problem1 link
首先计算任意两点的距离。然后枚举选出的集合中的两个点，判断其他点是否可以即可。
problem2 link
假设字符串为$s$，长度为$n$。那么对于$SA$中的两个排名$SA_{i},SA_{i+1}$来说，应该尽量使得$s[SA_{i}]=s[SA_{i+1}]$。如果这个满足的话，那么需要两个后缀满足$s[SA_{i}+1\sim n-1]<s[SA_{i+1}+1\sim n-1]$,设他们的排名分别为$SA_{r},SA_{k}$，也就是说$r<k$即可。
problem3 link
这个可以转化为2-SAT问题。
首先将每个点拆成202个点，分别表示$\leq 0,\leq 1,...,\leq 100,>0,>1,...,>100$.然后就是每一个等式可以转化为一些2-SAT中的限制。
比如对于$g_{x}+g_{y}<50$来说.比如$g_{x}> 19,g_{y}> 29$是冲突的，那么在2-SAT可以描述为$g_{x}> 19\rightarrow g_{y}\leq 29$，以及$g_{y}> 29\rightarrow g_{x}\leq 19$
最后就是得到2_SAT的一组解。
code for problem1
#include <algorithm>
#include <unordered_set>
#include <vector> class Egalitarianism3 {
public:
int maxCities(int n, const std::vector<int> &a, const std::vector<int> &b,
const std::vector<int> &len) {
if (n == 1) {
return 1;
}
std::vector<std::vector<int>> g(n, std::vector<int>(n, -1));
for (int i = 0; i < n - 1; ++i) {
g[a[i] - 1][b[i] - 1] = g[b[i] - 1][a[i] - 1] = len[i];
}
for (int i = 0; i < n; ++i) {
g[i][i] = 0;
}
for (int i = 0; i < n; ++i) {
for (int u = 0; u < n; ++u) {
if (g[u][i] != -1) {
for (int v = 0; v < n; ++v) {
if (g[i][v] != -1) {
if (g[u][v] == -1 || g[u][v] > g[u][i] + g[i][v]) {
g[u][v] = g[u][i] + g[i][v];
}
}
}
}
}
}
auto Compute = [&](int s, int t) {
std::unordered_set<int> all;
all.insert(s);
all.insert(t);
const int d = g[s][t];
for (int i = 0; i < n; ++i) {
if (all.find(i) == all.end()) {
bool tag = true;
for (auto e : all) {
if (g[i][e] != d) {
tag = false;
break;
}
}
if (tag) {
all.insert(i);
}
}
}
return static_cast<int>(all.size());
};
int result = 0;
for (int i = 0; i < n; ++i) {
for (int j = i + 1; j < n; ++j) {
result = std::max(result, Compute(i, j));
}
}
return result;
}
};
code for problem2
#include <vector>
class SuffixArrayDiv1 {
public:
int minimalCharacters(const std::vector<int> &SA) {
int n = static_cast<int>(SA.size());
std::vector<int> s(n + 1, -1);
for (int i = 0; i < n; ++i) {
s[SA[i]] = i;
}
int result = 1;
for (int i = 0; i + 1 < n; ++i) {
if (s[SA[i] + 1] > s[SA[i + 1] + 1]) {
++result;
}
}
return result;
}
};
code for problem3
#include <algorithm>
#include <stack>
#include <unordered_map>
#include <unordered_set>
#include <vector> class StronglyConnectedComponentSolver {
public:
StronglyConnectedComponentSolver() = default; void Initialize(int n) { edges_.resize(n); } std::vector<int> Solve() {
total_ = static_cast<int>(edges_.size());
if (total_ == 0) {
return {};
}
visited_.resize(total_, false);
low_indices_.resize(total_, 0);
dfs_indices_.resize(total_, 0);
connected_component_indices_.resize(total_, 0);
for (int i = 0; i < total_; ++i) {
if (0 == dfs_indices_[i]) {
Dfs(i);
}
}
return connected_component_indices_;
} int VertexNumber() const { return static_cast<int>(edges_.size()); } inline void AddEdge(int from, int to) { edges_[from].push_back(to); } const std::vector<int> &Tos(int u) const { return edges_[u]; } private:
void Dfs(const int u) {
low_indices_[u] = dfs_indices_[u] = ++index_;
stack_.push(u);
visited_[u] = true;
for (auto v : edges_[u]) {
if (0 == dfs_indices_[v]) {
Dfs(v);
low_indices_[u] = std::min(low_indices_[u], low_indices_[v]);
} else if (visited_[v]) {
low_indices_[u] = std::min(low_indices_[u], dfs_indices_[v]);
}
}
if (dfs_indices_[u] == low_indices_[u]) {
int v = 0;
do {
v = stack_.top();
stack_.pop();
visited_[v] = false;
connected_component_indices_[v] = connected_component_index_;
} while (u != v);
++connected_component_index_;
}
} std::vector<std::vector<int>> edges_;
int total_ = 0;
std::vector<bool> visited_;
std::vector<int> low_indices_;
std::vector<int> dfs_indices_;
std::stack<int> stack_;
int index_ = 0;
int connected_component_index_ = 0;
std::vector<int> connected_component_indices_;
}; class TwoSatisfiabilitySolver {
public:
void Initialize(int total_vertex_number) {
scc_solver_.Initialize(total_vertex_number);
} // If idx1 is type1, then idx2 must be type2.
