Library
This documentation is automatically generated by online-judge-tools/verification-helper
Extended Euclidean Algorithm
(src/number_theory/ext_gcd.hpp)
- View this file on GitHub
- Last update: 2022-04-06 15:02:09+09:00
- Include:
#include "src/number_theory/ext_gcd.hpp"
Depends on
Required by
- Berlekamp-Massey Algorithm
(src/algebra/berlekamp_massey.hpp)
- Fast Fourier Transform
(src/algebra/fft.hpp)
- Formal Power Series
(src/algebra/fps.hpp)
- Polynomial Interpolation
(src/algebra/interpolation.hpp)
Number Theoretic Transform
(src/algebra/ntt.hpp)
- Polynomial
(src/algebra/polynomial.hpp)
- Order
(src/number_theory/order.hpp)
Verified with
- test/aizu-online-judge/NTL_1_E.test.cpp
- test/library-checker/convolution_mod.test.cpp
- test/library-checker/division_of_polynomials.test.cpp
- test/library-checker/exp_of_formal_power_series.test.cpp
- test/library-checker/find_linear_recurrence.test.cpp
- test/library-checker/inv_of_formal_power_series.test.cpp
- test/library-checker/log_of_formal_power_series.test.cpp
- test/library-checker/multipoint_evaluation.test.cpp
- test/library-checker/polynomial_interpolation.test.cpp
- test/library-checker/polynomial_taylor_shift.test.cpp
- test/library-checker/pow_of_formal_power_series.test.cpp
- test/library-checker/sqrt_of_formal_power_series.test.cpp
Code
#pragma once
/**
* @file ext_gcd.hpp
* @brief Extended Euclidean Algorithm
*/
#include <tuple>
#include "src/utils/sfinae.hpp"
namespace workspace {
/**
* @param __a Integer
* @param __b Integer
* @return Pair of integers (x, y) s.t. ax + by = g = gcd(a, b) and (b = 0 or 0
* <= x < |b/g|) and (a = 0 or -|a/g| < y <= 0). Return (0, 0) if (a, b) = (0,
* 0).
*/
template <typename _T1, typename _T2>
constexpr auto ext_gcd(_T1 __a, _T2 __b) noexcept {
static_assert(is_integral_ext<_T1>::value);
static_assert(is_integral_ext<_T2>::value);
using value_type = typename std::make_signed<
typename std::common_type<_T1, _T2>::type>::type;
using result_type = std::pair<value_type, value_type>;
value_type a{__a}, b{__b}, p{1}, q{}, r{}, s{1};
while (b != value_type(0)) {
auto t = a / b;
r ^= p ^= r ^= p -= t * r;
s ^= q ^= s ^= q -= t * s;
b ^= a ^= b ^= a -= t * b;
}
if (a < 0) p = -p, q = -q, a = -a;
if (p < 0) {
__a /= a, __b /= a;
if (__b > 0)
p += __b, q -= __a;
else
p -= __b, q += __a;
}
return result_type{p, q};
}
/**
* @param __a Integer
* @param __b Integer
* @param __c Integer
* @return Pair of integers (x, y) s.t. ax + by = c and (b = 0 or 0 <= x <
* |b/g|). Return (0, 0) if there is no solution.
