#937544 (C++17) No.2580 Hyperinflation

• 問題: No.2580 Hyperinflation
• ユーザー: tute7627
• 提出日時: 2023-12-08 22:03:13
• 言語: C++17 (gcc 13.3.0 + boost 1.87.0)
• 結果: AC
• 実行時間: 1,208 ms / 4,000 ms
• コード長: 18,422 bytes
• コンパイル時間: 5,318 ms
• コンパイル使用メモリ: 261,876 KB
• 最終ジャッジ日時: 2025-02-18 09:38:09
• ジャッジサーバーID （参考情報）: judge4 / judge1

ファイルパターン	結果
sample	AC * 3
other	AC * 31

ソースコード

// https://atcoder.jp/contests/arc118/submissions/24453032

#include <bits/stdc++.h>

#include <atcoder/convolution>

using Fp = atcoder::modint998244353;
std::ostream& operator<<(std::ostream& os, Fp a) { return os << a.val(); }

using Fps = std::vector<Fp>;
int sz(const Fps& a) { return a.size(); }
Fps operator-(Fps a) {
  for (auto&& e : a) e = -e;
  return a;
}
Fps& operator+=(Fps& a, const Fps& b) {
  if (sz(a) < sz(b)) a.reserve(sz(b)), a.resize(sz(b));
  for (int i = 0; i < sz(b); ++i) a[i] += b[i];
  return a;
}
Fps operator+(Fps a, const Fps& b) { return std::move(a += b); }
Fps& operator-=(Fps& a, const Fps& b) {
  if (sz(a) < sz(b)) a.reserve(sz(b)), a.resize(sz(b));
  for (int i = 0; i < sz(b); ++i) a[i] -= b[i];
  return a;
}
Fps operator-(Fps a, const Fps& b) { return std::move(a -= b); }
Fps& operator*=(Fps& a, Fp b) {
  for (auto&& e : a) e *= b;
  return a;
}
Fps operator*(Fps a, Fp b) { return std::move(a *= b); }
Fps operator*(Fp a, Fps b) { return std::move(b *= a); }
Fps& operator/=(Fps& a, Fp b) {
  b = b.inv();
  for (auto&& e : a) e *= b;
  return a;
}
Fps operator/(Fps a, Fp b) { return std::move(a /= b); }
Fps fft(const Fps& a, int n) {
  Fps res(n);
  std::copy(a.begin(), a.begin() + std::min(n, sz(a)), res.begin());
  atcoder::internal::butterfly(res);
  return res;
}
Fps circ(Fps&& a, const Fps& b) {
  if (sz(a) < sz(b)) a.reserve(sz(b)), a.resize(sz(b));
  for (int i = 0; i < sz(b); ++i) a[i] *= b[i];
  return a;
}
Fps circ(Fps&& a) {
  for (auto&& e : a) e *= e;
  return a;
}
Fps ifft(Fps&& a, int size) {
  int n = sz(a);
  atcoder::internal::butterfly_inv(a);
  a.resize(size);
  a *= (1 - Fp::mod()) / n;
  return a;
}
int ceil_pow2(int n) {
    int x = 0;
    while ((1U << x) < (unsigned int)(n)) x++;
    return x;
}
Fps operator*(const Fps& a, const Fps& b) {
  if (a.empty() || b.empty()) return {};
  if (std::min(sz(a), sz(b)) <= 1) {
    Fps res(std::max(sz(a), sz(b)));
    for (int i = 0; i < sz(a); ++i)
      for (int j = 0; j < sz(b); ++j) {
        if (i + j == sz(res)) break;
        res[i + j] += a[i] * b[j];
      }
    return res;
  }
  int n = 1 << ceil_pow2(sz(a) + sz(b) - 1);
  auto buf = fft(a, n);
  if (&a == &b)
    buf = circ(std::move(buf));
  else
    buf = circ(std::move(buf), fft(b, n));
  return ifft(std::move(buf), std::max(sz(a), sz(b)));
}
Fps& operator*=(Fps& a, const Fps& b) { return a = a * b; }
Fps inv(const Fps& a) {
  Fps res{a[0].inv()};
  for (int n = 1; n < sz(a); n *= 2) {
    auto f_res = fft(res, 2 * n);
    Fps buf = ifft(circ(fft(a, 2 * n), f_res), 2 * n);
    std::fill(buf.begin(), buf.begin() + n, 0);
    buf = ifft(circ(fft(buf, 2 * n), f_res), std::min(2 * n, sz(a)));
    for (int i = n; i < sz(buf); ++i) res.push_back(-buf[i]);
  }
  return res;
}

