Max of Medians

AtCoder

IDabc304_g

Time5000ms

Memory256MB

Difficulty—

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: $N$ $A_1$ $A_2$ $\ldots$ $A_{2N}$ [samples]

Samples

Input #1

4
4 0 0 11 2 7 9 5

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 $14$.

Input #2

1
998244353 1000000007

Output #2

Input #3

5
1 2 4 8 16 32 64 128 256 512

Output #3

API Response (JSON)

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      "content": "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",
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    "difficulty": "None",
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      "statement_type": "Markdown",
      "content": "You are given a sequence of length $2N$: $A = (A_1, A_2, \\ldots, A_{2N})$.\nFind 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...",
      "is_translate": false,
      "language": "English"
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