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ext_gcd.hpp
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/
ext_gcd.hpp
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#pragma once
#include "../template/template.hpp"
// https://github.com/maspypy/library/blob/main/nt/extgcd.hpp
namespace lib {
tuple<ll, ll, ll, ll> Farey_lr(ll a, ll b) {
assert(a > 0 && b > 0);
/*
Farey 数列で、a/b が最初に現れるときの、左右を求める。
a/b = 19/12 → (x1/y1, x2/y2) = (11/7, 8/5) → (11,7,8,5) を返す。
*/
if (a == b) return {0, 1, 1, 0};
ll q = (a - 1) / b;
auto [x1, y1, x2, y2] = Farey_lr(b, a - q * b);
return {q * x2 + y2, x2, q * x1 + y1, x1};
}
tuple<ll, ll, ll> extgcd(ll a, ll b) {
// ax + by = d の最小解 (x, y, d) を返す。
// (|x|+|y|, x) に関する辞書順最小とする。
auto [x1, y1, x2, y2] = Farey_lr(a, b);
tie(x1, y1) = make_pair(y1, -x1);
tie(x2, y2) = make_pair(-y2, x2);
ll g = a * x1 + b * y1;
pair<ll,ll> key1 = make_pair(abs(x1) + abs(y1), x1);
pair<ll,ll> key2 = make_pair(abs(x2) + abs(y2), x2);
return (key1 < key2 ? make_tuple(x1, y1, g) : make_tuple(x2, y2, g));
}
} // namespace lib