#line 1 "combinatorial_opt/test/mincostflow.yuki1288.test.cpp" #define PROBLEM "https://yukicoder.me/problems/no/1288" #line 2 "combinatorial_opt/mincostflow_nonegativeloop.hpp" #include #include #include #include /* // CUT begin // Minimum cost flow WITH NO NEGATIVE CYCLE (just negative cost edge is allowed) // Verified: // - SRM 770 Div1 Medium https://community.topcoder.com/stat?c=problem_statement&pm=15702 // - CodeChef LTIME98 Ancient Magic https://www.codechef.com/problems/ANCT template ::max() / 2> struct MinCostFlow { struct _edge { int to, rev; Cap cap; Cost cost; template friend Ostream &operator<<(Ostream &os, const _edge &e) { return os << '(' << e.to << ',' << e.rev << ',' << e.cap << ',' << e.cost << ')'; } }; bool _is_dual_infeasible; int V; std::vector> g; std::vector dist; std::vector prevv, preve; std::vector dual; // dual[V]: potential std::vector> pos; bool _initialize_dual_dag() { std::vector deg_in(V); for (int i = 0; i < V; i++) { for (const auto &e : g[i]) deg_in[e.to] += (e.cap > 0); } std::vector st; st.reserve(V); for (int i = 0; i < V; i++) { if (!deg_in[i]) st.push_back(i); } for (int n = 0; n < V; n++) { if (int(st.size()) == n) return false; // Not DAG int now = st[n]; for (const auto &e : g[now]) { if (!e.cap) continue; deg_in[e.to]--; if (deg_in[e.to] == 0) st.push_back(e.to); if (dual[e.to] >= dual[now] + e.cost) dual[e.to] = dual[now] + e.cost; } } return true; } bool _initialize_dual_spfa() { // Find feasible dual's when negative cost edges exist dual.assign(V, 0); std::queue q; std::vector in_queue(V); std::vector nvis(V); for (int i = 0; i < V; i++) q.push(i), in_queue[i] = true; while (q.size()) { int now = q.front(); q.pop(), in_queue[now] = false; if (nvis[now] > V) return false; // Negative cycle exists nvis[now]++; for (const auto &e : g[now]) { if (!e.cap) continue; if (dual[e.to] > dual[now] + e.cost) { dual[e.to] = dual[now] + e.cost; if (!in_queue[e.to]) in_queue[e.to] = true, q.push(e.to); } } } return true; } bool initialize_dual() { return !_is_dual_infeasible or _initialize_dual_dag() or _initialize_dual_spfa(); } template void _dijkstra(int s) { // O(ElogV) prevv.assign(V, -1); preve.assign(V, -1); dist.assign(V, INF_COST); dist[s] = 0; heap q; q.emplace(0, s); while (!q.empty()) { auto p = q.top(); q.pop(); int v = p.second; if (dist[v] < Cost(p.first)) continue; for (int i = 0; i < (int)g[v].size(); i++) { _edge &e = g[v][i]; auto c = dist[v] + e.cost + dual[v] - dual[e.to]; if (e.cap > 0 and dist[e.to] > c) { dist[e.to] = c, prevv[e.to] = v, preve[e.to] = i; q.emplace(dist[e.to], e.to); } } } } MinCostFlow(int V = 0) : _is_dual_infeasible(false), V(V), g(V), dual(V, 0) { static_assert(INF_COST > 0, "INF_COST must be positive"); } int add_edge(int from, int to, Cap cap, Cost cost) { assert(0 <= from and from < V); assert(0 <= to and to < V); assert(cap >= 0); if (cost < 0) _is_dual_infeasible = true; pos.emplace_back(from, g[from].size()); g[from].push_back({to, (int)g[to].size() + (from == to), cap, cost}); g[to].push_back({from, (int)g[from].size() - 1, (Cap)0, -cost}); return int(pos.size()) - 1; } // Flush flow f from s to t. Graph must not have negative cycle. using Pque = std::priority_queue, std::vector>, std::greater>>; template std::pair flow(int s, int t, const Cap &flow_limit) { // You can also use radix_heap::type, int> as prique if (!initialize_dual()) throw; // Fail to find feasible dual Cost cost = 0; Cap flow_rem = flow_limit; while (flow_rem > 0) { _dijkstra(s); if (dist[t] == INF_COST) break; for (int v = 0; v < V; v++) dual[v] = std::min(dual[v] + dist[v], INF_COST); Cap d = flow_rem; for (int v = t; v != s; v = prevv[v]) d = std::min(d, g[prevv[v]][preve[v]].cap); flow_rem -= d; cost += d * (dual[t] - dual[s]); for (int v = t; v != s; v = prevv[v]) { _edge &e = g[prevv[v]][preve[v]]; e.cap -= d; g[v][e.rev].cap += d; } } return std::make_pair(flow_limit - flow_rem, cost); } struct edge { int from, to; Cap cap, flow; Cost cost; template friend Ostream &operator<<(Ostream &os, const edge &e) { return os << '(' << e.from << "->" << e.to << ',' << e.flow << '/' << e.cap << ",$" << e.cost << ')'; } }; edge get_edge(int edge_id) const { int m = int(pos.size()); assert(0 <= edge_id and edge_id < m); auto _e = g[pos[edge_id].first][pos[edge_id].second]; auto _re = g[_e.to][_e.rev]; return {pos[edge_id].first, _e.to, _e.cap + _re.cap, _re.cap, _e.cost}; } std::vector edges() const { std::vector ret(pos.size()); for (int i = 0; i < int(pos.size()); i++) ret[i] = get_edge(i); return ret; } template friend Ostream &operator<<(Ostream &os, const MinCostFlow &mcf) { os << "[MinCostFlow]V=" << mcf.V << ":"; for (int i = 0; i < mcf.V; i++) { for (auto &e : mcf.g[i]) os << "\n" << i << "->" << e.to << ":cap" << e.cap << ",$" << e.cost; } return os; } }; */ template ::max() / 2> struct MinCostFlow { template struct csr { std::vector start; std::vector elist; explicit csr(int n, const std::vector> &edges) : start(n + 1), elist(edges.size()) { for (auto e : edges) { start[e.first + 1]++; } for (int i = 1; i <= n; i++) { start[i] += start[i - 1]; } auto counter = start; for (auto e : edges) { elist[counter[e.first]++] = e.second; } } }; public: MinCostFlow() {} explicit MinCostFlow(int n) : is_dual_infeasible(false), _n(n) { static_assert(std::numeric_limits::max() > 0, "max() must be greater than 0"); } int add_edge(int from, int to, Cap cap, Cost cost) { assert(0 <= from && from < _n); assert(0 <= to && to < _n); assert(0 <= cap); // assert(0 <= cost); if (cost < 0) is_dual_infeasible = true; int m = int(_edges.size()); _edges.push_back({from, to, cap, 0, cost}); return m; } struct edge { int from, to; Cap cap, flow; Cost cost; }; edge get_edge(int i) { int m = int(_edges.size()); assert(0 <= i && i < m); return _edges[i]; } std::vector edges() { return _edges; } std::pair flow(int s, int t) { return flow(s, t, std::numeric_limits::max()); } std::pair flow(int s, int t, Cap flow_limit) { return slope(s, t, flow_limit).back(); } std::vector> slope(int s, int t) { return slope(s, t, std::numeric_limits::max()); } std::vector> slope(int s, int t, Cap flow_limit) { assert(0 <= s && s < _n); assert(0 <= t && t < _n); assert(s != t); int m = int(_edges.size()); std::vector edge_idx(m); auto g = [&]() { std::vector degree(_n), redge_idx(m); std::vector> elist; elist.reserve(2 * m); for (int i = 0; i < m; i++) { auto e = _edges[i]; edge_idx[i] = degree[e.from]++; redge_idx[i] = degree[e.to]++; elist.push_back({e.from, {e.to, -1, e.cap - e.flow, e.cost}}); elist.push_back({e.to, {e.from, -1, e.flow, -e.cost}}); } auto _g = csr<_edge>(_n, elist); for (int i = 0; i < m; i++) { auto e = _edges[i]; edge_idx[i] += _g.start[e.from]; redge_idx[i] += _g.start[e.to]; _g.elist[edge_idx[i]].rev = redge_idx[i]; _g.elist[redge_idx[i]].rev = edge_idx[i]; } return _g; }(); auto result = slope(g, s, t, flow_limit); for (int i = 0; i < m; i++) { auto e = g.elist[edge_idx[i]]; _edges[i].flow = _edges[i].cap - e.cap; } return result; } private: bool is_dual_infeasible; int _n; std::vector _edges; // inside edge struct _edge { int to, rev; Cap cap; Cost cost; }; std::vector> slope(csr<_edge> &g, int s, int t, Cap flow_limit) { // variants (C = maxcost): // -(n-1)C <= dual[s] <= dual[i] <= dual[t] = 0 // reduced cost (= e.cost + dual[e.from] - dual[e.to]) >= 0 for all edge // dual_dist[i] = (dual[i], dist[i]) std::vector> dual_dist(_n); if (is_dual_infeasible) { auto check_dag = [&]() { std::vector deg_in(_n); for (int v = 0; v < _n; v++) { for (int i = g.start[v]; i < g.start[v + 1]; i++) { deg_in[g.elist[i].to] += g.elist[i].cap > 0; } } std::vector st; st.reserve(_n); for (int i = 0; i < _n; i++) { if (!deg_in[i]) st.push_back(i); } for (int n = 0; n < _n; n++) { if (int(st.size()) == n) return false; // Not DAG int now = st[n]; for (int i = g.start[now]; i < g.start[now + 1]; i++) { const auto &e = g.elist[i]; if (!e.cap) continue; deg_in[e.to]--; if (deg_in[e.to] == 0) st.push_back(e.to); if (dual_dist[e.to].first >= dual_dist[now].first + e.cost) dual_dist[e.to].first = dual_dist[now].first + e.cost; } } return true; }(); if (!check_dag) throw; } std::vector prev_e(_n); std::vector vis(_n); struct Q { Cost key; int to; bool operator<(Q r) const { return key > r.key; } }; std::vector que_min; std::vector que; auto dual_ref = [&]() { for (int i = 0; i < _n; i++) { dual_dist[i].second = std::numeric_limits::max(); } std::fill(vis.begin(), vis.end(), false); que_min.clear(); que.clear(); // que[0..heap_r) was heapified unsigned heap_r = 0; dual_dist[s].second = 0; que_min.push_back(s); while (!que_min.empty() || !que.empty()) { int v; if (!que_min.empty()) { v = que_min.back(); que_min.pop_back(); } else { while (heap_r < que.size()) { heap_r++; std::push_heap(que.begin(), que.begin() + heap_r); } v = que.front().to; std::pop_heap(que.begin(), que.end()); que.pop_back(); heap_r--; } if (vis[v]) continue; vis[v] = true; if (v == t) break; // dist[v] = shortest(s, v) + dual[s] - dual[v] // dist[v] >= 0 (all reduced cost are positive) // dist[v] <= (n-1)C Cost dual_v = dual_dist[v].first, dist_v = dual_dist[v].second; for (int i = g.start[v]; i < g.start[v + 1]; i++) { auto e = g.elist[i]; if (!e.cap) continue; // |-dual[e.to] + dual[v]| <= (n-1)C // cost <= C - -(n-1)C + 0 = nC Cost cost = e.cost - dual_dist[e.to].first + dual_v; if (dual_dist[e.to].second - dist_v > cost) { Cost dist_to = dist_v + cost; dual_dist[e.to].second = dist_to; prev_e[e.to] = e.rev; if (dist_to == dist_v) { que_min.push_back(e.to); } else { que.push_back(Q{dist_to, e.to}); } } } } if (!vis[t]) { return false; } for (int v = 0; v < _n; v++) { if (!vis[v]) continue; // dual[v] = dual[v] - dist[t] + dist[v] // = dual[v] - (shortest(s, t) + dual[s] - dual[t]) + // (shortest(s, v) + dual[s] - dual[v]) = - shortest(s, // t) + dual[t] + shortest(s, v) = shortest(s, v) - // shortest(s, t) >= 0 - (n-1)C dual_dist[v].first -= dual_dist[t].second - dual_dist[v].second; } return true; }; Cap flow = 0; Cost cost = 0, prev_cost_per_flow = -1; std::vector> result = {{Cap(0), Cost(0)}}; while (flow < flow_limit) { if (!dual_ref()) break; Cap c = flow_limit - flow; for (int v = t; v != s; v = g.elist[prev_e[v]].to) { c = std::min(c, g.elist[g.elist[prev_e[v]].rev].cap); } for (int v = t; v != s; v = g.elist[prev_e[v]].to) { auto &e = g.elist[prev_e[v]]; e.cap += c; g.elist[e.rev].cap -= c; } Cost d = -dual_dist[s].first; flow += c; cost += c * d; if (prev_cost_per_flow == d) { result.pop_back(); } result.push_back({flow, cost}); prev_cost_per_flow = d; } return result; } }; #line 3 "combinatorial_opt/test/mincostflow.yuki1288.test.cpp" #include #include #include #line 7 "combinatorial_opt/test/mincostflow.yuki1288.test.cpp" using namespace std; int main() { int N; string S; cin >> N >> S; vector V(N); for (auto &x : V) cin >> x; const int s = N * 5, t = s + 1; MinCostFlow graph(t + 1); for (int d = 0; d < 5; d++) { for (int i = 0; i < N - 1; i++) graph.add_edge(d * N + i, d * N + i + 1, N / 4, 0); } graph.add_edge(s - 1, 0, N / 4, 0); for (int i = 0; i < N; i++) { int b = 0; if (S[i] == 'u') b = N * 1; if (S[i] == 'k') b = N * 2; if (S[i] == 'i') b = N * 3; int fr = b + i + N, to = b + i; graph.add_edge(s, fr, 1, 0); graph.add_edge(fr, to, 1, V[i]); graph.add_edge(to, t, 1, 0); } auto cost = graph.flow(s, t, N).second; cout << accumulate(V.begin(), V.end(), 0LL) - cost << '\n'; }