test/yosupo-judge/sqrt_of_formal_power_series/main.test.cpp

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• Last update: 2021-04-23 23:44:44+09:00
• Problem: https://judge.yosupo.jp/problem/sqrt_of_formal_power_series
• Link: View error logs on GitHub Actions

Depends on

• Number theoretic transform (Mylib/Convolution/ntt_convolution.cpp)
• Input vector (Mylib/IO/input_vector.cpp)
• join function (Mylib/IO/join.cpp)
• Formal power series (Mylib/Math/formal_power_series.cpp)
• Formal power series (Sqrt) (Mylib/Math/fps_sqrt.cpp)
• Modint (Mylib/Number/Mint/mint.cpp)
• Mod pow (Mylib/Number/Mod/mod_pow.cpp)
• Mod sqrt (Mylib/Number/Mod/mod_sqrt.cpp)
• Primitive root (Mylib/Number/Prime/primitive_root.cpp)

Code

#define PROBLEM "https://judge.yosupo.jp/problem/sqrt_of_formal_power_series"

#include <functional>
#include <iostream>
#include <vector>
#include "Mylib/Convolution/ntt_convolution.cpp"
#include "Mylib/IO/input_vector.cpp"
#include "Mylib/IO/join.cpp"
#include "Mylib/Math/formal_power_series.cpp"
#include "Mylib/Math/fps_sqrt.cpp"
#include "Mylib/Number/Mint/mint.cpp"
#include "Mylib/Number/Mod/mod_sqrt.cpp"
#include "Mylib/Number/Prime/primitive_root.cpp"

namespace hl = haar_lib;

constexpr int mod       = 998244353;
constexpr int prim_root = hl::primitive_root(mod);
using mint              = hl::modint<mod>;
using NTT               = hl::number_theoretic_transform<mint, prim_root, 1 << 21>;
const static auto ntt   = NTT();
using FPS               = hl::formal_power_series<mint, ntt>;

int main() {
  std::cin.tie(0);
  std::ios::sync_with_stdio(false);

  int N;
  std::cin >> N;
  auto a   = hl::input_vector<mint>(N);
  auto ans = FPS(a).sqrt();

  if (ans) {
    std::cout << hl::join((*ans).begin(), (*ans).begin() + N) << "\n";
  } else {
    std::cout << -1 << "\n";
  }

  return 0;
}

#line 1 "test/yosupo-judge/sqrt_of_formal_power_series/main.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/sqrt_of_formal_power_series"

#include <functional>
#include <iostream>
#include <vector>
#line 2 "Mylib/Convolution/ntt_convolution.cpp"
#include <algorithm>
#include <cassert>
#include <utility>
#line 4 "Mylib/Number/Mint/mint.cpp"

namespace haar_lib {
  template <int32_t M>
  class modint {
    uint32_t val_;

  public:
    constexpr static auto mod() { return M; }

    constexpr modint() : val_(0) {}
    constexpr modint(int64_t n) {
      if (n >= M)
        val_ = n % M;
      else if (n < 0)
        val_ = n % M + M;
      else
        val_ = n;
    }

    constexpr auto &operator=(const modint &a) {
      val_ = a.val_;
      return *this;
    }
    constexpr auto &operator+=(const modint &a) {
      if (val_ + a.val_ >= M)
        val_ = (uint64_t) val_ + a.val_ - M;
      else
        val_ += a.val_;
      return *this;
    }
    constexpr auto &operator-=(const modint &a) {
      if (val_ < a.val_) val_ += M;
      val_ -= a.val_;
      return *this;
    }
    constexpr auto &operator*=(const modint &a) {
      val_ = (uint64_t) val_ * a.val_ % M;
      return *this;
    }
    constexpr auto &operator/=(const modint &a) {
      val_ = (uint64_t) val_ * a.inv().val_ % M;
      return *this;
    }

