big_integer_multiplication.hpp¶
用卷积乘任意长有符号整数，支持 2 到 36 进制输入输出。
\[
\displaystyle C = A B,\qquad A=\sum_{i=0}^{n-1}a_i B^i,\qquad B=\sum_{j=0}^{m-1}b_j B^j,\qquad C=\sum_{k=0}^{n+m-2}\left(\sum_{i+j=k}a_i b_j\right)B^k
\]
Complexity: Time: O(n log n), where n is the larger digit count. Space: O(n).
AC 记录：multiplication_of_big_integers, multiplication_of_hex_big_integers。
Implementation¶
当前头文件，省略 include guard；依赖见 #include。
/// @complexity Time: O(n log n), where n is the larger digit count.
/// Space: O(n).
#include "noya/big_integer_addition.hpp"
#include "noya/convolution_mod.hpp"
#include <algorithm>
#include <cassert>
#include <cstdint>
#include <string>
#include <string_view>
#include <vector>
namespace noya {
namespace big_integer_multiplication_internal {
inline std::vector<std::uint64_t> read_limbs(std::string_view mag, int bas,
int dig) {
std::vector<std::uint64_t> lim;
for (std::size_t r = mag.size(); r > 0;) {
std::size_t l = r > std::size_t(dig) ? r - dig : 0;
std::uint64_t val = 0;
for (std::size_t pos = l; pos < r; pos++) {
val = val * bas + big_integer_addition_internal::digit_value(mag[pos]);
}
lim.push_back(val);
r = l;
}
return lim;
}
inline std::vector<std::uint64_t>
multiply_limbs(const std::vector<std::uint64_t> &a,
const std::vector<std::uint64_t> &b, std::uint64_t bs) {
if (std::min(a.size(), b.size()) <= 32) {
std::vector<std::uint64_t> res(a.size() + b.size());
for (int l = 0; l < int(a.size()); l++) {
unsigned __int128 car = 0;
for (int r = 0; r < int(b.size()); r++) {
unsigned __int128 val =
res[l + r] + car + static_cast<unsigned __int128>(a[l]) * b[r];
res[l + r] = std::uint64_t(val % bs);
car = val / bs;
}
int pos = l + int(b.size());
while (car > 0) {
unsigned __int128 val = res[pos] + car;
res[pos] = std::uint64_t(val % bs);
car = val / bs;
pos++;
}
}
return res;
}
std::uint64_t ma = *std::max_element(a.begin(), a.end());
std::uint64_t mb = *std::max_element(b.begin(), b.end());
unsigned __int128 cb =
static_cast<unsigned __int128>(std::min(a.size(), b.size())) * ma * mb;
assert(cb < static_cast<unsigned __int128>(UINT64_MAX));
// Every raw coefficient is below this modulus, so reduction after CRT does
// not discard information.
std::uint64_t mod = std::uint64_t(cb) + 1;
std::vector<std::uint64_t> prd = convolution_mod(a, b, mod);
std::uint64_t car = 0;
for (std::uint64_t &val : prd) {
unsigned __int128 cur = static_cast<unsigned __int128>(val) + car;
val = std::uint64_t(cur % bs);
car = std::uint64_t(cur / bs);
}
while (car > 0) {
prd.push_back(car % bs);
car /= bs;
}
return prd;
}
inline std::string write_limbs(std::vector<std::uint64_t> lim, int bas, int dig,
bool cap) {
while (lim.size() > 1 && lim.back() == 0) {
lim.pop_back();
}
std::string res;
res.reserve(lim.size() * dig);
for (int idx = int(lim.size()) - 1; idx >= 0; idx--) {
std::uint64_t val = lim[idx];
std::string blk;
do {
blk.push_back(
big_integer_addition_internal::digit_character(int(val % bas), cap));
val /= bas;
} while (val > 0);
if (idx + 1 != int(lim.size())) {
blk.resize(dig, '0');
}
std::reverse(blk.begin(), blk.end());
res += blk;
}
return res;
}
} // namespace big_integer_multiplication_internal
/// @brief Multiply two arbitrarily long signed integers in base 2..36. Digits
/// are packed into limbs whose square fits exact three-prime CRT convolution;
/// the product coefficients are then normalized by a linear carry pass.
