CF1883G2.Dances (Hard Version)

This is the hard version of the problem. The only difference is that in this version $m \leq 10^9$.

You are given two arrays of integers $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$. Before applying any operations, you can reorder the elements of each array as you wish. Then, in one operation, you will perform both of the following actions, if the arrays are not empty:

• Choose any element from array $a$ and remove it (all remaining elements are shifted to a new array $a$),
• Choose any element from array $b$ and remove it (all remaining elements are shifted to a new array $b$).

Let $k$ be the final size of both arrays. You need to find the minimum number of operations required to satisfy $a_i \lt b_i$ for all $1 \leq i \leq k$.

This problem was too easy, so the problem author decided to make it more challenging. You are also given a positive integer $m$. Now, you need to find the sum of answers to the problem for $m$ pairs of arrays $(c[i], b)$, where $1 \leq i \leq m$. Array $c[i]$ is obtained from $a$ as follows:

• $c[i]_1 = i$,
• $c[i]_j = a_j$, for $2 \leq j \leq n$.

这是该问题的困难版本。唯一的区别在于本版本中 $m \leq 10^9$。

给你两个整数数组 $a_1, a_2, \ldots, a_n$ 和 $b_1, b_2, \ldots, b_n$。在执行任何操作之前，你可以任意重排每个数组中的元素。然后，在一次操作中（当两个数组均非空时），你需同时执行以下两个动作：

• 从数组 $a$ 中任选一个元素并将其删除（其余所有元素向左移动，构成新的数组 $a$）；
• 从数组 $b$ 中任选一个元素并将其删除（其余所有元素向左移动，构成新的数组 $b$）。

设 $k$ 为最终两个数组的长度。你需要找出满足对所有 $1 \leq i \leq k$ 均有 $a_i \lt b_i$ 所需的最少操作次数。

本题原本过于简单，因此出题人决定提升其难度。现在你还被给定一个正整数 $m$。你需要计算该问题在 $m$ 对数组 $(c[i], b)$（其中 $1 \leq i \leq m$）上的答案之和。数组 $c[i]$ 由 $a$ 按如下方式构造：

• $c[i]_1 = i$，
• $c[i]_j = a_j$，其中 $2 \leq j \leq n$。

Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 10^4$) - the number of sets of input data. This is followed by their description.

The first line of each test case contains two integers $n$ and $m$ ($2 \leq n \leq 10^5$, $1 \leq m \leq 10^9$) - the size of arrays $a$ and $b$ and the constraints on the value of element $a_1$.

The second line of each test case contains $n - 1$ integers $a_2, \ldots, a_n$ ($1 \leq a_i \leq 10^9$).

The third line of each test case contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1 \leq b_i \leq 10^9$).

It is guaranteed that the sum of $n$ over all test cases does not exceed $10^5$.

每个测试包含多个测试用例。第一行包含一个整数 $t$（$1 \leq t \leq 10^4$），表示输入数据组数。随后是各组数据的描述。

每个测试用例的第一行包含两个整数 $n$ 和 $m$（$2 \leq n \leq 10^5$，$1 \leq m \leq 10^9$），分别表示数组 $a$ 和 $b$ 的长度，以及对元素 $a_1$ 的取值约束。

每个测试用例的第二行包含 $n - 1$ 个整数 $a_2, \ldots, a_n$（$1 \leq a_i \leq 10^9$）。

每个测试用例的第三行包含 $n$ 个整数 $b_1, b_2, \ldots, b_n$（$1 \leq b_i \leq 10^9$）。

保证所有测试用例中 $n$ 的总和不超过 $10^5$。

For each test case, output the total number of minimum operations for all pairs of arrays $(c_i, b)$.

对于每个测试用例，输出所有数组对 $(c_i, b)$ 所需的最少操作总数。

In the first test case:

• For the pair of arrays $([1, 1], [3, 2])$, the answer is $0$. No operations or reordering of elements are needed.
• For the pair of arrays $([2, 1], [3, 2])$, the answer is $0$. The elements of the first array can be rearranged to obtain $[1, 2)$. No operations are needed.
• For the pair of arrays $([3, 1], [3, 2])$, the answer is $1$. The element $3$ can be removed from the first array and the element $2$ can be removed from the second array.
• For the pair of arrays $([4, 1], [3, 2])$, the answer is $1$. The element $4$ can be removed from the first array and the element $3$ can be removed from the second array.

在第一个测试用例中：

• 对于数组对 $([1, 1], [3, 2])$，答案为 $0$。无需执行任何操作，也无需重排元素。
• 对于数组对 $([2, 1], [3, 2])$，答案为 $0$。第一个数组的元素可重排为 $[1, 2)$，无需执行任何操作。
• 对于数组对 $([3, 1], [3, 2])$，答案为 $1$。可从第一个数组中删除元素 $3$，并从第二个数组中删除元素 $2$。
• 对于数组对 $([4, 1], [3, 2])$，答案为 $1$。可从第一个数组中删除元素 $4$，并从第二个数组中删除元素 $3$。