/** * date : 2024-05-03 00:07:27 * author : Nyaan */ #define NDEBUG using namespace std; // intrinstic #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include // utility namespace Nyaan { using ll = long long; using i64 = long long; using u64 = unsigned long long; using i128 = __int128_t; using u128 = __uint128_t; template using V = vector; template using VV = vector>; using vi = vector; using vl = vector; using vd = V; using vs = V; using vvi = vector>; using vvl = vector>; template using minpq = priority_queue, greater>; template struct P : pair { template P(Args... args) : pair(args...) {} using pair::first; using pair::second; P &operator+=(const P &r) { first += r.first; second += r.second; return *this; } P &operator-=(const P &r) { first -= r.first; second -= r.second; return *this; } P &operator*=(const P &r) { first *= r.first; second *= r.second; return *this; } template P &operator*=(const S &r) { first *= r, second *= r; return *this; } P operator+(const P &r) const { return P(*this) += r; } P operator-(const P &r) const { return P(*this) -= r; } P operator*(const P &r) const { return P(*this) *= r; } template P operator*(const S &r) const { return P(*this) *= r; } P operator-() const { return P{-first, -second}; } }; using pl = P; using pi = P; using vp = V; constexpr int inf = 1001001001; constexpr long long infLL = 4004004004004004004LL; template int sz(const T &t) { return t.size(); } template inline bool amin(T &x, U y) { return (y < x) ? (x = y, true) : false; } template inline bool amax(T &x, U y) { return (x < y) ? (x = y, true) : false; } template inline T Max(const vector &v) { return *max_element(begin(v), end(v)); } template inline T Min(const vector &v) { return *min_element(begin(v), end(v)); } template inline long long Sum(const vector &v) { return accumulate(begin(v), end(v), 0LL); } template int lb(const vector &v, const T &a) { return lower_bound(begin(v), end(v), a) - begin(v); } template int ub(const vector &v, const T &a) { return upper_bound(begin(v), end(v), a) - begin(v); } constexpr long long TEN(int n) { long long ret = 1, x = 10; for (; n; x *= x, n >>= 1) ret *= (n & 1 ? x : 1); return ret; } template pair mkp(const T &t, const U &u) { return make_pair(t, u); } template vector mkrui(const vector &v, bool rev = false) { vector ret(v.size() + 1); if (rev) { for (int i = int(v.size()) - 1; i >= 0; i--) ret[i] = v[i] + ret[i + 1]; } else { for (int i = 0; i < int(v.size()); i++) ret[i + 1] = ret[i] + v[i]; } return ret; }; template vector mkuni(const vector &v) { vector ret(v); sort(ret.begin(), ret.end()); ret.erase(unique(ret.begin(), ret.end()), ret.end()); return ret; } template vector mkord(int N, F f) { vector ord(N); iota(begin(ord), end(ord), 0); sort(begin(ord), end(ord), f); return ord; } template vector mkinv(vector &v) { int max_val = *max_element(begin(v), end(v)); vector inv(max_val + 1, -1); for (int i = 0; i < (int)v.size(); i++) inv[v[i]] = i; return inv; } vector mkiota(int n) { vector ret(n); iota(begin(ret), end(ret), 0); return ret; } template T mkrev(const T &v) { T w{v}; reverse(begin(w), end(w)); return w; } template bool nxp(T &v) { return next_permutation(begin(v), end(v)); } // 返り値の型は入力の T に依存 // i 要素目 : [0, a[i]) template