結果
| 問題 |
No.980 Fibonacci Convolution Hard
|
| ユーザー |
 hashiryo
|
| 提出日時 | 2020-04-09 09:58:13 |
| 言語 | C++14 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 1,193 ms / 2,000 ms |
| コード長 | 15,928 bytes |
| コンパイル時間 | 2,247 ms |
| コンパイル使用メモリ | 184,480 KB |
| 実行使用メモリ | 126,152 KB |
| 最終ジャッジ日時 | 2024-10-05 13:37:44 |
| 合計ジャッジ時間 | 24,658 ms |
|
ジャッジサーバーID （参考情報） |
judge2 / judge4 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| other | AC * 17 |
ソースコード
#include <bits/stdc++.h>
#define debug(x) cerr << #x << ": " << x << endl
#define debugArray(x, n) \
for (long long hoge = 0; (hoge) < (n); ++(hoge)) \
cerr << #x << "[" << hoge << "]: " << x[hoge] << endl
using namespace std;
double tick() {
static clock_t oldtick;
clock_t newtick = clock();
double diff = 1.0 * (newtick - oldtick) / CLOCKS_PER_SEC;
oldtick = newtick;
return diff;
}
using u32 = unsigned;
using u64 = unsigned long long;
namespace ntt {
template <u64 mod, u64 prim_root>
class Mod64 {
private:
using u128 = __uint128_t;
static constexpr u64 mul_inv(u64 n, int e = 6, u64 x = 1) {
return e == 0 ? x : mul_inv(n, e - 1, x * (2 - x * n));
}
public:
static constexpr u64 inv = mul_inv(mod, 6, 1);
static constexpr u64 r2 = -u128(mod) % mod;
static constexpr int level = __builtin_ctzll(mod - 1);
static_assert(inv * mod == 1, "invalid 1/M modulo 2^64.");
Mod64() {}
Mod64(u64 n) : x(init(n)){};
static u64 modulo() { return mod; }
static u64 init(u64 w) { return reduce(u128(w) * r2); }
static u64 reduce(const u128 w) {
return u64(w >> 64) + mod - ((u128(u64(w) * inv) * mod) >> 64);
}
static Mod64 omega() { return Mod64(prim_root).pow((mod - 1) >> level); }
Mod64 &operator+=(Mod64 rhs) {
this->x += rhs.x;
return *this;
}
Mod64 &operator-=(Mod64 rhs) {
this->x += 2 * mod - rhs.x;
return *this;
}
Mod64 &operator*=(Mod64 rhs) {
this->x = reduce(u128(this->x) * rhs.x);
return *this;
}
Mod64 operator+(Mod64 rhs) const { return Mod64(*this) += rhs; }
Mod64 operator-(Mod64 rhs) const { return Mod64(*this) -= rhs; }
Mod64 operator*(Mod64 rhs) const { return Mod64(*this) *= rhs; }
u64 get() const { return reduce(this->x) % mod; }
void set(u64 n) const { this->x = n; }
Mod64 pow(u64 exp) const {
Mod64 ret = Mod64(1);
for (Mod64 base = *this; exp; exp >>= 1, base *= base)
if (exp & 1) ret *= base;
return ret;
}
Mod64 inverse() const { return pow(mod - 2); }
friend ostream &operator<<(ostream &os, const Mod64 &m) {
return os << m.get();
}
u64 x;
};
template <typename mod_t>
void convolute(mod_t *A, int s1, mod_t *B, int s2, bool cyclic = false) {
int s = (cyclic ? max(s1, s2) : s1 + s2 - 1);
int size = 1;
while (size < s) size <<= 1;
mod_t roots[mod_t::level] = {mod_t::omega()};
for (int i = 1; i < mod_t::level; i++) roots[i] = roots[i - 1] * roots[i - 1];
fill(A + s1, A + size, 0);
ntt_dit4(A, size, 1, roots);
if (A == B && s1 == s2) {
for (int i = 0; i < size; i++) A[i] *= A[i];
} else {
fill(B + s2, B + size, 0);
