#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
#include "../../template/template.hpp"
//
#include "../../data-structure/union-find.hpp"
#include "../../misc/rng.hpp"
#include "../../tree/convert-tree.hpp"
#include "../../tree/pruefer-code.hpp"
#include "../../tree/tree-query.hpp"
using namespace Nyaan;
template <typename G>
bool is_tree(G& g, bool directed = false) {
int n = g.size();
UnionFind uf(n);
rep(i, n) each(j, g[i]) {
if (!directed and i > j) continue;
if (!uf.unite(i, j)) return false;
}
return uf.size(0) == n;
}
template <typename G>
bool is_rooted_tree(G& g, int root) {
int n = g.size();
vector<char> vis(n, false);
auto dfs = [&](auto rc, int c) -> void {
vis[c] = true;
each(d, g[c]) {
assert(vis[d] == false);
rc(rc, d);
}
};
dfs(dfs, root);
int sm = 0;
each(b, vis) sm += !!b;
return sm == n;
}
template <typename G>
bool is_inverse_tree(G& g, G& rg) {
set<pair<int, int>> s, t;
int n = g.size();
for (int i = 0; i < n; i++) {
for (auto& j : g[i]) s.emplace(i, int(j));
for (auto& j : rg[i]) t.emplace(j, int(i));
}
return s == t;
}
void test_tree_query(vvi& g, int root = 0) {
Tree<vvi> tree(g, root);
int N = sz(g);
rep(i, N) rep(j, N) {
vi p1 = tree.path(i, j);
vi p2{int(i)};
for (int k = i; k != j;) {
p2.push_back(k = tree.nxt(k, j));
}
assert(p1 == p2);
int l = i, ld = tree.depth(i);
each(x, p1) if (amin(ld, tree.depth(x))) l = x;
assert(l == tree.lca(i, j));
}
}
using namespace Nyaan;
void Nyaan::solve() {
// Random Tree
for (int N : vi{2, 3, 4, 5, 10, 100}) {
vvi g = random_tree(N);
assert(is_tree(g) && "random tree");
test_tree_query(g);
int root = randint(0, N);
vvi rg = rooted_tree(g, root);
assert(is_tree(rg, true) && "rooted tree");
test_tree_query(rg, root);
vvi rh = inverse_tree(rg);
assert(is_inverse_tree(rg, rh) && "inverse tree");
}
for (int N : vi{2, 3, 4, 5, 10, 100}) {
vvi _g = random_tree(N);
assert(is_tree(_g) && "random tree");
WeightedGraph<ll> g(N);
rep(i, N) each(j, _g[i]) { g[i].emplace_back(i, j, rng() & 15); }
int root = randint(0, N);
auto rg = rooted_tree(g, root);
assert(is_tree(rg, true) && "rooted tree");
auto rh = inverse_tree(rg);
assert(is_inverse_tree(rg, rh) && "inverse tree");
}
// Line Tree
for (int N : vi{2, 3, 4, 5, 10, 100}) {
vvi g(N);
rep1(i, N - 1) g[i - 1].push_back(i);
test_tree_query(g);
}
// Preufer Code
{
V<vvi> gs(125);
rep(i, 125) {
vi pr(3);
for (int j = i, k = 0; k < 3; j /= 5, k++) {
pr[k] = j % 5;
}
gs[i] = pruefer_code(pr);
for (auto& es : gs[i]) sort(begin(es), end(es));
}
rep(i, 125) assert(is_tree(gs[i]));
rep(i, 125) rep(j, i) assert(gs[i] != gs[j]);
}
int a, b;
cin >> a >> b;
cout << a + b << endl;
}
#line 1 "verify/verify-unit-test/tree-path.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
#line 2 "template/template.hpp"
using namespace std;
// intrinstic
#include <immintrin.h>
#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <cctype>
#include <cfenv>
#include <cfloat>
#include <chrono>
#include <cinttypes>
#include <climits>
#include <cmath>
#include <complex>
#include <cstdarg>
#include <cstddef>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <deque>
#include <fstream>
#include <functional>
#include <initializer_list>
#include <iomanip>
