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#include "tree/lca_based_auxiliary_tree.hpp"
$k$ 頂点 $vs$ を与え、その頂点とそれらのLCAからなる補助的な木を構築する。
返り値として、auxiliary treeのノードとその親の組をdfs順で返す。根の親は $-1$ としている。
返り値で補助的な木での頂点番号と元の木の頂点番号の対応を格納した配列と、補助的な木のグラフを返す。
#pragma once #include <algorithm> #include <cassert> #include <ranges> #include <stack> #include <vector> #include "../tree/heavy_light_decomposition.hpp" namespace ebi { template <class T> std::vector<std::pair<int, int>> heavy_light_decomposition<T>::lca_based_auxiliary_tree_dfs_order( std::vector<int> vs) const { if (vs.empty()) return {}; std::sort(vs.begin(), vs.end(), [&](int u, int v) -> bool { return in[u] < in[v]; }); auto s = vs; for (int i = 1; i < int(vs.size()); i++) { s.emplace_back(lca(vs[i - 1], vs[i])); } std::sort(s.begin(), s.end(), [&](int u, int v) -> bool { return in[u] < in[v]; }); s.erase(std::unique(s.begin(), s.end()), s.end()); std::stack<int> stack; stack.push(s[0]); int sz = s.size(); std::vector<std::pair<int, int>> dfs_order(sz); dfs_order[0] = {s[0], -1}; for (int i = 1; i < int(s.size()); i++) { int v = s[i]; while (!stack.empty()) { int u = stack.top(); if (in[u] <= in[v] && in[v] < out[u]) { break; } else { stack.pop(); } } assert(!stack.empty()); int par = stack.top(); dfs_order[i] = {v, par}; stack.push(v); } return dfs_order; } template <class T> std::pair<std::vector<int>, Graph<T>> heavy_light_decomposition<T>::lca_based_auxiliary_tree( std::vector<int> vs) const { static std::vector<int> a(1'000'000, -1), p(1'000'000, -1); int k = vs.size(); if (k == 1) { return {vs, Graph<T>(1)}; } std::sort(vs.begin(), vs.end(), [&](int v, int u) { return in[v] < in[u]; }); std::stack<int> stack; std::vector<int> s; stack.push(vs[0]); for (int i : std::views::iota(1, k)) { int w = lca(vs[i - 1], vs[i]); int prev = -1; while (!stack.empty() && depth_[w] <= depth_[stack.top()]) { if (prev != -1) { s.emplace_back(prev); p[prev] = stack.top(); } prev = stack.top(); stack.pop(); } if (prev != w) { assert(prev != -1); s.emplace_back(prev); p[prev] = w; } stack.push(w); stack.push(vs[i]); } { int prev = -1; while (!stack.empty()) { int v = stack.top(); s.emplace_back(v); if (prev != -1) p[prev] = v; prev = v; stack.pop(); } } std::reverse(s.begin(), s.end()); int m = s.size(); for (int i : std::views::iota(0, m)) { a[s[i]] = i; } Graph<T> tree(m); for (auto v : s) { if (p[v] < 0) continue; T cost = distance(p[v], v); tree.add_edge(a[p[v]], a[v], cost); tree.add_edge(a[v], a[p[v]], cost); } tree.build(); for (auto v : s) { a[v] = -1; p[v] = -1; } return {s, tree}; } } // namespace ebi
#line 2 "tree/lca_based_auxiliary_tree.hpp" #include <algorithm> #include <cassert> #include <ranges> #include <stack> #include <vector> #line 2 "tree/heavy_light_decomposition.hpp" #line 6 "tree/heavy_light_decomposition.hpp" #line 2 "graph/base.hpp" #line 4 "graph/base.hpp" #include <iostream> #line 7 "graph/base.hpp" #line 2 "data_structure/simple_csr.hpp" #line 4 "data_structure/simple_csr.hpp" #include <utility> #line 6 "data_structure/simple_csr.hpp" namespace ebi { template <class E> struct simple_csr { simple_csr() = default; simple_csr(int n, const