1988E · Range Minimum Sum

Problem: For an array $[a_1,a_2,\ldots,a_n]$ of length $n$, define $f(a)$ as the sum of the minimum element over all subsegments. That is, $f(a)=\sum_{l=1}^n\sum_{r=l}^n \min_{l\le i\le r}a_i.$

A permutation is a sequence of integers from $1$ to $n$ of length $n$ containing each number exactly once. You are given a permutation $[a_1,a_2,\ldots,a_n]$. For each $i$, solve the following problem independently:

• Erase $a_i$ from $a$, concatenating the remaining parts, resulting in $b = [a_1,a_2,\ldots,a_{i-1},\;a_{i+1},\ldots,a_{n}]$.
• Calculate $f(b)$.

Input Format: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^5$). Description of the test cases follows.

The first line of each test case contains an integer $n$ ($1\le n\le 5\cdot 10^5$).

The second line of each test case contains $n$ distinct integers $a_1,\ldots,a_n$ ($1\le a_i\le n$).

It is guaranteed that the sum of $n$ over all test cases does not exceed $10^6$.

Output Format: For each test case, print one line containing $n$ integers. The $i$-th integer should be the answer when erasing $a_i$.

Note: In the second test case, $a=[3,1,2]$.

• When removing $a_1$, $b=[1,2]$. $f(b)=1+2+\min\{1,2\}=4$.
• When removing $a_2$, $b=[3,2]$. $f(b)=3+2+\min\{3,2\}=7$.
• When removing $a_3$, $b=[3,1]$. $f(b)=3+1+\min\{3,1\}=5$.