#include <ios>
#include <iostream>
#include <istream>
#include <iterator>
#include <limits>
#include <list>
#include <map>
#include <memory>
#include <new>
#include <numeric>
#include <ostream>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <streambuf>
#include <string>
#include <tuple>
#include <type_traits>
#include <typeinfo>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>
// utility
#line 1 "template/util.hpp"
namespace Nyaan {
using ll = long long;
using i64 = long long;
using u64 = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
template <typename T>
using V = vector<T>;
template <typename T>
using VV = vector<vector<T>>;
using vi = vector<int>;
using vl = vector<long long>;
using vd = V<double>;
using vs = V<string>;
using vvi = vector<vector<int>>;
using vvl = vector<vector<long long>>;
template <typename T>
using minpq = priority_queue<T, vector<T>, greater<T>>;
template <typename T, typename U>
struct P : pair<T, U> {
template <typename... Args>
P(Args... args) : pair<T, U>(args...) {}
using pair<T, U>::first;
using pair<T, U>::second;
P &operator+=(const P &r) {
first += r.first;
second += r.second;
return *this;
}
P &operator-=(const P &r) {
first -= r.first;
second -= r.second;
return *this;
}
P &operator*=(const P &r) {
first *= r.first;
second *= r.second;
return *this;
}
template <typename S>
P &operator*=(const S &r) {
first *= r, second *= r;
return *this;
}
P operator+(const P &r) const { return P(*this) += r; }
P operator-(const P &r) const { return P(*this) -= r; }
P operator*(const P &r) const { return P(*this) *= r; }
template <typename S>
P operator*(const S &r) const {
return P(*this) *= r;
}
P operator-() const { return P{-first, -second}; }
};
using pl = P<ll, ll>;
using pi = P<int, int>;
using vp = V<pl>;
constexpr int inf = 1001001001;
constexpr long long infLL = 4004004004004004004LL;
template <typename T>
int sz(const T &t) {
return t.size();
}
template <typename T, typename U>
inline bool amin(T &x, U y) {
return (y < x) ? (x = y, true) : false;
}
template <typename T, typename U>
inline bool amax(T &x, U y) {
return (x < y) ? (x = y, true) : false;
}
template <typename T>
inline T Max(const vector<T> &v) {
return *max_element(begin(v), end(v));
}
template <typename T>
inline T Min(const vector<T> &v) {
return *min_element(begin(v), end(v));
}
template <typename T>
inline long long Sum(const vector<T> &v) {
return accumulate(begin(v), end(v), 0LL);
}
template <typename T>
int lb(const vector<T> &v, const T &a) {
return lower_bound(begin(v), end(v), a) - begin(v);
}
template <typename T>
int ub(const vector<T> &v, const T &a) {
return upper_bound(begin(v), end(v), a) - begin(v);
}
constexpr long long TEN(int n) {
long long ret = 1, x = 10;
for (; n; x *= x, n >>= 1) ret *= (n & 1 ? x : 1);
return ret;
}
template <typename T, typename U>
pair<T, U> mkp(const T &t, const U &u) {
return make_pair(t, u);
}
template <typename T>
vector<T> mkrui(const vector<T> &v, bool rev = false) {
vector<T> ret(v.size() + 1);
if (rev) {
for (int i = int(v.size()) - 1; i >= 0; i--) ret[i] = v[i] + ret[i + 1];
} else {
for (int i = 0; i < int(v.size()); i++) ret[i + 1] = ret[i] + v[i];
}
return ret;
