Tree and XOR
题意翻译
题意
给一棵$n$个点的树,有边权,你要求出$n^2$个点对中第$k$小的路径异或和(也就是说,$(a,b)$和$(b,a)$算不同的点对),如果点对形如$(a,a)$的话异或和为0。
数据范围
$n<=1e6,k<=n^2,$边权$<2^{62}$
题目描述
You are given a connected undirected graph without cycles (that is, a tree) of $ n $ vertices, moreover, there is a non-negative integer written on every edge.
Consider all pairs of vertices $ (v, u) $ (that is, there are exactly $ n^2 $ such pairs) and for each pair calculate the bitwise exclusive or (xor) of all integers on edges of the simple path between $ v $ and $ u $ . If the path consists of one vertex only, then xor of all integers on edges of this path is equal to $ 0 $ .
Suppose we sorted the resulting $ n^2 $ values in non-decreasing order. You need to find the $ k $ -th of them.
The definition of xor is as follows.
Given two integers $ x $ and $ y $ , consider their binary representations (possibly with leading zeros): $ x_k \dots x_2 x_1 x_0 $ and $ y_k \dots y_2 y_1 y_0 $ (where $ k $ is any number so that all bits of $ x $ and $ y $ can be represented). Here, $ x_i $ is the $ i $ -th bit of the number $ x $ and $ y_i $ is the $ i $ -th bit of the number $ y $ . Let $ r = x \oplus y $ be the result of the xor operation of $ x $ and $ y $ . Then $ r $ is defined as $ r_k \dots r_2 r_1 r_0 $ where:
$ $$$ r_i = \left\{ \begin{aligned} 1, ~ \text{if} ~ x_i \ne y_i \\ 0, ~ \text{if} ~ x_i = y_i \end{aligned} \right. $ $$$
输入输出格式
输入格式
The first line contains two integers $ n $ and $ k $ ( $ 2 \le n \le 10^6 $ , $ 1 \le k \le n^2 $ ) — the number of vertices in the tree and the number of path in the list with non-decreasing order.
Each of the following $ n - 1 $ lines contains two integers $ p_i $ and $ w_i $ ( $ 1 \le p_i \le i $ , $ 0 \le w_i < 2^{62} $ ) — the ancestor of vertex $ i + 1 $ and the weight of the corresponding edge.
输出格式
Print one integer: $ k $ -th smallest xor of a path in the tree.
输入输出样例
输入样例 #1
2 1
1 3
输出样例 #1
0
输入样例 #2
3 6
1 2
1 3
输出样例 #2
2
说明
The tree in the second sample test looks like this:
![](https://cdn.luogu.com.cn/upload/vjudge_pic/CF1055F/5bfa2eab3ef44ea89fb9c60d427736329daa0bdd.png)
For such a tree in total $ 9 $ paths exist:
1. $ 1 \to 1 $ of value $ 0 $
2. $ 2 \to 2 $ of value $ 0 $
3. $ 3 \to 3 $ of value $ 0 $
4. $ 2 \to 3 $ (goes through $ 1 $ ) of value $ 1 = 2 \oplus 3 $
5. $ 3 \to 2 $ (goes through $ 1 $ ) of value $ 1 = 2 \oplus 3 $
6. $ 1 \to 2 $ of value $ 2 $
7. $ 2 \to 1 $ of value $ 2 $
8. $ 1 \to 3 $ of value $ 3 $
9. $ 3 \to 1 $ of value $ 3 $