B. Wonderful Gloves
greedy; math; sortings
You are the proud owner of many colorful gloves, and you keep them in a drawer. Each glove is in one of \(n\) colors numbered \(1\) to \(n\). Specifically, for each \(i\) from \(1\) to \(n\), you have \(l_i\) left gloves and \(r_i\) right gloves with color \(i\).Unfortunately, it's late at night, so you can't see any of your gloves. In other words, you will only know the color and the type (left or right) of a glove after you take it out of the drawer.A matching pair of gloves with color \(i\) consists of exactly one left glove and one right glove with color \(i\). Find the minimum number of gloves you need to take out of the drawer to guarantee that you have at least \(k\) matching pairs of gloves with different colors.Formally, find the smallest positive integer \(x\) such that: For any set of \(x\) gloves you take out of the drawer, there will always be at least \(k\) matching pairs of gloves with different colors.
Each test contains multiple test cases. The first line contains the number of test cases \(t\) (\(1 \le t \le 10^4\)). The description of the test cases follows. The first line of each test case contains two integers \(n\), \(k\) (\(1 \leq k \leq n \leq 2 \cdot 10^5\)) — the number of different colors, and the minimum number of required matching pairs of gloves.The second line of each test case contains \(n\) integers \(l_1, l_2, \ldots, l_n\) (\(1 \leq l_i \leq 10^9\)) — the number of left gloves with color \(i\) for each \(i\) from \(1\) to \(n\).The third line of each test case contains \(n\) integers \(r_1, r_2, \ldots, r_n\) (\(1 \leq r_i \leq 10^9\)) — the number of right gloves with color \(i\) for each \(i\) from \(1\) to \(n\).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 single integer — the minimum number of gloves you need to take out of the drawer.
In the first test case, you must take out all of the gloves, so the answer is \(6\).In the second test case, the answer is \(101\). If you take out \(100\) gloves or fewer, then it is possible that all of them are left gloves, which means you won't have a matching pair of gloves.In the third test case, the answer is \(303\). If you only take out \(302\) gloves, then one possible scenario is as follows: Color \(1\): \(100\) left gloves, \(200\) right gloves Color \(2\): \(1\) left glove, \(0\) right gloves Color \(3\): \(0\) left gloves, \(1\) right glove You only have multiple matching pairs of gloves with color \(1\). So you won't have at least \(2\) matching pairs of gloves with different colors.