2-SAT
(Mylib/Graph/two_sat.cpp)

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• Last update: 2021-04-23 23:44:44+09:00
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Operations

• two_sat(int n)
• add_if(int a, int b)
  • $a \rightarrow b$を導入する。
• add_or(int a, int b)
  • $a \lor b$を導入する。
• not_coexist(int a, int b)
  • $\lnot (a \land b)$を導入する。
• solve()
  • 与えられた論理式をすべてTrueにするような変数の真偽値の割り当てを返す。
  • Time complexity $O(V + E)$

Requirements

• add_if, add_or, not_coexistの引数は[1, n]または[-n, -1]を満たさなければならない。
  • 正数iは$x_i$を、負数-iは$\lnot x_i$を表す。

Notes

Problems

• yukicoder No.274 The Wall
• yukicoder No.470 Inverse S+T Problem
• ARC 069 F - Flags

References

Depends on

• Strongly connected components (Mylib/Graph/GraphUtils/strongly_connected_components.cpp)
• Basic graph (Mylib/Graph/Template/graph.cpp)

Verified with

• test/yosupo-judge/two_sat/main.test.cpp

Code

#pragma once
#include <cassert>
#include <optional>
#include <vector>
#include "Mylib/Graph/GraphUtils/strongly_connected_components.cpp"
#include "Mylib/Graph/Template/graph.cpp"

namespace haar_lib {
  class two_sat {
    int n_;
    graph<int> g_;

    int f(int i) {
      assert(i != 0);
      assert(std::abs(i) <= n_);
      if (i > 0)
        return i - 1;
      else
        return std::abs(i) - 1 + n_;
    }

  public:
    two_sat() {}
    two_sat(int n) : n_(n), g_(2 * n) {}

    /**
     * @note a→bを導入する
     */
    void add_if(int a, int b) {
      g_.add_edge(f(a), f(b), 1);
    }

    /**
     * @note a∨bを導入する
     * @note a ∨ b <=> (!a => b) ∧ (!b => a)
     */
    void add_or(int a, int b) {
      add_if(-a, b);
      add_if(-b, a);
    }

    /**
     * @note ¬(a∧b)を導入する
     * @note !(A ∧ B) <=> (!A ∨ !B)
     */
    void not_coexist(int a, int b) {
      add_or(-a, -b);
    }

  public:
    std::optional<std::vector<bool>> solve() const {
      auto [scc, m] = strongly_connected_components(g_);

      for (int i = 0; i < n_; ++i) {
        if (scc[i] == scc[i + n_]) return std::nullopt;
      }

      std::vector<bool> ret(n_);
      for (int i = 0; i < n_; ++i) ret[i] = scc[i] > scc[i + n_];

      return ret;
    }
  };
}  // namespace haar_lib

#line 2 "Mylib/Graph/two_sat.cpp"
#include <cassert>
#include <optional>
#include <vector>
#line 2 "Mylib/Graph/GraphUtils/strongly_connected_components.cpp"
#include <algorithm>
#line 2 "Mylib/Graph/Template/graph.cpp"
#include <iostream>
#line 4 "Mylib/Graph/Template/graph.cpp"

namespace haar_lib {
  template <typename T>
  struct edge {
    int from, to;
    T cost;
    int index = -1;
    edge() {}
    edge(int from, int to, T cost) : from(from), to(to), cost(cost) {}
    edge(int from, int to, T cost, int index) : from(from), to(to), cost(cost), index(index) {}
  };

  template <typename T>
  struct graph {
    using weight_type = T;
    using edge_type   = edge<T>;

    std::vector<std::vector<edge<T>>> data;

    auto& operator[](size_t i) { return data[i]; }
    const auto& operator[](size_t i) const { return data[i]; }

    auto begin() const { return data.begin(); }
    auto end() const { return data.end(); }

    graph() {}
    graph(int N) : data(N) {}

    bool empty() const { return data.empty(); }
    int size() const { return data.size(); }

    void add_edge(int i, int j, T w, int index = -1) {
      data[i].emplace_back(i, j, w, index);
    }

    void add_undirected(int i, int j, T w, int index = -1) {
      add_edge(i, j, w, index);
      add_edge(j, i, w, index);
    }

    template <size_t I, bool DIRECTED = true, bool WEIGHTED = true>
    void read(int M) {
      for (int i = 0; i < M; ++i) {
        int u, v;
        std::cin >> u >> v;
        u -= I;
        v -= I;
        T w = 1;
        if (WEIGHTED) std::cin >> w;
        if (DIRECTED)
          add_edge(u, v, w, i);
        else
          add_undirected(u, v, w, i);
      }
    }
  };

  template <typename T>
  using tree = graph<T>;
}  // namespace haar_lib
#line 5 "Mylib/Graph/GraphUtils/strongly_connected_components.cpp"

namespace haar_lib {
  template <typename T>
  auto strongly_connected_components(const graph<T> &g) {
    const int n = g.size();

    std::vector<int> ret(n), low(n, -1), ord(n, -1), S;
    std::vector<bool> check(n);
    S.reserve(n);
    int t = 0;
    int k = 0;

    auto dfs =
        [&](auto &dfs, int cur) -> void {
      low[cur] = ord[cur] = t++;
      S.push_back(cur);
      check[cur] = true;

      for (auto &e : g[cur]) {
        if (ord[e.to] == -1) {
          dfs(dfs, e.to);
          low[cur] = std::min(low[cur], low[e.to]);
        } else if (check[e.to]) {
          low[cur] = std::min(low[cur], low[e.to]);
        }
      }

      if (low[cur] == ord[cur]) {
        while (true) {
          int u = S.back();
          S.pop_back();
          check[u] = false;
          ret[u]   = k;
          if (cur == u) break;
        }
        ++k;
      }
    };

    for (int i = 0; i < n; ++i) {
      if (ord[i] == -1) {
        t = 0;
        dfs(dfs, i);
      }
    }

    for (auto &x : ret) x = k - 1 - x;

    return std::make_pair(ret, k);
  }
}  // namespace haar_lib
#line 7 "Mylib/Graph/two_sat.cpp"

namespace haar_lib {
  class two_sat {
    int n_;
    graph<int> g_;

    int f(int i) {
      assert(i != 0);
      assert(std::abs(i) <= n_);
      if (i > 0)
        return i - 1;
      else
        return std::abs(i) - 1 + n_;
    }

  public:
    two_sat() {}
    two_sat(int n) : n_(n), g_(2 * n) {}

    /**
     * @note a→bを導入する
     */
    void add_if(int a, int b) {
      g_.add_edge(f(a), f(b), 1);
    }

    /**
     * @note a∨bを導入する
     * @note a ∨ b <=> (!a => b) ∧ (!b => a)
     */
    void add_or(int a, int b) {
      add_if(-a, b);
      add_if(-b, a);
    }

    /**
     * @note ¬(a∧b)を導入する
     * @note !(A ∧ B) <=> (!A ∨ !B)
     */
    void not_coexist(int a, int b) {
      add_or(-a, -b);
    }

  public:
    std::optional<std::vector<bool>> solve() const {
      auto [scc, m] = strongly_connected_components(g_);

      for (int i = 0; i < n_; ++i) {
        if (scc[i] == scc[i + n_]) return std::nullopt;
      }

      std::vector<bool> ret(n_);
      for (int i = 0; i < n_; ++i) ret[i] = scc[i] > scc[i + n_];

      return ret;
    }
  };
}  // namespace haar_lib

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