結果

問題 No.2062 Sum of Subset mod 999630629
ユーザー kwm_t
提出日時 2022-08-27 18:10:38
言語 C++17
(gcc 13.3.0 + boost 1.87.0)
結果
AC  
実行時間 239 ms / 5,000 ms
コード長 15,266 bytes
コンパイル時間 6,188 ms
コンパイル使用メモリ 289,608 KB
最終ジャッジ日時 2025-01-31 06:29:13
ジャッジサーバーID
（参考情報） 		judge2 / judge2
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ファイルパターン 結果
sample AC * 3
other AC * 29
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ソースコード

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プレゼンテーションモードにする

#include <bits/stdc++.h>
#include <atcoder/all>
using namespace std;
using namespace atcoder;
//using mint = modint1000000007;
//const int mod = 1000000007;
using mint = modint998244353;
const int mod = 998244353;
//const int INF = 1e9;
//const long long LINF = 1e18;
#define rep(i, n) for (int i = 0; i < (n); ++i)
#define rep2(i,l,r)for(int i=(l);i<(r);++i)
#define rrep(i, n) for (int i = (n-1); i >= 0; --i)
#define rrep2(i,l,r)for(int i=(r-1);i>=(l);--i)
#define all(x) (x).begin(),(x).end()
#define allR(x) (x).rbegin(),(x).rend()
#define endl "\n"
#define P pair<int,int>
template<typename A, typename B> inline bool chmax(A & a, const B & b) { if (a < b) { a = b; return true; } return false; }
template<typename A, typename B> inline bool chmin(A & a, const B & b) { if (a > b) { a = b; return true; } return false; }
// verified by:
// https://judge.yosupo.jp/problem/convolution_mod
// https://judge.yosupo.jp/problem/inv_of_formal_power_series
// https://judge.yosupo.jp/problem/log_of_formal_power_series
// https://judge.yosupo.jp/problem/exp_of_formal_power_series
// https://judge.yosupo.jp/problem/pow_of_formal_power_series
// https://judge.yosupo.jp/problem/polynomial_taylor_shift
// https://judge.yosupo.jp/problem/bernoulong longi_number
// https://judge.yosupo.jp/problem/sharp_p_subset_sum
//using mint = modint998244353;
template<typename T> struct Factorial {
    int MAX;
    vector<T> fac, finv;
    Factorial(int m = 0) : MAX(m), fac(m + 1, 1), finv(m + 1, 1) {
        rep2(i, 2, MAX + 1) fac[i] = fac[i - 1] * i;
        finv[MAX] /= fac[MAX];
        rrep2(i, MAX + 1, 3) finv[i - 1] = finv[i] * i;
    }
    T binom(int n, int k) {
        if (k < 0 || n < k) return 0;
        return fac[n] * finv[k] * finv[n - k];
    }
    T perm(int n, int k) {
        if (k < 0 || n < k) return 0;
        return fac[n] * finv[n - k];
    }
};
Factorial<mint> fc;
istream &operator>>(istream &is, modint998244353 &a) { long long v; is >> v; a = v; return is; }
ostream &operator<<(ostream &os, const modint998244353 &a) { return os << a.val(); }
istream &operator>>(istream &is, modint1000000007 &a) { long long v; is >> v; a = v; return is; }
ostream &operator<<(ostream &os, const modint1000000007 &a) { return os << a.val(); }
template<int m> istream &operator>>(istream &is, static_modint<m> &a) { long long v; is >> v; a = v; return is; }
template<int m> istream &operator>>(istream &is, dynamic_modint<m> &a) { long long v; is >> v; a = v; return is; }
template<int m> ostream &operator<<(ostream &os, const static_modint<m> &a) { return os << a.val(); }
template<int m> ostream &operator<<(ostream &os, const dynamic_modint<m> &a) { return os << a.val(); }
template<class T> istream &operator>>(istream &is, vector<T> &v) { for (auto &e : v) is >> e; return is; }
template<class T> ostream &operator<<(ostream &os, const vector<T> &v) { for (auto &e : v) os << e << ' '; return os; }
template<class T>
struct FormalPowerSeries : vector<T> {
    using vector<T>::vector;
    using vector<T>::operator=;
    using F = FormalPowerSeries;
    F operator-() const {
        F res(*this);
