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:  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 $