#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
#include "template/template.hpp"
#include "algebraic-structure/monoid.hpp"
#include "graph/graph.hpp"
#include "tree/static-tree-path-prod.hpp"
#include "util/xorshift.hpp"
struct StringMonoid {
using value_type = string;
static string op(const string& x, const string& y) { return x + y; }
static string e() { return ""; }
};
vector<int> get_path(const GraphUnweighted& g, int s, int t) {
int n = g.size();
vector<int> p(n, -1), st = {s};
p[s] = s;
for (int i = 0; i < (int)st.size(); i++) {
int x = st[i];
for (auto e : g[x]) {
if (p[e.to] != -1) continue;
p[e.to] = x;
st.push_back(e.to);
}
}
vector<int> path;
for (int x = t;; x = p[x]) {
path.push_back(x);
if (x == s) break;
}
reverse(path.begin(), path.end());
return path;
}
int main() {
rep(_, 0, 100) {
int n = XORShift::xor32() % 24 + 1;
int root = XORShift::xor32() % n;
GraphUnweighted g(n);
GraphWeighted<string> gs(n);
GraphWeighted<long long> gl(n);
vector<vector<string>> es(n, vector<string>(n));
vector<vector<long long>> el(n, vector<long long>(n));
rep(x, 1, n) {
int p = XORShift::xor32() % x;
string s(1, 'a' + XORShift::xor32() % 26);
long long v = XORShift::xor32() % 100;
g.add_edge(p, x);
gs.add_edge(p, x, s);
gl.add_edge(p, x, v);
es[p][x] = es[x][p] = s;
el[p][x] = el[x][p] = v;
}
vector<string> sv(n);
vector<long long> lv(n);
rep(x, 0, n) {
sv[x] = string(1, 'A' + XORShift::xor32() % 26);
lv[x] = XORShift::xor32() % 100;
}
StaticTreePathProd<StringMonoid, true, false> vs(g, sv, root);
StaticTreePathProd<StringMonoid, false, true> es_ds(gs, root);
StaticTreePathProd<StringMonoid, true, true> bs(gs, sv, root);
StaticTreePathProdCommutative<AddMonoid<long long>, true, false> vl(g, lv, root);
StaticTreePathProdCommutative<AddMonoid<long long>, false, true> el_ds(gl, root);
StaticTreePathProdCommutative<AddMonoid<long long>, true, true> bl(gl, lv, root);
vector<int> par(n, -1), dep(n), st = {root};
for (int i = 0; i < (int)st.size(); i++) {
int x = st[i];
for (auto e : g[x]) {
if (e.to == par[x]) continue;
par[e.to] = x;
dep[e.to] = dep[x] + 1;
st.push_back(e.to);
}
}
rep(s, 0, n) rep(t, 0, n) {
vector<int> path = get_path(g, s, t);
string ev, ee, eb = sv[path[0]];
long long av = 0, ae = 0, ab = lv[path[0]];
for (int i = 0; i < (int)path.size(); i++) {
ev += sv[path[i]];
av += lv[path[i]];
if (i == 0) continue;
int x = path[i - 1], y = path[i];
ee += es[x][y];
eb += es[x][y] + sv[y];
ae += el[x][y];
ab += el[x][y] + lv[y];
}
assert(vs.prod(s, t) == ev);
assert(es_ds.prod(s, t) == ee);
assert(bs.prod(s, t) == eb);
assert(vl.prod(s, t) == av);
assert(el_ds.prod(s, t) == ae);
assert(bl.prod(s, t) == ab);
int x = s, y = t;
while (dep[x] > dep[y]) x = par[x];
while (dep[y] > dep[x]) y = par[y];
while (x != y) x = par[x], y = par[y];
assert(vs.lca(s, t) == x);
assert(vl.lca(s, t) == x);
}
}
int a, b;
in(a, b);
out(a + b);
}
#line 1 "verify/tree/UNIT_static_tree_path_prod.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
#line 2 "template/template.hpp"
// repo: https://github.com/kumacs/library-cpp
// docs: https://kumacs.github.io/library-cpp