inline std::string multiply_big_integers(std::string_view a, std::string_view b,
int bas = 10, bool cap = false) {
using namespace big_integer_addition_internal;
using namespace big_integer_multiplication_internal;
parsed_integer l = parse(a, bas);
parsed_integer r = parse(b, bas);
if (l.mag == "0" || r.mag == "0") {
return "0";
}
constexpr std::uint64_t mbs = 10'000;
int dig = 1;
std::uint64_t bs = bas;
while (bs <= mbs / std::uint64_t(bas)) {
bs *= bas;
dig++;
}
auto ll = read_limbs(l.mag, bas, dig);
auto rl = read_limbs(r.mag, bas, dig);
std::string res = write_limbs(multiply_limbs(ll, rl, bs), bas, dig, cap);
if (l.neg != r.neg) {
res.insert(res.begin(), '-');
}
return res;
}
} // namespace noya
#ifndef NOYA_BIG_INTEGER_MULTIPLICATION_HPP
#define NOYA_BIG_INTEGER_MULTIPLICATION_HPP 1
/// @complexity Time: O(n log n), where n is the larger digit count.
/// Space: O(n).
#include "noya/big_integer_addition.hpp"
#include "noya/convolution_mod.hpp"
#include <algorithm>
#include <cassert>
#include <cstdint>
#include <string>
#include <string_view>
#include <vector>
namespace noya {
namespace big_integer_multiplication_internal {
inline std::vector<std::uint64_t> read_limbs(std::string_view mag, int bas,
int dig) {
std::vector<std::uint64_t> lim;
for (std::size_t r = mag.size(); r > 0;) {
std::size_t l = r > std::size_t(dig) ? r - dig : 0;
std::uint64_t val = 0;
for (std::size_t pos = l; pos < r; pos++) {
val = val * bas + big_integer_addition_internal::digit_value(mag[pos]);
}
lim.push_back(val);
r = l;
}
return lim;
}
inline std::vector<std::uint64_t>
multiply_limbs(const std::vector<std::uint64_t> &a,
const std::vector<std::uint64_t> &b, std::uint64_t bs) {
if (std::min(a.size(), b.size()) <= 32) {
std::vector<std::uint64_t> res(a.size() + b.size());
for (int l = 0; l < int(a.size()); l++) {
unsigned __int128 car = 0;
for (int r = 0; r < int(b.size()); r++) {
unsigned __int128 val =
res[l + r] + car + static_cast<unsigned __int128>(a[l]) * b[r];
res[l + r] = std::uint64_t(val % bs);
car = val / bs;
}
int pos = l + int(b.size());
while (car > 0) {
unsigned __int128 val = res[pos] + car;
res[pos] = std::uint64_t(val % bs);
car = val / bs;
pos++;
}
}
return res;
}
std::uint64_t ma = *std::max_element(a.begin(), a.end());
std::uint64_t mb = *std::max_element(b.begin(), b.end());
unsigned __int128 cb =
static_cast<unsigned __int128>(std::min(a.size(), b.size())) * ma * mb;
assert(cb < static_cast<unsigned __int128>(UINT64_MAX));
// Every raw coefficient is below this modulus, so reduction after CRT does
// not discard information.
std::uint64_t mod = std::uint64_t(cb) + 1;
std::vector<std::uint64_t> prd = convolution_mod(a, b, mod);
std::uint64_t car = 0;
for (std::uint64_t &val : prd) {
unsigned __int128 cur = static_cast<unsigned __int128>(val) + car;
val = std::uint64_t(cur % bs);
car = std::uint64_t(cur / bs);
}
while (car > 0) {
prd.push_back(car % bs);
car /= bs;
}
return prd;
}
inline std::string write_limbs(std::vector<std::uint64_t> lim, int bas, int dig,
bool cap) {
while (lim.size() > 1 && lim.back() == 0) {
lim.pop_back();
}
std::string res;
res.reserve(lim.size() * dig);
for (int idx = int(lim.size()) - 1; idx >= 0; idx--) {
std::uint64_t val = lim[idx];
std::string blk;
do {
blk.push_back(
big_integer_addition_internal::digit_character(int(val % bas), cap));
val /= bas;
} while (val > 0);
if (idx + 1 != int(lim.size())) {
blk.resize(dig, '0');
}
std::reverse(blk.begin(), blk.end());
res += blk;
}
return res;
}
} // namespace big_integer_multiplication_internal
/// @brief Multiply two arbitrarily long signed integers in base 2..36. Digits
/// are packed into limbs whose square fits exact three-prime CRT convolution;
/// the product coefficients are then normalized by a linear carry pass.