    constexpr auto operator+(const modint &a) const { return modint(*this) += a; }
    constexpr auto operator-(const modint &a) const { return modint(*this) -= a; }
    constexpr auto operator*(const modint &a) const { return modint(*this) *= a; }
    constexpr auto operator/(const modint &a) const { return modint(*this) /= a; }

    constexpr bool operator==(const modint &a) const { return val_ == a.val_; }
    constexpr bool operator!=(const modint &a) const { return val_ != a.val_; }

    constexpr auto &operator++() {
      *this += 1;
      return *this;
    }
    constexpr auto &operator--() {
      *this -= 1;
      return *this;
    }

    constexpr auto operator++(int) {
      auto t = *this;
      *this += 1;
      return t;
    }
    constexpr auto operator--(int) {
      auto t = *this;
      *this -= 1;
      return t;
    }

    constexpr static modint pow(int64_t n, int64_t p) {
      if (p < 0) return pow(n, -p).inv();

      int64_t ret = 1, e = n % M;
      for (; p; (e *= e) %= M, p >>= 1)
        if (p & 1) (ret *= e) %= M;
      return ret;
    }

    constexpr static modint inv(int64_t a) {
      int64_t b = M, u = 1, v = 0;

      while (b) {
        int64_t t = a / b;
        a -= t * b;
        std::swap(a, b);
        u -= t * v;
        std::swap(u, v);
      }

      u %= M;
      if (u < 0) u += M;

      return u;
    }

    constexpr static auto frac(int64_t a, int64_t b) { return modint(a) / modint(b); }

    constexpr auto pow(int64_t p) const { return pow(val_, p); }
    constexpr auto inv() const { return inv(val_); }

    friend constexpr auto operator-(const modint &a) { return modint(M - a.val_); }

    friend constexpr auto operator+(int64_t a, const modint &b) { return modint(a) + b; }
    friend constexpr auto operator-(int64_t a, const modint &b) { return modint(a) - b; }
    friend constexpr auto operator*(int64_t a, const modint &b) { return modint(a) * b; }
    friend constexpr auto operator/(int64_t a, const modint &b) { return modint(a) / b; }

    friend std::istream &operator>>(std::istream &s, modint &a) {
      s >> a.val_;
      return s;
    }
    friend std::ostream &operator<<(std::ostream &s, const modint &a) {
      s << a.val_;
      return s;
    }

    template <int N>
    static auto div() {
      static auto value = inv(N);
      return value;
    }

    explicit operator int32_t() const noexcept { return val_; }
    explicit operator int64_t() const noexcept { return val_; }
  };
}  // namespace haar_lib
#line 7 "Mylib/Convolution/ntt_convolution.cpp"

namespace haar_lib {
  template <typename T, int PRIM_ROOT, int MAX_SIZE>
  class number_theoretic_transform {
  public:
    using value_type                    = T;
    constexpr static int primitive_root = PRIM_ROOT;
    constexpr static int max_size       = MAX_SIZE;

  private:
    const int MAX_POWER_;
    std::vector<T> BASE_, INV_BASE_;

  public:
    number_theoretic_transform() : MAX_POWER_(__builtin_ctz(MAX_SIZE)),
                                   BASE_(MAX_POWER_ + 1),
                                   INV_BASE_(MAX_POWER_ + 1) {
      static_assert((MAX_SIZE & (MAX_SIZE - 1)) == 0, "MAX_SIZE must be power of 2.");
      static_assert((T::mod() - 1) % MAX_SIZE == 0);

