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Binomial Coefficient Small R Offline
(library/sequence/binomial_coefficient_small_r_offline.hpp)
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#include "library/sequence/binomial_coefficient_small_r_offline.hpp"
Binomial Coefficient Small R Offline
Code
#ifndef SUISEN_BINOMIAL_COEFFICIENT_SMALL_R_OFFLINE
#define SUISEN_BINOMIAL_COEFFICIENT_SMALL_R_OFFLINE
#include <algorithm>
#include <cassert>
#include <iostream>
#include <vector>
#include <atcoder/modint>
#include <atcoder/convolution>
namespace suisen {
namespace internal::binom_small_r_offline {
template <typename mint>
using Polynomial = std::vector<mint>;
// g s.t. fg=1 mod x^n
// O(nlog n)
template <typename mint>
Polynomial<mint> inv(Polynomial<mint> f, int n) {
const int size_f = f.size();
Polynomial<mint> f_fft, g_fft;
Polynomial<mint> g{ f[0].inv() };
for (int k = 1; k < n; k *= 2) {
f_fft = Polynomial<mint>(f.begin(), f.begin() + std::min(size_f, 2 * k));
g_fft = g;
f_fft.resize(2 * k);
g_fft.resize(2 * k);
atcoder::internal::butterfly(f_fft);
atcoder::internal::butterfly(g_fft);
Polynomial<mint> fg(2 * k);
for (int i = 0; i < 2 * k; ++i) {
fg[i] = f_fft[i] * g_fft[i];
}
atcoder::internal::butterfly_inv(fg);
for (int i = 0; i < k; ++i) {
fg[i] = std::exchange(fg[k + i], 0);
}
atcoder::internal::butterfly(fg);
for (int i = 0; i < 2 * k; ++i) {
fg[i] *= g_fft[i];
}
atcoder::internal::butterfly_inv(fg);
const mint inv_scale = mint(2 * k).inv(), c = -inv_scale * inv_scale;
g.resize(2 * k);
for (int i = 0; i < k; ++i) {
g[k + i] = fg[i] * c;
}
}
g.resize(n);
return g;
}
// q s.t. qg=f-(f mod g)
// O(k log k) where k:=|f|+|g|
template <typename mint>
Polynomial<mint> poly_div(Polynomial<mint> f, Polynomial<mint> g) {
while (not f.empty() and f.back() == 0) {
f.pop_back();
}
while (not g.empty() and g.back() == 0) {
g.pop_back();
}
const int deg_f = static_cast<int>(f.size()) - 1, deg_g = static_cast<int>(g.size()) - 1;
assert(deg_g >= 0);
if (deg_f < deg_g) {
return {};
}
std::reverse(f.begin(), f.end());
std::reverse(g.begin(), g.end());
const int deg_q = deg_f - deg_g;
Polynomial<mint> q = atcoder::convolution(f, inv(g, deg_q + 1));
q.resize(deg_q + 1);
std::reverse(q.begin(), q.end());
return q;
}
// f mod g
// O(k log k) where k:=|f|+|g|
template <typename mint>
Polynomial<mint> poly_mod(Polynomial<mint> f, const Polynomial<mint>& g) {
Polynomial<mint> q = poly_div(f, g);
Polynomial<mint> qg = atcoder::convolution(q, g);
for (std::size_t i = 0; i < std::min(f.size(), qg.size()); ++i) {
f[i] -= qg[i];
}
while (not f.empty() and f.back() == 0) {
f.pop_back();
}
return f;
}
// result[i][j] = S1[2^i,j] * (is_signed ? (-1)^(2^i+j) : +1) for 0<=i<t, 0<=j<=2^i
// O(t*2^t)
template <bool is_signed, typename mint>
std::vector<Polynomial<mint>> stirling_number1_doubling(int t) {
const int n = 1 << t;
std::vector<mint> fac(n + 1), fac_inv(n + 1);
fac[0] = 1;
for (int i = 1; i <= n; ++i) {
fac[i] = fac[i - 1] * i;
}
