topcoder srm 535 div1
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problem1 link
对于每个质因子$p$,枚举其出现的最少次数以及最多次数分别在哪个数字中.
problem2 link
分数规划.题目是求$\\frac{3600K+\\sum_{i=0}^{K-1}a_{c_{i}}p_{c_{i}}}{\\sum_{i=0}^{K-1}a_{c_{i}}}*totalWork$.其中$c_{i}$是选出的第$i$个元素.二分$\\frac{3600K+\\sum_{i=0}^{K-1}a_{c_{i}}p_{c_{i}}}{\\sum_{i=0}^{K-1}a_{c_{i}}}$的值,然后选择$a_{i}(result-p_{i})$中最大的$K$个,看是否大于$3600K$.
problem3 link
到达最下面一行或者最右边一列时,只能朝一个方向走.设$f(i)$表示从(0,0)走到(n-1,i)的方案数(最后一步是从(n-2,i)到(n-1,i)),以及$g(i)$表示从$(0,0)$走到(i,m-1)的方案数(最后一步从$(i,m-2)$到(i,m-1)).这两个是类似的.考虑$f(i)$.
如下面的图示($n=6,m=8,i=4$)的两种走法,可以发现,横向走了红色的四个以及纵向走了黄色的5个,且最后一步到达$(n-1,i)$是黄色.
每次横向走时,需要对应的红色大于等于对应的蓝色;纵向走时,需要对应的黄色大于对应的绿色.
从(0,0)走到$(n-1,i)$一共走了$(i+n-1)$个格子,其中红色可以选择在这个序列的前$(i+n-2)$个中出现(最后一个一定是黄色).
code for problem1
#include <vector> using namespace std; class FoxAndGCDLCM { using Type = std::pair<long long, int>; public: long long get(long long G, long long L) { if (G > L) { return -1; } std::vector<Type> a = split(G); std::vector<Type> b = split(L); merge(a, b); for (size_t i = 0; i < a.size(); ++i) { if (a[i] > b[i]) { return -1; } } min_value = -1; dfs(0, a, b, 1ll, 1ll); return min_value; } private: void dfs(int dep, const std::vector<Type> &a, std::vector<Type> &b, long long x, long long y) { if(dep == a.size()) { if (min_value == -1 || min_value > x + y) { min_value = x + y; } return; } const long long p = pow(a[dep].first, a[dep].second); const long long q = pow(b[dep].first, b[dep].second); dfs(dep + 1, a, b, x * p, y * q); dfs(dep + 1, a, b, x * q, y * p); } std::vector<Type> split(long long x) { std::vector<Type> result; for (long long t = 2; t * t <= x; ++t) { if (x % t == 0) { int num = 0; while (x % t == 0) { x /= t; ++num; } result.emplace_back(t, num); } } if (x > 1) { result.emplace_back(x, 1); } return result; } void merge(std::vector<Type> &a, std::vector<Type> &b) { std::vector<Type> ra, rb; std::size_t i = 0, j = 0; while (i < a.size() || j < b.size()) { if (i == a.size() || (j < b.size() && a[i].first > b[j].first)) { rb.emplace_back(b[j]); ra.emplace_back(b[j].first, 0); ++j; } else if (j == b.size() || (i < a.size() && a[i].first < b[j].first)) { ra.emplace_back(a[i]); rb.emplace_back(a[i].first, 0); ++i; } else { ra.emplace_back(a[i++]); rb.emplace_back(b[j++]); } } a = ra; b = rb; } long long pow(long long a, int b) { long long result = 1; while (b > 0) { if (b & 1) { result *= a; } a *= a; b >>= 1; } return result; } long long min_value; };
code for problem2
#include <algorithm> #include <iostream> class FoxAndBusiness { public: double minimumCost(int K, int totalWork, std::vector<int> a, std::vector<int> p) { const size_t n = a.size(); double low = 0.0, high = 1e10; for (int i = 0; i < 200; ++i) { double mid = (low + high) * 0.5; std::vector<double> det(n); for (size_t j = 0; j < n; ++j) { det[j] = a[j] * (mid - p[j]); } std::sort(det.rbegin(), det.rend()); double sum = 0; for (int j = 0; j < K; ++j) { sum += det[j]; } if (sum >= K * 3600) { high = mid; } else { low = mid; } } return low * totalWork; } };
code for problem3
#include <algorithm> #include <iostream> using namespace std; class FoxAndGreed { private: static const int MOD = 10007; static void Add(int &x, int y) { x += y; if (x >= MOD) { x -= MOD; } } public: int count(int H, int W, int S) { Init(H, W, S); if (std::min(H, W) == 1) { return f3[W * H - 1][S]; } int result = 0; for (int i = 0; i <= W - 2; ++i) { int h = i; int v = H - 1; int r = W - i - 1; Add(result, Binomial(h + v - 1, h) * Pow(S + 1, W * H - 1 - h * 2 - v * 2 - r) % MOD * Compute(h, v, r, S) % MOD); } for (int i = 0; i <= H - 2; ++i) { int h = W - 1; int v = i; int r = H - i - 1; Add(result, Binomial(h + v - 1, v) * Pow(S + 1, W * H - 1 - h * 2 - v * 2 - r) % MOD * Compute(h, v, r, S) % MOD); } return result; } private: int Compute(const int h, const int v, int r, const int S) { int result = 0; for (int x = 0; x <= S; ++x) { for (int y = 0; y + x <= S; ++y) { int z = S - x - y; Add(result, f1[h][x] * f2[v][y] % MOD * f3[r][z] % MOD); } } return result; } void Init(const int H, const int W, const int S) { auto SetSize = [H, W, S](std::vector<std::vector<int>> &f) { f.resize(H + W + 2); for (size_t i = 0; i < f.size(); ++i) { f[i].resize(S + 1); } }; SetSize(f1); SetSize(f2); SetSize(f3); f1[0][0] = f2[0][0] = f3[0][0] = 1; for (int i = 1; i <= H + W + 1; ++i) { for (int j = 0; j <= S; ++j) { f1[i][j] = f2[i][j] = f3[i][j] = 0; for (int x = 0; x <= j; ++x) { Add(f1[i][j], f1[i - 1][j - x] * (x + 1) % MOD); Add(f2[i][j], f2[i - 1][j - x] * x % MOD); Add(f3[i][j], f3[i - 1][j - x]); } } } fact.resize(H + W + 2); invFact.resize(H + W + 2); fact[0] = invFact[0] = 1; for (size_t i = 1; i < fact.size(); ++i) { fact[i] = fact[i - 1] * i % MOD; invFact[i] = Pow(fact[i], MOD - 2); } } int Binomial(int n, int m) { return fact[n] * invFact[m] % MOD * invFact[n - m] % MOD; } static int Pow(int a, int b) { int r = 1; while (b > 0) { if (b & 1) { r = r * a % MOD; } a = a * a % MOD; b >>= 1; } return r; } std::vector<std::vector<int>> f1, f2, f3; std::vector<int> fact, invFact; };
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