      T t = T::pow(PRIM_ROOT, (T::mod() - 1) >> (MAX_POWER_ + 2));
      T s = t.inv();

      for (int i = MAX_POWER_; --i >= 0;) {
        t *= t;
        s *= s;
        BASE_[i]     = -t;
        INV_BASE_[i] = -s;
      }
    }

    void run(std::vector<T> &f, bool INVERSE = false) const {
      const int n = f.size();
      assert((n & (n - 1)) == 0 and n <= MAX_SIZE);  // データ数は2の冪乗個

      if (INVERSE) {
        for (int b = 1; b < n; b <<= 1) {
          T w = 1;
          for (int j = 0, k = 1; j < n; j += 2 * b, ++k) {
            for (int i = 0; i < b; ++i) {
              const auto s = f[i + j];
              const auto t = f[i + j + b];

              f[i + j]     = s + t;
              f[i + j + b] = (s - t) * w;
            }
            w *= INV_BASE_[__builtin_ctz(k)];
          }
        }

        const T t = T::inv(n);
        for (auto &x : f) x *= t;
      } else {
        for (int b = n >> 1; b; b >>= 1) {
          T w = 1;
          for (int j = 0, k = 1; j < n; j += 2 * b, ++k) {
            for (int i = 0; i < b; ++i) {
              const auto s = f[i + j];
              const auto t = f[i + j + b] * w;

              f[i + j]     = s + t;
              f[i + j + b] = s - t;
            }
            w *= BASE_[__builtin_ctz(k)];
          }
        }
      }
    }

    template <typename U>
    std::vector<T> convolve(std::vector<U> f, std::vector<U> g, bool is_same = false) const {
      const int m = f.size() + g.size() - 1;
      int n       = 1;
      while (n < m) n *= 2;

      std::vector<T> f2(n);
      for (int i = 0; i < (int) f.size(); ++i) f2[i] = (int64_t) f[i];
      run(f2);

      if (is_same) {
        for (int i = 0; i < n; ++i) f2[i] *= f2[i];
        run(f2, true);
      } else {
        std::vector<T> g2(n);
        for (int i = 0; i < (int) g.size(); ++i) g2[i] = (int64_t) g[i];
        run(g2);

        for (int i = 0; i < n; ++i) f2[i] *= g2[i];
        run(f2, true);
      }

      return f2;
    }

    template <typename U>
    std::vector<T> operator()(std::vector<U> f, std::vector<U> g, bool is_same = false) const {
      return convolve(f, g, is_same);
    }
  };

  template <typename T>
  std::vector<T> convolve_general_mod(std::vector<T> f, std::vector<T> g) {
    static constexpr int M1 = 167772161, P1 = 3;
    static constexpr int M2 = 469762049, P2 = 3;
    static constexpr int M3 = 1224736769, P3 = 3;

    auto res1 = number_theoretic_transform<modint<M1>, P1, 1 << 20>().convolve(f, g);
    auto res2 = number_theoretic_transform<modint<M2>, P2, 1 << 20>().convolve(f, g);
    auto res3 = number_theoretic_transform<modint<M3>, P3, 1 << 20>().convolve(f, g);

    const int n = res1.size();

    std::vector<T> ret(n);

    const int64_t M12 = (int64_t) modint<M2>::inv(M1);
    const int64_t M13 = (int64_t) modint<M3>::inv(M1);
    const int64_t M23 = (int64_t) modint<M3>::inv(M2);

    for (int i = 0; i < n; ++i) {
      const int64_t r[3] = {(int64_t) res1[i], (int64_t) res2[i], (int64_t) res3[i]};

      const int64_t t0 = r[0] % M1;
      const int64_t t1 = (r[1] - t0 + M2) * M12 % M2;
      const int64_t t2 = ((r[2] - t0 + M3) * M13 % M3 - t1 + M3) * M23 % M3;

      ret[i] = T(t0) + T(t1) * M1 + T(t2) * M1 * M2;
    }

    return ret;
  }
}  // namespace haar_lib
#line 4 "Mylib/IO/input_vector.cpp"

namespace haar_lib {
  template <typename T>
  std::vector<T> input_vector(int N) {
    std::vector<T> ret(N);
    for (int i = 0; i < N; ++i) std::cin >> ret[i];
    return ret;
  }

