number/test/zeta_moebius_transform.test.cpp

• View this file on GitHub
• Last update: 2023-12-26 21:26:22+09:00
• Problem: https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ITP1_1_A

Depends on

• modint.hpp
• Linear sieve （線形篩） (number/sieve.hpp)
• Zeta transform / Moebius transform （約数包除） (number/zeta_moebius_transform.hpp)

Code

#define PROBLEM "https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ITP1_1_A" // DUMMY
#include "../zeta_moebius_transform.hpp"
#include "../../modint.hpp"
#include <cassert>
#include <iostream>
#include <vector>
using namespace std;

using mint = ModInt<1000000007>;

long long state = 1;
mint rndgen() {
    state = (state * 1234567 + 890123) % mint::mod();
    return mint(state);
}
vector<mint> vecgen(int n) {
    vector<mint> ret(n + 1);
    for (int i = 1; i <= n; ++i) ret[i] = rndgen();
    return ret;
}

void test_divisor_zeta() {
    for (int n = 1; n <= 100; ++n) {
        for (int iter = 0; iter < 10; ++iter) {
            vector<mint> x = vecgen(n), y(n + 1);
            for (int i = 1; i <= n; ++i) {
                for (int j = 1; j <= i; ++j) {
                    if (i % j == 0) y[i] += x[j];
                }
            }
            divisor_zeta(x);
            assert(x == y);
        }
    }
}

void test_divisor_moebius() {
    for (int n = 1; n <= 100; ++n) {
        for (int iter = 0; iter < 10; ++iter) {
            vector<mint> x = vecgen(n), y = x;
            divisor_zeta(x);
            divisor_moebius(x);
            assert(x == y);
        }
    }
}

void test_multiple_zeta() {
    for (int n = 1; n <= 100; ++n) {
        for (int iter = 0; iter < 10; ++iter) {
            vector<mint> x = vecgen(n), y(n + 1);
            for (int i = 1; i <= n; ++i) {
                for (int j = i; j <= n; ++j) {
                    if (j % i == 0) y[i] += x[j];
                }
            }
            multiple_zeta(x);
            assert(x == y);
        }
    }
}

void test_multiple_moebius() {
    for (int n = 1; n <= 100; ++n) {
        for (int iter = 0; iter < 10; ++iter) {
            vector<mint> x = vecgen(n), y = x;
            multiple_zeta(x);
            multiple_moebius(x);
            assert(x == y);
        }
    }
}

void test_gcdconv() {
    for (int n = 1; n <= 100; ++n) {
        for (int iter = 0; iter < 10; ++iter) {
            vector<mint> x = vecgen(n), y = vecgen(n), z(n + 1);
            auto conv = gcdconv(x, y);
            for (int i = 1; i <= n; ++i) {
                for (int j = 1; j <= n; ++j) z[__gcd(i, j)] += x[i] * y[j];
            }
            assert(conv == z);
        }
    }
}

void test_lcmconv() {
    for (int n = 1; n <= 100; ++n) {
        for (int iter = 0; iter < 10; ++iter) {
            vector<mint> x = vecgen(n), y = vecgen(n), z(n + 1);
            auto conv = lcmconv(x, y);
            for (int i = 1; i <= n; ++i) {
                for (int j = 1; j <= n; ++j) {
                    int l = i * j / __gcd(i, j);
                    if (l <= n) z[l] += x[i] * y[j];
                }
            }
            assert(conv == z);
        }
    }
}

int main() {
    test_divisor_zeta();
    test_divisor_moebius();
    test_multiple_zeta();
    test_multiple_moebius();
    test_gcdconv();
    test_lcmconv();
    cout << "Hello World\n";
}

#line 1 "number/test/zeta_moebius_transform.test.cpp"
#define PROBLEM "https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ITP1_1_A" // DUMMY
#line 2 "number/sieve.hpp"
#include <cassert>
#include <map>
#include <vector>

