#460639 (C++14) No.658 テトラナッチ数列 Hard

提出ソース
結果

問題	No.658 テトラナッチ数列 Hard
ユーザー	hashiryo
提出日時	2020-04-08 19:55:32
言語	C++14 (gcc 12.3.0 + boost 1.83.0)
結果	TLE
実行時間	-
コード長	16,175 bytes
コンパイル時間	2,029 ms
コンパイル使用メモリ	187,000 KB
実行使用メモリ	17,024 KB
最終ジャッジ日時	2024-07-19 05:31:19
合計ジャッジ時間	6,053 ms
ジャッジサーバーID （参考情報）	judge1 / judge4
このコードへのチャレンジ
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テストケース

テストケース表示
入力	結果	実行時間実行使用メモリ
testcase_00	AC	18 ms 17,024 KB
testcase_01	AC	2 ms 5,376 KB
testcase_02	AC	43 ms 11,904 KB
testcase_03	AC	223 ms 11,648 KB
testcase_04	TLE	-
testcase_05	--	-
testcase_06	--	-
testcase_07	--	-
testcase_08	--	-
testcase_09	--	-
testcase_10	--	-
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ソースコード

raw source code
#include <bits/stdc++.h>

#define debug(x) cerr << #x << ": " << x << endl
#define debugArray(x, n)                           \
  for (long long hoge = 0; (hoge) < (n); ++(hoge)) \
  cerr << #x << "[" << hoge << "]: " << x[hoge] << endl
using namespace std;
double tick() {
  static clock_t oldtick;
  clock_t newtick = clock();
  double diff = 1.0 * (newtick - oldtick) / CLOCKS_PER_SEC;
  oldtick = newtick;
  return diff;
}
using u32 = unsigned;
using u64 = unsigned long long;
namespace ntt {
template <u64 mod, u64 prim_root>
class Mod64 {
 private:
  using u128 = __uint128_t;
  static constexpr u64 mul_inv(u64 n, int e = 6, u64 x = 1) {
    return e == 0 ? x : mul_inv(n, e - 1, x * (2 - x * n));
  }

 public:
  static constexpr u64 inv = mul_inv(mod, 6, 1);
  static constexpr u64 r2 = -u128(mod) % mod;
  static constexpr int level = __builtin_ctzll(mod - 1);
  static_assert(inv * mod == 1, "invalid 1/M modulo 2^64.");
  Mod64() {}
  Mod64(u64 n) : x(init(n)){};
  static u64 modulo() { return mod; }
  static u64 init(u64 w) { return reduce(u128(w) * r2); }
  static u64 reduce(const u128 w) {
    return u64(w >> 64) + mod - ((u128(u64(w) * inv) * mod) >> 64);
  }
  static Mod64 omega() { return Mod64(prim_root).pow((mod - 1) >> level); }
  Mod64 &operator+=(Mod64 rhs) {
    this->x += rhs.x;
    return *this;
  }
  Mod64 &operator-=(Mod64 rhs) {
    this->x += 2 * mod - rhs.x;
    return *this;
  }
  Mod64 &operator*=(Mod64 rhs) {
    this->x = reduce(u128(this->x) * rhs.x);
    return *this;
  }
  Mod64 operator+(Mod64 rhs) const { return Mod64(*this) += rhs; }
  Mod64 operator-(Mod64 rhs) const { return Mod64(*this) -= rhs; }
  Mod64 operator*(Mod64 rhs) const { return Mod64(*this) *= rhs; }
  u64 get() const { return reduce(this->x) % mod; }
  void set(u64 n) const { this->x = n; }
  Mod64 pow(u64 exp) const {
    Mod64 ret = Mod64(1);
    for (Mod64 base = *this; exp; exp >>= 1, base *= base)
      if (exp & 1) ret *= base;
    return ret;
  }
  Mod64 inverse() const { return pow(mod - 2); }
  friend ostream &operator<<(ostream &os, const Mod64 &m) {
    return os << m.get();
  }
  u64 x;
};

