Formal Power Series (Sparse)
(fps/fps_sparse.hpp)

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• Last update: 2024-02-16 18:48:16+09:00
• Include: #include "fps/fps_sparse.hpp"

説明

疎な形式的べき級数に対する処理を集めたもの。

mul_sparse(std::vector f, std::vector g)

ナイーブな多項式積。$f$ の非ゼロの項を $N$ 個、$g$ の非ゼロの項を $M$ 個として $O(NM)$。

inv_sparse(std::vector f, int d)

$f^{-1} \mod x^d$ を求める。$f$ の非ゼロの項を $M$ 個として $O(NM)$

exp_sparse(std::vector f, int d)

$\exp(f) \mod x^d$ を求める。$f$ の非ゼロの項を $M$ 個として $O(NM)$

log_sparse(std::vector f, int d)

$\log{f} \mod x^d$ を求める。$f$ の非ゼロの項を $M$ 個として $O(NM)$

pow_sparse(std::vector f, long long k, int d)

$f^k \mod x^d$ を求める。$k$ は非負整数のみ正常に動作する。$f$ の非ゼロの項を $M$ 個として $O(NM)$

pow_sparse_1(std::vector f, long long k, int d)

$[x^0]f = 1$ であるような $f$ に対して $f^k \mod x^d$ を求める。 $k$ が負でも有理数でも動作する。 $f$ の非ゼロの項を $M$ 個として $O(NM)$

Depends on

• math/factorial.hpp
• テンプレート (template/template.hpp)

Verified with

• test/polynomial/Exp_of_FPS_Sparse.test.cpp
• test/polynomial/Inv_of_FPS_Sparse.test.cpp
• test/polynomial/Log_of_FPS_Sparse.test.cpp
• test/polynomial/Pow_of_FPS_Sparse.test.cpp

Code

#pragma once

#include "../math/factorial.hpp"
#include "../template/template.hpp"

namespace lib {

template <class mint>
std::vector<mint> mul_sparse(const std::vector<mint> &f,
                             const std::vector<mint> &g) {
    int n = f.size();
    int m = g.size();
    std::vector<std::pair<int, mint>> cf, cg;
    for (int i = 0; i < n; i++) {
        if (f[i] != 0) cf.emplace_back(i, f[i]);
    }
    for (int i = 0; i < m; i++) {
        if (g[i] != 0) cg.emplace_back(i, g[i]);
    }
    std::vector<mint> h(n + m - 1);
    for (auto [i, p] : cf) {
        for (auto [j, q] : cg) {
            h[i + j] += p * q;
        }
    }
    return h;
}

template <class mint>
std::vector<mint> inv_sparse(const std::vector<mint> &f, int d = -1) {
    assert(f[0] != 0);
    if (d < 0) {
        d = f.size();
    }
    std::vector<std::pair<int, mint>> ret;
    for (int i = 1; i < int(f.size()); i++) {
        if (f[i] != 0) {
            ret.emplace_back(i, f[i]);
        }
    }
    std::vector<mint> g(d);
    g[0] = f[0].inv();
    for (int i = 1; i < d; i++) {
        for (auto [k, p] : ret) {
            if (i - k < 0) break;
            g[i] -= g[i - k] * p;
        }
        g[i] *= g[0];
    }
    return g;
}

template <class mint>
std::vector<mint> exp_sparse(const std::vector<mint> &f, int d = -1) {
    int n = f.size();
    if (d < 0) d = n;
    std::vector<std::pair<int, mint>> ret;
    for (int i = 1; i < n; i++) {
        if (f[i] != 0) {
            ret.emplace_back(i - 1, f[i] * i);
        }
    }
    std::vector<mint> g(d);
    g[0] = 1;
    Binom<mint> binom(d);
    for (int i = 0; i < d - 1; i++) {
        for (auto [k, p] : ret) {
            if (i - k < 0) break;
            g[i + 1] += g[i - k] * p;
        }
        g[i + 1] *= binom.Inv(i + 1);
    }
    return g;
}

template <class mint>
std::vector<mint> log_sparse(const std::vector<mint> &f, int d = -1) {
    int n = f.size();
    if (d < 0) d = n;
    std::vector<mint> df(d);
    for (int i = 0; i < std::min(d, n - 1); i++) {
        df[i] = f[i + 1] * (i + 1);
    }
    auto dg = mul_sparse(df, inv_sparse(f));
    dg.resize(d);
    std::vector<mint> g(d);
    Binom<mint> binom(d);
    for (int i = 0; i < d - 1; i++) {
        g[i + 1] = dg[i] * binom.Inv(i + 1);
    }
    return g;
}

