CP-Algorithms Library

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:heavy_check_mark: cp-algo/math/poly/impl/euclid.hpp

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#ifndef CP_ALGO_MATH_POLY_IMPL_EUCLID_HPP
#define CP_ALGO_MATH_POLY_IMPL_EUCLID_HPP
#include "../../affine.hpp"
#include "../../fft.hpp"
#include <functional>
#include <algorithm>
#include <numeric>
#include <cassert>
#include <vector>
#include <tuple>
#include <list>
// operations related to gcd and Euclidean algo
namespace cp_algo::math::poly::impl {
    template<typename poly>
    using gcd_result = std::pair<
        std::list<std::decay_t<poly>>,
        linfrac<std::decay_t<poly>>>;

    template<typename poly>
    gcd_result<poly> half_gcd(poly &&A, poly &&B) {
        assert(A.deg() >= B.deg());
        int m = size(A.a) / 2;
        if(B.deg() < m) {
            return {};
        }
        auto [ai, R] = A.divmod(B);
        std::tie(A, B) = {B, R};
        std::list a = {ai};
        auto T = -linfrac(ai).adj();

        auto advance = [&](int k) {
            auto [ak, Tk] = half_gcd(A.div_xk(k), B.div_xk(k));
            a.splice(end(a), ak);
            T.prepend(Tk);
            return Tk;
        };
        advance(m).apply(A, B);
        if constexpr (std::is_reference_v<poly>) {
            advance(2 * m - A.deg()).apply(A, B);
        } else {
            advance(2 * m - A.deg());
        }
        return {std::move(a), std::move(T)};
    }
    template<typename poly>
    gcd_result<poly> full_gcd(poly &&A, poly &&B) {
        using poly_t = std::decay_t<poly>;
        std::list<poly_t> ak;
        std::vector<linfrac<poly_t>> trs;
        while(!B.is_zero()) {
            auto [a0, R] = A.divmod(B);
            ak.push_back(a0);
            trs.push_back(-linfrac(a0).adj());
            std::tie(A, B) = {B, R};

            auto [a, Tr] = half_gcd(A, B);
            ak.splice(end(ak), a);
            trs.push_back(Tr);
        }
        return {ak, std::accumulate(rbegin(trs), rend(trs), linfrac<poly_t>{}, std::multiplies{})};
    }

    // computes product of linfrac on [L, R)
    auto convergent(auto L, auto R) {
        using poly = decltype(L)::value_type;
        if(R == next(L)) {
            return linfrac(*L);
        } else {
            int s = std::transform_reduce(L, R, 0, std::plus{}, std::mem_fn(&poly::deg));
            auto M = L;
            for(int c = M->deg(); 2 * c <= s; M++) {
                c += next(M)->deg();
            }
            return convergent(L, M) * convergent(M, R);
        }
    }
    template<typename poly>
    poly min_rec(poly const& p, size_t d) {
        auto R2 = p.mod_xk(d).reversed(d), R1 = poly::xk(d);
        if(R2.is_zero()) {
            return poly(1);
        }
        auto [a, Tr] = full_gcd(R1, R2);
        a.emplace_back();
        auto pref = begin(a);
        for(int delta = d - a.front().deg(); delta >= 0; pref++) {
            delta -= pref->deg() + next(pref)->deg();
        }
        return convergent(begin(a), pref).a;
    }

    template<typename poly>
    std::optional<poly> inv_mod(poly p, poly q) {
        assert(!q.is_zero());
        auto [a, Tr] = full_gcd(q, p);
        if(q.deg() != 0) {
            return std::nullopt;
        }
        return Tr.b / q[0];
    }
}
#endif // CP_ALGO_MATH_POLY_IMPL_EUCLID_HPP

#line 1 "cp-algo/math/poly/impl/euclid.hpp"


#line 1 "cp-algo/math/affine.hpp"


