C++ Library for Competitive Programming
#include "emthrm/math/formal_power_series/bostan-mori.hpp"
$d$-階線形回帰数列の第 $N$ 項を求めるアルゴリズムである。
$d$ 次多項式同士の乗算の算術計算量を $\mathsf{M}(d)$ とおくと $O(\mathsf{M}(d) \log{N})$
名前 | 戻り値 | 要件 |
---|---|---|
template <typename T> T bostan_mori(FormalPowerSeries<T> p, FormalPowerSeries<T> q, long long n);
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${\lbrack x^N \rbrack}\frac{P(x)}{Q(x)}$ | ${\lbrack x^0 \rbrack}Q = Q(0)$ は可逆元 (invertible element) である。 |
template <typename T> T nth_term_of_linear_recurrence_sequence(FormalPowerSeries<T> a, FormalPowerSeries<T> q, const long long n);
|
特性多項式 $Q(x)$ をもち、$A(x) = B(x) \bmod{x^{\mathrm{deg}(A) + 1}}$ を満たす線形回帰数列の母関数 $B(x)$ に対して ${\lbrack x^N \rbrack}B$ |
https://judge.yosupo.jp/submission/80098
#ifndef EMTHRM_MATH_FORMAL_POWER_SERIES_BOSTAN_MORI_HPP_
#define EMTHRM_MATH_FORMAL_POWER_SERIES_BOSTAN_MORI_HPP_
#include <cassert>
#include "emthrm/math/formal_power_series/formal_power_series.hpp"
namespace emthrm {
template <typename T>
T bostan_mori(FormalPowerSeries<T> p, FormalPowerSeries<T> q, long long n) {
q.shrink();
const int d = q.degree();
assert(d >= 0 && q[0] != 0);
T res = 0;
p.shrink();
if (p.degree() >= d) {
const FormalPowerSeries<T> quotient = p / q;
p -= quotient * q;
p.shrink();
if (n <= quotient.degree()) res += quotient[n];
}
if (d == 0 || (p.degree() == 0 && p[0] == 0)) return res;
p.resize(d - 1);
for (; n > 0; n >>= 1) {
FormalPowerSeries<T> tmp = q;
for (int i = 1; i <= d; i += 2) {
tmp[i] = -tmp[i];
}
p *= tmp;
if (n & 1) {
for (int i = 0; i < d; ++i) {
p[i] = p[(i << 1) + 1];
}
} else {
for (int i = 1; i < d; ++i) {
p[i] = p[i << 1];
}
}
p.resize(d - 1);
q *= tmp;
for (int i = 1; i <= d; ++i) {
q[i] = q[i << 1];
}
q.resize(d);
}
return res + p[0] / q[0];
}
} // namespace emthrm
#endif // EMTHRM_MATH_FORMAL_POWER_SERIES_BOSTAN_MORI_HPP_
#line 1 "include/emthrm/math/formal_power_series/bostan-mori.hpp"
#include <cassert>
#line 1 "include/emthrm/math/formal_power_series/formal_power_series.hpp"
#include <algorithm>
#line 6 "include/emthrm/math/formal_power_series/formal_power_series.hpp"
#include <functional>
#include <initializer_list>
#include <iterator>
#include <numeric>
#include <vector>
namespace emthrm {
template <typename T>
struct FormalPowerSeries {
std::vector<T> coef;
explicit FormalPowerSeries(const int deg = 0) : coef(deg + 1, 0) {}
explicit FormalPowerSeries(const std::vector<T>& coef) : coef(coef) {}
FormalPowerSeries(const std::initializer_list<T> init)
: coef(init.begin(), init.end()) {}
template <typename InputIter>
explicit FormalPowerSeries(const InputIter first, const InputIter last)
: coef(first, last) {}
inline const T& operator[](const int term) const { return coef[term]; }
inline T& operator[](const int term) { return coef[term]; }
using Mult = std::function<std::vector<T>(const std::vector<T>&,
const std::vector<T>&)>;
using Sqrt = std::function<bool(const T&, T*)>;
static void set_mult(const Mult mult) { get_mult() = mult; }
static void set_sqrt(const Sqrt sqrt) { get_sqrt() = sqrt; }
