CF1827B2.Range Sorting (Hard Version)

The only difference between this problem and the easy version is the constraints on $t$ and $n$ .

You are given an array $a$ , consisting of $n$ distinct integers $a_1, a_2, \ldots, a_n$ .

Define the beauty of an array $p_1, p_2, \ldots p_k$ as the minimum amount of time needed to sort this array using an arbitrary number of range-sort operations. In each range-sort operation, you will do the following:

• Choose two integers $l$ and $r$ ( $1 \le l < r \le k$ ).
• Sort the subarray $p_l, p_{l + 1}, \ldots, p_r$ in $r - l$ seconds.

Please calculate the sum of beauty over all subarrays of array $a$ .

A subarray of an array is defined as a sequence of consecutive elements of the array.

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 a single integer $n$ ( $1 \le n \le 3 \cdot 10^5$ ) — the length of the array $a$ .

The second line of each test case consists of $n$ integers $a_1,a_2,\ldots, a_n$ ( $1\le a_i\le 10^9$ ). It is guaranteed that all elements of $a$ are pairwise distinct.

It is guaranteed that the sum of $n$ over all test cases does not exceed $3 \cdot 10^5$ .

For each test case, output the sum of beauty over all subarrays of array $a$ .

In the first test case:

• The subarray $[6]$ is already sorted, so its beauty is $0$ .
• The subarray $[4]$ is already sorted, so its beauty is $0$ .
• You can sort the subarray $[6, 4]$ in one operation by choosing $l = 1$ and $r = 2$ . Its beauty is equal to $1$ .

The sum of beauty over all subarrays of the given array is equal to $0 + 0 + 1 = 1$ .In the second test case:

• The subarray $[3]$ is already sorted, so its beauty is $0$ .
• The subarray $[10]$ is already sorted, so its beauty is $0$ .
• The subarray $[6]$ is already sorted, so its beauty is $0$ .
• The subarray $[3, 10]$ is already sorted, so its beauty is $0$ .
• You can sort the subarray $[10, 6]$ in one operation by choosing $l = 1$ and $r = 2$ . Its beauty is equal to $2 - 1 = 1$ .
• You can sort the subarray $[3, 10, 6]$ in one operation by choosing $l = 2$ and $r = 3$ . Its beauty is equal to $3 - 2 = 1$ .

The sum of beauty over all subarrays of the given array is equal to $0 + 0 + 0 + 0 + 1 + 1 = 2$ .