結果

問題 No.2633 Subsequence Combination Score
ユーザー Aeren
提出日時 2024-02-16 23:21:15
言語 C++23
(gcc 13.3.0 + boost 1.87.0)
結果
AC  
実行時間 263 ms / 2,000 ms
コード長 16,251 bytes
コンパイル時間 3,451 ms
コンパイル使用メモリ 264,624 KB
実行使用メモリ 10,604 KB
最終ジャッジ日時 2024-09-28 22:04:32
合計ジャッジ時間 15,480 ms
ジャッジサーバーID
（参考情報） 		judge1 / judge5
このコードへのチャレンジ
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ファイルパターン 結果
sample AC * 3
other AC * 38
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ソースコード

diff #
プレゼンテーションモードにする

// #pragma GCC optimize("O3,unroll-loops")
#include <bits/stdc++.h>
// #include <x86intrin.h>
using namespace std;
#if __cplusplus >= 202002L
using namespace numbers;
#endif
template<class data_t, data_t _mod>
struct modular_fixed_base{
#define IS_INTEGRAL(T) (is_integral_v<T> || is_same_v<T, __int128_t> || is_same_v<T, __uint128_t>)
#define IS_UNSIGNED(T) (is_unsigned_v<T> || is_same_v<T, __uint128_t>)
    static_assert(IS_UNSIGNED(data_t));
    static_assert(_mod >= 1);
    static constexpr bool VARIATE_MOD_FLAG = false;
    static constexpr data_t mod(){
        return _mod;
    }
    template<class T>
    static vector<modular_fixed_base> precalc_power(T base, int SZ){
        vector<modular_fixed_base> res(SZ + 1, 1);
        for(auto i = 1; i <= SZ; ++ i) res[i] = res[i - 1] * base;
        return res;
    }   
    static vector<modular_fixed_base> _INV;
    static void precalc_inverse(int SZ){
        if(_INV.empty()) _INV.assign(2, 1);
        for(auto x = _INV.size(); x <= SZ; ++ x) _INV.push_back(_mod / x * -_INV[_mod % x]);
    }
    // _mod must be a prime
    static modular_fixed_base _primitive_root;
    static modular_fixed_base primitive_root(){
        if(_primitive_root) return _primitive_root;
        if(_mod == 2) return _primitive_root = 1;
        if(_mod == 998244353) return _primitive_root = 3;
        data_t divs[20] = {};
        divs[0] = 2;
        int cnt = 1;
        data_t x = (_mod - 1) / 2;
        while(x % 2 == 0) x /= 2;
        for(auto i = 3; 1LL * i * i <= x; i += 2){
            if(x % i == 0){
                divs[cnt ++] = i;
                while(x % i == 0) x /= i;
            }
        }
        if(x > 1) divs[cnt ++] = x;
        for(auto g = 2; ; ++ g){
            bool ok = true;
            for(auto i = 0; i < cnt; ++ i){
                if((modular_fixed_base(g).power((_mod - 1) / divs[i])) == 1){
                    ok = false;
                    break;
                }
            }
            if(ok) return _primitive_root = g;
        }
    }
    constexpr modular_fixed_base(){ }
    modular_fixed_base(const double &x){ data = _normalize(llround(x)); }
    modular_fixed_base(const long double &x){ data = _normalize(llround(x)); }
    template<class T, typename enable_if<IS_INTEGRAL(T)>::type* = nullptr> modular_fixed_base(const T &x){ data = _normalize(x); }
    template<class T, typename enable_if<IS_INTEGRAL(T)>::type* = nullptr> static data_t _normalize(const T &x){
        int sign = x >= 0 ? 1 : -1;
        data_t v =  _mod <= sign * x ? sign * x % _mod : sign * x;
        if(sign == -1 && v) v = _mod - v;
        return v;
    }
    template<class T, typename enable_if<IS_INTEGRAL(T)>::type* = nullptr> operator T() const{ return data; }
    modular_fixed_base &operator+=(const modular_fixed_base &otr){ if((data += otr.data) >= _mod) data -= _mod; return *this; }
    modular_fixed_base &operator-=(const modular_fixed_base &otr){ if((data += _mod - otr.data) >= _mod) data -= _mod; return *this; }
    template<class T, typename enable_if<IS_INTEGRAL(T)>::type* = nullptr> modular_fixed_base &operator+=(const T &otr){ return *this += 