std::vector<std::pair<int, E>>& elements) : start(n + 1, 0), elist(elements.size()) { for (auto e : elements) { start[e.first + 1]++; } for (auto i : std::views::iota(0, n)) { start[i + 1] += start[i]; } auto counter = start; for (auto [i, e] : elements) { elist[counter[i]++] = e; } } simple_csr(const std::vector<std::vector<E>>& es) : start(es.size() + 1, 0) { int n = es.size(); for (auto i : std::views::iota(0, n)) { start[i + 1] = (int)es[i].size() + start[i]; } elist.resize(start.back()); for (auto i : std::views::iota(0, n)) { std::copy(es[i].begin(), es[i].end(), elist.begin() + start[i]); } } int size() const { return (int)start.size() - 1; } const auto operator[](int i) const { return std::ranges::subrange(elist.begin() + start[i], elist.begin() + start[i + 1]); } auto operator[](int i) { return std::ranges::subrange(elist.begin() + start[i], elist.begin() + start[i + 1]); } const auto operator()(int i, int l, int r) const { return std::ranges::subrange(elist.begin() + start[i] + l, elist.begin() + start[i + 1] + r); } auto operator()(int i, int l, int r) { return std::ranges::subrange(elist.begin() + start[i] + l, elist.begin() + start[i + 1] + r); } private: std::vector<int> start; std::vector<E> elist; }; } // namespace ebi #line 9 "graph/base.hpp" namespace ebi { template <class T> struct Edge { int from, to; T cost; int id; }; template <class E> struct Graph { using cost_type = E; using edge_type = Edge<cost_type>; Graph(int n_) : n(n_) {} Graph() = default; void add_edge(int u, int v, cost_type c) { buff.emplace_back(u, edge_type{u, v, c, m}); edges.emplace_back(edge_type{u, v, c, m++}); } void add_undirected_edge(int u, int v, cost_type c) { buff.emplace_back(u, edge_type{u, v, c, m}); buff.emplace_back(v, edge_type{v, u, c, m}); edges.emplace_back(edge_type{u, v, c, m}); m++; } void read_tree(int offset = 1, bool is_weighted = false) { read_graph(n - 1, offset, false, is_weighted); } void read_parents(int offset = 1) { for (auto i : std::views::iota(1, n)) { int p; std::cin >> p; p -= offset; add_undirected_edge(p, i, 1); } build(); } void read_graph(int e, int offset = 1, bool is_directed = false, bool is_weighted = false) { for (int i = 0; i < e; i++) { int u, v; std::cin >> u >> v; u -= offset; v -= offset; if (is_weighted) { cost_type c; std::cin >> c; if (is_directed) { add_edge(u, v, c); } else { add_undirected_edge(u, v, c); } } else { if (is_directed) { add_edge(u, v, 1); } else { add_undirected_edge(u, v, 1); } } } build(); } void build() { assert(!prepared); csr = simple_csr<edge_type>(n, buff); buff.clear(); prepared = true; } int size() const { return n; } int node_number() const { return n; } int edge_number() const { return m; } edge_type get_edge(int i) const { return edges[i]; } std::vector<edge_type> get_edges() const { return edges; } const auto operator[](int i) const { return csr[i]; } auto operator[](int i) { return csr[i]; } private: int n, m = 0; std::vector<std::pair<int,edge_type>> buff; std::vector<edge_type> edges; simple_csr<edge_type> csr; bool prepared = false; }; } // namespace ebi #line 8 "tree/heavy_light_decomposition.hpp" namespace ebi { template <class T> struct heavy_light_decomposition { private: void dfs_sz(int v, Graph<T> &g) { for (auto &e : g[v]) { if (e.to == par[v]) continue; par[e.to] = v; depth_[e.to] = depth_[v] + 1; dist[e.to] = dist[v] + e.cost; dfs_sz(e.to, g); sz[v] += sz[e.to]; if (sz[e.to] > sz[g[v][0].to] || g[v][0].to == par[v]) std::swap(e, g[v][0]); } } void dfs_hld(int v, const Graph<T> &g) { in[v] = num++; rev[in[v]] = v; for (auto e : g[v]) { if (e.to == par[v]) continue; nxt[e.to] = (e.to == g[v][0].to ? nxt[v] : e.to); dfs_hld(e.to, g); } out[v] = num; } // [u, v) パスの取得 (v は u の祖先) std::vector<std::pair<int, int>> ascend(int u, int v) const { std::vector<std::pair<int, int>> res; while (nxt[u] != nxt[v]) { res.emplace_back(in[u], in[nxt[u]]); u = par[nxt[u]]; } if (u != v) res.emplace_back(in[u], in[v] + 1); return res; } // (u, v] パスの取得 (u は v の祖先) std::vector<std::pair<int, int>> descend(int u, int v) const { if (u == v) return {}; if (nxt[u] == nxt[v]) return {{in[u] + 1, in[v]}}; auto res = descend(u, par[nxt[v]]); res.emplace_back(in[nxt[v]], in[v]); return res; } public: heavy_light_decomposition(Graph<T> gh, int root = 0) : n(gh.size()), sz(n, 1), in(n), out(n), nxt(n), par(n, -1), depth_(n, 0), rev(n), dist(n, 0) { nxt[root] = root; dfs_sz(root, gh); dfs_hld(root, gh); } int idx(int u) const { return in[u]; } int rev_idx(int i) const { return rev[i]; } int la(int v, int k) const { while (1) { int u = nxt[v]; if (in[u] <= in[v] - k) return rev[in[v] - k]; k -= in[v] - in[u] + 1; v = par[u]; } } int lca(int u, int v) const { while (nxt[u] != nxt[v]) { if (in[u] < in[v]) std::swap(u, v); u = par[nxt[u]]; } return depth_[u] < depth_[v] ? u : v; } int jump(int s, int t, int i) const { if (i == 0) return s; int l = lca(s, t); int d = depth_[s] + depth_[t] - depth_[l] * 2; if (d < i) return -1; if (depth_[s] - depth_[l] >= i) return la(s, i); i = d - i; return la(t, i); } std::vector<int> path(int s, int t) const { int l = lca(s, t); std::vector<int> a, b; for (; s != l; s = par[s]) a.emplace_back(s); for (; t != l; t = par[t]) b.emplace_back(t); a.emplace_back(l); std::reverse(b.begin(), b.end()); a.insert(a.end(), b.begin(), b.end()); return a; } int root_of_heavy_path(int u) const { return nxt[u]; } int parent(int u) const { return par[u]; } T distance(int u, int v) const { return dist[u] + dist[v] - 2 * dist[lca(u, v)]; } T distance_from_root(int v) const { return dist[v]; } T depth(int v) const { return depth_[v]; } bool at_path(int u, int v, int s) const { return distance(u, v) == distance(u, s) + distance(s, v); } template <class F> void path_noncommutative_query(int u, int v, bool vertex, const F &f) const { int l = lca(u, v); for (auto [a, b] : ascend(u, l)) f(a + 1, b); if (vertex) f(in[l], in[l] + 1); for (auto [a, b] : descend(l, v)) f(a, b + 1); } std::vector<std::pair<int, int>> path_sections(int u, int v, bool vertex) const { int l = lca(u, v); std::vector<std::pair<int, int>> sections; for (auto [a, b] : ascend(u, l)) sections.emplace_back(a + 1, b); if (vertex) sections.emplace_back(in[l], in[l] + 1); for (auto [a, b] : descend(l, v)) sections.emplace_back(a, b + 1); return sections; } template <class F> int max_path(int u, int v, bool vertex, F