A12117 | Tree and XOR

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:

$$$$ r_i = \left\{ \begin{aligned} 1, ~ \text{if} ~ x_i \ne y_i \\ 0, ~ \text{if} ~ x_i = y_i \end{aligned} \right. $$$$

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.

Print one integer: $k$ -th smallest xor of a path in the tree.