fps/FPS_Power_Projection.hpp

• View this file on GitHub
• Last update: 2024-06-24 02:01:01+09:00
• Include: #include "fps/FPS_Power_Projection.hpp"

Depends on

• fps/FPS_inv.hpp
• fps/FPS_pick_even_odd.hpp

Required by

• fps/FPS_inverse.hpp

Verified with

• test/fps/comp_inverse.test.cpp

Code

#pragma once
#include <vector>
#include <atcoder/convolution>

#include "../fps/FPS_pick_even_odd.hpp"
#include "../fps/FPS_inv.hpp"

namespace po167{
// n = |g|
// return 
// for i = 0, 1, ... , m - 1
//     [x ^ {n - 1}] g(x) f(x) ^ i
template<class T>
std::vector<T> Power_Projection(std::vector<T> g, std::vector<T> f, int m){
    int ind = (int)g.size() - 1;
    int n = 1;
    while(n < (int)g.size()) n *= 2;
    f.reserve(4 * n);
    g.reserve(4 * n);
    g.resize(n, 0);
    f.resize(n, 0);
    std::vector<T> hold_f(n), hold_g(n);
    // g(x) / (y - f(x))
    for (auto &x : f) x *= -1;
    int nk = n;
    T iz = (T)(1) / (T)(2 * n);
    while (nk != 1){
        hold_g = g;
        hold_f = f;
        // n -> 4 * n
        g.resize(4 * n);
        f.resize(4 * n);
        for (int i = n / nk - 1; i >= 0; i--){
            for (int j = nk - 1; j >= 0; j--){
                g[i * nk * 2 + j] = g[i * nk + j];
                if (i) g[i * nk + j] = 0;
                f[i * nk * 2 + j] = f[i * nk + j];
                if (i) f[i * nk + j] = 0;
            }
        }
        // tran
        atcoder::internal::butterfly(g);
        atcoder::internal::butterfly(f);
        for (int i = 0; i < 2 * n; i++){
            g[i * 2] *= f[i * 2 + 1];
            g[i * 2 + 1] *= f[i * 2];
            f[i * 2] *= f[i * 2 + 1];
            f[i * 2 + 1] = f[i * 2];
        }
        FPS_pick_even_odd(g, (ind & 1));
        FPS_pick_even_odd(f, 0);
        atcoder::internal::butterfly_inv(g);
        atcoder::internal::butterfly_inv(f);
        for (auto &x : g) x *= iz;
        for (auto &x : f) x *= iz;
        // y ^ nk
        for (int i = 0; i < n; i++){
            if ((ind + i + 1) & 1)
                g[n + (i / nk) * nk + (i & (nk - 1)) / 2] += hold_g[i];
            if ((i & 1) == 0)
                f[n + (i / nk) * nk + (i & (nk - 1)) / 2] += hold_f[i] * 2;
        }
        nk /= 2;
        for (int i = 0; i < n; i++){
            g[i] = g[(i / nk) * nk * 2 + (i & (nk - 1))];
            f[i] = f[(i / nk) * nk * 2 + (i & (nk - 1))];
        }
        g.resize(n);
        f.resize(n);
        ind /= 2;
    }
    f.push_back(1);
    std::reverse(g.begin(), g.end());
    std::reverse(f.begin(), f.end());
    g.resize(m);
    std::vector<T> ans = atcoder::convolution(g, FPS_inv(f, m));
    ans.resize(m);
    return ans;
}
}

#line 2 "fps/FPS_Power_Projection.hpp"
#include <vector>
#include <atcoder/convolution>

#line 3 "fps/FPS_pick_even_odd.hpp"

namespace po167{
// s.t |v| = 2 ^ s (no assert)
template<class T>
void FPS_pick_even_odd(std::vector<T> &v, int odd){
    int z = v.size() / 2;
    T half = (T)(1) / (T)(2);
    if (odd == 0){
        for (int i = 0; i < z; i++){
            v[i] = (v[i * 2] + v[i * 2 + 1]) * half;
        }
        v.resize(z);
    } else {
        T e = (T(atcoder::internal::primitive_root_constexpr(T::mod()))).pow(T::mod() / (2 * z));
        T ie = T(1) / e;
        std::vector<T> es = {half};
        while ((int)es.size() != z){
            std::vector<T> n_es((int)es.size() * 2);
            for (int i = 0; i < (int)es.size(); i++){
                n_es[i * 2] = (es[i]);
                n_es[i * 2 + 1] = (es[i] * ie);
            }
            ie *= ie;
            std::swap(n_es, es);
        }
        for (int i = 0; i < z; i ++){
            v[i] = (v[i * 2] - v[i * 2 + 1]) * es[i];
        }
        v.resize(z);
    }
}
}
#line 4 "fps/FPS_inv.hpp"