        for (auto &e : res) e = -e;
        return res;
    }
    F &operator*=(const T &g) {
        for (auto &e : *this) e *= g;
        return *this;
    }
    F &operator/=(const T &g) {
        assert(g != T(0));
        *this *= g.inv();
        return *this;
    }
    F &operator+=(const F &g) {
        int n = this->size(), m = g.size();
        rep(i, min(n, m)) (*this)[i] += g[i];
        return *this;
    }
    F &operator-=(const F &g) {
        int n = this->size(), m = g.size();
        rep(i, min(n, m)) (*this)[i] -= g[i];
        return *this;
    }
    F &operator<<=(const int d) {
        int n = this->size();
        if (d >= n) *this = F(n);
        this->insert(this->begin(), d, 0);
        this->resize(n);
        return *this;
    }
    F &operator>>=(const int d) {
        int n = this->size();
        this->erase(this->begin(), this->begin() + min(n, d));
        this->resize(n);
        return *this;
    }
    // O(n log n)
    F inv(int d = -1) const {
        int n = this->size();
        assert(n != 0 && (*this)[0] != 0);
        if (d == -1) d = n;
        assert(d >= 0);
        F res{ (*this)[0].inv() };
        for (int m = 1; m < d; m *= 2) {
            F f(this->begin(), this->begin() + min(n, 2 * m));
            F g(res);
            f.resize(2 * m), internal::butterfly(f);
            g.resize(2 * m), internal::butterfly(g);
            rep(i, 2 * m) f[i] *= g[i];
            internal::butterfly_inv(f);
            f.erase(f.begin(), f.begin() + m);
            f.resize(2 * m), internal::butterfly(f);
            rep(i, 2 * m) f[i] *= g[i];
            internal::butterfly_inv(f);
            T iz = T(2 * m).inv(); iz *= -iz;
            rep(i, m) f[i] *= iz;
            res.insert(res.end(), f.begin(), f.begin() + m);
        }
        res.resize(d);
        return res;
    }
    // fast: FMT-friendly modulus only
    // O(n log n)
    F &multiply_inplace(const F &g, int d = -1) {
        int n = this->size();
        if (d == -1) d = n;
        assert(d >= 0);
        *this = convolution(move(*this), g);
        this->resize(d);
        return *this;
    }
    F multiply(const F &g, const int d = -1) const { return F(*this).multiply_inplace(g, d); }
    // O(n log n)
    F &divide_inplace(const F &g, int d = -1) {
        int n = this->size();
        if (d == -1) d = n;
        assert(d >= 0);
        *this = convolution(move(*this), g.inv(d));
        this->resize(d);
        return *this;
    }
    F divide(const F &g, const int d = -1) const { return F(*this).divide_inplace(g, d); }
    // // naive
    // // O(n^2)
    // F &multiply_inplace(const F &g) {
    //   int n = this->size(), m = g.size();
    //   rrep(i, n) {
    //     (*this)[i] *= g[0];
    //     rep2(j, 1, min(i+1, m)) (*this)[i] += (*this)[i-j] * g[j];
    //   }
    //   return *this;
    // }
    // F multiply(const F &g) const { return F(*this).multiply_inplace(g); }
    // // O(n^2)
    // F &divide_inplace(const F &g) {
    //   assert(g[0] != T(0));
    //   T ig0 = g[0].inv();
    //   int n = this->size(), m = g.size();
    //   rep(i, n) {
    //     rep2(j, 1, min(i+1, m)) (*this)[i] -= (*this)[i-j] * g[j];
    //     (*this)[i] *= ig0;
    //   }
    //   return *this;
    // }
    // F divide(const F &g) const { return F(*this).divide_inplace(g); }
    // sparse
    // O(nk)
    F &multiply_inplace(vector<pair<int, T>> g) {
        int n = this->size();
        auto[d, c] = g.front();
        if (d == 0) g.erase(g.begin());
        else c = 0;
        rrep(i, n) {
            (*this)[i] *= c;
