Score : $625$ points

问题描述

给定一个长度为 $2N$ 的序列：$A = (A_1, A_2, \ldots, A_{2N})$。

通过重新排列序列 $A$ 中的元素，找出长度为 $N$ 的序列 $(A_1 \oplus A_2, A_3 \oplus A_4, \ldots, A_{2N-1} \oplus A_{2N})$ 的中位数可能达到的最大值。

此处，$\oplus$ 表示按位异或运算。

什么是按位异或运算？非负整数 $A$ 和 $B$ 的按位异或（记作 $A \oplus B$）定义如下：

• 在 $A \oplus B$ 的二进制表示中，处于 $2^k$ 位置（$k \geq 0$）的数字是 $1$ 当且仅当在 $A$ 和 $B$ 的二进制表示中恰好有一个在 $2^k$ 位置的数字为 $1$，否则该位置数字为 $0$。

例如，$3 \oplus 5 = 6$（二进制表示：$011 \oplus 101 = 110$）。

另外，对于长度为 $L$ 的序列 $B$，其中位数是在将 $B$ 按升序排序后得到的序列 $B'$ 中位于 $\lfloor \frac{L}{2} \rfloor + 1$ 位置处的数值。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

You are given a sequence of length $2N$: $A = (A_1, A_2, \ldots, A_{2N})$.

Find the maximum value that can be obtained as the median of the length-$N$ sequence $(A_1 \oplus A_2, A_3 \oplus A_4, \ldots, A_{2N-1} \oplus A_{2N})$ by rearranging the elements of the sequence $A$.

Here, $\oplus$ represents bitwise exclusive OR.

What is bitwise exclusive OR? The bitwise exclusive OR of non-negative integers $A$ and $B$, denoted as $A \oplus B$, is defined as follows:

• The number at the $2^k$ position ($k \geq 0$) in the binary representation of $A \oplus B$ is $1$ if and only if exactly one of the numbers at the $2^k$ position in the binary representation of $A$ and $B$ is $1$, and it is $0$ otherwise.

For example, $3 \oplus 5 = 6$ (in binary notation: $011 \oplus 101 = 110$).

Also, for a sequence $B$ of length $L$, the median of $B$ is the value at the $\lfloor \frac{L}{2} \rfloor + 1$-th position of the sequence $B'$ obtained by sorting $B$ in ascending order.

Constraints

• $1 \leq N \leq 10^5$
• $0 \leq A_i < 2^{30}$
• All input values are integers.

Input

The input is given from Standard Input in the following format:

$N$

$A_1$ $A_2$ $\ldots$ $A_{2N}$

Output

Print the answer.

Sample Input 1

4
4 0 0 11 2 7 9 5

Sample Output 1

14

By rearranging $A$ as $(5, 0, 9, 7, 11, 4, 0, 2)$, we get $(A_1 \oplus A_2, A_3 \oplus A_4, A_5 \oplus A_6, A_7 \oplus A_8) = (5, 14, 15, 2)$, and the median of this sequence is $14$.
It is impossible to rearrange $A$ so that the median of $(A_1 \oplus A_2, A_3 \oplus A_4, A_5 \oplus A_6, A_7 \oplus A_8)$ is $15$ or greater, so we print $14$.

Sample Input 2

1
998244353 1000000007

Sample Output 2

1755654

Sample Input 3

5
1 2 4 8 16 32 64 128 256 512

Sample Output 3

192

update @ 2024/3/10 08:35:05