二部グラフの辺彩色 (src/Graph/bipartite_edge_coloring.hpp)

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• Last update: 2024-09-20 12:04:43+09:00
• Include: #include "src/Graph/bipartite_edge_coloring.hpp"

関数名	概要	計算量
bipartite_edge_coloring(bg)	二部グラフの辺彩色を構築する． 引数は BipartiteGraph クラス． 戻り値は辺のサイズの vector<int> で各辺への色の割り当てを表す．	$O(E\sqrt{V}\log \Delta)$ ただし頂点の次数のうち最大のものを $\Delta$ とおいた．

Verify

• AtCoder Grand Contest 037 D - Sorting a Grid (sp judge)

参考

https://ei1333.hateblo.jp/entry/2020/08/25/015955

Depends on

• Union-Find (src/DataStructure/UnionFind.hpp)
• 二部グラフ (src/Graph/BipartiteGraph.hpp)
• 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/yosupo/bipartite_edge_coloring.test.cpp

Code

#pragma once
#include <queue>
#include <numeric>
#include "src/DataStructure/UnionFind.hpp"
#include "src/Graph/BipartiteGraph.hpp"
std::vector<int> bipartite_edge_coloring(BipartiteGraph bg) {
 const int m= bg.edge_size();
 int L= bg.left_size(), n= bg.vertex_size(), D, col= 0;
 {
  std::vector<int> deg(n), id(n);
  for (auto [l, r]: bg) ++deg[l], ++deg[r];
  D= *std::max_element(deg.begin(), deg.end());
  UnionFind uf(n);
  for (int _: {0, n}) {
   auto [b, e]= std::minmax(_, L);
   std::priority_queue<std::pair<int, int>> pq;
   for (int i= b; i < e; ++i) pq.emplace(-deg[i], i);
   for (; pq.size() > 1;) {
    auto [a, v]= pq.top();
    pq.pop();
    auto [b, u]= pq.top();
    pq.pop();
    if (int sum= a + b; -sum <= D) uf.unite(v, u), pq.emplace(sum, v);
    else break;
   }
  }
  int i= 0, cl= 0, cr= 0;
  for (; i < L; ++i)
   if (uf.leader(i) == i) id[i]= cl++;
  for (; i < n; ++i)
   if (uf.leader(i) == i) id[i]= cr++;
  L= std::max(cl, cr), deg.assign(n= L + L, 0), bg.reserve(L * D);
  for (auto &[l, r]: bg) ++deg[l= id[uf.leader(l)]], ++deg[r= id[uf.leader(r)] + L];
  for (int l= 0, r= L; l < L; ++l)
   while (deg[l] < D) {
    while (r < n && deg[r] == D) ++r;
    int x= D - std::max(deg[l], deg[r]);
    for (int k= x; k--;) bg.add_edge(l, r);
    deg[l]+= x, deg[r]+= x;
   }
 }
 std::vector<int> color(m, -1);
 auto rc= [&](auto &&rc, int d, const std::vector<int> &idx) -> void {
  if (!d) return;
  if (d == 1) {
   for (int e: idx)
    if (e < m) color[e]= col;
   ++col;
   return;
  }
  if (d & 1) {
   CSRArray<int> adj{std::vector<int>(idx.size()), std::vector<int>(L + 1)};
   for (int e: idx) ++adj.p[bg[e].first];
   for (int i= 0; i < L; ++i) adj.p[i + 1]+= adj.p[i];
   for (int e: idx) {
    auto [l, r]= bg[e];
    adj.dat[--adj.p[l]]= r;
   }
   std::vector<int> mate(n, -1), rm;
   _bg_internal::_bm(L, adj, mate);
   for (int e: idx) {
    auto [l, r]= bg[e];
    if (mate[l] == r) {
     if (mate[l]= mate[r]= -1; e < m) color[e]= col;
    } else rm.push_back(e);
   }
   return ++col, rc(rc, d - 1, rm);
  }
  const int mm= idx.size();
  std::vector<int> circuit;
  {
   std::vector<int> c(mm * 2), p(n + 1);
   for (int e: idx) {
    auto [l, r]= bg[e];
    ++p[l], ++p[r];
   }
   for (int i= 0; i < n; ++i) p[i + 1]+= p[i];
   for (int i= mm; i--;) {
    auto [l, r]= bg[idx[i]];
    c[--p[l]]= i, c[--p[r]]= i;
   }
   std::vector<int> it(p.begin(), p.begin() + n);
   std::vector<char> used1(n), used2(mm);
   for (int v= n; v--;)
    if (!used1[v]) {
     for (std::vector<std::pair<int, int>> st= {{v, -1}}; st.size();) {
      auto [u, e]= st.back();
      if (used1[u]= 1; it[u] == p[u + 1]) circuit.push_back(e), st.pop_back();
      else if (int i= c[it[u]++]; !used2[i]) used2[i]= 1, st.emplace_back(bg[idx[i]].to(u), i);
     }
     circuit.pop_back();
    }
  }
  std::vector<int> half1(mm / 2), half2(mm / 2);
  for (int i= mm / 2; i--;) half1[i]= idx[circuit[i * 2]], half2[i]= idx[circuit[i * 2 + 1]];
  rc(rc, d / 2, half1), rc(rc, d / 2, half2);
 };
 std::vector<int> idx(L * D);
 return std::iota(idx.begin(), idx.end(), 0), rc(rc, D, idx), color;
}

