cplib
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math/fft.cpp
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- Last update: 2020-05-26 19:55:50+09:00
Depends on
template.cpp
Code
#pragma once
#include "geometry/Pt.cpp"
#include "template.cpp"
namespace FFT {
using dbl = double;
using complex = Pt<dbl>;
inline complex conj(complex a) { return complex(a.x, -a.y); }
int base = 1;
vector<complex> root = {{0, 0}, {1, 0}};
vector<int> rev = {0, 1};
const dbl PI = asinl(1) * 2;
void prepare(int nbase) {
if (nbase <= base) return;
rev.resize(1 << nbase);
for (int i = 0; i < (1 << nbase); i++)
rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
root.resize(1 << nbase);
rep(b, base, nbase) {
dbl theta = 2 * PI / (1 << (b + 1));
for (int i = 1 << (b - 1); i < (1 << b); i++) {
root[i << 1] = root[i];
dbl aug = theta * (2 * i + 1 - (1 << b));
root[(i << 1) + 1] = complex(cos(aug), sin(aug));
}
}
}
template <typename T, typename U>
vector<T> cast(const vector<U>& v) {
vector<T> res(v.size());
if (is_integral<T>::value)
rep(i, v.size()) res[i] = T(round(v[i]));
else
copy(all(v), res.begin());
return res;
}
void fft(vector<complex>& A) {
int n = A.size();
assert((n & (n - 1)) == 0);
int zeros = __builtin_ctz(n);
prepare(zeros + 1);
int shift = base - zeros;
rep(i, n) if (i < (rev[i] >> shift)) swap(A[i], A[rev[i] >> shift]);
for (int k = 1; k < n; k <<= 1) {
for (int i = 0; i < n; i += 2 * k) {
for (int j = 0; j < k; j++) {
complex z = A[i + j + k] * root[j + k];
A[i + j + k] = A[i + j] - z;
A[i + j] += z;
}
}
}
}
template <typename T>
vector<T> multiply(const vector<T>& A, const vector<T>& B) {
int need = A.size() + B.size() - 1;
int sz = need <= 1 ? 1 : 1 << (32 - __builtin_clz(need - 1));
vector<complex> fa(sz);
rep(i, A.size()) fa[i].x = dbl(A[i]);
rep(i, B.size()) fa[i].y = dbl(B[i]);
fft(fa);
complex r(0, -0.25 / sz);
for (int i = 0; i <= (sz >> 1); i++) {
int j = (sz - i) & (sz - 1);
complex z = (fa[j] * fa[j] - conj(fa[i] * fa[i])) * r;
if (i != j) fa[j] = (fa[i] * fa[i] - conj(fa[j] * fa[j])) * r;
fa[i] = z;
}
fft(fa);
vector<T> res(need);
rep(i, need) res[i] = fa[i].x;
return res;
}
template <typename T>
vector<dbl> div(vector<T>& A, vector<T>& B) {
static vector<dbl> q_rev;
static vector<vector<dbl>> t; // t[i] * B == 0 mod x^2^i
while (B.back() == 0) B.pop_back();
if (not equal(all(B), rall(q_rev))) {
q_rev.resize(B.size());
copy(rall(B), q_rev.begin());
t.clear();
t.emplace_back(vector<dbl>{1 / q_rev[0]});
}
int n = A.size(), m = B.size();
int k = 32 - __builtin_clz(n - m); // ceil(log2(n-m+1))
rep((1 << k) - n) A.emplace_back(dbl(0));
n = A.size();
rep(k + 1 - t.size()) {
vector<dbl> next = multiply(multiply(t.back(), t.back()), q_rev);
next.resize(1 << t.size());
for (auto& a : next) a *= -1;
rep(i, t.back().size()) next[i] += t.back()[i] * 2;
t.emplace_back(move(next));
}
reverse(all(A));
vector<dbl> res = multiply(t[k], vector<dbl>(all(A)));
res.resize(n - m + 1);
reverse(all(A));
reverse(all(res));
return res;
}
template <typename Vector>
vector<dbl> div(Vector&& A, Vector&& B) {
return div(A, B);
}
template <typename T>
