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#include "graph/ds/static_toptree.hpp"

#include "graph/tree.hpp" /* 参考 joitour tatyam クラスタは根が virtual なもののみであるような簡易版 N 個の (頂+辺) をマージしていって，木全体＋根から親への辺とする． single(v) : v とその親辺を合わせたクラスタ rake(L,R) : L の boundary を維持 compress(L,R) (top-down) 順に x,y */ template <typename TREE> struct Static_TopTree { int N; TREE &tree; vc<int> par, lch, rch, A, B; // A, B boundary (top-down) vc<bool> is_compress; Static_TopTree(TREE &tree) : tree(tree) { build(); } void build() { N = tree.N; par.assign(N, -1), lch.assign(N, -1), rch.assign(N, -1), A.assign(N, -1), B.assign(N, -1), is_compress.assign(N, 0); FOR(v, N) { A[v] = tree.parent[v], B[v] = v; } build_dfs(tree.V[0]); assert(len(par) == 2 * N - 1); } // 木全体での集約値を得る // single(v) : v とその親辺を合わせたクラスタ // rake(x, y, u, v) uv(top down) が boundary になるように rake (maybe v=-1) // compress(x,y,a,b,c) (top-down) 順に (a,b] + (b,c] template <typename TREE_DP, typename F> typename TREE_DP::value_type tree_dp(F single) { using Data = typename TREE_DP::value_type; auto dfs = [&](auto &dfs, int k) -> Data { if (0 <= k && k < N) return single(k); Data x = dfs(dfs, lch[k]), y = dfs(dfs, rch[k]); if (is_compress[k]) { assert(B[lch[k]] == A[rch[k]]); return TREE_DP::compress(x, y); } return TREE_DP::rake(x, y); }; return dfs(dfs, 2 * N - 2); } private: int new_node(int l, int r, int a, int b, bool c) { int v = len(par); par.eb(-1), lch.eb(l), rch.eb(r), A.eb(a), B.eb(b), is_compress.eb(c); par[l] = par[r] = v; return v; } // height, node idx // compress 参考：https://atcoder.jp/contests/abc351/editorial/9910 // ただし heavy path の選び方までは考慮しない pair<int, int> build_dfs(int v) { assert(tree.head[v] == v); auto path = tree.heavy_path_at(v); vc<pair<int, int>> stack; stack.eb(0, path[0]); auto merge_last_two = [&]() -> void { auto [h2, k2] = POP(stack); auto [h1, k1] = POP(stack); stack.eb(max(h1, h2) + 1, new_node(k1, k2, A[k1], B[k2], true)); }; FOR(i, 1, len(path)) { pqg<pair<int, int>> que; int k = path[i]; que.emplace(0, k); for (auto &c: tree.collect_light(path[i - 1])) { que.emplace(build_dfs(c)); } while (len(que) >= 2) { auto [h1, i1] = POP(que); auto [h2, i2] = POP(que); if (i2 == k) swap(i1, i2); int i3 = new_node(i1, i2, A[i1], B[i1], false); if (k == i1) k = i3; que.emplace(max(h1, h2) + 1, i3); } stack.eb(POP(que)); while (1) { int n = len(stack); if (n >= 3 && (stack[n - 3].fi == stack[n - 2].fi || stack[n - 3].fi <= stack[n - 1].fi)) { auto [h3, k3] = POP(stack); merge_last_two(), stack.eb(h3, k3); } elif (n >= 2 && stack[n - 2].fi <= stack[n - 1].fi) { merge_last_two(); } else break; } } while (len(stack) >= 2) { merge_last_two(); } return POP(stack); } };