#include <optional>
#include <utility>
#include <cassert>
#include <tuple>
namespace cp_algo::math {
    // a * x + b
    template<typename base>
    struct lin {
        base a = 1, b = 0;
        std::optional<base> c;
        lin() {}
        lin(base b): a(0), b(b) {}
        lin(base a, base b): a(a), b(b) {}
        lin(base a, base b, base _c): a(a), b(b), c(_c) {}

        // polynomial product modulo x^2 - c
        lin operator * (const lin& t) {
            assert(c && t.c && *c == *t.c);
            return {a * t.b + b * t.a, b * t.b + a * t.a * (*c), *c};
        }

        // a * (t.a * x + t.b) + b
        lin apply(lin const& t) const {
            return {a * t.a, a * t.b + b};
        }

        void prepend(lin const& t) {
            *this = t.apply(*this);
        }

        base eval(base x) const {
            return a * x + b;
        }
    };

    // (ax+b) / (cx+d)
    template<typename base>
    struct linfrac {
        base a, b, c, d;
        linfrac(): a(1), b(0), c(0), d(1) {} // x, identity for composition
        linfrac(base a): a(a), b(1), c(1), d(0) {} // a + 1/x, for continued fractions
        linfrac(base a, base b, base c, base d): a(a), b(b), c(c), d(d) {}

        // composition of two linfracs
        linfrac operator * (linfrac t) const {
            return t.prepend(linfrac(*this));
        }

        linfrac operator-() const {
            return {-a, -b, -c, -d};
        }

        linfrac adj() const {
            return {d, -b, -c, a};
        }
        
        linfrac& prepend(linfrac const& t) {
            t.apply(a, c);
            t.apply(b, d);
            return *this;
        }

        // apply linfrac to A/B
        void apply(base &A, base &B) const {
            std::tie(A, B) = std::pair{a * A + b * B, c * A + d * B};
        }
    };
}

#line 1 "cp-algo/math/fft.hpp"


#line 1 "cp-algo/math/common.hpp"


#include <functional>
#include <cstdint>
namespace cp_algo::math {
#ifdef CP_ALGO_MAXN
    const int maxn = CP_ALGO_MAXN;
#else
    const int maxn = 1 << 19;
#endif
    const int magic = 64; // threshold for sizes to run the naive algo

    auto bpow(auto const& x, int64_t n, auto const& one, auto op) {
        if(n == 0) {
            return one;
        } else {
            auto t = bpow(x, n / 2, one, op);
            t = op(t, t);
            if(n % 2) {
                t = op(t, x);
            }
            return t;
        }
    }
    auto bpow(auto x, int64_t n, auto ans) {
        return bpow(x, n, ans, std::multiplies{});
    }
    template<typename T>
    T bpow(T const& x, int64_t n) {
        return bpow(x, n, T(1));
    }
}

#line 1 "cp-algo/math/modint.hpp"


#line 4 "cp-algo/math/modint.hpp"
#include <iostream>
namespace cp_algo::math {
    template<typename modint>
    struct modint_base {
        static int64_t mod() {
            return modint::mod();
        }
        modint_base(): r(0) {}
        modint_base(int64_t rr): r(rr % mod()) {
            r = std::min(r, r + mod());
        }
        modint inv() const {
            return bpow(to_modint(), mod() - 2);
        }
        modint operator - () const {return std::min(-r, mod() - r);}
        modint& operator /= (const modint &t) {
            return to_modint() *= t.inv();
        }
        modint& operator *= (const modint &t) {
            if(mod() <= uint32_t(-1)) {
                r = r * t.r % mod();
            } else {
                r = __int128(r) * t.r % mod();
            }
            return to_modint();
        }
        modint& operator += (const modint &t) {
            r += t.r; r = std::min(r, r - mod());
            return to_modint();
        }
        modint& operator -= (const modint &t) {
            r -= t.r; r = std::min(r, r + mod());
            return to_modint();
        }
        modint operator + (const modint &t) const {return modint(to_modint()) += t;}
        modint operator - (const modint &t) const {return modint(to_modint()) -= t;}
        modint operator * (const modint &t) const {return modint(to_modint()) *= t;}
        modint operator / (const modint &t) const {return modint(to_modint()) /= t;}
        auto operator <=> (const modint_base &t) const = default;
        int64_t rem() const {return 2 * r > (uint64_t)mod() ? r - mod() : r;}

