Centroid(木の重心)
(graph/tree/centroid.hpp)

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• Last update: 2022-09-11 00:53:50+09:00
• Include: #include "graph/tree/centroid.hpp"

概要

木の重心を求める. その頂点を取り除いたときにできる部分木たちの頂点数がすべて半分以下になる頂点を木の重心と呼び, 重心は $1$ 個か $2$ 個存在する.

• centroid(g): 木 g の重心となる頂点をすべて返す.

計算量

• $O(V)$

Depends on

• Graph Template(グラフテンプレート) (graph/graph-template.hpp)

Required by

• Tree-Isomorphism(木の同型性判定) (graph/tree/tree-isomorphism.hpp)

Verified with

• test/verify/aoj-2821.test.cpp

Code

#pragma once

#include "../graph-template.hpp"

/**
 * @brief Centroid(木の重心)
 * @docs docs/centroid.md
 */
template< typename T >
vector< int > centroid(const Graph< T > &g) {
  const int N = (int) g.size();

  stack< pair< int, int > > st;
  st.emplace(0, -1);
  vector< int > sz(N), par(N);
  while(!st.empty()) {
    auto p = st.top();
    if(sz[p.first] == 0) {
      sz[p.first] = 1;
      for(auto &to : g[p.first]) if(to != p.second) st.emplace(to, p.first);
    } else {
      for(auto &to : g[p.first]) if(to != p.second) sz[p.first] += sz[to];
      par[p.first] = p.second;
      st.pop();
    }
  }

  vector< int > ret;
  int size = N;
  for(int i = 0; i < N; i++) {
    int val = N - sz[i];
    for(auto &to : g[i]) if(to != par[i]) val = max(val, sz[to]);
    if(val < size) size = val, ret.clear();
    if(val == size) ret.emplace_back(i);
  }

  return ret;
}

#line 2 "graph/tree/centroid.hpp"

#line 2 "graph/graph-template.hpp"

/**
 * @brief Graph Template(グラフテンプレート)
 */
template< typename T = int >
struct Edge {
  int from, to;
  T cost;
  int idx;

  Edge() = default;

  Edge(int from, int to, T cost = 1, int idx = -1) : from(from), to(to), cost(cost), idx(idx) {}

  operator int() const { return to; }
};

template< typename T = int >
struct Graph {
  vector< vector< Edge< T > > > g;
  int es;

  Graph() = default;

  explicit Graph(int n) : g(n), es(0) {}

  size_t size() const {
    return g.size();
  }

  void add_directed_edge(int from, int to, T cost = 1) {
    g[from].emplace_back(from, to, cost, es++);
  }

  void add_edge(int from, int to, T cost = 1) {
    g[from].emplace_back(from, to, cost, es);
    g[to].emplace_back(to, from, cost, es++);
  }

  void read(int M, int padding = -1, bool weighted = false, bool directed = false) {
    for(int i = 0; i < M; i++) {
      int a, b;
      cin >> a >> b;
      a += padding;
      b += padding;
      T c = T(1);
      if(weighted) cin >> c;
      if(directed) add_directed_edge(a, b, c);
      else add_edge(a, b, c);
    }
  }

  inline vector< Edge< T > > &operator[](const int &k) {
    return g[k];
  }

  inline const vector< Edge< T > > &operator[](const int &k) const {
    return g[k];
  }
};

template< typename T = int >
using Edges = vector< Edge< T > >;
#line 4 "graph/tree/centroid.hpp"

/**
 * @brief Centroid(木の重心)
 * @docs docs/centroid.md
 */
template< typename T >
vector< int > centroid(const Graph< T > &g) {
  const int N = (int) g.size();

  stack< pair< int, int > > st;
  st.emplace(0, -1);
  vector< int > sz(N), par(N);
  while(!st.empty()) {
    auto p = st.top();
    if(sz[p.first] == 0) {
      sz[p.first] = 1;
      for(auto &to : g[p.first]) if(to != p.second) st.emplace(to, p.first);
    } else {
      for(auto &to : g[p.first]) if(to != p.second) sz[p.first] += sz[to];
      par[p.first] = p.second;
      st.pop();
    }
  }

  vector< int > ret;
  int size = N;
  for(int i = 0; i < N; i++) {
    int val = N - sz[i];
    for(auto &to : g[i]) if(to != par[i]) val = max(val, sz[to]);
    if(val < size) size = val, ret.clear();
    if(val == size) ret.emplace_back(i);
  }

  return ret;
}

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