using Poly = std::vector<Fp>;
Fp eval(const Poly& a, Fp x) {
  Fp res;
  for (int i = sz(a); i--;) (res *= x) += a[i];
  return res;
}

std::vector<Fp> fact, ifact;
void reserve(int n) {
  fact.resize(n + 1), ifact.resize(n + 1);
  fact[0] = 1;
  for (int i = 1; i <= n; ++i) fact[i] = i * fact[i - 1];
  ifact[n] = fact[n].pow(Fp::mod() - 2);
  for (int i = n; i; --i) ifact[i - 1] = ifact[i] * i;
}

using namespace std;
using cpx = complex<double>;
const double PI = acos(-1);
vector<cpx> roots = {{0, 0},
                     {1, 0}};

void ensure_capacity(int min_capacity) {
    for (int len = roots.size(); len < min_capacity; len *= 2) {
        for (int i = len >> 1; i < len; i++) {
            roots.emplace_back(roots[i]);
            double angle = 2 * PI * (2 * i + 1 - len) / (len * 2);
            roots.emplace_back(cos(angle), sin(angle));
        }
    }
}

void fft(vector<cpx> &z, bool inverse) {
    int n = z.size();
    assert((n & (n - 1)) == 0);
    ensure_capacity(n);
    for (unsigned i = 1, j = 0; i < n; i++) {
        int bit = n >> 1;
        for (; j >= bit; bit >>= 1)
            j -= bit;
        j += bit;
        if (i < j)
            swap(z[i], z[j]);
    }
    for (int len = 1; len < n; len <<= 1) {
        for (int i = 0; i < n; i += len * 2) {
            for (int j = 0; j < len; j++) {
                cpx root = inverse ? conj(roots[j + len]) : roots[j + len];
                cpx u = z[i + j];
                cpx v = z[i + j + len] * root;
                z[i + j] = u + v;
                z[i + j + len] = u - v;
            }
        }
    }
    if (inverse)
        for (int i = 0; i < n; i++)
            z[i] /= n;
}

vector<int> multiply_bigint(const vector<int> &a, const vector<int> &b, int base) {
    int need = a.size() + b.size();
    int n = 1;
    while (n < need) n <<= 1;
    vector<cpx> p(n);
    for (size_t i = 0; i < n; i++) {
        p[i] = cpx(i < a.size() ? a[i] : 0, i < b.size() ? b[i] : 0);
    }
    fft(p, false);
    // a[w[k]] = (p[w[k]] + conj(p[w[n-k]])) / 2
    // b[w[k]] = (p[w[k]] - conj(p[w[n-k]])) / (2*i)
    vector<cpx> ab(n);
    cpx r(0, -0.25);
    for (int i = 0; i < n; i++) {
        int j = (n - i) & (n - 1);
        ab[i] = (p[i] * p[i] - conj(p[j] * p[j])) * r;
    }
    fft(ab, true);
    vector<int> result(need);
    long long carry = 0;
    for (int i = 0; i < need; i++) {
        long long d = (long long) (ab[i].real() + 0.5) + carry;
        carry = d / base;
        result[i] = d % base;
    }
    return result;
}

vector<int> multiply_mod(const vector<int> &a, const vector<int> &b, int m) {
    int need = a.size() + b.size() - 1;
    int n = 1;
    while (n < need) n <<= 1;
    vector<cpx> A(n);
    for (size_t i = 0; i < a.size(); i++) {
        int x = (a[i] % m + m) % m;
        A[i] = cpx(x & ((1 << 15) - 1), x >> 15);
    }
    fft(A, false);

    vector<cpx> B(n);
    for (size_t i = 0; i < b.size(); i++) {
        int x = (b[i] % m + m) % m;
        B[i] = cpx(x & ((1 << 15) - 1), x >> 15);
    }
    fft(B, false);

    vector<cpx> fa(n);
    vector<cpx> fb(n);
    for (int i = 0, j = 0; i < n; i++, j = n - i) {
        cpx a1 = (A[i] + conj(A[j])) * cpx(0.5, 0);
        cpx a2 = (A[i] - conj(A[j])) * cpx(0, -0.5);
        cpx b1 = (B[i] + conj(B[j])) * cpx(0.5, 0);
        cpx b2 = (B[i] - conj(B[j])) * cpx(0, -0.5);
        fa[i] = a1 * b1 + a2 * b2 * cpx(0, 1);
        fb[i] = a1 * b2 + a2 * b1;
    }