};
template <typename T>
vector<T> mkuni(const vector<T> &v) {
vector<T> ret(v);
sort(ret.begin(), ret.end());
ret.erase(unique(ret.begin(), ret.end()), ret.end());
return ret;
}
template <typename F>
vector<int> mkord(int N, F f) {
vector<int> ord(N);
iota(begin(ord), end(ord), 0);
sort(begin(ord), end(ord), f);
return ord;
}
template <typename T>
vector<int> mkinv(vector<T> &v) {
int max_val = *max_element(begin(v), end(v));
vector<int> inv(max_val + 1, -1);
for (int i = 0; i < (int)v.size(); i++) inv[v[i]] = i;
return inv;
}
vector<int> mkiota(int n) {
vector<int> ret(n);
iota(begin(ret), end(ret), 0);
return ret;
}
template <typename T>
T mkrev(const T &v) {
T w{v};
reverse(begin(w), end(w));
return w;
}
template <typename T>
bool nxp(vector<T> &v) {
return next_permutation(begin(v), end(v));
}
// 返り値の型は入力の T に依存
// i 要素目 : [0, a[i])
template <typename T>
vector<vector<T>> product(const vector<T> &a) {
vector<vector<T>> ret;
vector<T> v;
auto dfs = [&](auto rc, int i) -> void {
if (i == (int)a.size()) {
ret.push_back(v);
return;
}
for (int j = 0; j < a[i]; j++) v.push_back(j), rc(rc, i + 1), v.pop_back();
};
dfs(dfs, 0);
return ret;
}
// F : function(void(T&)), mod を取る操作
// T : 整数型のときはオーバーフローに注意する
template <typename T>
T Power(T a, long long n, const T &I, const function<void(T &)> &f) {
T res = I;
for (; n; f(a = a * a), n >>= 1) {
if (n & 1) f(res = res * a);
}
return res;
}
// T : 整数型のときはオーバーフローに注意する
template <typename T>
T Power(T a, long long n, const T &I) {
return Power(a, n, I, function<void(T &)>{[](T &) -> void {}});
}
} // namespace Nyaan
#line 58 "template/template.hpp"
// bit operation
#line 1 "template/bitop.hpp"
namespace Nyaan {
__attribute__((target("popcnt"))) inline int popcnt(const u64 &a) {
return _mm_popcnt_u64(a);
}
inline int lsb(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int ctz(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int msb(const u64 &a) { return a ? 63 - __builtin_clzll(a) : -1; }
template <typename T>
inline int gbit(const T &a, int i) {
return (a >> i) & 1;
}
template <typename T>
inline void sbit(T &a, int i, bool b) {
if (gbit(a, i) != b) a ^= T(1) << i;
}
constexpr long long PW(int n) { return 1LL << n; }
constexpr long long MSK(int n) { return (1LL << n) - 1; }
} // namespace Nyaan
#line 61 "template/template.hpp"
// inout
#line 1 "template/inout.hpp"
namespace Nyaan {
template <typename T, typename U>
ostream &operator<<(ostream &os, const pair<T, U> &p) {
os << p.first << " " << p.second;
return os;
}
template <typename T, typename U>
istream &operator>>(istream &is, pair<T, U> &p) {
is >> p.first >> p.second;
return is;
}
template <typename T>
ostream &operator<<(ostream &os, const vector<T> &v) {
int s = (int)v.size();
for (int i = 0; i < s; i++) os << (i ? " " : "") << v[i];
return os;
}
template <typename T>
istream &operator>>(istream &is, vector<T> &v) {
for (auto &x : v) is >> x;
return is;
}
istream &operator>>(istream &is, __int128_t &x) {
string S;
is >> S;
x = 0;
int flag = 0;
for (auto &c : S) {
if (c == '-') {
flag = true;
continue;
}
x *= 10;
x += c - '0';
}
if (flag) x = -x;
return is;
}
istream &operator>>(istream &is, __uint128_t &x) {
string S;
is >> S;