  template <typename T>
  std::vector<std::vector<T>> input_vector(int N, int M) {
    std::vector<std::vector<T>> ret(N);
    for (int i = 0; i < N; ++i) ret[i] = input_vector<T>(M);
    return ret;
  }
}  // namespace haar_lib
#line 3 "Mylib/IO/join.cpp"
#include <sstream>
#include <string>

namespace haar_lib {
  template <typename Iter>
  std::string join(Iter first, Iter last, std::string delim = " ") {
    std::stringstream s;

    for (auto it = first; it != last; ++it) {
      if (it != first) s << delim;
      s << *it;
    }

    return s.str();
  }
}  // namespace haar_lib
#line 4 "Mylib/Math/formal_power_series.cpp"
#include <initializer_list>
#line 6 "Mylib/Math/formal_power_series.cpp"

namespace haar_lib {
  template <typename T, const auto &convolve>
  class formal_power_series {
  public:
    using value_type = T;

  private:
    std::vector<T> data_;

  public:
    formal_power_series() {}
    explicit formal_power_series(int N) : data_(N) {}
    formal_power_series(const std::vector<T> &data_) : data_(data_) {}
    formal_power_series(std::initializer_list<T> init) : data_(init.begin(), init.end()) {}
    formal_power_series(const formal_power_series &a) : data_(a.data_) {}
    formal_power_series(formal_power_series &&a) noexcept { *this = std::move(a); }

    size_t size() const {
      return data_.size();
    }

    const T &operator[](int i) const {
      return data_[i];
    }

    T &operator[](int i) {
      return data_[i];
    }

    auto begin() { return data_.begin(); }
    auto end() { return data_.end(); }

    const auto &data() const { return data_; }

    void resize(int n) {
      data_.resize(n);
    }

    auto &operator=(formal_power_series &&rhs) noexcept {
      if (this != &rhs) {
        data_ = std::move(rhs.data_);
      }
      return *this;
    }

    auto &operator+=(const formal_power_series &rhs) {
      if (data_.size() < rhs.size()) data_.resize(rhs.size());
      for (int i = 0; i < rhs.size(); ++i) data_[i] += rhs[i];
      return *this;
    }

    auto &operator+=(T rhs) {
      data_[0] += rhs;
      return *this;
    }

    auto operator+(T rhs) const {
      auto ret = *this;
      return ret += rhs;
    }

    auto operator+(const formal_power_series &rhs) const {
      auto ret = *this;
      return ret += rhs;
    }

    auto &operator-=(const formal_power_series &rhs) {
      if (data_.size() < rhs.size()) data_.resize(rhs.size());
      for (int i = 0; i < rhs.size(); ++i) data_[i] -= rhs[i];
      return *this;
    }

    auto &operator-=(T rhs) {
      data_[0] -= rhs;
      return *this;
    }

    auto operator-(T rhs) const {
      auto ret = *this;
      return ret -= rhs;
    }

    auto operator-(const formal_power_series &rhs) const {
      auto ret = *this;
      return ret -= rhs;
    }

    auto operator-() const {
      auto ret = *this;
      for (auto &x : ret) x = -x;
      return ret;
    }

    auto &operator*=(const formal_power_series &rhs) {
      data_ = convolve(data_, rhs.data_);
      return *this;
    }

    auto operator*(const formal_power_series &rhs) const {
      auto ret = convolve(data_, rhs.data_);
      return formal_power_series(ret);
    }

    auto &operator*=(T rhs) {
      for (auto &x : data_) x *= rhs;
      return *this;
    }

    auto operator*(T rhs) const {
      auto ret = *this;
      return ret *= rhs;
    }

    auto differentiate() const {
      const int n = data_.size();
      std::vector<T> ret(n - 1);
      for (int i = 0; i < n - 1; ++i) {
        ret[i] = data_[i + 1] * (i + 1);
      }

