CF1553H 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)|, $ $

where $ \\oplus $ denotes the bitwise XOR operation.

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 $ .