pair<vector<dbl>, vector<dbl>> divmod(vector<T>& A, vector<T>& B,
dbl eps = 1e-9) {
vector<dbl> q = div(A, B);
vector<dbl> r = multiply(q, cast<dbl>(B));
r.resize(A.size());
rep(i, r.size()) r[i] = A[i] - r[i];
while (r.size() > 1 and abs(r.back()) < eps) r.pop_back();
return make_pair(q, r);
}
template <typename Vector>
pair<vector<dbl>, vector<dbl>> divmod(Vector&& A, Vector&& B, dbl eps = 1e-9) {
return divmod(A, B, eps);
}
template <typename Vector>
vector<dbl> mod(Vector&& A, Vector&& B, dbl eps = 1e-9) {
return divmod(A, B, eps).second;
}
}; // namespace FFT#line 2 "math/fft.cpp"
#line 2 "geometry/Pt.cpp"
#line 2 "template.cpp"
#ifndef LOCAL
#pragma GCC optimize("O3")
#pragma GCC optimize("unroll-loops")
#pragma GCC target("avx")
#endif
#include <algorithm>
#include <bitset>
#include <cassert>
#include <cmath>
#include <functional>
#include <iostream>
#include <map>
#include <numeric>
#include <queue>
#include <set>
#include <stack>
using namespace std;
using ll = long long;
using ull = unsigned long long;
using VI = vector<int>;
using VVI = vector<vector<int>>;
using VLL = vector<ll>;
using VVLL = vector<vector<ll>>;
using VB = vector<bool>;
using PII = pair<int, int>;
using PLL = pair<ll, ll>;
constexpr int INF = 1000000007;
constexpr ll INF_LL = 1'000'000'000'000'000'007;
#define all(x) begin(x), end(x)
#define rall(x) rbegin(x), rend(x)
#define newl '\n'
// loops rep(until) / rep(var, until) / rep(var, from, until) / repr (reversed order)
#define OVERLOAD3(_1, _2, _3, name, ...) name
#define rep(...) OVERLOAD3(__VA_ARGS__, REPEAT_FROM_UNTIL, REPEAT_UNTIL, REPEAT)(__VA_ARGS__)
#define REPEAT(times) REPEAT_CNT(_repeat, __COUNTER__, times)
#define REPEAT_CNT(_repeat, cnt, times) REPEAT_CNT_CAT(_repeat, cnt, times)
#define REPEAT_CNT_CAT(_repeat, cnt, times) REPEAT_FROM_UNTIL(_repeat ## cnt, 0, times)
#define REPEAT_UNTIL(name, times) REPEAT_FROM_UNTIL(name, 0, times)
#define REPEAT_FROM_UNTIL(name, from, until) for (int name = from, name ## __until = (until); name < name ## __until; name++)
#define repr(...) OVERLOAD3(__VA_ARGS__, REPR_FROM_UNTIL, REPR_UNTIL, REPEAT)(__VA_ARGS__)
#define REPR_UNTIL(name, times) REPR_FROM_UNTIL(name, 0, times)
#define REPR_FROM_UNTIL(name, from, until) for (int name = (until)-1, name ## __from = (from); name >= name ## __from; name--)
template <typename T, typename U>
bool chmin(T& var, U x) { if (var > x) { var = x; return true; } else return false; }
template <typename T, typename U>
bool chmax(T& var, U x) { if (var < x) { var = x; return true; } else return false; }
ll power(ll e, ll t, ll mod = INF_LL) {
ll res = 1; for (; t; t >>= 1, (e *= e) %= mod) if (t & 1) (res *= e) %= mod; return res;
}
ll choose(ll n, int r) {
chmin(r, n-r); if (r < 0) return 0; ll res = 1; rep(i, r) res *= n-i, res /= i+1; return res;
}
template <typename T, typename U> T divceil(T m, U d) { return (m + d - 1) / d; }
template <typename T> vector<T> make_v(size_t a, T b) { return vector<T>(a, b); }
template <typename... Ts> auto make_v(size_t a, Ts... ts) {
return vector<decltype(make_v(ts...))>(a, make_v(ts...));
}
// debugging stuff
