GCD Convolution
(lib/convolution/gcd_convolution.hpp)

• View this file on GitHub
• Last update: 2024-11-13 13:43:26+09:00
• Include: #include "lib/convolution/gcd_convolution.hpp"
• Link: https://noshi91.hatenablog.com/entry/2018/12/27/121649

Depends on

• Divisor Zeta/Mobius Transform (lib/convolution/divisor_zeta_mobius_transform.hpp)
• Prime Sieve (エラトステネスの篩) (lib/math/prime_sieve.hpp)

Verified with

• test/library_checker/convolution/gcd_convolution.test.cpp

Code

#pragma once

/**
 * @brief GCD Convolution
 * @see https://noshi91.hatenablog.com/entry/2018/12/27/121649
 */

#include <vector>
#include <cassert>

#include "../convolution/divisor_zeta_mobius_transform.hpp"

template <typename T>
std::vector<T> gcd_convolution(std::vector<T> f, std::vector<T> g){
    const int n = (int) f.size();
    assert(f.size() == g.size());
    assert(1 <= n);
    divisor_reversed_zeta_transform(f);
    divisor_reversed_zeta_transform(g);
    for(int i = 1; i < n; ++i){
        f[i] *= g[i];
    }
    divisor_reversed_mobius_transform(f);
    return f;
}

#line 2 "lib/convolution/gcd_convolution.hpp"

/**
 * @brief GCD Convolution
 * @see https://noshi91.hatenablog.com/entry/2018/12/27/121649
 */

#include <vector>
#include <cassert>

#line 2 "lib/convolution/divisor_zeta_mobius_transform.hpp"

/**
 * @brief Divisor Zeta/Mobius Transform
 */

#line 8 "lib/convolution/divisor_zeta_mobius_transform.hpp"

#line 2 "lib/math/prime_sieve.hpp"

/**
 * @brief Prime Sieve (エラトステネスの篩)
 * @docs docs/math/prime-sieve.md
 */

#line 9 "lib/math/prime_sieve.hpp"

template <typename T>
struct PrimeSieve{
    int n, half;
    std::vector<bool> sieve;
    std::vector<T> prime_list;
    // sieve[i] ... 2 * i + 1

    PrimeSieve(T _n) : n(_n){
        init();
    }

    void init(){
        if(n < 2){
            return;
        }
        half = (n + 1) / 2;
        sieve.assign(half, true);
        sieve[0] = false;
        prime_list.emplace_back(2);
        for(long long i = 1; 2 * i + 1 <= n; ++i){
            if(!sieve[i]) continue;
            T p = 2 * i + 1;
            prime_list.emplace_back(p);
            for(long long j = 2 * i * (i + 1); j < half; j += p){
                sieve[j] = false;
            }
        }
    }

    bool isPrime(T x){
        if(x == 2) return true;
        if(x % 2 == 0) return false;
        return sieve[x / 2];
    }

    T getPrimeCount(){
        return prime_list.size();
    }

    T getKthPrime(int k){
        return prime_list[k];
    }
};
#line 10 "lib/convolution/divisor_zeta_mobius_transform.hpp"

template <typename T>
void divisor_zeta_transform(std::vector<T> &a){
    int n = a.size() - 1;
    PrimeSieve<int> sieve(n);
    for(int d = 2; d <= n; d++){
        if(sieve.isPrime(d)){
            for(int i = 1; i * d <= n; i++){
                a[i * d] += a[i];
            }
        }
    }
}

template <typename T>
void divisor_reversed_zeta_transform(std::vector<T> &a){
    int n = a.size() - 1;
    PrimeSieve<int> sieve(n);
    for(int d = 2; d <= n; d++){
        if(sieve.isPrime(d)){
            for(int i = n / d; i >= 1; i--){
                a[i] += a[i * d];
            }
        }
    }
}

template <typename T>
void divisor_mobius_transform(std::vector<T> &a){
    int n = a.size() - 1;
    PrimeSieve<int> sieve(n);
    for(int d = 2; d <= n; d++){
        if(sieve.isPrime(d)){
            for(int i = n / d; i >= 1; i--){
                a[i * d] -= a[i];
            }
        }
    }
}

template <typename T>
void divisor_reversed_mobius_transform(std::vector<T> &a){
    int n = a.size() - 1;
    PrimeSieve<int> sieve(n);
    for(int d = 2; d <= n; d++){
        if(sieve.isPrime(d)){
            for(int i = 1; i * d <= n; i++){
                a[i] -= a[i * d];
            }
        }
    }
}
#line 12 "lib/convolution/gcd_convolution.hpp"

template <typename T>
std::vector<T> gcd_convolution(std::vector<T> f, std::vector<T> g){
    const int n = (int) f.size();
    assert(f.size() == g.size());
    assert(1 <= n);
    divisor_reversed_zeta_transform(f);
    divisor_reversed_zeta_transform(g);
    for(int i = 1; i < n; ++i){
        f[i] *= g[i];
    }
    divisor_reversed_mobius_transform(f);
    return f;
}

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