D. Co-growing Sequence
bitmasks; constructive algorithms; greedy
A sequence of non-negative integers \(a_1, a_2, \dots, a_n\) is called growing if for all \(i\) from \(1\) to \(n - 1\) all ones (of binary representation) in \(a_i\) are in the places of ones (of binary representation) in \(a_{i + 1}\) (in other words, \(a_i \:\&\: a_{i + 1} = a_i\), where \(\&\) denotes bitwise AND). If \(n = 1\) then the sequence is considered growing as well.For example, the following four sequences are growing: \([2, 3, 15, 175]\) — in binary it's \([10_2, 11_2, 1111_2, 10101111_2]\); \([5]\) — in binary it's \([101_2]\); \([1, 3, 7, 15]\) — in binary it's \([1_2, 11_2, 111_2, 1111_2]\); \([0, 0, 0]\) — in binary it's \([0_2, 0_2, 0_2]\). The following three sequences are non-growing: \([3, 4, 5]\) — in binary it's \([11_2, 100_2, 101_2]\); \([5, 4, 3]\) — in binary it's \([101_2, 100_2, 011_2]\); \([1, 2, 4, 8]\) — in binary it's \([0001_2, 0010_2, 0100_2, 1000_2]\). Consider two sequences of non-negative integers \(x_1, x_2, \dots, x_n\) and \(y_1, y_2, \dots, y_n\). Let's call this pair of sequences co-growing if the sequence \(x_1 \oplus y_1, x_2 \oplus y_2, \dots, x_n \oplus y_n\) is growing where \(\oplus\) denotes bitwise XOR.You are given a sequence of integers \(x_1, x_2, \dots, x_n\). Find the lexicographically minimal sequence \(y_1, y_2, \dots, y_n\) such that sequences \(x_i\) and \(y_i\) are co-growing.The sequence \(a_1, a_2, \dots, a_n\) is lexicographically smaller than the sequence \(b_1, b_2, \dots, b_n\) if there exists \(1 \le k \le n\) such that \(a_i = b_i\) for any \(1 \le i < k\) but \(a_k < b_k\).
The first line contains an integer \(t\) (\(1 \le t \le 10^4\)). Then \(t\) test cases follow.The first line of each test case contains an integer \(n\) (\(1 \le n \le 2 \cdot 10^5\)) — length of the sequence \(x_i\).The second line contains \(n\) integers \(x_1, x_2, \dots, x_n\) (\(0 \le x_i < 2^{30}\)) — elements of the sequence \(x_i\).It is guaranteed that the sum of \(n\) overall all test cases doesn't exceed \(2 \cdot 10^5\).
For each test case, print \(n\) integers \(y_1, y_2, \dots, y_n\) (\(0 \le y_i < 2^{30}\)) — lexicographically minimal sequence such that such that it's co-growing with given sequence \(x_i\).