// O(n^4 log n) weighted-linear-matroid-parity-based solution #include #include #include #include #include #include using namespace std; #include #include using mint = atcoder::static_modint<(1 << 30) + 3>; mt19937 mt(530629); uniform_int_distribution rndgen(0, mint::mod()); // Upper Hessenberg reduction of square matrices // Complexity: O(n^3) // Reference: // http://www.phys.uri.edu/nigh/NumRec/bookfpdf/f11-5.pdf template void hessenberg_reduction(std::vector> &M) { assert(M.size() == M[0].size()); const int N = M.size(); for (int r = 0; r < N - 2; r++) { int piv = -1; for (int j = r + 1; j < N; ++j) if (M[j][r] != 0) { piv = j; break; } if (piv < 0) continue; for (int i = 0; i < N; i++) std::swap(M[r + 1][i], M[piv][i]); for (int i = 0; i < N; i++) std::swap(M[i][r + 1], M[i][piv]); const auto rinv = Tp(1) / M[r + 1][r]; for (int i = r + 2; i < N; i++) { const auto n = M[i][r] * rinv; for (int j = 0; j < N; j++) M[i][j] -= M[r + 1][j] * n; for (int j = 0; j < N; j++) M[j][r + 1] += M[j][i] * n; } } } // Characteristic polynomial of matrix M (|xI - M|) // Complexity: O(n^3) // R. Rehman, I. C. Ipsen, "La Budde's Method for Computing Characteristic Polynomials," 2011. template std::vector characteristic_poly(std::vector> &M) { hessenberg_reduction(M); const int N = M.size(); std::vector> p(N + 1); // p[i + 1] = (Characteristic polynomial of i-th leading principal minor) p[0] = {1}; for (int i = 0; i < N; i++) { p[i + 1].assign(i + 2, 0); for (int j = 0; j < i + 1; j++) p[i + 1][j + 1] += p[i][j]; for (int j = 0; j < i + 1; j++) p[i + 1][j] -= p[i][j] * M[i][i]; Tp betas = 1; for (int j = i - 1; j >= 0; j--) { betas *= M[j + 1][j]; Tp hb = -M[j][i] * betas; for (int k = 0; k < j + 1; k++) p[i + 1][k] += hb * p[j][k]; } } return p[N]; } // (X, N), (Y, N + 1), (Z, N + 2) shortest S-paths, without using vertices in `used_vs` // return -1 if not found int shortest_len(int N, int X, int Y, int Z, const vector> &conn, const vector &used_vs) { vector is_alive(N + 5, 1); for (auto i : used_vs) is_alive[i] = 0; vector alive_vs; for (int i = 0; i < int(is_alive.size()); ++i) { if (is_alive[i]) alive_vs.push_back(i); } vector vid(N + 5, -1); for (int j = 0; j < int(alive_vs.size()); ++j) vid[alive_vs[j]] = j; const vector terminal_vs{X, Y, Z, N, N + 1, N + 2, N + 3, N + 4}; auto Label = [&](int i) -> int { if (i == X or i == N) return 1; if (i == Y or i == N + 1) return 2; if (i == Z or i == N + 2) return 3; if (i == N + 3) return 4; if (i == N + 4) return 5; return 0; }; // 論文通りに実装すると Z は 2(N + 5) * M 行列となるが,all zero な行が 8 つ発生して厄介 // なので,これらを潰して (2N + 2) * M 行列(feasible ならばこれは行フルランク)を構築. // 潰した行を飛ばした index を取得する関数 auto truncate = [&](int i) -> int { int red = 0; for (auto l : terminal_vs) { if (l * 2 < i) ++red; } return vid[i / 2] * 2 + (i % 2) - red; }; int r = alive_vs.size() * 2 - 8; // M(x) = mat0 + mat1 * x // mat = [mat1, mat0] を考える vector mat(r, vector(r * 2)); auto add_edge = [&](int u, int v, int w) { vector> bret, cret; if (Label(u)) { bret.emplace_back(truncate(u * 2 + 1), -Label(u)); } else { bret.emplace_back(truncate(u * 2), 1); } if (Label(v)) { bret.emplace_back(truncate(v * 2 + 1), Label(v)); } else { bret.emplace_back(truncate(v * 2), -1); } cret.emplace_back(truncate(u * 2 + 1), 1); cret.emplace_back(truncate(v * 2 + 1), -1); mint x = rndgen(mt); for (auto [i, bi] : bret) { for (auto [j, cj] : cret) { auto v = bi * cj * x; if (w == 0) { mat[i][r + j] += v; mat[j][r + i] -= v; mat[i][j] += v; mat[j][i] -= v; } if (w == 1) { mat[i][j] += v; mat[j][i] -= v; } } } }; for (int i = 0; i < N + 3; ++i) { if (!is_alive[i]) continue; for (int j = 0; j < N + 3; ++j) { if (!conn[i][j] or i > j or !is_alive[j]) continue; add_edge(i, j, 1); } if (!Label(i)) add_edge(i, N + 3, 0); } add_edge(N + 3, N + 4, 0); // det(x mat1 + mat0) を x の多項式として求めたい // mat1 を掃き出して det(x \lambda I - A) の形に帰着させる for (int i = 0; i < r; ++i) { int piv = -1; for (int h = i; h < r; ++h) { if (mat[h][i] != 0) piv = h; } if (piv < 0) return -1; assert(piv >= i); swap(mat[i], mat[piv]); mint inv = mat[i][i].inv(); for (auto &x : mat[i]) x *= inv; for (int h = 0; h < r; ++h) { if (h == i) continue; if (mat[h][i] == 0) continue; const mint coeff = mat[h][i]; for (int w = 0; w < r * 2; ++w) { mat[h][w] -= coeff * mat[i][w]; } } } // det(xI - A) : https://judge.yosupo.jp/problem/characteristic_polynomial for (auto &v : mat) { v.erase(v.begin(), v.begin() + r); for (auto &x : v) x = -x; } auto det_poly = characteristic_poly(mat); int ret = 0; while (ret < int(det_poly.size()) and det_poly[ret] == 0) ++ret; if (ret < int(det_poly.size())) { return ret / 2; } else { return -1; } } // used_vs に含まれる頂点は使わずに,from -> to1 と from->to2 の点素なパスを構成する. // 両方のパスが構築できなければ empty vector の組を返す. pair, vector> twopaths(int N, const vector> &to, const vector &used_vs, int from, int to1, int to2) { const int gt = N * 2; atcoder::mcf_graph graph(gt + 1); vector valid_v(N, 1); for (auto i : used_vs) valid_v[i] = 0; valid_v[to1] = valid_v[to2] = 1; for (int i = 0; i < N; ++i) { graph.add_edge(i, i + N, valid_v[i], 0); } for (int i = 0; i < N; ++i) { for (auto j : to[i]) { graph.add_edge(i + N, j, 1, 1); } } graph.add_edge(to1 + N, gt, 1, 0); graph.add_edge(to2 + N, gt, 1, 0); auto f = graph.flow(from + N, gt, 2); if (f.first < 2) return {{}, {}}; vector conn(N); for (auto e : graph.edges()) { if (e.flow) { if (e.to == gt) continue; int s = e.from % N, t = e.to % N; conn[s] ^= t; conn[t] ^= s; // ループがないので生えている辺の xor だけ持っておけば後で解が復元できる } } vector ret1, ret2; while (to1 != from) { ret1.push_back(to1); to1 = conn[to1]; conn[to1] ^= ret1.back(); } while (to2 != from) { ret2.push_back(to2); to2 = conn[to2]; conn[to2] ^= ret2.back(); } ret1.push_back(from); ret2.push_back(from); reverse(ret1.begin(), ret1.end()); reverse(ret2.begin(), ret2.end()); return {ret1, ret2}; } template ::max() / 2, int INVALID = -1> struct ShortestPath { int V, E; bool single_positive_weight; T wmin, wmax; std::vector>> to; ShortestPath(int V = 0) : V(V), E(0), single_positive_weight(true), wmin(0), wmax(0), to(V) {} void add_edge(int s, int t, T w) { assert(0 <= s and s < V); assert(0 <= t and t < V); to[s].emplace_back(t, w); E++; if (w > 0 and wmax > 0 and wmax != w) single_positive_weight = false; wmin = std::min(wmin, w); wmax = std::max(wmax, w); } std::vector dist; std::vector prev; void ZeroOneBFS(int s) { assert(0 <= s and s < V); dist.assign(V, INF), prev.assign(V, INVALID); dist[s] = 0; std::deque que; que.push_back(s); while (!que.empty()) { int v = que.front(); que.pop_front(); for (auto nx : to[v]) { T dnx = dist[v] + nx.second; if (dist[nx.first] > dnx) { dist[nx.first] = dnx, prev[nx.first] = v; if (nx.second) { que.push_back(nx.first); } else { que.push_front(nx.first); } } } } } }; // a から b を経由し c に辿り着く点素なパスで,banned にあるものを使わないもののうち // 最短で辞書順最小のものを求め,{[a, ..., b], [b, ..., c]} という pair of vector を返す. pair, vector> refine_path(int N, const vector> &to, int a, int b, int c, vector banned) { vector is_banned(N); for (auto i : banned) is_banned[i] = 1; auto [v1, v2] = twopaths(N, to, banned, b, a, c); const int len = v1.size() + v2.size(); vector ab{a}; banned.push_back(a); is_banned[a] = 1; while (ab.back() != b) { int cur = ab.back(); for (auto j : to[cur]) { if (j == b) { ab.push_back(b); break; } if (is_banned[j]) continue; if (j == c) continue; auto [p1, p2] = twopaths(N, to, banned, b, j, c); if (p1.empty()) continue; if (int(p1.size() + p2.size() + ab.size()) != len) continue; ab.push_back(j); banned.push_back(j); is_banned[j] = 1; break; } } ShortestPath sssp(N); for (int i = 0; i < N; ++i) { if (is_banned[i]) continue; for (auto j : to[i]) { if (is_banned[j]) continue; sssp.add_edge(i, j, 1); } } sssp.ZeroOneBFS(b); const auto Db = sssp.dist; sssp.ZeroOneBFS(c); const auto Dc = sssp.dist; vector bc{b}; int cur = b; while (cur != c) { for (auto nxt : to[cur]) { if (Db[nxt] == Db[cur] + 1 and Dc[nxt] == Dc[cur] - 1) { cur = nxt; bc.push_back(cur); break; } } } return {ab, bc}; } int main() { cin.tie(nullptr), ios::sync_with_stdio(false); int N, M, X, Y, Z; cin >> N >> M >> X >> Y >> Z; --X, --Y, --Z; vector conn(N + 3, vector(N + 3, 1)); for (int i = 0; i < N + 3; ++i) conn[i][i] = 0; while (M--) { int u, v; cin >> u >> v; --u, --v; vector us{u}, vs{v}; if (u == X) us.push_back(N); if (u == Y) us.push_back(N + 1); if (u == Z) us.push_back(N + 2); if (v == X) vs.push_back(N); if (v == Y) vs.push_back(N + 1); if (v == Z) vs.push_back(N + 2); for (auto i : us) { for (auto j : vs) conn[i][j] = conn[j][i] = 0; } } int ans_len = shortest_len(N, X, Y, Z, conn, {}); cout << ans_len << '\n'; if (ans_len < 0) return 0; vector ret; vector is_used(N + 3); int cur = X; while (true) { vector next_steps; for (int i = 0; i < N + 3; ++i) { if (conn[cur][i] and !is_used[i]) next_steps.push_back(i); } int ng = 0, ok = next_steps.size(); while (ok - ng > 1) { int c = (ok + ng) / 2; for (int i = c; i < int(next_steps.size()); ++i) { conn[cur][next_steps[i]] = conn[next_steps[i]][cur] = 0; } auto d = shortest_len(N, cur, Y, Z, conn, ret); if (d < 0 or d + int(ret.size()) > ans_len) { ng = c; } else { ok = c; } for (int i = c; i < int(next_steps.size()); ++i) { conn[cur][next_steps[i]] = conn[next_steps[i]][cur] = 1; } } int nxt = next_steps.at(ng); ret.push_back(cur); is_used[cur] = 1; cur = nxt; if (cur == Y or cur == Z) break; } if (cur != Z) swap(Y, Z); // Z -> Y -> X 辞書順最小 vector> to(N); for (int i = 0; i < N; ++i) { for (int j = 0; j < N; ++j) { if (conn[i][j]) to[i].push_back(j); } } auto [ZY, YX] = refine_path(N, to, Z, Y, X, vector(ret.begin() + 1, ret.end())); for (auto v : ZY) ret.push_back(v); ret.pop_back(); for (auto v : YX) ret.push_back(v); for (auto x : ret) cout << x + 1 << ' '; cout << '\n'; }