x = 0;
for (auto &c : S) {
x *= 10;
x += c - '0';
}
return is;
}
ostream &operator<<(ostream &os, __int128_t x) {
if (x == 0) return os << 0;
if (x < 0) os << '-', x = -x;
string S;
while (x) S.push_back('0' + x % 10), x /= 10;
reverse(begin(S), end(S));
return os << S;
}
ostream &operator<<(ostream &os, __uint128_t x) {
if (x == 0) return os << 0;
string S;
while (x) S.push_back('0' + x % 10), x /= 10;
reverse(begin(S), end(S));
return os << S;
}
void in() {}
template <typename T, class... U>
void in(T &t, U &...u) {
cin >> t;
in(u...);
}
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...);
}
struct IoSetupNya {
IoSetupNya() {
cin.tie(nullptr);
ios::sync_with_stdio(false);
cout << fixed << setprecision(15);
cerr << fixed << setprecision(7);
}
} iosetupnya;
} // namespace Nyaan
#line 64 "template/template.hpp"
// debug
#line 1 "template/debug.hpp"
namespace DebugImpl {
template <typename U, typename = void>
struct is_specialize : false_type {};
template <typename U>
struct is_specialize<
U, typename conditional<false, typename U::iterator, void>::type>
: true_type {};
template <typename U>
struct is_specialize<
U, typename conditional<false, decltype(U::first), void>::type>
: true_type {};
template <typename U>
struct is_specialize<U, enable_if_t<is_integral<U>::value, void>> : true_type {
};
void dump(const char& t) { cerr << t; }
void dump(const string& t) { cerr << t; }
void dump(const bool& t) { cerr << (t ? "true" : "false"); }
void dump(__int128_t t) {
if (t == 0) cerr << 0;
if (t < 0) cerr << '-', t = -t;
string S;
while (t) S.push_back('0' + t % 10), t /= 10;
reverse(begin(S), end(S));
cerr << S;
}
void dump(__uint128_t t) {
if (t == 0) cerr << 0;
string S;
while (t) S.push_back('0' + t % 10), t /= 10;
reverse(begin(S), end(S));
cerr << S;
}
template <typename U,
enable_if_t<!is_specialize<U>::value, nullptr_t> = nullptr>
void dump(const U& t) {
cerr << t;
}
template <typename T>
void dump(const T& t, enable_if_t<is_integral<T>::value>* = nullptr) {
string res;
if (t == Nyaan::inf) res = "inf";
if constexpr (is_signed<T>::value) {
if (t == -Nyaan::inf) res = "-inf";
}
if constexpr (sizeof(T) == 8) {
if (t == Nyaan::infLL) res = "inf";
if constexpr (is_signed<T>::value) {
if (t == -Nyaan::infLL) res = "-inf";
}
}
if (res.empty()) res = to_string(t);
cerr << res;
}
template <typename T, typename U>
void dump(const pair<T, U>&);
template <typename T>
void dump(const pair<T*, int>&);
template <typename T>
void dump(const T& t,
enable_if_t<!is_void<typename T::iterator>::value>* = nullptr) {
cerr << "[ ";
for (auto it = t.begin(); it != t.end();) {
dump(*it);
cerr << (++it == t.end() ? "" : ", ");
}
cerr << " ]";
}
template <typename T, typename U>
void dump(const pair<T, U>& t) {
cerr << "( ";
dump(t.first);
cerr << ", ";
dump(t.second);
cerr << " )";
}
template <typename T>
void dump(const pair<T*, int>& t) {
cerr << "[ ";
for (int i = 0; i < t.second; i++) {
dump(t.first[i]);
cerr << (i == t.second - 1 ? "" : ", ");
}
cerr << " ]";
}
void trace() { cerr << endl; }
template <typename Head, typename... Tail>
void trace(Head&& head, Tail&&... tail) {
cerr << " ";
dump(head);
if (sizeof...(tail) != 0) cerr << ",";
