large_factorial.hpp¶
不建立线性阶乘表，批量计算多个大参数的 \(n!\bmod p\)。
\[
\displaystyle n!=\prod_{i=1}^{n}i
\]
Complexity: Time: large_factorials uses \(O(M (\sqrt{N}) \log N + T \sqrt{N})\); many_factorials, with \(B = 2^{15}\), uses \(O(M (B) \log B + M (K) \log (K + 1) + T \log B + \sum_{b} \lceil Q_{b} / 2^{b}\rceil M (2^{b}) \log (2^{b}))\), where \(K = \lceil N / B\rceil\) and \(Q_{b}\) is the number of queries whose remainder has bit \(b\) set. Space: \(O(\sqrt{N} \log N + T)\) and \(O(B \log B + K \log (K + 1) + T)\), respectively. \(N \ge 2\) bounds the query arguments; \(T\) is the query count.
AC 记录：factorial, many_factorials。
Implementation¶
当前头文件，省略 include guard；依赖见 #include。
/// @complexity Time: `large_factorials` uses
/// O(M(sqrt N) log N + T sqrt N); `many_factorials`, with B=2^15, uses
/// O(M(B) log B + M(K) log(K+1) + T log B +
/// sum_b ceil(Q_b/2^b) M(2^b) log(2^b)), where K=ceil(N/B) and Q_b is
/// the number of queries whose remainder has bit b set.
/// Space: O(sqrt N log N + T) and O(B log B + K log(K+1) + T), respectively.
/// N >= 2 bounds the query arguments; T is the query count.
#include "noya/math/combinatorial_sequences.hpp"
#include "noya/math/polynomial_multipoint.hpp"
#include "noya/math/polynomial_special_points.hpp"
#include <algorithm>
#include <cassert>
#include <cmath>
#include <cstdint>
#include <utility>
#include <vector>
namespace noya {
namespace large_factorial_detail {
template <class Mint>
std::vector<Mint> factorial_block_prefix(std::uint64_t mx, int blk) {
int nb = int(mx / blk);
std::vector<Mint> pre(nb + 1, Mint(1));
if (nb == 0) {
return pre;
}
std::vector<std::vector<Mint>> fs;
fs.reserve(blk);
for (int i = 1; i <= blk; i++) {
fs.push_back({Mint(i), Mint(1)});
}
std::vector<Mint> bp = polynomial_product_sequence(std::move(fs));
std::vector<Mint> poi(nb);
for (int i = 0; i < nb; i++) {
poi[i] = Mint(std::uint64_t(i) * blk);
}
std::vector<Mint> prd = polynomial_multipoint_evaluation(bp, poi);
for (int i = 0; i < nb; i++) {
pre[i + 1] = pre[i] * prd[i];
}
return pre;
}
} // namespace large_factorial_detail
/// @brief Compute several factorials modulo a fixed prime without a linear
/// table. Split 1..N into blocks of length B about sqrt(N). The product inside
/// one block is the degree-B polynomial P(x)=prod_{i=1}^B(x+i); a product tree
/// constructs P and multipoint evaluation obtains P(0),P(B),P(2B),.... Prefix
/// products answer every full block, followed by at most B direct factors.
/// Requires each argument below Mint::mod(), a prime modulus and the
/// convolution lengths supported by ACL.
template <class Mint>
std::vector<Mint> large_factorials(const std::vector<std::uint64_t> &que) {
if (que.empty()) {
return {};
}
std::uint64_t mx = *std::max_element(que.begin(), que.end());
assert(mx < std::uint64_t(Mint::mod()));
int blk = int(std::sqrt(static_cast<long double>(mx + 1)));
blk = std::max(blk, 1);
while (std::uint64_t(blk) * blk < mx + 1) {
blk++;
}
std::vector<Mint> pre =
large_factorial_detail::factorial_block_prefix<Mint>(mx, blk);
std::vector<Mint> res;
res.reserve(que.size());
for (std::uint64_t n : que) {
std::uint64_t com = n / blk;
Mint val = pre[com];
for (std::uint64_t i = com * blk + 1; i <= n; i++) {
val *= Mint(i);
}
res.push_back(val);
}
return res;
}
/// @brief Compute a large batch of factorials modulo a fixed prime. Boundary
/// values (kB)! are obtained by evaluating the block-product polynomial
/// prod_{i=1}^B(x+i). For a query n=qB+r, split the remaining product
/// (qB+1)...n into power-of-two suffixes. A suffix of length 2^b is the falling
/// factorial polynomial x(x-1)...(x-2^b+1) evaluated at its current right
/// endpoint. Queries sharing b are evaluated in batches of at most 2^b points,
/// retaining only one batch of points and query indices at a time.
/// Requires each argument below Mint::mod(), a prime modulus and the
/// convolution lengths supported by ACL.
template <class Mint>
std::vector<Mint> many_factorials(const std::vector<std::uint32_t> &que) {
if (que.empty()) {
return {};
}
constexpr int lg = 15;
constexpr int blk = 1 << lg;
std::uint32_t mx = *std::max_element(que.begin(), que.end());
assert(mx < std::uint32_t(Mint::mod()));
std::vector<Mint> pre =
large_factorial_detail::factorial_block_prefix<Mint>(mx, blk);
std::vector<Mint> res(que.size());
std::uint32_t msk = 0;
for (int qry = 0; qry < int(que.size()); qry++) {
res[qry] = pre[que[qry] / blk];
msk |= que[qry] % blk;
}
for (int bit = 0; bit < lg; bit++) {
int len = 1 << bit;
if (!(msk & len))
continue;
std::vector<Mint> ff = stirling_first_kind_row<Mint>(len), poi;
std::vector<int> ids;
int cap = std::min(len, int(que.size()));
poi.reserve(cap);
ids.reserve(cap);
auto flush = [&] {
if (poi.empty())
return;
if (len <= 32) {
// Match the product tree's small-polynomial Horner cutoff.
for (int i = 0; i < int(ids.size()); ++i) {
Mint val = 0;
for (int j = len; j >= 0; --j)
val = val * poi[i] + ff[j];
res[ids[i]] *= val;
}
} else {
auto vs = polynomial_multipoint_evaluation(ff, poi);
for (int i = 0; i < int(ids.size()); ++i)
res[ids[i]] *= vs[i];
}
poi.clear();
ids.clear();
};
for (int qry = 0; qry < int(que.size()); ++qry) {
auto n = que[qry];
if (!(n & len))
continue;
// Lower suffixes have already consumed the low bits.
poi.emplace_back(n - (n & (len - 1)));
ids.push_back(qry);
if (int(poi.size()) == len)
flush();
}
flush();
}
return res;
}
} // namespace noya
// NOYA_LARGE_FACTORIAL_HPP
#ifndef NOYA_LARGE_FACTORIAL_HPP
#define NOYA_LARGE_FACTORIAL_HPP 1
/// @complexity Time: `large_factorials` uses
/// O(M(sqrt N) log N + T sqrt N); `many_factorials`, with B=2^15, uses
/// O(M(B) log B + M(K) log(K+1) + T log B +
/// sum_b ceil(Q_b/2^b) M(2^b) log(2^b)), where K=ceil(N/B) and Q_b is
/// the number of queries whose remainder has bit b set.
/// Space: O(sqrt N log N + T) and O(B log B + K log(K+1) + T), respectively.
/// N >= 2 bounds the query arguments; T is the query count.
#include "noya/math/combinatorial_sequences.hpp"
#include "noya/math/polynomial_multipoint.hpp"
#include "noya/math/polynomial_special_points.hpp"
#include <algorithm>
#include <cassert>
#include <cmath>
#include <cstdint>
#include <utility>
#include <vector>
namespace noya {
namespace large_factorial_detail {
template <class Mint>
std::vector<Mint> factorial_block_prefix(std::uint64_t mx, int blk) {
int nb = int(mx / blk);
std::vector<Mint> pre(nb + 1, Mint(1));
if (nb == 0) {
return pre;
}
std::vector<std::vector<Mint>> fs;
fs.reserve(blk);
for (int i = 1; i <= blk; i++) {
fs.push_back({Mint(i), Mint(1)});
}
std::vector<Mint> bp = polynomial_product_sequence(std::move(fs));
std::vector<Mint> poi(nb);
for (int i = 0; i < nb; i++) {
poi[i] = Mint(std::uint64_t(i) * blk);
}
std::vector<Mint> prd = polynomial_multipoint_evaluation(bp, poi);
for (int i = 0; i < nb; i++) {
pre[i + 1] = pre[i] * prd[i];
}
return pre;
}
} // namespace large_factorial_detail
/// @brief Compute several factorials modulo a fixed prime without a linear
/// table. Split 1..N into blocks of length B about sqrt(N). The product inside
/// one block is the degree-B polynomial P(x)=prod_{i=1}^B(x+i); a product tree
/// constructs P and multipoint evaluation obtains P(0),P(B),P(2B),.... Prefix
/// products answer every full block, followed by at most B direct factors.
/// Requires each argument below Mint::mod(), a prime modulus and the
/// convolution lengths supported by ACL.
template <class Mint>
std::vector<Mint> large_factorials(const std::vector<std::uint64_t> &que) {
if (que.empty()) {
return {};
}
std::uint64_t mx = *std::max_element(que.begin(), que.end());
assert(mx < std::uint64_t(Mint::mod()));
int blk = int(std::sqrt(static_cast<long double>(mx + 1)));
blk = std::max(blk, 1);
while (std::uint64_t(blk) * blk < mx + 1) {
blk++;
}
std::vector<Mint> pre =
large_factorial_detail::factorial_block_prefix<Mint>(mx, blk);
std::vector<Mint> res;
res.reserve(que.size());
for (std::uint64_t n : que) {
std::uint64_t com = n / blk;
Mint val = pre[com];
for (std::uint64_t i = com * blk + 1; i <= n; i++) {
val *= Mint(i);
}
res.push_back(val);
}
return res;
}
/// @brief Compute a large batch of factorials modulo a fixed prime. Boundary
/// values (kB)! are obtained by evaluating the block-product polynomial
/// prod_{i=1}^B(x+i). For a query n=qB+r, split the remaining product
/// (qB+1)...n into power-of-two suffixes. A suffix of length 2^b is the falling
/// factorial polynomial x(x-1)...(x-2^b+1) evaluated at its current right
/// endpoint. Queries sharing b are evaluated in batches of at most 2^b points,
/// retaining only one batch of points and query indices at a time.
