二点連結成分分解 (block-cut-tree (拡張)) (src/Graph/block_cut_tree.hpp)

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• Last update: 2024-02-19 15:31:52+09:00
• Include: #include "src/Graph/block_cut_tree.hpp"

関数名	概要	計算量
block_cut_tree(CSRArray<int> adj) block_cut_tree(Graph g)	無向グラフ g を二点連結成分分解して構築した block-cut-tree (拡張) を返す. 引数は頂点 → 頂点の隣接リスト(CSRArray<int>クラス) もしくは Graphクラス で無向グラフを渡す.　 返り値は Graph クラス．	$O(V+E)$

block-cut-tree (拡張) について

参考

頂点数 $N$ の無向グラフ $g$ の二点連結成分の個数を $C$ とする.
block-cut-tree (拡張) は頂点数が $N+C$ の森である.

• $N$ 個の頂点 $0,1,\dots,N-1$ は元のグラフ $g$ の頂点に対応.
  • 隣接頂点は属する二点連結成分を指す.
  • 隣接頂点が複数 $(>1)\rightarrow$ 関節点
• $C$ 個の頂点 $N,N+1,\dots,N+C-1$ は $g$ の二点連結成分に対応.
  • 隣接頂点はその二点連結成分を構成する頂点を指す.

参考

https://twitter.com/noshi91/status/1529858538650374144?s=20&t=eznpFbuD9BDhfTb4PplFUg
https://nachiavivias.github.io/cp-library/column/2022/01.html

Depends on

• グラフ (src/Graph/Graph.hpp)
• CSR 表現を用いた二次元配列 他 (src/Internal/ListRange.hpp)

Verified with

• test/aoj/3022.test.cpp
• test/aoj/GRL_3_A.test.cpp
• test/atcoder/arc062_f.test.cpp
• test/hackerrank/bonnie-and-clyde.test.cpp
• test/yosupo/biconnected_components.test.cpp
• test/yukicoder/1326.test.cpp

Code

#pragma once
#include "src/Graph/Graph.hpp"
// [0,n) : vertex
// [n,n+b) : block
// deg(v) > 1 -> articulation point
Graph block_cut_tree(const CSRArray<int> &adj) {
 const int n= adj.size();
 std::vector<int> ord(n), par(n, -2), dat(adj.p.begin(), adj.p.begin() + n);
 int k= 0;
 for (int s= n, p; s--;)
  if (par[s] == -2)
   for (par[p= s]= -1; p >= 0;) {
    if (dat[p] == adj.p[p]) ord[k++]= p;
    if (dat[p] == adj.p[p + 1]) p= par[p];
    else if (int q= adj.dat[dat[p]++]; par[q] == -2) par[q]= p, p= q;
   }
 for (int i= n; i--;) dat[ord[i]]= i;
 auto low= dat;
 for (int v= n; v--;)
  for (int u: adj[v]) low[v]= std::min(low[v], dat[u]);
 for (int i= n; i--;)
  if (int p= ord[i], pp= par[p]; pp >= 0) low[pp]= std::min(low[pp], low[p]);
 Graph ret(k);
 for (int p: ord)
  if (par[p] >= 0) {
   if (int pp= par[p]; low[p] < dat[pp]) ret.add_edge(low[p]= low[pp], p);
   else ret.add_vertex(), ret.add_edge(k, pp), ret.add_edge(k, p), low[p]= k++;
  }
 for (int s= 0; s < n; ++s)
  if (!adj[s].size()) ret.add_edge(ret.add_vertex(), s);
 return ret;
}
Graph block_cut_tree(const Graph &g) { return block_cut_tree(g.adjacency_vertex(0)); }