*/
template <typename _T1, typename _T2, typename _T3>
constexpr auto ext_gcd(_T1 __a, _T2 __b, _T3 __c) noexcept {
static_assert(is_integral_ext<_T1>::value);
static_assert(is_integral_ext<_T2>::value);
static_assert(is_integral_ext<_T3>::value);
using value_type = typename std::make_signed<
typename std::common_type<_T1, _T2, _T3>::type>::type;
using result_type = std::pair<value_type, value_type>;
value_type a{__a}, b{__b}, p{1}, q{}, r{}, s{1};
while (b != value_type(0)) {
auto t = a / b;
r ^= p ^= r ^= p -= t * r;
s ^= q ^= s ^= q -= t * s;
b ^= a ^= b ^= a -= t * b;
}
if (__c % a) return result_type{};
__a /= a, __b /= a, __c /= a;
p *= __c, q *= __c;
if (__b != value_type(0)) {
auto t = p / __b;
p -= __b * t;
q += __a * t;
if (p < 0) {
if (__b > 0)
p += __b, q -= __a;
else
p -= __b, q += __a;
}
}
return result_type{p, q};
}
} // namespace workspace#line 2 "src/number_theory/ext_gcd.hpp"
/**
* @file ext_gcd.hpp
* @brief Extended Euclidean Algorithm
*/
#include <tuple>
#line 2 "src/utils/sfinae.hpp"
/**
* @file sfinae.hpp
* @brief SFINAE
*/
#include <cstdint>
#include <iterator>
#include <type_traits>
#ifndef __INT128_DEFINED__
#ifdef __SIZEOF_INT128__
#define __INT128_DEFINED__ 1
#else
#define __INT128_DEFINED__ 0
#endif
#endif
namespace std {
#if __INT128_DEFINED__
template <> struct make_signed<__uint128_t> { using type = __int128_t; };
template <> struct make_signed<__int128_t> { using type = __int128_t; };
template <> struct make_unsigned<__uint128_t> { using type = __uint128_t; };
template <> struct make_unsigned<__int128_t> { using type = __uint128_t; };
template <> struct is_signed<__uint128_t> : std::false_type {};
template <> struct is_signed<__int128_t> : std::true_type {};
template <> struct is_unsigned<__uint128_t> : std::true_type {};
template <> struct is_unsigned<__int128_t> : std::false_type {};
#endif
} // namespace std
namespace workspace {
template <class Tp, class... Args> struct variadic_front { using type = Tp; };
template <class... Args> struct variadic_back;
template <class Tp> struct variadic_back<Tp> { using type = Tp; };
template <class Tp, class... Args> struct variadic_back<Tp, Args...> {
using type = typename variadic_back<Args...>::type;
};
template <class type, template <class> class trait>
using enable_if_trait_type = typename std::enable_if<trait<type>::value>::type;
/**
* @brief Return type of subscripting ( @c [] ) access.
*/
template <class _Tp>
using subscripted_type =
typename std::decay<decltype(std::declval<_Tp&>()[0])>::type;
template <class Container>
using element_type = typename std::decay<decltype(*std::begin(
std::declval<Container&>()))>::type;
template <class _Tp, class = void> struct has_begin : std::false_type {};
template <class _Tp>
struct has_begin<
_Tp, std::__void_t<decltype(std::begin(std::declval<const _Tp&>()))>>
: std::true_type {
using type = decltype(std::begin(std::declval<const _Tp&>()));
};
template <class _Tp, class = void> struct has_size : std::false_type {};
template <class _Tp>
struct has_size<_Tp, std::__void_t<decltype(std::size(std::declval<_Tp>()))>>
: std::true_type {};
template <class _Tp, class = void> struct has_resize : std::false_type {};
template <class _Tp>
struct has_resize<_Tp, std::__void_t<decltype(std::declval<_Tp>().resize(
std::declval<size_t>()))>> : std::true_type {};
template <class _Tp, class = void> struct has_mod : std::false_type {};
template <class _Tp>
struct has_mod<_Tp, std::__void_t<decltype(_Tp::mod)>> : std::true_type {};
template <class _Tp, class = void> struct is_integral_ext : std::false_type {};
template <class _Tp>
struct is_integral_ext<
_Tp, typename std::enable_if<std::is_integral<_Tp>::value>::type>
: std::true_type {};
#if __INT128_DEFINED__
template <> struct is_integral_ext<__int128_t> : std::true_type {};
template <> struct is_integral_ext<__uint128_t> : std::true_type {};
#endif
#if __cplusplus >= 201402
template <class _Tp>
constexpr static bool is_integral_ext_v = is_integral_ext<_Tp>::value;
