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‎docs/fps/sparse.md‎

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sparse な FPS についての各種 FPS 演算．
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与えられた FPS で非ゼロの項が $K$ 項で，$N$ 次まで求めるとき $O(NK)$ 時間．
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以下の等式から係数についての sparse な漸化式が得られる．
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- inv : $fg=1$
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- exp : $g'=f'g$
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- log : $fg'=f'$
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- pow : $fg'=Mf'g$
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- sqrt : $f=g^2$

‎fps/sparse.hpp‎

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#pragma once
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#include "modint/factorial.hpp"
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#include "modint/mod-sqrt.hpp"
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#include "fps/formal-power-series.hpp"
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namespace FPSSparse {
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template <class mint>
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FormalPowerSeries<mint> inv(map<int, mint> f, int n) {
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assert(f[0] != 0);
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if (n == 0) return {};
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mint c = f[0].inv();
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FormalPowerSeries<mint> g(n);
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g[0] = c;
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for (int i = 1; i < n; i++) {
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for (auto [j, v] : f)
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if (j <= i) g[i] -= v * g[i - j];
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g[i] *= c;
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}
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return g;
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}
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template <class mint>
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FormalPowerSeries<mint> exp(map<int, mint> f, int n) {
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assert(f[0] == 0);
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if (n == 0) return {};
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FormalPowerSeries<mint> g(n);
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g[0] = 1;
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for (int i = 1; i < n; i++) {
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for (auto [j, v] : f)
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if (j <= i) g[i] += j * v * g[i - j];
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g[i] *= Factorial<mint>::inv(i);
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}
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return g;
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}
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template <class mint>
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FormalPowerSeries<mint> log(map<int, mint> f, int n) {
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assert(f[0] == 1);
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if (n == 0) return {};
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FormalPowerSeries<mint> g(n);
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g[0] = 0;
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for (auto [j, v] : f)
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if (0 < j && j < n) g[j - 1] = v * j;
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for (int i = 1; i < n; i++) {
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for (auto [j, v] : f)
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if (0 < j && j <= i) g[i] -= v * g[i - j];
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}
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for (int i = n - 1; i > 0; i--) g[i] = g[i - 1] * Factorial<mint>::inv(i);
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g[0] = 0;
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return g;
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}
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template <class mint>
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FormalPowerSeries<mint> pow(map<int, mint> f, long long m, int n) {
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if (n == 0) return {};
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FormalPowerSeries<mint> g(n, 0);
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if (m == 0) {
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g[0] = 1;
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return g;
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}
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if (f[0] == 0) {
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if (m >= n) return g;
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int s = 1;
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while (s < n && f[s] == 0) s++;
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if (s * m >= n) return g;
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map<int, mint> f1;
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for (auto [i, v] : f) f1[i - s] = v;
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auto g1 = pow(f1, m, int(n - s * m));
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copy(g1.begin(), g1.end(), g.begin() + int(s * m));
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return g;
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}
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g[0] = f[0].pow(m);
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mint c = f[0].inv();
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for (int i = 1; i < n; i++) {
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for (auto [j, v] : f) {
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if (0 < j && j <= i) g[i] += j * f[j] * g[i - j] * m;
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if (0 < j && j < i) g[i] -= f[j] * (i - j) * g[i - j];
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}
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g[i] *= c * Factorial<mint>::inv(i);
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}
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return g;
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}
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template <class mint>
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FormalPowerSeries<mint> sqrt(map<int, mint> f, int n) {
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if (n == 0) return {};
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if (f.empty()) return FormalPowerSeries<mint>(n, 0);
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FormalPowerSeries<mint> g(n, 0);
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if (f[0] == 0) {
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int s = 1;
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while (s < n * 2 && f[s] == 0) s++;
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if (s & 1) return {};
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s /= 2;
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if (s >= n) return g;
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map<int, mint> f1;
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for (auto [i, v] : f) f1[i - s * 2] = v;
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auto g1 = sqrt(f1, int(n - s));
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if (g1.empty()) return {};
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copy(g1.begin(), g1.end(), g.begin() + s);
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return g;
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}
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long long sq = ModSqrt(f[0].val(), mint::get_mod());
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if (sq < 0) return {};
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g[0] = sq;
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mint c = f[0].inv(), inv2 = mint(2).inv();
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for (int i = 1; i < n; i++) {
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for (auto [j, v] : f) {
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if (0 < j && j <= i) g[i] += j * f[j] * g[i - j] * inv2;
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if (0 < j && j < i) g[i] -= f[j] * (i - j) * g[i - j];
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}
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g[i] *= c * Factorial<mint>::inv(i);
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}
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return g;
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}
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}; // namespace FPSSparse
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/**
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* @brief Sparse な FPS 演算
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* @docs docs/fps/sparse.md
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*/