template <typename mod_t>
void convolute(mod_t *A, int s1, mod_t *B, int s2, bool cyclic = false) {
  int s = (cyclic ? max(s1, s2) : s1 + s2 - 1);
  int size = 1;
  while (size < s) size <<= 1;
  mod_t roots[mod_t::level] = {mod_t::omega()};
  for (int i = 1; i < mod_t::level; i++) roots[i] = roots[i - 1] * roots[i - 1];
  fill(A + s1, A + size, 0);
  ntt_dit4(A, size, 1, roots);
  if (A == B && s1 == s2) {
    for (int i = 0; i < size; i++) A[i] *= A[i];
  } else {
    fill(B + s2, B + size, 0);
    ntt_dit4(B, size, 1, roots);
    for (int i = 0; i < size; i++) A[i] *= B[i];
  }
  ntt_dit4(A, size, -1, roots);
  mod_t inv = mod_t(size).inverse();
  for (int i = 0; i < (cyclic ? size : s); i++) A[i] *= inv;
}

template <typename mod_t>
void rev_permute(mod_t *A, int n) {
  int r = 0, nh = n >> 1;
  for (int i = 1; i < n; i++) {
    for (int h = nh; !((r ^= h) & h); h >>= 1)
      ;
    if (r > i) swap(A[i], A[r]);
  }
}

template <typename mod_t>
void ntt_dit4(mod_t *A, int n, int sign, mod_t *roots) {
  rev_permute(A, n);
  int logn = __builtin_ctz(n);
  if (logn & 1)
    for (int i = 0; i < n; i += 2) {
      mod_t a = A[i], b = A[i + 1];
      A[i] = a + b;
      A[i + 1] = a - b;
    }
  mod_t imag = roots[mod_t::level - 2];
  if (sign < 0) imag = imag.inverse();
  mod_t one = mod_t(1);
  for (int e = 2 + (logn & 1); e < logn + 1; e += 2) {
    const int m = 1 << e;
    const int m4 = m >> 2;
    mod_t dw = roots[mod_t::level - e];
    if (sign < 0) dw = dw.inverse();
    const int block_size = max(m, (1 << 15) / int(sizeof(A[0])));
    for (int k = 0; k < n; k += block_size) {
      mod_t w = one, w2 = one, w3 = one;
      for (int j = 0; j < m4; j++) {
        for (int i = k + j; i < k + block_size; i += m) {
          mod_t a0 = A[i + m4 * 0] * one, a2 = A[i + m4 * 1] * w2;
          mod_t a1 = A[i + m4 * 2] * w, a3 = A[i + m4 * 3] * w3;
          mod_t t02 = a0 + a2, t13 = a1 + a3;
          A[i + m4 * 0] = t02 + t13;
          A[i + m4 * 2] = t02 - t13;
          t02 = a0 - a2, t13 = (a1 - a3) * imag;
          A[i + m4 * 1] = t02 + t13;
          A[i + m4 * 3] = t02 - t13;
        }
        w *= dw;
        w2 = w * w;
        w3 = w2 * w;
      }
    }
  }
}

const int size = 1 << 20;
using m64_1 = ntt::Mod64<34703335751681, 3>;
using m64_2 = ntt::Mod64<35012573396993, 3>;
m64_1 f1[size], g1[size];
m64_2 f2[size], g2[size];

}  // namespace ntt

using R = u64;
class FormalPowerSeries {
  using FPS = FormalPowerSeries;

 public:
  FormalPowerSeries() {}
  FormalPowerSeries(int n) : coefs(n) {}
  FormalPowerSeries(int n, int c) : coefs(n, c % mod) {}
  FormalPowerSeries(const vector<R> &v) : coefs(v) {}
  FormalPowerSeries(const FPS &f, int beg, int end = -1) {
    if (end < 0) end = beg, beg = 0;
    resize(end - beg);
    for (int i = beg; i < end; i++)
      if (i < f.size()) coefs[i - beg] = f[i];
  }
  int size() const { return coefs.size(); }
  void resize(int s, R v = 0) { coefs.resize(s, v); }
  void push_back(R c) { coefs.push_back(c); }
  void shrink() {
    while (!coefs.empty() && !coefs.back()) coefs.pop_back();
  }
  const R *data() const { return coefs.data(); }
  R *data() { return coefs.data(); }
  const R &operator[](int i) const { return coefs[i]; }
  R &operator[](int i) { return coefs[i]; }

