Binomial Coefficient Small Prime Mod
(library/sequence/binomial_coefficient_small_prime_mod.hpp)

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• Last update: 2024-01-30 20:57:42+09:00
• Include: #include "library/sequence/binomial_coefficient_small_prime_mod.hpp"

Binomial Coefficient Small Prime Mod

以下に示す Lucas の定理を用いる。

$p$ が素数のとき、非負整数 $n, r$ に対して次が成り立つ: \(\binom{n}{r} \equiv \prod _ {i = 0} ^ {k - 1} \binom{n _ i}{r _ i} \pmod{p}.\) ここで、$n$ を $p$ 進表記したときの $i$ 桁目を $n_i$ とした ($r$ についても同様)。

全ての $0\leq n\lt p,0\leq r\lt p$ に対する $\displaystyle \binom{n}{r} \bmod p$ を $O(p ^ 2)$ 時間だけ掛けて前計算しておくことで、クエリあたり $O(\log _ p n)$ で計算できる。

Depends on

• 階乗テーブル (library/math/factorial.hpp)

Required by

• Stirling Number of the First Kind (Small Prime Mod) (library/sequence/stirling_number1_small_prime_mod.hpp)
• Stirling Number2 Small Prime Mod (library/sequence/stirling_number2_small_prime_mod.hpp)

Verified with

• test/src/sequence/stirling_number1_small_prime_mod/stirling_number_of_the_first_kind_small_p_large_n.test.cpp
• test/src/sequence/stirling_number2_small_prime_mod/stirling_number_of_the_second_kind_small_p_large_n.test.cpp

Code

#ifndef SUISEN_BINOMIAL_COEFFICIENT_SMALL_P
#define SUISEN_BINOMIAL_COEFFICIENT_SMALL_P

#include "library/math/factorial.hpp"

namespace suisen {
    template <typename mint>
    struct BinomialCoefficientSmallPrimeMod {
        mint operator()(long long n, long long r) const {
            return binom(n, r);
        }
        static mint binom(long long n, long long r) {
            factorial<mint> fac(mint::mod() - 1);

            if (r < 0 or n < r) return 0;
            r = std::min(r, n - r);
            // Lucas's theorem
            mint res = 1;
            while (r) {
                int ni = n % mint::mod(), ri = r % mint::mod();
                if (ni < ri) return 0;
                res *= fac.binom(ni, ri);
                n = n / mint::mod(), r = r / mint::mod();
            }
            return res;
        }
    };
} // namespace suisen


#endif // SUISEN_BINOMIAL_COEFFICIENT_SMALL_P

#line 1 "library/sequence/binomial_coefficient_small_prime_mod.hpp"



#line 1 "library/math/factorial.hpp"



#include <cassert>
#include <vector>

namespace suisen {
    template <typename T, typename U = T>
    struct factorial {
        factorial() = default;
        factorial(int n) { ensure(n); }

        static void ensure(const int n) {
            int sz = _fac.size();
            if (n + 1 <= sz) return;
            int new_size = std::max(n + 1, sz * 2);
            _fac.resize(new_size), _fac_inv.resize(new_size);
            for (int i = sz; i < new_size; ++i) _fac[i] = _fac[i - 1] * i;
            _fac_inv[new_size - 1] = U(1) / _fac[new_size - 1];
            for (int i = new_size - 1; i > sz; --i) _fac_inv[i - 1] = _fac_inv[i] * i;
        }

        T fac(const int i) {
            ensure(i);
            return _fac[i];
        }
        T operator()(int i) {
            return fac(i);
        }
        U fac_inv(const int i) {
            ensure(i);
            return _fac_inv[i];
        }
        U binom(const int n, const int r) {
            if (n < 0 or r < 0 or n < r) return 0;
            ensure(n);
            return _fac[n] * _fac_inv[r] * _fac_inv[n - r];
        }
        template <typename ...Ds, std::enable_if_t<std::conjunction_v<std::is_integral<Ds>...>, std::nullptr_t> = nullptr>
        U polynom(const int n, const Ds& ...ds) {
            if (n < 0) return 0;
            ensure(n);
            int sumd = 0;
            U res = _fac[n];
            for (int d : { ds... }) {
                if (d < 0 or d > n) return 0;
                sumd += d;
                res *= _fac_inv[d];
            }
            if (sumd > n) return 0;
            res *= _fac_inv[n - sumd];
            return res;
        }
        U perm(const int n, const int r) {
            if (n < 0 or r < 0 or n < r) return 0;
            ensure(n);
            return _fac[n] * _fac_inv[n - r];
        }
    private:
        static std::vector<T> _fac;
        static std::vector<U> _fac_inv;
    };
    template <typename T, typename U>
    std::vector<T> factorial<T, U>::_fac{ 1 };
    template <typename T, typename U>
    std::vector<U> factorial<T, U>::_fac_inv{ 1 };
} // namespace suisen


#line 5 "library/sequence/binomial_coefficient_small_prime_mod.hpp"

namespace suisen {
    template <typename mint>
    struct BinomialCoefficientSmallPrimeMod {
        mint operator()(long long n, long long r) const {
            return binom(n, r);
        }
        static mint binom(long long n, long long r) {
            factorial<mint> fac(mint::mod() - 1);

            if (r < 0 or n < r) return 0;
            r = std::min(r, n - r);
            // Lucas's theorem
            mint res = 1;
            while (r) {
                int ni = n % mint::mod(), ri = r % mint::mod();
                if (ni < ri) return 0;
                res *= fac.binom(ni, ri);
                n = n / mint::mod(), r = r / mint::mod();
            }
            return res;
        }
    };
} // namespace suisen

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