‎verify/fps/LC_exp_of_formal_power_series_sparse.test.cpp‎

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#define PROBLEM "https://judge.yosupo.jp/problem/exp_of_formal_power_series_sparse"
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#include "template/template.hpp"
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#include "modint/modint.hpp"
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using mint = ModInt<998244353>;
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#include "fps/fps-ntt-friendly.hpp"
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using fps = FormalPowerSeries<mint>;
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#include "fps/sparse.hpp"
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int main() {
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int n, k;
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in(n, k);
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map<int, mint> f;
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rep(i, 0, k) {
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int j;
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mint a;
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in(j, a);
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f[j] = a;
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}
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out(FPSSparse::exp(f, n));
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}

‎verify/fps/LC_inv_of_formal_power_series_sparse.test.cpp‎

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#define PROBLEM "https://judge.yosupo.jp/problem/inv_of_formal_power_series_sparse"
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#include "template/template.hpp"
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#include "modint/modint.hpp"
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using mint = ModInt<998244353>;
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#include "fps/fps-ntt-friendly.hpp"
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using fps = FormalPowerSeries<mint>;
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#include "fps/sparse.hpp"
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int main() {
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int n, k;
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in(n, k);
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map<int, mint> f;
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rep(i, 0, k) {
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int j;
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mint a;
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in(j, a);
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f[j] = a;
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}
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out(FPSSparse::inv(f, n));
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}

‎verify/fps/LC_log_of_formal_power_series_sparse.test.cpp‎

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#define PROBLEM "https://judge.yosupo.jp/problem/log_of_formal_power_series_sparse"
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#include "template/template.hpp"
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#include "modint/modint.hpp"
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using mint = ModInt<998244353>;
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#include "fps/fps-ntt-friendly.hpp"
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using fps = FormalPowerSeries<mint>;
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#include "fps/sparse.hpp"
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int main() {
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int n, k;
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in(n, k);
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map<int, mint> f;
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rep(i, 0, k) {
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int j;
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mint a;
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in(j, a);
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f[j] = a;
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}
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out(FPSSparse::log(f, n));
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}

‎verify/fps/LC_pow_of_formal_power_series_sparse.test.cpp‎

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#define PROBLEM "https://judge.yosupo.jp/problem/pow_of_formal_power_series_sparse"
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#include "template/template.hpp"
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#include "modint/modint.hpp"
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using mint = ModInt<998244353>;
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#include "fps/fps-ntt-friendly.hpp"
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using fps = FormalPowerSeries<mint>;
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#include "fps/sparse.hpp"
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int main() {
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int n, k;
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ll m;
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in(n, k, m);
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map<int, mint> f;
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rep(i, 0, k) {
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int j;
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mint a;
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in(j, a);
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f[j] = a;
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}
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out(FPSSparse::pow(f, m, n));
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}

‎verify/fps/LC_sqrt_of_formal_power_series_sparse.test.cpp‎

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#define PROBLEM "https://judge.yosupo.jp/problem/pow_of_formal_power_series_sparse"
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#include "template/template.hpp"
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#include "modint/modint.hpp"
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using mint = ModInt<998244353>;
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#include "fps/fps-ntt-friendly.hpp"
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using fps = FormalPowerSeries<mint>;
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#include "fps/sparse.hpp"
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#include "fps/fps-sqrt.hpp"
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int main() {
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int n, k;
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in(n, k);
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map<int, mint> f;
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rep(i, 0, k) {
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int j;
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mint a;
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in(j, a);
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f[j] = a;
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}
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auto g = FPSSparse::sqrt(f, n);
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if (g.empty())
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out(-1);
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else
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out(g);
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}

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