graph/bipartite_balanced_edge_coloring.hpp

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• Last update: 2025-03-31 01:16:57+09:00
• Include: #include "graph/bipartite_balanced_edge_coloring.hpp"
• Link: https://codeforces.com/contest/212/problem/A

Depends on

• ds/hashmap.hpp
• ds/unionfind/unionfind.hpp
• flow/bipartite.hpp
• graph/base.hpp
• graph/bipartite_edge_coloring.hpp
• graph/bipartite_vertex_coloring.hpp
• graph/strongly_connected_component.hpp

Code

#include "graph/bipartite_edge_coloring.hpp"

// 辺を K 色で塗る. 各点のまわりで色が均等（max-min <= 1）にせよ.
// return : color[eid].
// https://codeforces.com/contest/212/problem/A
vc<int> bipartite_balanced_edge_coloring(Graph<int, 0> G, int K) {
  int N = G.N, M = G.M;
  vc<int> A, B;
  vc<int> cnt(N);
  vc<int> now(N);
  int nxt = 0;

  auto get = [&](int v) -> int {
    if (cnt[v] % K == 0) { now[v] = nxt++; }
    cnt[v]++;
    return now[v];
  };
  for (auto& e: G.edges) {
    int a = e.frm, b = e.to;
    a = get(a), b = get(b);
    A.eb(a), B.eb(b);
  }
  Graph<int, 0> H(nxt);
  FOR(i, M) H.add(A[i], B[i]);
  H.build();
  return bipartite_edge_coloring(H).se;
}

#line 2 "graph/bipartite_vertex_coloring.hpp"

#line 2 "ds/hashmap.hpp"

// u64 -> Val

template <typename Val>
struct HashMap {
  // n は入れたいものの個数で ok

  HashMap(u32 n = 0) { build(n); }
  void build(u32 n) {
    u32 k = 8;
    while (k < n * 2) k *= 2;
    cap = k / 2, mask = k - 1;
    key.resize(k), val.resize(k), used.assign(k, 0);
  }

  // size を保ったまま. size=0 にするときは build すること.

  void clear() {
    used.assign(len(used), 0);
    cap = (mask + 1) / 2;
  }
  int size() { return len(used) / 2 - cap; }

  int index(const u64& k) {
    int i = 0;
    for (i = hash(k); used[i] && key[i] != k; i = (i + 1) & mask) {}
    return i;
  }

  Val& operator[](const u64& k) {
    if (cap == 0) extend();
    int i = index(k);
    if (!used[i]) { used[i] = 1, key[i] = k, val[i] = Val{}, --cap; }
    return val[i];
  }

  Val get(const u64& k, Val default_value) {
    int i = index(k);
    return (used[i] ? val[i] : default_value);
  }

  bool count(const u64& k) {
    int i = index(k);
    return used[i] && key[i] == k;
  }

  // f(key, val)

  template <typename F>
  void enumerate_all(F f) {
    FOR(i, len(used)) if (used[i]) f(key[i], val[i]);
  }

private:
  u32 cap, mask;
  vc<u64> key;
  vc<Val> val;
  vc<bool> used;

  u64 hash(u64 x) {
    static const u64 FIXED_RANDOM = std::chrono::steady_clock::now().time_since_epoch().count();
    x += FIXED_RANDOM;
    x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
    x = (x ^ (x >> 27)) * 0x94d049bb133111eb;
    return (x ^ (x >> 31)) & mask;
  }

  void extend() {
    vc<pair<u64, Val>> dat;
    dat.reserve(len(used) / 2 - cap);
    FOR(i, len(used)) {
      if (used[i]) dat.eb(key[i], val[i]);
    }
    build(2 * len(dat));
    for (auto& [a, b]: dat) (*this)[a] = b;
  }
};
#line 3 "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;
  }

  HashMap<int> MP_FOR_EID;

  int get_eid(u64 a, u64 b) {
    if (len(MP_FOR_EID) == 0) {
      MP_FOR_EID.build(N - 1);
      for (auto& e: edges) {
        u64 a = e.frm, b = e.to;
        u64 k = to_eid_key(a, b);
        MP_FOR_EID[k] = e.id;
      }
    }
    return MP_FOR_EID.get(to_eid_key(a, b), -1);
  }

  u64 to_eid_key(u64 a, u64 b) {
    if (!directed && a > b) swap(a, b);
    return N * a + b;
  }

