T b){if(b<a){a=b;return 1;}else return 0;}
template<class T> bool chmax(T &a,T b){if(a<b){a=b;return 1;}else return 0;}
template<class T> void So(vector<T> &v) {sort(v.begin(),v.end());}
template<class T> void Sore(vector<T> &v) {sort(v.begin(),v.end(),[](T x,T y){return x>y;});}
bool yneos(bool a,bool upp=false){if(a){cout<<(upp?"YES\n":"Yes\n");}else{cout<<(upp?"NO\n":"No\n");}return a;}
template<class T> void vec_out(vector<T> &p,int ty=0){
if(ty==2){cout<<'{';for(int i=0;i<(int)p.size();i++){if(i){cout<<",";}cout<<'"'<<p[i]<<'"';}cout<<"}\n";}
else{if(ty==1){cout<<p.size()<<"\n";}for(int i=0;i<(int)(p.size());i++){if(i) cout<<" ";cout<<p[i];}cout<<"\n";}}
template<class T> T vec_min(vector<T> &a){assert(!a.empty());T ans=a[0];for(auto &x:a) chmin(ans,x);return ans;}
template<class T> T vec_max(vector<T> &a){assert(!a.empty());T ans=a[0];for(auto &x:a) chmax(ans,x);return ans;}
template<class T> T vec_sum(vector<T> &a){T ans=T(0);for(auto &x:a) ans+=x;return ans;}
int pop_count(long long a){int res=0;while(a){res+=(a&1),a>>=1;}return res;}
template<class T> T square(T a){return a * a;}
#line 3 "/Users/Shared/po167_library/fps/FPS_Boston_Mori.hpp"
#include <atcoder/convolution>
#line 4 "/Users/Shared/po167_library/fps/FPS_extend.hpp"
namespace po167{
// in : DFT(v) (len(v) = z)
// out : DFT(v) (len(v) = 2 * z)
template<class T>
void FPS_extend(std::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]);
};
}
#line 3 "/Users/Shared/po167_library/fps/FPS_pick_even_odd.hpp"
namespace po167{
// s.t |v| = 2 ^ s (no assert)
template<class T>
void FPS_pick_even_odd(std::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;
std::vector<T> es = {half};
while ((int)es.size() != z){
std::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;
std::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);
}
}
}
#line 7 "/Users/Shared/po167_library/fps/FPS_Boston_Mori.hpp"
namespace po167{
// return [x^k] P(x) / Q(x)
template<class T>
T Boston_Mori(long long k, std::vector<T> P, std::vector<T> Q){
assert(!Q.empty() && Q[0] != 0);
int z = 1;
while (z < (int)std::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)
std::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];
}
FPS_pick_even_odd(P, k & 1);
FPS_pick_even_odd(Q, 0);
k /= 2;
if (k == 0) break;
FPS_extend(P);
FPS_extend(Q);
}
T SP = 0, SQ = 0;
for (int i = 0; i < z; i++) SP += P[i], SQ += Q[i];
return SP / SQ;
// simple
/*
while (k){
auto n_Q = Q;
for (int i = 0; i < int(Q.size()); i++){
if (i & 1) n_Q[i] *= -1;
}
auto n_P = atcoder::convolution(P, n_Q);
n_Q = atcoder::convolution(Q, n_Q);
for (int i = 0; i < int(Q.size()); i++){
Q[i] = n_Q[i * 2];
}
P.clear();
for (int i = (k & 1); i < int(n_P.size()); i += 2){
P.push_back(n_P[i]);
}
k >>= 1;
}
return P[0] / Q[0];
*/
}
template<class 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 Kth_Linear(long long k, std::vector<T> a, std::vector<T> c){
int d = a.size();
assert(d + 1 == int(c.size()));
std::vector<T> P = atcoder::convolution(a, c);
P.resize(d);
return Boston_Mori(k, P, c);
}
};
#line 2 "/Users/Shared/po167_library/math/Binomial.hpp"
#line 5 "/Users/Shared/po167_library/math/Binomial.hpp"
namespace po167{
template<class T>
struct Binomial{
