P11677 [USACO25JAN] Shock Wave P

Bessie is experimenting with a powerful hoof implant that has the ability to create massive shock waves. She has $N$ ($2 \leq N \leq 10^5$) tiles lined up in front of her, which require powers of at least $p_0,p_1,\dots,p_{N-1}$ to break, respectively ($0 \leq p_i \leq 10^{18}$). Bessie can apply power by punching a specific tile but due to the strange nature of her implant, it will not apply any power to the tile she punches. Instead, if she chooses to punch tile $x$ once, where $x$ is an integer in $[0,N-1]$, it applies $|i-x|$ power to tile $i$ for all integers $i$ in the range $[0,N-1]$. This power is also cumulative, so applying $2$ power twice to a tile will apply a total of $4$ power to the tile. Please determine the fewest number of punches required to break all the tiles.

The first line contains $T$ ($1 \leq T \leq 100$) representing the number of test cases. Line $2t$ contains a single integer $N$, the number of tiles in test case $t$. Line $2t+1$ contains $N$ space separated numbers $p_0,p_1, \ldots, p_{N-1}$ representing that tile $i$ takes $p_i$ power to be broken. It is guaranteed that the sum of all $N$ in a single input does not exceed $5\cdot 10^5$.

$T$ lines, the $i$th line representing the answer to the $i$th test case.

For the first test, the only way for Bessie to break all the tiles with two punches is to punch $0$ twice, applying total powers of $[0,2,4,6,8]$ respectively. For the second test, one way for Bessie to break all the tiles with three punches is to punch $0$, $2$, and $4$ one time each, applying total powers of $[6,5,4,5,6]$ respectively. For the third test, one way for Bessie to break all the tiles with two punches is to punch $0$ and $1$ one time each, applying total powers of $[1,1,3,5,7]$ respectively. #### SCORING: - Input 2: All $p_i$ are equal. - Inputs 3-6: $N\le 100$ - Inputs 7-14: No additional constraints.