Score : $500$ points

问题描述

给定一棵有 $N$ 个顶点的根树，其中根节点为顶点 $1$。
对于每个 $i = 1, 2, \ldots, N-1$，第 $i$ 条边连接顶点 $u_i$ 和顶点 $v_i$。

对于每个 $i = 1, 2, \ldots, N$，令 $S_i$ 表示以顶点 $i$ 为根的子树中所有顶点的集合。（每个顶点都在以其自身为根的子树中，即 $i \in S_i$。）

另外，对于整数 $l$ 和 $r$，令 $[l, r]$ 表示从 $l$ 到 $r$ 之间的所有整数集合，即 $[l, r] = \lbrace l, l+1, l+2, \ldots, r \rbrace$。

考虑一个由 $N$ 对整数组成的序列 $\big((L_1, R_1), (L_2, R_2), \ldots, (L_N, R_N)\big)$，满足以下条件：

• 对于每一个整数 $i$（满足 $1 \leq i \leq N$），都有 $1 \leq L_i \leq R_i$。
• 对于每一对整数 $(i, j)$（满足 $1 \leq i, j \leq N$），满足以下情况：
  • 若 $S_i \subseteq S_j$，则有 $[L_i, R_i] \subseteq [L_j, R_j]$；
  • 若 $S_i \cap S_j = \emptyset$，则有 $[L_i, R_i] \cap [L_j, R_j] = \emptyset$。

可以证明至少存在一个序列 $\big((L_1, R_1), (L_2, R_2), \ldots, (L_N, R_N)\big)$。在这些序列中，找出并输出一个使得 $\max \lbrace L_1, L_2, \ldots, L_N, R_1, R_2, \ldots, R_N \rbrace$ 达到最小值的序列，即使用的最大整数最小化。（如果有多个这样的序列，你可以任选其中一个打印。）

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

You are given a rooted tree with $N$ vertices. The root is Vertex $1$.
For each $i = 1, 2, \ldots, N-1$, the $i$-th edge connects Vertex $u_i$ and Vertex $v_i$.

For each $i = 1, 2, \ldots, N$, let $S_i$ denote the set of all vertices in the subtree rooted at Vertex $i$. (Each vertex is in the subtree rooted at itself, that is, $i \in S_i$.)

Additionally, for integers $l$ and $r$, let $[l, r]$ denote the set of all integers between $l$ and $r$, that is, $[l, r] = \lbrace l, l+1, l+2, \ldots, r \rbrace$.

Consider a sequence of $N$ pairs of integers $\big((L_1, R_1), (L_2, R_2), \ldots, (L_N, R_N)\big)$ that satisfies the conditions below.

• $1 \leq L_i \leq R_i$ for every integer $i$ such that $1 \leq i \leq N$.
• The following holds for every pair of integers $(i, j)$ such that $1 \leq i, j \leq N$.
  • $[L_i, R_i] \subseteq [L_j, R_j]$ if $S_i \subseteq S_j$
  • $[L_i, R_i] \cap [L_j, R_j] = \emptyset$ if $S_i \cap S_j = \emptyset$

It can be shown that there is at least one sequence $\big((L_1, R_1), (L_2, R_2), \ldots, (L_N, R_N)\big)$. Among those sequences, print one that minimizes $\max \lbrace L_1, L_2, \ldots, L_N, R_1, R_2, \ldots, R_N \rbrace$, the maximum integer used. (If there are multiple such sequences, you may print any of them.)

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $1 \leq u_i, v_i \leq N$
• All values in input are integers.
• The given graph is a tree.

Input

Input is given from Standard Input in the following format:

$N$

$u_1$ $v_1$

$u_2$ $v_2$

$\vdots$

$u_{N-1}$ $v_{N-1}$

Output

Print $N$ lines in the format below. That is, for each $i = 1, 2, \ldots, N$, the $i$-th line should contain $L_i$ and $R_i$ separated by a space.

$L_1$ $R_1$

$L_2$ $R_2$

$\vdots$

$L_N$ $R_N$

Sample Input 1

3
2 1
3 1

Sample Output 1

1 2
2 2
1 1

$(L_1, R_1) = (1, 2), (L_2, R_2) = (2, 2), (L_3, R_3) = (1, 1)$ satisfies the conditions.
Indeed, we have $[L_2, R_2] \subseteq [L_1, R_1], [L_3, R_3] \subseteq [L_1, R_1], [L_2, R_2] \cap [L_3, R_3] = \emptyset$.
Additionally, $\max \lbrace L_1, L_2, L_3, R_1, R_2, R_3 \rbrace = 2$ is the minimum possible value.

Sample Input 2

5
3 4
5 4
1 2
1 4

Sample Output 2

1 3
3 3
2 2
1 2
1 1

Sample Input 3

5
4 5
3 2
5 2
3 1

Sample Output 3

1 1
1 1
1 1
1 1
1 1

update @ 2024/3/10 10:21:31