        modular_fixed_base(otr); }
    template<class T, typename enable_if<IS_INTEGRAL(T)>::type* = nullptr> modular_fixed_base &operator-=(const T &otr){ return *this -= 
        modular_fixed_base(otr); }
    modular_fixed_base &operator++(){ return *this += 1; }
    modular_fixed_base &operator--(){ return *this += _mod - 1; }
    modular_fixed_base operator++(int){ modular_fixed_base result(*this); *this += 1; return result; }
    modular_fixed_base operator--(int){ modular_fixed_base result(*this); *this += _mod - 1; return result; }
    modular_fixed_base operator-() const{ return modular_fixed_base(_mod - data); }
    modular_fixed_base &operator*=(const modular_fixed_base &rhs){
        if constexpr(is_same_v<data_t, unsigned int>) data = (unsigned long long)data * rhs.data % _mod;
        else if constexpr(is_same_v<data_t, unsigned long long>){
            long long res = data * rhs.data - _mod * (unsigned long long)(1.L / _mod * data * rhs.data);
            data = res + _mod * (res < 0) - _mod * (res >= (long long)_mod);
        }
        else data = _normalize(data * rhs.data);
        return *this;
    }
    template<class T, typename enable_if<IS_INTEGRAL(T)>::type* = nullptr>
    modular_fixed_base &inplace_power(T e){
        if(e == 0) return *this = 1;
        if(data == 0) return *this = {};
        if(data == 1) return *this;
        if(data == mod() - 1) return e % 2 ? *this : *this = -*this;
        if(e < 0) *this = 1 / *this, e = -e;
        modular_fixed_base res = 1;
        for(; e; *this *= *this, e >>= 1) if(e & 1) res *= *this;
        return *this = res;
    }
    template<class T, typename enable_if<IS_INTEGRAL(T)>::type* = nullptr>
    modular_fixed_base power(T e) const{
        return modular_fixed_base(*this).inplace_power(e);
    }
    modular_fixed_base &operator/=(const modular_fixed_base &otr){
        make_signed_t<data_t> a = otr.data, m = _mod, u = 0, v = 1;
        if(a < _INV.size()) return *this *= _INV[a];
        while(a){
            make_signed_t<data_t> t = m / a;
            m -= t * a; swap(a, m);
            u -= t * v; swap(u, v);
        }
        assert(m == 1);
        return *this *= u;
    }
#define ARITHMETIC_OP(op, apply_op)\
modular_fixed_base operator op(const modular_fixed_base &x) const{ return modular_fixed_base(*this) apply_op x; }\
template<class T, typename enable_if<IS_INTEGRAL(T)>::type* = nullptr>\
modular_fixed_base operator op(const T &x) const{ return modular_fixed_base(*this) apply_op modular_fixed_base(x); }\
template<class T, typename enable_if<IS_INTEGRAL(T)>::type* = nullptr>\
friend modular_fixed_base operator op(const T &x, const modular_fixed_base &y){ return modular_fixed_base(x) apply_op y; }
    ARITHMETIC_OP(+, +=) ARITHMETIC_OP(-, -=) ARITHMETIC_OP(*, *=) ARITHMETIC_OP(/, /=)
#undef ARITHMETIC_OP
#define COMPARE_OP(op)\
bool operator op(const modular_fixed_base &x) const{ return data op x.data; }\
template<class T, typename enable_if<IS_INTEGRAL(T)>::type* = nullptr>\
bool operator op(const T &x) const{ return data op modular_fixed_base(x).data; }\
template<class T, typename enable_if<IS_INTEGRAL(T)>::type* = nullptr>\
friend bool operator op(const T &x, const modular_fixed_base &y){ return modular_fixed_base(x).data op y.data; }
    COMPARE_OP(==) COMPARE_OP(!=) COMPARE_OP(<) COMPARE_OP(<=) COMPARE_OP(>) COMPARE_OP(>=)
#undef COMPARE_OP
    friend istream &operator>>(istream &in, modular_fixed_base &number){
        long long x;
        in >> x;
        number.data = modular_fixed_base::_normalize(x);
        return in;
    }
#define _SHOW_FRACTION
    friend ostream &operator<<(ostream &out, const modular_fixed_base &number){