void resize(const int deg) { coef.resize(deg + 1, 0); }
void shrink() {
while (coef.size() > 1 && coef.back() == 0) coef.pop_back();
}
int degree() const { return std::ssize(coef) - 1; }
FormalPowerSeries& operator=(const std::vector<T>& coef_) {
coef = coef_;
return *this;
}
FormalPowerSeries& operator=(const FormalPowerSeries& x) = default;
FormalPowerSeries& operator+=(const FormalPowerSeries& x) {
const int deg_x = x.degree();
if (deg_x > degree()) resize(deg_x);
for (int i = 0; i <= deg_x; ++i) {
coef[i] += x[i];
}
return *this;
}
FormalPowerSeries& operator-=(const FormalPowerSeries& x) {
const int deg_x = x.degree();
if (deg_x > degree()) resize(deg_x);
for (int i = 0; i <= deg_x; ++i) {
coef[i] -= x[i];
}
return *this;
}
FormalPowerSeries& operator*=(const T x) {
for (T& e : coef) e *= x;
return *this;
}
FormalPowerSeries& operator*=(const FormalPowerSeries& x) {
return *this = get_mult()(coef, x.coef);
}
FormalPowerSeries& operator/=(const T x) {
assert(x != 0);
return *this *= static_cast<T>(1) / x;
}
FormalPowerSeries& operator/=(const FormalPowerSeries& x) {
const int n = degree() - x.degree() + 1;
if (n <= 0) return *this = FormalPowerSeries();
const std::vector<T> tmp = get_mult()(
std::vector<T>(coef.rbegin(), std::next(coef.rbegin(), n)),
FormalPowerSeries(
x.coef.rbegin(),
std::next(x.coef.rbegin(), std::min(x.degree() + 1, n)))
.inv(n - 1).coef);
return *this = FormalPowerSeries(std::prev(tmp.rend(), n), tmp.rend());
}
FormalPowerSeries& operator%=(const FormalPowerSeries& x) {
if (x.degree() == 0) return *this = FormalPowerSeries{0};
*this -= *this / x * x;
resize(x.degree() - 1);
return *this;
}
FormalPowerSeries& operator<<=(const int n) {
coef.insert(coef.begin(), n, 0);
return *this;
}
FormalPowerSeries& operator>>=(const int n) {
if (degree() < n) return *this = FormalPowerSeries();
coef.erase(coef.begin(), coef.begin() + n);
return *this;
}
bool operator==(FormalPowerSeries x) const {
x.shrink();
FormalPowerSeries y = *this;
y.shrink();
return x.coef == y.coef;
}
FormalPowerSeries operator+() const { return *this; }
FormalPowerSeries operator-() const {
FormalPowerSeries res = *this;
for (T& e : res.coef) e = -e;
return res;
}
FormalPowerSeries operator+(const FormalPowerSeries& x) const {
return FormalPowerSeries(*this) += x;
}
FormalPowerSeries operator-(const FormalPowerSeries& x) const {
return FormalPowerSeries(*this) -= x;
}
FormalPowerSeries operator*(const T x) const {
return FormalPowerSeries(*this) *= x;
}
FormalPowerSeries operator*(const FormalPowerSeries& x) const {
return FormalPowerSeries(*this) *= x;
}
FormalPowerSeries operator/(const T x) const {
return FormalPowerSeries(*this) /= x;
}
FormalPowerSeries operator/(const FormalPowerSeries& x) const {
return FormalPowerSeries(*this) /= x;
}
FormalPowerSeries operator%(const FormalPowerSeries& x) const {
return FormalPowerSeries(*this) %= x;
}
FormalPowerSeries operator<<(const int n) const {
return FormalPowerSeries(*this) <<= n;
}
FormalPowerSeries operator>>(const int n) const {
return FormalPowerSeries(*this) >>= n;