binary_search) const { int prev = -1; int l = lca(u, v); for (auto [a, b] : ascend(u, l)) { a++; int m = binary_search(a, b); if (m == b) { prev = rev[b]; } else { return (m == a ? prev : rev[m]); } } if (vertex) { int m = binary_search(in[l], in[l] + 1); if (m == in[l]) { return prev; } else { prev = l; } } for (auto [a, b] : descend(l, v)) { b++; int m = binary_search(a, b); if (m == b) { prev = rev[b - 1]; } else { return m == a ? prev : rev[m - 1]; } } return v; } template <class F> void subtree_query(int u, bool vertex, const F &f) { f(in[u] + int(!vertex), out[u]); } const std::vector<int> &dfs_order() const { return rev; } std::vector<std::pair<int, int>> lca_based_auxiliary_tree_dfs_order( std::vector<int> vs) const; std::pair<std::vector<int>, Graph<T>> lca_based_auxiliary_tree( std::vector<int> vs) const; private: int n; std::vector<int> sz, in, out, nxt, par, depth_, rev; std::vector<T> dist; int num = 0; }; } // namespace ebi #line 10 "tree/lca_based_auxiliary_tree.hpp" namespace ebi { template <class T> std::vector<std::pair<int, int>> heavy_light_decomposition<T>::lca_based_auxiliary_tree_dfs_order( std::vector<int> vs) const { if (vs.empty()) return {}; std::sort(vs.begin(), vs.end(), [&](int u, int v) -> bool { return in[u] < in[v]; }); auto s = vs; for (int i = 1; i < int(vs.size()); i++) { s.emplace_back(lca(vs[i - 1], vs[i])); } std::sort(s.begin(), s.end(), [&](int u, int v) -> bool { return in[u] < in[v]; }); s.erase(std::unique(s.begin(), s.end()), s.end()); std::stack<int> stack; stack.push(s[0]); int sz = s.size(); std::vector<std::pair<int, int>> dfs_order(sz); dfs_order[0] = {s[0], -1}; for (int i = 1; i < int(s.size()); i++) { int v = s[i]; while (!stack.empty()) { int u = stack.top(); if (in[u] <= in[v] && in[v] < out[u]) { break; } else { stack.pop(); } } assert(!stack.empty()); int par = stack.top(); dfs_order[i] = {v, par}; stack.push(v); } return dfs_order; } template <class T> std::pair<std::vector<int>, Graph<T>> heavy_light_decomposition<T>::lca_based_auxiliary_tree( std::vector<int> vs) const { static std::vector<int> a(1'000'000, -1), p(1'000'000, -1); int k = vs.size(); if (k == 1) { return {vs, Graph<T>(1)}; } std::sort(vs.begin(), vs.end(), [&](int v, int u) { return in[v] < in[u]; }); std::stack<int> stack; std::vector<int> s; stack.push(vs[0]); for (int i : std::views::iota(1, k)) { int w = lca(vs[i - 1], vs[i]); int prev = -1; while (!stack.empty() && depth_[w] <= depth_[stack.top()]) { if (prev != -1) { s.emplace_back(prev); p[prev] = stack.top(); } prev = stack.top(); stack.pop(); } if (prev != w) { assert(prev != -1); s.emplace_back(prev); p[prev] = w; } stack.push(w); stack.push(vs[i]); } { int prev = -1; while (!stack.empty()) { int v = stack.top(); s.emplace_back(v); if (prev != -1) p[prev] = v; prev = v; stack.pop(); } } std::reverse(s.begin(), s.end()); int m = s.size(); for (int i : std::views::iota(0, m)) { a[s[i]] = i; } Graph<T> tree(m); for (auto v : s) { if (p[v] < 0) continue; T cost = distance(p[v], v); tree.add_edge(a[p[v]], a[v], cost); tree.add_edge(a[v], a[p[v]], cost); } tree.build(); for (auto v : s) { a[v] = -1; p[v] = -1; } return {s, tree}; } } // namespace ebi