inline std::string multiply_big_integers(std::string_view a, std::string_view b,
int bas = 10, bool cap = false) {
using namespace big_integer_addition_internal;
using namespace big_integer_multiplication_internal;
parsed_integer l = parse(a, bas);
parsed_integer r = parse(b, bas);
if (l.mag == "0" || r.mag == "0") {
return "0";
}
constexpr std::uint64_t mbs = 10'000;
int dig = 1;
std::uint64_t bs = bas;
while (bs <= mbs / std::uint64_t(bas)) {
bs *= bas;
dig++;
}
auto ll = read_limbs(l.mag, bas, dig);
auto rl = read_limbs(r.mag, bas, dig);
std::string res = write_limbs(multiply_limbs(ll, rl, bs), bas, dig, cap);
if (l.neg != r.neg) {
res.insert(res.begin(), '-');
}
return res;
}
} // namespace noya
#endif // NOYA_BIG_INTEGER_MULTIPLICATION_HPP
#include <algorithm>
#include <array>
#include <cassert>
#include <cstdint>
#include <numeric>
#include <string>
#include <string_view>
#include <type_traits>
#include <utility>
#include <vector>
/// @complexity Time: O(n log n), where n is the larger digit count.
/// Space: O(n).
/// @complexity Time: O(|a| + |b|).
/// Space: O(max(|a|, |b|)) for the returned representation.
namespace noya {
namespace big_integer_addition_internal {
struct parsed_integer {
bool neg = false;
std::string_view mag;
};
inline int digit_value(char dig) {
if ('0' <= dig && dig <= '9') {
return dig - '0';
}
if ('a' <= dig && dig <= 'z') {
return dig - 'a' + 10;
}
if ('A' <= dig && dig <= 'Z') {
return dig - 'A' + 10;
}
return -1;
}
inline parsed_integer parse(std::string_view val, int bas) {
assert(2 <= bas && bas <= 36);
assert(!val.empty());
bool neg = val.front() == '-';
if (neg || val.front() == '+') {
val.remove_prefix(1);
}
assert(!val.empty());
for (char dig : val) {
assert(0 <= digit_value(dig) && digit_value(dig) < bas);
}
while (val.size() > 1 && val.front() == '0') {
val.remove_prefix(1);
}
if (val == "0") {
neg = false;
}
return {neg, val};
}
inline int compare_magnitude(std::string_view a, std::string_view b) {
if (a.size() != b.size()) {
return a.size() < b.size() ? -1 : 1;
}
for (std::size_t idx = 0; idx < a.size(); idx++) {
int l = digit_value(a[idx]);
int r = digit_value(b[idx]);
if (l != r) {
return l < r ? -1 : 1;
}
}
return 0;
}
inline char digit_character(int val, bool cap) {
assert(0 <= val && val < 36);
if (val < 10) {
return char('0' + val);
}
return char((cap ? 'A' : 'a') + val - 10);
}
inline std::string add_magnitudes(std::string_view a, std::string_view b,
int bas, bool cap) {
std::string res;
res.reserve(std::max(a.size(), b.size()) + 1);
int car = 0;
std::size_t l = a.size();
std::size_t r = b.size();
while (l > 0 || r > 0 || car != 0) {
int val = car;
if (l > 0) {
val += digit_value(a[--l]);
}
if (r > 0) {
val += digit_value(b[--r]);
}
res.push_back(digit_character(val % bas, cap));
car = val / bas;
}
std::reverse(res.begin(), res.end());
return res;
}
// Precondition: a >= b as unsigned magnitudes.
inline std::string subtract_magnitudes(std::string_view a, std::string_view b,
int bas, bool cap) {
std::string res;
res.reserve(a.size());
int bor = 0;
std::size_t l = a.size();
std::size_t r = b.size();
while (l > 0) {
int val = digit_value(a[--l]) - bor;
if (r > 0) {
val -= digit_value(b[--r]);
}
if (val < 0) {
val += bas;
bor = 1;
} else {
bor = 0;
}
res.push_back(digit_character(val, cap));
}
assert(bor == 0);
while (res.size() > 1 && res.back() == '0') {
res.pop_back();
}
std::reverse(res.begin(), res.end());
return res;
}
} // namespace big_integer_addition_internal
/// @brief Add two arbitrarily long signed integers represented in base 2..36.