#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 10 "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 12 "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 14 "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 "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 "algebraic-structure/monoid.hpp"
#ifdef __cpp_concepts
template <class M>
concept Monoid = Magma<M> && requires {
{ M::e() } -> same_as<typename M::value_type>;
};
#endif
template <class T>
struct AddMonoid {
using value_type = T;
static T op(T x, T y) { return x + y; }
static T e() { return T(0); }
};
template <class T>
struct MulMonoid {
using value_type = T;
static T op(T x, T y) { return x * y; }
static T e() { return T(1); }
};
template <class T, T id>
struct MaxMonoid {
using value_type = T;
static T op(T x, T y) { return x > y ? x : y; }
static T e() { return id; }
};
template <class T, T id>
struct MinMonoid {
using value_type = T;
static T op(T x, T y) { return x < y ? x : y; }
static T e() { return id; }
};
#line 2 "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}); }
};
struct EdgeUnweightedIndexed {
int to, id;
};
struct GraphUnweightedIndexed : GraphBase<EdgeUnweightedIndexed> {
using base = GraphBase<EdgeUnweightedIndexed>;
public:
GraphUnweightedIndexed() : GraphUnweightedIndexed(0) {}
GraphUnweightedIndexed(int size) : base(size), m(0) {}
int edge_count() const { return m; }
int add_edge(int x, int y) {
int id = m++;
(*this)[x].push_back({y, id});
(*this)[y].push_back({x, id});
return id;
}
int add_edge_directed(int from, int to) {
int id = m++;
(*this)[from].push_back({to, id});
return id;
}
private:
int m;
};
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 "tree/static-tree-path-prod.hpp"
#line 4 "tree/static-tree-path-prod.hpp"
template <class M, bool vertex, bool edge>
REQUIRES(Monoid<M>)
struct StaticTreePathProd {
static_assert(vertex || edge);
using T = M::value_type;
StaticTreePathProd() {}
template <class G>
StaticTreePathProd(const G& g, const vector<T>& vertex_value, int root = 0)
: n(g.size()), data(vertex_value) {
static_assert(vertex);
assert((int)data.size() == n);
build(g, root);
}
template <class G>
StaticTreePathProd(const G& g, int root = 0) : n(g.size()) {
static_assert(edge && !vertex);
build(g, root);
}
int lca(int x, int y) const {
assert(0 <= x && x < n && 0 <= y && y < n);
if (depth[x] < depth[y]) swap(x, y);
x = climb(x, depth[x] - depth[y]);
if (x == y) return x;
for (int k = log - 1; k >= 0; k--) {
if (parent[k][x] != parent[k][y]) x = parent[k][x], y = parent[k][y];
}
return parent[0][x];
}
T prod(int x, int y) const {
assert(0 <= x && x < n && 0 <= y && y < n);
int z = lca(x, y);
T pl = M::e(), pr = M::e();
for (int d = depth[x] - depth[z]; d > 0;) {
int k = topbit(d);
pl = M::op(pl, prod_up[k][x]);
x = parent[k][x];
d -= 1 << k;
}
for (int d = depth[y] - depth[z]; d > 0;) {
int k = topbit(d);
pr = M::op(prod_down[k][y], pr);
y = parent[k][y];
d -= 1 << k;
}
if constexpr (vertex) pl = M::op(pl, data[z]);
return M::op(pl, pr);
}
private:
int n = 0, log = 0;
vector<T> data;
vector<int> depth;
vector<vector<int>> parent;
vector<vector<T>> prod_up, prod_down;
int climb(int x, int d) const {
for (int k = 0; d > 0; k++, d >>= 1)
if (d & 1) x = parent[k][x];
return x;
}
template <class G>
void build(const G& g, int root) {
assert(n > 0);