    fft(fa, true);
    fft(fb, true);
    vector<int> res(need);
    for (int i = 0; i < need; i++) {
        long long aa = (long long) (fa[i].real() + 0.5);
        long long bb = (long long) (fb[i].real() + 0.5);
        long long cc = (long long) (fa[i].imag() + 0.5);
        res[i] = (aa % m + (bb % m << 15) + (cc % m << 30)) % m;
    }
    return res;
}
constexpr int digits(int base) noexcept {
    return base <= 1 ? 0 : 1 + digits(base / 10);
}
constexpr int base = 1000'000'000;
constexpr int base_digits = digits(base);

constexpr int fft_base = 10'000; // fft_base^2 * n / fft_base_digits <= 10^15 for double
constexpr int fft_base_digits = digits(fft_base);

struct bigint {
    // value == 0 is represented by empty z
    vector<int> z; // digits

    // sign == 1 <==> value >= 0
    // sign == -1 <==> value < 0
    int sign;

    bigint(long long v = 0) {
        *this = v;
    }

    bigint &operator=(long long v) {
        sign = v < 0 ? -1 : 1;
        v *= sign;
        z.clear();
        for (; v > 0; v = v / base)
            z.push_back((int) (v % base));
        return *this;
    }

    bigint(const string &s) {
        read(s);
    }

    bigint &operator+=(const bigint &other) {
        if (sign == other.sign) {
            for (int i = 0, carry = 0; i < other.z.size() || carry; ++i) {
                if (i == z.size())
                    z.push_back(0);
                z[i] += carry + (i < other.z.size() ? other.z[i] : 0);
                carry = z[i] >= base;
                if (carry)
                    z[i] -= base;
            }
        } else if (other != 0 /* prevent infinite loop */) {
            *this -= -other;
        }
        return *this;
    }

    friend bigint operator+(bigint a, const bigint &b) {
        a += b;
        return a;
    }

    bigint &operator-=(const bigint &other) {
        if (sign == other.sign) {
            if ((sign == 1 && *this >= other) || (sign == -1 && *this <= other)) {
                for (int i = 0, carry = 0; i < other.z.size() || carry; ++i) {
                    z[i] -= carry + (i < other.z.size() ? other.z[i] : 0);
                    carry = z[i] < 0;
                    if (carry)
                        z[i] += base;
                }
                trim();
            } else {
                *this = other - *this;
                this->sign = -this->sign;
            }
        } else {
            *this += -other;
        }
        return *this;
    }

    friend bigint operator-(bigint a, const bigint &b) {
        a -= b;
        return a;
    }

    bigint &operator*=(int v) {
        if (v < 0)
            sign = -sign, v = -v;
        for (int i = 0, carry = 0; i < z.size() || carry; ++i) {
            if (i == z.size())
                z.push_back(0);
            long long cur = (long long) z[i] * v + carry;
            carry = (int) (cur / base);
            z[i] = (int) (cur % base);
        }
        trim();
        return *this;
    }

    bigint operator*(int v) const {
        return bigint(*this) *= v;
    }

    friend pair<bigint, bigint> divmod(const bigint &a1, const bigint &b1) {
        int norm = base / (b1.z.back() + 1);
        bigint a = a1.abs() * norm;
        bigint b = b1.abs() * norm;
        bigint q, r;
        q.z.resize(a.z.size());

        for (int i = (int) a.z.size() - 1; i >= 0; i--) {
            r *= base;
            r += a.z[i];
            int s1 = b.z.size() < r.z.size() ? r.z[b.z.size()] : 0;
            int s2 = b.z.size() - 1 < r.z.size() ? r.z[b.z.size() - 1] : 0;
            int d = (int) (((long long) s1 * base + s2) / b.z.back());
            r -= b * d;
            while (r < 0)
                r += b, --d;
            q.z[i] = d;
        }

        q.sign = a1.sign * b1.sign;
        r.sign = a1.sign;
        q.trim();
        r.trim();
        return {q, r / norm};
    }

    friend bigint sqrt(const bigint &a1) {
        bigint a = a1;
        while (a.z.empty() || a.z.size() % 2 == 1)
            a.z.push_back(0);

