seq/reeds_sloane.hpp

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• Last update: 2026-06-23 23:59:02+09:00
• Include: #include "seq/reeds_sloane.hpp"

Depends on

• mod/mod_inv.hpp
• mod/mongomery_modint.hpp
• nt/factor.hpp
• nt/primetest.hpp
• random/base.hpp

Verified with

• test/1_mytest/reeds_sloane.test.cpp

Code

#include "nt/factor.hpp"
#include "mod/mod_inv.hpp"

vc<int> Reeds_Sloane_Prime_Power(vc<int> S, int p, int e) {
  int N = len(S);
  if (N == 0) return {1};
  int M = 1;
  FOR(e) M *= p;

  using mint = Dynamic_Modint<20260623>;
  mint::set_mod(M);

  auto decompose = [&](mint x) -> pair<mint, int> {
    // x = tp^u
    int t = x.val, u = 0;
    if (t == 0) return {1, e};
    while (t % p == 0) t /= p, ++u;
    return {t, u};
  };

  using poly = vc<mint>;
  vc<poly> Q(e);
  vc<int> L(e);

  vc<poly> B(e);
  vc<int> LB(e);
  vc<int> nB(e);
  vc<mint> tB(e);

  mint pw = 1;
  for (int i = 0; i < e; ++i, pw *= p) {
    Q[i] = {pw};
    L[i] = 0;
    nB[i] = -1;
  }

  for (int n = 0; n < N; ++n) {
    // delta=tp^u
    vc<mint> t(e);
    vc<int> u(e);
    FOR(i, e) {
      mint delta = 0;
      assert(len(Q[i]) <= 1 + n);
      FOR(k, len(Q[i])) delta += Q[i][k] * S[n - k];
      tie(t[i], u[i]) = decompose(delta);
    }

    vc<poly> Q_next = Q;
    vc<int> L_next = L;

    FOR(i, e) {
      if (u[i] == e) continue;

      int j = e - 1 - u[i];
      if (nB[j] == -1) {
        Q_next[i].resize(n + 2);
        L_next[i] = n + 1;
      } else {
        L_next[i] = max(L[i], LB[j] + n - nB[j]);
        Q_next[i].resize(L_next[i] + 1);
        mint c = t[i] / tB[j];
        FOR(k, len(B[j])) Q_next[i][k + n - nB[j]] -= c * B[j][k];
      }
    }
    FOR(i, e) {
      if (L[i] < L_next[i]) {
        int j = e - 1 - u[i];
        B[i] = Q[j];
        LB[i] = L[j];
        nB[i] = n;
        tB[i] = t[j];
      }
    }
    swap(Q, Q_next);
    swap(L, L_next);
  }
  vc<int> res;
  for (auto& x : Q[0]) res.eb(x.val);
  assert(len(res) == L[0] + 1);
  return res;
}

/*
return {P(x),Q(x)} such that
S(x)=P(x)/Q(x) mod x^N, [x^0]Q=1
minimize L=max(deg(P)+1,deg(Q))
*/
template <typename mint>
pair<vc<mint>, vc<mint>> Reeds_Sloane(vc<mint> S, vc<pair<ll, int>> pfs = {}) {
  int mod = mint::get_mod();
  if (mod > 1 && pfs.empty()) {
    pfs = factor(mod);
  }
  {
    int check = mod;
    for (auto [p, e] : pfs) {
      FOR(e) {
        assert(check % p == 0);
        check /= p;
      }
    }
    assert(check == 1);
  }

  if (mod == 1) return {{}, {1}};

  int n = len(pfs);
  vi coef(n);
  FOR(i, n) {
    auto [p, e] = pfs[i];
    int a = 1, b = mod;
    FOR(e) a *= p, b /= p;
    ll c = mod_inv(b, a);
    coef[i] = c * b % mod;
  }
  vc<mint> Q;
  FOR(k, n) {
    auto [p, e] = pfs[k];
    int a = 1;
    FOR(e) a *= p;
    vc<int> T(len(S));
    FOR(i, len(S)) T[i] = (S[i].val) % a;
    auto Qk = Reeds_Sloane_Prime_Power(T, p, e);
    if (len(Q) < len(Qk)) Q.resize(len(Qk));
    FOR(i, len(Qk)) Q[i] += Qk[i] * coef[k];
  }
  vc<mint> P(len(Q) - 1);
  FOR(i, len(P)) FOR(j, i + 1) P[i] += Q[j] * S[i - j];
  return {P, Q};
}

#line 2 "nt/factor.hpp"

#line 2 "random/base.hpp"

u64 RNG_64() {
  static u64 x_ = u64(chrono::duration_cast<chrono::nanoseconds>(chrono::high_resolution_clock::now().time_since_epoch()).count()) * 10150724397891781847ULL;
  x_ ^= x_ << 7;
  return x_ ^= x_ >> 9;
}

u64 RNG(u64 lim) { return RNG_64() % lim; }

ll RNG(ll l, ll r) { return l + RNG_64() % (r - l); }
#line 2 "mod/mongomery_modint.hpp"

