Bakry and Partitioning

题意翻译

### 题目背景 Bakry 遇到了一道题，但他懒得做了，于是他想让你帮他做。 ### 题目描述 一棵树有 $n$ 个节点，第 $i$ 个节点的点权为 $a_i$ 。（注：树是一个有 $n$ 个节点、$n-1$ 条边的连通图） 你需要回答：能不能选择这棵树中的至少 $1$ 条边、至多 $k-1$ 条边删除，使得删除完这些边的树满足以下条件： - 每个联通块的点权异或和相等 ### 输入格式 每个测试点包含多个询问，故第一行输入询问数 $t\ (1\leq t\leq 5\cdot 10^4)$ 。每个询问的输入格式如下： 每个询问的第一行包含两个正整数 $n$ 和 $k\ (2\leq k \leq n \leq 10^5)$ 。 每个询问的第二行包含 $n$ 个整数，分别为 $a_1, a_2, ..., a_n\ (1\leq a_i \leq 10^9)$ 。 接下来的 $n-1$ 行中的第 $i$ 行包含两个整数 $u_i$ 和 $v_i\ (1\leq u_i, v_i\leq n, u_i\neq v_i)$，代表 $u_i$ 和 $v_i$ 之间有一条边。 保证给定的图是一棵树。 保证所有询问里的 $n$ 的和不超过 $2\cdot 10^5$ 。 ### 输出格式 对于每个询问，你需要在单独的一行中输出一个字符串。如果这个询问给出的数据满足题目条件，输出 `YES` 。否则，输出 `NO` 。 温馨小提示：本题输出不区分大小写。 ###### Translated by [\_FILARET\_](https://www.luogu.com.cn/user/84121)

题目描述

Bakry faced a problem, but since he's lazy to solve it, he asks for your help. You are given a tree of $ n $ nodes, the $ i $ -th node has value $ a_i $ assigned to it for each $ i $ from $ 1 $ to $ n $ . As a reminder, a tree on $ n $ nodes is a connected graph with $ n-1 $ edges. You want to delete at least $ 1 $ , but at most $ k-1 $ edges from the tree, so that the following condition would hold: - For every connected component calculate the [bitwise XOR](https://en.wikipedia.org/wiki/Bitwise_operation#XOR) of the values of the nodes in it. Then, these values have to be the same for all connected components. Is it possible to achieve this condition?

输入输出格式

输入格式

Each test contains multiple test cases. The first line contains the number of test cases $ t $ $ (1 \leq t \leq 5 \cdot 10^4) $ . Description of the test cases follows. The first line of each test case contains two integers $ n $ and $ k $ $ (2 \leq k \leq n \leq 10^5) $ . The second line of each test case contains $ n $ integers $ a_1, a_2, ..., a_n $ $ (1 \leq a_i \leq 10^9) $ . The $ i $ -th of the next $ n-1 $ lines contains two integers $ u_i $ and $ v_i $ ( $ 1 \leq u_i, v_i \leq n $ , $ u_i\neq v_i $ ), which means that there's an edge between nodes $ u_i $ and $ v_i $ . It is guaranteed that the given graph is a tree. It is guaranteed that the sum of $ n $ over all test cases doesn't exceed $ 2 \cdot 10^5 $ .

输出格式

For each test case, you should output a single string. If you can delete the edges according to the conditions written above, output "YES" (without quotes). Otherwise, output "NO" (without quotes). You can print each letter of "YES" and "NO" in any case (upper or lower).

输入输出样例

输入样例 #1

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

输出样例 #1

NO
YES
NO
YES
NO

说明

It can be shown that the objection is not achievable for first, third, and fifth test cases. In the second test case, you can just remove all the edges. There will be $ 5 $ connected components, each containing only one node with value $ 3 $ , so the bitwise XORs will be $ 3 $ for all of them. In the fourth test case, this is the tree: . You can remove an edge $ (4,5) $ The bitwise XOR of the first component will be, $ a_1 \oplus a_2 \oplus a_3 \oplus a_4 = 1 \oplus 6 \oplus 4 \oplus 1 = 2 $ (where $ \oplus $ denotes the bitwise XOR). The bitwise XOR of the second component will be, $ a_5 = 2 $ .