/// The result is canonical (no leading zeroes and no negative zero).
inline std::string add_big_integers(std::string_view a, std::string_view b,
int bas = 10, bool cap = false) {
using namespace big_integer_addition_internal;
parsed_integer l = parse(a, bas);
parsed_integer r = parse(b, bas);
if (l.neg == r.neg) {
std::string res = add_magnitudes(l.mag, r.mag, bas, cap);
if (l.neg) {
res.insert(res.begin(), '-');
}
return res;
}
int ord = compare_magnitude(l.mag, r.mag);
if (ord == 0) {
return "0";
}
bool neg = ord > 0 ? l.neg : r.neg;
std::string res = ord > 0 ? subtract_magnitudes(l.mag, r.mag, bas, cap)
: subtract_magnitudes(r.mag, l.mag, bas, cap);
if (neg) {
res.insert(res.begin(), '-');
}
return res;
}
} // namespace noya
/// @complexity Time: O(n log n) for result length n.
/// Space: O(n).
#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
#ifdef _MSC_VER
#include <intrin.h>
#endif
#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
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
namespace noya {
namespace convolution_mod_internal {
using u64 = std::uint64_t;
using u128 = unsigned __int128;
inline u64 power(u64 vl, u64 exp, u64 mod) {
u64 res = 1;
while (exp > 0) {
if (exp & 1) {
res = u64(u128(res) * vl % mod);
}
vl = u64(u128(vl) * vl % mod);
exp >>= 1;
}
return res;
}
inline u64 inverse_prime(u64 vl, u64 p) { return power(vl, p - 2, p); }
template <int Mod>
std::vector<int> run_ntt(const std::vector<std::uint64_t> &lhs,
const std::vector<std::uint64_t> &rhs) {
using mint = atcoder::static_modint<Mod>;
std::vector<mint> a(lhs.size());
std::vector<mint> b(rhs.size());
for (int idx = 0; idx < int(lhs.size()); idx++) {
a[idx] = lhs[idx] % Mod;
}
for (int idx = 0; idx < int(rhs.size()); idx++) {
b[idx] = rhs[idx] % Mod;
}
auto prd = atcoder::convolution(a, b);
std::vector<int> res(prd.size());
for (int idx = 0; idx < int(prd.size()); idx++) {
res[idx] = prd[idx].val();
}
return res;
}
} // namespace convolution_mod_internal
/// @brief Convolve nonnegative residues modulo an arbitrary modulus using
/// three NTT primes. Requires result length <= 2^24 and each integer
/// coefficient before reduction to be smaller than the product of the primes.
inline std::vector<std::uint64_t>
convolution_mod(const std::vector<std::uint64_t> &lhs,
const std::vector<std::uint64_t> &rhs, std::uint64_t mod) {
using namespace convolution_mod_internal;
assert(mod >= 1);
if (lhs.empty() || rhs.empty()) {
return {};
}
constexpr u64 p1 = 167772161;
constexpr u64 p2 = 469762049;
constexpr u64 p3 = 1224736769;
constexpr u128 pp = u128(p1) * p2 * p3;
std::size_t nr = lhs.size() + rhs.size() - 1;
assert(nr <= (std::size_t(1) << 24));
u64 ma = 0;
u64 mb = 0;
for (u64 vl : lhs) {
assert(vl < mod);
ma = std::max(ma, vl);
}
for (u64 vl : rhs) {
assert(vl < mod);
mb = std::max(mb, vl);
}
assert(u128(std::min(lhs.size(), rhs.size())) * ma * mb < pp);
auto rsd = run_ntt<p1>(lhs, rhs);
auto re0 = run_ntt<p2>(lhs, rhs);
auto re1 = run_ntt<p3>(lhs, rhs);
const u64 i12 = inverse_prime(p1 % p2, p2);
const u64 p12 = u64(u128(p1) * p2 % p3);
const u64 i13 = inverse_prime(p12, p3);
std::vector<u64> res(nr);
for (int idx = 0; idx < int(nr); idx++) {
u64 x1 = rsd[idx];
u64 x2 = u64(u128((re0[idx] + p2 - x1 % p2) % p2) * i12 % p2);
u64 x12 = u64((u128(x1) + u128(p1) * x2) % p3);
u64 x3 = u64(u128((re1[idx] + p3 - x12) % p3) * i13 % p3);
res[idx] = u64((u128(x1 % mod) + u128(p1 % mod) * x2 +
u128(u64(u128(p1) * p2 % mod)) * x3) %