 public:
  static void mod_add(R &a, R b) {
    if ((a += b) >= mod) a -= mod;
  }
  static void mod_sub(R &a, R b) {
    if (int(a -= b) < 0) a += mod;
  }
  static R mod_mul(R a, R b) { return a * b % fast_mod; }
  static R mod_pow(R v, u64 exp) {
    R ret = 1;
    for (R base = v; exp; exp >>= 1, base = mod_mul(base, base))
      if (exp & 1) ret = mod_mul(ret, base);
    return ret;
  }
  static R mod_inverse(R v) { return mod_pow(v, mod - 2); }
  static R mod_sqrt(R x) {
    if (x == 0 || mod == 2) return x;
    if (mod_pow(x, (mod - 1) >> 1) != 1) return 0;  // no solution
    R b = 2;
    R w = (b * b + mod - x) % fast_mod;
    while (mod_pow(w, (mod - 1) >> 1) == 1) {
      b++;
      w = (b * b + mod - x) % fast_mod;
    }
    auto mul = [&](pair<R, R> u, pair<R, R> v) {
      R a = (u.first * v.first + u.second * v.second % fast_mod * w) % fast_mod;
      R b = (u.first * v.second + u.second * v.first) % fast_mod;
      return make_pair(a, b);
    };
    u32 exp = (mod + 1) >> 1;
    pair<R, R> ret = make_pair(1, 0);
    for (auto base = make_pair(b, 1); exp; exp >>= 1, base = mul(base, base)) {
      if (exp & 1) ret = mul(ret, base);
    }
    return ret.first;
  }

 public:
  struct fast_div {
    using u128 = __uint128_t;
    fast_div(){};
    fast_div(u64 n) : m(n) {
      s = (n == 1) ? 0 : 127 - __builtin_clzll(n - 1);
      x = ((u128(1) << s) + n - 1) / n;
    }
    friend u64 operator/(u64 n, fast_div d) { return u128(n) * d.x >> d.s; }
    friend u64 operator%(u64 n, fast_div d) { return n - n / d * d.m; }
    u64 m, s, x;
  };

  static FPS mul_crt(int beg, int end) {
    using namespace ntt;
    auto inv = m64_2(m64_1::modulo()).inverse();
    auto mod1 = m64_1::modulo() % fast_mod;
    FPS ret(end - beg);
    for (int i = 0; i < (int)ret.size(); i++) {
      u64 r1 = f1[i + beg].get(), r2 = f2[i + beg].get();
      ret[i]
          = (r1
             + (m64_2(r2 + m64_2::modulo() - r1) * inv).get() % fast_mod * mod1)
            % fast_mod;
    }
    return ret;
  }

  static void mul2(const FPS &f, const FPS &g, bool cyclic = false) {
    using namespace ntt;
    if (&f == &g) {
      for (int i = 0; i < (int)f.size(); i++) f1[i] = f[i];
      convolute(f1, f.size(), f1, f.size(), cyclic);
      for (int i = 0; i < (int)f.size(); i++) f2[i] = f[i];
      convolute(f2, f.size(), f2, f.size(), cyclic);
    } else {
      for (int i = 0; i < (int)f.size(); i++) f1[i] = f[i];
      for (int i = 0; i < (int)g.size(); i++) g1[i] = g[i];
      convolute(f1, f.size(), g1, g.size(), cyclic);
      for (int i = 0; i < (int)f.size(); i++) f2[i] = f[i];
      for (int i = 0; i < (int)g.size(); i++) g2[i] = g[i];
      convolute(f2, f.size(), g2, g.size(), cyclic);
    }
  }

 private:
  FPS mul(const FPS &g) const {
    const auto &f = *this;
    if (f.size() == 0 || g.size() == 0) return FPS();
    mul2(f, g, false);
    return mul_crt(0, f.size() + g.size() - 1);
  }

  FPS middle_product(const FPS &g) const {
    const FPS &f = *this;
    if (f.size() == 0 || g.size() == 0) return FPS();
    mul2(f, g, true);
    return mul_crt(f.size(), g.size());
  }

  FPS inverse(int deg = -1) const {
    if (deg < 0) deg = size();
    FPS ret(1, mod_inverse((*this)[0]));
    for (int e = 1, ne; e < deg; e = ne) {
      ne = min(2 * e, deg);
      FPS h = FPS(ret, ne - e) * -ret.middle_product(FPS(*this, ne));
      for (int i = e; i < ne; i++) ret.push_back(h[i - e]);
    }
    return ret;
  }
  FPS differentiation() const {
    FPS ret(max(0, size() - 1));
    for (int i = 1; i < size(); i++) ret[i - 1] = mod_mul(i, (*this)[i]);
    return ret;
  }

  FPS integral() const {
    FPS ret(size() + 1);
    ret[0] = 0;
    for (int i = 0; i < size(); i++)
      ret[i + 1] = mod_mul(inve[i + 1], (*this)[i]);
    return ret;
  }

  FPS logarithm(int deg = -1) const {
    assert((*this)[0] == 1);
    if (deg < 0) deg = size();
    return FPS(differentiation() * inverse(deg), deg - 1).integral();
  }

