$\sqrt{f}$
(fps/fps_sqrt.hpp)

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• Last update: 2024-05-24 14:53:16+09:00
• Include: #include "fps/fps_sqrt.hpp"

説明

形式的べき級数 $f$ について $\sqrt{f}$ が存在するなら求める。存在しない場合は std::nulloptを返す。 疎な場合は非負の要素数を $M$ として $O(NM)$。密な場合は $O(N\log N)$

Depends on

• Formal Power Series (fps/fps.hpp)
• Formal Power Series (Sparse) (fps/fps_sparse.hpp)
• Mod Inv (math/mod_inv.hpp)
• Mod Sqrt (math/mod_sqrt.hpp)
• modint/base.hpp
• modint/dynamic_modint.hpp

Verified with

• test/polynomial/Sqrt_of_Formal_Power_Series.test.cpp
• test/polynomial/Sqrt_of_Formal_Power_Series_Sparse.test.cpp

Code

#pragma once

#include "../fps/fps.hpp"
#include "../fps/fps_sparse.hpp"
#include "../math/mod_sqrt.hpp"
#include "../modint/base.hpp"

namespace ebi {

template <Modint mint>
std::optional<FormalPowerSeries<mint>> FormalPowerSeries<mint>::sqrt(
    int d) const {
    using FPS = FormalPowerSeries<mint>;
    if (d < 0) d = deg();
    if ((*this)[0] == 0) {
        for (int i = 1; i < this->deg(); i++) {
            if ((*this)[i] != 0) {
                if (i & 1) return std::nullopt;
                if (d - i / 2 <= 0) break;
                auto opt = ((*this) >> i).sqrt(d - i / 2);
                if (!opt) return std::nullopt;
                auto ret = opt.value() << (i / 2);
                if ((int)ret.deg() < d) ret.resize(d);
                return ret;
            }
        }
        return FPS(d, 0);
    }
    auto s = mod_sqrt((*this)[0].val(), mint::mod());
    if (!s) {
        return std::nullopt;
    }
    if (this->count_terms() <= 200) {
        mint y = s.value();
        std::vector<mint> sqrt_f =
            pow_sparse_1(*this / (*this)[0], mint(2).inv().val(), d);
        FPS g(d);
        for (int i = 0; i < d; i++) g[i] = sqrt_f[i] * y;
        return g;
    }
    int n = 1;
    FPS g(n);
    g[0] = s.value();
    mint inv_two = mint(2).inv();
    while (n < d) {
        n <<= 1;
        g = (g + this->pre(n) * g.inv(n)).pre(n) * inv_two;
    }
    g.resize(d);
    return g;
}

}  // namespace ebi

#line 2 "fps/fps_sqrt.hpp"

#line 2 "fps/fps.hpp"

#include <algorithm>
#include <cassert>
#include <optional>
#include <vector>

#line 2 "modint/base.hpp"

#include <concepts>
#include <iostream>
#include <utility>

namespace ebi {

template <class T>
concept Modint = requires(T a, T b) {
    a + b;
    a - b;
    a * b;
    a / b;
    a.inv();
    a.val();
    a.pow(std::declval<long long>());
    T::mod();
};

template <Modint mint> std::istream &operator>>(std::istream &os, mint &a) {
    long long x;
    os >> x;
    a = x;
    return os;
}

template <Modint mint>
std::ostream &operator<<(std::ostream &os, const mint &a) {
    return os << a.val();
}

}  // namespace ebi
#line 9 "fps/fps.hpp"

namespace ebi {

template <Modint mint> struct FormalPowerSeries : std::vector<mint> {
  private:
    using std::vector<mint>::vector;
    using std::vector<mint>::vector::operator=;
    using FPS = FormalPowerSeries;

  public:
    FormalPowerSeries(const std::vector<mint> &a) {
        *this = a;
    }

    FPS operator+(const FPS &rhs) const noexcept {
        return FPS(*this) += rhs;
    }
    FPS operator-(const FPS &rhs) const noexcept {
        return FPS(*this) -= rhs;
    }
    FPS operator*(const FPS &rhs) const noexcept {
        return FPS(*this) *= rhs;
    }
    FPS operator/(const FPS &rhs) const noexcept {
        return FPS(*this) /= rhs;
    }
    FPS operator%(const FPS &rhs) const noexcept {
        return FPS(*this) %= rhs;
    }

