lowest_common_ancestor.hpp¶
用单整数 RMQ 表预处理有根树，以 \(O(1)\) 回答最近公共祖先与距离；向上跳祖先和路径拆分使用树链剖分。
Complexity: Time: O(n log n) build, O(1) LCA/distance/ancestor checks. Space: O(n log n). Path listing is linear in the returned path length.
AC 记录：lca。
Implementation¶
当前头文件，省略 include guard；依赖见 #include。
/// @complexity Time: O(n log n) build, O(1) LCA/distance/ancestor checks.
/// Space: O(n log n). Path listing is linear in the returned path length.
#include "noya/ds/sparse_table.hpp"
#include "noya/graph/tree_preprocessing.hpp"
#include <algorithm>
#include <cassert>
#include <cstdint>
#include <limits>
#include <type_traits>
#include <utility>
#include <vector>
namespace noya {
/// @brief RMQ LCA with O(1) queries, without heavy paths or a binary jump table.
/// RMQ stores only parent DFS indices; tl maps the minimum back to a vertex.
/// Weighted adjacency automatically enables weighted distance; rebuilding from
/// unweighted adjacency resets it. Use hld for kth_ancestor/kth_on_path queries.
/// Weights must be integers (including signed/unsigned __int128). Each weight,
/// root distance, and returned distance must be representable as int64_t.
struct fastlca : tree_queries<fastlca> {
struct minimum_index_semilattice {
using value_type = int;
static value_type unit() { return std::numeric_limits<int>::max(); }
static value_type op(value_type a, value_type b) { return std::min(a, b); }
};
sparse_table<minimum_index_semilattice> ST;
std::vector<std::int64_t> len;
bool wt = false;
fastlca() = default;
template <class Edge>
explicit fastlca(const std::vector<std::vector<Edge>> &g, int root = 0) {
build(g, root);
}
template <class Edge>
void build(const std::vector<std::vector<Edge>> &g, int root = 0) {
tree_preprocessing::build(g, root);
std::vector<int> rmq(n);
for (int u : tl) {
rmq[dfn[u]] = fa[u] == -1 ? -1 : dfn[fa[u]];
}
ST.build(std::move(rmq));
wt = !std::is_integral_v<Edge>;
len.clear();
if constexpr (!std::is_integral_v<Edge>) {
using Weight = std::remove_cvref_t<decltype(std::get<1>(std::declval<Edge>()))>;
static_assert(std::numeric_limits<Weight>::is_integer,
"fastlca requires integer edge weights");
static_assert(sizeof(Weight) <= sizeof(__int128),
"fastlca supports integer weights up to 128 bits");
len.assign(n, 0);
for (int u : tl) {
for (const auto &edge : g[u]) {
int v = endpoint(edge);
if (fa[v] == u) {
const auto weight = std::get<1>(edge);
if constexpr (std::numeric_limits<Weight>::is_signed) {
assert(__int128(weight) >= std::numeric_limits<std::int64_t>::lowest() &&
__int128(weight) <= std::numeric_limits<std::int64_t>::max());
} else {
assert(__uint128_t(weight) <=
__uint128_t(std::numeric_limits<std::int64_t>::max()));
}
__int128 value = __int128(len[u]) + __int128(weight);
assert(std::numeric_limits<std::int64_t>::lowest() <= value &&
value <= std::numeric_limits<std::int64_t>::max());
len[v] = std::int64_t(value);
}
}
}
}
}
void build(const std::vector<std::vector<int>> &g = {}, int root = 0) {
build<int>(g, root);
}
int lca(int u, int v) const {
assert(0 <= u && u < n && 0 <= v && v < n);
if (u == v) {
return u;
}
int a = dfn[u], b = dfn[v];
if (a > b) {
std::swap(a, b);
}
return tl[ST.prod(a + 1, b + 1)];
}
std::int64_t distance(int a, int b) const {
if (!wt) {
return tree_queries<fastlca>::distance(a, b);
}
int c = lca(a, b);
__int128 value = __int128(len[a]) + len[b] - 2 * __int128(len[c]);
assert(std::numeric_limits<std::int64_t>::lowest() <= value &&
value <= std::numeric_limits<std::int64_t>::max());
return std::int64_t(value);
}
};
} // namespace noya
// NOYA_LOWEST_COMMON_ANCESTOR_HPP
#ifndef NOYA_LOWEST_COMMON_ANCESTOR_HPP
#define NOYA_LOWEST_COMMON_ANCESTOR_HPP 1
/// @complexity Time: O(n log n) build, O(1) LCA/distance/ancestor checks.
/// Space: O(n log n). Path listing is linear in the returned path length.
#include "noya/ds/sparse_table.hpp"
#include "noya/graph/tree_preprocessing.hpp"
#include <algorithm>
#include <cassert>
#include <cstdint>
#include <limits>
#include <type_traits>
#include <utility>
#include <vector>
namespace noya {
/// @brief RMQ LCA with O(1) queries, without heavy paths or a binary jump table.