namespace po167{
// return 1 / f
template <class T>
std::vector<T> FPS_inv(std::vector<T> f, int len = -1){
    if (len == -1) len = f.size();
    assert(f[0] != 0);
    std::vector<T> g = {1 / f[0]};
    int s = 1;
    while(s < len){
        // g = 2g_s - f(g_s)^2 (mod x ^ (2 * s))
        // g = g - (fg - 1)g
        // (fg - 1) = 0 (mod x ^ (s))
        std::vector<T> n_g(s * 2, 0);
        std::vector<T> f_s(s * 2, 0);
        g.resize(s * 2);
        for (int i = 0; i < s * 2; i++){
            if (int(f.size()) > i) f_s[i] = f[i];
            n_g[i] = g[i];
        }
        atcoder::internal::butterfly(g);
        atcoder::internal::butterfly(f_s);
        for (int i = 0; i < s * 2; i++){
            f_s[i] *= g[i];
        }
        atcoder::internal::butterfly_inv(f_s);
        T iz = 1 / (T)(s * 2);
        for (int i = s; i < s * 2; i++){
            f_s[i] *= iz;
        }
        for (int i = 0; i < s; i++){
            f_s[i] = 0;
        }
        atcoder::internal::butterfly(f_s);
        for (int i = 0; i < s * 2; i++){
            f_s[i] *= g[i];
        }
        atcoder::internal::butterfly_inv(f_s);
        for (int i = s; i < s * 2; i++){
            n_g[i] -= f_s[i] * iz;
        }
        std::swap(n_g, g);
        s *= 2;
    }
    g.resize(len);
    return g;
}
}
#line 7 "fps/FPS_Power_Projection.hpp"

namespace po167{
// n = |g|
// return 
// for i = 0, 1, ... , m - 1
//     [x ^ {n - 1}] g(x) f(x) ^ i
template<class T>
std::vector<T> Power_Projection(std::vector<T> g, std::vector<T> f, int m){
    int ind = (int)g.size() - 1;
    int n = 1;
    while(n < (int)g.size()) n *= 2;
    f.reserve(4 * n);
    g.reserve(4 * n);
    g.resize(n, 0);
    f.resize(n, 0);
    std::vector<T> hold_f(n), hold_g(n);
    // g(x) / (y - f(x))
    for (auto &x : f) x *= -1;
    int nk = n;
    T iz = (T)(1) / (T)(2 * n);
    while (nk != 1){
        hold_g = g;
        hold_f = f;
        // n -> 4 * n
        g.resize(4 * n);
        f.resize(4 * n);
        for (int i = n / nk - 1; i >= 0; i--){
            for (int j = nk - 1; j >= 0; j--){
                g[i * nk * 2 + j] = g[i * nk + j];
                if (i) g[i * nk + j] = 0;
                f[i * nk * 2 + j] = f[i * nk + j];
                if (i) f[i * nk + j] = 0;
            }
        }
        // tran
        atcoder::internal::butterfly(g);
        atcoder::internal::butterfly(f);
        for (int i = 0; i < 2 * n; i++){
            g[i * 2] *= f[i * 2 + 1];
            g[i * 2 + 1] *= f[i * 2];
            f[i * 2] *= f[i * 2 + 1];
            f[i * 2 + 1] = f[i * 2];
        }
        FPS_pick_even_odd(g, (ind & 1));
        FPS_pick_even_odd(f, 0);
        atcoder::internal::butterfly_inv(g);
        atcoder::internal::butterfly_inv(f);
        for (auto &x : g) x *= iz;
        for (auto &x : f) x *= iz;
        // y ^ nk
        for (int i = 0; i < n; i++){
            if ((ind + i + 1) & 1)
                g[n + (i / nk) * nk + (i & (nk - 1)) / 2] += hold_g[i];
            if ((i & 1) == 0)
                f[n + (i / nk) * nk + (i & (nk - 1)) / 2] += hold_f[i] * 2;
        }
        nk /= 2;
        for (int i = 0; i < n; i++){
            g[i] = g[(i / nk) * nk * 2 + (i & (nk - 1))];
            f[i] = f[(i / nk) * nk * 2 + (i & (nk - 1))];
        }
        g.resize(n);
        f.resize(n);
        ind /= 2;
    }
    f.push_back(1);
    std::reverse(g.begin(), g.end());
    std::reverse(f.begin(), f.end());
    g.resize(m);
    std::vector<T> ans = atcoder::convolution(g, FPS_inv(f, m));
    ans.resize(m);
    return ans;
}
}

Back to top page