// odd mod.
// x の代わりに rx を持つ
template <int id, typename U1, typename U2>
struct Mongomery_modint {
  using mint = Mongomery_modint;
  inline static U1 m, r, n2;
  static constexpr int W = numeric_limits<U1>::digits;

  static void set_mod(U1 mod) {
    assert(mod & 1 && mod <= U1(1) << (W - 2));
    m = mod, n2 = -U2(m) % m, r = m;
    FOR(5) r *= 2 - m * r;
    r = -r;
    assert(r * m == U1(-1));
  }
  static U1 reduce(U2 b) { return (b + U2(U1(b) * r) * m) >> W; }

  U1 x;
  Mongomery_modint() : x(0) {}
  Mongomery_modint(U1 x) : x(reduce(U2(x) * n2)){};
  U1 val() const {
    U1 y = reduce(x);
    return y >= m ? y - m : y;
  }
  mint &operator+=(mint y) {
    x = ((x += y.x) >= m ? x - m : x);
    return *this;
  }
  mint &operator-=(mint y) {
    x -= (x >= y.x ? y.x : y.x - m);
    return *this;
  }
  mint &operator*=(mint y) {
    x = reduce(U2(x) * y.x);
    return *this;
  }
  mint operator+(mint y) const { return mint(*this) += y; }
  mint operator-(mint y) const { return mint(*this) -= y; }
  mint operator*(mint y) const { return mint(*this) *= y; }
  bool operator==(mint y) const {
    return (x >= m ? x - m : x) == (y.x >= m ? y.x - m : y.x);
  }
  bool operator!=(mint y) const { return not operator==(y); }
  mint pow(ll n) const {
    assert(n >= 0);
    mint y = 1, z = *this;
    for (; n; n >>= 1, z *= z)
      if (n & 1) y *= z;
    return y;
  }
};

template <int id>
using Mongomery_modint_32 = Mongomery_modint<id, u32, u64>;
template <int id>
using Mongomery_modint_64 = Mongomery_modint<id, u64, u128>;
#line 3 "nt/primetest.hpp"

bool primetest(const u64 x) {
  assert(x < u64(1) << 62);
  if (x == 2 or x == 3 or x == 5 or x == 7) return true;
  if (x % 2 == 0 or x % 3 == 0 or x % 5 == 0 or x % 7 == 0) return false;
  if (x < 121) return x > 1;
  const u64 d = (x - 1) >> lowbit(x - 1);

  using mint = Mongomery_modint_64<202311020>;

  mint::set_mod(x);
  const mint one(u64(1)), minus_one(x - 1);
  auto ok = [&](u64 a) -> bool {
    auto y = mint(a).pow(d);
    u64 t = d;
    while (y != one && y != minus_one && t != x - 1) y *= y, t <<= 1;
    if (y != minus_one && t % 2 == 0) return false;
    return true;
  };
  if (x < (u64(1) << 32)) {
    for (u64 a: {2, 7, 61})
      if (!ok(a)) return false;
  } else {
    for (u64 a: {2, 325, 9375, 28178, 450775, 9780504, 1795265022}) {
      if (!ok(a)) return false;
    }
  }
  return true;
}
#line 5 "nt/factor.hpp"

template <typename mint>
ll rho(ll n, ll c) {
  assert(n > 1);
  const mint cc(c);
  auto f = [&](mint x) { return x * x + cc; };
  mint x = 1, y = 2, z = 1, q = 1;
  ll g = 1;
  const ll m = 1LL << (__lg(n) / 5);
  for (ll r = 1; g == 1; r <<= 1) {
    x = y;
    FOR(r) y = f(y);
    for (ll k = 0; k < r && g == 1; k += m) {
      z = y;
      FOR(min(m, r - k)) y = f(y), q *= x - y;
      g = gcd(q.val(), n);
    }
  }
  if (g == n) do {
      z = f(z);
      g = gcd((x - z).val(), n);
    } while (g == 1);
  return g;
}

ll find_prime_factor(ll n) {
  assert(n > 1);
  if (primetest(n)) return n;
  FOR(100) {
    ll m = 0;
    if (n < (1 << 30)) {
      using mint = Mongomery_modint_32<20231025>;
      mint::set_mod(n);
      m = rho<mint>(n, RNG(0, n));
    } else {
      using mint = Mongomery_modint_64<20231025>;
      mint::set_mod(n);
      m = rho<mint>(n, RNG(0, n));
    }
    if (primetest(m)) return m;
    n = m;
  }
  assert(0);
  return -1;
}

// ソートしてくれる
vc<pair<ll, int>> factor(ll n) {
  assert(n >= 1);
  vc<pair<ll, int>> pf;
  FOR(p, 2, 100) {
    if (p * p > n) break;
    if (n % p == 0) {
      ll e = 0;
      do { n /= p, e += 1; } while (n % p == 0);
      pf.eb(p, e);
    }
  }
  while (n > 1) {
    ll p = find_prime_factor(n);
    ll e = 0;
    do { n /= p, e += 1; } while (n % p == 0);
    pf.eb(p, e);
  }
  sort(all(pf));
  return pf;
}

vc<pair<ll, int>> factor_by_lpf(ll n, vc<int>& lpf) {
  vc<pair<ll, int>> res;
  while (n > 1) {
    int p = lpf[n];
    int e = 0;
    while (n % p == 0) {
      n /= p;
      ++e;
    }
    res.eb(p, e);
  }
  return res;
}
#line 2 "mod/mod_inv.hpp"

