Submission #1355172

#TimeUsernameProblemLanguageResultExecution timeMemory
1355172edoTrains (BOI24_trains)C++20
100 / 100
258 ms126828 KiB
#include <bits/stdc++.h>

using namespace std;
using ll = long long;

template <typename T> T inverse(T a, T m) {
  T u = 0, v = 1;
  while (a != 0) {
    T t = m / a;
    m -= t * a;
    swap(a, m);
    u -= t * v;
    swap(u, v);
  }
  assert(m == 1);
  return u;
}

template <typename T> class Modular {
public:
  using Type = typename std::decay<decltype(T::value)>::type;

  constexpr Modular() : value() {}
  template <typename U> Modular(const U &x) { value = normalize(x); }

  template <typename U> static Type normalize(const U &x) {
    Type v;
    if (-mod() <= x && x < mod())
      v = static_cast<Type>(x);
    else
      v = static_cast<Type>(x % mod());
    if (v < 0)
      v += mod();
    return v;
  }

  const Type &operator()() const { return value; }
  template <typename U> explicit operator U() const {
    return static_cast<U>(value);
  }
  constexpr static Type mod() { return T::value; }

  Modular &operator+=(const Modular &other) {
    value += other.value;
    value -= (value >= mod()) * mod();
    return *this;
  }
  Modular &operator-=(const Modular &other) {
    value -= other.value;
    value += (value < 0) * mod();
    return *this;
  }
  template <typename U> Modular &operator+=(const U &other) {
    return *this += Modular(other);
  }
  template <typename U> Modular &operator-=(const U &other) {
    return *this -= Modular(other);
  }
  Modular &operator++() { return *this += 1; }
  Modular &operator--() { return *this -= 1; }
  Modular operator++(int) {
    Modular result(*this);
    *this += 1;
    return result;
  }
  Modular operator--(int) {
    Modular result(*this);
    *this -= 1;
    return result;
  }
  Modular operator-() const { return Modular(-value); }

  template <typename U = T>
  typename std::enable_if<std::is_same<typename Modular<U>::Type, int>::value,
                          Modular>::type &
  operator*=(const Modular &rhs) {
    value = normalize(static_cast<int64_t>(value) *
                      static_cast<int64_t>(rhs.value));
    return *this;
  }
  template <typename U = T>
  typename std::enable_if<
      std::is_same<typename Modular<U>::Type, int64_t>::value, Modular>::type &
  operator*=(const Modular &rhs) {
    int64_t q = int64_t(static_cast<long double>(value) * rhs.value / mod());
    value = normalize(value * rhs.value - q * mod());
    return *this;
  }
  template <typename U = T>
  typename std::enable_if<!std::is_integral<typename Modular<U>::Type>::value,
                          Modular>::type &
  operator*=(const Modular &rhs) {
    value = normalize(value * rhs.value);
    return *this;
  }

  Modular &operator/=(const Modular &other) {
    return *this *= Modular(inverse(other.value, mod()));
  }

  friend const Type &abs(const Modular &x) { return x.value; }

  template <typename U>
  friend bool operator==(const Modular<U> &lhs, const Modular<U> &rhs);

  template <typename U>
  friend bool operator<(const Modular<U> &lhs, const Modular<U> &rhs);

  template <typename V, typename U>
  friend V &operator>>(V &stream, Modular<U> &number);

private:
  Type value;
};

template <typename T>
bool operator==(const Modular<T> &lhs, const Modular<T> &rhs) {
  return lhs.value == rhs.value;
}
template <typename T, typename U>
bool operator==(const Modular<T> &lhs, U rhs) {
  return lhs == Modular<T>(rhs);
}
template <typename T, typename U>
bool operator==(U lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) == rhs;
}

template <typename T>
bool operator!=(const Modular<T> &lhs, const Modular<T> &rhs) {
  return !(lhs == rhs);
}
template <typename T, typename U>
bool operator!=(const Modular<T> &lhs, U rhs) {
  return !(lhs == rhs);
}
template <typename T, typename U>
bool operator!=(U lhs, const Modular<T> &rhs) {
  return !(lhs == rhs);
}

template <typename T>
bool operator<(const Modular<T> &lhs, const Modular<T> &rhs) {
  return lhs.value < rhs.value;
}

template <typename T>
Modular<T> operator+(const Modular<T> &lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) += rhs;
}
template <typename T, typename U>
Modular<T> operator+(const Modular<T> &lhs, U rhs) {
  return Modular<T>(lhs) += rhs;
}
template <typename T, typename U>
Modular<T> operator+(U lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) += rhs;
}

template <typename T>
Modular<T> operator-(const Modular<T> &lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) -= rhs;
}
template <typename T, typename U>
Modular<T> operator-(const Modular<T> &lhs, U rhs) {
  return Modular<T>(lhs) -= rhs;
}
template <typename T, typename U>
Modular<T> operator-(U lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) -= rhs;
}