/// Requires each argument below Mint::mod(), a prime modulus and the
/// convolution lengths supported by ACL.
template <class Mint>
std::vector<Mint> many_factorials(const std::vector<std::uint32_t> &que) {
if (que.empty()) {
return {};
}
constexpr int lg = 15;
constexpr int blk = 1 << lg;
std::uint32_t mx = *std::max_element(que.begin(), que.end());
assert(mx < std::uint32_t(Mint::mod()));
std::vector<Mint> pre =
large_factorial_detail::factorial_block_prefix<Mint>(mx, blk);
std::vector<Mint> res(que.size());
std::uint32_t msk = 0;
for (int qry = 0; qry < int(que.size()); qry++) {
res[qry] = pre[que[qry] / blk];
msk |= que[qry] % blk;
}
for (int bit = 0; bit < lg; bit++) {
int len = 1 << bit;
if (!(msk & len))
continue;
std::vector<Mint> ff = stirling_first_kind_row<Mint>(len), poi;
std::vector<int> ids;
int cap = std::min(len, int(que.size()));
poi.reserve(cap);
ids.reserve(cap);
auto flush = [&] {
if (poi.empty())
return;
if (len <= 32) {
// Match the product tree's small-polynomial Horner cutoff.
for (int i = 0; i < int(ids.size()); ++i) {
Mint val = 0;
for (int j = len; j >= 0; --j)
val = val * poi[i] + ff[j];
res[ids[i]] *= val;
}
} else {
auto vs = polynomial_multipoint_evaluation(ff, poi);
for (int i = 0; i < int(ids.size()); ++i)
res[ids[i]] *= vs[i];
}
poi.clear();
ids.clear();
};
for (int qry = 0; qry < int(que.size()); ++qry) {
auto n = que[qry];
if (!(n & len))
continue;
// Lower suffixes have already consumed the low bits.
poi.emplace_back(n - (n & (len - 1)));
ids.push_back(qry);
if (int(poi.size()) == len)
flush();
}
flush();
}
return res;
}
} // namespace noya
#endif // NOYA_LARGE_FACTORIAL_HPP
/// @complexity Time: `large_factorials` uses
/// O(M(sqrt N) log N + T sqrt N); `many_factorials`, with B=2^15, uses
/// O(M(B) log B + M(K) log(K+1) + T log B +
/// sum_b ceil(Q_b/2^b) M(2^b) log(2^b)), where K=ceil(N/B) and Q_b is
/// the number of queries whose remainder has bit b set.
/// Space: O(sqrt N log N + T) and O(B log B + K log(K+1) + T), respectively.
/// N >= 2 bounds the query arguments; T is the query count.
/// @complexity Time: O(M(n) log n + n log mod) as a common upper bound;
/// factorials O(n + log mod), first-kind row O(M(n) + log n log mod).
/// Space: O(n) coefficients. M(n) is polynomial multiplication time.
/// Requires n >= 0, prime-modulus Mint and the convolution lengths supported
/// by ACL. Factorial-based formulas require n < mod (n + 1 < mod for Bernoulli).
/// @complexity Time: O(M(n) log n) for inverse/log/exp with convolution cost M(n).
/// Space: O(n) temporary coefficients.
/// @complexity Time: O(c log p + log^2 p) for c tested nonresidue candidates.
/// Space: O(1).
/// @complexity Time: O(log n) modular multiplications for primality testing;
/// Pollard-rho factorization is expected about O(n^(1/4)).
/// Space: O(log n) recursion and factors.
#include <algorithm>
#include <array>
#include <cassert>
#include <cstdint>
#include <numeric>
#include <utility>
#include <vector>
namespace noya {
namespace factorize_internal {
using u64 = std::uint64_t;
using u128 = unsigned __int128;
inline u64 multiply_mod(u64 a, u64 b, u64 mod) {
return u64(u128(a) * b % mod);
}
inline u64 power_mod(u64 a, u64 exp, u64 mod) {
u64 res = 1;
while (exp > 0) {
if (exp & 1) {
res = multiply_mod(res, a, mod);
}
a = multiply_mod(a, a, mod);
exp >>= 1;
}
return res;
}
inline bool miller_rabin(u64 n) {
if (n < 2) {
return false;
}
for (u64 p :
std::array<u64, 12>{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37}) {
if (n % p == 0) {
return n == p;
}
}
int shf = __builtin_ctzll(n - 1);
u64 odd = (n - 1) >> shf;
for (u64 bas :
std::array<u64, 7>{2, 325, 9375, 28178, 450775, 9780504, 1795265022}) {
if (bas % n == 0) {
continue;
}
u64 val = power_mod(bas % n, odd, n);
if (val == 1 || val == n - 1) {
continue;
}
bool cmp = true;
for (int i = 1; i < shf; i++) {
val = multiply_mod(val, val, n);
if (val == n - 1) {
cmp = false;
break;
}
}
if (cmp) {
return false;
}
}
return true;
}
inline u64 splitmix64(u64 &st) {
u64 z = (st += 0x9e3779b97f4a7c15ULL);
z = (z ^ (z >> 30)) * 0xbf58476d1ce4e5b9ULL;
z = (z ^ (z >> 27)) * 0x94d049bb133111ebULL;
return z ^ (z >> 31);
}
inline u64 pollard_rho(u64 n) {
if (n % 2 == 0) {
return 2;
}
if (n % 3 == 0) {
return 3;
}
static u64 st = 0x123456789abcdef0ULL;
while (true) {
u64 y = splitmix64(st) % (n - 1) + 1;
u64 c = splitmix64(st) % (n - 1) + 1;
constexpr u64 blk = 128;
u64 g = 1;
u64 r = 1;
u64 q = 1;
u64 x = 0;
u64 sy = 0;
auto nxt = [&](u64 val) {
val = multiply_mod(val, val, n);
// Add c modulo n without overflowing 64 bits.
return val >= n - c ? val - (n - c) : val + c;
};
while (g == 1) {
x = y;
for (u64 i = 0; i < r; i++) {
y = nxt(y);
}
for (u64 off = 0; off < r && g == 1; off += blk) {
sy = y;
for (u64 i = 0; i < std::min(blk, r - off); i++) {
y = nxt(y);
u64 dif = x > y ? x - y : y - x;
q = multiply_mod(q, dif, n);
}
g = std::gcd(q, n);
}
r <<= 1;
}
if (g == n) {
do {
sy = nxt(sy);
u64 dif = x > sy ? x - sy : sy - x;
g = std::gcd(dif, n);
} while (g == 1);
}
if (g != n) {
return g;
}
}
}
inline void collect_factors(u64 n, std::vector<u64> &res) {
if (n == 1) {
return;
}
if (miller_rabin(n)) {
res.push_back(n);
return;
}
u64 fct = pollard_rho(n);
collect_factors(fct, res);
collect_factors(n / fct, res);
}
} // namespace factorize_internal
/// @brief Deterministic Miller-Rabin primality test for unsigned 64-bit
/// integers.
inline bool is_prime(std::uint64_t n) {
return factorize_internal::miller_rabin(n);
}
/// @brief Return the prime factors of n with multiplicity in increasing order.
inline std::vector<std::uint64_t> prime_factors(std::uint64_t n) {
assert(n >= 1);
std::vector<std::uint64_t> res;
factorize_internal::collect_factors(n, res);
std::sort(res.begin(), res.end());
return res;
}
/// @brief Return the prime factorization of n as (prime, exponent) pairs.
inline std::vector<std::pair<std::uint64_t, int>> factorize(std::uint64_t n) {
std::vector<std::pair<std::uint64_t, int>> res;
for (std::uint64_t p : prime_factors(n)) {
if (res.empty() || res.back().first != p) {
res.emplace_back(p, 1);
} else {
res.back().second++;
}
}
return res;
}
} // namespace noya
#include <algorithm>
#include <cassert>
#include <cstdint>
#include <optional>
namespace noya {
/// @brief Compute the smaller square root modulo a prime, or nullopt if no
/// square root exists.
inline std::optional<std::uint64_t> mod_sqrt(std::uint64_t val,
std::uint64_t mod) {
assert(mod >= 2 && is_prime(mod));
val %= mod;
if (mod == 2 || val == 0) {
return val;
}
using factorize_internal::multiply_mod;
using factorize_internal::power_mod;
if (mod % 4 == 3) {
std::uint64_t rt = power_mod(val, (mod + 1) / 4, mod);
if (multiply_mod(rt, rt, mod) != val) return std::nullopt;
return std::min(rt, mod - rt);
}
if (power_mod(val, (mod - 1) / 2, mod) != 1) {
return std::nullopt;
}
std::uint64_t odd = mod - 1;
int exp = 0;
while ((odd & 1) == 0) {
odd >>= 1;
exp++;
}
std::uint64_t nqr = 2;
while (power_mod(nqr, (mod - 1) / 2, mod) != mod - 1) {
nqr++;
}
std::uint64_t rt = power_mod(val, (odd + 1) / 2, mod);
std::uint64_t rem = power_mod(val, odd, mod);
std::uint64_t ste = power_mod(nqr, odd, mod);
int rmn = exp;
while (rem != 1) {
std::uint64_t squ = rem;
int shf = 0;
while (squ != 1 && shf < rmn) {
squ = multiply_mod(squ, squ, mod);
shf++;
}
assert(shf < rmn);
std::uint64_t mul = ste;
for (int i = 0; i < rmn - shf - 1; ++i)
mul = multiply_mod(mul, mul, mod);
rt = multiply_mod(rt, mul, mod);
ste = multiply_mod(mul, mul, mod);
rem = multiply_mod(rem, ste, mod);
rmn = shf;
}
return std::min(rt, mod - rt);
}
} // namespace noya
/// @complexity Time: O(M(n) log n) inverse/division and O(M(n)) Taylor shift.
/// Space: O(n) temporaries.