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wmisleading-indentation"
#define repi(it, ds) for (auto it = ds.begin(); it != ds.end(); it++)
class DebugPrint { public: template <typename T> DebugPrint& operator <<(const T& v) {
#ifdef LOCAL
cerr << v;
#endif
return *this; } } debugos; template <typename T> DebugPrint& operator<<(DebugPrint& os, const
vector<T>& vec) { os << "{"; for (int i = 0; i < vec.size(); i++) os << vec[i] << (i + 1 ==
vec.size() ? "" : ", "); os << "}"; return os; } template <typename T, typename U> DebugPrint&
operator<<(DebugPrint& os, const map<T, U>& map_var) { os << "{"; repi(itr, map_var) { os << *
itr; itr++; if (itr != map_var.end()) os << ", "; itr--; } os << "}"; return os; } template <
typename T> DebugPrint& operator<<(DebugPrint& os, const set<T>& set_var) { os << "{"; repi(
itr, set_var) { os << *itr; itr++; if (itr != set_var.end()) os << ", "; itr--; } os << "}";
return os; } template <typename T, typename U> DebugPrint& operator<<(DebugPrint& os, const
pair<T, U>& p) { os << "(" << p.first << ", " << p.second << ")"; return os; } void dump_func(
) { debugos << newl; } template <class Head, class... Tail> void dump_func(Head &&head, Tail
&&... tail) { debugos << head; if (sizeof...(Tail) > 0) { debugos << ", "; } dump_func(forward
<Tail>(tail)...); }
#ifdef LOCAL
#define dump(...) debugos << " " << string(#__VA_ARGS__) << ": " << "[" << to_string(__LINE__) \
<< ":" << __FUNCTION__ << "]" << newl << " ", dump_func(__VA_ARGS__)
#else
#define dump(...) ({})
#endif
#pragma GCC diagnostic pop
#line 4 "geometry/Pt.cpp"
template <typename T = int>
struct Pt {
T x, y;
Pt(T x_ = 0, T y_ = 0): x(x_), y(y_) { }
Pt operator +(const Pt<T>& rhs) const {
return Pt(x+rhs.x, y+rhs.y);
}
Pt operator -(const Pt<T>& rhs) const {
return Pt(x-rhs.x, y-rhs.y);
}
Pt operator -() const {
return Pt(-x, -y);
}
Pt operator *(const Pt<T>& rhs) const {
return Pt(x*rhs.x - y*rhs.y, x*rhs.y + y*rhs.x);
}
Pt operator *(const T rhs) const {
return Pt(x*rhs, y*rhs);
}
Pt& operator +=(const Pt<T>& rhs) {
return *this = *this + rhs;
}
Pt& operator -=(const Pt<T>& rhs) {
return *this = *this - rhs;
}
Pt& operator *=(const Pt<T>& rhs) {
return *this = *this * rhs;
}
Pt& operator *=(const T rhs) {
return *this = *this * rhs;
}
bool operator ==(const Pt<T>& rhs) const {
return x == rhs.x and y == rhs.y;
}
bool operator !=(const Pt<T>& rhs) const {
return not (*this == rhs);
}
double abs() const {
return hypot(x, y);
}
T dot(const Pt<T>& rhs) const {
return x*rhs.x + y*rhs.y;
}
T det(const Pt<T>& rhs) const {
return x*rhs.y - y*rhs.x;
}
};
template <typename OutStream, typename T>
OutStream& operator<<(OutStream& out, const Pt<T>& var) {
return out << var.x << " " << var.y;
}
template <typename InStream, typename T>
InStream& operator>>(InStream& in, Pt<T>& var) {
return in >> var.x >> var.y;
}
#line 5 "math/fft.cpp"
namespace FFT {
using dbl = double;
using complex = Pt<dbl>;
inline complex conj(complex a) { return complex(a.x, -a.y); }
int base = 1;
vector<complex> root = {{0, 0}, {1, 0}};
vector<int> rev = {0, 1};
const dbl PI = asinl(1) * 2;
void prepare(int nbase) {
if (nbase <= base) return;
