CF2250B String Construction

You are given two integers $ n $ and $ k $ . Construct a binary string $ ^{\text{∗}} $ $ s $ of length $ n $ , such that both of the following conditions hold: - The absolute difference between the number of characters $ \mathtt{0} $ and the number of characters $ \mathtt{1} $ in $ s $ is at most $ 1 $ . - There are exactly $ k $ pairs of adjacent equal characters in $ s $ . Formally, there are exactly $ k $ indices $ i $ ( $ 1 \le i \le n-1 $ ) satisfying $ s_i = s_{i + 1} $ . Or determine that no such string exists. $ ^{\text{∗}} $ A binary string is a string where each character is either $ \mathtt{0} $ or $ \mathtt{1} $ .

Each test contains multiple test cases. The first line contains the number of test cases $ t $ ( $ 1 \le t \le 1000 $ ). The description of the test cases follows. The only line of each test case contains two integers $ n $ and $ k $ ( $ 2 \le n \le 2 \cdot 10^5 $ , $ 0 \le k \le n-1 $ ). It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 2 \cdot 10^5 $ .

For each test case, output a binary string $ s $ of length $ n $ — the string you constructed. Print $ -1 $ if such a string does not exist. If there are multiple answers, you may output any of them.

In the first test case, one possible answer is $ s=\mathtt{01110} $ . It contains three characters $ \mathtt{1} $ and two characters $ \mathtt{0} $ , and there are exactly $ 2 $ adjacent equal pairs in $ s $ : $ (s_2, s_3) $ and $ (s_3, s_4) $ . In the second test case, $ k=n-1 $ . All characters in $ s $ should be equal, so the numbers of characters $ \mathtt{0} $ and $ \mathtt{1} $ could not differ by at most $ 1 $ . Thus, the answer is $ -1 $ . In the third test case, note that $ \mathtt{010110} $ is also a possible answer.