#include #include #include #include using namespace std; int N, M, K; vector> C; // C[b][p] = bin b stakes with DSU parent p unordered_set S; // Set of currently uniform bin numbers void update_parent(int i, int r_old, int r_new) { if (r_old == r_new) { return; } auto &bin = C[i / K]; if (--bin[r_old] == 0) { bin.erase(r_old); } bin[r_new]++; if (bin.size() == 1) { S.insert(i / K); } else { S.erase(i / K); } } struct disjoint_sets { vector> values; vector parent, color_of_parent, parent_of_color; public: disjoint_sets(int N) : values(N), parent(N), color_of_parent(N), parent_of_color(N) { for (int i = 0; i < N; i++) { values[i] = {i}; parent[i] = i; color_of_parent[i] = i; parent_of_color[i] = i; } } // O(M + N log N) across M calls to unite(). void unite(int a, int b) { a = parent[a]; b = parent[b]; if (a != b) { if (values[a].size() < values[b].size()) { swap(a, b); } while (!values[b].empty()) { int v = values[b].back(); values[b].pop_back(); update_parent(v, parent[v], a); parent[v] = a; color_of_parent[v] = color_of_parent[a]; values[a].push_back(v); } } } void repaint(int c1, int c2) { int pa = parent_of_color[c1]; if (pa == -1) { return; } color_of_parent[pa] = c2; parent_of_color[c1] = -1; int pb = parent_of_color[c2]; if (pb == -1) { parent_of_color[c2] = pa; return; } unite(pa, pb); color_of_parent[pb] = c2; parent_of_color[c2] = parent[pb]; } }; int solve() { cin >> N >> M >> K; int nbins = (N + K - 1) / K; // ceil(N / K) // Initialize parent counts. C.assign(nbins, {}); for (int i = 0; i < N; i++) { C[i / K][i]++; } S.clear(); for (int b = 0; b < nbins; b++) { if (C[b].size() == 1) { S.insert(b); } } int ans = (S.size() == nbins ? 0 : -1); disjoint_sets DS(N); for (int i = 0, a, b; i < M; i++) { cin >> a >> b; DS.repaint(--a, --b); if (S.size() == nbins && ans < 0) { ans = i + 1; } } return ans; } int main() { ios_base::sync_with_stdio(false); cin.tie(nullptr); int T; cin >> T; for (int t = 1; t <= T; t++) { cout << "Case #" << t << ": " << solve() << endl; } return 0; }