#include "bits/extc++.h"
using namespace std;
template <typename T, typename... U>
void dbgh(const T& t, const U&... u) {
cerr << t;
((cerr << " | " << u), ...);
cerr << endl;
}
#ifdef DEBUG
#define dbg(...) \
cerr << "L" << __LINE__ << " [" << #__VA_ARGS__ << "]: "; \
dbgh(__VA_ARGS__)
#else
#define dbg(...)
#define cerr \
if (false) \
cerr
#endif
#define endl "\n"
#define long int64_t
#define sz(x) int(std::size(x))
struct DSU {
vector<int> p;
DSU(int n) : p(n, -1) {}
int find(int u) {
return p[u] < 0 ? u : (p[u] = find(p[u]));
}
bool merge(int u, int v) {
u = find(u);
v = find(v);
if (u == v) {
return false;
}
if (p[u] < p[v]) {
swap(u, v);
}
p[v] += p[u];
p[u] = v;
return true;
}
};
vector<pair<int, int>> spanning_tree(int n,
const vector<pair<int, int>>& edges) {
DSU dsu(n);
vector<pair<int, int>> ans;
for (auto& [u, v] : edges) {
if (dsu.merge(u, v)) {
ans.emplace_back(u, v);
}
}
return ans;
}
struct Centroid {
int n, cu;
vector<int> siz;
vector<vector<int>> graph, comps;
Centroid(int n, const vector<pair<int, int>>& edges)
: n(n), siz(n), graph(n) {
for (auto& [u, v] : edges) {
graph[u].push_back(v);
graph[v].push_back(u);
}
pdfs(0, -1);
cu = 0;
while (true) {
pair<int, int> opt {-1, -1};
for (auto& v : graph[cu]) {
opt = max(opt, {siz[v], v});
}
if (opt.first <= n / 2) {
break;
}
int nu = opt.second;
siz[cu] -= siz[nu];
siz[nu] += siz[cu];
cu = nu;
}
for (auto& v : graph[cu]) {
comps.emplace_back();
dfs(v, cu, comps.back());
dbg(v, siz[v], sz(comps.back()));
}
}
void pdfs(int u, int p) {
siz[u] = 1;
for (auto& v : graph[u]) {
if (v == p) {
continue;
}
pdfs(v, u);
siz[u] += siz[v];
}
}
void dfs(int u, int p, vector<int>& out) {
out.push_back(u);
for (auto& v : graph[u]) {
if (v == p) {
continue;
}
dfs(v, u, out);
}
}
};
struct Solver1 {
bool found = false;
int n, kv, csum;
vector<char> vis;
vector<int> arr, st;
vector<vector<int>> graph;
Solver1(int n,
int kv,
const vector<int>& arr,
const vector<pair<int, int>>& edges)
: n(n), kv(kv), vis(n), arr(arr), graph(n) {
assert(*max_element(begin(arr), end(arr)) < kv);
for (auto& [u, v] : edges) {
graph[u].push_back(v);
graph[v].push_back(u);
}
for (int i = 0; i < n && !found; i++) {
if (vis[i]) {
continue;
}
st.clear();
csum = 0;
dfs(i);
found |= csum >= kv;
}
}
void dfs(int u) {
if (found || vis[u]) {
return;
} else if (csum >= kv) {
assert(csum <= 2 * kv - 2);
found = true;
return;
}
vis[u] = true;
st.push_back(u);
csum += arr[u];
for (auto& v : graph[u]) {
dfs(v);
}
}
};
struct Solver2 {
int n, kv;
vector<char> vis;
vector<int> st;
vector<vector<int>> graph;
Solver2(int n,
int kv,
const vector<int>& nodes,
const vector<pair<int, int>>& edges)
: n(n), kv(kv), vis(n), graph(n) {
bool inode[n] {};
for (auto& a : nodes) {
inode[a] = true;
}
assert(kv <= n);
for (auto& [u, v] : edges) {
if (!inode[u] || !inode[v]) {
continue;
}
graph[u].push_back(v);
graph[v].push_back(u);
}
dfs(nodes[0]);
assert(sz(st) == kv);
}
void dfs(int u) {
if (vis[u] || sz(st) == kv) {
return;
}
vis[u] = true;
st.push_back(u);
for (auto& v : graph[u]) {
dfs(v);
}
}
};
vector<int> find_split(int n,
int kv1,
int kv2,
int kv3,
vector<int> edges_u,
vector<int> edges_v) {
int m = sz(edges_u);
vector<pair<int, int>> edges(m);
for (int i = 0; i < m; i++) {
edges[i] = {edges_u[i], edges_v[i]};
}
array<int, 3> ikv {kv1, kv2, kv3};
{
array<int, 3> ckv {kv1, kv2, kv3};
sort(begin(ckv), end(ckv));
kv1 = ckv[0];
kv2 = ckv[1];
kv3 = ckv[2];
}
auto repermute = [&](vector<int> arr) -> vector<int> {
int cnt[3] {};
for (auto& a : arr) {
cnt[a - 1]++;
}
int perm[3];
iota(begin(perm), end(perm), 0);
do {
bool ok = true;
for (int i = 0; i < 3; i++) {
ok &= ikv[perm[i]] == cnt[i];
}
if (!ok) {
continue;
}
for (auto& a : arr) {
a = perm[a - 1] + 1;
}
return arr;
} while (next_permutation(begin(perm), end(perm)));
assert(false);
};
dbg(kv1, kv2, kv3);
Centroid centroid(n, spanning_tree(n, edges));
auto& comps = centroid.comps;
int i_comp[n];
i_comp[centroid.cu] = -1;
for (int i = 0; i < sz(comps); i++) {
for (auto& a : comps[i]) {
i_comp[a] = i;
}
}
auto not_nodes = [&](const vector<int>& arr) -> vector<int> {
bool vis[n] {};
for (auto& a : arr) {
vis[a] = true;
}
vector<int> ans;
for (int i = 0; i < n; i++) {
if (!vis[i]) {
ans.push_back(i);
}
}
return ans;
};
auto answer = [&](const vector<int>& arra,
const vector<int>& arrb) -> vector<int> {
dbg(sz(arra), sz(arrb));
assert(kv1 <= sz(arra) && kv2 <= sz(arrb));
vector<int> ans(n, 3);
auto go = [&](const vector<int>& nodes, int kv, int val) -> void {
auto c_comp = Solver2(n, kv, nodes, edges).st;
for (auto& a : c_comp) {
ans[a] = val;
}
};
go(arra, kv1, 1);
go(arrb, kv2, 2);
return repermute(ans);
};
for (int i = 0; i < sz(comps); i++) {
if (sz(comps[i]) < kv1) {
continue;
}
auto arra = comps[i];
dbg("centroid quick");
return answer(arra, not_nodes(arra));
}
vector<int> c_sizes;
for (auto& a : comps) {
c_sizes.push_back(sz(a));
}
vector<pair<int, int>> c_edges;
for (auto& [u, v] : edges) {
if (u == centroid.cu || v == centroid.cu) {
continue;
}
dbg(i_comp[u], i_comp[v]);
c_edges.emplace_back(i_comp[u], i_comp[v]);
}
Solver1 s1(sz(comps), kv1, c_sizes, c_edges);
if (!s1.found) {
// assert(false);
return vector<int>(n);
}
vector<int> arra;
for (auto& a : s1.st) {
arra.insert(arra.end(), begin(comps[a]), end(comps[a]));
}
return answer(arra, not_nodes(arra));
}