trace(forward<Tail>(tail)...);
}
} // namespace DebugImpl
#ifdef NyaanDebug
#define trc(...) \
do { \
cerr << "## " << #__VA_ARGS__ << " = "; \
DebugImpl::trace(__VA_ARGS__); \
} while (0)
#else
#define trc(...) (void(0))
#endif
#ifdef NyaanLocal
#define trc2(...) \
do { \
cerr << "## " << #__VA_ARGS__ << " = "; \
DebugImpl::trace(__VA_ARGS__); \
} while (0)
#else
#define trc2(...) (void(0))
#endif
#line 67 "template/template.hpp"
// macro
#line 1 "template/macro.hpp"
#define each(x, v) for (auto&& x : v)
#define each2(x, y, v) for (auto&& [x, y] : v)
#define all(v) (v).begin(), (v).end()
#define rep(i, N) for (long long i = 0; i < (long long)(N); i++)
#define repr(i, N) for (long long i = (long long)(N)-1; i >= 0; i--)
#define rep1(i, N) for (long long i = 1; i <= (long long)(N); i++)
#define repr1(i, N) for (long long i = (N); (long long)(i) > 0; i--)
#define reg(i, a, b) for (long long i = (a); i < (b); i++)
#define regr(i, a, b) for (long long i = (b)-1; i >= (a); i--)
#define fi first
#define se second
#define ini(...) \
int __VA_ARGS__; \
in(__VA_ARGS__)
#define inl(...) \
long long __VA_ARGS__; \
in(__VA_ARGS__)
#define ins(...) \
string __VA_ARGS__; \
in(__VA_ARGS__)
#define in2(s, t) \
for (int i = 0; i < (int)s.size(); i++) { \
in(s[i], t[i]); \
}
#define in3(s, t, u) \
for (int i = 0; i < (int)s.size(); i++) { \
in(s[i], t[i], u[i]); \
}
#define in4(s, t, u, v) \
for (int i = 0; i < (int)s.size(); i++) { \
in(s[i], t[i], u[i], v[i]); \
}
#define die(...) \
do { \
Nyaan::out(__VA_ARGS__); \
return; \
} while (0)
#line 70 "template/template.hpp"
namespace Nyaan {
void solve();
}
int main() { Nyaan::solve(); }
#line 4 "verify/verify-unit-test/tree-path.test.cpp"
//
#line 2 "data-structure/union-find.hpp"
struct UnionFind {
vector<int> data;
UnionFind(int N) : data(N, -1) {}
int find(int k) { return data[k] < 0 ? k : data[k] = find(data[k]); }
int unite(int x, int y) {
if ((x = find(x)) == (y = find(y))) return false;
if (data[x] > data[y]) swap(x, y);
data[x] += data[y];
data[y] = x;
return true;
}
// f ... merge function
template<typename F>
int unite(int x, int y,const F &f) {
if ((x = find(x)) == (y = find(y))) return false;
if (data[x] > data[y]) swap(x, y);
data[x] += data[y];
data[y] = x;
f(x, y);
return true;
}
int size(int k) { return -data[find(k)]; }
int same(int x, int y) { return find(x) == find(y); }
};
/**
* @brief Union Find(Disjoint Set Union)
* @docs docs/data-structure/union-find.md
*/
#line 2 "misc/rng.hpp"
#line 2 "internal/internal-seed.hpp"
#line 4 "internal/internal-seed.hpp"
using namespace std;
namespace internal {
unsigned long long non_deterministic_seed() {
unsigned long long m =
chrono::duration_cast<chrono::nanoseconds>(
chrono::high_resolution_clock::now().time_since_epoch())
.count();
m ^= 9845834732710364265uLL;
m ^= m << 24, m ^= m >> 31, m ^= m << 35;
return m;
}
unsigned long long deterministic_seed() { return 88172645463325252UL; }
// 64 bit の seed 値を生成 (手元では seed 固定)
// 連続で呼び出すと同じ値が何度も返ってくるので注意
// #define RANDOMIZED_SEED するとシードがランダムになる
unsigned long long seed() {
#if defined(NyaanLocal) && !defined(RANDOMIZED_SEED)
return deterministic_seed();
#else
return non_deterministic_seed();
#endif
