#653770 (C++17) No.665 Bernoulli Bernoulli

提出ソース

結果

問題	No.665 Bernoulli Bernoulli
ユーザー	hitonanode
提出日時	2021-05-03 18:21:21
言語	C++17 (gcc 12.3.0 + boost 1.83.0)
結果	AC
実行時間	80 ms / 2,000 ms
コード長	11,276 bytes
コンパイル時間	2,617 ms
コンパイル使用メモリ	213,888 KB
実行使用メモリ	44,576 KB
最終ジャッジ日時	2024-07-22 09:10:44
合計ジャッジ時間	3,576 ms
ジャッジサーバーID （参考情報）	judge1 / judge2

このコードへのチャレンジ
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テストケース

テストケース表示

入力	結果	実行時間実行使用メモリ
testcase_00	AC	2 ms 5,248 KB
testcase_01	AC	3 ms 5,376 KB
testcase_02	AC	3 ms 5,376 KB
testcase_03	AC	4 ms 5,376 KB
testcase_04	AC	80 ms 44,576 KB
testcase_05	AC	4 ms 5,376 KB
testcase_06	AC	4 ms 5,376 KB
testcase_07	AC	4 ms 5,376 KB
testcase_08	AC	4 ms 5,376 KB
testcase_09	AC	3 ms 5,376 KB
testcase_10	AC	4 ms 5,376 KB
testcase_11	AC	4 ms 5,376 KB
testcase_12	AC	4 ms 5,376 KB
testcase_13	AC	4 ms 5,376 KB
testcase_14	AC	3 ms 5,376 KB
testcase_15	AC	4 ms 5,376 KB
testcase_16	AC	4 ms 5,376 KB
testcase_17	AC	4 ms 5,376 KB
testcase_18	AC	3 ms 5,376 KB

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ソースコード

raw source code

#include <bits/stdc++.h>
using namespace std;

template <int mod> struct ModInt {
#if __cplusplus >= 201402L
#define MDCONST constexpr
#else
#define MDCONST
#endif
    using lint = long long;
    MDCONST static int get_mod() { return mod; }
    static int get_primitive_root() {
        static int primitive_root = 0;
        if (!primitive_root) {
            primitive_root = [&]() {
                std::set<int> fac;
                int v = mod - 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 < mod; g++) {
                    bool ok = true;
                    for (auto i : fac)
                        if (ModInt(g).pow((mod - 1) / i) == 1) {
                            ok = false;
                            break;
                        }
                    if (ok) return g;
                }
                return -1;
            }();
        }
        return primitive_root;
    }
    int val;
    MDCONST ModInt() : val(0) {}
    MDCONST ModInt &_setval(lint v) { return val = (v >= mod ? v - mod : v), *this; }
    MDCONST ModInt(lint v) { _setval(v % mod + mod); }
    MDCONST explicit operator bool() const { return val != 0; }
    MDCONST ModInt operator+(const ModInt &x) const { return ModInt()._setval((lint)val + x.val); }
    MDCONST ModInt operator-(const ModInt &x) const { return ModInt()._setval((lint)val - x.val + mod); }
    MDCONST ModInt operator*(const ModInt &x) const { return ModInt()._setval((lint)val * x.val % mod); }
    MDCONST ModInt operator/(const ModInt &x) const { return ModInt()._setval((lint)val * x.inv() % mod); }
    MDCONST ModInt operator-() const { return ModInt()._setval(mod - val); }
    MDCONST ModInt &operator+=(const ModInt &x) { return *this = *this + x; }
    MDCONST ModInt &operator-=(const ModInt &x) { return *this = *this - x; }
    MDCONST ModInt &operator*=(const ModInt &x) { return *this = *this * x; }
    MDCONST ModInt &operator/=(const ModInt &x) { return *this = *this / x; }
    friend MDCONST ModInt operator+(lint a, const ModInt &x) { return ModInt()._setval(a % mod + x.val); }
    friend MDCONST ModInt operator-(lint a, const ModInt &x) { return ModInt()._setval(a % mod - x.val + mod); }
    friend MDCONST ModInt operator*(lint a, const ModInt &x) { return ModInt()._setval(a % mod * x.val % mod); }
    friend MDCONST ModInt operator/(lint a, const ModInt &x) { return ModInt()._setval(a % mod * x.inv() % mod); }
    MDCONST bool operator==(const ModInt &x) const { return val == x.val; }
    MDCONST bool operator!=(const ModInt &x) const { return val != x.val; }
    MDCONST 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;
    }
    MDCONST friend std::ostream &operator<<(std::ostream &os, const ModInt &x) { return os << x.val; }
    MDCONST ModInt pow(lint n) const {
        lint ans = 1, tmp = this->val;
        while (n) {
            if (n & 1) ans = ans * tmp % mod;
            tmp = tmp * tmp % mod, n /= 2;
        }
        return ans;
    }

