XOR Partition
Description
For a set of integers $S$, let's define its cost as the minimum value of $x \oplus y$ among all pairs of different integers from the set (here, $\oplus$ denotes bitwise XOR). If there are less than two elements in the set, its cost is equal to $2^{30}$.
You are given a set of integers $\{a_1, a_2, \dots, a_n\}$. You have to partition it into two sets $S_1$ and $S_2$ in such a way that every element of the given set belongs to exactly one of these two sets. The value of the partition is the minimum among the costs of $S_1$ and $S_2$.
Find the partition with the maximum possible value.
The first line contains $n$ ($2 \le n \le 200000$) — the number of elements in the set.
The second line contains $n$ distinct integers $a_1$, $a_2$, ..., $a_n$ ($0 \le a_i < 2^{30}$) — the elements of the set.
Print a string $n$ characters 0 and/or 1 describing the partition as follows: the $i$-th character should be 0 if the element $a_i$ belongs to $S_1$, otherwise, that character should be 1.
If there are multiple optimal answers, print any of them.
Input
The first line contains $n$ ($2 \le n \le 200000$) — the number of elements in the set.
The second line contains $n$ distinct integers $a_1$, $a_2$, ..., $a_n$ ($0 \le a_i < 2^{30}$) — the elements of the set.
Output
Print a string $n$ characters 0 and/or 1 describing the partition as follows: the $i$-th character should be 0 if the element $a_i$ belongs to $S_1$, otherwise, that character should be 1.
If there are multiple optimal answers, print any of them.
5
42 13 1337 37 152
4
1 2 3 4
2
1 2
8
7 6 5 4 3 2 1 0
10001
1100
10
10010110