ntt_dit4(B, size, 1, roots);
for (int i = 0; i < size; i++) A[i] *= B[i];
}
ntt_dit4(A, size, -1, roots);
mod_t inv = mod_t(size).inverse();
for (int i = 0; i < (cyclic ? size : s); i++) A[i] *= inv;
}
template <typename mod_t>
void rev_permute(mod_t *A, int n) {
int r = 0, nh = n >> 1;
for (int i = 1; i < n; i++) {
for (int h = nh; !((r ^= h) & h); h >>= 1)
;
if (r > i) swap(A[i], A[r]);
}
}
template <typename mod_t>
void ntt_dit4(mod_t *A, int n, int sign, mod_t *roots) {
rev_permute(A, n);
int logn = __builtin_ctz(n);
if (logn & 1)
for (int i = 0; i < n; i += 2) {
mod_t a = A[i], b = A[i + 1];
A[i] = a + b;
A[i + 1] = a - b;
}
mod_t imag = roots[mod_t::level - 2];
if (sign < 0) imag = imag.inverse();
mod_t one = mod_t(1);
for (int e = 2 + (logn & 1); e < logn + 1; e += 2) {
const int m = 1 << e;
const int m4 = m >> 2;
mod_t dw = roots[mod_t::level - e];
if (sign < 0) dw = dw.inverse();
const int block_size = max(m, (1 << 15) / int(sizeof(A[0])));
for (int k = 0; k < n; k += block_size) {
mod_t w = one, w2 = one, w3 = one;
for (int j = 0; j < m4; j++) {
for (int i = k + j; i < k + block_size; i += m) {
mod_t a0 = A[i + m4 * 0] * one, a2 = A[i + m4 * 1] * w2;
mod_t a1 = A[i + m4 * 2] * w, a3 = A[i + m4 * 3] * w3;
mod_t t02 = a0 + a2, t13 = a1 + a3;
A[i + m4 * 0] = t02 + t13;
A[i + m4 * 2] = t02 - t13;
t02 = a0 - a2, t13 = (a1 - a3) * imag;
A[i + m4 * 1] = t02 + t13;
A[i + m4 * 3] = t02 - t13;
}
w *= dw;
w2 = w * w;
w3 = w2 * w;
}
}
}
}
const int size = 1 << 22;
using m64_1 = ntt::Mod64<34703335751681, 3>;
using m64_2 = ntt::Mod64<35012573396993, 3>;
m64_1 f1[size], g1[size];
m64_2 f2[size], g2[size];
} // namespace ntt
using R = u64;
class FormalPowerSeries {
using FPS = FormalPowerSeries;
public:
FormalPowerSeries() {}
FormalPowerSeries(int n) : coefs(n) {}
FormalPowerSeries(int n, int c) : coefs(n, c % mod) {}
FormalPowerSeries(const vector<R> &v) : coefs(v) {}
FormalPowerSeries(const FPS &f, int beg, int end = -1) {
if (end < 0) end = beg, beg = 0;
resize(end - beg);
for (int i = beg; i < end; i++)
if (i < f.size()) coefs[i - beg] = f[i];
}
int size() const { return coefs.size(); }
void resize(int s, R v = 0) { coefs.resize(s, v); }
void push_back(R c) { coefs.push_back(c); }
void shrink() {
while (!coefs.empty() && !coefs.back()) coefs.pop_back();
}
void pop_back() { coefs.pop_back(); }
const R *data() const { return coefs.data(); }
R *data() { return coefs.data(); }
const R &operator[](int i) const { return coefs[i]; }
R &operator[](int i) { return coefs[i]; }
public:
static void mod_add(R &a, R b) {
if ((a += b) >= mod) a -= mod;
}
static void mod_sub(R &a, R b) {
if (int(a -= b) < 0) a += mod;
}
static R mod_mul(R a, R b) { return a * b % fast_mod; }
static R mod_pow(R v, u64 exp) {
R ret = 1;
for (R base = v; exp; exp >>= 1, base = mod_mul(base, base))
if (exp & 1) ret = mod_mul(ret, base);
return ret;
}
static R mod_inverse(R v) { return mod_pow(v, mod - 2); }
