#include #include #include #include #include using namespace std; typedef long long int64; const int64 kInf = 12345678912345LL; const int64 MAXK = 50, KLIM = (MAXK + 1) << 1, NLIM = 1000; int N, M, K; vector weights; vector> adj; int64 f[2][KLIM][KLIM][NLIM]; int timer; vector tin, tout, par; vector cycle_ids, cycle_lens, cycle_rev, dist_to_par; vector> edge_lnks; vector> dp_edges; void dfs(int x) { tin[x] = timer++; for (int y : adj[x]) { if (tin[y] != -1) { continue; } dfs(y); } tout[x] = timer++; } int cycle_counter; vector dfs_stack; void dfs_cycles(int x, int p) { tin[x] = timer++; dfs_stack.push_back(x); for (int y : adj[x]) { if (y == p) { continue; } if (y == par[x]) { int cur_len = 1; int rev_cycle = -1; for (int z = (int)dfs_stack.size() - 1; dfs_stack[z] != par[x]; z--) { int edge_id = edge_lnks[dfs_stack[z - 1]][dfs_stack[z]]; int rev_edge_id = edge_lnks[dfs_stack[z]][dfs_stack[z - 1]]; if (rev_edge_id != -1 && cycle_ids[rev_edge_id] != -1) { rev_cycle = cycle_ids[rev_edge_id]; cycle_rev[rev_cycle] = cycle_counter; } cycle_ids[edge_id] = cycle_counter; dist_to_par[edge_id] = cur_len++; } cycle_lens.push_back(cur_len); cycle_rev.push_back(rev_cycle); cycle_counter++; } else { if (tin[y] != -1) { continue; } if (edge_lnks[x][y] == -1) { edge_lnks[x][y] = dp_edges.size(); dp_edges.emplace_back(x, y); cycle_ids.push_back(-1); dist_to_par.push_back(-1); } dfs_cycles(y, x); } } dfs_stack.pop_back(); } int lazy_max_dist; int64 lazy_dp(int edge_id, int dist_to_last, int par_dist, int promised) { if (dist_to_last > K + 1) { promised = 1; } if (f[promised][dist_to_last][par_dist][edge_id] != -1LL) { return f[promised][dist_to_last][par_dist][edge_id]; } const int base_cycle_id = cycle_ids[edge_id]; if (!dist_to_last && base_cycle_id != -1 && cycle_lens[base_cycle_id] - dist_to_par[edge_id] < par_dist) { return f[promised][dist_to_last][par_dist][edge_id] = lazy_dp( edge_id, dist_to_last, cycle_lens[base_cycle_id] - dist_to_par[edge_id], promised); } if (!promised && base_cycle_id != -1 && dist_to_par[edge_id] + par_dist < dist_to_last) { return f[promised][dist_to_last][par_dist][edge_id] = lazy_dp( edge_id, dist_to_par[edge_id] + par_dist, par_dist, promised); } const int x = dp_edges[edge_id].second; int64 res = kInf; if (dist_to_last) { res = min(res, lazy_dp(edge_id, 0, par_dist, 0) + weights[x]); } if (dist_to_last == lazy_max_dist) { return f[promised][dist_to_last][par_dist][edge_id] = res; } const int p = dp_edges[edge_id].first; vector edge_list; vector> edge_groups; map cycle_to_ind; for (int y : adj[x]) { if (y == p) { continue; } int new_edge_id = edge_lnks[x][y]; if (new_edge_id == -1) { continue; } int cycle_id = cycle_ids[new_edge_id]; if (cycle_id != -1) { auto mit = cycle_to_ind.find(cycle_id); if (mit != cycle_to_ind.end()) { edge_groups[mit->second].second = edge_list.size(); edge_list.push_back(new_edge_id); continue; } } int rev_cycle_id = (cycle_id == -1 ? -1 : cycle_rev[cycle_id]); cycle_to_ind[rev_cycle_id] = edge_groups.size(); edge_groups.emplace_back(edge_list.size(), -1); edge_list.push_back(new_edge_id); } const int esz = edge_list.size(); if (!esz) { return f[promised][dist_to_last][par_dist][edge_id] = (promised || dist_to_last > K) ? res : 0LL; } const int egsz = edge_groups.size(); vector edge_f(esz), edge_f2(esz); for (int new_dist_to_last = dist_to_last + 1, new_promised = promised; new_dist_to_last <= lazy_max_dist && res; ++new_dist_to_last, new_promised = 1) { const int inv_dist_to_last = lazy_max_dist - new_dist_to_last; const int min_dist_to_last = min(dist_to_last + 1, 2 + inv_dist_to_last); for (int i = 0; i < esz; ++i) { const int new_edge_id = edge_list[i]; const int new_cycle_id = cycle_ids[new_edge_id]; int new_par_dist = lazy_max_dist; if (new_cycle_id != -1) { if (new_cycle_id == base_cycle_id) { new_par_dist = par_dist; new_par_dist = min(new_par_dist, 1 + cycle_lens[base_cycle_id] - dist_to_par[edge_id] + inv_dist_to_last); } else { new_par_dist = min_dist_to_last - 1; } } edge_f[i] = lazy_dp(new_edge_id, min_dist_to_last, new_par_dist, 0); } for (int i = 0; i < esz; i++) { const int new_edge_id = edge_list[i]; const int new_cycle_id = cycle_ids[new_edge_id]; int new_par_dist = lazy_max_dist; if (new_cycle_id != -1) { if (new_cycle_id == base_cycle_id) { new_par_dist = par_dist; } else { new_par_dist = dist_to_last; } } edge_f2[i] = lazy_dp(new_edge_id, new_dist_to_last, new_par_dist, new_promised); } int64 f_sum = 0LL, min_diff = kInf; for (int i = 0; i < egsz; i++) { const int ind1 = edge_groups[i].first; int64 edge_group_f = edge_f[ind1], edge_group_f2 = edge_f2[ind1]; const int ind2 = edge_groups[i].second; if (ind2 != -1) { edge_group_f = min(edge_group_f, edge_f[ind2]); edge_group_f2 = min(edge_group_f2, edge_f2[ind2]); } min_diff = min(min_diff, edge_group_f2 - edge_group_f); f_sum += edge_group_f; } res = min(res, f_sum + min_diff); } return f[promised][dist_to_last][par_dist][edge_id] = res; } int64 solve() { vector> edges; cin >> N >> M >> K; weights.resize(N); for (int i = 0; i < N; ++i) { cin >> weights[i]; } edges.resize(M); for (int i = 0; i < M; ++i) { cin >> edges[i].first >> edges[i].second; --edges[i].first; --edges[i].second; } int root = N; edges.emplace_back(N++, 0); M++; adj.assign(N, {}); for (int i = 0; i < M; i++) { adj[edges[i].first].push_back(edges[i].second); adj[edges[i].second].push_back(edges[i].first); } timer = 0; tin.assign(N, -1); tout.resize(N, -1); dfs(root); vector tin0 = tin, tout0 = tout; for (int x = 0; x < N; x++) { reverse(adj[x].begin(), adj[x].end()); } timer = 0; tin.assign(N, -1); tout.resize(N, -1); dfs(root); par.assign(N, -1); for (int x = 0; x < N; x++) { for (int y : adj[x]) { if (tin0[y] < tin0[x] && tin[y] < tin[x]) { par[x] = y; } } } cycle_counter = 0; dp_edges.clear(); edge_lnks.assign(N, vector(N, -1)); cycle_ids.clear(); cycle_lens.clear(); cycle_rev.clear(); dist_to_par.clear(); timer = 0; tin.assign(N, -1); dfs_stack.clear(); dfs_cycles(root, root); for (int x = 0; x < N; x++) { reverse(adj[x].begin(), adj[x].end()); } timer = 0; tin.assign(N, -1); dfs_cycles(root, root); memset(f, -1, sizeof f); lazy_max_dist = ((K + 1) << 1) - 1; int64 res = kInf; for (int i = 0; i <= K; i++) { res = min(res, lazy_dp(edge_lnks[root][0], i + K + 1, lazy_max_dist, 1)); } return res; } int main() { int T; cin >> T; for (int t = 1; t <= T; t++) { cout << "Case #" << t << ": " << solve() << endl; } return 0; }