assert(0 <= root && root < n);
log = 1;
while ((1 << log) <= n) log++;
depth.assign(n, 0);
parent.assign(log, vector<int>(n, root));
prod_up.assign(log, vector<T>(n, M::e()));
prod_down.assign(log, vector<T>(n, M::e()));
vector<int> order = {root};
for (int i = 0; i < (int)order.size(); i++) {
int x = order[i];
for (const auto& e : g[x]) {
int y = e.to;
if (y == parent[0][x]) continue;
parent[0][y] = x;
depth[y] = depth[x] + 1;
if constexpr (edge) {
prod_up[0][y] = vertex ? M::op(data[y], e.weight) : e.weight;
prod_down[0][y] = vertex ? M::op(e.weight, data[y]) : e.weight;
} else {
prod_up[0][y] = prod_down[0][y] = data[y];
}
order.push_back(y);
}
}
assert((int)order.size() == n);
for (int k = 0; k + 1 < log; k++) {
for (int x = 0; x < n; x++) {
int p = parent[k][x];
parent[k + 1][x] = parent[k][p];
prod_up[k + 1][x] = M::op(prod_up[k][x], prod_up[k][p]);
prod_down[k + 1][x] = M::op(prod_down[k][p], prod_down[k][x]);
}
}
}
};
template <class M, bool vertex, bool edge>
REQUIRES(Monoid<M>)
struct StaticTreePathProdCommutative {
static_assert(vertex || edge);
using T = M::value_type;
StaticTreePathProdCommutative() {}
template <class G>
StaticTreePathProdCommutative(const G& g, const vector<T>& vertex_value, int root = 0)
: n(g.size()), data(vertex_value) {
static_assert(vertex);
assert((int)data.size() == n);
build(g, root);
}
template <class G>
StaticTreePathProdCommutative(const G& g, int root = 0) : n(g.size()) {
static_assert(edge && !vertex);
build(g, root);
}
int lca(int x, int y) const {
assert(0 <= x && x < n && 0 <= y && y < n);
if (depth[x] < depth[y]) swap(x, y);
x = climb(x, depth[x] - depth[y]);
if (x == y) return x;
for (int k = log - 1; k >= 0; k--) {
if (parent[k][x] != parent[k][y]) x = parent[k][x], y = parent[k][y];
}
return parent[0][x];
}
T prod(int x, int y) const {
assert(0 <= x && x < n && 0 <= y && y < n);
int z = lca(x, y);
T p = M::e();
for (int d = depth[x] - depth[z]; d > 0;) {
int k = topbit(d);
p = M::op(p, prod_db[k][x]);
x = parent[k][x];
d -= 1 << k;
}
for (int d = depth[y] - depth[z]; d > 0;) {
int k = topbit(d);
p = M::op(p, prod_db[k][y]);
y = parent[k][y];
d -= 1 << k;
}
if constexpr (vertex) p = M::op(p, data[z]);
return p;
}
private:
int n = 0, log = 0;
vector<T> data;
vector<int> depth;
vector<vector<int>> parent;
vector<vector<T>> prod_db;
int climb(int x, int d) const {
for (int k = 0; d > 0; k++, d >>= 1)
if (d & 1) x = parent[k][x];
return x;
}
template <class G>
void build(const G& g, int root) {
assert(n > 0);
assert(0 <= root && root < n);
log = 1;
while ((1 << log) <= n) log++;
depth.assign(n, 0);
parent.assign(log, vector<int>(n, root));
prod_db.assign(log, vector<T>(n, M::e()));
vector<int> order = {root};
for (int i = 0; i < (int)order.size(); i++) {
int x = order[i];
for (const auto& e : g[x]) {
int y = e.to;
if (y == parent[0][x]) continue;
parent[0][y] = x;
depth[y] = depth[x] + 1;
if constexpr (edge)
prod_db[0][y] = vertex ? M::op(data[y], e.weight) : e.weight;
else
prod_db[0][y] = data[y];
order.push_back(y);
}
}
assert((int)order.size() == n);
for (int k = 0; k + 1 < log; k++) {
for (int x = 0; x < n; x++) {
int p = parent[k][x];
parent[k + 1][x] = parent[k][p];
prod_db[k + 1][x] = M::op(prod_db[k][x], prod_db[k][p]);