#line 2 "src/Internal/ListRange.hpp"
#include <vector>
#include <iostream>
#include <iterator>
#include <type_traits>
#define _LR(name, IT, CT) \
 template <class T> struct name { \
  using Iterator= typename std::vector<T>::IT; \
  Iterator bg, ed; \
  Iterator begin() const { return bg; } \
  Iterator end() const { return ed; } \
  size_t size() const { return std::distance(bg, ed); } \
  CT &operator[](int i) const { return bg[i]; } \
 }
_LR(ListRange, iterator, T);
_LR(ConstListRange, const_iterator, const T);
#undef _LR
template <class T> struct CSRArray {
 std::vector<T> dat;
 std::vector<int> p;
 size_t size() const { return p.size() - 1; }
 ListRange<T> operator[](int i) { return {dat.begin() + p[i], dat.begin() + p[i + 1]}; }
 ConstListRange<T> operator[](int i) const { return {dat.cbegin() + p[i], dat.cbegin() + p[i + 1]}; }
};
template <template <class> class F, class T> std::enable_if_t<std::disjunction_v<std::is_same<F<T>, ListRange<T>>, std::is_same<F<T>, ConstListRange<T>>, std::is_same<F<T>, CSRArray<T>>>, std::ostream &> operator<<(std::ostream &os, const F<T> &r) {
 os << '[';
 for (int _= 0, __= r.size(); _ < __; ++_) os << (_ ? ", " : "") << r[_];
 return os << ']';
}
#line 3 "src/Graph/Graph.hpp"
struct Edge: std::pair<int, int> {
 using std::pair<int, int>::pair;
 Edge &operator--() { return --first, --second, *this; }
 int to(int v) const { return first ^ second ^ v; }
 friend std::istream &operator>>(std::istream &is, Edge &e) { return is >> e.first >> e.second, is; }
};
struct Graph: std::vector<Edge> {
 size_t n;
 Graph(size_t n= 0, size_t m= 0): vector(m), n(n) {}
 size_t vertex_size() const { return n; }
 size_t edge_size() const { return size(); }
 size_t add_vertex() { return n++; }
 size_t add_edge(int s, int d) { return emplace_back(s, d), size() - 1; }
 size_t add_edge(Edge e) { return emplace_back(e), size() - 1; }
#define _ADJ_FOR(a, b) \
 for (auto [u, v]: *this) a; \
 for (size_t i= 0; i < n; ++i) p[i + 1]+= p[i]; \
 for (int i= size(); i--;) { \
  auto [u, v]= (*this)[i]; \
  b; \
 }
#define _ADJ(a, b) \
 vector<int> p(n + 1), c(size() << !dir); \
 if (!dir) { \
  _ADJ_FOR((++p[u], ++p[v]), (c[--p[u]]= a, c[--p[v]]= b)) \
 } else if (dir > 0) { \
  _ADJ_FOR(++p[u], c[--p[u]]= a) \
 } else { \
  _ADJ_FOR(++p[v], c[--p[v]]= b) \
 } \
 return {c, p}
 CSRArray<int> adjacency_vertex(int dir) const { _ADJ(v, u); }
 CSRArray<int> adjacency_edge(int dir) const { _ADJ(i, i); }
#undef _ADJ
#undef _ADJ_FOR
};
#line 3 "src/Graph/block_cut_tree.hpp"
// [0,n) : vertex
// [n,n+b) : block
// deg(v) > 1 -> articulation point
Graph block_cut_tree(const CSRArray<int> &adj) {
 const int n= adj.size();
 std::vector<int> ord(n), par(n, -2), dat(adj.p.begin(), adj.p.begin() + n);
 int k= 0;
 for (int s= n, p; s--;)
  if (par[s] == -2)
   for (par[p= s]= -1; p >= 0;) {
    if (dat[p] == adj.p[p]) ord[k++]= p;
    if (dat[p] == adj.p[p + 1]) p= par[p];
    else if (int q= adj.dat[dat[p]++]; par[q] == -2) par[q]= p, p= q;
   }
 for (int i= n; i--;) dat[ord[i]]= i;
 auto low= dat;
 for (int v= n; v--;)
  for (int u: adj[v]) low[v]= std::min(low[v], dat[u]);
 for (int i= n; i--;)
  if (int p= ord[i], pp= par[p]; pp >= 0) low[pp]= std::min(low[pp], low[p]);
 Graph ret(k);
 for (int p: ord)
  if (par[p] >= 0) {
   if (int pp= par[p]; low[p] < dat[pp]) ret.add_edge(low[p]= low[pp], p);
   else ret.add_vertex(), ret.add_edge(k, pp), ret.add_edge(k, p), low[p]= k++;
  }
 for (int s= 0; s < n; ++s)
  if (!adj[s].size()) ret.add_edge(ret.add_vertex(), s);
 return ret;
}
Graph block_cut_tree(const Graph &g) { return block_cut_tree(g.adjacency_vertex(0)); }

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