    FPS operator+(const mint &rhs) const noexcept {
        return FPS(*this) += rhs;
    }
    FPS operator-(const mint &rhs) const noexcept {
        return FPS(*this) -= rhs;
    }
    FPS operator*(const mint &rhs) const noexcept {
        return FPS(*this) *= rhs;
    }
    FPS operator/(const mint &rhs) const noexcept {
        return FPS(*this) /= rhs;
    }

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

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

    FPS &operator*=(const FPS &);

    FPS &operator/=(const FPS &rhs) noexcept {
        int n = deg() - 1;
        int m = rhs.deg() - 1;
        if (n < m) {
            *this = {};
            return *this;
        }
        *this = (*this).rev() * rhs.rev().inv(n - m + 1);
        (*this).resize(n - m + 1);
        std::reverse((*this).begin(), (*this).end());
        return *this;
    }

    FPS &operator%=(const FPS &rhs) noexcept {
        *this -= *this / rhs * rhs;
        shrink();
        return *this;
    }

    FPS &operator+=(const mint &rhs) noexcept {
        if (this->empty()) this->resize(1);
        (*this)[0] += rhs;
        return *this;
    }

    FPS &operator-=(const mint &rhs) noexcept {
        if (this->empty()) this->resize(1);
        (*this)[0] -= rhs;
        return *this;
    }

    FPS &operator*=(const mint &rhs) noexcept {
        for (int i = 0; i < deg(); ++i) {
            (*this)[i] *= rhs;
        }
        return *this;
    }
    FPS &operator/=(const mint &rhs) noexcept {
        mint inv_rhs = rhs.inv();
        for (int i = 0; i < deg(); ++i) {
            (*this)[i] *= inv_rhs;
        }
        return *this;
    }

    FPS operator>>(int d) const {
        if (deg() <= d) return {};
        FPS f = *this;
        f.erase(f.begin(), f.begin() + d);
        return f;
    }

    FPS operator<<(int d) const {
        FPS f = *this;
        f.insert(f.begin(), d, 0);
        return f;
    }

    FPS operator-() const {
        FPS g(this->size());
        for (int i = 0; i < (int)this->size(); i++) g[i] = -(*this)[i];
        return g;
    }

    FPS pre(int sz) const {
        return FPS(this->begin(), this->begin() + std::min(deg(), sz));
    }

    FPS rev() const {
        auto f = *this;
        std::reverse(f.begin(), f.end());
        return f;
    }

    FPS differential() const {
        int n = deg();
        FPS g(std::max(0, n - 1));
        for (int i = 0; i < n - 1; i++) {
            g[i] = (*this)[i + 1] * (i + 1);
        }
        return g;
    }

    FPS integral() const {
        int n = deg();
        FPS g(n + 1);
        g[0] = 0;
        if (n > 0) g[1] = 1;
        auto mod = mint::mod();
        for (int i = 2; i <= n; i++) g[i] = (-g[mod % i]) * (mod / i);
        for (int i = 0; i < n; i++) g[i + 1] *= (*this)[i];
        return g;
    }

    FPS inv(int d = -1) const {
        int n = 1;
        if (d < 0) d = deg();
        FPS g(n);
        g[0] = (*this)[0].inv();
        while (n < d) {
            n <<= 1;
            g = (g * 2 - g * g * this->pre(n)).pre(n);
        }
        g.resize(d);
        return g;
    }

    FPS log(int d = -1) const {
        assert((*this)[0].val() == 1);
        if (d < 0) d = deg();
        return ((*this).differential() * (*this).inv(d)).pre(d - 1).integral();
    }

    FPS exp(int d = -1) const {
        assert((*this)[0].val() == 0);
        int n = 1;
        if (d < 0) d = deg();
        FPS g(n);
        g[0] = 1;
        while (n < d) {
            n <<= 1;
            g = (g * (this->pre(n) - g.log(n) + 1)).pre(n);
        }
        g.resize(d);
        return g;
    }