fac_inv[n] = fac[n].inv();
for (int i = n; i >= 1; --i) {
fac_inv[i - 1] = fac_inv[i] * i;
}
// S1[i][j] = S1[2^i,j] for 0<=i<t, 0<=j<=2^i
std::vector<Polynomial<mint>> S1(t);
S1[0] = { 1, 0 };
for (int l = 0; l < t - 1; ++l) {
const int m = 1 << l;
Polynomial<mint> f(m + 1), g(m + 1);
mint pow_m = 1;
for (int i = 0; i <= m; ++i) {
f[i] = pow_m * fac_inv[i];
g[i] = S1[l][i] * fac[m - i];
pow_m *= (is_signed ? -m : +m);
}
f = atcoder::convolution(f, g);
f.resize(m + 1);
for (int i = 0; i <= m; ++i) {
f[i] *= fac_inv[m - i];
}
S1[l + 1] = atcoder::convolution(S1[l], f);
}
for (auto& S1_i : S1) {
std::reverse(S1_i.begin(), S1_i.end());
}
return S1;
}
// O(|xs|*log(t)^2) where t:=min(|f|,|xs|)
template <typename mint>
std::vector<mint> multipoint_evaluation(const Polynomial<mint>& f, const std::vector<mint>& xs) {
const int n = f.size();
const int m = xs.size();
if (n == 0) {
return std::vector<mint>(m, 0);
}
auto impl = [&f](const std::vector<mint>& xs) {
const int n = xs.size();
std::vector<Polynomial<mint>> seg(2 * n);
for (int i = 0; i < n; ++i) {
seg[n + i] = Polynomial<mint>{ -xs[i], 1 };
}
for (int i = n - 1; i > 0; --i) {
seg[i] = atcoder::convolution(seg[i * 2], seg[i * 2 + 1]);
}
seg[1] = poly_mod(f, seg[1]);
for (int i = 2; i < 2 * n; ++i) {
seg[i] = poly_mod(seg[i / 2], seg[i]);
}
std::vector<mint> ys(n);
for (int i = 0; i < n; ++i) {
ys[i] = seg[n + i].size() ? seg[n + i][0] : 0;
}
return ys;
};
std::vector<mint> ys(m);
for (int l = 0; l < m; l += n) {
int r = std::min(m, l + n);
std::vector<mint> ys_lr = impl(std::vector<mint>(xs.begin() + l, xs.begin() + r));
std::move(ys_lr.begin(), ys_lr.end(), ys.begin() + l);
}
return ys;
}
}
// O(QlogR(log min(Q,R))^2 + RlogR)
template <typename mint>
std::vector<mint> binomial_coefficient_small_r_offline(std::vector<std::pair<long long, int>> queries) {
using namespace internal::binom_small_r_offline;
const int q = queries.size();
const int max_r = [&] {
int max_r = 0;
for (auto [n, r] : queries) {
max_r = std::max(max_r, r);
}
return max_r;
}(); // r <= max_r
const int log_r = [&] {
int log_r = 0;
while (1 << log_r <= max_r) {
++log_r;
}
return log_r;
}(); // r < 2^(log_r)
const std::vector<mint> fac_inv = [&] {
mint fac_max_r = 1;
for (int i = 1; i <= max_r; ++i) {
fac_max_r = fac_max_r * i;
}
std::vector<mint> fac_inv(max_r + 1);
fac_inv[max_r] = fac_max_r.inv();
for (int i = max_r; i >= 1; --i) {
fac_inv[i - 1] = fac_inv[i] * i;
}
return fac_inv;
}();
std::vector<mint> ans(q);
for (int i = 0; i < q; ++i) {
ans[i] = fac_inv[queries[i].second];
}
// O(R log R)
auto S1 = stirling_number1_doubling<true, mint>(log_r);
// sum[bit=0,log R] O(Q*bit^2) = O(Q(log R)^3)
for (int bit = 0; bit < log_r; ++bit) {
// x (x - 1) (x - 2) ... (x - 2^bit + 1)
std::vector<mint> xs;
std::vector<int> ids;
// O(Q)
for (int i = 0; i < q; ++i) {
auto& [n, r] = queries[i];
if ((r >> bit) & 1) {
xs.push_back(n);
ids.push_back(i);
n -= 1 << bit;
r -= 1 << bit;
}
}
const int size = xs.size();
// O(min(Q*bit^2, Q(logQ)^2 + bit 2^bit))