// CUT begin
// Linear sieve algorithm for fast prime factorization
// Complexity: O(N) time, O(N) space:
// - MAXN = 10^7:  ~44 MB,  80~100 ms (Codeforces / AtCoder GCC, C++17)
// - MAXN = 10^8: ~435 MB, 810~980 ms (Codeforces / AtCoder GCC, C++17)
// Reference:
// [1] D. Gries, J. Misra, "A Linear Sieve Algorithm for Finding Prime Numbers,"
//     Communications of the ACM, 21(12), 999-1003, 1978.
// - https://cp-algorithms.com/algebra/prime-sieve-linear.html
// - https://37zigen.com/linear-sieve/
struct Sieve {
    std::vector<int> min_factor;
    std::vector<int> primes;
    Sieve(int MAXN) : min_factor(MAXN + 1) {
        for (int d = 2; d <= MAXN; d++) {
            if (!min_factor[d]) {
                min_factor[d] = d;
                primes.emplace_back(d);
            }
            for (const auto &p : primes) {
                if (p > min_factor[d] or d * p > MAXN) break;
                min_factor[d * p] = p;
            }
        }
    }
    // Prime factorization for 1 <= x <= MAXN^2
    // Complexity: O(log x)           (x <= MAXN)
    //             O(MAXN / log MAXN) (MAXN < x <= MAXN^2)
    template <class T> std::map<T, int> factorize(T x) const {
        std::map<T, int> ret;
        assert(x > 0 and
               x <= ((long long)min_factor.size() - 1) * ((long long)min_factor.size() - 1));
        for (const auto &p : primes) {
            if (x < T(min_factor.size())) break;
            while (!(x % p)) x /= p, ret[p]++;
        }
        if (x >= T(min_factor.size())) ret[x]++, x = 1;
        while (x > 1) ret[min_factor[x]]++, x /= min_factor[x];
        return ret;
    }
    // Enumerate divisors of 1 <= x <= MAXN^2
    // Be careful of highly composite numbers https://oeis.org/A002182/list
    // https://gist.github.com/dario2994/fb4713f252ca86c1254d#file-list-txt (n, (# of div. of n)):
    // 45360->100, 735134400(<1e9)->1344, 963761198400(<1e12)->6720
    template <class T> std::vector<T> divisors(T x) const {
        std::vector<T> ret{1};
        for (const auto p : factorize(x)) {
            int n = ret.size();
            for (int i = 0; i < n; i++) {
                for (T a = 1, d = 1; d <= p.second; d++) {
                    a *= p.first;
                    ret.push_back(ret[i] * a);
                }
            }
        }
        return ret; // NOT sorted
    }
    // Euler phi functions of divisors of given x
    // Verified: ABC212 G https://atcoder.jp/contests/abc212/tasks/abc212_g
    // Complexity: O(sqrt(x) + d(x))
    template <class T> std::map<T, T> euler_of_divisors(T x) const {
        assert(x >= 1);
        std::map<T, T> ret;
        ret[1] = 1;
        std::vector<T> divs{1};
        for (auto p : factorize(x)) {
            int n = ret.size();
            for (int i = 0; i < n; i++) {
                ret[divs[i] * p.first] = ret[divs[i]] * (p.first - 1);
                divs.push_back(divs[i] * p.first);
                for (T a = divs[i] * p.first, d = 1; d < p.second; a *= p.first, d++) {
                    ret[a * p.first] = ret[a] * p.first;
                    divs.push_back(a * p.first);
                }
            }
        }
        return ret;
    }
    // Moebius function Table, (-1)^{# of different prime factors} for square-free x
    // return: [0=>0, 1=>1, 2=>-1, 3=>-1, 4=>0, 5=>-1, 6=>1, 7=>-1, 8=>0, ...] https://oeis.org/A008683
    std::vector<int> GenerateMoebiusFunctionTable() const {
        std::vector<int> ret(min_factor.size());
        for (unsigned i = 1; i < min_factor.size(); i++) {
            if (i == 1) {
                ret[i] = 1;
            } else if ((i / min_factor[i]) % min_factor[i] == 0) {
                ret[i] = 0;
            } else {
                ret[i] = -ret[i / min_factor[i]];
            }
        }
        return ret;
    }
    // Calculate [0^K, 1^K, ..., nmax^K] in O(nmax)
    // Note: **0^0 == 1**
    template <class MODINT> std::vector<MODINT> enumerate_kth_pows(long long K, int nmax) const {
        assert(nmax < int(min_factor.size()));
        assert(K >= 0);
        if (K == 0) return std::vector<MODINT>(nmax + 1, 1);
        std::vector<MODINT> ret(nmax + 1);
        ret[0] = 0, ret[1] = 1;
        for (int n = 2; n <= nmax; n++) {
            if (min_factor[n] == n) {
                ret[n] = MODINT(n).pow(K);
            } else {
                ret[n] = ret[n / min_factor[n]] * ret[min_factor[n]];
            }
        }
        return ret;
    }
};
// Sieve sieve((1 << 20));
#line 3 "number/zeta_moebius_transform.hpp"
#include <algorithm>
#line 5 "number/zeta_moebius_transform.hpp"
#include <utility>
#line 7 "number/zeta_moebius_transform.hpp"

