graph/maximum_matching.hpp

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• Last update: 2024-05-29 22:32:29+09:00
• Include: #include "graph/maximum_matching.hpp"

Depends on

• graph/base.hpp

Verified with

• test/1_mytest/matching.test.cpp
• test/2_library_checker/graph/general_matching.test.cpp

Code

#include "graph/base.hpp"

// return : (match size, match)
// match[v] : マッチング相手 OR 0
// O(N^3)
// 「組合せ最適化」第2版, アルゴリズム 10.2
template <typename GT>
pair<int, vc<int>> maximum_matching(const GT& G) {
  const int N = G.N;
  vc<int> mu(N), phi(N), rho(N);
  vc<bool> scanned(N);
  FOR(v, N) mu[v] = v;
  ll ans = 0;
  for (auto&& e: G.edges) {
    if (e.frm != e.to && mu[e.frm] == e.frm && mu[e.to] == e.to) {
      mu[e.frm] = e.to, mu[e.to] = e.frm, ++ans;
    }
  }

  auto odd = [&](int x) -> bool {
    return mu[x] != x && phi[mu[x]] == mu[x] && mu[x] != x;
  };
  auto out_of_forest = [&](int x) -> bool {
    return mu[x] != x && phi[mu[x]] == mu[x] && phi[x] == x;
  };
  auto P = [&](int x) -> vc<int> {
    vc<int> P;
    P.eb(x);
    while (mu[x] != x) {
      P.eb(mu[x]);
      P.eb(phi[mu[x]]);
      x = phi[mu[x]];
    }
    return P;
  };

  vc<bool> on_path(N);
  while (1) {
    FOR(v, N) phi[v] = rho[v] = v, scanned[v] = 0;
    bool aug = 0;
    while (1) {
      bool upd = 0;
      FOR(x, N) {
        if (upd) break;
        if (scanned[x] || odd(x)) continue;
        for (auto&& e: G[x]) {
          int y = e.to;
          if (out_of_forest(y)) {
            upd = 1;
            // grow
            phi[y] = x;
          }
          elif (rho[y] != rho[x] && !odd(y)) {
            vc<int> F;
            FOR(v, N) if (!out_of_forest(v)) F.eb(v);
            upd = 1;
            // augument OR shrink
            vc<int> Px = P(x);
            vc<int> Py = P(y);
            if (Px.back() != Py.back()) {
              aug = 1;
              // augument
              FOR(2) {
                swap(Px, Py);
                for (int i = 1; i < len(Px); i += 2) {
                  int v = Px[i];
                  mu[phi[v]] = v, mu[v] = phi[v];
                }
              }
              mu[x] = y, mu[y] = x, ++ans;
              break;
            } else {
              // shrink
              int r = -1;
              int Nx = len(Px), Ny = len(Py);
              for (int i = 0; i < Nx; i += 2) {
                int v = Px[i];
                int j = i + Ny - Nx;
                if (0 <= j && j < Ny && Py[j] == v && rho[v] == v) {
                  r = v;
                  break;
                }
              }
              while (Px.back() != r) Px.pop_back();
              while (Py.back() != r) Py.pop_back();
              vc<int> change;
              FOR(2) {
                swap(Px, Py);
                for (int i = 1; i < len(Px); i += 2) {
                  int v = Px[i];
                  if (rho[phi[v]] != r) change.eb(v);
                }
              }
              for (auto&& v: change) phi[phi[v]] = v;

              if (rho[x] != r) phi[x] = y;
              if (rho[y] != r) phi[y] = x;
              for (auto&& v: Px) on_path[v] = 1;
              for (auto&& v: Py) on_path[v] = 1;
              FOR(v, N) if (on_path[rho[v]]) { rho[v] = r; }
              fill(all(on_path), 0);
            }
          }
        }
        scanned[x] = 1;
      }
      if (!upd || aug) break;
    }
    if (!aug) break;
  }
  FOR(v, N) if (mu[v] == v) mu[v] = -1;
  return {ans, mu};
}

