# Submit Info #46741

Problem Lang User Status Time Memory
Polynomial Interpolation cpp14 hos AC 268 ms 39.78 MiB

ケース詳細
Name Status Time Memory
example_00 AC 20 ms 12.68 MiB
example_01 AC 21 ms 12.61 MiB
max_random_00 AC 268 ms 39.78 MiB
max_random_01 AC 265 ms 39.70 MiB
random_00 AC 266 ms 39.70 MiB
random_01 AC 250 ms 39.06 MiB
random_02 AC 130 ms 25.63 MiB

#include <assert.h> #include <string.h> #include <algorithm> #include <initializer_list> #include <iostream> #include <vector> using std::max; using std::min; using std::vector; //////////////////////////////////////////////////////////////////////////////// template <unsigned M_> struct ModInt { static constexpr unsigned M = M_; unsigned x; constexpr ModInt() : x(0) {} constexpr ModInt(unsigned x_) : x(x_ % M) {} constexpr ModInt(unsigned long long x_) : x(x_ % M) {} constexpr ModInt(int x_) : x(((x_ %= static_cast<int>(M)) < 0) ? (x_ + static_cast<int>(M)) : x_) {} constexpr ModInt(long long x_) : x(((x_ %= static_cast<long long>(M)) < 0) ? (x_ + static_cast<long long>(M)) : x_) {} ModInt &operator+=(const ModInt &a) { x = ((x += a.x) >= M) ? (x - M) : x; return *this; } ModInt &operator-=(const ModInt &a) { x = ((x -= a.x) >= M) ? (x + M) : x; return *this; } ModInt &operator*=(const ModInt &a) { x = (static_cast<unsigned long long>(x) * a.x) % M; return *this; } ModInt &operator/=(const ModInt &a) { return (*this *= a.inv()); } ModInt pow(long long e) const { if (e < 0) return inv().pow(-e); ModInt a = *this, b = 1; for (; e; e >>= 1) { if (e & 1) b *= a; a *= a; } return b; } ModInt inv() const { unsigned a = M, b = x; int y = 0, z = 1; for (; b; ) { const unsigned q = a / b; const unsigned c = a - q * b; a = b; b = c; const int w = y - static_cast<int>(q) * z; y = z; z = w; } assert(a == 1); return ModInt(y); } ModInt operator+() const { return *this; } ModInt operator-() const { ModInt a; a.x = x ? (M - x) : 0; return a; } ModInt operator+(const ModInt &a) const { return (ModInt(*this) += a); } ModInt operator-(const ModInt &a) const { return (ModInt(*this) -= a); } ModInt operator*(const ModInt &a) const { return (ModInt(*this) *= a); } ModInt operator/(const ModInt &a) const { return (ModInt(*this) /= a); } template <class T> friend ModInt operator+(T a, const ModInt &b) { return (ModInt(a) += b); } template <class T> friend ModInt operator-(T a, const ModInt &b) { return (ModInt(a) -= b); } template <class T> friend ModInt operator*(T a, const ModInt &b) { return (ModInt(a) *= b); } template <class T> friend ModInt operator/(T a, const ModInt &b) { return (ModInt(a) /= b); } explicit operator bool() const { return x; } bool operator==(const ModInt &a) const { return (x == a.x); } bool operator!=(const ModInt &a) const { return (x != a.x); } friend std::ostream &operator<<(std::ostream &os, const ModInt &a) { return os << a.x; } }; //////////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////////// constexpr unsigned MO = 998244353U; constexpr unsigned MO2 = 2U * MO; constexpr int FFT_MAX = 23; using Mint = ModInt<MO>; constexpr Mint FFT_ROOTS[FFT_MAX + 1] = {1U, 998244352U, 911660635U, 372528824U, 929031873U, 452798380U, 922799308U, 781712469U, 476477967U, 166035806U, 258648936U, 584193783U, 63912897U, 350007156U, 666702199U, 968855178U, 629671588U, 24514907U, 996173970U, 363395222U, 565042129U, 733596141U, 267099868U, 15311432U}; constexpr Mint INV_FFT_ROOTS[FFT_MAX + 1] = {1U, 998244352U, 86583718U, 509520358U, 337190230U, 87557064U, 609441965U, 135236158U, 304459705U, 685443576U, 381598368U, 335559352U, 129292727U, 358024708U, 814576206U, 708402881U, 283043518U, 3707709U, 121392023U, 704923114U, 950391366U, 428961804U, 382752275U, 469870224U}; constexpr Mint FFT_RATIOS[FFT_MAX - 1] = {911660635U, 509520358U, 369330050U, 332049552U, 983190778U, 123842337U, 238493703U, 975955924U, 603855026U, 856644456U, 131300601U, 842657263U, 730768835U, 942482514U, 806263778U, 151565301U, 510815449U, 503497456U, 743006876U, 741047443U, 56250497U, 867605899U}; constexpr Mint INV_FFT_RATIOS[FFT_MAX - 1] = {86583718U, 372528824U, 373294451U, 645684063U, 112220581U, 692852209U, 155456985U, 797128860U, 90816748U, 860285882U, 927414960U, 354738543U, 109331171U, 293255632U, 535113200U, 308540755U, 121186627U, 608385704U, 438932459U, 359477183U, 824071951U, 103369235U}; // as[rev(i)] <- \sum_j \zeta^(ij) as[j] void fft(Mint *as, int n) { assert(!