      return formal_power_series(ret);
    }

    auto integrate() const {
      const int n = data_.size();
      std::vector<T> ret(n + 1), invs(n + 1, 1);
      const int p = T::mod();
      for (int i = 2; i <= n; ++i) invs[i] = -invs[p % i] * (p / i);
      for (int i = 0; i < n; ++i) {
        ret[i + 1] = data_[i] * invs[i + 1];
      }

      return formal_power_series(ret);
    }

    auto inv() const {
      assert(data_[0] != 0);
      const int n = data_.size();

      int t              = 1;
      std::vector<T> ret = {data_[0].inv()};
      ret.reserve(n * 2);

      while (t <= n * 2) {
        std::vector<T> c(data_.begin(), data_.begin() + std::min(t, n));
        auto a = convolve(ret, ret, true);
        if ((int) a.size() > t) a.resize(t);

        c = convolve(c, a);

        if ((int) c.size() > t) c.resize(t);
        if ((int) ret.size() > t) ret.resize(t);

        for (int i = 0; i < (int) ret.size(); ++i) ret[i] = ret[i] * 2;

        if (ret.size() < c.size()) ret.resize(std::min<int>(c.size(), t));

        for (int i = 0; i < (int) c.size(); ++i) {
          ret[i] -= c[i];
        }

        t <<= 1;
      }

      ret.resize(n);

      return formal_power_series(ret);
    }

    auto log() const {
      assert(data_[0] == 1);
      const int n = data_.size();
      auto ret    = (differentiate() * inv()).integrate();
      ret.resize(n);
      return ret;
    }

    auto exp() const {
      const int n = data_.size();

      int t = 1;
      formal_power_series b({1});

      while (t <= n * 2) {
        t <<= 1;
        auto temp = b.log();
        temp.resize(t);

        for (int i = 0; i < t; ++i) temp[i] = -temp[i];
        temp[0] += 1;
        for (int i = 0; i < std::min(t, n); ++i) temp[i] += data_[i];

        b = b * temp;
        b.resize(t);
      }

      b.resize(n);

      return b;
    }

    auto shift(int64_t k) const {
      const int64_t n = data_.size();
      formal_power_series ret(n);

      if (k >= 0) {
        for (int64_t i = k; i < n; ++i) {
          ret[i] = data_[i - k];
        }
      } else {
        for (int64_t i = 0; i < n + k; ++i) {
          ret[i] = data_[i - k];
        }
      }

      return ret;
    }

    auto pow(int64_t M) const {
      assert(M >= 0);

      const int n = data_.size();
      int k       = 0;
      for (; k < n; ++k) {
        if (data_[k] != 0) {
          break;
        }
      }

      if (k >= n) return *this;

      T a = data_[k];

      formal_power_series ret = *this;
      ret                     = (ret.shift(-k)) * a.inv();
      ret                     = (ret.log() * (T) M).exp();
      ret                     = (ret * a.pow(M)).shift(M * k);

      return ret;
    }

    std::optional<formal_power_series> sqrt() const;
  };
}  // namespace haar_lib
#line 2 "Mylib/Number/Mod/mod_sqrt.cpp"
#include <optional>
#include <random>
#line 2 "Mylib/Number/Mod/mod_pow.cpp"
#include <cstdint>

namespace haar_lib {
  constexpr int64_t mod_pow(int64_t n, int64_t p, int64_t m) {
    int64_t ret = 1;
    while (p > 0) {
      if (p & 1) (ret *= n) %= m;
      (n *= n) %= m;
      p >>= 1;
    }
    return ret;
  }
}  // namespace haar_lib
#line 5 "Mylib/Number/Mod/mod_sqrt.cpp"

namespace haar_lib {
  std::optional<int64_t> mod_sqrt(int64_t a, int64_t p) {
    if (p == 2) return a % 2;
    if (a == 0) return 0;

    int64_t b = mod_pow(a, (p - 1) / 2, p);