  FPS exponent(int deg = -1) const {
    assert((*this)[0] == 0);
    if (deg < 0) deg = size();
    FPS ret(vector<R>({1, 1 < size() ? (*this)[1] : 0})), retinv(1, 1);
    FPS f = differentiation();
    FPS retdif = ret.differentiation();
    for (int e = 1, ne = 2, nne; ne < deg; e = ne, ne = nne) {
      nne = min(2 * ne, deg);
      FPS h = FPS(retinv, ne - e) * -retinv.middle_product(ret);
      for (int i = e; i < ne; i++) retinv.push_back(h[i - e]);
      FPS a = ret * FPS(f, nne) - retdif;
      FPS c = (retinv * FPS(a, nne)).integral();
      h = ret.middle_product(FPS(c, nne));
      for (int i = ne; i < nne; i++) {
        ret.push_back(h[i - ne]);
        retdif.push_back(mod_mul(i, h[i - ne]));
      }
    }
    return ret;
  }

  FPS square_root(int deg = -1) const {
    if (deg < 0) deg = size();
    if ((*this)[0] == 0) {
      for (int i = 1; i < size(); i++) {
        if ((*this)[i] != 0) {
          if (i & 1) return FPS();  // no solution
          if (deg - i / 2 <= 0) break;
          auto ret = (*this >> i).square_root(deg - i / 2);
          if (!ret.size()) return FPS();  // no solution
          ret = ret << (i / 2);
          if (ret.size() < deg) ret.resize(deg, 0);
          return ret;
        }
      }
      return FPS(deg, 0);
    }
    R sqr = mod_sqrt((*this)[0]);
    if (sqr * 2 > mod) sqr = mod - sqr;
    if (mod_mul(sqr, sqr) != (*this)[0]) return FPS();  // no solution
    FPS ret(1, sqr);
    for (int i = 1; i < deg; i <<= 1) {
      ret = FPS(ret + (FPS(*this, i << 1) * ret.inverse(i << 1)), i << 1);
      ret *= inve[2];
    }
    return ret;
  }

  FPS power(u64 k, int deg = -1) const {
    if (deg < 0) deg = size();
    for (int i = 0; i < size(); i++) {
      if ((*this)[i] != 0) {
        if (i * k > deg) return FPS(deg, 0);
        R rev = mod_inverse((*this)[i]);
        FPS ret = (((*this * rev) >> i).logarithm() * k).exponent()
                  * mod_pow((*this)[i], k);
        return FPS(ret << (i * k), deg);
      }
    }
    return *this;
  }

 public:
  FPS rev() const {
    FPS ret((*this), size());
    reverse(ret.coefs.begin(), ret.coefs.end());
    return ret;
  }
  FPS operator-() {
    FPS ret = *this;
    for (int i = 0; i < (int)ret.size(); i++)
      ret[i] = (ret[i] == 0 ? 0 : mod - ret[i]);
    return ret;
  }
  FPS &operator+=(const FPS &rhs) {
    if (size() < rhs.size()) resize(rhs.size());
    for (int i = 0; i < (int)rhs.size(); i++) mod_add((*this)[i], rhs[i]);
    return *this;
  }
  FPS &operator-=(const FPS &rhs) {
    if (size() < rhs.size()) resize(rhs.size());
    for (int i = 0; i < (int)rhs.size(); i++) mod_sub((*this)[i], rhs[i]);
    return *this;
  }
  FPS &operator*=(const FPS &rhs) { return *this = *this * rhs; }
  FPS &operator/=(const FPS &rhs) {
    if (size() < rhs.size()) return *this = FPS();
    int sq = size() - rhs.size() + 1;
    return *this
           = FPS(FPS(rev(), sq) * FPS(rhs.rev().inverse(sq), sq), sq).rev();
  }
  FPS &operator%=(const FPS &rhs) {
    *this -= (*this / rhs) * rhs;
    shrink();
    return *this;
  }
  FPS &operator+=(const R &v) {
    mod_add((*this)[0], v);
    return *this;
  }
  FPS &operator-=(const R &v) {
    mod_sub((*this)[0], v);
    return *this;
  }
  FPS &operator*=(const R &v) {
    for (int k = 0; k < size(); k++) (*this)[k] = mod_mul((*this)[k], v);
    return *this;
  }
  FPS operator>>(int sz) const {
    if (size() <= sz) return {};
    return FPS(*this, sz, size());
  }
  FPS operator<<(int sz) const {
    FPS ret(size() + sz, 0);
    for (int i = 0; i < size(); i++) ret[sz + i] = (*this)[i];
    return ret;
  }
  FPS operator+(const FPS &rhs) const { return FPS(*this) += rhs; }  // O(N)
  FPS operator-(const FPS &rhs) const { return FPS(*this) -= rhs; }  // O(N)
  FPS operator*(const FPS &rhs) const { return this->mul(rhs); }     // O(NlogN)
  FPS operator/(const FPS &rhs) const { return FPS(*this) /= rhs; }  // O(NlogN)
  FPS operator%(const FPS &rhs) const { return FPS(*this) %= rhs; }  // O(NlogN)
  FPS operator+(const R &v) const { return FPS(*this) += v; }        // O(1)
  FPS operator-(const R &v) const { return FPS(*this) -= v; }        // O(1)
  FPS operator*(const R &v) const { return FPS(*this) *= v; }        // O(N)
  FPS diff() const { return differentiation(); }                     // O(N)
  FPS inte() const { return integral(); }                            // O(N)
  FPS inv(int deg = -1) const { return inverse(deg); }               // O(NlogN)
  FPS log(int deg = -1) const { return logarithm(deg); }             // O(NlogN)
  FPS exp(int deg = -1) const { return exponent(deg); }              // O(NlogN)
  FPS sqrt(int deg = -1) const { return square_root(deg); }          // O(NlogN)
  FPS pow(u64 k, int deg = -1) const { return power(k, deg); }       // O(NlogN)