#line 2 "graph/tree.hpp" #line 2 "graph/base.hpp" template <typename T> struct Edge { int frm, to; T cost; int id; }; template <typename T = int, bool directed = false> struct Graph { static constexpr bool is_directed = directed; int N, M; using cost_type = T; using edge_type = Edge<T>; vector<edge_type> edges; vector<int> indptr; vector<edge_type> csr_edges; vc<int> vc_deg, vc_indeg, vc_outdeg; bool prepared; class OutgoingEdges { public: OutgoingEdges(const Graph* G, int l, int r) : G(G), l(l), r(r) {} const edge_type* begin() const { if (l == r) { return 0; } return &G->csr_edges[l]; } const edge_type* end() const { if (l == r) { return 0; } return &G->csr_edges[r]; } private: const Graph* G; int l, r; }; bool is_prepared() { return prepared; } Graph() : N(0), M(0), prepared(0) {} Graph(int N) : N(N), M(0), prepared(0) {} void build(int n) { N = n, M = 0; prepared = 0; edges.clear(); indptr.clear(); csr_edges.clear(); vc_deg.clear(); vc_indeg.clear(); vc_outdeg.clear(); } void add(int frm, int to, T cost = 1, int i = -1) { assert(!prepared); assert(0 <= frm && 0 <= to && to < N); if (i == -1) i = M; auto e = edge_type({frm, to, cost, i}); edges.eb(e); ++M; } #ifdef FASTIO // wt, off void read_tree(bool wt = false, int off = 1) { read_graph(N - 1, wt, off); } void read_graph(int M, bool wt = false, int off = 1) { for (int m = 0; m < M; ++m) { INT(a, b); a -= off, b -= off; if (!wt) { add(a, b); } else { T c; read(c); add(a, b, c); } } build(); } #endif void build() { assert(!prepared); prepared = true; indptr.assign(N + 1, 0); for (auto&& e: edges) { indptr[e.frm + 1]++; if (!directed) indptr[e.to + 1]++; } for (int v = 0; v < N; ++v) { indptr[v + 1] += indptr[v]; } auto counter = indptr; csr_edges.resize(indptr.back() + 1); for (auto&& e: edges) { csr_edges[counter[e.frm]++] = e; if (!directed) csr_edges[counter[e.to]++] = edge_type({e.to, e.frm, e.cost, e.id}); } } OutgoingEdges operator[](int v) const { assert(prepared); return {this, indptr[v], indptr[v + 1]}; } vc<int> deg_array() { if (vc_deg.empty()) calc_deg(); return vc_deg; } pair<vc<int>, vc<int>> deg_array_inout() { if (vc_indeg.empty()) calc_deg_inout(); return {vc_indeg, vc_outdeg}; } int deg(int v) { if (vc_deg.empty()) calc_deg(); return vc_deg[v]; } int in_deg(int v) { if (vc_indeg.empty()) calc_deg_inout(); return vc_indeg[v]; } int out_deg(int v) { if (vc_outdeg.empty()) calc_deg_inout(); return vc_outdeg[v]; } #ifdef FASTIO void debug() { print("Graph"); if (!prepared) { print("frm to cost id"); for (auto&& e: edges) print(e.frm, e.to, e.cost, e.id); } else { print("indptr", indptr); print("frm to cost id"); FOR(v, N) for (auto&& e: (*this)[v]) print(e.frm, e.to, e.cost, e.id); } } #endif vc<int> new_idx; vc<bool> used_e; // G における頂点 V[i] が、新しいグラフで i になるようにする // {G, es} // sum(deg(v)) の計算量になっていて、 // 新しいグラフの n+m より大きい可能性があるので注意 Graph<T, directed> rearrange(vc<int> V, bool keep_eid = 0) { if (len(new_idx) != N) new_idx.assign(N, -1); int n = len(V); FOR(i, n) new_idx[V[i]] = i; Graph<T, directed> G(n); vc<int> history; FOR(i, n) { for (auto&& e: (*this)[V[i]]) { if (len(used_e) <= e.id) used_e.resize(e.id + 1); if (used_e[e.id]) continue; int a = e.frm, b = e.to; if (new_idx[a] != -1 && new_idx[b] != -1) { history.eb(e.id); used_e[e.id] = 1; int eid = (keep_eid ? e.id : -1); G.add(new_idx[a], new_idx[b], e.cost, eid); } } } FOR(i, n) new_idx[V[i]] = -1; for (auto&& eid: history) used_e[eid] = 0; G.build(); return