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#include "graph/minimum_hamiltonian_cycle.hpp"
#pragma once #include "graph/base.hpp" #include "enumerate/bits.hpp" /* return [cost, cycle] cycle なしの場合:{-1, {}} */ template <typename T, typename GT> pair<T, vc<int>> minimum_hamiltonian_cycle(GT& G) { assert(G.is_prepared()); int n = G.N; vv(T, dist, n, n, infty<T>); FOR(v, n) { for (auto&& e: G[v]) chmin(dist[v][e.to], e.cost); } n -= 1; const int full = (1 << n) - 1; vv(T, dp, 1 << n, n, infty<T>); FOR(v, n) chmin(dp[1 << v][v], dist[n][v]); for (int s = 0; s < (1 << n); ++s) { FOR(frm, n) if (dp[s][frm] < infty<T>) { enumerate_bits_32(full - s, [&](int to) -> void { int t = s | 1 << to; T cost = dist[frm][to]; if (cost < infty<T>) chmin(dp[t][to], dp[s][frm] + cost); }); } } int s = (1 << n) - 1; T res = infty<T>; int best_v = -1; FOR(v, n) if (dist[v][n] < infty<T> && dp[s][v] < infty<T>) { if (chmin(res, dp[s][v] + dist[v][n])) best_v = v; } if (res == infty<T>) return {-1, {}}; vc<int> C = {n, best_v}; int t = s; while (len(C) <= n) { int to = C.back(); int frm = [&]() -> int { FOR(frm, n) { int s = t ^ (1 << to); T inf = infty<T>; if (dp[s][frm] < inf && dist[frm][to] < inf && dp[s][frm] + dist[frm][to] == dp[t][to]) return frm; } return -1; }(); C.eb(frm); t ^= 1 << to; } reverse(all(C)); return {res, C}; }
#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 "enumerate/bits.hpp" template <typename F> void enumerate_bits_32(u32 s, F f) { while (s) { int i = __builtin_ctz(s); f(i); s ^= 1 << i; } } template <typename F> void enumerate_bits_64(u64 s, F f) { while (s) { int i = __builtin_ctzll(s); f(i); s ^= u64(1) << i; } } template <typename BS, typename F> void enumerate_bits_bitset(BS& b, int L, int R, F f) { int p = (b[L] ? L : b._Find_next(L)); while (p < R) { f(p); p = b._Find_next(p); } } #line 4 "graph/minimum_hamiltonian_cycle.hpp" /* return [cost, cycle] cycle なしの場合:{-1, {}} */ template <typename T, typename GT> pair<T, vc<int>> minimum_hamiltonian_cycle(GT& G) { assert(G.is_prepared()); int n = G.N; vv(T, dist, n, n, infty<T>); FOR(v, n) { for (auto&& e: G[v]) chmin(dist[v][e.to], e.cost); } n -= 1; const int full = (1 << n) - 1; vv(T, dp, 1 << n, n, infty<T>); FOR(v, n) chmin(dp[1 << v][v], dist[n][v]); for (int s = 0; s < (1 << n); ++s) { FOR(frm, n) if (dp[s][frm] < infty<T>) { enumerate_bits_32(full - s, [&](int to) -> void { int t = s | 1 << to; T cost = dist[frm][to]; if (cost < infty<T>) chmin(dp[t][to], dp[s][frm] + cost); }); } } int s = (1 << n) - 1; T res = infty<T>; int best_v = -1; FOR(v, n) if (dist[v][n] < infty<T> && dp[s][v] < infty<T>) { if (chmin(res, dp[s][v] + dist[v][n])) best_v = v; } if (res == infty<T>) return {-1, {}}; vc<int> C = {n, best_v}; int t = s; while (len(C) <= n) { int to = C.back(); int frm = [&]() -> int { FOR(frm, n) { int s = t ^ (1 << to); T inf = infty<T>; if (dp[s][frm] < inf && dist[frm][to] < inf && dp[s][frm] + dist[frm][to] == dp[t][to]) return frm; } return -1; }(); C.eb(frm); t ^= 1 << to; } reverse(all(C)); return {res, C}; }