math/lcm_convolution.hpp

• View this file on GitHub
• Last update: 2024-07-01 23:39:10+09:00
• Include: #include "math/lcm_convolution.hpp"

Depends on

• math/sieve.hpp
• template/const.hpp
• template/inout_old.hpp
• template/template.hpp
• template/utils.hpp

Verified with

• test/math/Lcm_Convolution.test.cpp

Code

#pragma once

#include"../math/sieve.hpp"

namespace noya2 {

template <typename T>
vector<T> divisor_zeta_transform(const vector<T> &f){
    int n = f.size() - 1;
    sieve.request(n);
    auto F = f;
    for (const auto &p : sieve.primes){
        if (n < p) break;
        for (int i = 1; i * p <= n; i++) F[i * p] += F[i];
    }
    return F;
}

template <typename  T>
vector<T> divisor_mobius_transform(const vector<T> &F){
    int n = F.size() - 1;
    sieve.request(n);
    auto f = F;
    for (const auto &p : sieve.primes){
        if (n < p) break;
        for (int i = n / p; i >= 1; i--) f[i * p] -= f[i];
    }
    return f;
}

template <typename T>
vector<T> lcm_convolution(const vector<T> &a, const vector<T> &b){
    assert(a.size() == b.size());
    int n = a.size();
    auto ra = divisor_zeta_transform(a);
    auto rb = divisor_zeta_transform(b);
    for (int i = 0; i < n; i++) ra[i] *= rb[i];
    return divisor_mobius_transform(ra);
}

} // namespace noya2

#line 2 "math/lcm_convolution.hpp"

#line 2 "math/sieve.hpp"

#line 2 "template/template.hpp"
using namespace std;

#include<bits/stdc++.h>
#line 1 "template/inout_old.hpp"
namespace noya2 {

template <typename T, typename U>
ostream &operator<<(ostream &os, const pair<T, U> &p){
    os << p.first << " " << p.second;
    return os;
}
template <typename T, typename U>
istream &operator>>(istream &is, pair<T, U> &p){
    is >> p.first >> p.second;
    return is;
}

template <typename T>
ostream &operator<<(ostream &os, const vector<T> &v){
    int s = (int)v.size();
    for (int i = 0; i < s; i++) os << (i ? " " : "") << v[i];
    return os;
}
template <typename T>
istream &operator>>(istream &is, vector<T> &v){
    for (auto &x : v) is >> x;
    return is;
}

void in() {}
template <typename T, class... U>
void in(T &t, U &...u){
    cin >> t;
    in(u...);
}

void out() { cout << "\n"; }
template <typename T, class... U, char sep = ' '>
void out(const T &t, const U &...u){
    cout << t;
    if (sizeof...(u)) cout << sep;
    out(u...);
}

template<typename T>
void out(const vector<vector<T>> &vv){
    int s = (int)vv.size();
    for (int i = 0; i < s; i++) out(vv[i]);
}

struct IoSetup {
    IoSetup(){
        cin.tie(nullptr);
        ios::sync_with_stdio(false);
        cout << fixed << setprecision(15);
        cerr << fixed << setprecision(7);
    }
} iosetup_noya2;

} // namespace noya2
#line 1 "template/const.hpp"
namespace noya2{

const int iinf = 1'000'000'007;
const long long linf = 2'000'000'000'000'000'000LL;
const long long mod998 =  998244353;
const long long mod107 = 1000000007;
const long double pi = 3.14159265358979323;
const vector<int> dx = {0,1,0,-1,1,1,-1,-1};
const vector<int> dy = {1,0,-1,0,1,-1,-1,1};
const string ALP = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
const string alp = "abcdefghijklmnopqrstuvwxyz";
const string NUM = "0123456789";

void yes(){ cout << "Yes\n"; }
void no(){ cout << "No\n"; }
void YES(){ cout << "YES\n"; }
void NO(){ cout << "NO\n"; }
void yn(bool t){ t ? yes() : no(); }
void YN(bool t){ t ? YES() : NO(); }

} // namespace noya2
#line 2 "template/utils.hpp"

#line 6 "template/utils.hpp"

namespace noya2{

unsigned long long inner_binary_gcd(unsigned long long a, unsigned long long b){
    if (a == 0 || b == 0) return a + b;
    int n = __builtin_ctzll(a); a >>= n;
    int m = __builtin_ctzll(b); b >>= m;
    while (a != b) {
        int mm = __builtin_ctzll(a - b);
        bool f = a > b;
        unsigned long long c = f ? a : b;
        b = f ? b : a;
        a = (c - b) >> mm;
    }
    return a << std::min(n, m);
}

template<typename T> T gcd_fast(T a, T b){ return static_cast<T>(inner_binary_gcd(std::abs(a),std::abs(b))); }

long long sqrt_fast(long long n) {
    if (n <= 0) return 0;
    long long x = sqrt(n);
    while ((x + 1) * (x + 1) <= n) x++;
    while (x * x > n) x--;
    return x;
}