static R mod_sqrt(R x) {
if (x == 0 || mod == 2) return x;
if (mod_pow(x, (mod - 1) >> 1) != 1) return 0; // no solution
R b = 2;
R w = (b * b + mod - x) % fast_mod;
while (mod_pow(w, (mod - 1) >> 1) == 1) {
b++;
w = (b * b + mod - x) % fast_mod;
}
auto mul = [&](pair<R, R> u, pair<R, R> v) {
R a = (u.first * v.first + u.second * v.second % fast_mod * w) % fast_mod;
R b = (u.first * v.second + u.second * v.first) % fast_mod;
return make_pair(a, b);
};
u32 exp = (mod + 1) >> 1;
pair<R, R> ret = make_pair(1, 0);
for (auto base = make_pair(b, 1); exp; exp >>= 1, base = mul(base, base)) {
if (exp & 1) ret = mul(ret, base);
}
return ret.first;
}
public:
struct fast_div {
using u128 = __uint128_t;
fast_div(){};
fast_div(u64 n) : m(n) {
s = (n == 1) ? 0 : 127 - __builtin_clzll(n - 1);
x = ((u128(1) << s) + n - 1) / n;
}
friend u64 operator/(u64 n, fast_div d) { return u128(n) * d.x >> d.s; }
friend u64 operator%(u64 n, fast_div d) { return n - n / d * d.m; }
u64 m, s, x;
};
static FPS mul_crt(int beg, int end) {
using namespace ntt;
auto inv = m64_2(m64_1::modulo()).inverse();
auto mod1 = m64_1::modulo() % fast_mod;
FPS ret(end - beg);
for (int i = 0; i < (int)ret.size(); i++) {
u64 r1 = f1[i + beg].get(), r2 = f2[i + beg].get();
ret[i]
= (r1
+ (m64_2(r2 + m64_2::modulo() - r1) * inv).get() % fast_mod * mod1)
% fast_mod;
}
return ret;
}
static void mul2(const FPS &f, const FPS &g, bool cyclic = false) {
using namespace ntt;
if (&f == &g) {
for (int i = 0; i < (int)f.size(); i++) f1[i] = f[i];
convolute(f1, f.size(), f1, f.size(), cyclic);
for (int i = 0; i < (int)f.size(); i++) f2[i] = f[i];
convolute(f2, f.size(), f2, f.size(), cyclic);
} else {
for (int i = 0; i < (int)f.size(); i++) f1[i] = f[i];
for (int i = 0; i < (int)g.size(); i++) g1[i] = g[i];
convolute(f1, f.size(), g1, g.size(), cyclic);
for (int i = 0; i < (int)f.size(); i++) f2[i] = f[i];
for (int i = 0; i < (int)g.size(); i++) g2[i] = g[i];
convolute(f2, f.size(), g2, g.size(), cyclic);
}
}
private:
FPS mul_n(const FPS &g) const {
if (size() == 0 || g.size() == 0) return FPS();
FPS ret(size() + g.size() - 1, 0);
for (int i = 0; i < size(); i++)
for (int j = 0; j < g.size(); j++)
mod_add(ret[i + j], mod_mul((*this)[i], g[j]));
return ret;
}
FPS mul(const FPS &g) const {
if (size() + g.size() < 80) return mul_n(g);
const auto &f = *this;
mul2(f, g, false);
return mul_crt(0, f.size() + g.size() - 1);
}
pair<FPS, FPS> div_n(const FPS &g) const {
FPS f(*this, size());
if (f.size() < g.size()) return make_pair(FPS(), f);
FPS u(f.size() - g.size() + 1, 0);
R inv = mod_inverse(g[g.size() - 1]);
for (int i = u.size() - 1; i >= 0; --i) {
u[i] = mod_mul(f[f.size() - 1], inv);
for (int j = 0; j < g.size(); ++j)
mod_sub(f[j + f.size() - g.size()], mod_mul(g[j], u[i]));
f.pop_back();
}
return {u, f};
}
FPS middle_product(const FPS &g) const {
const FPS &f = *this;