#include <algorithm>
#include <array>
#include <cassert>
#include <type_traits>
#include <vector>
#ifdef _MSC_VER
#include <intrin.h>
#endif
#if __cplusplus >= 202002L
#include <bit>
#endif
namespace atcoder {
namespace internal {
#if __cplusplus >= 202002L
using std::bit_ceil;
#else
// @return same with std::bit::bit_ceil
unsigned int bit_ceil(unsigned int n) {
unsigned int x = 1;
while (x < (unsigned int)(n)) x *= 2;
return x;
}
#endif
// @param n `1 <= n`
// @return same with std::bit::countr_zero
int countr_zero(unsigned int n) {
#ifdef _MSC_VER
unsigned long index;
_BitScanForward(&index, n);
return index;
#else
return __builtin_ctz(n);
#endif
}
// @param n `1 <= n`
// @return same with std::bit::countr_zero
constexpr int countr_zero_constexpr(unsigned int n) {
int x = 0;
while (!(n & (1 << x))) x++;
return x;
}
} // namespace internal
} // namespace atcoder
#include <cassert>
#include <numeric>
#include <type_traits>
#ifdef _MSC_VER
#include <intrin.h>
#endif
#include <utility>
#ifdef _MSC_VER
#include <intrin.h>
#endif
namespace atcoder {
namespace internal {
// @param m `1 <= m`
// @return x mod m
constexpr long long safe_mod(long long x, long long m) {
x %= m;
if (x < 0) x += m;
return x;
}
// Fast modular multiplication by barrett reduction
// Reference: https://en.wikipedia.org/wiki/Barrett_reduction
// NOTE: reconsider after Ice Lake
struct barrett {
unsigned int _m;
unsigned long long im;
// @param m `1 <= m`
explicit barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}
// @return m
unsigned int umod() const { return _m; }
// @param a `0 <= a < m`
// @param b `0 <= b < m`
// @return `a * b % m`
unsigned int mul(unsigned int a, unsigned int b) const {
// [1] m = 1
// a = b = im = 0, so okay
// [2] m >= 2
// im = ceil(2^64 / m)
// -> im * m = 2^64 + r (0 <= r < m)
// let z = a*b = c*m + d (0 <= c, d < m)
// a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im
// c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2
// ((ab * im) >> 64) == c or c + 1
unsigned long long z = a;
z *= b;
#ifdef _MSC_VER
unsigned long long x;
_umul128(z, im, &x);
#else
unsigned long long x =
(unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
unsigned long long y = x * _m;
return (unsigned int)(z - y + (z < y ? _m : 0));
}
};
// @param n `0 <= n`
// @param m `1 <= m`
// @return `(x ** n) % m`
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
if (m == 1) return 0;
unsigned int _m = (unsigned int)(m);
unsigned long long r = 1;
unsigned long long y = safe_mod(x, m);
while (n) {
if (n & 1) r = (r * y) % _m;
y = (y * y) % _m;
n >>= 1;
}
return r;
}
// Reference:
// M. Forisek and J. Jancina,
// Fast Primality Testing for Integers That Fit into a Machine Word
// @param n `0 <= n`
constexpr bool is_prime_constexpr(int n) {
if (n <= 1) return false;
if (n == 2 || n == 7 || n == 61) return true;
if (n % 2 == 0) return false;
long long d = n - 1;
while (d % 2 == 0) d /= 2;
constexpr long long bases[3] = {2, 7, 61};
for (long long a : bases) {
long long t = d;
long long y = pow_mod_constexpr(a, t, n);
while (t != n - 1 && y != 1 && y != n - 1) {
y = y * y % n;
t <<= 1;
}
if (y != n - 1 && t % 2 == 0) {
return false;
}
}
return true;
}
template <int n> constexpr bool is_prime = is_prime_constexpr(n);
// @param b `1 <= b`
// @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
a = safe_mod(a, b);
if (a == 0) return {b, 0};
// Contracts:
// [1] s - m0 * a = 0 (mod b)
// [2] t - m1 * a = 0 (mod b)
// [3] s * |m1| + t * |m0| <= b
long long s = b, t = a;
long long m0 = 0, m1 = 1;
while (t) {
long long u = s / t;
s -= t * u;
m0 -= m1 * u; // |m1 * u| <= |m1| * s <= b
// [3]:
// (s - t * u) * |m1| + t * |m0 - m1 * u|
// <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)
// = s * |m1| + t * |m0| <= b
auto tmp = s;
s = t;
t = tmp;
tmp = m0;
m0 = m1;
m1 = tmp;
}
// by [3]: |m0| <= b/g
// by g != b: |m0| < b/g
if (m0 < 0) m0 += b / s;
return {s, m0};
}
// Compile time primitive root
// @param m must be prime
// @return primitive root (and minimum in now)
constexpr int primitive_root_constexpr(int m) {
if (m == 2) return 1;
if (m == 167772161) return 3;
if (m == 469762049) return 3;
if (m == 754974721) return 11;
if (m == 998244353) return 3;
int divs[20] = {};
divs[0] = 2;
int cnt = 1;
int x = (m - 1) / 2;
while (x % 2 == 0) x /= 2;
for (int i = 3; (long long)(i)*i <= x; i += 2) {
if (x % i == 0) {
divs[cnt++] = i;
while (x % i == 0) {
x /= i;
}
}
}
if (x > 1) {
divs[cnt++] = x;
}
for (int g = 2;; g++) {
bool ok = true;
for (int i = 0; i < cnt; i++) {
if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
ok = false;
break;
}
}
if (ok) return g;
}
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);
// @param n `n < 2^32`
// @param m `1 <= m < 2^32`
// @return sum_{i=0}^{n-1} floor((ai + b) / m) (mod 2^64)
unsigned long long floor_sum_unsigned(unsigned long long n,
unsigned long long m,
unsigned long long a,
unsigned long long b) {
unsigned long long ans = 0;
while (true) {
if (a >= m) {
ans += n * (n - 1) / 2 * (a / m);
a %= m;
}
if (b >= m) {
ans += n * (b / m);
b %= m;
}
unsigned long long y_max = a * n + b;
if (y_max < m) break;
// y_max < m * (n + 1)
// floor(y_max / m) <= n
n = (unsigned long long)(y_max / m);
b = (unsigned long long)(y_max % m);
std::swap(m, a);
}
return ans;
}
} // namespace internal
} // namespace atcoder
#include <cassert>
#include <numeric>
#include <type_traits>
namespace atcoder {
namespace internal {
#ifndef _MSC_VER
template <class T>
using is_signed_int128 =
typename std::conditional<std::is_same<T, __int128_t>::value ||
std::is_same<T, __int128>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int128 =
typename std::conditional<std::is_same<T, __uint128_t>::value ||
std::is_same<T, unsigned __int128>::value,
std::true_type,
std::false_type>::type;
template <class T>
using make_unsigned_int128 =
typename std::conditional<std::is_same<T, __int128_t>::value,
__uint128_t,
unsigned __int128>;
template <class T>
using is_integral = typename std::conditional<std::is_integral<T>::value ||
is_signed_int128<T>::value ||
is_unsigned_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_signed_int = typename std::conditional<(is_integral<T>::value &&
std::is_signed<T>::value) ||
is_signed_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int =
typename std::conditional<(is_integral<T>::value &&
std::is_unsigned<T>::value) ||
is_unsigned_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using to_unsigned = typename std::conditional<
is_signed_int128<T>::value,
make_unsigned_int128<T>,
typename std::conditional<std::is_signed<T>::value,
std::make_unsigned<T>,
std::common_type<T>>::type>::type;
#else
template <class T> using is_integral = typename std::is_integral<T>;
template <class T>
using is_signed_int =
typename std::conditional<is_integral<T>::value && std::is_signed<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int =
typename std::conditional<is_integral<T>::value &&
std::is_unsigned<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using to_unsigned = typename std::conditional<is_signed_int<T>::value,
std::make_unsigned<T>,
std::common_type<T>>::type;
#endif
template <class T>
using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;
template <class T>
using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;
template <class T> using to_unsigned_t = typename to_unsigned<T>::type;
} // namespace internal
} // namespace atcoder
namespace atcoder {
namespace internal {
struct modint_base {};
struct static_modint_base : modint_base {};
template <class T> using is_modint = std::is_base_of<modint_base, T>;
template <class T> using is_modint_t = std::enable_if_t<is_modint<T>::value>;
} // namespace internal
template <int m, std::enable_if_t<(1 <= m)>* = nullptr>
struct static_modint : internal::static_modint_base {
using mint = static_modint;
public:
static constexpr int mod() { return m; }
static mint raw(int v) {
mint x;
x._v = v;
return x;
}
static_modint() : _v(0) {}
template <class T, internal::is_signed_int_t<T>* = nullptr>
static_modint(T v) {
long long x = (long long)(v % (long long)(umod()));
if (x < 0) x += umod();
_v = (unsigned int)(x);
}
template <class T, internal::is_unsigned_int_t<T>* = nullptr>
static_modint(T v) {
_v = (unsigned int)(v % umod());
}
int val() const { return _v; }
mint& operator++() {
_v++;
if (_v == umod()) _v = 0;
return *this;
}
mint& operator--() {
if (_v == 0) _v = umod();
_v--;
return *this;
}
mint operator++(int) {
mint result = *this;
++*this;
return result;
}
mint operator--(int) {
mint result = *this;
--*this;
return result;
}
mint& operator+=(const mint& rhs) {
_v += rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
mint& operator-=(const mint& rhs) {
_v -= rhs._v;
if (_v >= umod()) _v += umod();
return *this;
}
mint& operator*=(const mint& rhs) {
unsigned long long z = _v;
z *= rhs._v;
_v = (unsigned int)(z % umod());
return *this;
}
mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }
mint operator+() const { return *this; }
mint operator-() const { return mint() - *this; }
mint pow(long long n) const {
assert(0 <= n);
mint x = *this, r = 1;
while (n) {
if (n & 1) r *= x;
x *= x;
n >>= 1;
}
return r;
}
mint inv() const {
if (prime) {
assert(_v);
return pow(umod() - 2);
} else {
auto eg = internal::inv_gcd(_v, m);
assert(eg.first == 1);
return eg.second;
}
}
friend mint operator+(const mint& lhs, const mint& rhs) {
return mint(lhs) += rhs;
}
friend mint operator-(const mint& lhs, const mint& rhs) {
return mint(lhs) -= rhs;
}
friend mint operator*(const mint& lhs, const mint& rhs) {
return mint(lhs) *= rhs;
}
friend mint operator/(const mint& lhs, const mint& rhs) {
return mint(lhs) /= rhs;
}
friend bool operator==(const mint& lhs, const mint& rhs) {
return lhs._v == rhs._v;
}
friend bool operator!=(const mint& lhs, const mint& rhs) {
return lhs._v != rhs._v;
}
private:
unsigned int _v;
static constexpr unsigned int umod() { return m; }
static constexpr bool prime = internal::is_prime<m>;
};
template <int id> struct dynamic_modint : internal::modint_base {
using mint = dynamic_modint;
public:
static int mod() { return (int)(bt.umod()); }
static void set_mod(int m) {
assert(1 <= m);
bt = internal::barrett(m);
}
static mint raw(int v) {
mint x;
x._v = v;
return x;
}
dynamic_modint() : _v(0) {}
template <class T, internal::is_signed_int_t<T>* = nullptr>
dynamic_modint(T v) {
long long x = (long long)(v % (long long)(mod()));
if (x < 0) x += mod();
_v = (unsigned int)(x);
}
template <class T, internal::is_unsigned_int_t<T>* = nullptr>
dynamic_modint(T v) {
_v = (unsigned int)(v % mod());
}
int val() const { return _v; }
mint& operator++() {
_v++;
if (_v == umod()) _v = 0;
