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#include "../linear_recurrence.hpp" #include "../../modint.hpp" #define PROBLEM "https://judge.yosupo.jp/problem/find_linear_recurrence" #include <iostream> using namespace std; using mint = ModInt<998244353>; int main() { cin.tie(nullptr), ios::sync_with_stdio(false); int N; cin >> N; vector<mint> A(N); for (auto &a : A) cin >> a; auto L_poly = find_linear_recurrence(A); auto L = L_poly.first; auto poly = L_poly.second; cout << L << '\n'; for (int i = 1; i <= L; i++) cout << -poly[i] << ' '; cout << '\n'; }
#line 2 "formal_power_series/linear_recurrence.hpp" #include <algorithm> #include <cassert> #include <utility> #include <vector> // CUT begin // Berlekamp–Massey algorithm // https://en.wikipedia.org/wiki/Berlekamp%E2%80%93Massey_algorithm // Complexity: O(N^2) // input: S = sequence from field K // return: L = degree of minimal polynomial, // C_reversed = monic min. polynomial (size = L + 1, reversed order, C_reversed[0] = 1)) // Formula: convolve(S, C_reversed)[i] = 0 for i >= L // Example: // - [1, 2, 4, 8, 16] -> (1, [1, -2]) // - [1, 1, 2, 3, 5, 8] -> (2, [1, -1, -1]) // - [0, 0, 0, 0, 1] -> (5, [1, 0, 0, 0, 0, 998244352]) (mod 998244353) // - [] -> (0, [1]) // - [0, 0, 0] -> (0, [1]) // - [-2] -> (1, [1, 2]) template <typename Tfield> std::pair<int, std::vector<Tfield>> find_linear_recurrence(const std::vector<Tfield> &S) { int N = S.size(); using poly = std::vector<Tfield>; poly C_reversed{1}, B{1}; int L = 0, m = 1; Tfield b = 1; // adjust: C(x) <- C(x) - (d / b) x^m B(x) auto adjust = [](poly C, const poly &B, Tfield d, Tfield b, int m) -> poly { C.resize(std::max(C.size(), B.size() + m)); Tfield a = d / b; for (unsigned i = 0; i < B.size(); i++) C[i + m] -= a * B[i]; return C; }; for (int n = 0; n < N; n++) { Tfield d = S[n]; for (int i = 1; i <= L; i++) d += C_reversed[i] * S[n - i]; if (d == 0) m++; else if (2 * L <= n) { poly T = C_reversed; C_reversed = adjust(C_reversed, B, d, b, m); L = n + 1 - L; B = T; b = d; m = 1; } else C_reversed = adjust(C_reversed, B, d, b, m++); } return std::make_pair(L, C_reversed); } // Calculate $x^N \bmod f(x)$ // Known as `Kitamasa method` // Input: f_reversed: monic, reversed (f_reversed[0] = 1) // Complexity: $O(K^2 \log N)$ ($K$: deg. of $f$) // Example: (4, [1, -1, -1]) -> [2, 3] // ( x^4 = (x^2 + x + 2)(x^2 - x - 1) + 3x + 2 ) // Reference: http://misawa.github.io/others/fast_kitamasa_method.html // http://sugarknri.hatenablog.com/entry/2017/11/18/233936 template <typename Tfield> std::vector<Tfield> monomial_mod_polynomial(long long N, const std::vector<Tfield> &f_reversed) { assert(!f_reversed.empty() and f_reversed[0] == 1); int K = f_reversed.size() - 1; if (!K) return {}; int D = 64 - __builtin_clzll(N); std::vector<Tfield> ret(K, 0); ret[0] = 1; auto self_conv = [](std::vector<Tfield> x) -> std::vector<Tfield> { int d = x.size(); std::vector<Tfield> ret(d * 2 - 1); for (int i = 0; i < d; i++) { ret[i * 2] += x[i] * x[i]; for (int j = 0; j < i; j++) ret[i + j] += x[i] * x[j] * 2; } return ret; }; for (int d = D; d--;) { ret = self_conv(ret); for (int i = 2 * K - 2; i >= K; i--) { for (int j = 1; j <= K; j++) ret[i - j] -= ret[i] * f_reversed[j]; } ret.resize(K); if ((N >> d) & 1) { std::vector<Tfield> c(K); c[0] = -ret[K - 1] * f_reversed[K]; for (int i = 1; i < K; i++) { c[i] = ret[i - 1] - ret[K - 1] * f_reversed[K - i]; } ret = c; } } return ret; } // Guess k-th element of the sequence, assuming