#include "passageconstruction.h"

#include <cassert>
#include <cmath>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <algorithm>
#include <bitset>
#include <complex>
#include <deque>
#include <functional>
#include <iostream>
#include <limits>
#include <map>
#include <numeric>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <string>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>

using namespace std;

using Int = long long;

template <class T1, class T2> ostream &operator<<(ostream &os, const pair<T1, T2> &a) { return os << "(" << a.first << ", " << a.second << ")"; };
template <class T> ostream &operator<<(ostream &os, const vector<T> &as) { const int sz = as.size(); os << "["; for (int i = 0; i < sz; ++i) { if (i >= 256) { os << ", ..."; break; } if (i > 0) { os << ", "; } os << as[i]; } return os << "]"; }
template <class T> void pv(T a, T b) { for (T i = a; i != b; ++i) cerr << *i << " "; cerr << endl; }
template <class T> bool chmin(T &t, const T &f) { if (t > f) { t = f; return true; } return false; }
template <class T> bool chmax(T &t, const T &f) { if (t < f) { t = f; return true; } return false; }
#define COLOR(s) ("\x1b[" s "m")


mt19937_64 rng(58);

int N;
vector<vector<int>> graph;
vector<pair<int, int>> ans;

int dfs(int u) {
  int ret = -1;
  for (const int v : graph[u]) {
    const int res = dfs(v);
    if (~res) {
      if (~ret) {
        ans.emplace_back(ret, res);
        ret = -1;
      } else {
        ret = res;
      }
    }
  }
  if (~ret) {
    ans.emplace_back(ret, u);
    return -1;
  } else {
    return u;
  }
}

vector<pair<int, int>> ConstructPassages(int N_, const vector<pair<int, int>> &E) {
  N = N_;
  const int rt = 1 + rng() % (2 * N);
  vector<vector<int>> uss(2 * N);
  for (int u = 1; u <= 2 * N; ++u) {
    uss[GetDistance(rt, u)].push_back(u);
  }
  for (int d = 0; d < 2 * N; ++d) {
    shuffle(uss[d].begin(), uss[d].end(), rng);
  }
// cerr<<"uss = "<<uss<<endl;
  graph.assign(2 * N + 1, {});
  for (int d = 0; d < 2 * N - 1; ++d) if (uss[d + 1].size()) {
    const int len = uss[d].size();
    assert(len);
    for (const int v : uss[d + 1]) {
      for (int j = 0; j < len; ++j) {
        const int u = uss[d][j];
        if (j == len - 1 || QueryLCA({rt}, {u, v}, u)[0]) {
// cerr<<"ae "<<u<<" "<<v<<endl;
          graph[u].push_back(v);
          break;
        }
      }
    }
  }
  dfs(rt);
  return ans;
}