return *this;
}
mint& operator--() {
if (_v == 0) _v = umod();
_v--;
return *this;
}
mint operator++(int) {
mint result = *this;
++*this;
return result;
}
mint operator--(int) {
mint result = *this;
--*this;
return result;
}
mint& operator+=(const mint& rhs) {
_v += rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
mint& operator-=(const mint& rhs) {
_v += mod() - rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
mint& operator*=(const mint& rhs) {
_v = bt.mul(_v, rhs._v);
return *this;
}
mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }
mint operator+() const { return *this; }
mint operator-() const { return mint() - *this; }
mint pow(long long n) const {
assert(0 <= n);
mint x = *this, r = 1;
while (n) {
if (n & 1) r *= x;
x *= x;
n >>= 1;
}
return r;
}
mint inv() const {
auto eg = internal::inv_gcd(_v, mod());
assert(eg.first == 1);
return eg.second;
}
friend mint operator+(const mint& lhs, const mint& rhs) {
return mint(lhs) += rhs;
}
friend mint operator-(const mint& lhs, const mint& rhs) {
return mint(lhs) -= rhs;
}
friend mint operator*(const mint& lhs, const mint& rhs) {
return mint(lhs) *= rhs;
}
friend mint operator/(const mint& lhs, const mint& rhs) {
return mint(lhs) /= rhs;
}
friend bool operator==(const mint& lhs, const mint& rhs) {
return lhs._v == rhs._v;
}
friend bool operator!=(const mint& lhs, const mint& rhs) {
return lhs._v != rhs._v;
}
private:
unsigned int _v;
static internal::barrett bt;
static unsigned int umod() { return bt.umod(); }
};
template <int id> internal::barrett dynamic_modint<id>::bt(998244353);
using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;
namespace internal {
template <class T>
using is_static_modint = std::is_base_of<internal::static_modint_base, T>;
template <class T>
using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>;
template <class> struct is_dynamic_modint : public std::false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint<id>> : public std::true_type {};
template <class T>
using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>;
} // namespace internal
} // namespace atcoder
namespace atcoder {
namespace internal {
template <class mint,
int g = internal::primitive_root<mint::mod()>,
internal::is_static_modint_t<mint>* = nullptr>
struct fft_info {
static constexpr int rank2 = countr_zero_constexpr(mint::mod() - 1);
std::array<mint, rank2 + 1> root; // root[i]^(2^i) == 1
std::array<mint, rank2 + 1> iroot; // root[i] * iroot[i] == 1
std::array<mint, std::max(0, rank2 - 2 + 1)> rate2;
std::array<mint, std::max(0, rank2 - 2 + 1)> irate2;
std::array<mint, std::max(0, rank2 - 3 + 1)> rate3;
std::array<mint, std::max(0, rank2 - 3 + 1)> irate3;
fft_info() {
root[rank2] = mint(g).pow((mint::mod() - 1) >> rank2);
iroot[rank2] = root[rank2].inv();
for (int i = rank2 - 1; i >= 0; i--) {
root[i] = root[i + 1] * root[i + 1];
iroot[i] = iroot[i + 1] * iroot[i + 1];
}
{
mint prod = 1, iprod = 1;
for (int i = 0; i <= rank2 - 2; i++) {
rate2[i] = root[i + 2] * prod;
irate2[i] = iroot[i + 2] * iprod;
prod *= iroot[i + 2];
iprod *= root[i + 2];
}
}
{
mint prod = 1, iprod = 1;
for (int i = 0; i <= rank2 - 3; i++) {
rate3[i] = root[i + 3] * prod;
irate3[i] = iroot[i + 3] * iprod;
prod *= iroot[i + 3];
iprod *= root[i + 3];
}
}
}
};
template <class mint, internal::is_static_modint_t<mint>* = nullptr>
void butterfly(std::vector<mint>& a) {
int n = int(a.size());
int h = internal::countr_zero((unsigned int)n);
static const fft_info<mint> info;
int len = 0; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
while (len < h) {
if (h - len == 1) {
int p = 1 << (h - len - 1);
mint rot = 1;
for (int s = 0; s < (1 << len); s++) {
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
auto l = a[i + offset];
auto r = a[i + offset + p] * rot;
a[i + offset] = l + r;
a[i + offset + p] = l - r;
}
if (s + 1 != (1 << len))
rot *= info.rate2[countr_zero(~(unsigned int)(s))];
}
len++;
} else {
// 4-base
int p = 1 << (h - len - 2);
mint rot = 1, imag = info.root[2];
for (int s = 0; s < (1 << len); s++) {
mint rot2 = rot * rot;
mint rot3 = rot2 * rot;
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
auto mod2 = 1ULL * mint::mod() * mint::mod();
auto a0 = 1ULL * a[i + offset].val();
auto a1 = 1ULL * a[i + offset + p].val() * rot.val();
auto a2 = 1ULL * a[i + offset + 2 * p].val() * rot2.val();
auto a3 = 1ULL * a[i + offset + 3 * p].val() * rot3.val();
auto a1na3imag =
1ULL * mint(a1 + mod2 - a3).val() * imag.val();
auto na2 = mod2 - a2;
a[i + offset] = a0 + a2 + a1 + a3;
a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3));
a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag);
}
if (s + 1 != (1 << len))
rot *= info.rate3[countr_zero(~(unsigned int)(s))];
}
len += 2;
}
}
}
template <class mint, internal::is_static_modint_t<mint>* = nullptr>
void butterfly_inv(std::vector<mint>& a) {
int n = int(a.size());
int h = internal::countr_zero((unsigned int)n);
static const fft_info<mint> info;
int len = h; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
while (len) {
if (len == 1) {
int p = 1 << (h - len);
mint irot = 1;
for (int s = 0; s < (1 << (len - 1)); s++) {
int offset = s << (h - len + 1);
for (int i = 0; i < p; i++) {
auto l = a[i + offset];
auto r = a[i + offset + p];
a[i + offset] = l + r;
a[i + offset + p] =
(unsigned long long)((unsigned int)(l.val() - r.val()) + mint::mod()) *
irot.val();
;
}
if (s + 1 != (1 << (len - 1)))
irot *= info.irate2[countr_zero(~(unsigned int)(s))];
}
len--;
} else {
// 4-base
int p = 1 << (h - len);
mint irot = 1, iimag = info.iroot[2];
for (int s = 0; s < (1 << (len - 2)); s++) {
mint irot2 = irot * irot;
mint irot3 = irot2 * irot;
int offset = s << (h - len + 2);
for (int i = 0; i < p; i++) {
auto a0 = 1ULL * a[i + offset + 0 * p].val();
auto a1 = 1ULL * a[i + offset + 1 * p].val();
auto a2 = 1ULL * a[i + offset + 2 * p].val();
auto a3 = 1ULL * a[i + offset + 3 * p].val();
auto a2na3iimag =
1ULL *
mint((mint::mod() + a2 - a3) * iimag.val()).val();
a[i + offset] = a0 + a1 + a2 + a3;
a[i + offset + 1 * p] =
(a0 + (mint::mod() - a1) + a2na3iimag) * irot.val();
a[i + offset + 2 * p] =
(a0 + a1 + (mint::mod() - a2) + (mint::mod() - a3)) *
irot2.val();
a[i + offset + 3 * p] =
(a0 + (mint::mod() - a1) + (mint::mod() - a2na3iimag)) *
irot3.val();
}
if (s + 1 != (1 << (len - 2)))
irot *= info.irate3[countr_zero(~(unsigned int)(s))];
}
len -= 2;
}
}
}
template <class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution_naive(const std::vector<mint>& a,
const std::vector<mint>& b) {
int n = int(a.size()), m = int(b.size());
std::vector<mint> ans(n + m - 1);
if (n < m) {
for (int j = 0; j < m; j++) {
for (int i = 0; i < n; i++) {
ans[i + j] += a[i] * b[j];
}
}
} else {
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) {
ans[i + j] += a[i] * b[j];
}
}
}
return ans;
}
template <class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution_fft(std::vector<mint> a, std::vector<mint> b) {
int n = int(a.size()), m = int(b.size());
int z = (int)internal::bit_ceil((unsigned int)(n + m - 1));
a.resize(z);
internal::butterfly(a);
b.resize(z);
internal::butterfly(b);
for (int i = 0; i < z; i++) {
a[i] *= b[i];
}
internal::butterfly_inv(a);
a.resize(n + m - 1);
mint iz = mint(z).inv();
for (int i = 0; i < n + m - 1; i++) a[i] *= iz;
return a;
}
} // namespace internal
template <class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution(std::vector<mint>&& a, std::vector<mint>&& b) {
int n = int(a.size()), m = int(b.size());
if (!n || !m) return {};
int z = (int)internal::bit_ceil((unsigned int)(n + m - 1));
assert((mint::mod() - 1) % z == 0);
if (std::min(n, m) <= 60) return convolution_naive(std::move(a), std::move(b));
return internal::convolution_fft(std::move(a), std::move(b));
}
template <class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution(const std::vector<mint>& a,
const std::vector<mint>& b) {
int n = int(a.size()), m = int(b.size());
if (!n || !m) return {};
int z = (int)internal::bit_ceil((unsigned int)(n + m - 1));
assert((mint::mod() - 1) % z == 0);
if (std::min(n, m) <= 60) return convolution_naive(a, b);
return internal::convolution_fft(a, b);
}
template <unsigned int mod = 998244353,
class T,
std::enable_if_t<internal::is_integral<T>::value>* = nullptr>
std::vector<T> convolution(const std::vector<T>& a, const std::vector<T>& b) {
int n = int(a.size()), m = int(b.size());
if (!n || !m) return {};
using mint = static_modint<mod>;
int z = (int)internal::bit_ceil((unsigned int)(n + m - 1));
assert((mint::mod() - 1) % z == 0);
std::vector<mint> a2(n), b2(m);
for (int i = 0; i < n; i++) {
a2[i] = mint(a[i]);
}
for (int i = 0; i < m; i++) {
b2[i] = mint(b[i]);
}
auto c2 = convolution(std::move(a2), std::move(b2));
std::vector<T> c(n + m - 1);
for (int i = 0; i < n + m - 1; i++) {
c[i] = c2[i].val();
}
return c;
}
std::vector<long long> convolution_ll(const std::vector<long long>& a,
const std::vector<long long>& b) {
int n = int(a.size()), m = int(b.size());
if (!n || !m) return {};
static constexpr unsigned long long MOD1 = 754974721; // 2^24
static constexpr unsigned long long MOD2 = 167772161; // 2^25
static constexpr unsigned long long MOD3 = 469762049; // 2^26
static constexpr unsigned long long M2M3 = MOD2 * MOD3;
static constexpr unsigned long long M1M3 = MOD1 * MOD3;
static constexpr unsigned long long M1M2 = MOD1 * MOD2;
static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3;
static constexpr unsigned long long i1 =
internal::inv_gcd(MOD2 * MOD3, MOD1).second;
static constexpr unsigned long long i2 =
internal::inv_gcd(MOD1 * MOD3, MOD2).second;
static constexpr unsigned long long i3 =
internal::inv_gcd(MOD1 * MOD2, MOD3).second;
static constexpr int MAX_AB_BIT = 24;
static_assert(MOD1 % (1ull << MAX_AB_BIT) == 1, "MOD1 isn't enough to support an array length of 2^24.");
static_assert(MOD2 % (1ull << MAX_AB_BIT) == 1, "MOD2 isn't enough to support an array length of 2^24.");
static_assert(MOD3 % (1ull << MAX_AB_BIT) == 1, "MOD3 isn't enough to support an array length of 2^24.");
assert(n + m - 1 <= (1 << MAX_AB_BIT));
auto c1 = convolution<MOD1>(a, b);
auto c2 = convolution<MOD2>(a, b);
auto c3 = convolution<MOD3>(a, b);
std::vector<long long> c(n + m - 1);
for (int i = 0; i < n + m - 1; i++) {
unsigned long long x = 0;
x += (c1[i] * i1) % MOD1 * M2M3;
x += (c2[i] * i2) % MOD2 * M1M3;
x += (c3[i] * i3) % MOD3 * M1M2;
// B = 2^63, -B <= x, r(real value) < B
// (x, x - M, x - 2M, or x - 3M) = r (mod 2B)
// r = c1[i] (mod MOD1)
// focus on MOD1
// r = x, x - M', x - 2M', x - 3M' (M' = M % 2^64) (mod 2B)
// r = x,
// x - M' + (0 or 2B),
// x - 2M' + (0, 2B or 4B),
// x - 3M' + (0, 2B, 4B or 6B) (without mod!)