// f[n] に対して、全ての n の倍数 n*i に対する f[n*i] の和が出てくる 計算量 O(N loglog N)
// 素数p毎に処理する高速ゼータ変換
// 使用例 https://yukicoder.me/submissions/385043
template <class T> void multiple_zeta(std::vector<T> &f) {
    int N = int(f.size()) - 1;
    std::vector<int> is_prime(N + 1, 1);
    for (int p = 2; p <= N; p++) {
        if (is_prime[p]) {
            for (int q = p * 2; q <= N; q += p) is_prime[q] = 0;
            for (int j = N / p; j > 0; --j) f[j] += f[j * p];
        }
    }
}

// inverse of multiple_zeta O(N loglog N)
// 使用例 https://yukicoder.me/submissions/385120
template <class T> void multiple_moebius(std::vector<T> &f) {
    int N = int(f.size()) - 1;
    std::vector<int> is_prime(N + 1, 1);
    for (int p = 2; p <= N; p++) {
        if (is_prime[p]) {
            for (int q = p * 2; q <= N; q += p) is_prime[q] = 0;
            for (int j = 1; j * p <= N; ++j) f[j] -= f[j * p];
        }
    }
}

// f[n] に関して、全ての n の約数 m に対する f[m] の総和が出てくる 計算量 O(N loglog N)
template <class T> void divisor_zeta(std::vector<T> &f) {
    int N = int(f.size()) - 1;
    std::vector<int> is_prime(N + 1, 1);
    for (int p = 2; p <= N; ++p) {
        if (is_prime[p]) {
            for (int q = p * 2; q <= N; q += p) is_prime[q] = 0;
            for (int j = 1; j * p <= N; ++j) f[j * p] += f[j];
        }
    }
}
// inverse of divisor_zeta()
// Verified: https://codeforces.com/contest/1630/problem/E
template <class T> void divisor_moebius(std::vector<T> &f) {
    int N = int(f.size()) - 1;
    std::vector<int> is_prime(N + 1, 1);
    for (int p = 2; p <= N; ++p) {
        if (is_prime[p]) {
            for (int q = p * 2; q <= N; q += p) is_prime[q] = 0;
            for (int j = N / p; j > 0; --j) f[j * p] -= f[j];
        }
    }
}

// GCD convolution
// ret[k] = \sum_{gcd(i, j) = k} f[i] * g[j]
template <class T> std::vector<T> gcdconv(std::vector<T> f, std::vector<T> g) {
    assert(f.size() == g.size());
    multiple_zeta(f);
    multiple_zeta(g);
    for (int i = 0; i < int(g.size()); ++i) f[i] *= g[i];
    multiple_moebius(f);
    return f;
}

// LCM convolution
// ret[k] = \sum_{lcm(i, j) = k} f[i] * g[j]
template <class T> std::vector<T> lcmconv(std::vector<T> f, std::vector<T> g) {
    assert(f.size() == g.size());
    divisor_zeta(f);
    divisor_zeta(g);
    for (int i = 0; i < int(g.size()); ++i) f[i] *= g[i];
    divisor_moebius(f);
    return f;
}