        int n = a.z.size();

        int firstDigit = (int) ::sqrt((double) a.z[n - 1] * base + a.z[n - 2]);
        int norm = base / (firstDigit + 1);
        a *= norm;
        a *= norm;
        while (a.z.empty() || a.z.size() % 2 == 1)
            a.z.push_back(0);

        bigint r = (long long) a.z[n - 1] * base + a.z[n - 2];
        firstDigit = (int) ::sqrt((double) a.z[n - 1] * base + a.z[n - 2]);
        int q = firstDigit;
        bigint res;

        for (int j = n / 2 - 1; j >= 0; j--) {
            for (;; --q) {
                bigint r1 = (r - (res * 2 * base + q) * q) * base * base +
                            (j > 0 ? (long long) a.z[2 * j - 1] * base + a.z[2 * j - 2] : 0);
                if (r1 >= 0) {
                    r = r1;
                    break;
                }
            }
            res *= base;
            res += q;

            if (j > 0) {
                int d1 = res.z.size() + 2 < r.z.size() ? r.z[res.z.size() + 2] : 0;
                int d2 = res.z.size() + 1 < r.z.size() ? r.z[res.z.size() + 1] : 0;
                int d3 = res.z.size() < r.z.size() ? r.z[res.z.size()] : 0;
                q = (int) (((long long) d1 * base * base + (long long) d2 * base + d3) / (firstDigit * 2));
            }
        }

        res.trim();
        return res / norm;
    }

    bigint operator/(const bigint &v) const {
        return divmod(*this, v).first;
    }

    bigint operator%(const bigint &v) const {
        return divmod(*this, v).second;
    }

    bigint &operator/=(int v) {
        if (v < 0)
            sign = -sign, v = -v;
        for (int i = (int) z.size() - 1, rem = 0; i >= 0; --i) {
            long long cur = z[i] + rem * (long long) base;
            z[i] = (int) (cur / v);
            rem = (int) (cur % v);
        }
        trim();
        return *this;
    }

    bigint operator/(int v) const {
        return bigint(*this) /= v;
    }

    int operator%(int v) const {
        if (v < 0)
            v = -v;
        int m = 0;
        for (int i = (int) z.size() - 1; i >= 0; --i)
            m = (int) ((z[i] + m * (long long) base) % v);
        return m * sign;
    }

    bigint &operator*=(const bigint &v) {
        *this = *this * v;
        return *this;
    }

    bigint &operator/=(const bigint &v) {
        *this = *this / v;
        return *this;
    }

    bigint &operator%=(const bigint &v) {
        *this = *this % v;
        return *this;
    }

    bool operator<(const bigint &v) const {
        if (sign != v.sign)
            return sign < v.sign;
        if (z.size() != v.z.size())
            return z.size() * sign < v.z.size() * v.sign;
        for (int i = (int) z.size() - 1; i >= 0; i--)
            if (z[i] != v.z[i])
                return z[i] * sign < v.z[i] * sign;
        return false;
    }

    bool operator>(const bigint &v) const {
        return v < *this;
    }

    bool operator<=(const bigint &v) const {
        return !(v < *this);
    }

    bool operator>=(const bigint &v) const {
        return !(*this < v);
    }

    bool operator==(const bigint &v) const {
        return !(*this < v) && !(v < *this);
    }

    bool operator!=(const bigint &v) const {
        return *this < v || v < *this;
    }

    void trim() {
        while (!z.empty() && z.back() == 0)
            z.pop_back();
        if (z.empty())
            sign = 1;
    }

    bool isZero() const {
        return z.empty();
    }

    friend bigint operator-(bigint v) {
        if (!v.z.empty())
            v.sign = -v.sign;
        return v;
    }

    bigint abs() const {
        return sign == 1 ? *this : -*this;
    }

    long long longValue() const {
        long long res = 0;
        for (int i = (int) z.size() - 1; i >= 0; i--)
            res = res * base + z[i];
        return res * sign;
    }

    friend bigint gcd(const bigint &a, const bigint &b) {
        return b.isZero() ? a : gcd(b, a % b);
    }

    friend bigint lcm(const bigint &a, const bigint &b) {
        return a / gcd(a, b) * b;
    }

