A14398. XOR and Distance

You are given an array $a$ consisting of $n$ distinct elements and an integer $k$ . Each element in the array is a non-negative integer not exceeding $2^k-1$ .

Let's define the XOR distance for a number $x$ as the value of

$$$$f(x) = \min\limits_{i = 1}^{n} \min\limits_{j = i + 1}^{n} |(a_i \oplus x) - (a_j \oplus x)|, $$ </p><p>where $\\oplus$ denotes <a href="https://en.wikipedia.org/wiki/Bitwise_operation#XOR">the bitwise XOR operation</a>.</p><p>For every integer $x$ from $0$ to $2^k-1$ , you have to calculate $f(x)$$$.

The first line contains two integers $n$ and $k$ ( $1 \le k \le 19$ ; $2 \le n \le 2^k$ ).

The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ( $0 \le a_i \le 2^k-1$ ). All these integers are distinct.

Print $2^k$ integers. The $i$ -th of them should be equal to $f(i-1)$ .

Consider the first example:

- for $x = 0$ , if we apply bitwise XOR to the elements of the array with $x$ , we get the array $[6, 0, 3]$ , and the minimum absolute difference of two elements is $3$ ;
- for $x = 1$ , if we apply bitwise XOR to the elements of the array with $x$ , we get the array $[7, 1, 2]$ , and the minimum absolute difference of two elements is $1$ ;
- for $x = 2$ , if we apply bitwise XOR to the elements of the array with $x$ , we get the array $[4, 2, 1]$ , and the minimum absolute difference of two elements is $1$ ;
- for $x = 3$ , if we apply bitwise XOR to the elements of the array with $x$ , we get the array $[5, 3, 0]$ , and the minimum absolute difference of two elements is $2$ ;
- for $x = 4$ , if we apply bitwise XOR to the elements of the array with $x$ , we get the array $[2, 4, 7]$ , and the minimum absolute difference of two elements is $2$ ;
- for $x = 5$ , if we apply bitwise XOR to the elements of the array with $x$ , we get the array $[3, 5, 6]$ , and the minimum absolute difference of two elements is $1$ ;
- for $x = 6$ , if we apply bitwise XOR to the elements of the array with $x$ , we get the array $[0, 6, 5]$ , and the minimum absolute difference of two elements is $1$ ;
- for $x = 7$ , if we apply bitwise XOR to the elements of the array with $x$ , we get the array $[1, 7, 4]$ , and the minimum absolute difference of two elements is $3$ .