std::vector<mint> ys = multipoint_evaluation<mint>(S1[bit], xs);
for (int i = 0; i < size; ++i) {
ans[ids[i]] *= ys[i];
}
}
return ans;
}
}
#endif // SUISEN_BINOMIAL_COEFFICIENT_SMALL_R_OFFLINE#line 1 "library/sequence/binomial_coefficient_small_r_offline.hpp"
#include <algorithm>
#include <cassert>
#include <iostream>
#include <vector>
#include <atcoder/modint>
#include <atcoder/convolution>
namespace suisen {
namespace internal::binom_small_r_offline {
template <typename mint>
using Polynomial = std::vector<mint>;
// g s.t. fg=1 mod x^n
// O(nlog n)
template <typename mint>
Polynomial<mint> inv(Polynomial<mint> f, int n) {
const int size_f = f.size();
Polynomial<mint> f_fft, g_fft;
Polynomial<mint> g{ f[0].inv() };
for (int k = 1; k < n; k *= 2) {
f_fft = Polynomial<mint>(f.begin(), f.begin() + std::min(size_f, 2 * k));
g_fft = g;
f_fft.resize(2 * k);
g_fft.resize(2 * k);
atcoder::internal::butterfly(f_fft);
atcoder::internal::butterfly(g_fft);
Polynomial<mint> fg(2 * k);
for (int i = 0; i < 2 * k; ++i) {
fg[i] = f_fft[i] * g_fft[i];
}
atcoder::internal::butterfly_inv(fg);
for (int i = 0; i < k; ++i) {
fg[i] = std::exchange(fg[k + i], 0);
}
atcoder::internal::butterfly(fg);
for (int i = 0; i < 2 * k; ++i) {
fg[i] *= g_fft[i];
}
atcoder::internal::butterfly_inv(fg);
const mint inv_scale = mint(2 * k).inv(), c = -inv_scale * inv_scale;
g.resize(2 * k);
for (int i = 0; i < k; ++i) {
g[k + i] = fg[i] * c;
}
}
g.resize(n);
return g;
}
// q s.t. qg=f-(f mod g)
// O(k log k) where k:=|f|+|g|
template <typename mint>
Polynomial<mint> poly_div(Polynomial<mint> f, Polynomial<mint> g) {
while (not f.empty() and f.back() == 0) {
f.pop_back();
}
while (not g.empty() and g.back() == 0) {
g.pop_back();
}
const int deg_f = static_cast<int>(f.size()) - 1, deg_g = static_cast<int>(g.size()) - 1;
assert(deg_g >= 0);
if (deg_f < deg_g) {
return {};
}
std::reverse(f.begin(), f.end());
std::reverse(g.begin(), g.end());
const int deg_q = deg_f - deg_g;
Polynomial<mint> q = atcoder::convolution(f, inv(g, deg_q + 1));
q.resize(deg_q + 1);
std::reverse(q.begin(), q.end());
return q;
}
// f mod g
// O(k log k) where k:=|f|+|g|
template <typename mint>
Polynomial<mint> poly_mod(Polynomial<mint> f, const Polynomial<mint>& g) {
Polynomial<mint> q = poly_div(f, g);
Polynomial<mint> qg = atcoder::convolution(q, g);
for (std::size_t i = 0; i < std::min(f.size(), qg.size()); ++i) {
f[i] -= qg[i];
}
while (not f.empty() and f.back() == 0) {
f.pop_back();
}
return f;
}
// result[i][j] = S1[2^i,j] * (is_signed ? (-1)^(2^i+j) : +1) for 0<=i<t, 0<=j<=2^i
// O(t*2^t)
template <bool is_signed, typename mint>
std::vector<Polynomial<mint>> stirling_number1_doubling(int t) {
const int n = 1 << t;
std::vector<mint> fac(n + 1), fac_inv(n + 1);
fac[0] = 1;
for (int i = 1; i <= n; ++i) {
fac[i] = fac[i - 1] * i;
}
fac_inv[n] = fac[n].inv();
for (int i = n; i >= 1; --i) {
fac_inv[i - 1] = fac_inv[i] * i;
}
// S1[i][j] = S1[2^i,j] for 0<=i<t, 0<=j<=2^i
std::vector<Polynomial<mint>> S1(t);