        // Only use if you really know what you're doing!
        uint64_t modmod() const {return 8ULL * mod() * mod();};
        void add_unsafe(uint64_t t) {r += t;}
        void pseudonormalize() {r = std::min(r, r - modmod());}
        modint const& normalize() {
            if(r >= (uint64_t)mod()) {
                r %= mod();
            }
            return to_modint();
        }
        uint64_t& setr() {return r;}
        uint64_t getr() const {return r;}
    private:
        uint64_t r;
        modint& to_modint() {return static_cast<modint&>(*this);}
        modint const& to_modint() const {return static_cast<modint const&>(*this);}
    };
    template<typename modint>
    std::istream& operator >> (std::istream &in, modint_base<modint> &x) {
        return in >> x.setr();
    }
    template<typename modint>
    std::ostream& operator << (std::ostream &out, modint_base<modint> const& x) {
        return out << x.getr();
    }

    template<typename modint>
    concept modint_type = std::is_base_of_v<modint_base<modint>, modint>;

    template<int64_t m>
    struct modint: modint_base<modint<m>> {
        static constexpr int64_t mod() {return m;}
        using Base = modint_base<modint<m>>;
        using Base::Base;
    };

    struct dynamic_modint: modint_base<dynamic_modint> {
        static int64_t mod() {return m;}
        static void switch_mod(int64_t nm) {m = nm;}
        using Base = modint_base<dynamic_modint>;
        using Base::Base;

        // Wrapper for temp switching
        auto static with_mod(int64_t tmp, auto callback) {
            struct scoped {
                int64_t prev = mod();
                ~scoped() {switch_mod(prev);}
            } _;
            switch_mod(tmp);
            return callback();
        }
    private:
        static int64_t m;
    };
    int64_t dynamic_modint::m = 0;
}

#line 5 "cp-algo/math/fft.hpp"
#include <algorithm>
#include <complex>
#line 8 "cp-algo/math/fft.hpp"
#include <ranges>
#include <vector>
#include <bit>

namespace cp_algo::math::fft {
    using ftype = double;
    static constexpr size_t bytes = 32;
    static constexpr size_t flen = bytes / sizeof(ftype);
    using point = std::complex<ftype>;
    using vftype [[gnu::vector_size(bytes)]] = ftype;
    using vpoint = std::complex<vftype>;

#define WITH_IV(...)                             \
  [&]<size_t ... i>(std::index_sequence<i...>) { \
      return __VA_ARGS__;                        \
  }(std::make_index_sequence<flen>());

    template<typename ft>
    constexpr ft to_ft(auto x) {
        return ft{} + x;
    }
    template<typename pt>
    constexpr pt to_pt(point r) {
        using ft = std::conditional_t<std::is_same_v<point, pt>, ftype, vftype>;
        return {to_ft<ft>(r.real()), to_ft<ft>(r.imag())};
    }
    struct cvector {
        static constexpr size_t pre_roots = 1 << 17;
        std::vector<vftype> x, y;
        cvector(size_t n) {
            n = std::max(flen, std::bit_ceil(n));
            x.resize(n / flen);
            y.resize(n / flen);
        }
        template<class pt = point>
        void set(size_t k, pt t) {
            if constexpr(std::is_same_v<pt, point>) {
                x[k / flen][k % flen] = real(t);
                y[k / flen][k % flen] = imag(t);
            } else {
                x[k / flen] = real(t);
                y[k / flen] = imag(t);
            }
        }
        template<class pt = point>
        pt get(size_t k) const {
            if constexpr(std::is_same_v<pt, point>) {
                return {x[k / flen][k % flen], y[k / flen][k % flen]};
            } else {
                return {x[k / flen], y[k / flen]};
            }
        }
        vpoint vget(size_t k) const {
            return get<vpoint>(k);
        }

