setfunc/transposed_sps_composition.hpp

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• Last update: 2024-04-19 02:20:22+09:00
• Include: #include "setfunc/transposed_sps_composition.hpp"

Depends on

• poly/convolution_naive.hpp
• setfunc/ranked_zeta.hpp
• setfunc/subset_convolution.hpp

Verified with

• test/library_checker/math/powproj_sps.test.cpp
• test_atcoder/abc253h.test.cpp

Code

#include "setfunc/subset_convolution.hpp"
#include "poly/convolution_naive.hpp"

// for fixed sps s, consider linear map F:a->b = subset-conv(a,s)
// given x, calculate transpose(F)(x)
template <typename mint, int LIM>
vc<mint> transposed_subset_convolution(vc<mint> s, vc<mint> x) {
  /*
  sum_{j}x_jb_j = sum_{i subset j}x_ja_is_{j-i} = sum_{i}y_ia_i.
  y_i = sum_{j supset i}x_js_{j-i}
  (rev y)_i = sum_{j subset i}(rev x)_js_{i-j}
  y = rev(conv(rev x), s)
  */
  reverse(all(x));
  x = subset_convolution<mint, LIM>(x, s);
  reverse(all(x));
  return x;
}

// for fixed sps s s.t. s[0] == 0.
// consider linear map F:f->t=f(s) for egf f.
// given x, calcuate transpose(F)(x)
// equivalent: calculate sum_i x_i(s^k/k!)_i for k=0,1,...,N
template <typename mint, int LIM>
vc<mint> transposed_sps_composition_egf(vc<mint>& s, vc<mint> x) {
  const int N = topbit(len(s));
  assert(len(s) == (1 << N) && len(x) == (1 << N) && s[0] == mint(0));
  vc<mint> y(N + 1);
  y[0] = x[0];
  auto& dp = x;
  FOR(i, N) {
    vc<mint> newdp(1 << (N - 1 - i));
    FOR(j, N - i) {
      vc<mint> a = {s.begin() + (1 << j), s.begin() + (2 << j)};
      vc<mint> b = {dp.begin() + (1 << j), dp.begin() + (2 << j)};
      b = transposed_subset_convolution<mint, LIM>(a, b);
      FOR(k, len(b)) newdp[k] += b[k];
    }
    swap(dp, newdp);
    y[1 + i] = dp[0];
  }
  return y;
}

// for fixed sps s s.t. s[0] == 0.
// consider linear map F:f->t=f(s) for polynomial f.
// given x, calcuate transpose(F)(x)
// equivalent: calculate sum_i x_i(s^k/k!)_i for k=0,1,...,M-1
template <typename mint, int LIM>
vc<mint> transposed_sps_composition_poly(vc<mint> s, vc<mint> x, int M) {
  const int N = topbit(len(s));
  assert(len(s) == (1 << N) && len(x) == (1 << N));
  mint c = s[0];
  s[0] -= c;
  x = transposed_sps_composition_egf<mint, LIM>(s, x);
  vc<mint> g(M);
  mint pow = 1;
  FOR(i, M) { g[i] = pow * fact_inv<mint>(i), pow *= c; }
  x = convolution_naive<mint>(x, g);
  x.resize(M);
  FOR(i, M) x[i] *= fact<mint>(i);
  return x;
}

#line 2 "setfunc/subset_convolution.hpp"

#line 2 "setfunc/ranked_zeta.hpp"

template <typename T, int LIM>
vc<array<T, LIM + 1>> ranked_zeta(const vc<T>& f) {
  int n = topbit(len(f));
  assert(n <= LIM);
  assert(len(f) == 1 << n);
  vc<array<T, LIM + 1>> Rf(1 << n);
  for (int s = 0; s < (1 << n); ++s) Rf[s][popcnt(s)] = f[s];
  for (int i = 0; i < n; ++i) {
    int w = 1 << i;
    for (int p = 0; p < (1 << n); p += 2 * w) {
      for (int s = p; s < p + w; ++s) {
        int t = s | 1 << i;
        for (int d = 0; d <= n; ++d) Rf[t][d] += Rf[s][d];
      }
    }
  }
  return Rf;
}

template <typename T, int LIM>
vc<T> ranked_mobius(vc<array<T, LIM + 1>>& Rf) {
  int n = topbit(len(Rf));
  assert(len(Rf) == 1 << n);
  for (int i = 0; i < n; ++i) {
    int w = 1 << i;
    for (int p = 0; p < (1 << n); p += 2 * w) {
      for (int s = p; s < p + w; ++s) {
        int t = s | 1 << i;
        for (int d = 0; d <= n; ++d) Rf[t][d] -= Rf[s][d];
      }
    }
  }
  vc<T> f(1 << n);
  for (int s = 0; s < (1 << n); ++s) f[s] = Rf[s][popcnt(s)];
  return f;
}
#line 4 "setfunc/subset_convolution.hpp"

template <typename T, int LIM = 20>
vc<T> subset_convolution_square(const vc<T>& A) {
  auto RA = ranked_zeta<T, LIM>(A);
  int n = topbit(len(RA));
  FOR(s, len(RA)) {
    auto& f = RA[s];
    FOR_R(d, n + 1) {
      T x = 0;
      FOR(i, d + 1) x += f[i] * f[d - i];
      f[d] = x;
    }
  }
  return ranked_mobius<T, LIM>(RA);
}