    FPS pow(long long k, int d = -1) const {
        assert(k >= 0);
        int n = deg();
        if (d < 0) d = n;
        if (k == 0) {
            FPS f(d);
            if (d > 0) f[0] = 1;
            return f;
        }
        int low = d;
        for (int i = n - 1; i >= 0; i--)
            if ((*this)[i] != 0) low = i;
        if (low >= (d + k - 1) / k) return FPS(d, 0);
        int offset = k * low;
        mint c = (*this)[low];
        FPS g(d - offset);
        for (int i = 0; i < std::min(n - low, d - offset); i++) {
            g[i] = (*this)[i + low];
        }
        g /= c;
        g = g.pow_1(k);
        return (g << offset) * c.pow(k);
    }

    FPS pow_1(mint k, int d = -1) const {
        assert((*this)[0] == 1);
        return ((*this).log(d) * k).exp(d);
    }

    FPS pow_newton(long long k, int d = -1) const {
        assert(k >= 0);
        const int n = deg();
        if (d < 0) d = n;
        if (k == 0) {
            FPS f(d);
            if (d > 0) f[0] = 1;
            return f;
        }
        for (int i = 0; i < n; i++) {
            if ((*this)[i] != 0) {
                mint rev = (*this)[i].inv();
                FPS f = (((*this * rev) >> i).log(d) * k).exp(d);
                f *= (*this)[i].pow(k);
                f = (f << (i * k)).pre(d);
                if (f.deg() < d) f.resize(d);
                return f;
            }
            if (i + 1 >= (d + k - 1) / k) break;
        }
        return FPS(d);
    }

    int deg() const {
        return (*this).size();
    }

    void shrink() {
        while ((!this->empty()) && this->back() == 0) this->pop_back();
    }

    int count_terms() const {
        int c = 0;
        for (int i = 0; i < deg(); i++) {
            if ((*this)[i] != 0) c++;
        }
        return c;
    }

    std::optional<FPS> sqrt(int d = -1) const;

    static FPS exp_x(int n) {
        FPS f(n);
        mint fact = 1;
        for (int i = 1; i < n; i++) fact *= i;
        f[n - 1] = fact.inv();
        for (int i = n - 1; i >= 0; i--) f[i - 1] = f[i] * i;
        return f;
    }

    void fft();
    void ifft();
};

}  // namespace ebi
#line 2 "fps/fps_sparse.hpp"

#line 5 "fps/fps_sparse.hpp"

#line 2 "math/mod_inv.hpp"

#line 5 "math/mod_inv.hpp"

#line 7 "math/mod_inv.hpp"

namespace ebi {

template <Modint mint> mint inv(int n) {
    static const int mod = mint::mod();
    static std::vector<mint> dat = {0, 1};
    assert(0 <= n);
    if (n >= mod) n -= mod;
    while (int(dat.size()) <= n) {
        int num = dat.size();
        int q = (mod + num - 1) / num;
        dat.emplace_back(dat[num * q - mod] * mint(q));
    }
    return dat[n];
}

}  // namespace ebi
#line 8 "fps/fps_sparse.hpp"

namespace ebi {

template <Modint mint>
std::vector<mint> mul_sparse(const std::vector<mint> &f,
                             const std::vector<mint> &g) {
    int n = f.size();
    int m = g.size();
    std::vector<std::pair<int, mint>> cf, cg;
    for (int i = 0; i < n; i++) {
        if (f[i] != 0) cf.emplace_back(i, f[i]);
    }
    for (int i = 0; i < m; i++) {
        if (g[i] != 0) cg.emplace_back(i, g[i]);
    }
    std::vector<mint> h(n + m - 1);
    for (auto [i, p] : cf) {
        for (auto [j, q] : cg) {
            h[i + j] += p * q;
        }
    }
    return h;
}

template <Modint mint>
std::vector<mint> inv_sparse(const std::vector<mint> &f, int d = -1) {
    assert(f[0] != 0);
    if (d < 0) {
        d = f.size();
    }
    std::vector<std::pair<int, mint>> ret;
    for (int i = 1; i < int(f.size()); i++) {
        if (f[i] != 0) {
            ret.emplace_back(i, f[i]);
        }
    }
    std::vector<mint> g(d);
    g[0] = f[0].inv();
    for (int i = 1; i < d; i++) {
        for (auto [k, p] : ret) {
            if (i - k < 0) break;
            g[i] -= g[i - k] * p;
        }
        g[i] *= g[0];
    }
    return g;
}