/// RMQ stores only parent DFS indices; tl maps the minimum back to a vertex.
/// Weighted adjacency automatically enables weighted distance; rebuilding from
/// unweighted adjacency resets it. Use hld for kth_ancestor/kth_on_path queries.
/// Weights must be integers (including signed/unsigned __int128). Each weight,
/// root distance, and returned distance must be representable as int64_t.
struct fastlca : tree_queries<fastlca> {
struct minimum_index_semilattice {
using value_type = int;
static value_type unit() { return std::numeric_limits<int>::max(); }
static value_type op(value_type a, value_type b) { return std::min(a, b); }
};
sparse_table<minimum_index_semilattice> ST;
std::vector<std::int64_t> len;
bool wt = false;
fastlca() = default;
template <class Edge>
explicit fastlca(const std::vector<std::vector<Edge>> &g, int root = 0) {
build(g, root);
}
template <class Edge>
void build(const std::vector<std::vector<Edge>> &g, int root = 0) {
tree_preprocessing::build(g, root);
std::vector<int> rmq(n);
for (int u : tl) {
rmq[dfn[u]] = fa[u] == -1 ? -1 : dfn[fa[u]];
}
ST.build(std::move(rmq));
wt = !std::is_integral_v<Edge>;
len.clear();
if constexpr (!std::is_integral_v<Edge>) {
using Weight = std::remove_cvref_t<decltype(std::get<1>(std::declval<Edge>()))>;
static_assert(std::numeric_limits<Weight>::is_integer,
"fastlca requires integer edge weights");
static_assert(sizeof(Weight) <= sizeof(__int128),
"fastlca supports integer weights up to 128 bits");
len.assign(n, 0);
for (int u : tl) {
for (const auto &edge : g[u]) {
int v = endpoint(edge);
if (fa[v] == u) {
const auto weight = std::get<1>(edge);
if constexpr (std::numeric_limits<Weight>::is_signed) {
assert(__int128(weight) >= std::numeric_limits<std::int64_t>::lowest() &&
__int128(weight) <= std::numeric_limits<std::int64_t>::max());
} else {
assert(__uint128_t(weight) <=
__uint128_t(std::numeric_limits<std::int64_t>::max()));
}
__int128 value = __int128(len[u]) + __int128(weight);
assert(std::numeric_limits<std::int64_t>::lowest() <= value &&
value <= std::numeric_limits<std::int64_t>::max());
len[v] = std::int64_t(value);
}
}
}
}
}
void build(const std::vector<std::vector<int>> &g = {}, int root = 0) {
build<int>(g, root);
}
int lca(int u, int v) const {
assert(0 <= u && u < n && 0 <= v && v < n);
if (u == v) {
return u;
}
int a = dfn[u], b = dfn[v];
if (a > b) {
std::swap(a, b);
}
return tl[ST.prod(a + 1, b + 1)];
}
std::int64_t distance(int a, int b) const {
if (!wt) {
return tree_queries<fastlca>::distance(a, b);
}
int c = lca(a, b);
__int128 value = __int128(len[a]) + len[b] - 2 * __int128(len[c]);
assert(std::numeric_limits<std::int64_t>::lowest() <= value &&
value <= std::numeric_limits<std::int64_t>::max());
return std::int64_t(value);
}
};
} // namespace noya
#endif // NOYA_LOWEST_COMMON_ANCESTOR_HPP
/// @complexity Time: O(n log n) build, O(1) LCA/distance/ancestor checks.
/// Space: O(n log n). Path listing is linear in the returned path length.
/// @complexity Time: O(n log n) build and O(1) idempotent range query.
/// Space: O(n log n).
#include <bit>
#include <cassert>
#include <utility>
#include <vector>
namespace noya {
/// @brief Static O(1) range product for an idempotent semilattice.
/// Monoid provides value_type, unit(), and an associative, idempotent op().
/// Empty ranges return unit(); input arrays are owned by the table.
/// @details For k = floor(log2(r-l)), [l, r) is covered by the length-2^k
/// blocks beginning at l and ending at r. They may overlap, so combining them
/// is correct precisely because op(x, x) = x.
template <class Monoid> class sparse_table {
public:
using value_type = typename Monoid::value_type;
sparse_table() = default;
explicit sparse_table(std::vector<value_type> a) { build(std::move(a)); }
template <class F> sparse_table(int n_, F f) { build(n_, std::move(f)); }
template <class F> void build(int n_, F f) {
assert(n_ >= 0);
std::vector<value_type> a;
a.reserve(n_);
for (int i = 0; i < n_; i++) {
a.push_back(f(i));
}
build(std::move(a));
}
/// @brief Rebuild from a static array.