// (r - x) = 0, (0)
// - M' + (0 or 2B), (1)
// -2M' + (0 or 2B or 4B), (2)
// -3M' + (0 or 2B or 4B or 6B) (3) (mod MOD1)
// we checked that
// ((1) mod MOD1) mod 5 = 2
// ((2) mod MOD1) mod 5 = 3
// ((3) mod MOD1) mod 5 = 4
long long diff =
c1[i] - internal::safe_mod((long long)(x), (long long)(MOD1));
if (diff < 0) diff += MOD1;
static constexpr unsigned long long offset[5] = {
0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3};
x -= offset[diff % 5];
c[i] = x;
}
return c;
}
} // namespace atcoder
#include <algorithm>
#include <cassert>
#include <utility>
#include <vector>
namespace noya {
/// @brief Remove trailing zero coefficients from a polynomial.
template <class T> void polynomial_trim(std::vector<T> &pol) {
while (!pol.empty() && pol.back() == T{}) {
pol.pop_back();
}
}
/// @brief Return the formal derivative of a polynomial.
template <class T>
std::vector<T> polynomial_derivative(const std::vector<T> &pol) {
if (pol.size() <= 1) {
return {};
}
std::vector<T> res(pol.size() - 1);
for (int i = 1; i < int(pol.size()); i++) {
res[i - 1] = pol[i] * T(i);
}
return res;
}
/// @brief Return the formal integral with constant coefficient zero;
/// coefficient division by each integer 1 through pol.size() must be valid.
template <class T>
std::vector<T> polynomial_integral(const std::vector<T> &pol) {
std::vector<T> res(pol.size() + 1);
for (int i = 0; i < int(pol.size()); i++) {
res[i + 1] = pol[i] / T(i + 1);
}
return res;
}
/// @brief Return the first n coefficients of 1/f using Newton iteration.
/// Requires invertible f[0] and convolution lengths supported by ACL.
template <class Mint>
std::vector<Mint> polynomial_inverse_series(const std::vector<Mint> &f, int n) {
assert(n >= 0);
if (n == 0) {
return {};
}
assert(!f.empty() && f[0] != Mint{});
std::vector<Mint> inv = {Mint(1) / f[0]};
while (int(inv.size()) < n) {
int tar = std::min(n, int(inv.size()) * 2);
std::vector<Mint> pre(tar);
for (int i = 0; i < std::min(tar, int(f.size())); i++) {
pre[i] = f[i];
}
std::vector<Mint> cor =
atcoder::convolution(std::move(pre), std::vector<Mint>(inv));
cor.resize(tar);
for (Mint &val : cor) {
val = -val;
}
cor[0] += Mint(2);
inv = atcoder::convolution(std::move(inv), std::move(cor));
inv.resize(tar);
}
return inv;
}
/// @brief Divide f by nonzero g and return (quotient, remainder).
/// Requires invertible leading coefficient of g and ACL-supported convolutions.
template <class Mint>
std::pair<std::vector<Mint>, std::vector<Mint>>
polynomial_divmod(std::vector<Mint> f, std::vector<Mint> g) {
polynomial_trim(f);
polynomial_trim(g);
assert(!g.empty());
if (f.size() < g.size()) {
return {{}, f};
}
if (g.size() == 1) {
Mint iv = Mint(1) / g[0];
for (auto &v : f) v *= iv;
return {std::move(f), {}};
}
int nq = int(f.size() - g.size() + 1);
std::vector<Mint> rf(f.rbegin(), f.rbegin() + nq);
std::vector<Mint> rg(g.rbegin(), g.rbegin() + std::min(int(g.size()), nq));
std::vector<Mint> inv = polynomial_inverse_series(rg, nq);
std::vector<Mint> quo = atcoder::convolution(std::move(rf), std::move(inv));
quo.resize(nq);
std::reverse(quo.begin(), quo.end());
// Higher coefficients cancel by construction; only degree < deg(g) remains.
g.pop_back();
f.resize(g.size());
std::vector<Mint> low(quo.begin(), quo.begin() + std::min(quo.size(), g.size()));
std::vector<Mint> prd = atcoder::convolution(std::move(low), std::move(g));
for (int i = 0; i < int(f.size()); i++) {
f[i] -= prd[i];
}
polynomial_trim(f);
return {quo, f};
}
/// @brief Return f(x + shf) in O(M(n)) time. Factorials through degree(f)
/// must be invertible, and convolution lengths must be supported by ACL.
template <class Mint>
std::vector<Mint> polynomial_taylor_shift(const std::vector<Mint> &f,
Mint shf) {
int sz = int(f.size());
if (sz == 0) {
return {};
}
std::vector<Mint> rev(sz);
Mint fac = 1;
for (int i = 0; i < sz; i++) {
if (i > 0) fac *= Mint(i);
rev[sz - 1 - i] = f[i] * fac;
}
std::vector<Mint> ifc(sz, Mint(1));
ifc.back() = Mint(1) / fac;
for (int i = sz - 1; i > 0; i--) {
ifc[i - 1] = ifc[i] * Mint(i);
}
std::vector<Mint> pow(sz);
Mint pw = Mint(1);
for (int i = 0; i < sz; i++) {
pow[i] = pw * ifc[i];
pw *= shf;
}
std::vector<Mint> res = atcoder::convolution(std::move(rev), std::move(pow));
res.resize(sz);
std::reverse(res.begin(), res.end());
for (int i = 0; i < sz; i++) {
res[i] *= ifc[i];
}
return res;
}
} // namespace noya
#include <algorithm>
#include <cassert>
#include <cstdint>
#include <optional>
#include <vector>
namespace noya {
/// @brief Return the first n coefficients of 1/f; requires f[0] != 0.
template <class Mint>
std::vector<Mint> fps_inverse(const std::vector<Mint> &f, int n) {
return polynomial_inverse_series(f, n);
}
/// @brief Return log(f) modulo x^n; requires f[0] = 1 and n <= Mint::mod()
/// over a prime field. Convolution lengths must be supported by ACL.
template <class Mint>
std::vector<Mint> fps_logarithm(const std::vector<Mint> &f, int n) {
assert(n >= 0);
if (n == 0) {
return {};
}
assert(!f.empty() && f[0] == Mint(1));
assert(n <= Mint::mod());
int len = std::min(n, int(f.size()));
std::vector<Mint> df(len - 1);
for (int i = 1; i < len; i++) df[i - 1] = f[i] * Mint(i);
std::vector<Mint> inv = fps_inverse(f, n);
std::vector<Mint> prd = atcoder::convolution(df, inv);
prd.resize(n - 1);
// Recover all reciprocal integers from factorial prefixes with one inverse.
std::vector<Mint> res(n);
Mint fac = 1;
for (int i = 1; i < n; i++) {
res[i] = fac;
fac *= Mint(i);
}
fac = Mint(1) / fac;
for (int i = n - 1; i > 0; i--) {
res[i] *= fac * prd[i - 1];
fac *= Mint(i);
}
return res;
}
/// @brief Return exp(f) modulo x^n; requires f[0] = 0 and n <= Mint::mod()
/// over a prime field, with ACL-supported convolution lengths.