#endif
template <typename _Tp, typename = void> struct multiplicable_uint {
using type = uint_least32_t;
};
template <typename _Tp>
struct multiplicable_uint<
_Tp,
typename std::enable_if<(2 < sizeof(_Tp)) &&
(!__INT128_DEFINED__ || sizeof(_Tp) <= 4)>::type> {
using type = uint_least64_t;
};
#if __INT128_DEFINED__
template <typename _Tp>
struct multiplicable_uint<_Tp,
typename std::enable_if<(4 < sizeof(_Tp))>::type> {
using type = __uint128_t;
};
#endif
template <typename _Tp> struct multiplicable_int {
using type =
typename std::make_signed<typename multiplicable_uint<_Tp>::type>::type;
};
template <typename _Tp> struct multiplicable {
using type = std::conditional_t<
is_integral_ext<_Tp>::value,
std::conditional_t<std::is_signed<_Tp>::value,
typename multiplicable_int<_Tp>::type,
typename multiplicable_uint<_Tp>::type>,
_Tp>;
};
template <class> struct first_arg { using type = void; };
template <class _R, class _Tp, class... _Args>
struct first_arg<_R(_Tp, _Args...)> {
using type = _Tp;
};
template <class _R, class _Tp, class... _Args>
struct first_arg<_R (*)(_Tp, _Args...)> {
using type = _Tp;
};
template <class _G, class _R, class _Tp, class... _Args>
struct first_arg<_R (_G::*)(_Tp, _Args...)> {
using type = _Tp;
};
template <class _G, class _R, class _Tp, class... _Args>
struct first_arg<_R (_G::*)(_Tp, _Args...) const> {
using type = _Tp;
};
template <class _Tp, class = void> struct parse_compare : first_arg<_Tp> {};
template <class _Tp>
struct parse_compare<_Tp, std::__void_t<decltype(&_Tp::operator())>>
: first_arg<decltype(&_Tp::operator())> {};
template <class _Container, class = void> struct get_dimension {
static constexpr size_t value = 0;
};
template <class _Container>
struct get_dimension<_Container,
std::enable_if_t<has_begin<_Container>::value>> {
static constexpr size_t value =
1 + get_dimension<typename std::iterator_traits<
typename has_begin<_Container>::type>::value_type>::value;
};
} // namespace workspace
#line 11 "src/number_theory/ext_gcd.hpp"
namespace workspace {
/**
* @param __a Integer
* @param __b Integer
* @return Pair of integers (x, y) s.t. ax + by = g = gcd(a, b) and (b = 0 or 0
* <= x < |b/g|) and (a = 0 or -|a/g| < y <= 0). Return (0, 0) if (a, b) = (0,
* 0).
*/
template <typename _T1, typename _T2>
constexpr auto ext_gcd(_T1 __a, _T2 __b) noexcept {
static_assert(is_integral_ext<_T1>::value);
static_assert(is_integral_ext<_T2>::value);
using value_type = typename std::make_signed<
typename std::common_type<_T1, _T2>::type>::type;
using result_type = std::pair<value_type, value_type>;
value_type a{__a}, b{__b}, p{1}, q{}, r{}, s{1};
while (b != value_type(0)) {
auto t = a / b;
r ^= p ^= r ^= p -= t * r;
s ^= q ^= s ^= q -= t * s;
b ^= a ^= b ^= a -= t * b;
}
if (a < 0) p = -p, q = -q, a = -a;
if (p < 0) {
__a /= a, __b /= a;
if (__b > 0)
p += __b, q -= __a;
else
p -= __b, q += __a;
}
return result_type{p, q};
}
/**
* @param __a Integer
* @param __b Integer
* @param __c Integer
* @return Pair of integers (x, y) s.t. ax + by = c and (b = 0 or 0 <= x <
* |b/g|). Return (0, 0) if there is no solution.
*/
template <typename _T1, typename _T2, typename _T3>
constexpr auto ext_gcd(_T1 __a, _T2 __b, _T3 __c) noexcept {
static_assert(is_integral_ext<_T1>::value);
static_assert(is_integral_ext<_T2>::value);
static_assert(is_integral_ext<_T3>::value);
using value_type = typename std::make_signed<
typename std::common_type<_T1, _T2, _T3>::type>::type;
using result_type = std::pair<value_type, value_type>;
value_type a{__a}, b{__b}, p{1}, q{}, r{}, s{1};
while (b != value_type(0)) {
auto t = a / b;
r ^= p ^= r ^= p -= t * r;
s ^= q ^= s ^= q -= t * s;
b ^= a ^= b ^= a -= t * b;
}
if (__c % a) return result_type{};
__a /= a, __b /= a, __c /= a;
p *= __c, q *= __c;
if (__b != value_type(0)) {
auto t = p / __b;
p -= __b * t;
q += __a * t;
if (p < 0) {
if (__b > 0)
p += __b, q -= __a;
else
p -= __b, q += __a;
}
}
return result_type{p, q};
}
} // namespace workspace