XOR and Distance

Misaka 给了你一个长度为 $n$ 的序列 $\{a_i\}$，保证 $\forall i\in[1,n],a_i\in[0,2^k)$，现在她要你对所有 $x\in[0,2^k)$，求出 $\min_{i,j\in[1,n]\land i\not=j}\left|(a_i\oplus x)-(a_j\oplus x)\right|$

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) $ .

3 3
6 0 3

3 1 1 2 2 1 1 3

3 4
13 4 2

2 2 6 6 3 1 2 2 2 2 1 3 6 6 2 2

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 $ .