            for (auto &[j, b] : g) {
                if (j > i) break;
                (*this)[i] += (*this)[i - j] * b;
            }
        }
        return *this;
    }
    F multiply(const vector<pair<int, T>> &g) const { return F(*this).multiply_inplace(g); }
    // O(nk)
    F &divide_inplace(vector<pair<int, T>> g) {
        int n = this->size();
        auto[d, c] = g.front();
        assert(d == 0 && c != T(0));
        T ic = c.inv();
        g.erase(g.begin());
        rep(i, n) {
            for (auto &[j, b] : g) {
                if (j > i) break;
                (*this)[i] -= (*this)[i - j] * b;
            }
            (*this)[i] *= ic;
        }
        return *this;
    }
    F divide(const vector<pair<int, T>> &g) const { return F(*this).divide_inplace(g); }
    // multiply and divide (1 + cz^d)
    // O(n)
    void multiply_inplace(const int d, const T c) {
        int n = this->size();
        if (c == T(1)) rrep(i, n - d) (*this)[i + d] += (*this)[i];
        else if (c == T(-1)) rrep(i, n - d) (*this)[i + d] -= (*this)[i];
        else rrep(i, n - d) (*this)[i + d] += (*this)[i] * c;
    }
    // O(n)
    void divide_inplace(const int d, const T c) {
        int n = this->size();
        if (c == T(1)) rep(i, n - d) (*this)[i + d] -= (*this)[i];
        else if (c == T(-1)) rep(i, n - d) (*this)[i + d] += (*this)[i];
        else rep(i, n - d) (*this)[i + d] -= (*this)[i] * c;
    }
    // O(n)
    T eval(const T &a) const {
        T x(1), res(0);
        for (auto e : *this) res += e * x, x *= a;
        return res;
    }
    // O(n)
    F &integ_inplace() {
        int n = this->size();
        assert(n > 0);
        if (n == 1) return *this = F{ 0 };
        this->insert(this->begin(), 0);
        this->pop_back();
        vector<T> inv(n);
        inv[1] = 1;
        int p = T::mod();
        rep2(i, 2, n) inv[i] = -inv[p%i] * (p / i);
        rep2(i, 2, n) (*this)[i] *= inv[i];
        return *this;
    }
    F integ() const { return F(*this).integ_inplace(); }
    // O(n)
    F &deriv_inplace() {
        int n = this->size();
        assert(n > 0);
        rep2(i, 2, n) (*this)[i] *= i;
        this->erase(this->begin());
        this->push_back(0);
        return *this;
    }
    F deriv() const { return F(*this).deriv_inplace(); }
    // O(n log n)
    F &log_inplace(int d = -1) {
        int n = this->size();
        assert(n > 0 && (*this)[0] == 1);
        if (d == -1) d = n;
        assert(d >= 0);
        if (d < n) this->resize(d);
        F f_inv = this->inv();
        this->deriv_inplace();
        this->multiply_inplace(f_inv);
        this->integ_inplace();
        return *this;
    }
    F log(const int d = -1) const { return F(*this).log_inplace(d); }
    // O(n log n)
    // https://arxiv.org/abs/1301.5804 (Figure 1, right)
    F &exp_inplace(int d = -1) {
        int n = this->size();
        assert(n > 0 && (*this)[0] == 0);
        if (d == -1) d = n;
        assert(d >= 0);
        F g{ 1 }, g_fft{ 1, 1 };
        (*this)[0] = 1;
        this->resize(d);
        F h_drv(this->deriv());
        for (int m = 2; m < d; m *= 2) {
            // prepare
            F f_fft(this->begin(), this->begin() + m);
            f_fft.resize(2 * m), internal::butterfly(f_fft);
            // Step 2.a'
            // {
            F _g(m);
            rep(i, m) _g[i] = f_fft[i] * g_fft[i];
            internal::butterfly_inv(_g);
            _g.erase(_g.begin(), _g.begin() + m / 2);
            _g.resize(m), internal::butterfly(_g);
            rep(i, m) _g[i] *= g_fft[i];
            internal::butterfly_inv(_g);
