結果
問題 | No.886 Direct |
ユーザー | hitonanode |
提出日時 | 2022-01-29 16:15:50 |
言語 | C++23 (gcc 12.3.0 + boost 1.83.0) |
結果 |
AC
|
実行時間 | 173 ms / 4,000 ms |
コード長 | 15,280 bytes |
コンパイル時間 | 1,219 ms |
コンパイル使用メモリ | 115,468 KB |
実行使用メモリ | 26,752 KB |
最終ジャッジ日時 | 2024-06-10 15:43:55 |
合計ジャッジ時間 | 4,451 ms |
ジャッジサーバーID (参考情報) |
judge4 / judge3 |
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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
6,816 KB |
testcase_01 | AC | 1 ms
6,816 KB |
testcase_02 | AC | 1 ms
6,944 KB |
testcase_03 | AC | 2 ms
6,940 KB |
testcase_04 | AC | 2 ms
6,940 KB |
testcase_05 | AC | 1 ms
6,944 KB |
testcase_06 | AC | 2 ms
6,944 KB |
testcase_07 | AC | 2 ms
6,940 KB |
testcase_08 | AC | 1 ms
6,940 KB |
testcase_09 | AC | 1 ms
6,940 KB |
testcase_10 | AC | 1 ms
6,944 KB |
testcase_11 | AC | 1 ms
6,944 KB |
testcase_12 | AC | 1 ms
6,940 KB |
testcase_13 | AC | 1 ms
6,944 KB |
testcase_14 | AC | 2 ms
6,940 KB |
testcase_15 | AC | 2 ms
6,940 KB |
testcase_16 | AC | 2 ms
6,944 KB |
testcase_17 | AC | 2 ms
6,940 KB |
testcase_18 | AC | 2 ms
6,944 KB |
testcase_19 | AC | 2 ms
6,940 KB |
testcase_20 | AC | 2 ms
6,940 KB |
testcase_21 | AC | 3 ms
6,944 KB |
testcase_22 | AC | 2 ms
6,940 KB |
testcase_23 | AC | 72 ms
14,212 KB |
testcase_24 | AC | 80 ms
15,324 KB |
testcase_25 | AC | 49 ms
10,616 KB |
testcase_26 | AC | 64 ms
12,968 KB |
testcase_27 | AC | 153 ms
25,040 KB |
testcase_28 | AC | 147 ms
24,420 KB |
testcase_29 | AC | 173 ms
26,752 KB |
testcase_30 | AC | 162 ms
26,660 KB |
testcase_31 | AC | 163 ms
26,656 KB |
testcase_32 | AC | 164 ms
26,700 KB |
testcase_33 | AC | 167 ms
26,692 KB |
testcase_34 | AC | 164 ms
26,596 KB |
testcase_35 | AC | 171 ms
26,616 KB |
ソースコード
#line 1 "number/test/multiple_moebius.yuki886.test.cpp" #define PROBLEM "https://yukicoder.me/problems/no/886" #line 2 "modint.hpp" #include <iostream> #include <set> #include <vector> // CUT begin template <int md> struct ModInt { #if __cplusplus >= 201402L #define MDCONST constexpr #else #define MDCONST #endif using lint = long long; MDCONST 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; MDCONST ModInt() : val(0) {} MDCONST ModInt &_setval(lint v) { return val = (v >= md ? v - md : v), *this; } MDCONST ModInt(lint v) { _setval(v % md + md); } 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 + md); } MDCONST ModInt operator*(const ModInt &x) const { return ModInt()._setval((lint)val * x.val % md); } MDCONST ModInt operator/(const ModInt &x) const { return ModInt()._setval((lint)val * x.inv() % md); } MDCONST ModInt operator-() const { return ModInt()._setval(md - 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 % md + x.val); } friend MDCONST ModInt operator-(lint a, const ModInt &x) { return ModInt()._setval(a % md - x.val + md); } friend MDCONST ModInt operator*(lint a, const ModInt &x) { return ModInt()._setval(a % md * x.val % md); } friend MDCONST ModInt operator/(lint a, const ModInt &x) { return ModInt()._setval(a % md * x.inv() % md); } 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 { ModInt ans = 1, tmp = *this; while (n) { if (n & 1) ans *= tmp; tmp *= tmp, n >>= 1; } return ans; } static std::vector<ModInt> facs, facinvs, invs; MDCONST static void _precalculation(int N) { 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]; } MDCONST lint inv() const { if (this->val < std::min(md >> 1, 1 << 21)) { while (this->val >= int(facs.size())) _precalculation(facs.size() * 2); return invs[this->val].val; } else { return this->pow(md - 2).val; } } MDCONST ModInt fac() const { while (this->val >= int(facs.size())) _precalculation(facs.size() * 2); return facs[this->val]; } MDCONST ModInt facinv() const { while (this->val >= int(facs.size())) _precalculation(facs.size() * 2); return facinvs[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).facinv() * r.facinv(); } MDCONST ModInt nPr(const ModInt &r) const { return (this->val < r.val) ? 0 : this->fac() * (*this - r).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 mint = ModInt<998244353>; // using mint = ModInt<1000000007>; #line 2 "number/sieve.hpp" #include <cassert> #include <map> #line 5 "number/sieve.hpp" // 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() 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 7 "number/test/multiple_moebius.yuki886.test.cpp" using namespace std; using mint = ModInt<1000000007>; int main() { int H, W; cin >> H >> W; int N = max(H, W); vector<mint> dp(N); auto calc = [&](int n, int W) -> mint { // 2 * ((W - n) + (W - 2 * n) + ...) mint k = W / n; return W * k * 2 - k * (k + 1) * n; }; for (int n = 1; n < N; ++n) { mint n1 = calc(n, H), n2 = calc(n, W); dp[n] = (n1 + H) * (n2 + W) - mint(H) * W; } multiple_moebius(dp); cout << dp[1] / 2 << '\n'; }