}
} // namespace internal
#line 4 "misc/rng.hpp"
namespace my_rand {
using i64 = long long;
using u64 = unsigned long long;
// [0, 2^64 - 1)
u64 rng() {
static u64 _x = internal::seed();
return _x ^= _x << 7, _x ^= _x >> 9;
}
// [l, r]
i64 rng(i64 l, i64 r) {
assert(l <= r);
return l + rng() % u64(r - l + 1);
}
// [l, r)
i64 randint(i64 l, i64 r) {
assert(l < r);
return l + rng() % u64(r - l);
}
// choose n numbers from [l, r) without overlapping
vector<i64> randset(i64 l, i64 r, i64 n) {
assert(l <= r && n <= r - l);
unordered_set<i64> s;
for (i64 i = n; i; --i) {
i64 m = randint(l, r + 1 - i);
if (s.find(m) != s.end()) m = r - i;
s.insert(m);
}
vector<i64> ret;
for (auto& x : s) ret.push_back(x);
return ret;
}
// [0.0, 1.0)
double rnd() { return rng() * 5.42101086242752217004e-20; }
// [l, r)
double rnd(double l, double r) {
assert(l < r);
return l + rnd() * (r - l);
}
template <typename T>
void randshf(vector<T>& v) {
int n = v.size();
for (int i = 1; i < n; i++) swap(v[i], v[randint(0, i + 1)]);
}
} // namespace my_rand
using my_rand::randint;
using my_rand::randset;
using my_rand::randshf;
using my_rand::rnd;
using my_rand::rng;
#line 2 "tree/convert-tree.hpp"
#line 2 "graph/graph-template.hpp"
template <typename T>
struct edge {
int src, to;
T cost;
edge(int _to, T _cost) : src(-1), to(_to), cost(_cost) {}
edge(int _src, int _to, T _cost) : src(_src), to(_to), cost(_cost) {}
edge &operator=(const int &x) {
to = x;
return *this;
}
operator int() const { return to; }
};
template <typename T>
using Edges = vector<edge<T>>;
template <typename T>
using WeightedGraph = vector<Edges<T>>;
using UnweightedGraph = vector<vector<int>>;
// Input of (Unweighted) Graph
UnweightedGraph graph(int N, int M = -1, bool is_directed = false,
bool is_1origin = true) {
UnweightedGraph g(N);
if (M == -1) M = N - 1;
for (int _ = 0; _ < M; _++) {
int x, y;
cin >> x >> y;
if (is_1origin) x--, y--;
g[x].push_back(y);
if (!is_directed) g[y].push_back(x);
}
return g;
}
// Input of Weighted Graph
template <typename T>
WeightedGraph<T> wgraph(int N, int M = -1, bool is_directed = false,
bool is_1origin = true) {
WeightedGraph<T> g(N);
if (M == -1) M = N - 1;
for (int _ = 0; _ < M; _++) {
int x, y;
cin >> x >> y;
T c;
cin >> c;
if (is_1origin) x--, y--;
g[x].emplace_back(x, y, c);
if (!is_directed) g[y].emplace_back(y, x, c);
}
return g;
}
// Input of Edges
template <typename T>
Edges<T> esgraph(int N, int M, int is_weighted = true, bool is_1origin = true) {
Edges<T> es;
for (int _ = 0; _ < M; _++) {
int x, y;
cin >> x >> y;
T c;
if (is_weighted)
cin >> c;
else
c = 1;
if (is_1origin) x--, y--;
es.emplace_back(x, y, c);
}
return es;
}
// Input of Adjacency Matrix
template <typename T>
vector<vector<T>> adjgraph(int N, int M, T INF, int is_weighted = true,
bool is_directed = false, bool is_1origin = true) {
vector<vector<T>> d(N, vector<T>(N, INF));
for (int _ = 0; _ < M; _++) {
int x, y;
cin >> x >> y;
T c;
if (is_weighted)
cin >> c;
else
c = 1;
if (is_1origin) x--, y--;
d[x][y] = c;
if (!is_directed) d[y][x] = c;
}
return d;
}
/**
* @brief グラフテンプレート
* @docs docs/graph/graph-template.md
*/
#line 4 "tree/convert-tree.hpp"
template <typename T>
struct has_cost {