    void read(const string &s) {
        sign = 1;
        z.clear();
        int pos = 0;
        while (pos < s.size() && (s[pos] == '-' || s[pos] == '+')) {
            if (s[pos] == '-')
                sign = -sign;
            ++pos;
        }
        for (int i = (int) s.size() - 1; i >= pos; i -= base_digits) {
            int x = 0;
            for (int j = max(pos, i - base_digits + 1); j <= i; j++)
                x = x * 10 + s[j] - '0';
            z.push_back(x);
        }
        trim();
    }

    friend istream &operator>>(istream &stream, bigint &v) {
        string s;
        stream >> s;
        v.read(s);
        return stream;
    }

    friend ostream &operator<<(ostream &stream, const bigint &v) {
        if (v.sign == -1)
            stream << '-';
        stream << (v.z.empty() ? 0 : v.z.back());
        for (int i = (int) v.z.size() - 2; i >= 0; --i)
            stream << setw(base_digits) << setfill('0') << v.z[i];
        return stream;
    }

    static vector<int> convert_base(const vector<int> &a, int old_digits, int new_digits) {
        vector<long long> p(max(old_digits, new_digits) + 1);
        p[0] = 1;
        for (int i = 1; i < p.size(); i++)
            p[i] = p[i - 1] * 10;
        vector<int> res;
        long long cur = 0;
        int cur_digits = 0;
        for (int v : a) {
            cur += v * p[cur_digits];
            cur_digits += old_digits;
            while (cur_digits >= new_digits) {
                res.push_back(int(cur % p[new_digits]));
                cur /= p[new_digits];
                cur_digits -= new_digits;
            }
        }
        res.push_back((int) cur);
        while (!res.empty() && res.back() == 0)
            res.pop_back();
        return res;
    }

    bigint operator*(const bigint &v) const {
        if (min(z.size(), v.z.size()) < 150)
            return mul_simple(v);
        bigint res;
        res.sign = sign * v.sign;
        res.z = multiply_bigint(convert_base(z, base_digits, fft_base_digits),
                                convert_base(v.z, base_digits, fft_base_digits), fft_base);
        res.z = convert_base(res.z, fft_base_digits, base_digits);
        res.trim();
        return res;
    }

    bigint mul_simple(const bigint &v) const {
        bigint res;
        res.sign = sign * v.sign;
        res.z.resize(z.size() + v.z.size());
        for (int i = 0; i < z.size(); ++i)
            if (z[i])
                for (int j = 0, carry = 0; j < v.z.size() || carry; ++j) {
                    long long cur = res.z[i + j] + (long long) z[i] * (j < v.z.size() ? v.z[j] : 0) + carry;
                    carry = (int) (cur / base);
                    res.z[i + j] = (int) (cur % base);
                }
        res.trim();
        return res;
    }
};

mt19937 rng(1);

bigint random_bigint(int n) {
    string s;
    for (int i = 0; i < n; i++) {
        s += uniform_int_distribution<int>('0', '9')(rng);
    }
    return bigint(s);
}
int main() {
  using namespace std;
  cin.tie(nullptr)->sync_with_stdio(false);
  int n;
  bigint m;
  cin >> n;n--;
  vector<int> a(n);
  bigint mult = 1;
  for (auto&& e : a){
    cin >> e;
    mult *= bigint(e);
  }
  reverse(a.begin(),a.end());
  cin >> m;
  m += mult;
  reserve(n + 1);
  auto calc=[&](bigint m){
    Fps bernoulli(n + 1);
    for (int i = 0; i <= n; ++i) bernoulli[i] = ifact[i + 1];
    bernoulli = inv(bernoulli);
    auto faulhaber = [&](Poly f) -> Poly {
      for (int i = 0; i < sz(f); ++i) f[i] *= fact[i];
      reverse(begin(f), end(f));
      f *= Fps(begin(bernoulli), begin(bernoulli) + sz(f));
      f.push_back(0);
      reverse(begin(f), end(f));
      for (int i = 0; i < sz(f); ++i) f[i] *= ifact[i];
      return f;
    };
    Poly f{1};
    for (int i = n; i--;) {
      Poly g = faulhaber(f);
      f = -g;
      Fp coeff = 1;
      for (int j = 1; j < sz(f); ++j) f[j] *= coeff *= a[i];
      f[0] = eval(g, ((m + 1)%bigint(998244353)).longValue());
      m /= bigint(a[i]);
    }
    f = faulhaber(f);
    return eval(f, ((m + 1)%bigint(998244353)).longValue()) - eval(f, 1);
  };
  cout << calc(m) - calc(m-1) << '\n';
}