codeforces#P1827B2. Range Sorting (Hard Version)
Range Sorting (Hard Version)
Description
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 $a$.
Input
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 $3 \cdot 10^5$.
Output
For each test case, output the sum of beauty over all subarrays of array $a$.
5
2
6 4
3
3 10 6
4
4 8 7 2
5
9 8 2 4 6
12
2 6 13 3 15 5 10 8 16 9 11 18
1
2
8
16
232
Note
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 $1$.
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 $3 - 2 = 1$.