        out << number.data;
    #if defined(LOCAL) && defined(_SHOW_FRACTION)
        cerr << "(";
        for(auto d = 1; ; ++ d){
            if((number * d).data <= 1000000){
                cerr << (number * d).data;
                if(d != 1) cerr << "/" << d;
                break;
            }
            else if((-number * d).data <= 1000000){
                cerr << "-" << (-number * d).data;
                if(d != 1) cerr << "/" << d;
                break;
            }
        }
        cerr << ")";
    #endif
        return out;
    }
    data_t data = 0;
#undef _SHOW_FRACTION
#undef IS_INTEGRAL
#undef IS_SIGNED
};
template<class data_t, data_t _mod> vector<modular_fixed_base<data_t, _mod>> modular_fixed_base<data_t, _mod>::_INV;
template<class data_t, data_t _mod> modular_fixed_base<data_t, _mod> modular_fixed_base<data_t, _mod>::_primitive_root;
const unsigned int mod = (119 << 23) + 1; // 998244353
// const unsigned int mod = 1e9 + 7; // 1000000007
// const unsigned int mod = 1e9 + 9; // 1000000009
// const unsigned long long mod = (unsigned long long)1e18 + 9;
using modular = modular_fixed_base<decay_t<decltype(mod)>, mod>;
template<class T>
struct combinatorics{
    // O(n)
    static vector<T> precalc_fact(int n){
        vector<T> f(n + 1, 1);
        for(auto i = 1; i <= n; ++ i) f[i] = f[i - 1] * i;
        return f;
    }
    // O(n * m)
    static vector<vector<T>> precalc_C(int n, int m){
        vector<vector<T>> c(n + 1, vector<T>(m + 1));
        for(auto i = 0; i <= n; ++ i) for(auto j = 0; j <= min(i, m); ++ j) c[i][j] = i && j ? c[i - 1][j - 1] + c[i - 1][j] : T(1);
        return c;
    }
    int SZ = 0;
    vector<T> inv, fact, invfact;
    combinatorics(){ }
    // O(SZ)
    combinatorics(int SZ): SZ(SZ), inv(SZ + 1, 1), fact(SZ + 1, 1), invfact(SZ + 1, 1){
        for(auto i = 1; i <= SZ; ++ i) fact[i] = fact[i - 1] * i;
        invfact[SZ] = 1 / fact[SZ];
        for(auto i = SZ - 1; i >= 0; -- i){
            invfact[i] = invfact[i + 1] * (i + 1);
            inv[i + 1] = invfact[i + 1] * fact[i];
        }
    }
    // O(1)
    T C(int n, int k) const{
        assert(0 <= min(n, k) && max(n, k) <= SZ);
        return n >= k ? fact[n] * invfact[k] * invfact[n - k] : T{0};
    }
    // O(1)
    T P(int n, int k) const{
        assert(0 <= min(n, k) && max(n, k) <= SZ);
        return n >= k ? fact[n] * invfact[n - k] : T{0};
    }
    // O(1)
    T H(int n, int k) const{
        assert(0 <= min(n, k));
        if(n == 0) return 0;
        return C(n + k - 1, k);
    }
    // O(min(k, n - k))
    T naive_C(long long n, long long k) const{
        assert(0 <= min(n, k));
        if(n < k) return 0;
        T res = 1;
        k = min(k, n - k);
        assert(k <= SZ);
        for(auto i = n; i > n - k; -- i) res *= i;
        return res * invfact[k];
    }
    // O(k)
    T naive_P(long long n, int k) const{
        assert(0 <= min<long long>(n, k));
        if(n < k) return 0;
        T res = 1;
        for(auto i = n; i > n - k; -- i) res *= i;
        return res;
    }
    // O(k)
    T naive_H(long long n, int k) const{
        assert(0 <= min<long long>(n, k));
        return naive_C(n + k - 1, k);
    }
    // O(1)
    bool parity_C(long long n, long long k) const{
        assert(0 <= min(n, k));
        return n >= k ? (n & k) == k : false;
    }
    // Number of ways to place n '('s and k ')'s starting with m copies of '(' such that in each prefix, number of '(' is equal or greater than ')'
    // Catalan(n, n, 0): n-th catalan number
    // Catalan(s, s+k-1, k-1): sum of products of k catalan numbers where the index of product sums up to s.