template <class Mint>
std::vector<Mint> fps_exponential(const std::vector<Mint> &f, int n) {
assert(n >= 0);
if (n == 0) {
return {};
}
assert(f.empty() || f[0] == Mint{});
assert(n <= Mint::mod());
std::vector<Mint> res = {Mint(1)};
while (int(res.size()) < n) {
int tar = std::min(n, int(res.size()) * 2);
std::vector<Mint> lg = fps_logarithm(res, tar);
std::vector<Mint> cor(tar);
for (int i = 0; i < tar; i++) {
if (i < int(f.size())) {
cor[i] += f[i];
}
cor[i] -= lg[i];
}
cor[0] += Mint(1);
res = atcoder::convolution(res, cor);
res.resize(tar);
}
return res;
}
/// @brief Compute vl^exp by binary exponentiation.
template <class Mint> Mint fps_scalar_power(Mint vl, std::uint64_t exp) {
Mint res = Mint(1);
while (exp > 0) {
if (exp & 1) {
res *= vl;
}
vl *= vl;
exp >>= 1;
}
return res;
}
/// @brief Return f^exp modulo x^n for a nonnegative exponent. The log/exp
/// reduction requires a prime field and n <= Mint::mod(), with convolution
/// lengths supported by ACL.
template <class Mint>
std::vector<Mint> fps_power(const std::vector<Mint> &f, std::uint64_t exp,
int n) {
assert(n >= 0);
if (n == 0) {
return {};
}
std::vector<Mint> zer(n);
if (exp == 0) {
zer[0] = Mint(1);
return zer;
}
int a = 0;
int len = std::min(n, int(f.size()));
while (a < len && f[a] == Mint{}) {
a++;
}
if (a == len || (a > 0 && exp > std::uint64_t((n - 1) / a))) {
return zer;
}
int shf = int(std::uint64_t(a) * exp);
int tar = n - shf;
Mint ld = f[a];
Mint il = Mint(1) / ld;
std::vector<Mint> nrm(tar);
for (int i = 0; i < tar && a + i < int(f.size()); i++) {
nrm[i] = f[a + i] * il;
}
std::vector<Mint> lg = fps_logarithm(nrm, tar);
Mint e = Mint(exp);
for (Mint &vl : lg) {
vl *= e;
}
std::vector<Mint> pow = fps_exponential(lg, tar);
Mint lp = fps_scalar_power(ld, exp);
for (Mint &vl : pow) {
vl *= lp;
}
std::vector<Mint> res(n);
for (int i = 0; i < tar; i++) {
res[shf + i] = pow[i];
}
return res;
}
/// @brief Return a formal square root of f modulo x^n over a prime field,
/// if one exists. Odd characteristics require ACL-supported convolution lengths.
template <class Mint>
std::optional<std::vector<Mint>> fps_square_root(const std::vector<Mint> &f,
int n) {
assert(n >= 0);
if (n == 0) {
return std::vector<Mint>{};
}
if (Mint::mod() == 2) {
std::vector<Mint> res(n);
for (int i = 0; i < std::min(n, int(f.size())); i++) {
if (i & 1) {
if (f[i] != Mint{}) return std::nullopt;
} else {
res[i / 2] = f[i];
}
}
return res;
}
int a = 0;
while (a < std::min(n, int(f.size())) && f[a] == Mint{}) {
a++;
}
if (a == std::min(n, int(f.size()))) {
return std::vector<Mint>(n);
}
if (a & 1) {
return std::nullopt;
}
if (a > 0) {
int shf = a / 2;
int tar = n - a;
std::vector<Mint> red(tar);
for (int i = 0; i < tar && a + i < int(f.size()); i++) {
red[i] = f[a + i];
}
auto rt = fps_square_root(red, tar);
if (!rt) {
return std::nullopt;
}
std::vector<Mint> res(n);
for (int i = 0; i < int(rt->size()) && shf + i < n; i++) {
res[shf + i] = (*rt)[i];
}
return res;
}
auto c0 = mod_sqrt(std::uint64_t(f[0].val()), std::uint64_t(Mint::mod()));
if (!c0) {
return std::nullopt;
}
std::vector<Mint> res = {Mint(*c0)};
Mint iv2 = Mint(1) / Mint(2);
while (int(res.size()) < n) {
int tar = std::min(n, int(res.size()) * 2);
std::vector<Mint> pre(tar);
for (int i = 0; i < tar && i < int(f.size()); i++) {
pre[i] = f[i];
}
std::vector<Mint> quo =
atcoder::convolution(pre, polynomial_inverse_series(res, tar));
quo.resize(tar);
res.resize(tar);
for (int i = 0; i < tar; i++) {
res[i] = (res[i] + quo[i]) * iv2;
}
}
return res;
}
} // namespace noya
#include <algorithm>
#include <utility>
#include <vector>
namespace noya {
template <class Mint>
std::pair<std::vector<Mint>, std::vector<Mint>> factorials_and_inverses(int n) {
std::vector<Mint> fac(n + 1, Mint(1));
std::vector<Mint> ifc(n + 1, Mint(1));
for (int i = 1; i <= n; i++) {
fac[i] = fac[i - 1] * Mint(i);
}
ifc[n] = Mint(1) / fac[n];
for (int i = n; i > 0; i--) {
ifc[i - 1] = ifc[i] * Mint(i);
}
return {std::move(fac), std::move(ifc)};
}
/// @brief Return B_0 through B_n. Their exponential generating function is
/// exp(exp(x)-1), so one FPS exponential followed by factorial scaling yields
/// all Bell numbers simultaneously.
template <class Mint> std::vector<Mint> bell_numbers(int n) {
auto [fac, ifc] = factorials_and_inverses<Mint>(n);
ifc[0] = Mint{};
auto res = fps_exponential(ifc, n + 1);
for (int i = 0; i <= n; i++) {
res[i] *= fac[i];
}
return res;
}
/// @brief Return B_0 through B_n with B_1=-1/2. Since
/// x/(exp(x)-1)=1/(sum_{i>=0} x^i/(i+1)!), a series inverse and factorial
/// scaling produce all Bernoulli numbers.
template <class Mint> std::vector<Mint> bernoulli_numbers(int n) {
auto [fac, ifc] = factorials_and_inverses<Mint>(n + 1);
for (int i = 0; i <= n; i++) {
ifc[i] = ifc[i + 1];
}
ifc.resize(n + 1);
auto res = fps_inverse(ifc, n + 1);
for (int i = 0; i <= n; i++) {
res[i] *= fac[i];
}
return res;
}
/// @brief Return p(0) through p(n). Euler's pentagonal theorem gives
/// prod_{j>=1}(1-x^j) = 1 + sum_{k>=1} (-1)^k (x^{k(3k-1)/2}+x^{k(3k+1)/2}).
/// Its inverse is the partition generating function; no divisions by indices.
template <class Mint> std::vector<Mint> partition_numbers(int n) {
std::vector<Mint> den(n + 1);
den[0] = Mint(1);
for (long long k = 1; k * (3 * k - 1) / 2 <= n; k++) {
auto i = k * (3 * k - 1) / 2;
Mint sign = (k & 1) ? Mint(-1) : Mint(1);
den[i] = sign;
if (i + k <= n) {
den[i + k] = sign;
}
}
return fps_inverse(den, n + 1);
}
namespace combinatorial_sequences_detail {
template <class Mint>
std::vector<Mint> consecutive_linear_product(int l, int r) {
int n = r - l;
// Small products use the same naive cutoff as ACL convolution.
if (n <= 60) {
std::vector<Mint> res(n + 1);
res[0] = Mint(1);
for (int j = 0; j < n; j++) {
Mint root = Mint(l + j);
for (int i = j + 1; i > 0; i--) {
res[i] = res[i - 1] - root * res[i];
}
res[0] *= -root;
}
return res;
}
int half = n / 2;
auto a = consecutive_linear_product<Mint>(l, l + half);
// The next half has the same roots translated by half.
auto b = polynomial_taylor_shift(a, -Mint(half));
auto res = atcoder::convolution(std::move(a), std::move(b));
if (n & 1) {
res.push_back(Mint{});
Mint last = Mint(r - 1);
for (int i = n; i > 0; i--) {
res[i] = res[i - 1] - last * res[i];
}
res[0] *= -last;
}
return res;
}
} // namespace combinatorial_sequences_detail
/// @brief Return the signed first-kind Stirling row s(n,0..n) by building the
/// product x(x-1)...(x-n+1), reusing each half by Taylor shift.
template <class Mint> std::vector<Mint> stirling_first_kind_row(int n) {
return combinatorial_sequences_detail::consecutive_linear_product<Mint>(0, n);
}
/// @brief Return the second-kind Stirling row S(n,0..n). Expanding
/// S(n,k)=1/k! sum_i (-1)^(k-i) binom(k,i)i^n turns the whole row into one
/// convolution of the sequences (-1)^i/i! and i^n/i!.
template <class Mint> std::vector<Mint> stirling_second_kind_row(int n) {
auto [fac, ifc] = factorials_and_inverses<Mint>(n);
std::vector<Mint> sig(n + 1), pw(n + 1);
for (int i = 0; i <= n; i++) {
sig[i] = (i & 1) ? -ifc[i] : ifc[i];
pw[i] = Mint(i).pow(n) * ifc[i];
}
auto res = atcoder::convolution(std::move(sig), std::move(pw));
res.resize(n + 1);
return res;
}
/// @brief Return s(k,k) through s(n,k). The exponential generating function
/// for a fixed column is log(1+x)^k/k!. Remove x^k first, then compute
/// exp(k*log(log(1+x)/x)) to n-k+1 terms and scale by i!/k!. Requires 0 <= k <= n.
template <class Mint>
std::vector<Mint> stirling_first_kind_fixed_column(int n, int k) {
if (k == 0) {
std::vector<Mint> res(n + 1);
res[0] = Mint(1);
return res;
}
if (k == n) {
return {Mint(1)};
}
if (k == 1) {
std::vector<Mint> res(n, Mint(1));
for (int i = 1; i < n; i++) {
res[i] = -Mint(i) * res[i - 1];
}
return res;
}
auto [fac, ifc] = factorials_and_inverses<Mint>(n);
int len = n - k + 1;
std::vector<Mint> series(len);
for (int i = 0; i < len; i++) {
series[i] = fac[i] * ifc[i + 1];
if (i & 1) {
series[i] = -series[i];
}
}
auto lg = fps_logarithm(series, len);
for (auto &x : lg) {
x *= Mint(k);
}
auto res = fps_exponential(lg, len);
for (int i = 0; i < len; i++) {
res[i] *= fac[i + k] * ifc[k];
}
return res;
}
/// @brief Return S(k,k) through S(n,k). The fixed-column exponential
/// generating function is (exp(x)-1)^k/k!. Remove x^k first, then compute
/// exp(k*log((exp(x)-1)/x)) to n-k+1 terms and scale by i!/k!. Requires 0 <= k <= n.
template <class Mint>
std::vector<Mint> stirling_second_kind_fixed_column(int n, int k) {
if (k == 0) {
std::vector<Mint> res(n + 1);
res[0] = Mint(1);
return res;
}
if (k == n) {
return {Mint(1)};
}
if (k == 1) {
std::vector<Mint> res(n, Mint(1));
return res;
}
auto [fac, ifc] = factorials_and_inverses<Mint>(n);
int len = n - k + 1;
std::vector<Mint> series(len);
for (int i = 0; i < len; i++) {
series[i] = ifc[i + 1];
}
auto lg = fps_logarithm(series, len);
for (auto &x : lg) {
x *= Mint(k);
}
auto res = fps_exponential(lg, len);
for (int i = 0; i < len; i++) {
res[i] *= fac[i + k] * ifc[k];
}
return res;
}
} // namespace noya
/// @complexity Time: O(M(N) log(N+1)) evaluation/interpolation,
/// where N is the larger of the coefficient and point counts.