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;
  }

  vc<int> get_all() {
    vc<int> A(n);
    FOR(i, n) A[i] = (*this)[i];
    return A;
  }
};
#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 "graph/strongly_connected_component.hpp"

template <typename GT>
pair<int, vc<int>> strongly_connected_component(GT& G) {
  static_assert(GT::is_directed);
  assert(G.is_prepared());
  int N = G.N;
  int C = 0;
  vc<int> comp(N), low(N), ord(N, -1), path;
  int now = 0;

  auto dfs = [&](auto& dfs, int v) -> void {
    low[v] = ord[v] = now++;
    path.eb(v);
    for (auto&& [frm, to, cost, id]: G[v]) {
      if (ord[to] == -1) {
        dfs(dfs, to), chmin(low[v], low[to]);
      } else {
        chmin(low[v], ord[to]);
      }
    }
    if (low[v] == ord[v]) {
      while (1) {
        int u = POP(path);
        ord[u] = N, comp[u] = C;
        if (u == v) break;
      }
      ++C;
    }
  };
  FOR(v, N) {
    if (ord[v] == -1) dfs(dfs, v);
  }
  FOR(v, N) comp[v] = C - 1 - comp[v];
  return {C, comp};
}

template <typename GT>
Graph<int, 1> scc_dag(GT& G, int C, vc<int>& comp) {
  Graph<int, 1> DAG(C);
  vvc<int> edges(C);
  for (auto&& e: G.edges) {
    int x = comp[e.frm], y = comp[e.to];
    if (x == y) continue;
    edges[x].eb(y);
  }
  FOR(c, C) {
    UNIQUE(edges[c]);
    for (auto&& to: edges[c]) DAG.add(c, to);
  }
  DAG.build();
  return DAG;
}
#line 4 "flow/bipartite.hpp"

template <typename GT>
struct BipartiteMatching {
  int N;
  GT& G;
  vc<int> color;
  vc<int> dist, match;
  vc<int> vis;

  BipartiteMatching(GT& G) : N(G.N), G(G), dist(G.N, -1), match(G.N, -1) {
    color = bipartite_vertex_coloring(G);
    if (N > 0) assert(!color.empty());
    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;
    }
  }

  BipartiteMatching(GT& G, vc<int> color)
      : N(G.N), G(G), color(color), dist(G.N, -1), match(G.N, -1) {
    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;
    }
  }

  void bfs() {
    dist.assign(N, -1);
    queue<int> que;
    FOR(v, N) if (!color[v] && match[v] == -1) que.emplace(v), dist[v] = 0;
    while (!que.empty()) {
      int v = que.front();
      que.pop();
      for (auto&& e: G[v]) {
        dist[e.to] = 0;
        int w = match[e.to];
        if (w != -1 && dist[w] == -1) dist[w] = dist[v] + 1, que.emplace(w);
      }
    }
  }

  bool dfs(int v) {
    vis[v] = 1;
    for (auto&& e: G[v]) {
      int w = match[e.to];
      if (w == -1 || (!vis[w] && dist[w] == dist[v] + 1 && dfs(w))) {
        match[e.to] = v, match[v] = e.to;
        return true;
      }
    }
    return false;
  }

  vc<pair<int, int>> matching() {
    vc<pair<int, int>> res;
    FOR(v, N) if (v < match[v]) res.eb(v, match[v]);
    return res;
  }

  vc<int> vertex_cover() {
    vc<int> res;
    FOR(v, N) if (color[v] ^ (dist[v] == -1)) { res.eb(v); }
    return res;
  }

  vc<int> independent_set() {
    vc<int> res;
    FOR(v, N) if (!(color[v] ^ (dist[v] == -1))) { res.eb(v); }
    return res;
  }

  vc<int> edge_cover() {
    vc<bool> done(N);
    vc<int> res;
    for (auto&& e: G.edges) {
      if (done[e.frm] || done[e.to]) continue;
      if (match[e.frm] == e.to) {
        res.eb(e.id);
        done[e.frm] = done[e.to] = 1;
      }
    }
    for (auto&& e: G.edges) {
      if (!done[e.frm]) {
        res.eb(e.id);
        done[e.frm] = 1;
      }
      if (!done[e.to]) {
        res.eb(e.id);
        done[e.to] = 1;
      }
    }
    sort(all(res));
    return res;
  }

