problem1 link
如果两个循环之内可以跳完，那么我们只要让这些步数之内的数字组成两个数字$p,q,p\leq q$，使得$p,q,x$组成三角形即可($p+q\geq x,p+x\geq q$)。
否则，若$x$是所有数字之和的很多倍，则一开始是一直直着向前跳$m$次，剩下$r=x-m\sum_{i=0}^{n-1}t_{i}$，然后找到一个前缀和大于等于$r$即可。
problem2 link
对于某个节点$u$，假如最后一定选择该节点，那么对于节点$p$，若选它，那么$p$到$u$路径上的点都必须选，这样就成了一个最大权闭合图问题。可以用最小割来计算。
problem3 link
对于不同的素数，可以分开考虑。
对于某一个素数p：用$min[i],max[i]$计算$n$个数字中每个数字最少最多含有多少个$p$。然后第$x$个数字含有的$p$要么取$min[x]$，要么取$max[x]$，因此可用2sat解决。
code for problem1
#include <algorithm>
#include <vector>
class PeriodicJumping {
public:
int minimalTime(int x, const std::vector<int> &jumps) {
std::vector<int> copys = jumps;
int n = static_cast<int>(copys.size());
for (int i = 0; i < n; ++i) {
copys.emplace_back(copys[i]);
}
n *= 2;
if (x < 0) {
x *= -1;
}
if (x == 0) {
return 0;
}
long long s = 0;
int max_jump = 0;
for (int i = 0; i < n; ++i) {
max_jump = std::max(max_jump, copys[i]);
s += copys[i];
long long min = std::max(0ll, max_jump - (s - max_jump));
if (min <= x && x <= s) return i + 1;
}
int result = static_cast<int>(x / s * n);
int remain = x % s;
for (int i = 0; i < n && remain > 0; ++i) {
remain -= copys[i];
++result;
}
return result;
}
};
code for problem2
#include <limits>
#include <memory>
#include <unordered_map>
#include <vector>
template <typename FlowType>
class MaxFlowSolver {
static constexpr FlowType kMaxFlow = std::numeric_limits<FlowType>::max();
static constexpr FlowType kZeroFlow = static_cast<FlowType>(0);
struct node {
int v;
int next;
FlowType cap;
};
public:
int VertexNumber() const { return used_index_; }
FlowType MaxFlow(int source, int sink) {
source = GetIndex(source);
sink = GetIndex(sink);
int n = VertexNumber();
std::vector<int> pre(n);
std::vector<int> cur(n);
std::vector<int> num(n);
std::vector<int> h(n);
for (int i = 0; i < n; ++i) {
cur[i] = head_[i];
num[i] = 0;
h[i] = 0;
}
int u = source;
FlowType result = 0;
while (h[u] < n) {
if (u == sink) {
FlowType min_cap = kMaxFlow;
int v = -1;
for (int i = source; i != sink; i = edges_[cur[i]].v) {
int k = cur[i];
if (edges_[k].cap < min_cap) {
min_cap = edges_[k].cap;
v = i;
}
}
result += min_cap;
u = v;
for (int i = source; i != sink; i = edges_[cur[i]].v) {
int k = cur[i];
edges_[k].cap -= min_cap;
edges_[k ^ 1].cap += min_cap;
}
}
int index = -1;
for (int i = cur[u]; i != -1; i = edges_[i].next) {
if (edges_[i].cap > 0 && h[u] == h[edges_[i].v] + 1) {
index = i;
break;
}
}
if (index != -1) {
cur[u] = index;
pre[edges_[index].v] = u;
u = edges_[index].v;
} else {
if (--num[h[u]] == 0) {
break;
}
int k = n;
cur[u] = head_[u];
for (int i = head_[u]; i != -1; i = edges_[i].next) {
if (edges_[i].cap > 0 && h[edges_[i].v] < k) {
k = h[edges_[i].v];
}
}
if (k + 1 < n) {
num[k + 1] += 1;
}
h[u] = k + 1;
if (u != source) {
u = pre[u];
}
}
}
return result;
}
MaxFlowSolver() = default;
void Clear() {
edges_.clear();
head_.clear();
vertex_indexer_.clear();
used_index_ = 0;
}
void InsertEdge(int from, int to, FlowType cap) {
from = GetIndex(from);
to = GetIndex(to);
AddEdge(from, to, cap);
AddEdge(to, from, kZeroFlow);
}
private:
int GetIndex(int idx) {
auto iter = vertex_indexer_.find(idx);
if (iter != vertex_indexer_.end()) {
return iter->second;
}
int map_idx = used_index_++;
head_.push_back(-1);
return vertex_indexer_[idx] = map_idx;
}
void AddEdge(int from, int to, FlowType cap) {
node p;
p.v = to;
p.cap = cap;
p.next = head_[from];
head_[from] = static_cast<int>(edges_.size());
edges_.emplace_back(p);
}
std::vector<node> edges_;
std::vector<int> head_;
