graph/steiner_tree.hpp

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• Last update: 2025-04-06 22:14:02+09:00
• Include: #include "graph/steiner_tree.hpp"

Depends on

• ds/hashmap.hpp
• enumerate/bits.hpp
• graph/base.hpp

Verified with

• test/3_yukicoder/114.test.cpp

Code

#include "graph/base.hpp"
#include "enumerate/bits.hpp"

// 辺重みは e.cost、頂点重みは vector で渡す。返り値：{cost, vs, es}
// O(3^kn + 2^k(n+m)log n), k: terminal size
template <typename T, typename GT>
tuple<T, vc<int>, vc<int>> steiner_tree(GT& G, vc<int> S, vc<T> v_wt = {}) {
  static_assert(!GT::is_directed);
  assert(!S.empty());
  const int N = G.N, M = G.M, K = len(S);
  if (v_wt.empty()) v_wt.assign(N, 0);

  // ターミナル集合, root -> cost
  vv(T, DP, 1 << K, N, infty<T>);
  FOR(v, N) DP[0][v] = v_wt[v];

  // 2 * t or 2 * eid + 1
  vv(int, par, 1 << K, N, -1);

  for (int s = 1; s < (1 << K); ++s) {
    auto& dp = DP[s];
    for (int k: all_bit<u32>(s)) {
      int v = S[k];
      chmin(dp[v], DP[s ^ 1 << k][v]);
    }
    for (u32 t: all_subset<u32>(s)) {
      if (t == 0 || t == s) continue;
      FOR(v, N) {
        if (chmin(dp[v], DP[t][v] + DP[s ^ t][v] - v_wt[v])) par[s][v] = 2 * t;
      }
    }
    // 根の移動を dijkstra で
    pqg<pair<T, int>> que;
    FOR(v, N) que.emplace(dp[v], v);
    while (!que.empty()) {
      auto [dv, v] = POP(que);
      if (dv != dp[v]) continue;
      for (auto&& e: G[v]) {
        if (chmin(dp[e.to], dv + e.cost + v_wt[e.to])) {
          par[s][e.to] = 2 * e.id + 1;
          que.emplace(dp[e.to], e.to);
        }
      }
    }
  }

  // 復元する
  vc<bool> used_v(N), used_e(M);
  vc<int> v_to_k(N, -1);
  FOR(k, K) v_to_k[S[k]] = k;

  vc<pair<int, int>> que;
  int root = min_element(all(DP.back())) - DP.back().begin();
  que.eb((1 << K) - 1, root);
  used_v[root] = 1;

  while (len(que)) {
    auto [s, v] = POP(que);
    if (s == 0) { continue; }
    if (par[s][v] == -1) {
      int k = v_to_k[v];
      assert(k != -1 && s >> k & 1);
      que.eb(s ^ 1 << k, v);
      continue;
    }
    elif (par[s][v] & 1) {
      int eid = par[s][v] / 2;
      auto& e = G.edges[eid];
      int w = v ^ e.frm ^ e.to;
      used_v[w] = used_e[eid] = 1;
      que.eb(s, w);
      continue;
    }
    else {
      int t = par[s][v] / 2;
      que.eb(t, v), que.eb(s ^ t, v);
    }
  }
  vc<int> vs, es;
  FOR(v, N) if (used_v[v]) vs.eb(v);
  FOR(e, M) if (used_e[e]) es.eb(e);
  T cost = 0;
  for (auto&& v: vs) cost += v_wt[v];
  for (auto&& e: es) cost += G.edges[e].cost;
  assert(cost == DP.back()[root]);
  return {cost, vs, es};
}

#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() {
#ifdef LOCAL
    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
  }
#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 1 "enumerate/bits.hpp"
template <typename BS, typename F>
void enumerate_bits_bitset(BS& b, int L, int R, F f) {
  if (L >= len(b)) return;
  int p = (b[L] ? L : b._Find_next(L));
  while (p < R) {
    f(p);
    p = b._Find_next(p);
  }
}
#line 3 "graph/steiner_tree.hpp"

// 辺重みは e.cost、頂点重みは vector で渡す。返り値：{cost, vs, es}
// O(3^kn + 2^k(n+m)log n), k: terminal size
template <typename T, typename GT>
tuple<T, vc<int>, vc<int>> steiner_tree(GT& G, vc<int> S, vc<T> v_wt = {}) {
  static_assert(!GT::is_directed);
  assert(!S.empty());
  const int N = G.N, M = G.M, K = len(S);
  if (v_wt.empty()) v_wt.assign(N, 0);

  // ターミナル集合, root -> cost
  vv(T, DP, 1 << K, N, infty<T>);
  FOR(v, N) DP[0][v] = v_wt[v];

  // 2 * t or 2 * eid + 1
  vv(int, par, 1 << K, N, -1);

  for (int s = 1; s < (1 << K); ++s) {
    auto& dp = DP[s];
    for (int k: all_bit<u32>(s)) {
      int v = S[k];
      chmin(dp[v], DP[s ^ 1 << k][v]);
    }
    for (u32 t: all_subset<u32>(s)) {
      if (t == 0 || t == s) continue;
      FOR(v, N) {
        if (chmin(dp[v], DP[t][v] + DP[s ^ t][v] - v_wt[v])) par[s][v] = 2 * t;
      }
    }
    // 根の移動を dijkstra で
    pqg<pair<T, int>> que;
    FOR(v, N) que.emplace(dp[v], v);
    while (!que.empty()) {
      auto [dv, v] = POP(que);
      if (dv != dp[v]) continue;
      for (auto&& e: G[v]) {
        if (chmin(dp[e.to], dv + e.cost + v_wt[e.to])) {
          par[s][e.to] = 2 * e.id + 1;
          que.emplace(dp[e.to], e.to);
        }
      }
    }
  }

  // 復元する
  vc<bool> used_v(N), used_e(M);
  vc<int> v_to_k(N, -1);
  FOR(k, K) v_to_k[S[k]] = k;

  vc<pair<int, int>> que;
  int root = min_element(all(DP.back())) - DP.back().begin();
  que.eb((1 << K) - 1, root);
  used_v[root] = 1;

  while (len(que)) {
    auto [s, v] = POP(que);
    if (s == 0) { continue; }
    if (par[s][v] == -1) {
      int k = v_to_k[v];
      assert(k != -1 && s >> k & 1);
      que.eb(s ^ 1 << k, v);
      continue;
    }
    elif (par[s][v] & 1) {
      int eid = par[s][v] / 2;
      auto& e = G.edges[eid];
      int w = v ^ e.frm ^ e.to;
      used_v[w] = used_e[eid] = 1;
      que.eb(s, w);
      continue;
    }
    else {
      int t = par[s][v] / 2;
      que.eb(t, v), que.eb(s ^ t, v);
    }
  }
  vc<int> vs, es;
  FOR(v, N) if (used_v[v]) vs.eb(v);
  FOR(e, M) if (used_e[e]) es.eb(e);
  T cost = 0;
  for (auto&& v: vs) cost += v_wt[v];
  for (auto&& e: es) cost += G.edges[e].cost;
  assert(cost == DP.back()[root]);
  return {cost, vs, es};
}

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