private:
template <typename U>
static auto confirm(U u) -> decltype(u.cost, std::true_type());
static auto confirm(...) -> std::false_type;
public:
enum : bool { value = decltype(confirm(std::declval<T>()))::value };
};
template <typename T>
vector<vector<T>> inverse_tree(const vector<vector<T>>& g) {
int N = (int)g.size();
vector<vector<T>> rg(N);
for (int i = 0; i < N; i++) {
for (auto& e : g[i]) {
if constexpr (is_same<T, int>::value) {
rg[e].push_back(i);
} else if constexpr (has_cost<T>::value) {
rg[e].emplace_back(e.to, i, e.cost);
} else {
assert(0);
}
}
}
return rg;
}
template <typename T>
vector<vector<T>> rooted_tree(const vector<vector<T>>& g, int root = 0) {
int N = (int)g.size();
vector<vector<T>> rg(N);
vector<char> v(N, false);
v[root] = true;
queue<int> que;
que.emplace(root);
while (!que.empty()) {
auto p = que.front();
que.pop();
for (auto& e : g[p]) {
if (v[e] == false) {
v[e] = true;
que.push(e);
rg[p].push_back(e);
}
}
}
return rg;
}
/**
* @brief 根付き木・逆辺からなる木への変換
*/
#line 2 "tree/pruefer-code.hpp"
#line 4 "tree/pruefer-code.hpp"
// input: [c \in [0, n)] * (n-2), n>=3
vector<vector<int>> pruefer_code(const vector<int>& code) {
int n = code.size() + 2;
assert(n > 2);
vector<vector<int>> g(n);
vector<int> deg(n, 1);
int e = 0;
for (auto& x : code) deg[x]++;
set<int> ps;
for (int j = 0; j < n; j++) {
if (deg[j] == 1) ps.insert(j);
}
for (auto& i : code) {
if (ps.empty()) break;
int j = *begin(ps);
ps.erase(j);
g[i].push_back(j);
g[j].push_back(i);
if (deg[i] == 1) ps.erase(i);
deg[i]--, deg[j]--;
if (deg[i] == 1) ps.insert(i);
e++;
}
int u = -1, v = -1;
for (int i = 0; i < n; i++) {
if (deg[i] == 1) (u == -1 ? u : v) = i;
}
assert(u != -1 and v != -1);
g[u].push_back(v);
g[v].push_back(u);
e++;
assert(e == n - 1);
return g;
}
vector<vector<int>> random_tree(int n) {
if (n <= 2) {
vector<vector<int>> g(n);
if (n == 2) {
g[0].push_back(1);
g[1].push_back(0);
}
return g;
}
vector<int> pruefer(n - 2);
for (auto& x : pruefer) x = randint(0, n);
return pruefer_code(pruefer);
}
/**
* @brief Pruefer Code
*/
#line 3 "tree/tree-query.hpp"
template <typename G>
struct Tree {
private:
G& g;
int root;
vector<array<int, 24>> bl;
vector<int> dp;
void build() {
bl.resize(g.size());
dp.resize(g.size());
for (auto& v : bl) fill(begin(v), end(v), -1);
dfs(root, -1, 0);
}
void dfs(int c, int p, int _dp) {
dp[c] = _dp;
for (int i = p, x = 0; i != -1;) {
bl[c][x] = i;
i = bl[i][x], x++;
}
for (auto& d : g[c]) {
if (d == p) continue;
dfs(d, c, _dp + 1);
}
}
public:
Tree(G& _g, int _r = 0) : g(_g), root(_r) { build(); }
int depth(int u) const { return dp[u]; }
int par(int u) const { return u == root ? -1 : bl[u][0]; }
int kth_ancestor(int u, int k) const {
if (dp[u] < k) return -1;
while (k) {
int t = __builtin_ctz(k);
u = bl[u][t], k ^= 1 << t;
}
return u;
}
int nxt(int s, int t) const {
if (dp[s] >= dp[t]) return par(s);
int u = kth_ancestor(t, dp[t] - dp[s] - 1);
return bl[u][0] == s ? u : bl[s][0];
}
vector<int> path(int s, int t) const {
vector<int> pre, suf;
while (dp[s] > dp[t]) {
pre.push_back(s);
s = bl[s][0];
}
while (dp[s] < dp[t]) {
suf.push_back(t);
t = bl[t][0];
}
while (s != t) {
pre.push_back(s);
suf.push_back(t);
