graph/dag_path_cover.hpp

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• Last update: 2024-04-19 22:50:36+09:00
• Include: #include "graph/dag_path_cover.hpp"

Depends on

• ds/unionfind/unionfind.hpp
• flow/maxflow.hpp
• graph/base.hpp

Verified with

• test/aoj/2251_1.test.cpp

Code

#include "graph/base.hpp"
#include "flow/maxflow.hpp"
#include "ds/unionfind/unionfind.hpp"

// 各頂点の色をかえす。各色はひとつのパス上にあるようにする
template <typename DAG>
vc<int> dag_path_cover(DAG& G) {
  static_assert(DAG::is_directed);
  for (auto&& e: G.edges) assert(e.frm < e.to);

  int N = G.N;
  int source = 2 * N, sink = 2 * N + 1;
  MaxFlow<int> F(2 * N + 2, source, sink);
  FOR(v, N) {
    F.add(source, 2 * v + 1, 1);
    F.add(2 * v + 0, sink, 1);
    F.add(2 * v + 0, 2 * v + 1, infty<int>);
  }
  for (auto&& e: G.edges) F.add(2 * e.frm + 1, 2 * e.to + 0, infty<int>);

  F.flow();
  auto paths = F.path_decomposition();

  UnionFind uf(N);
  for (auto& P: paths) {
    int a = P[1], b = P[len(P) - 2];
    uf.merge(a / 2, b / 2);
  }

  vc<int> ANS(N, -1);
  int p = 0;
  FOR(v, N) if (uf[v] == v) ANS[v] = p++;
  FOR(v, N) if (uf[v] != v) ANS[v] = ANS[uf[v]];
  return ANS;
};

#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);
    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;
  }

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 1 "flow/maxflow.hpp"
// incremental に辺を追加してよい
// 辺の容量の変更が可能
// 変更する capacity が F のとき、O((N+M)|F|) 時間で更新
template <typename Cap>
struct MaxFlow {
  struct Edge {
    int to, rev;
    Cap cap; // 残っている容量. したがって cap+flow が定数.
    Cap flow = 0;
  };

  const int N, source, sink;
  vvc<Edge> edges;
  vc<pair<int, int>> pos;
  vc<int> prog, level;
  vc<int> que;
  bool calculated;

  MaxFlow(int N, int source, int sink)
      : N(N),
        source(source),
        sink(sink),
        edges(N),
        calculated(0),
        flow_ans(0) {}

  void add(int frm, int to, Cap cap, Cap rev_cap = 0) {
    calculated = 0;
    assert(0 <= frm && frm < N);
    assert(0 <= to && to < N);
    assert(Cap(0) <= cap);
    int a = len(edges[frm]);
    int b = (frm == to ? a + 1 : len(edges[to]));
    pos.eb(frm, a);
    edges[frm].eb(Edge{to, b, cap, 0});
    edges[to].eb(Edge{frm, a, rev_cap, 0});
  }

  void change_capacity(int i, Cap after) {
    auto [frm, idx] = pos[i];
    auto& e = edges[frm][idx];
    Cap before = e.cap + e.flow;
    if (before < after) {
      calculated = (e.cap > 0);
      e.cap += after - before;
      return;
    }
    e.cap = after - e.flow;
    // 差分を押し戻す処理発生
    if (e.cap < 0) flow_push_back(e);
  }

  void flow_push_back(Edge& e0) {
    auto& re0 = edges[e0.to][e0.rev];
    int a = re0.to;
    int b = e0.to;
    /*
    辺 e0 の容量が正になるように戻す
    path-cycle 分解を考えれば、
    - uv 辺を含むサイクルを消す
    - suvt パスを消す
    前者は残余グラフで ab パス（flow_ans が変わらない）
    後者は残余グラフで tb, as パス
    */

    auto find_path = [&](int s, int t, Cap lim) -> Cap {
      vc<bool> vis(N);
      prog.assign(N, 0);
      auto dfs = [&](auto& dfs, int v, Cap f) -> Cap {
        if (v == t) return f;
        for (int& i = prog[v]; i < len(edges[v]); ++i) {
          auto& e = edges[v][i];
          if (vis[e.to] || e.cap <= Cap(0)) continue;
          vis[e.to] = 1;
          Cap a = dfs(dfs, e.to, min(f, e.cap));
          assert(a >= 0);
          if (a == Cap(0)) continue;
          e.cap -= a, e.flow += a;
          edges[e.to][e.rev].cap += a, edges[e.to][e.rev].flow -= a;
          return a;
        }
        return 0;
      };
      return dfs(dfs, s, lim);
    };

    while (e0.cap < 0) {
      Cap x = find_path(a, b, -e0.cap);
      if (x == Cap(0)) break;
      e0.cap += x, e0.flow -= x;
      re0.cap -= x, re0.flow += x;
    }
    Cap c = -e0.cap;
    while (c > 0 && a != source) {
      Cap x = find_path(a, source, c);
      assert(x > 0);
      c -= x;
    }
    c = -e0.cap;
    while (c > 0 && b != sink) {
      Cap x = find_path(sink, b, c);
      assert(x > 0);
      c -= x;
    }
    c = -e0.cap;
    e0.cap += c, e0.flow -= c;
    re0.cap -= c, re0.flow += c;
    flow_ans -= c;
  }

