Tiles Placement

题意翻译

给出一个$n$个点的树，给出颜色种类数$k$，现在要给每一个点染色，每一个点只能染一种颜色，求一种染色方案，使得树上任意一条点数为$k$的简单路径包含$k$种颜色，若没有输出$No$，若存在第一行输出$Yes$，第二行输出$n$个整数，第$i$个数字表示$i$号点的颜色

题目描述

The new pedestrian zone in Moscow city center consists of $ n $ squares connected with each other by $ n - 1 $ footpaths. We define a simple path as a sequence of squares such that no square appears in this sequence twice and any two adjacent squares in this sequence are directly connected with a footpath. The size of a simple path is the number of squares in it. The footpaths are designed in a such a way that there is exactly one simple path between any pair of different squares. During preparations for Moscow City Day the city council decided to renew ground tiles on all $ n $ squares. There are $ k $ tile types of different colors, numbered from $ 1 $ to $ k $ . For each square exactly one tile type must be selected and then used to cover this square surface. To make walking through the city center more fascinating, it was decided to select tiles types for each square in such a way that any possible simple path of size exactly $ k $ contains squares with all $ k $ possible tile colors. You need to find out whether it is possible to place the tiles this way or not.

输入输出格式

输入格式

The first line contains two integers $ n $ , $ k $ ( $ 2 \le k \le n \le 200\,000 $ ) — the number of squares in the new pedestrian zone, the number of different tile colors. Each of the following $ n - 1 $ lines contains two integers $ v_i $ and $ u_i $ ( $ 1 \le v_i, u_i \le n $ ) — numbers of the squares connected by the corresponding road. It's guaranteed, that it's possible to go from any square to any other square, moreover there is exactly one such simple path.

输出格式

Print "Yes" if it is possible to assign tile colors this way and "No" otherwise. In case your answer is "Yes", print $ n $ integers from $ 1 $ to $ k $ each, the color of the tile for every square.

输入输出样例

输入样例 #1

7 4
1 3
2 3
3 4
4 5
5 6
5 7

输出样例 #1

Yes
1 1 2 3 4 1 1

输入样例 #2

7 3
1 3
2 3
3 4
4 5
5 6
5 7

输出样例 #2

No

说明

The following pictures illustrate the pedestrian zone in first and second examples. The second picture also shows one possible distribution of colors among the squares for $ k = 4 $ .