    // O(1)
    T Catalan(int n, int k, int m = 0) const{
        assert(0 <= min({n, k, m}));
        return k <= m ? C(n + k, n) : k <= n + m ? C(n + k, n) - C(n + k, k - m - 1) : T();
    }
};
// T must be of modular type
// mod must be a prime
// Requires modular
template<class T>
struct number_theoric_transform_wrapper{
    // i \in [2^k, 2^{k+1}) holds w_{2^k+1}^{i-2^k}
    static vector<T> root, buffer1, buffer2;
    static void adjust_root(int n){
        if(root.empty()) root = {1, 1};
        for(auto k = (int)root.size(); k < n; k <<= 1){
            root.resize(n, 1);
            T w = T::primitive_root().power((T::mod() - 1) / (k << 1));
            for(auto i = k; i < k << 1; ++ i) root[i] = i & 1 ? root[i >> 1] * w : root[i >> 1];
        }
    }
    // n must be a power of two
    // p must have next n memories allocated
    // O(n * log(n))
    static void transform(int n, T *p, bool invert = false){
        assert(n && __builtin_popcount(n) == 1 && (T::mod() - 1) % n == 0);
        for(auto i = 1, j = 0; i < n; ++ i){
            int bit = n >> 1;
            for(; j & bit; bit >>= 1) j ^= bit;
            j ^= bit;
            if(i < j) swap(p[i], p[j]);
        }
        adjust_root(n);
        for(auto len = 1; len < n; len <<= 1) for(auto i = 0; i < n; i += len << 1) for(auto j = 0; j < len; ++ j){
            T x = p[i + j], y = p[len + i + j] * root[len + j];
            p[i + j] = x + y, p[len + i + j] = x - y;
        }
        if(invert){
            reverse(p + 1, p + n);
            T inv_n = T(1) / n;
            for(auto i = 0; i < n; ++ i) p[i] *= inv_n;
        }
    }
    static void transform(vector<T> &p, bool invert = false){
        transform((int)p.size(), p.data(), invert);
    }
    // Double the length of the ntt array
    // n must be a power of two
    // p must have next 2n memories allocated
    // O(n * log(n))
    static void double_up(int n, T *p){
        assert(n && __builtin_popcount(n) == 1 && (T().mod() - 1) % (n << 1) == 0);
        buffer1.resize(n << 1);
        for(auto i = 0; i < n; ++ i) buffer1[i << 1] = p[i];
        transform(n, p, true);
        adjust_root(n << 1);
        for(auto i = 0; i < n; ++ i) p[i] *= root[n | i];
        transform(n, p);
        for(auto i = 0; i < n; ++ i) buffer1[i << 1 | 1] = p[i];
        copy(buffer1.begin(), buffer1.begin() + 2 * n, p);
    }
    static void double_up(vector<T> &p){
        int n = (int)p.size();
        p.resize(n << 1);
        double_up(n, p.data());
    }
    // O(n * m)
    static vector<T> convolute_naive(const vector<T> &p, const vector<T> &q){
        vector<T> res(max((int)p.size() + (int)q.size() - 1, 0));
        for(auto i = 0; i < (int)p.size(); ++ i) for(auto j = 0; j < (int)q.size(); ++ j) res[i + j] += p[i] * q[j];
        return res;
    }
    // O((n + m) * log(n + m))
    static vector<T> convolute(const vector<T> &p, const vector<T> &q){
        if(min(p.size(), q.size()) < 55) return convolute_naive(p, q);
        int m = (int)p.size() + (int)q.size() - 1, n = 1 << __lg(m) + 1;
        buffer1.assign(n, 0);
        copy(p.begin(), p.end(), buffer1.begin());
        transform(buffer1);
        buffer2.assign(n, 0);
        copy(q.begin(), q.end(), buffer2.begin());
        transform(buffer2);
        for(auto i = 0; i < n; ++ i) buffer1[i] *= buffer2[i];
        transform(buffer1, true);
        return vector<T>(buffer1.begin(), buffer1.begin() + m);
    }
    // O((n + m) * log(n + m))
    static vector<T> square(const vector<T> &p){
        if((int)p.size() < 40) return convolute_naive(p, p);
        int m = 2 * (int)p.size() - 1, n = 1 << __lg(m) + 1;
        buffer1.assign(n, 0);
        copy(p.begin(), p.end(), buffer1.begin());
        transform(buffer1);
        for(auto i = 0; i < n; ++ i) buffer1[i] *= buffer1[i];
        transform(buffer1, true);
        return vector<T>(buffer1.begin(), buffer1.begin() + m);