}
T horner(const T x) const {
return std::accumulate(
coef.rbegin(), coef.rend(), static_cast<T>(0),
[x](const T l, const T r) -> T { return l * x + r; });
}
FormalPowerSeries differential() const {
const int deg = degree();
assert(deg >= 0);
FormalPowerSeries res(std::max(deg - 1, 0));
for (int i = 1; i <= deg; ++i) {
res[i - 1] = coef[i] * i;
}
return res;
}
FormalPowerSeries exp(const int deg) const {
assert(coef[0] == 0);
const int n = coef.size();
const FormalPowerSeries one{1};
FormalPowerSeries res = one;
for (int i = 1; i <= deg; i <<= 1) {
res *= FormalPowerSeries(coef.begin(),
std::next(coef.begin(), std::min(n, i << 1)))
- res.log((i << 1) - 1) + one;
res.coef.resize(i << 1);
}
res.resize(deg);
return res;
}
FormalPowerSeries exp() const { return exp(degree()); }
FormalPowerSeries inv(const int deg) const {
assert(coef[0] != 0);
const int n = coef.size();
FormalPowerSeries res{static_cast<T>(1) / coef[0]};
for (int i = 1; i <= deg; i <<= 1) {
res = res + res - res * res * FormalPowerSeries(
coef.begin(), std::next(coef.begin(), std::min(n, i << 1)));
res.coef.resize(i << 1);
}
res.resize(deg);
return res;
}
FormalPowerSeries inv() const { return inv(degree()); }
FormalPowerSeries log(const int deg) const {
assert(coef[0] == 1);
FormalPowerSeries integrand = differential() * inv(deg - 1);
integrand.resize(deg);
for (int i = deg; i > 0; --i) {
integrand[i] = integrand[i - 1] / i;
}
integrand[0] = 0;
return integrand;
}
FormalPowerSeries log() const { return log(degree()); }
FormalPowerSeries pow(long long exponent, const int deg) const {
const int n = coef.size();
if (exponent == 0) {
FormalPowerSeries res(deg);
if (deg != -1) [[unlikely]] res[0] = 1;
return res;
}
assert(deg >= 0);
for (int i = 0; i < n; ++i) {
if (coef[i] == 0) continue;
if (i > deg / exponent) break;
const long long shift = exponent * i;
T tmp = 1, base = coef[i];
for (long long e = exponent; e > 0; e >>= 1) {
if (e & 1) tmp *= base;
base *= base;
}
const FormalPowerSeries res = ((*this >> i) / coef[i]).log(deg - shift);
return ((res * exponent).exp(deg - shift) * tmp) << shift;
}
return FormalPowerSeries(deg);
}
FormalPowerSeries pow(const long long exponent) const {
return pow(exponent, degree());
}
FormalPowerSeries mod_pow(long long exponent,
const FormalPowerSeries& md) const {
const int deg = md.degree() - 1;
if (deg < 0) [[unlikely]] return FormalPowerSeries(-1);
const FormalPowerSeries inv_rev_md =
FormalPowerSeries(md.coef.rbegin(), md.coef.rend()).inv();
const auto mod_mult = [&md, &inv_rev_md, deg](
FormalPowerSeries* multiplicand, const FormalPowerSeries& multiplier)
-> void {
*multiplicand *= multiplier;
if (deg < multiplicand->degree()) {
const int n = multiplicand->degree() - deg;
const FormalPowerSeries quotient =
FormalPowerSeries(multiplicand->coef.rbegin(),
std::next(multiplicand->coef.rbegin(), n))
* FormalPowerSeries(
inv_rev_md.coef.begin(),
std::next(inv_rev_md.coef.begin(), std::min(deg + 2, n)));
*multiplicand -=
FormalPowerSeries(std::prev(quotient.coef.rend(), n),
