problem1 link
从大到小贪心,较大的数字应该放置在较浅的位置。
problem2 link
最后的位置要么都是整数(经过偶数次变换),要么是$(p.5, q.5)$这种位置(奇数次变换)。
先假设是偶数次变换。那么可以从最终的点向前枚举变换的次数,可以发现,每两次变换相等于向外扩展一圈。假设向外扩展了$k$圈(经过$2k$步变换),那么如果每个最终的点向外扩展$k$圈后,由于这些正方形相交导致出现了一个新的正方形(这个正方形边长也是$2k$),那么就出现了错误(因为这将导致最后的答案多了一个点)。所以答案是最大的不导致出现错误的值。
如果答案是奇数,那么可以先看答案是1的时候是否可以。然后就变成跟答案是偶数类似的问题。
problem3 link
首先,这个可以看做一个无向图,给每个节点赋一个值,边的权值为两个顶点权值乘积。最后使得所有边权值和最大。
最后肯定是一些节点的值是最小值,一些是最大值,剩下的是中间值。
假设是中间值的顶点集合是$S$。如果$S$中存在两个节点$u,v$,它们之间没有边相连,设它们现在的权值为$p_{u},p_{v}$,与它们相连的点的权值和分别为$w_{u},w_{v}$。设$w_{u} ge w_{v}$.令$x=min(upper[u]-p_{u},p[v]-lower[v])$,那么将$u$的权值设为$p_{u}+x$,$v$的权值设为$p_{v}-x$,这样所有节点总的权值不变,但是最后的答案不会变小。但是这样的结果是又多了一个节点是最大值或者最小值。所以,按照这个思路一直做下去,那么最后$S$中的节点一定是两两相连的。
设$|S|=k$,其中的节点为$x_{1},x_{2},...,x_{k}$,$S$外与$x_{i}$相连的节点总和为$a_{i}$,$S$中所有点的权值和为$m$.那么$S$中与$x_{i}$相连的点的总权值为$m-x_{i}$。所以最后$S$中的点对答案的贡献为$sum_{i=1}^{k}x_{i}(a_{i}+frac{m-x_{i}}{2})$.除以2是因为出现了重复计算。
假设现在为$k$个点随机赋一个值为$x_{1},x_{2},...,x_{k}$。
考虑其中的两个点$x_{i},x_{j}$,令$T=x_{i}+x_{j}$,那么如果$x_{i}+x_{j}$是个定值的话,下面看$x_{i},x_{j}$分别取什么值时答案是最大值。这两个节点对答案的贡献为
$x_{i}x_{j}+x_{i}(m-T+a_{i})+x_{j}(m-T+a_{j})$
=$x_{i}x_{j}+x_{i}(m-T+a_{i})+(T-x_{i})(m-T+a_{j})$
=$-x_{i}^{2}+(a_{i}-a_{j}+T)x_{i}+T(m-T+a_{j})$
所以当$x_{i}=frac{a_{i}-a_{j}+T}{2},x_{j}=frac{a_{j}-a_{i}+T}{2}$时值最大。
此时有$x_{i}-a_{i}=x_{j}-a_{j}=frac{-a_{i}-a_{j}+T}{2}$
从两个推广到$k$个,所以存在一个常数$c$,使得对于所有的$i$,满足$x_{i}-a_{i}=c$,由于$sum_{i=1}^{k}x_{i}=m$,所以$c=frac{m-sum_{i=1}^{k}a_{i}}{k}$,所以$x_{t}=a_{t}+frac{m-sum_{i=1}^{k}a_{i}}{k}$
code for problem1
#include <vector> #include <algorithm> class TomekPhone { public: int minKeystrokes(std::vector<int> frequencies, std::vector<int> keySizes) { int total = 0; for (auto x : keySizes) { total += x; } int N = static_cast<int>(frequencies.size()); if (total < N) { return -1; } int M = static_cast<int>(keySizes.size()); int result = 0; std::sort(frequencies.begin(), frequencies.end()); std::vector<int> added(M, 0); int idx = 0; for (int i = N - 1; i >= 0; --i) { result += (added[idx] + 1) * frequencies[i]; if (i == 0) { break; } added[idx] += 1; idx = (idx + 1) % M; while (idx < M && added[idx] >= keySizes[idx]) { idx = (idx + 1) % M; } } return result; } }; int main() {}
code for problem2
#include <memory.h> #include <algorithm> #include <functional> #include <iostream> #include <limits> #include <set> #include <vector> const int MAX = 427; int g[MAX][MAX]; class DrawingPointsDivOne { public: int maxSteps(std::vector<int> x, std::vector<int> y) { if (x.size() == 1) { return -1; } auto EvenCheck = [&](const Range &range, int radius) -> bool { int num = 0; for (int x = range.min_x; x <= range.max_x; ++x) { for (int y = range.min_y; y <= range.max_y; ++y) { if (RangeCheck(x - radius, y - radius, x + radius, y + radius)) { ++num; } } } return num == static_cast<int>(x.size()); }; return std::max(CheckOld(x, y), CheckEven(x, y, EvenCheck)); } private: struct Range { int min_x; int max_x; int min_y; int max_y; Range() { Initialize(); } void Initialize() { min_x = min_y = std::numeric_limits<int>::max(); max_x = max_y = std::numeric_limits<int>::min(); } void Update(int x, int y) { min_x = std::min(min_x, x); max_x = std::max(max_x, x); min_y = std::min(min_y, y); max_y = std::max(max_y, y); } void Move(int detx, int dety) { min_x += detx; max_x += detx; min_y += dety; max_y += dety; } }; int Get(int x, int y) { if (x < 0 || y < 0) { return 0; } return g[x][y]; } bool RangeCheck(int x1, int y1, int x2, int y2) { return Get(x2, y2) - Get(x2, y1 - 1) - Get(x1 - 1, y2) + Get(x1 - 1, y1 - 1) == (x2 - x1 + 1) * (y2 - y1 + 1); } int CheckOld(const std::vector<int> &x, const std::vector<int> &y) { const int N = static_cast<int>(x.size()); std::set<std::pair<int, int>> points; for (int i = 0; i < N; ++i) { points.insert({x[i], y[i]}); points.insert({x[i] + 1, y[i]}); points.insert({x[i], y[i] + 1}); points.insert({x[i] + 1, y[i] + 1}); } auto Contains = [&](int x, int y) { return points.find({x, y}) != points.end(); }; auto Check = [&Contains](int x, int y) { return Contains(x, y) && Contains(x + 1, y) && Contains(x, y + 1) && Contains(x + 1, y + 1); }; int num = 0; for (auto element : points) { if (Check(element.first, element.second)) { ++num; } } if (num != N) { return 0; } std::vector<int> px, py; for (auto element : points) { px.push_back(element.first); py.push_back(element.second); } auto OldCheck = [&](const Range &range, int radius) -> bool { int num = 0; int dx[] = {0, 1, 0, 1}; int dy[] = {0, 0, 1, 1}; for (int x = range.min_x; x <= range.max_x; ++x) { for (int y = range.min_y; y <= range.max_y; ++y) { bool tag = true; for (int k = 0; k < 4; ++k) { if (!RangeCheck(x + dx[k] - radius, y + dy[k] - radius, x + dx[k] + radius, y + dy[k] + radius)) { tag = false; break; } } if (tag) { ++num; } } } return num == N; }; int even_result = CheckEven(px, py, OldCheck); if (even_result == -1) { return -1; } return even_result + 1; } int CheckEven(const std::vector<int> &x, const std::vector<int> &y, std::function<bool(const Range &, int)> Check) { Range search_range; for (size_t i = 0; i < x.size(); ++i) { search_range.Update(x[i], y[i]); } const int MAX_EXTEND = 141; int low = 0, high = MAX_EXTEND + 1; int result = 0; while (low <= high) { int radius = (low + high) >> 1; memset(g, 0, sizeof(g)); Range range; for (size_t i = 0; i < x.size(); ++i) { range.Update(x[i] - radius, y[i] - radius); range.Update(x[i] + radius, y[i] + radius); } int detx = -range.min_x; int dety = -range.min_y; range.Move(detx, dety); search_range.Move(detx, dety); for (size_t i = 0; i < x.size(); ++i) { Update(x[i] - radius + detx, y[i] - radius + dety, x[i] + radius + detx, y[i] + radius + dety); } Initialize(); if (Check(search_range, radius)) { result = std::max(result, radius * 2); low = radius + 1; } else { high = radius - 1; } search_range.Move(-detx, -dety); } if (result == (MAX_EXTEND + 1) * 2) { return -1; } return result; } void Update(int x1, int y1, int x2, int y2) { g[x1][y1] += 1; g[x1][y2 + 1] -= 1; g[x2 + 1][y1] -= 1; g[x2 + 1][y2 + 1] += 1; } void Initialize() { for (int i = 1; i < MAX; ++i) { g[0][i] += g[0][i - 1]; g[i][0] += g[i - 1][0]; } for (int i = 1; i < MAX; ++i) { for (int j = 1; j < MAX; ++j) { g[i][j] += g[i - 1][j] + g[i][j - 1] - g[i - 1][j - 1]; } } if (g[0][0] != 0) { g[0][0] = 1; } for (int i = 1; i < MAX; ++i) { if (g[0][i] != 0) { g[0][i] = 1; } if (g[i][0] != 0) { g[i][0] = 1; } g[0][i] += g[0][i - 1]; g[i][0] += g[i - 1][0]; } for (int i = 1; i < MAX; ++i) { for (int j = 1; j < MAX; ++j) { if (g[i][j] != 0) { g[i][j] = 1; } g[i][j] += g[i - 1][j] + g[i][j - 1] - g[i - 1][j - 1]; } } } };
code for problem3
#include <string> #include <vector> class BoundedOptimization { public: double maxValue(std::vector<std::string> expr, std::vector<int> lowerBound, std::vector<int> upperBound, int maxSum) { n = static_cast<int>(lowerBound.size()); Initialize(expr); int max_mask = 1; for (int i = 0; i < n; ++i) { max_mask *= 3; } double result = 0.0; for (int mask = 0; mask < max_mask; ++mask) { std::vector<int> lower, upper, middle; Split(lower, upper, middle, mask); if (!CheckClique(middle)) { continue; } std::vector<double> value(n, 0.0); int m = maxSum; for (auto x : lower) { value[x] = lowerBound[x]; m -= lowerBound[x]; } for (auto x : upper) { value[x] = upperBound[x]; m -= upperBound[x]; } if (m >= 0 && ComputeMiddleValue(lowerBound, upperBound, middle, m, value)) { double current_result = 0.0; for (int i = 0; i < n; ++i) { for (int j = i + 1; j < n; ++j) { current_result += graph[i][j] * value[i] * value[j]; } } result = std::max(result, current_result); } } return result; } private: bool ComputeMiddleValue(const std::vector<int> &lowerBound, const std::vector<int> &upperBound, const std::vector<int> &middle, int m, std::vector<double> &value) { if (middle.empty()) { return true; } int k = static_cast<int>(middle.size()); std::vector<double> a(k, 0.0); double total_a = 0.0; for (int i = 0; i < k; ++i) { int p = middle[i]; for (int j = 0; j < n; ++j) { if (graph[p][j] != 0) { a[i] += value[j]; } } total_a += a[i]; } double c = (m - (total_a + k * m / 2.0)) / k; bool tag = true; for (int i = 0; i < k; ++i) { int p = middle[i]; double val = a[i] + m / 2.0 + c; if (lowerBound[p] <= val && val <= upperBound[p]) { value[p] = val; } else { tag = false; break; } } return tag; } void Split(std::vector<int> &lower, std::vector<int> &upper, std::vector<int> &middle, int mask) { for (int i = 0; i < n; ++i) { int t = mask % 3; if (t == 0) { lower.push_back(i); } else if (t == 2) { upper.push_back(i); } else { middle.push_back(i); } mask /= 3; } } bool CheckClique(const std::vector<int> &clique) { for (size_t i = 0; i < clique.size(); ++i) { for (size_t j = i + 1; j < clique.size(); ++j) { if (graph[clique[i]][clique[j]] != 1) { return false; } } } return true; } void Initialize(const std::vector<std::string> &expr) { graph.resize(n); for (int i = 0; i < n; ++i) { graph[i].resize(n); } std::string all; for (auto &s : expr) { all += s; } for (size_t i = 0; i < all.size(); i += 3) { int u = all[i] - 'a'; int v = all[i + 1] - 'a'; graph[u][v] = graph[v][u] = 1; } } int n; std::vector<std::vector<int>> graph; };