    }
    // O((n + m) * log(n + m))
    static vector<T> arbitrarily_convolute(const vector<T> &p, const vector<T> &q){
        using modular0 = modular_fixed_base<unsigned int, 1045430273>;
        using modular1 = modular_fixed_base<unsigned int, 1051721729>;
        using modular2 = modular_fixed_base<unsigned int, 1053818881>;
        using ntt0 = number_theoric_transform_wrapper<modular0>;
        using ntt1 = number_theoric_transform_wrapper<modular1>;
        using ntt2 = number_theoric_transform_wrapper<modular2>;
        vector<modular0> p0((int)p.size()), q0((int)q.size());
        for(auto i = 0; i < (int)p.size(); ++ i) p0[i] = p[i].data;
        for(auto i = 0; i < (int)q.size(); ++ i) q0[i] = q[i].data;
        auto xy0 = ntt0::convolute(p0, q0);
        vector<modular1> p1((int)p.size()), q1((int)q.size());
        for(auto i = 0; i < (int)p.size(); ++ i) p1[i] = p[i].data;
        for(auto i = 0; i < (int)q.size(); ++ i) q1[i] = q[i].data;
        auto xy1 = ntt1::convolute(p1, q1);
        vector<modular2> p2((int)p.size()), q2((int)q.size());
        for(auto i = 0; i < (int)p.size(); ++ i) p2[i] = p[i].data;
        for(auto i = 0; i < (int)q.size(); ++ i) q2[i] = q[i].data;
        auto xy2 = ntt2::convolute(p2, q2);
        static const modular1 r01 = 1 / modular1(modular0::mod());
        static const modular2 r02 = 1 / modular2(modular0::mod());
        static const modular2 r12 = 1 / modular2(modular1::mod());
        static const modular2 r02r12 = r02 * r12;
        static const T w1 = modular0::mod();
        static const T w2 = w1 * modular1::mod();
        int n = (int)p.size() + (int)q.size() - 1;
        vector<T> res(n);
        for(auto i = 0; i < n; ++ i){
            using ull = unsigned long long;
            ull a = xy0[i].data;
            ull b = (xy1[i].data + modular1::mod() - a) * r01.data % modular1::mod();
            ull c = ((xy2[i].data + modular2::mod() - a) * r02r12.data + (modular2::mod() - b) * r12.data) % modular2::mod();
            res[i] = xy0[i].data + w1 * b + w2 * c;
        }
        return res;
    }
};
template<class T> vector<T> number_theoric_transform_wrapper<T>::root;
template<class T> vector<T> number_theoric_transform_wrapper<T>::buffer1;
template<class T> vector<T> number_theoric_transform_wrapper<T>::buffer2;
using ntt = number_theoric_transform_wrapper<modular>;
template<class F>
struct y_combinator_result{
    F f;
    template<class T> explicit y_combinator_result(T &&f): f(forward<T>(f)){ }
    template<class ...Args> decltype(auto) operator()(Args &&...args){ return f(ref(*this), forward<Args>(args)...); }
};
template<class F>
decltype(auto) y_combinator(F &&f){
    return y_combinator_result<decay_t<F>>(forward<F>(f));
}
int main(){
    cin.tie(0)->sync_with_stdio(0);
    cin.exceptions(ios::badbit | ios::failbit);
    const int mx = 1e5;
    combinatorics<modular> C(mx);
    int n;
    cin >> n;
    vector<int> cnt(mx + 1);
    for(auto i = 0; i < n; ++ i){
        int x;
        cin >> x;
        ++ cnt[x];
    }
    auto power = modular::precalc_power(2, n);
    vector<modular> f(mx + 1);
    y_combinator([&](auto self, int l, int r)->void{
        if(r - l == 1){
            f[l] = (f[l] + C.invfact[l]) * (power[cnt[l]] - 1);
            return;
        }
        int m = l + r >> 1;
        self(l, m);
        vector<modular> p(f.begin() + l, f.begin() + m);
        vector<modular> q(r - l);
        for(auto i = 1; i < r - l; ++ i){
            q[i] = C.invfact[i];
        }
        p = ntt::convolute(p, q);
        for(auto i = m; i < r; ++ i){
            if(i - l < (int)p.size()){
                f[i] += p[i - l];
            }
        }
        self(m, r);
    })(0, mx + 1);
    modular res = 0;
    for(auto x = 0; x <= mx; ++ x){
        res += C.fact[x] * f[x];
    }
    cout << res << "\n";
    return 0;
}
/*
*/
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