two_sat.hpp¶
Incremental 2-SAT formula. A clause is (x_i == first_value) OR (x_j == second_value).
Verified by two_sat.
求解每个变量二选一、约束为二元析取的可满足性，并构造一组布尔赋值。
Implementation¶
#ifndef NOYA_TWO_SAT_HPP
#define NOYA_TWO_SAT_HPP 1
/// @complexity Time: O(variables + clauses) to solve.
/// Space: O(variables + clauses).
#include "atcoder/twosat.hpp"
#include <cassert>
#include <optional>
#include <vector>
namespace noya {
/// @brief Incremental 2-SAT formula. A clause is (x_i == first_value) OR
/// (x_j == second_value).
class two_sat {
public:
explicit two_sat(int variable_count)
: variable_count_(variable_count), implementation_(variable_count) {
assert(variable_count >= 0);
}
void add_clause(int first, bool first_value, int second, bool second_value) {
implementation_.add_clause(first, first_value, second, second_value);
}
/// @brief Return one satisfying assignment, or nullopt if none exists.
std::optional<std::vector<bool>> solve() {
if (!implementation_.satisfiable()) {
return std::nullopt;
}
return implementation_.answer();
}
int variable_count() const { return variable_count_; }
private:
int variable_count_;
atcoder::two_sat implementation_;
};
} // namespace noya
#endif // NOYA_TWO_SAT_HPP
#include <algorithm>
#include <cassert>
#include <optional>
#include <utility>
#include <vector>
/// @complexity Time: O(variables + clauses) to solve.
/// Space: O(variables + clauses).
namespace atcoder {
namespace internal {
template <class E> struct csr {
std::vector<int> start;
std::vector<E> elist;
explicit csr(int n, const std::vector<std::pair<int, E>>& edges)
: start(n + 1), elist(edges.size()) {
for (auto e : edges) {
start[e.first + 1]++;
}
for (int i = 1; i <= n; i++) {
start[i] += start[i - 1];
}
auto counter = start;
for (auto e : edges) {
elist[counter[e.first]++] = e.second;
}
}
};
} // namespace internal
} // namespace atcoder
namespace atcoder {
namespace internal {
// Reference:
// R. Tarjan,
// Depth-First Search and Linear Graph Algorithms
struct scc_graph {
public:
explicit scc_graph(int n) : _n(n) {}
int num_vertices() { return _n; }
void add_edge(int from, int to) { edges.push_back({from, {to}}); }
// @return pair of (# of scc, scc id)
std::pair<int, std::vector<int>> scc_ids() {
auto g = csr<edge>(_n, edges);
int now_ord = 0, group_num = 0;
std::vector<int> visited, low(_n), ord(_n, -1), ids(_n);
visited.reserve(_n);
auto dfs = [&](auto self, int v) -> void {
low[v] = ord[v] = now_ord++;
visited.push_back(v);
for (int i = g.start[v]; i < g.start[v + 1]; i++) {
auto to = g.elist[i].to;
if (ord[to] == -1) {
self(self, to);
low[v] = std::min(low[v], low[to]);
} else {
low[v] = std::min(low[v], ord[to]);
}
}
if (low[v] == ord[v]) {
while (true) {
int u = visited.back();
visited.pop_back();
ord[u] = _n;
ids[u] = group_num;
if (u == v) break;
}
group_num++;
}
};
for (int i = 0; i < _n; i++) {
if (ord[i] == -1) dfs(dfs, i);
}
for (auto& x : ids) {
x = group_num - 1 - x;
}
return {group_num, ids};
}
std::vector<std::vector<int>> scc() {
auto ids = scc_ids();
int group_num = ids.first;
std::vector<int> counts(group_num);
for (auto x : ids.second) counts[x]++;
std::vector<std::vector<int>> groups(ids.first);
for (int i = 0; i < group_num; i++) {
groups[i].reserve(counts[i]);
}
for (int i = 0; i < _n; i++) {
groups[ids.second[i]].push_back(i);
}
return groups;
}
private:
int _n;
struct edge {
int to;
};
std::vector<std::pair<int, edge>> edges;
};
} // namespace internal
} // namespace atcoder
namespace atcoder {
// Reference:
// B. Aspvall, M. Plass, and R. Tarjan,
// A Linear-Time Algorithm for Testing the Truth of Certain Quantified Boolean
// Formulas
struct two_sat {
public:
two_sat() : _n(0), scc(0) {}
explicit two_sat(int n) : _n(n), _answer(n), scc(2 * n) {}
void add_clause(int i, bool f, int j, bool g) {
assert(0 <= i && i < _n);
assert(0 <= j && j < _n);
scc.add_edge(2 * i + (f ? 0 : 1), 2 * j + (g ? 1 : 0));
scc.add_edge(2 * j + (g ? 0 : 1), 2 * i + (f ? 1 : 0));
}
bool satisfiable() {
auto id = scc.scc_ids().second;
for (int i = 0; i < _n; i++) {
if (id[2 * i] == id[2 * i + 1]) return false;
_answer[i] = id[2 * i] < id[2 * i + 1];
}
return true;
}
std::vector<bool> answer() { return _answer; }
private:
int _n;
std::vector<bool> _answer;
internal::scc_graph scc;
};
} // namespace atcoder
namespace noya {
/// @brief Incremental 2-SAT formula. A clause is (x_i == first_value) OR
/// (x_j == second_value).
class two_sat {
public:
explicit two_sat(int variable_count)
: variable_count_(variable_count), implementation_(variable_count) {
assert(variable_count >= 0);
}
void add_clause(int first, bool first_value, int second, bool second_value) {
implementation_.add_clause(first, first_value, second, second_value);
}
/// @brief Return one satisfying assignment, or nullopt if none exists.
std::optional<std::vector<bool>> solve() {
if (!implementation_.satisfiable()) {
return std::nullopt;
}
return implementation_.answer();
}
int variable_count() const { return variable_count_; }
private:
int variable_count_;
atcoder::two_sat implementation_;
};
} // namespace noya