}
}
}
};
/**
* @brief Static Tree Path Product
* @docs docs/tree/static-tree-path-prod.md
*/
#line 2 "util/xorshift.hpp"
namespace XORShift {
unsigned int xor32() {
static unsigned int x = 123456789u;
x ^= x << 13, x ^= x >> 17, x ^= x << 5;
return x;
}
unsigned long long xor64() {
static unsigned long long x = 123456789ull;
x ^= x << 13, x ^= x >> 7, x ^= x << 17;
return x;
}
}; // namespace XORShift
/**
* @brief XOR shift
*/
#line 8 "verify/tree/UNIT_static_tree_path_prod.test.cpp"
struct StringMonoid {
using value_type = string;
static string op(const string& x, const string& y) { return x + y; }
static string e() { return ""; }
};
vector<int> get_path(const GraphUnweighted& g, int s, int t) {
int n = g.size();
vector<int> p(n, -1), st = {s};
p[s] = s;
for (int i = 0; i < (int)st.size(); i++) {
int x = st[i];
for (auto e : g[x]) {
if (p[e.to] != -1) continue;
p[e.to] = x;
st.push_back(e.to);
}
}
vector<int> path;
for (int x = t;; x = p[x]) {
path.push_back(x);
if (x == s) break;
}
reverse(path.begin(), path.end());
return path;
}
int main() {
rep(_, 0, 100) {
int n = XORShift::xor32() % 24 + 1;
int root = XORShift::xor32() % n;
GraphUnweighted g(n);
GraphWeighted<string> gs(n);
GraphWeighted<long long> gl(n);
vector<vector<string>> es(n, vector<string>(n));
vector<vector<long long>> el(n, vector<long long>(n));
rep(x, 1, n) {
int p = XORShift::xor32() % x;
string s(1, 'a' + XORShift::xor32() % 26);
long long v = XORShift::xor32() % 100;
g.add_edge(p, x);
gs.add_edge(p, x, s);
gl.add_edge(p, x, v);
es[p][x] = es[x][p] = s;
el[p][x] = el[x][p] = v;
}
vector<string> sv(n);
vector<long long> lv(n);
rep(x, 0, n) {
sv[x] = string(1, 'A' + XORShift::xor32() % 26);
lv[x] = XORShift::xor32() % 100;
}
StaticTreePathProd<StringMonoid, true, false> vs(g, sv, root);
StaticTreePathProd<StringMonoid, false, true> es_ds(gs, root);
StaticTreePathProd<StringMonoid, true, true> bs(gs, sv, root);
StaticTreePathProdCommutative<AddMonoid<long long>, true, false> vl(g, lv, root);
StaticTreePathProdCommutative<AddMonoid<long long>, false, true> el_ds(gl, root);
StaticTreePathProdCommutative<AddMonoid<long long>, true, true> bl(gl, lv, root);
vector<int> par(n, -1), dep(n), st = {root};
for (int i = 0; i < (int)st.size(); i++) {
int x = st[i];
for (auto e : g[x]) {
if (e.to == par[x]) continue;
par[e.to] = x;
dep[e.to] = dep[x] + 1;
st.push_back(e.to);
}
}
rep(s, 0, n) rep(t, 0, n) {
vector<int> path = get_path(g, s, t);
string ev, ee, eb = sv[path[0]];
long long av = 0, ae = 0, ab = lv[path[0]];
for (int i = 0; i < (int)path.size(); i++) {
ev += sv[path[i]];
av += lv[path[i]];
if (i == 0) continue;
int x = path[i - 1], y = path[i];
ee += es[x][y];
eb += es[x][y] + sv[y];
ae += el[x][y];
ab += el[x][y] + lv[y];
}
assert(vs.prod(s, t) == ev);
assert(es_ds.prod(s, t) == ee);
assert(bs.prod(s, t) == eb);
assert(vl.prod(s, t) == av);
assert(el_ds.prod(s, t) == ae);
assert(bl.prod(s, t) == ab);
int x = s, y = t;
while (dep[x] > dep[y]) x = par[x];
while (dep[y] > dep[x]) y = par[y];
while (x != y) x = par[x], y = par[y];
assert(vs.lca(s, t) == x);
assert(vl.lca(s, t) == x);
}
}
int a, b;
in(a, b);
out(a + b);
}
verify/tree/UNIT_static_tree_path_prod.test.cpp