#line 1 "verify/tree/UNIT_lca_auxiliary_tree.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
#line 2 "template/template.hpp"
#include <bits/stdc++.h>
using namespace std;
#line 2 "template/macro.hpp"
#define rep(i, a, b) for (int i = (a); i < (int)(b); i++)
#define rrep(i, a, b) for (int i = (int)(b) - 1; i >= (a); i--)
#define ALL(v) (v).begin(), (v).end()
#define UNIQUE(v) sort(ALL(v)), (v).erase(unique(ALL(v)), (v).end())
#define SZ(v) (int)v.size()
#define MIN(v) *min_element(ALL(v))
#define MAX(v) *max_element(ALL(v))
#define LB(v, x) int(lower_bound(ALL(v), (x)) - (v).begin())
#define UB(v, x) int(upper_bound(ALL(v), (x)) - (v).begin())
#define YN(b) cout << ((b) ? "YES" : "NO") << "\n";
#define Yn(b) cout << ((b) ? "Yes" : "No") << "\n";
#define yn(b) cout << ((b) ? "yes" : "no") << "\n";
#line 6 "template/template.hpp"
#line 2 "template/util.hpp"
using uint = unsigned int;
using ll = long long int;
using ull = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
template <class T>
using priority_queue_asc = priority_queue<T, vector<T>, greater<T>>;
template <class T, class S = T>
S SUM(const vector<T>& a) {
return accumulate(ALL(a), S(0));
}
template <class T1, class T2>
inline bool chmin(T1& a, T2 b) {
if (a > b) {
a = b;
return true;
}
return false;
}
template <class T1, class T2>
inline bool chmax(T1& a, T2 b) {
if (a < b) {
a = b;
return true;
}
return false;
}
template <class T1, class T2>
inline bool chmin_opt(optional<T1>& a, T2 b) {
if (!a || a > b) {
a = b;
return true;
}
return false;
}
template <class T1, class T2>
inline bool chmax_opt(optional<T1>& a, T2 b) {
if (!a || a < b) {
a = b;
return true;
}
return false;
}
template <class T>
int popcnt(T x) {
return __builtin_popcountll(x);
}
template <class T>
int topbit(T x) {
return (x == 0 ? -1 : 63 - __builtin_clzll(x));
}
template <class T>
int lowbit(T x) {
return (x == 0 ? -1 : __builtin_ctzll(x));
}
#line 8 "template/template.hpp"
#line 2 "template/inout.hpp"
struct Fast {
Fast() {
cin.tie(nullptr);
ios_base::sync_with_stdio(false);
cout << fixed << setprecision(15);
}
} fast;
ostream& operator<<(ostream& os, __uint128_t x) {
char buf[40];
size_t k = 0;
while (x > 0) buf[k++] = (char)(x % 10 + '0'), x /= 10;
if (k == 0) buf[k++] = '0';
while (k) os << buf[--k];
return os;
}
ostream& operator<<(ostream& os, __int128_t x) {
return x < 0 ? (os << '-' << (__uint128_t)(-x)) : (os << (__uint128_t)x);
}
template <class T, size_t N>
ostream& operator<<(ostream& os, const array<T, N>& a);
template <class T1, class T2>
istream& operator>>(istream& is, pair<T1, T2>& p) {
return is >> p.first >> p.second;
}
template <class T1, class T2>
ostream& operator<<(ostream& os, const pair<T1, T2>& p) {
return os << p.first << " " << p.second;
}
template <class T>
istream& operator>>(istream& is, vector<T>& a) {
for (auto& v : a) is >> v;
return is;
}
template <class T>
ostream& operator<<(ostream& os, const vector<T>& a) {
for (auto it = a.begin(); it != a.end();) {
os << *it;
if (++it != a.end()) os << " ";
}
return os;
}
template <class T, size_t N>
ostream& operator<<(ostream& os, const array<T, N>& a) {
for (auto it = a.begin(); it != a.end();) {
os << *it;
if (++it != a.end()) os << " ";
}
return os;
}
template <class T>
ostream& operator<<(ostream& os, const set<T>& st) {
os << "{";
for (auto it = st.begin(); it != st.end();) {
os << *it;
if (++it != st.end()) os << ",";