std::vector<T> fact_vec, fact_inv_vec;
void extend(int m = -1){
int n = fact_vec.size();
if (m == -1) m = n * 2;
if (n >= m) return;
fact_vec.resize(m);
fact_inv_vec.resize(m);
for (int i = n; i < m; i++){
fact_vec[i] = fact_vec[i - 1] * T(i);
}
fact_inv_vec[m - 1] = T(1) / fact_vec[m - 1];
for (int i = m - 1; i > n; i--){
fact_inv_vec[i - 1] = fact_inv_vec[i] * T(i);
}
}
Binomial(int MAX = 0){
fact_vec.resize(1, T(1));
fact_inv_vec.resize(1, T(1));
extend(MAX + 1);
}
T fact(int i){
if (i < 0) return 0;
while (int(fact_vec.size()) <= i) extend();
return fact_vec[i];
}
T invfact(int i){
if (i < 0) return 0;
while (int(fact_inv_vec.size()) <= i) extend();
return fact_inv_vec[i];
}
T C(int a, int b){
if (a < b || b < 0) return 0;
return fact(a) * invfact(b) * invfact(a - b);
}
T invC(int a, int b){
if (a < b || b < 0) return 0;
return fact(b) * fact(a - b) *invfact(a);
}
T P(int a, int b){
if (a < b || b < 0) return 0;
return fact(a) * invfact(a - b);
}
T inv(int a){
if (a < 0) return inv(-a) * T(-1);
if (a == 0) return 1;
return fact(a - 1) * invfact(a);
}
T Catalan(int n){
if (n < 0) return 0;
return fact(2 * n) * invfact(n + 1) * invfact(n);
}
T narayana(int n, int k){
if (n <= 0 || n < k || k < 1) return 0;
return C(n, k) * C(n, k - 1) * inv(n);
}
T Catalan_pow(int n,int d){
if (n < 0 || d < 0) return 0;
if (d == 0){
if (n == 0) return 1;
return 0;
}
return T(d) * inv(d + n) * C(2 * n + d - 1, n);
}
// retrun [x^a] 1/(1-x)^b
T ruiseki(int a,int b){
if (a < 0 || b < 0) return 0;
if (a == 0){
return 1;
}
return C(a + b - 1, b - 1);
}
// (a, b) -> (c, d)
// always x + e >= y
T mirror(int a, int b, int c, int d, int e = 0){
if (a + e < b || c + e < d) return 0;
if (a > c || b > d) return 0;
a += e;
c += e;
return C(c + d - a - b, c - a) - C(c + d - a - b, c - b + 1);
}
// return sum_{i = 0, ... , a} sum_{j = 0, ... , b} C(i + j, i)
// return C(a + b + 2, a + 1) - 1;
T gird_sum(int a, int b){
if (a < 0 || b < 0) return 0;
return C(a + b + 2, a + 1) - 1;
}
// return sum_{i = a, ..., b - 1} sum_{j = c, ... , d - 1} C(i + j, i)
// AGC 018 E
T gird_sum_2(int a, int b, int c, int d){
if (a >= b || c >= d) return 0;
a--, b--, c--, d--;
return gird_sum(a, c) - gird_sum(a, d) - gird_sum(b, c) + gird_sum(b, d);
}
// the number of diagonal dissections of a convex n-gon into k+1 regions.
// OEIS A033282
// AGC065D
T diagonal(int n, int k){
if (n <= 2 || n - 3 < k || k < 0) return 0;
return C(n - 3, k) * C(n + k - 1, k) * inv(k + 1);
}
};
}
#line 27 "e.cpp"
using mint = atcoder::modint998244353;
void solve();
// CITRUS CURIO CITY / FREDERIC
int main() {
ios::sync_with_stdio(false);
cin.tie(nullptr);
int t = 1;
// cin >> t;
rep(i, 0, t) solve();
}
void solve(){
ll N, K;
cin >> N >> K;
string S;
cin >> S;
int C = 0;
rep(i, 0, K){
if (S[i] == ')') break;
C++;
}
reverse(all(S));
rep(i, 0, K) {
if (S[i] == '(') break;
C++;
}
po167::Binomial<mint> table;
if (C != K){
mint ans = 1;
rep(i, 0, C + 1) ans *= (N - K + C + 1 - i);
ans *= table.invfact(C + 1);
cout << ans.val() << "\n";
return;
}
vector<mint> p(K + 1);
rep(i, 0, K + 1) p[i] = table.C(K, i) * (i & 1 ? -1 : 1);
p.push_back(0);
for (int i = K; i >= 0; i--) p[i + 1] -= p[i] * 2;
auto ans = po167::Boston_Mori(N - K, {1}, p);
cout << ans.val() << "\n";
}