C. Fenwick Tree
bitmasks; brute force; combinatorics; data structures; dp; math; trees
Let \(\operatorname{lowbit}(x)\) denote the value of the lowest binary bit of \(x\), e.g. \(\operatorname{lowbit}(12)=4\), \(\operatorname{lowbit}(8)=8\).For an array \(a\) of length \(n\), if an array \(s\) of length \(n\) satisfies \(s_k=\left(\sum\limits_{i=k-\operatorname{lowbit}(k)+1}^{k}a_i\right)\bmod 998\,244\,353\) for all \(k\), then \(s\) is called the Fenwick Tree of \(a\). Let's denote it as \(s=f(a)\).For a positive integer \(k\) and an array \(a\), \(f^k(a)\) is defined as follows:$$$\( f^k(a)= \begin{cases} f(a)&\textrm{if }k=1\\ f(f^{k-1}(a))&\textrm{otherwise.}\\ \end{cases} \)\(You are given an array \)b\( of length \)n\( and a positive integer \)k\(. Find an array \)a\( that satisfies \)0\le a_i < 998\,244\,353\( and \)f^k(a)=b$$$. It can be proved that an answer always exists. If there are multiple possible answers, you may print any of them.
Each test contains multiple test cases. The first line contains the number of test cases \(t\) (\(1\le t\le 10^4\)). The description of the test cases follows.The first line of each test case contains two positive integers \(n\) (\(1 \leq n \leq 2\cdot 10^5\)) and \(k\) (\(1\le k\le 10^9\)), representing the length of the array and the number of times the function \(f\) is performed.The second line of each test case contains an array \(b_1, b_2, \ldots, b_n\) (\(0\le b_i < 998\,244\,353\)).It is guaranteed that the sum of \(n\) over all test cases does not exceed \(2\cdot 10^5\).
For each test case, print a single line, containing a valid array \(a\) of length \(n\).
In the first test case, it can be seen that \(f^1([1,1,1,1,1,1,1,1])=[1,2,1,4,1,2,1,8]\).In the second test case, it can be seen that \(f^2([1,2,3,4,5,6])=f^1([1,3,3,10,5,11])=[1,4,3,17,5,16]\).