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:heavy_check_mark: flow/rank_maximal_bipartite_matching.hpp

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#include "graph/bipartite_vertex_coloring.hpp"

// (N1, N2) bipartite graph with edge-weight 0, 1, ..., K - 1.
// find a matching s.t. (# of K-1, # of K-2, ...) is lex max.
// https://yukicoder.me/problems/no/1615
// https://qoj.ac/contest/1388/problem/6546

template <typename GT>
struct Rank_Maximal_Bipartite_Matching {
  int N, K;
  GT& G;
  vc<int> color;
  vc<int> dist, match;
  vc<int> que;
  vc<bool> vis;
  vc<bool> vcover;

  // edge の管理
  // [L,M) : active, [M,R) : inactive
  vc<pair<int, int>> dat;
  vc<int> LID, MID, RID;

  Rank_Maximal_Bipartite_Matching(GT& G) : N(G.N), G(G) {
    color = bipartite_vertex_coloring(G);
    if (N > 0) assert(!color.empty());
    dist.assign(N, -1), match.assign(N, -1);
    que.assign(N, -1), vis.assign(N, -1);
    vcover.assign(N, 0);
    K = 0;
    for (auto& e: G.edges) chmax(K, e.cost);
    ++K;
    build();
    FOR_R(k, K) solve(k);
  }

  void build() {
    FOR(v, N) {
      LID.eb(len(dat));
      if (color[v] == 0) {
        for (auto& e: G[v]) { dat.eb(e.to, e.cost); }
      }
    }
    LID.eb(len(dat));
    MID.resize(N), RID.resize(N);
    FOR(v, N) MID[v] = LID[v], RID[v] = LID[v + 1];
  }

  void solve(int k) {
    // weight k の edge を active にする
    FOR(v, N) {
      if (vcover[v]) continue;
      FOR(i, MID[v], RID[v]) {
        auto [to, cost] = dat[i];
        if (cost != k) continue;
        swap(dat[MID[v]], dat[i]), ++MID[v];
      }
    }
    while (1) {
      bfs();
      vis.assign(N, false);
      int flow = 0;
      FOR(v, N) if (!color[v] && match[v] == -1 && dfs(v))++ flow;
      if (!flow) break;
    }

    // update vertex cover
    FOR(v, N) { vcover[v] = (color[v] ^ (dist[v] == -1)); }
    // active な辺のうち両端点が vcover に触れているものを削除
    FOR(v, N) {
      if (!vcover[v]) continue;
      FOR_R(i, LID[v], MID[v]) {
        auto [to, cost] = dat[i];
        if (vcover[to]) {
          swap(dat[i], dat[MID[v] - 1]);
          swap(dat[MID[v] - 1], dat[RID[v] - 1]);
          --MID[v], --RID[v];
        }
      }
    }
    // inactive な辺のうち少なくとも一方が vcover に触れているものを削除
    FOR(v, N) {
      if (vcover[v]) {
        RID[v] = MID[v];
        continue;
      }
      FOR_R(i, MID[v], RID[v]) {
        auto [to, cost] = dat[i];
        if (vcover[to]) { swap(dat[i], dat[RID[v] - 1]), --RID[v]; }
      }
    }
  }

  void bfs() {
    dist.assign(N, -1);
    int ql = 0, qr = 0;
    FOR(v, N) if (!color[v] && match[v] == -1) que[qr++] = v, dist[v] = 0;
    while (ql < qr) {
      int v = que[ql++];
      FOR(i, LID[v], MID[v]) {
        auto [to, cost] = dat[i];
        dist[to] = 0;
        int w = match[to];
        if (w != -1 && dist[w] == -1) dist[w] = dist[v] + 1, que[qr++] = w;
      }
    }
  }

  bool dfs(int v) {
    vis[v] = 1;
    FOR(i, LID[v], MID[v]) {
      auto [to, cost] = dat[i];
      int w = match[to];
      if (w == -1 || (!vis[w] && dist[w] == dist[v] + 1 && dfs(w))) {
        match[to] = v, match[v] = to;
        return true;
      }
    }
    return false;
  }

  vc<int> get_matching_edges() {
    vc<int> match_wt(N, -1);
    vc<int> match_e(N, -1);
    for (auto& e: G.edges) {
      int a = e.frm, b = e.to;
      if (color[a]) swap(a, b);
      if (match[a] == b && chmax(match_wt[a], e.cost)) match_e[a] = e.id;
    }
    vc<int> res;
    FOR(v, N) if (match_e[v] != -1) res.eb(match_e[v]);
    return res;
  }
};
#line 1 "flow/rank_maximal_bipartite_matching.hpp"

#line 2 "graph/bipartite_vertex_coloring.hpp"

#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}
  Graph<T, directed> rearrange(vc<int> V, bool keep_eid = 0) {
    if (len(new_idx) != N) new_idx.assign(N, -1);
    if (len(used_e) != M) used_e.assign(M, 0);
    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 (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;
  }

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 "ds/unionfind/unionfind.hpp"

struct UnionFind {
  int n, n_comp;
  vc<int> dat; // par or (-size)
  UnionFind(int n = 0) { build(n); }

  void build(int m) {
    n = m, n_comp = m;
    dat.assign(n, -1);
  }

  void reset() { build(n); }

  int operator[](int x) {
    while (dat[x] >= 0) {
      int pp = dat[dat[x]];
      if (pp < 0) { return dat[x]; }
      x = dat[x] = pp;
    }
    return x;
  }

  ll size(int x) {
    x = (*this)[x];
    return -dat[x];
  }

  bool merge(int x, int y) {
    x = (*this)[x], y = (*this)[y];
    if (x == y) return false;
    if (-dat[x] < -dat[y]) swap(x, y);
    dat[x] += dat[y], dat[y] = x, n_comp--;
    return true;
  }
};
#line 5 "graph/bipartite_vertex_coloring.hpp"

