CF1450D Rating Compression

On the competitive programming platform CodeCook, every person has a rating graph described by an array of integers $ a $ of length $ n $ . You are now updating the infrastructure, so you've created a program to compress these graphs. The program works as follows. Given an integer parameter $ k $ , the program takes the minimum of each contiguous subarray of length $ k $ in $ a $ . More formally, for an array $ a $ of length $ n $ and an integer $ k $ , define the $ k $ -compression array of $ a $ as an array $ b $ of length $ n-k+1 $ , such that $$ b_j =\min_{j\le i\le j+k-1}a_i $$ For example, the $ 3 $ -compression array of $ [1, 3, 4, 5, 2] $ is $ [\min{1, 3, 4}, \min{3, 4, 5}, \min{4, 5, 2}]=[1, 3, 2]. $ A permutation of length $ m $ is an array consisting of $ m $ distinct integers from $ 1 $ to $ m $ in arbitrary order. For example, $ [2,3,1,5,4] $ is a permutation, but $ [1,2,2] $ is not a permutation ( $ 2 $ appears twice in the array) and $ [1,3,4] $ is also not a permutation ( $ m=3 $ but there is $ 4 $ in the array). A $ k $ -compression array will make CodeCook users happy if it will be a permutation. Given an array $ a $ , determine for all $ 1\leq k\leq n $ if CookCook users will be happy after a $ k$-compression of this array or not.

The first line contains a single integer $ t $ ( $ 1\leq t\leq 10^4 $ ) — the number of test cases. The first line of the description of each test case contains a single integer $ n $ ( $ 1\leq n\leq 3\cdot 10^5 $ ) — the length of the array. The second line of the description of each test case contains $ n $ integers $ a_1,\ldots,a_n $ ( $ 1\leq a_i\leq n $ ) — the elements of the array. It is guaranteed, that the sum of $ n $ for all test cases does not exceed $ 3\cdot 10^5 $ .

For each test case, print a binary string of length $ n $ . The $ k $ -th character of the string should be $ 1 $ if CookCook users will be happy after a $ k $ -compression of the array $ a $ , and $ 0 $ otherwise.

In the first test case, $ a=[1, 5, 3, 4, 2] $ . - The $ 1 $ -compression of $ a $ is $ [1, 5, 3, 4, 2] $ and it is a permutation. - The $ 2 $ -compression of $ a $ is $ [1, 3, 3, 2] $ and it is not a permutation, since $ 3 $ appears twice. - The $ 3 $ -compression of $ a $ is $ [1, 3, 2] $ and it is a permutation. - The $ 4 $ -compression of $ a $ is $ [1, 2] $ and it is a permutation. - The $ 5 $ -compression of $ a $ is $ [1] $ and it is a permutation.