}
os << "}";
return os;
}
template <class T1, class T2>
ostream& operator<<(ostream& os, const map<T1, T2>& mp) {
os << "{";
for (auto it = mp.begin(); it != mp.end();) {
os << it->first << ":" << it->second;
if (++it != mp.end()) os << ",";
}
os << "}";
return os;
}
void in() {}
template <typename T, class... U>
void in(T& t, U&... u) {
cin >> t;
in(u...);
}
template <class... T>
void in_zip(int n, T&... t) {
assert(n >= 0 && ((size(t) >= static_cast<size_t>(n)) && ...));
for (int i = 0; i < n; i++) in(t[i]...);
}
void out() { cout << "\n"; }
template <typename T, class... U, char sep = ' '>
void out(const T& t, const U&... u) {
cout << t;
if (sizeof...(u)) cout << sep;
out(u...);
}
template <class T, class U>
void out_opt(const optional<T>& opt, const U& fallback, ostream& os = cout) {
if (opt.has_value())
os << opt.value();
else
os << fallback;
os << "\n";
}
template <class T, class U>
void out_opt(const vector<optional<T>>& vec, const U& fallback, ostream& os = cout) {
for (auto it = vec.begin(); it != vec.end();) {
if ((*it).has_value())
os << (*it).value();
else
os << fallback;
if (++it != vec.end()) os << " ";
}
os << "\n";
}
namespace IO {
template <class T, class... U>
T read(U&&... u) {
T t = T(forward<U>(u)...);
in(t);
return t;
}
namespace Graph {
vector<vector<int>> unweighted(int n, int m, bool directed = false, int offset = 1) {
vector<vector<int>> g(n);
for (int i = 0; i < m; i++) {
int u, v;
cin >> u >> v;
u -= offset, v -= offset;
g[u].push_back(v);
if (!directed) g[v].push_back(u);
}
return g;
}
template <class T>
vector<vector<pair<int, T>>> weighted(int n, int m, bool directed = false, int offset = 1) {
vector<vector<pair<int, T>>> g(n);
for (int i = 0; i < m; i++) {
int u, v;
T w;
cin >> u >> v >> w;
u -= offset, v -= offset;
g[u].push_back({v, w});
if (!directed) g[v].push_back({u, w});
}
return g;
}
} // namespace Graph
namespace Tree {
vector<vector<int>> unweighted(int n, bool directed = false, int offset = 1) {
return Graph::unweighted(n, n - 1, directed, offset);
}
template <class T>
vector<vector<pair<int, T>>> weighted(int n, bool directed = false, int offset = 1) {
return Graph::weighted<T>(n, n - 1, directed, offset);
}
vector<vector<int>> rooted(int n, bool to_root = true, bool to_leaf = true, int offset = 1) {
vector<vector<int>> g(n);
for (int i = 1; i < n; i++) {
int p;
cin >> p;
p -= offset;
if (to_root) g[i].push_back(p);
if (to_leaf) g[p].push_back(i);
}
return g;
}
} // namespace Tree
} // namespace IO
#line 10 "template/template.hpp"
#line 2 "template/debug.hpp"
#ifdef LOCAL
#define debug 1
#define show(...) _show(0, #__VA_ARGS__, __VA_ARGS__)
#else
#define debug 0
#define show(...) true
#endif
template <class T>
void _show(int, T) {
cerr << '\n';
}
template <class T1, class T2, class... T3>
void _show(int i, const T1& a, const T2& b, const T3&... c) {
for (; a[i] != ',' && a[i] != '\0'; i++) cerr << a[i];
cerr << ":" << b << " ";
_show(i + 1, a, c...);
}
#line 2 "graph/graph.hpp"
#line 4 "graph/graph.hpp"
template <class E>
struct GraphBase {
public:
GraphBase() : GraphBase(0) {}
GraphBase(int size) : n(size) {
assert(size >= 0);
g.resize(size);
}
size_t size() const { return n; }
const vector<E>& operator[](int x) const {
assert(0 <= x && x < n);
return g[x];
}
vector<E>& operator[](int x) {
assert(0 <= x && x < n);
return g[x];
}
protected:
int n;
vector<vector<E>> g;
};
struct EdgeUnweighted {
