#include #define REP_(i, a_, b_, a, b, ...) \ for (int i = (a), END_##i = (b); i < END_##i; ++i) #define REP(i, ...) REP_(i, __VA_ARGS__, __VA_ARGS__, 0, __VA_ARGS__) #define ALL(x) std::begin(x), std::end(x) using i64 = long long; template inline bool chmax(T &a, U b) { return a < b and ((a = std::move(b)), true); } template inline bool chmin(T &a, U b) { return a > b and ((a = std::move(b)), true); } template inline int ssize(const T &a) { return (int) std::size(a); } template std::istream &operator>>(std::istream &is, std::vector &a) { for (auto &x: a) is >> x; return is; } template std::ostream &operator<<(std::ostream &os, const std::pair &a) { return os << "(" << a.first << ", " << a.second << ")"; } template std::ostream &print_seq(const Container &a, std::string_view sep = " ", std::string_view ends = "\n", std::ostream &os = std::cout) { auto b = std::begin(a), e = std::end(a); for (auto it = std::begin(a); it != e; ++it) { if (it != b) os << sep; os << *it; } return os << ends; } template struct is_iterable : std::false_type {}; template struct is_iterable())), decltype(std::end(std::declval()))>> : std::true_type { }; template::value && !std::is_same::value && !std::is_same::value>> std::ostream &operator<<(std::ostream &os, const T &a) { return print_seq(a, ", ", "", (os << "{")) << "}"; } void print() { std::cout << "\n"; } template void print(const T &x) { std::cout << x << "\n"; } template void print(const Head &head, Tail... tail) { std::cout << head << " "; print(tail...); } struct Input { template operator T() const { T x; std::cin >> x; return x; } } in; #ifdef MY_DEBUG #include "debug_dump.hpp" #else #define DUMP(...) #endif using namespace std; enum Objective { MINIMIZE = 1, MAXIMIZE = -1, }; enum class Status { OPTIMAL, INFEASIBLE, }; template class MinCostFlow { using V_id = uint32_t; using E_id = uint32_t; class Edge { friend class MinCostFlow; V_id src, dst; Flow flow, cap; Cost cost; E_id rev; public: Edge() = default; Edge(const V_id src, const V_id dst, const Flow cap, const Cost cost, const E_id rev) : src(src), dst(dst), flow(0), cap(cap), cost(cost), rev(rev) {} [[nodiscard]] Flow residual_cap() const { return cap - flow; } }; public: class EdgePtr { friend class MinCostFlow; const MinCostFlow *instance; V_id v; E_id e; EdgePtr(const MinCostFlow *const instance, const V_id v, const E_id e) : instance(instance), v(v), e(e) {} [[nodiscard]] const Edge &edge() const { return instance->g[v][e]; } [[nodiscard]] const Edge &rev() const { const Edge &e = edge(); return instance->g[e.dst][e.rev]; } public: EdgePtr() = default; [[nodiscard]] V_id src() const { return v; } [[nodiscard]] V_id dst() const { return edge().dst; } [[nodiscard]] Flow flow() const { return edge().flow; } [[nodiscard]] Flow lower() const { return -rev().cap; } [[nodiscard]] Flow upper() const { return edge().cap; } [[nodiscard]] Cost cost() const { return edge().cost; } [[nodiscard]] Cost gain() const { return -edge().cost; } }; private: V_id n; std::vector> g; std::vector b; public: MinCostFlow() : n(0) {} V_id add_vertex() { ++n; g.resize(n); b.resize(n); return n - 1; } std::vector add_vertices(const size_t size) { std::vector ret(size); std::iota(std::begin(ret), std::end(ret), n); n += size; g.resize(n); b.resize(n); return ret; } EdgePtr add_edge(const V_id src, const V_id dst, const Flow lower, const Flow upper, const Cost cost) { const E_id e = g[src].size(), re = src == dst ? e + 1 : g[dst].size(); assert(lower <= upper); g[src].emplace_back(Edge{src, dst, upper, cost * objective, re}); g[dst].emplace_back(Edge{dst, src, -lower, -cost * objective, e}); return EdgePtr{this, src, e}; } void add_supply(const V_id v, const Flow amount) { b[v] += amount; } void add_demand(const V_id v, const Flow amount) { b[v] -= amount; } private: // Variables used in calculation static Cost constexpr unreachable = std::numeric_limits::max(); Cost farthest; std::vector potential; std::vector dist; std::vector parent; // out-forrest. std::priority_queue, std::vector>, std::greater<>> pq; // should be empty outside of dual() std::vector excess_vs, deficit_vs; Edge &rev(const Edge &e) { return g[e.dst][e.rev]; } void push(Edge &e, const Flow amount) { e.flow += amount; g[e.dst][e.rev].flow -= amount; } Cost residual_cost(const V_id src, const V_id dst, const Edge &e) { return e.cost + potential[src] - potential[dst]; } bool dual() { dist.assign(n, unreachable); parent.assign(n, nullptr); excess_vs.erase(std::remove_if(std::begin(excess_vs), std::end(excess_vs), [&](const V_id v) { return b[v] <= 0; }), std::end(excess_vs)); deficit_vs.erase(std::remove_if(std::begin(deficit_vs), std::end(deficit_vs), [&](const V_id v) { return b[v] >= 0; }), std::end(deficit_vs)); for (const auto v: excess_vs) pq.emplace(dist[v] = 0, v); farthest = 0; std::size_t deficit_count = 0; while (!pq.empty()) { const auto[d, u] = pq.top(); pq.pop(); if (dist[u] < d) continue; farthest = d; if (b[u] < 0) ++deficit_count; if (deficit_count >= deficit_vs.size()) break; for (auto &e: g[u]) { if (e.residual_cap() <= 0) continue; const auto v = e.dst; const auto new_dist = d + residual_cost(u, v, e); if (new_dist >= dist[v]) continue; pq.emplace(dist[v] = new_dist, v); parent[v] = &e; } } pq = decltype(pq)(); // pq.clear() doesn't exist. for (V_id v = 0; v < n; ++v) { potential[v] += std::min(dist[v], farthest); } return deficit_count > 0; } void primal() { for (const auto t: deficit_vs) { if (dist[t] > farthest) continue; Flow f = -b[t]; V_id v; for (v = t; parent[v] != nullptr; v = parent[v]->src) { f = std::min(f, parent[v]->residual_cap()); } f = std::min(f, b[v]); if (f <= 0) continue; for (v = t; parent[v] != nullptr;) { auto &e = *parent[v]; push(e, f); int u = parent[v]->src; if (e.residual_cap() <= 0) parent[v] = nullptr; v = u; } b[t] += f; b[v] -= f; } } public: std::pair solve() { potential.resize(n); for (auto &es: g) for (auto &e: es) { const Flow rcap = e.residual_cap(); const Cost rcost = residual_cost(e.src, e.dst, e); if (rcost < 0 || rcap < 0) { push(e, rcap); b[e.src] -= rcap; b[e.dst] += rcap; } } for (V_id v = 0; v < n; ++v) if (b[v] != 0) { (b[v] > 0 ? excess_vs : deficit_vs).emplace_back(v); } while (dual()) primal(); Cost value = 0; for (const auto &es: g) for (const auto &e: es) { value += e.flow * e.cost; } value /= 2; if (excess_vs.empty() && deficit_vs.empty()) { return {Status::OPTIMAL, value / objective}; } else { return {Status::INFEASIBLE, value / objective}; } } std::tuple solve(const V_id s, const V_id t) { assert(s != t); Flow inf_flow = std::abs(b[s]); for (const auto &e: g[s]) inf_flow += std::max(e.cap, static_cast(0)); add_edge(t, s, 0, inf_flow, 0); const auto[status, circulation_value] = solve(); if (status == Status::INFEASIBLE) { g[s].pop_back(); g[t].pop_back(); return {status, circulation_value, 0}; } inf_flow = std::abs(b[s]); for (const auto &e: g[s]) inf_flow += e.residual_cap(); b[s] += inf_flow; b[t] -= inf_flow; const auto[mf_status, mf_value] = solve(); b[s] -= inf_flow; b[t] += inf_flow; g[s].pop_back(); g[t].pop_back(); return {Status::OPTIMAL, mf_value, b[t]}; } }; template using MaxGainFlow = MinCostFlow; auto solve() { const int n = in, K = in; MaxGainFlow g; const auto vs = g.add_vertices(n); g.add_supply(vs[0], K); g.add_demand(vs[n - 1], K); vector a(n); REP(i, n) { a[i] = in; int m = in; REP(j, m) { int b = in; --b; g.add_edge(vs[b], vs[i], 0, 1, a[i] - a[b]); } } REP(i, n - 1) { g.add_edge(vs[i], vs[i + 1], 0, K, 0); } const auto[mf_status, mf_value] = g.solve(); assert (mf_status == Status::OPTIMAL); return mf_value; } int main() { ios_base::sync_with_stdio(false), cin.tie(nullptr); cout << std::fixed << std::setprecision(18); const int t = 1; REP(test_case, t) { auto ans = solve(); print(ans); } }