/// Space: O(N log(N+1)).
/// @complexity Time: O(n) field operations, one division for nonempty input.
/// Space: O(n) output and O(1) additional workspace.
#include <array>
#include <cassert>
#include <vector>
namespace noya {
/// @brief Invert nonzero field elements with four independent product chains.
/// Each chain contains indices with the same remainder modulo four.
template <class T> std::vector<T> batch_inverse(const std::vector<T> &vs) {
if (vs.empty()) return {};
int n = int(vs.size());
std::vector<T> res(n);
std::array<T, 4> prod;
prod.fill(T(1));
#pragma GCC unroll 4
for (int i = 0; i < n; ++i) {
assert(vs[i] != T{});
res[i] = prod[i & 3];
prod[i & 3] *= vs[i];
}
// Invert the four chain products together, using just one division.
auto last = prod;
for (int i = 1; i < 4; ++i) prod[i] *= prod[i - 1];
T inv = T(1) / prod[3];
for (int i = 3; i > 0; --i) {
prod[i] = prod[i - 1] * inv;
inv *= last[i];
}
prod[0] = inv;
#pragma GCC unroll 4
for (int i = n - 1; i >= 0; --i) {
res[i] *= prod[i & 3];
prod[i & 3] *= vs[i];
}
return res;
}
} // namespace noya
#include <algorithm>
#include <cassert>
#include <utility>
#include <vector>
namespace noya {
namespace polynomial_multipoint_internal {
template <class Mint> struct product_tree {
int np = 0;
int sz = 1;
std::vector<std::vector<Mint>> prd;
explicit product_tree(const std::vector<Mint> &poi) : np(int(poi.size())) {
while (sz < np) {
sz *= 2;
}
prd.resize(sz * 2);
for (int i = 0; i < np; i++) {
prd[sz + i] = {-poi[i], Mint(1)};
}
for (int i = np; i < sz; i++) prd[sz + i] = {Mint(1)};
for (int id = sz - 1; id > 0; id--) {
prd[id] = atcoder::convolution(prd[id * 2], prd[id * 2 + 1]);
}
}
std::vector<Mint> evaluate(const std::vector<Mint> &pol) const {
if (np == 0) {
return {};
}
std::vector<Mint> res(np);
auto visit = [&](auto &&self, int id, int l, int r, std::vector<Mint> f) -> void {
if (r - l <= 32) {
// Small remainders are cheaper to evaluate than to divide repeatedly.
for (int i = l; i < std::min(r, np); i++) {
Mint x = -prd[sz + i][0];
for (int j = int(f.size()) - 1; j >= 0; j--) res[i] = res[i] * x + f[j];
}
return;
}
int mid = (l + r) / 2;
self(self, id * 2, l, mid, polynomial_divmod(f, prd[id * 2]).second);
if (mid < np)
self(self, id * 2 + 1, mid, r,
polynomial_divmod(std::move(f), prd[id * 2 + 1]).second);
};
visit(visit, 1, 0, sz, polynomial_divmod(pol, prd[1]).second);
return res;
}
};
template <class Mint>
std::vector<Mint> add_polynomials(std::vector<Mint> l,
const std::vector<Mint> &r) {
l.resize(std::max(l.size(), r.size()));
for (int i = 0; i < int(r.size()); i++) {
l[i] += r[i];
}
polynomial_trim(l);
return l;
}
} // namespace polynomial_multipoint_internal
/// @brief Evaluate a polynomial at all points using a product tree.
/// Duplicate points are allowed. Requires a field and ACL-supported convolutions.
template <class Mint>
std::vector<Mint>
polynomial_multipoint_evaluation(const std::vector<Mint> &pol,
const std::vector<Mint> &poi) {
return polynomial_multipoint_internal::product_tree<Mint>(poi).evaluate(pol);
}
/// @brief Interpolate the unique degree < n polynomial through n distinct
/// points in O(n log^2 n) field operations.
/// Requires a field and ACL-supported convolutions; the output is trimmed.
template <class Mint>
std::vector<Mint> polynomial_interpolation(const std::vector<Mint> &poi,
const std::vector<Mint> &vs) {
assert(poi.size() == vs.size());
int n = int(poi.size());
if (n == 0) {
return {};
}
polynomial_multipoint_internal::product_tree<Mint> tre(poi);
std::vector<Mint> df = polynomial_derivative(tre.prd[1]);
std::vector<Mint> ds = tre.evaluate(df);
ds = batch_inverse(ds);
std::vector<std::vector<Mint>> ntr(tre.sz * 2);
for (int i = 0; i < n; i++) {
ntr[tre.sz + i] = {vs[i] * ds[i]};
}
for (int i = n; i < tre.sz; i++) {
ntr[tre.sz + i] = {};
}
using polynomial_multipoint_internal::add_polynomials;
for (int id = tre.sz - 1; id > 0; id--) {
std::vector<Mint> l =
atcoder::convolution(std::move(ntr[id * 2]), std::move(tre.prd[id * 2 + 1]));
std::vector<Mint> r =
atcoder::convolution(std::move(ntr[id * 2 + 1]), std::move(tre.prd[id * 2]));
ntr[id] = add_polynomials(std::move(l), r);
}
ntr[1].resize(n);
polynomial_trim(ntr[1]);
return ntr[1];
}
} // namespace noya
/// @complexity Time: O(M(n + m)) for consecutive or geometric evaluation and
/// O(M(n)) for geometric interpolation, where M(n) is convolution time.
/// Product of k polynomials: O(M(s) log(k+1)) time for s input coefficients.
/// Space: O(n + m), or O(s+k) for the product.
#include <algorithm>
#include <cassert>
#include <functional>
#include <queue>
#include <utility>
#include <vector>
namespace noya {
namespace polynomial_special_points_detail {
template <class Mint> std::vector<Mint> factorial_inverses(int sz) {
Mint fac = 1;
for (int i = 1; i < sz; i++) {
fac *= Mint(i);
}
std::vector<Mint> ifc(sz, Mint(1));
if (sz > 0) {
ifc.back() = Mint(1) / fac;
for (int i = sz - 1; i > 0; i--) {
ifc[i - 1] = ifc[i] * Mint(i);
}
}
return ifc;
}
template <class Mint>
std::vector<Mint> inverses_allowing_zero(const std::vector<Mint> &vs) {
std::vector<Mint> non;
non.reserve(vs.size());
for (Mint val : vs) {
if (val != Mint{}) {
non.push_back(val);
}
}
std::vector<Mint> inv = batch_inverse(non);
std::vector<Mint> res(vs.size());
int at = 0;
for (int i = 0; i < int(vs.size()); i++) {
if (vs[i] != Mint{}) {
res[i] = inv[at++];
}
}
return res;
}
} // namespace polynomial_special_points_detail
/// @brief Recover f(c),...,f(c+m-1) from f(0),...,f(n-1). Lagrange weights
/// turn every non-sampled value into one convolution with 1/(c+k-i); a
/// sliding product supplies prod_j(c+k-j). Positions that coincide with an
/// original sample are copied directly.
/// Requires n distinct field elements 0,...,n-1 and ACL-supported convolutions.
template <class Mint>
std::vector<Mint> polynomial_shift_samples(const std::vector<Mint> &ys, Mint c,
int m) {
assert(m >= 0);
int n = int(ys.size());
if (m == 0) {
return {};
}
assert(n > 0);
auto ifc = polynomial_special_points_detail::factorial_inverses<Mint>(n);
std::vector<Mint> wgt(n);
for (int i = 0; i < n; i++) {
wgt[i] = ys[i] * ifc[i] * ifc[n - 1 - i];
if ((n - 1 - i) & 1) {
wgt[i] = -wgt[i];
}
}
std::vector<Mint> dif(n + m - 1);
for (int off = 1 - n; off < m; off++) {
dif[off + n - 1] = c + Mint(off);
}
auto idf = polynomial_special_points_detail::inverses_allowing_zero(dif);
std::vector<Mint> cnv = atcoder::convolution(std::move(wgt), std::vector<Mint>(idf));
std::vector<int> cs(m, -1);
for (int pos = 0; pos < int(dif.size()); pos++) {
if (dif[pos] != Mint{}) {
continue;
}
int lo = std::max(0, pos - n + 1);
int hi = std::min(m - 1, pos);
for (int k = lo; k <= hi; k++) {
cs[k] = k + n - 1 - pos;
}
}
int nz = 0;
Mint nzp = 1;
for (int i = 0; i < n; i++) {
Mint val = dif[n - 1 - i];
if (val == Mint{}) {
nz++;
} else {
nzp *= val;
}
}
std::vector<Mint> res(m);
for (int k = 0; k < m; k++) {
if (nz > 0) {
int sid = cs[k];
assert(sid >= 0);
res[k] = ys[sid];
} else {
res[k] = nzp * cnv[k + n - 1];
}
if (k + 1 == m) {
break;
}
Mint rem = dif[k];
if (rem == Mint{}) {
nz--;
} else {
nzp *= idf[k];
}
Mint add = dif[k + n];
if (add == Mint{}) {
nz++;
} else {
nzp *= add;
}
}
return res;
}
/// @brief Evaluate f(a r^k) for k=0..m-1. Writing
/// r^(ik)=q(i+k)/(q(i)q(k)), q(t)=r^(t(t-1)/2), changes the Hankel product
/// into one ordinary convolution.
