Problem F
Famous Pagoda

In order to build a staircase to a famous pagoda on top of a mountain, the local authority has identified $N$ positions along the mountain slope $a_1, a_2, \ldots , a_ N$ where $a_ i$ is the height of the $i$-th position and $a_ i \leq a_{i+1}$ for all $0<i<N$.

The cost to build a staircase from position $i$ to position $j$ is

$\min _{v \in \mathbb {Z}}{\sum _{s=i}^{j}{|a_ s-v|^ k}}$

To speed up the process of building the staircase from position $1$ to position $N$, the local authority has decided to give the job to $G$ builders to build the staircase in parallel. The sequence of $N$ positions will be divided into $G$ segments of consecutive positions where every position belongs to exactly one segment and each segment is managed by exactly one builder.

Given an integer $G$ ($1 \leq G \leq N$), your task is to identify a way to allocate jobs to $G$ builders to minimize their total building costs.


  • The first line contains $3$ integers $N$, $G$, $k$ $(1 \leq N \leq 2\, 000, \; 1 \leq G \leq N, \; 1 \leq k \leq 2)$.

  • The second line contains $N$ integers $a_1, a_2, \ldots , a_ N$ $(1 \leq a_ i \leq 10^6, \; a_ i \leq a_{i+1} \; \forall \; 0<i<N)$.


Print the minimum building cost required.

Sample Input 1 Sample Output 1
5 1 1
1 2 3 4 5
Sample Input 2 Sample Output 2
5 1 2
1 2 3 4 5
Sample Input 3 Sample Output 3
5 2 2
1 2 3 4 5

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