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#include "graph/steiner_tree.hpp"
#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() { 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 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}; }