virtual_tree.hpp¶
把一次询问涉及的若干关键点及其 LCA 压成小树；适合多组树上关键点 DP。
Complexity: Time: O(k log k + k L) for k key vertices and LCA query cost L. Space: O(k).
Implementation¶
当前头文件，省略 include guard；依赖见 #include。
/// @complexity Time: O(k log k + k L) for k key vertices and LCA query cost L.
/// Space: O(k).
#include "noya/graph/tree_preprocessing.hpp"
#include <algorithm>
#include <cassert>
#include <vector>
namespace noya {
/// @brief Vertices and parent edges of a compressed virtual tree.
struct virtual_tree_result {
std::vector<int> vs;
std::vector<int> fa;
std::vector<int> len;
std::vector<bool> key;
};
/// @brief Build the minimal rooted tree containing all key vertices and LCAs.
/// Accepts fastlca (L=O(1)) or hld (L=O(log n)) without copying the tree.
template <class Tree>
virtual_tree_result build_virtual_tree(const Tree &tr, std::vector<int> vs1) {
virtual_tree_result res;
if (vs1.empty()) {
return res;
}
for (int u : vs1) {
assert(0 <= u && u < tr.n);
}
std::sort(vs1.begin(), vs1.end(),
[&](int a, int b) { return tr.dfn[a] < tr.dfn[b]; });
vs1.erase(std::unique(vs1.begin(), vs1.end()), vs1.end());
res.vs = vs1;
for (int i = 1; i < int(vs1.size()); i++) {
res.vs.push_back(tr.lca(vs1[i - 1], vs1[i]));
}
std::sort(res.vs.begin(), res.vs.end(),
[&](int a, int b) { return tr.dfn[a] < tr.dfn[b]; });
res.vs.erase(std::unique(res.vs.begin(), res.vs.end()), res.vs.end());
int sz = int(res.vs.size());
res.fa.assign(sz, -1);
res.len.assign(sz, 0);
res.key.resize(sz);
std::vector<int> stk;
int next_key = 0;
for (int idx = 0; idx < sz; idx++) {
int u = res.vs[idx];
if (next_key < int(vs1.size()) && vs1[next_key] == u) {
res.key[idx] = true;
next_key++;
}
while (!stk.empty() && !tr.is_subtree(u, res.vs[stk.back()])) {
stk.pop_back();
}
if (!stk.empty()) {
res.fa[idx] = stk.back();
res.len[idx] = tr.d[u] - tr.d[res.vs[stk.back()]];
}
stk.push_back(idx);
}
return res;
}
} // namespace noya
// NOYA_VIRTUAL_TREE_HPP
#ifndef NOYA_VIRTUAL_TREE_HPP
#define NOYA_VIRTUAL_TREE_HPP 1
/// @complexity Time: O(k log k + k L) for k key vertices and LCA query cost L.
/// Space: O(k).
#include "noya/graph/tree_preprocessing.hpp"
#include <algorithm>
#include <cassert>
#include <vector>
namespace noya {
/// @brief Vertices and parent edges of a compressed virtual tree.
struct virtual_tree_result {
std::vector<int> vs;
std::vector<int> fa;
std::vector<int> len;
std::vector<bool> key;
};
/// @brief Build the minimal rooted tree containing all key vertices and LCAs.
/// Accepts fastlca (L=O(1)) or hld (L=O(log n)) without copying the tree.
