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#include "graph/minimum_cost_cycle.hpp"
#include "graph/base.hpp" // {wt, vs, es}, O(N * shortest path) template <typename T, typename GT> tuple<T, vc<int>, vc<int>> minimum_cost_cycle_directed(GT& G) { const int N = G.N; T mi = 0, ma = 0; for (auto& e: G.edges) chmin(mi, e.cost), chmax(ma, e.cost); assert(mi >= 0); T ans = infty<T>; vc<T> dist(N); vc<int> vs, es; vc<int> par_e(N, -1); pqg<pair<T, int>> que; deque<int> deq; FOR(r, N) { fill(dist.begin() + r, dist.end(), infty<T>); if (ma <= 1) { auto push = [&](int v, bool back) -> void { (back ? deq.eb(v) : deq.emplace_front(v)); }; for (auto& e: G[r]) { if (r <= e.to && chmin(dist[e.to], e.cost)) par_e[e.to] = e.id, push(e.to, e.cost); } while (len(deq)) { auto v = POP(deq); for (auto& e: G[v]) { if (r <= e.to && chmin(dist[e.to], dist[v] + e.cost)) { par_e[e.to] = e.id, push(e.to, e.cost); } } } } else { for (auto& e: G[r]) { if (r <= e.to && chmin(dist[e.to], e.cost)) { par_e[e.to] = e.id, que.emplace(e.cost, e.to); } } while (len(que)) { auto [dv, v] = POP(que); if (dist[v] != dv) continue; for (auto& e: G[v]) { T x = dv + e.cost; if (r <= e.to && chmin(dist[e.to], x)) { par_e[e.to] = e.id, que.emplace(x, e.to); } } } } if (chmin(ans, dist[r])) { vs.clear(), es.clear(); vs.eb(r); while (1) { int eid = par_e[vs.back()]; es.eb(eid); vs.eb(G.edges[eid].frm); if (vs.back() == r) break; } reverse(all(vs)); reverse(all(es)); }; } return {ans, vs, es}; } // {wt, vs, es}, O(N * shortest path) template <typename T, typename GT> tuple<T, vc<int>, vc<int>> minimum_cost_cycle_undirected(GT& G) { const int N = G.N; T ans = infty<T>; vc<T> dist(N); vc<int> par_e(N); vc<int> vs, es; FOR(r, N) { fill(dist.begin() + r, dist.end(), infty<T>); pqg<pair<T, int>> que; dist[r] = 0, que.emplace(0, r); while (len(que)) { auto [dv, v] = POP(que); if (dist[v] != dv) continue; for (auto& e: G[v]) { if (e.to < r) continue; T x = dv + e.cost; if (chmin(dist[e.to], x)) { par_e[e.to] = e.id; que.emplace(x, e.to); } } } int best_e = -1; for (auto& e: G.edges) { int a = e.frm, b = e.to; if (a < r || b < r || par_e[a] == e.id || par_e[b] == e.id) continue; if (chmin(ans, dist[a] + dist[b] + e.cost)) best_e = e.id; } if (best_e == -1) continue; vs.clear(), es.clear(); auto& e = G.edges[best_e]; int a = e.frm, b = e.to; // r -> a while (a != r) { int eid = par_e[a]; vs.eb(a), es.eb(eid); a = G.edges[eid].frm ^ G.edges[eid].to ^ a; } vs.eb(a); reverse(all(vs)), reverse(all(es)); es.eb(best_e); while (b != r) { int eid = par_e[b]; vs.eb(b), es.eb(eid); b = G.edges[eid].frm ^ G.edges[eid].to ^ b; } vs.eb(b); } return {ans, vs, es}; } // {wt, vs, es}, O(N * shortest path) template <typename T, typename GT> tuple<T, vc<int>, vc<int>> minimum_cost_cycle(GT& G) { for (auto& e: G.edges) assert(e.cost >= 0); if constexpr (GT::is_directed) { return minimum_cost_cycle_directed<T>(G); } else { return minimum_cost_cycle_undirected<T>(G); } }