            _g.resize(m / 2);
            _g /= T(-m) * m;
            g.insert(g.end(), _g.begin(), _g.begin() + m / 2);
            // }
            // Step 2.b'--d'
            F t(this->begin(), this->begin() + m);
            t.deriv_inplace();
            // {
              // Step 2.b'
            F r{ h_drv.begin(), h_drv.begin() + m - 1 };
            // Step 2.c'
            r.resize(m); internal::butterfly(r);
            rep(i, m) r[i] *= f_fft[i];
            internal::butterfly_inv(r);
            r /= -m;
            // Step 2.d'
            t += r;
            t.insert(t.begin(), t.back()); t.pop_back();
            // }
            // Step 2.e'
            if (2 * m < d) {
                t.resize(2 * m); internal::butterfly(t);
                g_fft = g; g_fft.resize(2 * m); internal::butterfly(g_fft);
                rep(i, 2 * m) t[i] *= g_fft[i];
                internal::butterfly_inv(t);
                t.resize(m);
                t /= 2 * m;
            }
            else { // この場合分けをしても数パーセントしか速くならない
                F g1(g.begin() + m / 2, g.end());
                F s1(t.begin() + m / 2, t.end());
                t.resize(m / 2);
                g1.resize(m), internal::butterfly(g1);
                t.resize(m), internal::butterfly(t);
                s1.resize(m), internal::butterfly(s1);
                rep(i, m) s1[i] = g_fft[i] * s1[i] + g1[i] * t[i];
                rep(i, m) t[i] *= g_fft[i];
                internal::butterfly_inv(t);
                internal::butterfly_inv(s1);
                rep(i, m / 2) t[i + m / 2] += s1[i];
                t /= m;
            }
            // Step 2.f'
            F v(this->begin() + m, this->begin() + min<int>(d, 2 * m)); v.resize(m);
            t.insert(t.begin(), m - 1, 0); t.push_back(0);
            t.integ_inplace();
            rep(i, m) v[i] -= t[m + i];
            // Step 2.g'
            v.resize(2 * m); internal::butterfly(v);
            rep(i, 2 * m) v[i] *= f_fft[i];
            internal::butterfly_inv(v);
            v.resize(m);
            v /= 2 * m;
            // Step 2.h'
            rep(i, min(d - m, m)) (*this)[m + i] = v[i];
        }
        return *this;
    }
    F exp(const int d = -1) const { return F(*this).exp_inplace(d); }
    // O(n log n)
    F &pow_inplace(const long long k, int d = -1) {
        int n = this->size();
        if (d == -1) d = n;
        assert(d >= 0 && k >= 0);
        if (k == 0) {
            *this = F(d);
            if (d > 0) (*this)[0] = 1;
            return *this;
        }
        int l = 0;
        while (l < n && (*this)[l] == 0) ++l;
        if (l > (d - 1) / k || l == n) return *this = F(d);
        T c = (*this)[l];
        this->erase(this->begin(), this->begin() + l);
        *this /= c;
        this->log_inplace(d - l * k);
        *this *= k;
        this->exp_inplace();
        *this *= c.pow(k);
        this->insert(this->begin(), l*k, 0);
        return *this;
    }
    F pow(const long long k, const int d = -1) const { return F(*this).pow_inplace(k, d); }
    // O(n log n)
    F &shift_inplace(const T c) {
        int n = this->size();
        fc = Factorial<T>(n);
        rep(i, n) (*this)[i] *= fc.fac[i];
        reverse(this->begin(), this->end());
        F g(n);
        T cp = 1;
        rep(i, n) g[i] = cp * fc.finv[i], cp *= c;
        this->multiply_inplace(g, n);
        reverse(this->begin(), this->end());
        rep(i, n) (*this)[i] *= fc.finv[i];
        return *this;
    }
    F shift(const T c) const { return F(*this).shift_inplace(c); }
    F operator*(const T &g) const { return F(*this) *= g; }