(n & (n - 1))); assert(1 <= n); assert(n <= 1 << FFT_MAX); int m = n; if (m >>= 1) { for (int i = 0; i < m; ++i) { const unsigned x = as[i + m].x; // < MO as[i + m].x = as[i].x + MO - x; // < 2 MO as[i].x += x; // < 2 MO } } if (m >>= 1) { Mint prod = 1; for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) { for (int i = i0; i < i0 + m; ++i) { const unsigned x = (prod * as[i + m]).x; // < MO as[i + m].x = as[i].x + MO - x; // < 3 MO as[i].x += x; // < 3 MO } prod *= FFT_RATIOS[__builtin_ctz(++h)]; } } for (; m; ) { if (m >>= 1) { Mint prod = 1; for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) { for (int i = i0; i < i0 + m; ++i) { const unsigned x = (prod * as[i + m]).x; // < MO as[i + m].x = as[i].x + MO - x; // < 4 MO as[i].x += x; // < 4 MO } prod *= FFT_RATIOS[__builtin_ctz(++h)]; } } if (m >>= 1) { Mint prod = 1; for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) { for (int i = i0; i < i0 + m; ++i) { const unsigned x = (prod * as[i + m]).x; // < MO as[i].x = (as[i].x >= MO2) ? (as[i].x - MO2) : as[i].x; // < 2 MO as[i + m].x = as[i].x + MO - x; // < 3 MO as[i].x += x; // < 3 MO } prod *= FFT_RATIOS[__builtin_ctz(++h)]; } } } for (int i = 0; i < n; ++i) { as[i].x = (as[i].x >= MO2) ? (as[i].x - MO2) : as[i].x; // < 2 MO as[i].x = (as[i].x >= MO) ? (as[i].x - MO) : as[i].x; // < MO } } // as[i] <- (1/n) \sum_j \zeta^(-ij) as[rev(j)] void invFft(Mint *as, int n) { assert(!(n & (n - 1))); assert(1 <= n); assert(n <= 1 << FFT_MAX); int m = 1; if (m < n >> 1) { Mint prod = 1; for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) { for (int i = i0; i < i0 + m; ++i) { const unsigned long long y = as[i].x + MO - as[i + m].x; // < 2 MO as[i].x += as[i + m].x; // < 2 MO as[i + m].x = (prod.x * y) % MO; // < MO } prod *= INV_FFT_RATIOS[__builtin_ctz(++h)]; } m <<= 1; } for (; m < n >> 1; m <<= 1) { Mint prod = 1; for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) { for (int i = i0; i < i0 + (m >> 1); ++i) { const unsigned long long y = as[i].x + MO2 - as[i + m].x; // < 4 MO as[i].x += as[i + m].x; // < 4 MO as[i].x = (as[i].x >= MO2) ? (as[i].x - MO2) : as[i].x; // < 2 MO as[i + m].x = (prod.x * y) % MO; // < MO } for (int i = i0 + (m >> 1); i < i0 + m; ++i) { const unsigned long long y = as[i].x + MO - as[i + m].x; // < 2 MO as[i].x += as[i + m].x; // < 2 MO as[i + m].x = (prod.x * y) % MO; // < MO } prod *= INV_FFT_RATIOS[__builtin_ctz(++h)]; } } if (m < n) { for (int i = 0; i < m; ++i) { const unsigned y = as[i].x + MO2 - as[i + m].x; // < 4 MO as[i].x += as[i + m].x; // < 4 MO as[i + m].x = y; // < 4 MO } } const Mint invN = Mint(n).inv(); for (int i = 0; i < n; ++i) { as[i] *= invN; } } void fft(vector<Mint> &as) { fft(as.data(), as.size()); } void invFft(vector<Mint> &as) { invFft(as.data(), as.size()); } vector<Mint> convolve(vector<Mint> as, vector<Mint> bs) { if (as.empty() || bs.empty()) return {}; const int len = as.size() + bs.size() - 1; int n = 1; for (; n < len; n <<= 1) {} as.resize(n); fft(as); bs.resize(n); fft(bs); for (int i = 0; i < n; ++i) as[i] *= bs[i]; invFft(as); as.resize(len); return as; } //////////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////////// // inv: log, exp, pow constexpr int LIM_INV = 1 << 20; // @ Mint inv[LIM_INV], fac[LIM_INV], invFac[LIM_INV]; struct ModIntPreparator { ModIntPreparator() { inv[1] = 1; for (int i = 2; i < LIM_INV; ++i) inv[i] = -((Mint::M / i) * inv[Mint::M % i]); fac[0] = 1; for (int i = 1; i < LIM_INV; ++i) fac[i] = fac[i - 1] * i; invFac[0] = 1; for (int i = 1; i < LIM_INV; ++i) invFac[i] = invFac[i - 1] * inv[i]; } } preparator; // polyWork0: *, inv, div, divAt, log, exp, pow, sqrt // polyWork1: inv, div, divAt, log, exp, pow, sqrt // polyWork2: divAt, exp, pow, sqrt // polyWork3: exp, pow, sqrt static constexpr int LIM_POLY = 1 << 20; // @ static_assert(LIM_POLY <= 1 << FFT_MAX); static Mint polyWork0[LIM_POLY], polyWork1[LIM_POLY], polyWork2[LIM_POLY], polyWork3[LIM_POLY]; struct Poly : public vector<Mint> { Poly() {} explicit Poly(int n) : vector<Mint>(n) {} Poly(const vector<Mint> &vec) : vector<Mint>(vec) {} Poly(std::initializer_list<Mint> il) : vector<Mint>(il) {} int size() const { return vector<Mint>::size(); } Mint at(long long k) const { return (0 <= k && k < size()) ? (*this)[k] : 0U; } int ord() const { for (int i = 0; i < size(); ++i) if ((*this)[i]) return i; return -1; } int deg() const { for (int i = size(); --i >= 0; ) if ((*this)[i]) return i; return -1; } Poly mod(int n) const { return Poly(vector<Mint>(data(), data() + min(n, size()))); } friend std::ostream &operator<<(std::ostream &os, const Poly &fs) { os << "["; for (int i = 0; i < fs.size(); ++i) { if (i > 0) os << ", "; os << fs[i]; } return os << "]"; } Poly &operator+=(const Poly &fs) { if (size() < fs.size()) resize(fs.size()); for (int i = 0; i < fs.size(); ++i) (*this)[i] += fs[i]; return *this; } Poly &operator-=(const Poly &fs) { if (size() < fs.size()) resize(fs.size()); for (int i = 0; i < fs.size(); ++i) (*this)[i] -= fs[i]; return *this; } // 3 E(|t| + |f|) Poly &operator*=(const Poly &fs) { if (empty() || fs.empty()) return *this = {}; const int nt = size(), nf = fs.size(); int n = 1; for (; n < nt + nf - 1; n <<= 1) {} assert(n <= LIM_POLY); resize(n); fft(data(), n); // 1 E(n) memcpy(polyWork0, fs.data(), nf * sizeof(Mint)); memset(polyWork0 + nf, 0, (n - nf) * sizeof(Mint)); fft(polyWork0, n); // 1 E(n) for (int i = 0; i < n; ++i) (*this)[i] *= polyWork0[i]; invFft(data(), n); // 1 E(n) resize(nt + nf - 1); return *this; } // 13 E(deg(t) - deg(f) + 1) // rev(t) = rev(f) rev(q) + x^(deg(t)-deg(f)+1) rev(r) Poly &operator/=(const Poly &fs) { const int m = deg(), n = fs.deg(); assert(n != -1); if (m < n) return *this = {}; Poly tsRev(m - n + 1), fsRev(min(m - n, n) + 1); for (int i = 0; i <= m - n; ++i) tsRev[i] = (*this)[m - i]; for (int i = 0, i0 = min(m - n, n); i <= i0; ++i) fsRev[i] = fs[n - i]; const Poly qsRev = tsRev.div(fsRev, m - n + 1); // 13 E(m - n + 1) resize(m - n + 1); for (int i = 0; i <= m - n; ++i) (*this)[i] = qsRev[m - n - i]; return *this; } // 13 E(deg(t) - deg(f) + 1) + 3 E(|t|) Poly &operator%=(const Poly &fs) { const Poly qs = *this / fs; // 13 E(deg(t) - deg(f) + 1) *this -= fs * qs; // 3 E(|t|) resize(deg() + 1); return *this; } Poly &operator*=(const Mint &a) { for (int i = 0; i < size(); ++i) (*this)[i] *= a; return *this; } Poly &operator/=(const Mint &a) { const Mint b = a.inv(); for (int i = 0; i < size(); ++i) (*this)[i] *= b; return *this; } Poly operator+() const { return *this; } Poly operator-() const { Poly fs(size()); for (int i = 0; i < size(); ++i) fs[i] = -(*this)[i]; return fs; } Poly operator+(const Poly &fs) const { return (Poly(*this) += fs); } Poly operator-(const Poly &fs) const { return (Poly(*this) -= fs); } Poly operator*(const Poly &fs) const { return (Poly(*this) *= fs); } Poly operator/(const Poly &fs) const { return (Poly(*this) /= fs); } Poly operator%(const Poly &fs) const { return (Poly(*this) %= fs); } Poly operator*(const Mint &a) const { return (Poly(*this) *= a); } Poly operator/(const Mint &a) const { return (Poly(*this) /= a); } friend Poly operator*(const Mint &a, const Poly &fs) { return fs * a; } // 10 E(n) // f <- f - (t f - 1) f Poly inv(int n) const { assert(!empty()); assert((*this)[0]); assert(1 <= n); assert(n == 1 || 1 << (32 - __builtin_clz(n - 1)) <= LIM_POLY); Poly fs(n); fs[0] = (*this)[0].inv(); for (int m = 1; m < n; m <<= 1) { memcpy(polyWork0, data(), min(m << 1, size()) * sizeof(Mint)); memset(polyWork0 + min(m << 1, size()), 0, ((m << 1) - min(m << 1, size())) * sizeof(Mint)); fft(polyWork0, m << 1); // 2 