        size_t size() const {
            return flen * std::size(x);
        }
        void dot(cvector const& t) {
            size_t n = size();
            for(size_t k = 0; k < n; k += flen) {
                set(k, get<vpoint>(k) * t.get<vpoint>(k));
            }
        }
        static const cvector roots;
        template<class pt = point>
        static pt root(size_t n, size_t k) {
            if(n < pre_roots) {
                return roots.get<pt>(n + k);
            } else {
                auto arg = std::numbers::pi / n;
                if constexpr(std::is_same_v<pt, point>) {
                    return {cos(k * arg), sin(k * arg)};
                } else {
                    return WITH_IV(pt{vftype{cos((k + i) * arg)...},
                                      vftype{sin((k + i) * arg)...}});
                }
            }
        }
        template<class pt = point>
        static void exec_on_roots(size_t n, size_t m, auto &&callback) {
            size_t step = sizeof(pt) / sizeof(point);
            pt cur;
            pt arg = to_pt<pt>(root<point>(n, step));
            for(size_t i = 0; i < m; i += step) {
                if(i % 64 == 0 || n < pre_roots) {
                    cur = root<pt>(n, i);
                } else {
                    cur *= arg;
                }
                callback(i, cur);
            }
        }

        void ifft() {
            size_t n = size();
            for(size_t i = 1; i < n; i *= 2) {
                for(size_t j = 0; j < n; j += 2 * i) {
                    auto butterfly = [&]<class pt>(size_t k, pt rt) {
                        k += j;
                        auto t = get<pt>(k + i) * conj(rt);
                        set(k + i, get<pt>(k) - t);
                        set(k, get<pt>(k) + t);
                    };
                    if(2 * i <= flen) {
                        exec_on_roots(i, i, butterfly);
                    } else {
                        exec_on_roots<vpoint>(i, i, butterfly);
                    }
                }
            }
            for(size_t k = 0; k < n; k += flen) {
                set(k, get<vpoint>(k) /= to_pt<vpoint>(n));
            }
        }
        void fft() {
            size_t n = size();
            for(size_t i = n / 2; i >= 1; i /= 2) {
                for(size_t j = 0; j < n; j += 2 * i) {
                    auto butterfly = [&]<class pt>(size_t k, pt rt) {
                        k += j;
                        auto A = get<pt>(k) + get<pt>(k + i);
                        auto B = get<pt>(k) - get<pt>(k + i);
                        set(k, A);
                        set(k + i, B * rt);
                    };
                    if(2 * i <= flen) {
                        exec_on_roots(i, i, butterfly);
                    } else {
                        exec_on_roots<vpoint>(i, i, butterfly);
                    }
                }
            }
        }
    };
    const cvector cvector::roots = []() {
        cvector res(pre_roots);
        for(size_t n = 1; n < res.size(); n *= 2) {
            auto base = std::polar(1., std::numbers::pi / n);
            point cur = 1;
            for(size_t k = 0; k < n; k++) {
                if((k & 15) == 0) {
                    cur = std::polar(1., std::numbers::pi * k / n);
                }
                res.set(n + k, cur);
                cur *= base;
            }
        }
        return res;
    }();

    template<typename base>
    struct dft {
        cvector A;
        
        dft(std::vector<base> const& a, size_t n): A(n) {
            for(size_t i = 0; i < std::min(n, a.size()); i++) {
                A.set(i, a[i]);
            }
            if(n) {
                A.fft();
            }
        }