template <Modint mint>
std::vector<mint> exp_sparse(const std::vector<mint> &f, int d = -1) {
    int n = f.size();
    if (d < 0) d = n;
    std::vector<std::pair<int, mint>> ret;
    for (int i = 1; i < n; i++) {
        if (f[i] != 0) {
            ret.emplace_back(i - 1, f[i] * i);
        }
    }
    std::vector<mint> g(d);
    g[0] = 1;
    for (int i = 0; i < d - 1; i++) {
        for (auto [k, p] : ret) {
            if (i - k < 0) break;
            g[i + 1] += g[i - k] * p;
        }
        g[i + 1] *= inv<mint>(i + 1);
    }
    return g;
}

template <Modint mint>
std::vector<mint> log_sparse(const std::vector<mint> &f, int d = -1) {
    int n = f.size();
    if (d < 0) d = n;
    std::vector<mint> df(d);
    for (int i = 0; i < std::min(d, n - 1); i++) {
        df[i] = f[i + 1] * (i + 1);
    }
    auto dg = mul_sparse(df, inv_sparse(f));
    dg.resize(d);
    std::vector<mint> g(d);
    for (int i = 0; i < d - 1; i++) {
        g[i + 1] = dg[i] * inv<mint>(i + 1);
    }
    return g;
}

template <Modint mint>
std::vector<mint> pow_sparse_1(const std::vector<mint> &f, long long k,
                               int d = -1) {
    int n = f.size();
    assert(n == 0 || f[0] == 1);
    std::vector<std::pair<int, mint>> ret;
    for (int i = 1; i < n; i++) {
        if (f[i] != 0) ret.emplace_back(i, f[i]);
    }
    std::vector<mint> g(d);
    g[0] = 1;
    for (int i = 0; i < d - 1; i++) {
        for (const auto &[j, cf] : ret) {
            if (i + 1 - j < 0) break;
            g[i + 1] +=
                (mint(k) * mint(j) - mint(i - j + 1)) * cf * g[i + 1 - j];
        }
        g[i + 1] *= inv<mint>(i + 1);
    }
    return g;
}

template <Modint mint>
std::vector<mint> pow_sparse(const std::vector<mint> &f, long long k,
                             int d = -1) {
    int n = f.size();
    if (d < 0) d = n;
    assert(k >= 0);
    if (k == 0) {
        std::vector<mint> g(d);
        if (d > 0) g[0] = 1;
        return g;
    }
    for (int i = 0; i < n; i++) {
        if (f[i] != 0) {
            mint rev = f[i].inv();
            std::vector<mint> f2(n - i);
            for (int j = i; j < n; j++) {
                f2[j - i] = f[j] * rev;
            }
            f2 = pow_sparse_1(f2, k, d);
            mint fk = f[i].pow(k);
            std::vector<mint> g(d);
            for (int j = 0; j < int(f2.size()); j++) {
                if (j + i * k >= d) break;
                g[j + i * k] = f2[j] * fk;
            }
            return g;
        }
        if (i >= (d + k - 1) / k) break;
    }
    return std::vector<mint>(d);
}

}  // namespace ebi
#line 2 "math/mod_sqrt.hpp"

#include <cstdint>
#line 5 "math/mod_sqrt.hpp"

#line 2 "modint/dynamic_modint.hpp"

#line 4 "modint/dynamic_modint.hpp"

#line 6 "modint/dynamic_modint.hpp"

namespace ebi {

template <int id> struct dynamic_modint {
  private:
    using modint = dynamic_modint;

  public:
    static void set_mod(int p) {
        assert(1 <= p);
        m = p;
    }

    static int mod() {
        return m;
    }

    modint raw(int v) {
        modint x;
        x._v = v;
        return x;
    }

    dynamic_modint() : _v(0) {}

    dynamic_modint(long long v) {
        v %= (long long)umod();
        if (v < 0) v += (long long)umod();
        _v = (unsigned int)v;
    }

    unsigned int val() const {
        return _v;
    }

    unsigned int value() const {
        return val();
    }

    modint &operator++() {
        _v++;
        if (_v == umod()) _v = 0;
        return *this;
    }
    modint &operator--() {
        if (_v == 0) _v = umod();
        _v--;
        return *this;
    }
    modint &operator+=(const modint &rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    modint &operator-=(const modint &rhs) {
        _v -= rhs._v;
        if (_v >= umod()) _v += umod();
        return *this;
    }
    modint &operator*=(const modint &rhs) {
        unsigned long long x = _v;
        x *= rhs._v;
        _v = (unsigned int)(x % (unsigned long long)umod());
        return *this;
    }
    modint &operator/=(const modint &rhs) {
        return *this = *this * rhs.inv();
    }