E(n) memcpy(polyWork1, fs.data(), min(m << 1, n) * sizeof(Mint)); memset(polyWork1 + min(m << 1, n), 0, ((m << 1) - min(m << 1, n)) * sizeof(Mint)); fft(polyWork1, m << 1); // 2 E(n) for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork1[i]; invFft(polyWork0, m << 1); // 2 E(n) memset(polyWork0, 0, m * sizeof(Mint)); fft(polyWork0, m << 1); // 2 E(n) for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork1[i]; invFft(polyWork0, m << 1); // 2 E(n) for (int i = m, i0 = min(m << 1, n); i < i0; ++i) fs[i] = -polyWork0[i]; } return fs; } // 9 E(n) // Need (4 m)-th roots of unity to lift from (mod x^m) to (mod x^(2m)). // f <- f - (t f - 1) f // (t f^2) mod ((x^(2m) - 1) (x^m - 1^(1/4))) /* Poly inv(int n) const { assert(!empty()); assert((*this)[0]); assert(1 <= n); assert(n == 1 || 3 << (31 - __builtin_clz(n - 1)) <= LIM_POLY); assert(n <= 1 << (FFT_MAX - 1)); Poly fs(n); fs[0] = (*this)[0].inv(); for (int h = 2, m = 1; m < n; ++h, m <<= 1) { const Mint a = FFT_ROOTS[h], b = INV_FFT_ROOTS[h]; memcpy(polyWork0, data(), min(m << 1, size()) * sizeof(Mint)); memset(polyWork0 + min(m << 1, size()), 0, ((m << 1) - min(m << 1, size())) * sizeof(Mint)); { Mint aa = 1; for (int i = 0; i < m; ++i) { polyWork0[(m << 1) + i] = aa * polyWork0[i]; aa *= a; } for (int i = 0; i < m; ++i) { polyWork0[(m << 1) + i] += aa * polyWork0[m + i]; aa *= a; } } fft(polyWork0, m << 1); // 2 E(n) fft(polyWork0 + (m << 1), m); // 1 E(n) memcpy(polyWork1, fs.data(), min(m << 1, n) * sizeof(Mint)); memset(polyWork1 + min(m << 1, n), 0, ((m << 1) - min(m << 1, n)) * sizeof(Mint)); { Mint aa = 1; for (int i = 0; i < m; ++i) { polyWork1[(m << 1) + i] = aa * polyWork1[i]; aa *= a; } for (int i = 0; i < m; ++i) { polyWork1[(m << 1) + i] += aa * polyWork1[m + i]; aa *= a; } } fft(polyWork1, m << 1); // 2 E(n) fft(polyWork1 + (m << 1), m); // 1 E(n) for (int i = 0; i < (m << 1) + m; ++i) polyWork0[i] *= polyWork1[i] * polyWork1[i]; invFft(polyWork0, m << 1); // 2 E(n) invFft(polyWork0 + (m << 1), m); // 1 E(n) // 2 f0 + (-f2), (-f1) + (-f3), 1^(1/4) (-f1) - (-f2) - 1^(1/4) (-f3) { Mint bb = 1; for (int i = 0, i0 = min(m, n - m); i < i0; ++i) { unsigned x = polyWork0[i].x + (bb * polyWork0[(m << 1) + i]).x + MO2 - (fs[i].x << 1); // < 4 MO fs[m + i] = Mint(static_cast<unsigned long long>(FFT_ROOTS[2].x) * x) - polyWork0[m + i]; fs[m + i].x = ((fs[m + i].x & 1) ? (fs[m + i].x + MO) : fs[m + i].x) >> 1; bb *= b; } } } return fs; } */ // 13 E(n) // g = (1 / f) mod x^m // h <- h - (f h - t) g Poly div(const Poly &fs, int n) const { assert(!fs.empty()); assert(fs[0]); assert(1 <= n); if (n == 1) return {at(0) / fs[0]}; // m < n <= 2 m const int m = 1 << (31 - __builtin_clz(n - 1)); assert(m << 1 <= LIM_POLY); Poly gs = fs.inv(m); // 5 E(n) gs.resize(m << 1); fft(gs.data(), m << 1); // 1 E(n) memcpy(polyWork0, data(), min(m, size()) * sizeof(Mint)); memset(polyWork0 + min(m, size()), 0, ((m << 1) - min(m, size())) * sizeof(Mint)); fft(polyWork0, m << 1); // 1 E(n) for (int i = 0; i < m << 1; ++i) polyWork0[i] *= gs[i]; invFft(polyWork0, m << 1); // 1 E(n) Poly hs(n); memcpy(hs.data(), polyWork0, m * sizeof(Mint)); memset(polyWork0 + m, 0, m * sizeof(Mint)); fft(polyWork0, m << 1); // 1 E(n) memcpy(polyWork1, fs.data(), min(m << 1, fs.size()) * sizeof(Mint)); memset(polyWork1 + min(m << 1, fs.size()), 0, ((m << 1) - min(m << 1, fs.size())) * sizeof(Mint)); fft(polyWork1, m << 1); // 1 E(n) for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork1[i]; invFft(polyWork0, m << 1); // 1 E(n) memset(polyWork0, 0, m * sizeof(Mint)); for (int i = m, i0 = min(m << 1, size()); i < i0; ++i) polyWork0[i] -= (*this)[i]; fft(polyWork0, m << 1); // 1 E(n) for (int i = 0; i < m << 1; ++i) polyWork0[i] *= gs[i]; invFft(polyWork0, m << 1); // 1 E(n) for (int i = m; i < n; ++i) hs[i] = -polyWork0[i]; return hs; } // (4 (floor(log_2 k) - ceil(log_2 |fs|)) + 16) E(|fs|) // [x^k] (t(x) / f(x)) = [x^k] ((t(x) f(-x)) / (f(x) f(-x)) // polyWork0: half of (2 m)-th roots of unity, inversed, bit-reversed