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重心分解(Centroid-Path-Decomposition)

説明

木を重心分解する。結構癖のある書き方みたい。

計算量

$O(V + E)$

実装例

struct CentroidPathDecomposition
{
  struct Centroid
  {
    int ParIndex, ParDepth, Deep;
    vector< int > node;
    inline int size()
    {
      return(node.size());
    }
    inline int &operator[](int k)
    {
      return(node[k]);
    }
    inline pair< int, int > Up()
    {
      return(make_pair(ParIndex, ParDepth));
    }
  };
    
  vector< int > SubTreeSize, NextPath;
  vector< int > TreeIndex, TreeDepth;
  vector< Centroid > Centroids;
    
  void BuildSubTreeSize()
  {
    stack< pair< int, int > > s;
    s.push({0, -1});
    while(!s.empty()) {
      auto p = s.top(); s.pop();
      if(~SubTreeSize[p.first]) {
        NextPath[p.first] = -1;
        for(auto& to : graph[p.first]) {
          if(p.second == to) continue;
          SubTreeSize[p.first] += SubTreeSize[to];
          if(NextPath[p.first] == -1 || SubTreeSize[NextPath[p.first]] < SubTreeSize[to]) {
            NextPath[p.first] = to;
          }
        }
      } else {
        s.push(p);
        SubTreeSize[p.first] = 1;
        for(auto& to : graph[p.first]) {
          if(p.second != to) s.push({to, p.first});
        }
      }
    }
  }
  void BuildPath()
  {
    stack< pair< int, int > > s;
    Centroids.push_back((Centroid){-1, -1, 0});
    s.push({0, -1});
    TreeIndex[0] = 0;
    while(!s.empty()) {
      auto p = s.top(); s.pop();
      TreeDepth[p.first] = Centroids[TreeIndex[p.first]].size();
      for(auto& to : graph[p.first]) {
        if(p.second != to) {
          if(to == NextPath[p.first]) { // Centroid-Path
            TreeIndex[to] = TreeIndex[p.first];
          } else {                  // Not Centroid-Path
            TreeIndex[to] = Centroids.size();
            Centroids.push_back((Centroid){TreeIndex[p.first], TreeDepth[p.first], Centroids[TreeIndex[p.first]].Deep + 1});
          }
          s.push({to, p.first});
        }
      }
      Centroids[TreeIndex[p.first]].node.push_back(p.first);
    }
  }
  void AddEdge(int x, int y)
  {
    graph[x].push_back(y);
    graph[y].push_back(x);
  }
  void Build()
  {
    BuildSubTreeSize();
    BuildPath();
  }
    
  inline int size()
  {
    return(Centroids.size());
  }
  inline pair< int, int > Information(int idx)
  {
    return(make_pair(TreeIndex[idx], TreeDepth[idx]));
  }
  inline Centroid &operator[](int k)
  {
    return(Centroids[k]);
  }
  inline int LCA(int a, int b)
  {
    int TreeIdxA, TreeDepthA, TreeIdxB, TreeDepthB;
    tie(TreeIdxA, TreeDepthA) = Information(a);
    tie(TreeIdxB, TreeDepthB) = Information(b);
    while(TreeIdxA != TreeIdxB) {
      if(Centroids[TreeIdxA].Deep > Centroids[TreeIdxB].Deep) {
        tie(TreeIdxA, TreeDepthA) = Centroids[TreeIdxA].Up();
      } else {
        tie(TreeIdxB, TreeDepthB) = Centroids[TreeIdxB].Up();
      }
    }
    if(TreeDepthA > TreeDepthB) swap(TreeDepthA, TreeDepthB);
    return(Centroids[TreeIdxA][TreeDepthA]);
  }
 
  CentroidPathDecomposition(int SZ)
  {
    graph.resize(SZ);
    SubTreeSize.assign(SZ, -1);
    NextPath.resize(SZ);
    TreeIndex.resize(SZ);
    TreeDepth.resize(SZ);
  }
};

問題例