G; } Graph<T, true> to_directed_tree(int root = -1) { if (root == -1) root = 0; assert(!is_directed && prepared && M == N - 1); Graph<T, true> G1(N); vc<int> par(N, -1); auto dfs = [&](auto& dfs, int v) -> void { for (auto& e: (*this)[v]) { if (e.to == par[v]) continue; par[e.to] = v, dfs(dfs, e.to); } }; dfs(dfs, root); for (auto& e: edges) { int a = e.frm, b = e.to; if (par[a] == b) swap(a, b); assert(par[b] == a); G1.add(a, b, e.cost); } G1.build(); return G1; } private: void calc_deg() { assert(vc_deg.empty()); vc_deg.resize(N); for (auto&& e: edges) vc_deg[e.frm]++, vc_deg[e.to]++; } void calc_deg_inout() { assert(vc_indeg.empty()); vc_indeg.resize(N); vc_outdeg.resize(N); for (auto&& e: edges) { vc_indeg[e.to]++, vc_outdeg[e.frm]++; } } }; #line 4 "graph/tree.hpp" // HLD euler tour をとっていろいろ。 template <typename GT> struct Tree { using Graph_type = GT; GT &G; using WT = typename GT::cost_type; int N; vector<int> LID, RID, head, V, parent, VtoE; vc<int> depth; vc<WT> depth_weighted; Tree(GT &G, int r = 0, bool hld = 1) : G(G) { build(r, hld); } void build(int r = 0, bool hld = 1) { if (r == -1) return; // build を遅延したいとき N = G.N; LID.assign(N, -1), RID.assign(N, -1), head.assign(N, r); V.assign(N, -1), parent.assign(N, -1), VtoE.assign(N, -1); depth.assign(N, -1), depth_weighted.assign(N, 0); assert(G.is_prepared()); int t1 = 0; dfs_sz(r, -1, hld); dfs_hld(r, t1); } void dfs_sz(int v, int p, bool hld) { auto &sz = RID; parent[v] = p; depth[v] = (p == -1 ? 0 : depth[p] + 1); sz[v] = 1; int l = G.indptr[v], r = G.indptr[v + 1]; auto &csr = G.csr_edges; // 使う辺があれば先頭にする for (int i = r - 2; i >= l; --i) { if (hld && depth[csr[i + 1].to] == -1) swap(csr[i], csr[i + 1]); } int hld_sz = 0; for (int i = l; i < r; ++i) { auto e = csr[i]; if (depth[e.to] != -1) continue; depth_weighted[e.to] = depth_weighted[v] + e.cost; VtoE[e.to] = e.id; dfs_sz(e.to, v, hld); sz[v] += sz[e.to]; if (hld && chmax(hld_sz, sz[e.to]) && l < i) { swap(csr[l], csr[i]); } } } void dfs_hld(int v, int ×) { LID[v] = times++; RID[v] += LID[v]; V[LID[v]] = v; bool heavy = true; for (auto &&e: G[v]) { if (depth[e.to] <= depth[v]) continue; head[e.to] = (heavy ? head[v] : e.to); heavy = false; dfs_hld(e.to, times); } } vc<int> heavy_path_at(int v) { vc<int> P = {v}; while (1) { int a = P.back(); for (auto &&e: G[a]) { if (e.to != parent[a] && head[e.to] == v) { P.eb(e.to); break; } } if (P.back() == a) break; } return P; } int heavy_child(int v) { int k = LID[v] + 1; if (k == N) return -1; int w = V[k]; return (parent[w] == v ? w : -1); } int e_to_v(int eid) { auto e = G.edges[eid]; return (parent[e.frm] == e.to ? e.frm : e.to); } int v_to_e(int v) { return VtoE[v]; } int get_eid(int u, int v) { if (parent[u] != v) swap(u, v); assert(parent[u] == v); return VtoE[u]; } int ELID(int v) { return 2 * LID[v] - depth[v]; } int ERID(int v) { return 2 * RID[v] - depth[v] - 1; } // 目標地点へ進む個数が k int LA(int v, int k) { assert(k <= depth[v]); while (1) { int u = head[v]; if (LID[v] - k >= LID[u]) return V[LID[v] - k]; k -= LID[v] - LID[u] + 1; v = parent[u]; } } int