Hashiryo's Library

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:question: 一般グラフの最大マッチング (src/Graph/general_matching.hpp)

GabowのEdmonds’ Algorithm

関数名 概要 計算量
generate_matching(Graph g, vector<int> partner = {}) 無向グラフ g の最大マッチングの一例を返す.
引数は Graphクラス.
第二引数は推論補助(※)
返り値は二つの要素を pair でラッピングしたものを返す．
一つ目は最大マッチングに使用する辺の番号の集合を表す vector<int>．
　二つ目は各頂点のマッチング相手が記録 (noマッチなら -1) されている vector<int> ．		 $O(VE \log V)$
はやい

※ 各頂点の（最大マッチングとは限らない）マッチング相手が記録されている vector<int>　( 返り値の二つ目の形式と同じ ) を渡す. マッチングとして矛盾している場合の挙動は未定義．一度この関数を実行した後，（辺を一本追加あるいは削除などの）少しだけ変化させた場合の再計算を効率よくするためのもの．

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Code

#pragma once
#include <cassert>
#include "src/Graph/Graph.hpp"
// {matching edge ids, partner(-1 if unmatched)}
std::pair<std::vector<int>, std::vector<int>> general_matching(const Graph &g, std::vector<int> partner= {}) {
 auto adj= g.adjacency_vertex(0);
 const int n= adj.size();
 std::vector<int> q, z(n), p(n);
 if (partner.empty()) partner.assign(n, -1);
 assert((int)partner.size() == n);
 std::vector<Edge> fs(n);
 auto rematch= [&](auto &&rc, int u, int v) -> void {
  int w= partner[u];
  if (partner[u]= v; w == -1 || partner[w] != u) return;
  if (auto [x, y]= fs[u]; y == -1) rc(rc, partner[w]= x, w);
  else rc(rc, x, y), rc(rc, y, x);
 };
 int t= 1;
 auto f= [&](auto &&rc, int x) -> int { return z[x] != t || p[x] == -1 ? x : (p[x]= rc(rc, p[x])); };
 auto check= [&](int r) {
  q.clear(), q.push_back(r), fs[r]= {-1, -1}, z[r]= t, p[r]= -1;
  for (int i= 0; i < (int)q.size(); ++i) {
   int x= q[i];
   for (int y: adj[x]) {
    if (y == r) continue;
    if (partner[y] == -1) return rematch(rematch, partner[y]= x, y), true;
    if (z[y] == t) {
     int u= f(f, x), v= f(f, y), w= r;
     if (u == v) continue;
     for (; u != r || v != r; fs[u]= {x, y}, u= f(f, fs[partner[u]].first)) {
      if (v != r) std::swap(u, v);
      if (fs[u].first == x && fs[u].second == y) {
       w= u;
       break;
      }
     }
     for (int a: {f(f, x), f(f, y)})
      for (; a != w; a= f(f, fs[partner[a]].first)) z[a]= t, p[a]= w, q.push_back(a);
    } else if (z[partner[y]] != t) fs[y]= {-1, -1}, fs[partner[y]]= {x, -1}, z[partner[y]]= t, p[partner[y]]= y, q.push_back(partner[y]);
   }
  }
  return false;
 };
 for (int r= n; r--;)
  if (partner[r] == -1) t+= check(r);
 q.clear();
 for (int i= 0, e= g.edge_size(); i < e; ++i)
  if (auto [u, v]= g[i]; partner[u] == v && z[u] >= 0) q.push_back(i), z[u]= z[v]= -1;
 return {q, partner};
}
#line 2 "src/Graph/general_matching.hpp"
#include <cassert>
#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 4 "src/Graph/general_matching.hpp"
// {matching edge ids, partner(-1 if unmatched)}
std::pair<std::vector<int>, std::vector<int>> general_matching(const Graph &g, std::vector<int> partner= {}) {
 auto adj= g.adjacency_vertex(0);
 const int n= adj.size();
 std::vector<int> q, z(n), p(n);
 if (partner.empty()) partner.assign(n, -1);
 assert((int)partner.size() == n);
 std::vector<Edge> fs(n);
 auto rematch= [&](auto &&rc, int u, int v) -> void {
  int w= partner[u];
  if (partner[u]= v; w == -1 || partner[w] != u) return;
  if (auto [x, y]= fs[u]; y == -1) rc(rc, partner[w]= x, w);
  else rc(rc, x, y), rc(rc, y, x);
 };
 int t= 1;
 auto f= [&](auto &&rc, int x) -> int { return z[x] != t || p[x] == -1 ? x : (p[x]= rc(rc, p[x])); };
 auto check= [&](int r) {
  q.clear(), q.push_back(r), fs[r]= {-1, -1}, z[r]= t, p[r]= -1;
  for (int i= 0; i < (int)q.size(); ++i) {
   int x= q[i];
   for (int y: adj[x]) {
    if (y == r) continue;
    if (partner[y] == -1) return rematch(rematch, partner[y]= x, y), true;
    if (z[y] == t) {
     int u= f(f, x), v= f(f, y), w= r;
     if (u == v) continue;
     for (; u != r || v != r; fs[u]= {x, y}, u= f(f, fs[partner[u]].first)) {
      if (v != r) std::swap(u, v);
      if (fs[u].first == x && fs[u].second == y) {
       w= u;
       break;
      }
     }
     for (int a: {f(f, x), f(f, y)})
      for (; a != w; a= f(f, fs[partner[a]].first)) z[a]= t, p[a]= w, q.push_back(a);
    } else if (z[partner[y]] != t) fs[y]= {-1, -1}, fs[partner[y]]= {x, -1}, z[partner[y]]= t, p[partner[y]]= y, q.push_back(partner[y]);
   }
  }
  return false;
 };
 for (int r= n; r--;)
  if (partner[r] == -1) t+= check(r);
 q.clear();
 for (int i= 0, e= g.edge_size(); i < e; ++i)
  if (auto [u, v]= g[i]; partner[u] == v && z[u] >= 0) q.push_back(i), z[u]= z[v]= -1;
 return {q, partner};
}
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