template <class mint>
std::vector<mint> pow_sparse_1(const std::vector<mint> &f, long long k,
                               int d = -1) {
    int n = f.size();
    assert(n == 0 || f[0] == 1);
    std::vector<std::pair<int, mint>> ret;
    for (int i = 1; i < n; i++) {
        if (f[i] != 0) ret.emplace_back(i, f[i]);
    }
    std::vector<mint> g(d);
    g[0] = 1;
    Binom<mint> binom(d);
    for (int i = 0; i < d - 1; i++) {
        for (const auto &[j, cf] : ret) {
            if (i + 1 - j < 0) break;
            g[i + 1] +=
                (mint(k) * mint(j) - mint(i - j + 1)) * cf * g[i + 1 - j];
        }
        g[i + 1] *= binom.Inv(i + 1);
    }
    return g;
}

template <class mint>
std::vector<mint> pow_sparse(const std::vector<mint> &f, long long k,
                             int d = -1) {
    int n = f.size();
    if (d < 0) d = n;
    assert(k >= 0);
    if (k == 0) {
        std::vector<mint> g(d);
        if (d > 0) g[0] = 1;
        return g;
    }
    for (int i = 0; i < n; i++) {
        if (f[i] != 0) {
            mint rev = f[i].inv();
            std::vector<mint> f2(n - i);
            for (int j = i; j < n; j++) {
                f2[j - i] = f[j] * rev;
            }
            f2 = pow_sparse_1(f2, k, d);
            mint fk = f[i].pow(k);
            std::vector<mint> g(d);
            for (int j = 0; j < int(f2.size()); j++) {
                if (j + i * k >= d) break;
                g[j + i * k] = f2[j] * fk;
            }
            return g;
        }
        if (i >= (d + k - 1) / k) break;
    }
    return std::vector<mint>(d);
}

}  // namespace lib

#line 2 "fps/fps_sparse.hpp"

#line 2 "math/factorial.hpp"

#line 2 "template/template.hpp"

#include <bits/stdc++.h>

#define rep(i, s, n) for (int i = (int)(s); i < (int)(n); i++)
#define rrep(i, s, n) for (int i = (int)(n)-1; i >= (int)(s); i--)
#define all(v) v.begin(), v.end()

using ll = long long;
using ld = long double;
using ull = unsigned long long;

template <typename T> bool chmin(T &a, const T &b) {
    if (a <= b) return false;
    a = b;
    return true;
}
template <typename T> bool chmax(T &a, const T &b) {
    if (a >= b) return false;
    a = b;
    return true;
}

namespace lib {

using namespace std;

}  // namespace lib

// using namespace lib;
#line 4 "math/factorial.hpp"

namespace lib {

template<typename T>
struct Binom{
    Binom(int lim = 300000){
        if (kaijo.empty()){
            kaijo = {1,1};
            kainv = {1,1};
        }
        extend(lim);
    }
    static T fact(int x) {
        if (x < 0) return T(0);
        return kaijo[x];
    }
    static T ifact(int x){
        if (x < 0) return T(0);
        return kainv[x];
    }
    static T C(int n, int r){
        if (n < 0 || n < r || r < 0) return T(0);
        return kaijo[n] * kainv[r] * kainv[n-r];
    }
    static T P(int n, int r){
        if (n < 0 || n < r || r < 0) return T(0);
        return kaijo[n] * kainv[n-r];
    }
    static T Inv(int n){
        assert(0 < n);
        return ifact(n) * fact(n-1);
    }
    T operator()(int n, int r){ return C(n,r); }
  private:
    static vector<T> kaijo, kainv;
    static void extend(int lim){
        if ((int)kaijo.size() > lim) return ;
        int pre = kaijo.size();
        kaijo.resize(lim+1);
        kainv.resize(lim+1);
        for (int i = pre; i <= lim; i++) kaijo[i] = kaijo[i-1] * T(i);
        kainv[lim] = kaijo[lim].inv();
        for (int i = lim-1; i >= pre; i--) kainv[i] = kainv[i+1] * T(i+1);
    }
};
template<typename T>
vector<T>Binom<T>::kaijo = vector<T>(2,T(1));
template<typename T>
vector<T>Binom<T>::kainv = vector<T>(2,T(1));