la(int u, int v) { return LA(u, v); } int LCA(int u, int v) { for (;; v = parent[head[v]]) { if (LID[u] > LID[v]) swap(u, v); if (head[u] == head[v]) return u; } } int meet(int a, int b, int c) { return LCA(a, b) ^ LCA(a, c) ^ LCA(b, c); } int lca(int u, int v) { return LCA(u, v); } int subtree_size(int v, int root = -1) { if (root == -1) return RID[v] - LID[v]; if (v == root) return N; int x = jump(v, root, 1); if (in_subtree(v, x)) return RID[v] - LID[v]; return N - RID[x] + LID[x]; } int dist(int a, int b) { int c = LCA(a, b); return depth[a] + depth[b] - 2 * depth[c]; } WT dist_weighted(int a, int b) { int c = LCA(a, b); return depth_weighted[a] + depth_weighted[b] - WT(2) * depth_weighted[c]; } // a is in b bool in_subtree(int a, int b) { return LID[b] <= LID[a] && LID[a] < RID[b]; } int jump(int a, int b, ll k) { if (k == 1) { if (a == b) return -1; return (in_subtree(b, a) ? LA(b, depth[b] - depth[a] - 1) : parent[a]); } int c = LCA(a, b); int d_ac = depth[a] - depth[c]; int d_bc = depth[b] - depth[c]; if (k > d_ac + d_bc) return -1; if (k <= d_ac) return LA(a, k); return LA(b, d_ac + d_bc - k); } vc<int> collect_child(int v) { vc<int> res; for (auto &&e: G[v]) if (e.to != parent[v]) res.eb(e.to); return res; } vc<int> collect_light(int v) { vc<int> res; bool skip = true; for (auto &&e: G[v]) if (e.to != parent[v]) { if (!skip) res.eb(e.to); skip = false; } return res; } vc<pair<int, int>> get_path_decomposition(int u, int v, bool edge) { // [始点, 終点] の"閉"区間列。 vc<pair<int, int>> up, down; while (1) { if (head[u] == head[v]) break; if (LID[u] < LID[v]) { down.eb(LID[head[v]], LID[v]); v = parent[head[v]]; } else { up.eb(LID[u], LID[head[u]]); u = parent[head[u]]; } } if (LID[u] < LID[v]) down.eb(LID[u] + edge, LID[v]); elif (LID[v] + edge <= LID[u]) up.eb(LID[u], LID[v] + edge); reverse(all(down)); up.insert(up.end(), all(down)); return up; } // 辺の列の情報 (frm,to,str) // str = "heavy_up", "heavy_down", "light_up", "light_down" vc<tuple<int, int, string>> get_path_decomposition_detail(int u, int v) { vc<tuple<int, int, string>> up, down; while (1) { if (head[u] == head[v]) break; if (LID[u] < LID[v]) { if (v != head[v]) down.eb(head[v], v, "heavy_down"), v = head[v]; down.eb(parent[v], v, "light_down"), v = parent[v]; } else { if (u != head[u]) up.eb(u, head[u], "heavy_up"), u = head[u]; up.eb(u, parent[u], "light_up"), u = parent[u]; } } if (LID[u] < LID[v]) down.eb(u, v, "heavy_down"); elif (LID[v] < LID[u]) up.eb(u, v, "heavy_up"); reverse(all(down)); concat(up, down); return up; } vc<int> restore_path(int u, int v) { vc<int> P; for (auto &&[a, b]: get_path_decomposition(u, v, 0)) { if (a <= b) { FOR(i, a, b + 1) P.eb(V[i]); } else { FOR_R(i, b, a + 1) P.eb(V[i]); } } return P; } // path [a,b] と [c,d] の交わり. 