  /* Dulmage–Mendelsohn decomposition
  https://en.wikipedia.org/wiki/Dulmage%E2%80%93Mendelsohn_decomposition
  http://www.misojiro.t.u-tokyo.ac.jp/~murota/lect-ouyousurigaku/dm050410.pdf
  https://hitonanode.github.io/cplib-cpp/graph/dulmage_mendelsohn_decomposition.hpp.html
  - 最大マッチングとしてありうる iff 同じ W を持つ
  - 辺 uv が必ず使われる：同じ W を持つ辺が唯一
  - color=0 から 1 への辺：W[l] <= W[r]
  - color=0 の点が必ず使われる：W=1,2,...,K
  - color=1 の点が必ず使われる：W=0,1,...,K-1
  */
  pair<int, vc<int>> DM_decomposition() {
    // 非飽和点からの探索

    vc<int> W(N, -1);
    vc<int> que;
    auto add = [&](int v, int x) -> void {
      if (W[v] == -1) {
        W[v] = x;
        que.eb(v);
      }
    };
    FOR(v, N) if (match[v] == -1 && color[v] == 0) add(v, 0);
    FOR(v, N) if (match[v] == -1 && color[v] == 1) add(v, infty<int>);
    while (len(que)) {
      auto v = POP(que);
      if (match[v] != -1) add(match[v], W[v]);
      if (color[v] == 0 && W[v] == 0) {
        for (auto&& e: G[v]) { add(e.to, W[v]); }
      }
      if (color[v] == 1 && W[v] == infty<int>) {
        for (auto&& e: G[v]) { add(e.to, W[v]); }
      }
    }
    // 残った点からなるグラフを作って強連結成分分解

    vc<int> V;
    FOR(v, N) if (W[v] == -1) V.eb(v);
    int n = len(V);
    Graph<bool, 1> DG(n);
    FOR(i, n) {
      int v = V[i];
      if (match[v] != -1) {
        int j = LB(V, match[v]);
        DG.add(i, j);
      }
      if (color[v] == 0) {
        for (auto&& e: G[v]) {
          if (W[e.to] != -1 || e.to == match[v]) continue;
          int j = LB(V, e.to);
          DG.add(i, j);
        }
      }
    }
    DG.build();
    auto [K, comp] = strongly_connected_component(DG);
    K += 1;
    // 答

    FOR(i, n) { W[V[i]] = 1 + comp[i]; }
    FOR(v, N) if (W[v] == infty<int>) W[v] = K;
    return {K, W};
  }

#ifdef FASTIO
  void debug() {
    print("match", match);
    print("min vertex covor", vertex_cover());
    print("max indep set", independent_set());
    print("min edge cover", edge_cover());
  }
#endif
};
#line 4 "graph/bipartite_edge_coloring.hpp"

struct RegularBipartiteColoring {
  using P = pair<int, int>;
  int N, M;
  vc<P> edges;

  vvc<int> solve(int n, int k, vc<P> G) {
    N = n;
    M = len(G);
    edges = G;
    vc<int> A(M);
    iota(all(A), 0);
    return solve_inner(M / N, A);
  }

  vvc<int> solve_inner(int k, vc<int> A) { return (k % 2 == 0 ? solve_even(k, A) : solve_odd(k, A)); }

  vvc<int> solve_even(int k, vc<int> A) {
    assert(k % 2 == 0);
    if (k == 0) return {};
    // 2^m <= k < 2^{m+1}
    int m = 0;
    while (1 << (m + 1) <= k) ++m;
    vvc<int> res;
    if (k != 1 << m) {
      auto [B, C] = split(k, A);
      auto dat = solve_inner(k / 2, C);
      FOR(j, k - (1 << m)) { res.eb(dat[j]); }
      FOR(j, k - (1 << m), len(dat)) {
        for (auto&& idx: dat[j]) B.eb(idx);
      }
      k = 1 << m;
      swap(A, B);
    }
    auto dfs = [&](auto& dfs, int K, vc<int> A) -> void {
      if (K == 1) {
        res.eb(A);
        return;
      }
      auto [B, C] = split(k, A);
      dfs(dfs, K / 2, B);
      dfs(dfs, K / 2, C);
    };
    dfs(dfs, k, A);
    return res;
  }

  vvc<int> solve_odd(int k, vc<int> A) {
    assert(k % 2 == 1);
    if (k == 1) { return {A}; }
    vc<bool> match = matching(k, A);
    vc<int> B;
    B.reserve(len(A) - N);
    vc<int> es;
    FOR(i, len(A)) {
      if (match[i]) es.eb(A[i]);
      if (!match[i]) B.eb(A[i]);
    }
    vvc<int> res = solve_inner(k - 1, B);
    res.eb(es);
    return res;
  }