        std::vector<base> operator *= (dft const& B) {
            assert(A.size() == B.A.size());
            size_t n = A.size();
            if(!n) {
                return std::vector<base>();
            }
            A.dot(B.A);
            A.ifft();
            std::vector<base> res(n);
            for(size_t k = 0; k < n; k++) {
                res[k] = A.get(k);
            }
            return res;
        }

        auto operator * (dft const& B) const {
            return dft(*this) *= B;
        }

        point operator [](int i) const {return A.get(i);}
    };

    template<modint_type base>
    struct dft<base> {
        int split;
        cvector A, B;
        
        dft(auto const& a, size_t n): A(n), B(n) {
            split = std::sqrt(base::mod());
            cvector::exec_on_roots(2 * n, size(a), [&](size_t i, point rt) {
                size_t ti = std::min(i, i - n);
                A.set(ti, A.get(ti) + ftype(a[i].rem() % split) * rt);
                B.set(ti, B.get(ti) + ftype(a[i].rem() / split) * rt);
    
            });
            if(n) {
                A.fft();
                B.fft();
            }
        }

        void mul(auto &&C, auto const& D, auto &res, size_t k) {
            assert(A.size() == C.size());
            size_t n = A.size();
            if(!n) {
                res = {};
                return;
            }
            for(size_t i = 0; i < n; i += flen) {
                auto tmp = A.vget(i) * D.vget(i) + B.vget(i) * C.vget(i);
                A.set(i, A.vget(i) * C.vget(i));
                B.set(i, B.vget(i) * D.vget(i));
                C.set(i, tmp);
            }
            A.ifft();
            B.ifft();
            C.ifft();
            auto splitsplit = (base(split) * split).rem();
            cvector::exec_on_roots(2 * n, std::min(n, k), [&](size_t i, point rt) {
                rt = conj(rt);
                auto Ai = A.get(i) * rt;
                auto Bi = B.get(i) * rt;
                auto Ci = C.get(i) * rt;
                int64_t A0 = llround(real(Ai));
                int64_t A1 = llround(real(Ci));
                int64_t A2 = llround(real(Bi));
                res[i] = A0 + A1 * split + A2 * splitsplit;
                if(n + i >= k) {
                    return;
                }
                int64_t B0 = llround(imag(Ai));
                int64_t B1 = llround(imag(Ci));
                int64_t B2 = llround(imag(Bi));
                res[n + i] = B0 + B1 * split + B2 * splitsplit;
            });
        }
        void mul_inplace(auto &&B, auto& res, size_t k) {
            mul(B.A, B.B, res, k);
        }
        void mul(auto const& B, auto& res, size_t k) {
            mul(cvector(B.A), B.B, res, k);
        }
        std::vector<base> operator *= (dft &B) {
            std::vector<base> res(2 * A.size());
            mul_inplace(B, res, size(res));
            return res;
        }
        std::vector<base> operator *= (dft const& B) {
            std::vector<base> res(2 * A.size());
            mul(B, res, size(res));
            return res;
        }
        auto operator * (dft const& B) const {
            return dft(*this) *= B;
        }
        
        point operator [](int i) const {return A.get(i);}
    };
    
    void mul_slow(auto &a, auto const& b, size_t k) {
        if(empty(a) || empty(b)) {
            a.clear();
        } else {
            int n = std::min(k, size(a));
            int m = std::min(k, size(b));
            a.resize(k);
            for(int j = k - 1; j >= 0; j--) {
                a[j] *= b[0];
                for(int i = std::max(j - n, 0) + 1; i < std::min(j + 1, m); i++) {
                    a[j] += a[j - i] * b[i];
                }
            }
        }
    }
    size_t com_size(size_t as, size_t bs) {
        if(!as || !bs) {
            return 0;
        }
        return std::max(flen, std::bit_ceil(as + bs - 1) / 2);
    }
    void mul_truncate(auto &a, auto const& b, size_t k) {
        using base = std::decay_t<decltype(a[0])>;
        if(std::min({k, size(a), size(b)}) < 64) {
            mul_slow(a, b, k);
            return;
        }
        auto n = std::max(flen, std::bit_ceil(
            std::min(k, size(a)) + std::min(k, size(b)) - 1
        ) / 2);
        a.resize(k);
        auto A = dft<base>(a, n);
        if(&a == &b) {
            A.mul(A, a, k);
        } else {
            A.mul_inplace(dft<base>(std::views::take(b, k), n), a, k);
        }
    }
    void mul(auto &a, auto const& b) {
        if(size(a)) {
            mul_truncate(a, b, size(a) + size(b) - 1);
        }
    }
}