    modint operator+() const {
        return *this;
    }
    modint operator-() const {
        return modint() - *this;
    }

    modint pow(long long n) const {
        assert(0 <= n);
        modint x = *this, res = 1;
        while (n) {
            if (n & 1) res *= x;
            x *= x;
            n >>= 1;
        }
        return res;
    }
    modint inv() const {
        assert(_v);
        return pow(umod() - 2);
    }

    friend modint operator+(const modint &lhs, const modint &rhs) {
        return modint(lhs) += rhs;
    }
    friend modint operator-(const modint &lhs, const modint &rhs) {
        return modint(lhs) -= rhs;
    }
    friend modint operator*(const modint &lhs, const modint &rhs) {
        return modint(lhs) *= rhs;
    }

    friend modint operator/(const modint &lhs, const modint &rhs) {
        return modint(lhs) /= rhs;
    }
    friend bool operator==(const modint &lhs, const modint &rhs) {
        return lhs.val() == rhs.val();
    }
    friend bool operator!=(const modint &lhs, const modint &rhs) {
        return !(lhs == rhs);
    }

  private:
    unsigned int _v = 0;
    static int m;

    static unsigned int umod() {
        return m;
    }
};

template <int id> int dynamic_modint<id>::m = 998244353;

}  // namespace ebi
#line 7 "math/mod_sqrt.hpp"

namespace ebi {

std::optional<std::int64_t> mod_sqrt(const std::int64_t &a,
                                     const std::int64_t &p) {
    if (a == 0 || a == 1) return a;
    using mint = dynamic_modint<100>;
    mint::set_mod(p);
    if (mint(a).pow((p - 1) >> 1) != 1) return std::nullopt;
    mint b = 1;
    while (b.pow((p - 1) >> 1) == 1) b += 1;
    std::int64_t m = p - 1, e = 0;
    while (m % 2 == 0) m >>= 1, e++;
    mint x = mint(a).pow((m - 1) >> 1);
    mint y = mint(a) * x * x;
    x *= a;
    mint z = b.pow(m);
    while (y != 1) {
        std::int64_t j = 0;
        mint t = y;
        while (t != 1) {
            j++;
            t *= t;
        }
        z = z.pow(1ll << (e - j - 1));
        x *= z;
        z *= z;
        y *= z;
        e = j;
    }
    return x.val();
}

}  // namespace ebi
#line 7 "fps/fps_sqrt.hpp"

namespace ebi {

template <Modint mint>
std::optional<FormalPowerSeries<mint>> FormalPowerSeries<mint>::sqrt(
    int d) const {
    using FPS = FormalPowerSeries<mint>;
    if (d < 0) d = deg();
    if ((*this)[0] == 0) {
        for (int i = 1; i < this->deg(); i++) {
            if ((*this)[i] != 0) {
                if (i & 1) return std::nullopt;
                if (d - i / 2 <= 0) break;
                auto opt = ((*this) >> i).sqrt(d - i / 2);
                if (!opt) return std::nullopt;
                auto ret = opt.value() << (i / 2);
                if ((int)ret.deg() < d) ret.resize(d);
                return ret;
            }
        }
        return FPS(d, 0);
    }
    auto s = mod_sqrt((*this)[0].val(), mint::mod());
    if (!s) {
        return std::nullopt;
    }
    if (this->count_terms() <= 200) {
        mint y = s.value();
        std::vector<mint> sqrt_f =
            pow_sparse_1(*this / (*this)[0], mint(2).inv().val(), d);
        FPS g(d);
        for (int i = 0; i < d; i++) g[i] = sqrt_f[i] * y;
        return g;
    }
    int n = 1;
    FPS g(n);
    g[0] = s.value();
    mint inv_two = mint(2).inv();
    while (n < d) {
        n <<= 1;
        g = (g + this->pre(n) * g.inv(n)).pre(n) * inv_two;
    }
    g.resize(d);
    return g;
}

}  // namespace ebi

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