int to;
};
struct GraphUnweighted : GraphBase<EdgeUnweighted> {
using base = GraphBase<EdgeUnweighted>;
public:
GraphUnweighted() : base(0) {}
GraphUnweighted(int size) : base(size) {}
GraphUnweighted(const vector<vector<int>>& graph) : base(graph.size()) {
for (int x = 0; x < n; x++) {
base::g[x].reserve(graph[x].size());
for (int y : graph[x]) base::g[x].push_back({y});
}
}
void add_edge(int x, int y) {
(*this)[x].push_back({y});
(*this)[y].push_back({x});
}
void add_edge_directed(int from, int to) { (*this)[from].push_back({to}); }
};
template <class T>
struct EdgeWeighted {
int to;
T weight;
};
template <class T>
struct GraphWeighted : GraphBase<EdgeWeighted<T>> {
using base = GraphBase<EdgeWeighted<T>>;
public:
GraphWeighted() : base(0) {}
GraphWeighted(int size) : base(size) {}
GraphWeighted(const vector<vector<pair<int, T>>>& graph) : base(graph.size()) {
for (int x = 0; x < base::n; x++) {
base::g[x].reserve(graph[x].size());
for (const auto& [y, w] : graph[x]) base::g[x].push_back({y, w});
}
}
void add_edge(int x, int y, T w) {
(*this)[x].push_back({y, w});
(*this)[y].push_back({x, w});
}
void add_edge_directed(int from, int to, T w) { (*this)[from].push_back({to, w}); }
};
inline GraphWeighted<int> GraphUnweightedToWeighted(const GraphUnweighted& g) {
GraphWeighted<int> h(g.size());
for (int x = 0; x < static_cast<int>(g.size()); x++) {
h[x].reserve(g[x].size());
for (const auto& e : g[x]) h[x].push_back({e.to, 1});
}
return h;
}
/**
* @brief Graph
* @docs docs/graph/graph.md
*/
#line 2 "algebraic-structure/util.hpp"
#ifdef __cpp_concepts
#define REQUIRES(...) requires __VA_ARGS__
#else
#define REQUIRES(...)
#endif
#line 3 "algebraic-structure/magma.hpp"
#ifdef __cpp_concepts
template <class M>
concept Magma = requires(typename M::value_type x, typename M::value_type y) {
typename M::value_type;
{ M::op(x, y) } -> same_as<typename M::value_type>;
};
#endif
template <class T>
struct AddMagma {
using value_type = T;
static T op(T x, T y) { return x + y; }
};
template <class T>
struct MulMagma {
using value_type = T;
static T op(T x, T y) { return x * y; }
};
template <class T, T id>
struct MaxMagma {
using value_type = T;
static T op(T x, T y) { return x > y ? x : y; }
};
template <class T, T id>
struct MinMagma {
using value_type = T;
static T op(T x, T y) { return x < y ? x : y; }
};
#line 3 "data-structure/sparse-table.hpp"
template <class M>
REQUIRES(Magma<M>)
struct SparseTable {
using T = typename M::value_type;
private:
int n;
vector<vector<T>> st;
public:
SparseTable() : n(0) {}
SparseTable(const vector<T>& arr) {
n = arr.size();
int log = 1;
while (n >> log) log++;
st = vector<vector<T>>(log);
st[0] = vector<T>(arr.begin(), arr.end());
for (int k = 1; k < log; k++) {
const auto& stp = st[k - 1];
auto sti = vector<T>(n - (1 << k) + 1);
for (int i = 0; i < (int)sti.size(); i++)
sti[i] = M::op(stp[i], stp[i + (1 << (k - 1))]);
st[k] = move(sti);
}
}
T prod(int l, int r) const // [l,r)
{
assert(0 <= l && l < r && r <= n);
int j = bit_width(static_cast<unsigned int>(r - l)) - 1;
return M::op(st[j][l], st[j][r - (1 << j)]);
}
};
#line 3 "tree/lowest-common-ancestor.hpp"
struct LowestCommonAncestor {
using P = pair<int, int>;
struct LcaMagma {
using value_type = P;
static P op(P x, P y) { return x.second <= y.second ? x : y; }
};
protected:
int n, r;
SparseTable<LcaMagma> st;
vector<int> in_time, depth, parent, euler_tour;
