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Merge pull request #91 from rainbou-kpr/89-combination
crt, combinatorics
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| Original file line number | Diff line number | Diff line change |
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| #pragma once | ||
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| #include <vector> | ||
| #include "number-theory.hpp" | ||
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| /** | ||
| * @brief 組み合わせ | ||
| */ | ||
| template <typename Modint> | ||
| class Combination { | ||
| static std::vector<Modint> fact, inv_fact; | ||
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| public: | ||
| /** | ||
| * @brief n までの階乗とその逆元を前計算する | ||
| * @param n | ||
| * @note O(n) 必要になったら呼び出されるが、予め大きなnに対して呼び出しておくことで逆元の直接計算を減らせる | ||
| */ | ||
| inline static void extend(int n) { | ||
| int m = fact.size(); | ||
| if (n < m) return; | ||
| fact.resize(n + 1); | ||
| inv_fact.resize(n + 1); | ||
| for (int i = m; i <= n; ++i) { | ||
| fact[i] = fact[i - 1] * i; | ||
| } | ||
| inv_fact[n] = fact[n].inv(); | ||
| for (int i = n; i > m; --i) { | ||
| inv_fact[i - 1] = inv_fact[i] * i; | ||
| } | ||
| } | ||
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| /** | ||
| * @brief n の階乗を返す | ||
| * @param n | ||
| * @return n! | ||
| * @note extend(n), O(1) | ||
| */ | ||
| inline static Modint factorial(int n) { | ||
| extend(n); | ||
| return fact[n]; | ||
| } | ||
| /** | ||
| * @brief n の階乗の逆元を返す | ||
| * @param n | ||
| * @return n!^-1 | ||
| * @note extend(n), O(1) | ||
| */ | ||
| inline static Modint inverse_factorial(int n) { | ||
| extend(n); | ||
| return inv_fact[n]; | ||
| } | ||
| /** | ||
| * @brief n の逆元を返す | ||
| * @param n | ||
| * @return n^-1 | ||
| * @note extend(n), O(1) | ||
| */ | ||
| inline static Modint inverse(int n) { | ||
| extend(n); | ||
| return inv_fact[n] * fact[n - 1]; | ||
| } | ||
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| /** | ||
| * @brief nPr を返す | ||
| * @param n | ||
| * @param r | ||
| * @return nPr | ||
| * @note extend(n), O(1) | ||
| */ | ||
| inline static Modint P(int n, int r) { | ||
| if (r < 0 || n < r) return 0; | ||
| extend(n); | ||
| return fact[n] * inv_fact[n - r]; | ||
| } | ||
| /** | ||
| * @brief nCr を返す | ||
| * @param n | ||
| * @param r | ||
| * @return nCr | ||
| * @note extend(n), O(1) | ||
| */ | ||
| inline static Modint C(int n, int r) { | ||
| if (r < 0 || n < r) return 0; | ||
| extend(n); | ||
| return fact[n] * inv_fact[r] * inv_fact[n - r]; | ||
| } | ||
| /** | ||
| * @brief nHr を返す | ||
| * @param n | ||
| * @param r | ||
| * @return nHr | ||
| * @note extend(n+r-1), O(1) | ||
| */ | ||
| inline static Modint H(int n, int r) { | ||
| if (n < 0 || r < 0) return 0; | ||
| if (n == 0 && r == 0) return 1; | ||
| return C(n + r - 1, r); | ||
| } | ||
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| /** | ||
| * @brief nPr を定義どおり計算する | ||
| * @param n | ||
| * @param r | ||
| * @return nPr | ||
| * @note O(r) | ||
| */ | ||
| inline static Modint P_loop(long long n, int r) { | ||
| if (r < 0 || n < r) return 0; | ||
| Modint res = 1; | ||
| for (int i = 0; i < r; ++i) { | ||
| res *= n - i; | ||
| } | ||
| return res; | ||
| } | ||
| /** | ||
| * @brief nCr を定義どおり計算する | ||
| * @param n | ||
| * @param r | ||
| * @return nCr | ||
| * @note O(min(r, n-r)) | ||
| */ | ||
| inline static Modint C_loop(long long n, long long r) { | ||
| if (r < 0 || n < r) return 0; | ||
| if(r > n - r) r = n - r; | ||
| extend(r); | ||
| return P_loop(n, r) * inv_fact[r]; | ||
| } | ||
| /** | ||
| * @brief nHr を定義どおり計算する | ||
| * @param n | ||
| * @param r | ||
| * @return nHr | ||
| * @note O(r) | ||
| */ | ||
| inline static Modint H_loop(long long n, long long r) { | ||
| if (n < 0 || r < 0) return 0; | ||
| if (n == 0 && r == 0) return 1; | ||
| return C_loop(n + r - 1, r); | ||
| } | ||
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| /** | ||
| * @brief nCr を Lucas の定理を用いて計算する | ||
| * @param n | ||