s = bl[s][0];
t = bl[t][0];
}
pre.push_back(s);
reverse(begin(suf), end(suf));
copy(begin(suf), end(suf), back_inserter(pre));
return pre;
}
int lca(int u, int v) {
if (dp[u] != dp[v]) {
if (dp[u] > dp[v]) swap(u, v);
v = kth_ancestor(v, dp[v] - dp[u]);
}
if (u == v) return u;
for (int i = __lg(dp[u]); i >= 0; --i) {
if (dp[u] < (1 << i)) continue;
if (bl[u][i] != bl[v][i]) u = bl[u][i], v = bl[v][i];
}
return bl[u][0];
}
// u - v 間のパス上の頂点のうち u から距離 i の頂点
// (dist(u, v) < i のとき -1)
int jump(int u, int v, int i) {
int lc = lca(u, v);
int d1 = dp[u] - dp[lc];
if (i <= d1) return kth_ancestor(u, i);
int d = d1 + dp[v] - dp[lc];
if (i <= d) return kth_ancestor(v, d - i);
return -1;
}
};
/**
* @brief 木に対する一般的なクエリ
* @docs docs/tree/tree-query.md
*/
#line 10 "verify/verify-unit-test/tree-path.test.cpp"
using namespace Nyaan;
template <typename G>
bool is_tree(G& g, bool directed = false) {
int n = g.size();
UnionFind uf(n);
rep(i, n) each(j, g[i]) {
if (!directed and i > j) continue;
if (!uf.unite(i, j)) return false;
}
return uf.size(0) == n;
}
template <typename G>
bool is_rooted_tree(G& g, int root) {
int n = g.size();
vector<char> vis(n, false);
auto dfs = [&](auto rc, int c) -> void {
vis[c] = true;
each(d, g[c]) {
assert(vis[d] == false);
rc(rc, d);
}
};
dfs(dfs, root);
int sm = 0;
each(b, vis) sm += !!b;
return sm == n;
}
template <typename G>
bool is_inverse_tree(G& g, G& rg) {
set<pair<int, int>> s, t;
int n = g.size();
for (int i = 0; i < n; i++) {
for (auto& j : g[i]) s.emplace(i, int(j));
for (auto& j : rg[i]) t.emplace(j, int(i));
}
return s == t;
}
void test_tree_query(vvi& g, int root = 0) {
Tree<vvi> tree(g, root);
int N = sz(g);
rep(i, N) rep(j, N) {
vi p1 = tree.path(i, j);
vi p2{int(i)};
for (int k = i; k != j;) {
p2.push_back(k = tree.nxt(k, j));
}
assert(p1 == p2);
int l = i, ld = tree.depth(i);
each(x, p1) if (amin(ld, tree.depth(x))) l = x;
assert(l == tree.lca(i, j));
}
}
using namespace Nyaan;
void Nyaan::solve() {
// Random Tree
for (int N : vi{2, 3, 4, 5, 10, 100}) {
vvi g = random_tree(N);
assert(is_tree(g) && "random tree");
test_tree_query(g);
int root = randint(0, N);
vvi rg = rooted_tree(g, root);
assert(is_tree(rg, true) && "rooted tree");
test_tree_query(rg, root);
vvi rh = inverse_tree(rg);
assert(is_inverse_tree(rg, rh) && "inverse tree");
}
for (int N : vi{2, 3, 4, 5, 10, 100}) {
vvi _g = random_tree(N);
assert(is_tree(_g) && "random tree");
WeightedGraph<ll> g(N);
rep(i, N) each(j, _g[i]) { g[i].emplace_back(i, j, rng() & 15); }
int root = randint(0, N);
auto rg = rooted_tree(g, root);
assert(is_tree(rg, true) && "rooted tree");
auto rh = inverse_tree(rg);
assert(is_inverse_tree(rg, rh) && "inverse tree");
}
// Line Tree
for (int N : vi{2, 3, 4, 5, 10, 100}) {
vvi g(N);
rep1(i, N - 1) g[i - 1].push_back(i);
test_tree_query(g);
}
// Preufer Code
{
V<vvi> gs(125);
rep(i, 125) {
vi pr(3);
for (int j = i, k = 0; k < 3; j /= 5, k++) {
pr[k] = j % 5;
}
gs[i] = pruefer_code(pr);
for (auto& es : gs[i]) sort(begin(es), end(es));
}
rep(i, 125) assert(is_tree(gs[i]));
rep(i, 125) rep(j, i) assert(gs[i] != gs[j]);
}
int a, b;
cin >> a >> b;
cout << a + b << endl;
}
verify/verify-unit-test/tree-path.test.cpp