Garner's algorithm
(modulus/garner.hpp)

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Last update: 2021-08-30 04:35:37+09:00
Include: #include "modulus/garner.hpp"

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#pragma once
#include <cassert>
#include <cstdint>
#include <tuple>
#include <utility>
#include <vector>
#include "../modulus/mint.hpp"
#include "../utils/macros.hpp"

/**
 * @brief Garner's algorithm
 * @arg eqns is equations like x = a_i (mod m_i)
 * @return the minimal solution of given equations
 */
int32_t garner_algorithm(std::vector<std::pair<int32_t, int32_t> > eqns, int32_t MOD) {
    eqns.emplace_back(0, MOD);
    std::vector<int64_t> k(eqns.size(), 1);
    std::vector<int64_t> c(eqns.size(), 0);
    REP (i, eqns.size() - 1) {
        int32_t a_i, m_i; std::tie(a_i, m_i) = eqns[i];

        int32_t x = (a_i - c[i]) * modinv(k[i], m_i) % m_i;
        if (x < 0) x += m_i;
        assert (a_i == (k[i] * x + c[i]) % m_i);

        REP3 (j, i + 1, eqns.size()) {
            int32_t a_j, m_j; std::tie(a_j, m_j) = eqns[j];
            (c[j] += k[j] * x) %= m_j;
            (k[j] *= m_i) %= m_j;
        }
    }
    return c.back();
}

#line 2 "modulus/garner.hpp"
#include <cassert>
#include <cstdint>
#include <tuple>
#include <utility>
#include <vector>
#line 3 "modulus/mint.hpp"
#include <iostream>
#line 4 "modulus/modpow.hpp"

inline int32_t modpow(uint_fast64_t x, uint64_t k, int32_t MOD) {
    assert (/* 0 <= x and */ x < (uint_fast64_t)MOD);
    uint_fast64_t y = 1;
    for (; k; k >>= 1) {
        if (k & 1) (y *= x) %= MOD;
        (x *= x) %= MOD;
    }
    assert (/* 0 <= y and */ y < (uint_fast64_t)MOD);
    return y;
}
#line 2 "modulus/modinv.hpp"
#include <algorithm>
#line 5 "modulus/modinv.hpp"

inline int32_t modinv_nocheck(int32_t value, int32_t MOD) {
    assert (0 <= value and value < MOD);
    if (value == 0) return -1;
    int64_t a = value, b = MOD;
    int64_t x = 0, y = 1;
    for (int64_t u = 1, v = 0; a; ) {
        int64_t q = b / a;
        x -= q * u; std::swap(x, u);
        y -= q * v; std::swap(y, v);
        b -= q * a; std::swap(b, a);
    }
    if (not (value * x + MOD * y == b and b == 1)) return -1;
    if (x < 0) x += MOD;
    assert (0 <= x and x < MOD);
    return x;
}

inline int32_t modinv(int32_t x, int32_t MOD) {
    int32_t y = modinv_nocheck(x, MOD);
    assert (y != -1);
    return y;
}
#line 6 "modulus/mint.hpp"

/**
 * @brief quotient ring / 剰余環 $\mathbb{Z}/n\mathbb{Z}$
 */
template <int32_t MOD>
struct mint {
    int32_t value;
    mint() : value() {}
    mint(int64_t value_) : value(value_ < 0 ? value_ % MOD + MOD : value_ >= MOD ? value_ % MOD : value_) {}
    mint(int32_t value_, std::nullptr_t) : value(value_) {}
    explicit operator bool() const { return value; }
    inline mint<MOD> operator + (mint<MOD> other) const { return mint<MOD>(*this) += other; }
    inline mint<MOD> operator - (mint<MOD> other) const { return mint<MOD>(*this) -= other; }
    inline mint<MOD> operator * (mint<MOD> other) const { return mint<MOD>(*this) *= other; }
    inline mint<MOD> & operator += (mint<MOD> other) { this->value += other.value; if (this->value >= MOD) this->value -= MOD; return *this; }
    inline mint<MOD> & operator -= (mint<MOD> other) { this->value -= other.value; if (this->value <    0) this->value += MOD; return *this; }
    inline mint<MOD> & operator *= (mint<MOD> other) { this->value = (uint_fast64_t)this->value * other.value % MOD; return *this; }
    inline mint<MOD> operator - () const { return mint<MOD>(this->value ? MOD - this->value : 0, nullptr); }
    inline bool operator == (mint<MOD> other) const { return value == other.value; }
    inline bool operator != (mint<MOD> other) const { return value != other.value; }
    inline mint<MOD> pow(uint64_t k) const { return mint<MOD>(modpow(value, k, MOD), nullptr); }
    inline mint<MOD> inv() const { return mint<MOD>(modinv(value, MOD), nullptr); }
    inline mint<MOD> operator / (mint<MOD> other) const { return *this * other.inv(); }
    inline mint<MOD> & operator /= (mint<MOD> other) { return *this *= other.inv(); }
};
template <int32_t MOD> mint<MOD> operator + (int64_t value, mint<MOD> n) { return mint<MOD>(value) + n; }
template <int32_t MOD> mint<MOD> operator - (int64_t value, mint<MOD> n) { return mint<MOD>(value) - n; }
template <int32_t MOD> mint<MOD> operator * (int64_t value, mint<MOD> n) { return mint<MOD>(value) * n; }
template <int32_t MOD> mint<MOD> operator / (int64_t value, mint<MOD> n) { return mint<MOD>(value) / n; }
template <int32_t MOD> std::istream & operator >> (std::istream & in, mint<MOD> & n) { int64_t value; in >> value; n = value; return in; }
template <int32_t MOD> std::ostream & operator << (std::ostream & out, mint<MOD> n) { return out << n.value; }
#line 2 "utils/macros.hpp"
#define REP(i, n) for (int i = 0; (i) < (int)(n); ++ (i))
#define REP3(i, m, n) for (int i = (m); (i) < (int)(n); ++ (i))
#define REP_R(i, n) for (int i = (int)(n) - 1; (i) >= 0; -- (i))
#define REP3R(i, m, n) for (int i = (int)(n) - 1; (i) >= (int)(m); -- (i))
#define ALL(x) std::begin(x), std::end(x)
#line 9 "modulus/garner.hpp"

/**
 * @brief Garner's algorithm
 * @arg eqns is equations like x = a_i (mod m_i)
 * @return the minimal solution of given equations
 */
int32_t garner_algorithm(std::vector<std::pair<int32_t, int32_t> > eqns, int32_t MOD) {
    eqns.emplace_back(0, MOD);
    std::vector<int64_t> k(eqns.size(), 1);
    std::vector<int64_t> c(eqns.size(), 0);
    REP (i, eqns.size() - 1) {
        int32_t a_i, m_i; std::tie(a_i, m_i) = eqns[i];

        int32_t x = (a_i - c[i]) * modinv(k[i], m_i) % m_i;
        if (x < 0) x += m_i;
        assert (a_i == (k[i] * x + c[i]) % m_i);

        REP3 (j, i + 1, eqns.size()) {
            int32_t a_j, m_j; std::tie(a_j, m_j) = eqns[j];
            (c[j] += k[j] * x) %= m_j;
            (k[j] *= m_i) %= m_j;
        }
    }
    return c.back();
}

Garner's algorithm (modulus/garner.hpp)

Depends on

Code

Garner's algorithm
(modulus/garner.hpp)