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#include "graph/functional.hpp"
#include "graph/tree.hpp" #include "ds/unionfind/unionfind.hpp" // N が根となる木を新たに作る template <typename T = int> struct FunctionalGraph { int N, M; vc<int> TO; vc<T> wt; vc<int> root; Graph<T, 1> G; FunctionalGraph() {} FunctionalGraph(int N) : N(N), M(0), TO(N, -1), wt(N), root(N, -1) {} void add(int a, int b, T c = 1) { assert(0 <= a && a < N); assert(TO[a] == -1); ++M; TO[a] = b; wt[a] = c; } pair<Graph<T, 1>, Tree<Graph<T, 1>>> build() { assert(N == M); UnionFind uf(N); FOR(v, N) if (!uf.merge(v, TO[v])) { root[v] = v; } FOR(v, N) if (root[v] == v) root[uf[v]] = v; FOR(v, N) root[v] = root[uf[v]]; G.build(N + 1); FOR(v, N) { if (root[v] == v) G.add(N, v, wt[v]); else G.add(TO[v], v, wt[v]); } G.build(); Tree<Graph<T, 1>> tree(G, N); return {G, tree}; } // a -> b にかかる回数. 不可能なら infty<int>. O(1). template <typename TREE> int dist(TREE& tree, int a, int b) { if (tree.in_subtree(a, b)) return tree.depth[a] - tree.depth[b]; int r = root[a]; int btm = TO[r]; // a -> r -> btm -> b if (tree.in_subtree(btm, b)) { int x = tree.depth[a] - tree.depth[r]; x += 1; x += tree.depth[btm] - tree.depth[b]; return x; } return infty<int>; } // functional graph に向かって進む template <typename TREE> int jump(TREE& tree, int v, ll step) { int d = tree.depth[v]; if (step <= d - 1) return tree.jump(v, N, step); v = root[v]; step -= d - 1; int bottom = TO[v]; int c = tree.depth[bottom]; step %= c; if (step == 0) return v; return tree.jump(bottom, N, step - 1); } // functional graph に step 回進む template <typename TREE> vc<int> jump_all(TREE& tree, ll step) { vc<int> res(N, -1); // v の k 個先を res[w] に入れる vvc<pair<int, int>> query(N); FOR(v, N) { int d = tree.depth[v]; int r = root[v]; if (d - 1 > step) { query[v].eb(v, step); } if (d - 1 <= step) { ll k = step - (d - 1); int bottom = TO[r]; int c = tree.depth[bottom]; k %= c; if (k == 0) { res[v] = r; continue; } query[bottom].eb(v, k - 1); } } vc<int> path; auto dfs = [&](auto& dfs, int v) -> void { path.eb(v); for (auto&& [w, k]: query[v]) { res[w] = path[len(path) - 1 - k]; } for (auto&& e: G[v]) dfs(dfs, e.to); path.pop_back(); }; for (auto&& e: G[N]) { dfs(dfs, e.to); } return res; } template <typename TREE> bool in_cycle(TREE& tree, int v) { int r = root[v]; int bottom = TO[r]; return tree.in_subtree(bottom, v); } vc<int> collect_cycle(int r) { assert(r == root[r]); vc<int> cyc = {TO[r]}; while (cyc.back() != r) cyc.eb(TO[cyc.back()]); return cyc; } };
#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}; } }; #line 2 "ds/unionfind/unionfind.hpp" struct UnionFind { int n, n_comp; vc<int> dat; // par or (-size) UnionFind(int n = 0) { build(n); } void build(int m) { n = m, n_comp = m; dat.assign(n, -1); } void reset() { build(n); } int operator[](int x) { while (dat[x] >= 0) { int pp = dat[dat[x]]; if (pp < 0) { return dat[x]; } x = dat[x] = pp; } return x; } ll size(int x) { x = (*this)[x]; return -dat[x]; } bool merge(int x, int y) { x = (*this)[x], y = (*this)[y]; if (x == y) return false; if (-dat[x] < -dat[y]) swap(x, y); dat[x] += dat[y], dat[y] = x, n_comp--; return true; } vc<int> get_all() { vc<int> A(n); FOR(i, n) A[i] = (*this)[i]; return A; } }; #line 3 "graph/functional.hpp" // N が根となる木を新たに作る template <typename T = int> struct FunctionalGraph { int N, M; vc<int> TO; vc<T> wt; vc<int> root; Graph<T, 1> G; FunctionalGraph() {} FunctionalGraph(int N) : N(N), M(0), TO(N, -1), wt(N), root(N, -1) {} void add(int a, int b, T c = 1) { assert(0 <= a && a < N); assert(TO[a] == -1); ++M; TO[a] = b; wt[a] = c; } pair<Graph<T, 1>, Tree<Graph<T, 1>>> build() { assert(N == M); UnionFind uf(N); FOR(v, N) if (!uf.merge(v, TO[v])) { root[v] = v; } FOR(v, N) if (root[v] == v) root[uf[v]] = v; FOR(v, N) root[v] = root[uf[v]]; G.build(N + 1); FOR(v, N) { if (root[v] == v) G.add(N, v, wt[v]); else G.add(TO[v], v, wt[v]); } G.build(); Tree<Graph<T, 1>> tree(G, N); return {G, tree}; } // a -> b にかかる回数. 不可能なら infty<int>. O(1). template <typename TREE> int dist(TREE& tree, int a, int b) { if (tree.in_subtree(a, b)) return tree.depth[a] - tree.depth[b]; int r = root[a]; int btm = TO[r]; // a -> r -> btm -> b if (tree.in_subtree(btm, b)) { int x = tree.depth[a] - tree.depth[r]; x += 1; x += tree.depth[btm] - tree.depth[b]; return x; } return infty<int>; } // functional graph に向かって進む template <typename TREE> int jump(TREE& tree, int v, ll step) { int d = tree.depth[v]; if (step <= d - 1) return tree.jump(v, N, step); v = root[v]; step -= d - 1; int bottom = TO[v]; int c = tree.depth[bottom]; step %= c; if (step == 0) return v; return tree.jump(bottom, N, step - 1); } // functional graph に step 回進む template <typename TREE> vc<int> jump_all(TREE& tree, ll step) { vc<int> res(N, -1); // v の k 個先を res[w] に入れる vvc<pair<int, int>> query(N); FOR(v, N) { int d = tree.depth[v]; int r = root[v]; if (d - 1 > step) { query[v].eb(v, step); } if (d - 1 <= step) { ll k = step - (d - 1); int bottom = TO[r]; int c = tree.depth[bottom]; k %= c; if (k == 0) { res[v] = r; continue; } query[bottom].eb(v, k - 1); } } vc<int> path; auto dfs = [&](auto& dfs, int v) -> void { path.eb(v); for (auto&& [w, k]: query[v]) { res[w] = path[len(path) - 1 - k]; } for (auto&& e: G[v]) dfs(dfs, e.to); path.pop_back(); }; for (auto&& e: G[N]) { dfs(dfs, e.to); } return res; } template <typename TREE> bool in_cycle(TREE& tree, int v) { int r = root[v]; int bottom = TO[r]; return tree.in_subtree(bottom, v); } vc<int> collect_cycle(int r) { assert(r == root[r]); vc<int> cyc = {TO[r]}; while (cyc.back() != r) cyc.eb(TO[cyc.back()]); return cyc; } };