mod);
}
return res;
}
} // namespace noya
namespace noya {
namespace big_integer_multiplication_internal {
inline std::vector<std::uint64_t> read_limbs(std::string_view mag, int bas,
int dig) {
std::vector<std::uint64_t> lim;
for (std::size_t r = mag.size(); r > 0;) {
std::size_t l = r > std::size_t(dig) ? r - dig : 0;
std::uint64_t val = 0;
for (std::size_t pos = l; pos < r; pos++) {
val = val * bas + big_integer_addition_internal::digit_value(mag[pos]);
}
lim.push_back(val);
r = l;
}
return lim;
}
inline std::vector<std::uint64_t>
multiply_limbs(const std::vector<std::uint64_t> &a,
const std::vector<std::uint64_t> &b, std::uint64_t bs) {
if (std::min(a.size(), b.size()) <= 32) {
std::vector<std::uint64_t> res(a.size() + b.size());
for (int l = 0; l < int(a.size()); l++) {
unsigned __int128 car = 0;
for (int r = 0; r < int(b.size()); r++) {
unsigned __int128 val =
res[l + r] + car + static_cast<unsigned __int128>(a[l]) * b[r];
res[l + r] = std::uint64_t(val % bs);
car = val / bs;
}
int pos = l + int(b.size());
while (car > 0) {
unsigned __int128 val = res[pos] + car;
res[pos] = std::uint64_t(val % bs);
car = val / bs;
pos++;
}
}
return res;
}
std::uint64_t ma = *std::max_element(a.begin(), a.end());
std::uint64_t mb = *std::max_element(b.begin(), b.end());
unsigned __int128 cb =
static_cast<unsigned __int128>(std::min(a.size(), b.size())) * ma * mb;
assert(cb < static_cast<unsigned __int128>(UINT64_MAX));
// Every raw coefficient is below this modulus, so reduction after CRT does
// not discard information.
std::uint64_t mod = std::uint64_t(cb) + 1;
std::vector<std::uint64_t> prd = convolution_mod(a, b, mod);
std::uint64_t car = 0;
for (std::uint64_t &val : prd) {
unsigned __int128 cur = static_cast<unsigned __int128>(val) + car;
val = std::uint64_t(cur % bs);
car = std::uint64_t(cur / bs);
}
while (car > 0) {
prd.push_back(car % bs);
car /= bs;
}
return prd;
}
inline std::string write_limbs(std::vector<std::uint64_t> lim, int bas, int dig,
bool cap) {
while (lim.size() > 1 && lim.back() == 0) {
lim.pop_back();
}
std::string res;
res.reserve(lim.size() * dig);
for (int idx = int(lim.size()) - 1; idx >= 0; idx--) {
std::uint64_t val = lim[idx];
std::string blk;
do {
blk.push_back(
big_integer_addition_internal::digit_character(int(val % bas), cap));
val /= bas;
} while (val > 0);
if (idx + 1 != int(lim.size())) {
blk.resize(dig, '0');
}
std::reverse(blk.begin(), blk.end());
res += blk;
}
return res;
}
} // namespace big_integer_multiplication_internal
/// @brief Multiply two arbitrarily long signed integers in base 2..36. Digits
/// are packed into limbs whose square fits exact three-prime CRT convolution;
/// the product coefficients are then normalized by a linear carry pass.
inline std::string multiply_big_integers(std::string_view a, std::string_view b,
int bas = 10, bool cap = false) {
using namespace big_integer_addition_internal;
using namespace big_integer_multiplication_internal;
parsed_integer l = parse(a, bas);
parsed_integer r = parse(b, bas);
if (l.mag == "0" || r.mag == "0") {
return "0";
}
constexpr std::uint64_t mbs = 10'000;
int dig = 1;
std::uint64_t bs = bas;
while (bs <= mbs / std::uint64_t(bas)) {
bs *= bas;
dig++;
}
auto ll = read_limbs(l.mag, bas, dig);
auto rl = read_limbs(r.mag, bas, dig);
std::string res = write_limbs(multiply_limbs(ll, rl, bs), bas, dig, cap);
if (l.neg != r.neg) {
res.insert(res.begin(), '-');
}
return res;
}
} // namespace noya