#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 2 "graph/minimum_cost_cycle.hpp" // {wt, vs, es}, O(N * shortest path) template <typename T, typename GT> tuple<T, vc<int>, vc<int>> minimum_cost_cycle_directed(GT& G) { const int N = G.N; T mi = 0, ma = 0; for (auto& e: G.edges) chmin(mi, e.cost), chmax(ma, e.cost); assert(mi >= 0); T ans = infty<T>; vc<T> dist(N); vc<int> vs, es; vc<int> par_e(N, -1); pqg<pair<T, int>> que; deque<int> deq; FOR(r, N) { fill(dist.begin() + r, dist.end(), infty<T>); if (ma <= 1) { auto push = [&](int v, bool back) -> void { (back ? deq.eb(v) : deq.emplace_front(v)); }; for (auto& e: G[r]) { if (r <= e.to && chmin(dist[e.to], e.cost)) par_e[e.to] = e.id, push(e.to, e.cost); } while (len(deq)) { auto v = POP(deq); for (auto& e: G[v]) { if (r <= e.to && chmin(dist[e.to], dist[v] + e.cost)) { par_e[e.to] = e.id, push(e.to, e.cost); } } } } else { for (auto& e: G[r]) { if (r <= e.to && chmin(dist[e.to], e.cost)) { par_e[e.to] = e.id, que.emplace(e.cost, e.to); } } while (len(que)) { auto [dv, v] = POP(que); if (dist[v] != dv) continue; for (auto& e: G[v]) { T x = dv + e.cost; if (r <= e.to && chmin(dist[e.to], x)) { par_e[e.to] = e.id, que.emplace(x, e.to); } } } } if (chmin(ans, dist[r])) { vs.clear(), es.clear(); vs.eb(r); while (1) { int eid = par_e[vs.back()]; es.eb(eid); vs.eb(G.edges[eid].frm); if (vs.back() == r) break; } reverse(all(vs)); reverse(all(es)); }; } return {ans, vs, es}; } // {wt, vs, es}, O(N * shortest path) template <typename T, typename GT> tuple<T, vc<int>, vc<int>> minimum_cost_cycle_undirected(GT& G) { const int N = G.N; T ans = infty<T>; vc<T> dist(N); vc<int> par_e(N); vc<int> vs, es; FOR(r, N) { fill(dist.begin() + r, dist.end(), infty<T>); pqg<pair<T, int>> que; dist[r] = 0, que.emplace(0, r); while (len(que)) { auto [dv, v] = POP(que); if (dist[v] != dv) continue; for (auto& e: G[v]) { if (e.to < r) continue; T x = dv + e.cost; if (chmin(dist[e.to], x)) { par_e[e.to] = e.id; que.emplace(x, e.to); } } } int best_e = -1; for (auto& e: G.edges) { int a = e.frm, b = e.to; if (a < r || b < r || par_e[a] == e.id || par_e[b] == e.id) continue; if (chmin(ans, dist[a] + dist[b] + e.cost)) best_e = e.id; } if (best_e == -1) continue; vs.clear(), es.clear(); auto& e = G.edges[best_e]; int a = e.frm, b = e.to; // r -> a while (a != r) { int eid = par_e[a]; vs.eb(a), es.eb(eid); a = G.edges[eid].frm ^ G.edges[eid].to ^ a; } vs.eb(a); reverse(all(vs)), reverse(all(es)); es.eb(best_e); while (b != r) { int eid = par_e[b]; vs.eb(b), es.eb(eid); b = G.edges[eid].frm ^ G.edges[eid].to ^ b; } vs.eb(b); } return {ans, vs, es}; } // {wt, vs, es}, O(N * shortest path) template <typename T, typename GT> tuple<T, vc<int>, vc<int>> minimum_cost_cycle(GT& G) { for (auto& e: G.edges) assert(e.cost >= 0); if constexpr (GT::is_directed) { return minimum_cost_cycle_directed<T>(G); } else { return minimum_cost_cycle_undirected<T>(G); } }