if (f.size() == 0 || g.size() == 0) return FPS();
mul2(f, g, true);
return mul_crt(f.size(), g.size());
}
FPS inverse(int deg = -1) const {
if (deg < 0) deg = size();
FPS ret(1, mod_inverse((*this)[0]));
for (int e = 1, ne; e < deg; e = ne) {
ne = min(2 * e, deg);
FPS h = FPS(ret, ne - e) * -ret.middle_product(FPS(*this, ne));
for (int i = e; i < ne; i++) ret.push_back(h[i - e]);
}
return ret;
}
FPS differentiation() const {
FPS ret(max(0, size() - 1));
for (int i = 1; i < size(); i++) ret[i - 1] = mod_mul(i, (*this)[i]);
return ret;
}
FPS integral() const {
FPS ret(size() + 1);
ret[0] = 0;
for (int i = 0; i < size(); i++)
ret[i + 1] = mod_mul(inve[i + 1], (*this)[i]);
return ret;
}
FPS logarithm(int deg = -1) const {
assert((*this)[0] == 1);
if (deg < 0) deg = size();
return FPS(differentiation() * inverse(deg), deg - 1).integral();
}
FPS exponent(int deg = -1) const {
assert((*this)[0] == 0);
if (deg < 0) deg = size();
FPS ret(vector<R>({1, 1 < size() ? (*this)[1] : 0})), retinv(1, 1);
FPS f = differentiation();
FPS retdif = ret.differentiation();
for (int e = 1, ne = 2, nne; ne < deg; e = ne, ne = nne) {
nne = min(2 * ne, deg);
FPS h = FPS(retinv, ne - e) * -retinv.middle_product(ret);
for (int i = e; i < ne; i++) retinv.push_back(h[i - e]);
FPS a = ret * FPS(f, nne) - retdif;
FPS c = (retinv * FPS(a, nne)).integral();
h = ret.middle_product(FPS(c, nne));
for (int i = ne; i < nne; i++) {
ret.push_back(h[i - ne]);
retdif.push_back(mod_mul(i, h[i - ne]));
}
}
return ret;
}
FPS square_root(int deg = -1) const {
if (deg < 0) deg = size();
if ((*this)[0] == 0) {
for (int i = 1; i < size(); i++) {
if ((*this)[i] != 0) {
if (i & 1) return FPS(); // no solution
if (deg - i / 2 <= 0) break;
auto ret = (*this >> i).square_root(deg - i / 2);
if (!ret.size()) return FPS(); // no solution
ret = ret << (i / 2);
if (ret.size() < deg) ret.resize(deg, 0);
return ret;
}
}
return FPS(deg, 0);
}
R sqr = mod_sqrt((*this)[0]);
if (sqr * 2 > mod) sqr = mod - sqr;
if (mod_mul(sqr, sqr) != (*this)[0]) return FPS(); // no solution
FPS ret(1, sqr);
for (int i = 1; i < deg; i <<= 1) {
ret = FPS(ret + (FPS(*this, i << 1) * ret.inverse(i << 1)), i << 1);
ret *= inve[2];
}
return ret;
}
FPS power(u64 k, int deg = -1) const {
if (deg < 0) deg = size();
for (int i = 0; i < size(); i++) {
if ((*this)[i] != 0) {
if (i * k > deg) return FPS(deg, 0);
R rev = mod_inverse((*this)[i]);
FPS ret = (((*this * rev) >> i).logarithm() * k).exponent()
* mod_pow((*this)[i], k);
return FPS(ret << (i * k), deg);
}
}
return *this;
}
public:
FPS rev() const {
FPS ret((*this), size());
reverse(ret.coefs.begin(), ret.coefs.end());
return ret;
}
FPS operator-() {
FPS ret = *this;
for (int i = 0; i < (int)ret.size(); i++)
ret[i] = (ret[i] == 0 ? 0 : mod - ret[i]);
return ret;
}
FPS &operator+=(const FPS &rhs) {
if (size() < rhs.size()) resize(rhs.size());
for (int i = 0; i < (int)rhs.size(); i++) mod_add((*this)[i], rhs[i]);
