B. Koxia and Permutation
constructive algorithms
Reve has two integers \(n\) and \(k\).Let \(p\) be a permutation\(^\dagger\) of length \(n\). Let \(c\) be an array of length \(n - k + 1\) such that $$$\(c_i = \max(p_i, \dots, p_{i+k-1}) + \min(p_i, \dots, p_{i+k-1}).\)\( Let the cost of the permutation \)p\( be the maximum element of \)c\(.Koxia wants you to construct a permutation with the minimum possible cost.\)^\dagger\( A permutation of length \)n\( is an array consisting of \)n\( distinct integers from \)1\( to \)n\( in arbitrary order. For example, \)[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]\( is also not a permutation (\)n=3\( but there is \)4$$$ in the array).
Each test consists of multiple test cases. The first line contains a single integer \(t\) (\(1 \leq t \leq 2000\)) — the number of test cases. The description of test cases follows.The first line of each test case contains two integers \(n\) and \(k\) (\(1 \leq k \leq n \leq 2 \cdot 10^5\)).It is guaranteed that the sum of \(n\) over all test cases does not exceed \(2 \cdot 10^5\).
For each test case, output \(n\) integers \(p_1,p_2,\dots,p_n\), which is a permutation with minimal cost. If there is more than one permutation with minimal cost, you may output any of them.
In the first test case, \(c_1 = \max(p_1,p_2,p_3) + \min(p_1,p_2,p_3) = 5 + 1 = 6\). \(c_2 = \max(p_2,p_3,p_4) + \min(p_2,p_3,p_4) = 3 + 1 = 4\). \(c_3 = \max(p_3,p_4,p_5) + \min(p_3,p_4,p_5) = 4 + 2 = 6\). Therefore, the cost is \(\max(6,4,6)=6\). It can be proven that this is the minimal cost.