// fast_integer_moebius の高速化（登場しない素因数が多ければ計算量改善）
// Requirement: f の key として登場する正整数の全ての約数が key として登場
// Verified: https://toph.co/p/height-of-the-trees
template <typename Int, typename Val>
void sparse_fast_integer_moebius(std::vector<std::pair<Int, Val>> &f, const Sieve &sieve) {
    if (f.empty()) return;
    std::sort(f.begin(), f.end());
    assert(f.back().first < Int(sieve.min_factor.size()));
    std::vector<Int> primes;
    for (auto p : f) {
        if (sieve.min_factor[p.first] == p.first) primes.push_back(p.first);
    }
    std::vector<std::vector<int>> p2is(primes.size());
    for (int i = 0; i < int(f.size()); i++) {
        Int a = f[i].first, pold = 0;
        int k = 0;
        while (a > 1) {
            auto p = sieve.min_factor[a];
            if (p != pold) {
                while (primes[k] != p) k++;
                p2is[k].emplace_back(i);
            }
            pold = p, a /= p;
        }
    }
    for (int d = 0; d < int(primes.size()); d++) {
        Int p = primes[d];
        for (auto i : p2is[d]) {
            auto comp = [](const std::pair<Int, Val> &l, const std::pair<Int, Val> &r) {
                return l.first < r.first;
            };
            auto itr = std::lower_bound(f.begin(), f.end(), std::make_pair(f[i].first / p, 0), comp);
            itr->second -= f[i].second;
        }
    }
}
#line 3 "modint.hpp"
#include <iostream>
#include <set>
#line 6 "modint.hpp"

template <int md> struct ModInt {
    using lint = long long;
    constexpr static int mod() { return md; }
    static int get_primitive_root() {
        static int primitive_root = 0;
        if (!primitive_root) {
            primitive_root = [&]() {
                std::set<int> fac;
                int v = md - 1;
                for (lint i = 2; i * i <= v; i++)
                    while (v % i == 0) fac.insert(i), v /= i;
                if (v > 1) fac.insert(v);
                for (int g = 1; g < md; g++) {
                    bool ok = true;
                    for (auto i : fac)
                        if (ModInt(g).pow((md - 1) / i) == 1) {
                            ok = false;
                            break;
                        }
                    if (ok) return g;
                }
                return -1;
            }();
        }
        return primitive_root;
    }
    int val_;
    int val() const noexcept { return val_; }
    constexpr ModInt() : val_(0) {}
    constexpr ModInt &_setval(lint v) { return val_ = (v >= md ? v - md : v), *this; }
    constexpr ModInt(lint v) { _setval(v % md + md); }
    constexpr explicit operator bool() const { return val_ != 0; }
    constexpr ModInt operator+(const ModInt &x) const {
        return ModInt()._setval((lint)val_ + x.val_);
    }
    constexpr ModInt operator-(const ModInt &x) const {
        return ModInt()._setval((lint)val_ - x.val_ + md);
    }
    constexpr ModInt operator*(const ModInt &x) const {
        return ModInt()._setval((lint)val_ * x.val_ % md);
    }
    constexpr ModInt operator/(const ModInt &x) const {
        return ModInt()._setval((lint)val_ * x.inv().val() % md);
    }
    constexpr ModInt operator-() const { return ModInt()._setval(md - val_); }
    constexpr ModInt &operator+=(const ModInt &x) { return *this = *this + x; }
    constexpr ModInt &operator-=(const ModInt &x) { return *this = *this - x; }
    constexpr ModInt &operator*=(const ModInt &x) { return *this = *this * x; }
    constexpr ModInt &operator/=(const ModInt &x) { return *this = *this / x; }
    friend constexpr ModInt operator+(lint a, const ModInt &x) { return ModInt(a) + x; }
    friend constexpr ModInt operator-(lint a, const ModInt &x) { return ModInt(a) - x; }
    friend constexpr ModInt operator*(lint a, const ModInt &x) { return ModInt(a) * x; }
    friend constexpr ModInt operator/(lint a, const ModInt &x) { return ModInt(a) / x; }
    constexpr bool operator==(const ModInt &x) const { return val_ == x.val_; }
    constexpr bool operator!=(const ModInt &x) const { return val_ != x.val_; }
    constexpr bool operator<(const ModInt &x) const {
        return val_ < x.val_;
    } // To use std::map<ModInt, T>
    friend std::istream &operator>>(std::istream &is, ModInt &x) {
        lint t;
        return is >> t, x = ModInt(t), is;
    }
    constexpr friend std::ostream &operator<<(std::ostream &os, const ModInt &x) {
        return os << x.val_;
    }