void AddConstraint(int idx1, bool type1, int idx2, bool type2) {
int from = idx1 * 2 + (type1 ? 1 : 0);
int to = idx2 * 2 + (type2 ? 1 : 0);
scc_solver_.AddEdge(from, to);
} void AddConflict(int idx1, bool type1, int idx2, bool type2) {
AddConstraint(idx1, type1, idx2, !type2);
AddConstraint(idx2, type2, idx1, !type1);
} void AddLead(int idx1, bool type1, int idx2, bool type2) {
AddConstraint(idx1, type1, idx2, type2);
AddConstraint(idx2, !type2, idx1, !type1);
} // The idx must not be type
void SetFalse(int idx, bool type) { SetTrue(idx, !type); } // The idx must be type
void SetTrue(int idx, bool type) { AddConstraint(idx, !type, idx, type); } bool ExistSolution() {
if (scc_indices_.empty()) {
scc_indices_ = scc_solver_.Solve();
total_scc_number_ =
*std::max_element(scc_indices_.begin(), scc_indices_.end()) + 1;
}
for (int i = 0; i < scc_solver_.VertexNumber() / 2; ++i) {
if (scc_indices_[i * 2] == scc_indices_[i * 2 + 1]) {
return false;
}
}
return true;
} std::vector<bool> GetOneSolution() {
if (!ExistSolution()) {
return {};
}
BuildNewGraph();
TopSort();
int total = scc_solver_.VertexNumber();
std::vector<bool> result(total / 2);
for (int e = 0; e < total / 2; ++e) {
if (last_color_[scc_indices_[e * 2]] == 0) {
result[e] = false;
} else {
result[e] = true;
}
}
return std::move(result);
} private:
void BuildNewGraph() {
new_edges_.resize(total_scc_number_);
new_graph_node_in_degree_.resize(total_scc_number_, 0);
int total = scc_solver_.VertexNumber();
for (int i = 0; i < total; ++i) {
int scc0 = scc_indices_[i];
for (auto e : scc_solver_.Tos(i)) {
int scc1 = scc_indices_[e];
if (scc0 != scc1 &&
new_edges_[scc1].find(scc0) == new_edges_[scc1].end()) {
new_edges_[scc1].insert(scc0);
++new_graph_node_in_degree_[scc0];
}
}
}
} void TopSort() {
std::vector<int> conflict(total_scc_number_);
int total = scc_solver_.VertexNumber() / 2;
for (int i = 0; i < total; ++i) {
conflict[scc_indices_[i * 2]] = scc_indices_[i * 2 + 1];
conflict[scc_indices_[i * 2 + 1]] = scc_indices_[i * 2];
}
last_color_.resize(total_scc_number_, -1);
std::stack<int> st;
for (int i = 0; i < total_scc_number_; ++i) {
if (0 == new_graph_node_in_degree_[i]) {
st.push(i);
}
}
while (!st.empty()) {
int u = st.top();
st.pop();
if (last_color_[u] == -1) {
last_color_[u] = 0;
last_color_[conflict[u]] = 1;
}
for (auto e : new_edges_[u]) {
int cur = --new_graph_node_in_degree_[e];
if (cur == 0) {
st.push(e);
}
}
}
} std::vector<int> scc_indices_;
int total_scc_number_ = 0;
std::vector<std::unordered_set<int>> new_edges_;
std::vector<int> new_graph_node_in_degree_;
std::vector<int> last_color_; StronglyConnectedComponentSolver scc_solver_;
}; class NeverAskHerAge {
public:
std::vector<int> possibleSolution(int n, const std::vector<int> &id1,
const std::vector<int> &id2,
const std::vector<std::string> &op,
const std::vector<std::string> &rl,
const std::vector<int> &val) {
solver.Initialize(n * 101 * 2);
int m = static_cast<int>(id1.size());
// 0: <= j
// 1: > j
for (int i = 1; i <= n; ++i) {
for (int j = 0; j <= 100; ++j) {
if (j > 0) {
solver.AddLead(GetIndex(i, j), 1, GetIndex(i, j - 1), 1);