quotient.coef.rend()) * md;
multiplicand->resize(deg);
}
multiplicand->shrink();
};
FormalPowerSeries res{1}, base = *this;
for (; exponent > 0; exponent >>= 1) {
if (exponent & 1) mod_mult(&res, base);
mod_mult(&base, base);
}
return res;
}
FormalPowerSeries sqrt(const int deg) const {
const int n = coef.size();
if (coef[0] == 0) {
for (int i = 1; i < n; ++i) {
if (coef[i] == 0) continue;
if (i & 1) return FormalPowerSeries(-1);
const int shift = i >> 1;
if (deg < shift) break;
FormalPowerSeries res = (*this >> i).sqrt(deg - shift);
if (res.coef.empty()) return FormalPowerSeries(-1);
res <<= shift;
res.resize(deg);
return res;
}
return FormalPowerSeries(deg);
}
T s;
if (!get_sqrt()(coef.front(), &s)) return FormalPowerSeries(-1);
FormalPowerSeries res{s};
const T half = static_cast<T>(1) / 2;
for (int i = 1; i <= deg; i <<= 1) {
res = (FormalPowerSeries(coef.begin(),
std::next(coef.begin(), std::min(n, i << 1)))
* res.inv((i << 1) - 1) + res) * half;
}
res.resize(deg);
return res;
}
FormalPowerSeries sqrt() const { return sqrt(degree()); }
FormalPowerSeries translate(const T c) const {
const int n = coef.size();
std::vector<T> fact(n, 1), inv_fact(n, 1);
for (int i = 1; i < n; ++i) {
fact[i] = fact[i - 1] * i;
}
inv_fact[n - 1] = static_cast<T>(1) / fact[n - 1];
for (int i = n - 1; i > 0; --i) {
inv_fact[i - 1] = inv_fact[i] * i;
}
std::vector<T> g(n), ex(n);
for (int i = 0; i < n; ++i) {
g[i] = coef[i] * fact[i];
}
std::reverse(g.begin(), g.end());
T pow_c = 1;
for (int i = 0; i < n; ++i) {
ex[i] = pow_c * inv_fact[i];
pow_c *= c;
}
const std::vector<T> conv = get_mult()(g, ex);
FormalPowerSeries res(n - 1);
for (int i = 0; i < n; ++i) {
res[i] = conv[n - 1 - i] * inv_fact[i];
}
return res;
}
private:
static Mult& get_mult() {
static Mult mult = [](const std::vector<T>& a, const std::vector<T>& b)
-> std::vector<T> {
const int n = a.size(), m = b.size();
std::vector<T> res(n + m - 1, 0);
for (int i = 0; i < n; ++i) {
for (int j = 0; j < m; ++j) {
res[i + j] += a[i] * b[j];
}
}
return res;
};
return mult;
}
static Sqrt& get_sqrt() {
static Sqrt sqrt = [](const T&, T*) -> bool { return false; };
return sqrt;
}
};
} // namespace emthrm
#line 7 "include/emthrm/math/formal_power_series/bostan-mori.hpp"
namespace emthrm {
template <typename T>
T bostan_mori(FormalPowerSeries<T> p, FormalPowerSeries<T> q, long long n) {
q.shrink();
const int d = q.degree();
assert(d >= 0 && q[0] != 0);
T res = 0;
p.shrink();
if (p.degree() >= d) {
const FormalPowerSeries<T> quotient = p / q;
p -= quotient * q;
p.shrink();
if (n <= quotient.degree()) res += quotient[n];
}
if (d == 0 || (p.degree() == 0 && p[0] == 0)) return res;
p.resize(d - 1);
for (; n > 0; n >>= 1) {
FormalPowerSeries<T> tmp = q;
for (int i = 1; i <= d; i += 2) {
tmp[i] = -tmp[i];
}
p *= tmp;
if (n & 1) {
for (int i = 0; i < d; ++i) {
p[i] = p[(i << 1) + 1];
}
} else {
for (int i = 1; i < d; ++i) {
p[i] = p[i << 1];
}
}
p.resize(d - 1);
q *= tmp;
for (int i = 1; i <= d; ++i) {
q[i] = q[i << 1];
}
q.resize(d);
}
return res + p[0] / q[0];
}
} // namespace emthrm