public:
size_t size() const { return n; }
int root() const { return r; }
LowestCommonAncestor() : n(0), r(0) {}
template <class G>
LowestCommonAncestor(const G& g, int root = 0) {
build(g, root);
}
template <class G>
void build(const G& g, int root = 0) {
n = g.size(), r = root;
assert(n > 0);
assert(0 <= r && r < n);
parent.assign(n, -1);
depth.assign(n, 0);
in_time.assign(n, 0);
euler_tour.clear();
euler_tour.reserve(2 * n - 1);
{
stack<int> dfs;
dfs.push(r);
vector<int> idx(n);
while (!dfs.empty()) {
int x = dfs.top();
dfs.pop();
if (idx[x] == 0) in_time[x] = euler_tour.size();
euler_tour.push_back(x);
if (idx[x] < static_cast<int>(g[x].size())) {
dfs.push(x);
int y = g[x][idx[x]++].to;
if (y != parent[x]) {
parent[y] = x;
depth[y] = depth[x] + 1;
dfs.push(y);
}
}
}
}
vector<P> data;
data.reserve(euler_tour.size());
for (auto v : euler_tour)
data.push_back({v, depth[v]});
st = SparseTable<LcaMagma>(data);
}
int lca(int u, int v) const {
assert(0 <= u && u < n);
assert(0 <= v && v < n);
int x = in_time[u], y = in_time[v];
if (x > y) swap(x, y);
return st.prod(x, y + 1).first;
}
};
/**
* @brief Lowest Common Ancestor
* @docs docs/tree/lowest-common-ancestor.md
*/
#line 3 "tree/lca-auxiliary-tree.hpp"
struct LCAAuxiliaryTree : LowestCommonAncestor {
using base = LowestCommonAncestor;
LCAAuxiliaryTree() = default;
template <class G>
LCAAuxiliaryTree(const G& g, int root = 0) : base(g, root) {}
template <class G>
pair<int, vector<int>> calc(vector<int> vs, G& g) const {
if (vs.empty()) return {-1, vector<int>{}};
assert(static_cast<int>(g.size()) == n);
for (int x : vs) assert(0 <= x && x < n);
sort(vs.begin(), vs.end(), [&](int x, int y) { return in_time[x] < in_time[y]; });
vs.erase(unique(vs.begin(), vs.end()), vs.end());
int m = vs.size();
stack<int> path;
path.push(vs[0]);
g[vs[0]] = {};
for (int i = 0; i < m - 1; i++) {
int x = vs[i], y = vs[i + 1];
int w = lca(x, y);
if (w != x) {
int last = path.top();
path.pop();
while (!path.empty() && depth[w] < depth[path.top()]) {
g[path.top()].push_back({last});
last = path.top();
path.pop();
}
if (path.empty() || path.top() != w) {
path.push(w);
vs.push_back(w);
g[w] = {{last}};
} else
g[w].push_back({last});
}
path.push(y);
g[y] = {};
}
int prv = path.top();
path.pop();
while (!path.empty()) {
g[path.top()].push_back({prv});
prv = path.top();
path.pop();
}
sort(vs.begin(), vs.end(), [&](int x, int y) { return in_time[x] < in_time[y]; });
return {prv, vs};
}
};
/**
* @brief LCA ベースの Auxiliary Tree
* @docs docs/tree/lca-auxiliary-tree.md
*/
#line 6 "verify/tree/UNIT_lca_auxiliary_tree.test.cpp"
void test_auxiliary_tree() {
GraphUnweighted g(8);
g.add_edge(0, 1);
g.add_edge(0, 2);
g.add_edge(1, 3);
g.add_edge(1, 4);
g.add_edge(2, 5);
g.add_edge(2, 6);
g.add_edge(5, 7);
const LCAAuxiliaryTree auxiliary_tree(g);
GraphUnweighted auxiliary(g.size());
auto [root, vertices] = auxiliary_tree.calc({7, 3, 4, 3}, auxiliary);
assert(root == 0);
assert(vertices == vector<int>({0, 1, 3, 4, 7}));
assert(auxiliary[0].size() == 2);
assert(auxiliary[0][0].to == 1 && auxiliary[0][1].to == 7);
assert(auxiliary[1].size() == 2);
assert(auxiliary[1][0].to == 3 && auxiliary[1][1].to == 4);
for (int x : vector<int>{3, 4, 7}) assert(auxiliary[x].empty());
}
int main() {
test_auxiliary_tree();
int a, b;
in(a, b);
out(a + b);
}
verify/tree/UNIT_lca_auxiliary_tree.test.cpp
algebraic-structure/magma.hpp