空ならば {-1,-1}. // https://codeforces.com/problemset/problem/500/G pair<int, int> path_intersection(int a, int b, int c, int d) { int ab = lca(a, b), ac = lca(a, c), ad = lca(a, d); int bc = lca(b, c), bd = lca(b, d), cd = lca(c, d); int x = ab ^ ac ^ bc, y = ab ^ ad ^ bd; // meet(a,b,c), meet(a,b,d) if (x != y) return {x, y}; int z = ac ^ ad ^ cd; if (x != z) x = -1; return {x, x}; } // uv path 上で check(v) を満たす最後の v // なければ （つまり check(v) が ng ）-1 template <class F> int max_path(F check, int u, int v) { if (!check(u)) return -1; auto pd = get_path_decomposition(u, v, false); for (auto [a, b]: pd) { if (!check(V[a])) return u; if (check(V[b])) { u = V[b]; continue; } int c = binary_search([&](int c) -> bool { return check(V[c]); }, a, b, 0); return V[c]; } return u; } }; #line 2 "graph/ds/static_toptree.hpp" /* 参考 joitour tatyam クラスタは根が virtual なもののみであるような簡易版 N 個の (頂+辺) をマージしていって，木全体＋根から親への辺とする． single(v) : v とその親辺を合わせたクラスタ rake(L,R) : L の boundary を維持 compress(L,R) (top-down) 順に x,y */ template <typename TREE> struct Static_TopTree { int N; TREE &tree; vc<int> par, lch, rch, A, B; // A, B boundary (top-down) vc<bool> is_compress; Static_TopTree(TREE &tree) : tree(tree) { build(); } void build() { N = tree.N; par.assign(N, -1), lch.assign(N, -1), rch.assign(N, -1), A.assign(N, -1), B.assign(N, -1), is_compress.assign(N, 0); FOR(v, N) { A[v] = tree.parent[v], B[v] = v; } build_dfs(tree.V[0]); assert(len(par) == 2 * N - 1); } // 木全体での集約値を得る // single(v) : v とその親辺を合わせたクラスタ // rake(x, y, u, v) uv(top down) が boundary になるように rake (maybe v=-1) // compress(x,y,a,b,c) (top-down) 順に (a,b] + (b,c] template <typename TREE_DP, typename F> typename TREE_DP::value_type tree_dp(F single) { using Data = typename TREE_DP::value_type; auto dfs = [&](auto &dfs, int k) -> Data { if (0 <= k && k < N) return single(k); Data x = dfs(dfs, lch[k]), y = dfs(dfs, rch[k]); if (is_compress[k]) { assert(B[lch[k]] == A[rch[k]]); return TREE_DP::compress(x, y); } return TREE_DP::rake(x, y); }; return dfs(dfs, 2 * N - 2); } private: int new_node(int l, int r, int a, int b, bool c) { int v = len(par); par.eb(-1), lch.eb(l), rch.eb(r), A.eb(a), B.eb(b), is_compress.eb(c); par[l] = par[r] = v; return v; } // height, node idx // compress 参考：https://atcoder.jp/contests/abc351/editorial/9910 // ただし heavy path の選び方までは考慮しない pair<int, int> build_dfs(int v) { assert(tree.head[v] == v); auto path = tree.heavy_path_at(v); vc<pair<int, int>> stack; stack.eb(0, path[0]); auto merge_last_two = [&]() -> void { auto [h2, k2] = POP(stack); auto [h1, k1] = POP(stack); stack.eb(max(h1, h2) + 1, new_node(k1, k2, A[k1], B[k2], true)); }; FOR(i, 1, len(path)) { pqg<pair<int, int>> que; int k = path[i]; que.emplace(0, k); for (auto &c: tree.collect_light(path[i - 1])) { que.emplace(build_dfs(c)); } while (len(que) >= 2) { auto [h1, i1] = POP(que); auto [h2, i2] = POP(que); if (i2 == k) swap(i1, i2); int i3 = new_node(i1, i2, A[i1], B[i1], false); if (k == i1) k = i3; que.emplace(max(h1, h2) + 1, i3); } stack.eb(POP(que)); while (1) { int n = len(stack); if (n >= 3 && (stack[n - 3].fi == stack[n - 2].fi || stack[n - 3].fi <= stack[n - 1].fi)) { auto [h3, k3] = POP(stack); merge_last_two(), stack.eb(h3, k3); } elif (n >= 2 && stack[n - 2].fi <= stack[n - 1].fi) { merge_last_two(); } else break; } } while (len(stack) >= 2) { merge_last_two(); } return POP(stack); } };