/// Requires a field and ACL-supported convolutions; repeated points are allowed.
template <class Mint>
std::vector<Mint> polynomial_evaluate_geometric(const std::vector<Mint> &pol,
int m, Mint a, Mint r) {
assert(m >= 0);
int n = int(pol.size());
if (m == 0) {
return {};
}
if (n == 0) {
return std::vector<Mint>(m);
}
if (r == Mint{}) {
std::vector<Mint> res(m, pol[0]);
Mint val = 0;
for (int i = n - 1; i >= 0; i--) {
val = val * a + pol[i];
}
res[0] = val;
return res;
}
std::vector<Mint> q(n + m - 1);
q[0] = 1;
Mint pw = 1;
for (int i = 1; i < int(q.size()); i++) {
q[i] = q[i - 1] * pw;
pw *= r;
}
std::vector<Mint> iq(q.begin(), q.begin() + std::max(n, m));
iq = batch_inverse(iq);
std::vector<Mint> l(n);
Mint apo = 1;
for (int i = 0; i < n; i++) {
l[n - 1 - i] = pol[i] * apo * iq[i];
apo *= a;
}
std::vector<Mint> prd = atcoder::convolution(std::move(l), std::move(q));
std::vector<Mint> res(m);
for (int k = 0; k < m; k++) {
res[k] = prd[n - 1 + k] * iq[k];
}
return res;
}
/// @brief Interpolate from the distinct points a,ar,...,ar^(n-1). Closed
/// forms for the product polynomial and its derivatives give barycentric
/// weights in linear time; a geometric evaluation computes all needed
/// moments, and one final convolution recovers the monomial coefficients.
/// Requires a field and ACL-supported convolutions. Returns exactly n coefficients.
template <class Mint>
std::vector<Mint> polynomial_interpolate_geometric(const std::vector<Mint> &vs,
Mint a, Mint r) {
int n = int(vs.size());
if (n == 0) {
return {};
}
if (n == 1) {
return vs;
}
assert(a != Mint{});
assert(r != Mint{});
std::vector<Mint> omp(n);
Mint rpo = r;
for (int i = 1; i <= n; i++) {
omp[i - 1] = Mint(1) - rpo;
if (i < n) {
assert(omp[i - 1] != Mint{});
}
rpo *= r;
}
std::vector<Mint> nzd(omp.begin(), omp.end() - 1);
auto iom = batch_inverse(nzd);
std::vector<Mint> pre(n, Mint(1));
for (int i = 1; i < n; i++) {
pre[i] = pre[i - 1] * omp[i - 1];
}
// Inverse prefix products need only one division.
pre.back() = Mint(1) / pre.back();
for (int i = n - 1; i > 0; i--) {
pre[i - 1] = pre[i] * omp[i - 1];
}
Mint ap = 1, rp = 1;
for (int i = 1; i < n; i++) ap *= a;
for (int i = 0; i < n - 2; i++) rp *= r;
Mint iap = Mint(1) / ap, step = Mint(1) / rp, irp = 1;
std::vector<Mint> wgt(n);
for (int i = 0; i < n; i++) {
Mint idv = iap * irp * pre[i] * pre[n - 1 - i];
if (i & 1) {
idv = -idv;
}
wgt[i] = vs[i] * idv;
// E(i+1)-E(i)=n-2-i for the exponent in the derivative weight.
irp *= step;
step *= r;
}
std::vector<Mint> mom = polynomial_evaluate_geometric(wgt, n, Mint(1), r);
ap = 1;
for (int i = 0; i < n; i++) {
mom[i] *= ap;
ap *= a;
}
std::vector<Mint> qbc(n + 1, Mint(1));
for (int t = 1; t < n; t++) {
qbc[t] = qbc[t - 1] * omp[n - t] * iom[t - 1];
}
std::vector<Mint> pf(n + 1);
Mint map = 1;
Mint qf = 1;
Mint qs = 1;
for (int t = 0; t <= n; t++) {
int cf = n - t;
pf[cf] = qbc[t] * map * qf;
map *= -a;
qf *= qs;
qs *= r;
}
std::vector<Mint> rev(n);
for (int i = 0; i < n; i++) {
rev[i] = pf[n - i];
}
std::vector<Mint> cnv = atcoder::convolution(std::move(rev), std::move(mom));
std::vector<Mint> res(n);
for (int k = 0; k < n; k++) {
res[k] = cnv[n - 1 - k];
}
return res;
}
/// @brief Multiply a sequence of polynomials by always combining the two
/// currently shortest factors. The Huffman-style merge order keeps the total
/// convolution work O(M(D) log n), where D is the final degree.
template <class Mint>
std::vector<Mint>
polynomial_product_sequence(std::vector<std::vector<Mint>> fs) {
using item = std::pair<int, int>;
std::priority_queue<item, std::vector<item>, std::greater<item>> que;
for (int i = 0; i < int(fs.size()); i++) {
if (fs[i].empty()) return {};
que.emplace(int(fs[i].size()), i);
}
if (que.empty()) {
return {Mint(1)};
}
while (que.size() > 1) {
int lhs = que.top().second;
que.pop();
int b = que.top().second;
que.pop();
fs.push_back(atcoder::convolution(std::move(fs[lhs]), std::move(fs[b])));
que.emplace(int(fs.back().size()), int(fs.size()) - 1);
}
return fs[que.top().second];
}
} // namespace noya
#include <algorithm>
#include <cassert>
#include <cmath>
#include <cstdint>
#include <utility>
#include <vector>
namespace noya {
namespace large_factorial_detail {
template <class Mint>
std::vector<Mint> factorial_block_prefix(std::uint64_t mx, int blk) {
int nb = int(mx / blk);
std::vector<Mint> pre(nb + 1, Mint(1));
if (nb == 0) {
return pre;
}
std::vector<std::vector<Mint>> fs;
fs.reserve(blk);
for (int i = 1; i <= blk; i++) {
fs.push_back({Mint(i), Mint(1)});
}
std::vector<Mint> bp = polynomial_product_sequence(std::move(fs));
std::vector<Mint> poi(nb);
for (int i = 0; i < nb; i++) {
poi[i] = Mint(std::uint64_t(i) * blk);
}
std::vector<Mint> prd = polynomial_multipoint_evaluation(bp, poi);
for (int i = 0; i < nb; i++) {
pre[i + 1] = pre[i] * prd[i];
}
return pre;
}
} // namespace large_factorial_detail
/// @brief Compute several factorials modulo a fixed prime without a linear
/// table. Split 1..N into blocks of length B about sqrt(N). The product inside
/// one block is the degree-B polynomial P(x)=prod_{i=1}^B(x+i); a product tree
/// constructs P and multipoint evaluation obtains P(0),P(B),P(2B),.... Prefix
/// products answer every full block, followed by at most B direct factors.
/// Requires each argument below Mint::mod(), a prime modulus and the
/// convolution lengths supported by ACL.
template <class Mint>
std::vector<Mint> large_factorials(const std::vector<std::uint64_t> &que) {
if (que.empty()) {
return {};
}
std::uint64_t mx = *std::max_element(que.begin(), que.end());
assert(mx < std::uint64_t(Mint::mod()));
int blk = int(std::sqrt(static_cast<long double>(mx + 1)));
blk = std::max(blk, 1);
while (std::uint64_t(blk) * blk < mx + 1) {
blk++;
}
std::vector<Mint> pre =
large_factorial_detail::factorial_block_prefix<Mint>(mx, blk);
std::vector<Mint> res;
res.reserve(que.size());
for (std::uint64_t n : que) {
std::uint64_t com = n / blk;
Mint val = pre[com];
for (std::uint64_t i = com * blk + 1; i <= n; i++) {
val *= Mint(i);
}
res.push_back(val);
}
return res;
}
/// @brief Compute a large batch of factorials modulo a fixed prime. Boundary
/// values (kB)! are obtained by evaluating the block-product polynomial
/// prod_{i=1}^B(x+i). For a query n=qB+r, split the remaining product
/// (qB+1)...n into power-of-two suffixes. A suffix of length 2^b is the falling
/// factorial polynomial x(x-1)...(x-2^b+1) evaluated at its current right
/// endpoint. Queries sharing b are evaluated in batches of at most 2^b points,
/// retaining only one batch of points and query indices at a time.
/// Requires each argument below Mint::mod(), a prime modulus and the
/// convolution lengths supported by ACL.
template <class Mint>
std::vector<Mint> many_factorials(const std::vector<std::uint32_t> &que) {
if (que.empty()) {
return {};
}
constexpr int lg = 15;
constexpr int blk = 1 << lg;
std::uint32_t mx = *std::max_element(que.begin(), que.end());
assert(mx < std::uint32_t(Mint::mod()));
std::vector<Mint> pre =
large_factorial_detail::factorial_block_prefix<Mint>(mx, blk);
std::vector<Mint> res(que.size());
std::uint32_t msk = 0;
for (int qry = 0; qry < int(que.size()); qry++) {
res[qry] = pre[que[qry] / blk];
msk |= que[qry] % blk;
}
for (int bit = 0; bit < lg; bit++) {
int len = 1 << bit;
if (!(msk & len))
continue;
std::vector<Mint> ff = stirling_first_kind_row<Mint>(len), poi;
std::vector<int> ids;
int cap = std::min(len, int(que.size()));
poi.reserve(cap);
ids.reserve(cap);
auto flush = [&] {
if (poi.empty())
return;
if (len <= 32) {
// Match the product tree's small-polynomial Horner cutoff.
for (int i = 0; i < int(ids.size()); ++i) {
Mint val = 0;
for (int j = len; j >= 0; --j)
val = val * poi[i] + ff[j];
res[ids[i]] *= val;
}
} else {
auto vs = polynomial_multipoint_evaluation(ff, poi);
for (int i = 0; i < int(ids.size()); ++i)
res[ids[i]] *= vs[i];
}
poi.clear();
ids.clear();
};
for (int qry = 0; qry < int(que.size()); ++qry) {
auto n = que[qry];
if (!(n & len))
continue;
// Lower suffixes have already consumed the low bits.
poi.emplace_back(n - (n & (len - 1)));
ids.push_back(qry);
if (int(poi.size()) == len)
flush();
}
flush();
}
return res;
}
} // namespace noya
// NOYA_LARGE_FACTORIAL_HPP