vector> product(const vector &a) { vector> ret; vector v; auto dfs = [&](auto rc, int i) -> void { if (i == (int)a.size()) { ret.push_back(v); return; } for (int j = 0; j < a[i]; j++) v.push_back(j), rc(rc, i + 1), v.pop_back(); }; dfs(dfs, 0); return ret; } // F : function(void(T&)), mod を取る操作 // T : 整数型のときはオーバーフローに注意する template T Power(T a, long long n, const T &I, const function &f) { T res = I; for (; n; f(a = a * a), n >>= 1) { if (n & 1) f(res = res * a); } return res; } // T : 整数型のときはオーバーフローに注意する template T Power(T a, long long n, const T &I = T{1}) { return Power(a, n, I, function{[](T &) -> void {}}); } template T Rev(const T &v) { T res = v; reverse(begin(res), end(res)); return res; } template vector Transpose(const vector &v) { using U = typename T::value_type; int H = v.size(), W = v[0].size(); vector res(W, T(H, U{})); for (int i = 0; i < H; i++) { for (int j = 0; j < W; j++) { res[j][i] = v[i][j]; } } return res; } template vector Rotate(const vector &v, int clockwise = true) { using U = typename T::value_type; int H = v.size(), W = v[0].size(); vector res(W, T(H, U{})); for (int i = 0; i < H; i++) { for (int j = 0; j < W; j++) { if (clockwise) { res[W - 1 - j][i] = v[i][j]; } else { res[j][H - 1 - i] = v[i][j]; } } } return res; } } // namespace Nyaan // bit operation namespace Nyaan { __attribute__((target("popcnt"))) inline int popcnt(const u64 &a) { return __builtin_popcountll(a); } inline int lsb(const u64 &a) { return a ? __builtin_ctzll(a) : 64; } inline int ctz(const u64 &a) { return a ? __builtin_ctzll(a) : 64; } inline int msb(const u64 &a) { return a ? 63 - __builtin_clzll(a) : -1; } template inline int gbit(const T &a, int i) { return (a >> i) & 1; } template inline void sbit(T &a, int i, bool b) { if (gbit(a, i) != b) a ^= T(1) << i; } constexpr long long PW(int n) { return 1LL << n; } constexpr long long MSK(int n) { return (1LL << n) - 1; } } // namespace Nyaan // inout namespace Nyaan { template ostream &operator<<(ostream &os, const pair &p) { os << p.first << " " << p.second; return os; } template istream &operator>>(istream &is, pair &p) { is >> p.first >> p.second; return is; } template ostream &operator<<(ostream &os, const vector &v) { int s = (int)v.size(); for (int i = 0; i < s; i++) os << (i ? " " : "") << v[i]; return os; } template istream &operator>>(istream &is, vector &v) { for (auto &x : v) is >> x; return is; } istream &operator>>(istream &is, __int128_t &x) { string S; is >> S; x = 0; int flag = 0; for (auto &c : S) { if (c == '-') { flag = true; continue; } x *= 10; x += c - '0'; } if (flag) x = -x; return is; } istream &operator>>(istream &is, __uint128_t &x) { string S; is >> S; x = 0; for (auto &c : S) { x *= 10; x += c - '0'; } return is; } ostream &operator<<(ostream &os, __int128_t x) { if (x == 0) return os << 0; if (x < 0) os << '-', x = -x; string S; while (x) S.push_back('0' + x % 10), x /= 10; reverse(begin(S), end(S)); return os << S; } ostream &operator<<(ostream &os, __uint128_t x) { if (x == 0) return os << 0; string S; while (x) S.push_back('0' + x % 10), x /= 10; reverse(begin(S), end(S)); return os << S; } void in() {} template void in(T &t, U &...u) { cin >> t; in(u...); } void out() { cout << "\n"; } template void out(const T &t, const U &...u) { cout << t; if (sizeof...