CF1659D Reverse Sort Sum

Suppose you had an array $ A $ of $ n $ elements, each of which is $ 0 $ or $ 1 $ . Let us define a function $ f(k,A) $ which returns another array $ B $ , the result of sorting the first $ k $ elements of $ A $ in non-decreasing order. For example, $ f(4,[0,1,1,0,0,1,0]) = [0,0,1,1,0,1,0] $ . Note that the first $ 4 $ elements were sorted. Now consider the arrays $ B_1, B_2,\ldots, B_n $ generated by $ f(1,A), f(2,A),\ldots,f(n,A) $ . Let $ C $ be the array obtained by taking the element-wise sum of $ B_1, B_2,\ldots, B_n $ . For example, let $ A=[0,1,0,1] $ . Then we have $ B_1=[0,1,0,1] $ , $ B_2=[0,1,0,1] $ , $ B_3=[0,0,1,1] $ , $ B_4=[0,0,1,1] $ . Then $ C=B_1+B_2+B_3+B_4=[0,1,0,1]+[0,1,0,1]+[0,0,1,1]+[0,0,1,1]=[0,2,2,4] $ . You are given $ C $ . Determine a binary array $ A $ that would give $ C $ when processed as above. It is guaranteed that an array $ A $ exists for given $ C $ in the input.

The first line contains a single integer $ t $ ( $ 1 \leq t \leq 1000 $ ) — the number of test cases. Each test case has two lines. The first line contains a single integer $ n $ ( $ 1 \leq n \leq 2 \cdot 10^5 $ ). The second line contains $ n $ integers $ c_1, c_2, \ldots, c_n $ ( $ 0 \leq c_i \leq n $ ). It is guaranteed that a valid array $ A $ exists for the given $ C $ . The sum of $ n $ over all test cases does not exceed $ 2 \cdot 10^5 $ .

For each test case, output a single line containing $ n $ integers $ a_1, a_2, \ldots, a_n $ ( $ a_i $ is $ 0 $ or $ 1 $ ). If there are multiple answers, you may output any of them.

Here's the explanation for the first test case. Given that $ A=[1,1,0,1] $ , we can construct each $ B_i $ : - $ B_1=[\color{blue}{1},1,0,1] $ ; - $ B_2=[\color{blue}{1},\color{blue}{1},0,1] $ ; - $ B_3=[\color{blue}{0},\color{blue}{1},\color{blue}{1},1] $ ; - $ B_4=[\color{blue}{0},\color{blue}{1},\color{blue}{1},\color{blue}{1}] $ And then, we can sum up each column above to get $ C=[1+1+0+0,1+1+1+1,0+0+1+1,1+1+1+1]=[2,4,2,4] $ .