    static std::vector<long long> facs, invs;
    MDCONST static void _precalculation(int N) {
        int l0 = facs.size();
        if (N <= l0) return;
        facs.resize(N), invs.resize(N);
        for (int i = l0; i < N; i++) facs[i] = facs[i - 1] * i % mod;
        long long facinv = ModInt(facs.back()).pow(mod - 2).val;
        for (int i = N - 1; i >= l0; i--) {
            invs[i] = facinv * facs[i - 1] % mod;
            facinv = facinv * i % mod;
        }
    }
    MDCONST lint inv() const {
        if (this->val < std::min(mod >> 1, 1 << 21)) {
            while (this->val >= int(facs.size())) _precalculation(facs.size() * 2);
            return invs[this->val];
        } else {
            return this->pow(mod - 2).val;
        }
    }
    MDCONST ModInt fac() const {
        while (this->val >= int(facs.size())) _precalculation(facs.size() * 2);
        return facs[this->val];
    }

    MDCONST 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);
    }
    MDCONST ModInt nCr(const ModInt &r) const { return (this->val < r.val) ? 0 : this->fac() / ((*this - r).fac() * r.fac()); }

    ModInt sqrt() const {
        if (val == 0) return 0;
        if (mod == 2) return val;
        if (pow((mod - 1) / 2) != 1) return 0;
        ModInt b = 1;
        while (b.pow((mod - 1) / 2) == 1) b += 1;
        int e = 0, m = mod - 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, mod - x.val));
    }
};
template <int mod> std::vector<long long> ModInt<mod>::facs = {1};
template <int mod> std::vector<long long> ModInt<mod>::invs = {0};

template <typename MODINT> MODINT interpolate_iota(const std::vector<MODINT> ys, MODINT x_eval) {
    const int N = ys.size();
    if (x_eval.val < N) return ys[x_eval.val];
    std::vector<MODINT> facinv(N);
    facinv[N - 1] = MODINT(N - 1).fac().inv();
    for (int i = N - 1; i > 0; i--) facinv[i - 1] = facinv[i] * i;
    std::vector<MODINT> numleft(N);
    MODINT numtmp = 1;
    for (int i = 0; i < N; i++) {
        numleft[i] = numtmp;
        numtmp *= x_eval - i;
    }
    numtmp = 1;
    MODINT ret = 0;
    for (int i = N - 1; i >= 0; i--) {
        MODINT tmp = ys[i] * numleft[i] * numtmp * facinv[i] * facinv[N - 1 - i];
        ret += ((N - 1 - i) & 1) ? (-tmp) : tmp;
        numtmp *= x_eval - i;
    }
    return ret;
}

template <typename MODINT> MODINT sum_of_exponential_times_polynomial_limit(MODINT r, std::vector<MODINT> init) {
    assert(r != 1);
    if (init.empty()) return 0;
    if (init.size() == 1) return init[0] / (1 - r);
    auto &bs = init;
    const int d = int(bs.size()) - 1;
    MODINT rp = 1;
    for (int i = 1; i <= d; i++) rp *= r, bs[i] = bs[i] * rp + bs[i - 1];
    MODINT ret = 0;
    rp = 1;
    for (int i = 0; i <= d; i++) {
        ret += bs[d - i] * MODINT(d + 1).nCr(i) * rp;
        rp *= -r;
    }
    return ret / MODINT(1 - r).pow(d + 1);
};
template <typename MODINT> MODINT sum_of_exponential_times_polynomial(MODINT r, const std::vector<MODINT> &init, long long N) {
    if (N <= 0) return 0;
    if (init.size() == 1) return init[0] * (r == 1 ? MODINT(N) : (1 - r.pow(N)) / (1 - r));

    std::vector<MODINT> S(init.size() + 1);
    MODINT rp = 1;
    for (int i = 0; i < int(init.size()); i++) {
        S[i + 1] = S[i] + init[i] * rp;
        rp *= r;
    }
    if (N < (long long)S.size()) return S[N];

    if (r == 1) return interpolate_iota<MODINT>(S, N);

    const MODINT Sinf = sum_of_exponential_times_polynomial_limit<MODINT>(r, init);
    MODINT rinv = r.inv(), rinvp = 1;
    for (int i = 0; i < int(S.size()); i++) {
        S[i] = (S[i] - Sinf) * rinvp;
        rinvp *= rinv;
    }
    return interpolate_iota<MODINT>(S, N) * r.pow(N) + Sinf;
};

// 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 <typename T> std::map<T, int> factorize(T x) {
        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 <typename T> std::vector<T> divisors(T x) {
        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
    }
    // 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() {
        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 <typename MODINT> std::vector<MODINT> enumerate_kth_pows(long long K, int nmax) {
        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 << 15);  // (can factorize n <= 10^9)

using mint = ModInt<1000000007>;

int main() {
    long long N;
    int K;
    cin >> N >> K;
    cout << sum_of_exponential_times_polynomial<mint>(1, Sieve(K).enumerate_kth_pows<mint>(K, K), N + 1) << '\n';
}

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結果

テストケース

ソースコード