#line 2 "src/Graph/bipartite_edge_coloring.hpp"
#include <queue>
#include <numeric>
#line 2 "src/DataStructure/UnionFind.hpp"
#include <vector>
#include <algorithm>
class UnionFind {
 std::vector<int> par;
public:
 UnionFind(int n): par(n, -1) {}
 int leader(int u) { return par[u] < 0 ? u : par[u]= leader(par[u]); }
 bool unite(int u, int v) {
  if ((u= leader(u)) == (v= leader(v))) return false;
  if (par[u] > par[v]) std::swap(u, v);
  return par[u]+= par[v], par[v]= u, true;
 }
 bool connected(int u, int v) { return leader(u) == leader(v); }
 int size(int u) { return -par[leader(u)]; }
};
#line 2 "src/Graph/BipartiteGraph.hpp"
#include <cassert>
#include <tuple>
#line 3 "src/Internal/ListRange.hpp"
#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 6 "src/Graph/BipartiteGraph.hpp"
// [0, L) is left, [L, n) is right
struct BipartiteGraph: Graph {
 size_t L;
 BipartiteGraph() {}
 BipartiteGraph(size_t L, size_t R, size_t m= 0): Graph(L + R, m), L(L) {}
 size_t left_size() const { return L; }
 size_t right_size() const { return this->n - L; }
};
std::vector<int> paint_two_colors(const CSRArray<int> &adj) {
 const int n= adj.size();
 std::vector<int> col(n, -1);
 for (int s= n; s--;)
  if (col[s] == -1) {
   std::vector<int> q= {s};
   for (int i= col[s]= 0, v; i < (int)q.size(); ++i)
    for (int u: adj[v= q[i]])
     if (int c= col[v]; col[u] == c) return {};
     else if (col[u] == -1) col[u]= c ^ 1, q.push_back(u);
  }
 return col;
}
std::vector<int> paint_two_colors(const Graph &g) { return paint_two_colors(g.adjacency_vertex(0)); }
// { BipartiteGraph , original to new, new to original }
// {{},{},{}} if not bipartite
std::tuple<BipartiteGraph, std::vector<int>, std::vector<int>> graph_to_bipartite(const Graph &g, std::vector<int> color= {}) {
 if (color.empty()) color= paint_two_colors(g);
 if (color.empty()) return {};
 const int n= g.vertex_size(), m= g.edge_size();
 std::vector<int> a(n), b(n);
 int l= 0, r= n;
 for (int i= n; i--;) b[a[i]= color[i] ? --r : l++]= i;
 BipartiteGraph bg(l, n - l, m);
 for (int i= m; i--;) {
  auto [u, v]= g[i];
  bg[i]= std::minmax(a[u], a[v]);
 }
 return {bg, a, b};
}
namespace _bg_internal {
std::vector<int> _bm(int L, const CSRArray<int> &adj, std::vector<int> &m) {
 std::vector<int> a, p, q(L);
 for (bool u= true; u;) {
  u= false, a.assign(L, -1), p.assign(L, -1);
  int t= 0;
  for (int l= L; l--;)
   if (m[l] == -1) q[t++]= a[l]= p[l]= l;
  for (int i= 0; i < t; ++i)
   if (int l= q[i], x; m[a[l]] == -1)
    for (int r: adj[l]) {
     if (x= m[r]; x == -1) {
      for (u= true; r != -1; l= p[l]) m[r]= l, std::swap(m[l], r);
      break;
     }
     if (p[x] == -1) a[q[t++]= x]= a[p[x]= l];
    }
 }
 return a;
}
}
template <bool lexical= false> std::pair<std::vector<int>, std::vector<int>> bipartite_matching(const BipartiteGraph &bg, std::vector<int> partner= {}) {
 const int L= bg.left_size(), M= bg.edge_size();
 if (partner.empty()) partner.assign(bg.vertex_size(), -1);
 assert(partner.size() == bg.vertex_size());
 {
  CSRArray<int> adj{std::vector<int>(M), std::vector<int>(L + 1)};
  for (auto [l, r]: bg) ++adj.p[l];
  for (int i= 0; i < L; ++i) adj.p[i + 1]+= adj.p[i];
  for (auto [l, r]: bg) adj.dat[--adj.p[l]]= r;
  if constexpr (lexical) {
   for (int l= L; l--;) std::sort(adj[l].begin(), adj[l].end());
   _bg_internal::_bm(L, adj, partner);
   std::vector<char> a(L, 1);
   for (int l= 0; l < L; ++l)
    if (int r= partner[l], v= l; r != -1) {
     std::vector<int> p(L, partner[v]= partner[r]= -1), c(adj.p.begin(), adj.p.begin() + L);
     for (p[v]= -2;;) {