  // frm, to, flow
  vc<tuple<int, int, Cap>> get_flow_edges() {
    vc<tuple<int, int, Cap>> res;
    FOR(frm, N) {
      for (auto&& e: edges[frm]) {
        if (e.flow <= 0) continue;
        res.eb(frm, e.to, e.flow);
      }
    }
    return res;
  }

  vc<bool> vis;

  // 差分ではなくこれまでの総量
  Cap flow() {
    if (calculated) return flow_ans;
    calculated = true;
    while (set_level()) {
      prog.assign(N, 0);
      while (1) {
        Cap x = flow_dfs(source, infty<Cap>);
        if (x == 0) break;
        flow_ans += x;
        chmin(flow_ans, infty<Cap>);
        if (flow_ans == infty<Cap>) return flow_ans;
      }
    }
    return flow_ans;
  }

  // 最小カットの値および、カットを表す 01 列を返す
  pair<Cap, vc<int>> cut() {
    flow();
    vc<int> res(N);
    FOR(v, N) res[v] = (level[v] >= 0 ? 0 : 1);
    return {flow_ans, res};
  }

  // O(F(N+M)) くらい使って経路復元
  // simple path になる
  vvc<int> path_decomposition() {
    flow();
    auto edges = get_flow_edges();
    vvc<int> TO(N);
    for (auto&& [frm, to, flow]: edges) { FOR(flow) TO[frm].eb(to); }
    vvc<int> res;
    vc<int> vis(N);

    FOR(flow_ans) {
      vc<int> path = {source};
      vis[source] = 1;
      while (path.back() != sink) {
        int to = POP(TO[path.back()]);
        while (vis[to]) { vis[POP(path)] = 0; }
        path.eb(to), vis[to] = 1;
      }
      for (auto&& v: path) vis[v] = 0;
      res.eb(path);
    }
    return res;
  }

  void debug() {
    print("source", source);
    print("sink", sink);
    print("edges (frm, to, cap, flow)");
    FOR(v, N) {
      for (auto& e: edges[v]) {
        if (e.cap == 0 && e.flow == 0) continue;
        print(v, e.to, e.cap, e.flow);
      }
    }
  }

private:
  Cap flow_ans;

  bool set_level() {
    que.resize(N);
    level.assign(N, -1);
    level[source] = 0;
    int l = 0, r = 0;
    que[r++] = source;
    while (l < r) {
      int v = que[l++];
      for (auto&& e: edges[v]) {
        if (e.cap > 0 && level[e.to] == -1) {
          level[e.to] = level[v] + 1;
          if (e.to == sink) return true;
          que[r++] = e.to;
        }
      }
    }
    return false;
  }

  Cap flow_dfs(int v, Cap lim) {
    if (v == sink) return lim;
    Cap res = 0;
    for (int& i = prog[v]; i < len(edges[v]); ++i) {
      auto& e = edges[v][i];
      if (e.cap > 0 && level[e.to] == level[v] + 1) {
        Cap a = flow_dfs(e.to, min(lim, e.cap));
        if (a > 0) {
          e.cap -= a, e.flow += a;
          edges[e.to][e.rev].cap += a, edges[e.to][e.rev].flow -= a;
          res += a;
          lim -= a;
          if (lim == 0) break;
        }
      }
    }
    return res;
  }
};
#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 4 "graph/dag_path_cover.hpp"

// 各頂点の色をかえす。各色はひとつのパス上にあるようにする
template <typename DAG>
vc<int> dag_path_cover(DAG& G) {
  static_assert(DAG::is_directed);
  for (auto&& e: G.edges) assert(e.frm < e.to);

  int N = G.N;
  int source = 2 * N, sink = 2 * N + 1;
  MaxFlow<int> F(2 * N + 2, source, sink);
  FOR(v, N) {
    F.add(source, 2 * v + 1, 1);
    F.add(2 * v + 0, sink, 1);
    F.add(2 * v + 0, 2 * v + 1, infty<int>);
  }
  for (auto&& e: G.edges) F.add(2 * e.frm + 1, 2 * e.to + 0, infty<int>);

  F.flow();
  auto paths = F.path_decomposition();

  UnionFind uf(N);
  for (auto& P: paths) {
    int a = P[1], b = P[len(P) - 2];
    uf.merge(a / 2, b / 2);
  }

  vc<int> ANS(N, -1);
  int p = 0;
  FOR(v, N) if (uf[v] == v) ANS[v] = p++;
  FOR(v, N) if (uf[v] != v) ANS[v] = ANS[uf[v]];
  return ANS;
};

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