#line 2 "graph/base.hpp"

template <typename T>
struct Edge {
  int frm, to;
  T cost;
  int id;
};

template <typename T = int, bool directed = false>
struct Graph {
  static constexpr bool is_directed = directed;
  int N, M;
  using cost_type = T;
  using edge_type = Edge<T>;
  vector<edge_type> edges;
  vector<int> indptr;
  vector<edge_type> csr_edges;
  vc<int> vc_deg, vc_indeg, vc_outdeg;
  bool prepared;

  class OutgoingEdges {
  public:
    OutgoingEdges(const Graph* G, int l, int r) : G(G), l(l), r(r) {}

    const edge_type* begin() const {
      if (l == r) { return 0; }
      return &G->csr_edges[l];
    }

    const edge_type* end() const {
      if (l == r) { return 0; }
      return &G->csr_edges[r];
    }

  private:
    const Graph* G;
    int l, r;
  };

  bool is_prepared() { return prepared; }

  Graph() : N(0), M(0), prepared(0) {}
  Graph(int N) : N(N), M(0), prepared(0) {}

  void build(int n) {
    N = n, M = 0;
    prepared = 0;
    edges.clear();
    indptr.clear();
    csr_edges.clear();
    vc_deg.clear();
    vc_indeg.clear();
    vc_outdeg.clear();
  }

  void add(int frm, int to, T cost = 1, int i = -1) {
    assert(!prepared);
    assert(0 <= frm && 0 <= to && to < N);
    if (i == -1) i = M;
    auto e = edge_type({frm, to, cost, i});
    edges.eb(e);
    ++M;
  }

#ifdef FASTIO
  // wt, off
  void read_tree(bool wt = false, int off = 1) { read_graph(N - 1, wt, off); }

  void read_graph(int M, bool wt = false, int off = 1) {
    for (int m = 0; m < M; ++m) {
      INT(a, b);
      a -= off, b -= off;
      if (!wt) {
        add(a, b);
      } else {
        T c;
        read(c);
        add(a, b, c);
      }
    }
    build();
  }
#endif

  void build() {
    assert(!prepared);
    prepared = true;
    indptr.assign(N + 1, 0);
    for (auto&& e: edges) {
      indptr[e.frm + 1]++;
      if (!directed) indptr[e.to + 1]++;
    }
    for (int v = 0; v < N; ++v) { indptr[v + 1] += indptr[v]; }
    auto counter = indptr;
    csr_edges.resize(indptr.back() + 1);
    for (auto&& e: edges) {
      csr_edges[counter[e.frm]++] = e;
      if (!directed)
        csr_edges[counter[e.to]++] = edge_type({e.to, e.frm, e.cost, e.id});
    }
  }

  OutgoingEdges operator[](int v) const {
    assert(prepared);
    return {this, indptr[v], indptr[v + 1]};
  }

  vc<int> deg_array() {
    if (vc_deg.empty()) calc_deg();
    return vc_deg;
  }

  pair<vc<int>, vc<int>> deg_array_inout() {
    if (vc_indeg.empty()) calc_deg_inout();
    return {vc_indeg, vc_outdeg};
  }

  int deg(int v) {
    if (vc_deg.empty()) calc_deg();
    return vc_deg[v];
  }

  int in_deg(int v) {
    if (vc_indeg.empty()) calc_deg_inout();
    return vc_indeg[v];
  }

  int out_deg(int v) {
    if (vc_outdeg.empty()) calc_deg_inout();
    return vc_outdeg[v];
  }

#ifdef FASTIO
  void debug() {
    print("Graph");
    if (!prepared) {
      print("frm to cost id");
      for (auto&& e: edges) print(e.frm, e.to, e.cost, e.id);
    } else {
      print("indptr", indptr);
      print("frm to cost id");
      FOR(v, N) for (auto&& e: (*this)[v]) print(e.frm, e.to, e.cost, e.id);
    }
  }
#endif

  vc<int> new_idx;
  vc<bool> used_e;

  // G における頂点 V[i] が、新しいグラフで i になるようにする
  // {G, es}
  // sum(deg(v)) の計算量になっていて、
  // 新しいグラフの n+m より大きい可能性があるので注意
  Graph<T, directed> rearrange(vc<int> V, bool keep_eid = 0) {
    if (len(new_idx) != N) new_idx.assign(N, -1);
    int n = len(V);
    FOR(i, n) new_idx[V[i]] = i;
    Graph<T, directed> G(n);
    vc<int> history;
    FOR(i, n) {
      for (auto&& e: (*this)[V[i]]) {
        if (len(used_e) <= e.id) used_e.resize(e.id + 1);
        if (used_e[e.id]) continue;
        int a = e.frm, b = e.to;
        if (new_idx[a] != -1 && new_idx[b] != -1) {
          history.eb(e.id);
          used_e[e.id] = 1;
          int eid = (keep_eid ? e.id : -1);
          G.add(new_idx[a], new_idx[b], e.cost, eid);
        }
      }
    }
    FOR(i, n) new_idx[V[i]] = -1;
    for (auto&& eid: history) used_e[eid] = 0;
    G.build();
    return G;
  }