template <typename T>
Modular<T> operator*(const Modular<T> &lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) *= rhs;
}
template <typename T, typename U>
Modular<T> operator*(const Modular<T> &lhs, U rhs) {
  return Modular<T>(lhs) *= rhs;
}
template <typename T, typename U>
Modular<T> operator*(U lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) *= rhs;
}

template <typename T>
Modular<T> operator/(const Modular<T> &lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) /= rhs;
}
template <typename T, typename U>
Modular<T> operator/(const Modular<T> &lhs, U rhs) {
  return Modular<T>(lhs) /= rhs;
}
template <typename T, typename U>
Modular<T> operator/(U lhs, const Modular<T> &rhs) {
  return Modular<T>(lhs) /= rhs;
}

template <typename T, typename U>
Modular<T> power(const Modular<T> &a, const U &b) {
  assert(b >= 0);
  Modular<T> x = a, res = 1;
  U p = b;
  while (p > 0) {
    if (p & 1)
      res *= x;
    x *= x;
    p >>= 1;
  }
  return res;
}

template <typename T> bool IsZero(const Modular<T> &number) {
  return number() == 0;
}

template <typename T> std::string to_string(const Modular<T> &number) {
  return std::to_string(number());
}

// U == std::ostream? but done this way because of fastoutput
template <typename U, typename T>
U &operator<<(U &stream, const Modular<T> &number) {
  return stream << number();
}

// U == std::istream? but done this way because of fastinput
template <typename U, typename T> U &operator>>(U &stream, Modular<T> &number) {
  typename std::common_type<typename Modular<T>::Type, int64_t>::type x;
  stream >> x;
  number.value = Modular<T>::normalize(x);
  return stream;
}

// constexpr int md = 998244353;
constexpr int md = 1e9 + 7;
using Mint =
    Modular<std::integral_constant<std::decay<decltype(md)>::type, md>>;

struct Comb {
  int n;
  std::vector<Mint> _fac;
  std::vector<Mint> _invfac;
  std::vector<Mint> _inv;

  Comb() : n{0}, _fac{Mint(1)}, _invfac{Mint(1)}, _inv{Mint(0)} {}
  Comb(int n_) : Comb() { init(n_); }

  void init(int m) {
    if (m <= n)
      return;
    _fac.resize(m + 1);
    _invfac.resize(m + 1);
    _inv.resize(m + 1);

    for (int i = n + 1; i <= m; i++) {
      _fac[i] = _fac[i - 1] * i;
    }

    _invfac[m] = power(_fac[m], md - 2);
    for (int i = m; i > n; i--) {
      _invfac[i - 1] = _invfac[i] * i;
      _inv[i] = _invfac[i] * _fac[i - 1];
    }
    n = m;
  }

  Mint fac(int m) {
    if (m > n)
      init(max(2 * m, m));
    return _fac[m];
  }
  Mint invfac(int m) {
    if (m > n)
      init(max(2 * m, m));
    return _invfac[m];
  }
  Mint inv(int m) {
    if (m > n)
      init(max(2 * m, m));
    return _inv[m];
  }
  Mint binom(int nn, int kk) {
    if (kk < 0 || kk > nn)
      return Mint(0);
    if (nn > n)
      init(max(2 * nn, nn));
    return _fac[nn] * _invfac[kk] * _invfac[nn - kk];
  }
} comb;

Mint C(int n, int k) { return comb.binom(n, k); }
Mint fac(int n) { return comb.fac(n); }
Mint Inv(int n) { return comb.inv(n); }
Mint Cat(int n) { return n == 0 ? 1 : C(2 * n, n) / (n + 1); }

int main() {
  ios::sync_with_stdio(false);
  cin.tie(nullptr);

  int n;
  cin >> n;
  vector<int> d(n + 1), x(n + 1);
  for (int i = 1; i <= n; ++i) {
    cin >> d[i] >> x[i];
  }
  vector<Mint> dp(n + 2);
  // dp[i] = 1 + sum(dp[i + d[i]], dp[[i + 2 * dp[i]]] ..., dp[i + x[i] * d[i]])
  int B = sqrt(n) + 1;
  vector<vector<Mint>> pref(B + 1, vector<Mint>(n + B + 2));
  for (int i = n; i; --i) {
    dp[i] = 1;
    if (d[i]) {
      if (d[i] <= B) {
        int step = d[i];
        int l = i + step;
        ll r = 1ll * i + 1ll * (x[i] + 1) * d[i];
        if (l <= n) {
          dp[i] += pref[step][l];
          if (r <= n)
            dp[i] -= pref[step][r];
        }
      } else {
        for (ll j = 1ll * i + d[i], cnt = 1; j <= n && cnt <= x[i];
             j += d[i], ++cnt)
          dp[i] += dp[j];
      }
    }
    for (int step = 1; step <= B; ++step) {
      pref[step][i] = dp[i];
      if (i + step <= n) {
        pref[step][i] += pref[step][i + step];
      }
    }
  }

  cout << dp[1];
  return 0;
}
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