#line 7 "cp-algo/math/poly/impl/euclid.hpp"
#include <numeric>
#line 11 "cp-algo/math/poly/impl/euclid.hpp"
#include <list>
// operations related to gcd and Euclidean algo
namespace cp_algo::math::poly::impl {
    template<typename poly>
    using gcd_result = std::pair<
        std::list<std::decay_t<poly>>,
        linfrac<std::decay_t<poly>>>;

    template<typename poly>
    gcd_result<poly> half_gcd(poly &&A, poly &&B) {
        assert(A.deg() >= B.deg());
        int m = size(A.a) / 2;
        if(B.deg() < m) {
            return {};
        }
        auto [ai, R] = A.divmod(B);
        std::tie(A, B) = {B, R};
        std::list a = {ai};
        auto T = -linfrac(ai).adj();

        auto advance = [&](int k) {
            auto [ak, Tk] = half_gcd(A.div_xk(k), B.div_xk(k));
            a.splice(end(a), ak);
            T.prepend(Tk);
            return Tk;
        };
        advance(m).apply(A, B);
        if constexpr (std::is_reference_v<poly>) {
            advance(2 * m - A.deg()).apply(A, B);
        } else {
            advance(2 * m - A.deg());
        }
        return {std::move(a), std::move(T)};
    }
    template<typename poly>
    gcd_result<poly> full_gcd(poly &&A, poly &&B) {
        using poly_t = std::decay_t<poly>;
        std::list<poly_t> ak;
        std::vector<linfrac<poly_t>> trs;
        while(!B.is_zero()) {
            auto [a0, R] = A.divmod(B);
            ak.push_back(a0);
            trs.push_back(-linfrac(a0).adj());
            std::tie(A, B) = {B, R};

            auto [a, Tr] = half_gcd(A, B);
            ak.splice(end(ak), a);
            trs.push_back(Tr);
        }
        return {ak, std::accumulate(rbegin(trs), rend(trs), linfrac<poly_t>{}, std::multiplies{})};
    }

    // computes product of linfrac on [L, R)
    auto convergent(auto L, auto R) {
        using poly = decltype(L)::value_type;
        if(R == next(L)) {
            return linfrac(*L);
        } else {
            int s = std::transform_reduce(L, R, 0, std::plus{}, std::mem_fn(&poly::deg));
            auto M = L;
            for(int c = M->deg(); 2 * c <= s; M++) {
                c += next(M)->deg();
            }
            return convergent(L, M) * convergent(M, R);
        }
    }
    template<typename poly>
    poly min_rec(poly const& p, size_t d) {
        auto R2 = p.mod_xk(d).reversed(d), R1 = poly::xk(d);
        if(R2.is_zero()) {
            return poly(1);
        }
        auto [a, Tr] = full_gcd(R1, R2);
        a.emplace_back();
        auto pref = begin(a);
        for(int delta = d - a.front().deg(); delta >= 0; pref++) {
            delta -= pref->deg() + next(pref)->deg();
        }
        return convergent(begin(a), pref).a;
    }

    template<typename poly>
    std::optional<poly> inv_mod(poly p, poly q) {
        assert(!q.is_zero());
        auto [a, Tr] = full_gcd(q, p);
        if(q.deg() != 0) {
            return std::nullopt;
        }
        return Tr.b / q[0];
    }
}
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