    if (b == p - 1) return {};
    if (p % 4 == 3) return mod_pow(a, (p + 1) / 4, p);

    int64_t q = p - 1, s = 0;
    while (q % 2 == 0) {
      q /= 2;
      s += 1;
    }

    static std::mt19937_64 rand(time(0));
    std::uniform_int_distribution<> dist(0, p - 1);

    int64_t z;
    while (1) {
      z = dist(rand);
      if (mod_pow(z, (p - 1) / 2, p) == p - 1) break;
    }

    int64_t m = s;
    int64_t c = mod_pow(z, q, p);
    int64_t t = mod_pow(a, q, p);
    int64_t r = mod_pow(a, (q + 1) / 2, p);

    while (1) {
      if (t == 0) return 0;
      if (t == 1) return r;

      int i = 1;
      for (int64_t T = t; i < m; ++i) {
        (T *= T) %= p;
        if (T == 1) break;
      }

      int64_t b = mod_pow(c, 1LL << (m - i - 1), p);

      m = i;
      c = b * b % p;
      (t *= b * b % p) %= p;
      (r *= b) %= p;
    }
  }
}  // namespace haar_lib
#line 4 "Mylib/Math/fps_sqrt.cpp"

namespace haar_lib {
  template <typename T, const auto &convolve>
  auto formal_power_series<T, convolve>::sqrt() const -> std::optional<formal_power_series<T, convolve>> {
    constexpr int mod = value_type::mod();

    const int n = data_.size();
    int k       = 0;
    for (; k < n; ++k)
      if (data_[k] != 0) break;

    if (k >= n) return *this;
    if (k % 2 != 0) return {};

    auto x = mod_sqrt((int64_t) data_[k], mod);

    if (not x) return {};

    const int m = n - k;

    auto it = data_.begin() + k;
    formal_power_series ret({*x});

    int t = 1;
    while (t <= m * 2) {
      formal_power_series f(std::vector<T>(it, it + std::min(t, m)));
      ret.resize(t);
      f.resize(t);
      ret = (ret + f * ret.inv()) * T(2).inv();
      t <<= 1;
    }

    ret.resize(n);
    ret = ret.shift(k / 2);

    return ret;
  }
}  // namespace haar_lib
#line 3 "Mylib/Number/Prime/primitive_root.cpp"

namespace haar_lib {
  constexpr int primitive_root(int p) {
    int pf[30] = {};
    int k      = 0;
    {
      int n = p - 1;
      for (int64_t i = 2; i * i <= p; ++i) {
        if (n % i == 0) {
          pf[k++] = i;
          while (n % i == 0) n /= i;
        }
      }
      if (n != 1)
        pf[k++] = n;
    }

    for (int g = 2; g <= p; ++g) {
      bool ok = true;
      for (int i = 0; i < k; ++i) {
        if (mod_pow(g, (p - 1) / pf[i], p) == 1) {
          ok = false;
          break;
        }
      }

      if (not ok) continue;

      return g;
    }
    return -1;
  }
}  // namespace haar_lib
#line 14 "test/yosupo-judge/sqrt_of_formal_power_series/main.test.cpp"

namespace hl = haar_lib;

constexpr int mod       = 998244353;
constexpr int prim_root = hl::primitive_root(mod);
using mint              = hl::modint<mod>;
using NTT               = hl::number_theoretic_transform<mint, prim_root, 1 << 21>;
const static auto ntt   = NTT();
using FPS               = hl::formal_power_series<mint, ntt>;

int main() {
  std::cin.tie(0);
  std::ios::sync_with_stdio(false);

  int N;
  std::cin >> N;
  auto a   = hl::input_vector<mint>(N);
  auto ans = FPS(a).sqrt();

  if (ans) {
    std::cout << hl::join((*ans).begin(), (*ans).begin() + N) << "\n";
  } else {
    std::cout << -1 << "\n";
  }

  return 0;
}

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