return *this;
}
FPS &operator-=(const FPS &rhs) {
if (size() < rhs.size()) resize(rhs.size());
for (int i = 0; i < (int)rhs.size(); i++) mod_sub((*this)[i], rhs[i]);
return *this;
}
FPS &operator*=(const FPS &rhs) { return *this = *this * rhs; }
FPS &operator/=(const FPS &rhs) {
if (size() < rhs.size()) return *this = FPS();
if (rhs.size() < 250) return *this = div_n(rhs).first;
int sq = size() - rhs.size() + 1;
return *this
= FPS(FPS(rev(), sq) * FPS(rhs.rev().inverse(sq), sq), sq).rev();
}
FPS &operator%=(const FPS &rhs) {
if (rhs.size() < 250) return *this = div_n(rhs).second;
*this -= (*this / rhs) * rhs;
shrink();
return *this;
}
FPS &operator+=(const R &v) {
mod_add((*this)[0], v);
return *this;
}
FPS &operator-=(const R &v) {
mod_sub((*this)[0], v);
return *this;
}
FPS &operator*=(const R &v) {
for (int k = 0; k < size(); k++) (*this)[k] = mod_mul((*this)[k], v);
return *this;
}
FPS operator>>(int sz) const {
if (size() <= sz) return {};
return FPS(*this, sz, size());
}
FPS operator<<(int sz) const {
FPS ret(size() + sz, 0);
for (int i = 0; i < size(); i++) ret[sz + i] = (*this)[i];
return ret;
}
FPS operator+(const FPS &rhs) const { return FPS(*this) += rhs; } // O(N)
FPS operator-(const FPS &rhs) const { return FPS(*this) -= rhs; } // O(N)
FPS operator*(const FPS &rhs) const { return this->mul(rhs); } // O(NlogN)
FPS operator/(const FPS &rhs) const { return FPS(*this) /= rhs; } // O(NlogN)
FPS operator%(const FPS &rhs) const { return FPS(*this) %= rhs; } // O(NlogN)
FPS operator+(const R &v) const { return FPS(*this) += v; } // O(1)
FPS operator-(const R &v) const { return FPS(*this) -= v; } // O(1)
FPS operator*(const R &v) const { return FPS(*this) *= v; } // O(N)
FPS diff() const { return differentiation(); } // O(N)
FPS inte() const { return integral(); } // O(N)
FPS inv(int deg = -1) const { return inverse(deg); } // O(NlogN)
FPS log(int deg = -1) const { return logarithm(deg); } // O(NlogN)
FPS exp(int deg = -1) const { return exponent(deg); } // O(NlogN)
FPS sqrt(int deg = -1) const { return square_root(deg); } // O(NlogN)
FPS pow(u64 k, int deg = -1) const { return power(k, deg); } // O(NlogN)
public:
vector<R> coefs;
static R mod;
static fast_div fast_mod;
static constexpr R SIZE = 1 << 20;
static R inve[SIZE];
static void init(R m) {
mod = m;
fast_mod = fast_div(m);
inve[1] = 1;
for (int i = 2; i < SIZE; ++i)
inve[i] = inve[mod % i] * (mod - mod / i) % fast_mod;
}
};
using FPS = FormalPowerSeries;
R FPS::mod;
FPS::fast_div FPS::fast_mod;
R FPS::inve[];
signed main() {
cin.tie(0);
ios::sync_with_stdio(0);
// tick();
FPS::init(1e9 + 7);
R p;
cin >> p;
int q_max = 2000000;
FPS a(q_max);
a[0] = 0;
a[1] = 1;
for (int i = 2; i < q_max; i++) {
a[i] = FPS::mod_mul(p, a[i - 1]);
FPS::mod_add(a[i], a[i - 2]);
}
FPS b = a * a;
int Q;
cin >> Q;
while (Q--) {
int q;
cin >> q;
cout << b[q - 2] << endl;
}
// cerr << tick() << endl;
return 0;
}