topcoder srm 630 div1 (2-SAT and SCC template)

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;
};