template<typename T> T floor_div(const T n, const T d) {
    assert(d != 0);
    return n / d - static_cast<T>((n ^ d) < 0 && n % d != 0);
}

template<typename T> T ceil_div(const T n, const T d) {
    assert(d != 0);
    return n / d + static_cast<T>((n ^ d) >= 0 && n % d != 0);
}

template<typename T> void uniq(std::vector<T> &v){
    std::sort(v.begin(),v.end());
    v.erase(unique(v.begin(),v.end()),v.end());
}

template <typename T, typename U> inline bool chmin(T &x, U y) { return (y < x) ? (x = y, true) : false; }

template <typename T, typename U> inline bool chmax(T &x, U y) { return (x < y) ? (x = y, true) : false; }

template<typename T> inline bool range(T l, T x, T r){ return l <= x && x < r; }

} // namespace noya2
#line 8 "template/template.hpp"

#define rep(i,n) for (int i = 0; i < (int)(n); i++)
#define repp(i,m,n) for (int i = (m); i < (int)(n); i++)
#define reb(i,n) for (int i = (int)(n-1); i >= 0; i--)
#define all(v) (v).begin(),(v).end()

using ll = long long;
using ld = long double;
using uint = unsigned int;
using ull = unsigned long long;
using pii = pair<int,int>;
using pll = pair<ll,ll>;
using pil = pair<int,ll>;
using pli = pair<ll,int>;

namespace noya2{

/*　~ (. _________ . /)　*/

}

using namespace noya2;


#line 4 "math/sieve.hpp"

namespace noya2{

struct Sieve {
    vector<int> primes, factor, mu;
    Sieve (int N = 1024){
        build(N);
    }
    void request(int N){
        int len = n_max();
        if (len >= N) return ;
        while (len < N) len <<= 1;
        build(len);
    }
    int n_max(){ return factor.size()-1; }
  private:
    void build (int N){
        primes.clear();
        factor.resize(N+1); fill(factor.begin(),factor.end(),0);
        mu.resize(N+1); fill(mu.begin(),mu.end(),1);

        for(int n = 2; n <= N; n++) {
            if (factor[n] == 0){
                primes.push_back(n);
                factor[n] = n;
                mu[n] = -1;
            }
            for (int p : primes){
                if(n * p > N || p > factor[n]) break;
                factor[n * p] = p;
                mu[n * p] = p == factor[n] ? 0 : -mu[n];
            }
        }
    }
} sieve;

int mobius_sieve(int n){
    assert(1 <= n && n <= sieve.n_max());
    return sieve.mu[n];
}
bool is_prime_sieve(int n){
    if (n <= 2) return n == 2;
    assert(n <= sieve.n_max());
    return sieve.factor[n] == n;
}

vector<pair<int,int>> prime_factorization_sieve(int n){
    assert(1 <= n && n <= sieve.n_max());
    vector<int> facts;
    while (n > 1){
        int p = sieve.factor[n];
        facts.push_back(p);
        n /= p;
    }
    vector<pair<int,int>> pes;
    int siz = facts.size();
    for (int l = 0, r = 0; l < siz; l = r){
        while (r < siz && facts[r] == facts[l]) r++;
        pes.emplace_back(facts[l],r-l);
    }
    return pes;
}

vector<int> divisor_enumeration_sieve(int n){
    auto pes = prime_factorization_sieve(n);
    vector<int> divs = {1};
    for (auto [p, e] : pes){
        vector<int> nxt; nxt.reserve(divs.size() * (e+1));
        for (auto x : divs){
            for (int tt = 0; tt <= e; tt++){
                nxt.push_back(x);
                x *= p;
            }
        }
        swap(divs,nxt);
    }
    return divs;
}

} // namespace noya2
#line 4 "math/lcm_convolution.hpp"

namespace noya2 {

template <typename T>
vector<T> divisor_zeta_transform(const vector<T> &f){
    int n = f.size() - 1;
    sieve.request(n);
    auto F = f;
    for (const auto &p : sieve.primes){
        if (n < p) break;
        for (int i = 1; i * p <= n; i++) F[i * p] += F[i];
    }
    return F;
}

template <typename  T>
vector<T> divisor_mobius_transform(const vector<T> &F){
    int n = F.size() - 1;
    sieve.request(n);
    auto f = F;
    for (const auto &p : sieve.primes){
        if (n < p) break;
        for (int i = n / p; i >= 1; i--) f[i * p] -= f[i];
    }
    return f;
}

template <typename T>
vector<T> lcm_convolution(const vector<T> &a, const vector<T> &b){
    assert(a.size() == b.size());
    int n = a.size();
    auto ra = divisor_zeta_transform(a);
    auto rb = divisor_zeta_transform(b);
    for (int i = 0; i < n; i++) ra[i] *= rb[i];
    return divisor_mobius_transform(ra);
}

} // namespace noya2

Back to top page