std::unordered_map<int, int> vertex_indexer_;
int used_index_ = 0;
};
class DoubleTree {
public:
int maximalScore(const std::vector<int> &a, const std::vector<int> &b,
const std::vector<int> &c, const std::vector<int> &d,
const std::vector<int> &score) {
int n = static_cast<int>(a.size() + 1);
std::vector<std::vector<int>> g1(n);
std::vector<std::vector<int>> g2(n);
for (int i = 0; i < n - 1; ++i) {
g1[a[i]].push_back(b[i]);
g1[b[i]].push_back(a[i]);
g2[c[i]].push_back(d[i]);
g2[d[i]].push_back(c[i]);
}
constexpr int kInfiniteFlow = 1000000;
std::unique_ptr<MaxFlowSolver<int>> solver(new MaxFlowSolver<int>());
int result = 0;
for (int root = 0; root < n; ++root) {
std::vector<int> father1(n);
std::vector<int> father2(n);
Dfs(root, -1, father1, g1);
Dfs(root, -1, father2, g2);
solver->Clear();
int source = -1;
int sink = -2;
int s = 0;
for (int i = 0; i < n; ++i) {
if (score[i] > 0) {
s += score[i];
solver->InsertEdge(source, i, score[i]);
} else {
solver->InsertEdge(i, sink, -score[i]);
}
if (i != root) {
solver->InsertEdge(i, father1[i], kInfiniteFlow);
solver->InsertEdge(i, father2[i], kInfiniteFlow);
}
}
result = std::max(result, s - solver->MaxFlow(source, sink));
}
return result;
}
private:
void Dfs(int u, int pre, std::vector<int> &father,
const std::vector<std::vector<int>> &g) {
father[u] = pre;
for (auto e : g[u]) {
if (e != pre) {
Dfs(e, u, father, g);
}
}
}
};
code for problem3
#include <algorithm>
#include <memory>
#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 GCDLCM {
public:
std::string possible(int n, const std::string &type,
const std::vector<int> &A, const std::vector<int> &B,
const std::vector<int> &C) {
std::unordered_set<int> primes;
for (auto c : C) {
for (int i = 2; i * i <= c; ++i) {
if (c % i == 0) {
primes.insert(i);
while (c % i == 0) {
c /= i;
}
}
}
if (c > 1) {
primes.insert(c);
}
}
int m = static_cast<int>(C.size());
auto Check = [&](int p) {
std::vector<int> min(n, 0);
std::vector<int> max(n, 1000);
std::vector<int> number(m, 0);
for (int i = 0; i < m; ++i) {
int t = C[i];
while (t % p == 0) {
++number[i];
t /= p;
}
if (type[i] == 'G') {
min[A[i]] = std::max(min[A[i]], number[i]);
min[B[i]] = std::max(min[B[i]], number[i]);
} else {
max[A[i]] = std::min(max[A[i]], number[i]);
max[B[i]] = std::min(max[B[i]], number[i]);
}
}
for (int i = 0; i < n; ++i) {
if (min[i] > max[i]) {
return false;
}
}
std::unique_ptr<TwoSatisfiabilitySolver> solver(
new TwoSatisfiabilitySolver());
solver->Initialize(n * 2);
for (int i = 0; i < m; i++) {
int u = A[i];
int v = B[i];
if (type[i] == 'G') {
bool x = min[u] > number[i];
bool y = min[v] > number[i];
if (x && y) {
return false;
} else if (x) {
if (min[v] != max[v]) {
solver->SetTrue(v, false);
}
} else if (y) {
if (min[u] != max[u]) {
solver->SetTrue(u, false);
}
} else {
if (min[v] != max[v] && min[u] != max[u]) {
solver->AddConstraint(v, true, u, false);
solver->AddConstraint(u, true, v, false);
}
}
} else {
bool x = max[u] < number[i];
bool y = max[v] < number[i];
if (x && y) {
return false;
} else if (x) {
if (min[v] != max[v]) {
solver->SetTrue(v, true);
}
} else if (y) {
if (min[u] != max[u]) {
solver->SetTrue(u, true);
}
} else {
if (min[v] != max[v] && min[u] != max[u]) {
solver->AddConstraint(v, false, u, true);
solver->AddConstraint(u, false, v, true);
}
}
}
}
return solver->ExistSolution();
};
for (auto p : primes) {
if (!Check(p)) {
return "Solution does not exist";
}
}
return "Solution exists";
}
};
