CF1768B Quick Sort

You are given a permutation $ ^\dagger $ $ p $ of length $ n $ and a positive integer $ k \le n $ . In one operation, you: - Choose $ k $ distinct elements $ p_{i_1}, p_{i_2}, \ldots, p_{i_k} $ . - Remove them and then add them sorted in increasing order to the end of the permutation. For example, if $ p = [2,5,1,3,4] $ and $ k = 2 $ and you choose $ 5 $ and $ 3 $ as the elements for the operation, then $ [2, \color{red}{5}, 1, \color{red}{3}, 4] \rightarrow [2, 1, 4, \color{red}{3},\color{red}{5}] $ . Find the minimum number of operations needed to sort the permutation in increasing order. It can be proven that it is always possible to do so. $ ^\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).

The first line contains a single integer $ t $ ( $ 1 \le t \le 10^4 $ ) — the number of test cases. The description of test cases follows. The first line of each test case contains two integers $ n $ and $ k $ ( $ 2 \le n \le 10^5 $ , $ 1 \le k \le n $ ). The second line of each test case contains $ n $ integers $ p_1,p_2,\ldots, p_n $ ( $ 1 \le p_i \le n $ ). It is guaranteed that $ p $ is a permutation. It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 10^5 $ .

For each test case output a single integer — the minimum number of operations needed to sort the permutation. It can be proven that it is always possible to do so.

In the first test case, the permutation is already sorted. In the second test case, you can choose element $ 3 $ , and the permutation will become sorted as follows: $ [\color{red}{3}, 1, 2] \rightarrow [1, 2, \color{red}{3}] $ . In the third test case, you can choose elements $ 3 $ and $ 4 $ , and the permutation will become sorted as follows: $ [1, \color{red}{3}, 2, \color{red}{4}] \rightarrow [1, 2, \color{red}{3},\color{red}{4}] $ . In the fourth test case, it can be shown that it is impossible to sort the permutation in $ 1 $ operation. However, if you choose elements $ 2 $ and $ 1 $ in the first operation, and choose elements $ 3 $ and $ 4 $ in the second operation, the permutation will become sorted as follows: $ [\color{red}{2}, 3, \color{red}{1}, 4] \rightarrow [\color{blue}{3}, \color{blue}{4}, \color{red}{1}, \color{red}{2}] \rightarrow [1,2, \color{blue}{3}, \color{blue}{4}] $ .