}
if (j + 1 < 100) {
solver.AddLead(GetIndex(i, j), 0, GetIndex(i, j + 1), 0);
}
}
solver.SetFalse(GetIndex(i, 0), 0);
solver.SetFalse(GetIndex(i, 100), 1);
}
for (int i = 0; i < m; ++i) {
if (rl[i] == "=") {
Add(id1[i], id2[i], op[i][0], ">=", val[i]);
Add(id1[i], id2[i], op[i][0], "<=", val[i]);
} else {
Add(id1[i], id2[i], op[i][0], rl[i], val[i]);
}
}
if (!solver.ExistSolution()) {
return {};
}
std::vector<int> result(n);
auto sol = solver.GetOneSolution();
for (int i = 0; i < n; ++i) {
for (int j = 1; j < 101; ++j) {
int t = GetIndex(i + 1, j);
if (!sol[t]) {
result[i] = j;
break;
}
}
}
return result;
} private:
void Add(int x, int y, char op, const std::string &rl, int z) {
if (op == '+' || op == '*') {
if (rl[0] == '<') {
AddMulLess(x, y, op, rl, z);
} else {
AddMulGreater(x, y, op, rl, z);
}
} else {
if (rl[0] == '<') {
SubDivLess(x, y, op, rl, z);
} else {
SubDivGreater(x, y, op, rl, z);
}
}
} void AddMulLess(int g1, int g2, char op, const std::string &rl, int z) {
// Assume g2 > i - 1, (i, i+1, i+2, ..., 100)
for (int i = 1; i <= 101; ++i) {
// If rl is '<', then 1000(i + j) < z.
// Here consider opposite: 1000(i + j) >= z.
// Get j >= ceil((z - 1000i) / 1000) = (z - 1000i + 999) / 1000
// So g2 >= i and g1 >= j conflicts
// If rl is '<=', then 1000(i + j) <= z.
// Here consider opposite: 1000(i + j) > z.
// Get j >= floor((z - 1000i) / 1000) + 1 = (z - 1000i + 1000) / 1000
// So g2 >= i and g1 >= j conflicts
int j = op == '+' ? Ceil(z - 1000 * i, 1000, EqualTag(rl))
: Ceil(z, i * 1000, EqualTag(rl));
if (j < 1) {
solver.SetFalse(GetIndex(g2, i - 1), 1);
} else if (j <= 101) {
solver.AddConflict(GetIndex(g2, i - 1), 1, GetIndex(g1, j - 1), 1);
}
}
} void AddMulGreater(int g1, int g2, char op, const std::string &rl, int z) {
for (int i = 0; i < 101; ++i) {
int j = op == '+' ? Floor(z - 1000 * i, 1000, EqualTag(rl))
: Floor(z, i * 1000, EqualTag(rl));
if (j >= 101) {
solver.SetFalse(GetIndex(g2, i), 0);
} else if (j >= 0) {
solver.AddConflict(GetIndex(g2, i), 0, GetIndex(g1, j), 0);
}
}
} void SubDivGreater(int g1, int g2, char op, const std::string &rl, int z) {
for (int i = 1; i <= 101; ++i) {
int j = op == '-' ? Floor(z + 1000 * i, 1000, EqualTag(rl))
: Floor(z * i, 1000, EqualTag(rl));
if (j >= 101) {
solver.SetFalse(GetIndex(g2, i - 1), 1);
} else if (j >= 0) {
solver.AddConflict(GetIndex(g2, i - 1), 1, GetIndex(g1, j), 0);
}
}
} void SubDivLess(int g1, int g2, char op, const std::string &rl, int z) {
for (int i = 0; i < 101; ++i) {
int j = op == '-' ? Ceil(z + 1000 * i, 1000, EqualTag(rl))
: Ceil(z * i, 1000, EqualTag(rl));
if (j < 1) {
solver.SetFalse(GetIndex(g2, i), 0);
} else if (j <= 101) {
solver.AddConflict(GetIndex(g2, i), 0, GetIndex(g1, j - 1), 1);
}
}
} bool EqualTag(const std::string &rl) { return rl.length() < 2; } int Ceil(int x, int y, bool tag) {
if (x < 0) {
return -1;
}
return (x + y - (tag ? 1 : 0)) / y;
} int Floor(int x, int y, bool tag) {
if (y == 0) {
return 101;
}
if (x <= 0) {
return -1;
}
return (x - (tag ? 0 : 1)) / y;
} int GetIndex(int i, int j) { return (i - 1) * 101 + j; } TwoSatisfiabilitySolver solver;
};