Mint divAt(const Poly &fs, long long k) const { assert(k >= 0); if (size() >= fs.size()) { // TODO: operator% assert(false); } int h = 0, m = 1; for (; m < fs.size(); ++h, m <<= 1) {} if (k < m) { const Poly gs = fs.inv(k + 1); // 10 E(|fs|) Mint sum; for (int i = 0, i0 = min<int>(k + 1, size()); i < i0; ++i) sum += (*this)[i] * gs[k - i]; return sum; } assert(m << 1 <= LIM_POLY); polyWork0[0] = Mint(2U).inv(); for (int hh = 0; hh < h; ++hh) for (int i = 0; i < 1 << hh; ++i) polyWork0[1 << hh | i] = polyWork0[i] * INV_FFT_ROOTS[hh + 2]; const Mint a = FFT_ROOTS[h + 1]; memcpy(polyWork2, data(), size() * sizeof(Mint)); memset(polyWork2 + size(), 0, ((m << 1) - size()) * sizeof(Mint)); fft(polyWork2, m << 1); // 2 E(|fs|) memcpy(polyWork1, fs.data(), fs.size() * sizeof(Mint)); memset(polyWork1 + fs.size(), 0, ((m << 1) - fs.size()) * sizeof(Mint)); fft(polyWork1, m << 1); // 2 E(|fs|) for (; ; ) { if (k & 1) { for (int i = 0; i < m; ++i) polyWork2[i] = polyWork0[i] * (polyWork2[i << 1 | 0] * polyWork1[i << 1 | 1] - polyWork2[i << 1 | 1] * polyWork1[i << 1 | 0]); } else { for (int i = 0; i < m; ++i) { polyWork2[i] = polyWork2[i << 1 | 0] * polyWork1[i << 1 | 1] + polyWork2[i << 1 | 1] * polyWork1[i << 1 | 0]; polyWork2[i].x = ((polyWork2[i].x & 1) ? (polyWork2[i].x + MO) : polyWork2[i].x) >> 1; } } for (int i = 0; i < m; ++i) polyWork1[i] = polyWork1[i << 1 | 0] * polyWork1[i << 1 | 1]; if ((k >>= 1) < m) { invFft(polyWork2, m); // 1 E(|fs|) invFft(polyWork1, m); // 1 E(|fs|) // Poly::inv does not use polyWork2 const Poly gs = Poly(vector<Mint>(polyWork1, polyWork1 + k + 1)).inv(k + 1); // 10 E(|fs|) Mint sum; for (int i = 0; i <= k; ++i) sum += polyWork2[i] * gs[k - i]; return sum; } memcpy(polyWork2 + m, polyWork2, m * sizeof(Mint)); invFft(polyWork2 + m, m); // (floor(log_2 k) - ceil(log_2 |fs|)) E(|fs|) memcpy(polyWork1 + m, polyWork1, m * sizeof(Mint)); invFft(polyWork1 + m, m); // (floor(log_2 k) - ceil(log_2 |fs|)) E(|fs|) Mint aa = 1; for (int i = m; i < m << 1; ++i) { polyWork2[i] *= aa; polyWork1[i] *= aa; aa *= a; } fft(polyWork2 + m, m); // (floor(log_2 k) - ceil(log_2 |fs|)) E(|fs|) fft(polyWork1 + m, m); // (floor(log_2 k) - ceil(log_2 |fs|)) E(|fs|) } } // 13 E(n) // D log(t) = (D t) / t Poly log(int n) const { assert(!empty()); assert((*this)[0].x == 1U); assert(n <= LIM_INV); Poly fs = mod(n); for (int i = 0; i < fs.size(); ++i) fs[i] *= i; fs = fs.div(*this, n); for (int i = 1; i < n; ++i) fs[i] *= ::inv[i]; return fs; } // (16 + 1/2) E(n) // f = exp(t) mod x^m ==> (D f) / f == D t (mod x^m) // g = (1 / exp(t)) mod x^m // f <- f - (log f - t) / (1 / f) // = f - (I ((D f) / f) - t) f // == f - (I ((D f) / f + (f g - 1) ((D f) / f - D (t mod x^m))) - t) f (mod x^(2m)) // = f - (I (g (D f - f D (t mod x^m)) + D (t mod x^m)) - t) f // g <- g - (f g - 1) g // polyWork1: DFT(f, 2 m), polyWork2: g, polyWork3: DFT(g, 2 m) Poly exp(int n) const { assert(!empty()); assert(!(*this)[0]); assert(1 <= n); assert(n == 1 || 1 << (32 - __builtin_clz(n - 1)) <= min(LIM_INV, LIM_POLY)); if (n == 1) return {1U}; if (n == 2) return {1U, at(1)}; Poly fs(n); fs[0].x = polyWork1[0].x = polyWork1[1].x = polyWork2[0].x = 1U; int m; for (m = 1; m << 1 < n; m <<= 1) { for (int i = 0, i0 = min(m, size()); i < i0; ++i) polyWork0[i] = i * (*this)[i]; memset(polyWork0 + min(m, size()), 0, (m - min(m, size())) * sizeof(Mint)); fft(polyWork0, m); // (1/2) E(n) for (int i = 0; i < m; ++i) polyWork0[i] *= polyWork1[i]; invFft(polyWork0, m); // (1/2) E(n) for (int i = 0; i < m; ++i) polyWork0[i] -= i * fs[i]; memset(polyWork0 + m, 0, m * sizeof(Mint)); fft(polyWork0, m << 1); // 1 E(n) memcpy(polyWork3, polyWork2, m * sizeof(Mint)); memset(polyWork3 + m, 0, m * sizeof(Mint)); fft(polyWork3, m << 1); // 1 E(n) for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork3[i]; invFft(polyWork0, m << 1); // 1 E(n) for (int i = 0; i < m; ++i) polyWork0[i] *= ::inv[m + i]; for (int i = 0, i0 = min(m, size() - m); i < i0; ++i) polyWork0[i] += (*this)[m + i]; memset(polyWork0 + m, 0, m * sizeof(Mint)); fft(polyWork0, m << 1); // 1 E(n) for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork1[i]; invFft(polyWork0, m << 1); // 1 E(n) memcpy(fs.data() + m, polyWork0, m * sizeof(Mint)); memcpy(polyWork1, fs.data(), (m << 1) * sizeof(Mint)); memset(polyWork1 + (m << 1), 0, (m << 1) * sizeof(Mint)); fft(polyWork1, m << 2); // 2 E(n) for (int i = 0; i < m << 1; ++i) polyWork0[i] = polyWork1[i] * polyWork3[i]; invFft(polyWork0, m << 1); // 1 E(n) memset(polyWork0, 0, m * sizeof(Mint)); fft(polyWork0, m << 1); // 1 E(n) for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork3[i]; invFft(polyWork0, m << 1); // 1 E(n) for (int i = m; i < m << 1; ++i) polyWork2[i] = -polyWork0[i]; } for (int i = 0, i0 = min(m, size()); i < i0; ++i) polyWork0[i] = i * (*this)[i]; memset(polyWork0 + min(m, size()), 0, (m - min(m, size())) * sizeof(Mint)); fft(polyWork0, m); // (1/2) E(n) for (int i = 0; i < m; ++i) polyWork0[i] *= polyWork1[i]; invFft(polyWork0, m); // (1/2) E(n) for (int i = 0; i < m; ++i) polyWork0[i] -= i * fs[i]; memcpy(polyWork0 + m, polyWork0 + (m >> 1), (m >> 1) * sizeof(Mint)); memset(polyWork0 + (m >> 1), 0, (m >> 1) * sizeof(Mint)); memset(polyWork0 + m + (m >> 1), 0, (m >> 1) * sizeof(Mint)); fft(polyWork0, m); // (1/2) E(n) fft(polyWork0 + m, m); // (1/2) E(n) memcpy(polyWork3 + m, polyWork2 + (m >> 1), (m >> 1) * sizeof(Mint)); memset(polyWork3 + m + (m >> 1), 0, (m >> 1) * sizeof(Mint)); fft(polyWork3 + m, m); // (1/2) E(n) for (int i = 0; i < m; ++i) polyWork0[m + i] = polyWork0[i] * polyWork3[m + i] + polyWork0[m + i] * polyWork3[i]; for (int i = 0; i < m; ++i) polyWork0[i] *= polyWork3[i]; invFft(polyWork0, m); // (1/2) E(n) invFft(polyWork0 + m, m); // (1/2) E(n) for (int i = 0; i < m >> 1; ++i) polyWork0[(m >> 1) + i] += polyWork0[m + i]; for (int i = 0; i < m; ++i) polyWork0[i] *= ::inv[m + i]; for (int i = 0, i0 = min(m, size() - m); i < i0; ++i) polyWork0[i] += (*this)[m + i]; memset(polyWork0 + m, 0, m * sizeof(Mint)); fft(polyWork0, m << 1); // 1 E(n) for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork1[i]; invFft(polyWork0, m << 1); // 1 E(n) memcpy(fs.data() + m, polyWork0, (n - m) * sizeof(Mint)); return fs; } // (29 + 1/2) E(n) // g <- g - (log g - a log t) g Poly pow(Mint a, int n) const { assert(!empty()); assert((*this)[0].x == 1U); assert(1 <= n); return (a * log(n)).exp(n); // 13 E(n) + (16 + 1/2) E(n) } // (29 + 1/2) E(n - a ord(t)) Poly pow(long long a, int n) const { assert(a >= 0); assert(1 <= n); if (a == 0) { Poly gs(n); gs[0].x = 1U; return gs; } const int o = ord(); if (o == -1 || o > (n - 1) / a) return Poly(n); const Mint b = (*this)[o].inv(), c = (*this)[o].pow(a); const int ntt = min<int>(n - a * o, size() - o); Poly tts(ntt); for (int i = 0; i < ntt; ++i) tts[i] = b * (*this)[o + i]; tts = tts.pow(Mint(a), n - a * o); // (29 + 1/2) E(n - a ord(t)) Poly gs(n); for (int i = 0; i < n - a * o; ++i) gs[a * o + i] = c * tts[i]; return gs; } // (10 + 1/2) E(n) // f = t^(1/2) mod x^m, g = 1 / t^(1/2) mod x^m // f <- f - (f^2 - h) g / 2 // g <- g - (f g - 1) g // polyWork1: DFT(f, m), polyWork2: g, polyWork3: DFT(g, 2 m) Poly sqrt(int n) const { assert(!empty()); assert((*this)[0].x == 1U); assert(1 <= n); assert(n == 1 || 1 << (32 - __builtin_clz(n - 1)) <= LIM_POLY); if (n == 1) return {1U}; if (n == 2) return {1U, at(1) / 2}; Poly fs(n); fs[0].x = polyWork1[0].x = polyWork2[0].x = 1U; int m; for (m = 1; m << 1 < n; m <<= 1) { for (int i = 0; i < m; ++i) polyWork1[i] *= polyWork1[i]; invFft(polyWork1, m); // (1/2) E(n) for (int i = 0, i0 = min(m, size()); i < i0; ++i) polyWork1[i] -= (*this)[i]; for (int i = 0, i0 = min(m, size() - m); i < i0; ++i) polyWork1[i] -= (*this)[m + i]; memset(polyWork1 + m, 0, m * sizeof(Mint)); fft(polyWork1, m << 1); // 1 E(n) memcpy(polyWork3, polyWork2, m * sizeof(Mint)); memset(polyWork3 + m, 0, m * sizeof(Mint)); fft(polyWork3, m << 1); // 1 E(n) for (int i = 0; i < m << 1; ++i) polyWork1[i] *= polyWork3[i]; invFft(polyWork1, m << 1); // 1 E(n) for (int i = 0; i < m; ++i) { polyWork1[i] = -polyWork1[i]; fs[m + i].x = ((polyWork1[i].x & 1) ? (polyWork1[i].x + MO) : polyWork1[i].x) >> 1; } memcpy(polyWork1, fs.data(), (m << 1) * sizeof(Mint)); fft(polyWork1, m << 1); // 1 E(n) for (int i = 0; i < m << 1; ++i) polyWork0[i] = polyWork1[i] * polyWork3[i]; invFft(polyWork0, m << 1); // 1 E(n) memset(polyWork0, 0, m * sizeof(Mint)); fft(polyWork0, m << 1); // 1 E(n) for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork3[i]; invFft(polyWork0, m << 1); // 1 E(n) for (int i = m; i < m << 1; ++i) polyWork2[i] = -polyWork0[i]; } for (int i = 0; i < m; ++i) polyWork1[i] *= polyWork1[i]; invFft(polyWork1, m); // (1/2) E(n) for (int i = 0, i0 = min(m, size()); i < i0; ++i) polyWork1[i] -= (*this)[i]; for (int i = 0, i0 = min(m, size() - m); i < i0; ++i) polyWork1[i] -= (*this)[m + i]; memcpy(polyWork1 + m, polyWork1 + (m >> 1), (m >> 1) * sizeof(Mint)); memset(polyWork1 + (m >> 1), 0, (m >> 1) * sizeof(Mint)); memset(polyWork1 + m + (m >> 1), 0, (m >> 1) * sizeof(Mint)); fft(polyWork1, m); // (1/2) E(n) fft(polyWork1 + m, m); // (1/2) E(n) memcpy(polyWork3 + m, polyWork2 + (m >> 1), (m >> 1) * sizeof(Mint)); memset(polyWork3 + m + (m >> 1), 0, (m >> 1) * sizeof(Mint)); fft(polyWork3 + m, m); // (1/2) E(n) // for (int i = 0; i < m << 1; ++i) polyWork1[i] *= polyWork3[i]; for (int i = 0; i < m; ++i) polyWork1[m + i] = polyWork1[i] * polyWork3[m + i] + polyWork1[m + i] * polyWork3[i]; for (int i = 0; i < m; ++i) polyWork1[i] *= polyWork3[i]; invFft(polyWork1, m); // (1/2) E(n) invFft(polyWork1 + m, m); // (1/2) E(n) for (int i = 0; i < m >> 1; ++i) polyWork1[(m >> 1) + i] += polyWork1[m + i]; for (int i = 0; i < n - m; ++i) { polyWork1[i] = -polyWork1[i]; fs[m + i].x = ((polyWork1[i].x & 1) ? (polyWork1[i].x + MO) : polyWork1[i].x) >> 1; } return fs; } // (10 + 1/2) E(n) // modSqrt must return a quadratic residue if exists, or anything otherwise. // Return {} if *this does not have a square root. template <class F> Poly sqrt(int n, F modSqrt) const { assert(1 <= n); const int o = ord(); if (o == -1) return Poly(n); if (o & 1) return {}; const Mint c = modSqrt((*this)[o]); if (c * c != (*this)[o]) return {}; if (o >> 1 >= n) return Poly(n); const Mint b = (*this)[o].inv(); const int ntt = min(n - (o >> 1), size() - o); Poly tts(ntt); for (int i = 0; i < ntt; ++i) tts[i] = b * (*this)[o + i]; tts = tts.sqrt(n - (o >> 1)); // (10 + 1/2) E(n) Poly gs(n); for (int i = 0; i < n - (o >> 1); ++i) gs[(o >> 1) + i] = c * tts[i]; return gs; } }; Mint linearRecurrenceAt(const vector<Mint> &as, const vector<Mint> &cs, long long k) { assert(!cs.empty()); assert(cs[0]); const int d = cs.size() - 1; assert(as.size() >= static_cast<size_t>(d)); return (Poly(vector<Mint>(as.begin(), as.begin() + d)) * cs).mod(d).divAt(cs, k); } struct SubproductTree { int logN, n, nn; vector<Mint> xs; // [DFT_4((X-xs[0])(X-xs[1])(X-xs[2])(X-xs[3]))] [(X-xs[0])(X-xs[1])(X-xs[2])(X-xs[3])mod X^4] // [ DFT_4((X-xs[0])(X-xs[1])) ] [ DFT_4((X-xs[2])(X-xs[3])) ] // [ DFT_2(X-xs[0]) ] [ DFT_2(X-xs[1]) ] [ DFT_2(X-xs[2]) ] [ DFT_2(X-xs[3]) ] vector<Mint> buf; vector<Mint *> gss; // (1 - xs[0] X) ... (1 - xs[nn-1] X) Poly all; // (ceil(log_2 n) + O(1)) E(n) SubproductTree(const vector<Mint> &xs_) { n = xs_.size(); for (logN = 0, nn = 1; nn < n; ++logN, nn <<= 1) {} xs.assign(nn, 0U); memcpy(xs.data(), xs_.data(), n * sizeof(Mint)); buf.assign((logN + 1) * (nn << 1), 0U); gss.assign(nn << 1, nullptr); for (int h = 0; h <= logN; ++h) for (int u = 1 << h; u < 1 << (h + 1); ++u) { gss[u] = buf.data() + (h * (nn << 1) + ((u - (1 << h)) << (logN - h + 1))); } for (int i = 0; i < nn; ++i) { gss[nn + i][0] = -xs[i] + 1; gss[nn + i][1] = -xs[i] - 1; } if (nn == 1) gss[1][1] += 2; for (int h = logN; --h >= 0; ) { const int m = 1 << (logN - h); for (int u = 1 << (h + 1); --u >= 1 << h; ) { for (int i = 0; i < m; ++i) gss[u][i] = gss[u << 1][i] * gss[u << 1 | 1][i]; memcpy(gss[u] + m, gss[u], m * sizeof(Mint)); invFft(gss[u] + m, m); // ((1/2) ceil(log_2 n) + O(1)) E(n) if (h > 0) { gss[u][m] -= 2; const Mint a = FFT_ROOTS[logN - h + 1]; Mint aa = 1; for (int i = m; i < m << 1; ++i) { gss[u][i] *= aa; aa *= a; }; fft(gss[u] + m, m); // ((1/2) ceil(log_2 n) + O(1)) E(n) } } } all.resize(nn + 1); all[0] = 1; for (int i = 1; i < nn; ++i) all[i] = gss[1][nn + nn - i]; all[nn] = gss[1][nn] - 1; } // ((3/2) ceil(log_2 n) + O(1)) E(n) + 10 E(|fs|) + 3 E(|fs| + 2^(ceil(log_2 n))) vector<Mint> multiEval(const Poly &fs) const { vector<Mint> work0(nn), work1(nn), work2(nn); { const int m = max(fs.size(), 1); auto invAll = all.inv(m); // 10 E(|fs|) std::reverse(invAll.begin(), invAll.end()); int mm; for (mm = 1; mm < m - 1 + nn; mm <<= 1) {} invAll.resize(mm, 0U); fft(invAll); // E(|fs| + 2^(ceil(log_2 n))) vector<Mint> ffs(mm, 0U); memcpy(ffs.data(), fs.data(), fs.size() * sizeof(Mint)); fft(ffs); // E(|fs| + 2^(ceil(log_2 n))) for (int i = 0; i < mm; ++i) ffs[i] *= invAll[i]; invFft(ffs); // E(|fs| + 2^(ceil(log_2 n))) memcpy(((logN & 1) ? work1 : work0).data(), ffs.data() + m - 1, nn * sizeof(Mint)); } for (int h = 0; h < logN; ++h) { const int m = 1 << (logN - h); for (int u = 1 << h; u < 1 << (h + 1); ++u) { Mint *hs = (((logN - h) & 1) ? work1 : work0).data() + ((u - (1 << h)) << (logN - h)); Mint *hs0 = (((logN - h) & 1) ? work0 : work1).data() + ((u - (1 << h)) << (logN - h)); Mint *hs1 = hs0 + (m >> 1); fft(hs, m); // ((1/2) ceil(log_2 n) + O(1)) E(n) for (int i = 0; i < m; ++i) work2[i] = gss[u << 1 | 1][i] * hs[i]; invFft(work2.data(), m); // ((1/2) ceil(log_2 n) + O(1)) E(n) memcpy(hs0, work2.data() + (m >> 1), (m >> 1) * sizeof(Mint)); for (int i = 0; i < m; ++i) work2[i] = gss[u << 1][i] * hs[i]; invFft(work2.data(), m); // ((1/2) ceil(log_2 n) + O(1)) E(n) memcpy(hs1, work2.data() + (m >> 1), (m >> 1) * sizeof(Mint)); } } work0.resize(n); return work0; } // ((5/2) ceil(log_2 n) + O(1)) E(n) Poly interpolate(const vector<Mint> &ys) const { assert(static_cast<int>(ys.size()) == n); Poly gs(n); for (int i = 0; i < n; ++i) gs[i] = (i + 1) * all[n - (i + 1)]; const vector<Mint> denoms = multiEval(gs); // ((3/2) ceil(log_2 n) + O(1)) E(n) vector<Mint> work(nn << 1, 0U); for (int i = 0; i < n; ++i) { // xs[0], ..., xs[n - 1] are not distinct assert(denoms[i]); work[i << 1] = work[i << 1 | 1] = ys[i] / denoms[i]; } for (int h = logN; --h >= 0; ) { const int m = 1 << (logN - h); for (int u = 1 << (h + 1); --u >= 1 << h; ) { Mint *hs = work.data() + ((u - (1 << h)) << (logN - h + 1)); for (int i = 0; i < m; ++i) hs[i] = gss[u << 1 | 1][i] * hs[i] + gss[u << 1][i] * hs[m + i]; if (h > 0) { memcpy(hs + m, hs, m * sizeof(Mint)); invFft(hs + m, m); // ((1/2) ceil(log_2 n) + O(1)) E(n) const Mint a = FFT_ROOTS[logN - h + 1]; Mint aa = 1; for (int i = m; i < m << 1; ++i) { hs[i] *= aa; aa *= a; }; fft(hs + m, m); // ((1/2) ceil(log_2 n) + O(1)) E(n) } } } invFft(work.data(), nn); // E(n) return Poly(vector<Mint>(work.data() + nn - n, work.data() + nn)); } }; //////////////////////////////////////////////////////////////////////////////// #include <stdio.h> int readInt() { int c; for (; ; ) { c = getchar(); if ('0' <= c && c <= '9') break; if (c == -1) throw -1; if (c == '-') return -readInt(); } int x = c - '0'; for (; ; ) { c = getchar(); if (!('0' <= c && c <= '9')) return x; x = x * 10 + (c - '0'); } } char writeIntBuffer[10]; void writeInt(int x) { if (x < 0) { putchar('-'); x = -x; } int i = 0; do { writeIntBuffer[i++] = '0' + (x % 10); x /= 10; } while (x != 0); for (; i > 0; ) { putchar(writeIntBuffer[--i]); } } // https://judge.yosupo.jp/problem/polynomial_interpolation int main() { try { for (; ; ) { const int N = readInt(); vector<Mint> X(N), Y(N); for (int i = 0; i < N; ++i) { X[i].x = readInt(); } for (int i = 0; i < N; ++i) { Y[i].x = readInt(); } const Poly ans = SubproductTree(X).interpolate(Y); for (int j = 0; j < N; ++j) { if (j > 0) putchar(' '); writeInt(ans[j].x); } putchar('\n'); } } catch (int) { } return 0; }