template <class Tree>
virtual_tree_result build_virtual_tree(const Tree &tr, std::vector<int> vs1) {
virtual_tree_result res;
if (vs1.empty()) {
return res;
}
for (int u : vs1) {
assert(0 <= u && u < tr.n);
}
std::sort(vs1.begin(), vs1.end(),
[&](int a, int b) { return tr.dfn[a] < tr.dfn[b]; });
vs1.erase(std::unique(vs1.begin(), vs1.end()), vs1.end());
res.vs = vs1;
for (int i = 1; i < int(vs1.size()); i++) {
res.vs.push_back(tr.lca(vs1[i - 1], vs1[i]));
}
std::sort(res.vs.begin(), res.vs.end(),
[&](int a, int b) { return tr.dfn[a] < tr.dfn[b]; });
res.vs.erase(std::unique(res.vs.begin(), res.vs.end()), res.vs.end());
int sz = int(res.vs.size());
res.fa.assign(sz, -1);
res.len.assign(sz, 0);
res.key.resize(sz);
std::vector<int> stk;
int next_key = 0;
for (int idx = 0; idx < sz; idx++) {
int u = res.vs[idx];
if (next_key < int(vs1.size()) && vs1[next_key] == u) {
res.key[idx] = true;
next_key++;
}
while (!stk.empty() && !tr.is_subtree(u, res.vs[stk.back()])) {
stk.pop_back();
}
if (!stk.empty()) {
res.fa[idx] = stk.back();
res.len[idx] = tr.d[u] - tr.d[res.vs[stk.back()]];
}
stk.push_back(idx);
}
return res;
}
} // namespace noya
#endif // NOYA_VIRTUAL_TREE_HPP
/// @complexity Time: O(k log k + k L) for k key vertices and LCA query cost L.
/// Space: O(k).
/// @complexity Time: O(n) build; common queries add O(1) work to their LCA calls,
/// except path(), which takes O(path length). Space: O(n).
#include <algorithm>
#include <array>
#include <cassert>
#include <cstdint>
#include <tuple>
#include <type_traits>
#include <utility>
#include <vector>
namespace noya {
/// @brief Iterative rooted-tree preprocessing shared by RMQ LCA and HLD.
/// The graph is a connected tree, with integer endpoints or (endpoint, weight)
/// pairs. Root depth is zero, root parent is -1, and subtree ranges are [l, r).
/// No adjacency-list copy, RMQ table, jump table, or heavy paths are retained.
struct tree_preprocessing {
int n = 0;
int rt = -1;
std::vector<int> fa, d, dfn, siz, tl;
tree_preprocessing() = default;
template <class Edge>
explicit tree_preprocessing(const std::vector<std::vector<Edge>> &g,
int root = 0) {
build(g, root);
}
template <class Edge> static int endpoint(const Edge &edge) {
if constexpr (std::is_integral_v<Edge>) {
return int(edge);
} else {
return int(std::get<0>(edge));
}
}
template <class Edge>
void build(const std::vector<std::vector<Edge>> &g, int root = 0) {
n = int(g.size());
rt = n == 0 ? -1 : root;
fa.assign(n, -1);
d.assign(n, 0);
dfn.assign(n, -1);
siz.assign(n, 1);
tl.clear();
tl.reserve(n);
if (n == 0) {
return;
}
assert(0 <= root && root < n);
std::vector<int> stack{root};
while (!stack.empty()) {
int u = stack.back();
stack.pop_back();
assert(dfn[u] == -1);
dfn[u] = int(tl.size());
tl.push_back(u);
for (auto it = g[u].rbegin(); it != g[u].rend(); ++it) {
int v = endpoint(*it);
assert(0 <= v && v < n);
if (v != fa[u]) {
fa[v] = u;
d[v] = d[u] + 1;
stack.push_back(v);
}
}
}
assert(int(tl.size()) == n);
for (auto it = tl.rbegin(); it != tl.rend(); ++it) {
if (fa[*it] != -1) {
siz[fa[*it]] += siz[*it];
}
}
}
void build(const std::vector<std::vector<int>> &g = {}, int root = 0) {
build<int>(g, root);
}
bool is_subtree(int a, int b) const {
assert(0 <= a && a < n && 0 <= b && b < n);
return dfn[b] <= dfn[a] && dfn[a] < dfn[b] + siz[b];
}
std::array<int, 2> subtree(int u) const {
assert(0 <= u && u < n);
return {dfn[u], dfn[u] + siz[u]};
}
};
/// @brief Shared tree queries with compile-time dispatch to Derived::lca().
