// O(n^3) weighted-linear-matroid-parity-based solution #include #include #include #include #include #include #include #include using namespace std; #include using mint = atcoder::static_modint<(1 << 30) + 3>; uint32_t rand_int() { static uint32_t x = 123456789, y = 362436069, z = 521288629, w = 88675123; uint32_t t = x ^ (x << 11); x = y; y = z; z = w; return w = (w ^ (w >> 19)) ^ (t ^ (t >> 8)); } tuple>> read_input() { int N, M, X, Y, Z; cin >> N >> M >> X >> Y >> Z; --X, --Y, --Z; vector mat(N, vector(N)); for (int i = 0; i < N; ++i) { for (int j = 0; j < N; ++j) mat[i][j] = (i != j); } while (M--) { int a, b; cin >> a >> b; --a, --b; mat[a][b] = mat[b][a] = 0; } vector> to(N); for (int i = 0; i < N; ++i) { for (int j = 0; j < N; ++j) { if (mat[i][j]) to[i].push_back(j); } } return {N, X, Y, Z, to}; } // 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]; } int main() { cin.tie(nullptr), ios::sync_with_stdio(false); auto Graph = read_input(); const int X = get<1>(Graph), Y = get<2>(Graph), Z = get<3>(Graph); auto to = get<4>(Graph); const int N = to.size(); const int V = N + 5, r = V * 2 - 8; // https://drops.dagstuhl.de/opus/volltexte/2016/6832/ // 2.1 の Remark に従って行列を構築 // X を (X, N) に,Y を (Y, N + 1) に,Z を (Z, N + 2) にそれぞれ分裂 // 論文 Fig. 1 の b1, b2 が N + 3, N + 4 に対応 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 i - red; }; // Z の各列ベクトルを生成 // 各列はスパースなので,ノンゼロな要素番号とその値のペアの列を返す auto gen_edge_vec = [&](int i, int j) -> pair>, vector>> { vector> bret, cret; if (Label(i)) { bret.emplace_back(truncate(i * 2 + 1), -Label(i)); } else { bret.emplace_back(truncate(i * 2), 1); } if (Label(j)) { bret.emplace_back(truncate(j * 2 + 1), Label(j)); } else { bret.emplace_back(truncate(j * 2), -1); } cret.emplace_back(truncate(i * 2 + 1), 1); cret.emplace_back(truncate(j * 2 + 1), -1); return {bret, cret}; }; to.resize(N + 3); for (auto i : to[X]) { to[N].push_back(i); to[i].push_back(N); } for (auto i : to[Y]) { to[N + 1].push_back(i); to[i].push_back(N + 1); } for (auto i : to[Z]) { to[N + 2].push_back(i); to[i].push_back(N + 2); } // 論文通りに辺を張る vector>, vector>, int>> bcws; for (int i = 0; i < N + 3; ++i) { for (int j : to[i]) { if (i > j) continue; auto [b, c] = gen_edge_vec(i, j); bcws.emplace_back(b, c, 1); } if (!Label(i)) { auto [b, c] = gen_edge_vec(i, N + 3); bcws.emplace_back(b, c, 0); } } { auto [b, c] = gen_edge_vec(N + 3, N + 4); bcws.emplace_back(b, c, 0); } // M(x) = mat0 + mat1 * x vector mat0(r, vector(r)); vector mat1(r, vector(r)); for (const auto &[b, c, w] : bcws) { mint x = rand_int() % mint::mod(); mint y = rand_int() % mint::mod(); for (auto [i, bi] : b) { for (auto [j, cj] : c) { auto v = bi * cj * x; if (w == 0) { mat0[i][j] += v; mat0[j][i] -= v; mat1[i][j] += v * y; mat1[j][i] -= v * y; } if (w == 1) { mat1[i][j] += v; mat1[j][i] -= v; } } } } // 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(\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 (mat1[h][i] != 0) piv = h; } if (piv < 0) { cout << "-1\n"; return 0; } assert(piv >= i); swap(mat0[i], mat0[piv]); swap(mat1[i], mat1[piv]); if (i != piv) { for (int w = 0; w < r; ++w) { mat0[i][w] *= -1; mat1[i][w] *= -1; } } mint inv = mat1[i][i].inv(); for (int w = 0; w < r; ++w) { mat0[i][w] *= inv; mat1[i][w] *= inv; } for (int h = 0; h < r; ++h) { if (h == i) continue; if (mat1[h][i] == 0) continue; const mint coeff = mat1[h][i]; for (int w = 0; w < r; ++w) { mat1[h][w] -= coeff * mat1[i][w]; mat0[h][w] -= coeff * mat0[i][w]; } } } for(auto &v : mat0) for (auto &x : v) x = -x; auto det_poly = characteristic_poly(mat0); int ret = 0; while (ret < int(det_poly.size()) and det_poly[ret] == 0) ++ret; if (ret < int(det_poly.size())) { cout << ret / 2 << '\n'; } else { cout << "-1\n"; } }