linear recurrence // initial_elements: 0-ORIGIN // Verify: abc198f https://atcoder.jp/contests/abc198/submissions/21837815 template <typename Tfield> Tfield guess_kth_term(const std::vector<Tfield> &initial_elements, long long k) { assert(k >= 0); if (k < static_cast<long long>(initial_elements.size())) return initial_elements[k]; const auto f = find_linear_recurrence<Tfield>(initial_elements).second; const auto g = monomial_mod_polynomial<Tfield>(k, f); Tfield ret = 0; for (unsigned i = 0; i < g.size(); i++) ret += g[i] * initial_elements[i]; return ret; } #line 3 "modint.hpp" #include <iostream> #include <set> #line 6 "modint.hpp" template <int md> struct ModInt { using lint = long long; constexpr static int mod() { return md; } static int get_primitive_root() { static int primitive_root = 0; if (!primitive_root) { primitive_root = [&]() { std::set<int> fac; int v = md - 1; for (lint i = 2; i * i <= v; i++) while (v % i == 0) fac.insert(i), v /= i; if (v > 1) fac.insert(v); for (int g = 1; g < md; g++) { bool ok = true; for (auto i : fac) if (ModInt(g).pow((md - 1) / i) == 1) { ok = false; break; } if (ok) return g; } return -1; }(); } return primitive_root; } int val_; int val() const noexcept { return val_; } constexpr ModInt() : val_(0) {} constexpr ModInt &_setval(lint v) { return val_ = (v >= md ? v - md : v), *this; } constexpr ModInt(lint v) { _setval(v % md + md); } constexpr explicit operator bool() const { return val_ != 0; } constexpr ModInt operator+(const ModInt &x) const { return ModInt()._setval((lint)val_ + x.val_); } constexpr ModInt operator-(const ModInt &x) const { return ModInt()._setval((lint)val_ - x.val_ + md); } constexpr ModInt operator*(const ModInt &x) const { return ModInt()._setval((lint)val_ * x.val_ % md); } constexpr ModInt operator/(const ModInt &x) const { return ModInt()._setval((lint)val_ * x.inv().val() % md); } constexpr ModInt operator-() const { return ModInt()._setval(md - val_); } constexpr ModInt &operator+=(const ModInt &x) { return *this = *this + x; } constexpr ModInt &operator-=(const ModInt &x) { return *this = *this - x; } constexpr ModInt &operator*=(const ModInt &x) { return *this = *this * x; } constexpr ModInt &operator/=(const ModInt &x) { return *this = *this / x; } friend constexpr ModInt operator+(lint a, const ModInt &x) { return ModInt(a) + x; } friend constexpr ModInt operator-(lint a, const ModInt &x) { return ModInt(a) - x; } friend constexpr ModInt operator*(lint a, const ModInt &x) { return ModInt(a) * x; } friend constexpr ModInt operator/(lint a, const ModInt &x) { return ModInt(a) / x; } constexpr bool operator==(const ModInt &x) const { return val_ == x.val_; } constexpr bool operator!=(const ModInt &x) const { return val_ != x.val_; } constexpr bool operator<(const ModInt &x) const { return val_ < x.val_; } // To use std::map<ModInt, T> friend std::istream &operator>>(std::istream &is, ModInt &x) { lint t; return is >> t, x = ModInt(t), is; } constexpr friend std::ostream &operator<<(std::ostream &os, const ModInt &x) { return os << x.val_; } constexpr ModInt pow(lint n) const { ModInt ans = 1, tmp = *this; while (n) { if (n & 1) ans *= tmp; tmp *= tmp, n >>= 1; } return ans; } static constexpr int cache_limit = std::min(md, 1 << 21); static std::vector<ModInt> facs, facinvs, invs; constexpr static void _precalculation(int N) { const int l0 = facs.size(); if (N > md) N = md; if (N <= l0) return; facs.resize(N), facinvs.resize(N), invs.resize(N); for (int i = l0; i < N; i++) facs[i] = facs[i - 1] * i; facinvs[N - 1] = facs.back().pow(md - 2); for (int i = N - 2; i >= l0; i--) facinvs[i] = facinvs[i + 1] * (i + 1); for (int i = N - 1; i >= l0; i--) invs[i] = facinvs[i] * facs[i - 1]; } constexpr ModInt inv() const { if (this->val_ < cache_limit) { if (facs.empty()) facs = {1}, facinvs = {1}, invs = {0}; while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2); return invs[this->val_]; } else { return this->pow(md - 2); } } constexpr ModInt fac() const { while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2); return facs[this->val_]; } constexpr ModInt facinv() const { while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2); return facinvs[this->val_]; } constexpr ModInt doublefac() const { lint k = (this->val_ + 1) / 2; return (this->val_ & 1) ? ModInt(k * 2).fac() / (ModInt(2).pow(k) * ModInt(k).fac()) : ModInt(k).fac() * ModInt(2).pow(k); } constexpr ModInt nCr(int r) const { if (r < 0 or this->val_ < r) return ModInt(0); return this->fac() * (*this - r).facinv() * ModInt(r).facinv(); } constexpr ModInt nPr(int r) const { if (r < 0 or this->val_ < r) return ModInt(0); return this->fac() * (*this - r).facinv(); } static ModInt binom(int n, int r) { static long long bruteforce_times = 0; if (r < 0 or n < r) return ModInt(0); if (n <= bruteforce_times or n < (int)facs.size()) return ModInt(n).nCr(r); r = std::min(r, n - r); ModInt ret = ModInt(r).facinv(); for (int i = 0; i < r; ++i) ret *= n - i; bruteforce_times += r; return ret; } // Multinomial coefficient, (k_1 + k_2 + ... + k_m)! / (k_1! k_2! ... k_m!) // Complexity: O(sum(ks)) template <class Vec> static ModInt multinomial(const Vec &ks) { ModInt ret{1}; int sum = 0; for (int k : ks) { assert(k >= 0); ret *= ModInt(k).facinv(), sum += k; } return ret * ModInt(sum).fac(); } // Catalan number, C_n = binom(2n, n) / (n + 1) // C_0 = 1, C_1 = 1, C_2 = 2, C_3 = 5, C_4 = 14, ... // https://oeis.org/A000108 // Complexity: O(n) static ModInt catalan(int n) { if (n < 0) return ModInt(0); return ModInt(n * 2).fac() * ModInt(n + 1).facinv() * ModInt(n).facinv(); } ModInt sqrt() const { if (val_ == 0) return 0; if (md == 2) return val_; if (pow((md - 1) / 2) != 1) return 0; ModInt b = 1; while (b.pow((md - 1) / 2) == 1) b += 1; int e = 0, m = md - 1; while (m % 2 == 0) m >>= 1, e++; ModInt x = pow((m - 1) / 2), y = (*this) * x * x; x *= (*this); ModInt z = b.pow(m); while (y != 1) { int j = 0; ModInt t = y; while (t != 1) j++, t *= t; z = z.pow(1LL << (e - j - 1)); x *= z, z *= z, y *= z; e = j; } return ModInt(std::min(x.val_, md - x.val_)); } }; template <int md> std::vector<ModInt<md>> ModInt<md>::facs = {1}; template <int md> std::vector<ModInt<md>> ModInt<md>::facinvs = {1}; template <int md> std::vector<ModInt<md>> ModInt<md>::invs = {0}; using ModInt998244353 = ModInt<998244353>; // using mint = ModInt<998244353>; // using mint = ModInt<1000000007>; #line 3 "formal_power_series/test/linear_recurrence.test.cpp" #define PROBLEM "https://judge.yosupo.jp/problem/find_linear_recurrence" #line 5 "formal_power_series/test/linear_recurrence.test.cpp" using namespace std; using mint = ModInt<998244353>; int main() { cin.tie(nullptr), ios::sync_with_stdio(false); int N; cin >> N; vector<mint> A(N); for (auto &a : A) cin >> a; auto L_poly = find_linear_recurrence(A); auto L = L_poly.first; auto poly = L_poly.second; cout << L << '\n'; for (int i = 1; i <= L; i++) cout << -poly[i] << ' '; cout << '\n'; }