G - Max of Medians
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G - Max of Medians
Score : $625$ points
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.
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:
Output
Print the answer.
4
4 0 0 11 2 7 9 5
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 $14$.
1
998244353 1000000007
1755654
5
1 2 4 8 16 32 64 128 256 512
192