#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 #include #include #include #include #include template class radix_heap_array { int sz; Uint last; std::array>, std::numeric_limits::digits + 1> v; struct smallpii { unsigned b : 7; int j : 25; }; std::vector i2bj; template ::type * = nullptr> static inline unsigned bucket(U x) noexcept { return x ? 32 - __builtin_clz(x) : 0; } template ::type * = nullptr> static inline unsigned bucket(U x) noexcept { return x ? 64 - __builtin_clzll(x) : 0; } void pull() { if (!v[0].empty()) return; int b = 1; while (v[b].empty()) ++b; last = v[b].back().first; for (int j = 0; j < int(v[b].size()); j++) last = std::min(last, v[b][j].first); for (int j = 0; j < int(v[b].size()); j++) { int i = v[b][j].second; auto bnxt = bucket(v[b][j].first ^ last); i2bj[i] = {bnxt, int(v[bnxt].size())}, v[bnxt].emplace_back(std::move(v[b][j])); } v[b].clear(); } public: radix_heap_array() : sz(0), last(0) {} bool empty() const noexcept { return sz == 0; } int argmin_pop() { pull(), --sz; int i = v[0].back().second; i2bj[i].j = -1; v[0].pop_back(); return i; } void chmin(Uint vnew, int i) { if (i >= int(i2bj.size())) i2bj.resize(i + 1, {0, -1}); if (i2bj[i].j < 0) { auto b = bucket(vnew ^ last); ++sz, i2bj[i] = {b, int(v[b].size())}, v[b].emplace_back(vnew, i); } else if (v[i2bj[i].b][i2bj[i].j].first > vnew) { auto bold = i2bj[i].b, bnew = bucket(vnew ^ last); if (bnew < bold) { int ilast = v[bold].back().second, j = i2bj[i].j; std::swap(v[bold][j], v[bold].back()); i2bj[ilast].j = j, i2bj[i] = {bnew, int(v[bnew].size())}; v[bnew].emplace_back(vnew, i), v[bold].pop_back(); } else { v[bold][i2bj[i].j].first = vnew; } } } void pop() { argmin_pop(); } std::pair top() { return pull(), v[0].back(); } [[deprecated("NOT usual emplace() opeation!")]] void emplace(Uint vnew, int i) { chmin(vnew, i); } void clear() noexcept { sz = 0, last = 0, i2bj.clear(); } }; // 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 { 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; auto dt = dual_dist[t].first; for (int v = 0; v < _n; v++) dual_dist[v].first -= dt; is_dual_infeasible = false; } 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); radix_heap_array heap; heap.chmin(0, s); // while (!que_min.empty() || !que.empty()) { while (!heap.empty()) { int v = heap.argmin_pop(); // 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; heap.chmin(dist_to, e.to); // 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'; }