template <class Derived> struct tree_queries : tree_preprocessing {
int rooted_lca(int a, int b, int root) const {
const auto &self = static_cast<const Derived &>(*this);
return self.lca(a, b) ^ self.lca(a, root) ^ self.lca(b, root);
}
std::int64_t distance(int a, int b) const {
const auto &self = static_cast<const Derived &>(*this);
int c = self.lca(a, b);
return std::int64_t(d[a]) + d[b] - 2 * std::int64_t(d[c]);
}
std::pair<int, int> intersection(int a, int b, int c, int e) const {
const auto &self = static_cast<const Derived &>(*this);
int ab = self.lca(a, b), ac = self.lca(a, c), ae = self.lca(a, e);
int bc = self.lca(b, c), be = self.lca(b, e), ce = self.lca(c, e);
int x = ab ^ ac ^ bc;
int y = ab ^ ae ^ be;
if (x != y) {
return {x, y};
}
int z = ac ^ ae ^ ce;
return x == z ? std::pair{x, x} : std::pair{-1, -1};
}
std::pair<int, int> intersection(std::pair<int, int> a,
std::pair<int, int> b) const {
return intersection(a.first, a.second, b.first, b.second);
}
std::vector<int> path(int a, int b) const {
const auto &self = static_cast<const Derived &>(*this);
int c = self.lca(a, b);
std::vector<int> left, right;
while (a != c) {
left.push_back(a);
a = fa[a];
}
while (b != c) {
right.push_back(b);
b = fa[b];
}
left.push_back(c);
left.insert(left.end(), right.rbegin(), right.rend());
return left;
}
};
} // namespace noya
#include <algorithm>
#include <cassert>
#include <vector>
namespace noya {
/// @brief Vertices and parent edges of a compressed virtual tree.
struct virtual_tree_result {
std::vector<int> vs;
std::vector<int> fa;
std::vector<int> len;
std::vector<bool> key;
};
/// @brief Build the minimal rooted tree containing all key vertices and LCAs.
/// Accepts fastlca (L=O(1)) or hld (L=O(log n)) without copying the tree.
template <class Tree>
virtual_tree_result build_virtual_tree(const Tree &tr, std::vector<int> vs1) {
virtual_tree_result res;
if (vs1.empty()) {
return res;
}
for (int u : vs1) {
assert(0 <= u && u < tr.n);
}
std::sort(vs1.begin(), vs1.end(),
[&](int a, int b) { return tr.dfn[a] < tr.dfn[b]; });
vs1.erase(std::unique(vs1.begin(), vs1.end()), vs1.end());
res.vs = vs1;
for (int i = 1; i < int(vs1.size()); i++) {
res.vs.push_back(tr.lca(vs1[i - 1], vs1[i]));
}
std::sort(res.vs.begin(), res.vs.end(),
[&](int a, int b) { return tr.dfn[a] < tr.dfn[b]; });
res.vs.erase(std::unique(res.vs.begin(), res.vs.end()), res.vs.end());
int sz = int(res.vs.size());
res.fa.assign(sz, -1);
res.len.assign(sz, 0);
res.key.resize(sz);
std::vector<int> stk;
int next_key = 0;
for (int idx = 0; idx < sz; idx++) {
int u = res.vs[idx];
if (next_key < int(vs1.size()) && vs1[next_key] == u) {
res.key[idx] = true;
next_key++;
}
while (!stk.empty() && !tr.is_subtree(u, res.vs[stk.back()])) {
stk.pop_back();
}
if (!stk.empty()) {
res.fa[idx] = stk.back();
res.len[idx] = tr.d[u] - tr.d[res.vs[stk.back()]];
}
stk.push_back(idx);
}
return res;
}
} // namespace noya
// NOYA_VIRTUAL_TREE_HPP