(u)) cout << sep; out(u...); } struct IoSetupNya { IoSetupNya() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(15); cerr << fixed << setprecision(7); } } iosetupnya; } // namespace Nyaan // debug #ifdef NyaanDebug #define trc(...) (void(0)) #else #define trc(...) (void(0)) #endif #ifdef NyaanLocal #define trc2(...) (void(0)) #else #define trc2(...) (void(0)) #endif // macro #define each(x, v) for (auto&& x : v) #define each2(x, y, v) for (auto&& [x, y] : v) #define all(v) (v).begin(), (v).end() #define rep(i, N) for (long long i = 0; i < (long long)(N); i++) #define repr(i, N) for (long long i = (long long)(N)-1; i >= 0; i--) #define rep1(i, N) for (long long i = 1; i <= (long long)(N); i++) #define repr1(i, N) for (long long i = (N); (long long)(i) > 0; i--) #define reg(i, a, b) for (long long i = (a); i < (b); i++) #define regr(i, a, b) for (long long i = (b)-1; i >= (a); i--) #define fi first #define se second #define ini(...) \ int __VA_ARGS__; \ in(__VA_ARGS__) #define inl(...) \ long long __VA_ARGS__; \ in(__VA_ARGS__) #define ins(...) \ string __VA_ARGS__; \ in(__VA_ARGS__) #define in2(s, t) \ for (int i = 0; i < (int)s.size(); i++) { \ in(s[i], t[i]); \ } #define in3(s, t, u) \ for (int i = 0; i < (int)s.size(); i++) { \ in(s[i], t[i], u[i]); \ } #define in4(s, t, u, v) \ for (int i = 0; i < (int)s.size(); i++) { \ in(s[i], t[i], u[i], v[i]); \ } #define die(...) \ do { \ Nyaan::out(__VA_ARGS__); \ return; \ } while (0) namespace Nyaan { void solve(); } int main() { Nyaan::solve(); } // struct UnionFind { vector data; UnionFind(int N) : data(N, -1) {} int find(int k) { return data[k] < 0 ? k : data[k] = find(data[k]); } int unite(int x, int y) { if ((x = find(x)) == (y = find(y))) return false; if (data[x] > data[y]) swap(x, y); data[x] += data[y]; data[y] = x; return true; } // f ... merge function template int unite(int x, int y,const F &f) { if ((x = find(x)) == (y = find(y))) return false; if (data[x] > data[y]) swap(x, y); data[x] += data[y]; data[y] = x; f(x, y); return true; } int size(int k) { return -data[find(k)]; } int same(int x, int y) { return find(x) == find(y); } }; /** * @brief Union Find(Disjoint Set Union) * @docs docs/data-structure/union-find.md */ template struct edge { int src, to; T cost; edge(int _to, T _cost) : src(-1), to(_to), cost(_cost) {} edge(int _src, int _to, T _cost) : src(_src), to(_to), cost(_cost) {} edge &operator=(const int &x) { to = x; return *this; } operator int() const { return to; } }; template using Edges = vector>; template using WeightedGraph = vector>; using UnweightedGraph = vector>; // Input of (Unweighted) Graph UnweightedGraph graph(int N, int M = -1, bool is_directed = false, bool is_1origin = true) { UnweightedGraph g(N); if (M == -1) M = N - 1; for (int _ = 0; _ < M; _++) { int x, y; cin >> x >> y; if (is_1origin) x--, y--; g[x].push_back(y); if (!is_directed) g[y].push_back(x); } return g; } // Input of Weighted Graph template WeightedGraph wgraph(int N, int M = -1, bool is_directed = false, bool is_1origin = true) { WeightedGraph g(N); if (M == -1) M = N - 1; for (int _ = 0; _ < M; _++) { int x, y; cin >> x >> y; T c; cin >> c; if (is_1origin) x--, y--; g[x].emplace_back(x, y, c); if (!is_directed) g[y].emplace_back(y, x, c); } return g; } // Input of Edges template Edges esgraph([[maybe_unused]] int N, int M, int is_weighted = true, bool is_1origin = true) { Edges es; for (int _ = 0; _ < M; _++) { int x, y; cin >> x >> y; T c; if (is_weighted) cin >> c; else c = 1; if (is_1origin) x--, y--; es.emplace_back(x, y, c); } return es; } // Input of Adjacency Matrix template vector> adjgraph(int N, int M, T INF, int is_weighted = true, bool is_directed = false, bool is_1origin = true) { vector> d(N, vector(N, INF)); for (int _ = 0; _ < M; _++) { int x, y; cin >> x >> y; T c; if (is_weighted) cin >> c; else c = 1; if (is_1origin) x--, y--; d[x][y] = c; if (!is_directed) d[y][x] = c; } return d; } /** * @brief グラフテンプレート * @docs docs/graph/graph-template.md */ using namespace std; namespace internal { template using is_broadly_integral = typename conditional_t || is_same_v || is_same_v, true_type, false_type>::type; template using is_broadly_signed = typename conditional_t || is_same_v, true_type, false_type>::type; template using is_broadly_unsigned = typename conditional_t || is_same_v, true_type, false_type>::type; #define ENABLE_VALUE(x) \ template \ constexpr bool x##_v = x::value; ENABLE_VALUE(is_broadly_integral); ENABLE_VALUE(is_broadly_signed); ENABLE_VALUE(is_broadly_unsigned); #undef ENABLE_VALUE #define ENABLE_HAS_TYPE(var) \ template \ struct has_##var : false_type {}; \ template \ struct has_##var> : true_type {}; \ template \ constexpr auto has_##var##_v = has_##var::value; #define ENABLE_HAS_VAR(var) \ template \ struct has_##var : false_type {}; \ template \ struct has_##var> : true_type {}; \ template \ constexpr auto has_##var##_v = has_##var::value; } // namespace internal namespace FunctionalGraphImpl { ENABLE_HAS_VAR(cost) template struct FunctionalGraph { int N; WeightedGraph g; vector to, represent; vector weight; FunctionalGraph() = default; FunctionalGraph(int n, const vector& adj, const vector& w = vector{}) : N(n), g(N + 1), to(N + 1, -1), represent(N + 1, -1), weight(N + 1) { assert((int)adj.size() == N); assert((int)w.size() == N or w.empty()); for (auto& x : adj) assert(0 <= x and x < N); UnionFind uf(N); for (int i = 0; i < N; i++) { int j = adj[i]; to[i] = j, weight[i] = w.empty() ? T{1} : w[i]; if (uf.same(i, j)) { g[N].emplace_back(N, i, weight[i]); } else { uf.unite(i, j); g[j].emplace_back(j, i, weight[i]); } } calc_represent(N, -1); } // _g は無向グラフ template FunctionalGraph(int n, const G& _g) : N(n), g(N + 1), to(N + 1, -1), represent(N + 1, -1), weight(N + 1) { constexpr bool cost_flag = has_cost_v; WeightedGraph h(n); vector roots; UnionFind uf(N); for (int i = 0; i < N; i++) { for (auto& j : _g[i]) { if (i > j) continue; T cost; if constexpr (cost_flag) { cost = j.cost; } else { cost = 1; } if (uf.same(i, j)) { // i -> j 向きということにして, i を根とする roots.push_back(i); g[N].emplace_back(N, i, 0); to[i] = j, weight[i] = cost; } else { uf.unite(i, j); h[i].emplace_back(i, j, cost); h[j].emplace_back(j, i, cost); } } } auto dfs = [&](auto rc, int c, int p) -> void { for (auto& d : h[c]) { if (d == p) continue; T cost; if constexpr (cost_flag) { cost = d.cost; } else { cost = 1; } to[d] = c, weight[d] = cost; g[c].emplace_back(c, d, cost); rc(rc, d, c); } }; for (auto& r : roots) dfs(dfs, r, -1); calc_represent(N, -1); } vector> get_loops() const { vector> res; for (auto r : g[N]) { vector cur{r}; for (int i = to[r]; i != r; i = to[i]) { cur.push_back(i); } res.push_back(cur); } return res; } private: void calc_represent(int c, int r) { represent[c] = r; for (auto& d : g[c]) calc_represent(d, r == -1 ? d : r); } }; } // namespace FunctionalGraphImpl using FunctionalGraphImpl::FunctionalGraph; template struct HeavyLightDecomposition { private: void dfs_sz(int cur) { size[cur] = 1; for (auto& dst : g[cur]) { if (dst == par[cur]) { if (g[cur].size() >= 2 && int(dst) == int(g[cur][0])) swap(g[cur][0], g[cur][1]); else continue; } depth[dst] = depth[cur] + 1; par[dst] = cur; dfs_sz(dst); size[cur] += size[dst]; if (size[dst] > size[g[cur][0]]) { swap(dst, g[cur][0]); } } } void dfs_hld(int cur) { down[cur] = id++; for (auto dst : g[cur]) { if (dst == par[cur]) continue; nxt[dst] = (int(dst) == int(g[cur][0]) ? nxt[cur] : int(dst)); dfs_hld(dst); } up[cur] = id; } // [u, v) vector> ascend(int u, int v) const { vector> res; while (nxt[u] != nxt[v]) { res.emplace_back(down[u], down[nxt[u]]); u = par[nxt[u]]; } if (u != v) res.emplace_back(down[u], down[v] + 1); return res; } // (u, v] vector> descend(int u, int v) const { if (u == v) return {}; if (nxt[u] == nxt[v]) return {{down[u] + 1, down[v]}}; auto res = descend(u, par[nxt[v]]); res.emplace_back(down[nxt[v]], down[v]); return res; } public: G& g; int root, id; vector size, depth, down, up, nxt, par; HeavyLightDecomposition(G& _g, int _root = 0) : g(_g), root(_root), id(0), size(g.size(), 0), depth(g.size(), 0), down(g.size(), -1), up(g.size(), -1), nxt(g.size(), root), par(g.size(), root) { dfs_sz(root); dfs_hld(root); } pair idx(int i) const { return make_pair(down[i], up[i]); } template void path_query(int u, int v, bool vertex, const F& f) { int l = lca(u, v); for (auto&& [a, b] : ascend(u, l)) { int s = a + 1, t = b; s > t ? f(t, s) : f(s, t); } if (vertex) f(down[l], down[l] + 1); for (auto&& [a, b] : descend(l, v)) { int s = a, t = b + 1; s > t ? f(t, s) : f(s, t); } } template void path_noncommutative_query(int u, int v, bool vertex, const F& f) { int l = lca(u, v); for (auto&& [a, b] : ascend(u, l)) f(a + 1, b); if (vertex) f(down[l], down[l] + 1); for (auto&& [a, b] : descend(l, v)) f(a, b + 1); } template void subtree_query(int u, bool vertex, const F& f) { f(down[u] + int(!vertex), up[u]); } int lca(int a, int b) { while (nxt[a] != nxt[b]) { if (down[a] < down[b]) swap(a, b); a = par[nxt[a]]; } return depth[a] < depth[b] ? a : b; } int dist(int a, int b) { return depth[a] + depth[b] - depth[lca(a, b)] * 2; } }; /** * @brief Heavy Light Decomposition(重軽分解) * @docs docs/tree/heavy-light-decomposition.md */ using namespace Nyaan; void q() { ini(N); vvi g(N); map, int> es; rep1(i, N) { ini(u, v); --u, --v; es[minmax(u, v)] = i; g[u].push_back(v); g[v].push_back(u); } FunctionalGraph fg{N, g}; auto loops = fg.get_loops(); assert(sz(loops) == 1); auto loop = loops[0]; vi ans; rep(i, sz(loop)) { int u = loop[i]; int v = loop[(i + 1) % sz(loop)]; ans.push_back(es[minmax(u, v)]); } sort(all(ans)); out(ans); } void Nyaan::solve() { int t = 1; // in(t); while (t--) q(); }