| * @param r | ||
| * @return nCr | ||
| * @note expand(Mod), O(log(r)) | ||
| */ | ||
| inline static Modint C_lucas(long long n, long long r) { | ||
| if (r < 0 || n < r) return 0; | ||
| if (r == 0 || n == r) return 1; | ||
| Modint res = 1; | ||
| while(r > 0) { | ||
| int ni = n % Modint::mod(), ri = r % Modint::mod(); | ||
| if (ni < ri) return 0; | ||
| res *= C(ni, ri); | ||
| n /= Modint::mod(); | ||
| r /= Modint::mod(); | ||
| } | ||
| return res; | ||
| } | ||
| }; | ||
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| template <typename Modint> | ||
| std::vector<Modint> Combination<Modint>::fact{1, 1}; | ||
| template <typename Modint> | ||
| std::vector<Modint> Combination<Modint>::inv_fact{1, 1}; | ||
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| /** | ||
| * @brief mod p^q での二項係数を求める構造 | ||
| * @note 前計算O(p^q) 参考: https://nyaannyaan.github.io/library/modulo/arbitrary-mod-binomial.hpp | ||
| */ | ||
| struct CombinationPQ { | ||
| int p, q; | ||
| int pq; | ||
| std::vector<int> fact_p, inv_fact_p; | ||
| int delta; | ||
| CombinationPQ(int p, int q) : p(p), q(q) { | ||
| pq = 1; | ||
| for(int i = 0; i < q; i++) pq *= p; | ||
| fact_p.resize(pq); | ||
| fact_p[0] = 1; | ||
| for(int i = 1; i < pq; i++) { | ||
| if(i % p == 0) fact_p[i] = fact_p[i - 1]; | ||
| else fact_p[i] = (long long)fact_p[i - 1] * i % pq; | ||
| } | ||
| inv_fact_p.resize(pq); | ||
| inv_fact_p[pq - 1] = modinv(fact_p[pq - 1], pq); | ||
| for(int i = pq - 1; i > 0; i--) { | ||
| if(i % p == 0) inv_fact_p[i - 1] = inv_fact_p[i]; | ||
| else inv_fact_p[i - 1] = (long long)inv_fact_p[i] * i % pq; | ||
| } | ||
| if(p == 2 && q >= 3) delta = 1; | ||
| else delta = -1; | ||
| } | ||
| /** | ||
| * @brief nCr mod p^q を返す | ||
| * @param n long long | ||
| * @param r long long | ||
| * @return nCr mod p^q | ||
| * @note O(log(n)) | ||
| */ | ||
| int C(long long n, long long r) { | ||
| if(r < 0 || n < r) return 0; | ||
| long long m = n - r; | ||
| int ans = 1; | ||
| std::vector<int> epsilon; | ||
| while(n > 0) { | ||
| ans = (long long)ans * fact_p[n % pq] % pq; | ||
| ans = (long long)ans * inv_fact_p[m % pq] % pq; | ||
| ans = (long long)ans * inv_fact_p[r % pq] % pq; | ||
| n /= p; | ||
| m /= p; | ||
| r /= p; | ||
| epsilon.push_back(n - m - r); | ||
| } | ||
| if(delta == -1 && epsilon.size() >= q && accumulate(epsilon.begin()+q-1, epsilon.end(), 0) % 2 == 1) ans = pq - ans; | ||
| if(ans == pq) ans = 0; | ||
| int e = accumulate(epsilon.begin(), epsilon.end(), 0); | ||
| if(e >= q) ans = 0; | ||
| else { | ||
| for(int i = 0; i < e; i++) { | ||
| ans = (long long)ans * p % pq; | ||
| } | ||
| } | ||
| return ans; | ||
| } | ||
| }; |
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,40 @@ | ||
| #define PROBLEM "https://judge.yosupo.jp/problem/binomial_coefficient" | ||
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| #include "../cpp/combinatorics.hpp" | ||
| #include "../cpp/number-theory.hpp" | ||
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| #include <iostream> | ||
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| int main() { | ||
| int t, m; std::cin >> t >> m; | ||
| std::vector<std::pair<int, int>> factors; | ||
| { | ||
| int mm = m; | ||
| for(int i = 2; i * i <= mm; ++i) { | ||
| if(mm % i == 0) { | ||
| int cnt = 0; | ||
| while(mm % i == 0) { | ||
| mm /= i; | ||
| cnt++; | ||
| } | ||
| factors.emplace_back(i, cnt); | ||
| } | ||
| } | ||
| if(mm > 1) factors.emplace_back(mm, 1); | ||
| } | ||
| std::vector<CombinationPQ> combpq; | ||
| for(int i = 0; i < factors.size(); ++i) { | ||
| auto [p, e] = factors[i]; | ||
| combpq.emplace_back(p, e); | ||
| } | ||
| while(t--) { | ||
| long long n, k; std::cin >> n >> k; | ||
| std::vector<long long> x, y; | ||
| for(int i = 0; i < factors.size(); ++i) { | ||
| auto [p, e] = factors[i]; | ||
| x.push_back(combpq[i].C(n, k)); | ||
| y.push_back(combpq[i].pq); | ||
| } | ||
| std::cout << crt(x, y).first << std::endl; | ||
| } | ||
| } |
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