    F operator/(const T &g) const { return F(*this) /= g; }
    F operator+(const F &g) const { return F(*this) += g; }
    F operator-(const F &g) const { return F(*this) -= g; }
    F operator<<(const int d) const { return F(*this) <<= d; }
    F operator>>(const int d) const { return F(*this) >>= d; }
};
// template<class T>
// FormalPowerSeries<T> interpolation(const vector<pair<T, T>> &xy) {
// }
using fps = FormalPowerSeries<mint>;
// https://judge.yosupo.jp/problem/inv_of_formal_power_series
// f(x)に対して 1/f(x)の先頭n項を求める
void yosupo_inv() {
    int n; cin >> n;
    fps a(n); cin >> a;
    cout << a.inv() << '\n';
}
// https://judge.yosupo.jp/problem/log_of_formal_power_series
// f(x)に対して log(f(x))の先頭n項を求める
void yosupo_log() {
    int n; cin >> n;
    fps a(n); cin >> a;
    cout << a.log_inplace() << '\n';
}
// https://judge.yosupo.jp/problem/exp_of_formal_power_series
// f(x)に対して exp(f(x))の先頭n項を求める
void yosupo_exp() {
    int n; cin >> n;
    fps a(n); cin >> a;
    cout << a.exp_inplace() << '\n';
}
// https://judge.yosupo.jp/problem/pow_of_formal_power_series
// f(x)に対してf(x)^mの先頭n項を求める
void yosupo_pow() {
    int n, m; cin >> n >> m;
    fps a(n); cin >> a;
    cout << a.pow_inplace(m) << '\n';
}
// https://judge.yosupo.jp/problem/polynomial_taylor_shift
// f(x)に対してf(x+c)の先頭n項を求める
void yosupo_shift() {
    int n, c; cin >> n >> c;
    fps a(n); cin >> a;
    cout << a.shift_inplace(c) << '\n';
}
// https://judge.yosupo.jp/problem/bernoulli_number
// ベルヌーイ数の先頭n項を求める
void yosupo_bernoulli() {
    int n; cin >> n;
    ++n;
    fc = Factorial<mint>(n);
    fps f((fc.finv).begin() + 1, (fc.finv).end());
    f = f.inv();
    rep(i, n) f[i] *= fc.fac[i];
    cout << f << '\n';
}
// https://judge.yosupo.jp/problem/sharp_p_subset_sum
// Π(1+x^a)の先頭t+1項を求める
void yosupo_count_subset_sum() {
    int n, t; cin >> n >> t;
    ++t;
    vector<int> c(t);
    rep(i, n) {
        int s; cin >> s;
        ++c[s];
    }
    vector<mint> inv(t);
    inv[1] = 1;
    int p = mint::mod();
    rep2(i, 2, t) inv[i] = -inv[p%i] * (p / i);
    fps f(t);
    rep(i, t) {
        if (c[i] == 0) continue;
        for (int j = 1, d = i; d < t; ++j, d += i) {
            if (j & 1) f[d] += c[i] * inv[j];
            else     f[d] -= c[i] * inv[j];
        }
    }
    f.exp_inplace();
    f.erase(f.begin());
    cout << f << '\n';
}
vector<mint> yosupo_count_subset_sum(int t, vector<int>a) {
    ++t;
    vector<int> c(t);
    rep(i, a.size()) {
        ++c[a[i]];
    }
    vector<mint> inv(t);
    inv[1] = 1;
    int p = mint::mod();
    rep2(i, 2, t) inv[i] = -inv[p%i] * (p / i);
    fps f(t);
    rep(i, t) {
        if (c[i] == 0) continue;
        for (int j = 1, d = i; d < t; ++j, d += i) {
            if (j & 1) f[d] += c[i] * inv[j];
            else     f[d] -= c[i] * inv[j];
        }
    }
    f.exp_inplace();
    return f;
    //vector<mint>ret(t);
    //rep(i, f.size()) ret[i] = f[i];
    //return ret;
}
int main() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);
    int n; cin >> n;
    vector<int>a(n);
    int sum = 0;
    rep(i, n) {
        cin >> a[i];
        sum += a[i];
    }
    mint ans = 0;
    rep(i, n) {
        mint sub = pow_mod(2, n - 1, mod);
        sub *= a[i];
        ans += sub;
    }
    if (sum >= 999630629) {
        int over = sum - 999630629;
        map<int, mint>mp;
        mp[0] = 1;
        auto get = yosupo_count_subset_sum(over, a);
        rep(i, get.size())ans -= get[i] * 999630629;
    }
    cout << ans.val() << endl;
    return 0;
}
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