  Graph<T, true> to_directed_tree(int root = -1) {
    if (root == -1) root = 0;
    assert(!is_directed && prepared && M == N - 1);
    Graph<T, true> G1(N);
    vc<int> par(N, -1);
    auto dfs = [&](auto& dfs, int v) -> void {
      for (auto& e: (*this)[v]) {
        if (e.to == par[v]) continue;
        par[e.to] = v, dfs(dfs, e.to);
      }
    };
    dfs(dfs, root);
    for (auto& e: edges) {
      int a = e.frm, b = e.to;
      if (par[a] == b) swap(a, b);
      assert(par[b] == a);
      G1.add(a, b, e.cost);
    }
    G1.build();
    return G1;
  }

private:
  void calc_deg() {
    assert(vc_deg.empty());
    vc_deg.resize(N);
    for (auto&& e: edges) vc_deg[e.frm]++, vc_deg[e.to]++;
  }

  void calc_deg_inout() {
    assert(vc_indeg.empty());
    vc_indeg.resize(N);
    vc_outdeg.resize(N);
    for (auto&& e: edges) { vc_indeg[e.to]++, vc_outdeg[e.frm]++; }
  }
};
#line 2 "graph/maximum_matching.hpp"

// return : (match size, match)
// match[v] : マッチング相手 OR 0
// O(N^3)
// 「組合せ最適化」第2版, アルゴリズム 10.2
template <typename GT>
pair<int, vc<int>> maximum_matching(const GT& G) {
  const int N = G.N;
  vc<int> mu(N), phi(N), rho(N);
  vc<bool> scanned(N);
  FOR(v, N) mu[v] = v;
  ll ans = 0;
  for (auto&& e: G.edges) {
    if (e.frm != e.to && mu[e.frm] == e.frm && mu[e.to] == e.to) {
      mu[e.frm] = e.to, mu[e.to] = e.frm, ++ans;
    }
  }

  auto odd = [&](int x) -> bool {
    return mu[x] != x && phi[mu[x]] == mu[x] && mu[x] != x;
  };
  auto out_of_forest = [&](int x) -> bool {
    return mu[x] != x && phi[mu[x]] == mu[x] && phi[x] == x;
  };
  auto P = [&](int x) -> vc<int> {
    vc<int> P;
    P.eb(x);
    while (mu[x] != x) {
      P.eb(mu[x]);
      P.eb(phi[mu[x]]);
      x = phi[mu[x]];
    }
    return P;
  };

  vc<bool> on_path(N);
  while (1) {
    FOR(v, N) phi[v] = rho[v] = v, scanned[v] = 0;
    bool aug = 0;
    while (1) {
      bool upd = 0;
      FOR(x, N) {
        if (upd) break;
        if (scanned[x] || odd(x)) continue;
        for (auto&& e: G[x]) {
          int y = e.to;
          if (out_of_forest(y)) {
            upd = 1;
            // grow
            phi[y] = x;
          }
          elif (rho[y] != rho[x] && !odd(y)) {
            vc<int> F;
            FOR(v, N) if (!out_of_forest(v)) F.eb(v);
            upd = 1;
            // augument OR shrink
            vc<int> Px = P(x);
            vc<int> Py = P(y);
            if (Px.back() != Py.back()) {
              aug = 1;
              // augument
              FOR(2) {
                swap(Px, Py);
                for (int i = 1; i < len(Px); i += 2) {
                  int v = Px[i];
                  mu[phi[v]] = v, mu[v] = phi[v];
                }
              }
              mu[x] = y, mu[y] = x, ++ans;
              break;
            } else {
              // shrink
              int r = -1;
              int Nx = len(Px), Ny = len(Py);
              for (int i = 0; i < Nx; i += 2) {
                int v = Px[i];
                int j = i + Ny - Nx;
                if (0 <= j && j < Ny && Py[j] == v && rho[v] == v) {
                  r = v;
                  break;
                }
              }
              while (Px.back() != r) Px.pop_back();
              while (Py.back() != r) Py.pop_back();
              vc<int> change;
              FOR(2) {
                swap(Px, Py);
                for (int i = 1; i < len(Px); i += 2) {
                  int v = Px[i];
                  if (rho[phi[v]] != r) change.eb(v);
                }
              }
              for (auto&& v: change) phi[phi[v]] = v;

              if (rho[x] != r) phi[x] = y;
              if (rho[y] != r) phi[y] = x;
              for (auto&& v: Px) on_path[v] = 1;
              for (auto&& v: Py) on_path[v] = 1;
              FOR(v, N) if (on_path[rho[v]]) { rho[v] = r; }
              fill(all(on_path), 0);
            }
          }
        }
        scanned[x] = 1;
      }
      if (!upd || aug) break;
    }
    if (!aug) break;
  }
  FOR(v, N) if (mu[v] == v) mu[v] = -1;
  return {ans, mu};
}

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