} // namespace lib
#line 5 "fps/fps_sparse.hpp"

namespace lib {

template <class mint>
std::vector<mint> mul_sparse(const std::vector<mint> &f,
                             const std::vector<mint> &g) {
    int n = f.size();
    int m = g.size();
    std::vector<std::pair<int, mint>> cf, cg;
    for (int i = 0; i < n; i++) {
        if (f[i] != 0) cf.emplace_back(i, f[i]);
    }
    for (int i = 0; i < m; i++) {
        if (g[i] != 0) cg.emplace_back(i, g[i]);
    }
    std::vector<mint> h(n + m - 1);
    for (auto [i, p] : cf) {
        for (auto [j, q] : cg) {
            h[i + j] += p * q;
        }
    }
    return h;
}

template <class mint>
std::vector<mint> inv_sparse(const std::vector<mint> &f, int d = -1) {
    assert(f[0] != 0);
    if (d < 0) {
        d = f.size();
    }
    std::vector<std::pair<int, mint>> ret;
    for (int i = 1; i < int(f.size()); i++) {
        if (f[i] != 0) {
            ret.emplace_back(i, f[i]);
        }
    }
    std::vector<mint> g(d);
    g[0] = f[0].inv();
    for (int i = 1; i < d; i++) {
        for (auto [k, p] : ret) {
            if (i - k < 0) break;
            g[i] -= g[i - k] * p;
        }
        g[i] *= g[0];
    }
    return g;
}

template <class mint>
std::vector<mint> exp_sparse(const std::vector<mint> &f, int d = -1) {
    int n = f.size();
    if (d < 0) d = n;
    std::vector<std::pair<int, mint>> ret;
    for (int i = 1; i < n; i++) {
        if (f[i] != 0) {
            ret.emplace_back(i - 1, f[i] * i);
        }
    }
    std::vector<mint> g(d);
    g[0] = 1;
    Binom<mint> binom(d);
    for (int i = 0; i < d - 1; i++) {
        for (auto [k, p] : ret) {
            if (i - k < 0) break;
            g[i + 1] += g[i - k] * p;
        }
        g[i + 1] *= binom.Inv(i + 1);
    }
    return g;
}

template <class mint>
std::vector<mint> log_sparse(const std::vector<mint> &f, int d = -1) {
    int n = f.size();
    if (d < 0) d = n;
    std::vector<mint> df(d);
    for (int i = 0; i < std::min(d, n - 1); i++) {
        df[i] = f[i + 1] * (i + 1);
    }
    auto dg = mul_sparse(df, inv_sparse(f));
    dg.resize(d);
    std::vector<mint> g(d);
    Binom<mint> binom(d);
    for (int i = 0; i < d - 1; i++) {
        g[i + 1] = dg[i] * binom.Inv(i + 1);
    }
    return g;
}

template <class mint>
std::vector<mint> pow_sparse_1(const std::vector<mint> &f, long long k,
                               int d = -1) {
    int n = f.size();
    assert(n == 0 || f[0] == 1);
    std::vector<std::pair<int, mint>> ret;
    for (int i = 1; i < n; i++) {
        if (f[i] != 0) ret.emplace_back(i, f[i]);
    }
    std::vector<mint> g(d);
    g[0] = 1;
    Binom<mint> binom(d);
    for (int i = 0; i < d - 1; i++) {
        for (const auto &[j, cf] : ret) {
            if (i + 1 - j < 0) break;
            g[i + 1] +=
                (mint(k) * mint(j) - mint(i - j + 1)) * cf * g[i + 1 - j];
        }
        g[i + 1] *= binom.Inv(i + 1);
    }
    return g;
}

template <class mint>
std::vector<mint> pow_sparse(const std::vector<mint> &f, long long k,
                             int d = -1) {
    int n = f.size();
    if (d < 0) d = n;
    assert(k >= 0);
    if (k == 0) {
        std::vector<mint> g(d);
        if (d > 0) g[0] = 1;
        return g;
    }
    for (int i = 0; i < n; i++) {
        if (f[i] != 0) {
            mint rev = f[i].inv();
            std::vector<mint> f2(n - i);
            for (int j = i; j < n; j++) {
                f2[j - i] = f[j] * rev;
            }
            f2 = pow_sparse_1(f2, k, d);
            mint fk = f[i].pow(k);
            std::vector<mint> g(d);
            for (int j = 0; j < int(f2.size()); j++) {
                if (j + i * k >= d) break;
                g[j + i * k] = f2[j] * fk;
            }
            return g;
        }
        if (i >= (d + k - 1) / k) break;
    }
    return std::vector<mint>(d);
}

}  // namespace lib

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