// 二部グラフでなかった場合には empty

template <typename GT>
vc<int> bipartite_vertex_coloring(GT& G) {
  assert(!GT::is_directed);
  assert(G.is_prepared());

  int n = G.N;
  UnionFind uf(2 * n);
  for (auto&& e: G.edges) {
    int u = e.frm, v = e.to;
    uf.merge(u + n, v), uf.merge(u, v + n);
  }

  vc<int> color(2 * n, -1);
  FOR(v, n) if (uf[v] == v && color[uf[v]] < 0) {
    color[uf[v]] = 0;
    color[uf[v + n]] = 1;
  }
  FOR(v, n) color[v] = color[uf[v]];
  color.resize(n);
  FOR(v, n) if (uf[v] == uf[v + n]) return {};
  return color;
}
#line 3 "flow/rank_maximal_bipartite_matching.hpp"

// (N1, N2) bipartite graph with edge-weight 0, 1, ..., K - 1.
// find a matching s.t. (# of K-1, # of K-2, ...) is lex max.
// https://yukicoder.me/problems/no/1615
// https://qoj.ac/contest/1388/problem/6546

template <typename GT>
struct Rank_Maximal_Bipartite_Matching {
  int N, K;
  GT& G;
  vc<int> color;
  vc<int> dist, match;
  vc<int> que;
  vc<bool> vis;
  vc<bool> vcover;

  // edge の管理
  // [L,M) : active, [M,R) : inactive
  vc<pair<int, int>> dat;
  vc<int> LID, MID, RID;

  Rank_Maximal_Bipartite_Matching(GT& G) : N(G.N), G(G) {
    color = bipartite_vertex_coloring(G);
    if (N > 0) assert(!color.empty());
    dist.assign(N, -1), match.assign(N, -1);
    que.assign(N, -1), vis.assign(N, -1);
    vcover.assign(N, 0);
    K = 0;
    for (auto& e: G.edges) chmax(K, e.cost);
    ++K;
    build();
    FOR_R(k, K) solve(k);
  }

  void build() {
    FOR(v, N) {
      LID.eb(len(dat));
      if (color[v] == 0) {
        for (auto& e: G[v]) { dat.eb(e.to, e.cost); }
      }
    }
    LID.eb(len(dat));
    MID.resize(N), RID.resize(N);
    FOR(v, N) MID[v] = LID[v], RID[v] = LID[v + 1];
  }

  void solve(int k) {
    // weight k の edge を active にする
    FOR(v, N) {
      if (vcover[v]) continue;
      FOR(i, MID[v], RID[v]) {
        auto [to, cost] = dat[i];
        if (cost != k) continue;
        swap(dat[MID[v]], dat[i]), ++MID[v];
      }
    }
    while (1) {
      bfs();
      vis.assign(N, false);
      int flow = 0;
      FOR(v, N) if (!color[v] && match[v] == -1 && dfs(v))++ flow;
      if (!flow) break;
    }

    // update vertex cover
    FOR(v, N) { vcover[v] = (color[v] ^ (dist[v] == -1)); }
    // active な辺のうち両端点が vcover に触れているものを削除
    FOR(v, N) {
      if (!vcover[v]) continue;
      FOR_R(i, LID[v], MID[v]) {
        auto [to, cost] = dat[i];
        if (vcover[to]) {
          swap(dat[i], dat[MID[v] - 1]);
          swap(dat[MID[v] - 1], dat[RID[v] - 1]);
          --MID[v], --RID[v];
        }
      }
    }
    // inactive な辺のうち少なくとも一方が vcover に触れているものを削除
    FOR(v, N) {
      if (vcover[v]) {
        RID[v] = MID[v];
        continue;
      }
      FOR_R(i, MID[v], RID[v]) {
        auto [to, cost] = dat[i];
        if (vcover[to]) { swap(dat[i], dat[RID[v] - 1]), --RID[v]; }
      }
    }
  }

  void bfs() {
    dist.assign(N, -1);
    int ql = 0, qr = 0;
    FOR(v, N) if (!color[v] && match[v] == -1) que[qr++] = v, dist[v] = 0;
    while (ql < qr) {
      int v = que[ql++];
      FOR(i, LID[v], MID[v]) {
        auto [to, cost] = dat[i];
        dist[to] = 0;
        int w = match[to];
        if (w != -1 && dist[w] == -1) dist[w] = dist[v] + 1, que[qr++] = w;
      }
    }
  }

  bool dfs(int v) {
    vis[v] = 1;
    FOR(i, LID[v], MID[v]) {
      auto [to, cost] = dat[i];
      int w = match[to];
      if (w == -1 || (!vis[w] && dist[w] == dist[v] + 1 && dfs(w))) {
        match[to] = v, match[v] = to;
        return true;
      }
    }
    return false;
  }

  vc<int> get_matching_edges() {
    vc<int> match_wt(N, -1);
    vc<int> match_e(N, -1);
    for (auto& e: G.edges) {
      int a = e.frm, b = e.to;
      if (color[a]) swap(a, b);
      if (match[a] == b && chmax(match_wt[a], e.cost)) match_e[a] = e.id;
    }
    vc<int> res;
    FOR(v, N) if (match_e[v] != -1) res.eb(match_e[v]);
    return res;
  }
};
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