template <typename T, int LIM = 20>
vc<T> subset_convolution(const vc<T>& A, const vc<T>& B) {
  if (A == B) return subset_convolution_square(A);
  auto RA = ranked_zeta<T, LIM>(A);
  auto RB = ranked_zeta<T, LIM>(B);
  int n = topbit(len(RA));
  FOR(s, len(RA)) {
    auto &f = RA[s], &g = RB[s];
    FOR_R(d, n + 1) {
      T x = 0;
      FOR(i, d + 1) x += f[i] * g[d - i];
      f[d] = x;
    }
  }
  return ranked_mobius<T, LIM>(RA);
}
#line 2 "poly/convolution_naive.hpp"

template <class T, typename enable_if<!has_mod<T>::value>::type* = nullptr>
vc<T> convolution_naive(const vc<T>& a, const vc<T>& b) {
  int n = int(a.size()), m = int(b.size());
  if (n > m) return convolution_naive<T>(b, a);
  if (n == 0) return {};
  vector<T> ans(n + m - 1);
  FOR(i, n) FOR(j, m) ans[i + j] += a[i] * b[j];
  return ans;
}

template <class T, typename enable_if<has_mod<T>::value>::type* = nullptr>
vc<T> convolution_naive(const vc<T>& a, const vc<T>& b) {
  int n = int(a.size()), m = int(b.size());
  if (n > m) return convolution_naive<T>(b, a);
  if (n == 0) return {};
  vc<T> ans(n + m - 1);
  if (n <= 16 && (T::get_mod() < (1 << 30))) {
    for (int k = 0; k < n + m - 1; ++k) {
      int s = max(0, k - m + 1);
      int t = min(n, k + 1);
      u64 sm = 0;
      for (int i = s; i < t; ++i) { sm += u64(a[i].val) * (b[k - i].val); }
      ans[k] = sm;
    }
  } else {
    for (int k = 0; k < n + m - 1; ++k) {
      int s = max(0, k - m + 1);
      int t = min(n, k + 1);
      u128 sm = 0;
      for (int i = s; i < t; ++i) { sm += u64(a[i].val) * (b[k - i].val); }
      ans[k] = T::raw(sm % T::get_mod());
    }
  }
  return ans;
}
#line 3 "setfunc/transposed_sps_composition.hpp"

// for fixed sps s, consider linear map F:a->b = subset-conv(a,s)
// given x, calculate transpose(F)(x)
template <typename mint, int LIM>
vc<mint> transposed_subset_convolution(vc<mint> s, vc<mint> x) {
  /*
  sum_{j}x_jb_j = sum_{i subset j}x_ja_is_{j-i} = sum_{i}y_ia_i.
  y_i = sum_{j supset i}x_js_{j-i}
  (rev y)_i = sum_{j subset i}(rev x)_js_{i-j}
  y = rev(conv(rev x), s)
  */
  reverse(all(x));
  x = subset_convolution<mint, LIM>(x, s);
  reverse(all(x));
  return x;
}

// for fixed sps s s.t. s[0] == 0.
// consider linear map F:f->t=f(s) for egf f.
// given x, calcuate transpose(F)(x)
// equivalent: calculate sum_i x_i(s^k/k!)_i for k=0,1,...,N
template <typename mint, int LIM>
vc<mint> transposed_sps_composition_egf(vc<mint>& s, vc<mint> x) {
  const int N = topbit(len(s));
  assert(len(s) == (1 << N) && len(x) == (1 << N) && s[0] == mint(0));
  vc<mint> y(N + 1);
  y[0] = x[0];
  auto& dp = x;
  FOR(i, N) {
    vc<mint> newdp(1 << (N - 1 - i));
    FOR(j, N - i) {
      vc<mint> a = {s.begin() + (1 << j), s.begin() + (2 << j)};
      vc<mint> b = {dp.begin() + (1 << j), dp.begin() + (2 << j)};
      b = transposed_subset_convolution<mint, LIM>(a, b);
      FOR(k, len(b)) newdp[k] += b[k];
    }
    swap(dp, newdp);
    y[1 + i] = dp[0];
  }
  return y;
}

// for fixed sps s s.t. s[0] == 0.
// consider linear map F:f->t=f(s) for polynomial f.
// given x, calcuate transpose(F)(x)
// equivalent: calculate sum_i x_i(s^k/k!)_i for k=0,1,...,M-1
template <typename mint, int LIM>
vc<mint> transposed_sps_composition_poly(vc<mint> s, vc<mint> x, int M) {
  const int N = topbit(len(s));
  assert(len(s) == (1 << N) && len(x) == (1 << N));
  mint c = s[0];
  s[0] -= c;
  x = transposed_sps_composition_egf<mint, LIM>(s, x);
  vc<mint> g(M);
  mint pow = 1;
  FOR(i, M) { g[i] = pow * fact_inv<mint>(i), pow *= c; }
  x = convolution_naive<mint>(x, g);
  x.resize(M);
  FOR(i, M) x[i] *= fact<mint>(i);
  return x;
}

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