// long でも大丈夫

// (val * x - 1) が mod の倍数になるようにする

// 特に mod=0 なら x=0 が満たす

ll mod_inv(ll val, ll mod) {
  if (mod == 0) return 0;
  mod = abs(mod);
  val %= mod;
  if (val < 0) val += mod;
  ll a = val, b = mod, u = 1, v = 0, t;
  while (b > 0) {
    t = a / b;
    swap(a -= t * b, b), swap(u -= t * v, v);
  }
  if (u < 0) u += mod;
  return u;
}
#line 3 "seq/reeds_sloane.hpp"

vc<int> Reeds_Sloane_Prime_Power(vc<int> S, int p, int e) {
  int N = len(S);
  if (N == 0) return {1};
  int M = 1;
  FOR(e) M *= p;

  using mint = Dynamic_Modint<20260623>;
  mint::set_mod(M);

  auto decompose = [&](mint x) -> pair<mint, int> {
    // x = tp^u
    int t = x.val, u = 0;
    if (t == 0) return {1, e};
    while (t % p == 0) t /= p, ++u;
    return {t, u};
  };

  using poly = vc<mint>;
  vc<poly> Q(e);
  vc<int> L(e);

  vc<poly> B(e);
  vc<int> LB(e);
  vc<int> nB(e);
  vc<mint> tB(e);

  mint pw = 1;
  for (int i = 0; i < e; ++i, pw *= p) {
    Q[i] = {pw};
    L[i] = 0;
    nB[i] = -1;
  }

  for (int n = 0; n < N; ++n) {
    // delta=tp^u
    vc<mint> t(e);
    vc<int> u(e);
    FOR(i, e) {
      mint delta = 0;
      assert(len(Q[i]) <= 1 + n);
      FOR(k, len(Q[i])) delta += Q[i][k] * S[n - k];
      tie(t[i], u[i]) = decompose(delta);
    }

    vc<poly> Q_next = Q;
    vc<int> L_next = L;

    FOR(i, e) {
      if (u[i] == e) continue;

      int j = e - 1 - u[i];
      if (nB[j] == -1) {
        Q_next[i].resize(n + 2);
        L_next[i] = n + 1;
      } else {
        L_next[i] = max(L[i], LB[j] + n - nB[j]);
        Q_next[i].resize(L_next[i] + 1);
        mint c = t[i] / tB[j];
        FOR(k, len(B[j])) Q_next[i][k + n - nB[j]] -= c * B[j][k];
      }
    }
    FOR(i, e) {
      if (L[i] < L_next[i]) {
        int j = e - 1 - u[i];
        B[i] = Q[j];
        LB[i] = L[j];
        nB[i] = n;
        tB[i] = t[j];
      }
    }
    swap(Q, Q_next);
    swap(L, L_next);
  }
  vc<int> res;
  for (auto& x : Q[0]) res.eb(x.val);
  assert(len(res) == L[0] + 1);
  return res;
}

/*
return {P(x),Q(x)} such that
S(x)=P(x)/Q(x) mod x^N, [x^0]Q=1
minimize L=max(deg(P)+1,deg(Q))
*/
template <typename mint>
pair<vc<mint>, vc<mint>> Reeds_Sloane(vc<mint> S, vc<pair<ll, int>> pfs = {}) {
  int mod = mint::get_mod();
  if (mod > 1 && pfs.empty()) {
    pfs = factor(mod);
  }
  {
    int check = mod;
    for (auto [p, e] : pfs) {
      FOR(e) {
        assert(check % p == 0);
        check /= p;
      }
    }
    assert(check == 1);
  }

  if (mod == 1) return {{}, {1}};

  int n = len(pfs);
  vi coef(n);
  FOR(i, n) {
    auto [p, e] = pfs[i];
    int a = 1, b = mod;
    FOR(e) a *= p, b /= p;
    ll c = mod_inv(b, a);
    coef[i] = c * b % mod;
  }
  vc<mint> Q;
  FOR(k, n) {
    auto [p, e] = pfs[k];
    int a = 1;
    FOR(e) a *= p;
    vc<int> T(len(S));
    FOR(i, len(S)) T[i] = (S[i].val) % a;
    auto Qk = Reeds_Sloane_Prime_Power(T, p, e);
    if (len(Q) < len(Qk)) Q.resize(len(Qk));
    FOR(i, len(Qk)) Q[i] += Qk[i] * coef[k];
  }
  vc<mint> P(len(Q) - 1);
  FOR(i, len(P)) FOR(j, i + 1) P[i] += Q[j] * S[i - j];
  return {P, Q};
}

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