sps_in_poly.hpp

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• Last update: 2025-01-19 15:28:01+08:00
• Include: #include "sps_in_poly.hpp"
• Link: https://codeforces.com/blog/entry/92183

Depends on

• subset_conv.hpp

Verified with

• test/set_power_series/polynomial_composite_set_power_series.0.test.cpp

Code

#pragma once

#include "subset_conv.hpp"
#include <algorithm>
#include <cassert>
#include <vector>

// returns F(G)
// requires deg(F)<=n, G(0)=0
// see:
// [1]: Elegia. Optimal Algorithm on Polynomial Composite Set Power Series.
//      https://codeforces.com/blog/entry/92183
template<typename Tp>
inline std::vector<Tp> sps_in_egf(const std::vector<Tp> &F, const std::vector<Tp> &G) {
    const int N = (int)F.size() - 1;
    assert((int)G.size() == (1 << N));
    assert(G[0] == 0);

    auto conv_ranked = [](const auto &rankedA, const auto &rankedB, int LogN, auto res) {
        const int N = (1 << LogN);
        std::vector<int> map(LogN + 1);
        for (int i = 0; i <= LogN; ++i) map[i] = (i & 1) ? map[i / 2] : i / 2;
        std::vector rankedAB(LogN / 2 + 1, std::vector<Tp>(N));
        for (int i = 0; i <= LogN; ++i)
            for (int j = 0; i + j <= LogN; ++j)
                for (int k = (1 << j) - 1; k < N; ++k)
                    rankedAB[map[i + j]][k] += rankedA[i][k] * rankedB[j][k];
        for (int i = 0; i <= LogN / 2; ++i) subset_moebius(rankedAB[i]);
        for (int i = 0; i < N; ++i) res[i] = rankedAB[map[__builtin_popcount(i)]][i];
    };

    std::vector<std::vector<std::vector<Tp>>> rankedG;
    std::vector res = {F[N]};
    for (int i = 0; i < N; ++i) {
        auto &&rankedGi = rankedG.emplace_back(
            to_ranked(std::vector(G.begin() + (1 << i), G.begin() + (2 << i))));
        auto rankedRes = to_ranked(res);
        for (int j = 0; j <= i; ++j) {
            subset_zeta(rankedGi[j]);
            subset_zeta(rankedRes[j]);
        }
        res.resize(1 << (i + 1));
        res[0] = F[N - (i + 1)];
        for (int j = 0; j <= i; ++j) conv_ranked(rankedG[j], rankedRes, j, res.begin() + (1 << j));
    }

    return res;
}

template<typename Tp> inline std::vector<Tp> sps_in_poly(std::vector<Tp> F, std::vector<Tp> G) {
    const int N = G.size();
    int LogN    = 0;
    while ((1 << LogN) != N) ++LogN;

    if (G[0] != 0) {
        std::vector<Tp> bin(LogN + 1), pw(F.size() + 1), FF(LogN + 1);
        pw[0] = 1;
        for (int i = 1; i < (int)pw.size(); ++i) pw[i] = pw[i - 1] * G[0];
        G[0]   = 0;
        bin[0] = 1;
        for (int i = 0; i < (int)F.size(); ++i) {
            for (int j = 0; j <= std::min(LogN, i); ++j) FF[j] += F[i] * bin[j] * pw[i - j];
            for (int j = LogN; j > 0; --j) bin[j] += bin[j - 1];
        }
        FF.swap(F);
    }

    F.resize(LogN + 1);
    Tp c = 1;                                       // factorial
    for (int i = 1; i <= LogN; ++i) F[i] *= c *= i; // to EGF
    return sps_in_egf(F, G);
}

#line 2 "sps_in_poly.hpp"

#line 2 "subset_conv.hpp"

#include <cassert>
#include <vector>

template<typename Tp> inline std::vector<std::vector<Tp>> to_ranked(const std::vector<Tp> &A) {
    const int N = A.size();
    int LogN    = 0;
    while ((1 << LogN) != N) ++LogN;
    std::vector res(LogN + 1, std::vector<Tp>(N));
    for (int i = 0; i < N; ++i) res[__builtin_popcount(i)][i] = A[i];
    return res;
}

template<typename Tp> inline std::vector<Tp> from_ranked(const std::vector<std::vector<Tp>> &A) {
    const int N = A[0].size();
    std::vector<Tp> res(N);
    for (int i = 0; i < N; ++i) res[i] = A[__builtin_popcount(i)][i];
    return res;
}

template<typename Iterator> inline void subset_zeta_n(Iterator a, int n) {
    assert((n & (n - 1)) == 0);
    for (int i = 2; i <= n; i *= 2)
        for (int j = 0; j < n; j += i)
            for (int k = j; k < j + i / 2; ++k) a[k + i / 2] += a[k];
}

template<typename Tp> inline void subset_zeta(std::vector<Tp> &a) {
    subset_zeta_n(a.begin(), a.size());
}