void build(std::vector<value_type> a) {
n = int(a.size());
int lg = int(std::bit_width(static_cast<unsigned>(n)));
st.resize(lg);
if (n == 0) {
return;
}
st[0] = std::move(a);
for (int dep = 1; dep < lg; dep++) {
int w = 1 << dep;
int hf = w >> 1;
st[dep].resize(n - w + 1, Monoid::unit());
for (int l = 0; l + w <= n; l++) {
st[dep][l] = Monoid::op(st[dep - 1][l], st[dep - 1][l + hf]);
}
}
}
int size() const { return n; }
bool empty() const { return n == 0; }
value_type get(int p) const {
assert(0 <= p && p < n);
return st[0][p];
}
value_type all_prod() const { return prod(0, n); }
/// @brief Return the idempotent product over [l, r).
value_type prod(int l, int r) const {
assert(0 <= l && l <= r && r <= n);
if (l == r) {
return Monoid::unit();
}
int dep = int(std::bit_width(static_cast<unsigned>(r - l))) - 1;
int w = 1 << dep;
return Monoid::op(st[dep][l], st[dep][r - w]);
}
private:
int n = 0;
std::vector<std::vector<value_type>> st;
};
} // namespace noya
/// @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 <cstdint>
#include <limits>
#include <type_traits>
#include <utility>
#include <vector>
namespace noya {
/// @brief RMQ LCA with O(1) queries, without heavy paths or a binary jump table.
/// RMQ stores only parent DFS indices; tl maps the minimum back to a vertex.
/// Weighted adjacency automatically enables weighted distance; rebuilding from
/// unweighted adjacency resets it. Use hld for kth_ancestor/kth_on_path queries.
/// Weights must be integers (including signed/unsigned __int128). Each weight,
/// root distance, and returned distance must be representable as int64_t.
struct fastlca : tree_queries<fastlca> {
struct minimum_index_semilattice {
using value_type = int;
static value_type unit() { return std::numeric_limits<int>::max(); }
static value_type op(value_type a, value_type b) { return std::min(a, b); }
};
sparse_table<minimum_index_semilattice> ST;
std::vector<std::int64_t> len;
bool wt = false;
fastlca() = default;
template <class Edge>
explicit fastlca(const std::vector<std::vector<Edge>> &g, int root = 0) {
build(g, root);
}
template <class Edge>
void build(const std::vector<std::vector<Edge>> &g, int root = 0) {
tree_preprocessing::build(g, root);
std::vector<int> rmq(n);
for (int u : tl) {
rmq[dfn[u]] = fa[u] == -1 ? -1 : dfn[fa[u]];
}
ST.build(std::move(rmq));
wt = !std::is_integral_v<Edge>;
len.clear();
if constexpr (!std::is_integral_v<Edge>) {
using Weight = std::remove_cvref_t<decltype(std::get<1>(std::declval<Edge>()))>;
static_assert(std::numeric_limits<Weight>::is_integer,
"fastlca requires integer edge weights");
static_assert(sizeof(Weight) <= sizeof(__int128),
"fastlca supports integer weights up to 128 bits");
len.assign(n, 0);
for (int u : tl) {
for (const auto &edge : g[u]) {
int v = endpoint(edge);
if (fa[v] == u) {
const auto weight = std::get<1>(edge);
if constexpr (std::numeric_limits<Weight>::is_signed) {
assert(__int128(weight) >= std::numeric_limits<std::int64_t>::lowest() &&
__int128(weight) <= std::numeric_limits<std::int64_t>::max());
} else {
assert(__uint128_t(weight) <=
__uint128_t(std::numeric_limits<std::int64_t>::max()));
}
__int128 value = __int128(len[u]) + __int128(weight);
assert(std::numeric_limits<std::int64_t>::lowest() <= value &&
value <= std::numeric_limits<std::int64_t>::max());
len[v] = std::int64_t(value);
}
}
}
}
}
void build(const std::vector<std::vector<int>> &g = {}, int root = 0) {
build<int>(g, root);
}
int lca(int u, int v) const {
assert(0 <= u && u < n && 0 <= v && v < n);
if (u == v) {
return u;
}
int a = dfn[u], b = dfn[v];
if (a > b) {
std::swap(a, b);
}
return tl[ST.prod(a + 1, b + 1)];
}
std::int64_t distance(int a, int b) const {
if (!wt) {
return tree_queries<fastlca>::distance(a, b);
}
int c = lca(a, b);
__int128 value = __int128(len[a]) + len[b] - 2 * __int128(len[c]);
assert(std::numeric_limits<std::int64_t>::lowest() <= value &&
value <= std::numeric_limits<std::int64_t>::max());
return std::int64_t(value);
}
};
} // namespace noya
// NOYA_LOWEST_COMMON_ANCESTOR_HPP