Two Edge Connected Components(二重辺連結成分分解) (graph/connected-components/two-edge-connected-components.hpp)

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Last update: 2024-05-01 23:44:47+09:00
Include: #include "graph/connected-components/two-edge-connected-components.hpp"

概要

二辺連結成分分解とも. 二重辺連結成分とは, $1$ 本の辺を取り除いても連結である部分グラフである. つまり, 橋を含まない部分グラフなので, 橋を列挙することで二重辺連結成分を列挙できる.

二重辺連結成分で縮約後の頂点と橋からなるグラフは森になっている.

build(): 二重辺連結成分分解する. tree には縮約後の頂点からなる森が格納される. comp には各頂点が属する二重辺連結成分の頂点番号が格納される. group には各二重辺連結成分について, それに属する頂点が格納される.

計算量

$O(E + V)$

Depends on

Verified with

test/verify/yosupo-two-edge-connected-components.test.cpp

Code

#pragma once

#include "../graph-template.hpp"
#include "../others/low-link.hpp"

template <typename T = int>
struct TwoEdgeConnectedComponents : LowLink<T> {
 public:
  using LowLink<T>::LowLink;
  using LowLink<T>::g;
  using LowLink<T>::ord;
  using LowLink<T>::low;
  using LowLink<T>::bridge;

  vector<int> comp;
  Graph<T> tree;
  vector<vector<int> > group;

  int operator[](const int &k) const { return comp[k]; }

  void build() override {
    LowLink<T>::build();
    comp.assign(g.size(), -1);
    int k = 0;
    for (int i = 0; i < (int)comp.size(); i++) {
      if (comp[i] == -1) dfs(i, -1, k);
    }
    group.resize(k);
    for (int i = 0; i < (int)g.size(); i++) {
      group[comp[i]].emplace_back(i);
    }
    tree = Graph<T>(k);
    for (auto &e : bridge) {
      tree.add_edge(comp[e.from], comp[e.to], e.cost);
    }
  }

  explicit TwoEdgeConnectedComponents(const Graph<T> &g) : Graph<T>(g) {}

 private:
  void dfs(int idx, int par, int &k) {
    if (par >= 0 && ord[par] >= low[idx])
      comp[idx] = comp[par];
    else
      comp[idx] = k++;
    for (auto &to : g[idx]) {
      if (comp[to] == -1) dfs(to, idx, k);
    }
  }
};

#line 2 "graph/connected-components/two-edge-connected-components.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 2 "graph/others/low-link.hpp"

#line 4 "graph/others/low-link.hpp"

/**
 * @brief Low Link(橋/関節点)
 * @see http://kagamiz.hatenablog.com/entry/2013/10/05/005213
 *
 */
template <typename T = int>
struct LowLink : Graph<T> {
 public:
  using Graph<T>::Graph;
  vector<int> ord, low, articulation;
  vector<Edge<T> > bridge;
  using Graph<T>::g;

  virtual void build() {
    used.assign(g.size(), 0);
    ord.assign(g.size(), 0);
    low.assign(g.size(), 0);
    int k = 0;
    for (int i = 0; i < (int)g.size(); i++) {
      if (!used[i]) k = dfs(i, k, -1);
    }
  }

  explicit LowLink(const Graph<T> &g) : Graph<T>(g) {}

 private:
  vector<int> used;

  int dfs(int idx, int k, int par) {
    used[idx] = true;
    ord[idx] = k++;
    low[idx] = ord[idx];
    bool is_articulation = false, beet = false;
    int cnt = 0;
    for (auto &to : g[idx]) {
      if (to == par && !exchange(beet, true)) {
        continue;
      }
      if (!used[to]) {
        ++cnt;
        k = dfs(to, k, idx);
        low[idx] = min(low[idx], low[to]);
        is_articulation |= par >= 0 && low[to] >= ord[idx];
        if (ord[idx] < low[to]) bridge.emplace_back(to);
      } else {
        low[idx] = min(low[idx], ord[to]);
      }
    }
    is_articulation |= par == -1 && cnt > 1;
    if (is_articulation) articulation.push_back(idx);
    return k;
  }
};
#line 5 "graph/connected-components/two-edge-connected-components.hpp"

template <typename T = int>
struct TwoEdgeConnectedComponents : LowLink<T> {
 public:
  using LowLink<T>::LowLink;
  using LowLink<T>::g;
  using LowLink<T>::ord;
  using LowLink<T>::low;
  using LowLink<T>::bridge;

  vector<int> comp;
  Graph<T> tree;
  vector<vector<int> > group;

  int operator[](const int &k) const { return comp[k]; }

  void build() override {
    LowLink<T>::build();
    comp.assign(g.size(), -1);
    int k = 0;
    for (int i = 0; i < (int)comp.size(); i++) {
      if (comp[i] == -1) dfs(i, -1, k);
    }
    group.resize(k);
    for (int i = 0; i < (int)g.size(); i++) {
      group[comp[i]].emplace_back(i);
    }
    tree = Graph<T>(k);
    for (auto &e : bridge) {
      tree.add_edge(comp[e.from], comp[e.to], e.cost);
    }
  }

  explicit TwoEdgeConnectedComponents(const Graph<T> &g) : Graph<T>(g) {}

 private:
  void dfs(int idx, int par, int &k) {
    if (par >= 0 && ord[par] >= low[idx])
      comp[idx] = comp[par];
    else
      comp[idx] = k++;
    for (auto &to : g[idx]) {
      if (comp[to] == -1) dfs(to, idx, k);
    }
  }
};