  vc<bool> matching(int k, vc<int> A) {
    Graph<bool, 0> G(N + N);
    vc<int> color(N + N);
    FOR(v, N) color[v] = 0;
    for (auto&& eid: A) {
      auto [a, b] = edges[eid];
      G.add(a, b);
    }
    G.build();
    BipartiteMatching<decltype(G)> BM(G);
    auto& match = BM.match;
    vc<bool> res(len(A));
    FOR(i, len(A)) {
      auto idx = A[i];
      auto [a, b] = edges[idx];
      if (match[a] == b) {
        match[a] = -1;
        res[i] = 1;
      }
    }
    return res;
  }

  pair<vc<int>, vc<int>> split(int k, vc<int> A) {
    assert(k % 2 == 0);
    // 2 つの k/2 - regular に分割する。
    int n = len(A);
    vc<bool> rest(n);
    vc<int> A0, A1;
    A0.reserve(n / 2), A1.reserve(n / 2);
    vvc<P> G(N + N);
    FOR(i, n) {
      rest[i] = 1;
      auto [a, b] = edges[A[i]];
      G[a].eb(i, b);
      G[b].eb(i, a);
    }
    auto dfs = [&](auto& dfs, int v, int color) -> void {
      while (len(G[v])) {
        auto [i, to] = POP(G[v]);
        if (!rest[i]) continue;
        rest[i] = 0;
        if (color == 0) A0.eb(A[i]);
        if (color == 1) A1.eb(A[i]);
        dfs(dfs, to, 1 ^ color);
      }
    };
    FOR(v, N) dfs(dfs, v, 0);
    return {A0, A1};
  }
};

template <typename GT>
pair<int, vc<int>> bipartite_edge_coloring(GT& G) {
  if (G.M == 0) { return {0, {}}; }
  auto vcolor = bipartite_vertex_coloring<GT>(G);
  auto deg = G.deg_array();
  int D = MAX(deg);

  UnionFind uf(G.N);
  FOR(c, 2) {
    pqg<pair<int, int>> que;
    FOR(v, G.N) {
      if (vcolor[v] == c) que.emplace(deg[v], v);
    }
    while (len(que) > 1) {
      auto [d1, v1] = POP(que);
      auto [d2, v2] = POP(que);
      if (d1 + d2 > D) break;
      uf.merge(v1, v2);
      int r = uf[v1];
      que.emplace(d1 + d2, r);
    }
  }

  vc<int> LV, RV;
  FOR(v, G.N) if (uf[v] == v) {
    if (vcolor[v] == 0) LV.eb(v);
    if (vcolor[v] == 1) RV.eb(v);
  }
  int X = max(len(LV), len(RV));
  vc<int> degL(X), degR(X);

  vc<pair<int, int>> edges;
  for (auto&& e: G.edges) {
    int a = e.frm, b = e.to;
    a = uf[a], b = uf[b];
    a = LB(LV, a);
    b = LB(RV, b);
    degL[a]++, degR[b]++;
    edges.eb(a, X + b);
  }
  int p = 0, q = 0;
  while (p < X && q < X) {
    if (degL[p] == D) {
      ++p;
      continue;
    }
    if (degR[q] == D) {
      ++q;
      continue;
    }
    edges.eb(p, X + q);
    degL[p]++, degR[q]++;
  }
  RegularBipartiteColoring RBC;
  vvc<int> res = RBC.solve(X, D, edges);
  vc<int> ecolor(len(edges));
  FOR(i, len(res)) {
    for (auto&& j: res[i]) ecolor[j] = i;
  }
  ecolor.resize(G.M);
  return {D, ecolor};
}
#line 2 "graph/bipartite_balanced_edge_coloring.hpp"

// 辺を K 色で塗る. 各点のまわりで色が均等（max-min <= 1）にせよ.
// return : color[eid].
// https://codeforces.com/contest/212/problem/A
vc<int> bipartite_balanced_edge_coloring(Graph<int, 0> G, int K) {
  int N = G.N, M = G.M;
  vc<int> A, B;
  vc<int> cnt(N);
  vc<int> now(N);
  int nxt = 0;

  auto get = [&](int v) -> int {
    if (cnt[v] % K == 0) { now[v] = nxt++; }
    cnt[v]++;
    return now[v];
  };
  for (auto& e: G.edges) {
    int a = e.frm, b = e.to;
    a = get(a), b = get(b);
    A.eb(a), B.eb(b);
  }
  Graph<int, 0> H(nxt);
  FOR(i, M) H.add(A[i], B[i]);
  H.build();
  return bipartite_edge_coloring(H).se;
}

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