rev.resize(1 << nbase);
for (int i = 0; i < (1 << nbase); i++)
rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
root.resize(1 << nbase);
rep(b, base, nbase) {
dbl theta = 2 * PI / (1 << (b + 1));
for (int i = 1 << (b - 1); i < (1 << b); i++) {
root[i << 1] = root[i];
dbl aug = theta * (2 * i + 1 - (1 << b));
root[(i << 1) + 1] = complex(cos(aug), sin(aug));
}
}
}
template <typename T, typename U>
vector<T> cast(const vector<U>& v) {
vector<T> res(v.size());
if (is_integral<T>::value)
rep(i, v.size()) res[i] = T(round(v[i]));
else
copy(all(v), res.begin());
return res;
}
void fft(vector<complex>& A) {
int n = A.size();
assert((n & (n - 1)) == 0);
int zeros = __builtin_ctz(n);
prepare(zeros + 1);
int shift = base - zeros;
rep(i, n) if (i < (rev[i] >> shift)) swap(A[i], A[rev[i] >> shift]);
for (int k = 1; k < n; k <<= 1) {
for (int i = 0; i < n; i += 2 * k) {
for (int j = 0; j < k; j++) {
complex z = A[i + j + k] * root[j + k];
A[i + j + k] = A[i + j] - z;
A[i + j] += z;
}
}
}
}
template <typename T>
vector<T> multiply(const vector<T>& A, const vector<T>& B) {
int need = A.size() + B.size() - 1;
int sz = need <= 1 ? 1 : 1 << (32 - __builtin_clz(need - 1));
vector<complex> fa(sz);
rep(i, A.size()) fa[i].x = dbl(A[i]);
rep(i, B.size()) fa[i].y = dbl(B[i]);
fft(fa);
complex r(0, -0.25 / sz);
for (int i = 0; i <= (sz >> 1); i++) {
int j = (sz - i) & (sz - 1);
complex z = (fa[j] * fa[j] - conj(fa[i] * fa[i])) * r;
if (i != j) fa[j] = (fa[i] * fa[i] - conj(fa[j] * fa[j])) * r;
fa[i] = z;
}
fft(fa);
vector<T> res(need);
rep(i, need) res[i] = fa[i].x;
return res;
}
template <typename T>
vector<dbl> div(vector<T>& A, vector<T>& B) {
static vector<dbl> q_rev;
static vector<vector<dbl>> t; // t[i] * B == 0 mod x^2^i
while (B.back() == 0) B.pop_back();
if (not equal(all(B), rall(q_rev))) {
q_rev.resize(B.size());
copy(rall(B), q_rev.begin());
t.clear();
t.emplace_back(vector<dbl>{1 / q_rev[0]});
}
int n = A.size(), m = B.size();
int k = 32 - __builtin_clz(n - m); // ceil(log2(n-m+1))
rep((1 << k) - n) A.emplace_back(dbl(0));
n = A.size();
rep(k + 1 - t.size()) {
vector<dbl> next = multiply(multiply(t.back(), t.back()), q_rev);
next.resize(1 << t.size());
for (auto& a : next) a *= -1;
rep(i, t.back().size()) next[i] += t.back()[i] * 2;
t.emplace_back(move(next));
}
reverse(all(A));
vector<dbl> res = multiply(t[k], vector<dbl>(all(A)));
res.resize(n - m + 1);
reverse(all(A));
reverse(all(res));
return res;
}
template <typename Vector>
vector<dbl> div(Vector&& A, Vector&& B) {
return div(A, B);
}
template <typename T>
pair<vector<dbl>, vector<dbl>> divmod(vector<T>& A, vector<T>& B,
dbl eps = 1e-9) {
vector<dbl> q = div(A, B);
vector<dbl> r = multiply(q, cast<dbl>(B));
r.resize(A.size());
rep(i, r.size()) r[i] = A[i] - r[i];
while (r.size() > 1 and abs(r.back()) < eps) r.pop_back();
return make_pair(q, r);
}
template <typename Vector>
pair<vector<dbl>, vector<dbl>> divmod(Vector&& A, Vector&& B, dbl eps = 1e-9) {
return divmod(A, B, eps);
}
template <typename Vector>
vector<dbl> mod(Vector&& A, Vector&& B, dbl eps = 1e-9) {
return divmod(A, B, eps).second;
}
}; // namespace FFT