    constexpr ModInt pow(lint n) const {
        ModInt ans = 1, tmp = *this;
        while (n) {
            if (n & 1) ans *= tmp;
            tmp *= tmp, n >>= 1;
        }
        return ans;
    }

    static constexpr int cache_limit = std::min(md, 1 << 21);
    static std::vector<ModInt> facs, facinvs, invs;

    constexpr static void _precalculation(int N) {
        const int l0 = facs.size();
        if (N > md) N = md;
        if (N <= l0) return;
        facs.resize(N), facinvs.resize(N), invs.resize(N);
        for (int i = l0; i < N; i++) facs[i] = facs[i - 1] * i;
        facinvs[N - 1] = facs.back().pow(md - 2);
        for (int i = N - 2; i >= l0; i--) facinvs[i] = facinvs[i + 1] * (i + 1);
        for (int i = N - 1; i >= l0; i--) invs[i] = facinvs[i] * facs[i - 1];
    }

    constexpr ModInt inv() const {
        if (this->val_ < cache_limit) {
            if (facs.empty()) facs = {1}, facinvs = {1}, invs = {0};
            while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
            return invs[this->val_];
        } else {
            return this->pow(md - 2);
        }
    }
    constexpr ModInt fac() const {
        while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
        return facs[this->val_];
    }
    constexpr ModInt facinv() const {
        while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
        return facinvs[this->val_];
    }
    constexpr ModInt doublefac() const {
        lint k = (this->val_ + 1) / 2;
        return (this->val_ & 1) ? ModInt(k * 2).fac() / (ModInt(2).pow(k) * ModInt(k).fac())
                                : ModInt(k).fac() * ModInt(2).pow(k);
    }

    constexpr ModInt nCr(int r) const {
        if (r < 0 or this->val_ < r) return ModInt(0);
        return this->fac() * (*this - r).facinv() * ModInt(r).facinv();
    }

    constexpr ModInt nPr(int r) const {
        if (r < 0 or this->val_ < r) return ModInt(0);
        return this->fac() * (*this - r).facinv();
    }

    static ModInt binom(int n, int r) {
        static long long bruteforce_times = 0;

        if (r < 0 or n < r) return ModInt(0);
        if (n <= bruteforce_times or n < (int)facs.size()) return ModInt(n).nCr(r);

        r = std::min(r, n - r);

        ModInt ret = ModInt(r).facinv();
        for (int i = 0; i < r; ++i) ret *= n - i;
        bruteforce_times += r;

        return ret;
    }

    // Multinomial coefficient, (k_1 + k_2 + ... + k_m)! / (k_1! k_2! ... k_m!)
    // Complexity: O(sum(ks))
    template <class Vec> static ModInt multinomial(const Vec &ks) {
        ModInt ret{1};
        int sum = 0;
        for (int k : ks) {
            assert(k >= 0);
            ret *= ModInt(k).facinv(), sum += k;
        }
        return ret * ModInt(sum).fac();
    }

    // Catalan number, C_n = binom(2n, n) / (n + 1)
    // C_0 = 1, C_1 = 1, C_2 = 2, C_3 = 5, C_4 = 14, ...
    // https://oeis.org/A000108
    // Complexity: O(n)
    static ModInt catalan(int n) {
        if (n < 0) return ModInt(0);
        return ModInt(n * 2).fac() * ModInt(n + 1).facinv() * ModInt(n).facinv();
    }

