graph/two_sat.hpp

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• Last update: 2022-07-30 11:55:56+09:00
• Include: #include "graph/two_sat.hpp"

Depends on

• graph/strongly_connected_components.hpp

Verified with

• graph/test/two_sat.test.cpp

Code

#pragma once

#include "strongly_connected_components.hpp"

#include <optional>

class TwoSat {
public:
    struct Variable {
    private:
        int idx;
            
        Variable(int i) : idx(i) {}
        
    public:
        Variable operator!() const noexcept {
            return Variable(idx ^ 1);
        }
        
        int index() const {
            return idx / 2;
        }
        
        friend class TwoSat;
    };
    
private:
    std::vector<std::vector<int>> graph;
    
public:
    TwoSat() : graph() {}
    
    TwoSat::Variable add_variable() {
        Variable var(graph.size());
        graph.resize(graph.size() + 2, std::vector<int>());
        return var;
    }
    
    std::vector<TwoSat::Variable> add_variables(int num) {
        std::vector<TwoSat::Variable> vars;
        vars.reserve(num);
        for (int i = 0; i < num; ++i) {
            vars.emplace_back(Variable((int) graph.size() + 2 * i));
        }
        graph.resize((int) graph.size() + 2 * num, std::vector<int>());
        return vars;
    }
    
    // x or y
    void add_clause(Variable x, Variable y) {
        graph[(!x).idx].push_back(y.idx);
        graph[(!y).idx].push_back(x.idx);
    }
    
    // x implies y
    void implies(Variable x, Variable y) {
        add_clause(!x, y);
    }
    
    void at_most_one(const std::vector<Variable> &vars) {
        if (vars.size() <= 1) {
            return;
        }
        std::vector<Variable> sum = add_variables((int) vars.size() - 1);
        for (int i = 0; i < (int) sum.size(); ++i) {
            implies(vars[i], sum[i]);
            implies(sum[i], !vars[i + 1]);
        }
        for (int i = 0; i < (int) sum.size() - 1; ++i) {
            implies(sum[i], sum[i + 1]);
        }
    }
    
    std::optional<std::vector<bool>> solve() {
        StronglyConnectedComponents scc(graph);
        std::vector<bool> ans(graph.size() / 2, false);
        for (int i = 0; i < (int) graph.size() / 2; ++i) {
            if (scc[2 * i] == scc[2 * i + 1]) {
                return std::nullopt;
            }
            if (scc[2 * i] > scc[2 * i + 1]) {
                ans[i] = true;
            }
        }
        return ans;
    }
};

#line 2 "graph/two_sat.hpp"

#line 2 "graph/strongly_connected_components.hpp"

#include <vector>

template <typename G>
class StronglyConnectedComponents {    
    std::vector<int> comp_id;
    int comp_num;
    
public:
    StronglyConnectedComponents(const G &g) : comp_id(g.size(), -1), comp_num(0) {
        int now = 0;
        std::vector<int> vs;
        std::vector<int> ord(g.size(), -1);
        std::vector<int> low(g.size(), -1);
        
        const auto dfs = [&](const auto &dfs, int v) -> void {
            vs.push_back(v);
            ord[v] = now++;
            low[v] = ord[v];
            for (int u : g[v]) {
                if (comp_id[u] != -1) {
                    continue;
                }
                if (ord[u] == -1) {
                    dfs(dfs, u);
                }
                low[v] = std::min(low[v], low[u]);
            }
            if (low[v] == ord[v]) {
                while (true) {
                    int u = vs.back();
                    vs.pop_back();
                    comp_id[u] = comp_num;
                    if (u == v) {
                        break;
                    }
                }
                ++comp_num;
            }
        };
        
        for (int v = 0; v < (int) g.size(); ++v) {
            if (ord[v] == -1) {
                dfs(dfs, v);
            }
        }
        
        for (int v = 0; v < (int) g.size(); ++v) {
            comp_id[v] = comp_num - 1 - comp_id[v];
        }
    }
    
    int comps() const {
        return comp_num;
    }

    int operator[](int v) const {
        assert(v >= 0 && v < (int) comp_id.size());
        return comp_id[v];
    }

    std::vector<std::vector<int>> groups() const {
        std::vector<std::vector<int>> ret(comp_num);
        for (int v = 0; v < (int) comp_id.size(); ++v) {
            ret[comp_id[v]].push_back(v);
        }
        return ret;
    }
};

#line 4 "graph/two_sat.hpp"

#include <optional>

class TwoSat {
public:
    struct Variable {
    private:
        int idx;
            
        Variable(int i) : idx(i) {}
        
    public:
        Variable operator!() const noexcept {
            return Variable(idx ^ 1);
        }
        
        int index() const {
            return idx / 2;
        }
        
        friend class TwoSat;
    };
    
private:
    std::vector<std::vector<int>> graph;
    
public:
    TwoSat() : graph() {}
    
    TwoSat::Variable add_variable() {
        Variable var(graph.size());
        graph.resize(graph.size() + 2, std::vector<int>());
        return var;
    }
    
    std::vector<TwoSat::Variable> add_variables(int num) {
        std::vector<TwoSat::Variable> vars;
        vars.reserve(num);
        for (int i = 0; i < num; ++i) {
            vars.emplace_back(Variable((int) graph.size() + 2 * i));
        }
        graph.resize((int) graph.size() + 2 * num, std::vector<int>());
        return vars;
    }
    
    // x or y
    void add_clause(Variable x, Variable y) {
        graph[(!x).idx].push_back(y.idx);
        graph[(!y).idx].push_back(x.idx);
    }
    
    // x implies y
    void implies(Variable x, Variable y) {
        add_clause(!x, y);
    }
    
    void at_most_one(const std::vector<Variable> &vars) {
        if (vars.size() <= 1) {
            return;
        }
        std::vector<Variable> sum = add_variables((int) vars.size() - 1);
        for (int i = 0; i < (int) sum.size(); ++i) {
            implies(vars[i], sum[i]);
            implies(sum[i], !vars[i + 1]);
        }
        for (int i = 0; i < (int) sum.size() - 1; ++i) {
            implies(sum[i], sum[i + 1]);
        }
    }
    
    std::optional<std::vector<bool>> solve() {
        StronglyConnectedComponents scc(graph);
        std::vector<bool> ans(graph.size() / 2, false);
        for (int i = 0; i < (int) graph.size() / 2; ++i) {
            if (scc[2 * i] == scc[2 * i + 1]) {
                return std::nullopt;
            }
            if (scc[2 * i] > scc[2 * i + 1]) {
                ans[i] = true;
            }
        }
        return ans;
    }
};

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