 public:
  vector<R> coefs;
  static R mod;
  static fast_div fast_mod;
  static constexpr R SIZE = 1 << 20;
  static R inve[SIZE];
  static void init(R m) {
    mod = m;
    fast_mod = fast_div(m);
    inve[1] = 1;
    for (int i = 2; i < SIZE; ++i)
      inve[i] = inve[mod % i] * (mod - mod / i) % fast_mod;
  }
};
using FPS = FormalPowerSeries;
R FPS::mod;
FPS::fast_div FPS::fast_mod;
R FPS::inve[];

// b[0] = a[0], b[1] = a[1], ..., b[N-1] = a[N-1]
// b[n] = c[0] * b[n-N] + c[1] * b[n-N+1] + ... + c[N-1] * b[n-1] (n >= N)
// calc b[k]
R kitamasa(const vector<R> &c, const vector<R> &a, u64 k, R mod) {
  assert(a.size() == c.size());
  int N = a.size();
  if (k < N) return a[k];
  if (FPS::mod != mod) FPS::init(mod);
  auto rem_pre = [&](const FPS &f, const FPS &b, const FPS &inv) {
    if (f.size() < b.size()) return f;
    int sq = f.size() - b.size() + 1;
    FPS q = FPS(FPS(f, sq) * FPS(inv, sq), sq);
    FPS::mul2(q, b, true);
    int s = max(q.size(), b.size()), size = 1;
    while (size < s) size <<= 1;
    FPS p = FPS::mul_crt(0, size);
    int mask = p.size() - 1;
    for (int i = 0; i < sq; i++) FPS::mod_sub(p[i & mask], f[i & mask]);
    FPS r = FPS(f, sq, f.size());
    for (int i = 0; i < (int)r.size(); i++)
      FPS::mod_sub(r[i], p[(sq + i) & mask]);
    return r;
  };
  FPS f(N + 1);
  f[0] = 1;
  for (int i = 0; i < N; i++) f[N - i] = mod - c[i];
  FPS r = FPS(vector<R>({1, 0}));
  FPS inv = f.inv(N);
  r = rem_pre(r, f, inv);
  u64 mask = (u64(1) << (63 - __builtin_clzll(k))) >> 1;
  while (mask) {
    r *= r;
    if (k & mask) r.push_back(0);
    r = rem_pre(r, f, inv);
    mask >>= 1;
  }
  R ret = 0;
  for (int i = 0; i < N; i++)
    FPS::mod_add(ret, FPS::mod_mul(r[N - i - 1], a[i]));
  return ret;
}

signed main() {
  cin.tie(0);
  ios::sync_with_stdio(0);
  int Q;
  cin >> Q;
  while (Q--) {
    long long n;
    cin >> n;
    cout << kitamasa({1, 1, 1, 1}, {0, 0, 0, 1}, n - 1, 17) << "\n";
  }
  cout << flush;
  return 0;
}
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結果

テストケース

ソースコード