    ModInt sqrt() const {
        if (val_ == 0) return 0;
        if (md == 2) return val_;
        if (pow((md - 1) / 2) != 1) return 0;
        ModInt b = 1;
        while (b.pow((md - 1) / 2) == 1) b += 1;
        int e = 0, m = md - 1;
        while (m % 2 == 0) m >>= 1, e++;
        ModInt x = pow((m - 1) / 2), y = (*this) * x * x;
        x *= (*this);
        ModInt z = b.pow(m);
        while (y != 1) {
            int j = 0;
            ModInt t = y;
            while (t != 1) j++, t *= t;
            z = z.pow(1LL << (e - j - 1));
            x *= z, z *= z, y *= z;
            e = j;
        }
        return ModInt(std::min(x.val_, md - x.val_));
    }
};
template <int md> std::vector<ModInt<md>> ModInt<md>::facs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::facinvs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::invs = {0};

using ModInt998244353 = ModInt<998244353>;
// using mint = ModInt<998244353>;
// using mint = ModInt<1000000007>;
#line 7 "number/test/zeta_moebius_transform.test.cpp"
using namespace std;

using mint = ModInt<1000000007>;

long long state = 1;
mint rndgen() {
    state = (state * 1234567 + 890123) % mint::mod();
    return mint(state);
}
vector<mint> vecgen(int n) {
    vector<mint> ret(n + 1);
    for (int i = 1; i <= n; ++i) ret[i] = rndgen();
    return ret;
}

void test_divisor_zeta() {
    for (int n = 1; n <= 100; ++n) {
        for (int iter = 0; iter < 10; ++iter) {
            vector<mint> x = vecgen(n), y(n + 1);
            for (int i = 1; i <= n; ++i) {
                for (int j = 1; j <= i; ++j) {
                    if (i % j == 0) y[i] += x[j];
                }
            }
            divisor_zeta(x);
            assert(x == y);
        }
    }
}

void test_divisor_moebius() {
    for (int n = 1; n <= 100; ++n) {
        for (int iter = 0; iter < 10; ++iter) {
            vector<mint> x = vecgen(n), y = x;
            divisor_zeta(x);
            divisor_moebius(x);
            assert(x == y);
        }
    }
}

void test_multiple_zeta() {
    for (int n = 1; n <= 100; ++n) {
        for (int iter = 0; iter < 10; ++iter) {
            vector<mint> x = vecgen(n), y(n + 1);
            for (int i = 1; i <= n; ++i) {
                for (int j = i; j <= n; ++j) {
                    if (j % i == 0) y[i] += x[j];
                }
            }
            multiple_zeta(x);
            assert(x == y);
        }
    }
}

void test_multiple_moebius() {
    for (int n = 1; n <= 100; ++n) {
        for (int iter = 0; iter < 10; ++iter) {
            vector<mint> x = vecgen(n), y = x;
            multiple_zeta(x);
            multiple_moebius(x);
            assert(x == y);
        }
    }
}

void test_gcdconv() {
    for (int n = 1; n <= 100; ++n) {
        for (int iter = 0; iter < 10; ++iter) {
            vector<mint> x = vecgen(n), y = vecgen(n), z(n + 1);
            auto conv = gcdconv(x, y);
            for (int i = 1; i <= n; ++i) {
                for (int j = 1; j <= n; ++j) z[__gcd(i, j)] += x[i] * y[j];
            }
            assert(conv == z);
        }
    }
}

void test_lcmconv() {
    for (int n = 1; n <= 100; ++n) {
        for (int iter = 0; iter < 10; ++iter) {
            vector<mint> x = vecgen(n), y = vecgen(n), z(n + 1);
            auto conv = lcmconv(x, y);
            for (int i = 1; i <= n; ++i) {
                for (int j = 1; j <= n; ++j) {
                    int l = i * j / __gcd(i, j);
                    if (l <= n) z[l] += x[i] * y[j];
                }
            }
            assert(conv == z);
        }
    }
}

int main() {
    test_divisor_zeta();
    test_divisor_moebius();
    test_multiple_zeta();
    test_multiple_moebius();
    test_gcdconv();
    test_lcmconv();
    cout << "Hello World\n";
}

Back to top page