S1[0] = { 1, 0 };
for (int l = 0; l < t - 1; ++l) {
const int m = 1 << l;
Polynomial<mint> f(m + 1), g(m + 1);
mint pow_m = 1;
for (int i = 0; i <= m; ++i) {
f[i] = pow_m * fac_inv[i];
g[i] = S1[l][i] * fac[m - i];
pow_m *= (is_signed ? -m : +m);
}
f = atcoder::convolution(f, g);
f.resize(m + 1);
for (int i = 0; i <= m; ++i) {
f[i] *= fac_inv[m - i];
}
S1[l + 1] = atcoder::convolution(S1[l], f);
}
for (auto& S1_i : S1) {
std::reverse(S1_i.begin(), S1_i.end());
}
return S1;
}
// O(|xs|*log(t)^2) where t:=min(|f|,|xs|)
template <typename mint>
std::vector<mint> multipoint_evaluation(const Polynomial<mint>& f, const std::vector<mint>& xs) {
const int n = f.size();
const int m = xs.size();
if (n == 0) {
return std::vector<mint>(m, 0);
}
auto impl = [&f](const std::vector<mint>& xs) {
const int n = xs.size();
std::vector<Polynomial<mint>> seg(2 * n);
for (int i = 0; i < n; ++i) {
seg[n + i] = Polynomial<mint>{ -xs[i], 1 };
}
for (int i = n - 1; i > 0; --i) {
seg[i] = atcoder::convolution(seg[i * 2], seg[i * 2 + 1]);
}
seg[1] = poly_mod(f, seg[1]);
for (int i = 2; i < 2 * n; ++i) {
seg[i] = poly_mod(seg[i / 2], seg[i]);
}
std::vector<mint> ys(n);
for (int i = 0; i < n; ++i) {
ys[i] = seg[n + i].size() ? seg[n + i][0] : 0;
}
return ys;
};
std::vector<mint> ys(m);
for (int l = 0; l < m; l += n) {
int r = std::min(m, l + n);
std::vector<mint> ys_lr = impl(std::vector<mint>(xs.begin() + l, xs.begin() + r));
std::move(ys_lr.begin(), ys_lr.end(), ys.begin() + l);
}
return ys;
}
}
// O(QlogR(log min(Q,R))^2 + RlogR)
template <typename mint>
std::vector<mint> binomial_coefficient_small_r_offline(std::vector<std::pair<long long, int>> queries) {
using namespace internal::binom_small_r_offline;
const int q = queries.size();
const int max_r = [&] {
int max_r = 0;
for (auto [n, r] : queries) {
max_r = std::max(max_r, r);
}
return max_r;
}(); // r <= max_r
const int log_r = [&] {
int log_r = 0;
while (1 << log_r <= max_r) {
++log_r;
}
return log_r;
}(); // r < 2^(log_r)
const std::vector<mint> fac_inv = [&] {
mint fac_max_r = 1;
for (int i = 1; i <= max_r; ++i) {
fac_max_r = fac_max_r * i;
}
std::vector<mint> fac_inv(max_r + 1);
fac_inv[max_r] = fac_max_r.inv();
for (int i = max_r; i >= 1; --i) {
fac_inv[i - 1] = fac_inv[i] * i;
}
return fac_inv;
}();
std::vector<mint> ans(q);
for (int i = 0; i < q; ++i) {
ans[i] = fac_inv[queries[i].second];
}
// O(R log R)
auto S1 = stirling_number1_doubling<true, mint>(log_r);
// sum[bit=0,log R] O(Q*bit^2) = O(Q(log R)^3)
for (int bit = 0; bit < log_r; ++bit) {
// x (x - 1) (x - 2) ... (x - 2^bit + 1)
std::vector<mint> xs;
std::vector<int> ids;
// O(Q)
for (int i = 0; i < q; ++i) {
auto& [n, r] = queries[i];
if ((r >> bit) & 1) {
xs.push_back(n);
ids.push_back(i);
n -= 1 << bit;
r -= 1 << bit;
}
}
const int size = xs.size();
// O(min(Q*bit^2, Q(logQ)^2 + bit 2^bit))
std::vector<mint> ys = multipoint_evaluation<mint>(S1[bit], xs);
for (int i = 0; i < size; ++i) {
ans[ids[i]] *= ys[i];
}
}
return ans;
}
}