E. Tests for problem D
time limit per test
2 seconds
memory limit per test
256 megabytes
standard input
standard output

We had a really tough time generating tests for problem D. In order to prepare strong tests, we had to solve the following problem.

Given an undirected labeled tree consisting of $$$n$$$ vertices, find a set of segments such that:

  1. both endpoints of each segment are integers from $$$1$$$ to $$$2n$$$, and each integer from $$$1$$$ to $$$2n$$$ should appear as an endpoint of exactly one segment;
  2. all segments are non-degenerate;
  3. for each pair $$$(i, j)$$$ such that $$$i \ne j$$$, $$$i \in [1, n]$$$ and $$$j \in [1, n]$$$, the vertices $$$i$$$ and $$$j$$$ are connected with an edge if and only if the segments $$$i$$$ and $$$j$$$ intersect, but neither segment $$$i$$$ is fully contained in segment $$$j$$$, nor segment $$$j$$$ is fully contained in segment $$$i$$$.

Can you solve this problem too?


The first line contains one integer $$$n$$$ ($$$1 \le n \le 5 \cdot 10^5$$$) — the number of vertices in the tree.

Then $$$n - 1$$$ lines follow, each containing two integers $$$x_i$$$ and $$$y_i$$$ ($$$1 \le x_i, y_i \le n$$$, $$$x_i \ne y_i$$$) denoting the endpoints of the $$$i$$$-th edge.

It is guaranteed that the given graph is a tree.


Print $$$n$$$ pairs of integers, the $$$i$$$-th pair should contain two integers $$$l_i$$$ and $$$r_i$$$ ($$$1 \le l_i < r_i \le 2n$$$) — the endpoints of the $$$i$$$-th segment. All $$$2n$$$ integers you print should be unique.

It is guaranteed that the answer always exists.

1 2
1 3
3 4
3 5
2 6
9 12
7 10
3 11
1 5
2 4
6 8
1 2