      if (c[v] == adj.p[v + 1]) v= p[v];
      else if (int u= partner[r= adj.dat[c[v]++]]; u == -1) {
       for (; r != -1; v= p[v]) partner[r]= v, std::swap(partner[v], r);
       break;
      } else if (a[u] && p[u] == -1) p[u]= v, v= u;
     }
     a[l]= 0;
    }
  } else _bg_internal::_bm(L, adj, partner);
 }
 std::vector<int> c;
 std::vector<char> p(L);
 for (int i= 0; i < M; ++i)
  if (auto [l, r]= bg[i]; partner[l] == r && !p[l]) c.push_back(i), p[l]= 1;
 return {c, partner};
}
#line 6 "src/Graph/bipartite_edge_coloring.hpp"
std::vector<int> bipartite_edge_coloring(BipartiteGraph bg) {
 const int m= bg.edge_size();
 int L= bg.left_size(), n= bg.vertex_size(), D, col= 0;
 {
  std::vector<int> deg(n), id(n);
  for (auto [l, r]: bg) ++deg[l], ++deg[r];
  D= *std::max_element(deg.begin(), deg.end());
  UnionFind uf(n);
  for (int _: {0, n}) {
   auto [b, e]= std::minmax(_, L);
   std::priority_queue<std::pair<int, int>> pq;
   for (int i= b; i < e; ++i) pq.emplace(-deg[i], i);
   for (; pq.size() > 1;) {
    auto [a, v]= pq.top();
    pq.pop();
    auto [b, u]= pq.top();
    pq.pop();
    if (int sum= a + b; -sum <= D) uf.unite(v, u), pq.emplace(sum, v);
    else break;
   }
  }
  int i= 0, cl= 0, cr= 0;
  for (; i < L; ++i)
   if (uf.leader(i) == i) id[i]= cl++;
  for (; i < n; ++i)
   if (uf.leader(i) == i) id[i]= cr++;
  L= std::max(cl, cr), deg.assign(n= L + L, 0), bg.reserve(L * D);
  for (auto &[l, r]: bg) ++deg[l= id[uf.leader(l)]], ++deg[r= id[uf.leader(r)] + L];
  for (int l= 0, r= L; l < L; ++l)
   while (deg[l] < D) {
    while (r < n && deg[r] == D) ++r;
    int x= D - std::max(deg[l], deg[r]);
    for (int k= x; k--;) bg.add_edge(l, r);
    deg[l]+= x, deg[r]+= x;
   }
 }
 std::vector<int> color(m, -1);
 auto rc= [&](auto &&rc, int d, const std::vector<int> &idx) -> void {
  if (!d) return;
  if (d == 1) {
   for (int e: idx)
    if (e < m) color[e]= col;
   ++col;
   return;
  }
  if (d & 1) {
   CSRArray<int> adj{std::vector<int>(idx.size()), std::vector<int>(L + 1)};
   for (int e: idx) ++adj.p[bg[e].first];
   for (int i= 0; i < L; ++i) adj.p[i + 1]+= adj.p[i];
   for (int e: idx) {
    auto [l, r]= bg[e];
    adj.dat[--adj.p[l]]= r;
   }
   std::vector<int> mate(n, -1), rm;
   _bg_internal::_bm(L, adj, mate);
   for (int e: idx) {
    auto [l, r]= bg[e];
    if (mate[l] == r) {
     if (mate[l]= mate[r]= -1; e < m) color[e]= col;
    } else rm.push_back(e);
   }
   return ++col, rc(rc, d - 1, rm);
  }
  const int mm= idx.size();
  std::vector<int> circuit;
  {
   std::vector<int> c(mm * 2), p(n + 1);
   for (int e: idx) {
    auto [l, r]= bg[e];
    ++p[l], ++p[r];
   }
   for (int i= 0; i < n; ++i) p[i + 1]+= p[i];
   for (int i= mm; i--;) {
    auto [l, r]= bg[idx[i]];
    c[--p[l]]= i, c[--p[r]]= i;
   }
   std::vector<int> it(p.begin(), p.begin() + n);
   std::vector<char> used1(n), used2(mm);
   for (int v= n; v--;)
    if (!used1[v]) {
     for (std::vector<std::pair<int, int>> st= {{v, -1}}; st.size();) {
      auto [u, e]= st.back();
      if (used1[u]= 1; it[u] == p[u + 1]) circuit.push_back(e), st.pop_back();
      else if (int i= c[it[u]++]; !used2[i]) used2[i]= 1, st.emplace_back(bg[idx[i]].to(u), i);
     }
     circuit.pop_back();
    }
  }
  std::vector<int> half1(mm / 2), half2(mm / 2);
  for (int i= mm / 2; i--;) half1[i]= idx[circuit[i * 2]], half2[i]= idx[circuit[i * 2 + 1]];
  rc(rc, d / 2, half1), rc(rc, d / 2, half2);
 };
 std::vector<int> idx(L * D);
 return std::iota(idx.begin(), idx.end(), 0), rc(rc, D, idx), color;
}

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