template<typename Iterator> inline void subset_moebius_n(Iterator a, int n) {
    assert((n & (n - 1)) == 0);
    for (int i = 2; i <= n; i *= 2)
        for (int j = 0; j < n; j += i)
            for (int k = j; k < j + i / 2; ++k) a[k + i / 2] -= a[k];
}

template<typename Tp> inline void subset_moebius(std::vector<Tp> &a) {
    subset_moebius_n(a.begin(), a.size());
}

template<typename Tp>
inline std::vector<Tp> subset_convolution(const std::vector<Tp> &A, const std::vector<Tp> &B) {
    assert(A.size() == B.size());
    const int N = A.size();
    int LogN    = 0;
    while ((1 << LogN) != N) ++LogN;
    auto rankedA = to_ranked(A);
    auto rankedB = to_ranked(B);

    for (int i = 0; i <= LogN; ++i) {
        subset_zeta(rankedA[i]);
        subset_zeta(rankedB[i]);
    }

    // see: https://codeforces.com/blog/entry/126418
    // see: https://oeis.org/A025480
    std::vector<int> map(LogN + 1);
    for (int i = 0; i <= LogN; ++i) map[i] = (i & 1) ? map[i / 2] : i / 2;

    std::vector rankedAB(LogN / 2 + 1, std::vector<Tp>(N));
    for (int i = 0; i <= LogN; ++i)
        for (int j = 0; i + j <= LogN; ++j)
            for (int k = (1 << j) - 1; k < N; ++k)
                rankedAB[map[i + j]][k] += rankedA[i][k] * rankedB[j][k];

    for (int i = 0; i <= LogN / 2; ++i) subset_moebius(rankedAB[i]);

    std::vector<Tp> res(N);
    for (int i = 0; i < N; ++i) res[i] = rankedAB[map[__builtin_popcount(i)]][i];
    return res;
}
#line 4 "sps_in_poly.hpp"
#include <algorithm>
#line 7 "sps_in_poly.hpp"

// returns F(G)
// requires deg(F)<=n, G(0)=0
// see:
// [1]: Elegia. Optimal Algorithm on Polynomial Composite Set Power Series.
//      https://codeforces.com/blog/entry/92183
template<typename Tp>
inline std::vector<Tp> sps_in_egf(const std::vector<Tp> &F, const std::vector<Tp> &G) {
    const int N = (int)F.size() - 1;
    assert((int)G.size() == (1 << N));
    assert(G[0] == 0);

    auto conv_ranked = [](const auto &rankedA, const auto &rankedB, int LogN, auto res) {
        const int N = (1 << LogN);
        std::vector<int> map(LogN + 1);
        for (int i = 0; i <= LogN; ++i) map[i] = (i & 1) ? map[i / 2] : i / 2;
        std::vector rankedAB(LogN / 2 + 1, std::vector<Tp>(N));
        for (int i = 0; i <= LogN; ++i)
            for (int j = 0; i + j <= LogN; ++j)
                for (int k = (1 << j) - 1; k < N; ++k)
                    rankedAB[map[i + j]][k] += rankedA[i][k] * rankedB[j][k];
        for (int i = 0; i <= LogN / 2; ++i) subset_moebius(rankedAB[i]);
        for (int i = 0; i < N; ++i) res[i] = rankedAB[map[__builtin_popcount(i)]][i];
    };

    std::vector<std::vector<std::vector<Tp>>> rankedG;
    std::vector res = {F[N]};
    for (int i = 0; i < N; ++i) {
        auto &&rankedGi = rankedG.emplace_back(
            to_ranked(std::vector(G.begin() + (1 << i), G.begin() + (2 << i))));
        auto rankedRes = to_ranked(res);
        for (int j = 0; j <= i; ++j) {
            subset_zeta(rankedGi[j]);
            subset_zeta(rankedRes[j]);
        }
        res.resize(1 << (i + 1));
        res[0] = F[N - (i + 1)];
        for (int j = 0; j <= i; ++j) conv_ranked(rankedG[j], rankedRes, j, res.begin() + (1 << j));
    }

    return res;
}

template<typename Tp> inline std::vector<Tp> sps_in_poly(std::vector<Tp> F, std::vector<Tp> G) {
    const int N = G.size();
    int LogN    = 0;
    while ((1 << LogN) != N) ++LogN;

    if (G[0] != 0) {
        std::vector<Tp> bin(LogN + 1), pw(F.size() + 1), FF(LogN + 1);
        pw[0] = 1;
        for (int i = 1; i < (int)pw.size(); ++i) pw[i] = pw[i - 1] * G[0];
        G[0]   = 0;
        bin[0] = 1;
        for (int i = 0; i < (int)F.size(); ++i) {
            for (int j = 0; j <= std::min(LogN, i); ++j) FF[j] += F[i] * bin[j] * pw[i - j];
            for (int j = LogN; j > 0; --j) bin[j] += bin[j - 1];
        }
        FF.swap(F);
    }

    F.resize(LogN + 1);
    Tp c = 1;                                       // factorial
    for (int i = 1; i <= LogN; ++i) F[i] *= c *= i; // to EGF
    return sps_in_egf(F, G);
}

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